Tensor Basis Gaussian Process Models of Hyperelastic Materials
This work addresses the problem of improving material modeling accuracy and efficiency for researchers in computational mechanics, though it is incremental as it builds on existing physics-informed machine learning methods.
The paper tackled modeling hyperelastic materials using Gaussian process regression by embedding physical constraints like rotational invariance, resulting in models that require fewer training examples and achieve higher accuracy while maintaining exact invariance.
In this work, we develop Gaussian process regression (GPR) models of hyperelastic material behavior. First, we consider the direct approach of modeling the components of the Cauchy stress tensor as a function of the components of the Finger stretch tensor in a Gaussian process. We then consider an improvement on this approach that embeds rotational invariance of the stress-stretch constitutive relation in the GPR representation. This approach requires fewer training examples and achieves higher accuracy while maintaining invariance to rotations exactly. Finally, we consider an approach that recovers the strain-energy density function and derives the stress tensor from this potential. Although the error of this model for predicting the stress tensor is higher, the strain-energy density is recovered with high accuracy from limited training data. The approaches presented here are examples of physics-informed machine learning. They go beyond purely data-driven approaches by embedding the physical system constraints directly into the Gaussian process representation of materials models.