The elasticity theory of inclusions surrounded by a foreign matrix plays an essential role in the mechanical modelling [1-4] and mathematical analysis [5-7] of particle-filled composites. One of the fundamental problems in this research area is concerned with the macroscopic response of the composite inclusion-matrix system to external loadings, and therefore focuses on optimal bounds of the overall properties of the composite system for a given volume fraction of the inclusions. Obviously, the overall properties of the composite system are strongly related to the corresponding microstructure, i.e., the configuration and arrangement of the inclusions. It has been shown by Liu [8] and Silvestre [9] that extremal overall properties of the composite system correspond to special configurations of the inclusions requiring that a constant stress/strain is achieved inside each inclusion for certain external loadings imposed on the composite system. Consequently, the problem of the uniformity of the stress in inclusions is essentially associated with the design of extremal particulate composites. On the other hand, it was shown in Ref. [10] that the shape of inclusions holding uniform stress minimizes the stress concentration in the entire composite, leading to an increase in the strength of the composite. Additionally, the problems of inclusions with uniform stress are found to have direct links with a broad class of energy minimization problems in multiphase composites [11]. We particularly mention that the problem of the uniformity of the stress within inclusions is closely related to (sometimes equivalent to) two specific kinds of inverse problems of holes embedded in an elastic solid, i.e., the design of equally strong holes [12-14] and that of harmonic holes [15,16].

One of the classical results established in two-dimensional linearly elastic deformations of either plane strain/stress or anti-plane shear states that an isolated elliptical inclusion enveloped by an infinite elastic matrix could acquire a constant internal stress field if the loading applied to the matrix remotely is uniform or the eigenstrain exerted on the inclusion is uniform [17,18], and any isolated non-elliptical inclusion cannot share this property of constant stress [19,20]. Similar results of an isolated ellipsoidal inclusion in three-dimensional linear isotropic elasticity were identified by Eshelby [21], and the related uniqueness of the ellipsoidal shape was verified (except for the strong version) by Liu [22], Kang and Milton [23], Ammari et al. [24] and Yuan et al. [25,26]. Here, the validity of this result requires that the infinite matrix surrounding the inclusion is homogeneous. However, the fact is that a bulk material may contain many internal microdefects (like dislocations, micro-cracks or grain boundaries), and it is reasonably anticipated that an elliptical inclusion no longer enjoys the constant-stress feature if a microdefect in the matrix appears near it. A question then arises: how does an elliptical inclusion evolve to maintain its internal constant stress when encountering a microdefect in the surrounding environment? To answer this question, one would inevitably encounter some inverse problems of the parameterization and identification of the domain of definition of certain potential functions. Such inverse problems are usually much more challenging than their forward counterparts, in which the objective is to determine the stress field for given domain of definition of the potential functions, especially when analytic or semi-analytic solutions are required to ensure a high accuracy of recovering the configuration of the corresponding domain of definition.

In the literature, there have been only few pieces of work on the identification of the inclusion shape allowing for a uniform internal stress field in the presence of a nearby microdefect. For example, in the context of anti-plane shear deformations, Wang et al. [27] devised an ingenious mapping function to derive explicit solutions for a certain family of the geometry of a uniformly stressed inclusion interacting with a screw dislocation, and they subsequently extended their essential idea to develop an analytic procedure of determining the shape of an inclusion with constant stress nearby a straight mode-III crack [28], while Dai et al. [29] resorted to the existence theorem of holomorphic functions with specified-form boundary values and proposed a unified analytic algorithm for identifying the shape of an inclusion with uniform stress in the vicinity of a screw dislocation whether the inclusion-matrix interface is treated as being perfectly or imperfectly bonded.

