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If an elastic body is defined as one that does not dissipate energy into heat, the classes of elastic bodies not only include the Green elastic solid, but also some types of implicit constitutive relations recently presented in the literature. In this paper one of such new implicit relations is studied in detail, wherein the energy function depend on the second Piola–Kirchhoff stress tensor and the Green Saint-Venant strain tensor. It is assumed that the function is anisotropic having two directions of anisotropy, thus the case of a transversely isotropic body and an isotropic body are special cases of the above function. Spectral invariants are used and explicit expressions for some second derivatives of the energy function are found. Such second derivatives appear in the implicit constitutive relation. [ABSTRACT FROM AUTHOR]

Recently, Arumugam et al. (2023) developed a constitutive relation for the response of isotropic inhomogeneous compressible elastic solids in order to describe the response of the trabecular bone. Since porous solids such as bones, cement concrete, rocks, metallic alloys, etc., are anisotropic, in this short note we develop a constitutive relation for such bodies that exhibit transverse isotropy and also having two preferred directions of symmetry. Another characteristic of bones is that they exhibit different response characteristics in tension and compression, and hence any constitutive relation that is developed has to be capable of describing this. Also, the material moduli depend on both the density and the mean value of the stress (mechanical pressure), as is to be expected in a porous solid. In the constitutive relation that is developed in this paper, though the stress and the linearized strain appear linearly in the constitutive relation, the relationship is nonlinear. We also derive the response of such solids when undergoing uniaxial extension and compression, simple shear and torsion. [ABSTRACT FROM AUTHOR]

For a relatively new class of constitutive equation for incompressible elastic bodies, wherein the Hencky strain tensor is a function of the Cauchy stress tensor, incremental equations are obtained, where it is assumed the presence of an initial time-independent stress that causes large deformations, to which a small time-dependent stress tensor is added. That stress tensor is assumed to cause a small (incremental) time-dependent deformation. For the case of infinite media the incremental equations are solved assuming travelling waves, and the speed of such waves is obtained for different problems considering initial homogeneous distributions of stresses and strains. Some numerical calculations of the speed of such waves are presented, considering a particular constitutive equation for rubber published recently in the literature. The speed of such small amplitude waves are compared with the predictions of the classical theory of nonlinear elasticity for similar problems. [ABSTRACT FROM AUTHOR]

Some new classes of constitutive relations for elastic bodies have been proposed in the literature, wherein the stresses and strains are obtained from implicit constitutive relations. A special case of the above relations corresponds to a class of constitutive equations where the linearized strain tensor is given as a nonlinear function of the stresses. For such constitutive equations we consider the problem of decomposing the stresses into two parts: one corresponds to a time-independent solution of the boundary value problem, plus a small (in comparison with the above) time-dependent stress tensor. The effect of this initial time-independent stress in the propagation of a small wave motion is studied for an infinite medium. [ABSTRACT FROM AUTHOR]

A relatively new type of constitutive equation for elastic bodies, wherein the linearized strain tensor is a function of the Cauchy stress tensor, is studied for the particular case the stresses are divided into two parts. One is an initial time-independent stress to which is added an incremental (small in comparison with that initial stress) time-dependent stress. Incremental equations are derived that can be used to study the propagation of small-amplitude waves. The behaviour of such waves is studied for three boundary value problems, where the distributions of the initial stresses are inhomogeneous. The problems are: the inflation of a hollow sphere, the inflation of an infinitely long cylindrical annulus, and the circumferential shear and inflation of a cylindrical annulus. The behaviour of the small-amplitude waves is compared for different external loads, investigating how the initial stress affects such waves. A nonlinear model for a class of isotropic dry elastic rock is considered. • Small waves are studied for elastic bodies. • Infinitesimal strains are functions of the stresses. • A nonlinear constitutive equation for rock is considered. • The effect of an initial stress is analysed. • The case of initial non-homogeneous distributions for the stress is considered. [ABSTRACT FROM AUTHOR]

It is often assumed in the literature that the nine classical strain invariants, which are used to characterize the strain energy of a compressible anisotropic elastic solid with two preferred non-orthogonal directions are independent. In this paper, it is shown that only six of the classical strain invariants are independent, and syzygies exist between the classical invariants. Alternatively, using principal axis techniques, it is simply proven that, only six of the classical strain invariants are independent and syzygies exist between the principal axis strain invariants. Consequently, all other sets of strain invariants, proposed in the literature, which are uniquely related to the set of principal axis strain invariants, have only six independent invariants. Due to syzygies, it is shown that the number of ground state constants required to fully describe the quadratic linear strain energy function of two-fibre solids is fourteen, not thirteen, as assumed in the literature. [ABSTRACT FROM AUTHOR]

Abstract: In this paper the problem of modeling three-dimensional residual stress distributions in hyperelastic tubes is addressed. First, the problem of a radially opened straight and bent tube, where the opening angle depends on the axial position, is explored with the semi-inverse method. As a result a rather complicated system of nonlinear partial differential equations is achieved which is difficult to solve. Second, a different approximate method considers the tube as a composition of two, three, four or more rings in the axial direction. Also here the opening angle of the tube depends on the axial position. Some numerical solutions for the stress components in the radial, circumferential and axial directions are analyzed in more detail. Third, the tube wall is divided into a number of radial layers, with different mechanical properties, and an approximate method to treat that problems is presented. It is emphasized that the proposed approach can also be used to compute 3D residual stress distributions in arterial walls. A final conclusion points to possible future research directions. [ABSTRACT FROM AUTHOR]

Abstract: We review in detail two concepts that appear in the theory that governs the interaction between electric fields and deformable media, namely the ‘Maxwell stress’ and the ‘total stress’. We focus on the quasi-static context, based on the force, couple and energy balance equations, with particular reference to boundary conditions. Equivalent formulations of the governing equations expressed in terms of different measures of stress are highlighted and connections with a principle of virtual work and a variational principle are noted. [Copyright &y& Elsevier]

Recently, among other smart and multifunctional materials, magneto-electric soft materials are expected to open a new horizon with myriad of potential applications such as wireless energy harvesting, spintronics and nonvolatile memories, magneto-electric random access memory, to mention a few. Magneto-electric coupling can be defined as the ability of a material to electrically polarize upon the application of a magnetic field and conversely, to magnetize under the application of an electric field. In contrast to traditional multi-ferroic hard materials, magneto-electric soft materials are of largely deformable where electric and magnetic fields and mechanical deformations are intricately coupled at finite strians. In this contribution, we will emphasis to formulate generalised mathematical frameworks of finitely deformed magneto-electric soft materials. After elaborating fundamental and governing equations, some homogeneous and non-homogeneous classical boundary value problems are studied under magneto-electrically coupled loads. [ABSTRACT FROM AUTHOR]

Universal solutions are found for isotropic elastic bodies, for a class of constitutive equation that is not Green elastic, wherein the linearized strain is assumed to be a function of the Cauchy stress. The structure of the solutions is compared with one example from the classical theory of nonlinear elasticity, namely the case of the inflation and uniform extension/compression of a cylindrical annulus. [ABSTRACT FROM AUTHOR]