# Higher-Order Material Constants: Symmetry Relations from Tensor Invariance

Nico F. Declercq

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Higher-Order Material Constants

*Symmetry Relations from Tensor Invariance*

Nico F. Declercq

*Professor*

The Georgia Institute of Technology
Higher-Order Material Constants
Symmetry Relations from Tensor Invariance

Nico F. Declercq

Professor

George W. Woodruff School of Mechanical Engineering

© IRL 2958 Georgia Tech – CNRS (Metz, France)

The Georgia Institute of Technology

Atlanta

United States of America
Higher-Order Material Constants: Symmetry Relations from Tensor In-
variance

Copyright © 2026 Nico F. Declercq

Author affiliation:

George W. Woodruff School of Mechanical Engineering & IRL 2958 Georgia Tech –
CNRS (Metz, France), The Georgia Institute of Technology, Atlanta, United States of
America

ISBN (print): 9798247330448

DOI (Zenodo): doi:10.5281/zenodo.18517782

Edition: First edition, February 7, 2026

Authoritative version and updates:

The authoritative version of this book is published on CERN's Zenodo
at https://doi.org/10.5281/zenodo.18517782.
**License:**

This work is licensed under the Creative Commons Attribution 4.0 International Li-cense (CC BY 4.0).

**Suggested citation:**

N. F. Declercq, *Higher-Order Material Constants: Symmetry Relations from Tensor Invariance*,

First edition, 2026. Zenodo. doi:10.5281/zenodo.18517782.

**Disclaimer:**

The author has made every effort to ensure correctness. If you find an error, please report it at: declercq@gatech.edu.
# Preface

This book is the culmination of a long intellectual trajectory, one that spans formal education, independent study, professional research, and a sustained fascination with symmetry as a unifying principle in physics. While its pages are largely composed of tables, tensor relations, and explicit symmetry reductions, the motivation behind the work is deeply conceptual: to render the role of symmetry in higher-order material behavior transparent, systematic, and reliable.

My earliest encounter with symmetry was not mathematical, but visual. As a child, I was captivated by the intricate geometric patterns of the Alhambra in Granada, where repetition, rotation, reflection, and hierarchy combine into structures of striking coherence and beauty. Long before I could articulate the language of group theory, these patterns conveyed a powerful intuition: that symmetry is not decoration, but structure — a principle that constrains form while simultaneously enabling richness.

This intuition later found formal expression during my studies in physics and crystallography. As a Master's student, I took a course on symmetry groups, much of which I studied independently while residing in Sri Lanka in 1997. In parallel, I encountered the profound role of symmetry in particle physics, where group representations classify fundamental particles and dictate allowable interactions. During my early doctoral years at the Catholic University of Leuven and Ghent University, crystallography further deepened this perspective: here, symmetry does not merely classify matter, but directly determines its physical response to mechanical, electrical, and electromagnetic fields.

A decisive influence on my thinking came from the work of Emmy Noether. Her theorem — establishing that every continuous symmetry of the action corresponds to a conserved quantity — provided a conceptual bridge between abstract invariance and concrete physical law. In mechanics, symmetry under time translation yields energy conservation; in electromagnetism, gauge symmetry constrains allowable field dynamics; in modern theoretical physics, symmetry principles often precede and dictate the form of the equations themselves. This viewpoint strongly informed my later work on Trembling Spacetime Relativity, but it also shaped my approach to much more applied questions: how symmetry governs material behavior, and how violations or reductions of symmetry give rise to new physical effects.

During my doctoral research, this perspective became operational. My PhD work focused on acoustic wave propagation in piezoelectric crystals subjected to bias stress and bias electric fields, a setting in which higher-order elastic, piezoelectric, dielec-
tric, and electromechanical constants become unavoidable. Theoretical formulations of such problems quickly lead to tensors of third order and higher, whose components are severely constrained by crystal symmetry. It was here that a practical difficulty emerged: while second-order symmetry reductions are well known, higher-order results are scattered, incomplete, or entirely absent from the literature, and are often reproduced without derivation.

The real challenge was not formulating higher-order constitutive theories, but determining — rigorously and without error — which tensor components are allowed, which are forced to vanish, and which are related by symmetry. Trial-and-error approaches proved unreliable, especially when multiple coupled fields and tensor symmetries were involved. Small mistakes propagate easily, and inconsistencies between sources are common.

To overcome this, I developed a strictly invariance-based, constructive procedure. Starting from the full tensor transformation laws, symmetry generators of each point group are applied systematically. Each invariance condition is written explicitly, component by component, and the resulting linear constraints are resolved exhaustively. No heuristic assumptions are made, and no results are imported without verification. Although conceptually simple, this approach demands discipline, consistency of conventions, and meticulous bookkeeping. To make this feasible, I developed structured LATEX workflows that allowed the derivations to be carried out, checked, and cross-checked with minimal risk of transcription errors.

This methodology was first validated in a demanding context: trigonal piezoelectric crystals with higher-order elastic and electromechanical couplings, published in *Annalen der Physik* [1]. That paper confirmed that the method reproduces all known lower-order results and yields internally consistent higher-order extensions. However, it also made clear that the published results represented only a fragment of a much larger body of work.

In 2006, shortly after joining the faculty at the Georgia Institute of Technology, I discussed with my doctoral advisor, Distinguished Professor Mack A. Breazeale, the idea of writing either an extended article or a book devoted to higher-order material constants and their symmetry relations. His response was unambiguous: such a work would be of lasting value, precisely because it would spare researchers the need to repeatedly rediscover symmetry constraints through ad hoc calculations.

Over the years that followed, professional responsibilities — teaching, advising, and broader research — prevented the immediate completion of this project. Nevertheless, the work never stopped entirely. During extended periods spent teaching in Atlanta, I returned to these symmetry derivations during evenings and weekends, gradually expanding the coverage to additional tensor ranks and symmetry classes. On several occasions, I provided colleagues and external research groups with specific higher-order symmetry results upon request, though the full procedures remained unpublished, with the intention of eventually presenting them coherently and completely.

Only recently did the opportunity arise to revisit the entire framework, verify every derivation afresh, resolve remaining gaps, and unify the results into a single, consistent
reference. This book is the result.

Its purpose is twofold. First, it provides complete and systematically derived symmetry reductions for higher-order elastic, piezoelectric, dielectric, electrostrictive, and electromechanical tensors across all crystallographic point groups, including the isotropic limit. Second — and more importantly — it documents a reproducible method by which these results are obtained. Every table in this book is the endpoint of an explicit invariance calculation, not a compilation of unchecked formulas.

The relevance of such information has only increased. Higher-order material constants play a central role in modern applications, including nonlinear acoustics and ultrasonics, sonar and underwater sensing, energy harvesting and piezoelectric power generation, high-field electromechanical devices, precision metrology, space and aerospace technologies, and medical ultrasound and elastography. In many of these areas, materials operate under strong bias fields, prestress, or large amplitudes, where linear theories are insufficient and symmetry considerations become decisive.

This book is therefore not intended as a narrative text, but as a durable technical reference — one that can be trusted, cited, and built upon. At the same time, it is written with the conviction that rigor and transparency need not exclude accessibility. Readers familiar with group theory and tensor calculus will find the structure natural; those approaching higher-order constitutive modeling for the first time will find a clear path from symmetry principles to concrete results.

Finally, this work is offered in the spirit of openness. Foundational tools that underpin wide areas of science and engineering should be available without unnecessary barriers. It is my hope that this book will serve not only as a reference, but as an invitation: to verify, to extend, and to apply symmetry-based reasoning wherever complex material behavior demands clarity and precision.
# Introduction

The purpose of this book is to provide a complete, systematic, and rigorously derived account of symmetry relations among material constants in crystalline media, with particular emphasis on higher-order constitutive tensors. While symmetry-based arguments have long played a central role in crystallography, elasticity, and electromechanics—most notably through the tensorial formulation of physical properties in crystalline media [2]—their application to tensors beyond second order remains fragmented in the literature and, in many cases, undocumented or inconsistently presented.

At the heart of continuum descriptions of solids lies the principle that material response must be invariant under the symmetry operations of the crystal’s point group. This principle is most naturally formulated in the language of tensors and linear transformations, where matrices represent symmetry operations and tensors encode material response in a coordinate-independent manner [2, 3]. This invariance imposes powerful constraints on the admissible components of constitutive tensors, drastically reducing the number of independent material constants. For linear theories, such as classical elasticity or linear piezoelectricity, these reductions are well known and widely tabulated, and they form the basis of crystal acoustics and anisotropic wave propagation theory as developed in classical treatments [4]. However, modern applications increasingly operate outside the strictly linear regime, where higher-order elastic, dielectric, piezoelectric, and electromechanical couplings become essential.

Such higher-order effects arise naturally in the presence of large strains, strong electric fields, prestress, bias fields, or finite-amplitude wave propagation. In nonlinear elasticity, this is most systematically described by expanding the strain-energy density in a Taylor series about a stressed or biased equilibrium configuration, leading to effective elastic constants that depend explicitly on higher-order material parameters [5]. They play a decisive role in nonlinear acoustics, ultrasonic wave distortion, harmonic generation, energy harvesting, high-power piezoelectric devices, and field-biased sensing technologies. In these regimes, constitutive descriptions involve third-order and higher tensors whose symmetry reductions are far more intricate than their second-order counterparts.

The determination of these symmetry reductions is, in principle, straightforward: each tensor must remain invariant under every symmetry operation of the crystal’s point group. In practice, however, the complexity grows rapidly with tensor rank. The number of components increases combinatorially, multiple intrinsic index symmetries must be respected, and different physical fields may be coupled. As a result, higher-
order symmetry relations are often derived ad hoc, incompletely, or not at all, and published results are difficult to verify independently.

This book addresses that gap by adopting a strictly constructive and invariance-based approach. The methodological foundations of this approach were developed during the author's doctoral research on nonlinear wave propagation in anisotropic and piezoelectric media [6], where higher-order constitutive effects first emerged as a practical necessity rather than a purely formal extension. Starting from the full tensor transformation laws, symmetry generators of each point group are applied explicitly, one by one. The resulting invariance conditions are written component-wise and resolved exhaustively as linear constraints. No symmetry relations are assumed a priori beyond the intrinsic index symmetries dictated by the physical nature of each tensor. In this way, every reduction presented in this book follows transparently from first principles.

The scope of the book is deliberately comprehensive. It covers all crystallographic point groups and treats, within a unified framework, elastic, piezoelectric, dielectric, electrostrictive, and higher-order electromechanical tensors. The progression through symmetry classes is systematic, beginning with the triclinic case, where no symmetry constraints beyond intrinsic ones exist, and advancing through increasing levels of symmetry to the isotropic limit. This organization mirrors the logical structure of crystallographic classification and allows the reader to see explicitly how additional symmetry operations progressively restrict material behavior.

The foundational concepts and invariance principles underlying all subsequent developments are introduced in Chapter 1. There, the general notion of material constants, tensorial constitutive laws, and symmetry invariance is established independently of any particular crystal class. This chapter serves as the conceptual anchor for the remainder of the book.

Chapters 2 through 8 are devoted to individual symmetry systems. Each chapter treats a single crystal family in detail, deriving the allowed tensor components and their relations for the relevant point groups. The presentation is intentionally uniform across chapters, so that results for different symmetries can be compared directly and patterns in symmetry reduction become apparent.

Chapter 9 considers isotropic media as a limiting case, showing how full rotational invariance further restricts the constitutive tensors obtained for cubic symmetry. This chapter provides a bridge between crystallographic media and classical isotropic continuum mechanics.

To facilitate practical use, Chapter 10 collects the results of the preceding chapters into consolidated summary tables, allowing rapid identification of the number of independent constants for each tensor and symmetry class. These tables are intended as a reference for researchers and engineers who require reliable symmetry information without repeating lengthy derivations.

The book concludes in Chapter 11 with a discussion of implications, limitations, and possible extensions, including connections to computational implementations and emerging applications where higher-order material behavior is expected to play an
increasingly important role.

The appendices provide essential technical support for the main text. Appendix A summarizes tensor transformation rules; Appendix B details Voigt and generalized Voigt notation; Appendix C lists the point-group generators used throughout the book; and Appendix D presents worked invariance calculations that illustrate the methodology step by step. Together, these appendices ensure that all results in the book can be independently reproduced and verified.

In this way, the book is intended to function both as a rigorous reference and as a methodological guide. While it does not aim to be a tutorial on group theory or continuum mechanics, it seeks to make the symmetry-based derivation of higher-order material constants transparent, reproducible, and dependable—qualities that are essential when such constants are used as the foundation for theoretical modeling and experimental interpretation.

*Scope.* This book treats crystallographic (non-magnetic) point groups acting on polar and axial tensors in the usual spatial sense; magnetic point groups (Shubnikov groups) and explicit time-reversal symmetry constraints are not considered. Accordingly, magneto-electric couplings and time-reversal-odd constitutive tensors are outside the present scope.
# Notation and Conventions

This section defines the notation and conventions used consistently throughout the book. Because symmetry derivations are highly sensitive to sign conventions, index orderings, and representation choices, all conventions are fixed explicitly here and adhered to without exception in the main text and appendices.

## Coordinate System and Indices

A fixed right-handed Cartesian coordinate system ($x_1, x_2, x_3$) is used throughout. All tensors are expressed with respect to this basis unless explicitly stated otherwise.

* Latin indices i, j, k, l, m, n, p, q, r, s, t, u take values in {1, 2, 3}.

* Repeated indices imply summation (Einstein summation convention).

* Index position (upper or lower) is not distinguished, since all transformations are orthogonal.

## Tensor Character

Unless explicitly noted otherwise:

* All material tensors are polar tensors.

* Axial (pseudo-)tensor behavior is stated explicitly when relevant.

* Time-reversal symmetry is not invoked unless explicitly stated.

## Orthogonal Transformations

Point-group symmetry operations are represented by orthogonal matrices $\mathbf{a} \in O(3)$ with components $a_{ij}$ satisfying

$$a_{ip}a_{jp} = \delta_{ij}, \quad a_{pi}a_{pj} = \delta_{ij}. \qquad (0.1)$$

* Proper rotations satisfy $\det \mathbf{a} = +1$.

* Improper operations (reflections, inversion) satisfy $\det \mathbf{a} = -1$.

All symmetry generators are listed explicitly in Appendix C.
Tensor Transformation Convention

A Cartesian tensor $T_{i_1i_2...i_n}$ transforms under a symmetry operation a as

$$T'_{i_1i_2...i_n} = a_{i_1j_1}a_{i_2j_2}...a_{i_nj_n}T_{j_1j_2...j_n} \quad (0.2)$$

This convention corresponds to an active transformation acting on the physical tensor components in a fixed coordinate frame.

Invariance Condition

A tensor is said to be invariant under a symmetry operation a if

$$T'_{i_1i_2...i_n} = T_{i_1i_2...i_n} \quad (0.3)$$

Invariance under all generators of a point group is both necessary and sufficient for invariance under the full group.

Intrinsic Tensor Symmetries

Intrinsic (permutation) symmetries arising from the constitutive framework are always enforced before crystal symmetry is applied. Examples include: symmetry of strain and stress tensors, symmetry within index pairs of energy-derived tensors, full symmetry of higher-order elastic tensors under permutation of index pairs.

This separation of intrinsic symmetries from crystal symmetries is essential for clarity and correctness.

Voigt and Generalized Voigt Notation

Voigt notation is used exclusively for symmetric index pairs, following the mapping

$$(11, 22, 33, 23, 13, 12) \leftrightarrow (1, 2, 3, 4, 5, 6). \quad (0.4)$$

* No engineering factors (such as factors of 2 in shear components) are introduced.

* Voigt notation is treated as a representational tool only; all tensorial meaning resides in the underlying Cartesian tensor.

* Generalized Voigt notation is used when multiple symmetric index pairs occur.

Full details are provided in Appendix B.

Constitutive Tensors Considered

The following classes of material tensors are treated systematically in this book:
• second- and third-order elastic tensors,

• linear and higher-order piezoelectric tensors,

• linear and nonlinear dielectric tensors,

• electrostrictive tensors.

All tensors are assumed to be defined with respect to a reference configuration and small-signal expansions, unless stated otherwise.

## Sign Conventions

All signs follow directly from:

• the chosen Cartesian basis,

• the explicit generator matrices,

• the tensor transformation law.

No additional sign conventions are introduced. In particular:

• shear components are not rescaled,

• no hidden minus signs are absorbed into Voigt mappings,

• all sign changes arise explicitly from transformation matrices.

## Consistency Across the Book

The conventions defined in this chapter are used without exception in: all subsequent symmetry chapters, the summary tables, all appendices.

Any deviation from these conventions would invalidate cross-chapter comparisons; none are made.
# List of Symbols

This list collects the symbols used throughout the book, together with their meaning. Symbols are grouped by category. When a symbol is used with additional indices, its meaning follows directly from the base definition given here.

## Indices and Sets

<table><tr><td>$i, j, k, l, m, n, p, q, r, s, t, u$</td><td>Cartesian indices, values in $\{1, 2, 3\}$</td></tr><tr><td>$I, J, K, P, Q, R$</td><td>Voigt or generalized Voigt indices, values in $\{1, \dots, 6\}$</td></tr><tr><td>$\alpha$</td><td>Index labeling point-group generators</td></tr><tr><td>$\mathcal{G}$</td><td>Point group or symmetry group</td></tr></table>

## Matrices and Group Elements

<table><tr><td>a</td><td>Orthogonal transformation (point-group operation)</td></tr><tr><td>$a_{ij}$</td><td>Cartesian components of a</td></tr><tr><td>I</td><td>Identity matrix</td></tr><tr><td>i</td><td>Inversion operator, $-I$</td></tr><tr><td>det a</td><td>Determinant of an orthogonal transformation</td></tr><tr><td>SO(3)</td><td>Group of proper rotations in three dimensions</td></tr><tr><td>O(3)</td><td>Group of all orthogonal transformations in three dimensions</td></tr></table>

## Tensor Transformation and Invariance

<table><tr><td>$T_{i_1 \dots i_n}$</td><td>Cartesian tensor of rank n</td></tr><tr><td>$T'_{i_1 \dots i_n}$</td><td>Transformed tensor under a symmetry operation</td></tr><tr><td>$K_{I,J}(a)$</td><td>Voigt transformation matrix induced by a</td></tr><tr><td>$K(a)$</td><td>$6 \times 6$ Voigt transformation matrix</td></tr><tr><td>$\delta_{ij}$</td><td>Kronecker delta</td></tr><tr><td>$\epsilon_{ijk}$</td><td>Levi-Civita tensor</td></tr></table>
Elasticity

$c_{ijkl}$ Second-order elastic stiffness tensor

$C_{IJ}$ Voigt representation of $c_{ijkl}$

$c_{ijklmn}$ Third-order elastic stiffness tensor

$C_{IJK}$ Generalized Voigt representation of $c_{ijklmn}$

$\lambda, \mu$ Lamé constants (isotropic elasticity)

$l, m, n$ Murnaghan third-order elastic constants

Piezoelectricity

$\xi_{kij}$ Linear piezoelectric tensor

$\xi_{kI}$ Voigt representation of $\xi_{kij}$

$\xi_{kijmn}$ Fifth-order piezoelectric tensor

$\xi_{kIJ}$ Generalized Voigt representation of $\xi_{kijmn}$

Dielectric Response

$\epsilon_{ij}$ Second-order dielectric tensor

$\epsilon_{ijk}$ Third-order dielectric tensor

$\epsilon$ Scalar dielectric constant (isotropic case)

Electrostriction

$\gamma_{ijkl}$ Electrostrictive tensor

$\Gamma_{IJ}$ Voigt representation of $\gamma_{ijkl}$

Strain and Energy

E Green-Lagrange strain tensor

W Strain energy density

tr(·) Trace of a tensor

Symmetry Classes

1, 2, m, 222, mm2, mmm Triclinic, monoclinic, and orthorhombic classes

4, 422, 4mm, 4/m, 4/mmm Tetragonal classes

3, 32, 3m, 3m Trigonal classes

6, 622, 6mm, 6/m, 6/mmm Hexagonal classes

23, m3, 432, 43m, m3m Cubic classes

Miscellaneous

$N_{(i)}$ Number of independent material constants

≡ Identical vanishing or identity by symmetry
All symbols are used consistently throughout the book in accordance with the conventions defined in the preceding section.
# List of Symmetry Classes and Point Groups

This section provides a consolidated overview of all crystallographic symmetry classes and point groups considered in the book. It serves as a quick reference linking the standard crystallographic notation to the chapter in which each class is treated in detail, together with its defining generators.

