rheological modeling of randomly packed granules with visco-elastic binders of maxwell type

ELSEVIER

Computers and Geotechnics, Vol. 21, No. 1, pp. 4143, 1997 0 1997 Else-&r Science Ltd. All rights reserved

Printed in Great Britain PII: SO266-352X(97)00012-8 0266-352X/97 $17.00+0.00

Rheological Modeling of Randomly Packed Granules With Visco-Elastic Binders of Maxwell Type

C. S. Chang & J. Gao

Department of Civil and Environmental Engineering, University of Massachusetts, Amherst MA, U.S.A.

(Received 15 April 1996; revised version received 21 January 1997; accepted 14 March 1997)

ABSTRACT

This paper focuses attention on the modeling of a particle assembly bonded by a visco-elastic binder material. The inter-particle behavior for two particles with a visco-elastic binder is modeled based on two basic elements, a spring and a dashpot. Dtflerent combinations of the basic elements charac- terize direrent inter-particle behavior. Here, we mainly consider the visco- elastic binder of Maxwell type. Thus, granular materials can be viewed as a discrete system of particles connected with springs and dashpots. The discrete system can be simulated by an equivalent visco-elastic continuum. The macrorheological behavior of granular materials is derived based on both kinemetic and static methods. 0 1997 Elsevier Science Ltd.

INTRODUCTION

Granular materials such as cemented particulate material or asphalt concrete can be perceived as a collection of particles bonded with a binder. The material is a two-phase media consisting of the particles and the binder matrix. It is desirable to derive the mechanical behavior of such materials based on the microscale behavior of two different phases.

Along this line of approach in granular micromechanics, the microscale behavior is now characterized as the inter-particle behavior of two bonded particles which depends significantly on the properties of particles and matrix media. The inter-particle stiffness for two elastic particles in direct contact can be found from earlier work by Duffy and Mindlin [l]. Inter- particle behavior for two elastic particles bonded by an elastic binder can be found from recent work by Dvorkin et al. [2] and Zhu et al. [3]. Inter-particle

41

42 C. S. Chang and J. Gao

behavior for two elastic particles bonded by a visco-elastic binder can be found in the work by Zhu et al. [4].

The macroscale behavior of a granular material is characterized as the mechanical behavior of granular assembly. The relationships between the mechanical behavior of assembly and the inter-particle behavior have been studied for particles with direct frictional contact, such as sands [5-81. In these approaches, the material constants for granular materials can be explicitly determined from the material constants of particles. In the deriva- tion process, it is necessary to link the macroscale continuum variables of stress and strain to microscale discrete variables of force and displacement. The methods can be classified into two categories, the kinematic method [5, 7, 9, lo] and the static method [ 1, 111.

In this paper, we focus our attention on the modeling of particles bonded by visco-elastic binder. For particles with a visco-elastic binder, the contact behavior of two particles can be modeled based on two basic elements, a spring and a dashpot. Different combinations of the basic elements charac- terize different inter-particle behavior [12, 131. Here, we mainly consider the visco-elastic binder of Maxwell type. Thus, granular materials can be viewed as a discrete system of particles which are connected with springs and dashpots. The system can be simulated by an equivalent visco-elastic continuum. In this paper, the macrorheological behavior of granular materials is derived based on both kinemetic and static methods.

MICROSCALE INTER-PARTICLE BEHAVIOR

According to the work by Zhu et al. [4], the compliance of a system of two elastic particles with a viscous binder of Maxwell type can be modeled by a system of two rigid particles connected by a serial connection of a spring and a dashpot (Fig. 1). The time-dependent compliance relationship can be

Fig. 1. Schematic plot for the inter-particle model of a system consisting of two particles with a binder.

Rheological modeling of granules 43

expressed in terms of the inter-particle force, the rate of inter-particle force, and the rate of relative displacement of the two particles, given by

fn+t”fn=C&; fs+tsj‘s=csSs; ft+ttf;=ctSt (1)

where the subscript “n” represents the direction normal to the contact area, denoted by a unit vector ni; while the subscripts “s” and “t” represent the two orthogonal directions tangential to the contact area, denoted by unit vectors si and ti. The vector of a contact force or a relative displacement in the x-y-z coordinates have the following relationships with their three components in the b, s and t directions:

or

sf, = $nf ; q = s;s;; 8; = s;t;

f; = ffnf; f; = f;s;; f; = fftf

In Eqn (l), the time scale constants t,, t,, and tt have the are defined as:

(3)

units of time and

where k,, k,, kt are the spring constants and c,, c,, ct are the dashpot constants, respectively, in the n, s, t directions. The material constants t,, t, and tt have significant effects on the rate of deformation. When the viscosity of the dashpots cn, c, and ct, approaches infinity, the time scales of the Maxwell units, t,, t, and tt also approaches infinity. Under this condition, the Maxwell visco-elastic model in Eqn (1) is reduced to an elastic model.

