B.M. Singh

Axisymmetric contact and crack problems for an initially stressed neo-Hookean elastic layer

Abstract A solution of the contact problem for an initially stressed neo-Hookean infinite layer is obtained. The layer is assumed to be bonded to a rigid foundation. The punch is taken to be axisymmetric and in particular, the conical and cylindrical shapes of the punch are considered in detail. The expression for the total load applied to the punch to maintain a given displacement is obtained. We have also considered the penny-shaped crack in the layer under an initial stress. The crack is taken to lie in the central plane of the layer. The surfaces of the layer are considered to be either stress free or fixed. Expressions for the stress intensity factors are obtained. Numerical values of the physical quantities are exhibited graphically.

A problem of Reissner-Sagoci type for an elastic cylinder embedded in an elastic half-space☆

Abstract The problem considered is that of the torsion of an elastic cylinder which is embedded in an elastic half-space of different rigidity modulus. It is assumed that there is perfect bonding at the common cylindrical surface and also that the torque is applied to the cylinder through a rigid disk bonded to its flat surface. The problem is reduced, by means of the use of integral transforms and the theory of dual integral equations to that of solving a Fredholm integral equation of the second kind. The results obtained by solving this equation are exhibited graphically in Fig. 2.

Scattering of anti-plane shear waves by an interface crack between two bonded dissimilar functionally graded piezoelectric materials

In the present paper, the dynamic behaviour of a Griffith crack situated at the interface of two bonded dissimilar functionally graded piezoelectric materials (FGPMs) is considered. It is assumed that the elastic stiffness, piezoelectric constant, dielectric permittivity and mass density of the FGPMs vary continuously as an exponential function of the x and y coordinates, and that the FGPMs are under anti-plane mechanical loading and in-plane electrical loading. By using an integral transform technique the problem is reduced to four pairs of dual integral equations, which are transformed into four simultaneous Fredholm integral equations with four unknown functions. By solving the four simultaneous Fredholm integral equations numerically the effects of the material properties on the stress and electric displacement intensity factors are calculated and displayed graphically.

Diffraction of SH waves by a moving crack

SummaryThe problem of diffraction of anti-plane shear waves by a running crack of finite length is investigated analytically. Fourier transform method is used to solve the mixed boundary value problem which reduces to two pairs of dual integral equations. These dual integral equations are further reduced to a pair of Fredholm integral equations of the second kind. The iterative solution of the integral equations has been obtained for small wave number. The solution is used to calculate the dynamic stress intensity factor at the edge of the crack.

Diffraction of torsional wave or plane harmonic compressional wave by an annular rigid disc

Abstract In this paper we have considered the following two problems. Firstly the diffraction of normally incident SH waves by a rigid annular disc situated at the interface of two elastic half-spaces is considered. The solution of the problem is reduced into the solution of triple integral equations involving Bessel functions. The solution of the triple integral equations is reduced into Fredholm integral equations of th second kind. By finding the solution of the Fredholm integral equation, the numerical values for the moment required to produce the rotation of disc are obtained. Secondly, the problem of diffraction of plane harmonic compressional wave by an annular circular disc embedded in an infinite elastic space is considered. The annular disc is assumed to be perfectly welded with the infinite solid. The solution of the problem is reduced into the solution of Fredholm integral equation of the second kind. The Fredholm integral equation is solved numerically and the numerical values for the couple applied on the disc are obtained.

Axisymmetric contact problem for an elastic layer on a rigid foundation with a cylindrical hole

Abstract A solution of an axisymmetric contact problem for an isotropie homogeneous, elastic layer bonded to the rigid foundation with a cylindrical hole, is obtained. The effect of the weight of the layer is also taken into account. The problem is reduced to the solution of two simultaneous Fredholm integral equations of the second kind whose iterative solution is obtained for h > max (a, b) where h is the layer thickness and a and b are respectively the radius of contact of the axisymmetrie punch and the radius of the cylindrical hole. Simple expressions are obtained for the total pressure on the punch, the distribution of normal pressure under the punch, the shape of the deformed surface outside the punch and the normal displacement in the cylindrical hole.

Closed-form solution for piezoelectric layer with two collinear cracks parallel to the boundaries

We consider the problem of determining the stress distribution in an infinitely long piezoelectric layer of finite width, with two collinear cracks of equal length and parallel to the layer boundaries. Within the framework of reigning piezoelectric theory under mode III, the cracked piezoelectric layer subjected to combined electromechanical loading is analyzed. The faces of the layers are subjected to electromechanical loading. The collinear cracks are located at the middle plane of the layer parallel to its face. By the use of Fourier transforms we reduce the problem to solving a set of triple integral equations with cosine kernel and a weight function. The triple integral equations are solved exactly. Closed form analytical expressions for stress intensity factors, electric displacement intensity factors, and shape of crack and energy release rate are derived. As the limiting case, the solution of the problem with one crack in the layer is derived. Some numerical results for the physical quantities are obtained and displayed graphically.

Two coplanar Griffith cracks under shear loading in an infinitely long elastic layer

Abstract We consider the problem of determining the stress distribution in an infinitely long isotropic homogeneous elastic layer containing two coplanar Griffith cracks which are opened by internal shear stress acting along the lengths of the cracks. The faces of the layer are rigidly fixed. The cracks are located in the middle plane of the layer parallel to its faces. By the use of Fourier transforms we reduce the problem to solving a set of triple integral equations with a cosine kernel and a weight function. The triple integral equations are solved exactly. Closed form analytical expressions are derived for the stress intensity factors, shape of the deformed crack, and the crack energy. Solutions to some particular problems are derived as limiting cases. Numerical results are presented in the form of graphs.

Thermal stresses near a penny-shaped crack in an elastic sphere embedded in an infinite elastic space

Abstract This paper gives an analysis of the distribution of thermal stresses in a sphere which is bonded to an infinite elastic medium. The thermal and the elastic properties of the sphere and the elastic infinite medium are assumed to be different. The penny-shaped crack lies on the diametral plane of the sphere and the centre of the crack is the centre of the sphere. By making a suitable representation of the temperature function, the heat conduction problem is reduced to the solution of a Fredholm integral equation of the second kind. Using suitable solution of the thermoelastic displacement differential equation, the problem is then reduced to the solution of a Fredholm integral equation, in which the solution of the earlier integral equation arising from heat conduction problem occurs as a known function. Numerical solutions of these two Fredholm integral equations are obtained. These solutions are used to evaluate numerical values for the stress intensity factors. These values are displayed graphically.

The study of dynamic behavior of functionally graded piezoelectric materials and an application to a contact problem

In the present paper, the dynamic behavior of functionally graded piezoelectric materials is investigated when it is under anti-plane mechanical loading and in-plane electrical loading. It is assumed that the shear modulus, the piezoelectric modulus, the dielectric modulus and mass density of FGPM vary continuously as functions of X and Y. By using Fourier transforms the solution of equilibrium equations is obtained in closed form. The expressions for displacement and electrical potential are obtained in terms of one unknown function. Finally the results are applied to obtain a solution of the moving contact problem on the surface of the functionally graded piezoelectric material (FGPM).