Simulations of focused shear shock waves in soft solids and the brain

Abstract

Because of a very small speed, shear waves in soft solids are extremely nonlinear, with nonlinearities four orders of magnitude larger than in classical solids. Consequently, these nonlinear shear waves can transition from a smooth to a shock profile in less than one wavelength. We hypothesize that traumatic brain injuries (TBI) could be caused by the sharp gradients resulting from shear shock waves. However, shear shock waves are not currently modeled by simulations of TBI. The objective of this paper is to describe shear shock wave propagation in soft solids within the brain, with source geometry determined by the skull. A 2D nonlinear paraxial equation with cubic nonlinearities is used as a starting point. We present a numerical scheme based on a second order operator splitting which allows the application of optimized numerical methods for each terms. We then validate the scheme with Guiraud's nonlinear self-similarity law applied to cusped caustics. Once validated, the numerical scheme is then applied to a blast wave problem. A CT measurement of the human skull is used to determine the initial conditions and shear shock wave simulations are presented to demonstrate the focusing effects of the skull geometry.