Eshelby's problem of non-elliptical inclusions

The Eshelby problem consists in determining the strain field of an infinite linearly elastic homogeneous medium due to a uniform eigenstrain prescribed over a subdomain, called inclusion, of the medium. The salient feature of Eshelby's solution for an ellipsoidal inclusion is that the strain tensor field inside the latter is uniform. This uniformity has the important consequence that the solution to the fundamental problem of determination of the strain field in an infinite linearly elastic homogeneous medium containing an embedded ellipsoidal inhomogeneity and subjected to remote uniform loading can be readily deduced from Eshelby's solution for an ellipsoidal inclusion upon imposing appropriate uniform eigenstrains. Based on this result, most of the existing micromechanics schemes dedicated to estimating the effective properties of inhomogeneous materials have been nevertheless applied to a number of materials of practical interest where inhomogeneities are in reality non-ellipsoidal. Aiming to examine the validity of the ellipsoidal approximation of inhomogeneities underlying various micromechanics schemes, we first derive a new boundary integral expression for calculating Eshelby's tensor field (ETF) in the context of two-dimensional isotropic elasticity. The simple and compact structure of the new boundary integral expression leads us to obtain the explicit expressions of ETF and its average for a wide variety of non-elliptical inclusions including arbitrary polygonal ones and those characterized by the finite Laurent series. In light of these new analytical results, we show that: (i) the elliptical approximation to the average of ETF is valid for a convex non-elliptical inclusion but becomes inacceptable for a non-convex non-elliptical inclusion; (ii) in general, the Eshelby tensor field inside a non-elliptical inclusion is quite non-uniform and cannot be replaced by its average; (iii) the substitution of the generalized Eshelby tensor involved in various micromechanics schemes by the average Eshelby tensor for non-elliptical inhomogeneities is in general inadmissible.     

Finite-amplitude shear horizontal waves propagating in a pre-stressed layer between two half-spaces

The propagation of finite-amplitude time-harmonic shear horizontal waves, in a pre-stressed compressible elastic layer of finite thickness embedded between two identical compressible elastic half-spaces, is investigated. This is accomplished by combining finite-amplitude linearly polarized inhomogeneous transverse plane wave solutions in the half-spaces and finite-amplitude linearly polarized unattenuated transverse plane wave solutions in the layer. The layer and half-spaces are made of different pre-stressed compressible neo-Hookean materials. The dispersion relation which relates wave speed and wavenumber is obtained in explicit form. The special case where the interfaces between the layer and the half-spaces are principal planes of the left Cauchy–Green deformation tensor is also investigated. Numerical results are presented showing the variation of the shear horizontal wave speed with the pre-stress and the propagation angle.     

Fracture detection problems:: applications and limitations of the energy momentum tensor and related invariants

We show that under certain circumstances, if displacement measurements are made inside and⧸or outside a body, it is possible to use two in variants based on the energy momentum tensor to determine (to some extent) the crack direction and length for cracks in bidimensional problems or the crack direction and area for cracks in three dimensional problems. This is done for a certain family of non-linear materials with a given toughness which includes linearly elastic materials with quadratic strain energy and power law elastic. One of the limitations is that the crack must be straight in 2D or planar in 3D.     

Strong ellipticity for tetragonal system in linearly elastic solids

The present paper studies the strong ellipticity for all crystal classes of tetragonal system in a linearly elastic material. Explicit conditions characterizing the strong ellipticity of the elasticity tensor are established for the tetragonal system with six elasticities (that is tetragonal–scalenohedral, ditetragonal–pyramidal, tetragonal–trapezohedral, ditetragonal–dipyramidal crystal classes) as well as for the tetragonal system with seven elasticities (tetragonal–disphenoidal, tetragonal–pyramidal and tetragonal–dipyramidal crystal classes).     

Plane-Strain Problems for a Class of Gradient Elasticity Models��A Stress Function Approach

The plane strain problem is analyzed in detail for a class of isotropic, compressible, linearly elastic materials with a strain energy density function that depends on both the strain tensor ?? and its spatial gradient ???. The appropriate Airy stress-functions and double-stress-functions are identified and the corresponding boundary value problem is formulated. The problem of an annulus loaded by an internal and an external pressure is solved.     