The inter-particle law in Eqn (1) is expressed in a differential form. Its integral form can be obtained by solving the ordinary differential equations in Eqn (1) as follows:

(5)

C. S. Chang and J. Gao

MACROSCALE CONSTITUTIVE RELATIONSHIP

The rheological behavior of a visco-elastic continuum can be expressed in the form of the following general differential equation:

where Bgkh D’vk,C’gk, are the constitutive tensors of the system. In some cases, it is more convenient to represent the differential rheological relationship in an equivalent integral form. Two types of integral equations exist, namely, the modulus function and the compliance function.

Modulus function

The macroscale rheological behavior of a granular assembly can be expressed by the time-dependent stress as an integral function of strain rate, given by:

The kernel of the integral in Eqn (7) E+, is termed as the modulus function.

Compliance function

The macroscale rheological behavior of a granular assembly can also be expressed by the time-dependent strain as an integral function of stress rate, given by:

The kernel of the integral in Eqn (8), Jijkl, is termed as the compliance function. The compliance function is the resolvent of the modulus function and vice versa. Therefore, either one of Eqns (7) or (8) is sufficient to define the rheological behavior of the material.

In what follows, we aim to derive the modulus function and compliance function representing the rheological behavior of the granular assembly


based on the time-dependent inter-particle behavior. Two analytical methods are adopted, namely, the kinematic method and the static method.

KINEMATIC FORMULATION

Stress-strain relationship for a general random packing

The global mean stress can be expressed as the average of contact forces within the particle assembly, given by

q(t) = $2 lffi(t) c=l

(9)

Substituting Eqn (5) into Eqn (9), and assuming that k, = kt and c, = ct, the constitutive equations of granular materials with a viscous binder can be expressed as

j (n:k,,exp(- y) 2

0 (10)

Here, we use the kinematic hypothesis in which the displacement in Eqn (10) is linked to the strain by a linear relationship, 6i= sijlj. This leads to an expression of constitutive equation similar to the form given in Eqn (7), where the modulus function is

+ k, exp

It is noted that in the micromechanics of granular materials, the kinematic hypothesis imposes the condition of uniform strain, thus the estimated stiffness in Eqn (11) is greater than the true stiffness of the granular material.

46 C. S. Chap and J. Gao

Closed-form stress-strain relationship for an isotropic random packing

For a representative volume with a suitably large number of particles, the summation over all contacts can be expressed in terms of an integral over all contact orientations. In a spherical polar coordinate system, let P(y, 4) be a quantity dependent on the orientation of inter-particle contact, where y is the angle between the contact orientation and the z-axis, and 4 is the angle between the x-axis and the contact orientation projected on the x-y plane. If the distribution of P is isotropic, the summation of such a function over all contacts can be written as

M 2n II

c r’=E

ss r’(y, 4) sin ydyd4

c=l 0 0

(12)

where M is the total number of inter-particle contacts within the assembly. For an isotropic granular structure with equal size granules, the modulus

function in Eqn (11) can be expressed as

(13) TQFZ,(S~S~ + tjtk) sin ydyd@

0 0

)

Modulus function After the integrals in Eqn (13) have been carried out, the 81 components of the modulus tensor can be reduced to two independent modulus constants for the isotropic material. It leads to

where the modulus functions i(t - t) and p(t - t), as a result of the inte- gration of Eqn (13), are related to the inter-particle properties by

l(t - T) = $ k, exp - ( ( y) -kexp(-7))

p(t - r) = - li* ((2k,,rrp(-y) +3kexp(-y)) (15)


where the packing parameter $ is equal to 3 V/4Mr2 in which V is the representative volume, M is the total number of contacts in the volume, and Y is the average particle radius.

Thus, the rheological behavior of the granular assembly in Eqn (7) is reduced to the following form:

t

oii(t) = 1

(1(t - t)&& + 2& - t)&&,) gdr (16)

0

The corresponding bulk modulus function can be derived from Eqn (15) and is given by

t--t K(t-t) =ik,exp -?

