An integral representation for the solution of the inclusion problem in the theory of antiplane micropolar elasticity

1. Introduction

The theory of Cosserat (micropolar, asymmetric) elasticity was developed as an extension of the classical theory in order to accommodate the effect of material microstructure on the overall deformations of elastic bodies [1–3]. The general stipulations of the theory imply that the transfer of loadings between the two neighboring elements of an elastic body occur not only by means of the force vector but also by means of the independent moment vector which leads to the description of the elastic behavior in terms of the both asymmetric stress and couple stress tensors.

The micropolar theory of elasticity is in good agreement with experiments performed on many modern day advanced materials [4], such as foams, synthetic polymers, bones and the others for which the effect of material microstructure has been found to be significant and where the classical theory fails to predict adequately satisfactory results.

Kupradze [5] has established the boundary integral equation method and applied it for the rigorous treatment of the boundary value problems of three dimensional Cosserat elasticity. More recent works by Schiavone [6] and Potapenko [7–9] provided an elegant approach for the treatment of various interesting problems arising in plane theories of micropolar elasticity. However, the analysis presented in these works is confined only to the consideration of Dirichlet, Neumann, Robin and mixed boundary value problems and does not mention the class of transmission (inclusion) boundary value problems [10–12].

Inclusion problems have been receiving an increasing amount of attention in literature. The interest is attributed to the fact that the interaction of fibers with the surrounding matrix can be modeled as an inclusion problem which leads to numerous applications in composite mechanics, the area which currently attracts significant interest from engineering scientists and industrial practitioners due to the rapid development and growing use of composite materials.

The first solution for the problem of a circular inclusion embedded in an infinite elastic plate under the assumptions of the classical elasticity was derived by Muskhelishvili [13] using the complex variable method. In the work by Ru and Schiavone [14], this method was applied to obtain rigorous solutions for the problems associated with a circular elastic inclusion in antiplane shear under the assumptions of circumferentially homogeneous interface. In the work by Shen et al. [15, 16] the method was extended for the treatment of elliptic inclusions for which the corresponding semi-analytic solutions in the form of series have been derived. Martin reduced the problem of an inclusion of arbitrary shape in plane strain elasticity to the system of singular boundary integral equations and proved the solvability results [17].

A significant amount of research has been conducted to extend the methods used in classical elasticity for the treatment of inclusion problems arising in micropolar and couple-stress elasticity. For example, in the work by Hartranft and Sih, Lubarda, Weitsmann and Banks and Sokolowski [18–21] the authors analyzed the influence of couple stresses on stress concentrations around a circular inclusion. However, the solutions available in the literature are restricted to either the consideration of inclusions of a particular shape or to a particular form of the applied loading. For instance, Cheng and He [22, 23] and Ma and Hu [24] considered a spherical, cylindrical and ellipsoidal inclusions, respectively, and derived the analytical expressions of the corresponding micropolar Eshelby tensors [25].

Recently, Atroshchenko and Potapenko [26] proposed the extension of the boundary integral equation method formulated in the work by Constanda and Thomson [27, 28] for the treatment of the boundary value problems relating to the inclusion of arbitrary shape with an imperfect interface under the provisions of antiplane micropolar elasticity and proved the existence and uniqueness of the solution results for this boundary value problem. In the present paper, we find the exact analytical solution in the form of integral potentials to the boundary value problem for the inclusion of the most general form of inhomogeneous interface arising from the work by Atroshchenko and Potapenko [26]. Further we demonstrate that the integral densities of the potentials can easily be approximated using the boundary element method to represent the solution in the applicable form for the treatment of practical engineering problems arising in composite mechanics. We validate the method using an example of elliptical inclusion with a homogeneously imperfect interface embedded in an infinite matrix under the assumptions of anti-plane shear in micropolar elasticity. We find shear stresses along the inclusion interface and show the discrepancies between the results of the classical and micropolar theory. The method developed herein is universal and applicable to inclusions of arbitrary shape and a general form of the applied loading.