Throughout the book, only *point-group* symmetries are considered; translational symmetries of space groups play no role in the tensorial invariance of homogeneous material constants.

## Triclinic System

<table><thead><tr><td>Crystal system</td><td>Point group</td><td>Chapter</td></tr></thead><tbody><tr><td>Triclinic</td><td>1</td><td>Chapter 2</td></tr></tbody></table>

*Characteristics:* No symmetry operations beyond the identity. All material constants allowed subject only to intrinsic tensor symmetries.

## Monoclinic System

<table><thead><tr><td>Crystal system</td><td>Point group</td><td>Chapter</td></tr></thead><tbody><tr><td>Monoclinic</td><td>2</td><td>Chapter 3</td></tr><tr><td>Monoclinic</td><td>m</td><td>Chapter 3</td></tr></tbody></table>

*Characteristics:* One nontrivial symmetry operation (twofold rotation or mirror plane), introducing the first systematic reductions relative to triclinic symmetry.

## Orthorhombic System

<table><thead><tr><td>Crystal system</td><td>Point group</td><td>Chapter</td></tr></thead><tbody><tr><td>Orthorhombic</td><td>222</td><td>Chapter 4</td></tr><tr><td>Orthorhombic</td><td>mm2</td><td>Chapter 4</td></tr><tr><td>Orthorhombic</td><td>mmm</td><td>Chapter 4</td></tr></tbody></table>

*Characteristics:* Three mutually orthogonal symmetry directions. Strong reduction of tensor components through commuting symmetry operations.
## Tetragonal System

<table><thead><tr><th>Crystal system</th><th>Point group</th><th>Chapter</th></tr></thead><tbody><tr><td>Tetragonal</td><td>4</td><td>Chapter 5</td></tr><tr><td>Tetragonal</td><td>422</td><td>Chapter 5</td></tr><tr><td>Tetragonal</td><td>4mm</td><td>Chapter 5</td></tr><tr><td>Tetragonal</td><td>4/m</td><td>Chapter 5</td></tr><tr><td>Tetragonal</td><td>4mmm</td><td>Chapter 5</td></tr></tbody></table>

*Characteristics:* One principal fourfold axis, inducing rotational degeneracies in the transverse plane.

## Trigonal System

<table><thead><tr><th>Crystal system</th><th>Point group</th><th>Chapter</th></tr></thead><tbody><tr><td>Trigonal</td><td>3</td><td>Chapter 6</td></tr><tr><td>Trigonal</td><td>32</td><td>Chapter 6</td></tr><tr><td>Trigonal</td><td>3m</td><td>Chapter 6</td></tr><tr><td>Trigonal</td><td>3&#773;m</td><td>Chapter 6</td></tr></tbody></table>

*Characteristics:* Threefold rotational symmetry. This system serves as a benchmark in the book, with complete higher-order results verified against the author's Annalen der Physik publication.

## Hexagonal System

<table><thead><tr><th>Crystal system</th><th>Point group</th><th>Chapter</th></tr></thead><tbody><tr><td>Hexagonal</td><td>6</td><td>Chapter 7</td></tr><tr><td>Hexagonal</td><td>622</td><td>Chapter 7</td></tr><tr><td>Hexagonal</td><td>6mm</td><td>Chapter 7</td></tr><tr><td>Hexagonal</td><td>6/m</td><td>Chapter 7</td></tr><tr><td>Hexagonal</td><td>6mmm</td><td>Chapter 7</td></tr></tbody></table>

*Characteristics:* Sixfold rotational symmetry. Additional constraints relative to trigonal symmetry lead to further collapse of admissible tensor structures.

## Cubic System

<table><thead><tr><th>Crystal system</th><th>Point group</th><th>Chapter</th></tr></thead><tbody><tr><td>Cubic</td><td>23</td><td>Chapter 8</td></tr><tr><td>Cubic</td><td>m&#773;3</td><td>Chapter 8</td></tr><tr><td>Cubic</td><td>432</td><td>Chapter 8</td></tr><tr><td>Cubic</td><td>4&#773;3m</td><td>Chapter 8</td></tr><tr><td>Cubic</td><td>m&#773;3m</td><td>Chapter 8</td></tr></tbody></table>

*Characteristics:* Highest crystallographic symmetry. All coordinate axes are equivalent, yielding highly constrained and elegant tensor forms.
## Isotropic Limit

<table><thead><tr><td>System</td><td>Symmetry group</td><td>Chapter</td></tr></thead><tbody><tr><td>Isotropic</td><td>SO(3)</td><td>Chapter 9</td></tr></tbody></table>

*Characteristics:* Full rotational invariance. This is not a crystallographic point group but the continuous-symmetry limit of increasing rotational symmetry.

## Concluding Remark

This list provides the structural map for the entire book. Each symmetry class listed here is treated using the same invariant-based procedure, ensuring consistency, completeness, and direct comparability across crystal systems and tensor ranks.
# Contents

<table><tr><td>Preface</td><td>vii</td></tr><tr><td>Introduction</td><td>xi</td></tr><tr><td>Notation and Conventions</td><td>xv</td></tr><tr><td>List of Symbols</td><td>xix</td></tr><tr><td>List of Symmetry Classes and Point Groups</td><td>xxiii</td></tr><tr><td>1 Material Constants &amp; Symmetry Invariance</td><td>1</td></tr><tr><td>1.1 Material Constants as Tensorial Objects</td><td>1</td></tr><tr><td>1.2 Intrinsic Permutation Symmetries</td><td>1</td></tr><tr><td>1.3 Active and Passive Symmetry Transformations</td><td>2</td></tr><tr><td>1.4 Point Groups and Generators</td><td>2</td></tr><tr><td>1.5 Tensor Transformation Laws</td><td>2</td></tr><tr><td>1.6 Invariance as a Necessary and Sufficient Condition</td><td>3</td></tr><tr><td>1.7 Voigt and Generalized Voigt Notation</td><td>3</td></tr><tr><td>1.8 Systematic Derivation Strategy</td><td>3</td></tr><tr><td>1.9 Error Sources and Constructive Robustness</td><td>4</td></tr><tr><td>1.10 Scope of Applicability</td><td>4</td></tr><tr><td>1.11 Perspective</td><td>4</td></tr><tr><td>2 Triclinic Symmetry</td><td>5</td></tr><tr><td>2.1 Definition of Triclinic Symmetry (Class 1)</td><td>5</td></tr><tr><td>2.2 Absence of Symmetry Constraints Beyond Intrinsic Symmetries</td><td>5</td></tr><tr><td>2.3 Second-Order Elastic Constants</td><td>6</td></tr><tr><td>2.3.1 Tensor and Intrinsic Symmetries</td><td>6</td></tr><tr><td>2.3.2 Number of Independent Constants</td><td>6</td></tr><tr><td>2.3.3 General Triclinic Form</td><td>6</td></tr><tr><td>2.4 Third-Order Elastic Constants</td><td>6</td></tr><tr><td>2.4.1 Tensor and Intrinsic Symmetries</td><td>6</td></tr><tr><td>2.4.2 Number of Independent Constants</td><td>7</td></tr><tr><td>2.4.3 Indexing Convention for Independent Components</td><td>7</td></tr><tr><td>2.5 Piezoelectric Constants</td><td>7</td></tr></table>
<table><tr><td>2.5.1</td><td>Second-Order (Linear) Piezoelectric Tensor</td><td>7</td></tr><tr><td>2.5.2</td><td>Higher-Order Piezoelectric Tensor (Fifth Order)</td><td>7</td></tr><tr><td>2.6</td><td>Dielectric Constants</td><td>8</td></tr><tr><td>2.6.1</td><td>Second-Order Dielectric Tensor</td><td>8</td></tr><tr><td>2.6.2</td><td>Third-Order Dielectric Tensor (quadratic electric-field response)</td><td>8</td></tr><tr><td>2.7</td><td>Electrostrictive Tensor</td><td>9</td></tr><tr><td>2.8</td><td>Practical relevance &amp; typical material examples</td><td>9</td></tr><tr><td>2.9</td><td>Baseline Summary</td><td>9</td></tr></table>

3 Monoclinic Symmetry 11

<table><tr><td>3.1</td><td>Monoclinic Point Groups and Coordinate Convention</td><td>11</td></tr><tr><td>3.2</td><td>Transformation of Symmetric Second-Rank Tensors and Voigt Mapping</td><td>11</td></tr><tr><td>3.3</td><td>Second-Order Elastic Constants (Classes 2 &amp; m)</td><td>12</td></tr><tr><td>3.3.1</td><td>Invariance Equation</td><td>12</td></tr><tr><td>3.3.2</td><td>Resulting Form and Independent Constants</td><td>12</td></tr><tr><td>3.4</td><td>Third-Order Elastic Constants (Classes 2 &amp; m)</td><td>13</td></tr><tr><td>3.4.1</td><td>Invariance Equation</td><td>13</td></tr><tr><td>3.4.2</td><td>Parity Rule and Independent Count</td><td>13</td></tr><tr><td>3.5</td><td>Linear Piezoelectric Constants</td><td>13</td></tr><tr><td>3.5.1</td><td>Class 2</td><td>13</td></tr><tr><td>3.5.2</td><td>Class m</td><td>14</td></tr><tr><td>3.5.3</td><td>Class 2/m</td><td>14</td></tr><tr><td>3.6</td><td>Fifth-Order Piezoelectric Constants</td><td>14</td></tr><tr><td>3.6.1</td><td>Classes 2 &amp; m: component rule and counts</td><td>14</td></tr><tr><td>3.7</td><td>Dielectric Constants</td><td>14</td></tr><tr><td>3.7.1</td><td>Second-Order Dielectric Tensor</td><td>14</td></tr><tr><td>3.7.2</td><td>Third-Order Dielectric Tensor (quadratic electric-field response)</td><td>15</td></tr><tr><td>3.8</td><td>Electrostrictive Tensor (Classes 2, m, and 2/m)</td><td>15</td></tr><tr><td>3.9</td><td>Practical relevance &amp; typical material examples</td><td>16</td></tr><tr><td>3.10</td><td>Comparison with the Triclinic Baseline</td><td>16</td></tr></table>

4 Orthorhombic Symmetry 19

<table><tr><td>4.1</td><td>Orthorhombic Point Groups and Generators</td><td>19</td></tr><tr><td>4.2</td><td>Voigt Transformation of Symmetric Index Pairs</td><td>20</td></tr><tr><td>4.3</td><td>Sequential Invariance and Cumulative Constraint</td><td>20</td></tr><tr><td>4.4</td><td>Second-Order Elastic Constants</td><td>21</td></tr><tr><td>4.4.1</td><td>Invariance Equations</td><td>21</td></tr><tr><td>4.4.2</td><td>Resulting Orthorhombic Form and Independent Constants</td><td>21</td></tr><tr><td>4.5</td><td>Third-Order Elastic Constants</td><td>21</td></tr><tr><td>4.5.1</td><td>Invariance Equations</td><td>21</td></tr><tr><td>4.5.2</td><td>Allowed Components and Independent Count</td><td>22</td></tr><tr><td>4.6</td><td>Linear Piezoelectric Constants</td><td>22</td></tr><tr><td>4.6.1</td><td>Class 222</td><td>22</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>4.6.2</td>
      <td>Class mm2 (polar axis x<sub>3</sub>)</td>
      <td>23</td>
    </tr>
    <tr>
      <td>4.6.3</td>
      <td>Class mmm</td>
      <td>23</td>
    </tr>
    <tr>
      <td>4.7</td>
      <td>Fifth-Order Piezoelectric Constants</td>
      <td>23</td>
    </tr>
    <tr>
      <td>4.7.1</td>
      <td>Class 222</td>
      <td>23</td>
    </tr>
    <tr>
      <td>4.7.2</td>
      <td>Class mm2 (polar axis x<sub>3</sub>)</td>
      <td>23</td>
    </tr>
    <tr>
      <td>4.7.3</td>
      <td>Class mmm</td>
      <td>24</td>
    </tr>
    <tr>
      <td>4.8</td>
      <td>Dielectric Constants</td>
      <td>24</td>
    </tr>
    <tr>
      <td>4.8.1</td>
      <td>Second-Order Dielectric Tensor</td>
      <td>24</td>
    </tr>
    <tr>
      <td>4.8.2</td>
      <td>Third-Order Dielectric Tensor (quadratic electric-field response)</td>
      <td>24</td>
    </tr>
    <tr>
      <td>4.9</td>
      <td>Electrostrictive Tensor</td>
      <td>25</td>
    </tr>
    <tr>
      <td>4.10</td>
      <td>Practical relevance &amp; typical material examples</td>
      <td>25</td>
    </tr>
    <tr>
      <td>4.11</td>
      <td>Summary and Emerging Structural Patterns</td>
      <td>25</td>
    </tr>
    <tr>
      <td><b>5</b></td>
      <td><b>Tetragonal Symmetry</b></td>
      <td><b>27</b></td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Tetragonal Point Groups and Principal-Axis Convention</td>
      <td>27</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Voigt Transformation under Fourfold Rotation</td>
      <td>28</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>Second-Order Elastic Constants</td>
      <td>28</td>
    </tr>
    <tr>
      <td>5.3.1</td>
      <td>Invariance under a<sub>4</sub></td>
      <td>28</td>
    </tr>
    <tr>
      <td>5.3.2</td>
      <td>Additional Orthogonal Symmetries: 422, 4mm, and 4/mmm</td>
      <td>29</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>Third-Order Elastic Constants</td>
      <td>29</td>
    </tr>
    <tr>
      <td>5.4.1</td>
      <td>Invariance Equations</td>
      <td>29</td>
    </tr>
    <tr>
      <td>5.4.2</td>
      <td>Independent Counts</td>
      <td>30</td>
    </tr>
    <tr>
      <td>5.5</td>
      <td>Linear Piezoelectric Constants</td>
      <td>30</td>
    </tr>
    <tr>
      <td>5.5.1</td>
      <td>Class 4</td>
      <td>30</td>
    </tr>
    <tr>
      <td>5.5.2</td>
      <td>Class 422</td>
      <td>30</td>
    </tr>
    <tr>
      <td>5.5.3</td>
      <td>Class 4mm</td>
      <td>31</td>
    </tr>
    <tr>
      <td>5.5.4</td>
      <td>Centrosymmetric tetragonal classes</td>
      <td>31</td>
    </tr>
    <tr>
      <td>5.6</td>
      <td>Fifth-Order Piezoelectric Constants</td>
      <td>31</td>
    </tr>
    <tr>
      <td>5.7</td>
      <td>Dielectric Constants</td>
      <td>31</td>
    </tr>
    <tr>
      <td>5.7.1</td>
      <td>Second-Order Dielectric Tensor</td>
      <td>31</td>
    </tr>
    <tr>
      <td>5.7.2</td>
      <td>Third-Order Dielectric Tensor (quadratic electric-field response)</td>
      <td>32</td>
    </tr>
    <tr>
      <td>5.8</td>
      <td>Electrostrictive Tensor</td>
      <td>32</td>
    </tr>
    <tr>
      <td>5.8.1</td>
      <td>Class 4 (fourfold rotation only)</td>
      <td>33</td>
    </tr>
    <tr>
      <td>5.8.2</td>
      <td>Higher tetragonal classes: 422, 4mm, 4/mmm</td>
      <td>33</td>
    </tr>
    <tr>
      <td>5.9</td>
      <td>Practical relevance &amp; typical material examples</td>
      <td>33</td>
    </tr>
    <tr>
      <td>5.10</td>
      <td>Summary and Practical Relevance</td>
      <td>34</td>
    </tr>
    <tr>
      <td><b>6</b></td>
      <td><b>Trigonal Symmetry</b></td>
      <td><b>35</b></td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Trigonal 3m and Generator Choice</td>
      <td>35</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Invariance Enforcement (Operational Form)</td>
      <td>35</td>
    </tr>
    <tr>
      <td>6.3</td>
      <td>Second-Order Elastic Constants c<sub>ijkl</sub></td>
      <td>36</td>
    </tr>
    <tr>
      <td>6.4</td>
      <td>Third-Order Elastic Constants c<sub>ijklmn</sub></td>
      <td>36</td>
    </tr>
    <tr>
      <td>6.4.1</td>
      <td>Forced Zeros</td>
      <td>36</td>
    </tr>
  </tbody>
</table>
<table><tbody><tr><td>6.4.2</td><td>Linear Relations</td><td>36</td></tr><tr><td>6.5</td><td>Second-Order Piezoelectric Constants ξ<sub>kij</sub></td><td>37</td></tr><tr><td>6.6</td><td>Third-Order Piezoelectric Constants ξ<sub>kijmn</sub></td><td>37</td></tr><tr><td>6.6.1</td><td>Forced Zeros</td><td>37</td></tr><tr><td>6.6.2</td><td>Linear Relations</td><td>37</td></tr><tr><td>6.7</td><td>Dielectric Constants</td><td>38</td></tr><tr><td>6.7.1</td><td>Second-Order Dielectric Tensor ε<sub>ij</sub></td><td>38</td></tr><tr><td>6.7.2</td><td>Third-Order Dielectric Tensor: symmetry-allowed form</td><td>38</td></tr><tr><td>6.8</td><td>Electrostrictive Constants γ<sub>ijkl</sub></td><td>39</td></tr><tr><td>6.8.1</td><td>Forced Zeros</td><td>39</td></tr><tr><td>6.8.2</td><td>Identifications and Linear Relations</td><td>39</td></tr><tr><td>6.9</td><td>Benchmark Tables (Independent vs Dependent Sets)</td><td>39</td></tr><tr><td>6.10</td><td>Consistency with Literature and Benchmark Role</td><td>40</td></tr><tr><td>6.11</td><td>Practical relevance &amp; typical material examples</td><td>40</td></tr><tr><td><b>7</b></td><td><b>Hexagonal Symmetry</b></td><td><b>41</b></td></tr><tr><td>7.1</td><td>Hexagonal Point Groups and Generator Choice</td><td>41</td></tr><tr><td>7.2</td><td>Relation to Trigonal Symmetry</td><td>41</td></tr><tr><td>7.3</td><td>Voigt Transformation under Sixfold Rotation</td><td>42</td></tr><tr><td>7.4</td><td>Second-Order Elastic Constants</td><td>42</td></tr><tr><td>7.4.1</td><td>Sixfold Rotational Invariance</td><td>42</td></tr><tr><td>7.5</td><td>Third-Order Elastic Constants</td><td>43</td></tr><tr><td>7.5.1</td><td>Generalized Voigt Form and Invariance</td><td>43</td></tr><tr><td>7.5.2</td><td>Independent Parameter Set</td><td>43</td></tr><tr><td>7.6</td><td>Linear Piezoelectric Constants</td><td>44</td></tr><tr><td>7.6.1</td><td>Class 6 (sixfold rotation only)</td><td>44</td></tr><tr><td>7.6.2</td><td>Class 622</td><td>44</td></tr><tr><td>7.6.3</td><td>Class 6mm</td><td>45</td></tr><tr><td>7.6.4</td><td>Centrosymmetric classes 6/m and 6/mmm</td><td>45</td></tr><tr><td>7.7</td><td>Fifth-Order Piezoelectric Constants</td><td>45</td></tr><tr><td>7.7.1</td><td>Class 6mm</td><td>45</td></tr><tr><td>7.7.2</td><td>Class 622</td><td>46</td></tr><tr><td>7.7.3</td><td>Classes 6, 6/m, and 6/mmm</td><td>46</td></tr><tr><td>7.8</td><td>Dielectric Constants</td><td>47</td></tr><tr><td>7.8.1</td><td>Second-Order Dielectric Tensor</td><td>47</td></tr><tr><td>7.8.2</td><td>Third-Order Dielectric Tensor (quadratic electric-field response)</td><td>47</td></tr><tr><td>7.9</td><td>Electrostrictive Tensor</td><td>47</td></tr><tr><td>7.10</td><td>Summary and Physical Interpretation</td><td>48</td></tr><tr><td>7.11</td><td>Practical relevance &amp; typical material examples</td><td>48</td></tr><tr><td><b>8</b></td><td><b>Cubic Symmetry</b></td><td><b>49</b></td></tr><tr><td>8.1</td><td>Cubic Point Groups and Generator Choice</td><td>49</td></tr><tr><td>8.2</td><td>Strong Invariance Constraints and Axis Equivalence</td><td>50</td></tr></tbody></table>
<table><tr><td>8.3</td><td>Second-Order Elastic Constants</td><td>50</td></tr><tr><td>8.3.1</td><td>Cubic reduction of C<sub>IJ</sub></td><td>50</td></tr><tr><td>8.4</td><td>Third-Order Elastic Constants</td><td>50</td></tr><tr><td>8.4.1</td><td>Independent set in generalized Voigt form</td><td>50</td></tr><tr><td>8.4.2</td><td>Complete cubic identifications</td><td>51</td></tr><tr><td>8.5</td><td>Linear Piezoelectric Constants</td><td>51</td></tr><tr><td>8.6</td><td>Second- and Third-Order Dielectric Constants</td><td>52</td></tr><tr><td>8.6.1</td><td>Second-order dielectric tensor</td><td>52</td></tr><tr><td>8.6.2</td><td>Third-order dielectric tensor (quadratic electric-field response)</td><td>52</td></tr><tr><td>8.7</td><td>Electrostrictive Tensor</td><td>52</td></tr><tr><td>8.8</td><td>Comparison with the Isotropic Limit</td><td>53</td></tr><tr><td>8.9</td><td>Practical relevance &amp; typical material examples</td><td>53</td></tr></table>

