A note on the growth of trajectories in nonlinear elastodynamics (Q1202237)

The paper is concerned with the growth of time-dependent solutions in a nonlinear hyperelastic body under zero displacement conditions on some part of its boundary in its reference configuration which may not be stress-free. It is further assumed any nonconservative loadings and shock waves dissipate the mechanical energy so that the total energy, namely, the sum of kinetic and potential energy, is a non-increasing function of time. By employing the results of a growth lemma in a real Hilbert space whose vectors satisfy an energy inequality, it is shown that the norm of the displacement vector (density-weighted \(L_ 2\) norm) never becomes unbounded in a finite time. Hence under these conditions a blow-up of solutions in nonlinear elastodynamics is ruled out. The apparent contradiction of this result with those obtained previously is discussed at length.