2. Preliminaries

In what follows Greek and Latin indices take the values 1, 2 and 1, 2, 3, respectively, the convention of summation over repeated indices is understood, M m × n is the space of ( m × n ) - matrices, E n is the identity element in M m × n , a superscript T indicates matrix transposition and ( … ) , α = ∂ ( … ) / ∂ x α . Also, if X is a space of scalar functions and v a matrix, v ∈ X means that every component of v belongs to X.

Let S be a domain in R 2 bounded by a closed C 2 -curve ∂ S and occupied by a homogeneous and isotropic linearly elastic micropolar material with elastic constants μ , α , β , γ , ε . The state of micropolar antiplane is characterized by a displacement field u = ( u 1 , u 2 , u 3 ) T and a microrotation field Φ = ( φ 1 , φ 2 , φ 3 ) T of the form

u 3 ( x 1 , x 2 ) = u 3 , u α = 0 , φ α ( x 1 , x 2 ) = φ α , φ 3 = 0 ,

(1)

where x = ( x 1 , x 2 ) is a generic point in R 2 . In the absence of body forces and couples, (1) gives rise to the system of equilibrium equations

L ( ∂ x ) u ( x ) = 0 ,

(2)

in which now, denoting φ α by u α , we obtain u = ( u 1 , u 2 , u 3 ) T . The matrix partial differential operator L ( ∂ x ) = L ( ∂ / ∂ x 1 , ∂ / ∂ x 2 ) = L ( ξ 1 , ξ 2 ) , is defined by Potapenko et al. [7]

L ( ξ 1 , ξ 2 ) = ( ( γ + ε ) Δ − 4 α + ( β + γ − ε ) ξ 1 2 ( β + γ − ε ) ξ 1 ξ 2 2 α ξ 2 ( β + γ − ε ) ξ 1 ξ 2 ( γ + ε ) Δ − 4 α + ( β + γ − ε ) ξ 2 2 − 2 α ξ 1 − 2 α ξ 2 2 α ξ 1 ( μ + α ) Δ ) ,

where Δ = ξ α ξ α .

Together with L we consider the boundary stress operator T ( ∂ x ) = T ( ∂ / ∂ x 1 , ∂ / ∂ x 2 ) defined by Potapenko et al. [7]

T ( ξ 1 , ξ 2 ) = ( ( 2 γ + β ) ξ 1 n 1 + ( γ + ε ) ξ 2 n 2 ( γ − ε ) ξ 2 n 1 + β ξ 1 n 2 2 α n 2 ( γ − ε ) ξ 1 n 2 + β ξ 2 n 1 ( γ + ε ) ξ 1 n 1 + ( 2 γ + β ) ξ 2 n 2 2 α n 1 0 0 ( μ + α ) ξ α n α ) ,

where n = ( n 1 , n 2 ) T is the unit outward normal to ∂ S .

The internal energy density is given by

E ( u , u ) = E 1 ( u , u ) + E 2 ( u , u ) + 1 4 [ 2 μ u 3 , α 2 + 2 α [ ( 2 u 2 + u 3 , 1 ) 2 + ( 2 u 1 − u 3 , 2 ) 2 ] ,

where

E 1 ( u , u ) = γ + κ 2 u 1 , 2 2 + γ + κ 2 u 2 , 1 2 + ( γ − κ ) u 1 , 2 u 2 , 1 , E 2 ( u , u ) = ( γ + β 2 ) u 1 , 1 2 + ( γ + β 2 ) u 2 , 2 2 + β u 1 , 1 u 2 , 2 .

Throughout what follows we assume that

2 γ + β > 0 , ε , α , γ , μ > 0 ,

(3)

in which case it is easy to see that the operator L is elliptic and E ( u , u ) is a positive quadratic form. In fact, E ( u , u ) = 0 if and only if

u ( x ) = ( 0 , 0 , c ) T ,

(4)

where c is an arbitrary constant. This is the most general rigid displacement and microrotation associated with equation (1). We take { F ( i ) } to be a basis for the space of such rigid displacements and microrotations, where F ( i ) are the columns of the matrix

F = ( 0 0 0 0 0 0 0 0 1 ) .