<table><tr><td>9</td><td>Isotropic Media as a Limiting Case</td><td>55</td></tr><tr><td>9.1</td><td>Isotropy as Full Rotational Invariance</td><td>55</td></tr><tr><td>9.2</td><td>Recovery of Lamé Form for Linear Elasticity</td><td>55</td></tr><tr><td>9.2.1</td><td>Fourth-rank isotropic stiffness tensor</td><td>55</td></tr><tr><td>9.2.2</td><td>Voigt form and isotropic condition as a cubic limit</td><td>56</td></tr><tr><td>9.3</td><td>Isotropic Higher-Order Elasticity</td><td>56</td></tr><tr><td>9.3.1</td><td>Energy-based characterization and invariant basis</td><td>56</td></tr><tr><td>9.3.2</td><td>Sixth-rank third-order stiffness and independent count</td><td>56</td></tr><tr><td>9.3.3</td><td>Practical representation</td><td>57</td></tr><tr><td>9.4</td><td>Isotropic Dielectric Response &amp; Odd-Rank Polar Tensors</td><td>57</td></tr><tr><td>9.4.1</td><td>Second-order dielectric tensor</td><td>57</td></tr><tr><td>9.4.2</td><td>Third-order dielectric tensor (quadratic electric-field response)</td><td>57</td></tr><tr><td>9.5</td><td>Isotropic Piezoelectricity</td><td>57</td></tr><tr><td>9.6</td><td>Isotropic Electrostriction</td><td>58</td></tr><tr><td>9.7</td><td>Connection to Nonlinear Continuum Mechanics</td><td>58</td></tr></table>

<table><tr><td>10</td><td>Summary Tables for All Symmetry Classes</td><td>59</td></tr><tr><td>10.1</td><td>Purpose and Scope</td><td>59</td></tr><tr><td>10.2</td><td>Symmetry Classes Considered</td><td>59</td></tr><tr><td>10.3</td><td>Second-Order Elasticity c<sub>ijkl</sub></td><td>60</td></tr><tr><td>10.4</td><td>Third-Order Elasticity c<sub>ijklmn</sub></td><td>60</td></tr><tr><td>10.5</td><td>Linear Piezoelectricity ξ<sub>kij</sub></td><td>60</td></tr><tr><td>10.6</td><td>Higher-Order Piezoelectricity ξ<sub>kijmn</sub></td><td>60</td></tr><tr><td>10.7</td><td>Dielectric Tensors</td><td>60</td></tr><tr><td>10.8</td><td>Independent Constants: Electrostriction</td><td>60</td></tr><tr><td>10.9</td><td>Concluding Remarks</td><td>61</td></tr></table>

<table><tr><td>11</td><td>Conclusions and Outlook</td><td>63</td></tr><tr><td>11.1</td><td>Summary of the Invariance-Based Framework</td><td>63</td></tr><tr><td>11.2</td><td>Achievements Across Tensor Rank and Symmetry Class</td><td>63</td></tr><tr><td>11.3</td><td>Robustness, Transparency, and Reproducibility</td><td>64</td></tr></table>
<table><tbody><tr><td>11.4 Position Within the Literature</td><td>65</td></tr><tr><td>11.5 Relevance to Modern Research and Applications</td><td>65</td></tr><tr><td>11.6 Outlook and Future Directions</td><td>65</td></tr><tr><td>11.7 Final Perspective</td><td>66</td></tr><tr><td><b>A Tensor Transformation Rules</b></td><td><b>67</b></td></tr><tr><td>A.1 Purpose and Scope of This Appendix</td><td>67</td></tr><tr><td>A.2 Orthogonal Transformations and Notation</td><td>67</td></tr><tr><td>A.3 Transformation of Vectors and Covectors</td><td>67</td></tr><tr><td>A.4 General Transformation Law for Cartesian Tensors</td><td>68</td></tr><tr><td>A.5 Invariance Condition</td><td>68</td></tr><tr><td>A.6 Transformation of Second-Rank Tensors</td><td>68</td></tr><tr><td>A.6.1 Symmetric and Antisymmetric Parts</td><td>68</td></tr><tr><td>A.7 Transformation of Fourth-Rank Tensors</td><td>69</td></tr><tr><td>A.8 Transformation of Higher-Rank Tensors</td><td>69</td></tr><tr><td>A.8.1 Sixth-Rank Tensors</td><td>69</td></tr><tr><td>A.8.2 Fifth-Rank Tensors</td><td>69</td></tr><tr><td>A.9 Voigt and Generalized Voigt Transformation Rules</td><td>69</td></tr><tr><td>A.9.1 Voigt Mapping for Symmetric Index Pairs</td><td>69</td></tr><tr><td>A.9.2 Generalized Voigt Notation</td><td>70</td></tr><tr><td>A.10 Mixed Polar–Tensor Transformations</td><td>70</td></tr><tr><td>A.11 Summary</td><td>70</td></tr><tr><td><b>B Voigt and Generalized Voigt Notation</b></td><td><b>71</b></td></tr><tr><td>B.1 Purpose of This Appendix</td><td>71</td></tr><tr><td>B.2 Symmetric Second-Rank Tensors</td><td>71</td></tr><tr><td>B.2.1 Intrinsic Symmetry</td><td>71</td></tr><tr><td>B.2.2 Voigt Index Mapping</td><td>72</td></tr><tr><td>B.3 Transformation Behavior in Voigt Form</td><td>72</td></tr><tr><td>B.3.1 Diagonal Orthogonal Transformations</td><td>72</td></tr><tr><td>B.3.2 Non-Diagonal Rotations</td><td>73</td></tr><tr><td>B.4 Fourth-Rank Tensors in Voigt Form</td><td>73</td></tr><tr><td>B.4.1 Intrinsic Pair Symmetries</td><td>73</td></tr><tr><td>B.4.2 Voigt Representation</td><td>73</td></tr><tr><td>B.5 Generalized Voigt Notation</td><td>73</td></tr><tr><td>B.5.1 Motivation</td><td>73</td></tr><tr><td>B.5.2 Third-Order Elasticity</td><td>73</td></tr><tr><td>B.5.3 Mixed Cartesian–Voigt Tensors</td><td>74</td></tr><tr><td>B.6 Distinction Between Tensor and Representation</td><td>74</td></tr><tr><td>B.7 Common Pitfalls and How They Are Avoided Here</td><td>74</td></tr><tr><td>B.8 Summary</td><td>75</td></tr></tbody></table>
C Point-Group Generators 77

C.1 Purpose of This Appendix 77

C.2 General Principles for Generator Selection 77

C.3 Triclinic Class 78

C.3.1 Class 1 78

C.4 Monoclinic Classes 78

C.4.1 Class 2 78

C.4.2 Class m 78

C.5 Orthorhombic Classes 78

C.5.1 Class 222 78

C.5.2 Class mm2 78

C.5.3 Class mmm 79

C.6 Tetragonal Classes 79

C.6.1 Fourfold Rotation 79

C.6.2 Class 4 79

C.6.3 Class 422 79

C.6.4 Class 4mm 79

C.6.5 Class 4/mmm 79

C.7 Trigonal Classes 79

C.7.1 Threefold Rotation 80

C.7.2 Class 3 80

C.7.3 Class 32 80

C.7.4 Class 3m 80

C.7.5 Class 3m 80

C.8 Hexagonal Classes 80

C.8.1 Sixfold Rotation 80

C.8.2 Class 6 80

C.8.3 Class 622 80

C.8.4 Class 6mm 81

C.8.5 Class 6/mmm 81

C.9 Cubic Classes 81

C.9.1 Proper Cubic Rotation Generators 81

C.9.2 Non-centrosymmetric cubic classes 81

C.9.3 Centrosymmetric cubic classes 81

C.10 Isotropic Case 81

C.11 Summary 82

D Worked Invariance Examples 83

D.1 Purpose of This Appendix 83

D.2 Example 1: Second-Order Elasticity in Monoclinic Class 2 83

D.2.1 Problem Setup 83

D.2.2 Voigt Transformation Matrix 84

D.2.3 Invariance Equation 84
<table><tbody><tr><td>D.2.4 Component Analysis</td><td>84</td></tr><tr><td>D.3 Example 2: Linear Piezoelectricity in mm2</td><td>84</td></tr><tr><td>D.3.1 Problem Setup</td><td>84</td></tr><tr><td>D.3.2 Transformation Law</td><td>84</td></tr><tr><td>D.3.3 Invariance under m<sub>x</sub></td><td>85</td></tr><tr><td>D.3.4 Invariance under m<sub>y</sub></td><td>85</td></tr><tr><td>D.4 Example 3: 4-fold Rotational Invariance of a 2nd-Rank Tensor</td><td>85</td></tr><tr><td>D.4.1 Problem Setup</td><td>85</td></tr><tr><td>D.4.2 Direct Cartesian Evaluation</td><td>85</td></tr><tr><td>D.4.3 Final Tetragonal Form</td><td>86</td></tr><tr><td>D.5 Example 4: Third-Order Elasticity in Cubic Symmetry</td><td>86</td></tr><tr><td>D.5.1 Problem Setup</td><td>86</td></tr><tr><td>D.5.2 Structural Consequences</td><td>86</td></tr><tr><td>D.5.3 Interpretation</td><td>86</td></tr><tr><td>D.6 Example 5: Centrosymmetry & Odd-Rank Tensors</td><td>86</td></tr><tr><td>D.6.1 Problem Setup</td><td>86</td></tr><tr><td>D.6.2 Transformation and Invariance</td><td>86</td></tr><tr><td>D.6.3 Conclusion</td><td>87</td></tr><tr><td><strong>Acknowledgments</strong></td><td><strong>89</strong></td></tr><tr><td><strong>About the Author</strong></td><td><strong>91</strong></td></tr></tbody></table>
# Chapter 1

# Material Constants & Symmetry Invariance

## 1.1 Material Constants as Tensorial Objects

Material constants arise as coefficients in constitutive relations that connect physically observable fields such as stress, strain, electric field, electric displacement, polarization, and higher-order gradients thereof. In crystalline solids, these coefficients encode the material response to mechanical, electrical, and coupled electromechanical excitations.

From a mathematical standpoint, material constants are *tensors of fixed rank*, defined with respect to a chosen orthonormal basis in physical space. For example:

* the linear elastic response is governed by a fourth-order tensor $c_{ijkl}$,

* the second-order elastic (nonlinear) response by a sixth-order tensor $c_{ijklmn}$,

* the linear piezoelectric effect by a third-order tensor $\xi_{kij}$,

* higher-order piezoelectric effects by fifth-order tensors $\xi_{kijmn}$,

* the dielectric response by second- and third-order tensors $\epsilon_{ij}$ and $\epsilon_{ijk}$,

* electrostriction by a fourth-order tensor $\gamma_{ijkl}$.

In all cases, the tensorial character of these quantities is not optional but essential: it reflects the directional dependence of the material response and ensures covariance under changes of reference frame.

## 1.2 Intrinsic Permutation Symmetries

Before considering crystal symmetry, many material tensors already possess *intrinsic permutation symmetries* imposed by the nature of the physical fields they connect.

As a representative example, the elastic stiffness tensor satisfies

$$c_{ijkl} = c_{jikl} = c_{ijk} = c_{klij}, \qquad (1.1)$$
which follows from the symmetry of stress and strain tensors and from the existence of an elastic energy density.

These intrinsic symmetries are *independent of crystal symmetry*. They reduce the number of independent components but do not, by themselves, encode any information about the underlying lattice structure. In what follows, intrinsic permutation symmetries are always assumed to be enforced *prior* to the application of crystal symmetry constraints.

## 1.3 Active and Passive Symmetry Transformations

Symmetry operations may be interpreted either passively (as changes of coordinate system) or actively (as physical transformations applied to the material). In this work, an *active interpretation* is adopted throughout.

Let $\mathbf{a} \in O(3)$ be an orthogonal transformation representing a symmetry operation of the crystal point group. Under an active transformation, physical vectors and tensors are transformed while the coordinate system is held fixed.

For a vector $v_i$, the transformation reads

$$v'_{i} = a_{ij}v_{j}. \qquad (1.2)$$

Higher-rank tensors transform by repeated application of the same rule to each index.

## 1.4 Point Groups and Generators

The symmetry of a crystal is described by a *point group*, i.e. a finite subgroup of the orthogonal group $O(3)$ that leaves at least one point fixed.

Every point group can be generated by a finite set of symmetry operations, called its *generators*. Invariance under the full group is guaranteed if and only if invariance holds under each generator.

This observation is central to the methodology of this book: instead of enforcing invariance under all symmetry operations, it suffices to enforce invariance under a minimal generating set, dramatically simplifying the derivation of symmetry relations.

## 1.5 Tensor Transformation Laws

Let $T_{i_1 i_2 \dots i_n}$ be a tensor of rank $n$. Under an orthogonal transformation $a_{ij}$, it transforms as

$$T'_{i_1 i_2 \dots i_n} = a_{i_1 j_1} a_{i_2 j_2} \dots a_{i_n j_n} T_{j_1 j_2 \dots j_n}. \qquad (1.3)$$

This transformation law is exact and universal. All symmetry relations derived in this book follow directly from Eq. (1.3) together with the invariance requirement introduced next.
## 1.6 Invariance as a Necessary and Sufficient Condition

A material tensor is compatible with a given symmetry operation if and only if it is invariant under that operation. Mathematically, invariance is expressed as

$$T'_{i_1 i_2 \dots i_n} = T_{i_1 i_2 \dots i_n} \quad \text{for all indices.} \qquad (1.4)$$

Substituting Eq. (1.3) into Eq. (1.4) yields a system of linear equations relating the tensor components. These equations have two possible consequences:

certain components must vanish identically; certain components must be equal or linearly related.

Equation (1.4) is both *necessary and sufficient*. No additional assumptions are required, and no symmetry relations exist beyond those implied by invariance.

## 1.7 Voigt and Generalized Voigt Notation

To make the invariance equations tractable, tensors with intrinsic permutation symmetries are systematically reduced using Voigt and generalized Voigt notation.

For example, a symmetric index pair $(ij)$ is mapped to a single index $I \in \{1, \dots, 6\}$, allowing fourth-order tensors such as $c_{ijkl}$ to be represented as matrices $C_{IJ}$.

Higher-order tensors are treated analogously using generalized Voigt mappings, which preserve linear independence and symmetry while enabling compact algebraic manipulation.

Throughout this book, Voigt mappings are used purely as *representational tools*. All symmetry relations ultimately originate from the full tensorial invariance condition and are therefore exact.

## 1.8 Systematic Derivation Strategy

The derivation of symmetry relations follows a fixed, reproducible procedure:

1. Selection of a minimal set of point-group generators.

2. Application of the tensor transformation law to each generator.

3. Mapping of tensor components into Voigt or generalized Voigt notation.

4. Enforcement of the invariance condition for each generator.

5. Identification of vanishing components and linear relations among the remaining ones.

Because each step is explicit and algebraic, the procedure is immune to hidden assumptions and can be applied uniformly to tensors of arbitrary rank.
## 1.9 Error Sources and Constructive Robustness

Manual derivations of symmetry relations are notoriously prone to errors, including missed sign changes, inconsistent index permutations, and incomplete enforcement of invariance.

