The study of metamaterials and architected materials has increased interest in continuum mechanics models that capture size-dependent microstructure interactions. Among these, Consistent Couple-Stress Theory (C-CST) accounts for mechanical interactions at the microscale adding higher-order derivatives in the strain energy. Previous studies have explored the dynamics of C-CST using methods such as convolutional principles or inverse Laplace transforms to obtain time-dependent solutions. In contrast, our results show that implicit time integration, applied to a mixed finite element method with a Lagrange multiplier, provides direct time-domain solutions for dynamic C-CST modeling and ensures solution stability. In particular, the proposed finite element scheme is verified in comparison with results from previous literature, by employing the Method of Manufactured Solutions in the static case and dynamic simulations of simple cases. The approach presented introduces new tools for studying size-dependent material behavior, laying the groundwork for further applications in advanced material design and wave propagation studies.

翻译：超构材料与架构材料的研究提升了人们对能够捕捉尺寸依赖性微结构相互作用的连续介质力学模型的兴趣。其中，一致偶应力理论通过在本构能中引入高阶导数，描述了微观尺度的力学相互作用。先前的研究已采用卷积原理或逆拉普拉斯变换等方法探索一致偶应力理论的动力学行为，以获得时间相关的解。与之相对，我们的研究结果表明，将隐式时间积分应用于含拉格朗日乘子的混合有限元方法，可为动态一致偶应力建模提供直接的时域解，并确保解的稳定性。具体而言，所提出的有限元方案通过静态情形下的制造解方法及简单案例的动态仿真，与已有文献结果进行了对比验证。本方法为研究尺寸依赖性材料行为提供了新工具，为先进材料设计与波传播研究的进一步应用奠定了基础。