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A Doctoral Thesis.
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Elaheh Ghassemieh
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Finite Element Modelling Of Mechanical Properties
Of Polymer Composites
By ELAHEH GHASSEMIEH
A Doctoral Thesis
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To those whoselove kept my heart warm and encouraged me all through my life, during the period of this study and in near or far, especially my dear mother Parichehr and my father Mehdi
Acknowledgements
HerebyI would like to thank my supervisor,Dr. V.Nassehi,for his assistanceand advice during the courseof this researchand I wish to to all the members of staff and students in the Chemical Engineering Department of Loughborough University who showmygratitude
accompanied me allover the way. Ilearned a lot from them and I being enjoyed with them. A door to a new world was opened to me, an experience that will be always remembered.
ABSTRACT
Polymeric composites are used widely in modern industry. The prediction behaviour is loadings different these of mechanical of material under therefore of vital importance in many applications. Mathematical modelling offers a robust and cost effective method to satisfy this for finite In this objective. element model project a comprehensive particulate and fibre reinforced compositesis developed.
The most significant features of this model are:
" The inclusion of slip boundary conditions. boundaries interphase to take The inclusion flux the terms " across of the discontinuity of the material properties into account in the model. Stokes flow in The equations with " use of penalty method conjunction developed to the the model solid elasticity of application which allow flows. analysis as well as creeping viscous
The predictions of this model are compared with available theoretical data. These comparisons show that the models and experimental developed model yields accurate and reliable data for composite deformation.
TABLE OF CONTENTS
Chapter One
INTRODUCTION
.....................................................................................................
I
Chapter Two LITERATURE REVIEW 2.1 INTRODUCTION
..........................................................................................
8
8 ........................................................................................................
9 2.2 TYPES OF COMPOSITES ........................................................................................... 9 2.2.1 Polymeric composites filled with fibres .................................................................................. 10 2.2.2 Polymeric composites filled with particulate fillers .............................................................. COMPOSITES 2.3 MECHANICAL PROPERTIES OF POLYMERIC ........................ 2.3.1 Mechanical properties of unfilled polymers ......................................................................... 2.3.2 Mechanical properties of particulate filled polymers ........................................................... 2.3.3 Nonlinear behaviour of polymeric composites ....................................................................
10 11 12 14
2.4 THEORETICAL MODELS FOR DETERMINATION OF THE MODULUS OF 15 COMPOSITES .................................................................................................................. 2.4.1 Theories of rigid inclusions in a nonrigid matrix ................................................................ 2.4.1.1 Einstein equation and its modifications ..................................................................... 2.4.1.2 The Kerner equation and its modifications ................................................................ 2.4.2 Theories of nonrigid inclusions in a nonrigid matrix ........................................................ 2.4.2.1 Series and parallel models ......................................................................................... 2.4.2.2 The Hashin and Shtrikman model ............................................................................. 2.4.2.3 The Hirsch model ..................................................................................................... 2.4.2.4 The Takayanagi model .............................................................................................. 2.4.2.5 The Counto model .................................................................................................... 2.4.3 Llimitations of the theoretical models ..................................................................................
16 16 18 19 20 20 21 22 22 23
ANALYSIS OF POLYMERIC COMPOSITES 2.5 MICROMECHANICAL 24 .............. 2.5.1 Qualitativedescriptionof the microstructure 25 ...................................................................... 2.5.1.1 Geometric 25 models.....................................................................................................
I
27 2.5.1.2 Geometric model used in finite element modelling of composites ................................ 28 2.5.1.3 Structural descriptors ................................................................................................ 30 2.5.1.4 Size Of Particles ........................................................................................................ 30 2.5.2 Interfaceadhesion .............................................................................................................
PROPERTIES OF 2.6. FINITE ELEMENT MODELLING OF THE MECHANICAL COMPOSITES .................................................................................................................. 31
2.6.1 Finite element methods basedon variational principles ....................................................... 31 32 2.6.1.1 Displacement method ................................................................................................. 2.6.2 Leastsquareand Weighted residual method ........................................................................ 37 38 2.6.2.1 A brief outline of the Galerkin method ........................................................................
Chapter Three
DEVELOPMENT OF THE PREDICTIVE MODEL ............................................. 39 39 EQUATIONS 3.1 MODEL .............................................................................................. 39 3.1.1 Axisymmetric stresscondition ............................................................................................ 44 3.1.2 Plain strain condition .........................................................................................................
47 CONDITIONS AND BOUNDARY GEOMETRY 3.2 MODEL ................................... 47 filled 3.2.1 Modelling of the particulate composites..................................................................... 50 3.2.2 Fibre reinforced composite ................................................................................................. 51 3.2.1.1 Tensile loading .......................................................................................................... 52 3.2.2.2 Shear loading ............................................................................................................. 52 3.2.3 Slip boundary conditions .................................................................................................... 55 3.3 CALCULATIONS .............................................. ......................................................... 55 Poisson's 3.3.1 Composite modulus of elasticity and ratio ........................................................... 56 3.3.2 Composite strength .............................................................................................................. 56 3.3.3 Stress calculations( Variational recovery) ...........................................................................
ChapterFour 60 DISCUSSION AND RESULTS .............................................................................. 63 4.1 MODULUS .................................................................................................................. 4.1.1 Equal Stressand Equal Strain Bounds................................................................................. 63 4.1.2 Model predictions and experimental results ......................................................................... 67 4.1.2.1 Agglomeration ............................................................................................................ 67 67 4.1.2.2 Dewetting ................................................................................................................... 69 4.1.2.3 Adhesion and bonding ................................................................................................ 4.1.2.4 Arrangement .............................................................................................................. 69 71 4.1.2.5 Filler particle shape .................................................................................................... 72 4.2.3. Boundary conditions used in the finite element model ........................................................
11
4.2 COMPOSITES FILLED WITH RIGID PARTICLES ............................................. 4.2.1. Young's modulus ............................................................................................................... 4.2.2 Stress distribution ................................................................................................................ 4.2.2.1 Concentration of direct stress ...................................................................................... 4.2.2.2 Stressesat the interface ............................................................................................... 4.2.3. Compression ...................................................................................................................... 4.2.4 Concentration of yield stress ................................................................................................ 4.2.5 Fracture behaviour ...............................................................................................................
73 73 73 73 76 78 78 83
90 FILLED WITH DEBONDED RIGID PARTICLES 4.3 COMPOSITES ..................... 90 4.3.1 Boundary conditions at the interface ................................................................................... 92 4.3.2 Displacement ...................................................................................................................... 92 4.3.3 Interfacial stresses ............................................................................................................... 102 4.3.4 Slipping of the particle at part of the interface ................................................................... 102 4.3.5 Strength ............................................................................................................................ 105 4.4 COMPOSITES FILLED WITH SOFT PARTICLES ............................................ 105 4.4.1 Material properties ............................................................................................................ 105 4.4.2 Young's modulus ............................................................................................................. 109 4.4.3 Stress distribution .............................................................................................................. 109 4.4.3.1 Concentration of direct stress .................................................................................... 114 4.4.3.2 Stressesat the interface ............................................................................................. 114 4.4.3.3 Hydrostatic stressin the rubber particle .................................................................... 116 4.4.3.4 Concentration of yield stress ..................................................................................... 116 4.4.3.5 Stressesin the matrix ............................................................................................... 118 4.4.4 Fracture behaviour ............................................................................................................ 4.5 FIBRE REINFORCED COMPOSITES ................................................................... 4.5.1 Transverse tensile loading ................................................................................................. 4.5.1.1 Modulus ................................................................................................................... 4.5.1.2 Concentration of applied stress ................................................................................ 4.5.1.3 Stressesat interface ................................................................................................... 4.5.1.4 Fracture strength ...................................................................................................... 4.5.2 Transverse shear loading .................................................................................................. 4.5.2.1 Modulus .................................................................................................................. 4.5.2.2 Stress distribution and strength .................................................................................
123 123 123 123 128 130 134 134 134
COMPOSITES 4.6 SHORT FIBRE REINFORCED ................................................... 4.6.1 Modulus ............................................................................................................................ 4.6.2 Stress distribution .............................................................................................................. 4.6.2.1 Interfacial shear stressdistribution ............................................................................ 4.6.2.2 Interfacial tensile stressdistribution ......................................................................... 4.6.2.3 Tensile stress in the fibre .......................................................................................... 4.6.3 Critical length ................................................................................................................... 4.6.4 Strength of short fibre composites ......................................................................................
139 139 147 147 147 147 150 154
UI
Chapter Five CONCLUSION
.................................................................................................................
159
159 Of THE PRESENT WORK 5.1 CONCLUSIONS ........................................................ 159 5.1.1 Modulus of particulate filled composites ............................................................................ 160 5.1.2 composites filled with hard particles .................................................................................. 162 5.1.3 Composite filled with partially bonded particles ................................................................ 163 5.1.4 Composite filled with soft particles .................................................................................... 165 5.1.5 Composites reinforced with continuous fibres .................................................................... 167 5.1.6 Composites reinforced with short fibres .............................................................................
5.2 SUGGESTIONS FOR FURTHER WORK
IV
169 ............................................................
Chapter One
INTRODUCTION
Polymeric composites are amongst the most important new material resources.This is becauseof the relative low cost of manufacturing of thesematerials and the possibility of obtaining improved and unique properties from them. The bulk properties of polymeric composites obviously depend on their microstructure. Therefore the composite properties and their behaviour should be assessedand analysedconsidering microstructural responsesof compositesto external loading and conditions. The main objectives of this project have been the analysis of the micromechanical behaviour of different types of polymeric composites using mathematical modelling. We have considered continuous fibre reinforced composite, short fibre reinforced composite and composites with hard or soft particles.
To understand the mechanical behaviour of composites, the mechanical behaviour of the constituent materials and the interactions of these constituents should be investigated. Mathematical modelling offers a powerful prediction tool to carry out such investigation. In recent years a number of mathematicalmodels for the microstructural behaviour of composites have been proposed. Most of these models are formulated in terms of mathematical equations, which cannot be solved analytically. Therefore in most casesthe use of a numerical technique for the quantitative analysisof composite behaviour is required.
Chanter One
Introduction
Finite element method combines robustnesswith flexibility and hence it is the method finite In have in this the element applications. project we considered of choice most is It `displacement `weighted based the the method'. method' and residual on schemes for latter the important that the to using model a note method mathematical especially is based be developed on equations can which microstructural analysisof composites has dynamics. This fundamental fluid the the to approach of equations similar in liquid for the or solid state. single of composites model analysis advantageof using a This thesis consists of five chapters. The introductory chapter describesthe outline of the thesis and includes a summary of the properties of various composites investigated in the present work. In chapter two a thorough literature survey is given and the background of the present project is presented.In chapter three the developmentof the This described. domain chapter geometry are working equations of the model and the boundary the finite includes conditions the mesh, element also explanation regarding is devoted four the to Chapter computational the and postprocessingcalculations used. data the and discussion the experimental available the and comparisonwith results and following four the sections: Chapter by of consists results generated other models. In section one of this chapter the most important bulk property of composites, the boundary the on conditions is The and effect of model assumptions modulus studied. discussed. are results obtained In section two the mechanicalbehaviour of epoxy resin filled with hard glass particles is studied as a typical composite filled with rigid particles. The addition of rigid improvement in in the the of a significant properties to result can resins epoxy particles in in is invariably There increase the cost. an reduction considerable a and resin is fracture behaviour but the the the particles upon effect of stiffness of the resin behaviour has been fracture there The of multiphase polymers and reviewed complex. has been considerable interest over the years in crack propagation in brittle materials filler is found It in that, particles. general, both the critical stress reinforced with rigid intensity factor and fracture energy increasewith the addition of rigid particles, at least for low volume fractions of filler. The most generally accepted explanation of this
2
Chapter One
Introduction
behaviour, using the analogy of a dislocation moving through a crystal, is that a crack in a body possesses "line tension" and that when it meets an array of a impenetrable have In it becomes to the the order move crack would pinned. past obstacles obstacles to bow out and this leadsto an increasein fracture energy.The fracture energy reaches a maximum at a particular value of filler volume fraction and then falls with the further addition of particles, implying that there may be another mechanismwhich competes with crack front pinning at high volume fractions of filler.
Particulate filled composites are used in applications ranging from everyday usage, like automobile tires, to specific, such as solid rocket propellants. These materials exhibit interesting failure properties. Phenomenasuch as cavitation (the appearanceof voids) lead filler to gross debonding (adhesive failure between particles) and matrix and large to in Particulate behaviour. subjected their composites nonlinearities stressstrain The this that debonding. large degrees affect parameters strains generally exhibit of failure include particle size, filler concentration, surface treatments, matrix and filler filler Debonding particles of properties, superimposed pressure, and strain rate. both influencing factor be dominant stressstrain and volumetric to the appears behaviour of particulate materials. In section three the stress field in a composite with partially or fully debonded rigid factor in determining is the degradation is Bond a critical often particles analysed. its fatigue impact as well resistance, as ultimate strength of a composite material, filler bonding between The important the of strength properties. resistance, and other deflect bridge in the to the composite of ability cracks or and matrix plays a major role fracture interface to thereby, toughness. the contribute composite and, cracks along For such fracture toughening to occur, the filler matrix interface must exhibit just the is bonding like If bonding. behaves degree the too the a strong, composite of right in through the material generally resulting monolithic material and cracks propagate brittle fracture.
3
Chapter One
Introduction
Improving adhesionat the interface increasesthe fracture strength of the composite it , is not entirely clear how this affects crack propagation. There have been reports of improving adhesionat the interface both increasingand decreasingthe fracture energy, for crack propagation in particle reinforced composites. With good adhesion it is found that the fracture strength of the composites is approximately the same as that of the unfilled matrices. On the other hand, with no surface pretreatments,or releaseagents,applied to the particles the strength decreases with increasingvolume fractions of filler particles.
In the fourth section of chapter four the effects of adding rubbery particles on the The are studied. mechanical characteristic of a matrix material such as epoxy resin improvement of the impact properties of polymers is possible through incorporation of in increase The the toughness domains into brittle of matrix. rubbery phase a polymer induced due be is believed to to wideglassypolymers with addition of rubber particles in deformation processes,such as crazing and shear yielding, spread energy absorbing factor it is firstly, important; is the fracture. Shear during the matrix material yielding A be fracture brittle if suppressed. limits can the strength of the composite which if bulk, homogeneous be in to high have and strong yield stress order composite must a Secondly, be is likely tough. to does recent evidence the polymer yielding occur in key bands, the form in a role plays the microshear of that suggests shear yielding, initiation of cracks. Shear yielding and crazing, both involve localised, or inhomogeneous,plastic deformation of the material which arises from strain softening between is difference The the that shear mechanisms and geometric considerations. increase whereas crazing occurs volume with an yielding occurs essentiallyat constant in volume. Thus, unlike shear yielding, crazing is a cavitation process in which the initiation step requires the presenceof a dilatational component to the stresstensor and hydrostatic but by in inhibited be the presence of pressure applying enhanced may triaxial tensile stresses.
A craze is initiated when an applied tensile stress causes microvoids to nucleate at by high in These concentrations stress points of polymer. microvoids are created
4
Chapter One
Introduction
scratches, flaws, cracks, dust particles, molecular heterogeneities. In general the microvoids develop in a plane perpendicular to the maximum principal stress but do not coalesce to form a true crack. Thus a microvoid is capable of transmitting loads across its faces. However, when cracks do initiate and grow they do so by means of the breakdown of the fibrillar structure in a craze. The importance of crazing is that it is frequently a precursor to brittle fracture. This is because, although considerable deformation plastic and local energy adsorption are involved in craze initiation, growth and breakdown, this micromechanismis often highly localised and confined to a very small volume of the material. However, it should be recognised that if stable crazes can be initiated in a comparatively large volume of the polymer, i.e. a multiple deformation mechanism is induced, then such multiple crazing may lead to a tough, and possibly even a ductile, material response. Crazes are formed at the rubber particles whereas shear yielding takes place between the modifier particles. The rubber inclusions cause a local stress magnification in the matrix material immediately surrounding the inclusions. This local stress magnification is believed to initiate crazing and shear yielding.
A great deal of controversy still exists on the
issues dispute the Much the of surrounds of nature of the toughening mechanisms. whether the rubber or the matrix absorbs most of the energy and whether the matrix undergoes massive crazing or simple voiding.
Presumably, once the mechanisms
identified, increased then the material for toughness clearly the are responsible be for these can enhancedor modified to produce mechanisms parameters responsible an optimal combination of properties.
The behaviour of the continuous fibre reinforced composites under tensile and shear loading is studied in section five of chapter four. High specific strength and stiffness fibre have composite materials reinforced properties of resulted in their widespread use in load bearing structures. These structures have complex geometries and are often loadings. Monolithic material mechanical behaviour can be to subjected multiaxial adequatelydescribed by a limited number of material properties and strength criteria as these materials present simple failure modes under different loading and boundary
5
Chapter One
Introduction
conditions. Strength characterisationof composite laminate structures is more difficult to estimate becauseof the variety of failure modes and failure mode interactions. For a from is derived fibre the the composite, strength continuous reinforced of composite the strength of the fibres, but this strength is highly directional in nature.
The
longitudinal strength of the continuous fibre reinforced composites is much greater than the transverse strength. The compressive strengths associated with these directions may be different from the corresponding tensile strengths.
Failure of composite materialsare determined not only by their internal properties such by but external also as properties of constituents and microstructural parameters conditions such as geometric variables, type of loading and boundary conditions. Critical failure modes for each composite material system under various loading for be identified failure be each established criterion should conditions must and a failure mode.
Finally the last section of chapter four presentsthe results of the analysisof the short fibre reinforced composites. Short fibre reinforced composites are not as strong or as be in likely to critical fibre used stiff as continuous reinforced composites and are not have do fibre However, several attractive composites short structural applications. For for other applications. them worthy of consideration characteristics that make fibres having in continuous may contours, complex geometrical example, components desired because to the be they conform shape without may not not of practical use being damaged or distorted from the desired pattern. On the other hand, short fibres be liquid be the the and matrix resin, resin/fibre can mixture with can easily mixed injection or compression moulded to produce parts having complex shapes. Such inexpensive, fast for them also and which are makes very attractive processing methods high volume applications. Composites having randomly oriented short fibre isotropic, whereasunidirectional continuous fibre composites are nearly reinforcement are highly anisotropic. In many applications the advantages of low cost, ease of fabricating geometrically complex parts, and isotropic behaviour are enough to make short fibre composites the material of choice.
6
Chanter One
Introduction
Since the elastic modulus of the fibre is typically much larger than that of the matrix, the axial elastic displacementsof the two componentscan be very different. In order to rationalise the design of reinforced materials, it is thus of primary importance to have a detailed knowledge of stress distribution induced by the applied load. Indeed, when discontinuous fibres are used, the attainment of good mechanical properties depends critically upon the efficiency of stress transfer between matrix and the fibres. That is efficiency often characterisedby the critical length required of the fibre to build up a maximum stressequal to that of an infinitely long fibre.
The effective properties of fibre reinforced composites strongly depend on the geometrical arrangement of the fibres within the matrix. This
arrangement is
characterised by the volume fraction of fibres, the fibre aspect ratio and the fibre spacing parameter. Analytical equations for the variation of stress along discontinuous fibres in a cylindrically symmetrical model have been derived by Cox. In this approach the adhesion across the end face of the fibres is neglected and the local stress concentration effects near fibre ends have not been taken into account. The importance of these assumptionshas been demonstratedby finite elementapproaches.
The overall conclusions of the present project are discussedin chapter five. The list of referencesquoted in the text is included at the end of the thesis.
The main objectives of this project can be summarisedas:
Developing a model to predict the mechanicalproperties of polymer composite Developing a code that can be used for studying the behaviour of the polymer composites in the both solid and liquid states. Including the boundary line integral terms in the model and investigating the effect of that on the final results of the computations for different shapeof the fillers and composites.
7
Chapter One
Introduction
Imposing the slip boundary condition at the interface of the filler and matrix in order to simulate the level of adhesionat the boundary of a debondedfiller particle. Applying the developed model for different types of composites such as composites filled with hard particles or soft particles, composite reinforced with Using the proper geometry model and boundary
continuous or short fibres.
condition for each case. Validating the results of the computation by comparing in in data order to case each model and other well established with experimental evaluate our model in qualitative and quantitative analysis.
ý
Chapter Two
LITERATURE
REVIEW
2.1 INTRODUCTION A composite material is a combination of at least two chemically distinct materials with a
distinct interface separating the components. Composites can offer a
combination of properties and a diversity of applications unobtainable with metals, ceramics or polymers alone. Composites are also used when it is necessary to substitute the traditional materials. Substitution can be the result of legislation, performance improvement, cost reduction and expansion of product demand. For legislation impact damagehas provided motivation for the trend of on minor example widespread substitution of plastics with metals in automobile bumpers. Improvement of mechanicalproperties such as load bearing and transfer, creep, fatigue strength and high temperature strength is also regarded as an important reason for designing and manufacturing of polymer composites. The enhancementof heat, abrasion, oxidation, corrosion and wear resistance which can protect
the objects confronting
environmental attacks is another reason for the increaseduse of composite materials. Improved electrical, magnetic and thermal conductivity properties can also be considered as objectives of the design of composite materials.
q
CIL
Chapter Two
Literature Review
2.2 TYPES OF COMPOSITES In polymeric composites the baseor matrix is a polymer and the filler is an inorganic or organic material in either fibre or particulate form. The properties of an advancedcomposite are shapednot only by the kind of matrix and reinforcing materials it contains but also by another factor which is distinct from composition. This factor is the geometry of reinforcement. Geometrically, composites can be grouped roughly by the shape of the reinforcing elements as particulate, continuous fibre or short fibre composites.
2.2.1 Polymeric composites filled with fibres Addition of
fibres to a polymer matrix enhancesits stiffness, strength, hardness,
its lubricity deflection heat temperature while reducing and abrasion resistance, length L fibre (i. The the the characteristic ratio of aspectratio e. shrinkage and creep. to the characteristic diameter D of a typical fibre) and the orientation of fibres have profound influence on the properties
of the composites. The most effective
high intrinsically have fibres fillers a elastic modulus and tensile which are reinforcing is A a material which can effectively transfer the applied composite superior strength. load to the fibres. Fibres can be made of organic material such as: Cellulose, Wood, Carbon / Graphite, Nylons and Polyester. At the present time however, for reasons of constitutional inorganic fibres thermal cost, are the most and sometimes stability, strength, stiffness, important reinforcement materials which are compounded with polymers. Recent developments in high modulus and thermallyresistant organic fibres are creating interest, especially where light weight materials are needed
as in aerospace
applications. Inorganic materials such as Asbestos, Glass, Boron, Ceramic, Metal filaments are commonly used in the production of fibres. Similar to organic fibres, these are produced using both natural and synthetic raw materials. 2.2.2 Polymeric composites filled with particulate fillers This kind of filler embracesnot only fillers with regular shapes,such as spheres, but also many of irregular shapespossibly having extensive convolution and porosity in
9
Chapter Two
Literature Review
addition. However, the use of these types of fillers does not improve the ultimate tensile strength of the composites. In fact the tensile, flexural and impact strength of these composites are lowered, especially at higher filler contents. On the other hand hardness, heat deflection temperature and surface finish of particulate filled composites
may be enhanced and their stiffness is
mainly improved. Thermal
expansion, mold shrinkage extendibility and creep are reduced too. The main advantage of using these fillers is that regardlessof the bonding efficiency between particulate filler and matrix, the properties remain consistent. In addition the elastic modulus and the heat distortion temperature increases while structural strength decreases. Wood flour, Cork, Nutshell, Starch, Polymers, Carbon and Protein are some of the commonly used organic particulate fillers. Despite limited thermal stability, organic fillers have' an advantage of being of low density and many have a valuable role as a cheap extender for the more expensive base polymer, as well as providing some incidental property such as reduction of is important in polymer processing. mould shrinkage which Glass, Calcium carbonate, Alumina, Metal oxides, Silica and Metal powder are fillers. inorganic the used as particulate materials which are examplesof This class of fillers constitutes the more important group of particulate fillers in view for basis Thus low their a reducing the cost providing price and ready availability. of if loss, filled too without composites with particulate made much articles of moulded (Sheldon, 1982). desired properties any, of
2.3 MECHANICAL PROPERTIES OF POLYMERIC COMPOSITES In many respects the mechanical properties of different polymers are their most important characteristics. Since whatever may be the reason for the choice of a for an application, (whether it be thermal, electrical or even particular polymer aesthetic grounds), it must have certain characteristicsof shape, rigidity and strength. For polymer composites, improvement of mechanical behaviour is a prime requirement. Since by definition the polymer constitutes the continuous phase, the filler acts essentially through a modification of the intrinsic mechanical properties of
10
Chapter Two
Literature Review
the polymer. Factors such as concentration, type, shape and geometrical arrangement of the filler within the matrix are the main contributors to the modification of the mechanical properties of polymer composites.
2.3.1 Mechanical properties of unfilled polymers The most common way of recording mechanicalproperties of polymers is to carry out stressstrain, or more precisely loadextension tests. For amorphous polymers the finite natural relaxation rate of polymer chains results in the viscoelasticbehaviour of these materials.The viscoelasticity of the polymers is itself a varying quantity relevant to the spectrum of chain movementswithin a particular polymer. Results of stressfigure 2.1. in for tests types strain various of polymersare shown schematically Factors which affect the movementof polymer chains are responsible for changes in be listed as: of and can mechanicalproperties polymers "
Molecular weight: The first parameter to be considered is the molecular
weight, which reflects in many casesthe method of polymerisation of polymer and its origin.
A lower
in invariably a softer results average molecular weight
polymer.
N fn ui cr FN
STRAIN
Fig2.1 StressStrain behaviour of polymers: (a) hard, brittle; (b) hard, strong (c)hard, tough; (d) soft, tough; (e) soft, weak
II
Chapter Two
"
Literature Review
Branching or crosslinking: Any crosslinking between polymer chains will
push up the transition temperature. "
Crystallinity: Crystallinity in the polymer presentsan intermediate case, with
some of rigidity being retained through the stabilising effect of the crystalline regions which themselvesonly fail when the melting point is reached.The overall relaxation behaviour is affected by the restricted movement of those chains which are in crystalline regions, and a new type of time dependent response can arise through structural slippagewithin theseregions. "
Impurities: The presenceof impurities or low molecular weight additives such
as moisture or organic liquids, will produce a softening as well as a weakening effect. "
Temperature: It is not exceptional for a polymer to transverseall five of the
above classesof mechanical properties in a temperature range of no more than hundred degrees.For an amorphouspolymer, the biggest transition in mechanical properties takes place at the glasstransition temperature. "
Strain rate: The response of a polymer is affected by the rate of applying
strain especially just above the glass transition temperature, when an increase of deformation rate causesan increasein apparentmodulus and also usually gives rise to a more brittlelike failure of the polymer.
2.3.2 Mechanical properties of particulate filled polymers A filler may be primarily used as an inexpensiveextender, pigment or UV stabiliser. The produced composite must still have suitable mechanicalproperties for its intended application. Based on a simple analysisit may be thought that the influence of a particulate filler on the deformability of a polymer may be purely hydrodynamic. This means that, it merely distorts the molecular flow pattern during the deformation of polymer, in practice however, specific interactions very often produce enhancedstiffening effects beyond what is expected from an unfilled polymer. Figure 2.2a shows the stressstraincurves for a filled elastomer, while figure 2.2b and 2.2c show typical data for a particulatefilled composite having a rigid matrix. In
12
Chapter Two
Literature Review
both cases,as expected,the modulus increaseswith increasedfiller concentration.This may not always be the case, since if fabrication is accompaniedby extensive void formation then the modulus of the producedcomposite may decrease.It can be seen in figures 2.2a2c that the modulus increasesfor both soft and rigid matrices, while the tensile strength and elongation at break do not follow the same relationship. The tensile strength, however, particularly for a rigid matrix with rigid filler decreaseswith the increase of filler concentration. This is attributed to an increasedconcentration in interface locally formation the the the or at effect as well as of microcracks either is In the capable of which composite, case of a soft elastomericbased matrix. dispersing stress more effectively, the tensile strength may very well increase as a if in latter be filler higher A this case, reached concentrations. maximum will result of
STRAIN
STRAIN
(01
(bI
N
O O
CONCENTRATION (c)
Fig. 2.2 Stressstrain behaviour of polymerparticulate filler composites (a) soft matrix, hard filler; (b) hard matrix, soft filler; (c) hard matrix, hard filler
13
Chanter Two
Literature Review
for no other reason than that eventually the matrix continuity will be replaced by particle/particle contact. Thus mechanicalcoherence except for some agglomeration will be greatly reduced. On the latter point, it follows that improved dispersion can have an opposite effect on strength compared to modulus. Considering the other extreme property, i.e. elongation at break, it might be expected that this quantity will fall with increasing filler concentration. This is becauseproportionally, more strain is being applied to less polymer. However the use of soft fillers in a rigid matrix can give rise to an increasein elongation at break and also in impact strength. Part of this in deformed derived formation be from fillers the to may ability of some promote craze polymer prior to fracture. Debonding and crack formation generally lower the strength of composites. In certain caseshowever, where crosslinked low energy brittle plastics such as polyesters and from fracture is distinct involved, the strength, may actual energy which epoxides are be increasedby the presenceof filler. The fracture energy increasesup to quite high it decreases filler again. after which of concentrations
2.3.3 Nonlinear behaviour of polymeric composites It is usually assumedthat compositeshave a linear mechanicalbehaviour to avoid the for be hardly in This involved a can observed even nonlinear analysis. complexities is due following behaviour The to the material normally of a nonlinear pure polymer. factors: "
Non linearity due to material behaviour: Although the first approximation of
based linear is behaviour on a usually relationship between stress and constitutive strain, many common engineering thermoplastics exhibit a very nonlinear stressstrain relationship. In order to define a nonlinear stressstrainrelationship, a finite data by generally provided experiment, is considered. A set of stressstrain is interpolate then to used software all other stressstrainvalues that mathematical are required during a broad analysis. "
Non linearity due to large displacements: A customary assumption in most
is engineering analysis that the displacements are small. The elastic moduli of polymers are, however, usually as much as two orders of magnitude less than
14
Chapter Two
Literature Review
those of materials with simpler behaviour such as metals. Furthermore, plastics will undergo as much as an order of magnitude more strain before incurring damage. These phenomenacan often result in larger rotations and displacementsin plastic structures than in metals.
"
Non linearity due to the loaddeformation interaction: A third type of non
linear behaviour is the result of the interaction of deformation with the application in load distribution location load. Linear a that the a of and analysis assumes of is its deformation. This during do assumption not always valid, system not change become large. deformations especially when
2.4 THEORETICAL
MODELS FOR DETERMINATION
OF THE
MODULUS OF COMPOSITES The micromechanical analysis of the mechanical behaviour in terms of the separate inclusion) to the and two matrix the polymeric e. components(i. contributions of from This is only not arises complication complicated. mechanical properties in from but the filler uncertainties also the concentration, of recognised complexities itself interaction Especially interaction. vary the might of magnitude as magnitude of during into forced filler greater contact are mechanically as the polymer and deformation. In addition there are uncertainties in filler size distribution, complicated by possible agglomeration, and the extent of
void formation occurred during
fabrication, and the related problem of imperfect interfacial contact between the describe Nevertheless, the mechanical properties of filler. theories which matrix and developed. Often have been better filled theory one account gives polymers particulate is bulk depends Modulus than composites a of property which other. of one situation distribution, the the of particle size and modulus, concentration geometry, on primarily filler and has been representedby a large number of theoretically derived equations.
15
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2.4.1 Theories of rigid inclusions in a nonrigid matrix 2.4.1.1 Einstein equation and its modifications Einstein equation: One of the earliest theories for the description of the behaviour of a composite system was developed for elastomersand is basedon Einstein's equation (Einstein, 1956) for the viscosity of a suspensionof rigid spherical inclusions and is given by: (2.1)
"IC=llm(I+KEVP)
where 17, and r). are the viscosity of suspensionand the matrix respectively. KE is known as the Einstein coefficient and its value dependson the shape of the filler (it equals 2.5 for spheres)and VP is the volume fraction of particulate filler. It has been assumedthat a similar equation can be written for the shear modulus of composites (Smallwood, 1944; Hanson, 1965; Hashin, 1955). Thus we have:
G, =Gm(1+2.5vP)
(2.2)
is G the shearmodulus and p, m and c refer to particle, matrix and composite, where is filler be Einstein's In to the the action of a stiffening assumed equation, respectively. independentof its size. The equation also implies that it is the volume occupied by the filler, not its weight which is the important factor affecting the property of the composite. This equation is useful for low concentration of filler because it neglects the fact that by increasingthe volume fraction of filler the flow or strain fields around particles interact. The difficulties associatedwith defining these interactions has led to Einstein of equation. several modifications
Mooney equation: Mooney (Mooney, 1951) made use of a functional equation which must be satisfied if the final viscosity of a suspension to be independent of the sequenceof stepwise additions of partial volume fractions of the spherical particles to the suspension.For a monodisperse systemthe solution of his functional equation is:
16
Chanter Two
Literature Review
2.5 VP
Gc=GmeXp( 1SVP
ý
(2.3)
where S is the crowding factor that shows the volume occupied by the filler/true volume of the filler. Guth equation: Guth's equation (Guth, 1951) is an expansionof Einstein's equation to take into account the interparticle interactions at higher filler concentrations.
G2)
VP)
(2.4)
Thomas equation: Thomas's equation (Thomas, 1965) is an empirical relationship based on data generated with monodispersedspherical particles. The coefficients of different power series relating relative viscosity and volume fraction of solids were determined using a nonlinear least squareprocedure.
Gc=G, (1+2.5Vp+10.05Vp+Aexp(BVp)) where A= 0.00273 and
(2.5)
B= 16.6.
Quemada equation: Quemeda (Quemeda, 1977) introduces a variable coefficient to account for interparticle interactions and differencesin particle geometry. Thus:
G, =G,. (10.5K VP)2
(2.6)
where K is usually 2.5.
Frankel and Acrivos equation:
An asymptotic expansion technique was used by
Frankel and Acrivos (Frankle and Acrivos, 1977) to derive the functional dependence of effective viscosity on concentration for a suspensionof uniform solid spheres.The result containing no empirical constant, is intended to complement the classical Einstein formula which is valid only at infinite dilution and is given by:
17
Chapter Two
Literature Review
^^ l7c
y (`'/)
1ý(i,
=llmll+9o /t
ýý
/)V
(2.7)
where (Pis the maximum packing fraction of filler.
2.4.1.2 The Kerner equation and its modifications One of the most versatile and elaborate equations for determining modulus of
a
composite material consisting of spherical particles in a matrix, is due to Kerner. For Gp > G.
Kerner equation (Kerner, 1956) simplifies to : the ,
ý
11t=`jmýl
Nielsen
VPl5(lym)` vm(8lOvm)
(2.8)
have Tsai Halpin, 1969) Halpin (Tsai, 1968; shown that and equation:
be into for the the an can put elastic moduli of composite materials of equations most form the general as: equation of
(2.9)
G=Gm(1+ABVpý 1BVP
where A and B are constants for any given composite. The constant A takes into account such factors as geometry of the filler size and the Poisson'sratio of the matrix. The constant B takes into account the relative moduli of the filler and matrix phases, and is defined as: B_
(GmlGp)I (GmlGp)
+A
(2.10)
Neilsen (Neilsen, 1970) has extended this equation to take the maximum packing volume fraction into account and to point out the relation between the constant A and the generalisedEinstein coefficient k which depends on the Poisson's ratio. The equation that he finally derives is given as:
18
Chapter Two
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(2.11)
Gc=Gm(1+(k1)BVP) 1BVP
2.4.2 Theories of nonrigid inclusions in a nonrigid matrix The random distribution of the constituent phases in a filled system demands a statistical approach, but this requires a knowledge of the distribution of the individual phases.Consequently,the problem is usually simplified to a two phasemodel in which average stressesand strains are consideredto govern the behaviour of each phase. The average behaviour of the composite is defined in terms of a representative volume element. When subjected to a gross uniform stressor strain, a uniform strain field is induced in the composite which can be used to estimate the elastic constant. The other approachesconsist of the establishmentof bounds for the moduli by the use in theory of elasticity. criteria of energy
(b)
(a)
(c)
(a)
Fig. 2.3 Models for particulate filled composites (a) Parallel model (b)series model (c) Hirsch's model (d) Counto's model
2.4.2.1 Series and parallel models In the simplest case for a twophase material, the arrangementof the phasesis shown in figures 3.a and 3.b. For the parallel arrangement (case a), the uniform strain is
19
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Krock, in bound is by (Broutman the two the and assumed phase and upper given 1967): Gc= GmVm+ GPVP
ý2.12ý
in be two in b) the (case to the uniform stress assumed series arrangement whereas phases.The lower bound is G,
GPGm
(2.13)
GPVm+G. VP
In equation 2.12 it is assumedthat the Poisson'sratios of constituent phasesare equal. Whereas using equation 2.13 vv , the corresponding Poisson'sratio should be given by:
(vpVPGm+umVmGp) (VpGm+VmGp)
(2.14)
do 2.13 2.12 from bounds lower not often and The upper and equations obtained implies data. This of the a state either that of assumption the experimental represent is filled individual in the not system the phases of uniform strain or uniform stress describe the modulus. to sufficient
2.4.2.2 The Hashin and Shtrikman model Improved bounds for the modulus of twophase media were obtained by Hashin and Shtrikman who took into account the Poisson contraction of the constituent phases. The overall responseof the composite was assumedto be isotropic and linearly elastic. The equations for the lower and upper bounds of the composite modulus (Hashin, Shtrikman, 1963) are given, respectively, as:
9(K. +(VKrK.
)+(3V.J(3K. +4G.))xG'+TV(GrG.))+(6(K.+2G.)VJS(3K.,+4G.)G. )) E,3(K, VP V, +((Gr +(VK, K. )+(3V. 15(3K. )) )))+(G. 1(3K. Y +4G., )V. ))+(6(K. +4Gý)G. +2G. G. (2.15)
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V. V. 9(K' + + (yK.,K, )+(3V,/(3K, +4G,)))(G, (i/(G,G. ))+(6(j{,+2G,)VP/s(3K, +4G,)G,)) V. V) 3(K,,+ + l(3K. +4G,)))+(G, (V(G, G,))+cýK,+2G,)V,/s(3K.+4G,)G,,)., cyK.K, )+(3V
.
(2.16)
where K and G are the bulk and shear moduli, respectively. The corresponding Poisson's ratio of the composite in this caseis given by:
3K, 2G, V. =  2(G, +3K, )
(2.17)
The separation of the Hashin upper and lower bound is dependentupon the modular ratio of particle to the matrix(m= EP = E. ). When the moduli of the constituent phasesare closely matched, the bounds predict values within 10%.
2.4.2.3 The Hirsch model, Hirsch (Hirsch, 1962) proposed a relation for G, which is a summation of equations 2.12 and 2.13 and is written as: GPGm
Gt=X(GmVm+GPVP) +(1X)
GPVm+GmVP
(2.18)
The model is illustrated in figure 2.2.c. Parameters x and 1x are the relative proportions
of material conforming to the upper and lower bound solutions,
respectively. When x=0 equation 2.18 reduces to equation 2.13
which can be
identified with a poorly bonded filler. For the perfectly bonded filler, (i. e. when x=1) the equation reducesto equation 2.12.
2.4.2.4 The Takayanagi model Takayanagi (Takayanagi et.al, 1964) combined equations 2.12 and 2.13 and proposed a seriesparallelmodel given by:
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Chanter Two
Literature Review
Gc=(a+ (1
(1 a) P'Gm+ßGP
GP
)ý
(2.19)
ß a where parameters and represent the state of parallel and series coupling in the composite, respectively. Equation 2.19 was developed to predict the modulus of a crystalline polymer. The basic problem with this model is the determination of values for a and P. The arrangement of the series and parallel element is, however, an inherent difficulty in all of the proceeding models and there are conceptual difficulties in relating these models to real systems.
2.4.2.5 The Counto model A simpler model, for a two phase system is proposed by Counto (Counto, 1964) and The bonding between the the modulus of the and matrix. perfect particle assumes composite is given by:
i G,
_1vý+ (1Vg)I G,
1
(2.20)
VgG, +Gr
This model predicts moduli which are in good agreement with a wide range of in 0.5 be data. It takes that a value of x equation when should noted experimental 2.18 it coincides with the values predicted from equation 2.20.
2.4.3 Limitations of the theoretical models Equations 2.12 and 2.13 assumethat the individual phasesare under uniform strain or stress. In practice, however, the filler particles may not be completely separatedfrom one another and the reinforcement element may, on the microlevel, effectively be an aggregate of smaller particles. Thus in responseto the applied load the stress will be distributed unevenly between the particles and aggregatesand the assumptionof either uniform strain or stress is clearly an oversimplification. To account for the complex stress and phase distribution, Hirsch
differing Takayanagi considered and
combinations of the upper and lower bounds of the laws of mixtures. All of these
22
Chapter Two
Literature Review
require an empirical factor which is determined by a curve fitting routine to furnish a phenomenologicaldescription of the experimentaldata. Theories which deal with filled systemsindicate that the elastic modulus for a given particle and matrix dependsonly upon the volume fraction of filler and not the particle size. The modulus however, increasesas the particle size decreases. The properties of the composites may also be affected by changesin particle shape. This effect is especially pronounced with larger or nonspherical particles where a behaviour. deformation the particle preferred orientation can modify The particle size distribution affects the maximum packing fraction 0'.. Mixtures of densely differing than monodispersed particles can pack sizes more with particles becausethe small ones can fill the spacebetween the closely packed large particles to form an agglomerate. These aggregated particles may be able to carry a large higher the load same the to at than modulus, a the yield primary particles proportion of volume fraction. Most of the theories which explain the reinforcing action of a filler assume perfect imperfect The filler between adhesion the the case of polymer matrix. and adhesion Furukawa, (Sato Furukawa by Sato discussed however, and theoretically and was, 1963). They assumed that the nonbonded particles act as holes and, therefore, One the filler that increasing in decrease can argue content. with modulus predicted a holes the do they matrix also restrain since as entirely not act nonbonded particles from collapsing. A changeof the matrixfiller adhesionhas a smaller effect on modulus fact, In dependent is latter The surface on pretreatment. than on strength. much more the degree of adhesion does not appear to be an important factor as long as the frictional forces between the phasesare not exceededby the applied stress. In most filled systems there is a mismatch in the coefficients of thermal expansion which is from induced Brassell bond thermally stresses. resulting reflected as a mechanical (Brassell and Wischmann, 1974) found that the degree of bonding between the phases does not appear to have any influence on mechanical properties at liquid nitrogen temperature and this was attributed to the compressivestresseson the filler particle. In most caseseven if the adhesion between phasesis poor the theories remain valid as
23
Chapter Two
Literature Review
long as there is not a relative motion across the fillermatrix interface(no slip case) (Ahmed and Jones, 1990).
2.5 MICROMECHANICAL ANALYSIS OF POLYMERIC COMPOSITES It would be an impossible task to analysecomposite materials behaviour by keeping track of the strains, strain rates and strain gradients within and around each and every inclusion in the material. At the other end of the scale, we could simply assumethat the individual phasesdo not exist, measurethe macroscopic properties and proceed This design task(macromechanical approach, while analysis). the structural with ignores the main opportunity and challenge of composite materials, namely practical, desired features to and optimal achieve the and characteristics to tailor microscale the led Thus to the behaviour. of averaging problem we are naturally macroscopic behaviour to the and to macroscopic microscale effects and characteristics predict investigate the effects of
distribution and microstructure, particle size, particle
interface on the final properties (micromechanicalanalysis). deterministic both involves and The microscale geometry of composite materials is selected In size features. a cell analysis, with micromechanical proceeding statistical different for The are is done methods this scales scale. on and averaging approximately:
atomic, molecular
108109 m
microscale
105 m
macroscale
101_102 m
The microscale dimension reflects the typical filler diameters, as well as being dimensions. The is inclusion thus nearly microscale characteristic of many particulate behaviour. Understanding between in the the and macroscale atomic equally spaced the behaviour of composites on the microscale offers considerable promise for improving their bulk material properties. Obviously the cell size to be used in a volume An dimension. be larger than the characteristic microscale averaging operation must in the for limit the to the gradients cell size strain must relate macroscopic upper is dimension that In be particular, the cell size must small compared with a material.
24
Chapter Two
Literature Review
characteristic of the inverse of the strain gradient. Averaging is done with regard to cell sizes on the scale of the inhomogeneity. Further to statistical averaging certain features of the composites such as isotropic behaviour of system with no preferred orientation are usually used to develop microstructural models (Christensen,1982).
2.5.1 Qualitative description of the microstructure 2.5.1.1 Geometric models The procedure typically used to determine macroscopic properties involves the Most the the cell of material. volume of cell or element of a representative analysis inclusions fibres to the when or particulate cases of geometries apply equally well The in composite sphere model was spherical coordinates. or viewed either cylindrical introduced by Hashin and the corresponding composite cylinders models by Hashin is that 1964). Rosen A (Hashin Rosen such assumed of cells of sizes gradation and and is is fixed A filling assumed radii ratio of cylinders' configuration obtained. a volume be be the the taken to of representative that the single sphere can of analysis such entire composite system. There is a more complicated model known as selfconsistent scheme.In this model, by determined in solution of a separate the average stress and strain each phase are have is inclusion the to The the effectively assumed outside material problem. in figure The the shown problems solution of unknown macroscopic properties. 2.4.a then allows us to determine the macroscopic properties. A third major type of model is that of the three phasemodel, shown in figure 2.4.b, be inclusion by involves to the taking surrounded as a system an annulus of which in turn matrix material
embedded in an infinite medium with unknown effective
macroscopic properties. In a fourth model type a regular arrangementof inclusions is considered. In this case, single size cylindrical inclusions are taken as arranged in regular patterns, usually with either square or hexagonal packing. These models are usually used in finite element by distribution investigated Although has been analysis. nonuniform of particles also Guild and Davy (Davy and Guild, 1989) with application of a combination of finite
25
Chapter Two
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element and statistical calculations. The describedgeometrical models are usually used to formulate the finite elementanalysisof stressstrainbehaviour of composites.
phase 1
phase 2
(a)
(b) Fig. 2.4 (a) Self consistent model (b) Three phase model
There is no limit in how many microscopic geometrical models can be designed for in For behaviour to second order example, results applicable analysis. composite inclusions in interacting just by be fraction two the an of analysis can obtained volume infinite medium. At dilute concentrations, ellipsoidal inclusions can be used to inclusions directly Also, are also ellipsoidal represent a variety of geometric shapes. 1982). (Christensen, to the scheme selfconsistent related
2.5.1.2 Geometric model used in finite element modelling of composites Although usually a periodic distribution of particles are chosen, there is possibility of distribution through using statistical calculations such as the nonuniform studying model due to Guild and Davy. In this model, spherical particles of equal diameter are is distributed infinite be Finite to within an randomly matrix. assumed element analysis performed for a cylinder of resin, radius equal halfheight, R, containing a single sphere at its centre, radius r. This cylinder can be represented by the plane ABCD using axisymmetric elements(figure 2.5).
26
Chapter Two
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The value of the sphere radius, r, was kept constant and cylinder radius was varied. The statistical model has been used in order to calculate the distribution of the distance from a sphere centre to boundary of its Voronoi cell which indicates the interparticle distance(Christensen 1982).
B
A
C
r Fig. 2.5 GuildDavy model
However in most of the finite element analysis,a specific packing geometry is assumed in formulating a micromechanical analysis, dictated by the necessity of establishing Two be typically boundary the to geometries analysed. on region conditions suitable hexagonal figures 2.6. 2.6b, in indicated array and a square array a a and assumedare being special cases of these two geometries. By utilising a regular periodic array, a for isolated lines be dashed by indicated (such the that can typical repeating unit as detailed analysis. In fact if symmetry with respect to both the x and y axes is be lines) (solid to this repeating element needs of maintained, only one quadrant be boundary Because the which conditions must satisfied even complex of analysed. form is analytical solutions are not suitable. closed when such symmetry maintained, Of the several numerical analysis procedures which are available, e.g., finite differences, finite elements, truncated series and boundary collacation, the finite element method has emergedas the most generally useful (Hashin and Rosen, 1964). The material region to be analysedis resolved into an array of subregions. According to the order of interpolation and the type of element considered the strain variation be can modelled. across an element
27
Chapter Two
Literature Review
000 0000 0000 QQQ2a 000 0000
2a
2b
2b
Rectangular array
Diamond array
Fig. 2.6 packing geometries
2.5.1.3 Structural descriptors Various types of composite materials, ranging from aligned continuous fibre laminates to particulate filled systems,may be distinguished as special casesof microstructures described in terms of (i) aspect ratios (a, ) and (ii) the orientation of the reinforcing agents(f). The following comparison points out the relatively simple microstructure of ) ( laminates. fibre For the this system, continuity special material co continuous a,, 4 of the collimated (f=
1) fibres assures the simplifying condition that the strain
longitudinal fibres is As the to the a consequence aligned essentiallyuniform. parallel however, be from the variation the of mixtures; simple rule predicted may properties into be fibre direction fields taken the transverse the must account to obtain of of relationships which predict the transverseproperties and shearmoduli. Particulate filled systemsrepresent the next level of complexity becauseunlike fibres, have low fillers a aspect ratios, often approximating those of spheres or particulate plates. In most particulate filled composites, the reinforcing agents are spherical (or is The that the aspect so ratio unity. marked discontinuity of particulate spherical) near filled systemsintroduces significant fluctuations in the internal fields which complicate the analysis of properties. On the other hand, simple reinforcing geometry precludes a dependenceon the orientation of the reinforcing agent so that the consideration of this structural feature is not required in the analysisof these materials.
28
Chapter Two
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Symmetry
Microstructure
........................
............................
Orthotropic Transversley Isotropic
Aspect ratio > Orientation Aligned
continlous fibres
Isotropic
Aspect ratio =1 Orientation: Independent
orthotropic Transversley
I< Aspect ratio< Planar Random (f=0) < Orientation (f)
particles
Isotropic
< Aligned (f=1) Short fibres Fig2.7 schematic definition of the structural features
Shortfibre reinforced materials represent yet a higher level of complexity since into be fibre taken in both the must orientation aspect ratio and variation distinguished further be the to fibre according 'Short can systems consideration. (Sato Furukawa, fabrication by induced features and procedures microstructural 1963).
2.5.1.4 Size of particles The use of smaller particles results in rapid increasein the value of modulus. This may be due to a greater total surface of interaction or to a change in the value of maximum packing fraction. Agglomeration of particles, arising from strong particleparticle interactions, might dissociate trapped polymer from the main polymer matrix and in so doing produce an effect Of increasedstressfor a given strain.
29
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Larger particles give rise to greater stress concentrations and lower tensile strength, than the smaller particles. Where bonding is weak, then at some critical strain, debonding takes place and the composite exhibits opacity. But where a suitable bonding agent has been employed, a greater level of stresswill be required to produce breakdown of interfacial adhesion. In fact, if the interaction is extremely strong, fracture of matrix or even filler may occur first.
2.5.2 Interfaceadhesion Recently, there has been much renewed interest regarding the role of the interface in composite material behaviour. This is due largely to the realisation that any interaction occurring between the primary constituents must propagate through a common interfacial boundary. Intuitively, it is reasonableto expect that a better understanding improved lead interfacial design to the the of could and preparation of region finite interphase interfacial The volume of region representsan composite structures. bulk between those the of matrix and vary continuously properties material wherein bulk filler. Such an interface might be the result of processingconditions, for example, which impart the unique material properties to the region. Also the morphology of a in fibre. This be different ' to the the adjacent region matrix polymer or resin may quite bulk from different interphase that the of quite region with properties can give an interlayer interphase Alternatively, an of some composition may an encompass matrix. in improve into introduced is deliberately to the the composite structure order which load transfer properties of the interface (Brassell and Wischmann, 1974). An inconsistent interphasecausesa poor distribution of stressconcentration centres which results in the premature failure of the composite or growing of cracks. An optimal interphase coating maximises the composite strength. The level of bonding of inclusions to the matrix is also one of the dictating aspectsin load transfer. Interfacial bonding in composites can be divided into three levels: weak, ideal, and strong. Factors leading to a good polymerfiller bonding are as follows: "
Low viscosity of resin at time of its application.
"
Increasedpressureto assistflow.
"
High viscosity after application.
30
Chapter Two
Literature Review
"
Clean and dust free surface on filler.
"
Absenceof cracks and pores on filler surface.
"
Moderate roughening of filler surface.
" "
For impermeablefiller solvent basedresinsshould be avoided Use of resins less rigid than filler.
"
Similarity of the coefficient of thermal expansionof components.
2.6 FINITE ELEMENT MODELLING
OF THE MECHANICAL
PROPERTIES OF COMPOSITES The finite element techniques which are commonly used in solid and fluid mechanics can be broadly categorised as: Leastsquare, Weighted residual and Variational methods.
2.6.1 Finite element methods based on variational principles Generally in these techniquesa variational principle characteristic of the system under study is formed and minimised to obtain a solution for the system unknowns. This is the oldest of the finite element methods and it is developed by the engineers who by to complex structural section approach. for solve problems using a section wanted formed be simple variational on the basis of load or force principles can a solid system displacement relationships.Depending on the form of the basic governing equation of different be derived. These are called: displacement approaches can a system equations (stiffness), force (flexibility) and hybrid (mixed) methods.
Each of these approaches is equivalent to a variational principle that is, the minimisation of an appropriate system property. The three most commonly used variational principles are the principle of minimum potential energy (displacement method), the principle of complementary energy (force method) and the Reissner principle (mixed method). In all of these methods, it is necessary to identify the physical condition of the system. In any physical situation then, an expression for the total energy could be obtained and minimised to find the equilibrium solution. Now,
31
Chapter Two
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we consider different steps of displacementmethod as the most commonly used one in this category. A brief outline of the most important techniques based on 2D domains is given in the following sections.
2.6.1.1 Displacement method In the most commonly used form of this method a general two dimensional structure is divided into a number of 2D, triangular finite elementsand associatednodes. When fashion in deform that is load the a to the must all of elements structure, applied a deformation In forces between the of the elements. addition, guaranteesequilibrium of discontinuities in that to ensure the modelled structure must remain compatible order in displacementdo not develop at elementboundaries. for is developing the in element first to these expression The establish equations step in The displacements forces sequence the nodesof an element. at and stiffness, relating this process is as follows: is function This for function displacement the (i)Assume an approximate element. defined in terms of the displacementsat the nodes of the element and should its displacements entire along elements with neighbouring ensure compatibility of boundary. (ii)Apply the kinematic equations defining strain in terms of the approximate displacementfunctions. determine for to the material (iii)Use the constitutive relationship appropriate stressesin terms of strains. (iv)Develop equilibrium equations relating internal element nodal forces to forces. externally applied nodal Displacement function The displacement function characterises the displacements within an element as a function of space. The choice of displacement function affects the accuracy of the the behaviour displacement, in over strain, actual and stress element approximating linear is displacements, first derivative Since a strain a order of volume of the element. displacement function leads to the approximation of constant strains and stresses
32
Chapter Two
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linear function displacement Similarly, strain the simulates within element. a quadratic For fields the three node, triangular element, we will stress within an element. and designatethe x axis to lie along one edge of the triangle. With displacements(Ui, vi) in two coordinate directions (x, y) at each node(i) there are a total of six nodal displacements (degrees of freedom) in terms of which the deformation field for the in define displacements the be defined. In element the within order to element can terms of these six nodal displacements,functions with a total of six coefficients are is for A this element required. natural set of choices
u=a+bx+cy
(2.21)
fy + v=d+ ex
(2.22).
f d, b, where a, c, e, and are unknown constants.
Element displacements in terms of nodal displacements Using equations 2.21 and 2.22 to evaluatethe displacementat each node (i),
Vk=d+eXk+f
Yk
Uk = a+bXk+cYk
(2.23) (2.24)
(k=1,2,3)
A locations. known define can set of equations similar Yk coordinate Xk, and where be written to define the coefficients d, e, and f in terms of the nodal y displacements. displacements it is the to within and v u Using matrix manipulation possible represent displacements in as terms of nodal an element U; V; {u(xY)} v(x, Y)
_
(N]
U1
=[N]{S} Vi Uk Vk
33
(2.25)
Chapter Two
Literature Review
[N] is the shape function matrix that can be formed by using equation 2.23 the where {8}is 2.24, the vector of nodal displacements. and equation and
Strain as a function of nodal displacements The two dimensionaldefinitions of strain in terms of displacementare:
4:
(2.26)
ax 9v
ýy 
(2.27)
dy
au
av Yxy= +ax 02y
(2.28)
These relations are then used to calculate the strain within the element in terms of its nodal displacements.Using matrix notation again, this relationship can be expressedas r 41
{4}
4y =[B]{S}
(2.29)
Yxy Yxy
Where [B] is a matrix that can be defined in terms of derivatives of the shapefunctions 2.26,2.27 2.28. equations and using
Stressesin terms of strains In order to relate stressesto strains, a material constitutive model is necessary.For simple linear elasticity, the planestressconstitutive relations are
Qx 1ý
V7y) `ýx+
(2.30) (2.31)
cry
34
Chapter Two
Literature Review
Ylly_E
(2.32)
Y
2(l+ v) ý
where E is the elastic modulus and v is Poisson'sratio. Using matrix notation,
ax
[D] =
ay T zXri xy
4y
} [D][B]{S =
(2.33)
4
14 xy Y
2.32. 2.30,2.31 formed is [D] and the using equations material matrix where
Nodal forces in terms of displacements Load is transmitted from one element to another through forces at nodes of the in forces {F}. These two be coordinate nodal as represented can which elements, directions are related to the nodal displacementsthrough a set of element equilibrium by defined be the work These external equating can equations equilibrium equations. forces virtual to by set of nodal arbitrary an the when subjected nodal accomplished its in internal stress d{S}, the as volume displacements, element's to the energy stored field subjected to the virtual strain field resulting from the same virtual nodal displacements.This relationship can be expressedas
L, {F} (d{8})T =
T{6}dVol
(2.34)
1df6j
Since the virtual strains can be related to the virtual nodal displacementsas
d{e} = [B]d{8}
(2.35)
the element equilibrium equation 2.32 takes the form
f1 181T 131T [B]T {F} [D][B]dVol d =d
35
(2.36)
Chapter Two
Literature Review
equation 2.36 now takes the form of a relationship between the nodal forces {F} and the nodal displacements{S},
{F} = [K]e {S}
(2.37)
Where [K]e is the elementalstiffnessmatrix defined as:
[K]e = fv",[B]T[D][B]dVol
(2.38)
Global equilibrium Equation 2.37 establishes the relationship between the nodal displacements of an element and the corresponding nodal forces. When individual elements are joined at common nodes to model a structure, global equilibrium must be ensuredat each node. This requirement meansthat the summation of the forces associatedwith all elements attached to that node must be equal in magnitude and opposite in direction to the externally applied force at that node. To construct these equations, individual element stiffnesses are assembled using algebraic techniques into a global stiffness matrix representing the stiffness of the entire structure. This global set of equationsrelatesall the nodal degreesof freedom in the structure to the externally applied nodal forces. If the externally applied nodal forces are known, a solution for the nodal degreesof freedom can be obtained using linear algebra once the required boundary conditions are applied. When the displacementsof all the nodes are known, the state of deformation of each element is also defined. Thus the state of the stresses and strains can be calculated using 2.33. 2.29 However, since equilibrium is only guaranteed at a finite and equations number of nodal points in the structure, the finite element method is a numerical approximation rather than an exact solution. (McCullough, 1982)
36
Chapter Two
Literature Review
2.6.2 Leastsquare and Weighted residual method In this approach, instead of fording the variational principle explicitly, we start with the governing partial differential equations derived from the conservation laws of physics. These equations are used to formulate a functional statement. In the least square method the functional is the least squareof errors generatedby the substitution of the unknown function in the governing equation with an approximation. In the weighted residual methods, on the other hand a more general approach based on projection methods is used to derive a functional as a weak statement.Generally these functional statementsare obtained without directly determining an expressionfor the total energy of the system. The advantagesof these methods over the simple variational techniques are: be independent The the technique of considered as can a procedure mathematical physics of the problem under study. being concerned deal different is flexible It can with without and conditions more about proving different physical relationships. Consequently, the nonisotherm in be and each case the can studied easily condition and nonlinear cases corresponding equation will be added to the set of equations.
2.6.2.1 A brief outline of Galerkin method The Galerkin method has been judged to be the most powerful technique for differential finite of nonlinear equations (Oden, element representation generating 1972). The method is a special caseof the method of weighted residuals.Consider the nonlinear equation, Y(f) =P
where (f)
(2.39)
is a solution of equation 2.39 and Y representsa nonlinear operator. The
approximate solution of this nonlinear equation is assumedto be a combination of trial functions, say r
f=
YO, f, , =,
37
(2.40)
Chapter Two
Literature Review
The substitution of f into equation 2.39 gives
Y(f )P=
r(f)
(2.41)
where r representsthe error or residual due to the approximate solution. The method of weighted residuals involves the identification of a set of weight functions in some weighted average sense over the WW(f ). that so r(f) vanishes x,=,._,P , is domain 92. Generally the written requirement solution
fn W"'(.f )r(f )dSZ =0
(2.42)
Galerkin suggesteda rational choice of weight functions. In this method the weighting functions are chosen to be the same as the shapefunctions ( or trial functions). Thus, becomes 2.42 equation each element over
P
Y(ý o; f; ) P de =0
10; e
1=1.... The P. system of where
(2.43)
i_1
P equations generated above are then solved to
determine values of f;
38
Chapter Three
DEVELOPMENT OF THE PREDICTIVE MODEL
This chapter deals with the derivation of the working equationsand different aspectsof the development of a mathematical model to predict the behaviour of composites under various types of loadings. The main advantageof the penalty method which is used in this study is that it can cope with both fluid and solid state behaviour of composites. This is particularly important considering that polymeric composites are in liquid state under high stressand temperaturecondition.
3.1 MODEL EQUATIONS 3.1.1 Axisymmetric stress condition The problem of stress distribution in bodies of revolution (axisymmetric solids) under is loading axisymmetric of considerablepractical interest. Axisymmetric formulation is expressed in terms of cylindrical coordinates, r the radial coordinate and z the axial coordinate and 0 the circumferential coordinate. The basic hypothesis of axisymmetry is that all functions under consideration are independentof 0. That is, they are functions of r and z only. Thus
three
dimensional
problem
is
reduced
to
a
two
dimensional
Chapter Three
Predictive Model
one.(Zienkiewicz and Taylor, 1988) In this form of stress distribution any radial displacementautomatically induces a strain in the circumferential direction, and as the stressesin this direction are certainly nonzero, this fourth component of strain and its associatedstress need to be considered. Here lies the essentialdifference in the treatment of the axisymmetric situation. Equilibrium equations in r and z directions in axisymmetriccase are:
ßrr+aQrz+Qrr(Tog
ar
dz
(3.1)
=o
r
aa,=+aatt+az ar dz r
(3.2)
The matrix equation which definesthe stressand strain relations can be written as follows :
(3.3)
6=DE is defined the as: stress matrix where 6u Qn
(3.4)
Q= Coo ZR
and
1' E(1  v) D= (1+v)(12v)
VVO
1v' 1y,
1v' 1v 10
0 12v 2(1v)
and the strain matrix can be presentedas follows:
40
(3.5)
Chapter Three
Predictive Model
t)V
öz au
Ezz
f
E?
Ego
0u
YR
r
au az
(3.6) av
+
ar
where v is the Poisson ratio, u and v are the displacement components in r and z directions. By substituting the stress terms in the equilibrium equations by the above constitutive relations the model equilibrium equations in terms of the strain in r and z directions are found as:
a  Tr
(aT av,
avz
ý, ++
dz
r
(
aVr+ + rz l ar _a r V,.
+
a
2µ
ar
av,
+
ar
dz
21/ l'
+a
rz
ar
l
2µv,
Tr 
r
aVrl
aVt
2µ av,
r2
ýOv' avl+ , +µ+
r(.
av, +
dz
µ+
aµa
dz J az
r
dz
avZ ]=O ar
Vr 2v')]
+ ar dz
(3.7)
= 0(3.8)
The Galerkin weighted residual method is used to write the weak formulation of the in is Green's theorem order to reduce the order of the and applied above equations integration. Then the weak forms are obtained in r direction as : f [)(NiNi 0
r2
aNf+aNj
äNr+Nr +N1 r
dr
r
dr
dr
aNrý ai
+µ( +ýý« n
r2
NJ aN; +aNjaN; )l U; rdrdz + ar är az ai
aNj aNi )I Vrrdrdz +r Ni +µýaz ar dz dz
aNj äN, ar
2Nj N;
1a
Ni)
(P r
rn, ý, NiaNi rn, AN, dr
dz
41
Nin, )Uiý
Chapter Three
Predictive Model
aNf ý(µN; A rnZ N; ar
aNJ dz
r
)V, ý'=0 n,
(3.9)
and in z direction as below:
aNiaNr (aNjaNi +1NjaN; )+ µ( )] V, rdrdz dz ar dz ar az r aN aN! aNf aNr )+ µ(2 Vr rdrdz ý ý)l
jýtc +n aZj +f (µNraý r
jrnrýNraä
az
jrnZýNsNjn=)UrdI'
+f (211Ni
aNi
r
a
aNf rn=+iuN;
dr
INr rn, ,
aNj rnZ)Vrdl=O dz
(3.10)
In the above equations µ is shearmodulus, X is penalty parameter,nZ,nr are components of unit vector, N1,Nj are the weight and shapefunctions. An equivalent formulation for the stress distribution can be derived starting from the Stokes flow equation for incompressible fluids in conjunction with so called Penalty method using appropriate penalty parameter.For steady state and axisymmetric flow the equations of motion and continuity are:
aP+µ laav, az
aP ar+µ
ar
rar(r
)+
a2vt+PgL=O
az2
alaa2v, ))+
ar(rar(rv,
j+Pgr a2
(3.11)
(3.1Z)
Z
avr+vr+avZ ar
r
dz
(3.13)
where v,, v, are velocity components,P is the pressure, u representsfluid viscosity, p is fluid density and g,, g, are the components of body force vector.(Huebner and Thornton, 1982)
42
Chapter Three
Predictive Model
In a penalty method approach the pressure is eliminated as an unknown field variable through the use of a penalty parameterand modified momentumequationsare solved for the velocity components.The pressureis representedby
P/1
aV+Vr+avZ Or
(3.14)
r}z
r
A> (i. is incompressibility 0 This that the e. the condition a parameter. means where In be treated the a can as constraint equation. a on momentum continuity equation) A in is flow if have large the the to value specified a numerical parameter viscous solution, the incompressibility condition will be satisfied. The main advantageof penalty formulation is that the additional variable P is eliminated and the working equation take final form be form. The matrix can written as: compact a more
ivZ
[[C]+[K]+A[L]]
vr
Rv` = Rv.
(3.15)
For an incompressible flow we must seek a solution to the above matrix equation as A.ý ca. Since the matrices [C] and [K] are finite, as A becomeslarge the solution tends to . AjL]
= Rv, Vv,
(3.16)
The special consideration that is required in the penalty function approach is that [L] Most be matrix. commonly used conforming elements produce a a singular must integrals if [L] the are evaluated exactly. The procedure used to make [L] nonsingular singular is to evaluate [L] approximately by using reduced Gaussintegration. Now if we define the penalty parameter as follows, the equation of flow will result in the equations similar to the equations of equilibrium. ý =2vµl(12v)
43
(3.17)
Chapter Three
Predictive Model
In our study we have basedour model on Stokes flow equation and the penalty method. This gives us the flexibility
to switch the model from the analysis of flow to solid
material deformation under applied load. Since most composite materials are in fluid state at the processing time and are in solid state when they are used, this approach offers the advantage of the ability of predicting the material behaviour in both cases through one model. When establishing the element mesh for axisymmetric problems, care should be taken to avoid positioning elements in such a way that two nodes have the same or nearly the difference If the two calculated radial coordinates are close, same radial coordinates. between them may be grossly in error, and if r; = rj some of the integrals in equations 2.9 and 2.10 becomeinfinite. Another problem can arise in casewhere nodes he on the z in in infinite because (i. terms the working the this appearance of result axis, e. r=0), hole be introducing This by the the along core can avoided a small scheme. of equations The be low to the that zero. would normally radial coordiantes values axis and assigning be the displacements to the to then actual simulate core are set zero along radial displacement at r=0. of zero radial condition 3.1.2 Plain strain condition In the plain strain problem the displacement field is uniquely given by the u and v displacementsin directions of the, orthogonal Cartesianx and y axes. It should be noted that the only strains and stressesthat have to be consideredare the three components in the xy plane. The stress in a direction perpendicular to the xy plane is not zero. However, by definition, the strain in that direction is zero, and therefore no contribution to internal work is made by this stress,which if desired can in fact be explicitly evaluated from the three main stresscomponentsat the end of all computations. Equilibrium equations in x and y directions in plain strain case: aý +a6xy
(3.18)
+aßyy dy
(3.19)
dy
aý
44
Chapter Three
Predictive Model
The matrix equation which definesthe stressand strain relations can be written as follows (3.20)
6=De where the stressmatrix can be defined as: Uzu 6=
(3.21)
6yy,
zz
1v0 D
E(1 v) (1+v)(12v)
1v
v10 1v
(3.22)
12v
00
2(1v)
follows: be as the strain matrix can presented and
,=ý
(3.23)
dy
au dv az
jý
By substituting the stress terms in the equilibrium equations by the above constitutive in in directions the terms strain the of x and are y equations equilibrium model relations resulted as:
a
ýý+ a
A aY
ý"
vy
ý'
ý+a +(2,4 a vy
vy + + 29 aY ý'
lft(avx ý'
ý
)+{x+x1 aY
45
(3.24)
+ý=0
=o
(3.25)
Chapter Three
Predictive Model
The Galerkin weighted residual method is used to derive the weak formulation of the above equations and Green's theorem is applied in order to reduce the order of the integration. Thus the weak variational forms
are obtained in x
and y directions,
respectively, as : ýJ.
ý`+µaNJ.
f [(ý+2µ)
ay
n
aN']V, dy
dxdy aNj
äNr+
aNi az ay t +J; n
ýJnxA
+f (11NraNJny+2µN;
Ni
ý'
r
ay ax
Urdxdy
ýJnx)UrdI'
1
+ý(µNiaa r
u
aNrI
VI
1 rnZ,
vL
(3.26)
)Vrý'=0 n.
and
dN'
Jµ ny
Ni d +ý a]
aýN'
d ayN'
Vi dxdy f [ttaNi. n
ý'+(A,
a1Vj PAM rnz+µNiaNj TrnrANi rý
aNj
Iý(ý.lNtaNýrnrýNi az r
aN']U; +2, u)aNJ ay
ay
dxdy
ýýrnz)Uidr'
46
rnz)Vidi'=0 dz
(3.27)
Chapter Three
3.2 MODEL
Predictive Model
GEOMETRY AND BOUNDARY CONDITIONS
3.2.1 Modelling of the particulate filled composites In the finite element approximation of axisymmetric solids, the continuous structure or interconnected by is at nodal of axisymmetric elements a system replaced medium filled is It that the with particles( assumed to possess composites assumed points. 3.1. When in figure by this be unit a. shown unit cell a approximated could symmetry) is in hemisphere AD, is 360° embedded a cylinder produced. a around axis cell rotated (Zienkiewics and Taylor, 1989) The interparticle spacing is equal to 2(rir2). The volume fraction of the filler can be For from the a square or cubic array the relation of volume r2/r1. ratio calculated fraction and the ratio r2/rl is defined as:
n r2 3 Vf= 6(rý)
(3.28)
is it hexagonal for modified as: array and
Vf _n 3ýýriý
rz 3
(3.29)
This form of axisymmetric representation of the composite only approximates its real but These actual are are not repetitive cells units axisymmetric packing and structure. fraction interparticle The dimensions in the to their spacing. maximum volume related for hexagonal and square array in this case is 0.74 and 0.52 respectively. (Agrawal and Broutman, 1973) For rod like particles, the unit cell is defined as shown in figure 3.2.a. When this unit cell is rotated 360° around axis AD, a cylinder embeddedin a cylinder is produced. Different parametersused are:
47
Chapter Three
If
Predictive Model
length, diameter fibre the and radius rf: of ,df,
1mdm,r, length, diameter and radius of the cell : af
fibre aspect ratio:
af=lfld
Sf
fibre tip spacing:
l,
xI
fibre spacingparameter:
l,
f
if=sflf If=zfdf
Then the dimensions of the cell are related to the volume fraction of the fibres in the composite through the expressions:(Berthelot et al, 1993) (rfr,) 2_1 V f(1+xf/af)
(3.30)
The boundary conditions imposed for analysing the tensile load applied on the follows: filled (Christman et al, 1989) are as composite particulate VZ=0
on Z=0
(3.31.a)
VZ=V
on Z=bo
(3.31.b)
Vr =U
on r=Ro
(3.31.c)
Vr=O
on r=0
(3.31.d)
Here V is a prescribed constant while U is determinedfrom the condition that the lateral i. traction rate vanishes, e. average b, Jo. dZ=O
on rR.
(3.32)
0 These conditions can be applied to unit cell of spherical particulate and rod shape particulate filled composites shown in figure 3.1.a and figure 3.2.a respectively. In addition to the boundary conditions 3.31.ad, there is the requirement that displacement components vanish on the surface of the rigid fibre.
48
('ir<
I 'i '(IR IIv'e Model
ýz A
u,
L3
vo 7
C
C I
Fig. 3. La: Axisymmetric unit cell for spherical particulate filled composite
Fig: 3.1. b: Mesh for spherical particulate filled composite
1II, 7 0
rm
II c
F
Fig. 3.2. a: Axisymmetric particulate
unit cell for rodlike
filled Composite
Fig. 3.2. b: Mesh for rodlike particulate filled composite
1O
Chapter Three
Predictive Model
In some calculations, a second set of boundary conditions is employed consisting of 3.31.a and 3.31.b, but with 3.31.c and 3.31.d replaced by aR =0 on r=R,. So that every does is free. Under the these the cell along r=Ro stress of point outer sidewall conditions not remain straight and vertical. Of course, for the entire tension specimen,aR =0 on the d 3.31. boundary Relaxing boundary 3.31. the c and of specimen. conditions outer highly from in deviations highly this a approximate manner, consequencesof permits, 3.31 based 3.31. be fibre distribution Predictions to are c and on explored. constrained .d based "with to on art =0 on r=Ro are constraint" and predictions as result referred referred to as results "without constraint". To satisfy the first set of boundary conditions (with constraint) , the following is procedure used. 1) The stressand displacementdistribution is found such that:
on Z0
(3.33.a)
=1 vzt
on Zb.
(3.33.b)
vr,=0
on r=Ro
(3.33.c)
on r=0
(3.33.d)
=0 vz,
vr1= 0
2) The stressand displacementis found such that : on Z0
(3.34.a)
vz2 =0
on Zbo
(3.34.b)
vr2 =1
on r=Ro
(3.34.c)
Vr2 =0
on r=0
(3.34.d)
vz2 =0
3) These stress and displacementdistributions are superimposedto obtain
6=a,
+k 62 50
(3.35)
Chapter Three
Predictive Model
and (3.36)
v=Vl+kV2
is be is determined force in k direction that the the zero. such along net r where (Agrawal et al 1971) thus ,
(a,, +k (Tr2)dZ= IBCJ(6º1+k ar2) BC=0
(Fr) BCf
(3.37)
BC
so that r1
k=(2
Qr2
(3.38) BC
The stresson AB is thus ßrl 
\QL/AB
l6zIJAB
(6z2)AB