The constructive strategy adopted here minimizes these risks by:

* relying exclusively on exact tensor transformation laws,

* reducing all symmetry enforcement to linear algebra,

* avoiding heuristic or pattern-based arguments,

* enabling systematic cross-checking generator by generator.

As a result, the derived symmetry relations are exhaustive and unambiguous.

## 1.10 Scope of Applicability

The methodology developed in this chapter applies equally to:

linear and nonlinear constitutive laws, static and dynamic regimes, purely mechanical, purely electrical, and coupled electromechanical responses.

No assumptions are made regarding microscopic models or specific materials. The only input is the crystal symmetry encoded in the point group.

## 1.11 Perspective

All subsequent chapters are direct applications of the invariant principle established here. Once the symmetry generators are specified, the structure of the material tensors follows uniquely and rigorously from invariance alone.

In this sense, the entire book is a systematic unfolding of the single idea developed in this chapter.
# Chapter 2

# Triclinic Symmetry

## 2.1 Definition of Triclinic Symmetry (Class 1)

The triclinic crystal class 1 (often denoted $C_1$) is the most general anisotropic case: it possesses no nontrivial point-group symmetries. Equivalently, its point group contains only the identity operation,

$$ \mathcal{G} = \{\mathbf{I}\}. \qquad (2.1) $$

Consequently, there exist no nontrivial generators and the invariance condition of Chapter 1 imposes no additional constraints beyond the intrinsic permutation symmetries of the constitutive tensors themselves.

This chapter therefore provides the reference case for the entire book: it defines the maximal number of independent material constants compatible with tensorial structure and intrinsic symmetries alone. Every reduction obtained in later chapters is measured against this baseline.

## 2.2 Absence of Symmetry Constraints Beyond Intrinsic Symmetries

Let $T$ be any material tensor. Since the only group element is $\mathbf{I}$, invariance is automatic:

$$ T' = T \quad \text{for } \mathbf{a} = \mathbf{I}. \qquad (2.2) $$

Hence, the component structure of triclinic tensors is fully determined by tensor rank and transformation character, and intrinsic permutation symmetries implied by the underlying constitutive form (e.g. existence of an energy density, symmetry of stress/strain, symmetry within quadratic products).
## 2.3 Second-Order Elastic Constants

### 2.3.1 Tensor and Intrinsic Symmetries

The linear elastic stiffness tensor $c_{ijkl}$ satisfies the standard intrinsic symmetries

$$c_{ijkl} = c_{jikl} = c_{ijkl} = c_{klij}. \qquad (2.3)$$

Using Voigt notation, symmetric index pairs $(ij)$ are mapped to $I \in \{1, \dots, 6\}$, and $c_{ijkl}$ is represented by a symmetric $6 \times 6$ matrix $C_{IJ}$,

$$C_{IJ} = C_{JI}. \qquad (2.4)$$

### 2.3.2 Number of Independent Constants

A symmetric $6 \times 6$ matrix contains

$$N_{C^{(2)}} = \frac{6(6+1)}{2} = 21 \qquad (2.5)$$

independent constants. This is the maximal number for linear elasticity.

### 2.3.3 General Triclinic Form

In triclinic symmetry, all 21 entries of $C_{IJ}$ may be nonzero:

$$[C_{IJ}] = \begin{pmatrix} C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ C_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\ C_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\ C_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\ C_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\ C_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66} \end{pmatrix}. \qquad (2.6)$$

## 2.4 Third-Order Elastic Constants

### 2.4.1 Tensor and Intrinsic Symmetries

The third-order elastic tensor $c_{ijklmn}$ (second-order nonlinearity in strain) possesses intrinsic symmetries corresponding to:

* symmetry within each index pair: $(ij)$, $(kl)$, $(mn)$,

* symmetry under permutation of the three index pairs, as implied by an elastic energy density.

In this book we adopt the *hyperelastic* convention: the third-order elastic constants are defined as third derivatives of a strain-energy density with respect to strain measures, so that the tensor is symmetric under any permutation of the three index pairs. If one instead adopts a more general non-potential nonlinear constitutive law, only the minor (pairwise) symmetries need hold; those broader cases are outside the present scope.
These symmetries imply that, in generalized Voigt notation,

$$C_{IJK} \text{ is fully symmetric in } (I, J, K), \quad C_{IJK} = C_{JIK} = C_{IKJ} = C_{KJI} = \dots \tag{2.7}$$

## 2.4.2 Number of Independent Constants

A fully symmetric third-order tensor in 6 dimensions has

$$N_{C^{(3)}} = \binom{6+3-1}{3} = \binom{8}{3} = 56 \tag{2.8}$$

independent constants. This is the maximal number for third-order elasticity.

## 2.4.3 Indexing Convention for Independent Components

A convenient independent set is given by the lexicographically ordered components

$$C_{IJK} \text{ with } 1 \le I \le J \le K \le 6, \tag{2.9}$$

which yields exactly 56 components.

# 2.5 Piezoelectric Constants

## 2.5.1 Second-Order (Linear) Piezoelectric Tensor

The linear piezoelectric coupling tensor $\xi_{kij}$ is symmetric in its last two indices,

$$\xi_{kij} = \xi_{kji}. \tag{2.10}$$

In Voigt notation this becomes a 3 × 6 array $\xi_{kI}$ with no further intrinsic symmetries:

$$[\xi_{kI}] = \begin{pmatrix} \xi_{11} & \xi_{12} & \xi_{13} & \xi_{14} & \xi_{15} & \xi_{16} \\ \xi_{21} & \xi_{22} & \xi_{23} & \xi_{24} & \xi_{25} & \xi_{26} \\ \xi_{31} & \xi_{32} & \xi_{33} & \xi_{34} & \xi_{35} & \xi_{36} \end{pmatrix}. \tag{2.11}$$

Hence, for triclinic class 1,

$$N_{\xi^{(2)}} = 3 \times 6 = 18 \tag{2.12}$$

independent linear piezoelectric constants.

## 2.5.2 Higher-Order Piezoelectric Tensor (Fifth Order)

A common higher-order piezoelectric coupling involves one electric-field index and two symmetric strain pairs, represented by a fifth-order tensor $\xi_{kijmn}$ with intrinsic symmetries

$$\xi_{kijmn} = \xi_{kjmn} = \xi_{kijnm} = \xi_{kmnj}. \tag{2.13}$$
In generalized Voigt notation this corresponds to $\xi_{kIJ}$ which is symmetric in $I, J$:

$$ \xi_{kIJ} = \xi_{kJI}. \qquad (2.14) $$

Therefore, for each fixed $k \in \{1, 2, 3\}$, $\xi_{kIJ}$ is a symmetric $6 \times 6$ matrix with 21 independent entries, giving

$$ N_{\xi^{(5)}} = 3 \times 21 = 63 \qquad (2.15) $$

independent fifth-order piezoelectric constants in triclinic class 1.

## 2.6 Dielectric Constants

### 2.6.1 Second-Order Dielectric Tensor

The dielectric tensor $\epsilon_{ij}$ is intrinsically symmetric,

$$ \epsilon_{ij} = \epsilon_{ji}. \qquad (2.16) $$

Thus it is a symmetric $3 \times 3$ tensor with

$$ N_{\epsilon^{(2)}} = \frac{3(3+1)}{2} = 6 \qquad (2.17) $$

independent constants.

### 2.6.2 Third-Order Dielectric Tensor (quadratic electric-field response)

In this book, the third-order dielectric response is represented by a polar rank-3 tensor

$$ \epsilon_{kij}^{(3)}, \qquad (2.18) $$

defined such that it multiplies a quadratic electric-field product $E_i E_j$ in the expansion of $D_k$. Therefore it is intrinsically symmetric in the last two indices:

$$ \epsilon_{kij}^{(3)} = \epsilon_{kji}^{(3)}. \qquad (2.19) $$

In Voigt notation this becomes a $3 \times 6$ array $\epsilon_{kI}^{(3)}$ with no additional intrinsic symmetries. Hence, in triclinic class 1,

$$ N_{\epsilon^{(3)}} = 3 \times 6 = 18 \qquad (2.20) $$

independent third-order dielectric constants.

*Convention note.* As defined here via the constitutive expansion $D_k = \epsilon_{ki}E_i + \frac{1}{2}\epsilon_{kij}^{(3)}E_iE_j + \dots$, the tensor $\epsilon_{kij}^{(3)}$ is required to be symmetric only in the last two indices, $\epsilon_{kij}^{(3)} = \epsilon_{kji}^{(3)}$, because it multiplies the symmetric product $E_i E_j$. If the dielectric nonlinearity is instead derived from an electrostatic energy (or electric enthalpy) expanded as $U(E) = \frac{1}{2}\epsilon_{ij}E_iE_j + \frac{1}{6}\epsilon_{ijk}^{(3)}E_iE_jE_k + \dots$, then the third-order dielectric tensor is fully symmetric in its indices; imposing that additional potential-based symmetry yields
further reductions beyond point-group invariance. The present book does not impose that extra symmetry unless explicitly stated.

## 2.7 Electrostrictive Tensor

Electrostriction is represented here as a fourth-rank tensor $\gamma_{ijkl}$ coupling a symmetric strain pair $(ij)$ to a symmetric quadratic electric-field pair $(kl)$. Consequently, the intrinsic symmetries are:

$$ \gamma_{ijkl} = \gamma_{jikl} = \gamma_{ijkl}, \qquad (2.21) $$

i.e. symmetry within each index pair, but no symmetry is assumed under interchange of the pairs $(ij) \leftrightarrow (kl)$ because they correspond to different physical factors in the constitutive expansion.

In Voigt notation this becomes a general $6 \times 6$ matrix $\Gamma_{IJ}$ with no intrinsic symmetry between $I$ and $J$:

$$ \Gamma_{IJ} \text{ arbitrary.} \qquad (2.22) $$

Therefore, in triclinic class 1,

$$ N_{\gamma} = 6 \times 6 = 36 \qquad (2.23) $$

independent electrostrictive constants.

## 2.8 Practical relevance & typical material examples

* **Where it appears:** Low-symmetry crystals, often in complex molecular or highly distorted lattices.

* **Why it matters:** Triclinic symmetry provides the maximal set of admissible tensor components (subject only to intrinsic symmetries), and is the correct baseline for validating any symbolic/computational reduction pipeline.

* **Typical examples:** Low-symmetry molecular crystals and certain feldspars / minerals in triclinic phases (material-specific identification depends on phase and temperature).

## 2.9 Baseline Summary

Table 2.1 summarizes the maximal numbers of independent constants in triclinic class 1, i.e. in the absence of any crystal symmetry constraints beyond intrinsic tensor symmetries.

This triclinic baseline will be used in every subsequent chapter to quantify the reduction of independent constants induced by increasing symmetry. In all higher symmetry classes, the results are obtained by enforcing invariance under a generator set, but the intrinsic symmetries and baseline counts established here remain unchanged.
<table><caption>Table 2.1: Maximal number of independent material constants in triclinic class 1 (baseline case).</caption><thead><tr><th>Property tensor</th><th>Intrinsic symmetry (Voigt form)</th><th>N</th></tr></thead><tbody><tr><td>Second-order elasticity c<sub>ijkl</sub></td><td>C<sub>IJ</sub> = C<sub>JI</sub></td><td>21</td></tr><tr><td>Third-order elasticity c<sub>ijklmn</sub></td><td>C<sub>ijk</sub> fully symmetric</td><td>56</td></tr><tr><td>Linear piezoelectricity ξ<sub>kij</sub></td><td>ξ<sub>kI</sub> arbitrary 3 × 6</td><td>18</td></tr><tr><td>Fifth-order piezoelectricity ξ<sub>kijmn</sub></td><td>ξ<sub>kIJ</sub> = ξ<sub>kJI</sub></td><td>63</td></tr><tr><td>Dielectric ε<sub>ij</sub></td><td>symmetric 3 × 3</td><td>6</td></tr><tr><td>Third-order dielectric ε<sub>kij</sub><sup>(3)</sup></td><td>ε<sub>kI</sub><sup>(3)</sup> arbitrary 3 × 6</td><td>18</td></tr><tr><td>Electrostriction γ<sub>ijkl</sub></td><td>Γ<sub>ij</sub> arbitrary 6 × 6</td><td>36</td></tr></tbody></table>
# Chapter 3

# Monoclinic Symmetry

## 3.1 Monoclinic Point Groups and Coordinate Convention

Monoclinic crystals possess exactly one nontrivial symmetry element: either a twofold rotation axis, or a mirror plane, or both (in which case the point group is centrosymmetric). The monoclinic point groups are therefore

$$ 2, \quad m, \quad 2/m. \qquad (3.1) $$

Throughout this chapter a standard unique-axis convention is adopted: the unique symmetry axis is taken along the $x_2$ direction, and the mirror plane (when present) is the $x_1x_3$-plane. With this choice, convenient generators are:

$$ \text{Class 2: } \mathbf{a}_2 = \text{diag}(-1, +1, -1), \qquad (3.2) $$

$$ \text{Class m: } \mathbf{a}_m = \text{diag}(+1, -1, +1), \qquad (3.3) $$

$$ \text{Class } 2/m: \langle \mathbf{a}_2, \mathbf{a}_m \rangle, \quad \mathbf{a}_2 \mathbf{a}_m = -\mathbf{I}. \qquad (3.4) $$

Thus $2/m$ contains inversion and is centrosymmetric.

## 3.2 Transformation of Symmetric Second-Rank Tensors and Voigt Mapping

For any symmetric second-rank tensor $S_{ij} = S_{ji}$, the transformation law is

$$ S'_{ij} = a_{ip}a_{jq}S_{pq}. \qquad (3.5) $$

Using the tensorial Voigt mapping

$$ (11, 22, 33, 23, 13, 12) \leftrightarrow (1, 2, 3, 4, 5, 6), \qquad (3.6) $$
the components of $S_{ij}$ are collected into a six-vector $S_I$. For diagonal generators (3.2)–(3.3), the induced transformation on $S_I$ is diagonal:

$$S'_I = K_{IJ}S_J, \quad \mathbf{K} = \mathrm{diag}(+1, +1, +1, -1, +1, -1). \quad (3.7)$$

Importantly, the same $\mathbf{K}$ applies to both monoclinic generators $\mathbf{a}_2$ and $\mathbf{a}_m$, because for symmetric pairs only the parity of the underlying sign flips matters.

A crucial consequence is that all constitutive tensors built purely from symmetric index pairs (e.g. $c_{ijkl}$, $c_{ijklmn}$, and any pair-pair tensor written in Voigt form) have identical structural restrictions in classes 2 and m. Differences between 2 and m arise only for tensors carrying at least one vector index (e.g. piezoelectric and third-order dielectric tensors), because vectors transform with $\mathbf{a}_2$ or $\mathbf{a}_m$, not with $\mathbf{K}$.

## 3.3 Second-Order Elastic Constants (Classes 2 & m)

### 3.3.1 Invariance Equation

In Voigt form, the stiffness matrix $C_{IJ}$ transforms as

$$C'_{IJ} = K_{IP}K_{JQ}C_{PQ}. \quad (3.8)$$

For diagonal $\mathbf{K}$, invariance $C'_{IJ} = C_{IJ}$ reduces componentwise to

$$C_{IJ} = K_{II}K_{JJ}C_{IJ}. \quad (3.9)$$

Thus, $C_{IJ}$ is permitted only when $K_{II}K_{JJ} = +1$; otherwise $C_{IJ} = 0$.

### 3.3.2 Resulting Form and Independent Constants

With $\mathbf{K}$ given by (3.7), the general monoclinic stiffness matrix (unique axis $x_2$) is

$$[C_{IJ}] = \begin{pmatrix} C_{11} & C_{12} & C_{13} & 0 & C_{15} & 0 \\ C_{12} & C_{22} & C_{23} & 0 & C_{25} & 0 \\ C_{13} & C_{23} & C_{33} & 0 & C_{35} & 0 \\ 0 & 0 & 0 & C_{44} & 0 & C_{46} \\ C_{15} & C_{25} & C_{35} & 0 & C_{55} & 0 \\ 0 & 0 & 0 & C_{46} & 0 & C_{66} \end{pmatrix}. \quad (3.10)$$

All displayed entries are independent, and the total number of independent second-order elastic constants is

$$N_{C(2)}^{(2)} = N_{C(2)}^{(m)} = N_{C(2)}^{(2/m)} = 13, \quad (3.11)$$

to be compared with the triclinic baseline 21.
## 3.4 Third-Order Elastic Constants (Classes 2 & m)

### 3.4.1 Invariance Equation

In generalized Voigt form, the third-order elastic tensor is represented by a fully symmetric $C_{IJK}$. Under a monoclinic generator,

$$C'_{IJK} = K_{IP}K_{JQ}K_{KR}C_{PQR}. \qquad (3.12)$$

Since $K$ is diagonal, invariance reduces componentwise to

$$C_{IJK} = K_{II}K_{JJ}K_{KK}C_{IJK}. \qquad (3.13)$$

Therefore $C_{IJK}$ may be nonzero if and only if $K_{II}K_{JJ}K_{KK} = +1$.

### 3.4.2 Parity Rule and Independent Count

Because $K_{44} = K_{66} = -1$ and the other $K_{II} = +1$, Eq. (3.13) implies:

**Monoclinic parity rule (classes 2, m, and 2/m).** A fully symmetric component $C_{IJK}$ is allowed if and only if the number of indices among {4, 6} appearing in $(I, J, K)$ is even (i.e. 0 or 2).

Consequently,

$$N_{C(3)}^{(2)} = N_{C(3)}^{(m)} = N_{C(3)}^{(2/m)} = 32, \qquad (3.14)$$

to be compared with the triclinic baseline 56.

## 3.5 Linear Piezoelectric Constants

The linear piezoelectric tensor $\xi_{kij} = \xi_{kji}$ is represented in Voigt form as $\xi_{kI}$. For a diagonal generator $\mathbf{a} = \text{diag}(a_{11}, a_{22}, a_{33})$, invariance reads

$$\xi_{kI} = a_{kk} K_{II} \xi_{kI}, \qquad (3.15)$$

so $\xi_{kI}$ is allowed only if $a_{kk}K_{II} = +1$.

### 3.5.1 Class 2

For $\mathbf{a}_2 = \text{diag}(-1, +1, -1)$, Eq. (3.15) yields

$$[\xi_{kI}]_{(2)} = \begin{pmatrix} 0 & 0 & 0 & \xi_{14} & 0 & \xi_{16} \\ \xi_{21} & \xi_{22} & \xi_{23} & 0 & \xi_{25} & 0 \\ 0 & 0 & 0 & \xi_{34} & 0 & \xi_{36} \end{pmatrix}, \qquad (3.16)$$

containing $N_{\xi(2)}^{(2)} = 8$ independent constants.
### 3.5.2 Class m

For $a_m = \text{diag}(+1, -1, +1)$, Eq. (3.15) yields

$$ [\xi_{kI}]_{(m)} = \begin{pmatrix} \xi_{11} & \xi_{12} & \xi_{13} & 0 & \xi_{15} & 0 \\ 0 & 0 & 0 & \xi_{24} & 0 & \xi_{26} \\ \xi_{31} & \xi_{32} & \xi_{33} & 0 & \xi_{35} & 0 \end{pmatrix}, \qquad (3.17) $$

containing $N_{\xi^{(2)}}^{(m)} = 10$ independent constants.

