Jtam-A4.dvi


JOURNAL OF THEORETICAL SHORT RESEARCH COMMUNICATION

AND APPLIED MECHANICS

53, 1, pp. 255-257, Warsaw 2015
DOI: 10.15632/jtam-pl.53.1.255

UNIQUENESS IN INCLUSION PROBLEMS WITH

IMPERFECT INTERFACE

Peter Schiavone

University of Alberta, Department of Mechanical Engineering, Edmonton, Canada

e-mail: p.schiavone@ualberta.ca

We consider a non-standard boundary value problem characterizing deformations of a com-
posite consisting of a arbitrarily-shaped elastic inclusion embedded in an infinite elastic
matrix subjected to uniform remote stresses. The interface between the inclusion and the
surroundingmatrix is taken tobe imperfectwith ’spring-like’ interfaceparametersdescribing
the properties of the interface layer. We show that any classical solution to the boundary
value problem is necessarily unique despite the fact that the asymptotic behaviour of the
solution is not accommodated by the corresponding classical results from the same theory
of elasticity.

Keywords: imperfect interface, inclusion, plane elasticity, uniqueness of solution

1. Introduction

Problems in compositemechanics involving elastic inclusions (inhomogeneities) embedded in an
infinite matrix of a separate elastic material have attracted considerable attention from both
theoreticians and practitioners alike. Earlier studies in this area focused on the assumption that
the inclusionwas perfectly bonded to the surroundingmatrix (tractions and displacements were
assumed to be continuous across the inclusion-matrix interface: see, for example, Eshelby, 1957,
1959). Recently, in an attempt to understand interface damage and its subsequent effect on the
effective properties of composites, there has been increased emphasis on a class of problems in
which the inclusion is imperfectly bonded to the matrix. Analyses in this context have been
performed for linear isotropic materials (see, for example, Hashin, 1991a; Gao, 1995; Ru and
Schiavone, 1997; Sudak et al., 1999, and the references contained therein), linear anisotropic
materials (for example, Ting and Schiavone, 2010) and for a class of materials undergoing finite
plane deformations (for example,Wang, 2012). Inmany of these cases, the (imperfect) interface
model used is the well-known ’spring-layer model’ (Hashin, 1991b) in which tractions are conti-
nuous but displacements are discontinuous across the interface with ’jumps’ in the displacement
components assumed to be proportional to their corresponding interface traction components.
Thismodel is particularly attractive in that it can be used to account formicro-voids andmicro-
-cracks almost always present in the interfacial region (Fan and Sze, 2001). Unfortunately, the
model is almost always associated with a non-standard ’transmission-type’ boundary value pro-
blemandalthough certain solutions for specific inclusion geometries and loading conditions have
recently been established (see, for example, Ru and Schiavone, 1997; Sudak et al., 1999; Ting
and Schiavone, 2010) no rigorous analysis of the well-posedness of the corresponding boundary
value problems has been undertaken. The question of the uniqueness of solution is particularly
important since it forms the basis of most constructive methods (numerical or otherwise) cur-
rently being used to solve this important class of problems. Unfortunately, uniqueness theorems
for this type of problem are not accommodated by those available for classical boundary value



256 P. Schiavone

problems, essentially because of the peculiar nature of the boundary condition across the inter-
face. In this paper, we establish uniqueness results for the problem describing the plane-strain
deformations of an inclusion-matrix composite subjected to plane-strain deformations.

2. Formulation

Consider a domain in R2, infinite in extent, containing a single internal elastic inhomogeneity
with elastic properties different than those of the surroundingmatrix.Thematrix and inhomoge-
neity occupy regions denoted by S(1) and S(2), respectively, while the inclusion-matrix interface
is denoted by the smooth curve Γ . In what follows, the superscripts (1), (2) refer, respectively,
to quantities corresponding to the regions S(1) and S(2).
The classical boundary-value problem describing the deformation of the composite in the

absence of body forces is given by Hashin (1991b) and Constanda (1995):

Find a regular solution pair {u(1),u(2)} such that u(2) ∈ C2(S(2)) ∩ C1(S
(2)
) and

u(1) ∈C2(S(1))∩C1(S
(1)
)∩A and such that

L
(2)
u
(2) =0 in S(2)

L(1)u(1) =0 in S(1)

T
(1)
u
(1) =T(2)u(2) on Γ

M(u(1)−u(2))=T(2)u(2)+Mu(0) on Γ

(2.1)

Here, u is the (2×1)-matrix describing the displacement field;L, T are the (2×2)-matrices of
operators representing the governing equations and stress operators, respectively, of the plane-

-strain (seeConstanda, 1995);M is thematrix

[

m 0
0 n

]

of (variable) interfaceparametersm,n> 0

prescribed on Γ (Hashin, 1991b; Sudak et al., 1999); u(0) represents an additional displacement
induced within the inhomogeneity by a uniform eigenstrain, and A represents the collection of
displacement fields with asymptotic behaviour described by

u(1)(x,y)= (c1x− c2y+o(1),c3x− c4y+o(1))

for (x,y)∈ R2 as x2+y2 →∞
(2.2)

with known constants ci, i = 1,2,3,4. We note that Eq. (2.2) corresponds to the case of pre-
scribed uniform remote stress and, as such, is not accommodated by the classical results of
Constanda (1995).

