The effect of rotation and initial stress on the propagation of waves in a transversely isotropic elastic solid

In this paper the equations governing small amplitude motions in a rotating transversely isotropic initially stressed elastic solid are derived, both for compressible and incompressible linearly elastic materials. The equations are first applied to study the effects of initial stress and rotation on the speed of homogeneous plane waves propagating in a configuration with uniform initial stress. The general forms of the constitutive law, stresses and the elasticity tensor are derived within the finite deformation context and then summarized for the considered transversely isotropic material with initial stress in terms of invariants, following which they are specialized for linear elastic response and, for an incompressible material, to the case of plane strain, which involves considerable simplification. The equations for two-dimensional motions in the considered plane are then applied to the study of Rayleigh waves in a rotating half-space with the initial stress parallel to its boundary and the preferred direction of transverse isotropy either parallel to or normal to the boundary within the sagittal plane. The secular equation governing the wave speed is then derived for a general strain–energy function in the plane strain specialization, which involves only two material parameters. The results are illustrated graphically, first by showing how the wave speed depends on the material parameters and the rotation without specifying the constitutive law and, second, for a simple material model to highlight the effects of the rotation and initial stress on the surface wave speed.     

Benchmark trigonometric and 3-D elasticity solutions for an exponentially graded thick rectangular plate

The bending problem of a transverse load acting on an isotropic inhomogeneous rectangular plate using both two-dimensional (2-D) trigonometric and three-dimensional (3-D) elasticity solutions is considered. In the present 2-D solution, trigonometric terms are used for the displacements in addition to the initial terms of a power series through the thickness. The effects due to transverse shear and normal deformations are both included. The form of the assumed 2-D displacements is simplified by enforcing traction-free boundary conditions at the faces of the plate. No transverse shear correction factors are needed because a correct representation of the transverse shearing strain is given. The plate material is exponentially graded, meaning that Lamé’s coefficients vary exponentially in a given fixed direction (the thickness direction). A wide variety of results for the displacements and stresses of an exponentially graded rectangular plate are presented. The validity of the present 2-D trigonometric solution is demonstrated by comparison with the 3-D elasticity solution. The influence of aspect ratio, side-to-thickness ratio and the exponentially graded parameter on the bending response are investigated.     

Spatial contact problems for a prestressed incompressible elastic layer

Two problems are considered on frictionless indentation of a stamp into the upper face of a layer with a homogeneous field of initial stresses present in the layer. The model of an isotropic incompressible nonlinearly-elastic material determined by the Mooney potential is used. The following two cases are studied: the lower face of the prestressed layer is rigidly fixed, and the lower face of a prestressed layer is supported by a rigid foundation without friction. It is assumed that the additional stresses due to the action of the stamp on the layer are small as compared with the initial stresses. This assumption makes it possible to linearize the problems of determining the additional stresses. In what follows, the problems are reduced to solving two-dimensional integral equations (IE) of the first kind with symmetric irregular kernels with respect to the pressure in the contact region. As an example, the case of an elliptic (in plan) stamp acting on a layer is considered. The spatial contact problem for a prestressed elastic half-space was first considered in [1].     

A Mode- II Griffith Crack in Decagonal Quasicrystals

By using the method of stress functions, the problem of mode-II Griffith crack in decagonal quasicrystals was solved. First, the crack problem of two-dimensional quasicrystals was decomposed into a plane strain state problem superposed on anti-plane state problem and secondly, by introducing stress functions, the18 basic elasticity equations on coupling phonon-phason field of decagonal quasicrystals were reduced to a single higher-order partial differential equations. The solution of this equation under mixed boundary conditions of mode-II Griffith crack was obtained in terms of Fourier transform and dual integral equations methods. All components of stresses and displacements can be expressed by elemental functions and the stress intensity factor and the strain energy release rate were determined. Biography: GUO Yu-cui (1962-), Associate professor, Doctor     

Green's function for incremental nonlinear elasticity: shear bands and boundary integral formulation

An elastic, incompressible, infinite body is considered subject to plane and homogeneous deformation. At a certain value of the loading, when the material is still in the elliptic range, an incremental concentrated line load is considered acting at an arbitrary location in the body and extending orthogonally to the plane of deformation. This plane strain problem is solved, so that a Green's function for incremental, nonlinear elastic deformation is obtained. This is used in two different ways: to quantify the decay rate of self-equilibrated loads in a homogeneously stretched elastic solid; and to give a boundary element formulation for incremental deformations superimposed upon a given homogeneous strain. The former result provides a perturbative approach to shear bands, which are shown to develop in the elliptic range, induced by self-equilibrated perturbations. The latter result lays the foundations for a rigorous approach to boundary element techniques in finite strain elasticity.     

Stress analysis for a layer of finite length bonded to a half-space of identical material

The problem in the plane theory of elasticity of an elastic layer bonded to an elastic half-space of the same material is considered. The formulation is achieved by means of integral transforms and the problem is reduced to the solution of a system of singular integral equations. A numerical solution is accomplished for the half-space and layer by use of the collocation scheme developed by Erdogan and Gupta.     