### 3.5.3 Class 2/m

Since $2/m$ contains inversion $-I$, all odd-rank polar couplings vanish:

$$ \xi_{kI} \equiv 0 \quad \text{for class } 2/m, \quad N_{\xi^{(2)}}^{(2/m)} = 0. \qquad (3.18) $$

## 3.6 Fifth-Order Piezoelectric Constants

For the fifth-order piezoelectric tensor $\xi_{kijmn}$, the generalized Voigt representation $\xi_{kIJ}$ is symmetric in $I, J$. Under a diagonal generator $a$, invariance reads

$$ \xi_{kIJ} = a_{kk} K_{II} K_{JJ} \xi_{kIJ}. \qquad (3.19) $$

Thus $\xi_{kIJ}$ is allowed only if $a_{kk}K_{II}K_{JJ} = +1$.

### 3.6.1 Classes 2 & m: component rule and counts

Using (3.7) and $\xi_{kIJ} = \xi_{kJI}$, one obtains:

$$ N_{\xi^{(5)}}^{(2)} = 29, \quad N_{\xi^{(5)}}^{(m)} = 34, \quad N_{\xi^{(5)}}^{(2/m)} = 0, \qquad (3.20) $$

where $N_{\xi^{(5)}}^{(2/m)} = 0$ follows again from centrosymmetry (odd-rank polar tensor).

## 3.7 Dielectric Constants

### 3.7.1 Second-Order Dielectric Tensor

The dielectric tensor $\epsilon_{ij} = \epsilon_{ji}$ transforms as a symmetric second-rank tensor:

$$ \epsilon'_{ij} = a_{ip}a_{jq}\epsilon_{pq}. \qquad (3.21) $$

For both generators $a_2$ and $a_m$, invariance implies

$$ \epsilon_{12} = \epsilon_{23} = 0, \qquad (3.22) $$
while $\epsilon_{13}$ is permitted. Thus, for classes 2, m, and 2/m,

$$[\epsilon_{ij}] = \begin{pmatrix} \epsilon_{11} & 0 & \epsilon_{13} \\ 0 & \epsilon_{22} & 0 \\ \epsilon_{13} & 0 & \epsilon_{33} \end{pmatrix}, \quad N_{\epsilon(2)}^{(2)} = N_{\epsilon(2)}^{(m)} = N_{\epsilon(2)}^{(2/m)} = 4. \quad (3.23)$$

### 3.7.2 Third-Order Dielectric Tensor (quadratic electric-field response)

In this book, the third-order dielectric response is represented by a polar rank-3 tensor $\epsilon_{kij}^{(3)}$ symmetric in the last two indices:

$$\epsilon_{kij}^{(3)} = \epsilon_{kji}^{(3)}, \quad (3.24)$$

and written in Voigt form as $\epsilon_{kI}^{(3)}$. For a diagonal generator, invariance is

$$\epsilon_{kI}^{(3)} = a_{kk} K_{II} \epsilon_{kI}^{(3)}. \quad (3.25)$$

**Class 2.** For $a_2 = \text{diag}(-1, +1, -1)$, allowed entries are:

$$[\epsilon_{kI}^{(3)}]_{(2)} = \begin{pmatrix} 0 & 0 & 0 & \epsilon_{14}^{(3)} & 0 & \epsilon_{16}^{(3)} \\ \epsilon_{21}^{(3)} & \epsilon_{22}^{(3)} & \epsilon_{23}^{(3)} & 0 & \epsilon_{25}^{(3)} & 0 \\ 0 & 0 & 0 & \epsilon_{34}^{(3)} & 0 & \epsilon_{36}^{(3)} \end{pmatrix}, \quad N_{\epsilon^{(3)}}^{(2)} = 8. \quad (3.26)$$

**Class m.** For $a_m = \text{diag}(+1, -1, +1)$,

$$[\epsilon_{kI}^{(3)}]_{(m)} = \begin{pmatrix} \epsilon_{11}^{(3)} & \epsilon_{12}^{(3)} & \epsilon_{13}^{(3)} & 0 & \epsilon_{15}^{(3)} & 0 \\ 0 & 0 & 0 & \epsilon_{24}^{(3)} & 0 & \epsilon_{26}^{(3)} \\ \epsilon_{31}^{(3)} & \epsilon_{32}^{(3)} & \epsilon_{33}^{(3)} & 0 & \epsilon_{35}^{(3)} & 0 \end{pmatrix}, \quad N_{\epsilon^{(3)}}^{(m)} = 10. \quad (3.27)$$

**Class 2/m.** Centrosymmetry forbids all odd-rank polar tensors:

$$\epsilon_{kI}^{(3)} \equiv 0, \quad N_{\epsilon^{(3)}}^{(2/m)} = 0. \quad (3.28)$$

## 3.8 Electrostrictive Tensor (Classes 2, m, and 2/m)

Electrostriction is represented by a fourth-rank tensor $\gamma_{ijkl}$ symmetric within each index pair:

$$\gamma_{ijkl} = \gamma_{jikl} = \gamma_{ijkl}, \quad (3.29)$$

but no intrinsic symmetry is assumed under interchange of the pairs $(ij) \leftrightarrow (kl)$. In Voigt form this corresponds to a general 6 × 6 matrix $\Gamma_{IJ}$ (not necessarily symmetric). Under a monoclinic operation,

$$\Gamma'_{IJ} = K_{II} K_{JJ} \Gamma_{IJ}, \quad (3.30)$$

hence invariance gives

$$\Gamma_{IJ} = K_{II} K_{JJ} \Gamma_{IJ}. \quad (3.31)$$
Therefore, $\Gamma_{IJ}$ can be nonzero only if $K_{II}K_{JJ} = +1$. With (3.7), the monoclinic electrostriction matrix has the block structure

$$[\Gamma_{IJ}] = \begin{pmatrix} \Gamma_{11} & \Gamma_{12} & \Gamma_{13} & 0 & \Gamma_{15} & 0 \\ \Gamma_{21} & \Gamma_{22} & \Gamma_{23} & 0 & \Gamma_{25} & 0 \\ \Gamma_{31} & \Gamma_{32} & \Gamma_{33} & 0 & \Gamma_{35} & 0 \\ 0 & 0 & 0 & \Gamma_{44} & 0 & \Gamma_{46} \\ \Gamma_{51} & \Gamma_{52} & \Gamma_{53} & 0 & \Gamma_{55} & 0 \\ 0 & 0 & 0 & \Gamma_{64} & 0 & \Gamma_{66} \end{pmatrix}, \qquad (3.32)$$

containing

$$N_{\gamma}^{(2)} = N_{\gamma}^{(m)} = N_{\gamma}^{(2/m)} = 20 \qquad (3.33)$$

independent constants (to be compared with the triclinic baseline 36).

## 3.9 Practical relevance & typical material examples

* **Where it appears:** Crystals with a single twofold axis or mirror plane; common in minerals and low-symmetry ferroelectrics.

* **Why it matters:** Monoclinic symmetry is the first step where systematic sign/zero constraints arise from a single nontrivial generator, making it a key “sanity check” class for higher-rank coupling tensors.

* **Typical examples:** Gypsum (monoclinic), many mica-group minerals (monoclinic), and numerous engineered low-symmetry functional oxides in monoclinic phases.

## 3.10 Comparison with the Triclinic Baseline

Monoclinic symmetry provides the first nontrivial reduction from the triclinic case. For tensors built solely from symmetric index pairs, classes 2 and m yield the same reductions; for tensors with a vector index, classes 2 and m differ; and for centrosymmetric class 2/m, all odd-rank polar tensors vanish.

The subsequent chapters apply the same invariance procedure to higher symmetry classes, producing additional vanishing components and, in many cases, nontrivial linear relations among the remaining constants. In all cases, the monoclinic results above remain a key intermediate benchmark between the fully general triclinic baseline and the more restrictive orthorhombic, tetragonal, trigonal, hexagonal, cubic, and isotropic cases.
<table><caption>Table 3.1: Independent constants in triclinic class 1 versus monoclinic classes 2, *m*, and 2/*m* (unique axis *x*₂).</caption><thead><tr><th>Property tensor</th><th>1</th><th>2</th><th>m</th><th>2/m</th></tr></thead><tbody><tr><td>Second-order elasticity C<sub>IJ</sub></td><td>21</td><td>13</td><td>13</td><td>13</td></tr><tr><td>Third-order elasticity C<sub>ijk</sub></td><td>56</td><td>32</td><td>32</td><td>32</td></tr><tr><td>Linear piezoelectricity ξ<sub>kI</sub></td><td>18</td><td>8</td><td>10</td><td>0</td></tr><tr><td>Fifth-order piezoelectricity ξ<sub>kIJ</sub></td><td>63</td><td>29</td><td>34</td><td>0</td></tr><tr><td>Dielectric ε<sub>ij</sub></td><td>6</td><td>4</td><td>4</td><td>4</td></tr><tr><td>Third-order dielectric ε<sub>kI</sub><sup>(3)</sup></td><td>18</td><td>8</td><td>10</td><td>0</td></tr><tr><td>Electrostriction Γ<sub>IJ</sub></td><td>36</td><td>20</td><td>20</td><td>20</td></tr></tbody></table>

Columns correspond to triclinic 1 and monoclinic 2, *m*, 2/*m* (unique axis *x*₂).
# Chapter 4

## Orthorhombic Symmetry

### 4.1 Orthorhombic Point Groups and Generators

Orthorhombic crystals possess three mutually orthogonal symmetry directions aligned with the crystallographic axes. The orthorhombic point groups are

$$ 222, \quad mm2, \quad mmm. \qquad (4.1) $$

All orthorhombic symmetry operations commute in a suitably chosen orthonormal basis. Throughout this chapter, the coordinate axes ($x_1, x_2, x_3$) are chosen along the orthorhombic principal axes.

A convenient generator set for class 222 is given by 180° rotations about the coordinate axes:

$$ \mathbf{a}_1 = \text{diag}(+1, -1, -1), \qquad (4.2) $$

$$ \mathbf{a}_2 = \text{diag}(-1, +1, -1), \qquad (4.3) $$

$$ \mathbf{a}_3 = \text{diag}(-1, -1, +1). \qquad (4.4) $$

Invariance under $\mathbf{a}_1$ and $\mathbf{a}_2$ already implies invariance under $\mathbf{a}_3 = \mathbf{a}_1\mathbf{a}_2$.

For class mm2, a standard setting takes two mutually orthogonal mirror planes and a twofold axis perpendicular to them. With the twofold axis along $x_3$, one may choose generators

$$ \mathbf{m}_1 = \text{diag}(-1, +1, +1), \qquad (4.5) $$

$$ \mathbf{m}_2 = \text{diag}(+1, -1, +1), \qquad (4.6) $$

$$ \mathbf{a}_3 = \text{diag}(-1, -1, +1), \qquad (4.7) $$

where $\mathbf{a}_3 = \mathbf{m}_1\mathbf{m}_2$ is again a 180° rotation about $x_3$.

Class mmm is centrosymmetric. For example, it may be generated by the three mirrors

$$ \mathbf{m}_1, \quad \mathbf{m}_2, \quad \mathbf{m}_3 = \text{diag}(+1, +1, -1), \qquad (4.8) $$

and it contains inversion $-\mathbf{I} = \mathbf{m}_1\mathbf{m}_2\mathbf{m}_3$. Centrosymmetry will be used explicitly
when discussing odd-rank polar tensors.

## 4.2 Voigt Transformation of Symmetric Index Pairs

Using the tensorial Voigt mapping

$$ (11, 22, 33, 23, 13, 12) \leftrightarrow (1, 2, 3, 4, 5, 6), \qquad (4.9) $$

a symmetric index pair (ij) transforms under $\mathbf{a} = \text{diag}(s_1, s_2, s_3)$ with signs

$$ K_I(\mathbf{a}) = (s_1^2, s_2^2, s_3^2, s_2s_3, s_1s_3, s_1s_2), \qquad (4.10) $$

i.e. the induced transformation on a symmetric second-rank tensor written as a 6-vector is diagonal:

$$ S'_I = K_I(\mathbf{a}) S_I. \qquad (4.11) $$

For the rotation generators (4.2)–(4.4), the corresponding Voigt sign diagonals are

$$ \mathbf{K}^{(1)} = \text{diag}(+1, +1, +1, +1, -1, -1), \qquad (4.12) $$

$$ \mathbf{K}^{(2)} = \text{diag}(+1, +1, +1, -1, +1, -1), \qquad (4.13) $$

$$ \mathbf{K}^{(3)} = \text{diag}(+1, +1, +1, -1, -1, +1). \qquad (4.14) $$

Note that the mirrors $\mathbf{m}_1$ and $\mathbf{m}_2$ induce exactly the same $\mathbf{K}^{(1)}$ and $\mathbf{K}^{(2)}$, respectively. Hence, any constitutive tensor built only from symmetric index pairs (i.e. tensors whose Voigt-form transformation uses only factors $K_I$) has the same orthorhombic structural restrictions in classes 222, mm2, and mmm. Differences between these classes arise only for tensors carrying at least one vector index (e.g. piezoelectric and third-order dielectric tensors).

## 4.3 Sequential Invariance and Cumulative Constraint

Orthorhombic symmetry may be viewed as enforcing invariance under multiple commuting operations. Algebraically, each additional independent generator imposes extra sign-consistency conditions of the type

$$ T = T' \Leftrightarrow (\text{sign factor}) \times T = T, \qquad (4.15) $$

which either preserves a component or forces it to zero. This produces a cumulative reduction in allowed tensor structure that is particularly transparent for diagonal-sign generators.
## 4.4 Second-Order Elastic Constants

### 4.4.1 Invariance Equations

In Voigt form, the stiffness transforms as

$$C'_{IJ} = K_I(\mathbf{a}) K_J(\mathbf{a}) C_{IJ}, \qquad (4.16)$$

so invariance under a generator $\mathbf{a}$ requires

$$C_{IJ} = K_I(\mathbf{a}) K_J(\mathbf{a}) C_{IJ}. \qquad (4.17)$$

Orthorhombic symmetry requires this simultaneously for $\mathbf{a}_1$ and $\mathbf{a}_2$ (and hence also $\mathbf{a}_3$).

### 4.4.2 Resulting Orthorhombic Form and Independent Constants

The simultaneous constraints from $\mathbf{K}^{(1)}$, $\mathbf{K}^{(2)}$, and $\mathbf{K}^{(3)}$ force all couplings between normal strains and shear strains, and all couplings between distinct shear strains, to vanish. The orthorhombic stiffness matrix is therefore

$$[C_{IJ}] = \begin{pmatrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{12} & C_{22} & C_{23} & 0 & 0 & 0 \\ C_{13} & C_{23} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66} \end{pmatrix}. \qquad (4.18)$$

All displayed constants are independent, giving

$$N_{C^{(2)}}^{(\text{orth})} = 9, \qquad (4.19)$$

to be compared with 21 (triclinic) and 13 (monoclinic).

## 4.5 Third-Order Elastic Constants

### 4.5.1 Invariance Equations

In generalized Voigt form, the third-order elastic tensor is represented by a fully symmetric $C_{ijk}$. Under a generator $\mathbf{a}$,

$$C'_{ijk} = K_I(\mathbf{a}) K_J(\mathbf{a}) K_K(\mathbf{a}) C_{ijk}, \qquad (4.20)$$

so invariance requires

$$C_{ijk} = K_I(\mathbf{a}) K_J(\mathbf{a}) K_K(\mathbf{a}) C_{ijk}. \qquad (4.21)$$

Orthorhombic symmetry requires (4.21) for $\mathbf{a}_1$ and $\mathbf{a}_2$ (and hence also $\mathbf{a}_3$).
### 4.5.2 Allowed Components and Independent Count

Let $I, J, K \in \{1, \dots, 6\}$ with $I \le J \le K$. Using (4.12)-(4.14), invariance yields exactly

$$N_{C^{(3)}}^{(\text{orth})} = 20 \qquad (4.22)$$

independent third-order elastic constants.

A convenient independent set is the collection of $C_{IJK}$ with $1 \le I \le J \le K \le 6$ belonging to the following list:

$$\begin{aligned} C_{111}, C_{112}, C_{113}, C_{122}, C_{123}, C_{133}, C_{222}, C_{223}, C_{233}, C_{333}, \\ C_{144}, C_{155}, C_{166}, C_{244}, C_{255}, C_{266}, C_{344}, C_{355}, C_{366}, C_{456}. \end{aligned} \qquad (4.23)$$

All other components are either equal by full symmetry in $(I, J, K)$ or vanish by orthorhombic invariance.

## 4.6 Linear Piezoelectric Constants

The linear piezoelectric tensor $\xi_{kij}$ is symmetric in $(ij)$ and represented by $\xi_{kI}$. Under a generator $\mathbf{a}$,

$$\xi_{kI} = a_{kk} K_I(\mathbf{a}) \xi_{kI}, \qquad (4.24)$$

so invariance requires

$$\xi_{kI} = a_{kk} K_I(\mathbf{a}) \xi_{kI}. \qquad (4.25)$$

Orthorhombic symmetry requires (4.25) for a generator set.

### 4.6.1 Class 222

Using the rotation generators (4.2)-(4.4), the only permitted components are

$$[\xi_{kI}]_{(222)} = \begin{pmatrix} 0 & 0 & 0 & \xi_{14} & 0 & 0 \\ 0 & 0 & 0 & 0 & \xi_{25} & 0 \\ 0 & 0 & 0 & 0 & 0 & \xi_{36} \end{pmatrix}, \qquad (4.26)$$

so

$$N_{\xi^{(2)}}^{(222)} = 3. \qquad (4.27)$$

*Remark.* The count $N_{\xi^{(2)}}^{(222)} = 3$ follows directly from point-group invariance for the third-rank piezoelectric tensor (with only the intrinsic symmetry in its last index pair as encoded in Voigt form). Apparent “smaller” counts occasionally encountered in secondary tables typically reflect different coefficient definitions, directional effective moduli, or convention-dependent shear scalings rather than additional symmetry constraints.
### 4.6.2 Class mm2 (polar axis $x_3$)

Using the generators (4.5)-(4.7), the permitted components are

$$ [\xi_{kI}]_{(mm2)} = \begin{pmatrix} 0 & 0 & 0 & 0 & \xi_{15} & 0 \\ 0 & 0 & 0 & \xi_{24} & 0 & 0 \\ \xi_{31} & \xi_{32} & \xi_{33} & 0 & 0 & 0 \end{pmatrix}, \quad (4.28) $$

so

$$ N_{\xi^{(2)}}^{(mm2)} = 5. \quad (4.29) $$

### 4.6.3 Class mmm

Class mmm is centrosymmetric and therefore forbids all odd-rank polar couplings:

$$ \xi_{kI} \equiv 0, \quad N_{\xi^{(2)}}^{(mmm)} = 0. \quad (4.30) $$

## 4.7 Fifth-Order Piezoelectric Constants

For the fifth-order piezoelectric tensor in generalized Voigt form $\xi_{kIJ} = \xi_{kJJ}$, invariance under a generator $a$ requires

$$ \xi_{kIJ} = a_{kk} K_I(\mathbf{a}) K_J(\mathbf{a}) \xi_{kIJ}. \quad (4.31) $$

### 4.7.1 Class 222

Simultaneous invariance under (4.2)-(4.4) yields

$$ N_{\xi^{(5)}}^{(222)} = 12. \quad (4.32) $$

A convenient independent set is

$$ \begin{aligned} k &= 1 : \xi_{1,14}, \xi_{1,24}, \xi_{1,34}, \xi_{1,56}, \\ k &= 2 : \xi_{2,15}, \xi_{2,25}, \xi_{2,35}, \xi_{2,46}, \\ k &= 3 : \xi_{3,16}, \xi_{3,26}, \xi_{3,36}, \xi_{3,45}, \end{aligned} \quad (4.33) $$

where the generalized Voigt pair $(IJ)$ is understood in symmetric form.

