Transportation
Kentucky Transportation Center Research Report University of Kentucky
Year 1977
Stiffness of SolidLiquid Mixtures: Theoretical Considerations David L. Allen Kentucky Department of Transportation,
[email protected]
This paper is posted at UKnowledge. https://uknowledge.uky.edu/ktc researchreports/1067
COMMONWEALTH OF KENTUCKY DEPARTMENT OF TRANSPORTATION
JOHN C. ROBERTS
JULIAN M. CARROLL GOVERNOR
BUREAU OF HIGHWAYS
SECRETARY
JOHN C. ROBERTS
H220
COMMISSIONER Division of Research 533 South Limestone L.exlngton, KY 40508
January 21, 1977
MEMORANDUM:
G. F. Kemper
State Highway Engineer Chairman, Research Committee SUBJECT:
Research
Report
464;
"Stiffness
of
SolidLiquid
Mixtures;
Theoretical
Considerations;" KYHPR6420; HPR1(8), Part II. KYHPR6420 has been a longtime, involved study (entitled "Flexible Pavement Study Using Viscoelastic Principles11). Phase I was devoted to the flow properties (rheology) of asphalt cements. Progress reports on Phase I were issued in 1964 and 1967 rationalization
of
pavement
design
criteria.
In
(I, 2). Phases III, IV, and V were directed toward these
latter
phases,
temperaturedeflection
and
deflectionstrain relationships were derived, ambient temperatures were analyzed, and mechanistic models and analogies of empirical design procedures were developed. Phase II was conspicuously bypassed. As originally conceived, it was to be an indepth study of the basic, physical properties of asphaltic concrete and the influence of the asphaltic binder. The objective was to correlate, and thUs predict, the viscoelastic properties of materials in a pavement structure and, thereby, to determine the applicability of elastic theory to pavement design. In the interim, developments from research elsewhere provided the inputs sought in Phase II; and we proceeded toward the larger objective. Phase II was deleted, with
FHWA
approval, July 25, 1969. In Phase I, asphalts were chosen on the basis of foreknowledge of their performance and research histories. Some had been studied previously by us (1) and by others (2). Those were the asphalts reported in 1964
(3) and 1967 (4). Before concluding Phase I, it seemed appropriate to investigate a more random
set of asphalts  more especially those supplied to construction projects in Kentucky. The Division of Materials furnished an array of samples taken from construction projects during one construction season. A side benefit from this set of samples was foreseen in connection with the thenimminent, natiorral trend toward specifying asphalts according to viscosity grades. (Note:
1
Viscositygrading was adopted April 21, !971; Special Provision No. 91; amended January 18, 1973,  S.P. No. 91A)
Report 333 covered the aforementioned set of construction samples only  and supplemented the two, previous reports. The analyses, there, were simplified considerably. The Phase II question was:
How do properties of the asphalt affect the properties of an asphaltic
concrete in a pavement structure? From elastic theory analyses, we deduced that a typical value of Young1s
Modulus
of
bituminous
concrete
at
64°F
was
480,000
psi.
Having
determined
the
viscositytemperature relationships for asphalts, it should be possible to relate the stiffness ur modulus of asphaltic concrete to the stiffness (or viscosity) of the asphalt cement. Although we were aware of
difficulties other have encountered in solving this problem, we have endeavored to synthcsi;.c an equation which will fulfill the original objectives of Phase II.