In this paper, we are mainly concerned with the existence of a constant stress field inside an inclusion when it verges on a material interface. From a technical point of view, we need to classify the corresponding inverse problem into two categories. In the first category, the geometry of the material interface is not fixed, and both it and that of the inclusion-matrix interface are adjustable to make the stress inside the inclusion uniform. In the second category, however, the geometry of the material interface is specified in advance and that of the inclusion-matrix is then the only variable in the process of attaining the uniformity of the stress inside the inclusion. In short, the second category contains fewer degrees of freedom than the first one in designing a uniformly stressed inclusion nearby a material interface, and in this case, to the authors’ experience in analytically solving similar inverse problems, it would be more intractable than the first one. For example, Wang et al. [30] invented a special mapping function for the first-category design of the shape of an inclusion ensuring the uniformity of the stress within it when it is embedded in a bi-material composite composed of two infinite elastic half-planes with a certain wavy interface, although if one requires that the interface between the two half-planes is flat, the mapping function would be not applicable and particularly it remains unclear how to devise an appropriate mapping function. More typical examples for the first-category problems involving the interaction between a closed material interface and an inclusion with constant stress can be found in Refs. [22,31-35], in which two or more adjacent inclusions were investigated and all of the inclusion-matrix interfaces were treated as variables that were identified to guarantee a constant internal stress field for each inclusion. So far in the literature, however, we have not found any piece of research on the strict second-category problems. Nevertheless, one may find in the literature some results for the limiting case of a second-category problem in which the material interface reduces to a traction-free material boundary. For example, in Refs. [36,37], the authors developed analytic procedures to determine the configuration of multiple inclusions each enveloping a constant stress field in the vicinity of a flat traction-free surface of an elastic half-plane where the inclusions are embedded. It is also worth mentioning some pieces of work concentrating on certain weak versions of the second-category problems in which the material interface of specified shape in a matrix is roughly modelled by applying an eigenstrain to a subdomain of the matrix. For example, in the deformations of anti-plane shear, Wang et al. [38,39] determined the shape for a single inclusion holding a uniform stress in an infinite matrix when a circular Eshelby inclusion (of the same shear modulus as that of the matrix) with either a constant or linear eigenstrain appears in its neighborhood. From a theoretical point of view, a general circular inclusion (of distinct elastic constants from those of the matrix) may be equivalently replaced by a circular Eshelby inclusion with a certain eigenstrain, although the specific eigenstrain ensuring the equivalence of replacement depends strongly on the details of external loadings and especially on the geometry and distribution of other defects or inclusions (if any). Consequently, for the problems addressed in Refs. [38,39], the introduction of either constant or linear eigenstrain could never guarantee the exact equivalence between a general circular inclusion and a circular Eshelby inclusion, and particularly appropriate eigenstrains allowing for the exact equivalence should be treated as unknowns to be ascertained with the determination of the shape of the uniformly stressed inclusion. Recent literature reported also a particular weak version of the second-category problems in which the material interface of specified shape (the Booth’s lemniscate or cardioid) nearby the inclusion (designed to admit a constant internal field) is degenerated to such an extent that the materials on the two sides of the interface have the same shear modulus but distinct Poisson’s ratios [40]. In this paper, we endeavor to extend the results in Ref. [36] to the case of an inclusion with constant stress embedded in two bonded elastic half-planes with a straight interface, which forms a strict, non-degenerate second-category problem.

We arrange the following content of the paper into four parts. In the first one, we introduce basic equations for the anti-plane shear deformation of an inclusion with a uniform anti-plane shear eigenstrain placed into an infinite bi-material system made up of two semi-infinite elastic half-planes, and give a mathematical description for the corresponding inverse problem of identifying the geometry of the inclusion when the eigenstrain-induced stress inside it is required to be a constant. In the second one, we present two methods to derive the essential equation that the geometry of the inclusion needs to satisfy to attain the desired constant internal stress, and explain the solution procedure for such an equation. In the third one, numerical results for some admissible shapes of the inclusion are illustrated, and the influence of the stiffness of the bi-material interface on the shape of the inclusion is discussed. Finally, the main points of the problem, the solution and results are mentioned in the last part.

By referring to the Cartesian x₁-x₂ coordinate system, we consider a bi-material structure composed of two elastic half-planes bonded together perfectly via an infinitely long straight interface, in the context of the theory of linear isotropic elasticity. An inclusion with a uniform anti-plane shear eigenstrain is introduced into the lower half-plane, and the entire composite system, as diagramed in Fig. 1, is assumed to be under the anti-plane shear deformation and free of any other external loadings except for the eigenstrain exerted on the inclusion. Here, for the lower half-plane, we assume theoretically that the inclusion has the same shear modulus as that of its surrounding lower matrix (i.e., the entire lower half-plane has a unified shear modulus) since the eigenstrain contained in it may represent to a considerable extent the difference between the shear moduli of the inclusion and matrix in practical situations, while for the upper half-plane its shear modulus is allowed to be distinct from that of the lower half-plane. We denote the upper half-plane region and the complete lower half-plane region by S_U and S_L, the inclusion region and the lower matrix region by S_in and S_LM, and the bi-material interface between the two half-planes and the inclusion-matrix interface by L_b and L_in. We also introduce in the upper half-plane an auxiliary region denoted by S_Uin and an auxiliary closed curve denoted by L_Uin, which are symmetric, respectively, to S_in and L_in in the lower half-plane. The shear moduli of the upper and lower half-planes are denoted by G_U and G_L. We particularly use the index “3” to represent the x₃-diretion perpendicular to the x₁-x₂ plane, and therefore the Cartesian components of the uniform eigenstrain imposed on the inclusion are described by and

Distinct from many forward problems in which the main objective is to determine the stress field induced by the eigenstrain in the entire composite system when the geometry of the inclusion is given in advance, the current problem focuses on the inverse case in which the eigenstrain-induced stress is required to keep a constant within the entire inclusion and the geometry of the inclusion is identified to achieve this requirement.