### 4.7.2 Class mm2 (polar axis $x_3$)

For generators (4.5)-(4.7), one finds

$$ N_{\xi^{(5)}}^{(mm2)} = 17, \quad (4.34) $$
with a convenient independent set

$$
\begin{align}
k = 1: & \xi_{1,15}, \xi_{1,25}, \xi_{1,35}, \xi_{1,46}, \nonumber \\
k = 2: & \xi_{2,14}, \xi_{2,24}, \xi_{2,34}, \xi_{2,56}, \nonumber \\
k = 3: & \xi_{3,11}, \xi_{3,12}, \xi_{3,13}, \xi_{3,22}, \xi_{3,23}, \xi_{3,33}, \xi_{3,44}, \xi_{3,55}, \xi_{3,66}. \tag{4.35}
\end{align}
$$

4.7.3 Class mmm

Centrosymmetry forbids all odd-rank polar couplings:

$$
\xi_{kIJ} \equiv 0, \qquad N_{\xi^{(5)}}^{(mmm)} = 0. \tag{4.36}
$$

4.8 Dielectric Constants

4.8.1 Second-Order Dielectric Tensor

The dielectric tensor $\epsilon_{ij} = \epsilon_{ji}$ transforms as a symmetric second-rank tensor. In-
variance under the orthorhombic generator set forces all off-diagonal components to
vanish:

$$
[\epsilon_{ij}] = \begin{pmatrix} \epsilon_{11} & 0 & 0 \\ 0 & \epsilon_{22} & 0 \\ 0 & 0 & \epsilon_{33} \end{pmatrix}, \quad N_{\epsilon^{(2)}}^{(\text{orth})} = 3, \qquad (4.37)
$$

valid for 222, mm2, and mmm.

4.8.2 Third-Order Dielectric Tensor (quadratic electric-field response)

In this book, the third-order dielectric response is represented by a polar rank-3 tensor
$\epsilon_{kij}^{(3)}$ symmetric in the last two indices, written in Voigt form as $\epsilon_{kI}^{(3)}$. Its invariance
condition under $\mathbf{a}$ is identical in structure to the piezoelectric condition:

$$
\epsilon_{kI}^{(3)} = a_{kk} K_I(\mathbf{a}) \epsilon_{kI}^{(3)}. \quad (4.38)
$$

Class 222. Simultaneous invariance under (4.2)–(4.4) yields

$$
[\epsilon_{kI}^{(3)}]_{(222)} = \begin{pmatrix} 0 & 0 & 0 & \epsilon_{14}^{(3)} & 0 & 0 \\ 0 & 0 & 0 & 0 & \epsilon_{25}^{(3)} & 0 \\ 0 & 0 & 0 & 0 & 0 & \epsilon_{36}^{(3)} \end{pmatrix}, \quad N_{\epsilon^{(3)}}^{(222)} = 3. \quad (4.39)
$$

Class mm2 (polar axis $x_3$). Using (4.5)-(4.7),

$$
[\epsilon_{kI}^{(3)}]_{(mm2)} = \begin{pmatrix} 0 & 0 & 0 & 0 & \epsilon_{15}^{(3)} & 0 \\ 0 & 0 & 0 & \epsilon_{24}^{(3)} & 0 & 0 \\ \epsilon_{31}^{(3)} & \epsilon_{32}^{(3)} & \epsilon_{33}^{(3)} & 0 & 0 & 0 \end{pmatrix}, \quad N_{\epsilon^{(3)}}^{(mm2)} = 5. \quad (4.40)
$$
Class mmm. Centrosymmetry forbids all odd-rank polar tensors:

$$ \varepsilon_{kI}^{(3)} \equiv 0, \quad N_{\varepsilon^{(3)}}^{(mmm)} = 0. \qquad (4.41) $$

## 4.9 Electrostrictive Tensor

Electrostriction is represented by a fourth-rank tensor $\gamma_{ijkl}$ symmetric within each index pair:

$$ \gamma_{ijkl} = \gamma_{jikl} = \gamma_{ijkl}, \qquad (4.42) $$

but no intrinsic symmetry is assumed under interchange of the pairs $(ij) \leftrightarrow (kl)$. In Voigt form this corresponds to a general 6 × 6 matrix $\Gamma_{IJ}$.

Under a diagonal orthorhombic operation, invariance reduces to the same sign-product rule as for any pair-pair tensor:

$$ \Gamma_{IJ} = K_I(\mathbf{a}) K_J(\mathbf{a}) \Gamma_{IJ}, \qquad (4.43) $$

to hold simultaneously for $\mathbf{a}_1$ and $\mathbf{a}_2$. The resulting orthorhombic form is block-diagonal:

$$ [\Gamma_{IJ}] = \begin{pmatrix} \Gamma_{11} & \Gamma_{12} & \Gamma_{13} & 0 & 0 & 0 \\ \Gamma_{21} & \Gamma_{22} & \Gamma_{23} & 0 & 0 & 0 \\ \Gamma_{31} & \Gamma_{32} & \Gamma_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & \Gamma_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & \Gamma_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & \Gamma_{66} \end{pmatrix}, \quad N_{\gamma}^{(\text{orth})} = 12, \qquad (4.44) $$

valid for 222, mm2, and mmm.

## 4.10 Practical relevance & typical material examples

* **Where it appears:** Crystals with three mutually orthogonal symmetry directions; common in geophysics and structural materials.

* **Why it matters:** Orthorhombic symmetry produces strong, clean reductions while still allowing pronounced anisotropy, making it a frequent target for continuum modeling and wave-propagation calculations.

* **Typical examples:** Olivine (orthorhombic), aragonite (orthorhombic), many organic crystals and polymers in orthorhombic phases.

## 4.11 Summary and Emerging Structural Patterns

Orthorhombic symmetry illustrates clearly how multiple commuting operations progressively constrain tensor structure:
* For tensors built from symmetric index pairs, orthorhombic symmetry eliminates all normal-shear couplings and all shear-shear couplings between distinct shears, yielding the 9-parameter stiffness matrix (4.18).

* For fully symmetric third-order elasticity, invariance reduces the triclinic baseline 56 to 20 independent constants, including a single mixed-shear invariant $C_{456}$.

* For odd-rank polar tensors, non-centrosymmetric orthorhombic classes 222 and mm2 admit piezoelectric and third-order dielectric effects with class-dependent component structure, while centrosymmetric mmm forbids them entirely.

* For electrostriction viewed as a pair-pair coupling without imposed pair-interchange symmetry, orthorhombic invariance yields the block-diagonal 6 × 6 structure (4.44).

The reduction counts relative to triclinic and monoclinic baselines are summarized in Table 4.1.

<table><caption>Table 4.1: Independent constants: triclinic and monoclinic baselines versus orthorhombic classes 222, mm2, and mmm.</caption><thead><tr><th>Property tensor</th><th>1</th><th>2</th><th>m</th><th>222</th><th>mm2</th><th>mmm</th></tr></thead><tbody><tr><td>Second-order elasticity C<sub>IJ</sub></td><td>21</td><td>13</td><td>13</td><td>9</td><td>9</td><td>9</td></tr><tr><td>Third-order elasticity C<sub>ijk</sub></td><td>56</td><td>32</td><td>32</td><td>20</td><td>20</td><td>20</td></tr><tr><td>Linear piezoelectricity ξ<sub>kI</sub></td><td>18</td><td>8</td><td>10</td><td>3</td><td>5</td><td>0</td></tr><tr><td>Fifth-order piezoelectricity ξ<sub>kIJ</sub></td><td>63</td><td>29</td><td>34</td><td>12</td><td>17</td><td>0</td></tr><tr><td>Dielectric ε<sub>ij</sub></td><td>6</td><td>4</td><td>4</td><td>3</td><td>3</td><td>3</td></tr><tr><td>Third-order dielectric ε<sub>kI</sub><sup>(3)</sup></td><td>18</td><td>8</td><td>10</td><td>3</td><td>5</td><td>0</td></tr><tr><td>Electrostriction Γ<sub>IJ</sub></td><td>36</td><td>20</td><td>20</td><td>12</td><td>12</td><td>12</td></tr></tbody></table>
# Chapter 5

# Tetragonal Symmetry

## 5.1 Tetragonal Point Groups and Principal-Axis Convention

Tetragonal crystals possess a distinguished fourfold symmetry axis. In this chapter the fourfold axis is taken along $x_3$. The tetragonal point groups may be organized into families generated by a 90° rotation about $x_3$ (fourfold rotational invariance), and optional additional symmetries (twofold rotations and/or mirror planes), and possibly inversion.

A convenient generator for the fourfold rotation about $x_3$ is

$$ \mathbf{a}_4 = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \qquad (5.1) $$

corresponding to $x_1 \mapsto x_2$, $x_2 \mapsto -x_1$, $x_3 \mapsto x_3$.

To represent commonly used tetragonal settings, we will also employ:

* a twofold rotation about $x_1$,

$$ \mathbf{a}_{2x} = \mathrm{diag}(+1, -1, -1), \qquad (5.2) $$

which, together with $\mathbf{a}_4$, generates the nonpolar class 422;

* a mirror plane containing the fourfold axis, e.g. reflection in the $x_2x_3$-plane,

$$ \mathbf{m}_1 = \mathrm{diag}(-1, +1, +1), \qquad (5.3) $$

which, together with $\mathbf{a}_4$, yields a standard generator set for the polar class 4mm;

* inversion symmetry (when present), which eliminates all odd-rank polar tensors (e.g. piezoelectricity and third-order dielectric response).
## 5.2 Voigt Transformation under Fourfold Rotation

Using the tensorial Voigt mapping

$$ (11, 22, 33, 23, 13, 12) \leftrightarrow (1, 2, 3, 4, 5, 6), \qquad (5.4) $$

a symmetric second-rank tensor written as a Voigt 6-vector transforms under $\mathbf{a}_4$ via a $6 \times 6$ matrix $\mathbf{K}_4$ defined by $S'_I = (K_{IJ})_{IJ}S_J$. A direct evaluation gives

$$ \mathbf{K}_4 = \begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{pmatrix}, \qquad (5.5) $$

which encodes the mixing and sign changes

$$ 1 \leftrightarrow 2, \quad 4 \leftrightarrow -5, \quad 5 \leftrightarrow 4, \quad 6 \leftrightarrow -6, \quad 3 \leftrightarrow 3. \qquad (5.6) $$

For diagonal generators $\mathbf{a} = \text{diag}(s_1, s_2, s_3)$, the induced Voigt action is diagonal:

$$ \mathbf{K}(\mathbf{a}) = \text{diag}(1, 1, 1, s_2 s_3, s_1 s_3, s_1 s_2). \qquad (5.7) $$

In particular, for $\mathbf{a}_{2x}$ and $\mathbf{m}_1$ one obtains

$$ \mathbf{K}(\mathbf{a}_{2x}) = \mathbf{K}(\mathbf{m}_1) = \text{diag}(1, 1, 1, 1, -1, -1). \qquad (5.8) $$

## 5.3 Second-Order Elastic Constants

### 5.3.1 Invariance under $\mathbf{a}_4$

In Voigt form, the stiffness transforms as

$$ C' = \mathbf{K} \mathbf{C} \mathbf{K}^{\mathrm{T}}, \qquad (5.9) $$

where $\mathbf{K}$ is the induced $6 \times 6$ transformation (here $\mathbf{K}_4$). Fourfold invariance requires

$$ C = \mathbf{K}_4 C \mathbf{K}_4^{\mathrm{T}}. \qquad (5.10) $$

Solving (5.10) for a symmetric $6 \times 6$ matrix yields the general tetragonal form
compatible with fourfold rotation:

$$ [C_{IJ}]_{(4)} = \begin{pmatrix} C_{11} & C_{12} & C_{13} & 0 & 0 & C_{16} \\ C_{12} & C_{11} & C_{13} & 0 & 0 & -C_{16} \\ C_{13} & C_{13} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 \\ C_{16} & -C_{16} & 0 & 0 & 0 & C_{66} \end{pmatrix}. \quad (5.11) $$

Thus,

$$ N_{C(2)}^{(4)} = 7. \quad (5.12) $$

### 5.3.2 Additional Orthogonal Symmetries: 422, 4mm, and 4/mmm

Imposing, in addition to $\mathbf{a}_4$, invariance under either $\mathbf{a}_{2x}$ or $\mathbf{m}_1$ (or any equivalent symmetry that reverses $x_1$ while preserving $x_3$) eliminates the $C_{16}$ coupling:

$$ C_{16} = 0, \quad (5.13) $$

while all remaining constants in (5.11) are preserved. Hence, for tetragonal classes generated by $\{\mathbf{a}_4, \mathbf{a}_{2x}\}$ (e.g. 422) or $\{\mathbf{a}_4, \mathbf{m}_1\}$ (e.g. 4mm), and likewise for centrosymmetric 4/mmm,

$$ [C_{IJ}]_{\text{(high tetr)}} = \begin{pmatrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{13} & 0 & 0 & 0 \\ C_{13} & C_{13} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66} \end{pmatrix}, \quad (5.14) $$

with

$$ N_{C(2)}^{(422)} = N_{C(2)}^{(4mm)} = N_{C(2)}^{(4/mmm)} = 6. \quad (5.15) $$

For 4/m, the elastic tensor is even-rank and is not eliminated by inversion, hence $N_{C(2)}^{(4/m)} = N_{C(2)}^{(4)} = 7$.

## 5.4 Third-Order Elastic Constants

### 5.4.1 Invariance Equations

In generalized Voigt form, third-order elasticity is represented by a fully symmetric $C_{ijk}$. Under a generator with induced Voigt action $\mathbf{K}$,

$$ C'_{ijk} = K_{iP}K_{jQ}K_{KR}C_{PQR}, \quad (5.16) $$

and invariance requires $C'_{ijk} = C_{ijk}$ for all $I, J, K$.
### 5.4.2 Independent Counts

Enforcing invariance under $\mathbf{a}_4$ (equivalently $\mathbf{K}_4$) reduces the triclinic baseline 56 to

$$N_{C^{(3)}}^{(4)} = 16. \qquad (5.17)$$

If, in addition, invariance under $\mathbf{a}_{2x}$ or $\mathbf{m}_1$ is enforced (as in 422, 4mm, and 4/mmm), the independent count further reduces to

$$N_{C^{(3)}}^{(422)} = N_{C^{(3)}}^{(4mm)} = N_{C^{(3)}}^{(4/mmm)} = 12. \qquad (5.18)$$

A convenient independent set in each case is obtained by listing all $C_{ijk}$ with $1 \le I \le J \le K \le 6$ that satisfy the corresponding invariance relations under the chosen generator set.

## 5.5 Linear Piezoelectric Constants

The linear piezoelectric tensor $\xi_{kij}$ is symmetric in $(ij)$ and represented as $\xi_{kI}$. Under a generator $\mathbf{a}$ with induced $\mathbf{K}$,

$$\xi'_{kI} = a_{kp} K_{IJ} \xi_{pJ}, \qquad (5.19)$$

and invariance requires $\xi'_{kI} = \xi_{kI}$.

### 5.5.1 Class 4

Imposing invariance under $\mathbf{a}_4$ alone yields the tetragonal 4 pattern

$$[\xi_{kI}]_{(4)} = \begin{pmatrix} 0 & 0 & 0 & \xi_{14} & \xi_{15} & 0 \\ 0 & 0 & 0 & \xi_{15} & -\xi_{14} & 0 \\ \xi_{31} & \xi_{31} & \xi_{33} & 0 & 0 & 0 \end{pmatrix}, \qquad (5.20)$$

containing

$$N_{\xi^{(2)}}^{(4)} = 4 \qquad (5.21)$$

independent constants.

### 5.5.2 Class 422

Imposing invariance under $\mathbf{a}_4$ and $\mathbf{a}_{2x}$ yields

$$[\xi_{kI}]_{(422)} = \begin{pmatrix} 0 & 0 & 0 & \xi_{14} & 0 & 0 \\ 0 & 0 & 0 & 0 & -\xi_{14} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}, \qquad (5.22)$$

so

$$N_{\xi^{(2)}}^{(422)} = 1, \quad \xi_{25} = -\xi_{14}. \qquad (5.23)$$
### 5.5.3 Class 4mm

Imposing invariance under $\mathbf{a}_4$ and $\mathbf{m}_1$ yields

$$ [\xi_{kl}]_{(4mm)} = \begin{pmatrix} 0 & 0 & 0 & 0 & \xi_{15} & 0 \\ 0 & 0 & 0 & \xi_{15} & 0 & 0 \\ \xi_{31} & \xi_{31} & \xi_{33} & 0 & 0 & 0 \end{pmatrix}, \qquad (5.24) $$

with

$$ N_{\xi^{(2)}}^{(4mm)} = 3. \qquad (5.25) $$

### 5.5.4 Centrosymmetric tetragonal classes

Inversion symmetry forbids all odd-rank polar couplings. Therefore, for centrosymmetric tetragonal groups such as 4/m and 4/mmm,

$$ \xi_{kI} \equiv 0, \qquad N_{\xi^{(2)}}^{(4/m)} = N_{\xi^{(2)}}^{(4/mmm)} = 0. \qquad (5.26) $$

## 5.6 Fifth-Order Piezoelectric Constants

In generalized Voigt form, the fifth-order piezoelectric tensor is represented by $\xi_{kIJ} = \xi_{kJI}$. Under a generator ($\mathbf{a}, \mathbf{K}$),

$$ \xi'_{kIJ} = a_{kp} K_{IP} K_{JQ} \xi_{pPQ}, \qquad (5.27) $$

and invariance requires $\xi'_{kIJ} = \xi_{kIJ}$.