3ti ( > n (17)

It is noted that there is only a single time-scale for the relaxation of the bulk modulus function. This is similar to the classic visco-elastic continuum. However, there are two time-scales for the relaxation of shear modulus function which is different from the classic visco-elastic continuum. The two scales of relaxation are controlled by the inter-particle properties, respectively, in the directions normal and tangential to the contact area. When t,

= t,, the modulus function p(t - T) is reduced to a special case as follows:

p(t - r) = & (2k, + 3k,) exp (F) n

(18)

and the modulus function l(t - T) becomes zero. In this case, both bulk modulus and shear modulus share the same time scale for relaxation.

Compliance function Using the method of Laplace transformation outlined in Appendix A, the integral equation in Eqn (16) with modulus function can be rewritten as an integral equation with compliance function as follows:

t

q(t) = J

(l’(t - +SijSkL + 2/d(t - t)&&,) gdr (19)

0


where the compliance function L’(t - r) and /.~‘(t - r) are given by

p’(t - T) = w 2(2k, + 3kS) ( t--t

C+ I%

(t - 4 -i- (1 - C)exp -t’ ( 8 (20) n

(21)

where t,’ is the time scale for creep. A large value of the time scale indicates that creep deformation occurs very slowly. The derived time scale t L is related to the time scale t, by:

t:, =5 B

c=;(1+;(1-3)

(22)

(23)

and /3 is defined as a function of the shear to normal ratios of spring constants and dashpot constants as follows:

B= 42 + 34 k, CS

42+34 a! =k, a, =c,

The bulk compliance function K’(t - t) is derived to be

(24)

The bulk compliance function is linear with respect to time. No exponential decay is involved. This result is similar to that of the conventional visco- elastic model of Maxwell type.

However, for shear compliance function, it involves not only the linear function but also an exponential decay with the time scale t,‘. The time scale ratio of t,’ to t, depends greatly on the properties of spring and dashpot at


the microscale level. The effect of spring and dashpot constant on the time scale ratio is given in Fig. 2. According to this figure, when k, is small, the time scale t,’ is large and creep occurs slowly. On the other hand, when k, is large, creep occurs rapidly. The rate of creep is also affected by the ratio of c, to c,. Large c, corresponds to a fast rate of creep.

STATIC FORMULATION

In micromechanics of granular materials, static hypotheses is an alternative method for estimating the constitutive constants [lo]. In static hypotheses, it is assumed that the stresses of all particles are uniform and the static hypotheses often lead to a softer behavior than that of the kinematic method.

Stress-strain relationship for a general random packing

In the static method, the stress-strain relationship is based on the expression of mean strain of a representative volume as a function of relative displacement and fabric tensor A, given by [lo]:

1 M &ii = - V c Gjt$Ag

c=l

(26)

10

8

6 C

0 4 00 0.2 0.4 06 0.6 10

ct

Fig. 2. Effect of the spring and dashpot constants on the ratio of time scale.


When the granular material is under the condition of small deformation, the mean strain rate for a representative volume can be expressed as a summation of the rate of relative displacement between each pair of particles, given by:

(27)

Using the contact law given in Eqn (1) Eqn (27) becomes

Here, we use the static hypotheses that link the contact forces to the macroscale stress by the following relationship [lo]:

Substituting Eqn (29) into Eqn (28), we obtain the following differential constitutive equation:

where the viscous compliance IYi’jkl and the elastic compliance C’,, are given by

Closed-form stress-strain relationship for an isotropic random packing

(31)

For a representative volume with a suitably large number of particles, the summation over all contacts can be expressed in terms of an integral over all


contact orientations. For an isotropic granular structure with equal size granules, the fabric tensor A, can be given as follows [lo]:

Similarly, for the isotropic granular packing structure, the constitutive tensor in Eqn (30) can be expressed as

where

Modulus function Using the method outlined in Appendix B, the differential equation (30) can be transformed into an integral equation in a form similar to Eqn (15). The modulus functions A( t - T) and p(t - r) are related to the inter-particle properties by

(35)

The corresponding bulk modulus function derived from Eqn (35) is given by

t--t K(t-t) =sexp -?

3+ ( > n

This bulk modulus function is identical to that derived from the kinematic method in Eqn (17). It is noted that there is only a single time scale for relaxation in either the bulk modulus function or the shear modulus function,


although the two time-scales are different. This form is in agreement with the classic visco-elastic continuum. The two scales of relaxation time are controlled by the inter-particle properties, respectively, in the directions normal and tangential to the contact area. When k, = k, and c, = c, the modulus function n(t - r) becomes zero and the shear modulus function reduces to be identical to that derived from the kinematic method.

