Axisymmetric problem of a nonhomogeneous elastic layer

Summary The paper deals with a theoretical treatment of elastic behavior for a medium with nonhomogeneous materials property, which is defined by the relation , i.e., shear modulus of elasticity G varies with the dimensionless axial coordinate by the power product form, arbitrarily. Fundamental differential equation for such nonhomogeneous medium has been already proposed in [5]. It is given by a second-order partial differential equation. However, it was found that the fundamental equation is not sufficient in general to solve several kinds of boundary-value problems. On the other hand, it is shown in the present paper making use of the fundamental equations system for a nonhomogeneous medium, which has been proposed in our previous paper [7], it is possible to solve axisymmetric problems for a thick plate (layer) subjected to an arbitrarily distributed load or a concentrated load on its surfaces. Numerical calculations are carried out for several cases, taking into account the variation of the nonhomogeneous parameter m. The numerical results for displacements stress and components are shown in graphical form. Accepted for publication 25 March 1997     

Three-dimensional elastic behavior of a nonhomogeneous layer

Summary  This paper deals with the theoretical treatment of a three-dimensional elastic problem governed by a cylindrical coordinate system (r,θ,z) for a medium with nonhomogeneous material property. This property is defined by the relation G(z)=G ₀(1+z/a) ^m where G ₀,a and m are constants, i.e., shear modulus of elasticity G varies arbitrarily with the axial coordinate z by the power product form. We propose a fundamental equation system for such nonhomogeneous medium by using three kinds of displacement functions and, as an illustrative example, we apply them to an nonhomogeneous thick plate (layer) subjected to an arbitrarily distributed load (not necessarily axisymmetric) on its surfaces. Numerical calculations are carried out for several cases, taking into account the variation of the nonhomogeneous parameter m. The numerical results for displacement and stress components are shown graphically. Received 10 May 1999; accepted for publication 15 August 1999     

Effects of radially varying moduli on stress distribution of nonhomogeneous anisotropic cylindrical bodies

In this study, the stress distribution in a nonhomogeneous anisotropic cylindrical body is investigated. Using equilibrium equations, Hooke’s law and strain–displacement relations, a system of equations is obtained in cylindrical coordinates in terms of stress potentials where elastic properties change in radial direction. Young’s and shear moduli are expressed as power functions of r and Poisson’s ratios are kept constant. Closed-form solutions for stress potentials and stress distribution are obtained for an axisymmetric, orthotropic cylinder. Results are checked with FE results. A pressurized thick walled cylinder example is studied in details. Stresses in radial, tangential and axial directions and Von Mises stresses are plotted for different powers of r.     

Epstein transition zones in nonhomogeneous isotropic elastic media

A simple transformation in the independent variable makes it possible to obtain power series solutions of the stress equations of motion of elasticity for inhomogeneous elastic media whose refractive indices are represented by the Epstein profiles. Graphs of reflection and transmission coefficients versus angles of incidence are presented for different frequencies in the case of incident P as well as incident SV waves for some profiles.     

Finite element solutions for nonhomogeneous nonlocal elastic problems

A finite element procedure for analysing nonhomogeneous nonlocal elastic 2D problems is presented and discussed. The procedure grounds on a variationally consistent approach known, in the relevant literature, as Nonlocal Finite Element Method. The latter is recast making use of a recently theorized phenomenological strain-difference-based nonhomogeneous nonlocal elastic model. The peculiarities of the numerical procedure together with the pertinent nonlocal operators are expounded and discussed. Two simple numerical 2D examples close the paper.     

Coupled analysis of elastic plate-infinite nonhomogeneous layered medium

The static interaction problem between an elastic plate and infinite medium is studied by the semi-numerical (in one direction) and semi-analytical (in two directions) method-semianalytical element method, a simple practical and effective method for coupled analysis between above-ground structure and infinite soil medium with complex material character in engineering.     

Differential nonhomogeneous models for elastic randomly cracked solids

A differential scheme is developed to approximate the elastic behaviour of randomly cracked solids, accounting for possible locally nonhomogeneous distribution of the cracks. Certain inclusion or matrix spaces within the solids are modelled as forbidden regions for the cracks. At small to intermediate values of the crack density and proportion of forbidden regions, the effective elastic moduli of the models do not differ much from each other, but the differences become profound at higher values of those parameters: the effective moduli can be very small and large (toward those of the uncracked solids) depending upon the particular nonhomogeneous arrangements of the cracks.     

The problem of gas injection into an infinite nonhomogeneous aquifer

This paper considers the problem of water displacement by gas in an infinite aquifer which has nonuniform permeability.     

The stokes problem for a porous particle with radially nonuniform porosity

The flow past a nonuniform porous spherical particle immersed in a uniform steady-state stream is studied in the Stokes approximation. For a power-law radial dependence of the particle permeability coefficient, an analytical solution for the velocity and pressure fields outside and inside the particle is obtained. Volgograd, Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 179–184, January–February, 2000.     

Convergent shock wave in an ideal elastic nonhomogeneous medium

The boundary-value problem for symmetric focusing of a shock wave in a medium with variable density under a constant load (model of a porous body with variable initial velocity) is solved. The solution asymptotic is studied. Focusing in a homogeneous medium has been previously studied [1], One inverse problem related to the choice of the optimal pressure conditions is examined. Constraints on the applicability of the model are touched on.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 107–111, January–February, 1976.     

The remarkable nature of radially symmetric equilibrium states of aeolotropic nonlinearly elastic bodies

This paper treats the radially symmetric equilibrium states of aeolotropic nonlinearly elastic solid cylinders and balls under constant normal forces on their boundaries. It is shown that the aeolotropy gives rise to solutions describing both intact and cavitating states, which exhibit an array of remarkable new phenomena, not suggested by the solutions for isotropic bodies. E.g., it is shown that there are materials having a critical pressure such that for applied pressures on the boundary below the critical value, the normal pressures at the center of the body are zero and for applied pressures above the critical value, the normal pressures at the center are infinite. There are also materials for which there is no equilibrium state with center intact when the boundary is subjected to uniform tension. It is also shown that the equilibrium states treated here are the only radially symmetric equilibrium states. Thus the strange phenomena discovered here must be present in such stable equilibrium states.     

An elasticity solution of a nonhomogeneous half-plane problem

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