Construction of irreducible integrity basis for anisotropic hyperelasticity via structural tensors

TL;DR

This work develops a unified, structure-tensor–based framework to construct irreducible, polynomially complete scalar invariants for anisotropic hyperelasticity across all common anisotropy classes, including crystal and non-crystal systems such as icosahedral symmetry. It combines coordinate-dependent invariants (Smith, Boehler) with Xiao’s extension to higher-order structural tensors and employs an analytical–numerical elimination workflow to identify syzygies, proven via Molien series and cross-validation against known irreducible bases. The authors provide explicit irreducible integrity bases for monoclinic and cubic anisotropies and deliver complete invariant sets for 11 crystal and 4 non-crystal classes, including the icosahedral case, expressed in a coordinate-free form suitable for both classical constitutive modeling and data-driven approaches. The methodology enables direct differentiation in tensor form and supports deriving primary invariants and syzygies, with practical implications for physics-informed machine learning and robust material modeling across finite strains.

Abstract

We present a straightforward analytical-numerical methodology for determining polynomially complete and irreducible scalar-valued invariant sets for anisotropic hyperelasticity. By applying the proposed technique, we obtain irreducible integrity bases for all common anisotropies in hyperelasticity via the structural tensor concept, i.e., invariants are formed from a measure of deformation (symmetric 2nd order tensor) and a set of structural tensors describing the material's symmetry. Our work covers results for the 11 types of anisotropy that arise from the classical 7 crystal systems, as well as findings for 4 additional non-crystal anisotropies derived from the cylindrical, spherical, and icosahedral symmetry systems. Polynomial completeness and irreducibility of the proposed integrity bases are proven using the Molien series and, in addition, with established results for scalar-valued invariant sets from the literature. Furthermore, we derive relationships between a set of multiple structural tensors that specify a symmetry group and a description using only a single structural tensor. Both can be used to construct irreducible integrity bases by applying the proposed analytical-numerical method. The provided invariant sets are of great importance for modeling anisotropic materials via the structural tensor concept using both classical models as well as modern approaches based on machine learning. Alongside the results presented, this article also aims to provide an introductory overview of the complex field of modeling anisotropic materials.

Figures (2)

• Figure 1: Principle of material symmetry using coordinate-dependent invariants. The use of invariants in coordinate-dependent form requires specification in a fixed reference frame, as the orientation of the transformation $\bold {Q}\in\mathcal{G}$ is predefined. Therefore, a preliminary coordinate transformation using $\tilde{\bold {R}}\in\mathcal{SO}(3)$ is necessary to ensure that the reference frame $\tilde{\boldsymbol{e}}_i$ coincides with that of the invariants $\boldsymbol{e}_i$.
• Figure 2: Illustration of the structural tensor concept for obtaining isotropic tensor functions. The material is actively rotated by applying the transformation $\bold {Q}\in\mathcal{O}(3)$ to the structural tensors in $\mathcal{M}_\mathcal{G}$. This rotation is then reversed using $\bold {Q}^\top$, thereby ensuring that the principle of material symmetry is satisfied $\forall\:\bold {Q}\in\mathcal{O}(3)$. $\bold {R} = \bold {F}\cdot\bold {U}^{-1}$ denotes the rotational component of the deformation gradient.

Theorems & Definitions (36)

• Remark 1
• Definition 1: Scalar invariant
• Definition 2: Isotropic invariant
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7
• Remark 8