Page 2 January 21, 1977 An authority (5) on this subject stated several years ago: ",,the stiffness of a bitumenmineral mixture is determined only by the stiffness of the bitumen alone and the volume concentration of the mineral aggregate." The Einstein and EilersVan Dyck (or Van Dijck) equations fail in the regions of high concentrations of aggregates (6). The Einstein equation is:
where
viscosity of mixture, viscosity of solvent or liquid and �0 volume concentration of solids. Cv Eilers' and Van Dijck's equation is: [1 + 1.25 Cv/(1  1.28 C )]2 Sm /S b v
17
�
=
=
where
stiffness of the mixture and Sm stiffness of bitumen. s b Heukelom and Klomp (7) modified previous, empirical equations, as follows: �
�
stiffness 3G* (G* complex shear modulus), volume concentration of aggregate, 0.83 log saggi�bit), 6.7 x 10 pS!  Eag • and Sagg g 0.83 (6.83 log S bit!· n Huekelom also cited an empirical equation derived by Nijboer which related stiffness of a mixture to the Marshall stability test, as follows: where
S Cv n
=
�
=
�
�
·
2 S60oC; 4 sec. (in kg/cm )
�
1.6 x stability (kg)/fiow (mm)
or s1 40oF 4 sec (lbs/in 2) ; _
�
5.3 x stability (lbs)/ flow (in.)
The HeukelomKlomp equation could be employed with some minor adjustments to obtain desired values. However, there was a more challenging aspect of the problem: that was to find a prototype, mechanistic equation which yields at least a first approximation of the stiffness or rigidity of asphaltic concretes. McLeod's (8) graphical representation of the Heukelom equation is shown herein.
Page 3
January 21, 1977
'.
STIFFNESS
MO[)!J�!.ill_Qf.JlllJ.!..M..lli_!H_P.S.I
Fig. L. Relationships between Moduli of Stiffness of Asphalt Cements and of Paving Mixtures Containing the Same Asphalt Cements (Based on Heukelom and Klomp).
Report
464,
submitted herewith, presents a mechanistictype derivation which appears to fulfill the
stated objective. The whole study is, therefore, concluded  including Phase II, which was canceled or
voided quite some time ago. It seemed that Report 333 should await the issuance of a companion report
to explain, in some degree, the significance, continuity and application of the combined work.
Stiffness of a viscous material is related to Young's modulus of elasticity by the expression, E
where
!)
=
=
t
3n/t viscosity and
time.
Difficulties arise in deriving a rational equation for the modulus of elasticity of blends or mixtures of elastic solids. Simply adding together strain energies yields
where
E
E
A I
2 •c EcVc
and where
vc
vs1/Vc
=
=
=
=
=
=
=
engineering strain,
Young's modulus area,
length 2 2 + • s2Esz Vsz € S'ES,VS, + vs1 vs2 Cv; VdV c
=
1  C v
Unfortunately, it appears that this seemingly simple derivation cannot be extended.
Page 4 January 21, 1977
References: 1. Observations on the Constitution and Characterization Bulletin 118, Highway Research Board, 1956. 2. Public Roads, August 1959 and October 1960.
of Paving Asphalts,
J. Havens and W. Daniels;
W. A. Mossbarger, Jr., January 1964. W. A. Mossbarger, Jr., and J. A. Deacon, May 1967.
3.
A Rheological Investigation of Asphaltic Materials,
4. 5.
Flow Behavior of Asphalt Cements,
Time and Temperature Effects on the Deformation of Asphaltic Bitumens and BitumenMineral Mixtures,
C. Van der Poe!,
Journal,
Society of Plastics Engineers, Vol 11, No. 7, September 1955. C. Van der Poe!, Interscience
6.
Road Asphalt; Building Materials: Their Elasticity and Inelasticity,
7.
Road Design and Dynamic Loading, Proceedings,
Publishers, 1954. 8.
AAPT, 1964. N. W. McLeod, Discussion of Report by R. A. Burgess, 0. Kopvillem, and F. D. Young,
Ste. Anne
Rest Road  Relationships between Predicted Fracture Temperatures and Low Temperature Field Performance, Proceedings,
AAPT, Vol 40, 1971.
ZJ Clt/2y Respectfully su
Y"E'% JHH:gd Attachments cc's: Research Committee
ctor of Research
Technical keport Documentation Page 1.
Report No.
4.
Title and Subtitle
2.
Government Accession No.
3.
Recipient's Catalog No.
5.
Report Date
January 1977
Stiffness of SolidLiquid Mixtures: Theoretical Considerations
6.
