  We prove existence of $L^2$-weak solutions of a quasilinear wave equation
with boundary conditions. This describes the isothermal evolution of a one
dimensional non-linear elastic material, attached to a fixed point on one side
and subject to a force (tension) applied to the other side. The $L^2$-valued
solutions appear naturally when studying the hydrodynamic limit from a
microscopic dynamics of a chain of anharmonic springs connected to a thermal
bath. The proof of the existence is done using a vanishing viscosity
approximation with extra Neumann boundary conditions added. In this setting we
obtain a uniform a priori estimate in $L^2$, allowing us to use $L^2$ Young
measures, together with the classical tools of compensated compactness. We then
prove that the viscous solutions converge to weak solutions of the quasilinear
wave equation strongly in $L^p$, for any $p \in [1,2)$, that satisfy, in a weak
sense, the boundary conditions. Furthermore, these solutions satisfy the
Clausius inequality: the change of the free energy is bounded by the work done
by the boundary tension. In this sense they are the correct thermodynamic
solutions, and we conjecture their uniqueness.