On the stability of a plate under longitudinal compression on a two-layer half-space with lower layer prestressed by gravity forces

In the plane (plane strain) and axially symmetric statements, we study the problem of stability, under the action of longitudinal compressing forces, of an infinite elastic plate in two-sided contact with an elastic half-space. The upper layer of finite depth is described by the usual equations of linear theory of elasticity; the lower layer, which is geometrically nonlinear, incompressible, and infinite in depth, is prestressed by gravity forces. The total adhesion between the layer of finite depth and the lower half-space is realized. It is also assumed that the same adhesion takes place between the upper layer of the half-space and the plate with the contact tangential stresses taken into account.The results can be used to calculate the working capacity of coated bodies and layered composites and in problems of geophysics.The problem of stability of an infinite elastic plate under longitudinal compression under conditions of two-sided contact with an elastic base was studied earlier in the monograph [1] (Fuss-Winkler base) and in [2–4].     

Dynamics of a prestressed incompressible layered half-space under moving load

The paper addresses a plane problem: a concentrated force acts on a plate resting on an elastic half-space with homogeneous prestrain. The equations of motion of the plate incorporate shear and rotary inertia. The half-space is assumed to be incompressible and isotropic in the natural state. The elastic potential is given in general form and is only specified for numerical purposes. The dependence of the critical velocity of the load and the stress-strain state on the prestresses is analyzed for different ratios between the stiffnesses of the layer and half-space and different contact conditions. The calculations are carried out for a half-space with Bartenev-Khazanovich potential __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 36–54, March 2008.     

Stress-intensity factors for 3-D dynamic loading of a cracked half-space

A half-space containing a surface-breaking crack of uniform depth is subjected to three-dimensional dynamic loading. The elastodynamic stress-analysis problem has been decomposed into two problems, which are symmetric and antisymmetric, respectively, relative to the plane of the crack. The formulation of each problem has been reduced to a system of singular integral equations of the first kind. The symmetric problem is governed by a single integral equation for the opening-mode dislocation density. A pair of coupled integral equations for the two sliding-mode dislocation densities govern the antisymmetric problem. The systems of integral equations are solved numerically. The stress-intensity factors are obtained directly from the dislocation densities. The formulation is valid for arbitrary 3-D loading of the half-space. As an example, an applied stress field corresponding to an incident Rayleigh surface wave has been considered. The dependence of the stress-intensity factors on the frequency, and on the angle of incidence, is displayed in a set of figures.     

A state space formalism for anisotropic elasticity. Part I: Rectilinear anisotropy

A state space formalism for anisotropic elasticity including the thermal effect is developed. A salient feature of the formalism is that it bridges the compliance-based and stiffness-based formalisms in a natural way. The displacement and stress components and the thermoelastic constants of a general anisotropic elastic material appear explicitly in the formulation, yet it is simple and clear. This is achieved by using the matrix notation to express the basic equations and grouping the stress in such a way that it enables us to cast neatly the three-dimensional equations of anisotropic elasticity into a compact state equation and an output equation. The homogeneous solution to the state equation for the generalized plane problem leads naturally to the eigen relation and the sextic equation of Stroh. Extension, twisting, bending, temperature change and body forces are accounted for through the particular solution. Based on the formalism the general solution for generalized plane strain and generalized torsion of an anisotropic elastic body are determined in an elegant manner.     

The effect of initial stresses on harmonic stress fields within the stratified half plane

The effect of initial stresses on dynamic (harmonic) stress fields within an elastic stratified half plane is investigated. It is assumed that the point-located harmonic force acting on the free plane of the layer by which the half plane is stratified causes this stress field. By employing displacement potentials and the exponential Fourier transform the governing system of partial differential equations of motion is solved. The necessary inverse transformations including rigorous mathematical complexity is performed numerically. The analysis of the numerical results, which shows the influence of the homogeneous initial stresses on the distribution of the stresses on the inter-medium plane, is made. These analyses are examined for various problem parameters and it is assumed that the material of both the layer and the half plane is homogeneous, isotropic, compressible and linearly elastic. It has been observed that the initial stresses may change significantly the values of the superimposed harmonic stresses.     

On the theory of reflection from a wire grid parallel to an interface between homogeneous media

Summary The reflection from a wire grid parallel to a plane interface is considered. The respective media are homogeneous and either or both can be dissipative. The grid is composed of thin equi-spaced wires of finite conductivity. The plane wave solution for arbitrary incidence is then generalized for cylindricalwave excitation. The energy absorbed from a magnetic line source by a grid situated on the surface of a dissipative half-space is treated in some detail. This latter problem is a two-dimensional analogy of a vertical antenna with a radial wire ground system.     

