Brij Mohan Singh

On the theory of elssticity of a nonhomogeneous medium

In this paper the equation of equilibrium for a nonhomogeneous isotropic elastic solid under shear has been solved in rectangular Cartesian coordinates as well as in cylindrical polar coordinates. The modulus of rigidity of the material is assumed to vary in lateral as well as vertical directions. As an example, the above solution has been used to solve the problem of a Griffith crack in an infinite solid under shear.

The axisymmetric boussinesq problem of an initially stressed neo-hookean half-space for a punch of arbitrary profile☆

Abstract A solution of the axisymmetric Boussinesq problem for an initially stressed neo-Hookean half-space is obtained in the closed form, from which are deduced simple formulae for the depth of penetration of the tip of a rigid punch of arbitrary profile and the total load which must be applied to the punch to achieve this penetration. Simple closed form expressions are also deduced for the shape of the deformed surface outside the punch and the distribution of pressure under the punch. The corresponding results are obtained for cylindrical, conical, paraboloidal, ellipsoidal and spherical punch shapes.

Torsion by a circular die of a nonhomogeneous elastic layer bonded to a nonhomogeneous half-space

Abstract The title problem is equivalent to a torsional mixed boundary value problem in the theory of elasticity in which the only non-zero displacement component is specified inside the circular area r ⩽ l and the shearing stress σθz, is zero in the area r > 1 on the face z = 0; the continuity of the displacement and the shearing stress σθz is assumed at the interface z = h between the layer and the half-space having different modulii of rigidity. The modulii of rigidity are assumed to be continuous functions of 2. The problem is reduced to the solution of a Fredholm integral equation of the second kind whose iterative solution has been obtained valid for values of h >l, for a few particular cases. Expression for the torque T required to rotate the die through an angle ω has been obtained for the general as well as particular cases.

Penny-shaped crack in an infinite elastic cylinder bonded to an infinite elastic material surrounding it☆

Abstract This paper concerns with the state of stress in a long elastic cylinder, with a concentric penny-shaped crack, bonded to an infinite elastic medium. The crack is assumed to be opened by an internal pressure and that the plane of the crack is perpendicular to the axis of the cylinder. The elastic constants of the cylinder and the semi-infinite medium are assumed to be different. The problem is reduced to the solution of a Fredholm integral equation of the second kind. Closed form expressions are obtained for the stress-intensity factor and the crack energy. The integral equation is solved numerically and results are used to obtain the numerical values of the stress-intensity factor and the crack energy which are graphed.

MIXED BOUNDARY-VALUE PROBLEMS OF STEADY-STATE THERMOELASTICITY AND ELECTROSTATICS

We consider the following two mixed boundary-value problems: (1) The steady-state plane-strain thermoelastic problem of an elastic layer with one face stressfree and the other face resting on a rigid frictionless foundation; the free surface of the layer is subjected to arbitrary temperature on the part a < x < b, whereas the rest of the surface is insulated and the surface in contact with the foundation is insulated. (2) The two-dimensional electrostatic problem of the electrostatic potential due to two coplanar strips that are charged to equal and opposite potentials and that are parallel to and equidistant from a grounded strip. By the use of Fourier transforms, both problems are reduced to the solution of triple trigonometric integral equations. The closed-form solution of these triple-integral equations is obtained by using the finite Hilbert-transform technique. Closed-form expressions are obtained for the physical quantities in both problems.

Penny-shaped crack in a sphere embedded in an infinite medium☆

Abstract We consider the problem of determining the stress intensity factor and the crack energy in an Isotropie, homogeneous elastic sphere embedded in an infinite Isotropie, homogeneous elastic medium when there is a diametrical crack in the sphere. We assume that the crack is opened by an internal pressure and the sphere is bonded to the surrounding material. The problem is reduced to the solution of a Fredholm integral equation of the second kind in the auxiliary function φ( t ). Expressions for the stress intensity factor and the crack energy are obtained in terms of φ( t ). The integral equation is solved numerically and the numerical values of the stress intensity factor and the crack energy are graphed.

Three coplanar Griffith cracks in an infinite elastic strip

We consider the problem of determining the stress intensity factors and crack energy in an infinitely long isotropic, homogeneous elastic strip containing three coplanar Griffith cracks. The three coplanar Griffith cracks are situated symmetrically on a line perpendicular to the edges of the strip. We assume that the cracks are opened by an internal pressure and the edges of the strip are fixed. By using the theory of Fourier series we reduce the problem to solving a set of quadruple trigonometrical series equations with a cosine kernel. Closed form solution is obtained for the quadruple series equations. Closed form analytical expressions are derived for the stress intensity factors, the shape of the deformed cracks and the crack energy. Solutions to some particular problems are derived as limiting cases.

Torsion of an elastic layer by two circular dies

Abstract Two circular discs of rigid material and of different radii are bonded to the opposite faces of an infinite elastic layer. The torques T 1 and T 2 have been calculated when the discs are rotated through different angles.

Two coplanar Griffith cracks in an infinitely long elastic strip

We consider the problem of determining the stress intensity factors and the crack energy in an infinitely long isotropic, homogeneous elastic strip containing two coplanar Griffith cracks. We assume that the cracks are opened by an internal pressure and the edges of the strip are rigidly fixed. 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. Exact 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.

Dual series equations involving generalized laguerre polynomials

An exact solution is obtained for the dual series equations involving generalized Laguerre polynomials.