ar2
(3.39)
BC
is displacement the and ßrl2
(uz1)AB
(uz)AB
(UZ2)AB
Qr
since
(UZI)AB 
(3.40)
BC
(UZ2)AB =0.
3.2.2 Fibre reinforced composite The plain strain analysis can be used for the fibre reinforced composite. Figure 3.3.a The fibre for the composite. of reinforced the array configuration cubic unit cell shows following expression defines the relation between the unit cell dimensions and the volume fraction: Vf=
7i r2 4r l
(3.41)
Figure 3.4.a shows the unit cell employed for 'the hexagonal array configuration. Numerous choices exist for the unit cell geometry. The unit cell tested is rectangular in
51
Chapter Three
Predictive Model
shape with dimension 2L1*2L2, with L2= NF3L1. Then the volume fraction of the fibres is related to the cell geometry according to
nR3 Vf=V(L)
(3.42)
The maximum volume fractions for the square and hexagonal array in the fibre reinforced composite analysisare 0.907 and 0.785 respectively. (Gibson, 1994) Using the square array and hexagonal unit cells the composite response to transverse tensile and shearloading is examined.
2.2.1.1 Tensile loading The boundary conditions imposed for analysing the tensile load applied on the fibre follows: are as composite reinforced
on y=0
(3.43.a)
vy =V
on yYo
(3.43.b)
Vx =U
on x=Xo
(3.43.c)
Vx =0
on x=0
(3.43.d)
vy =0
Here V is a prescribed constant while U is determined from the condition that the i. lateral traction rate vanishes, e. average b. 1 Qxxdy=0
on x=Xo
(3.44)
0
These conditions can be applied to unit cell of square and hexagonal array of fibre reinforced composites shown infgure 3.3.a and figure 3.4.a.
52
('Iril)lrr'I'l
I'rrýlirlivi'
it
Mýx1ý I
`4i
A
13
Y
v C
0 x
Fig. 3.3. a: Unit cell of square array for fibre reinforced composite


