Physica 122A (1983) 105-113 North-Holland Publishing Co.

ISOTROPIC ELASTIC MEDIUM CONTAINING A SPHERICAL PARTICLE

I. INCOMPRESSIBLE MEDIA

R.B. JONES

Department of Physics, Queen Mary College, Mile End Road, London El 4NS, England

and

R. SCHMITZ

The Rockefeller University, 1230 York Avenue, New York, NY 10021, USA

Received 9 May 1983

For arbitrary incident displacement field we solve the linear elastic equilibrium equations for an infinite isotropic, incompressible medium containing a spherically symmetric, incompressible inclusion. Explicit results are obtained in the case that the inclusion is a sphere with uniform shear modulus. For more general particles the scattering coefficients are determined by perturbation theory.

1. Introduction

In this and the following article l) (hereafter referred to as II), we give the solution of the linear equilibrium equations of elasticity for an infinite isotropic medium containing an isotropic inclusion with spherically symmetric coefficients of elas- ticity. Our solution expresses the actual displacement field in terms of the incident displacement field which is a solution of the linear equations in the absence of the particle (i.e., when the particle is replaced by the uniform material surrounding it). We allow for a quite arbitrary incident solution, the only restriction being that it is expandable as a Taylor series in a region which contains the particle.

In solving the equations, we use methods that have been developed in earlier work 2-4) for a spherically symmetric polymer in viscous flow. The basic idea is to decompose the n th term in the Taylor series of the incident field into its irreducible parts. To each of these the corresponding scattered field is found by a simple ansatz involving still unknown radial functions. This procedure leads to ordinary differential equations for the radial functions which can be solved explicitly for some models. Here, we present the exact solution for a sphere with uniform elastic coefficients and give the scattered fields for general spherically symmetric inclusions in perturbation theory.

We expect that our results have wide applications in the theory of nondilute solid

0378-4371/83/0000-0000/$03.00 © 1983 North-Holland

106 R.B. JONES AND R. SCHMITZ

elastic suspensions, especially in the determination of the concentration dependence of the effective elastic coefficients beyond the linear order. There one has to consider clusters of inclusions and to compute how the strongly inhomogeneous field from a given cluster is scattered by a selected particle. Calculations of this kind have already been made for polarizable 5) and for fluid 6) suspensions.

A complete set of solutions of the linear elastic equilibrium equations for a uniform, isotropic medium has been found long ago by Lord Kelvin 7'8) in terms of spherical harmonics. In the case where the inclusion is a uniform sphere these solutions can be matched at the particle surface by the appropriate boundary conditions. When the expansion of the incident field in terms of the complete set is known, this way provides an alternative way of solving the scattering problem. Our solution is, however, directly applicable in the many body problem where the interactions between the particles are taken into account by the method of reflections9.~°).

In the first article, we restrict ourselves to the case where both the medium and the inclusion are incompressible. This should be a good approximation for mate- rials whose bulk modulus is much larger than their shear modulus. The general case of finite bulk modulus will be considered in II. In section 2 we summarize the equations of equilibrium for a linear elastic incompressible medium. In section 3 we decompose an arbitrary incident field into its irreducible parts, a technique which will be presented in greater detail in II. The displacement field in the presence of the inclusion is then obtained by a simple ansatz involving still unknown scalar functions for which ordinary differential equations are derived. In section 4 we introduce the scattering coefficients which determine the behavior of the perturbed displacement field in the region outside the particle. For a general spherical inclusion we give expressions for the scattering coefficients in perturbation theory. Finally, in section 5 we present exact expressions for the scattering coefficients for the case that the inclusion is a sphere with uniform shear modulus.

2. Basic equations

We consider a spherical particle of radius a immersed in an isotropic elastic medium. The particle may be any elastic inclusion which is locally isotropic and described by spherically symmetric elastic coefficients. Also the extreme case of a hard sphere is covered by this model. When the particle is centered at the origin the linear equations of elastic equilibrium for the displacement field u( r ) read u)

V ' a = 0 , (2.1)

a ~ = # (O~ut~ + tOBu~ - ~ V "u6~) + K V "u6~p, (2.2)

where a is the stress tensor with locally varying, spherically symmetric shear

ELASTIC MEDIUM CONTAINING A SPHERICAL PARTICLE. I 107

modulus/~(r) and bulk modulus x(r ) . We assume that #(r) and r ( r ) are uniform and equal to constants #0 and x0 outside the particle,

~ ( r ) = ~ , (2.3)

x ( r ) = r. o (r > a) .

