A Neutral Nonlinear Elastic Circular Inhomogeneity Obeying Neuber’s Stress–Strain Law in Anti-Plane Shear

A circular inhomogeneity with interface slip and diffusion under in-plane deformation

Uniform in-plane stresses and strains inside an incompressible nonlinear elastic elliptical inhomogeneity

We prove the uniformity of the in-plane and anti-plane elastic field of stresses and strains inside an incompressible nonlinear elastic elliptical inhomogeneity embedded in an infinite linear isotropic elastic matrix subjected to uniform remote in-plane and anti-plane stresses. The elastic material occupying the elliptical inhomogeneity obeys Neuber’s special nonlinear stress-strain law. The original boundary value problem is finally reduced to a single non-linear equation for the constant effective strain within the inhomogeneity, which is rigorously proved to have a unique solution. Once the non-linear equation is solved numerically, we establish the uniform elastic field within the elliptical inhomogeneity and the non-uniform elastic field in the linear elastic matrix.

The stress field inside a two-dimensional arbitrary-shape elastic inclusion bonded through an interphase layer to an infinite elastic matrix subjected to uniform stresses at infinity is analytically studied using the complex variable method in elasticity. Both in-plane and anti-plane shear loading cases are considered. It is shown that the stress field within the inclusion can be uniform and hydrostatic under remote constant in-plane stresses and can be uniform under remote constant anti-plane shear stresses. Both of these uniform stress states can be achieved when the shape of the inclusion, the elastic properties of each phase, and the thickness of the interphase layer are properly designed. Possible non-elliptical shapes of inclusions with uniform hydrostatic stresses induced by in-plane loading are identified and divided into three groups. For each group, two conditions that ensure a uniform hydrostatic stress state are obtained. One condition relates the thickness of the interphase layer to elastic properties of the composite phases, while the other links the remote stresses to geometrical and material parameters of the three-phase composite. Similar conditions are analytically obtained for enabling a uniform stress state inside an arbitrary-shape inclusion in a three-phase composite loaded by remote uniform anti-plane shear stresses.

Decagonal quasicrystalline elliptical inclusions under thermomechanical loading

We use complex variable methods to derive a closed-form solution to the generalized plane strain problem of an incompressible nonlinear elastic parabolic inhomogeneity obeying Neuber’s special nonlinear stress-strain law embedded in an infinite linear isotropic elastic matrix subjected to uniform remote in-plane and anti-plane stresses. We prove that the internal in-plane and anti-plane stresses and strains within the parabolic inhomogeneity remain uniform. The uniform effective strain within the parabolic inhomogeneity is obtained in closed-form. Consequently, the uniform elastic field within the parabolic inhomogeneity and the non-uniform elastic field in the matrix are completely determined.

In this paper, we consider the problem of a single elastic inhomogeneity embedded within an infinite elastic matrix in anti-plane shear. In particular, we examine the design of this inhomogeneity to achieve (stress) neutrality when a non-uniform stress field is prescribed in the surrounding matrix. Since it is known that neutral elastic inhomogeneities do not exist when the inhomogeneity is assumed to be perfectly bonded to the matrix, the design method presented here is based on the assumption of an imperfect interface and the appropriate choice of the (single) interface parameter (characterizing the imperfect interface) to achieve the desired neutrality. Specifically, in the case of a homogeneously imperfect interface, it is shown that the circular inhomogeneity is neutral if and only if the prescribed non-uniform stress field in the surrounding matrix belongs to a certain class of polynomial functions. In the case of an inhomogeneously imperfect interface, neutrality is established for circular and elliptic inhomogeneities for specific classes of prescribed states of stress in the surrounding matrix. The results in this paper affirm the feasibility of designing a neutral elastic inhomogeneity by controlling the (imperfect) interface parameter describing the inhomogeneity-matrix interface.

