Linear Theory Of Elastic Materials Research Articles

Existence and uniqueness of Rayleigh waves in isotropic elastic Cosserat materials and algorithmic aspects

We discuss the propagation of surface waves in an isotropic half space modelled with the linear Cosserat theory of isotropic elastic materials. To this aim we use a method based on the algebraic analysis of the surface impedance matrix and on the algebraic Riccati equation, and which is independent of the common Stroh formalism. Due to this method, a new algorithm which determines the amplitudes and the wave speed in the theory of isotropic elastic Cosserat materials is described. Moreover, this method allows us to prove the existence and uniqueness of a subsonic solution of the secular equation, a problem which remains unsolved in almost all generalized linear theories of elastic materials. Since the results are suitable to be used for numerical implementations, we propose two numerical algorithms which are viable for any elastic material. Explicit numerical calculations are made for aluminium-epoxy in the context of the Cosserat model. Since the novel form of the secular equation for isotropic elastic material has not been explicitly derived elsewhere, we establish it in this paper, too.

A plane strain problem in the theory of elastic materials with voids

This article is concerned with the linear theory of elastic materials with voids. With respect to the classical theory of elasticity, this model is characterized by four independent kinematic variables: the displacement field and the change in volume fraction . First, we present the field equations in the equilibrium theory and derive the equations of the plane strain problem. Then, the problem of a cylindrical rigid inclusion in an infinite body is investigated. The results are obtained in closed form. The solution can be considered as a generalization of the corresponding problem in the classical theory of elasticity. The displacement field and the stresses are expressed by mean of explicit formulas. The maximum tensile stress and the stress concentration factor are calculated.

Basic problems of thermoelasticity with microtemperatures for the half-space

The present paper deals with the three-dimensional linear theory of elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. In the half-space solutions of some basic boundary value problems are constructed in quadratures.

Some basic boundary value problems of the plane thermoelasticity with microtemperatures

The present paper is devoted to the two‐dimensional version of statics of the linear theory of elastic materials with inner structure whose particles, in addition to the classical displacement and temperature fields, possess microtemperatures. Some results of the classical theories of elasticity and thermoelasticity are generalized. The Green's formulas in the case under consideration are obtained, basic boundary value problems are formulated, and uniqueness theorems are proved. The fundamental matrix of solutions for the governing system of the model and the corresponding single and double layer thermoelastopotentials are constructed. Properties of the potentials are studied. Applying the potential method, for the first and second boundary value problems, we construct singular integral equations of the second kind and prove the existence theorems of solutions for the bounded and unbounded domains.This paper describes the use of the LaTeX2ϵ mmaauth.cls class file for setting papers for Mathematical Methods in the Applied Sciences. Copyright © 2012 John Wiley & Sons, Ltd.

Some theorems about spectrum and finite element approach for eigenvalue problems for elastic bodies with voids

The paper concerns eigenvalue problems for elastic bodies with voids in contact with massive rigid plane punches. The linear theory of elastic materials with voids according to the Cowin–Nunziato model is used. A variational principle is constructed which has the properties of minimality, similar to the well-known variational principle for problems with pure elastic media. The discreteness of the spectrum and completeness of the eigenfunctions are proved. As a consequence of variational principles, the properties of an increase or a decrease in the natural frequencies, when the mechanical and “porous” boundary conditions and the modulus of elastic solid with voids change, are established. A finite element method is proposed for numerical solution of eigenvalue problems for elastic media with voids. Some effective block algorithms for finite element eigenvalue problems with partial coupling are described. Numerical experiments are presented for determining the first eigenfrequencies of an axisymmetric elastic body with voids.

Generalized variational principles of the viscoelastic body with voids and their applications

From the Boltzmann’s constitutive law of viscoelastic materials and the linear theory of elastic materials with voids, a constitutive model of generalized force fields for viscoelastic solids with voids was given. By using the variational integral method, the convolution-type functional was given and the corresponding generalized variational principles and potential energy principle of viscoelastic solids with voids were presented. It can be shown that the variational principles correspond to the differential equations and the intitial and boundary conditions of viscoelastic body with voids. As an application, a generalized variational principle of viscoelastic Timoshenko beams with damage was obtained which corresponds to the differential equations of generalized motion and the intial and boundary conditions of beams. The variational principles provide a way for solving problems of viscoelastic solids with voids.

SPATIAL BEHAVIOR IN DYNAMICAL THERMOELASTICITY BACKWARD IN TIME FOR POROUS MEDIA

The aim of this paper is to study the spatial behavior of the solutions to the final-boundary-value problems associated with the linear theory of elastic materials with voids. More precisely, the present study is devoted to porous materials with a memory effect for the intrinsic equilibrated body forces. An appropriate time-weighted volume measure is associated with the backward-in-time thermoelastic processes. Then a first-order partial differential inequality in terms of such measure is established and it is further shown how the inequality implies the spatial exponential decay of the thermoelastic process in question.