Clearly, LF = 0 in R 2 , TF = 0 on ∂ S and a generic vector of the form of equation (4) can be written as Fk, where k ∈ M 3 × 1 is constant and arbitrary.

A Galerkin representation of the solution of equation (2) with the right-hand side replaced by the Dirac delta distribution δ ( | x − y | ) yields the matrix of fundamental solutions

D ( x , y ) = L * ( ∂ x ) t ( x , y ) ,

(5)

where

t ( x , y ) = − d 2 2 π { 1 c 1 2 c 2 2 ln | x − y | + 1 c 1 2 − c 2 2 [ 1 c 1 2 K 0 ( c 1 | x − y | ) + 1 c 2 2 K 0 ( c 2 | x − y | ) ] } ,

(6)

L * is the adjoint of L, K 0 the modified Bessel function of order zero and the constants c 1 2 , c 2 2 are defined by

c 1 2 = 8 α ( 2 γ + β ) , c 2 2 = 8 α μ ( μ + α ) ( γ + ε ) .

Finally, let A be the class of vectors u ∈ M 3 × 1 whose components in terms of polar coordinates, as r = | x | → ∞ , admit an asymptotic expansion of the form

u 1 ( r , θ ) = r − 2 [ m 0 sin 2 θ + m 1 ( 1 − cos 2 θ ) + m 2 ] + O ( r − 3 ) , u 2 ( r , θ ) = r − 2 [ − m 0 sin 2 θ − m 1 ( 1 − cos 2 θ ) + m 3 ] + O ( r − 3 ) , u 3 ( r , θ ) = r − 1 [ ( m 3 − m 0 ) cos θ − ( m 2 − m 1 ) sin θ ] + O ( r − 2 ) ,

(7)

where m 0 , … , m 3 are arbitrary constants.

We also introduce the set

A * = { u : u = Fk + s A } ,

(8)

where k is an arbitrary constant and s A ∈ M 3 × 1 ∩ A .

Let S + be the bounded domain enclosed by ∂ S and S − = R 2 \ ( S + ∪ ∂ S ) .

We recall and make use of the following well known facts for interior and exterior domains respectively [7].

Remark 1 The following two assertions can be proved without difficulty as in the work by Potapenko et al. [7] using classical techniques from the work by Kupradze et al. [5] and Constanda [27].

2 ∫ S + E ( u , u ) d σ = ∫ ∂ S u T Tu ds .

(9)

∫ S + ( v T Lu − u T Lv ) d σ = ∫ ∂ S ( v T Tu − u T Tv ) ds .

(10)

Remark 2 (Somigliana formulae). Using classical techniques, we can prove that if u ∈ C 2 ( S + ) ∩ C 1 ( S ¯ + ) is a solution of the homogeneous system given by equation (2) in S + , then

∫ ∂ S [ D ( x , y ) T ( ∂ y ) u ( y ) − P ( x , y ) u ( y ) ] d s y = { u ( x ) , x ∈ S + 1 2 u ( x ) , x ∈ ∂ S 0 , x ∈ S − .

(11)

Remark 3 The following assertions are proved using classical techniques from the work by Kupradze et al. [5] and Constanda [27].

2 ∫ S − E ( u , u ) d σ = − ∫ ∂ S u T Tu ds .

(12)

− ∫ ∂ S [ D ( x , y ) T ( ∂ y ) u ( y ) − P ( x , y ) u ( y ) ] d s y = { 0 , x ∈ S + 1 2 u ( x ) , x ∈ ∂ S u ( x ) , x ∈ S − .

(13)

3. Elastic potentials

We introduce the single layer potential

( V φ ) ( x ) = ∫ ∂ S D ( x , y ) φ ( y ) d s y ,

(14)

and the double layer potential

( W φ ) ( x ) = ∫ ∂ S P ( x , y ) φ ( y ) d s y ,

(15)

where φ ∈ M 3 × 1 is an unknown density matrix.

We recall the properties of single and double layer integral potentials formulating the following theorem which has been proved in the work by Potapenko et al. [7].