Fig. 3.3.b: Mesh for fibre reinforced composite (square array)
I..,
13
1
[_, Y
ý\ 0
C
x
Fig. 3.4. a: Unit cell of hexagonal array for fibre reinforced composite
Fig. 3.4. h: Mesh for fibre reinforced composite (hexagonal array)
Chapter Three
Predictive Model
In addition to the boundary conditions 3.31.ad, there is the requirement that displacementcomponents vanish on the surface of the rigid fibre. I.
3.2.2.2 Shear loading The responseof the fibre reinforced composite to transverseshear load is simulated with the following boundary conditions imposed on the unit cells shown in figure 3.3.a or figure 3.4.a:
vy=o
on y=0
(3.45.a)
vx =V
on yYo
(3.45.b)
Vr=0
on x=Xý
(3.45.c)
on x=0
(3.45.d)
vx_0
vy =0
Here V is a prescribed constant in addition to the boundary conditions 3.31.ad, there is the requirement that displacement components vanish on the surface of the rigid fibre. (Eischen and Torquato, 1993)
3.2.3 Slip boundary conditions When there is no slip at the interface of the matrix and filler, the relative displacementof the two phasesis the sameas zero. Whereas in the case of slip the relative displacement is described by Naviers boundary is the slip two condition the and not zero phase of boundary kind is (Robin third condition: convective) or relation, which a
0 t= (ßr.n+v).
(3.46)
Complementary with the above boundary condition is the following relation which displacement is there relative radial no of the two phase. ensures
(3.47)
vn=0
54
Chapter Three
Predictive Model
Where:
ß=
slip coefficient
ti = stresstensor n= Unit vector normal to the interface t= Unit vector tangential to the interface v=
relative displacementof the two phase
From the slip boundary equations and the usual stress and velocity relations the components of the slip velocity are obtained as follows:
19v1 av2 avl av2 +v1=µßn2[2(ax1 ax1)in2nl)] _ ax2)n1n2+(ax2
(3.48)
avl v2  µßnl[2(axl
(3.49)
_
av2
_ax2)nln2+(ax2
avl+ av2
axl)(n2
)l nl
We discretise the domain and write the weak formulation of the slip velocity relations. The interpolation points are chosen inside the elements adjacent to the slip boundary. The resulting stiffness matrices are assembledwith other elemental matrices obtained from the flow equations, to form the global stiffness matrix representing the entire domain. The matrix representation of these equations for an element located at
the slip
boundary results in the following equation:
aIl
a12
v1j
0
aql
v2j aY2
0
The membersof the abovestiffnessmatrix are givenas:
55
(3.50)
Chapter Three
Predictive Model
aV1 =ýNINjdS2+2n1n2
f, ußNiNj, Q
1dS2 2+(n
aV2=2n1n2
JußNiNl, S2
ný)1 uQ Ni N j, 2 dSZ
2dS2
+(n2ni)n2
ay1=2n2n1
JµßNiNj, S2
1dSZ
f µ/3NjN. j, 1dS2
(n2ni)n1 aZ2 =ýNiNjdS2+2n2n1
JµßNiNj, S2
2dS2
ýµßNiNj,
'(n2'ni)jýßNiN. 92
56
(3.51) 1,1dS2
Chapter Three
Predictive Model
3.3 CALCULATIONS 3.3.1 Composite modulus of elasticity and Poisson's ratio To calculate the stiffness or modulus of elasticity of the filled composite, the average stresson the boundary AB is calculated: (Agrawal et al., 1971)
Ja=d4 (Q: =A= Q= A
)Aa
(3.52)
where A is the area of the top of the cylinder in the finite element analysis and the integral is replaced as a summation as follows: rl
f
e
f
1/2(rirr1)Q:
QZdA=2n ßZrdr=2nE
(3.53)
! =1
A0
where r; and r1.1are the radii to the nodal circles that define the elementson the top of the cylinder, n is the number of such circles, and aZ is the corresponding normal stress in each element. The modulus is defined as E=aZ/e. where the strain used is calculated from the specified I. boundary displacement, sZ = (UZ)AB BC Poisson's ratio was calculated from the displacementsusing the equation u= ul + ku2. The displacement of boundary AB is (uZ)AB= (uz1)AB and the displacementof boundary BC is (Ur)BC= (url)BC+ k(u,2)BC but , , (ur,)Bc= 0; thus, (Ur)BC=k (ui2)BC.Poisson's ratio can be written as
I (Ur)BCI
l
AB
v= l(UZ)ABI l
_I
BC
andsince(un) _ (uZI)AB =I
57
M(Ur2)BCBC (uzl)AB
AB
(3.54)
Chapter Three
Predictive Model
(ký Jkj(BC= AB) vAB
(3.55)
3.3.2 Composite strength
In order to calculate the composite strength, it was assumedthat the composite would fracture as soon as an element of the matrix reached a large enough value of stress to cause fracture of the matrix. Since the matrix is subjectedto combined stresses(triaxial) a suitable failure criterion has to be used in order to predict matrix failure under combined stresses. The Von Mises failure criterion or distortion energy theory was Broutman, 1972) (Sahu and selected. This criterion is then applied by determining which element has the maximum value of distortion energy for the applied stress. This value'of energy may not exceed the value and thus the composite strength is calculated
(a') fail the to material matrix needed from
Sc=6Z
where U
ßys
(Uý)
v2
(3.56)
is the maximum value of distortion energy determined for the arbitrary
is S, displacement the composite the and stress which produces average a specified is by The the assumption the affected particularly of strength results accuracy strength. that composite failure occurs by the first matrix failure. 3.3.3 Stress calculations( Variational recovery)
The displacement formulation has frequently resulted in an unrealistic stress prediction, jumps if interelement the true stresseswere continuous. Resort to stress even giving is frequently stresses of element made in practice to make the results nodal averaging more meaningful to the user. However, it is possibleto obtain a better stresspicture by a projection or variational recovery process which in itself is another way of applying a mixed formulation.
58
Chapter Three
Predictive Model
In this method we obtain displacementsu by an irreducible formulation. The stressesa are approximated, (Zienkiewicz and Taylor, 1989) (3.57)
6= DBu We compute a set that is interpolated by
(3.58)
a= NQa We in to a. the write this approximation as sense approximates weak which faN
(3.59)
f Q)di2=0
or (JnNäNQdS )6=(fýNQDBu)dfl.
(3.60)
least is to the the It is interesting to note that the projection of above equation equivalent square fit or minimisation of II

