newtonian mechanics - Solving a continuum-mechanical model

A Body (density $\rho_1$, elasticity modulus $E_1$ and volume $V_1$) crashes with constant velocity $V$ into another resting Body (density $\rho_2$, elasticity modulus $E_2$ and volume $V_2$). Both bodies are described by the equations of Motion

$$\rho_{1,2} \frac{\partial^2 u(x,t)}{\partial t^2} = E_{1,2} \frac{\partial^2 u(x,t)}{\partial x^2}$$

where $t$ is time, $x$ is the coordinate (for simplicity I assume 1-dimensional model) and $u(x,t)$ is the field of displacements in the Body. It holds for the stress $\sigma_{1,2}(x,t)=E_{1,2} \frac{\partial u(x,t)}{\partial x}$. This description holds in the interior of $V_1$ or $V_2$. If These bodies collide, I have a contact surface, in which stress must be continuous. But how I can formulate proper Initial and boundary conditions?

How I determine the stress Distribution in These bodies for this case? I assume that everything is without external fields, friction, etc. But how I can determine stresses in a Body during collision???