Compliance function Using the method of Laplace transformation outlined in Appendix A, the integral constitutive equation of stress in Eqn (15) can be rewritten as an equivalent integral equation of strain similar to that given in Eqn (19). The derived compliance functions A’(t - T) and p’(t - r) are as follows:

The bulk compliance function K’(t - T) is derived to be

3@ 3qt-t F(t-r)=k+kT

n ” ”

(37)

(39)

This expression of bulk compliance is the same as that derived from the kinematic method. Using the static method, all derived compliance functions are linear with respect to time. This behavior is in agreement with the classic visco-elastic continuum of Maxwell type.

COMPARISON

Table 1 gives a comparison of the previously derived modulus and comph- ante functions from both the kinematic and the static methods. Table 1 shows that the bulk modulus and bulk compliance functions derived from both methods are identical while the shear modulus and shear compliance functions from the two methods are very different. The shear modulus function derived from the kinematic method exhibits two scales of relaxation time. The shear compliance function derived from the kinematic method shows exponential decay instead of a linear relationship with time.


TABLE 1 Comparison of the derived modulus and compliance functions from kinematic method and

static method

Kinematic method Static method

Bulk modulus K(t - r)

&exp ,. ( ) _1=I

$exlJ 1. ( ) _L=I

Bulk compliance Ir(t - t)

Shear modulus

wL(t - 5)

& 2k,exp (- y) + 3k,exp - 7 ( ))

2ti$+3j exp - Y ( ) ”

Shear compliance

z(C+g+(l-C)exp(-y)) &(t+$)+&($+t)(t--r) 2(2k,+3k,)

LL’lt - sj

Relaxation modulus function

Here, we derive the relaxation functions from both kinematic and static methods for the case of one-dimensional deformation in which a vertical strain is applied instantaneously to a specimen and remains constant there- after. That is, .srr(t) = s&(t), where H(t) is the Heaviside function. While the strain is constant, the stress encounters a relaxation with time. From the constitutive equation derived in “Kinematic Formulation” above, we obtain

t

(711 (t) = s (A(t - T) + q.4 - r>> 2 t (40) 0

The relaxation function derived from the kinematic method is given by

E-(t) = oll(t) -=$ (3k.erp(-t) +2k,exp(-t)) (41) so

and the relaxation function derived from the static method is given by

(42)


Figure 3 shows the numerical results for the relaxation function with k, / k, = c, / c, = 0.2. Throughout the relaxation process, the kinematic method shows a higher stress than does the static method under the same constant strain. The kinematic method also shows a faster rate of stress relaxation. The rate of relaxation is controlled by the values of time scales which are in turn functions of the properties of the spring and dashpot, as discussed previously in Fig. 2.

The relaxation functions are discussed for two special cases: (1) the case that k,+ co, and (2) the case that c,--t cc.

For the case that k,-+ 00 The inter-particle contacts become exclusively viscous in the tangential direction of the inter-particle contact. Thus the behavior for the granular packing is expected to be fluid-like. The relaxation function derived from the kinematic method is

E-(t) =$ 3k (

The relaxation function derived from the static method is

E_(t) = sexp -t +sexp 3* (1.) 3$ (f (2cs2Y”))

4,

(43)

(44)

Fig. 3. Comparison of relaxation functions derived from kinematic method and static method (k, 1 k, = c, / c, = 0.2).


The results of Eqns (43) and (44) are plotted, respectively, in Fig. 4(a) and (b). In both cases the stress relaxes to zero. This behavior is similar to that of visco-elastic continuum of Maxwell type. It is noted that, in the kinematic method, the initial stress is infinity but immediately drops to a finite number. In the static method the rate of relaxation is very fast during the relaxation process, the stress derived from kinematic method is higher than that from static method except in the short period of time in the initial stage.

For the case that c,+ cc The behavior is exclusively elastic in the normal direction of the inter-particle contact. However, the behavior is visco-elastic in the tangential direction of

(b) Cd)

Fig. 4. Relaxation functions in four cases: (a) kinematic method with k,+co, (b) static method with k,-+oo, (c) kinematic method with c, -00, and (d) static method with c,+co.

56 C. S. Chung and J. Gao

the inter-particle contact. Thus, the behavior for the granular packing is expected to be solid-like.

For the kinematic method, the relaxation function is

E-(t) = $ ( sexp(?))

