Pergamon Mechanics Research Communications, Vol. 21, No. 2, pp. 197-202, 1994

Copyright © 1994 Elseviex Science Ltd Printed in the USA. All rights ~ e d

0093-6413/94 $6.00 + .00

D E R I V I N G A V I S C O E L A S T I C C O N S T I T U T I V E E Q U A T I O N F R O M S O L I D - F L U I D I N T E R A C T I O N S

Xiang-Hua Xu *$& Xiang-Jing Xu* tDept.of Aerospace and Mechanical Engineering, Boston University, Boston, MA 02215 SDept. of Engineering Mechanics, Huazhong University of Science and Technology, Wuhan, PRC *Wuhan Institute of Textile, Wuhan, P.R.China

(Received 7 May 1993; accepted for print II November 1993)

1. Introduct ion

For linear viscoelastic solids there are many semiempirical models used for engi- neering problems. One particularly useful model is the so-called " s p r i n q - d a s h v o t " model ( Bland [1]), in which properties of viscoelastic materials are mo~lelled by'the combined effect of springs and dashpots.

An alternative way to derive the constitutive relations of the viscoelastic solids is to consider the model in which a dilute dispersion of drops in an elastic matrix. In fact, in the original papers by Frbhlich and Sack ([2]), and Oldroyd ([3]), the constitutive equations for viscoelastic fluids were obtained by considering the interaction between a viscous fluid and a suspension of elastic spheres.

In this paper, a homogenization method is used to calculate the constitutive equa- tion for the viscoelastic solid model in which a dilute dispersion of incompressible drops are dispersed in an incompressible elastic matrix.

Notations: E = Strain tensor; G= Shear modulus of solid; r = stress tensor in elastic medium; P = stress tensor in viscous drops; r /= viscous coefficient of drops.

2. M e t h o d

We consider a substance consisting of fluid spheres dispersed in an incompressible elastic medium.

197

198 X. XU and X. XU

It is assumed that the volume fraction of fluid drops is small, so that the interaction of different fluid drops can be omitted. If we use the notation ~b as the volume fraction of viscous fluid drops, then:

4)<<1

The method we use is similar to that used by Batchelor([4]). We consider the interaction of one viscous drop and the surrounding of incompressible elastic matrix. The effective constants are obtained by volume averaging the field quantities. The superposition of these individual contributions then gives the final results.

The equations governing fluid motion in the drop are written as:

~ V 2 v = V P , V . v = 0 , for r < a (1)

For the incompressible elastic matrix, the following equations are used:

V ' - r = 0 , V . p = 0 , for r > a (2)

Since the elastic medium considered is incompressible, the stress tensor can be ex- pressed as the sum of an isotropic part and a shear part ( Green, A. E. [5])

r = - h i + a ( V o + ( v p ) t)

where p is the displacement of the solid medium.

Boundary conditions are imposed at infinity for the elastic medium, and at the interface of the drop and elastic medium.

Suppose the far field displacement can be expressed as:

p - ~ r . E , r ---~ o z

where E is the strain tensor at far field.

At the interface of the elastic medium and the viscous fluid drop, the boundary conditions should impose continuous velocity and stress across the interface.

If we use a as the length scale, and ~/G as the time scale, the dimensionless equations governing motion of the drop are:

V*2v * = V - P • , V * - v * = 0 , for r* < 1 (3)

with v* = v/(aGfi/) and P* = P/G.

For the elastic medium, the nondimensional equations may be written as:

V*2p * = V'h*, V*. p* = 0, for r* > 1 (4)

p* --* r* - E

where p* = p/a and h* = h/G.

INCOMPRESSIBLE VISCOELASTIC CONSTITUTIVE EQUATIONS 199

We begin by solving the equations for the viscous drops and incompressible elastic mat r ix respectively, then we determine the solution for the whole field by matching solutions in different media on the interface. The solution for the elastic body has to satisfy the conditions at infinity.