It is convenient to set

/~(r) = p(r) -- U-o, (2.4)

~(r) = r ( r ) -- x0,

and to define the stress tensor s induced by the particle as

s~ = f~(O,u~ + ~9~u~ - 217" u ~ ) + ~17 • u r ~ . (2.5)

The equilibrium equations (2.1) and (2.2) can then be written as

laoV2U + (/~0/3 + Xo)VI7 "u = - F ( r ) , (2.6)

where

F(r ) = V " s ( r ) (2.7)

is the force density induced by the particle on the medium. Notice that F = 0 for r > a and that the total force and torque exerted by the particle are zero.

In this article, we will restrict ourselves to an incompressible medium and an incompressible inclusion. The general case will be dealt with in IIt).

Incompressibility is the limiting case where in (2.2) 17 • u---,O, x(r)---~oo such that

there remains a finite product p = - x l Z ' u which can be interpreted as a hydrostatic pressure fieldS,11). The stress tensor (2.2) now simplifies to

a~ = #(O~u~ + O#u~) - p 6 ~ (2.8)

and the equilibrium equations can be written in the form

~LOI72U - - V p = - F ( r ) , V • u = O, (2.9)

where the pressure is determined by the condition of incompressibility (V" u = O) and the induced force density is given by

F = V ' s , (2.10)

Eqs. (2.9) are formally identical to the linear Navier-Stokes equations for stationary incompressible flow. In this picture, (2.10) can be interpreted as the force density exerted by a fluid droplet whose viscosity differs from the surrounding fluid and which is assumed to remain spherical in shape under the action of shear.

1 0 8 R . B . J O N E S A N D R . S C H M I T Z

This model has been considered by Batchelor and Green t2) (assuming that fi is constant for r < a) and solved for the case that the flow at infinity is a pure strain field Uo(r)= o . r where e is a constant, symmetric and traceless tensor.

3. General solut ion of the equilibrium equations

In order to solve the equations of elastic equilibrium (2.9), (2.10) in general, we use techniques developed in earlier work 2,3) in the case of stationary incom- pressible flow. Let us consider a solution (u0(r),p0(r)) of (2.9) in the absence of the particle, i.e., for fi = 0 (and hence F = 0), and ask for the modification caused by the inclusion. Thus, we write

u ( r ) = Uo(r) + u,(r) , p (r) = po(r) + p , ( r ) (3.1)

and require for (u~(r), p~(r)) to go to zero at infinity. Inserting (3.1) into (2.9) yields

2 ] ~ V 2 U l - - Vpl + - d# r • (VuO S

r ~

where (2.10) has been used and

(Vu) s = l(a~u~ + a~u~).

= _ f i V 2 1 g 0 2 dp r ~ r r ' ( V u o ) s, V ' u , = O , (3.2)

(3.3)

Assuming now that the incident fields (uo, P0) are expandable as Taylor series in a region which contains the particle, we set

1 h(") = - - 8 ~ ~,oUo~(O) (3.4)

'~Yl. - - ~/n F / ! ; I - •

and have

to(,)= (3.5) n = 0 n = 2

where

u~)(r ) = h (") x x~, (3.6) ~tYl "" ")'n Yl ' " "

p(0,)(r) = Uo r " V2u(o") (n >~ 2), (3.6a) n - l

Expressing the tensor h (") in terms of irreducible pieces, we get a complete covariant decomposition of u(0 ") into three parts 1)

u(0 ") = u(0 "~s + u~0 ")T + u(0 ")P (3.7)

ELASTIC MEDIUM CONTAINING A SPHERICAL PARTICLE. I 109

These irreducible fields can be expressed in terms of u~ ") as follows:

1 1 = _ _ U (n) X#O~U~n ) u~")s n + l o, + - ~ - C i

(n + 1)(2n + 1) x~xaVZub~) + ~ r2VZu

U~o.)V= n .,(.) 1 xaO~u~.] + n + l "°" n + l

(n >f 0), (3.8a)

1 n(n + 1) [x 'x)72u~) -r2V2u~')] (n >~ 1),

(3.8b)

(3.8c) u~')P= n(2n + 1) x'xaV2u~) n 2rZV2 u (n >i 2).

Each of these fields satisfies the equations

V • U~o ")' = 0 , (3.9)

r • -,-orZ"(")' . . . . - ,,..o(")' ( I = S , T , P ) .

The advantage of the decomposit ion is that the irreducible fields have furthermore the properties

V2u~o")S = 0 , V x u~ ")s = 0 , (3.10a)

V2u~ n)T = 0 , r ° u~ ")r = 0, (3.10b)

1 r" U~o ~)P = r2r • V Zu(o") ,

2(2n + 1)

n 2 -- 2n -- 2

2n (2n + 1) [r x (r x VEU¢o"))] ~ .