3. Uniqueness Result

Consider the difference pair {w(1),w(2)} of any two solutions to boundary value problem (2.1)-
-(2.2). LetKR be a circle with smooth boundary ∂KR and radiusR sufficiently large so that Γ
lies inside KR. Writing the Betti formulae (Constanda, 1995) for {w

(1),w(2)} in the bounded
regionsS(1)∩KR andS

(2), making use of the homogeneous boundary conditions for {w(1),w(2)}
and asymptotic condition from (2.2) onw(1), we arrive at the equation

∫

S(2)

E(w(2),w(2)) dA+

∫

S(1)

E(w(1),w(1)) dA+

∫

Γ

{M−1Tw(2)}TTw(2) dS =0

where E is the internal energy density. Given that m,n > 0 on Γ , we deduce that w(1), w(2)

are arbitrary rigid displacements in each of S(1) and S(2), respectively. From the asymptotic



Short Research Communication – Uniqueness in inclusion problems... 257

conditions on w(1), we easily deduce that w(1) = 0 in S
(1)
; hence w(1) = T(1)w(1) = 0 on Γ .

Boundary conditions (2.1)2,3 then imply that −Mw
(2) = 0 on Γ , which, since M is invertible,

means that w(2) =0 on Γ . Since it has already been determined that w(2) is an arbitrary rigid
displacement in S(2), the continuity requirements of boundary value problem (2.1)-(2.2) mean

thatw(2) =0 inS
(2)
. Consequently, boundary value problem (2.1)-(2.2) has atmost one regular

solution pair {u(1),u(2)}which establishes the required uniqueness result.
It should be noted that particular solutions to (2.1)-(2.2) have been constructed by Sudak

et al. (1999). The formal analysis of the existence of solutions to (2.1)-(2.2) using the boundary
integral equationmethodwill be the subject of a future paper. The same technique as that used
above, with very little change in detail, can be applied to prove the uniqueness result for the
corresponding problem of anti-plane shear deformations studied by Ru and Schiavone (1997)
and Ting and Schiavone (2010).

References

1. Constanda C., 1995, The boundary integral equation method in plane elasticity, Proceedings of
the American Mathematical Society, 123, 11, 3385-3396

2. Eshelby J.D., 1957, The determination of the elastic field of an ellipsiodal inclusion and related
problems,Proceedings of the Royal Society of London A, 241, 376-396

3. Eshelby J.D., 1959, The elastic field outside an ellipsoidal inclusion, Proceedings of the Royal
Society of London A, 252, 561-569

4. Fan H., Sze K.Y., 2001, Amicro-mechanicsmodel for imperfect interface in dielectric materials.
Mechanics of Materials, 33, 363-370

5. Gao J., 1995,A circular inclusionwith imperfect interface: Eshelby’s tensor and related problems,
Transactions of the ASME, Journal of Applied Mechanics, 62, 860-866

6. Hashin Z., 1991a, The spherical inhomogeneity with imperfect interface, Transactions of the
ASME, Journal of Applied Mechanics, 58, 444-449

7. Hashin Z. 1991b, Thermoelastic properties of particulate composites with imperfect interface,
Journal of the Mechanics and Physics of Solids, 39, 6, 745-162

8. Ru C.-Q., Schiavone P., 1996, On the elliptic inclusion in antiplane shear, Mathematics and
Mechanics of Solids, 1, 327-333

9. Ru C.-Q., Schiavone P., 1997, A circular inclusion with circumferentially inhomogeneous inter-
face in antiplane shear,Proceedings of the Royal Society of London A, 453, 2552-2572

10. Sudak L.J., Ru C.-Q., Schiavone P., Mioduchowski A., 1999, A circular inclusion with
inhomogeneously imperfect interface in plane elasticity, Journal of Elasticity, 55, 19-41

11. Ting T.C.T., Schiavone P., 2010, Uniform antiplane shear stress inside an anisotropic elastic
inclusion of arbitrary shape with perfect or imperfect interface bonding, International Journal of
Engineering Science, 48, 67-77

12. Wang X., 2012, A circular inclusion with imperfect interface in finite plane elastostatics, Acta
Mechanica, 223, 481-491

Manuscript received May 30, 2014; accepted for print October 1, 2014