2.1 The Solution of The Viscous Drop

It can be shown tha t the general solution of equation (3) has the form:

v* = (2bl - 10b2r*2)r *. E + 4b2r*r*r* : E (5)

p = -42b~r*r* : E (6)

If we denote the stress tensor in the drop by P*, then the stress on the interface (r* = 1) can be written:

e . P* = (4ba - 32b2)e • E + 38b2eee : E (7)

where e = r/r is the unit vector on the interface.

2.2 Displacement o[ Elastic Medium

The solution for incompressible elastic med ium in which the boundary condition at infinity is satisfied is:

2 c i . . ( 5C 1 3C 2 ~ . . . p * = ( l + ~ - ~ ) r . E - ~ f + ~ - / g ) r r r : E

The stress distribution on the interface r* = 1 may be written:

e - r * = (2 - 1 6 0 - 6 c 2 ) e • E + ( 4 0 c , + 2 4 e 2 ) e e e : E

(8)

(9)

We denote t ime differentiation of p, cx and c2 by ~, fix and c2 respectively. Supposing the deformation of the drop is small compared with the original shape of drop, we m a y write the rate of the elastic deformation as:

p* = 2fir* • E - (5fl + 3f2)r*r*r* : E (10)

After we obtain the general solutions of both the interior (drop) and the exterior ( incompressible elastic medium) fields, we proceed to match the solutions in the two fields.

2.3 Matching Solutions by Boundary Conditions

It is required tha t the tangential stress, normal stress, normal velocity and tangen- tial velocity be continuous on both side of the interface. These boundary conditions thus can be given by:

38b2 = 40cl + 24c2 (11)

4bl - 32b2 = 2 - 16c1 - 6c2 (12)

f, = --~(b, - 5b2) (13)

200 X. XU and X. XU

(7, 5~, + 362 = -4--- b2 (14)

r/

The expressions of bl and b2 are derived in terms of Cl and c2, which are:

1 84 135 bl ---- ~ Jr- -i-~c1 -~- -~-c2 (15)

20 12 b2 = ]-~Cl -[- ~ c 2 (16)

2.4 The Constitutive Relation of viscoelastic solid

In this section, we use dimensional equations. When the non-dimensional quan- tities are used, an "*" appears.The quantities we have particular interest in are the "bulk" stress and "bulk" strain, in a viscoelastic solid. The constitutive equation is a relation between "bulk" stress, "bulk" strain, and their derivatives. By considering interactions between the drops and the elastic matrix, the "bulk" stress in the mixture is found first, then it is related to the "bulk" strain.

We begin the process of averaging all of our field quantities over a volume V of a size large enough to contain a representative number of particles but small enough comparing with any macroscopic length scales of interest. Denoting this average by angular brackets, we thus have the generic definition:

1; < q J > = ~ ~dV

with the integration carried out over both the liquid and solid volumes within V.

The "bulk" properties are obtained by a volume average. The volume average for the stress is expressed as:

1£ 1 £ Pdv (17) < r > = ~ _ ~ v o ( - h I + G ( V p + V p t ) ) d V + ~ Vo

where I is identical dyadic.

It is well known that the second integration on the right hand side of (17), can be changed to a surface integration ( Landau and Lifshiz [6]). The integration is 1 1 V f~Vo P d V = V fzs, r P . ndS. The symmetric and traceless part of the "bulk" stress

is of particular importance, since it is relevant to the "bulk" constitutive relations. This part can be expressed as:

<r>8= vG 2V1 [ L - ~ v 0 ( V P + V P t ) d V + ~ s ( r n . P + P . n r - 2 I n . P . r)dS] (18)

The first term on the right hand side of (18) can be expressed in terms of "bulk" strain tensor and an additional term which is "related to the integration of the strain field in the viscous drops.

