We applied physics-informed neural networks to solve the constitutive relations for nonlinear, path-dependent material behavior. As a result, the trained network not only satisfies all thermodynamic constraints but also instantly provides information about the current material state (i.e., free energy, stress, and the evolution of internal variables) under any given loading scenario without requiring initial data. One advantage of this work is that it bypasses the repetitive Newton iterations needed to solve nonlinear equations in complex material models. Additionally, strategies are provided to reduce the required order of derivative for obtaining the tangent operator. The trained model can be directly used in any finite element package (or other numerical methods) as a user-defined material model. However, challenges remain in the proper definition of collocation points and in integrating several non-equality constraints that become active or non-active simultaneously. We tested this methodology on rate-independent processes such as the classical von Mises plasticity model with a nonlinear hardening law, as well as local damage models for interface cracking behavior with a nonlinear softening law. In order to demonstrate the applicability of the methodology in handling complex path dependency in a three-dimensional (3D) scenario, we tested the approach using the equations governing a damage model for a three-dimensional interface model. Such models are frequently employed for intergranular fracture at grain boundaries. We have observed a perfect agreement between the results obtained through the proposed methodology and those obtained using the classical approach. Furthermore, the proposed approach requires significantly less effort in terms of implementation and computing time compared to the traditional methods.

翻译：我们应用基于物理信息的神经网络求解非线性、路径依赖材料行为的本构关系。训练后的网络不仅满足所有热力学约束，还能在任意荷载工况下即时提供当前材料状态信息（即自由能、应力和内变量的演化），无需初始数据。本工作的优势之一在于避免了求解复杂材料模型中非线性方程所需的重复牛顿迭代。此外，我们还提出了降低获取切线算子所需导数阶数的策略。训练后的模型可直接作为用户自定义材料模型用于任何有限元程序（或其他数值方法）。然而，在配点的合理定义以及整合多个同时激活或非激活的非等式约束方面仍存在挑战。我们在率无关过程上测试了本方法，例如具有非线性硬化律的经典冯·米塞斯塑性模型，以及具有非线性软化律的界面开裂行为局部损伤模型。为证明该方法处理三维场景中复杂路径依赖性的适用性，我们采用三维界面模型损伤控制方程进行了测试——此类模型常用于晶界沿晶断裂分析。结果表明，本方法所得结果与传统方法完美吻合。此外，与传统方法相比，本方法在实现难度和计算耗时方面显著降低。