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.

翻译：超材料和架构材料的研究，增强了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关注。其中，一致耦合应力理论通过引入应变能中的高阶导数，纳入了微观尺度力学相互作用。以往研究多依赖卷积原理或拉普拉斯逆变换来获取时变解，而本工作证明，将隐式时间积分应用于带有拉格朗日乘子的混合有限元方法，可为动态一致耦合应力理论建模提供稳定、直接的时域解。所提出的有限元方案通过制造解方法在静态案例和简单力学场景的动态模拟中进行了测试。我们的计算实验揭示了能量耗散现象，这强调了在未来工作中探索辛积分器以施加能量守恒的重要性。此外，需要进一步研究通过时域制造解方法验证时间稳定性，并探究复杂的力学场景，包括那些以往受限于现有动态方法、难以模拟或无法模拟的情形。本工作为研究尺寸依赖性材料行为奠定了基础，并为材料设计和波传播领域的先进应用提供了支撑。