Ballistic thermoelasticity of nonlinear chains under thermal shock

We investigate thermoelastic behavior of nonlinear chains with an instantaneously created temperature profile. We focus on a regime, further referred to as the ballistic thermoelasticity. In this regime, macroscopic strains are small, thermal expansion is linear, conversion of mechanical energy into heat is negligible, and heat transfer is almost ballistic. Equations describing ballistic thermoelasticity of the chains in the continuum limit are solved analytically. Several particular initial temperature profiles are considered to demonstrate peculiarities and limitations of these equations. The main feature of ballistic thermoelasticity is that wave propagation and ballistic heat transfer have the same characteristic timescale. Then, in particular, the quasistatic approximation, having reasonable accuracy in many problems of classical Fourier-law-based thermoelasticity, is irrelevant in the ballistic thermoelasticity. Another peculiarity is the ballistic resonance (BR), i.e., the emergence of mechanical vibrations with an infinitely growing amplitude in the case of sinusoidal initial temperature profile. Using BR as an example, we analyze the importance of dynamical terms in equations of ballistic thermoelasticity and discuss the possibility of observing the BR for initial conditions, corresponding to real experiments. We also show that the BR, previously observed only in the α-FPUT chain, is also present in the Lennard-Jones and Toda chains. To demonstrate limitations of the equations of ballistic thermoelasticity, we examine discontinuous and piecewise linear temperature profiles. We show that analytical solutions for strains, corresponding to discontinuous initial temperature profiles, have singularities. For piecewise linear profiles, the analytical solutions are finite and describe simulation results with reasonable accuracy. However the latter is true only if the characteristic length of temperature changes is much larger than the lattice constant.