Theorem 1 (1) If φ ∈ C ( ∂ S ) , then V φ , W φ are analytic and satisfy L ( V φ ) = L ( W φ ) = 0 in S + ∪ S − .

(2) If φ ∈ C 0 , α ( ∂ S ) , α ∈ ( 0 , 1 ) , then the direct values V 0 φ , W 0 φ of V φ , W φ on ∂ S exist (the latter as a principal value), the functions V + ( φ ) = ( V φ ) | S ¯ + , V − ( φ ) = ( V φ ) | S ¯ − are of class C 1 , α ( S ¯ + ) and C 1 , α ( S ¯ − ) , respectively and T V + ( φ ) = ( W 0 * + 1 2 I ) φ , T V − ( φ ) = ( W 0 * − 1 2 I ) φ on ∂ S ,

where W 0 * is the adjoint of W 0 and I - the identity operator.

(3) If φ ∈ C 1 , α ( ∂ S ) , α ∈ ( 0 , 1 ) , then the functions

W + ( φ ) = { ( W φ ) | S + , in S + , ( W 0 − 1 2 I ) φ , on ∂ S , , W − ( φ ) = { ( W φ ) | S − , in S + , ( W 0 + 1 2 I ) φ , on ∂ S ,

(16)

are of class C 1 , α ( S ¯ + ) and C 1 , α ( S ¯ − ) , respectively, and T W + ( φ ) = T W − ( φ ) on ∂ S .

We introduce operator N 0 : C 1 , α ( ∂ S ) → C 0 , α ( ∂ S ) as

N 0 φ = T W + ( φ ) = T W − ( φ ) .

(17)

Finally, the following theorem has been proved in [7].

Theorem 2 If φ ∈ C ( ∂ S ) then:

4. Inclusion problem

We consider an infinite plane with a bounded inclusion occupying the domain S ¯ + enclosed by C 2 -curve ∂ S . The inclusion is made of homogeneous and isotropic micropolar material with elastic constants μ i , α i , β i , γ i , ε i . The matrix, which lies in the domain S ¯ − , is also homogeneous and isotropic micropolar material with elastic constants μ e , α e , β e , γ e , ε e . Let L i ( ∂ x ) and L e ( ∂ x ) be the operators of the governing equations associated with the inclusion and the matrix respectively. The corresponding boundary stress operators T i ( ∂ x ) and T e ( ∂ x ) are defined in a similar way. We assume that both sets of the constants satisfy the conditions given by equation (3). The inclusion problem with an imperfect interface is formulated as follows:

Find u i ∈ C 2 ( S + ) ∩ C 1 ( S ¯ + ) and u e ∈ C 2 ( S − ) ∩ C 1 ( S ¯ − ) ∩ A such that

L i ( ∂ x ) u i ( x ) = 0 , x ∈ S + L e ( ∂ x ) u e ( x ) = 0 , x ∈ S −

(18)

and

u e − u i = f and T e u e − T i u i = g , x ∈ ∂ S .

(19)

The existence and uniqueness results for this problem have been discussed in details and proven in the work by Atroshchenko and Potapenko [26].

Theorem 3 The inclusion problem has at most one regular solution.

In order to investigate the solvability of the inclusion problem we can derive the integral representation of the solution using the direct boundary integral equation method based on the representation formulae given by equations (11) and (13).

Let

u i = V i + ( T i u i | ∂ S ) − W i + ( u i | ∂ S ) in S + ¯ u e = − V e − ( T e u e | ∂ S ) − W e − ( u e | ∂ S ) in S − ¯

(20)

Then

u i | ∂ S = V i 0 ( T i u i | ∂ S ) − ( W i 0 − 1 2 I ) ( u i | ∂ S ) , u e | ∂ S = − V e 0 ( g + T i u i | ∂ S ) + ( W e 0 + 1 2 I ) ( f + u i | ∂ S ) .

