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An HHT-$α$-based finite element framework for wave propagation in constitutively nonlinear elastic materials

This paper presents a computational framework for modeling wave propagation in geometrically linear elastic materials characterized by algebraically nonlinear constitutive relations. We derive a specific form of the nonlinear wave equation in which the nonlinearity explicitly appears in the time-derivative terms that govern the evolution of the mechanical fields. The numerical solution is established using a fully discrete formulation that combines the standard finite element method for spatial discretization with the implicit Hilber-Hughes-Taylor (HHT)-$α$ scheme for time integration. To address the nonlinear nature of the discrete system, we employ Newton's method to iteratively solve the linearized equations at each time step. The accuracy and robustness of the proposed framework are rigorously verified through convergence analyses, which demonstrate optimal convergence rates in both space and time. Furthermore, a detailed parametric study is conducted to elucidate the influence of the model's constitutive parameters. The results reveal that the magnitude parameter of the stress-dependent variation in wave speed leads to wavefront steepening and the formation of shock discontinuities. Conversely, the exponent parameter acts as a nonlinearity filter; high values suppress nonlinear effects in small-strain regimes, whereas low values allow significant dispersive behavior. This work provides a validated tool for analyzing shock formation in advanced nonlinear materials.