# Nonlinear elastodynamics

During my PhD studies, I also worked on the equations governing the motion of nonlinear elastic solids in one space dimension. The results are presented in a Wave Motion article (2017) link, where I fully detailed the derivation of the analytical solution to the Riemann problem

\begin{aligned} \partial_t \varepsilon &= \partial_x v\\ \rho_0 \partial_t v &= \partial_x \sigma(\varepsilon) \end{aligned} \qquad\text{with}\qquad (\varepsilon, v)|_{t=0} = \begin{cases} (\varepsilon_L, v_L), & x<0\\ (\varepsilon_R, v_R), & x>0 \end{cases} \notag

for various constitutive laws. The solution includes shock waves, rarefaction waves and compound waves, leading to an algorithm to solve the problem (development of a Matlab toolbox). The mathematical theory behind this kind of system of partial differential equations goes back to the 1970s.

During my first post-doctoral fellowship, I had the opportunity to investigate the diffraction of an acoustic beam propagating in a soft elastic solid along a given direction (Figure). This phenomenon is described by a system of coupled KZK-type equations, for which I developed a Finite Volume code from scratch. We published the results in Communications in Nonlinear Science and Numerical Simulation (2021) link.