Perform ing Organi:�:otion Code
Co;8. Performing
7.
Authorls)
464
David L. Allen
9.
10.
Performing Organization Nome and Address
Division of Research Kentucky Bureau of Highways 533 South Limestone Street Lexington, Kentucky 40508 12.
11. 13.
Sponsoring Agency Nome and Address
Work Unit No. (TRAIS)
Contract or Grant No.
KYHPR6420
Type of Report and Period Covered
Final
I
14. 1 s.

Organization Report No .
Sponsoring Agency Code
Supplementary Notes
Prepared in cooperation with the U.S. Department of Transportation Federal Highway Administration Study Title: Flexible Pavement Studies Using Viscoelastic Principles' 16.
Abstract
A rational approach to the solution of the stiffness of solidliquid mixtures is presented. The stiffness of such mixtures is dependent on the stiffness of the viscous medium and the volume concentration and elastic modulus of the solid portion. Finally, the general solution is applied, in particular, to bitumenaggregate mixtures; and the results are compared to experimental data.
17.
Key Words
19.
Security Clossif. {of this report)
18.
BitumenAggregate Mixtures Stiffness Viscosity
Form DOT F 1700.7
IB721
20.
Distribution Statement
Security Clossif. (of this page)
Reproduction of completed page authorized
21· No.
of P ages
22.
Price
Research Report 464
STIFFNESS OF SOLID·LIQUID MIXTURES: THEORETICAL CONSIDERATIONS
KYHPR 64·20; HPR·l(8), Part ll Final Report
by David L. Allen Research Engineer Principal
Division of Research Bureau of Highways DEPARTMENT OF TRANSPORTATION Commonwealth of Kentucky
in cooperation with Federal Highway Administration U.S. Department of Transportation
The contents of this report reflect the views of the
author who is responsible for the
facts and accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of the or
Federal
Bureau of
Highways
Highway Administration.
This report does not constitute a standard, specification, or regulation.
January 1977
STIFFNESS OF SOLIDLIQUID MIXTURES: THEORETICAL CONSIDERATIONS It has been stated by Van der Poe] (I) in a paper on the deformational characteristics of bitumen and bitumen�mineral mixtures that the stiffness of mixtures is determined only by the stiffness of the bitumen and the volume concentration of the mineral aggregate (assuming stresses and strains are small). Figure 1 is a summary of his data showing this relationship. The curves in Figure 1 appear to converge at a point where the stiffnesses of both the bitumen and the mixture are equal to approximately 6.89 x 1010 N/m2(6.89 X 1010 kPa). Van der Poe] says that value represents the stiffness (elastic modulus) of the mineral aggregate. Therefore, it would appear that, as the stiffness of the bitumen approaches the modulus of the aggregate, the deflection in the aggregate will become significant an.d play an important role in determining the stiffness of the mixture. This relationship or behavior can be visualized as shown in Figure 2. The stiffness of the liquid medium in Figure 2(A) is simply defined as s
1
�
s
stiffness. force, F radius, R strain, dv/10 'v deflection, and dv !0 � initial total height of specimen. In Figure 2(B), the stiffness of the solidliquid mixture is defmed as
where
�
2
s
Stefan (2), and the solution has been successfully applied to the parallel plate plastometer by Dienes and Klemm (3). The general equation of motion of the Newtonian fluid of viscosity '11 is given in cylindrical coordinates by the following: apjar
2 17V vr � p avrjat,
+
( 1/r)(ap;ae) ap;az
+
2 11V vz
+
2 11V v8 � p av8;at, and �
p
3
avz;at,
fluid pressure, velocity, density, and p vector operator. V To solve these equations, a number of assumptions must be made as follows: where
p v
I.
