Journal of Elasticity最新文献

The paper addresses the problem of finding the necessary and sufficient conditions to be satisfied by the engineering moduli of an anisotropic material for the elastic energy to be positive for each state of strain or stress. The problem is solved first in the most general case of a triclinic material and then each possible case of elastic syngony is treated as a special case. The method of analysis is based upon a rather forgotten theorem of linear algebra and, in the most general case, the calculations, too much involved, are carried out using a formal computation code. New, specific bounds, concerning some of the technical constants, are also found.

Motivated from standard procedures in linear wave equations, we write the equations of classical elastodynamics as a linear symmetric hyperbolic system in terms of the displacement gradient (({mathbf{u}}_{mathbf{x}})) and the velocity (({mathbf{u}}_{t})); this is in contrast with common practice, where the stress tensor and the velocity are used as the basic variables. We accomplish our goal by a judicious use of the compatibility equations. The approach using the stress tensor and the velocity requires use of the time differentiated constitutive law as a field equation; the present approach is devoid of this need. The symmetric form presented here is based on a Cartesian decomposition of the variables and the differential operators that does not alter the Hamiltonian structure of classical elastodynamics. We comment on the differences of our approach with that using the stress tensor in terms of the differentiability of the coefficients and the differentiability of the solution. Our analysis is confined to classical elastodynamics, namely geometrically and materially linear anisotropic elasticity which we treat as a linear theory per se and not as the linearization of the nonlinear theory. We, nevertheless, comment on the symmetrization processes of the nonlinear theories and the potential relation of them with the present approach.

We develop a numerical algorithm for simulation of wave propagation in anisotropic thermoelastic media, established with a generalized Fourier law of heat conduction. The wavefield is computed by using a grid method based on the Fourier differential operator and a first-order explicit Crank-Nicolson algorithm to compute the spatial derivatives and discretize the time variable (time stepping), respectively. The model predicts four propagation modes, namely, a fast compressional or (elastic) P wave, a slow thermal P diffusion/wave (the T wave), having similar characteristics to the fast and slow P waves of poroelasticity, respectively, and two shear waves, one of them coupled to the P wave and therefore affected by the heat flow. The thermal mode is diffusive for low values of the thermal conductivity and wave-like (it behaves as a wave) for high values of this property. As in the isotropic case, three velocities define the wavefront of the fast P wave, i.e, the isothermal velocity in the uncoupled case, the adiabatic velocity at low frequencies, and a higher velocity at high frequencies. The heat (thermal) wave shows an anisotropic behavior if the thermal conductivity is anisotropic, but an elastic source does not induce anisotropy in this wave if the thermal properties are isotropic.

In the constrained Cosserat theory of a rod with a rigid cross-section, the balance of angular momentum is satisfied by a restriction on the constitutive equations that requires a second order tensor to be symmetric. The kinematics of the rod are determined by satisfying the balances of linear and director momentum and kinematic constraints. In contrast, the Antman model for a special Cosserat theory of rods proposes constitutive equations directly for the force and mechanical moment applied to the rod and the kinematics are determined by the balances of linear and angular momentum. These two models differ by their treatment of angular momentum. This note poses and answers the question: Are the solutions of these two models identical for the same strain energy?

The analysis of plane stress problems has long been a topic of interest in linear elasticity. The corresponding problem for non-linearly elastic materials is considered here within the context of homogeneous incompressible isotropic elasticity. It is shown that when the problem is posed in terms of the Cauchy stress, a semi-inverse approach must be employed to obtain the displacement of a typical particle. If however the general plane stress problem is formulated in terms of the Piola-Kirchhoff stress, the deformation of a particle requires the solution of a non-linear partial differential equation for both simple tension and simple shear, the trivial solution of which yields a homogeneous deformation. It is also shown that the general plane stress problem can be solved for the special case of the neo-Hookean material.

