Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Representation of tensor functions using lower-order structural tensor set: three-dimensional theory

Mohammad Madadi, Pu Zhang

TL;DR

This work advances constitutive modeling of anisotropic materials by reformulating tensor-function representations for all 3D centrosymmetric point groups using only lower-order structural tensors. Building on the Man–Goddard framework, it constructs explicit representations for scalar-valued and 2nd-order symmetric tensor-valued functions across the 14 centrosymmetric groups, selecting suitable lower-order tensor sets and enforcing symmetry constraints via group generators. The paper demonstrates both single-tensor and multi-tensor approaches, with detailed treatment of several groups (e.g., Ci, D2h, D4h, Th, Oh, C2h, C4h, C3i, D3d, D6h, C6h) and continuous families, thereby enabling practical, data-supported constitutive modeling without higher-order tensors. By clarifying the distinctions between Boehler–Liu and Man–Goddard formulations and providing explicit bases and invariants, the framework broadens applicability to hyperelasticity, dielectric, and conductivity tensors in anisotropic materials, while remaining compatible with future data-driven extensions.

Abstract

The representation theory of tensor functions is a powerful mathematical tool for constitutive modeling of anisotropic materials. A major limitation of the traditional theory is that many point groups require fourth- or sixth-order structural tensors, which significantly impedes practical engineering applications. Recent advances have introduced a reformulated representation theory that enables the modeling of anisotropic materials using only lower-order structural tensors (i.e., second-order or lower). Building upon the reformulated theory, this work establishes the representations of tensor functions for three-dimensional centrosymmetric point groups. For each point group, we propose a lower-order structural tensor set and derive the representations of tensor functions explicitly. For scalar-valued and second-order symmetric tensor-valued functions, our theory is indeed applicable to all three-dimensional point groups because their representations are determined by the corresponding centrosymmetric groups. The representation theory presented here is broadly applicable for constitutive modeling of anisotropic materials.

Representation of tensor functions using lower-order structural tensor set: three-dimensional theory

TL;DR

This work advances constitutive modeling of anisotropic materials by reformulating tensor-function representations for all 3D centrosymmetric point groups using only lower-order structural tensors. Building on the Man–Goddard framework, it constructs explicit representations for scalar-valued and 2nd-order symmetric tensor-valued functions across the 14 centrosymmetric groups, selecting suitable lower-order tensor sets and enforcing symmetry constraints via group generators. The paper demonstrates both single-tensor and multi-tensor approaches, with detailed treatment of several groups (e.g., Ci, D2h, D4h, Th, Oh, C2h, C4h, C3i, D3d, D6h, C6h) and continuous families, thereby enabling practical, data-supported constitutive modeling without higher-order tensors. By clarifying the distinctions between Boehler–Liu and Man–Goddard formulations and providing explicit bases and invariants, the framework broadens applicability to hyperelasticity, dielectric, and conductivity tensors in anisotropic materials, while remaining compatible with future data-driven extensions.

Abstract

The representation theory of tensor functions is a powerful mathematical tool for constitutive modeling of anisotropic materials. A major limitation of the traditional theory is that many point groups require fourth- or sixth-order structural tensors, which significantly impedes practical engineering applications. Recent advances have introduced a reformulated representation theory that enables the modeling of anisotropic materials using only lower-order structural tensors (i.e., second-order or lower). Building upon the reformulated theory, this work establishes the representations of tensor functions for three-dimensional centrosymmetric point groups. For each point group, we propose a lower-order structural tensor set and derive the representations of tensor functions explicitly. For scalar-valued and second-order symmetric tensor-valued functions, our theory is indeed applicable to all three-dimensional point groups because their representations are determined by the corresponding centrosymmetric groups. The representation theory presented here is broadly applicable for constitutive modeling of anisotropic materials.
Paper Structure (18 sections, 65 equations, 2 figures, 4 tables)

This paper contains 18 sections, 65 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries of representation theory
  3. Lower-order structural tensor set for 3D point groups
  4. Group $\mathcal{C}_i$ ($\bar{1}$)
  5. Group $\mathcal{D}_{2h}$ ($mmm$)
  6. Using one structural tensor
  7. Using three structural tensors
  8. Group $\mathcal{D}_{4h}$ ($4/mmm$)
  9. Group $\mathcal{T}_{h}$ ($m\bar{3}$)
  10. Group $\mathcal{O}_h$ ($m\bar{3}m$)
  11. Group $\mathcal{C}_{2h}$ ($2/m$)
  12. Group $\mathcal{C}_{4h}$ ($4/m$)
  13. Group $\mathcal{C}_{3i}$ ($\bar{3}\ \text{or}\ \mathcal{S}_6$)
  14. Group $\mathcal{D}_{3d}$ ($\bar{3}m$)
  15. Group $\mathcal{D}_{6h}$ ($6/mmm$)
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Graphical illustration of 3D centrosymmetric point groups: 11 Laue groups and 3 continuous groups.
  • Figure 2: Illustration of various vectors used to define structural tensor set for point group $\mathcal{D}_{4h}$