A NOTE ON MORI-TANAKA＇S METHOD

Explicit expressions of Mori-Tanaka＇s tensor for a transversely isotropic fiber rein- forced UD composite are presented. Closed-form formulae for the effective elastic properties of the composite are obtained. In a 3D sense, the resulting compliance tensor of the composite is symmetric. Nevertheless, the 2D compliance tensor based on a deteriorated Mori-Tanaka＇s tensor is not symmetric. Nor is the compliance tensor defined upon a deteriorated 2D Eshelby＇s tensor. The in-plane effective elastic properties given by those three approaches are different. A detailed comparison between the predicted results obtained from those approaches with experimental data available for a number of UD composites is made.     

A novel internal dissipation inequality by isotropy and its implication for inelastic constitutive characterization

In the present work a novel inelastic deformation caused internal dissipation inequality by isotropy is revealed. This inequality has the most concise form among a variety of internal dissipation inequalities, including the one widely used in constitutive characterization of isotropic finite strain elastoplasticity and viscoelasticiy. Further, the evolution term describing the difference between the rate of deformation tensor and the “principal rate” of the elastic logarithmic strain tensor is set, according to the standard practice by isotropy, to equal a rank-two isotropic tensor function of the corresponding branch stress, with the tensor function having an eigenspace identical to the eigenspace of the branch stress tensor. Through that a general form of evolution equation for the elastic logarithmic strain is formulated and some interesting and important results are derived. Namely, by isotropy the evolution of the elastic logarithmic strain tensor is embodied separately by the evolutions of its eigenvalues and eigenprojections, with the evolution of the eigenprojections driven by the rate of deformation tensor and the evolution of the eigenvalues connected to specific material behavior. It can be proved that by isotropy the evolution term in the present dissipation inequality stands for the essential form of the evolution term in the extensively applied dissipation inequality.     

Coaxiality of strain and stress in anisotropic linear elasticity

For a linearly elastic anisotropic body there are at least two rotations of the principal axes of strain such that the stress and strain tensors become coaxial. These rotations correspond to critical points for the stored energy, viewed as a function of the relative orientation between the body and the strain tensor.Supported by Gruppo Nazionale per la Fisica Matematica of C.N.R. (Italy).     

Stress solutions to the three-dimensional problem of elasticity

New representations of the stress tensor in the linear theory of elasticity and thermoelasticity are proposed. These representations satisfy the equilibrium equations and the strain compatibility equation. The stress tensor is expressed in terms of a harmonic tensor or a harmonic vector. The second boundary-value problem for an elastic half-space and an elastic layer is solved as an example __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 8, pp. 3–35, August 2006.     

The evolution of Hooke's law due to texture development in FCC polycrystals

Initially isotropic aggregates of crystalline grains show a texture-induced anisotropy of both their inelastic and elastic behavior when submitted to large inelastic deformations. The latter, however, is normally neglected, although experiments as well as numerical simulations clearly show a strong alteration of the elastic properties for certain materials. The main purpose of the work is to formulate a phenomenological model for the evolution of the elastic properties of cubic crystal aggregates. The effective elastic properties are determined by orientation averages of the local elasticity tensors. Arithmetic, geometric, and harmonic averages are compared. It can be shown that for cubic crystal aggregates all of these averages depend on the same irreducible fourth-order tensor, which represents the purely anisotropic portion of the effective elasticity tensor. Coupled equations for the flow rule and the evolution of the anisotropic part of the elasticity tensor are formulated. The flow rule is based on an anisotropic norm of the stress deviator defined by means of the elastic anisotropy. In the evolution equation for the anisotropic part of the elasticity tensor the direction of the rate of change depends only on the inelastic rate of deformation. The evolution equation is derived according to the theory of isotropic tensor functions. The transition from an elastically isotropic initial state to a (path-dependent) final anisotropic state is discussed for polycrystalline copper. The predictions of the model are compared with micro–macro simulations based on the Taylor–Lin model and experimental data.     

Anisotropy of plane complex elastic bodies

The problem of the search of the invariants of an anisotropic elastic tensor representing the mechanical response of a complex elastic body in a two dimensional space is addressed, in particular for a tensor that does not possess all the tensor symmetries typical of classical elasticity. The invariants of the stiffness tensor are found and all the possible types of orthotropy are discussed.     

The Explicit Determination of the Logarithm of a Tensor and Its Derivatives

The logarithm of a tensor is often used in nonlinear constitutive relations of elastic materials. Here we show how the logarithm of an arbitrary tensor can be explicitly evaluated for any underlying space dimension n. We also present a method for the explicit evaluation of the derivatives of the logarithm of a tensor.      