_r~v0(Vp + Vpt )dV = G[< Vp > + < Vp t >] - p vo

INCOMPRESSIBLE VISCOELASTIC CONSTITUTIVE EQUATIONS 201

Gaa3~f~ = 2G < E > - ~ - v0.(V'v" + (V'v')*)dV'dt

By using relation (5), we can evaluate the following integrals:

" "

v0.(V v + (V'v*)*)dV*dt = ~r ( bi - --g--b2)dtE

= ~ i ( ~ + ?~c~)dtE (19)

~ V [ f E s ( r n . P + P . n r - ~ I n . P . r ) d S ] = ¢ G ( 2 + l - ~ c 2 ) E (20)

By substituting (19) and (20) into (18), the expression of < r >" is obtained:

< 7" >°= 2G < E > - G ¢ G (2 + c2)dtE + a¢(2 + -;-c2)E (21)

From relations (11),(12),(13) and (14), the relation between ~2 and c2 may be found. It is:

G 5 3 62 = ~ - ( - g - ~c2) (22)

Supposing c2 assumes the value c2(to) at initial time to, We can solve equation (23). The solution is:

5 5 _ ~ _ c2(t) = - ~ + (c2(to) + ~)e 2,(' ,0) (23)

Equation (21) can be solved by using (23). The result is:

12 < r > '= 2G < E > + T G ¢ ( c 2 ( t ) - c2(t0))E + G¢(2 + c2)E (24)

where c2(to) is chosen such that when q --* oo, the solution is identical to the cor- responding result of rigid spheres dispersed in an elastic medium. This gives the following equation:

< 7 " > ' = 2 G ( l + 5 ¢ ) < E > , y - - . ~ (25)

By using (23) and (24), when T/--. ~ , we can find:

< E > +G¢(2 + 1~85c2(to)) < E > (26) < T > s = 2G

Comparing (25) and (26), we specify the initial condition:

5 c2(to) = ~ (27)

202 X. XU and X. XU

By using (23), (24) and (27), we have the equation to relate < r >~ and < E >:

< > s = 2 c < E > + E > (28)

If we assume y/G << 1, we can have a relation between the stress tensor, the strain tensor and their derivatives. The expression is correct to the first order of volume fraction ¢ and ~]/G, and:

+ ~ - - ~ < 4 - > = 2 G ( 1 - ¢ ) < E > + ( 1 - ¢ ) < E > , for t > t o (29)

The equation (28) shows the effective modulus is a function of time, while (29) con- nects the stress and strain tensors to their first derivatives.

3. D i scuss ion

From a micromechanical model of viscoelastic body that assumes dilute viscous drops are dispersed in an incompressible elastic medium, we have derived the consti- tutive equations (28) and (29).

The relation (28) shows that the effective coefficient G* is larger than the corre- sponding G at the beginning, but becomes smaller than G very quickly. However, it approachs a limiting value when t - to becomes big enough (t - to >> 2r//3G). This behavior is typical for viscoelastic solids, since the shear stress doesn't disap- pear when deformation exist (Christensen [7]). The relation (29) has the same form as constitutive equation of a non-Newtonian fluid. However, < E > is the strain tensor in (29) in contrast to its being the rate-of-strain in the case of non-Newtonian fluids. Since we define G = E y ~ , the effective Young's modulus is expected to behave similarly to G.

The method we describe in this paper has been adopted to derive stress-strain relations of viscoelastic solid under periodic shear load (in the problem the unsteady effect of elasticity are included ). The result for non-Newtonian fluid has been ob- tained (Xu and Nadim [8]).

Acknowledgements The authors are grateful to the referees for their thoughtful and detailed comments, and special thanks to Ms. M. Pham for her help with the preparation of this paper.

REFERENCES 1. Bland, D. R., 1960, The theory of linear viscoelasticity, Pergamon Press 2. Fr/~hlich, H. and Sack, R., 1946, Proc. R. Soc. London A 185,415 3. Oldroyd, J. G., 1950, Proc. R. Soc. London A 200, 523 4. Batchelor, G. K., 1970, J. of Fluid Mechanics 41,545 5. Green, A. E., 1968, Theoretical elasticity, Clarendon P., 2nd ed. 6. Landau, L. D. and Lifshitz, E. M. 1987. Fluid Mechanics, Pergamon P., 2nd ed. 7. Christensen, R. M. 1971. Theory o£ Viscoelasticity, New York, Academic Press. 8. Xu, X. and Nadim, A. 1993 . The Society of Rheology 65th Annual Meeting, Boston.