To formulate the above-mentioned inverse problem mathematically, we recall the well-known complex formalism for anti-plane shear elasticity. In fact, since the intrinsic governing equation for the anti-plane shear deformation in elastostatics is a two-dimensional Laplace equation, the out-of-plane displacement u₃ and the anti-plane shear components (σ₁₃,σ₂₃) of the stress tensor could be represented in terms of an analytic function f (z) (z = x₁ + ix₂, i is the imaginary unit) defined in the elastic domain under consideration. The specific representation for each component of the composite system is organized aswhereand the indices “U”, “LM” and “in” are introduced to identify the corresponding quantities in the upper half-plane, the lower matrix and the inclusion, respectively. Here, the f-related functions introduced from Eq. (1) are all analytic functions in their respective domains of definition, and particularly we may stipulate in advance thatin order to make the solution for each of these analytic functions unique when the geometry of the inclusion is determined.

The boundary conditions on the bi-material interface L_b describing the continuity of the traction (i.e., the ₂₃-com-component) and displacement across the interface are expressed using f_U(z) and f_LM(z) as [34]

which combined with Eq. (3) indicates

On the inclusion-matrix interface L_in, similar boundary conditions are established as [36]where

Here, Eqs. (4) and (6) differ in the form mainly because the inclusion and lower matrix have the same shear modulus and specially the displacement in the inclusion should be interpreted as the combination of the stress-related displacement and the eigenstrain-caused displacement. Since a constant stress is required within the inclusion, f_in(z) has to bewhere Γ_in is a complex number characterizing the uniform stress-caused strain inside the inclusion, and C is a certain constant ensuring the fulfillment of all the boundary conditions. Substituting Eq. (8) into Eq. (6), we organize the boundary value of f_LM(z) on L_in intowith

Now in the context of the complex formalism and the boundary conditions presented above, we describe the current inverse problem as to find an appropriate geometry of L_in such that the analytic functions f_LM(z) and f_U(z) could exist (in their respective domains of definition) under the constraints given in Eqs. (3)-(5) and (9). In what follows, we show how to establish the equations with respect to the geometry of the inclusion, respectively, from analytic continuation techniques and existence theorems of analytic functions with specified-form boundary values, and from the Green’s function of a concentrated force imposed on the bimaterial structure of two bonded half-planes.

It follows from Eqs. (52), (48) and (49) that the shape parameters a_j (j = 1, 2, 3, ...) of the inclusion only depend on the choice of the dimensionless parameters z_in/R_in, g, αe^2iβ. This indicates that if we take the same combination of the dimensionless parameters in two different cases, then we would obtain identical shapes of the inclusion. Let us consider two cases (identified by I and II), in which the dimensionless parameters satisfy

It is not difficult to find the fact that Eq. (56) holds only when

Noting the specific definition of the parameters in Eq. (57), we read from Eq. (57) that

An interesting fact is suggested from Eq. (58) that if a certain inclusion shape allows for a uniform stress inside the inclusion for a material-loading combination described by (G_U/G_L)_I, and then it could still ensure the uniformity of the stress inside the inclusion when the material-loading combination changes to a different one identified by (G_U/G_L)_II, and provided the two different material-loading combinations fulfill Eq. (58).

It is implied in the above-mentioned fact that as the bimaterial interface becomes stiffer, the evolution of the inclusion’s shape that maintains a constant internal stress field depends strongly on the specific plane in which the eigenstrain is applied (hereinafter defined as the plane of the eigenstrain). Figures 3 and 4 illustrate how the shape of the inclusion occupying a uniform stress field changes when the stiffness of the upper half-plane increases. Here, α = -1 corresponds to the case, in which the upper half-plane becomes extremely soft and the bi-material interface is thus reduced to a traction-free boundary, while α = 1 denotes the case in which the shear modulus of the upper half-plane tends towards infinity and therefore the bi-material interface can be treated as a rigid boundary. It is demonstrated from Fig. 3 that as the bi-material interface becomes stiffer (relative to the inclusion) its attraction to the boundary of the nearby inclusion enclosing a constant internal stress increases if the plane of the eigenstrain applied is parallel to it. As observed in Fig. 4, however, the result turns out to be completely opposite (i.e., the attraction between the inclusion and the bi-material interface decreases with increasing relative stiffness of the interface) when the plane of the eigenstrain applied is perpendicular to the bi-material interface.

