Some theorems in the theory of elastic materials with voids

The linear theory of elastic materials with voids is considered. Some basic theorems concerning the existence and uniqueness of solution, the reciprocity relations and the variational characterization of the solution are presented.     

Energy principles in theory of elastic materials with voids

According to the basic idea of dual-complementarity, in a simple and unified way proposed by the author^[1], various energy principles in theory of elastic materials with voids can be established systematically. In this paper, an important integral relation is given, which can be considered essentially as the generalized pr. inciple of virtual work. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem of work in theory of elastic materials with voids, but also to derive systematically the complementary functionals for the eight-field, six-field, four-field and two-field generalized variational principles, and the principle of minimum potential and complementary energies. Furthermore, with this appro ach, the intrinsic relationship among various principles can be explained clearly. The project supported by the National Natural Science Foundation of China     

On some basic principles in dynamic theory of elastic materials with voids

According to the basic idea of dual-complementarity, in a simple and unified way proposed by the author^[1], some basic principles in dynamic theory of elastic materials with voids can be established systematically. In this paper, an important integral relation in terms of convolutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem in dynamic theory of elastic materials with voids, but also to derive systematically the complementary functionals for the eight-field, six-field, four-field and two-field simplified Gurtin-type variational principles. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly. The project supported by the Foundation of Zhongshan University Advanced Research Center     

Structural function theory of generalized variational principles for linear elastic materials with voids

In this paper,we have obtained generalized variational principles for linear elasticmaterials with voids from structural function theory.Correspondent relations betweenstructural functions and generalized variational principles are given.     

A uniqueness theorem in the theory of Cosserat surface

Within the scope of the non-isothermal theory of an elastic Cosserat surface and for a system of linear equations characterizing the initial mixed boundary-value problems of thermoelastic shells, a uniqueness theorem is obtained without the use of definiteness assumption for the free energy. The theorem holds for nonhomogeneous and anisotropic shells undergoing small motions (and small temperature change) superposed on a large deformation.

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Résumé Dans le cadre de la théorie d'une surface élastique de Cosserat et pour un système d'équations linéaires décrivant les problèmes à conditions initiales et à conditions aux frontières mixtes de coques thermoélastiques, on obtient un théorème d'unicité sans faire appel à une hypothèse de définition pour l'énergie libre. Le théorème est valable pour des coques non-homogènes et anisotropes soumises à de petits mouvements (et petits changements de température) superposés à une grande déformation.

     

Plane waves in linear elastic materials with voids

The behavior of plane harmonic waves in a linear elastic material with voids is analyzed. There are two dilational waves in this theory, one is predominantly the dilational wave of classical linear elasticity and the other is predominantly a wave carrying a change in the void volume fraction. Both waves are found to attenuate in their direction of propagation, to be dispersive and dissipative. At large frequencies the predominantly elastic wave propagates with the classical elastic dilational wave speed, but at low frequencies it propagates at a speed less than the classical speed. It makes a smooth but relatively distinct transition between these wave speeds in a relatively narrow range of frequency, the same range of frequency in which the specific loss has a relatively sharp peak. Dispersion curves and graphs of specific loss are given for four particular, but hypothetical, materials, corresponding to four cases of the solution.     

On the steady vibrations of elastic materials with voids

This paper is concerned with the linear elastodynamics of homogeneous and isotropic materials with voids. First, the singular solutions corresponding to concentrated forces in the case of steady vibrations are established. Then, representations of Somigliana type for the displacement field and the change in the volume fraction field are presented. Radiation conditions of Sommerfeld type are derived. The potentials of single layer and double layer are used to reduce the boundary value problems to singular integral equations for which Fredholm's basic theorems are valid. Existence and uniqueness results for exterior problems are established.     

On uniqueness and reciprocity in linear thermoelasticity of materials with voids

A linear thermoelastic theory of materials with voids is considered. First, we establish a uniqueness theorem with no definiteness assumption on the elasticities and in the absence of restriction that the conductivity tensor is positive definite. Then, we establish a basic relation which leads in a simple manner to the reciprocal theorem and to another uniqueness result. Some applications of the reciprocity relation are presented.     