fn(Q
Q)2dS2
(3.61)
fit least of square fact a thus are The that smoothed and more accurate, stresses location the or sampling which to the at of points as a clue provides stresses computed is evaluation of stress optimal. Stressesat the interface of the spherical filler and continous fibre are calculated from the following equations: CFO= QzCOS20 +a,
sin T, sin20 z
20 (hoop stress)
20+ 0+ 20 (radial stress) sin Cr cost zR a = aZsin T
cos = 'Crt rO
20 2
(o
(shear 20 stress) sin Qz) 
(tangentail stress) 6Wm= a'#
59
Chapter Four
RESULTS AND DISCUSSION
INTRODUCTION In this chapter the numerical analysisresults are presentedand the following topics are discussed:
"
Modulus of composites
A general review of modulus of composites found by numerical analysis and other different in The is and assumptions made strengths, weaknesses methods presented. methods are discussedand compared.
"
Composites filled with rigid particles
The imposition of the tensile load on a composite filled with particles much stiffer than the matrix is simulated. The modulus, stressdistribution and the strength of these types is The by tensile compared predicted. stress model are obtained our of composites diameter includes Papanicolaou the the taking the model which particle effects of with
Chapter Four
Results And Discussion
into account. The tensile strengthspredicted by different models are compared with the outcome of our model.
"
Composites filled with debonded rigid particles
Composites with a weak interface between the filler and matrix which are susceptible to interfacial crack formation are studied. This condition can be distinguished as between bonded inclusion. is bonding Another the there case when arises no partially inclusion and the matrix. In this latter case the slip boundary condition is imposed on the section of the interface which remains closed. The state of stress and displacement fields are obtained for both cases. The location of any further deformation through is identified Completely band formation tip. unbonded crack as a or shear crazing inclusion with the partial slip at a section of the interface reduces the concentration of the stress at the crack tip.
The effect of debondingon the strengthof the compositeis studied.
"
Composites filled with soft particle
The behaviour of the rubber filled composites under the tensile load is studied. The filler importance different the the use properties on the outcome of of of effect and the numerical analysesare discussed.Two types of toughening mechanisms,namely shear banding and cavitation,
are identified and the fracture behaviour of the
composites is studied.
"
Continuous fibre reinforced composites
The responses of the fibre reinforced composites under tensile and shear loads are investigated. The Young's modulus and shear modulus are calculated. The stress distribution at interface and throughout the matrix is analysed.Different failure modes are defined and discussed.
"
Short fibre reinforced composite
The effect of different geometrical parameterssuch as aspect ratio and fibre spacing on the modulus of the short fibre reinforced composites is examined. The critical fibre
61
Chapter Four
Results And Discussion
length for achieving the maximum reinforcement is described.The interfacial shear and tensile stress are predicted and compared with the shear lag and modified shear lag models. The model equations are modified by considering the boundary flux terms. The model predictions for the composite properties and stress fields are obtained with and without boundary fluxes. The possibility of the composite failure either by interfacial debonding and fibre breakageis investigated.
62
Chapter Four
Results And Discussion
4.1 MODULUS
4.1.1 Equal stress and equal strain bounds Modulus is a bulk property of composites that depends primarily on the geometry, modulus, particle size distribution, and concentration of the filler. The Hashin's bounds and rule of mixture bounds are compared with our prediction in figures 4.1ac for a particulate filled composite. The rule of mixture bounds predict that the modulus of a two phasecomposite should fall between the upper bound of.
E, =VmEm+VfEf (4.1) bound lower the of. and E, _
E, Ef
(4.2)
V,Ef+VfE,
The epoxy matrix and glass beads(filler) properties, which are used in our model, are 4.1. in table given Input Properties Phase
E (GPa)
Poisson's Ratio, v
Matrix
3.01
0.35
GlassSphere
76.0
0.21
Table 4.1. Elastic Material Properties Used in the Predictive Models
The modulus predictions agree well at low filler volume fractions. At
higher filler
volume fractions the Hashin's bound and rule of mixture bounds become widely limited they therefore are of and predictive value. However, these bounds can spaced, still serve as a useful test for the approximate theories, since any solution outside these bounds must be regarded as invalid. From a mechanical viewpoint, the upper bound in the situation represents a two phasematerial in which both phasesstrain equally.
63
('hahtcr Four
IE+IOI
kC. tillll.
l
And I )isru. ý.tiiýni
ýFEM
IF,+10
Mixtur© Upper Bound
IE+IO
Mixture Lower Bound
0
RE+09
1/
a 6E+09
E 0 U
4
2G+09
60 40 Filler Volume Fraction(9b)
20
0
SO
100
3E+10
3E+10
FEM
2F, +10
Mixture Upper Bound  Mixture Lower Bound
,a 0
2E+10 E IE+10 0 U
5E+09 0E+00 A
'T
100
60 40 Filler Volume Fraction(%)
20
0
8E+I0 af
FEM(Constrained)
<_ý  FEM(Unconstrained)
2
Mixture LLower Bound
Mixture Upper Bound Hashin lower bound

Hashin upper bound
6E+ 10 0
b 4E+I0 N 0
2E+I0
0E+00 0
20
4(l Filler Volume
60 Fraction( %)
90
Fig4.1: Comparison of the FEM modulus predictions of filled composite for different relative properties of filler and matrix with rule of mixture bounds and Hashin's bounds.
64
1(x)
Chapter Four
Results And Discussion
The lower bounds representthe casein which the phasesare stressedequally under an is load. This typified by particles in a matrix, with no hydrostatic situation applied stress present. The relative position of the modulus values in the bounds depends on the relative properties of the particle and the matrix. When a twophase particulate filled composite is deformed under an applied load, the matrix, which is usually softer than the particulate reinforcement, tends to deform at lower stressesthan the dispersed it is bonded harder However, the to the cannot since particles, matrix rigidly particles. deform in the samemanneras it would in the absenceof the filler. The restriction of matrix deformation imposed by the hard reinforcing particles results in the generation of a hydrostatic state of stress in the matrix. The magnitude of this hydrostatic constraint and the determination of whether the generated stress is purely hydrostatic depends on the relative clast properties of the two phases (i. e. elastic between hardness disparity As the Poisson's the and of phases). ratio and modulus in hydrostatic becomes the these the two composites greater, phases strength of it in be does factor the composites with phases as as cannot extended much constraint having closer properties. Therefore a greater relative modulus of particles and matrix ie; lower bound, in the to closer equal stressprediction. results a composite modulus In figures 4.1ac the modulus of composite has been predicted by the finite element It be different for three can and matrix. seen that as of particle relative moduli model the relative value increasesfrom 2.14 in figure 4.1a to 25 in figure 4.1c., the modulus from bound filler fraction to the the shift upper equalstrain of volume range values at the lower equalstressbound. The bounds are more widely spaced when the relative bounds in Hashin However, the are spacedcloser than rule of all cases modulus rises. bounds. mixture The advantageof the finite element prediction is that it can be used for any shape of interface conditions. particles and The lower and upper bound assumethat individual phasesare under uniform stress or from however, filler be In the particles may not completely strain. practice, separated be the the an reinforcement element may, on one another and microlevel, effectively be Thus in load to the the particles. aggregateof smaller response stresswill applied
65
Chapter Four
ResultsAnd Discussion
Modulus Volume fraction
2.0E+10 HexagonalArray(Constrained) A
ý1.5E+10ý
Experimental Data
ýý ^. PC
4
Hexagonal Array(Unconstrained)
m
A
0
:ý1.0E+10
0
Q ... ... 0 a
A
E S.OE+09
U
A, ýý 0.0E+00
1irý0
m 0
A
ý0 EFºrºri; 20 40 30 Filler Volume Fraction(%)
10
60
50
Fig. 4.2: Variation of the tensile modulus versus volume fraction for particulate filled composite. Results of FEM are for hexagonal array with constrained and unconstrained boundary conditions.
Ec/Em  Ef/Em
3.8 3.6 3.4 E 3.2 3 Ici 2.8 ez2.6 w 2.4 2.2 2
ý
0
ý
10
20
30
H40
i
50
Relative modulus (Ef/Em)
Fig. 4.3: Effect of Ef/Em on the EdEm for particulate filled composite. Volume fraction studied is
66
Chapter Four
Results And Discussion
distributed unevenly between the particles and aggregatesand the assumptionof either be (uniform distribution) an overstrain stress or particle size will uniform and simplification.
4.1.2 Model predictions and experimental results Accurately measured valuesof Young's modulus for epoxy resin reinforced with glass from literature. These experimental results agree well with our are available spheres finite element predictions as shown in figure 4.2. The differences can be attributed to the assumptions that have been made in the model. The discrepancy between model factors following ignoring be due the to the experimental results may predictions and in the model.
4.1.2.1 Agglomeration The filler particles may aggregate to form agglomeratesmuch larger than the filler filled their that Agglomerates tend to spaces so contain and air voids particles. filler is the the true than volume of material. greater apparent volume considerably If the agglomeratesare hard and have appreciablemechanicalstrength, they will not filled filler, the broken be the added as of material weight and considering easily Because have the volume of the than greater expected. a modulus material can filler Soft is larger in the true than the and this of particles. volume case aggregates be hand, to to disintegrating the would expected give rise on other aggregates, easily higher have The a considerable at effect agglomeration of particles an opposite effect. higher factor become filler fractions this therefore more significant at and of volume volume fractions.
4.1.2.2 Dewetting At high concentration of fillers, all the individual particles might not be wetted by in a poorer resulting matrix phase
dispersion of the filler within the composite.
Instead, the particles tend to aggregate.Dewetting and poor dispersion are amongst
67
('Imhtcr
Foui
Itcsults And Discussion
ModulusVolume
Fraction
Particulate Filled Comlmsite
2.0E+10
, 0 Square Array(Constraincd) 0n
,ä.
1.5E+10
Hexagonal Array(Constrained)
ý
(ID v O 1.0E+10
Square Array(Uncons(raincd)
.ý
aHexagonal Array(Unconstraincd)
40
LIN
.ý 0A
j 5.0E+09 V'R
.ý "o
O.0E+00 4 10
0
40 30 20 Filler Volume fraction(%)
Fig. 4.4: Comparison of FEM results for the modulusvolume fraction variations hexagonal arrays, constrained and unconstrained boundary conditions.
50
60
for square and
ModulusVolume Fraction Fibre Reinforced Composite
9.0E+09 ý Squarc Array(Unconstraincd)
8.0E+09
SquareArray(Constrained) 7.0E+09 HexagonalArray(Unconstrained) 6.0E+09 Hexagonal Array(Constraincd)
ý ý 5.0E+09 0 U
4.0E+09 {3.0E+09 W' 0
'i 10
20 30 40 Fibre Volume fraction(%)
50
60
Fig. 4.5: The variation of the continous fibre reinforced composite with volume fraction of the fibres for square and hexagonal array arrangement of the fibres. Comparion of constrained and unconstrained boundary conditions.
68
Chapter Four
Results And Discussion
the reasons that make the production of composites with
very high filler volume
fractions impractical.
4.1.2.3 Adhesion and bonding In theoretical analysisof the behaviour of the composites it is generally assumedthat the adhesionbetween the filler and the polymer matrix is perfect. By perfect adhesionit is meant that there is no relative movement of the phasesacross the interface under the higher loads. At applied stresses,however, the interfacial bond may break, and applied the adhesion is no longer perfect. Thus, the magnitude of the applied stressesoften determines whether there is a perfect adhesion or not. In reality, the degree of be important does factor forces long frictional to the not appear an as adhesion as between the phasesare not exceededby the applied stress. In most filled systemsthere is a mismatch between thermal expansioncoefficients of the phaseswhich results in a induced bond due to thermally stresses. mechanical In many practical casesit cannot be assumedthat there is a perfect adhesion between the filler and the matrix. In extreme casesof debondedparticles it can be assumedthat the particles act as holes and, therefore can seriously decreasethe composite modulus filler It be increasing do that the content. can unbonded argued particles not act with holes, from they the since also collapsing. as restrain matrix entirely If the bond or adhesionbetween filler and polymer is weak, the bond may break when is The deform load then applied. more than the filler so that polymer will a unsymmetrical cavities and voids develop around each filler particle. A change of the matrixfiller adhesionhas a smaller effect on composite modulus than its strength. The latter is much more dependent on the pretreatment of the filler is filler less Thus the efficient as a reinforcing agent, and in extreme casesthe surface. be being increased by the addition the than composite can of reduced rather modulus of filler.
4.1.2.4 Arrangement In practice the arrangementof the filler particles in a composite can never be uniform and the interparticle spacing varies in different parts of the composite. It has been
69
Chapter Four
ResultsAnd Discussion
ModulusVolume
fraction Particulate FilledFibre Reinforced
2.5E+10 i m
2.0E+10 ý
I
co
ParticulateFilled(Unconstrained) ED
ParticulateFilled(Constrained) e
G1.
ý U)
Fibre Reinforced(Unconstrained)
1.5E+10
A
Fibre Reinforced(Constrained) ö 1.0E+10 I aI=L4 Longitudinal Fibre (23%)
E 0
S.OE+09
I
0.0E+00 F 0
q
110
i
30 20 Volume fraction(%)
+ 40
ý
Fig. 4.6: Comparison of the stiffness obtained by pariculate filled and fibre reinforced composites. FEM results for square array arrangement and constrained and unconstrained boundary
70
50
Chapter Four
Results And Discussion
is distributed lower that when particles are modulus shown randomly, a composite finite A combination of element modelling and spatial statistical techniques obtained. has been used in the simulation of composites with randomly distributed
filler
particles. In this project two different types of particle arrangementsin the form of square and hexagonal arrays are examined to show the effect of the particle distribution on the final physical properties of the composite. The results are shown in figure 4.4 for a filled The composite. same comparison of the effect of the arrangementof particulate both In be in figure 4.5 for fibre seen a reinforced composite. reinforcing phase can for filler lower hexagonal the the volume same array gives composite modulus cases fraction and the hexagonal array gives a higher volume fraction for the same particle be be hexagonal The to a more random arrangement array can considered spacing. compared to the squarearray.
4.1.2.5 Filler particle shape In our model we assume that the particles are spherical and of the same size. Practically the filler particles are not uniform in size and only a few fillers are have dimensions However, in the nearly same since most particles spherical shape. in all directions, they can be approximated by sphere. Deviations of experimental by be from the model may caused results
more important factors than
small
deviations of the particles from spherical shapes.
Despite the described simplifying assumptionsour model predictions compare closely with the
between The model predictions and agreement experimental results.
boundary is unconstrained conditions more obvious when experimental results
is
imposed and hexagonalpacking is chosento show the distribution of the particles.
It can be shown that the addition of particles produces a substantial increase in However, 4.3 indicates in increase the asfigure particle composite modulus. effect of diminishes becoming infinitely to the asymptotic an modulus gradually modulus of rigid particle.
71
Chanter Four
Results And Discussion
4.2.3 Boundary conditions used in the finite element model The conditions at the boundary of the cell which is perpendicular to the boundary under tensile load can affect the result of the analysis.When boundary conditions with '. constraint are imposed, for each point of the modulusvolume fraction curve we need to follow two steps of loading. The results of these steps are superimposedand the total and final point is calculated. Under this set of boundary conditions we enforce geometric compatibility and therefore the outer sidewall of the cell remains straight. When the unconstrained boundary conditions is employed, every point along the sidewall is stressfree. However, since the constraints are released,the sidewalls do not remain straight. The results obtained for these sets of boundary conditions are compared for both particulate and fibre reinforced composites and are shown in figure 4.6. The results predicted using unconstrained boundary condition are lower than the results found for the constrained boundary conditions.
72
Chapter Four
Results And Discussion
4.2 COMPOSITES FILLED WITH RIGID PARTICLES
In this section the response of epoxy resin filled with glass particles to tensile and compressive loading is studied. The physical properties of the epoxy resin and glass filler used as the input values for the finite elementprogram are presentedin table 4.1.
4.2.1 Young's modulus The predicted values of the Young's modulus for compositesfilled with hard particles 4.4. in figure The data theoretical models which are are compared with experimental used to explain the variations of the modulus of polymer composites with volume fraction do not normally include the effect of filler particle size. However, experimental Young, 1984) Spanoudakis Young (Spanöudakis show that and and of observations there is a clear reduction in the modulus of a composite with increasingparticle size for fraction of particles. a given volume Relatively low values of Young's modulus for large particle sizes have been attributed to
the "skin" effect. Lewis and Neilsen suggestedthat in the moduli measurements,
the properties of the surface are emphasisedat the expense of the interior of the is for larger "skin" leads higher This the to where surface particles a error composite. depleted of particles. This could explain the drop in modulus with increasing particle by be due "skin" the The to can extrapolation of measured effect removed error size. (i. for with zero size e. points). particles values
4.2.2 Stress distribution 4.2.2.1 Concentration of direct stress The maximum direct stress concentration which is the maximum principle stress is found to be in the resin above the pole of the sphere.The precise position and value of fraction the the the maximum stress concentration of volume applied stress vary with increases The filler. the with the magnitude of stress concentration of applied stress of fraction is due increasing interactions to of added glass particles which volume
73
Chapter Four
Results And Discussion
Maximum Direct Stress Concentration
3.4
0.9 " Max. Direct stress
3.2 3
EB
0.7
0
%position(poleedge)
0 ...
.ý
03
0.8
2.8
0.6
0 EB
ý 2.6 0 U
... O h
0.5 0
.ý
0.4
2.4 ý ... `'o 2.2
40 ED E13
24
0.3
"
t0.2
"T
1.8 +I+ý+15 10
Iii0.1 25
20
30
35
40
Filler Volume Fraction(%) Fig. 4.7: Maximum direct stress concentration and position versus filler volume fraction.
Interfacial Normal Stress
2.5
ý 1 0
4
}20
ý
i
40
60
Angle(Degree) Fig.4.8a: Radial stressconcentration at the interface of the filler particle and matrix.
74
80
Chapter Four
ResultsAnd Discussion
Interfacial
Tangential Stress
2.5
ý 1 0
20
40 Angle(Degree)
60
80
Fig. 4.8b: Tangential stress concentration at the interface of the filler particle and matrix.
Interfacial Stress Concentration
0
20
60 40 Angle(Degree)
Fig. 4.8c: Shear stress concentration at the interface of the filler particle and matrix.
75
80
ResultsAnd Discussion
Chapter Four
Maximum
Interfacial
Shear Stress
50
0.65
.
t45 0.7 0 *Z
sý 0.8 a
0
®"
0.75
4ý
ý
""
+40 aý v
a
i
35 cn ý ö0 30 ý ý 25
"
0.85
ES
U
0.9 to
0® Max. Shear Stress
0.95
83 Position of Max.
1 410
 20
®
15
1.05 20
10
50
40
30 Filler Volume fraction
Fig. 4.9: Maximum Interfacial shear stress concentration and Its position.
Stress Concentration at Interface of Filler and Matrix
99.9 "
Radial stress
'9 Shear stress
e Tangential stress
79.9
cn
20
40 Angle(degree)
60
80
it'ig.4.10: Stress concentrations at the interface of the particle and matrix for the composite under compression.
76
Chapter Four
ResultsAnd Discussion
between the stressfields around the filler particles as their separationdecreases. Stress concentration in this work is defined as the stressvalue normalisedby the stress from is distance the the the pole to the composite and relative position applied by distance. the poleedge normalised The position of maximum stressconcentration factor of the applied stressvaries in the in figure 4.7. fractions is high filler At This the position shown volume cases. studied is at the edge of the grid, which is the midway between the particles. At lower volume fractions the position is relatively closer to the particle in the cylinder under analysis, into distances its from Taking is further the that adjacent particles. account away which between adjacent particles for different volume fractions is variable, the maximum distance between the position of stress concentration and adjacent particle was fraction. decreases increasing filler This distance volume rapidly with calculated.
4.2.2.2 Stresses at the interface Stressesaround the interface are transformed to polar coordinates. Radial and shear interface, is found in identical. It the the side of and resin are either glass on stresses that the stress transferred to the glass sphere is substantial. This leads to a sharp increasein the stiffness with increasingvolume fraction of particles. As figure 4.8a shows the maximum radial stress is at the pole of the particle sphere for all volume fractions. The variations of the value of this stress concentration with filler volume fraction is similar to that of the maximum stress concentration. Radial for At is the the tensile at equator particle. pole and compressive a spherical at stress 0= 22° position
the stress is zero for all volume fractions. The variation of the
tangential stress in the matrix at the interface is shown in figure 4.8b. There is a maximum in radialtangential shear stress as it can be seen in figure 4.8c. Both the value and the position of this stressconcentration vary with volume fraction. The absolute value of the stress concentration factor decreaseswith increasing volume fraction. This decrease is explained by considering the source of this stress its is difference between the the the and which particle moduli of concentration, As fraction difference increases the this volume material. of spheres surrounding decreases,and hence the rate of this decreasebecomessignificantly ( 4 times) smaller
77
Chapter Four
Results And Discussion
than the rate of the increaseof the concentrationof applied stress. Figure 4.9 indicates that at higher volume fractions the site of this maximum concentration of shear stress moves closer to the pole of the sphere. Stressesaround the interfacesof sphericalparticles have previously been considered by Dekkers and Heikens ( Dekkers and Heikens, 1983). For a poorly bonded interface they postulate that a crack will form around the interface becauseof the tensile radial have Using those to used, the we close constituent material properties very stress. 0= 68° 70°, interfacial to to the that of an angle up crack should grow they predicted depending on the remaining friction of the debondedinterface. The magnitude of the is the than the maximum smaller much not at pole concentration stress radial from debonding loading, for Therefore the tensile concentration of the applied stress. is Under the is the tensile at equator stress radial applied stress the pole expected. compressive.
4.2.3 Compression The radial stress would be tensile at the equator of the particle interface if the applied distribution 4.10 Figure the the is the stresses at particle of shows compressive. stress is interface. Although this the the almost stress at equator of absolute value and matrix it be however its to the than sufficient may value at pole smaller magnitude of order an from the the debonding the the condition of applied under of sphere equator cause compression. 4.2.4 Concentration of yield stress The contour diagrams of the Von Mises stress, direct stress and shear stress Von Mises 4.11ac. diagram In in figures the the contour of concentrations are shown first found: its the Von Mises stressreaches maximum are stress two positions where in the in the is the the position of the close of sphere vicinity of above pole resin point is direct the second stress concentration; point near to the position of the the maximum factors interface. The the these values of stress concentration maximum shear stress at in fraction is in figure 4.12. Their the same as vary shown volume positions vary with directions as the positions of the associated concentrations of the direct stresses. Figure 4.13 shows that the magnitudesof the maximum concentrations of
78
Chapter Four
Results And Discussion
I 1 U. 1441?7 q
2 0.2221?7 3 ().260F7 ý1 0.3131?7 5 0.1201?R 6 1).4RxF.7 7
O.8011{7
9 0.1 131?x 9 0.3351?x
0
Fig 4.1 la : Contour diagram of the direct stress field of the composite filled with 30% of hard filler particles
79
('h. ºhtcr JFour
Results And Discussion

