Journal of Elasticity最新文献

Having been studied for a long time, simple and pure shear deformation models are well known in elasticity. For small deformations, these two shear models differ only by a rigid rotation. On the contrary, for large deformations, the two models do not differ only by a rigid rotation. Therefore, in the latter situation, one cannot expect both models to fit the same constitutive properties for a prescribed form of the stored energy function. The kinematic and static differences between the two shear models, as well as their inadequacies for simulating experimental evidences, are discussed in this paper. To overcome these critical issues, a study was developed, which led quite naturally to the definition of a new shear deformation model, here called purely angular shear, based on the direct extension of the linearized pure shear model. The new model, characterized by a simple and immediate physical meaning, is particularly suitable for matching experimental tests.

A new formulation is proposed for linearized elastic solids, which can be used for the analysis of boundary value problems. This formulation is based on considering both the displacement field and the stress tensor as main variables for the problem, solving in parallel the equation of motion and the constitutive equation (expressing the linearized strain as a function of the stresses) to find such unknown variables. Some boundary value problems are solved using separation of variables for the fully dynamic case for isotropic bodies. The application of the above method is briefly considered for anisotropic bodies, and also for a relatively new class of constitutive equation, wherein the linearized strain is a nonlinear function of the stress.

We apply the formalism of analytical mechanics for constrained systems to reformulate the equilibrium theory of uniaxial nematic elastomers, allowing for constitutive dependence on the gradient (boldsymbol{G}) of the director (boldsymbol{n}). In this setting, inextensibility is enforced by requiring that (|boldsymbol{n}|^{2}=1) and that (boldsymbol{G}^{top }boldsymbol{n}=boldsymbol{0}). Starting from these constraints, and using the principle of virtual work within a thermomechanically consistent framework, we derive boundary-value problems for determining equilibrium configurations. We show that the original formulation yields an underdetermined system for the Lagrange multiplier fields unless ancillary gauge conditions are imposed. To resolve this indeterminacy, we introduce two effective Lagrange multiplier fields: one defined in the interior of the referential region and the other on that portion of the boundary where the director traction is prescribed.

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.