Propagation of a plane wave in an isotropic elastic material subjected to pure homogeneous deformation

Abstract

In the present paper, we apply the theory [1] of the superposition of infinitesimal deformations on finite deformations in an isotropic elastic material to the study of the propagation of a plane wave of small amplitude in an infinite body of the material which is subjected to a static, pure homogeneous deformation. It is seen that the secular equation for the determination of the square of the velocity of propagation in a given direction has three real eigen-values and correspondingly three real mutually perpendicular eigen-directions. Provided these three eigen-values are all positive, travelling waves may be propagated in the body in the direction considered with linear polarisations along each of these eigen-directions. If one or more of the eigen-values is negative for any direction of propagation, the body is inherently unstable in the state of pure homogeneous deformation considered.