A circular Eshelby inclusion interacting with a coated non-elliptical inhomogeneity with internal uniform stresses in anti-plane shear

We study the anti-plane strain problem associated with a p-Laplacian nonlinear elastic elliptical inhomogeneity embedded in an infinite linear elastic matrix subjected to uniform remote anti-plane stresses. A full-field exact solution is derived using complex variable techniques. It is proved that the stress field inside the elliptical inhomogeneity is nevertheless uniform. The uniformity of stresses is also observed inside a p-Laplacian nonlinear elastic parabolic inhomogeneity.

We analytically investigate a debonded arc-shaped anticrack lying on the interface between a circular elastic inhomogeneity and an infinite matrix when subjected to uniform remote in-plane stresses. One side of the anticrack is perfectly bonded to either the inhomogeneity or the matrix, whereas its other side has become fully debonded. Through the introduction of two sectionally holomorphic functions, the problem is reduced to a non-homogeneous Riemann–Hilbert problem of vector form that can be solved through a decoupling procedure and through evaluation of the Cauchy integrals. Solutions to both the non-degenerate case of distinct eigenvalues and the degenerate case of identical eigenvalues are derived.

Finite plane deformations of a three-phase circular inhomogeneity-matrix system

We examine the in-plane and anti-plane stress states inside a parabolic inhomogeneity which is bonded to an infinite matrix through an intermediate coating. The interfaces of the three-phase parabolic inhomogeneity are two confocal parabolas. The corresponding boundary value problems are studied in the physical plane rather than in the image plane. A simple condition is found that ensures that the internal stress state inside the parabolic inhomogeneity is uniform and hydrostatic. Furthermore, this condition is independent of the elastic properties of the coating and the two geometric parameters of the composite: in fact, the condition depends only on the elastic constants of the inhomogeneity and the matrix and the ratio between the two remote principal stresses. Once this condition is met, the mean stress in the coating is constant and the hoop stress on the coating side is also uniform along the entire inhomogeneity-coating interface. The unconditional uniformity of stresses inside a three-phase parabolic inhomogeneity is achieved when the matrix is subjected to uniform remote anti-plane shear stresses. The internal uniform anti-plane shear stresses inside the inhomogeneity are independent of the shear modulus of the coating and the two geometric parameters of the composite.

We consider the deformation of a nanocomposite by incorporating the effect of surface elasticity on the finite plane deformations of a circular inhomogeneity embedded in a particular class of compressible hyperelastic materials of harmonic type subjected to uniform remote Piola stresses. We incorporate the surface mechanics by using a version of the continuum-based surface/interface model of Gurtin and Murdoch. A complete solution is derived by reducing the original boundary value problem to two coupled first-order differential equations which can be solved analytically. The solution clearly demonstrates that the stress fields in the composite are size dependent and that the stress field inside a circular inhomogeneity is, in general, non-uniform. Two special cases which admit an internal uniform, yet size-dependent stress field, are discussed.

A circular inhomogeneity interacting with an open inhomogeneity designed to admit internal uniform hydrostatic stresses

Based on the Lekhnitskii‐Eshelby approach of two‐dimensional anisotropic elasticity, a semi‐analytical solution is derived for the problem associated with an anisotropic elliptical inhomogeneity embedded in an infinite anisotropic matrix subjected to remote uniform antiplane shear stresses. In this research, the linear spring type imperfect bonding conditions are imposed on the inhomogeneity‐matrix interface. We use a different approach than that developed by Shen et al. (2000) to expand the function encountered during the analysis for an imperfectly bonded interface. Our expansion method is in principle based on Isaac Newton's generalized binomial theorem. The solution is verified, both theoretically and numerically, by comparison with existing solution for a perfect interface. It is observed that the stress field inside an anisotropic elliptical inhomogeneity with a homogeneously imperfect interface is intrinsically nonuniform. The explicit expression of the nonuniform stress field within the inhomogeneity is presented. The nonuniform stress field inside the inhomogeneity is also graphically illustrated. A difference in internal stress distribution between a composite composed of anisotropic constituents and a composite composed of isotropic constituents is also observed. We finally extend the solution derived for a linear spring type imperfect interface to address an elliptical inhomogeneity with a viscous interface described by the linear law of rheology. It is observed that the stress field inside an elliptical inhomogeneity with a viscous interface is nonuniform and time‐dependent.