Elasticity Solution for a Radially Nonhomogeneous Hollow Circular Cylinder

An exact elasticity solution is developed for a radially nonhomogeneous hollow circular cylinder of exponential Young’s modulus and constant Poisson’s ratio. In the solution, the cylinder is first approximated by a piecewise homogeneous one, of the same overall dimension and composed of perfectly bonded constituent homogeneous hollow circular cylinders. For each of the constituent cylinders, the solution can be obtained from the theory of homogeneous elasticity in terms of several constants. In the limit case when the number of the constituent cylinders becomes unboundedly large and their thickness tends to infinitesimally small, the piecewise homogeneous hollow circular cylinder reverts to the original nonhomogeneous one, and the constants contained in the solutions for the constituent cylinders turn into continuous functions. These functions, governed by some systems of first-order ordinary differential equations with variable coefficients, stand for the exact elasticity solution of the nonhomogeneous cylinder. Rigorous and explicit solutions are worked out for the ordinary differential equation systems, and used to generate a number of numerical results. It is indicated in the discussion that the developed method can also be applied to hollow circular cylinders with arbitrary, continuous radial nonhomogeneity.

Surface instability due to initial compressive stress

Utilizing a linearized theory of isotropic, homogeneous elasticity that includes the effects of initial stress, it is demonstrated that Rayleigh and Love surface instabilities occur when the compressive stress reaches appropriate critical values. The Rayleigh surface instability exists for a half-space, as well as for a finite slab, and the critical stress value depends only on Poisson's ratio. The Love surface instability exists for a layer and a substratum, and the critical stress value depends on two parameters: the ratio of the shear moduli and a nondimensional number related to the geometry of the layer. It is suggested that these two instabilities may offer possible, although highly idealized, mechanisms for earthquake initiation and prehistoric land-mass formations such as mountain chains.