I
I, 9
ý
U. 9C,K1'A
2 O.21h1?4
',r !
3
U. 33(, l{0
4 0.452F6 5 0.452F6
R
r;
6
0.2191? 6
7 0.1621,17 9 t). 227F7 9
0.175FR
0 G
I
Fig 4.1 1b: Contour diagram of the shearstressfield of the composite filled with 30% of hard filler particles.
80
Chapter Four
_
Ilrsults And I)iscussion
I 2 ý 4 5
0.145E7 0.222E7 0.277E7 0.3(>4E7 O.11l ER 6 0.512E7 7 0.8RRE7
0.2(X)E8 0.421 Ex ýý
Fig 4.11c: Contour diagramof the Von Mises stressfield of the compositefilled with 3(YYo of hard filler particles
81
Chapter Four
ResultsAnd Discussion
Maximum VonMises Stress
6
5+ A 0
... .ý
ý `' 4 ýa
ý, U A
0
c) 3 ýý ý, w rn
2
m
A
m
1 20
10
30 Filler Volume fraction(%)
40
50
Fig. 4.12: Maximum Von Mises stress concentration at the range of filler volume fraction.
VonMises Stress at Interface at Angles 22 and 35 Degree
3.5 "
3+ ý 0
22 Degree 35 Degree
.... a w 2.5 ýu
93 E13
ý 0
U
ýrA w®""
9+"
PR "®
"
0
1.5+
0
Ell
1 0
10
20 Filler Volume Fraction
30
40
Fig. 4.13: VonMises stress concentration at angle 22 and 35 of the interface of the filler particle and matrix.
82
Chapter Four
Results And Discussion
the Von Mises stress vary with volume fraction in the same way as the associated stress concentrations, although the range of the variation of Von Mises stress is very small. The stress peak located above the pole is greater at higher volume fractions of the filler, but at volume fractions below about 15% the maximum located at the interface is greater. Dekkers and Heikens (Dekkers and Heikens, 1983) studied the for in band formation low fractions beads very of various polymer volume glass shear found interface, 0= They band formation 45°. The the shear at at results of matrices. for in figure 4.14 finite are element analysis similar composites given which show our that at low volume fractions the maximum concentration of Von Mises stress is at the interface, at 0= 40°. The Von Mises stress at which yield occurs is dependenton the hydrostatic stress as in hydrostatic The this chapter. stress values corresponding to the earlier explained Under Von Mises the application of tensile stress were calculated. maximum of points is hydrostatic The hydrostatic tensile. the stress stress concentration remains stress but it increases 20% fraction filler, for to volume up around of constant rapidly almost higher volume fractions. The maximum hydrostatic stress concentration occurs at the pole of the particle.
4.2.5 Fracture behaviour Crack growth is generally considered to move in the direction of
the position of
direct Our direct that the stress. predicts position of model maximum stress maximum is above the pole. These predictions correspond with the experimental observations (Spanoudakis and Young, 1984). For a well bonded sphere, at low volume fractions, crack growth is developed towards the resin above the pole of the sphere.Smearing of resin around the pole of the sphere is observed. This type of fracture appearancecontrasts with the observed crack growth in epoxy resin containing poorly bonded glass particles which is found to be attracted to the equator of the sphere. The stress distribution in composite with poorly bonded filler particle is similar to that of a composite containing holes. The site of the is found be to concentration stress at the equator of the sphere. maximum
83
pw
Chapter Four
ResultsAnd Discussion
Interfacial
VonMises Concentration
'+iii
r+
40 Angle(Degree)
20
0
80
60
Fig. 4.14: VonMises Stress concentration at the interface of the filler and matrix.
Tensile Strength FEMOther Models
ýý
0
FEM
e
Schrager
10
4
Leinder&Woodhams
Leinder&Piggot &,
Nicolais&Mashelkar
20 30 Filler Volume fraction(%)
40
50
Fig. 4.15: Prediction of the ratio of the tensile strength of composite to the tensile strength of matrix t the range of filler volume fraction by FEM and other models.
84
Chapter Four
Results And Discussion
The amount of resin smearing around the pole of a well bonded sphere would be dependent on the position of the crack with respect to the sphere. Our results show that at low volume fractions the maximum distance is much further away from the sphere than for high volume fractions. Thus the model predicts that more smearing if there are no spheresvisible in the fracture surface. Greater smearing at occur would low volume fractions has beenobserved.(Spanoudakisand Young, 1984)
It is found that besides matrix properties, strain rate and temperature, the degree of interfacial adhesionhas a profound effect on the competition between craze and shear band formation. In cases where the beads adhesion are perfect craze formation is favoured, whereas for poorly adhering beads shear band formation is dominant. This difference is by in local formation becomes the the craze caused stress situation, effect factor under a triaxial stress state and shear band formation becomes the controlling dominant factor under a biaxial stressstate. In the caseof an excellently adhering glass bead, the crazes form near the pole. Stress analysis shows that these are regions of dilatation At bonded bead and maximum stress. a principle perfectly glass maximum bands form that the observations show shear near the surface of the bead experimental defined by from 45° the the axis of symmetry maximum principal which are poles at density distortion the and of maximum energy. Our results confirm these shear stress observations. In the case of poor interfacial adhesionbetween glass beadsand matrix, both craze and interface between by dewetting bead and matrix. At band the along are preceded shear dewetting a curvilinear interfacial crack is formed, starting at the pole and propagating in the direction of the equator until, at an angle of about 60° from the pole, a craze or interfacial band the tip the at originates of crack. shear Neilsen suggesteda very simple model basedon the assumptionof Hookean behaviour In following breaking be this the to strains. case relation can up written:
(4.3)
Q, = EE,
85
,w
Chapter Four
Results And Discussion
be is E is breaking the tensile the c c modulus and may elongation and at point, where expressedas:
Ec=Em(1V
1/3 ) f
(4.4)
Using breaking is the above the the the matrix. c,,, of at point elongation where is found the the as: strength of composite relative relations
Eý
v3
f
ßnr=(1V
Em
)
(4.5)
The ratio Ec/Emcan be calculatedby the finite element model. There are other theoretical models which predict the strength of the filled composites finite for brought few the to of our them with results comparison attention are of and a filled for in following The the the are glass models given constants element model. epoxy composite.
1. Leidner and Woodham (Leidner and Woodham, 1978):
Qc=0.83paVf+Kam(1Vf)
(4.6)
K=0.8 0.9. or psi, where pa=1540 2. Piggot and Leidner (Piggot and Leidner, 1974):
6c=am(10.5Vj6)
(4.7)
3. Nicolais and Mashelkar (Nicolais and Mashelkar, 1976):
(4.8)
Q, =6mf%Vb
b=1.21 and n=0.66 with
86
Chapter Four
ResultsAnd Discussion
Tensile Stress Along Particle Diameter FEMPapanicolaouModels
1.8
I °ýýýý 1.5 a
0
1.2 C43
FEM(30%Vo1.Frac.) ýFEM(50%Vo1.Frac.) ý
0.9
Papanicolaou Model
u
0.6
. ýý. ý 1
Cl)
0.3
0 0
0.2
0.6 0.4 Relative Diameter Position
0.8
1
Fig. 4.16: Tensile stress along particle diameter predicted by FEM and Papanicolaou models.
Tensile Strength FEMExperimental Data
1
0.8 0 .., cl ý 0.6 u
0 u 0.4 ,, W
c, c,
Cl)
0.2 0 0
10
20 30 Filler Volume Fraction(%)
40
50
Fig. 4.17: Relative tensile strength by FEM and experimental results for different particle diameters.
87
Chapter Four
Results And Discussion
4. Schrager (Schrager, 1978):
Qý = Qmexp(2.66 Vb)
(4.9) ,
The results obtained using the above equations are given in figure 4.15 and are Neilsen's All the the these with results of our model. of models except compared in increasing decrease the the volume a strength composite with of model show fraction of the filler. The Neilsen's predictions depend strongly on the model which is fractions. filler different the the to modulus of composite at volume used obtain The finite element model gives rather low tensile strengths for casescorresponding to the described models. In the finite element analysis,it is assumedthat composite failure be first fails. A that the assumption would element more realistic. when occurs fail. This highly does failure the stressed elements not occur until several of composite into inclusions by for `crack takes account the also and arrest' neighbouring allows is in for increase the the strength of material which subjected small volume statistical to the stress concentration. The use of this averaging technique will results in an increasein the theoretical strength of the composite.
Papanicolaou et al (Papanicolaou and Bakos, 1992) have proposed a model for the filled have They the the tensile polymers. strength of particulate used a of prediction distribution diameter. load find The Cox to the a along particle stress model modified is by inclusions is to tensile the the calculated used and estimate strength carried which The the variations of the tensile stress concentration along the particle composite. of diameter of the inclusion are presented in figure 4.16. The trend of model predicted by Papaicolaou. Although by is the that to predicted curve obtained similar variations higher finite to values compared to the curve found the element model shows a shift Cox (Cox, 1952). model using modified
The experimental results show that the strength of the composite depends on the size (London for 1977). filler In figure 4.17 the the particles et al, results experimental of four different particle diameters, 216 µm, 147 µm, 77 µm and 21 µm are compared
88
I.
Chapter Four
Results And Discussion
with our model. It can be seen that for the range of particle diameters studied the composite strength diminishes at higher volume fractions. The results also indicate that smaller particles show higher strength. The reason for this phenomenon is not increase for but in interfacial filler the smaller unit volume of area per entirely clear, is important, important factor be be factor. A also an second which may particles must that the stress fields near a particle are independent of the size of the particle, nevertheless the volume of polymer that experiences a given stress concentration increaseswith particle size. Therefore the probability of finding a large flaw within this volume increases. If a large flaw exists within an area of stress concentration, the tensile strength will be reduced according to Griffith's theory. In
most of the
investigated. be In theoretical the size cannot models, of particle effect aforementioned is hence it flawless finite that the and composite we also assume model element our independent fields the the through the which are stress composite strength of predicts finite Figure 4.17 that our the again element model confirms once particle size. of for the the composite. strength conservative prediction of provides a somewhat
89
Chapter Four
Results And Discussion
4.3 COMPOSITES FILLED WITH
DEBONDED RIGID
PARTICLES Rigid spherical inclusions in glassy polymers induce inhomogeneousstress fields and thus act as stress concentrators. Consequently,plastic deformation processessuch as band formed inclusions. The mechanismsof craze these shear are at craze and formation
and shear band formation are
investigated for small glass spheres
in These have to tension. subjected matrices uniaxial studies microscopic embedded degree interfacial the the profound effect of of adhesionon the mechanismof revealed 1978; band Baer, Donald Kramer, 1982) (Wellinghoff and and and shear craze formation (Kinloch and Young, 1983). In the caseswhere the adhesionof glass sphere is and polystyrene matrix perfect, the crazes are found near the poles of the sphere. From stress analysis around a completely bonded sphere it appears that these are dilatation For and stress. maximum of maximum principal perfectly adhering of regions in bands found form the to a polycarbonate matrix shear are around the spheres glass 45° from defined by the the the axis of symmetry at an angle of sphere poles of surface distortional These density the are sphere. regions of maximum strain energy stressed of interfacial In the shear stress. principal case of poor adhesion between and maximum the glass sphere and the polymer matrix, both craze and shear band formation are found to be preceded by dewetting along the interface between sphere and matrix. During dewetting a curvilinear interfacial crack is formed, starting at the pole and interface in direction the the of the equator until, at an angle of along propagating band from 60° The the the tip this a craze or shear originates at pole, of crack. about results of these studies prompted us to analyse the stress conditions near the tip of a interfacial between crack a glass sphereand a polymer matrix. curvilinear
4.3.1 Boundary conditions at the interface Figure 4.8a indicates that for all volume fractions of the filler the normal stress has is in from interface the the the compressive and value part of which starts negative 0 At this point the normal stress is about zero and at about ends and =22°. equator from this point towards the pole of the particle it continues to rise to its maximum
90
Chapter Four
Results And Discussion
Interfacial radial displacement Filler Volume fraction (10%)
I E03 0 Matrix n Illb
Filler
C7
0
0
0 4p I 0 0 40
0
6E04
0 0 0
ev
0 0 41. 0 or 410
... A
P8 2E04
cl
Ppý, .
94 ,,;;,
6&A&
dwtýý6568&




A,
"A
I ý{
2E04
20
0
40 Angle(Degree)
60
80
jg. 4.18: Radial displacement of the matrix and filler at the interface of the partly debonded particle
Interfacial Radial Stress Filler Volume fraction(10%)
0.5 0
0
0.3
Matrix
0 Filler
0 .., cl 10.1
0
op
w 0000
ýoa1 aý c, +ý Ln
1 ý
m
0.3t
a
0.5 0
20
40 Angle(Degree)
60
80
Fig.4.19: Radial stressconcentration of matrix and filler at the interface of a partly debonded particle
91
Chapter Four
Results And Discussion
in debonded From it be this that the the case of pole. analysis can concluded at value filler or when dewetting occurs in a bonded filler, from the pole up to a length of the be 0 by the unbonded may regarded entire region =22°, unbonded region represented as an interfacial crack. Here we consider two cases. In the first casethe remainderof the interface is bonded investigate In is displacement the the case we there second matrix. of no relative and the situation in which all the interface is debondedand from 0 =22° to the equator of depending displacement be the tangential there on of matrix the particle can a relative interface is high force interface. frictional If force the frictional the the acting on the at have it tangential to the movement. relative matrix particle and prevent can enough Therefore under this condition the model will be similar to the first case. If the frictional force is weak the tangential displacementoccurs and hence slip boundary The for the is imposed this extent of relative controls coefficient slip case. condition filler the the particle. and matrix of movement
4.3.2 Displacement Figure 4.18 shows the normal displacementof the filler and the matrix at the interface. The radial displacement of the matrix and the filler from the equator to 0 =22° is displacement the the than the segment of of other smaller and much negative interface(nearly zero). This negative value confirms that even in the case of debonding be in the filler the stress can this two contact and at part, phases remain of the 8=22° filler. From from to the pole the radial the the to the point of transferred matrix displacement of the matrix is positive and increasesto its maximum at the pole. The filler displacementin this part is nearly zero.
4.3.3 Interfacial stresses The interfacial radial stress concentration in matrix and filler is shown in figure 4.19 for volume fraction of 10%. The results indicate that the normal stress is negative and interface from 0=0) higher (i. 0=22°. In the the to volume at equator e. compressive fractions this stressbecomecloser to zero. The normal stressconcentration has its
92
Chapter Four
Results And Discussion
Interfacial radial displacement Filler Volume fraction (10%)
I E03 0 Matrix
410
.
Filler 0
6E04 lb 
co
0
0
.
0
0
4D
0
0
0
410
a rA ...
0 '0 19
Pý ý ýö 2E04
r ..
cl
ts:
10 . ýý
1{
4 2E04 0
60 40 Angle(Degree)
20
80
debonded interface the filler the particle partly displacement the of Radial at matrix and of Fig. 4.18:
Interfacial Radial Stress Filler Volume fraction(10%)
o.s
0
Matrix 0.3 Filler
0 ýwv ý
.r
ýý
v
W 0.1 +
r.
u CýO00000000
ýo0
ýW cn
1 0.3
a
.. m
0.50
20
40 Angle(Degree)
Fig. 4.19: Radial stress concentration of matrix and filler particle
93 (:ý
60
80
at the interface of a partly debonded
Chapter Four
Results And Discussion
Tangential stress concentration Filler Volume fraction(10%)
5 0
4.5+
0 Matrix X
4. 3.5 I
p
Filler
0
0 3 cl u 2.5 uýo u2 0 J 1.5 W
C3
CROI
ý1 .r+ " 0.5 t
o
Q2
'°`
_ý
ol
Qýýgxxxx 0000
ý 0.5
X
1 0
60
40 Angle(Degree)
20
80
Fig.4.20: Tangential stress concentration of matrix and filler at the interface of the partly denonded Particle
VonMises Stress at the interface Filler Volume fraction(10%) 7
v 0
6.5
0
65.5
0 x
5 .., 4.A
CRI
4.5
Filler 0 x
4
0
3.5 0X
Ü3
c9l
X O
2.5 Ln
Matrix x
2 1.5 0
1
0
0.5
°
00
xxxQ25xxg25S2Pxxxx .
0 0
20
ýo..
40 60 Angle(Degree)
.ý
80
100
Fig. 4.21: Von Mises stress concentration of matrix and filler at the interface of the partly debonded particle
93
Chapter Four
ResultsAnd Discussion
Radial Displacement Filler Volume fraction(30%)
I E03
R
xx X
XX
ý
X
ý aý E u cts a ý ...
A
7E04
X
0 Matrix
I
X X X
X
Filler
x x X
X X
4E04 4
X

cý ýä 0
a1
E04
X
X
X
X
X
X
X
X
ý a000^00
1Fi&i. 2E04
21 20
0
40 Angle(Degree)
4
{80
60
Fig. 4.22: Radial displacement of the matrix and filler at the Interface of the partly debonded particle
Radial Stress Concentration Filler Volume fraction(30%)
0
Matrix
0.7 I
+. 
Filler
0.44
0.1
z
own"'ow` 0
0
a 0
40.2
0
a
20
60 40 Angle(Degree)
80
100
Fig. 4.23: Radial stress concentration of the matrix and filler at the interface of the partly debonded particle
94
Chapter Four
Results And Discussion
maximum value at the tip of the crack or unbonded region. This stressis nearly zero in both matrix and filler at the other part of the interface from the crack tip to the pole. The frictional force is proportional to the normal stress acting on the interface. Therefore in the segment closer to the pole, it is equal to zero which shows that no stress transfer occurs. In the other segment which experiencesa compressive stress, the frictional force rises. This frictional force acts in the opposite direction to the tangential stressand is proportional to the normal stressso that:
Ft = J.LF.
(4.10)
The value of frictional force determines whether there is any relative tangential displacementbetween filler and matrix or not.
The tangential stressconcentration distribution is shown in figure 4.20. The maximum is higher The is the tip the than the much normal stress. crack and value stress at is segment zero close.to the crack tip and shows a small tangential stress of cracked negative value close to the pole. In figure 4.21 the variation of Von Mises stress concentration is presented. The this is found The distribution tip. the stress at crack concentration stress of maximum Von for direct Mises interface the and stress at the crack values the maximum shows at deformation band further through This that crazing or shear plastic any tip. confirms formation starts from the crack tip.
Figures 4.22 and 4.23 show the radial displacementand radial stress concentration for have beads. The displacements 30% filled with glass radial an epoxy composite in bonded for These to the are negative near zero region. very values positive values filler volume fractions up to 12% (figure 4.19). As it can be seen in figure 4.23 the for Therefore be the this there part are at also positive. possibility can a stresses radial beads. The for higher fraction towards the equator crack volume a glass of growing of displacement fractions lower and of compressive stresses values at volume negative
95
Itrsulls
Chapter 1=our
L
And
I
I )iscUSsilln
0.3121?7
2 0.2x51i7 3 0.2KI1?7 4 0.27GV17 5 0.1051?5
(, O.5251;4 7 O. 1H2i?7 K 0.121F. 6 l)
0.1341`7
Fig.4.24a: Contour diagram of the direct stress field of the partly debondedcomposite with 30% of hard filler particles
96
Results And Discussion
ChapterFour
1 0.1(171{4 2 U.G441;3 3 U.2191;C, 4 t1.4471?5
9
5 0.102F5 6 0.2201; 7
7 0.KK9F6 8 0.1 Rfi1?C, 1%
().
1o"11:K
Fig.4.24b: Contour diagram of the shear stressfield of the partly debondedcomposite with 30% of hard filler particles
97
Chapter Foul.
Kxsulis And 1)isCussi0n
I
3.121'06
2 2. x51'+06 3 2. x5I'+05
H
4 2.851'106 5 5.251'+05 6 6.771'+06 7 2,241,1+06
x
4. x11'+05 3.2nI". M7
Fig.4.24c: Contour diagram of the Von Mises stress field of the partly debondedcomposite hard filler 30% particles of with
98
Chapter Four
ResultsAnd Discussion
Radial Stress Concentration Slip At The Interface(V10%)
0.5 0 Matrix
a
0.34
0 Filler
m
0.1 ý
.
ýý
apr.
ýo ý.
0
1
'ý
0
000000 v
C]
v
lJ
vv...

i
oýdýr
ý ab
0.3 ý 0
0.5
0
20
40 Angle(Degree)
80
60
Fig. 4.25: Radial displacement of the matrix and filler at the interface of the partly debonded particle with slip condition at the closed segment of the interface
Tangential stress concentration Slip At The Interface(Vf=10%)
5 4.5
0 Matrix
4
x
3.5 0 w3 ý
Filler CRI
2.5 0
u2 0
1.5 1
C; 0.5 4 r
ý0
o 0
fo
0.5
C) n
t
_°OOo
ii 0
20
40 60 Angle(Degree)
iý 80
Fig. 4.26: Tangential stress concentration at the interface of partly debonded particle with slip interface the the segment of closed condition at
99
100
Chapter Four
Results And Discussion
VonMises Stress Concentration Slip At The Interface(Vf=10ßö)
50
4.54
0 Matrix x
0
0
3.5
Filler
0
3ca
2.52
0 0 0
0
ý ý ý
1
O O 0
x
0.5
0000
ýo 00
0 } 0ý5 1 20
0
40 60 Angle(Degree)
100
80
Fig. 4.27: VonMises stress concentration at the interface of partly debonded partilce with slip condition at the closed segment of the interface.
Interfacial Radial Displacement Slip At Interface(Vf=10%)
1E03 d» 0
Matrix A Filler
6E04 ý
0 40
0
0 4D
a
0
0 00
a
0
5
0
u cl
0
a ý ..,
e 010
A 2E04 +
40 .
49
Ift
Ift
IN
N
2E04
0
20
40 60 Angle(Degree)
80
Fig. 4.28: Radial displacement of the matrix and filler at the interface of the partly debonded particle interface the the closed segment at of condition slip with
100
Chapter Four
ResultsAnd Discussion
Radial Stress Concentration Slip at the interface(Vf=30%)
Iw
0.5 0 Matrix
m
0.3 0 0
Filler
.., .r
ý 0.1 ,o
ý Fi
 