3k, + 2k

For the static method, the relaxation function is

E-(t) = $ 1+ 1 Ok, -3k,t

2k, + 3k, exp ts(2k, + 3k,)

(45)

The results of Eqns (45) and (46) are plotted in Fig. 4(c) and (d), respectively. It is interesting to note that, in both the kinematic and static methods, the stress relaxes to a finite value instead of zero. This behavior indicates that the granular material represents a visco-elastic continuum of three-element type.

Creep compliance function

Here, we derive the creep functions from both kinematic and static methods for the case of one-dimensional compression in which a vertical stress is applied instantaneously to a specimen and remained to be constant there- after. That is, 011 (t) = aoH( where H(t) is the Heaviside function. While the stress is constant, the sample creeps with time. From the constitutive equation given in the section on “Static Formulation” above, we obtain

I

r,,(t) = s @‘(t - r) +2&t - t))m(+t (47) 0

The creep function derived from the kinematic method is given by

5c 3(2 + 3a)

(48)

Xl

3&(2 + 3a) + V - c>

3(2 + 3~) exp

Rheological modeling of granules

and the creep function derived from the static method is given by

57

J-(t) =g (1+0.3(2+-g +;(1+0.3(2+..g)) (49)

Figure 5 shows the numerical results for the relaxation function with k, / k,

= cslcn = 0.5. The static method shows a faster rate of creep than the kinematic method. The rate of creep is controlled by the values of constants of the spring and dashpot. The creep function derived from the static method is linear, which represents a visco-elastic continuum of Maxwell type. The creep function from the kinematic method has an exponential term which represents a visco-elastic continuum of three-element type. However, the effect of this term is relatively small for the properties used in Fig. 5. The results for both the methods show a continuous creep with time.

The creep functions are discussed for two special cases, (1) the case that k, + co, and (2) the case that c, + cc.

For the case that k, + 0;) The creep function for kinematic method becomes

2o 9(2 + ~cx,)~

+I 1+ C 5

9(2 + 3u!,; )

.

Fig. 5. Comparison of creep functions derived from kinematic method and static method (k, / k, = c,/c, = 0.5).


and the creep function for static method becomes

The results of Eqns (50) and (51) are plotted in Fig. 6(a) and (b), respectively. In both cases, the sample creeps continuously with time. The rate of creep is higher for the static method.

For the case that c,--+ CC The creep function for kinematic method is

5a 2(2 + 34 cxp(-SJ)

0’ I 0 1 2 3 4 5

18

g 16

3

2

14

12

s 10 '=

g I.? 8

e 6

9 0 4

2

0

t/t,

0 1 2 3 4 5

tit,

0 1 2 3 4 5

ttts

/ Cd)

0 1 2 3 4 5

t/t,

(52)

Fig. 6. Creep functions in four cases: (a) kinematic method with k,--tc+ (b) static method with k,+co, (c) kinematic method with c, -00, and (d) static method with C,+CO.


The creep function for static method is

J-W =&(1+0.3(2+~) +0.9-L)

59

The results of Eqns (52) and (53) are plotted in Fig. 6(c) and (d), respectively. It is interesting to note that in the kinematic method, the strain creeps to a finite value instead of infinity. This behavior indicates that the granular material represents a visco-elastic continuum of three-element type as opposed to the Maxwell type.

SUMMARY

Granular materials with binder material can be viewed as a discrete system of particles which are connected with springs and dashpots. The system is simulated by an equivalent visco-elastic continuum. The macrorheological behavior of granular materials is derived, based on both kinemetic and static methods. Compared with the results from static method, the results from the kinematic method show stiffer behavior; the stress is always higher in a relaxation process and the strain is always smaller in a creep process.

The behavior of a granular assembly is not necessarily of Maxwell type even though the inter-particle behavior is of Maxwell type. The model derived from static method is in general of Maxwell type except for the case of c,-+ co. The model derived from the kinematic method is of a three-element type.

ACKNOWLEDGEMENTS

Research for this paper is supported by the U.S. Air Force Office of Scientific Research under a Grant No. F49620-95-1-0117. This support is gratefully acknowledged.

