Journal of Elasticity

This study is devoted to the development of a unified and explicit elastic solution to the problem of a spherical inhomogeneity with an imperfectly bonded interface. Both tangential and normal displacement discontinuities at the interface are considered and a linear interfacial condition, which assumes that the tangential and the normal displacement jumps are proportional to the associated tractions, is adopted. The elastic disturbance due to the presence of an imperfectly bonded inhomogeneity is decomposed into two parts: the first is formulated in terms of an equivalent nonuniform eigenstrain distributed over a perfectly bonded spherical inclusion, while the second is formulated in terms of an imaginary Somigliana dislocation field which models the interfacial sliding and normal separation. The exact form of the equivalent nonuniform eigenstrain and the imaginary Somigliana dislocation are fully determined in this paper.

One of the most cited papers in Applied Mechanics is the work of Eshelby from 1957 who showed that a homogeneous isotropic ellipsoidal inhomogeneity embedded in an unbounded (in all directions) homogeneous isotropic host would feel uniform strains and stresses when uniform strains or tractions are applied in the far-field. Of specific importance is the uniformity of Eshelby’s tensor $\mathbf{S}$ . Following Eshelby’s seminal work, a vast literature has been generated using and developing Eshelby’s result and ideas, leading to some beautiful mathematics and extremely useful results in a wide range of application areas. In 1961 Eshelby conjectured that for anisotropic materials only ellipsoidal inhomogeneities would lead to such uniform interior fields. Although much progress has been made since then, the quest to prove this conjecture is still not complete; numerous important problems remain open. Following a different approach to that considered by Eshelby, a closely related tensor $\mathbf{P}=\mathbf{S}\mathbf{D}^{0}$ arises, where $\mathbf{D}^{0}$ is the host medium compliance tensor. The tensor $\mathbf{P}$ is associated with Hill and is of course also uniform when ellipsoidal inhomogeneities are embedded in a homogeneous host phase. Two of the most fundamental and useful areas of applications of these tensors are in Newtonian potential problems such as heat conduction, electrostatics, etc. and in the vector problems of elastostatics. Knowledge of the Hill and Eshelby tensors permit a number of interesting aspects to be studied associated with inhomogeneity problems and more generally for inhomogeneous media. Micromechanical methods established mainly over the last half-century have enabled bounds on and predictions of the effective properties of composite media. In many cases such predictions can be explicitly written down in terms of the Hill tensor, or equivalently the Eshelby tensor and can be shown to provide excellent predictions in many cases. Of specific interest is that a number of important limits of the ellipsoidal inhomogeneity can be taken in order to be employed in predictions of the effective properties of, for example, layered media and fibre reinforced composites and also to the cases when voids and cracks are present. In the main, results for the Hill and Eshelby tensors are distributed over a wide range of articles and books, using different notation and terminology and so it is often difficult to extract the necessary information for the tensor that one requires. The case of an anisotropic host phase is also frequently non-trivial due to the requirement of the associated Green’s tensor. Here this classical problem is revisited and a large number of results for problems that are felt to be of great utility in a wide range of disciplines are derived or recalled. A scaling argument leads to the derivation of the Eshelby tensor for potential problems where the host phase is at most orthotropic, without the requirement of using the anisotropic Green’s function. The Concentration tensor $\boldsymbol{\mathcal{A}}$ linking interior fields to those imposed in the far-field is derived for a wide variety of problems. These results can therefore be used in the various micromechanical schemes. Directly related to the tensors of Eshelby and Hill is the so-called Moment tensor $\mathbf{M}$ . As well as arising in the literature on micromechanics, this tensor is important in the vast area of research associated with inverse problems and specifically with the problem of identifying an object inside some domain given the application of a specific set of boundary conditions. Due to its fundamental importance and direct link to the Eshelby and Hill tensors, here we state the connection between $\mathbf{M}, \mathbf{P}$ and $\mathbf{S}$ in an effort to ensure that the work is of use to as wide a community as possible. Both tensor and matrix formulations are considered and contrasted throughout. Appendices give various details that illustrate the implementation of both approaches.

We consider a family of deformations describing cylindrical inflations within the context of finite, compressible, isotropic elasticity. We pose the problem of finding the maximal class of materials for which these deformations are possible at equilibrium under surface tractions only. We solve this problem for families of cylindrical inflations whose principal strain invariants have a special dependence on the radius. These families comprise and extend all cases considered by Murphy [2].

Within the context of linear elastodynamics, the radiated fields (including inertia) for a plane inhomogeneous inclusion boundary with transformation strain (or eigenstrain), moving in general motion under applied loading, have been obtained on the basis of Eshelby’s equivalent inclusion method, by using the strain field of a moving homogeneous inclusion boundary previously obtained. This dynamic strain field, obtained from the dynamic Green’s function (for an isotropic material), is unique, and has as initial condition the limit of the spherical Eshelby inclusion, as the radius tends to infinity, which is the minimum energy solution for the half-space inclusion. With the equivalent dynamic eigenstrain (which is dependent on the velocity of the boundary), the radiated fields for the inhomogeneous plane inclusion boundary can be obtained, and from them the driving force on the moving boundary can be computed, consisting of a self-force (which is the rate of mechanical work (including inertia) required to create an incremental region of inhomogeneity with eigenstrain), and of a Peach-Koehler force associated with the external loading. While for an expanding plane homogeneous inclusion boundary the Peach-Koehler force is independent of the boundary velocity, in the case of an inhomogeneous one it is not.

The equilibrium theory of linear piezoelectricity is considered. Saint-Venant's problem for a homogeneous and anisotropic piezoelectric cylinder is studied.

Introducing a crack in an elastic plate is challenging from the mathematical point of view and relevant within an engineering context of evaluating strength and reliability of structures. Accordingly, a multitude of associated works is available to date, emanating from both applied mathematics and mechanics communities. Although considering the same problem, the given complex potentials prove to be different, revealing various inconsistencies in terms of resulting stresses and displacements. Essential information on crack near-tip fields and crack opening displacements is nonetheless available, while intuitive adaption is required to obtain the full-field solutions. Investigating the cause of prevailing deficiencies inevitably leads to a critical review of classical works by Muskhelishvili or Westergaard. Complex potentials of the mixed-mode loaded Griffith crack, sparing restrictive assumptions or limitations of validity, are finally provided, allowing for rigorous mathematical treatment. The entity of stresses and displacements in the whole plate is finally illustrated and the distributions in the crack plane are given explicitly.

Applying the theorem proved by the authors in Ndanou et al. (J. Elast. 115:1–15, 2014), we established the hyperbolicity of non-stationary equations of hyperelastic isotropic solids for a one-parameter family of equations of state containing, in particular, generalized neo-Hookean solids. The hyperbolicity is equivalent to the rank-one convexity of the corresponding stored energy (Dafermos, Hyperbolic Conservation Laws in Continuum Mechanics, Springer, Berlin, 2000; Šilhavý, The Mechanics and Thermodynamics of Continuous Media, Springer, Berlin, 1996). The influence of the parameter on the solution properties is shown in the case of a strong shear test.

In this paper we consider an alternative generalization of classical thermoelasticity to those already available. Restrictions on constitutive equations are discussed with the help of an entropy production inequality proposed by Green and Laws [4]. The work is closely related to that of Müller [3] but the final results are somewhat more explicit. The theory is linearized and a uniqueness theorem is stated. In agreement with Müller [3] it is shown that the linear heat conduction tensor is symmetric and that the theory allows for “second sound” effects.