(21)

If we set φ = u i | ∂ S and ψ = T i u i | ∂ S then equation (21) yields the following system of singular integral equations

( W i 0 + 1 2 I ) φ − V i 0 ψ = 0 , ( W e 0 − 1 2 I ) φ − V e 0 ψ = − ( W e 0 − 1 2 I ) f + V e 0 g

(22)

for unknown densities ( φ , ψ ) . We have shown that the system given by equation (22) has a solution ( φ , ψ ) = ( u i | ∂ S , T i u i | ∂ S ) so we only have to show that this solution is unique.

Theorem 4 For any f ∈ C 1 , α ( ∂ S ) and any g ∈ C 0 , α ( ∂ S ) , α ∈ ( 0 , 1 ) the system given by equation (22) has a unique solution pair φ ∈ C 0 , α ( ∂ S ) and ψ ∈ C 1 , α ( ∂ S ) if and only if

∫ ∂ S F T gdS = 0 .

(23)

This solution also satisfies

N i 0 φ − ( W i 0 * − 1 2 I ) ψ = 0 , N e 0 φ − ( W e 0 + 1 2 I ) ψ = − N e 0 f + ( W e 0 + 1 2 I ) g .

(24)

Proof. Consider the homogeneous system

( W i 0 + 1 2 I ) φ − V i 0 ψ = 0 , ( W e 0 − 1 2 I ) φ − V e 0 ψ = 0 .

(25)

Let ( φ , ψ ) be a solution of equation (25). Using the procedure described in the work by Martin [17] together with the results of Theorem 5 we arrive at φ = 0 and ψ = 0 . From Betti formulae given by equations (9) and (12) with T i u i | ∂ S = ψ

∫ ∂ S F T ψ dS = 0

and hence V e − ( g + ψ ) ∈ A if and only if ∫ ∂ S F T gdS = 0 by Theorem 4. Therefore equation (23) represents the necessary and sufficient condition for solution u e , given by equation (20), to belong to the asymptotic class A .

Finally, the proof that the unique solution ( φ , ψ ) of system given by equation (22) also satisfies equation (24) can be found in the work by Atroshchenko and Potapenko [26].

After the densities ( φ , ψ ) have been found from the system given by equation (24) the solution { u i , u e } of the problem given by equations (18) and (19) is given by

u i = V i + ( ψ ) − W i + ( φ ) , u e = − V e − ( ψ ) + W e − ( φ ) − V e − ( g ) + W e − ( f ) .

(26)

 ■

Remark 4 There exists a regular solution of equations (18) and (19) in A * unique up to an arbitrary rigid displacement z ∈ Z . This solution is given by

u i = V i + ( ψ ) − W i + ( φ ) + z , u e = − V e − ( ψ ) + W e − ( φ ) − V e − ( g ) + W e − ( f ) + z .

5. Boundary element method

The exact analytical solution to the boundary value problem given by equations (18) to (19) may be represented in the form of equation (26) and the corresponding boundary integral equations are uniquely solvable with respect to distributional densities φ and ψ . However, these densities cannot be found analytically. To approximate them numerically we use the boundary element method [29], which is based on Somigliana representation of the solution for the interior and exterior domains. In view of Somigliana formulae given by equations (11) and (13), taking into account the jump condition on the boundary in equation (19) for the boundary value problem given by equations (18) and (19) for x ∈ ∂ S we have

∫ ∂ S [ D i ( x , y ) ( T e ( ∂ y ) u e ( y ) ) − P i ( x , y ) u e ( y ) ] ds ( y ) + 1 2 u e ( x ) = 1 2 f ( x ) + ∫ ∂ S [ D i ( x , y ) g ( y ) − P i ( x , y ) f ( y ) ] ds ( y ) , x ∈ ∂ S ,

(27)

where y is a fixed point on the boundary and the matrices of fundamental and singular solutions D ( x , y ) and P ( x , y ) respectively allow expansions in the vicinity of point y [7]

D ( x , y ) = − 1 2 π ( b E γ γ + 1 μ + α E 33 ) ln | x − y | + Ω ( x , y )

and

P ( x , y ) = − 1 2 π ( ∂ ∂ n ( y ) ln | x − y | E kk + p ∂ ∂ s ( y ) ln | x − y | ε α β E α β ) + Π ( x , y ) .

