Journal of Elasticity最新文献

We investigate Hölder continuous dependence in theories of linear elastodynamics, assuming the elastic coefficients are not sign-definite. This is important with modern products such as auxetic materials where Poisson’s ratio may be negative. This study focusses on a class of linear elastic bodies where there are gradients of the strain and second gradients of the strain, and we also analyse a theory of elastodynamics with strain gradients where voids are also present in the body.

In this paper, we describe a uniform and standardized approach for analytically verifying the stability of isotropic, incompressible hyperelastic material models. Here, we address stability as fulfillment of the Hill condition – i.e. the positive definiteness of the material modulus in the Kirchhoff stress – log–strain relation. For incompressible material behavior, all mathematically and mechanically possible deformations lie within a range bounded, on the one hand, by uniaxial states and, on the other hand, by biaxial states; shear deformation states lie in between. This becomes particularly clear when the possible states are represented in the invariant plane. This very representation is now also used to visualize the regions of unstable material behavior depending on the selected strain energy function and the respective data set of material parameters. This demonstrates how, for some constellations of energy functions, with appropriate selection or calibration of parameters, stable and unstable regions can be observed. If such cases occur, it is no longer legitimate to use them to initiate, for example, finite element simulations. This is particularly striking when, for example, a fit appears stable in uniaxial tension, but the same parameter set for shear states results in unstable behavior without this being specifically investigated. The presented approach can reveal simple indicators for this. Nevertheless, the simple shear deformation, where the principal axes lag behind the deformation (gamma =tan alpha ) of the shear angle (alpha ), i.e. the rotation tensor (textbf{R} neq textbf{I}), still represents a special case that requires extra investigations. This is especially true given that all shear components of the logarithmic strains themselves exhibit a non–monotonic behavior with respect to the deformation angle.

The second-order differential equation for the non-linear radial oscillations of a transversely-isotropic hyperelastic tube has been postulated and derived based on certain requirements for the strain-energy function and the applied pressure. Lie point symmetry analysis has been used for the non-linear radial oscillatory system composed of neo-Hookean material to address the challenges in solving this equation. A comparison is conducted between the differential equations before and after the Lie transformation. Using Lie point symmetries, we demonstrate that the non-linear differential equations of the transversely-isotropic hyperelastic tube enhance non-periodic oscillations, which can contribute to the prediction of material reliability. This article aims to provide a comprehensive introduction and an application overview in the field of dynamical systems.

It has long been recognized that the theory of nonlinear elasticity provides a rich framework for a large variety of issues of interest to applied mathematicians. In particular, researchers with primary interest in nonlinear partial differential equations have been attracted to this area of continuum mechanics. However, the detailed theoretical background giving rise to the governing partial differential equations is not always familiar to non-specialists. The purpose of the present expository note is to attempt to alleviate this situation by describing a variety of nonlinear partial differential equations that have been found to govern the deformations of anti-plane shear and plane strain for isotropic incompressible hyperelastic solids in equilibrium.

We study the sense in which the continuum limit of a broad class of discrete materials with periodic structure can be viewed as a nonlinear elastic material. While we are not the first to consider this question, our treatment is more general and more physical than those in the literature. Indeed, it applies to a broad class of systems including ones that possess mechanisms; and we discuss how the degeneracy that plagues prior work in this area can be avoided by penalizing change of orientation. A key motivation for this work is its relevance to mechanism-based mechanical metamaterials. Such systems often have “soft modes”, achieved in typical examples by modulating mechanisms. Our results permit the following more general definition of a soft mode: it is a macroscopic deformation whose effective energy vanishes – in other words, one whose spatially-averaged elastic energy tends to zero in the continuum limit.

We describe via semigroup techniques the space decaying solutions to the system modeling an isotropic and homogeneous elastostatic semi-infinite band, both in the isothermal case and when thermal effects are present. The semigroup approach allows us to transfer some properties that typically occur in evolution problems to our model, such as analyticity and exponential decay at infinity. These results are closely related to the Saint-Venant principle. We conclude the article by recalling some consequences of the analyticity of the semigroup. The resulting properties are rather innovative compared to the usual results in the literature concerning the spatial decay of solutions.

By revisiting a model proposed in Zurlo and Truskinovsky (Mech. Res. Commun. 93:174–179, 2018), we address the accretive growth of a viscoelastic solid at large strains. The accreted material is assumed to accumulate at the boundary of the body in an unstressed state. The growth process is driven by the deformation state of the solid. The progressive build-up of incompatible strains in the material is modeled by considering an additional backstrain. The model is regularized by postulating the presence of a fictitious compliant material surrounding the accreting body. We show the existence of solutions to the coupled accretion and viscoelastic equilibrium problem.

In this work, we consider two dynamic systems arising in micropolar viscoelasticity. In this sense, the material structure is assumed to have macroscopic and microscopic levels. First, an existence and uniqueness result is proved by using the theory of linear semigroups and, secondly, the decay of the solutions to the equilibrium state is shown. Then, the polynomial energy decay is obtained applying a characterization of the system operator. In a second part, we consider the numerical approximation of a variational version of the above problem. This is done by using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. A discrete stability property is proved and an a priori error analysis is provided. The linear convergence of the approximations is deduced under some additional regularity conditions on the continuous solution. Finally, some numerical simulations are shown to demonstrate numerical convergence and the behavior of the discrete energy.

This paper is concerned with a linear theory of porous thermoelastic materials in which the second temperature gradient is included in the classical set of independent constitutive variables. The paper is based on the theory of microstretch thermoelastic solids, as well as on Green-Naghdi thermomechanics. We use thermal displacement and an entropy production inequality. The introduction of the entropy flux tensor allows the constitutive equations to depend on the second gradient of temperature. We first present the basic equations of the theory as well as the boundary conditions for this class of non-simple materials. We then study the case of isotropic and homogeneous materials and present a general solution of the field equations similar to that obtained by Mindlin in strain gradient elasticity. In the context of anisotropic solids we discuss the uniqueness question appropriate to the fundamental initial-boundary-value problems. The continuous dependence of solutions on initial data and body loads is established. The Mindlin-type solution is used to study the deformation produced by a concentrated heat source in a body occupying an unbounded region.