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Cracking induced by tensile wave at the free surface of an impacted target is an important issue in impact-resistant design.Here,we explore to use material nonlinearity to undermine the strength of the tensile wave.More specifically,we consider waves in a two-layer nonlinearly elastic composite bar induced by a compressive impact.Multiple reflections cause a tensile wave being transmitted into the second layer.The phenomenon that the tensile wave catches the first transmitted compressive wave is studied analytically and numerically.It is shown that there are two wave patterns in which catching-up phenomena can happen.