Acoustoelastic wave propagation in initially stressed solids using a pure stress formulation

Wave propagation in homogeneous elastic solids is governed by Cauchy's first law of motion, which relates the acceleration of material points to the divergence of stress. For Hookean solids, the stress is typically written in terms of displacements to form an equation of motion with displacement as the dependent variable, i.e., Navier's equation of elastodynamics. Previously, the authors considered wave propagation in anisotropic elastic solids by formulating the equation of motion using only stress as the dependent variable. Eigenvalue solutions provided the (stress) wave phase velocity in terms of elastic compliance constants. In this presentation, the stress formalism is extended and applied to the nonlinear problem of stress wave propagation in an initially stressed material (also known as acoustoelasticity). The stress formulation is viewed as a more natural fit over historical derivations, which were based on wave displacements superimposed on an initially strained material state. The stress formulation of elastodynamics is firstly reviewed. Then, the extension to stress wave propagation in an initially stressed material is given. The phase velocity of the stress wave is related to the initial stress through second- and third-order compliance constants (instead of elastic stiffnesses in traditional acoustoelasticity).