Since the influence of and on the evolution of the inclusion’s configuration allowing for a constant internal stress field with increasing stiffness of the bi-material interface are opposite, it is then of particular interest to see what would happen to the configuration of the inclusion as the bi-material interface becomes stiffer when a combined eigenstrain with its two components and having the same magnitude is imposed on the inclusion. An example is shown in Fig. 5. We find that when the two components of the eigenstrain applied have an identical magnitude, the varying stiffness of the bi-material interface does not have a great impact on the distance between the interface and the boundary of the inclusion, but instead it influences the orientation of the inclusion significantly.

To validate our theoretical analysis of the uniformly stressed inclusion near the bi-material interface, we conduct finite element simulations for the bi-material structure containing an inclusion whose shape is specified from the current algorithm, and check if the inclusion shape obtained from the theoretical analysis could actually ensure the uniformity of the internal stress inside the inclusion. As a representative example, we take

In this case, the configuration of the inclusion (for N = 10) is determined asand the corresponding desired uniform stress in the inclusion is

In the finite element simulations, we simply take G_U = 3, G_L = 1, , and R_in = 1, and particularly the eigenstrain inside the inclusion is achieved with the setting of anisotropic thermal expansion. The specific geometric model of the bi-material structure containing the inclusion defined geometrically by Eq. (60) is shown in Fig. 6. The simulation results for the two anti-plane shear stress components and the out-of-plane displacement component of the entire structure are illustrated in Figs. 7-9, respectively. As displayed from Figs. 7 and 8, the stress inside the inclusion defined by Eq. (60) is indeed quite uniform with its σ₁₃-component being around -1.21 and its σ₂₃-component being at the level 10^-4 (both are consistent with the theoretical results given in Eq. (61), verifying the accuracy of our algorithm). In addition, the contour plot for the displacement inside the inclusion in Fig. 9 exhibits a linear distribution in the x₁-direction, confirming that the stress in the inclusion is uniform with its σ₂₃-component approaching zero.

We consider the anti-plane shear deformation of an inclusion with a uniform eigenstrain when it is embedded in an infinite bi-material composite which incorporates two elastic half-planes (the upper one and the lower one) and an infinitely long straight interface between them. The inclusion is contained in the lower half-plane and has the same shear modulus as that of the lower half-plane. Distinct from the conventional forward problem in which one focuses on the determination of the eigenstrain-caused stress field in the entire composite for a given geometry of the inclusion, the current problem is defined as an inverse problem in which we aim at discovering the configuration of the inclusion that makes the eigenstrain-induced stress inside the inclusion uniform.

Instead of any purely numerical optimization algorithm, two analytic methods are used to examine such an inverse problem. One is based on the analytic continuation and the existence theorem of a holomorphic function involving specific-form boundary values, while the other concerns the Green’s function for a concentrated force acting on a similar bi-material composite but composed of two complete half-planes (involving no inclusion). As a consequence of employing either of the two methods, we obtain an exact integral equation of the boundary curve of the inclusion which is sufficiently and necessarily responsible for the uniformity of the eigenstrain-induced stress within the inclusion. By introducing a polynomial mapping function with a set of unknown coefficients to parameterize the shape of the inclusion, we then transform the integral equation into a group of nonlinear algebraic equations with respect to the unknown coefficients with the aid of the Faber polynomials and Faber series for a finite simply-connected domain. These equations are solved numerically by classical iterative algorithms, and some representative results of the shape of the inclusion corresponding to a uniform internal stress are analyzed and illustrated. Among others, the phenomena observed from the numerical results are summarized as follows:

(1) For the case in which the plane of the applied shear eigenstrain is parallel to the interface between the two half-planes, the increasing stiffness of the interface (relative to the shear modulus of the inclusion) raises its attraction to the inclusion possessing a constant stress, leading to that a sharp corner oriented to the interface may appear on the boundary of the inclusion as the interface becomes stiffer. In contrast, for the case in which the plane of the applied shear eigenstrain is perpendicular to the interface, the increasing stiffness of the interface decreases its attraction or equivalently increases its repulsion to the inclusion with a constant stress, resulting in that the boundary of the inclusion tends to be even as the interface turns stiffer.

(2) When two eigenstrains of the same magnitude, whose planes are respectively parallel and perpendicular to the interface between the two half-planes, are applied simultaneously to the inclusion in the design of a constant internal stress inside it, the varying stiffness of the interface has only a minor influence on the distance between the inclusion and the interface, but has a significant influence on the orientation of the configuration of the inclusion.