(3.10c) n

xaO~u~)" = 2(2n + 1) x'xaV2u~"B)

In Love's terminology s) u~ ")s, u~ ")r, u~ ")P are fields of type 4), X and w, respectively[ Due to linearity each of the irreducible fields #o"Y(r) can be considered as

incident field. Thus, we write for the solution of (3.2)

(3.11a) n=O n = l n = 2

p,(r) = ~. ptn)S(r) + ~. ptn)P(r) ( 3 . 1 1 b ) n=O n = 2

and make the ansatz

u~"~S(r ) = fS ( r )u~)s -- _ _ n + l

r- ' f~ ' (r)r × (r × u~n~),

p ~")S(r ) =/~ (r)g S(r)r" u~)n)S, (3.12a)

110 R.B. J O N E S A N D R. S C H M I T Z

u]")T(r) = f +.(r)u<o ")+ , (3.12b)

u~n)V(r ) =fV(r)u(o.)p 1 n + 2 r-~r'~'(r)r x (r x u(0")v),

P~")v(r) = I~(r)gP,(r)r " U~o "Iv , (3.12c)

wherefS(r) . . . . . gP,(r) are radial functions. The second terms in u~ ")s and u~ ")P have been chosen in such a way that V" u {")s'P = 0. Upon inserting (3.12) into (3.2) and using (3.9), (3.10), we find ordinary differential equations for the radial functions. For f~,(r) and gS(r) these read

I t (r)[fs"+ 2 ( n + 2)fS'r - r g S ' - ( n + 1)gS]

s, l gs+ = o r 2 '

[ I ~(r) i f .s '+ 3(n + 2)fs"+ 2(n + l)(n + 2) fs" - (n + 1)(n + 2)g s r

+ 2 N rf. - = o . r r

For f~(r) we find

" dp f ~ + = 0. (3.14) /~(r) ~ " + 2 ( n + l ) + d r ~ ' + ( n - 1 ) r r

Finally, for f~(r) and gP,(r) the equations are

It(r)[f~"+ 2(n + 2)f"Vr + 2(2n + D f " v - , r2 rgP.'--(n + 1)g~]

r - - - ~ r g ~ + -(n--~]~ = 0

and

i t ( r ) rfV '' 1~]~ + 3(n + 2)f.P" + 2(n 2 + 5n + 3) f"v + 2n(2n + -- n(n -- 1)g. v r J r 2

+ 2 ~ r r f P . " + ( n + l ) f ~ ' + ( n 2 - 1 ) J P " + = 0 . (3.15) r ?"

Since ul(r) and p~(r) tend to zero at infinity the asymptotic behavior of the solutions of (3.13)-(3.15) has to be

f~,(r)--+O, g~,(r)--+O (r--*~). (3.16)

E L A S T I C M E D I U M C O N T A I N I N G A S P H E R I C A L P A R T I C L E . I 111

4. Scattering coefficients

In the region outside the particle (r > a), we have/~(r) = go. From (3.13)-(3.15) and (3.16) follows that outside the particle the radial functions are of the form

A s B s 2n + 1 A s i s ( r ) = r~+~+ z,+s, g S ( r ) = - - 2 - - - -

r / / + 1 r 2n+3 '

f~(r ) = rZ.+,,

A P B. P 2n -- 3 A. P f P . ( r ) - rE._ l-~ r2.+l, gP.(r) = - 2 n rE.+ 1 (r > a ) .

(4.1)

The coefficients A s . . . . . B, P characterize the solution outside the particle and are thus called scattering coefficients. Their values follow from the detailed behavior of the function #(r) inside the particle. In general the scattering coefficients must be found by numerical integration of the differential equations (3.13)-(3.15).

If/~(r) =/~(r) - go is small for r < a, we can calculate the scattering coefficients in perturbation theory. Using that the scattered fields ut")'(r) are regular at the origin, we find

_ _ f o ~ s n(n + 2 ) Ia ' s _ n(n + 2) r2"ft(r) dr, B,ilj = - - r 2n +2/~(r) dr, A'tq -- go go do

A T n - - 1 t "a = r2"ft(r) dr , nil1 go 3 0

(4.2) p _ ( n - 2 ) n ( 2 n + l ) Io'

A ' n - (~n ~ ~ / ~ r ~ - 2/~(r) dr,

p 2n 3 - - n 2 - - 2n - 2 I , ' B"tu = ~-n ~ ~ ~v r ~ t ( r ) dr .

Here the subscript [1] indicates that the results are correct to first order in fi(r) only. Notice that the reciprocity relation

A p 2 n + 1 s . = ~ B._z (4.3)

is satisfied to first order. For a general proof of the reciprocity relation, we refer to ref. 9.