u
.
'
m
m o
N
19
ý
0.1 to) a) &.
0
En
0.3F
0.5
0
IIIII
ý I
20
0
60
40 Angle(Degree)
80
Fig.4.29: Radial displacementof the matrix and filler at the interface of the partly debondedparticle interface the the at closed segment of condition slip with
Modulus of Composite with Debonded Filler Particle or Void
3.5E+09 i 3.0E+09 ä, 2.5E+09 v h
0
b 2.0E+09 0
0
1.5E+09
E
1.0E+09 ul 5.0E+08
a}
Debonded particle filled composite 0
Porous Composite
O.0E+00 Iý+ 0
ý
10
f
}20
30
Volume Fraction(%) Fig. 4.30: Modulus of composite filled with debonded filler particle or void
101
H40
4
50
Chapter Four
Results And Discussion
exclude the possibility of growing or opening of the crack further than 0 =22° towards the equator.
The contour diagrams of the direct stress, shear stress and Von Mises stress in figure 4.24ac. Maximum stress concentration in all of are shown concentrations the above casescan be found at the crack tip.
4.3.3 Slipping of the particle at part of the interface Figures 4.2528 show the stress and displacementvariations at the interface for epoxy is imposed beads filled 10% of slip condition on the with glass while composite is 0 Since low from to the the a slip coefficient equator. particle =22° segment of less follow displacement fields the the trend the as more or same no and stress used, decrease Von Mises However, tangential the maximum radial, and stresses slip case. in Using 30% the the to slip case. same slip with non coefficient a composite compared 4.29. filler in figure The fraction the shown of results are stressconcentrations volume fraction. decrease this at volume shows noticeable In figure 4.30 the modulus of the composite filled with the unbonded particles and the for fractions. the of range volume voids are compared containing composite
4.3.4 Strength The tensile strength of the composites with perfectly bonded and unbonded interfaces finite The using element with voids are calculated model. results are and composites 4.31. increasing figure The decrease in tensile the strength volume of with compared fraction can be observed in all cases.The graph shows the strength of composite with is interface unbonded
between the upper bound curve which is for the bonded
interface and the lower bound curve which representsthe composites with voids. In figure 4.32 the strength predicted by the finite element analysis is compared with the following models proposed by Neilsen and Nicloais&Narkis. Different theoretical models are suggested for the unbonded or no adhesion case. A is due Nicolais Narkis Narkis, 1971) (Nicolais to model and and reported commonly that the based is the assumption unbonded particle cannot carry any of the on which 102
Chapter Four
Results And Discussion
load and the yielding occurs in the minimum cross section of the continuous phase. They presentedthe following equation for the yield stressof the composite. 6c =am(11.21Vf)
(4.11)
Another model suggested by Neilsen ( Neilsen, 1966) gives the yield stress in a filler, between and expressedas : polymer adhesion no assuming composite
f')S (1V 6c = Q.
(4.12)
by finite be determined function the is S the can which stress concentration where finite is Neilsen's The to the element prediction prediction very close element analysis. But it is the to the composite. for voided composites and unbonded also closer Nicolasis and Narkis's model gives a much higher values for the strength.
103
Chapter Four
ResultsAnd Discussion
Tensile Strength
5.0E+07 , . '4.0E+07 P= a a,
w 3.0E+07 cn d
r.. ..,
0
10
20 30 Volume fraction(%)
50
40
Fig. 4.31: Prediction of tensile strength of composite filled with perfectly bonded particles, debonded particles and voids.
Tensile strength FEMOther models
7.0E+07 i aFEM(Dcbondedparticle)
6.0E+07 5.0E+07 ý 4.0E+07 (V kg
3.0E+07
ý ... ýý F, 2.0E+07
1.0E+07 0.0E+00 E++i+i 05
II1 10
15 20 25 Volume fraction(%)
11
i1 30
ý 35
1 40
Fig. 4.32: Comparison of the predicted tensile strength of the composite with debonded particles and voids with other models predictions
104
Chapter Four
Results And Discussion
4.4 COMPOSITES FILLED WITH SOFT PARTICLES
4.4.1 Material properties The results presentedin this section include the linearelastic behaviour of the rubbery input The the values used to obtain stressconcentrations and epoxy matrix. and phase the composite toughnessvia a linear elasticity model are shown in table 4.2.
Input Properties
Derived Properties
Phase
E (GPa)
Poisson's Ratio, v
K (GPa)
G (GPa)
Epoxy
3.0
0.35
3.333
1.119
Rubber
0.0004
0.4900.4999
0.0060.667 0.0001340.00026
Table 4.2. Elastic Material Properties Used in the Predictive Models
Material properties for typical epoxy type polymer are wellknown.
However, the
is difficult A for E to the establish; sensible range of more rubbery phase value of for is for The to the v close upper value chosen very v rubber. selected was of values be finite 0.5. It that the theoretical noted should element value of the maximum if fails that v=0.5 was employed analysis package
is used. However, the maximum
Such high be is 0.4999. in the relatively values now work may present value of v used finite improved because the element code. of precision of used The most important advantageof being able to use values of the Poisson's ratio close high K(i. bulk in is this 0.5 values of that e. modulus of the way we can substitute to imply 0.4 MPa 0.4999 E input Indeed, the v a value of and values rubber particle). K of about 0.667 GPa, which is of the order expected for a rubbery polymer (Brandrup and Immergut, 1989). 4.4.2 Young's modulus As it is shown in figure 4.33 the relationship between the modulus of the composite linear. filler is The for Young's fraction the nearly of measured modulus and volume
105
Chapter Four
Results And Discussion
ModulusVolume fraction Rubber Filled Composite
Iw
3.5E+09
ý a' 2.5E+09 (I, ý ý. c
AsRubber Filled Composite APorousComposite 133 Experimentaldata
ED 03
2.0E+09 1.5E+09 1.0E+09
U
5.0E+08 0.0E+00 + 0
}
{10
{
t
f
ý
30 20 Rubber Volume Fraction
H
1
50
40
Fig4.33: The modulus of the rubber filled composite, porous composite and experimental data
ModulusVolume fraction Soft ParticlesFilled Composite
'2 2.8E+09 2.6E+09 ý
2.4E+09 2.2E+09 2.0E+09
1.8E+09
TIII111111III1
III HII111111III111111III111111
1E1
1E2
1E3
iI
1E4
1E5
I
Al 12 111E0 1E6
Relative Modulus(Er/Em) Fig4.34: Variation of the rubber filled composite modulus with the relative modulus(Er/Em)
106
Chapter Four
ResultsAnd Discussion
ModulusVolume fraction Poisson's Ratio 0.49990.49
2.0E+09 m
1.9E+09
Poisson'sRatio=0.4999
1.8E+09
_aPoisson'sRatio=0.49
.clI
ý 1.7E+09 0
ý
c
1.6E+09 t
ý
1.5E+09 1.4E+09 1.3E+09 20
i
H22
f
I
i
f
4
26 24 Volume Fraction(%)
28
30
Fig4.35: The effect of Poisson's ratio of rubber on the final result of FEM.
Maximum Applied Stress
6 at
5+
Poisson's ratio=0.49 , 
Poisson'sRatio=0.4999
0 ... cl
ý4+
m Q u 0 rý
r
2+
0
10
20 Volume Fraction(%)
30
Fig4.36: Concentration of maximum applied stress for different filler volume fractions.
107
40
Chapter Four
Results And Discussion
epoxy resin reinforced with 15% volume fraction of rubber spheresis also included. The measuredvalue is significantly higher than the predicted range by our model. The lack of precise agreement between experimental and numerical results could arise from the inaccuracy in description of the rubber properties and the assumptionthat the Beyond fraction of about 0.2, there are no separated. a volume are perfectly phases for data available comparison, because when higher concentrations of experimental inversion used, phase of the multiphasepolymer normally occurs. rubber are
Figure 4.34 shows predictions of our model for the Young's modulus of the function log(Em/Er), Ep Em epoxy as a of where and rubbertoughened multiphase fraction Young's A the the of modulus and volume rubber epoxy respectively. of are 20% has been assumedfor the rubber phase.In the calculations, the Poisson's ratio of the rubber is considered to be 0.49. Initially the Young's modulus of the two phase increase decreases in log(Em/Er). When this the with sharply an value of material Young's has 3.0, lower bound the modulus a constant of 1.87 GPa. value approaches The predicted values of the overall modulus are relatively insensitiveto the changesof the value of the rubber modulus.
The predicted overall modulus is found to increaseby 5% at 20% volume fraction and 7% at 30% of volume fraction if the Poisson's ratio of the rubber is increasedfrom the in figure 4.35. Therefore is 0.4999 0.49 to the properties of as shown value used Poisson's be best described but 0.5, this causes a using ratio probably of could rubber finite for the element analysis. a problem
The experimental values reported for rubber filled composites are in comparison higher than our model predictions, this is probably becauseof some stiffening of the rubber inadequate description Poisson's in the the the the of the ratio of rubber epoxy, via leading lower fraction incomplete to than a phase separation rubber volume model, and the expected theoretical value. The onset of plasticity during the experimental discrepancy as any well experimental as errors can also cause some measurements between the measured and the predicted values. In some cases the reason for the
108
Chapter Four
Results And Discussion
be higher than the predicted results can be the skin effect. The to value experimental by leads imposed to an excesspolymer at the surface of the walls of molds, restrictions the test specimens.Thus, in torsion or flexural tests where the maximum stress is at the surface, the properties of the surface is dominant and determine the behaviour of the whole sample.This error can be corrected by using thicker specimenswhich can be by infinite thickness, to or using particles of smaller size and extrapolation extrapolated to zero particle size. The skin effect can produce errors as large as ten to twenty depending on the thinnessof the specimen. percent
Despite the discrepancybetween the experimental and the model results, it should be is fraction between Young's linear volume modulus and relationship noted that almost by Yee Pearson. to the and measured experimentally relationship very similar
4.4.3 Stress distribution 4.4.3.1 Concentration of direct stress The contour diagrams of the direct stress, shear stress and Von Mises stress infgures 4.37ac. are given concentrations The contour diagram for the concentration of the applied stress indicates that the interface is found the the at equator of the sphere. at concentration stress maximum The examination of other contour diagramsconfirms that this is the maximum principle in The the the stress at of maximum concentration, system. position predicted stress in is theoretical the predictions and with previous sphere, agreement of equator it in figure 4.36 be 1984). As Young, (Spanoudakis can seen and experimental results the values of the maximum stress concentration, varies sharply with the volume fraction of rubber spheres.The stressconcentration decreaseswith increasing Poisson difference between the the of since effect matrix and physical ratio of rubber, figure 4.36 is In diminished. direct the the stress concentration particle properties of for be Poisson's fraction compared rubber can ratios, of versus volume 0.4999.
109
0.49 and
rw
Results And Discussion
Chapter Four
I
2.61E00
2 5.09F,OG 3 2.2RFO6
H
4
1.R1IA6 1.18F.05
6 (i. 33EO5 7 2.56E05 8
1.53E1)5
9
2.5(, E05
Fig. 4.37a: Contour diagram of direct stress field of composite with 30% of soft filler particles
110
hrýull`
Chapter Four
i
()
C1]
And I )iSruIIsi(, n
2 I7FO3 2
Ix
31:U4 ?05 LOX]
2xi; c)a I_6x1'05
o 1.701?05 7 1.4')º?02
x
2.091`.05
0 EL i 6
i ý,ýý.
I
Fig. 4.37b: Contour diagram of shear stress field of the composite with 30% of soft filler particles
III
IZrmill. ti And I )iu ns`ion
Cfiapter Four
L'J
I
0.2611?7
2 0.9221?(, 3 0.228V17 it 0. I 84V17 ý 0.411I; (, 6 0.6231', 6 7 0.25S1?6 8 0.462f; 6 1)
0 1)71)1": (,
I
Fig. 4.37c: Contour diagram of Von Mises stress field of composite with 30% of soft filler particles
112
Chapter Four
ResultsAnd Discussion
Interfacial Stress Concentration RubberFilled Composite
0.5
0
20
60 40 Angle(Degree)
80
100
Fig. 438: Stress concentration at the interface in the soft particle filled composites (Vf=20%).
Interfacial StressConcentration RubberVoidFilled Composite
Tangential(Rubberfilled) a Radial(Rubberfilled) Tangential(Porouscomposite) Radial(Porouscomposite)
0
20
40 Angle(Degree)
60
80
Fig.4.39: Comparison of interfacial stressconcentration of rubber filled and porous composites(Volume fraction of rubber or voids=30%).
113
Chapter Four
Results And Discussion
4.4.3.2 Stressesat the interface Stresses around the interface of the matrix and filler are converted to polar in both The the resin matrix and the rubber sphere are zero, shear stresses coordinates. The only high stress is the tangential stress in the resin matrix, which has its maximum 4.38 different Figure the the the sphere. shows concentration of of rubber equator at fraction. interface for 20% The the a volume composite with rubber radial at stresses interface; in the the the are and rubber equal and constant around resin stresses in identical the matrix and rubber particle. We note that the tangential stressesare also is found for for far interface that the the than the smaller soft particles at stress radial hard particles. This indicates that there is no tendencyfor debonding at the interface in the composites, which is in contrast to the tendency for debonding in the glass filled low the that The to the transmitted shows rubber rubber, of stress value material. is interface hole. The like is the compared concentration of stressesat a sphere acting in figure 4.39 for the matrix filled with 30% of rubber and matrix having 30% of filled behaviour The the the composites with rubber or voids of of similarity voidage. is confirmed by the values of matrix maximum stressconcentration, at the equator for by hardly holes The the values are changed or spheres. rubber containing resin epoxy presenceof rubber. 4.4.3.3 Hydrostatic stress in the rubber particle Examination of the stressesin the rubber shows that the whole particle is in uniform hydrostatic stress concentration factor The the hydrostatic tension. of variation pure 4.40. The is in figure is fraction this magnitude of uniform shown stress with volume Although the the value of this the prediction. of sphere within accuracy throughout is the two than of magnitudes smaller maximum about orders concentration stress be in it difference in the the significant may still since resin, stress concentration five hydrostatic is The two of magnitude. orders around of magnitude phases moduli of is increases its in fraction. Initially there with volume a rubber stress concentration little increaseby increasingvolume fraction up to around 20% and then it increasesby filler 50% 20% fraction. The between Poisson's 2 factor and of volume ratio of of a the has on predicted valuesof the effect significant a the rubber
114
"w
Chapter Four
Results And Discussion
Maximum Hydrostatic Concentration
0.026
Stress te.
0.024 0 ' 0.022 03
0.02
O
U W CA
0.018f ýWO Cl) 0.016+ 0.0 14 15
10
20
25 30 35 Rubber Volume Fraction
40
45
40.
45
Fig. 4.40: Maximum hydrostatic stress concentration in rubber particle.
Hydrostatic stressVolume fraction
0.8
00.6 ...
0.4 0 U W W rn 0.2
0 10
15
20
25 30 Rubber Volume fraction
35
FIG. 4.41: Effect of Poisson's ratio in FEM calculations of the hydrostatic stress.
115
Chapter Four
Results And Discussion
hydrostatic stress in the rubber. The variation of the hydrostatic stress concentrations fraction for Poisson's function the the two of volume rubber phase values of of as a 4.41. in figure These Poisson's 0.4999 0.49 the are presented values of and ratio, GPa GPa 0.006 0.667 bulk the the as and at the same of rubber modulus ratios gives Young's modulus.
4.4.3.4 Concentration of yield stress As it is shown in figure 4.37c the maximum Von Mises stressis found at the interface, direct identical is to the the the maximum position sphere, which at the equator of for different Von Mises The the unchanged stress remains maximum position of stress. increases factor but fractions the sharply the stressconcentration magnitude of volume 4.42. in f be fraction increased gure seen as can volume with As it is mentioned earlier in this chapter the magnitude of the hydrostatic stress at the in fraction. We dependent the is less the change can compare volume on equator fraction Our 20% 10% for between volume of rubber. and yield applied stressrequired from 1.50 2.00 in increase to this over range. stress concentration an show predictions Assuming that yield occurs when the stress within the material reaches a given level, increase hydrostatic in this this that bearing range changes over slowly, stress mind and in stress concentration predicts that applied stress at 20% rubber volume fraction fraction; 10% ' the be 85% volume the measured rubber at stress applied of should i.e. 91% (Yee and Pearson, 1986). Shear 50 MPa from 55 be found is to to reduction bands in the resin are expected to grow from the point of maximum concentration of Von Mises stress,which is predicted to be at the equator of the sphere.
4.4.3.5 Stresses in the matrix The stressesin the matrix on the lower side of the cell from the equator to the edge of Von 4.43a. for The direct in figure and concentrations stress stress the grid are shown Mises stress reach maximum at the equator and decreasetowards the edge of the cell. is hand The the almost zero. other variation of stressconcentration The shear stress on
116
Chapter Four
Rcsults And Discussion
Maximum Von Miscs Stress
6
5+
4+
3+
2+
1 35
25 20 ,30 Rubber Volume fraction
15
10
40
fraction. the Von Mises volume Maximum of rubber 4.42: range Fig. stress concentration at
Stress Concentrations Lower Side of The Cell
3.5
Radial
1 4I ýShear
o.5+ 0 s o.
1
1.1
1.2 Radial Distance
1.3
1.4
Fig. 4.43a: Stress concentration variation at the equator side wall of the rubber filled composite.
117
Chapter Four
Results And Discussion
follows diagonal the the of cell an opposite trend and decreasesfrom the particle on figure it be in the the towards upper corner right edge of cell, as can seen surface 4.43b. 4.4.4 Fracture behaviour Two important toughening mechanismshave been identified for the two phasematerial dispersed in of a matrix of crosslinked under study which consists a rubbery phase is banding, localised first The which occur shear yielding, or shear mechanism polymer. between rubber particles at an angle of approximately 45° to the direction of the involved, large Owing the to the tensile number of particles stress. maximum principal is thermoset undergo yielding effectively which can plastic matrix material volume of increased compared to the singlephasepolymer, consequently, far more irreversible is improved. The involved is dissipation the toughness the of material and energy interfacial debonding internal is the of rubbery cavitation or mechanism second formed by The the the voids plastic growth of subsequent mechanismenables particles. deformation of the epoxy matrix. This irreversible holegrowth in the epoxy matrix in in fracture to the the dissipates therefore enhancement contributes turn energy and toughness of the composite. Gent and Lindley (Gent and Lindley, 1959) have shown that cavitation of rubber can later include Their low to extended surface analysis was stresses. occur at relatively important for initial holes holes. Surface for are energy effects small energy effects for from hydrostatic The 0.1 stress required cavitation an than µm radius. smaller initial hole with radius greater than about 0.1 µm has been shown to be approximately is hole 0.4 MPa. For initial that the an of radius the about rubber modulus of close to 0.01 µm the stressrequired for cavitation increasesby a factor of 40. Internal cavitation of the rubbery particles or debonding of the particles from the (cavities) formed. being By leads to voids neglecting any plastic void growth at matrix this stage, the void can be treated as a particle with zero modulus, and a similar is described be done the that therefore to stress the to calculate one above can analysis Young's the the voids around and moduli of the voided epoxy polymer. concentration The formation of voids due to internal cavitation of the rubbery particles or debonding
118
Chapter Four
Results And Discussion
from does the the matrix particles not significantly change the level of the stress of in concentration the epoxy matrix. The described mechanisms are triggered by the different types of stressconcentrations that act within the overall stressfield in the twophase material. For example, initiation bands largely by Von Mises the the shear governed concentration of of are and growth interfacial debonding, in the of the rubber particles whereas cavitation, or matrix, stress is largely controlled by the hydrostatic (dilatational) tensile stresses. To analyse the it is finite field to technique the such a as employ numerical accurately, necessary stress element method. The yielding of the glassy polymers is usually dependent on the hydrostatic stress is Von Mises the therefore criterion not strictly satisfied. simple component, and Instead the Von Mises criterion should be modified as:
Zvm =
(4.13)
P µm Ty
is in 4.14, defined is Von Mises the 'ry the yield as r,,,. shear stress, equation where is hydrostatic P Thus is the µm stress, and a constant material stressunder pure shear,
r
2+
(o'2Q3)2
L(QIQ2)
(Q361)2 +
=2
Qvm2 =6
zvm2
(4.14)
and 3(o'I
P=
+(7s+Q3)
(4.15)
be has been Von Mises is to the tensile µm the and value of reported stress where a,,,,, between 0.175 and 0.225. In this study this value is taken to be 0.2. It is obvious from for tensile 4.13 the the that to under stress material required shear yield equation loading is reduced compared to the prediction of the unmodified Von Mises criteria. Hence, the relative size of the plastic zone will be increased.The tensile strength of the is in figure 4.44. toughened shown epoxy rubber
119
Chapter Four
Results And Discussion
Stress Concentration Cell Diagon
2
«Shear
o.5+
Von Mises
0
s 0. 1
1.1
1.2 Diagonal Distance
1.4
1.3
Fig. 4.43b: Stress Concentration on the diagonal line of the cell in the matrix.
Tensile StrengthVolume fraction Rubber filled composite
6.0E+07 5.0E+07 . .cC
a:' 4.0E+07 00
ý 3.0E+07 2.0E+07 H
1.0E+07 0.0E+00 0
10
20 30 Rubber Volume fraction(%)
40
Fig. 4.44: Tensile strength prediction using Von Mises stress comcentration criteria.
120
50
Chanter Four
Results And Discussion
Schwier et al (Schwier et al 1985) in their experimental investigation of polystyrene , filled with polybutadiene rubber spheres,concluded that intrinsic cavitation of rubber spheresoccurred at about an overall hydrostatic tension of 60 MPa. Our results predict that for 20% volume fraction of rubber the concentration factor of hydrostatic tension in the rubber is about 0.025. Shear band formation probably occurs at about 30 MPa stress for epoxy resins. Keeping the average applied tensile stress at this level would subject the rubber spheres to a further 1 MPa hydrostatic tension. Cavitation of rubber particles on fracture surfacesis generally found to start from the centre of the sphere. Our predictions do not indicate a preferred point of initiation but imposition the that show of an overall tensile stress of the order of the matrix clearly initiate in hydrostatic tension to the sufficient places rubber particles yield stress fully be flaws. flaws from These which are areas of not rubber small may cavitations There is be by techniques. microscopic observed conventional polymerised and cannot is the to the rubber particles necessary cavitation of some argument as whether or not for shear bands to form. The exact stresslevels for shearband formation and cavitation form bands depend the to values of stress matrix shear on required and cause will for the systemunder study. cavitation rubber Epoxy resin reinforced with rubber spheres may be described as an inherently tough be the of stress concentration, where a crack would since position maximum material likely to develop, is at the equator of the sphere. The sphere acts as a barrier to any further growth of the crack. However, the maximum stress concentration rise sharply description fractions. is A be high toughness that the simple of could yield volume at Considering high to the the the of crack. very stress growth preferred mechanism fraction, high become applied volume crack could of stress at growth concentration leading high fractions decreased dominant to the toughness of at volume more filled This by Kunz (Kunz with rubber was et al, particles. effect noted composites 1982) who found a rapid increase in toughness until around 5 pbw (parts by weight) 10% is fraction, found followed by A volume around effect a plateau. similar rubber, by Spanoudakis and Young (Spanoudakisand Young, 1984) for epoxy resin reinforced with glass spherestreated with a releaseagent; the spheremust be behaving as a stable hole, and stress distribution in the resin must be similar to our prediction here; the
121
r
I.
Chapter Four
Results And Discussion
observed crack growth for glass spheres treated with a release agent supported the expected maximum stressconcentration at the equator of the sphere.They found that fracture toughness decreased with volume fraction above 30% for this material. Although for glass spheres coupled to the resin matrix the toughness increased throughout the range of filler volume fraction. Stress distribution in the matrix for resin containing rubber sphereshas been found to be very similar to that of a resin containing holes. Shear band formation in the resin would occur in a similar way for the two materials. However, epoxy resin containing holes is known to be a very poor material. We therefore can postulate that shear band formation is not the only important fracture mechanismfor this material. Our results have shown that cavitation of the rubber particles is likely to occur, in agreement with be It the stretched. appears that may rubber experimental observations; after cavitation important contributions to the process of rubber cavitation and stretching may make the overall fracture energy. The magnitude of these contributions is dependent on the fracture Further to the the contributions energy of epoxy resin rubber. modulus of from the physical presence of the rubber which spheres may arise rubber containing would act as a crack stopper to a growing crack attracted to the equator of the sphere by either the concentration of direct stressor the initiation of shearbands.
122
Chapter Four
Results And Discussion
4.5 FIBRE REINFORCED COMPOSITES
In this section the response of epoxy resin reinforced with glass fibres to transverse tensile and shear loading is studied. The physical properties of the epoxy resin and glass fibres are presentedin table 4.1.
4.5.1 Transverse tensile loading
4.5.1.1 Modulus A tensile loading in the transverse direction of the fibres is applied to the composite. The modulus of particulate filled compositesis compared with the moduli of transverse in figure 4.6. having fibre longitudinal the same constituents reinforced composite and As expected the longitudinal modulus of the fibre composite is much higher than that is lower. Maximum the transverse the volume composite, while modulus particulate of fractions of the fibres are 0.785 and 0.907 for square array and hexagonal array, for hexagonal 0.74 0.57 in These to and square and array values reduce respectively. fraction Therefore higher be composites. a volume and modulus can particulate in fibre composites. reached It is observed that
the addition of fibres produces a substantial increase in the
transverse modulus of the binder material; however, the effect of an increase in fibre becoming diminishing is for to the one asymptotic value obtained a gradually modulus the infinitely rigid fibre. Therefore, at high values of the fibre modulus, The ratio of transverse Young's modulus to longitudinal Young's modulus becomesrelatively small high is the the associated use significant structural problems with of of one which If in filaments multiple oriented composite structures. modulus
fibre
arrays are
in low decrease direction, in the transverse the to a problem of stiffness solve utilised the major stiffness of the fibrous composite follows.
4.5.1.2 Concentration of applied stress Contours of the direct stress and Von Mises stress are obtained for fibre reinforced in figures 4.45ab. These contour diagrams indicate that the as shown composites
123
Chapter Four
IZrsu(s And I )iscussi
I
I 25I17
'
246117
Z
367117
4 F)
ý 6 7 8 9
. .
196117 x9F.7
.1 387117 403117 . 428I17 . 186I":7 .
L2
Fig 4.45a : Contour diagram of the direct stress field for the fibre reinforced composite
124
Results And Discussion
ChapterFour
F
1 0.889F6 {6 20
1741?7
3 U1371`.7 "1 n_I(141'17 5 U.2(,71;7 () 0 1971;7
9
7 03011`17
7 8 (1.3191: 1)
() 201? 7 .
ýý
Fig.4.45b: Contour diagram of the Von Mises stress field of the continuous fibre reinforced fibres. 30% of composite with
125
Chapter Four
Rcsults And Discussion
Maximum StressVolume Fraction
1.9 Maximum StressConc. eStressConc.at Pole
1.8
1.7 U H wl vN 1.6 cn ..,
f
1.4+
i20
10
{
4
III
30 Fibre Volume fraction(%)
50
40
Fig. 4.46: The maximum applied stress concentration in the matrix and at the pole of the fibre reinforced composite
Stress Concentration At Interface Fibre Volume Fraction 30%
0
20
60 40 Angle (Degree)
80
100
Fig. 4.47: Stress concentration at the interface of fibre and matrix for the 30% volume fraction of fibre reinforced composite.
126
Chapter Four
Results And Discussion
Interfacial Von Mises Concentration
1.2
0.2 4
0
20
60 40 Angle (Degree)
Fig. 4.48: Von Mises stress Concentration at interface of fibre and matrix fractions.
127
80
100
for different fibre volume
Chapter Four
Results And Discussion
concentration of applied stress in the matrix for a typical fibre reinforced composite is very similar to that of a particulate filled composite. Figure 4.46 indicates that the maximum concentration of the applied stress is in the matrix above the pole. The precise position of maximum concentration varies for different fibre volume fractions. For higher volume fractions of fibre the position of the maximum is at the edge of the between fibres. is higher However this neighbouring maximum not much centred grid, than the stress concentration at the pole. Since the failure strength of the interface is be lower failure than to strength of the resin, the stress concentration at the expected interface is considered to be more important for the study of the fracture behaviour than the stress concentration above the pole. The stress distribution 'in the fibre is fractions. for The stress concentration reaches a small all volume constant almost failure lead failure The to the the via of concentration of stressmay pole. maximum at the fibres in the transversedirection. The values of the stress concentration increase with increasing volume fraction of fibres for both matrix and fibre. This result is expected and may be attributed to increasing interactions between the stressfields as the interfibre separation decreases. It meansthat increasingdisorder increasesthe stressconcentration.
4.5.1.3 Stressesat interface Stressesin the xy plane around the interface within the fibre and matrix were extracted for 30% fraction fibres The to transformed of results volume polar coordinates. and 4.47. figure in are shown The trend of these results is a stringent check of the accuracy of the finite element in 4.47 Figure that the conditions are, shows necessary general, well satisfied. analysis. Shear stress is zero at the pole and equator. The maximum stress at the interface is the figure in 4.47 The the pole. results shown are comparable with the radial stress at for filled composites with sphericalparticles. results obtained The variations of Von Mises stress at the interface for different volume fractions of fibres are shown in figure 4.48.
The value of the maximum Von Mises stress
is increases for fraction the the slightly and range of volume studied concentration
128
Chapter Four
ResultsAnd Discussion
Interfacial Radial Stress Fibre ReinforcedComposite I.
0.5
rI
II1I
_0.5
60 40 Angle(Degree)
20
0
80
100
Fig. 4.49: Radial stress concentration at the interface of the fibre and matrix for composites with different fibre volume fractions
Interfacial Debonding Stress
4.2E+07
4.0E+07 ir r+ ý
on
3.8E+07
0
A 3.6E+07
3.4E+07 0
10
20 30 Fibre Volume Fraction(%)
40
Fig.4.S0: Interfacial debonding stress versus volume fraction of fibres for fibre reinforced composites
129
50
Chapter Four
Results And Discussion
from interface However, the the the moves position of maximum at almost constant. fractions. low fraction for higher 40° towards the the volume volume at pole about ýw
4.5.1.4 Fracture strength The overall stress distribution for the matrix and fibre may be used to deduce the failure mechanismsof this type of composites. It is assumedthat failure is essentially brittle, and takes place when local stress reachesa critical level. The concentration of fibres. The failure lead the fibre in transverse to splitting of the via may the stress interface lead in direction to tensile the the may at stress radial applied of concentration failure of the interface. The radial stress variations at the interface are shown for is figure 4.49. in The fractions the different volume maximum at the pole position of increasing fraction. The the values of stress concentration the volume range of over failure. for imply in fraction overall applied stress required a reduction with volume Thus the fracture strength of the material is predicted to fall with increasing fibre is Although fraction. the relatively the stress concentration maximum rise of volume slow. is higher is 2758 MPa fibres the much which The transverse strength of about glass Since 1994). 60 is MPa (Gibson, the about than that of the epoxy resin which in fibre is the the the the the matrix at and same pole at stress applied of concentration interface, failure is predicted to occur at the interface at the pole via failure of the (i. interface the the Assuming the of resin e. the strength equals strength of matrix. is interface failure for bonding), the the which overall applied stress assuming good from the be the pole. the at stress of radial concentration calculated expected can failure interfacial fibre/matrix debond is initiation criterion, The analysedusing an of a delamination for free 1988) Zhou Zhou, by Sun (Sun of edge and and proposed here is laminates, which modified to accommodatethe cylindrical geometry composite interface, fibre the matrix of 22 Cr
I
RS
Tre
1+
=1

