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

TL;DR

The paper develops an implicit time-marching scheme for Consistent Couple-Stress Theory (C-CST) elastodynamics using a mixed finite element method with a Lagrange multiplier to enforce compatibility. Time-domain solutions are obtained directly, avoiding convolutional or transform approaches, and the method is verified via static MMS, dynamic eigenstate analyses, and dispersion experiments, revealing energy dissipation and higher natural frequencies due to microstructure coupling. The results establish a robust framework for simulating size-dependent microstructural effects in metamaterials and wave propagation, while highlighting the need for symplectic integrators to preserve energy and for further MMS-based temporal verification. This work lays the groundwork for advanced dynamic analyses in C-CST and related non-classical continuum theories, with potential applications in material design and phononic wave control.

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.

Figures (13)

• Figure 1: Schematic of a CCST body, subject to body forces $f_i$ applied on the volume, and force and couple tractions $t_i$ and $m_i$ applied on the boundary. $n_i$ indicates the normal vector to the surface.
• Figure 2: Schematic representation of the domain and boundary conditions for the C-CST model. Known body forces $f_i$ are applied on the volume, and the boundary $S$ is split into boundaries where different conditions are prescribed.
• Figure 3: Finite element used for the finite element discretization of the C-CST material model. A second-order Lagrange interpolation is used for displacements $\vb{u}$ (left), a first-order Lagrange interpolation for rotations $\theta_z$ (center), and a piecewise constant is used for the skew-symmetric stresses $s_z$ (right).
• Figure 4: Schematic of the cantilever beam. A distributed load is applied on the right side, and the length scale ($l$) is varied to verify the behavior of the effective rigidity.
• Figure 5: Variation of a cantilever effective rigidity for different $h/l$ ratios. This behavior is congruent with previous literature results darrall_finite_2014.