Axially symmetric problems for a porous elastic solid

This paper is concerned with the linear theory of elastic materials with voids. The Dirichlet and Neumann problems for a half-space are studied by using the technique of integral transforms. The case of a concentrated body load is investigated in detail.

Spatial and temporal behavior in elastic materials with voids

The aim of this paper is to study the spatial and temporal behavior of the solution to the initial-boundary value problems associated with the linear theory of elastic materials with voids. As regards the spatial behavior, a domain of influence is established in the sense that for eacht ∈ [0,T] the whole activity is zero in the part of the body for whichr≥ct, wherer represents the distance from the support\(\hat D_T \) of the given data on [0,T] andc is a positive constant depending on the elastic properties of the material. Forr≤ct, a spatial decay estimate of Saint-Venant type is established for which the decay rate is independent of time. As regards the temporal behavior, some relations are established describing the asymptotic partition of the Cesaro mean of the total energy. When the dissipation is absent the above relations give an equipartition theorem of the Cesaro means of various energies.

Application of the theory of linear elastic materials with voids to NDT of interfaces of bilaminates

Non-destructive testing (NDT) of interfaces in composite structural elements is mandatory in several industries. This calls for modeling of composite interfaces incorporating real world conditions of imperfect bonding. Theories of materials that modern continuum mechanics offer can be profitably employed to develop these models. For instance, distributed voids are often present in the adhesive zone of a structural element and these voids may lead to a failure of the element due to the imperfect bond. As the theory of linear elastic materials with voids (LEMV) provides a model for solids with properties akin to viscoelastic materials, it is employed in the present work to represent the thin adhesive viscoelastic layer with its minutes pores spread over the interface region of a symmetric bilaminate. The bilaminate is assumed to be insonified in water and the governing leaky Lamb wave (LLW) frequency equation is derived. The dispersion and attenuation of LLWs in an Al/adhesive/Al bilaminate are graphically-presented for various porosity-coupling parameter values and are compared with those in perfectly-bonded Al plates. The results presented in this paper showing the influence of pore-infested adhesive layer on the early leaky Lamb modes promises to be useful in identifying interfacial porosity.

Domain of influence theorem in the theory of elastic materials with voids

A domain of influence theorem in the linear theory of elastic materials with voids, proposed by Cowin and Nunziato, has been established. The theorem shows that for a finite time t > 0, the displacement field u i and the change in volume fraction ϑ generate no disturbance outside a bounded domain Ω t .

Influence of voids in interface zones on Lamb-mode spectra in fiber-reinforced composite laminates

A recently formulated porous interface layer model is employed to study the influence of bond defects particularly due to the presence of voids in interface zones on Lamb wave propagation in a laminate made of two identical fiber-reinforced laminas. The model is developed making use of the mathematical theory of linear elastic materials with voids to describe the interface zone−the adhesive with its minute separation from the adherent laminas. Theoretical Lamb-mode dispersion spectra in a symmetric bilaminate made of boron fiber-reinforced aluminum sheets are obtained and compared with those obtained assuming perfect bond condition. It is hoped that the model will facilitate ultrasonic nondestructive evaluation of bond strengths in composite laminates.

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.

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.

The classical pressure vessel problems for linear elastic materials with voids

The traditional problems of the thick walled spherical and circular cylindrical shells under internal and external pressure are solved in the context of the theory of linear elastic materials with voids. The solutions are quasi-static. The stress distributions are those predicted by isotropic linear elasticity. The displacement and solid volume fraction charge fields exhibit a volumetric viscoelasticity induced by a rate dependence of the volume fraction change.

Linear elastic materials with voids

A linear theory of elastic materials with voids is presented. This theory differs significantly from classical linear elasticity in that the volume fraction corresponding to the void volume is taken as an independent kinematical variable. Following a discussion of the basic equations, boundary-value problems are formulated, and uniqueness and weak stability are established for the mixed problem. Then, several applications of the theory are considered, including the response to homogeneous deformations, pure bending of a beam, and small-amplitude acoustic waves. In each of these applications, the change in void volume induced by the deformation is determined. In the final section of the paper, the relationship between the theory presented and the effective moduli approach for porous materials is discussed. In the two year period between the submission of this manuscript and the receipt of the page proof, there have been some extensions of the results reported here. In the context of the theory described, the classical pressure vessel problems and the problem of the stress distribution around a circular hole in a field have uniaxial tension have been solved [19,22]. The solution given in the present paper for the pure bending of a beam when the rate effect of the theory is absent is extended to case when the rate effect is present in [21]. The various implications of the rate effect in the void volume deformation are pursued all the subsequent works [19,20,21,22].

Continuous dependence theorems in the theory of linear elastic materials with microstructure

It is shown that the solution to the equations of a linear micropolar elastic solid (see Eringen [1]), in an exterior domain in R 3, depends continuously on initial and boundary data, body forces and material coefficients. The linear dipolar elastic theory of Green and Rivlin [2] is also briefly examined.