Second-Order Elastic Deformation of Solids

Expressions for the velocities of elastic waves in stressed solids are derived using Murnaghan's theory of finite deformations and third-order terms in the energy. For isotropic materials, in addition to the Lam\'e constants $\ensuremath{\lambda}$ and $\ensuremath{\mu}$, three additional constants, $l$, $m$, and $n$, are required to describe the material.By measuring the transmission time of elastic pulses through the material, the velocities of longitudinal and shear waves are determined as a function of applied stress. By subjecting the material to hydrostatic pressure as well as simple compression, it is found that seven functions of the three constants $l$, $m$, and $n$ can be measured and thus numerical values calculated. Results are given for polystyrene, iron, and Pyrex glass.