Implicit Time-Marching for Lagrange Multiplier Formulation for Couple Stress Elastodynamics

TL;DR

This paper introduces a stable implicit time-marching finite element method for dynamic couple stress elastodynamics using a Lagrange multiplier, enabling direct time-domain solutions for size-dependent microstructured materials.

Contribution

It develops a novel implicit time integration scheme combined with a mixed finite element approach for dynamic C-CST, improving stability and directness over previous methods.

Findings

Finite element scheme is stable and accurate in static and dynamic tests.

Energy dissipation observed in simulations highlights the need for symplectic integrators.

Method provides a foundation for studying size-dependent behaviors and wave propagation.

Abstract

The study of metamaterials and architected materials has intensified interest in continuum mechanics models that capture size-dependent microstructure interactions. Among these, Consistent Couple-Stress Theory (C-CST) incorporates microscale mechanical interactions by introducing higher-order derivatives in the strain energy. While previous studies have relied on convolutional principles or inverse Laplace transforms to obtain time-dependent solutions, this work demonstrates that implicit time integration applied to a mixed finite element method with a Lagrange multiplier provides stable, direct time-domain solutions for dynamic C-CST modeling. The proposed finite element scheme is tested through the Method of Manufactured Solutions (MMS) for static cases and dynamic simulations of simple mechanical scenarios. Our computational experiments revealed energy dissipation, emphasizing the importance of exploring symplectic integrators in future work to impose energy conservation. Additionally, further research is required to verify temporal stability through time-domain MMS and to investigate complex mechanical scenarios, including those previously restrictive, challenging to simulate, or unfeasible with existing dynamic methods. This work lays the groundwork for studying size-dependent material behavior and provides the foundation for advanced applications in material design and wave propagation.