Variational multiscale enrichment method for dynamic response of hyperelastic materials at finite deformation

TL;DR

This paper extends the variational multiscale enrichment method to simulate the dynamic response of hyperelastic materials under large deformations, capturing wave propagation, nonlinearities, and micro-inertial effects in a multiscale framework.

Contribution

The paper introduces a multiscale variational enrichment approach that models wave propagation in hyperelastic materials considering nonlinearities and micro-inertial effects at finite deformation.

Findings

Multiscale dissipative schemes suppress spurious oscillations.

Material and geometric nonlinearities significantly influence wave dispersion and attenuation.

The framework effectively captures wave steepening and flattening in heterogeneous microstructures.

Abstract

In this manuscript, we extend the variational multiscale enrichment (VME) method to model the dynamic response of hyperelastic materials undergoing large deformations. This approach enables the simulation of wave propagation under scale-inseparable conditions, including short-wavelength regimes, while accounting for material and geometric nonlinearities that lead to wave steepening or flattening. By employing an additive decomposition of the displacement field, we derive multiscale governing equations for the coarse- and fine-scale problems, which naturally incorporate micro-inertial effects. The framework allows the discretization of each unit cell with a patch of coarse-scale elements, which is essential to accurately capture wave propagation in short-wavelength regimes. An operator-split procedure is used to iteratively solve the semi-discrete equations at both scales until convergence is achieved. The coarse-scale problem is integrated explicitly, while the fine-scale problem is solved using either explicit or implicit time integration schemes, including both dissipative and non-dissipative methods. Numerical examples demonstrate that multiscale dissipative schemes effectively suppress spurious oscillations. The multiscale framework was applied to investigate how material and geometric nonlinearities, along with elastic stiffness contrast in heterogeneous microstructures, influence key wave characteristics such as dispersion, attenuation, and steepening. This multiscale computational framework provides a foundation for studying the dynamic response of architected materials.