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，并指出了应变和应力、内能和耗散之间的关系。为了具有物理意义和数学上的良好定义，这些模型必须满足许多要求，如热力学一致性、材料框架无关性和对称性以及椭圆性。虽然这种材料模型已经开发了几十年，但还没有通用的方法可以同样有效地应用于各种材料类别，从而满足所需的特性。特别是由聚合物和生物材料制成的新型超材料和复合材料表现出高度的非线性、各向异性和非弹性的有效材料行为，这是现有的解析模型无法描述的。在本项目的范围内，将开发一种本构建模方法，该方法可以灵活地应用于强非线性、非弹性、耗散的大变形材料行为，从而满足所有基本的物理和数学要求。这将在“广义标准材料”概念的基础上实现，该概念确保热力学上的一致性，并通过人工神经网络(ANN)表示能量和耗散势。人工神经网络具有泛逼近性质，可以逼近任意的非线性函数。然而，与其他常见的数据驱动方法不同，它们可以以确保凸性和对称性等基本数学要求的方式来描述和构造。从解析形式的材料模型及其结构出发，我们将逐步建立基于物理信息的人工神经网络的粘超弹性和损伤的本构模型。通过选择内部状态变量和关于势和演化方程的表示的模型结构，保留了这些模型的物理可解释性。为了展示这种新方法的灵活性，我们将其应用于具有大变形、不稳定、粘性行为和Mullins效应的立方体三维梁格子结构的有效材料建模和多尺度模拟。为此，我们还将开发这些ANN模型的高效开源有限实现，它使用并行化和GPU计算。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、热力或多物理行为，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.