Longitudinal wave propagation in FG rods under impact force

A well-known fact is that one-dimensional wave analysis is the theoretical basis of the famous Hopkinson bar dynamic testing technique. The current one-dimensional wave theory is mostly confined to the slender rods of isotropic materials. It is not easy to obtain an analytical solution to the wave equation of an anisotropic rod. In this work, rods with both elastic modulus and density graded in the length direction are presented and analyzed. The one-dimensional variable coefficient wave equation corresponding to the functionally graded rod is constructed and converted into a second-order variable coefficient partial differential equation using the Laplace approach. Then, the details of solving the partial differential equation of the second-order variable coefficients are given. It is worth noting that here we construct a variable coefficient equation that satisfies the form of Euler's equation. It is still difficult to obtain analytical solutions for other equations that do not satisfy this form. Subsequently, studies of the wave propagation characteristics of rods with different graded configurations are carried out. The theoretical results show that the wave propagation behavior and post-impact vibration of the rod are significantly influenced by the graded configuration. It is possible to adjust not only the impact response at the end but also the impact response in the middle of the rod by optimizing the design of the rod's graded configuration.