Progressive Waves in Materials Exhibiting Long Range Memory

Source abstract

In this note, we establish the existence of traveling wave solutions to the equations of motions for materials which exhibit long range memory effects. These traveling waves are nontrivial solutions of the integral equation $$(IE)\qquad \sigma _1 (u(\zeta )) + \int_0^\infty {f(s,u(\zeta - s))\,ds = c^2 (u(\zeta ) - 1)}$$ and the boundary condition $$\mathop {\lim }\limits_{\zeta \to \infty } u(\zeta ) = 1$$. Subject to certain hypotheses on $\sigma _1$ and f, the functions which determine the mechanical properties of the material in question, it is shown that if $c^2 \geqq \sigma '_I (1)$, then (IE) has nontrivial solutions $u( \cdot )$ which are bounded and decreasing on $( - \infty ,\infty )$. These are the shock waves or structured shocks for the materials in question.