130
(4.16)
Chapter Four
ResultsAnd Discussion
Fibre Reinforced Composite Failure DebondingShear failure
6.5E+07 6.0E+07 5.5E+07
3.5E+07 Iý 0
10
ý
i4
ý1
40
30
20
4
50
Fibre volume fraction(%) Fig. 451: Comparison of the stresses required for interfaial debonding and matrix failure at different volume fractions of fibre.
Fibre Reinforced Composite Strength FEMCKM Models
8.0E+07 Matrix Failure s CKM
7.0E+07

Modified CKM(1) Debonding
Modified CKM(2)
6.0E+07 S.OE+07 rA Q 4.0E+07 *'a W 3.0E+07 2.0E+07 F 4
1.0E+07 0
10
20 30 Volume fraction
40
Fig. 4.52: Prediction of the failure of the composite by FEM and CKM models (1): Si =0.5.Sm , (2): Si=0.85. Sm
131
50
Chanter Four
ResultsAnd Discussion
where Cyr and rro are the normal (radial) and shear interfacial stresses,respectively, and R and S are the corresponding interfacial strengths with respect to tension and shear. Debond failure resulting from any combination of stressesis thus expected to occur when the left hand side of equation 4.16 is equal to or greater than unity. Since the values of R and S are not available, in order to define the failure criterion ideal bonding is assumedby taking the valuesof R and S as the tensile and shear strength of the matrix. A failure hypothesis may be defined by assumingthat the ultimate transversefailure of the composite occurs in response to an externally applied stress once an interfacial debond forms. A debond initiating stress,is the tensile stress necessaryto make the left hand side of the interfacial failure criterion equal to unity. The maximum interfacial in direction the stress occurs of the applied stress at the pole, while the radial interfacial shear stressis zero at this angle. Since the maximum interfacial shear stress is less than the maximum interfacial radial stress and the interface is expected to be in in than tension shear, the radial stresswill govern the initiation of a debond. weaker The applied stress capableof initiating interface debonding for different fibre volume fractions are calculated and presentedin figure 4.50. As it can be seenin this figure the initiate debonding decreases increasing to stress with required volume fraction applied of fibres.
In composites with a weak interface and/or ductile matrix, the transverse strength of the composite is expected to be more closely linked to the strength of the matrix. This is specially true if the failure of the matrix is analysedusing Von Mises failure theory is based definition the on of Von Mises stressexpressedas: which
[(QI62)2 6c = r
where a
+
(Q2Q3)2
+
(O'30ý1)2J
11/2 = Qm
(4.17)
is the effective stress, a, (1=1,2,3) are the principal stresses,'and am is the
tensile strength of the matrix. The Maximum Von Mises stress and the tensile strength of the matrix are used to calculate the strength of the composite for the range of volume fraction of the fibre. These results are shown in figure 4.51. In figure 4.51 a 132
Chanter Four
Results And Discussion
comparison of the applied stresses initiating the debond failure and the stresses calculated from the Von Mises criterion are made. The stress initiating the debonding calculated using equation 4.16 is less than the stressobtained by Von Mises criteria fo,; all volume fractions. Transverse failure stresses are also calculated using CooperKelly model (CKM) (Cooper and Kelly, 1969) developed for the prediction of transverse strength of the composite. For a composite with a weak interface, the matrix will fail at the minimum in the cross section at equator response to an external stress given by the matrix following equation:
Sc = .
vm
1
4VI
(4.18a)
n
I
is fraction is Vf SC transverse the the the volume of the strength of composite, where fibres and am is the tensile strength of the matrix. If the fibre/matrix interface strength is taken into account, the CKM is modified to .
ST =Qm 1
4Vt +O'r ný
4V
(4.18b)
is where a, the averagetensile stressnecessaryto separatethe fibre from the matrix. It 4.18b in limit be that the when equation a, am can be considered to noted = should be the upper limit of the transversestrength of the composite. In figure 4.52 the strength of the composite obtained by CooperKelly model can be for debonding failure and Von Mises failure. the of our model results compared with The curve obtained from equation 4.18a lies much lower than the other predictions on the same graph. When the value a, = 0.5am is used, the results of the equation 4.18b is close to the interface debonding curve. Raising the interfacial strength value to 4.18b 0.85 gives a variation which is in good agreementwith the via equation am a, = based Von Mises criteria. of our model on prediction The experimental transverse tensile strength at the 53% volume fraction is 37± 4.6 MPa (Pomies, 1992). The transversetensile strength is predicted by our model at 50%
133
Chapter Four
ResultsAnd Discussion
volume fraction as 35MPa for interface debonding and 56 MPa for matrix yielding. Our model predictions using interface debonding mechanismis in perfect agreement with the experimental results. This confirms that the interface debonding and brittle failure are the dominant mechanism governing the failure of the fibre reinforced composite in this case.
4.5.2 Transverse shear loading 4.5.2.1 Modulus A shear loading in the transversedirection of the fibre is applied to the composite. The transverse shear modulus of continuous glass fibre reinforced epoxy is given in figure 4.53. Both square and hexagonal arrays are examined.The results of square array are in close agreement with the results calculated from HalpinTsai model (Halpin and Tsai, 1969). In the case of hexagonal array, both constrained and unconstrained boundary conditions are investigated.The hexagonalarray shows less reinforcement at the same volume fraction compared to the square array. The unconstrained boundary lower for the transverseshearmodulus. values gives condition
4.5.2.2 Stress distribution and strength The stress distribution at the interface of the fibre and matrix is shown in figure 4.54 for the 30% volume fraction of continuous fibres. The maximums of the shear and Von Mises stress concentrations are found at the pole of the fibre. The maximum of the is 45°. at stress radial
The distribution of Von Mises stress concentration is presented in figure 4.55 for different volume fractions. The maximum of the Von Mises stress concentration The Von Mises distributions found for the pole. at stress occurs
different volume
fractions are very close to each other and a slight decreasein the maximum value is fraction increases. the volume as observed
134
Chapter Four
I: rýullý And I)i. tirii. ti.mun
Transverse
Shear Modulus
6.OE+09 m
Ilcxagonal(Unconstraincd)
+I
5.0E+09
ta Halpin
Icxagonal(Constraincd)
o
Syuarc(Unconstraincd)
..ý ý 4.0E+09 
ö 3.0E+09
2.0E+09 
0 Fig. 4.53: Transverse
10
20
shear modulus
30
40 Fibre volume Fraction(%)
at the range of volume fraction
70
60
50
for continuous
fibre reinforced
composites
Interfacial
3,
Stress Concentration Shear Loading
to
2+
; is
I
ýtr
_ ýýýaýý+
ýýý
4w"o'°
o +I+ Von Mises f
Radial
Shear
. " 4
20
0 Fig. 4.54: Interfacial reinforced
stress concentration composite.
40 Angle(Degree)
60
for 30% fibre volume fraction
135
_i
80 of continuous fibre
Chapter Four
Results And Discussion
Interfacial
Von Mises Concentration
3.5 
20% 
30% 
40% 
50%
3 ý C
ý 2.5 1.
ýa v ý2 0
U
Cl) C')
1.5 4. cn
I
0.5 20
0 Fig. 4.55: Interfacial reinforced
40 Angle(Degree)
Von Mises stress concentration
for different
80
60 fibre volume fractions
of fibre
composite.
Interfacial
Radial Concentration
1.2
I= 0.8 
0.6 U
"; 0.4 cdo
0.2
0+ýý 20
0 Fig. 4.56: Interfacial reinforced
F+
ý
radial
40 Angle(Degree)
stress concentration
for different
composite.
136
_{ 60 fibre

+_  _I 80
volume
fractions
of fibre
CIripterFuur
I. czttIts i1 n( 11 )i.ticuý,siun
Interfacial
Shear Stress Concentration
0.5 r
Lxs....,. ý 0
ý
ý 0.5 au U ý C
L) C,)
C) L
1
ý cn
r. A 1.5
aý _a_ n` ýs _ m0
ý
20%a30%A40%50%
t+, 20
2 0 Fig. 4.57: Interfacial reinforced
<.,
t1,9C>
?ý
11
ý I

Shear stress concentration
80
60
40 Angle(Degree)
f
for different fibre volume fractions of fibre
composite
Shear Strength
relative composite shear strength
0.6 ý ýn c aý c. ý ý L.
cý aý ý > 0.4 .ý ýý ý 9
0.2
ý
M
A
0
 
t
10
  
A
f
4
20 Fibre Volume
30 Fraction(%)
Fig. 4.58: Transverse shear strength of continuous fibre reinforced
137
._.
{{
40 composite.
50
Chapter Four
Results And Discussion
The distribution of the radial stress concentrations at the interface of the fibre and matrix for different volume fractions is shown in figure 4.56. In this case the maximum stress concentration occurs at about 45° of the pole for all the volume fractions. As the fibre volume fraction increases, the stress concentration at the interface becomes smaller.
The variation of the interfacial shear stress concentration with volume
fraction,
shown in figure 4.57, indicates
an insignificant increase of the stress
fibre. fraction increasing of volume with concentration The shear strength of composite obtained for different volume fractions is shown in figure 4.58. These values are calculated using the maximum Von Mises concentrations in the matrix. Although
for shear obtained good quantitative predictions are not
for the it the by strength of variation still can give a prediction our model, strength 4.58 in figure be it the As fraction shear seen can the qualitatively. volume with fraction. hardly the volume changeswith strength of the composite There is a considerable resistanceto shear fracture of the fibres. Hence this mode of fracture is unlikely to occur and matrix failure can be blunted by fibres.
138
"w
Chapter Four
ResultsAnd Discussion
4.6 SHORT FIBRE REINFORCED COMPOSITES
4.6.1 Modulus
1"
The model results obtained for Young's modulus of fibre reinforced composites show that it strongly dependson the geometrical arrangementof the fibres within the matrix. This arrangement is characterisedby the volume fraction of fibres, the fibre aspect fibres distributed Here fibre that the the are we assume spacing parameter. ratio and Under is fibre in the this there the of ends. no overlapping and matrix uniformly is between load through the transferred tensile end sections of the cells only condition the cells. In figure 4.59 the Young's modulus of glassepoxycomposite obtained using the finite from lag is the the shear model over values calculated element analysis compared with fibres fraction. The the and cell are equal the glass aspect ratios of volume a range of to 5 in these calculations.
In the shear lag model it is assumedthat the stress is transferred from matrix to fibre lag In transfer interfacial the model of normal shear stresses. modified the shear via is balance force is fibre the equation modified. ends also consideredand stress across The shear lag model gives an estimate for the tensile stressin the fibres as:
[1/ h(ns)] Ei sec cosh(nz r) 6; = E3c
(4.19)
is interface this the using model as: given the at stress shear and
nz ) h(ns) E3cE; sinh( sec z, =n 2 ro
(4.20)
defined is dimensional is the as: overall composite strain and n a constant where E3c
1/2
2 Em
E;(1+ vm)In(1/ f)
(4.21)
is from f distance fraction fibre, is is the the the the ratio, volume aspect of z where s distance. is the fibre radial r and the centre
139
Chapter Four
ResultsAnd Discussion
ModulusVolume
fraction 1w
1.4E+10 0 ti 0
ý 1.0E+10 ý 0 a ý
6.0E+09
30 20 Fibre Volume fraction(%)
10
0
40
50
Fig. 4.59: Modulus of short fibre filled compositepredicted by FEM and shear lag model
ModulusVolume fraction
1.8E+10
FEM(No Flux Term) 9
FEM(Flux Term) 1.4E+10 ý "O 0
.ýy
1.0E+10
0 a
E 6.0E+09
2.0E+09 + 0
10
20 30 Fibre Volume fraction(%)
40
Fig. 4.60: Modulus of short fibre filled composite by FEM taking flux terms into account and ignoring them.
140
50
Chapter Four
Results And I )isrussiun
ModulusVolume
fraction
Effccl OF Aspect Ratio
3.5E+10
J
O.0E+00
{
ý
10
0
20 Fibre Volume fraction(%)
40
30
Fig. 4.61: Comparison of the modulus of composite filled with fibres of 5,10 and 20 aspect ratio modulus of the continuous fibre reinforced composite and the longitudinal
ModulusVolume fraction Effect Of Fibre Spacing
1.6E+10 , 0Fibre Spacing= 2
1.4E+ 10
A Fibre Spacing= 6
. ,cý
411 _.
2.OC+09 I 0
+
j
  4
10
i
1 
I
20 30 Fibre Volume fraction(%)
Fig. 4.62: The effect of fibre spacing on the modulus
of the short fibre filled
141
40 composite.
50
Chapter Four
Results And Discussion
ModulusVolume fraction Fibre Spacing2
1.6E+10 ,
2.0E+09 10
0
30 20 Fibre Volume fraction(%)
40
50
Fig. 4.63a: Modulus of the composite with the fibre spacing 2 for constrained and unconstrained boundary conditions.
ModulusVolume fraction Fibre Spacing6
1.0E+10 ý Constrained
9.0E+09 .2.
e Unconstrained
8.0E+09
c 7.0E+09
ý0 a
6.0E+09
9 5.0E+09 Q U
0
10
Fig. 4.63b: Modulus of the composite boundary conditions. 
20 30 Fibre Volume fraction(%) with the fibre spacing 6 for constrained
142
40 and unconstrained
50
Results And Discussion
Chapter Four
ModulusVolume fraction Fibre Spacing 10
8.0E+09 Constrained ý
,..
0 Unconstrained
7.0E+09
6.0E+09 0 ý aý
ý 5.OE+09 O O
U 4.0E+09 ý
3.0E+09
0
5
20 10 15 Fibre Volume fraction(%)
25
30
Fig. 4.63c: Modulus of the composite with the fibre spacing 10 for constrained and unconstrained boundary conditions
Poisson's Ratio
0.2 +
0
10
20 30 Fibre Volume fraction(%)
40
Fig. 4.64: Poisson's ratio of short fibre reinforced composite calculated from FEM results
143
50
Chapter Four
ResultsAnd Discussion
Shear Stress At Interface
0.2
0 ý 0.2 0
.., ý .ý 0.4 ýý aý
ý 0
0.6
U ý 0.8 ý
ý'
1 1.2 1.4 012
4
3 Z along the fibre length
Fig. 4.65: Shear stress along the interface of the fibre and matrix for different fibre volume fractions
Shear Stress At The Interface of Fibre And Matrix
2.0E+06 0
1.5E+06
FEM ,ArShearLag Model a
1.0E+06
ModifiedShearLagModel
5.0E+05
0.0E+00
5.0E+05 012
3
4
Z along fibre Fig. 4.66: Shear stress at the interface of fibre and matrix predicted by FEM and Shear lag model
144
Chapter Four
Results And Discussion
The shear lag model gives much lower values for modulus. The values of moduli found by modified shearlag model are closer to the finite elementresults. In implementing the finite element analysisfirst the boundary tractions at the interface ignored. Then fibre the calculations are repeated and the the are matrix and of boundary flux terms are considered in the field equations. The moduli values found in boundary from lower traction the latter the than obtained no results the analysis are 4.60. is in figure This comparison shown case.
The effect of different fibre aspect ratios at constant fibre spacing on the composite 4.61 figure In the is investigated. the of modulus of glass epoxy variations modulus in is 6 fibre The 20 5,10 for all spacing are shown. the aspectratios of and composite increase be. the increase A of the with observed can modulus of these relative cases. of fibres is longitudinal limit the The continuous the of modulus upper the aspect ratio. high by fibres be aspect ratio. using with that can achieved fibre depends the filled on also The modulus of the composites with rod shapeparticles 2,6 10 fibre different for three The the of and spacing modulus variations of spacing. in is decrease 4.62. in figure There is 5 modulus as a shown at constant aspect ratio of increases. fibre the spacing
be In this the the with constraint. case boundary cell can of The wall side on condition be the distribution should of final composite and effective properties the stress in described the the two by the chapter steps are of results superimposing calculated 3. The unconstrainedboundary condition is also modelled. The results of the two sets difference 4.63ac. figures A in boundary remarkable conditions are compared of the boundary imposition The between the two of unconstrained cases. can be observed lower in the modulus. of values much results condition The Poisson's ratio values are given in figure 4.64 for both casesof with and without flux terms in the model equations.
145
ChapterFour
I; (stills And I )isru,,. tii, ui
Tensile
Stress at Interface of Fibre and Matrix
ý, QA
ý
2 Z along the fibre interface
0 Fig. 4.67: Tensile stress concentration fractions
3
4
at the interface along the fibre length for different
fibre volume
Tensile Stress at Interface of Fibre and Matrix
vý1 10% (No Flux) 0 
0
f 
10%(Flux) +
0 40%(No Flux)
a 40% (Flux)
iý
ý
234 Z along the fibre interface
Fig. 4.68: Tensile stress concentration at the interface along the fibre length volume fractions comparing the model with and without flux terms
I10
for the 10% and 40%
Chapter Four
Results And Discussion
4.6.2 Stress distribution 4.6.2.1 Interfacial shear stress distribution The normalised shear stress distributions at the interface of the fibre and matrix are in figure 4.65. fractions It indicates for different that the shear stress volume shown decreasesfrom the fibre end to the fibre centre where the stress is nearly zero. In figure 4.66 the finite element results for the interfacial shear stress are compared with the results of the shearlag and modified shear lag model. The shear lag analysisgives lag finite interfacial between The difference higher shear and model stress. shear much fraction. increases with volume element results
4.6.2.2 Interfacial tensile stress distribution The tensile stress concentration at the interface is maximum at the fibre end. A decrease of interfacial tensile stress concentration can be seen in figure 4.67 as the figure 4.68 interfacial increases. In fraction tensile stress concentration the volume interfacial flux finite from terms the are without analyses with and element resulted for fractions 10% 40% These the of and are made volume comparisons compared. higher for both flux from The terms the considering are analysis stresses respectively. distribution interface The fractions. the of the concentration at shear stress volume field including flux in by does filler the terms the the equations. not change matrix and The comparison is presentedin figure 4.69 for the volume fractions of 10% and 40% . 4.6.2.3 Tensile stress in the fibre The tensile stress distributions in the fibre are shown in figures 4.70ab for different finite The the fractions. the with are compared computations results of element volume lag in by fibre distribution the the shear model and the modified calculated tensile stress lag model. shear A comparison of the tensile stress distribution resulted from the finite element model flux boundary is 4.71. in figure the terms made without with and The modified shear lag model which considers the tensile stress transferred by the end higher The lag transfer fibres that the of values. simple shear model assumes gives of
147
ýw
Chapter Four
Results And Discussion
Shear Stress at Interface of Fibre and Matrix
0.2
0 0.2 O
 0.4 ýý u 0.6 ý
Ü 0.8 ý aý 1 ý. ý ý 1.2 1.4 1.6 1
0.5
0
2.5 2 1.5 Z along the fibre length
3.5
3
Fig. 3.69: Prediction of shear stress along the interface by FEM, Comparison of the results with and without flux boundary terms(Volume fraction 10% and 40%)
Tensile Stress In The Fibre Volume fractions(10%,20%)
4
3
2
012 1 Z along the fibre length
3
Fig. 3.70a: Tensile stress concentration in the fibre for 10% and 20% fibre volume fractions.
148
4
Chapter Four
ResultsAnd Discussion
Tensile Stress In The Fibre Volume Fractions(30%, 40% and 50%)
5.5E+06 5.0E+06 4.5E+06 ti h 4.0E+06 4)
.ý ...
Eý
3.5E+06 3.0E+06 ýý
50% 30% 40% +ýý IrIrIjI
2.5E+06 4
3
2
012 1 Z along the fibre length
4
3
Fig. 4.70b: Tensile stress concentration in the fibre for 30%, 40% and 50% fibre volume fractions
Tensile Stress In The Fibre
10%NoFlux
5.0E+06 F 10%Flux e. 50%NoFlux
RS
ý
W3
4.0E+06 f
CL)
50% Flux 3.0E+06
30% Flux
E+ 30%NoFlux
02 2 Z along the fibre length Fig. 4.71: Tensile stress concentration in the fibre. Comparison of the FEM predictions with and without flux boundary terms
149
Chapter Four
Results And Discussion
tensile stress to the fibres is only via shear stress at the interface of the fibre and the matrix gives the lower valuesfor the tensile stressin fibres. At low volume fractions up to 20% of filler the tensile stress in the fibre found by the finite element model is higher than the shearlag model prediction all over the length of the fibres as it is clear in figure 4.72a for 10% fibre volume fraction . Figure 4.72b indicates that at the 20% volume fraction of the fibres the finite element results are in lag When the volume the with modified shear model predictions. agreement close fraction rises to 30% the tensile stressdistribution of fibres obtained from our model is located between the simple shear lag and modified shearlag models as shown in figure 4.72c. For the higher volume fractions the result of our model for, tensile stress distribution conforms with the results of the modified shear lag model at the end fibre it However increases tensile the stress continuously. segment where
then moves
towards the simple shear lag model predictions and reaches a plateau at the central fibre. the of segment The tensile stress increasesfrom the fibre ends towards the centre of the fibre. For fibre length fraction tensile the the the stress of at a certain point along volume each length for the its the the of rest of constant and remains almost maximum reaches fibre. The point at which the tensile stressreachesits maximum value depends on the fraction fibres. higher The the fraction the the the peak point closer volume of volume is to the fibre end.
4.6.3 Critical length The minimum length of the fibre at each volume fraction through which the stress rises in length is length. The its the critical plays a great role critical to maximum called fibres in the order to provide the maximum strength or short characterising for in different There the theoretical composite. approaches are reinforcement fibre length. definition The length the the might change critical of of critical estimation from one theory to another. The model proposed by Rosen ( Rosen, 1964) defines the 0.9 fibre length length times the to minimum as or aspect reach required ratio critical fibre fibre about strain, each end in a long fibre. Rosen gives the critical the maximum fibre length as:
150
ý.
Chapter Four
Results And Discussion
Tensile Stress In Fibre Volume fraction(10%)
.'
3.0E+06
2.5E+06 2.0E+06 ýý ý 1.5E+06 ...
1.0E+06 5.0E+05 0.0E+00 2 Z along the fibre length
1
0
4
3
Fig. 4.72a: Tensile stress concentration in the fibre for 10% volume fraction. FEM and Shear lag models predictions.
Tensile Stress In Fibre Volume fraction(20%)
}
ý 2.8E+06
a ýýrA ýeLl2.1E+06 ý. .., F4
1.4E+06
"Modified Shear Lag Model aShear Lag Model
7.0E+05
E3FEM(With Flux Terms)
0.0E+00 0
1
2 Z along the fibre length
3
4
Fig. 4.72b: Tensile stress concentration in the fibre for volume fraction 20%. FEM and Shear lag models predictions.
151
Chapter Four
I. c.tiiil(s And I )isrus. tiioun
Tensile Stress in Fibre Volume fraction(30%)
5.5E+06 Modified Shear Lag Model m Shear Lag Model
r. ß
.+3_ FEM(With
4.0E+06
hlux Term)
ý. ý Fr 2.5E+06
1.0E+06 0
I
in the fibre 30% for volume fraction.
Fig. 4.72c: Tensile stress concentration predictions.
3
2 Z along the fibre length
4
FEM and Shear lag
Tensile stress in fibre Volume fraction(40%)
7.0E+06
5.5E+06 , .
4.0E+06
Shear Lag Model
2.5E+06
Modified Shear Lag Model FEM(With
1.0E+06
Flux Terms)
+U
i
Fig. 3.72d: Tensile stress concentration
2 Z along the fibre length in the fibre for 40% volume fraction.
predictions.
152
4 FENI and Shear lag model
Chapter Four
hrsuils
And I >isrussiun
Tensile Stress In Fibre Volume fraction(50%)
8.5E+06
" ý `ý\ýý
7.0E+06 ý
ý I
Shear Lag Model
Modified
H Shear lag Model
2.5E+06
1+
1.0E+06
i 3
012
4
Z along the fibre length Fig. 4.72e: Tensile stress concentration predictions.
in the fibre for volume fraction
Critical
50%. FRA1 and Shear lag
Length
FENIRosen models
12 0
Rosen
10
I&
FEM
8 I
6+
4
24i
i
0 Fig. 3.73: Critical
0.1
ii
iIi,
0.2 0.3 Fibre Volume fraction(%)
length of fibres versus volume fraction
153
0.4
i
0.5
Chapter Four
Results And Discussion
lý =dE
1/2 Vf2
V2 Ef(1+vm) Em mf
ýý
Vi` Vv2
(4.22)
where d is the fibre diameter, Ef and Vf are fibre Young's modulus and volume fraction, v, and E. are matrix Poisson'sratio and Young's modulus. Equation 4.22 shows that the critical fibre length depends on the fibre diameter, the square root of the ratio of the tensile moduli and on a term representing the fibre fraction. The lengths fibres critical of obtained by the finite element model and volume Rosen model are compared in figure 4.73 for a range of filler volume fractions. Our in its in fibre that the tensile the to stress rises peak a shorter length of model shows the fibre. Termonia (Termonia, 1990) concluded that the critical fibre length is a fibre But function the of and matrix modulus' ratio. our results show that the unique descirbes depends fraction. Another length the which model also on volume critical the critical length is developedby Kelly as:
d a. r 11= 2zw is interface diameter fibre, is d the the the shear strength of of or of the T. where fibre. is the the tensile strength of and matrix a,
4.6.4 Strength of short fibre composites Prediction of the strength of short fibre reinforced thermoplastics is a complex but industrially crucial problem. In the case of a composite containing short fibres, the fibre implies the that the average stress carried along a non uniform stress of existence by the fibres at the point of failure will be lessthan fibre ultimate tensile strength. Failure occurs by a different mechanismin the short fibre composite which depends on the mechanicalproperties of the constituents and the tensile and shear strength of the interfacial bonding of the fibres and the matrix. Another deterministic factor in the failure mechanismin short fibre reinforced composites is the fibre aspect ratio or fibre length. There is the possibility of failure in the fibre, in the matrix phaseor at the
154
Foul
('haI)lrr
I: r. iill.
Tensile
Stress At The Upper
2017, v 30% z
m 10% 
\n ;
l I1i`
Interface
40%
a
3.5 ý 0 ý cý ý. 4.
ý 2.5 0 U ýý 2
w
ý
1.5 
ý ý
__________________________ .
a0
.
ý
0
°+
.
O
m" i

