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Mathematical theory and analytical solutions for the wave catching-up phenomena in a nonlinearly elastic composite bar

Cracking induced by tensile wave at the free surface of an impacted target is an important issue in impact-resistant design. Here, we explore the use of material nonlinearity to undermine the strength of the tensile wave. More specifically, we consider waves in a two-material composite bar subjected to impact loading at one end. Multiple reflections cause a tensile wave being transmitted into the second material. The attention is on analytically and numerically studying the phenomenon that the tensile wave catches the first transmitted compressive wave. It turns out that, depending on the interval of the initial impact, catching-up phenomena can happen in two wave patterns. A general mathematical theory is provided to show the existence of these patterns together with some qualitative information. To gain more insights into such phenomena, asymptotic solutions are also constructed, which provide both qualitative and quantitative results on the requirement of the constitutive relation, the time and place at which the catching takes place, and how the initial impact, material and geometric parameters influence the solutions. Numerical simulations are also performed, confirming the validity of the analytical results. The analysis and results presented here could be useful for designing a composite structure that has a good impact-protection performance.