Polyconvex anisotropic hyperelasticity with neural networks

Abstract

In the present work, two machine learning based constitutive models for finite deformations are proposed. Using input convex neural networks, the models are hyperelastic, anisotropic and fulfill the polyconvexity condition, which implies ellipticity and thus ensures material stability. The first constitutive model is based on a set of polyconvex, anisotropic and objective invariants. The second approach is formulated in terms of the deformation gradient, its cofactor and determinant, uses group symmetrization to fulfill the material symmetry condition, and data augmentation to fulfill objectivity approximately. The extension of the dataset for the data augmentation approach is based on mechanical considerations and does not require additional experimental or simulation data. The models are calibrated with highly challenging simulation data of cubic lattice metamaterials, including finite deformations and lattice instabilities. A moderate amount of calibration data is used, based on deformations which are commonly applied in experimental investigations. While the invariant-based model shows drawbacks for several deformation modes, the model based on the deformation gradient alone is able to reproduce and predict the effective material behavior very well and exhibits excellent generalization capabilities. Thus, in particular the second model presents a highly flexible constitutive modeling approach, that leads to a mathematically well-posed problem.

Numerical Approaches for Investigating Quasiconvexity in the Context of Morrey’s Conjecture

“…Previous applications of machine learning in nonlinear elasticity, as we consider in Sect. 4.1, have mostly focused on the energy function itself (Fernández et al 2021;Klein et al 2022). The application to deformation functions presented here is based on the concept of physics-informed neural networks, which have recently been employed for finding approximate solutions to various partial differential equations (Raissi et al 2019;Karniadakis et al 2021).…”

Numerical approaches for investigating quasiconvexity in the context of Morrey's conjecture

“…Previous applications of machine learning in nonlinear elasticity, as we consider in Section 4.1, have mostly focused on the energy function itself [25,41]. The application to deformation functions presented here is based on the concept of physics-informed neural networks, which have recently been employed for finding approximate solutions to various partial differential equations [59,34].…”