.
+ý, z/
i=1..
T4
ý{
0.2
0.6 R along the upper interface
0.8
0.4
Fig. 4.74: Tensile stress concentration
_v
I
at the upper interface
Debonding Strength
3.0E+07
2.5E+07 4ý
an
2.0E+07 .C7
55 1.5E+07
ý an c
1.0E+07
Aspect Ratio5
0
" Aspect Katiw
OE+0G _5.
I(1
A
Aspect Ratio=20
O.0E+00
I
4
IO
Fig. 4.75: Debonding
20 strength
prediction
t   f
 
ý
30 Fibre Volume fraction
40
by FEM of the fibres with 5,10 and 20 aspect ratios.
i, ý
50
u.ti, i ui
Chapter Four
Itrsulls
And Uisrii,.
Debonding StressFibre Breaking Stress
3.5E+08 3.0E+08 E3
Debonding AR= 20
0  Debonding AR=5

 Fibreßreaking
Z
Fibreßreaking
AR=20
M
lýibrcßreaking
AR=201. M
V
Dctxmding AR201, M
AR=5
2.5E+08 r
Ä 2.0E+08
m
.ý
i
ý
dý
O.0E+00 Ti 10 Fig. 4.76: Stresses required (LM: Low Modulus)
up
i 20 for debonding
74 ii
30 Fibre Volume fraction(%) and breaking
156
of the fibre with
40 5 and 20 aspect ratios.
50
i
Chapter Four
Results,And Discussion
(I
fibre/matrix interface. A combination of these mechanismscan lead to the catastrophic failure in the composite. The significant stresson the lateral interface of the fibre/matrix along the fibre length is the shear stress.Even when the shear stress on this face exceedsthe shear strength of the bonding, the compressiveforces normal to this interface retain the fibre and lateral forces the The the the of contraction are of result compressive closed. matrix matrix.
Figure 4.74 shows the normal stress distribution at the upper interface of the fibre in be face this The tensile this seen and can are as normal stressesat and the matrix . figure the concentration of these stresses increasesfrom the axis towards the edge of fraction the The fibre. the tensile as volume reduces of stress the concentration increases. The debonding of the interface occurs when the tensile stress reaches the tensile by finite bonding. The debonding interfacial the the stresses obtained strength of in figure for 4.75. debonding The the aspect stresses element model are presented fraction. debonding The for 20 the 5 the volume of range are given and ratios of increase 5 is increases. At fraction the the of aspect ratio the volume stress rises as 20. the than of ratio aspect more significant The tensile stress required to be applied to the composite to bring the fibres to their failure tensile stress is given in figure 4.76. The applied stress to break the fibres is 20. An 5 the the for to of ratio aspect higher values of compared aspect ratio of fractions. The higher be breaking in volume increase the stressto point can observed at fibre for debonding the higher the breaking than stresses the are much point at stresses fibre lower differences aspect These at remarkable more are matrix. epoxy and glass ratios. fibres debonding the low acts be it the of Therefore can predicted that at aspect ratios failure by the it is followed for failure matrix the initiation of the and the mechanism as the is low, fibre Since of the fibre the concentration aspect ratio of short ends. near the direction from local the of is high transversely then the fibre ends cracks merge and the applied load.
157
Chapter Four
Results And Discussion
The catastrophic failure of the short fibre filled composite can occur without any breaking of the fibres. At higher aspect ratio, since the breaking stress of the fibres is lower, the failure occurs in two steps. In the first step the interface debonding and tensile failure of the matrix at the fibre ends is observed.However, in contrast to the low fibre aspect ratio case, growing matrix cracks at fibre ends are quickly blunted by failure hence Because fibres of the cracks catastrophic occurs. and no neighbouring drops debonding interfacial the significantly. modulus of composite and Our model shows that the breaking stress of the fibres reduces at low modulus. Therefore in the second step which corresponds to a decreasedmodulus the tensile leading fibres failure fibres to the in the the of the stress can easily exceed stress breakage of the fibres.
4
158
Iw
Chapter Five
CONCLUSION
5.1 CONCLUSIONS OF THE PRESENT WORK
The developed finite element model has proved to be a flexible and cost effective tool for the prediction of properties of polymer composites. The stiffness, stress different failure field, types of composites are of and strength mode concentration in The predictions are general more accurate than strength values stiffness studied. obtained. 5.1.1 Modulus Of Particulate Filled Composites It has been demonstrated that the modulus of elasticity of a spherical particle be finite using a can element stress analysis accurately predicted polymer reinforced technique. The strength of the composite can also be predicted, but further study is for interfacial bonding The to account effects. model used to calculate the required degree The be fracture to strength may also extended simulate mechanism. composite high influences but bonding the the volume strength modulus of elasticity, unless not of concentrations of particles are present.
4
The experimental results for effective modulus of the particulate composites agree well finite the element predictions. with
The differences can be attributed to the
have dewetting, been in that the made model, such as agglomeration, assumptions
Chapter Five
Conclusion
adhesion, arrangement and filler particle shape. Another reason for the discrepancy between experimental results and the model predictions can be due to skin effect, higher gives experimental values of modulus for soft particulate filled which composites. The same effect can result in lower experimental modulus values for composites filled with hard particles.
5.1.2 Composite Filled with Hard particles Our predictive model for particulate filled composite materials containing hard interesting The finite has has been and useful results. produced element model particles in between by the stiffness values of our predictions and agreement excellent validated distributions The stress predicted are validated by careful experimental measurements. interface. distributions for the predicted stress around epoxy exact matching of stresses have been filled successfullycorrelated with previous models spheres glass with resin description fracture The the of appearance successful and experimental observations. leads better to these various conditions understanding of their under materials of fracture behaviour, which will allow more confident use of these materials under different conditions.
The position and magnitude of the maximum direct stress concentration is found and this can be used to predict the fracture behaviour and the direction and amount of fractions, in bonded low For the sphere, at resin. a well volume crack crack growth growth is attracted to the resin above the pole of the sphere. Smearing of resin around the pole of the sphereis observed. The amount of resin smearing around the pole of a well bonded sphere would be dependent on the position of the crack with respect to the sphere. Our results show that at low volume fractions the maximum distance is much further away from the sphere than for high volume fractions. Thus the model predicts that more smearing low if in fracture Greater at the no spheres smearing occur are visible would surface. been fractions has observed. volume
160
Chapter Five
Conclusion
The concentration of the radial stress at the interface indicates the possibility of debonding at the interface and the extend of the opening up of the crack at the interface of the matrix and filler particle. Under compressionloading, the absolutevalue of the
is the at equator almost an order of magnitude smaller than its value at the radial stress pole it may however be sufficient to cause debonding from the equator of the sphere under the condition of the applied compression.
The concentration of the Von Mises stressesshows the position of the formation of determines bands. The the the governing mechanism of plastic state of stress shear deformation. The degree of interfacial adhesion has a profound effect on the formation. band between In cases where the beads shear craze and competition formation is favoured, for beads craze whereas adhering poorly perfect adhesion are in is local is dominant. This by difference formation band the stress effect caused shear situation, craze formation becomes controlling under a triaxial stress state and shear band formation is governing under a biaxial stress state. In the case of an excellently adhering glass bead, the crazesform near the pole. Stressanalysisshows that these are regions of maximum dilatation and maximum principle stress. At a perfectly bonded bands form bead the that shear near the surface experimental observations show glass of the bead at 45° from the poles defined by the symmetry axis of maximum principal shear stress and of maximum distortion energy density. Our results confirm these observations. The tensile strengths predicted by the finite element model and Von Mises criteria are in decrease All the strength with the with other models. compared models show a increasing filler volume fraction. However, since it is assumedthat composite failure is first fails, by the the rather when element model our occurs prediction of strength conservative. A more realistic assumption would be that composite failure does not highly by fail. for This the of arrest several until stressed crack elements allows occur inclusions in into increase takes the the and also account statistical neighbouring for the small volume of material which is subjectedto the stressconcentration. strength
161
Chapter Five
Conclusion
The experimental results show that the strength of the composite dependson the size is finite filler In the the composite flawless particles. our element model, we assume of and hence it predicts the strength of the composite through the stress fields which are independentof the particle size.
5.1.3 Composite Filled with Partially Bonded Particles An interesting result of the present study is that besidesmatrix properties, deformation heterogeneities the the temperature, also stress concentrating nature of rate and determines the mode of tensile deformation; the biaxial stress state induced by poorly brittle deformation ductile beads the of at expense shear promotes adhering glass in interest development be insight This the of new composite materials. of may crazing. By avoiding a triaxial stress state at the stress concentrators, a ductile response to tensile deformation might be achievedunder test conditions that otherwise would yield a brittle response. The present study provides a good insight into the factors that determine the stress inclusion between interfacial tip the crack a rigid spherical of a curvilinear state near and a polymer matrix. The maximum stress concentration near the tip of a curvilinear interfacial crack at a further bonded sphere does not simply increase with increasing is by but determined tip length, the the with regard to of crack orientation also crack the applied tension direction. The analysesfor a completely unbonded sphere have larger become interfacial that sphere cannot unbonded an crack at a completely shown is The 68° 70°. 0= the tip length by to near stress state than a critical represented biaxial and strongly determined by the extent of interfacial slip along that part of the failure interface that The the criteria that of elastic values remains closed. unbonded interfacial formation increases the slip reduces. of extent substantially as rule craze When relative tangential displacement of the particle and matrix is permitted and imposed interface is filler the the at slip of and matrix, the predicted stress partial decreased are concentrations .
162
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Conclusion
Comparison of the results of the completely unbonded spherewith the physical reality of craze and shear band formation at poorly adhering glass spheres has shown reasonable agreement with respect to the critical interfacial crack length that can maximally be reacheduntil a craze or shearband forms at the tip. Definite conclusion on whether craze and shear band formation occur more easily at the sphere than at an excellently adhering glass sphere could not be made becausethe extent and character of the interfacial slip between a poorly adhering glass sphereand a polymer matrix are not precisely known. For a debonded rigid particle the maximum values for direct and Von Mises stress are at the crack tip. This confirms that any further plastic deformation through crazing or shear band formation starts at the crack tip. Debonding of filler particles reducesthe strength of the composite, but the debonded keep from failure. in higher This'results the strength of the composite particles still can debonded filled particles compared to that of the porous composite. with composite
5.1.4 Composite Filled With Soft Particles Comparison of our predictions with experimental results for epoxy resin reinforced with soft particles provides valuable insight into the mechanical behaviour of these in design The terms of the of optimum particulate reinforced materials materials. desired volume fraction of filler and constituent material properties 'may now be a
considered. The experimental values of composite modulus are in comparison higher than our is because this predictions, probably of some stiffening of the rubber via the model in inadequate description Poisson's the the the rubber the model, and of ratio of epoxy, incomplete phase separation leading to a lower rubber volume fraction than the The during the experimental theoretical value. onset of plasticity expected discrepancy as well as any experimental errors some can also cause measurements between the measuredand the predicted values. In caseswhen the experjmental value is higher than the predicted results can be due to the skin effect. The restrictions imposed by the walls of molds, leads to an excess polymer at the surface of the test in is Thus, flexural torsion tests the or at the where stress maximum specimens.
163
1"
Chapter Five
Conclusion
is dominant and determinant the behaviour of the the the properties of surface surface, whole sample. This error can be corrected by using thicker specimenswhich can be extrapolated to infinite thickness, or by using particles of smaller size and extrapolation to zero particle size. The skin effect can produce errors as large as ten to twenty percent dependingon the thinnessof the specimen.
The effect of the accuracy of the rubber material properties such as the Poisson's finite is input for the the the of code, on result as element used which ratio, is calculations shown. Linear relationship between Young's modulus and volume fraction is found which is similar to the results of experimentalmeasurements. The maximum direct and Von Mises stress concentrations are found at the equator of the spherical filler particle. Shear bands in the resin are expected to grow from the point of the maximum is be Von Mises the to the of equator at which predicted stress, of concentration sphere. Initiation and growth of the shear bands are largely governed by the concentration of Von Mises stress in the matrix, whereas cavitation, or interfacial debonding, of the hydrostatic (dilatational) by is largely tensile the stresses. controlled particles rubber To analysethe stress field accurately, it is necessaryto employ a numerical technique finite the element method. such as
The radial stress at the interface for the soft particles is far smaller than that found for for debonding is indicates the This tendency hard there that at no the particles. interface in composites, which is in contrast to the tendency for debonding in the glass filled material.
Stress distribution in the matrix for resin containing rubber sphereshas been found to be very similar to that of a resin containing holes. The values of matrix maximum holes for the the or rubber equator epoxy resin containing stress concentration, at spheresare very close.
Shear band formation in the resin would occur in a similar
164
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Conclusion
way for the two materials. However, epoxy resin containing holes is known to be a very poor material. We therefore can postulate that shear band formation is not the only important fracture mechanism for this material. Our results have shown that cavitation of the rubber particles is likely to occur, in agreement with experimental observations; after cavitation the rubber may be stretched. It appearsthat the process of rubber cavitation and stretching may make important contributions to the overall fracture energy. The magnitude of thesecontributions is dependenton the modulus of the rubber. Further contributions to the fracture energy of epoxy resin containing from the physical presenceof the rubber which would act as a spheres may arise rubber crack stopper to a growing crack attracted to the equator of the sphere by either the initiation bands. direct the stress of shear of or concentration
Cavitation of rubber particles on fracture surfacesis generally found to start from the initiation but indicate Our do the of a preferred point predictions not sphere. centre of imposition the that of an overall tensile stress of the order of the matrix clearly show yield stress places the rubber particles in sufficient hydrostatic tension to initiate cavitations from small flaws. These flaws may be areas of rubber which are not fully polymerised and can not be observedby conventional microscopic techniques.There is some argument as to whether or not the cavitation of the rubber particles is necessary for shearbands to form. The exact stresslevels for shearband formation and cavitation will depend upon the values of stress required to form matrix shear bands and cause for the systemunder study. cavitation rubber 5.1.5 Composites Reinforced With Continuous Fibres It is observed that
the addition of fibres produces a substantial increase in the
transverse modulus of the binder material; however, the effect of an increase in fibre for is diminishing becoming the to obtained a gradually value asymptotic one modulus the infinitely rigid fibre. Therefore, at high values of the fibre modulus, The ratio of transverse Young's modulus to longitudinal Young's modulus becomesrelatively small high is the the of significant of one structural use problems associated with which in filaments composite structures. If multiple oriented modulus
165
fibre
arrays are
Chapter Five
Conclusion
in low in direction, decrease to the the transverse solve problem of stiffness utilised a the major stiffness of the fibrous composite follows. Contours of the direct stress and Von Mises stress for a typical fibre reinforced filled The to that maximum a composite. are very similar of particulate composite for fibre is in the the the glass applied above rigid of stress matrix pole concentration in epoxy resin.
The maximum stressat the interface is the radial stress at the pole. The position of the low fraction from 40° interface towards the the at volume about moves maximum at fractions. higher for the volume pole
Since the failure strength of the interface is expected to be lower than failure strength important is be interface to the considered more of the resin, the stressconcentration at for the study of the fracture behaviour than the stress concentration above the pole. The stress distribution in the fibre is almost constant for all volume fractions. The The the concentration of stress pole. a slight at maximum reaches stress concentration in direction. fibres failure failure transverse lead the the the to of via may
The concentration of the stressin the fibre may lead to failure via transversesplitting of interface in direction the fibres. The the at radial the concentration of applied stress the is fracture interface. The failure lead the to tensile strength of material of may fraction. fibre increasing fall to volume with predicted Interfacial debonding was approached by a quadratic failure criterion based on the finite by interfacial element radial and shear stresses computed magnitudes of the (CKM) The CooperKelly interfacial model the strength. corresponding method and interface is fibre/matrix generally overconservative. with weak The applied stressneededto causethe debonding of the fibre and matrix is significantly lower than the tensile stress necessaryto causeshearfailure of the matrix in this case. Our model predictions using interface debonding mechanism is in perfect agreement brittle debonding interface This the and that confirms with the experimental results.
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Conclusion
failure are the dominant mechanism governing the failure of the fibre reinforced composite in this case. The limited accuracy of the strength predictions is partly attributed to the simplified assumption of a regular microstructure. The sensitivity of the failure predictions to the fibre volume fraction indicates that failure may initiate locally in a region of closely packed fibres. To increasethe accuracyof the strength prediction, appropriate strength data for the matrix and interface is needed.
5.1.6 Composites Reinforced With Short Fibre The stress and strain distribution in unidirectional discontinuous fibre composites have been studied according to a fibre distribution without overlapping of fibres. The stress distributions obtained show that the tensile load applied io the cell is borne essentially by the fibre. Load is transferred between fibres by lateral shear of matrix. The shear load is localised at the fibre and inducing a stressconcentration. The results obtained for young's modulus show that modulus dependsstrongly on the load fibres, fibres. Without the tensile of of overlapping arrangement geometrical following is borne by layers to a alternate of matrix and matrixfibre, applied composite dependent So the strongly on the are effective moduli stress equality. scheme of thickness of the matrix layers. Although shear lag model is very appealing in this task becauseof its simplicity and into factors, its because to take such as the account critical ability of also it is based fraction on several crude and aspect ratio, reinforcement volume finite based the on element analysesshows that the shear stress models simplifications. is not constant along the reinforcement and matrix interface. The model also takes into fibre importance the the near ends. stress concentration of account
The brittle fracture stress of short glass fibre reinforced composite increases as the fibrematrix interface increases. Competitive deformation the processes at adhesion between interfacial debonding and matrix cracking at fibre ends is proposed. The increaseof interfacial shear strength by coupling prevents early failure at the interface, thus increasingthe tensile failure stressof short fibre composites.
167
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Conclusion
The deformation behaviour which shows a sharp drop in stress due to interfacial debonding followed by matrix yielding, is absentin the composites with good coupling is for higher interface. fracture fibrematrix The the yielding strain after upper shear at the composites with poor coupling.
The model results obtained for Young's modulus of fibre reinforced composites show that it strongly dependson the geometrical arrangementof the fibres within the matrix. This arrangement is characterisedby the volume fraction of fibres, the fibre aspect ratio and the fibre spacing parameter. the finite element results for the interfacial shearstressare compared with the results of the shear lag and modified shear lag model. The shear lag analysisgives much higher finite lag between difference The interfacial element and shear model stress. shear fraction. increases volume with results The
for both higher flux from terms the are tensile stresses analysis considering
interface distribution The fractions. the of the at shear stress concentration volume in field flux including does by filler the the terms the equations. not change matrix and The tensile stress increases from the fibre ends towards the centre of the fibre. For length fibre fraction tensile the the the stress of along a certain point at volume each length for its the the of the of rest maximum and remains almost constant reaches fibre. The point at which the tensile stressreachesits maximum value depends on the fraction higher fibres. The fraction the the closer the peak point the volume of volume is to the fibre end. The fibre critical lengths obtained by finite element model and Rosen model are Our fractions. filler that the tensile for shows model volume of a range compared Our length fibre. in in fibre its the the to results show that of peak shorter rises stress the critical length also dependson volume fraction. An increasein the stress to breaking point can be observed at higher volume fractions. stresses for the The stressesat the breaking point are much higher than the debonding fibre glass and epoxy matrix. These differences are more remarkable at lower fibre aspect ratios.
168
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Conclusion
Therefore it can be predicted that at low aspect ratios the debonding of the fibres acts as the initiation mechanismfor the failure and it is followed by the failure of the matrix is is fibre fibre Since low, the the the the of ends ends. aspect ratio concentration near high and then the local cracks merge transversely from the direction of the applied load. Catastrophic failure of the composite then occurs without any breaking of the fibres. At higher aspectratio, since the breaking stressof the fibres is lower, the failure occurs in two steps. In the first step the interface debonding and tensile failure of the matrix at the fibre ends is observed. However, in contrast to the low fibre aspectratio case, growing matrix cracks at fibre failure hence fibres by blunted catastrophic no and neighbouring ends are quickly debonding interfacial Because the modulus of composite the cracks and of occurs. drops significantly. Our model shows that the breaking stress of the fibres reduces at low modulus. Therefore in the secondstep which correspondsto a the decreasedmodulus the tensile leading fibres failure fibres to the in the the the of stress can exceed easily stress breakageof the fibres.
5.2 SUGGESTIONSFOR FURTHER WORK One basic assumptionmade in this analysisis that both the filaments and the matrix are linearly elastic and that no plastic or viscoelastic behaviour occurs. In an actual in does localised the typically weak occur yielding undoubtedly composite material, localised Depending high the on stresses. of matrix material, permitting a redistribution inelastic, being or type elastic, the considered, either nonlinear of matrix material viscoelastic behaviour may occur. Thus, a logical extension of the present analysis will be to study this nonconservativematerial behaviour.
Because strength is affected more than stiffness by material and geometric local in the the stress and strain and resulting perturbations eity nonhomogen distributions. The effects of such local stress and strain perturbations on stiffness are due integration in the effective modulus theories. the to smoothing effect of reduced
169
Chapter Five
Conclusion
On the other hand, material failure is often initiated at the sites of such stressand strain in is The the strength so effect on variability of strength concentrations, much greater. be fibres be used must alone may significant, statistical methods quite and reinforcing for accurate analysis.
The theoretical results obtained for Young's modulus of short fibre reinforced Young's fibres, depend thus the of strongly arrangement on geometrical composites isolated be derived from fibre discontinuous cell an composites cannot modulus of fibres. In to ignored have In order the the overlapping of effect scheme. our model we be fibre hexagonal this should of arrangement consider effect a modelled. The real arrangement of filler particles or fibres throughout the matrix is normally the is the of random to A effect technique consider needed statistical random. distribution of fibre in the matrix. In our model we assumedthat particles and the diameter of fibres are of the samesize. Whereasin practice the filler particles and fibre diameters is not the same. This means that in order to get closer to the real is the not enough. and a model geometry composite a unit cell microstructure of is different sizes needed of particles several containing
The findings of the project can be summarisedas:
A
FORTRAN code is developed that can be used for studying the behaviour of the in composites polymer
the both solid and liquid states. A convenient penalty
in is order to parameter used
obtain the equilibrium solid mechanic equations
starting from the Stokes flow equations. The boundary line integral terms are included in the model. Further investigation in do have the the these terms that result computational not much effect on shows case of the particulate filled composites. But for the short fibre composites taking these terms into account can affect the displacements and stresses values considerably.
170
Chapter Five
Conclusion
The slip boundary condition is imposed at the interface of the filler and matrix in order to simulate the level of adhesionat the boundary of a debonded filler particle and the stress field at the interface is studied. The result are used to predict the further crack propagation in the composite
The developed model
is applied for different types of composites such as
composites filled with hard particles or soft particles, composite reinforced with continuous or short fibres. The results are validated by comparing with be The data values can modulus well established model. and other experimental In the each case underestimated. are generally and strength values predicted closely the calculated stress fields are used to predict the failure mechanism.The dominant in For is the short the toughening the soft case of particles. predicted mechanismof fibre composites the critical length is found from the calculations and the effect of fibre aspect ratio and spacing is studied. The crack propagation is predicted for debondedfilled composites.
i
6
171
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