The viscoelastic behavior of linear elastic materials with voids

It is shown that the constitutive equations for a linear elastic material with voids imply a viscoelastic stress-strain relation known as the standard linear solid in the case of quasi-static, homogeneous deformations in the absence of self-equilibrated body forces. It is noted that, even for deformations that are dynamic and/or inhomogeneous the viscoelastic behavior is still qualitatively similar to that predicted by the standard linear solid model.     

Generalized variational principles for linear elastic materials with voids

In this paper, the idea of variational principles of linear elastic theory is used to establish generalized variational principles for linear elastic materials with voids. The fundamental equations of linear elastic materials with voids used have already been established in Ref. [5].     

Stability of bars made of linear elastic materials with voids

Summary Dynamic stability of an elastic bar with voids is considered. Using the Lyapunov approach some new sufficient stability conditions are obtained and explicit expressions for the critical load are derived.     

A note on the problem of pure bending for linear elastic materials with voids

This note concerns the problem of quasi-static pure bending of a beam in the context of the complete theory of linear elastic materials with voids presented in [1]. It is shown here that the solution in the context of the complete theory of [1] is coincident with the pure bending solution of classical elasticity for small time, and that the solution for large time is the bending solution given in [1], a solution which neglected the rate effect in the complete theory of [1]. In between these two limit solutions the rate effect moderates a monotonic transition.     

Rayleigh surface waves in the theory of thermoelastic materials with voids

In this paper we analyze the behavior of plane harmonic waves and Rayleigh waves in a linear thermoelastic material with voids. We take into account the damped effects of the thermal field upon the propagation waves. Consequently, the propagation condition is established in the form of an algebraic equation of 9th degree whose coefficients are complex numbers while the eigensolutions of the thermoelastodynamic with voids system are explicitly obtained in terms of the characteristic solutions. We show that the transverse waves are undamped in time and they are not influenced by the thermal and porous effects while the longitudinal waves are all damped in time and they are coupled with the thermal and porous effects. The related solution of the Rayleigh surface wave problem is expressed as a linear combination of the eigensolutions in concern. The secular equation is established in an implicit form and afterwards an explicit form is written for an isotropic and homogeneous thermoelastic with voids half-space. Furthermore, we use the numerical methods and computations to solve the secular equation for a specific material.     

On dynamic stress analysis for cracks in elastic materials with voids

The present paper is concerned with the development of a semi-analytical approach to the dynamic problem of the concentration of stresses near the edges of a crack located in a porous elastic space (two-dimensional problem) and subjected to a normal oscillating load applied to the crack faces. Our analysis is made in the context of the Goodman–Cowin–Nunziato (G–C–N) theory for porous media. In previous work we studied static crack problems for such materials; now we introduce an analysis of the relevant dynamic aspects. By using the Fourier transform, the problem is reduced in explicit form to a hyper-singular integral equation with a convolution kernel valid over the crack length. Then, we apply a collocation technique developed in our previous work to solve this equation, and study the stress intensity factor. The principal goal is to compare the stress intensity factor for the static and dynamic cases. We also compare our results with the case of an ordinary linear elastic medium.     

On the theory of elastic shells made from a material with voids

In this paper we present a theory for porous elastic shells using the model of Cosserat surfaces. We employ the Nunziato–Cowin theory of elastic materials with voids and introduce two scalar fields to describe the porosity of the shell: one field characterizes the volume fraction variations along the middle surface, while the other accounts for the changes in volume fraction along the shell thickness. Starting from the basic principles, we first deduce the equations of the nonlinear theory of Cosserat shells with voids. Then, in the context of the linear theory, we prove the uniqueness of solution for the boundary initial value problem. In the case of an isotropic and homogeneous material, we determine the constitutive coefficients for Cosserat shells, by comparison with the results derived from the three-dimensional theory of elastic media with voids. To this aim, we solve two elastostatic problems concerning rectangular plates with voids: the pure bending problem and the extensional deformation under hydrostatic pressure.     

On some theorems in the linear theory of viscoelastic materials with voids

A linear theory of viscoelasticity of materials with voids is considered. Some theorems concerning uniqueness and continuous dependence are established.