Evolution TANN and the discovery of the internal variables and evolution equations in solid mechanics

Data-driven and deep learning approaches have demonstrated to have the potential of replacing classical constitutive models for complex materials, displaying path-dependency and possessing multiple inherent scales. Yet, the necessity of structuring constitutive models with an incremental formulation has given rise to data-driven approaches where physical quantities, e.g. deformation, blend with artificial, non-physical ones, such as the increments in deformation and time. Neural networks and the consequent constitutive models depend, thus, on the particular incremental formulation, fail in identifying material representations locally in time, and suffer from poor generalization. Here, we propose a new approach which allows, for the first time, to decouple the material representation from the incremental formulation. Inspired by the Thermodynamics-based Artificial Neural Networks (TANN) and the theory of the internal variables, the evolution TANN (eTANN) are continuous-time, thus independent of the aforementioned artificial quantities. Key feature of the proposed approach is the discovery of the evolution equations of the internal variables in the form of ordinary differential equations, rather than in an incremental discrete-time form. In this work, we focus attention to juxtapose and show how the various general notions of solid mechanics are implemented in eTANN. The laws of thermodynamics are hardwired in the structure of the network and allow predictions which are always consistent. We propose a methodology that allows to discover, from data and first principles, admissible sets of internal variables from the microscopic fields in complex materials. The capabilities as well as the scalability of the proposed approach are demonstrated through several applications involving a broad spectrum of complex material behaviors, from plasticity to damage and viscosity.

Thermodynamics-based Artificial Neural Networks (TANN) for multiscale modeling of materials with inelastic microstructure

The mechanical behavior of inelastic materials with microstructure is very complex and hard to grasp with heuristic, empirical constitutive models. For this purpose, multiscale, homogenization approaches are often used for performing reliable, accurate predictions of the macroscopic mechanical behavior of microstructured solids. Nevertheless, the calculation cost of such approaches is extremely high and prohibitive for real-scale applications involving inelastic materials. Recently, data-driven approaches based on deep learning have risen as a promising alternative to replace ad-hoc constitutive laws and speed-up multiscale numerical methods. However, such approaches lack a rigorous frame based on the laws of physics. As a result, their application to model materials with complex microstructure in inelasticity is not yet established. Here, we propose Thermodynamics-based Artificial Neural Networks (TANN) for the constitutive modeling of materials with inelastic and complex microstructure. Our approach integrates thermodynamics-aware dimensionality reduction techniques and deep neural networks to identify the constitutive laws and the internal state variables of complex inelastic materials. The ability of TANN in delivering high-fidelity, physically consistent predictions is demonstrated through several examples both at the microscopic and macroscopic scale. In particular, we show the efficiency and accuracy of TANN in predicting the average and local stress-strain response, the internal energy and the dissipation of both regular and perturbed lattice microstructures in inelasticity. Finally, a double-scale homogenization scheme is used to solve a large scale boundary value problem. The high performance of the homogenized model using TANN is illustrated through detailed comparisons. An excellent agreement is shown for a variety of monotonous and cyclic stress-strain paths.

Thermodynamics-based Artificial Neural Networks for constitutive modeling

Machine Learning methods and, in particular, Artificial Neural Networks (ANNs) have demonstrated promising capabilities in material constitutive modeling. One of the main drawbacks of such approaches is the lack of a rigorous frame based on the laws of physics. This may render physically inconsistent the predictions of a trained network, which can be even dangerous for real applications. Here we propose a new class of data-driven, physics-based, neural networks for constitutive modeling of strain rate independent processes at the material point level, which we define as Thermodynamics-based Artificial Neural Networks (TANNs). The two basic principles of thermodynamics are encoded in the network's architecture by taking advantage of automatic differentiation to compute the numerical derivatives of a network with respect to its inputs. In this way, derivatives of the free-energy, the dissipation rate and their relation with the stress and internal state variables are hardwired in the network. Consequently, our network does not have to identify the underlying pattern of thermodynamic laws during training, reducing the need of large data-sets. Moreover the training is more efficient and robust, and the predictions more accurate. Finally and more important, the predictions remain thermodynamically consistent, even for unseen data. Based on these features, TANNs are a starting point for data-driven, physics-based constitutive modeling with neural networks. We demonstrate the wide applicability of TANNs for modeling elasto-plastic materials, with strain hardening and strain softening. Detailed comparisons show that the predictions of TANNs outperform those of standard ANNs. TANNs ' architecture is general, enabling applications to materials with different or more complex behavior, without any modification.