In this work, we study the mechanical behavior of solids with microstructure using the framework of Cosserat elasticity with a single unit director. This formulation captures the coupling between deformation and orientational fields that arises in many structured materials. To compute equilibrium configurations of such media, we develop two complementary computational approaches: a finite element formulation based on variational principles and a neural network-based solver that directly minimizes the total potential energy. The neural architecture is constructed to respect the fundamental kinematic structure of the theory. In particular, it enforces frame invariance of the energy, satisfies the unit-length constraint on the director field, and represents deformation and director fields through separate networks to preserve their kinematic independence in the variational setting. Beyond satisfying balance laws, however, physically admissible solutions must also correspond to stable energy minimizers. To assess this requirement, we derive the quasiconvexity condition, rank-one convexity condition, and the Legendre-Hadamard inequalities for the Cosserat model and formulate them in a manner suitable for evaluating neural network predictions. These necessary stability conditions provide a physics-based validation framework: network outputs that violate these necessary conditions cannot correspond to stable energy minimizers and can therefore be rejected. In this way, we integrate classical variational stability theory with modern machine-learning solvers, establishing a computational workflow in which equilibrium solutions are not only learned but also assessed for energetic consistency.

翻译：本研究采用具有单一单位指向矢的Cosserat弹性理论框架，探究具有微结构固体的力学行为。该理论框架能够捕捉多种结构化材料中变形场与取向场之间的耦合效应。为计算此类介质的平衡构型，我们开发了两种互补的计算方法：基于变分原理的有限元公式，以及通过直接最小化总势能实现的神经网络求解器。神经网络的架构设计遵循理论的基本运动学结构，具体而言：强制保证能量的框架不变性，满足指向矢场的单位长度约束，并通过独立网络分别表示变形场与指向矢场，以在变分框架中保持二者的运动学独立性。然而，物理可接受的解不仅需满足平衡定律，还必须对应稳定的能量最小化状态。为评估这一要求，我们推导了Cosserat模型的拟凸条件、秩一凸条件以及Legendre-Hadamard不等式，并将其表述为适用于评估神经网络预测的形式。这些必要的稳定性条件构成了基于物理的验证框架：违反这些必要条件的网络输出不可能对应稳定的能量最小化解，因而可予以排除。通过这种方式，我们将经典变分稳定性理论与现代机器学习求解器相结合，建立了一种计算工作流程，使得平衡解不仅能够被学习，还能接受能量一致性的评估。