Let the circular parallel plates in Figure 3 be the planes z � 0 and z h. 2. Let the ratio of h/r be small, that is, the velocity, vz, normal to the plates is negligible compared to the radial velocity, vr. Therefore, vz 0. 0. 3. Because of circular symmetry, v8 4. No slippage at the plates. Therefore, vr � 0 at z � 0 and z h. 5. Assume steady state flow. Therefore, avr /at � 0. With the above assumptions, the equation of motion reduces to the following form: �
�
:::
�
4
Integrating Equation 4 twice with respect to z and inserting the previously stated boundary conditions, vr � 0 at z � 0 and z h, gives �
(dv + ds)/10 �strain in the mixture, � deflection in the liquid phase, deflection in the solid phase, � 2 [(F/7TR )/M) [10  h0), and M � elastic modulus of the solid phase. Thus, to find the stiffness of both models in Figure 2, it is necessary to know or assume the modulus of the solid phase and, also, to know the deflection in the liquid phase after some time, t, for any set of conditions (load, area, and viscosity). These models present conditions closely approximating those encountered in paraliel plate plastometers (see Figure 3). Therefore, the problem is to find the equation of motion of two parallel plates, each moving towards the other under the action of an applied force, with a liquid medium between the plates. This problem has been solved by
where
vr � (1/2 'l)(ap;ar)(z2  zh).
5
�
From Equation 5, it is obvious that in any plane of angle 8 the velocity in the fluid varies parabolicallywith z. The flow, U, per unit arc length between the planes z �Oand z � h is h
U � f vrdz. 0
6
Substituting from Equation 5 into Equation 6, u
( 1/ 2'1)(apjar)
J (z 2 
0
(h3 1 12'1)(ap;ar).
zh)dz
7
8
Due to the action of a force, plate z h moves toward 0 at the rate of dh/dt. This will cause the plate z volume of the elemental prism, dr rdB h, to change at the rate of dr rdO dh/dt. Because the liquid is assumed here to be incompressible, the net rate of outward radial flow must equal the decrease of the volume of the element. Therefore,
Intregating Equation 17 with respect to
=
r
gives
=
·
d r rdO dh/dt
=
[(rdB U)dr]/ar.
a
9
F
18
Equation 18 integrated with respect to t gives 19 where C is the constant of integration. If z h0 at time zero, then z h at any time t can be found from =
=
Substituting Equation 8 into Equation 9 and performing the necessary manipulations gives 10 Intregrating twice with respect to r gives
where C1 and C are constants of integration. Since p must be finite for r 0, C must vanish. At the outer boundary of the specimen, r R, the pressure in the fluid must be in equilibrium with atmospheric pressure, p0. Therefore, =
=
12
c
or h
11
p
20
To find the stiffness of a solidliquid mixture, Equation 21 must first be solved to find the stiffness of the liquid phase alone (Cv 0) as in Figure 2(A). Therefore, given force, F; time, t; viscosity, 71; radius, R; and iilitial height of the liquid phase, h0; substituting into Equation 21 will give the height of the liquid phase, ha , after time t. In Figure 2(A) 10 and h0 are equivalent; therefore, ha subtracted from /0 or h0 gives the deflection, dv. Thus, =
and Equation 1 1 becomes 13
p
The total force acting on the plate i n the positive z direction is h
14
2rr f p rdr.