Enforcing fourfold invariance yields

$$ N_{\xi^{(5)}}^{(4)} = 15. \qquad (5.28) $$

Adding $\mathbf{a}_{2x}$ (class 422) or $\mathbf{m}_1$ (class 4mm) further reduces the independent count to

$$ N_{\xi^{(5)}}^{(422)} = 5, \qquad N_{\xi^{(5)}}^{(4mm)} = 10. \qquad (5.29) $$

For centrosymmetric tetragonal classes (e.g. 4/m, 4/mmm),

$$ \xi_{kIJ} \equiv 0, \qquad N_{\xi^{(5)}}^{(4/m)} = N_{\xi^{(5)}}^{(4/mmm)} = 0. \qquad (5.30) $$

## 5.7 Dielectric Constants

### 5.7.1 Second-Order Dielectric Tensor

The dielectric tensor $\epsilon_{ij} = \epsilon_{ji}$ transforms as a symmetric second-rank tensor. Fourfold invariance about $x_3$ enforces transverse isotropy in the $x_1x_2$-plane and eliminates all
off-diagonal components:

$$[\varepsilon_{ij}] = \begin{pmatrix} \varepsilon_{11} & 0 & 0 \\ 0 & \varepsilon_{11} & 0 \\ 0 & 0 & \varepsilon_{33} \end{pmatrix}, \quad N_{\varepsilon^{(2)}}^{(\text{tetr})} = 2, \qquad (5.31)$$

valid for all tetragonal classes.

### 5.7.2 Third-Order Dielectric Tensor (quadratic electric-field response)

In this book, the third-order dielectric response is represented by the polar rank-3 tensor $\varepsilon_{kij}^{(3)}$ symmetric in the last two indices, written in Voigt form as $\varepsilon_{kI}^{(3)}$. Under a generator a with induced K,

$$(\varepsilon^{(3)})'_{kI} = a_{kp} K_{IJ} \varepsilon_{pJ}^{(3)}, \qquad (5.32)$$

and invariance requires $(\varepsilon^{(3)})'_{kI} = \varepsilon_{kI}^{(3)}$. Therefore, the tetragonal patterns and counts are structurally identical to the linear piezoelectric patterns and counts of the same point group.

#### Class 4.

$$[\varepsilon_{kI}^{(3)}]_{(4)} = \begin{pmatrix} 0 & 0 & 0 & \varepsilon_{14}^{(3)} & \varepsilon_{15}^{(3)} & 0 \\ 0 & 0 & 0 & \varepsilon_{15}^{(3)} & -\varepsilon_{14}^{(3)} & 0 \\ \varepsilon_{31}^{(3)} & \varepsilon_{31}^{(3)} & \varepsilon_{33}^{(3)} & 0 & 0 & 0 \end{pmatrix}, \quad N_{\varepsilon^{(3)}}^{(4)} = 4. \qquad (5.33)$$

#### Class 422.

$$[\varepsilon_{kI}^{(3)}]_{(422)} = \begin{pmatrix} 0 & 0 & 0 & \varepsilon_{14}^{(3)} & 0 & 0 \\ 0 & 0 & 0 & 0 & -\varepsilon_{14}^{(3)} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}, \quad N_{\varepsilon^{(3)}}^{(422)} = 1. \qquad (5.34)$$

#### Class 4mm.

$$[\varepsilon_{kI}^{(3)}]_{(4mm)} = \begin{pmatrix} 0 & 0 & 0 & 0 & \varepsilon_{15}^{(3)} & 0 \\ 0 & 0 & 0 & \varepsilon_{15}^{(3)} & 0 & 0 \\ \varepsilon_{31}^{(3)} & \varepsilon_{31}^{(3)} & \varepsilon_{33}^{(3)} & 0 & 0 & 0 \end{pmatrix}, \quad N_{\varepsilon^{(3)}}^{(4mm)} = 3. \qquad (5.35)$$

Centrosymmetric tetragonal classes.

$$\varepsilon_{kI}^{(3)} \equiv 0, \quad N_{\varepsilon^{(3)}}^{(4/m)} = N_{\varepsilon^{(3)}}^{(4/mmm)} = 0. \qquad (5.36)$$

### 5.8 Electrostrictive Tensor

Electrostriction is represented by a fourth-rank tensor $\gamma_{ijkl}$ symmetric within each index pair:

$$\gamma_{ijkl} = \gamma_{jikl} = \gamma_{ijlk}, \qquad (5.37)$$
but no intrinsic symmetry is assumed under interchange of the pairs (ij) ↔ (kl). In Voigt form this corresponds to a general 6 × 6 matrix ΓIJ.

### 5.8.1 Class 4 (fourfold rotation only)

Under $\mathbf{a}_4$, invariance reads

$$ \Gamma = \mathbf{K}_4 \Gamma \mathbf{K}_4^\top. \qquad (5.38) $$

Solving (5.38) yields the general tetragonal-4 electrostriction form

$$ [\Gamma_{IJ}]_{(4)} = \begin{pmatrix} \Gamma_{11} & \Gamma_{12} & \Gamma_{13} & 0 & 0 & -\Gamma_{16} \\ \Gamma_{12} & \Gamma_{11} & \Gamma_{13} & 0 & 0 & \Gamma_{16} \\ \Gamma_{31} & \Gamma_{31} & \Gamma_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & \Gamma_{44} & -\Gamma_{45} & 0 \\ 0 & 0 & 0 & \Gamma_{45} & \Gamma_{44} & 0 \\ -\Gamma_{61} & \Gamma_{61} & 0 & 0 & 0 & \Gamma_{66} \end{pmatrix}, \qquad (5.39) $$

containing

$$ N_{\gamma}^{(4)} = 10 \qquad (5.40) $$

independent constants.

### 5.8.2 Higher tetragonal classes: 422, 4mm, 4/mmm

Imposing, in addition, invariance under either $\mathbf{a}_{2x}$ or $\mathbf{m}_1$ forces $\Gamma_{16} = \Gamma_{45} = \Gamma_{61} = 0$ and yields

$$ [\Gamma_{IJ}]_{\text{(high tetr)}} = \begin{pmatrix} \Gamma_{11} & \Gamma_{12} & \Gamma_{13} & 0 & 0 & 0 \\ \Gamma_{12} & \Gamma_{11} & \Gamma_{13} & 0 & 0 & 0 \\ \Gamma_{31} & \Gamma_{31} & \Gamma_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & \Gamma_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & \Gamma_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & \Gamma_{66} \end{pmatrix}, \qquad (5.41) $$

with

$$ N_{\gamma}^{(422)} = N_{\gamma}^{(4mm)} = N_{\gamma}^{(4/mmm)} = 7. \qquad (5.42) $$

For 4/m, electrostriction is even-rank and is not eliminated by inversion, hence $N_{\gamma}^{(4/m)} = N_{\gamma}^{(4)} = 10$.

## 5.9 Practical relevance & typical material examples

* **Where it appears:** Systems with a principal fourfold axis, including many technologically important oxides and ceramics.

* **Why it matters:** The fourfold axis enforces transverse-plane degeneracies and characteristic identifications among shear/normal couplings, strongly shaping admissible electromechanical couplings.
* Typical examples: Rutile TiO₂ (tetragonal), zircon ZrSiO₄ (tetragonal), several tetragonal ferroelectric phases in perovskite-derived systems.

## 5.10 Summary and Practical Relevance

Tetragonal symmetry illustrates clearly how rotational invariance about a principal axis induces degeneracy patterns:

* transverse degeneracy in the $x_1x_2$-plane, e.g. $C_{11} = C_{22}$, $C_{13} = C_{23}$, and $C_{44} = C_{55}$;

* the possible appearance of a characteristic coupling $C_{16} = -C_{26}$ in lower tetragonal classes compatible with fourfold rotation, which vanishes once additional orthogonal symmetries are imposed;

* strong constraints on odd-rank polar tensors, including complete suppression by inversion symmetry.

These results are particularly relevant for widely used tetragonal piezoelectric crystals (notably the polar class 4mm), for which the compact piezoelectric form (5.24) is the standard working representation.

<table><caption>Table 5.1: Independent constants for representative tetragonal classes (fourfold axis $x_3$).</caption><thead><tr><th>Property tensor</th><th>4</th><th>422</th><th>4mm</th><th>4/m</th><th>4/mmm</th></tr></thead><tbody><tr><td>Second-order elasticity C<sub>IJ</sub></td><td>7</td><td>6</td><td>6</td><td>7</td><td>6</td></tr><tr><td>Third-order elasticity C<sub>ijk</sub></td><td>16</td><td>12</td><td>12</td><td>16</td><td>12</td></tr><tr><td>Linear piezoelectricity ξ<sub>kI</sub></td><td>4</td><td>1</td><td>3</td><td>0</td><td>0</td></tr><tr><td>Fifth-order piezoelectricity ξ<sub>kIJ</sub></td><td>15</td><td>5</td><td>10</td><td>0</td><td>0</td></tr><tr><td>Dielectric ε<sub>ij</sub></td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td></tr><tr><td>Third-order dielectric ε<sub>kI</sub><sup>(3)</sup></td><td>4</td><td>1</td><td>3</td><td>0</td><td>0</td></tr><tr><td>Electrostriction Γ<sub>IJ</sub></td><td>10</td><td>7</td><td>7</td><td>10</td><td>7</td></tr></tbody></table>
# Chapter 6

## Trigonal Symmetry

### 6.1 Trigonal 3m and Generator Choice

This chapter presents a complete, fully worked symmetry reduction for the trigonal class 3m. It is used as a benchmark chapter because it corresponds to the trigonal symmetry setting treated in the author's earlier *Annalen der Physik* publication, while also making explicit (and separating) what is *purely symmetry-forced* from what may additionally follow from material-specific modeling assumptions.

A standard crystallographic convention is adopted: the threefold axis is aligned with $x_3$. For trigonal 3m, a convenient generator set consists of a 120° rotation about the $x_3$-axis, and a mirror reflection in a vertical plane containing $x_3$.

We use the following orthogonal generators:

$$ \mathbf{a}_1 = \begin{pmatrix} -\frac{1}{2} & \frac{\sqrt{3}}{2} & 0 \\ -\frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad \mathbf{a}_2 = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}. \qquad (6.1) $$

These generators produce the full trigonal 3m point group in the chosen setting.

### 6.2 Invariance Enforcement (Operational Form)

For a tensor $T_{i_1 \cdots i_n}$, invariance under a generator $\mathbf{a}$ means

$$ T_{i_1 \cdots i_n} = a_{i_1 j_1} \cdots a_{i_n j_n} T_{j_1 \cdots j_n}. \qquad (6.2) $$

In practice, one proceeds generator by generator:

$$ T \text{ is compatible with } 3m \iff T \text{ is invariant under both } \mathbf{a}_1 \text{ and } \mathbf{a}_2. \qquad (6.3) $$

For tensors built from symmetric index pairs, Voigt and generalized Voigt notation are used exactly as in Chapters 1–4, with the tensorial Voigt mapping

$$ (11, 22, 33, 23, 13, 12) \leftrightarrow (1, 2, 3, 4, 5, 6), \qquad (6.4) $$
and with no engineering shear factors.

## 6.3 Second-Order Elastic Constants $c_{ijkl}$

Using Voigt notation $C_{IJ}$, trigonal 3m invariance yields:

$$ 
\begin{aligned}
C_{15} &= C_{16} = C_{25} = C_{26} = C_{35} = C_{36} = C_{45} = C_{46} = C_{34} = 0, \\
C_{56} &= C_{14}, \quad C_{24} = -C_{14}, \quad C_{55} = C_{44}, \\
C_{23} &= C_{13}, \quad C_{22} = C_{11}, \quad C_{66} = \frac{C_{11} - C_{12}}{2}.
\end{aligned} 
\quad (6.5) $$

A standard matrix form consistent with (6.5) is therefore

$$ [C_{IJ}]_{(3m)} = \begin{pmatrix} C_{11} & C_{12} & C_{13} & C_{14} & 0 & 0 \\ C_{12} & C_{11} & C_{13} & -C_{14} & 0 & 0 \\ C_{13} & C_{13} & C_{33} & 0 & 0 & 0 \\ C_{14} & -C_{14} & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & C_{14} \\ 0 & 0 & 0 & 0 & C_{14} & \frac{1}{2}(C_{11} - C_{12}) \end{pmatrix}. \quad (6.6) $$

## 6.4 Third-Order Elastic Constants $c_{ijklm}$

In generalized Voigt notation, third-order elasticity is represented by a fully symmetric tensor $C_{IJK} = C_{(IJK)}$. Trigonal 3m invariance yields:

### 6.4.1 Forced Zeros

$$ 
\begin{aligned}
C_{236} &= C_{336} = C_{334} = C_{115} = C_{116} = C_{566} = C_{666} = C_{346} = C_{345} = C_{445} = C_{446} \\
&= C_{226} = C_{335} = C_{145} = C_{135} = C_{245} = C_{136} = C_{246} = C_{225} = C_{146} = C_{235} \\
&= C_{125} = C_{126} = C_{556} = C_{555} = 0.
\end{aligned} 
\quad (6.7) $$

### 6.4.2 Linear Relations

$$ 
\begin{aligned}
C_{233} &= C_{133}, & C_{255} &= C_{144}, & C_{356} &= C_{134}, \\
C_{234} &= -C_{134}, & C_{223} &= C_{113}, & & \\
C_{244} &= C_{155}, & C_{355} &= C_{344}, & C_{455} &= -C_{444}, & C_{466} &= C_{124}, \\
C_{266} &= -\frac{C_{112}}{4} + \frac{C_{111}}{2} - \frac{C_{222}}{4}, & C_{456} &= \frac{C_{155}}{2} - \frac{C_{144}}{2}, & & \\
C_{366} &= -\frac{C_{123}}{2} + \frac{C_{113}}{2}, & C_{224} &= -2C_{124} - C_{114}, & & \\
C_{156} &= \frac{C_{114}}{2} + \frac{3C_{124}}{2}, & C_{256} &= -\frac{C_{124}}{2} + \frac{C_{114}}{2}, & & \\
C_{166} &= -\frac{C_{111}}{2} - \frac{C_{112}}{4} + \frac{3C_{222}}{4}, & C_{122} &= C_{111} + C_{112} - C_{222}.
\end{aligned} 
\quad (6.8) $$
Equations (6.7)–(6.8) constitute a complete trigonal 3m symmetry reduction for third-order elasticity in generalized Voigt form (no additional assumptions beyond intrinsic symmetries and point-group invariance).

## 6.5 Second-Order Piezoelectric Constants $\xi_{kij}$

With $\xi_{kij} = \xi_{kji}$, Voigt notation yields $\xi_{kI}$. Trigonal 3m invariance gives:

$$ \begin{aligned} \xi_{34} &= \xi_{23} = \xi_{25} = \xi_{26} = \xi_{35} = \xi_{36} = \xi_{11} = \xi_{12} = \xi_{14} = \xi_{13} = 0, \\ \xi_{32} &= \xi_{31}, \quad \xi_{16} = -\xi_{22}, \quad \xi_{21} = -\xi_{22}, \quad \xi_{24} = \xi_{15}. \end{aligned} \quad (6.9) $$

A canonical 3 × 6 form consistent with (6.9) is:

$$ [\xi_{kI}]_{(3m)} = \begin{pmatrix} 0 & 0 & 0 & 0 & \xi_{15} & -\xi_{22} \\ -\xi_{22} & \xi_{22} & 0 & \xi_{15} & 0 & 0 \\ \xi_{31} & \xi_{31} & \xi_{33} & 0 & 0 & 0 \end{pmatrix}. \quad (6.10) $$

## 6.6 Third-Order Piezoelectric Constants $\xi_{kijmn}$

The higher-order piezoelectric tensor is taken as a polar rank-5 tensor symmetric within each strain-like index pair:

$$ \xi_{kijmn} = \xi_{kjnm} = \xi_{kijnm}, \quad (6.11) $$

and represented in generalized Voigt notation as $\xi_{kIJ} = \xi_{kJI}$. Trigonal 3m invariance yields:

### 6.6.1 Forced Zeros

$$ \begin{aligned} \xi_{346} &= \xi_{245} = \xi_{246} = \xi_{334} = \xi_{122} = \xi_{233} = \xi_{134} = \xi_{235} = \xi_{144} = \xi_{316} = \xi_{315} = \xi_{133} \\ &= \xi_{124} = \xi_{111} = \xi_{112} = \xi_{113} = \xi_{114} = \xi_{326} = \xi_{216} = \xi_{215} = \xi_{345} = \xi_{155} = \xi_{336} \\ &= \xi_{123} = \xi_{325} = \xi_{226} = \xi_{225} = \xi_{236} = \xi_{166} = \xi_{156} = \xi_{335} = 0. \end{aligned} \quad (6.12) $$

### 6.6.2 Linear Relations

$$ \begin{aligned} \xi_{255} &= \xi_{145}, & \xi_{214} &= \xi_{125}, & \xi_{244} &= -\xi_{145}, & \xi_{234} &= \xi_{135}, \\ \xi_{323} &= \xi_{313}, & \xi_{324} &= \xi_{314}, & \xi_{356} &= \xi_{314}, & \xi_{213} &= \xi_{136}, \\ \xi_{223} &= -\xi_{136}, & \xi_{224} &= \xi_{115}, & \xi_{355} &= \xi_{344}, & \xi_{322} &= \xi_{311}, \\ \xi_{146} &= \frac{\xi_{115}}{2} - \frac{\xi_{125}}{2}, & \xi_{366} &= -\frac{\xi_{312}}{2} + \frac{\xi_{311}}{2}, & \xi_{266} &= -\frac{\xi_{126}}{2} + \frac{\xi_{116}}{2}, \\ \xi_{222} &= -\frac{3\xi_{116}}{2} - \frac{\xi_{126}}{2}, & \xi_{211} &= \frac{3\xi_{126}}{2} + \frac{\xi_{116}}{2}, & \xi_{256} &= \frac{\xi_{115}}{2} - \frac{\xi_{125}}{2}, \\ \xi_{212} &= -\frac{\xi_{126}}{2} + \frac{\xi_{116}}{2}. \end{aligned} \quad (6.13) $$
Equations (6.12)–(6.13) provide a complete trigonal 3m symmetry reduction for the higher-order piezoelectric tensor in generalized Voigt form.

## 6.7 Dielectric Constants

### 6.7.1 Second-Order Dielectric Tensor $\epsilon_{ij}$

Trigonal 3m invariance yields transverse isotropy about the threefold axis:

$$ [\epsilon_{ij}]_{(3m)} = \begin{pmatrix} \epsilon_{11} & 0 & 0 \\ 0 & \epsilon_{11} & 0 \\ 0 & 0 & \epsilon_{33} \end{pmatrix}. \qquad (6.14) $$

Equivalently,

$$ \epsilon_{22} = \epsilon_{11}, \quad \epsilon_{12} = \epsilon_{13} = \epsilon_{23} = 0. \qquad (6.15) $$

### 6.7.2 Third-Order Dielectric Tensor: symmetry-allowed form

To avoid ambiguity, we define the third-order dielectric tensor here as a polar rank-3 tensor symmetric in its last two indices:

$$ \epsilon_{kij}^{(3)} = \epsilon_{kji}^{(3)}, \qquad (6.16) $$

represented in Voigt form by $\epsilon_{kI}^{(3)}$. This is the same index structure as the linear piezoelectric tensor $\xi_{kij}$ and is the natural object when writing field expansions of the form $D_k = \epsilon_{kI}E_j + \frac{1}{2}\epsilon_{kij}^{(3)}E_iE_j + \dots$.