Here, ∂ ∂ n ( y ) and ∂ ∂ s y represent derivatives in each of the normal and tangential directions respectively, that is, ∂ ∂ n ( y ) ≡ n . ∇ y , ∂ ∂ s y ≡ τ . ∇ y where τ = ( − n 2 , n 1 ) is the tangential direction at y, chosen so that { n ( y ) , τ ( y ) } is right-handed

b = ( 3 γ + β + ε ) 2 ( γ + ε ) ( 2 γ + β ) , p = ( γ + ε ) ( γ − ε ) − β ( 2 γ + β ) 2 ( γ + ε ) ( 2 γ + β ) ,

and Ω ( x , y ) , Π ( x , y ) are weakly singular in the sense of the work by Constanda [27] and { E ij } is the standard ordered basis for the vector space of ( 3 × 3 ) -matrices.

Following the standard procedure described in detail in the work by Shmoylova et al. [30] and using the standard approach employing linear expansions for the discretization of the boundary we solve equation (27) and construct the approximate densities φ and ψ in the form of series, which in turn can further be substituted in (26) to find the solutions inside and outside of the inclusion. The process of convergence of the approximate solution obtained by the boundary element method has been studied in detail in the work by Shmoylova et al. [30].

6. Example

In the previous section we formulated the most general form of the inclusion problem which assumes that the interface is imperfect, i.e. one can observe an arbitrary jump of tractions and displacements and microrotations across the interface boundary. So far, experimental results relating to the behavior of microrotations across the interface are absent from the literature. In classical elasticity one of the more widely adopted models of an imperfect interface is based on the assumptions of a “spring-like” interface model in the sense of Hashin (see, for example [11]). According to the model the tractions are continuous across the interface but displacements undergo a jump which is proportional to the tractions with the coefficient of proportionality h which is called an interface parameter. Such interface is called homogeneously imperfect. Obviously, an inclusion with a homogeneously imperfect interface in antiplane micropolar elasticity can be considered as a particular case of the boundary value problem given by equations (18) and (19) when the boundary condition in equation (19) takes the form

T e u e − T i u i = 0 and u e − u i = h T e u e , x ∈ ∂ S

(28)

where

h = ( h 1 , h 2 , h 3 ) T

(29)

is the interface parameter vector in which the first two components h 1 and h 2 are responsible for the jump of the microrotations φ 1 and φ 2 respectively and h 3 is the coefficient of proportionality in the expression for the jump of the out of plane displacement u 3 . In what follows we consider an example of an inclusion of elliptical shape with a homogeneously imperfect interface under the assumptions of anti-plane micropolar elasticity. The choice of the inclusion shape is explained by the opportunity of comparing the results with those published in an elegant paper by Shen et al. [16] for an elliptic inclusion in the field of anti plane shear under the provisions of the classical theory. Although it could be tempting in our example to assume that the interface does not allow the transfer of the microrotations across its boundary, i.e. h 1 = h 2 = 0 in equation (29) and consequently φ 1 = φ 2 =0, x ∈ ∂ S , the conditions which would make our example much easier from the computational point of view and might also be realistic from the point of view of the inclusion interface design. It can be shown that these assumptions are possible if and only if φ 1 = φ 2 =0 everywhere in both the interior and exterior domains so the problem reduces to the classical one considered in the work by Shen et al. [16]. The interface condition given by equation (29) with 3 interface parameters is supported by Videla and Atroshchenko [31] where a circular inclusion under the assumptions of plane micropolar elasticity is considered.

Consider an inclusion represented by an ellipse prescribed by the equations

x 1 = a cos θ , x 2 = b sin θ

in the case of remote loading S 0 under the assumptions of pure anti plane shear and in the absence of any eigenstrain inside the inclusion. By changing the values of the parameters a and b we can consider elliptic inclusions of different sizes and aspect ratios. Assume that h 1 = h 2 = 0 . 25 and h 3 = 0 . 5 and elastic constants take values μ i = 10 , α i = 60 , β i = 30 , γ i = 20 , ε i = 10 , μ e = 1 , α e = 6 , β e = 3 , γ e = 2 and ε e = 1 . Although micropolar elastic constants have been measured for several types of materials including polymers, foams, bone and graphite [4], we do not consider any specific material in this example but choose the constants only in a way so that the condition given by equation (3) is satisfied.

