Numerical simulation of linear and nonlinear waves in hypoelastic solids by the CESE method

In this paper, we present a comprehensive model for the numerical simulation of linear and nonlinear elastic waves in slender rods. The mathematical model, based on conservation of mass, conservation of linear momentum, and a hypoelastic constitutive equation, consists of a set of three first-order, fully-coupled, nonlinear, hyperbolic partial differential equations. Three forms of the model equations are presented: the conservative form, the non-conservative form, and the diagonal form. The conservative form is solved numerically using the Conservation Element and Solution Element (CESE) method, a highly-accurate explicit space–time finite-volume scheme for solving nonlinear hyperbolic systems. To demonstrate this numerical approach, two elastic wave problems are directly calculated: (i) resonant standing waves arising from a time-harmonic external axial load and (ii) propagating compression waves arising from a bi-material collinear impact. Simulations of the resonance problem illustrate a linear-to-nonlinear evolution of the resonant vibrations, the emergence of super-harmonics of the forcing frequency, and the distribution of wave energy among multiple modes. For the bi-material impact problem, the CESE method successfully captures the sharp traveling wavefronts, wave reflection and transmission, and the different wave propagation speeds in each material.