Stabilized mixed three‐field formulation for a generalized incompressible Oldroyd‐B model

This paper presents a generalization of the incompressible Oldroyd‐B model based on a thermodynamic framework within which the fluid can be viewed to exist in multiple natural configurations. The response of the fluid is viewed as a combination of an elastic component and a dissipative component. The dissipative component leads to the evolution of the underlying natural configurations, while the response from the natural configuration to the current configuration is considered elastic and therefore non‐dissipative. For an incompressible fluid, it is necessary that both the elastic behavior as well as the dissipative behavior is isochoric. This is achieved by ensuring that the determinant of the stretch tensor associated with the elastic response meets the constraint that its determinant is unity. A new stabilized mixed method is developed for the velocity, pressure and the kinematic tensor fields. Analytical models for fine scale fields are derived via the solution of the fine‐scale equations facilitated by the Variational Multiscale framework that are then variationally embedded in the coarse‐scale variational equations. The resulting method inherits the attributes of the classical SUPG and GLS methods, while a significant new contribution is that the form of the stabilization tensors is explicitly derived. A family of linear and quadratic tetrahedral and hexahedral elements is developed with equal‐order interpolations for the various fields. Numerical tests are presented that validate the incompressibility of the elastic stretch tensor, show optimal L₂ convergence for the conformation tensor field, and present stable response for high Weissenberg number flows. Copyright © 2016 John Wiley & Sons, Ltd.     

A class of universal relations in isotropic elasticity theory

A class of universal relations for isotropic elastic materials is described by the tensor equationTB = BT. This simple rule yields at most three component relations which are the generators of many known universal relations for isotropic elasticity theory, including the well-known universal rule for a simple shear. Universal relations for four families of nonhomogeneous deformations known to be controllable in every incompressible, homogeneous and isotropic elastic material are exhibited. These same universal relations may hold for special compressible materials. New universal relations for a homogeneous controllable shear, a nonhomogeneous shear, and a variable extension are derived. The general universal relation for an arbitrary isotropic tensor function of a symmetric tensor also is noted.     

An admissible form of an inelastic constitutive equation—I. General theory

From the viewpoint of irreversible thermodynamics an admissible form of rate-type constitutive equation of inelastic materials is given. The displacement gradient tensor F referred to the temporarily fixed reference frame which coincides with the Euler frame at the instant of the reference time is decomposed linearly into elastic and inelastic parts so that the procedure of formulation is simplified and clarified. The inelastic deformation rate is directly related to the internal production rate of entropy. The existence of an inelastic potential of the usual sense is not assumed, though the result can be understood to include the conventional flow theory based on an inelastic potential. An example of an elastoviscoplastic constitutive equation is given and some properties of yield surfaces are discussed.     

Orthotropic bone remodeling: case of plane stresses

Cancellous bone is constituted by a porous solid matrix filled with fluid. Matrix microstructure gives bone most of its mechanical strength properties. In our macroscopic approach, bone is seen as a continuous medium with a local (at our scale) time-dependent linearly elastic orthotropic behavior. Remodeling consists, by matrix material apposition or resorption, in microstructure modifications in order to optimize its mechanical characteristics. The proposed model is built on a time iterative procedure where the compliance tensor evolves such that, depending on the applied stresses, principal strains tend to fall within an admissible domain. The suggested remodeling laws in this work modify the elasticity “constants” as well as the orthotropy directions. The first results presented here correspond to the plane stresses case.     

Complete Asymptotic Expansions of Solutions of the System of Elastostatics in the Presence of an Inclusion of Small Diameter and Detection of an Inclusion

We consider the system of elastostatics for an elastic medium consisting of an imperfection of small diameter, embedded in a homogeneous reference medium. The Lamé constants of the imperfection are different from those of the background medium. We establish a complete asymptotic formula for the displacement vector in terms of the reference Lamé constants, the location of the imperfection and its geometry. Our derivation is rigorous, and based on layer potential techniques. The asymptotic expansions in this paper are valid for an elastic imperfection with Lipschitz boundaries. In the course of derivation of the asymptotic formula, we introduce the concept of (generalized) elastic moment tensors (Pólya–Szegö tensor) and prove that the first order elastic moment tensor is symmetric and positive (negative)-definite. We also obtain estimation of its eigenvalue. We then apply these asymptotic formulas for the purpose of identifying with high precision the order of magnitude of the diameter of the elastic inclusion, its location, and its elastic moment tensors.     

Strong Ellipticity of Transversely Isotropic Elasticity Tensors

The strong ellipticity of the elasticity tensor of a linearly hyperelastic, transversely isotropic material is investigated. The necessary and sufficient conditions for the elasticity tensor to be strongly elliptic are determined for the five constants characterizing it.