Bonded contact of a flexible elliptical disk with a transversely isotropic half-space

The article concerns the problem of bonded contact of a thin, flexible elliptical disk with a transversely isotropic half-space. Three different cases of loading have been considered: (a) the disk is loaded by a transverse force, whose line of action passes through the center of the disk and lies in the plane of the disk; (b) the disk is subjected to a rotation by a torque, whose axis is perpendicular to the surface of the half-space; (c) the half-space with the bonded disk is under uniform stress field at infinity. The problem corresponding to all three cases is reduced, in a unified manner, to a set of coupled two-dimensional integral equations. Closed-form solutions for these equations have been obtained by using Galins theorem.     

An antisymmetric problem of a penny-shaped crack in a piezoelectric medium

Summary  An exact, three-dimensional analysis is developed for a penny-shaped crack in an infinite transversely isotropic piezoelectric medium. The crack is assumed to be parallel to the plane of isotropy, with its faces subjected to a couple of concentrated normal forces and a couple of point electric charges that are antisymmetric with respect to the crack plane. The fundamental solution of a concentrated force and a point charge acting on the surface of a piezoelectric half-space is employed to derive the integral equations for the general boundary value problem. For the above antisymmetric crack problem, complete expressions for the elastoelectric field are obtained. A numerical calculation is finally performed to show the piezoelectric effect in piezoelectric materials. It is noted here that the present analysis is an extension of Fabrikant's theory for elasticity. Received 30 August 1999; accepted for publication 1 March 2000     

半无限大体广义电磁热弹耦合的二维问题

基于Lord和Shulman广义热弹性理论,研究了热、电可导的半无限大体电磁热弹耦合的二维问题。半无限大体受热和外加恒定磁场的作用,文中建立了电磁热弹性耦合的控制方程,利用正则模态法求解得到了所考虑物理量的解析解,并用图形反映了各物理量的分布规律,从分布图上可以看出,介质中出现了电磁热弹耦合效应,各物理量的非零值仅在一个有限的区域内。     

Crack bifurcation predicted for dynamic anti-plane collinear cracks in piezoelectric materials using a non-local theory

A non-local theory of elasticity is applied to obtain the dynamic interaction between two collinear cracks in the piezoelectric materials plane under anti-plane shear waves for the permeable crack surface boundary conditions. Unlike the classical elasticity solution, a lattice parameter enters into the problem that make the stresses and the electric displacements finite at the crack tip. A one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and electric displacement near the crack tips. By means of the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations in which the unknown variable is the jump of the displacement across the crack surface. The solutions are obtained by means of the Schmidt method. Crack bifurcation is predicted using the strain energy density criterion. Minimum values of the strain energy density functions are assumed to coincide with the possible locations of fracture initiation. Bifurcation angles of ±5° and ±175° are found. The result of possible crack bifurcation was not expected before hand.     

A receding contact plane problem between a functionally graded layer and a homogeneous substrate

In this paper, we consider the plane problem of a frictionless receding contact between an elastic functionally graded layer and a homogeneous half-space, when the two bodies are pressed together. The graded layer is modeled as a nonhomogeneous medium with an isotropic stress–strain law and over a certain segment of its top surface is subjected to normal tractions while the rest of this surface is free of tractions. Since the contact between the two bodies is assumed to be frictionless, then only compressive normal tractions can be transmitted in the contact area. Using integral transforms, the plane elasticity equations are converted analytically into a singular integral equation in which the unknowns are the contact pressure and the receding contact half-length. The global equilibrium condition of the layer is supplemented to solve the problem. The singular integral equation is solved numerically using Chebychev polynomials and an iterative scheme is employed to obtain the correct receding contact half-length that satisfies the global equilibrium condition. The main objective of the paper is to study the effect of the material nonhomogeneity parameter and the thickness of the graded layer on the contact pressure and on the length of the receding contact.     

Effect of rotation on plane waves in generalized thermo-microstretch elastic solid with a relaxation time

The present paper is aimed at studying the effect of rotation on the general model of the equations of generalized thermo-microstretch for a homogeneous isotropic elastic half-space solid whose surface is subjected to a Mode-I Crack problem considered. The problem is in the context of the generalized thermoelasticity Lord-?hulman??s (L-S) theory with one relaxation time, as well as the classical dynamical coupled theory (CD) The normal mode analysis is used to obtain the exact expressions for the displacement components, force stresses, temperature, couple stresses and microstress distribution. The variations of the considered variables through the horizontal distance are illustrated graphically. Comparisons are made with the results in the presence and absence of rotation and in the presence and absence of microstretch constants between the two theories.     

A general solution to some plane problems of micropolar elasticity

We obtain a general solution to the field equations of plane micropolar elasticity for materials characterized by a hexagonal or equilateral triangular structure. These materials exhibit 3-fold symmetry in the plane and the elastic response is isotropic. Utilizing two displacement potential functions, the solution is obtained in terms of two analytic functions and a third function satisfying the modified homogeneous Helmholtz equation. Expressions for the two-dimensional components of displacement, stress, and couple stress, along with the resultant force on a contour, are presented. We observe that micropolar effects are most significant in material regions subjected to large deformation gradients. Specific results are presented for the classical crack problem, the half plane loaded uniformly on the surface, Flamant's problem, and the circular cylinder compressed by equal and opposite concentrated forces.