Constitutive Modelling of Non-Linear Isotropic Elasticity Using Deep Regression Neural Networks

Deep neural networks (DNNs) have emerged as a promising approach for constitutive modelling of advanced materials in computational mechanics. However, achieving physically realistic and stable numerical simulations with DNNs can be challenging, especially when dealing with large deformations, that can lead to non-convergence effects in the presence of local stretch/stress peaks.
This PhD dissertation introduces a novel approach for data-driven modelling of non-linear compressible isotropic materials, focusing on predicting the large deformation response of 3D specimens. The proposed methodology formulates the underlying hyperelastic deformation problem in the principal space using principal stretches and principal stresses, in which the corresponding constitutive relation is captured by a deep neural network surrogate model. To ensure constitutive requirements of the model while preserving the robustness of underlying numerical solution schemes, several physics-motivated constraints are imposed on the architecture of the DNN, such as objectivity, growth condition, normalized condition, and Hill’s inequalities.
Furthermore, the prediction phase utilizes a constitutive blending approach to overcome divergence in the Newton-Raphson process, which can occur when solving boundary value problems using the Finite Element Method. The work also presents a machine learning finite element pipeline for modelling non-linear compressible isotropic materials, involving determining automatically the hyperparameters, training, and integrating the ANN operator into the finite element solver using symbolic representation.
The proposed formalism has been tested through numerical benchmarks, demonstrating its ability to describe non-trivial load-deformation trajectories of 3D test specimens accurately. Overall, the thesis presents a complete and general formalism for data-driven modelling of non-linear compressible isotropic materials that overcomes the limitations of existing approaches.