REFERENCES

Duffy, J. and Mindlin, R. D., Stress-strain relations and vibrations of granular media. Journal of Applied Mechanics, ASME, 1957, 24, 593-595. Dvorkin, J., Nur, A. and Yin, H., Contact laws for cemented grains: impli- cations for grain and cement failure material. Mechanics of Materials, Elsevier Science Publishers, Amsterdam, 1994, 18, 351-366. Zhu, H., Chang, C. S. and Rish, J. W., Normal and tangential compliance for conforming binder contact I: elastic binder. International Journal of Solids and Structures, Pergamon Press, 1996, 23(29), 43374349.

C. S. Chang and J. Gao 60

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

Zhu, H., Chang, C. S. and Rish, J. W., Normal and tangential compliance for conforming binder contact II: visco-elastic binder. International Journal of Solids and Structures, Pergamon Press, 1996, 23(29), 4351-4363. Jenkins, J. T., Volume change in small strain axisymmetric deformations of a granular material. In Micromechanics of Granular Materials, ed. M. Satake and J. T. Jenkins. Elsevier, Amsterdam, 1988, pp. 143-152. Chang, C. S. and Misra, A., Application of uniform strain theory to heterogeneous granular solids. Journal af Engineering Mechanics, ASCE, 1990, 116( lo), 23 1 CL 2328. Walton, K., The effective elastic moduli of a random packing of spheres. Jour- nal of the Mechanics and Physics of Solids, 1987, 35(3), 213-226. Cambou, B., Dubuiet, F. and Sidoroff, F., Homogenization for granular materials. European Journal of Mechanics A/Solids. 1995, 14(2), 51-61. Rothenberg, L. and Selvadurai, A. P. S., Micromechanical definition of the cauchy stress tensor for particulate media. In Mechanics of Structured Media, ed. A. P. S. Selvadurai. Elsevier, Amsterdam. 1981, pp. 4699486. Chang, C. S. and Gao, J., Kinematic and static hypotheses for constitutive modelling of granulates considering particle rotation. Acta Mechanica, 1996, 115(1-4), 213-229. Chang, C. S. and Liao, C. L., Estimates of elastic modulus for media of randomly packed granules. Applied Mechanics Review, ASME, 1994, 47(2-2) 1977 206. Fung, Y. C., Foundation of Solid Mechanics. Prentice-Hall Inc., New York, 1966, pp. 53-81. Christensen, R. M., Theory of Viscoelasticity: An Introduction. Academic Press, New York and London, 1971, pp. 2143.

APPENDIX A

Laplace transformation

The constitutive Eqn (16) derived from the kinematic method can be expressed separately in the following form:

First, we apply Laplace transformation to both sides of Eqn (Al) to trans-

form the functions from domain t into domain s. Using the convolution theorem and assuming zero initial conditions, we obtain


Then we rearrange the Eqn (A2) as follows:

Q2(S) = T (3+3)312(s)

&&) = -$ s +2 GC(S) n ( >

61

( w

W)

Applying inverse Laplace transform to both sides of Eqn (A3) leads to the following constitutive equation:

Eii = r (n’<t - qi&&) + &‘(t - ++&))d~ J ( fw 0

where the derived expressions of I’(t - r) and $(t - t) are shown in Eqns (20) and (21).

Similarly, the differential constitutive equation derived from the static method in the section on “Static formulation” can be expressed separately into the following form:

Applying Laplace transformation to both sides of the equations, the functions in domain t are transformed into domain s, thus:

se&) = (2P’S + 2/-4)55(s)

(A6) s&(s) = ((3X + 2P’)S + (34 + 2/4))%(4

We rearrange Eqn (A6) into the following form:


>

_ sakk

(A?

Applying inverse Laplace transformation to both sides of the equations above, leads to the following constitutive equation:

t

Eij = .I

(A’@ - r)&j&(t) + 2p’(f - t)+(t))dt

0

uw

where the derived expressions of J.‘(t - r) and ~‘(t - t) are shown in Eqns (37) and (38).

APPENDIX B

Transformation from differential form into integral form

The differential equation

a(t) = &(t) + cx(t)

has the following solution:

w

x(t) = ]kexp(-:(f - r))a(r)dr (B2)

0

The constitutive equation of differential form for static method in the section above on “Static formulation” can be separated into the following two equations:

& = (34 + 2&a/& + (32’ + 2&&k

(B3) El2 = 2/4a,2 + +2p’a,2

From the solution of these differential equations, we obtain


a,/& = s

(34t - t) + &.L(t - Z))&dt

0

63

034)

where

Substituting the expressions of coefficients with the spring and dashpot constants given in Eqn (34), the derived modulus functions are given in Eqn (35).

rheological modeling of randomly packed granules with visco-elastic binders of maxwell type

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