Nonlinear effects can enrich the propagation of elastic waves in mechanical metamaterials, which makes it possible to extend classical phenomena and functions in linear systems to nonlinear ones. In this work, rather than monochromatic waves in similar linear structures, the negative refraction is realized by mixing waves which are generated in nonlinear elastic wave metamaterials. Based on the stiffness matrix and plane wave expansion methods, dispersion curves of in–plane modes resulting from the collinear and non–linear mixings of two longitudinal waves are calculated. In the frequency spectrum, two propagating modes coalesce at exceptional points due to the coupling of in–plane modes, and those points at which the refraction type changes are also exceptional ones. Two kinds of negative refraction can be found in the mixing modes near exceptional points, but each of them needs to be induced in a specific configuration. Moreover, experiments are performed to support the pure negative refraction and beam splitting of the nonlinear elastic waves. Particularly, the parallel configuration is able to separate and extract the nonlinear mode when the single–mode negative refraction occurs, which shows the possibility to design elastic wave device by the negative refraction of nonlinear mixing waves.

An elastic map (mathbf{T}) associates stress with strain in some material. A symmetry of (mathbf{T}) is a rotation of the material that leaves (mathbf{T}) unchanged, and the symmetry group of (mathbf{T}) consists of all such rotations. The symmetry class of (mathbf{T}) describes the symmetry group but without the orientation information. With an eye toward geophysical applications, Browaeys & Chevrot developed a theory which, for any elastic map (mathbf{T}) and for each of six symmetry classes (Sigma ), computes the “(Sigma )-percentage” of (mathbf{T}). The theory also finds a “hexagonal approximation”—an approximation to (mathbf{T}) whose symmetry class is at least transverse isotropic. We reexamine their theory and recommend that the (Sigma )-percentages be abandoned. We also recommend that the hexagonal approximations to (mathbf{T}) be replaced with the closest transverse isotropic maps to (mathbf{T}).

A hyperelasticity modelling approach is employed for capturing various and complex mechanical behaviours exhibited by macroscopically isotropic polydomain liquid crystal elastomers (LCEs). These include the highly non-linear behaviour of nematic-genesis polydomain LCEs, and the soft elasticity plateau in isotropic-genesis polydomain LCEs, under finite multimodal deformations (uniaxial and pure shear) using in-house synthesised acrylate-based LCE samples. Examples of application to capturing continuous softening (i.e., in the primary loading path), discontinuous softening (i.e., in the unloading path) and auxetic behaviours are also demonstrated on using extant datasets. It is shown that our comparatively simple model, which breaks away from the neo-classical theory of liquid crystal elastomers, captures the foregoing behaviours favourably, simply as states of hyperelasticity. Improved modelling results obtained by our approach compared with the existing models are also discussed. Given the success of the considered model in application to these datasets and deformations, the simplicity of its functional form (and thereby its implementation), and comparatively low(er) number of parameters, the presented isotropic hyperelastic strain energy function here is suggested for: (i) modelling the general mechanical behaviour of LCEs, (ii) the backbone in the neo-classical theory, and/or (iii) the basic hyperelastic model in other frameworks where the incorporation of the director, anisotropy, viscoelasticity, temperature, softening etc parameters may be required.

An analytical expression for the strain energy of a constrained extensible Cosserat elastica is developed for general planar shapes and deformations of the rod. This strain energy function naturally couples tangential stretch and reference and current curvatures of the centroidal curve. The model considers a rigid rectangular cross-section of the rod which remains normal to the centroidal curve. The constitutive equations for the tangential force, shear force and bending moment are consistent with a restriction based on the balance of angular momentum that requires a stress-like tensor to be symmetric in a similar manner to the symmetry of the Cauchy stress in a three-dimensional continuum. Examples show that coupling of tangential stretch and reference and current curvatures of the centroidal curve in the new strain energy function can significantly influence predictions of tangential force, shear force and bending moments.

This paper develops an approximate stress distribution in a conical heap of jammed dry granular material loaded by gravity. An Eulerian formulation of elastic-inelastic response is used to explain why the residual stresses in the heap can be approximated by the current state of stress in the material. The proposed normalized stress components are functions of the normalized radial and vertical coordinates and are parameterized by only the angle of repose. It is shown that the vertical stress distribution applied to the base of the heap compares well with experiments using a rain procedure for sand deposition.