Data-driven constitutive laws for hyperelasticity in principal space: numerical challenges and remedies

This contribution discusses a formalism for data-driven modelling of advanced materials with a special interest in the large deformation response of three-dimensional specimens. The underlying hyperelastic deformation problem is formulated in the principal space using principal stretches and principal stresses. The associated constitutive relation is consequently using principal quantities and captured by the parameter-free representation using a deep neural network. The presentation investigates certain physics-motivated requirements imposed on the architecture of the artificial neural network such as symmetry and objectivity criteria. The training phase of the constitutive ANN operator employs a loss function which ensures the identified consistency conditions. The prediction phase exploits a constitutive blending approach to stabilise the numerical solution procedure in the presence of typically local stretch/stress extrema. The presented approach is implemented using FEniCS and builds on symbolic representation of the ANN operator based on the Unified Form Language (UFL). The neural network is constructed, trained, and tested using PyTorch. Numerical benchmarks demonstrate the ability of the presented formalism to describe non-trivial load-deformation trajectories of 3D test specimens.