A thermodynamically consistent, inelastic constitutive modeling framework based on artificial neural networks

本构模型描述了材料的力学行为，并表明应变和应力，内能和耗散之间的关系。为了在物理上有意义和数学上定义良好，这些模型必须满足许多要求，如热力学一致性，材料框架无差异性和对称性，以及椭圆率。虽然这种材料模型已经开发了几十年，但没有通用的方法可以同样有效地应用于各种材料类别，从而满足所需的性能。特别是，新型超材料和复合材料制成的聚合物和生物材料显示出高度非线性，各向异性，和非弹性有效材料的行为，不能用目前的解析制定的模型。在这个项目的范围内，本构建模方法将开发，可以灵活地应用于强非线性和非弹性，材料在大变形下的耗散行为，从而满足所有基本的物理和数学要求。这是要实现的基础上的“广义标准材料”的概念，它确保热力学的一致性，以及通过代表性的能量和耗散潜力与人工神经网络（ANN）的帮助。人工神经网络的特点是具有通用的逼近性质，因此它可以逼近任何非线性函数。然而，与其他常见的数据驱动方法相比，它们可以以确保凸性和对称性等基本数学要求的方式进行公式化和结构化。从解析公式化的材料模型及其结构开始，我们将逐步开发基于物理信息的ANN的粘滞超弹性和Mullins效应损伤的本构模型。这些模型的物理可解释性是通过选择内部状态变量和模型结构方面的潜力和演化方程的表示。为了证明这种新方法的灵活性，我们将把它应用到有效的材料建模和多尺度模拟立方三维梁格结构，其特点是大变形，不稳定性，粘性行为，和Mullins效应。为此，我们还将开发这些ANN模型的高效开源有限实现，该实现使用并行化和GPU评估。这种基于人工神经网络的本构模型的未来应用和扩展可以包括塑性、相变、热机械或多物理行为，从而显示出对任何材料类别和超材料的普遍适用性。

Constitutive models describe the mechanical behavior of materials and indicate the relationship between strains and stresses, internal energy, and dissipation. To be physically meaningful and mathematically well defined, these models have to meet numerous requirements, such as thermodynamic consistency, material frame indifference and symmetry, as well as ellipticity. Although such material models have been developed for decades, there are no universal approaches that can be applied equally effective to a wide variety of material classes and thereby meet the required properties. In particular, novel metamaterials and composites made of polymers and biomaterials show highly nonlinear, anisotropic, and inelastic effective material behavior that cannot be represented with the current analytically formulated models.Within the scope of this project, a constitutive modeling approach will be developed that can be flexibly applied to strongly nonlinear and inelastic, dissipative material behavior at large deformations and thereby fulfills all essential physical and mathematical requirements. This is to be realized on the basis of the "Generalized Standard Materials" concept, which ensures thermodynamic consistency, as well as through the representation of the energy and dissipation potentials with the aid of artificial neural networks (ANN). ANNs are characterized by their universal approximation property, so they can approximate any nonlinear functions. In contrast to other common data-driven approaches, however, they can be formulated and structured in such a way that essential mathematical requirements such as convexity and symmetries are ensured.Starting from analytically formulated material models and their structure, we will gradually develop constitutive models for visco-hyperelasticity and damage due to the Mullins effect, which will be based on physics-informed ANNs. The physical interpretability of these models is preserved through the choice of the internal state variables and the model structure with regard to the representation of the potentials and evolution equations. To demonstrate the flexibility of this novel approach, we will apply it to the effective material modeling and multiscale simulation of cubic 3D beam lattice structures, which are characterized by large deformations, instabilities, viscous behavior, and the Mullins effects. For this purpose, we will also develop an efficient open-source finite implementation of these ANN models, which uses parallelization and GPU evaluation. Future applications and extensions of such ANN-based constitutive models could then include plasticity, phase transformations, thermo-mechanical or multi-physical behavior and thus show the universal applicability to any material classes and metamaterials.