The numerical solution for the boundary tractions and moments has been found to approximate the exact solution to five decimal places for n = 44 elements of the boundary ∂ S . Figures 1 and 2 represent the values of shear stresses σ 13 and σ 23 along the inclusion interface based on the solution given by equation (26) with the integral density approximated numerically using the boundary element method based on equation (27) for the inclusions with aspect ratio a b = 3 in the cases when a = 3 , b = 1 and a = 1 . 5 , b = 0 . 5 respectively. We also compare these solutions with the results obtained by Shen et al [16] for an elliptic inclusion of aspect ratio a b = 3 in the classical case when micropolar elastic constants are taken to be zero.

OPEN IN VIEWER

Figure 1.

The shear stress distribution over the interface for remote stress S 0 .

OPEN IN VIEWER

Figure 2.

The stress distribution along the interface for remote stress S 0 .

The stresses presented in the figures are divided by the applied load S 0 to represent the data in non-dimensional values. One can draw a conclusion from Figures 1 and 2 that the shear stresses are significantly lower in the vicinity of the inclusion interface in the case when the microstructure is taken into an effect in comparison with the case when the microstructure is ignored (classical theory), particularly, for smaller inclusions. In the classical case the stresses are dependent only on the aspect ratio and independent of the actual values of the parameters a and b. Meanwhile, in the case where the stresses are evaluated using the provisions of micropolar elasticity, not only the aspect ratio but also the actual values of the parameters a and b do really matter. The results presented in Figures 1 and 2 may also be indirectly compared with the earlier investigations undertaken in the area of Cosserat solids. For example, Lakes et al. [32] performed an experimental study on stress concentrations around cracks in a human bone and Shmoylova et al. [30] conducted a theoretical study of the same problem involving the boundary integral equation method. Both of these studies revealed approximately a 20% difference in the results for stress concentrations around cracks in the classical and micropolar cases, which is approximately the same difference between the classical and micropolar cases observed in this study for an elliptic inclusion. Similar discrepancies between the results of the classical and micropolar theories have been found in other work [32–34] where both experimental and theoretical investigations of torsion of micropolar beams have been undertaken.

The explanation of discrepancies between our results and those presented in the work by Shen et al. [16] for the classical case in the vicinity of an inclusion interface can be explained by the fact that material particles at every point of the interface contour can rotate under the applied load and generate couple stresses. Consequently, the applied load is distributed between stresses and couple stresses, therefore, the resultant shear stress is reduced in comparison with the classical case. Consequently, couple stresses can absorb a part of the applied load and contribute into the decrease of the shear stress over the interface.

7. Conclusions

In this paper, we considered an inclusion boundary value problem under the assumptions of antiplane micropolar elasticity and applied the boundary integral equation to obtain its solution. The method is very general in nature so it allows us to find the solution for stress concentrations around an inclusion of any arbitrary shape with any interface design and any arbitrary type of prescribed initial loading. The exact analytical solution had been obtained in the form of an integral potential whose density was then approximated numerically using the boundary element method. The approximation of the density converges to its analytical representation so the solution for stress concentrations arising from the inclusion problem is almost as good as the analytical one so it can be applied for practical problems. The method can be used by researchers in materials science and engineers to study the properties of interface parameters necessary for the design of composite materials. To illustrate the effectiveness of the method we considered an inclusion of elliptical shape with a homogeneously imperfect interface under the constant antiplane shear loading prescribed at infinity. It has been found that the size scale of the inclusion and the presence of microstructure have an effect on the stress concentrations around the inclusion interface. The difference in the solutions obtained by the boundary integral equation method for an inclusion under the assumptions of micropolar elasticity and by the same method under the provisions of the classical elasticity is consistent with the experimental results.