Analytical solutions of the strain wave equation for an elastic rod with finite deformation

This study determines the exact solutions for a nonlinear strain wave equation, which characterizes wave propagation within circular elastic rods under finite deformation. While a previous study from the literature often simplifies such models by reducing the order of the partial differential equation (PDE) or neglecting specific physical parameters, a significant knowledge gap exists in obtaining analytical solutions that retain the full order of the system while simultaneously accounting for dispersion effects arising from transverse inertia and transverse shearing strain. The primary aim of this research is to resolve the nonlinear strain model without order reduction to ensure the physical integrity of the rod’s behavior is maintained. To achieve this, we find analytic solutions using hyperbolic tangent, extended hyperbolic tangent, and modified simple equation methods to the governing nonlinear PDE.