# Shear shock evolution in incompressible soft solids

## TL;DR

This paper investigates how shear waves evolve into shocks in incompressible soft solids, deriving formulas and phase maps to predict shock formation and applying findings to brain tissue to inform injury prevention.

## Contribution

It provides closed-form expressions for shock formation distance and non-dimensional phase maps, revealing the influence of loading and material parameters on shear shock evolution.

## Key findings

- Shock formation is sensitive to amplitude, shape, and ramp time of loading.
- Smaller brains are less susceptible to shear shocks under realistic loadings.
- Non-dimensional maps can guide protective structure design to prevent shocks.

## Abstract

Nonlinear evolution of shear waves into shocks in incompressible elastic materials is investigated using the framework of large deformation elastodynamics, for a family of loadings and commonly used hyperelastic material models. Closed form expressions for the shock formation distance are derived and used to construct non-dimensional phase maps that determine regimes in which a shock can be realized. These maps reveal the sensitivity of shock evolution to the amplitude, shape, and ramp time of the loading, and to the elastic material parameters. In light of a recent study (Espindola et al., 2017), which hypothesizes that shear shock formation could play a signicant role in Traumatic Brain Injury (TBI), application to brain tissue is considered and it is shown that the size matters in TBI research. Namely, for realistic loadings, smaller brains are less susceptible to formation of shear shocks. Furthermore, given the observed sensitivity to the imparted waveform and the constitutive properties, it is suggested that the non-dimensional maps can guide the design of protective structures by determining the combination of loading parameters, material dimensions, and elastic properties that can avoid shock formation.