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

TL;DR

This work develops a stress-based, second-order nonlinear wave equation for geometrically linear elastic materials with algebraically nonlinear constitutive relations within Rajagopal's implicit framework. The nonlinearity is embedded in the time derivatives, and the solution is obtained via a fully discrete finite element method in space coupled with the implicit $HHT$-$\alpha$ time integrator, using Newton-Raphson iterations for the nonlinear system. A rigorous verification against linear theory and a comprehensive parametric study reveal how constitutive parameters $a$ and $b$ govern stress-dependent wave speed, dispersion, and shock formation, providing a validated tool for analyzing nonlinear wave phenomena in advanced materials. The framework demonstrates robustness, adaptability (hp-refinement), and the ability to capture the transition from smooth wave propagation to steepened fronts and shocks, with implications for material design and nondestructive evaluation in biomedical, soft-tissue, and engineering contexts.

Abstract

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.

Figures (16)

• Figure 1: Comparison of experimental stress-strain data (markers) with the implicit strain-limiting model fits (lines). The model accurately captures the saturation behavior of the tissue.
• Figure 2: Adaptive polynomial degree distribution ($h$-$p$ refinement strategy). The domain is divided based on distance from the center ($L/2$). High-order $Q_3$ elements are localized in the center to resolve shock formation, transitioning to $Q_1$ at the boundaries.
• Figure 3: The stress $\sigma$ and strain $\epsilon$ for the linear case with $b=0.0$ and $a=1.5$. Note the identical wave profiles characteristic of linear elasticity.
• Figure 4: The displacement $u$ and wave speed $c$ for the linear case with $b=0.0$ and $a=1.5$. The constant wave speed $c=1$ confirms the linearity of the medium.
• Figure 5: The stress $\sigma$ and strain $\epsilon$ for the weakly nonlinear case with $b=1.0$ and $a=1.5$.

Theorems & Definitions (3)

• Remark 1: Stress-based formulation vs. standard elastodynamics
• Remark 2: Well-posedness and stability conditions
• Remark 3