From (3.2) follows that u, and Pl are zero for r > a in the case that the incident field represents just a rigid body motion of the whole system, i.e., when uo(r) = U + ~ x r where Uand 1'~ are constant.This implies that f0 s, g0 s and fT are all zero for r > a and from (4.1) we find that in general

A s = B s = A ~ = 0 . (4.4)

112 R.B. JONES AND R. SCHMITZ

5. Uniform sphere

For a uniform sphere one has

p ( r ) = {1~1' r ~< a , /6, r > a . (5.1)

In this case eqs. (3.13)-(3.15) can be solved explicitly and the scattering coefficients can be determined in closed form.

The general solution o f (3.13) which is regular at the origin and goes to zero at infinity is

s s 2 [ 2(2n + - C , + D , r , r<~a; 5 ) D , ,s r<~a, fS(r ) s s = A . , B. gS(r)= n + l

- r 2 ~ 5 + ~ , r > a ; _ 2 ( 2 n + 1 ) A s n + 2 r 2 n + 3 ' r > a .

(5.2)

The coefficients A s, B s, C s and D s must be found with aid o f the boundary

condit ions at the particle surface (r = a) which follow from (2.1) and (2.8). The

boundary condit ions are that u and a • n are cont inuous at r = a where n = r/r. From this follows

× ( . × . ) 1 = 0 ,

× [. × (, = 0 ,

where we have introduced the notat ion

(5.3)

~c~ = lim { c ( r = a + e ) - c ( r = a - e ) } . (5.4) c~0

Inserting (3.12a) and (2.8) into (5.3), we obtain four equations for the four unknown coefficients. After solving these we find

C s = I*, - / ~ D s = 0 , /a, + [1 + 3/(2n(n + 2))1#0'

2n + 3 2n + 1 A s - CSa2.+l, B s - _ C s a 2 . + 3 2 2

(5.5)

For f,~(r) we find f rom (3.14)

fT(r) =

- C , v, r ~<a,

r 2 n + P r > a .

(5.6)

ELASTIC MEDIUM CONTAINING A SPHERICAL PARTICLE. I 113

In order to determine the coefficients A T and C, r, we use the two normal boundary conditions on the left of (5.3). The result is

T ~ 1 - - / ' / ' 0 T ~ T 2 . + 1 (5.7) C. - A. = t~.a /~, + [1 + 3/(n - 1)]#0'

Finally, the funct ionsf P (r) and g~(r) are found from (3.15) to have the form

f C P 2(2n + 1)D P - -25 -+D, P, r ~<a; r ~<a, t / - - 1 r 2 '

f,P(r) = /

Ap " Bp " gP.(r) = 2(2n - 3) A. P

--r-WZS_l+r2-T; ~ , r > a ; n r 2"+1'

Applying the boundary conditions (5.3), we get for the coefficients

C. p ( 2 n - 1 ) ( 2 n + l ) E P a : , D, p s ( 2 n - 1 ) 2 p ~_- = - - C n _ 2 E n ,

4 4

2n + 1 2n - 1 A P = _ _ C s a ~ - l , p s . . - 2 B . : - - ( C n _ 2 .at- EP)a 2~+1 ,

2 2

where C s has been defined in (5.5) and

r > a .

(5.8)

(5.9)

12/~0(~, - ~0) (5.10) E P _- " [2(n 2 - l)p0 + (2n 2 + l)p~][(2n 2 - 4n + 3)#0 + 2n(n - 2)#~]"

One can easily check the reciprocity relation (4.3). Furthermore, after expan- ding (5.5), (5.7) and (5.9), (5.10) up to first order in (#l - ~ ) one finds agreement with (4.2). In the limit #~--~oo the scattering coefficients reduce to the results of a hard sphere with stick boundary conditions in a viscous fluid:) where the particle moves in such a way that the total force and torque exerted on it vanish3).

References

1) R.B. Jones and R. Schmitz, Physica 122A (1983) 114. 2) R. Schmitz and B.U. Felderhof, Physica 92A (1978) 423. 3) B.U. Felderhof and R.B. Jones, Physica 93A (1978) 457. 4) R.B. Jones, Physica 94A (1978) 271. 5) B.U. Felderhof, G.W. Ford and E.G.D. Cohen, J. Stat. Phys. 28 (1982) 649. 6) R. Schmitz, Dissertation, R.W.T.H. Aachen (1981). 7) Lord Kelvin, Phil. Trans. Roy. Soc. 153 (1863). 8) A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover, London, 1944) chap.

XI. 9) P. Reuland, B.U. Felderhof and R.B. Jones, Physica 93A (1978) 465.

10) P. Mazur and W. van Saarloos, Physica l lSA (1982) 21. 11) L.D. Landau and E.M. Lifschitz, Theory of Elasticity (Pergamon, London, 1959) chap. I. 12) G.K. Batchelor and J.T. Green, J. Fluid Mech. 56 (1972) 401.