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The Eshelby formalism for an inclusion in a solid is of significant theoretical and practical implications in mechanics and other fields of heterogeneous media. Eshelby's finding that a uniform eigenstrain prescribed in a solitary…
Eshelby's seminal work on the ellipsoidal inclusion problem leads to the conjecture that the ellipsoid is the only inclusion possessing the uniformity property that a uniform eigenstrain is transformed into a uniform elastic strain. For the…
We make progress towards proving the strong Eshelby's conjecture in three dimensions. We prove that if for a single nonzero uniform loading the strain inside inclusion is constant and further the eigenvalues of this strain are either all…
According to the Eshelby conjecture, an ellipse or ellipsoid is the only shape that induces an interior uniform strain under a uniform far-field loading. We extend the Eshelby conjecture to domains of general shape for anti-plane…
In the late 1950's, Eshelby's linear solutions for the deformation field inside an ellipsoidal inclusion and, subsequently, the infinite matrix in which it is embedded were published. The solutions' ability to capture the behavior of an…
A self-consistent extended Einstein-Maxwell model for relativistic non-stationary polarizable-magnetizable anisotropic media is presented. Based on the analogy with relativistic extended irreversible (transient) thermodynamics, the extended…
Polyconvexity is one of the known conditions which guarantee existence of solutions of boundary value problems in finite elasticity. In this work we propose a framework for development of polyconvex strain energy functions for hyperelastic…
Incompressibility is established for three-dimensional and two-dimensional deformations of an anisotropic linearly elastic material, as conditions to be satisfied by the elastic compliances. These conditions make it straightforward to…
Eshelby showed that if an inclusion is of elliptic or ellipsoidal shape then for any uniform elastic loading the field inside the inclusion is uniform. He then conjectured that the converse is true, i.e., that if the field inside an…
We prove that if the associated fourth order tensor of a quadratic form has a linear elastic cubic symmetry then it is quasiconvex if and only if it is polyconvex, i.e. a sum of convex and null-Lagrangian quadratic forms. We prove that…
An analytical solution in a closed form is obtained for the three-dimensional elastic strain distribution in an unlimited medium containing an inclusion with a coordinate-dependent lattice mismatch (an eigenstrain). Quantum dots consisting…
In 1957 Eshelby showed that a homogeneous isotropic ellipsoidal inhomogeneity embedded in a homogeneous isotropic host would feel uniform strains and stresses when uniform strains or stresses are applied in the far-field. Of specific…
Explicit expressions for inverse of Young's modulus E, inverse of shear modulus G, and Poisson's ratio for cubic media are considered. All these characteristics of elastic media depend on three components of the compliance tensor S, and on…
Eshelby's inclusion problems have been generalized to arbitrary shape of polygonal, polyhedral, and ellipsoidal inclusions embedded in an infinite isotropic domain under transient heat transfer, and Eshelby's tensors have been analytically…
We show that a complete, two-sided, stable immersed anisotropic minimal hypersurface in $\mathbf{R}^4$ has intrinsic cubic volume growth, provided the parametric elliptic integral is $C^2$-close to the area functional. We also obtain an…
We consider the time-harmonic elastic wave scattering from a general (possibly anisotropic) inhomogeneous medium with an embedded impenetrable obstacle. We show that the impenetrable obstacle can be effectively approximated by an isotropic…
Closed-form expressions were recently derived for depolarization dyadics for truncated spheres and truncated spheroids, and the formalism was extended to truncated ellipsoids. These results were exploited to develop an implementation of the…
The transformation theory of optics and acoustics is developed for the equations of linear anisotropic elasticity. The transformed equations correspond to non-unique material properties that can be varied for a given transformation by…
Transverse elastic waves behave differently in nonlinear isotropic and anisotropic media. Quadratically nonlinear coupling in the evolution equations for wave amplitudes is not possible in isotropic solids, but such a coupling may occur for…
For certain shapes of inclusions embedded in a body, the field inside the inclusion is uniform for some boundary condition. We provide a construction scheme for inclusions of general shapes having such a uniformity property in two…