0
The total force acting on the plate in the opposite direction is F
h
+
15
J p0 rdr, 0
where F is the applied, external force. Substituting for p in Expression 1 4 and equating to Expression 1 5 gives h
h
F + 2rr f p0 rdr 2rr f (37)/h3 ) (dh/dt) (R2 . " 0 r2) rdr +0 2tr f p0 rdr, 16 =
·
0
and F
h
=
2tr (dh/dt)(37)/h3 ) f (R2  r2) rdr. 1 7
Substituting Ev into Equation 1 gives the stiffness of the liquid phase. To solve for the stiffness of the mixture, Equation 21 must be solved using Figure 2(B). In this case, the volume concentration of the solid, Cv, equals (!0h0)/!0; therefore, 22 Using the same conditions of force, time, viscosity, and radius as previously used and solving Equation 21 using h0 as defined in Equation 22 gives a new height, h b , for the liquid phase of Figure 2(B). Thus, the defiection in the liquid phase is given by 23 Again, the deflection in the solid phase is given by 24
0
Inasmuch as the liquid completely fills the space between the plates, then r can be assumed equal to R at all times. 2
Therefore, the engineering strain in the mixture can be calculated from 25 Substituting Equation 25 into Equation 2 gives the stiffness of the mixture. This solution is general and should be applicable to solving for the stiffness of many different systems, including solidsolid or liquidliquid systems (assuming no movement at the interface between the two liquid media). It appears, therefore, to have applications to many types of industrial and engineering problems. This could include slurries (solid particles in liquids), emulsions, or liquids that are composed of two or more other liquids. This solution also could be used to solve such esoteric problems as the stiffness of blood (solids in a liquid). The general solution of solidwliquid mixtures can now be applied to the original problem of bitumenaggregate mixtures. Figures 8 and 9 of Report No. 333 (pages 8 and 9) give the apparent viscosities of several asphalts as a function of temperature (1/0R). The apparent viscosity (steadystate viscosity) was calculated as described in the body of that report and as shown there in Figure 7. For any set of conditions, this viscosity can be employed to solve for the stiffness of any of the asphalts tested in that study by using Equation 21 and Figure 2(A). Therefore, the stiffness of any mixture, using any of the asphalts of that study, can also be found from Equation 21 and Figure 2(B), assuming the modulus of the aggregate is known. To test this hypothesis, a number of calculations were made using the viscosities of Figures 8 and 9 of Report No. 333 and assuming the modulus of the aggregate was 3.45 x 1010 N/m 2 (3.45 x 1010 kPa).
Results are shown here as dashed lines in Figure 1. The calculated values correspond closely with Van der Poefs experimental data. This appears to support his conclusion that the stiffness of a biturnenwaggregate mixture is not a function of the type of bitumen, type of test, or time of loading, but is a function of the bitumen stiffness and the volume concentration of the aggregate. However, one other par�ameter should be added; i.e., the modulus of the aggregate. The solution of this problem should be very helpful to the design and analysis of pavements, in particular. For instance, if the properties of a mixture are known such as the viscositywtemperature relationship of th: asphalt and the percent, by volume, of the aggregate, the designer then has the tools available to obtain the stiffness modulus of the mixture. This value can then be used to calculate stress distributions withln the asphaltbound layers of a flexible pavement for any combination of load, time, and temperature. This stiffness modulus could be used with a number of theories such as linear elastic layer theory, linear viscoelastic layer theory, or finite elements. REFERENCES I.
Van der Poe!, C.,
Time and Temperature Effects
on the Deformation of Asphaltic Bitumens and BitumenMineral Mixtures, SPE Journal,
September
1955, pp 4753. 2.
3.
Stefan, M. J., Akad, Wiss. Wien. Math.  Natur. Klasse. Abt. 2., 69, pp 713735, 1874. Dienes, G. J. and Klemm, H. F., Theory and Application Journal of
of
the Parallel Plate Plastometer
;
Applied Physics, Vol 17, 1946, p
458471.
3
STIFFNESS OF BITUMEN, Figure
1.
Nlm
2
109 (kPa)
Relationships between Stiffness of the Mixture and Stiffness of Bitumen.
4
F
F
I I I
'
I I I I
I I
SOL I D
I I
L IQUID
�:���s:
I
h.,
I I I I I I
I I I
ER F
Rigid
1
.,
Plates
r
_,
'
LIQUID
I I
I I
R� F
( 8)
(A)
F =FO RCE R= RADIUS lo= TOTAL I NITIAL HEIGHT OF SPECI MEN ho= INITIAL HEIGHT OF LI QUID
PHASE
=
( 1 C11) 10
C11 =VOLUME CONCENTRATION OF SOLID
1 Cv " VOLUME CONCENTRATION OF L I QUID
Figure 2.
=
1, lh., o
MEDIUM
Basic Relationships.
5
z

Figure 3.


.....
Parallel Plate Plastometer.
6