Trigonal 3m invariance yields the canonical form

$$ [\epsilon_{kI}^{(3)}]_{(3m)} = \begin{pmatrix} 0 & 0 & 0 & 0 & \epsilon_{15}^{(3)} & -\epsilon_{22}^{(3)} \\ -\epsilon_{22}^{(3)} & \epsilon_{22}^{(3)} & 0 & \epsilon_{15}^{(3)} & 0 & 0 \\ \epsilon_{31}^{(3)} & \epsilon_{31}^{(3)} & \epsilon_{33}^{(3)} & 0 & 0 & 0 \end{pmatrix}, \qquad (6.17) $$

containing four independent constants

$$ \epsilon_{15}^{(3)}, \epsilon_{22}^{(3)}, \epsilon_{31}^{(3)}, \epsilon_{33}^{(3)}. \qquad (6.18) $$

**Remark (benchmark specialization).** In specific material models or datasets (including certain benchmark reductions presented historically in the literature), additional assumptions beyond point-group invariance may be imposed (e.g. model truncations or further constitutive symmetries). Such extra assumptions may reduce the effective independent set. In this book, however, (6.17) is the symmetry-complete trigonal 3m form.
## 6.8 Electrostrictive Constants $\gamma_{ijkl}$

Electrostriction is represented in Voigt form by a 6 × 6 matrix $\Gamma_{IJ}$ (with the intrinsic symmetries specified by the constitutive framework used in the main text). Trigonal 3m invariance yields:

### 6.8.1 Forced Zeros

$$ \begin{aligned} \gamma_{35} &= \gamma_{36} = \gamma_{52} = \gamma_{25} = \gamma_{61} = \gamma_{16} = \gamma_{26} = \gamma_{34} = \gamma_{15} = \gamma_{53} = \gamma_{54} = \gamma_{51} = \gamma_{63} \\ &= \gamma_{64} = \gamma_{62} = \gamma_{43} = \gamma_{45} = \gamma_{46} = 0. \end{aligned} \quad (6.19) $$

### 6.8.2 Identifications and Linear Relations

$$ \gamma_{32} = \gamma_{31}, \quad \gamma_{65} = \gamma_{14}, \quad \gamma_{42} = -\gamma_{41}, \quad \gamma_{56} = \gamma_{41}, \quad \gamma_{22} = \gamma_{11}, \quad (6.20) $$

and

$$ \gamma_{23} = \gamma_{13}, \quad \gamma_{55} = \gamma_{44}, \quad \gamma_{24} = -\gamma_{14}, \quad \gamma_{21} = \gamma_{12}, \quad \gamma_{66} = \frac{\gamma_{11} - \gamma_{12}}{2}. \quad (6.21) $$

Equations (6.19)–(6.21) provide the trigonal 3m symmetry reduction for electrostriction in the benchmark convention.

Equivalently, the number of independent electrostrictive constants for class 3m is

$$ N_{\gamma}^{(3m)} = 8. \quad (6.22) $$

One convenient independent parameter set is, for instance,

$$ \{\gamma_{11}, \gamma_{12}, \gamma_{13}, \gamma_{33}, \gamma_{44}, \gamma_{14}, \gamma_{31}, \gamma_{15}\}, $$

with all remaining components determined by Eqs. (6.19)–(6.21).

## 6.9 Benchmark Tables (Independent vs Dependent Sets)

For practical use, it is convenient to separate each tensor into a minimal independent parameter set and the dependent set generated by the symmetry relations above. One consistent choice of independent parameters is:

* Second-order elasticity: $C_{11}, C_{12}, C_{13}, C_{14}, C_{33}, C_{44}$.

* Third-order elasticity: a valid independent set is

$$ C_{111}, C_{112}, C_{113}, C_{114}, C_{123}, C_{124}, C_{133}, C_{134}, C_{144}, C_{155}, C_{222}, C_{333}, C_{344}, C_{444}. $$

with all remaining components fixed by (6.7)–(6.8) and full symmetry in (I, J, K).

* Second-order piezoelectricity: $\xi_{15}, \xi_{22}, \xi_{31}, \xi_{33}$.
* Higher-order piezoelectricity (rank 5): a valid independent set includes all right-hand-side constants appearing in (6.13), e.g.

$$ \xi_{115}, \xi_{116}, \xi_{125}, \xi_{126}, \xi_{135}, \xi_{136}, \xi_{145}, \xi_{311}, \xi_{312}, \xi_{313}, \xi_{314}, \xi_{344}, $$

together with any additional independent constants required by the precise constitutive definition adopted for the rank-5 coupling.

* Dielectric: $\epsilon_{11}, \epsilon_{33}$ (second order), and $\epsilon_{15}^{(3)}, \epsilon_{22}^{(3)}, \epsilon_{31}^{(3)}, \epsilon_{33}^{(3)}$ (third order as defined here).

* Electrostriction: $\gamma_{11}, \gamma_{12}, \gamma_{13}, \gamma_{14}, \gamma_{31}, \gamma_{33}, \gamma_{41}, \gamma_{44}$.

## 6.10 Consistency with Literature and Benchmark Role

The second-order elastic and piezoelectric reductions (6.5) and (6.9) coincide with standard trigonal 3m forms used in linear anisotropic elasticity and piezoelectricity, providing an essential validation check.

The higher-order reductions (third-order elasticity and rank-5 piezoelectricity), and the explicit trigonal 3m electrostriction structure, are obtained by the same invariance principle applied systematically to higher-rank tensors. This chapter therefore functions as the benchmark implementation of the method: it demonstrates that once generator matrices and intrinsic symmetries are fixed, all admissible tensor components and relations follow uniquely and reproducibly from invariance.

## 6.11 Practical relevance & typical material examples

* **Where it appears:** Threefold rotational symmetry; central in piezoelectric acoustics and wave devices.

* **Why it matters:** Trigonal symmetry is a common “benchmark” class because it admits nontrivial piezoelectricity with enough structure to strongly constrain higher-order couplings while still retaining rich anisotropic behavior.

* **Typical examples:** α-quartz (trigonal), lithium niobate LiNbO$_3$ (trigonal), lithium tantalate LiTaO$_3$ (trigonal).
# Chapter 7

## Hexagonal Symmetry

### 7.1 Hexagonal Point Groups and Generator Choice

Hexagonal crystals possess a distinguished sixfold symmetry axis. Throughout this chapter the sixfold axis is aligned with $x_3$. Representative hexagonal point groups considered here are

$$ 6, 622, 6mm, 6/m, 6mmm. \qquad (7.1) $$

All hexagonal classes share the fundamental sixfold rotation about $x_3$. A convenient generator for this rotation is

$$ a_6 = \begin{pmatrix} \cos(\pi/3) & \sin(\pi/3) & 0 \\ -\sin(\pi/3) & \cos(\pi/3) & 0 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} & 0 \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} & 0 \\ 0 & 0 & 1 \end{pmatrix}. \qquad (7.2) $$

To represent common subclasses, we additionally use:

* a twofold rotation about $x_1$, for 622,

$$ a_{2x} = \text{diag}(+1, -1, -1), \qquad (7.3) $$

* a vertical mirror plane containing $x_3$, for 6mm, e.g. reflection in the $x_2x_3$-plane,

$$ m_1 = \text{diag}(-1, +1, +1), \qquad (7.4) $$

* inversion symmetry (for 6/m and 6/mmm), which eliminates all odd-rank polar tensors.

### 7.2 Relation to Trigonal Symmetry

Hexagonal symmetry may be viewed as trigonal symmetry with an additional rotational constraint: trigonal classes require invariance under a 120° rotation about $x_3$, whereas hexagonal classes require invariance under a 60° rotation. Since 60° invariance
implies 120° invariance, hexagonal constraints are strictly stronger and therefore lead to a further collapse of admissible tensor structure relative to trigonal symmetry. In particular, couplings permitted in trigonal classes (such as the characteristic $C_{14}$ elastic coupling of 3m) are eliminated by sixfold rotational invariance.

## 7.3 Voigt Transformation under Sixfold Rotation

Using the tensorial Voigt mapping

$$ (11, 22, 33, 23, 13, 12) \leftrightarrow (1, 2, 3, 4, 5, 6), \qquad (7.5) $$

a symmetric second-rank tensor written as a 6-vector transforms under $a_6$ by a $6 \times 6$ matrix $K_6$, defined by $S'_I = (K_6)_{IJ}S_J$. A direct evaluation yields

$$ K_6 = \begin{pmatrix} \frac{1}{4} & \frac{3}{4} & 0 & 0 & 0 & \frac{\sqrt{3}}{2} \\ \frac{3}{4} & \frac{1}{4} & 0 & 0 & 0 & -\frac{\sqrt{3}}{2} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2} & -\frac{\sqrt{3}}{2} & 0 \\ 0 & 0 & 0 & \frac{\sqrt{3}}{2} & \frac{1}{2} & 0 \\ -\frac{\sqrt{3}}{4} & \frac{\sqrt{3}}{4} & 0 & 0 & 0 & -\frac{1}{2} \end{pmatrix}. \qquad (7.6) $$

For diagonal generators such as $a_{2x}$ and $m_1$, the induced Voigt action is diagonal. In particular,

$$ K(a_{2x}) = K(m_1) = \text{diag}(1, 1, 1, 1, -1, -1). \qquad (7.7) $$

## 7.4 Second-Order Elastic Constants

### 7.4.1 Sixfold Rotational Invariance

In Voigt form, the stiffness matrix transforms as

$$ C' = KCK^T. \qquad (7.8) $$

Imposing invariance under $a_6$ (i.e. $K = K_6$) yields the standard hexagonal (transversely isotropic) form:

$$ [C_{IJ}]_{\text{(hex)}} = \begin{pmatrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{13} & 0 & 0 & 0 \\ C_{13} & C_{13} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66} \end{pmatrix}, \qquad C_{66} = \frac{C_{11} - C_{12}}{2}. \qquad (7.9) $$

Thus, the independent second-order elastic constants are

$$ C_{11}, C_{12}, C_{13}, C_{33}, C_{44}, \qquad (7.10) $$
and the count is

$$N_{C^{(2)}}^{(hex)} = 5, \qquad (7.11)$$

valid for all hexagonal classes 6, 622, 6mm, 6/m, and 6/mmm, since these are all supergroups of the sixfold rotation.

# 7.5 Third-Order Elastic Constants

## 7.5.1 Generalized Voigt Form and Invariance

In generalized Voigt notation the third-order elastic constants are represented by a fully symmetric tensor $C_{IJK}$. Under the sixfold rotation $a_6$ with induced action $K_6$,

$$C'_{IJK} = (K_6)_{IP} (K_6)_{JQ} (K_6)_{KR} C_{PQR}, \quad C'_{IJK} = C_{IJK}. \qquad (7.12)$$

Enforcing this invariance yields a complete hexagonal reduction with

$$N_{C^{(3)}}^{(hex)} = 12 \qquad (7.13)$$

independent third-order elastic constants.

## 7.5.2 Independent Parameter Set

A convenient independent set is

$$\{C_{166}, C_{222}, C_{223}, C_{233}, C_{255}, C_{266}, C_{333}, C_{355}, C_{366}, C_{456}, C_{556}, C_{666}\}. \qquad (7.14)$$

All remaining components are fixed either by full symmetry in $(I, J, K)$ or by the sixfold invariance relations below.

### Forced zeros.

$$ \begin{aligned} & C_{114} = C_{115} = C_{124} = C_{125} = C_{134} = C_{135} = C_{136} \\ & = C_{146} = C_{156} = C_{224} = C_{225} = C_{234} = C_{235} = C_{236} = 0, \\ & C_{246} = C_{256} = C_{334} = C_{335} = C_{336} = C_{345} = C_{346} \\ & = C_{356} = C_{444} = C_{445} = C_{455} = C_{466} = C_{555} = C_{566} = 0. \end{aligned} \qquad (7.15) $$
Nontrivial relations.

$$C_{111} = -C_{166} + C_{222} + C_{266}, \quad C_{112} = -2C_{166} + C_{222} - 2C_{266}, \\ C_{113} = C_{223}, \tag{7.16}$$

$$C_{116} = -C_{666}, \quad C_{122} = -3C_{166} + C_{222} - C_{266}, \quad C_{123} = C_{223} - 2C_{366}, \\ C_{126} = C_{666}, \tag{7.17}$$

$$C_{133} = C_{233}, \quad C_{144} = C_{255}, \quad C_{145} = -C_{556}, \quad C_{155} = C_{255} + 2C_{456}, \tag{7.18}$$

$$C_{226} = -C_{666}, \quad C_{244} = C_{255} + 2C_{456}, \quad C_{245} = C_{556}, \quad C_{344} = C_{355}, \\ C_{446} = -C_{556}. \tag{7.19}$$

Equations (7.15)–(7.19) provide a complete and internally consistent hexagonal symmetry reduction for third-order elasticity.

## 7.6 Linear Piezoelectric Constants

The linear piezoelectric tensor $\xi_{kij}$ is symmetric in $(ij)$ and represented in Voigt form by $\xi_{kI}$. Under a generator $a$ with induced $K$,

$$\xi'_{kI} = a_{kp} K_{IJ} \xi_{pJ}, \quad \xi'_{kI} = \xi_{kI}. \tag{7.20}$$

### 7.6.1 Class 6 (sixfold rotation only)

Enforcing invariance under $a_6$ yields the standard class-6 form

$$[\xi_{kI}]_{(6)} = \begin{pmatrix} 0 & 0 & 0 & \xi_{14} & \xi_{15} & 0 \\ 0 & 0 & 0 & \xi_{15} & -\xi_{14} & 0 \\ \xi_{31} & \xi_{31} & \xi_{33} & 0 & 0 & 0 \end{pmatrix}, \tag{7.21}$$

containing

$$N_{\xi^{(2)}}^{(6)} = 4 \tag{7.22}$$

independent constants $\{\xi_{14}, \xi_{15}, \xi_{31}, \xi_{33}\}$.

### 7.6.2 Class 622

Adding invariance under $a_{2x}$ reduces (7.21) to a single independent constant:

$$[\xi_{kI}]_{(622)} = \begin{pmatrix} 0 & 0 & 0 & \xi_{14} & 0 & 0 \\ 0 & 0 & 0 & 0 & -\xi_{14} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}, \quad N_{\xi^{(2)}}^{(622)} = 1. \tag{7.23}$$
### 7.6.3 Class 6mm

Adding a vertical mirror plane $m_1$ eliminates the $\xi_{14}$ coupling and yields the standard polar hexagonal piezoelectric form:

$$ [\xi_{kI}]_{(6mm)} = \begin{pmatrix} 0 & 0 & 0 & 0 & \xi_{15} & 0 \\ 0 & 0 & 0 & \xi_{15} & 0 & 0 \\ \xi_{31} & \xi_{31} & \xi_{33} & 0 & 0 & 0 \end{pmatrix}, \quad N_{\xi^{(2)}}^{(6mm)} = 3. \quad (7.24) $$

### 7.6.4 Centrosymmetric classes 6/m and 6/mmm

Inversion symmetry forbids all odd-rank polar couplings:

$$ \xi_{kI} \equiv 0, \quad N_{\xi^{(2)}}^{(6/m)} = N_{\xi^{(2)}}^{(6/mmm)} = 0. \quad (7.25) $$

## 7.7 Fifth-Order Piezoelectric Constants

In generalized Voigt form, the fifth-order piezoelectric constants are represented by $\xi_{kIJ} = \xi_{kJI}$. Under a generator $(\mathbf{a}, \mathbf{K})$,

$$ \xi'_{kIJ} = a_{kp} K_{IP} K_{JQ} \xi_{pPQ}, \quad \xi'_{kIJ} = \xi_{kIJ}. \quad (7.26) $$

For hexagonal symmetry the resulting structure is compactly expressed by collecting, for each fixed $k$, the symmetric $6 \times 6$ matrix $\Xi_{I,J}^{(k)} := \xi_{kIJ}$.

### 7.7.1 Class 6mm

A complete 6mm reduction yields 8 independent constants. Defining the independent set

$$ \begin{align} x_{224} &:= \xi_{2,24}, & x_{234} &:= \xi_{2,34}, & x_{256} &:= \xi_{2,56}, \nonumber \\ x_{322} &:= \xi_{3,22}, & x_{323} &:= \xi_{3,23}, & x_{333} &:= \xi_{3,33}, \tag{7.27} \\ x_{355} &:= \xi_{3,55}, & x_{366} &:= \xi_{3,66}. \nonumber \end{align} $$

the three matrices $\Xi^{(k)}$ take the form

$$ \Xi^{(1)} = \begin{pmatrix} 0 & 0 & 0 & 0 & x_{224} & 0 \\ 0 & 0 & 0 & 0 & x_{224} - 2x_{256} & 0 \\ 0 & 0 & 0 & 0 & x_{234} & 0 \\ 0 & 0 & 0 & 0 & 0 & x_{256} \\ x_{224} & x_{224} - 2x_{256} & x_{234} & 0 & 0 & 0 \\ 0 & 0 & 0 & x_{256} & 0 & 0 \end{pmatrix}, \quad (7.28) $$
$$ \Xi^{(2)} = \begin{pmatrix} 0 & 0 & 0 & x_{224} - 2x_{256} & 0 & 0 \\ 0 & 0 & 0 & x_{224} & 0 & 0 \\ 0 & 0 & 0 & x_{234} & 0 & 0 \\ x_{224} - 2x_{256} & x_{224} & x_{234} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & x_{256} \\ 0 & 0 & 0 & 0 & x_{256} & 0 \end{pmatrix}, \quad (7.29) $$

$$ \Xi^{(3)} = \begin{pmatrix} x_{322} & x_{322} - 2x_{366} & x_{323} & 0 & 0 & 0 \\ x_{322} - 2x_{366} & x_{322} & x_{323} & 0 & 0 & 0 \\ x_{323} & x_{323} & x_{333} & 0 & 0 & 0 \\ 0 & 0 & 0 & x_{355} & 0 & 0 \\ 0 & 0 & 0 & 0 & x_{355} & 0 \\ 0 & 0 & 0 & 0 & 0 & x_{366} \end{pmatrix}. \quad (7.30) $$

Thus,

$$ N_{\xi^{(5)}}^{(6mm)} = 8. \quad (7.31) $$

### 7.7.2 Class 622

For 622, invariance under $a_6$ and $a_{2x}$ yields 3 independent constants. Using

$$ y_{225} := \xi_{2,25}, \quad y_{235} := \xi_{2,35}, \quad y_{246} := \xi_{2,46}, \quad (7.32) $$

the nonzero blocks are

$$ \Xi_{(622)}^{(1)} = \begin{pmatrix} 0 & 0 & 0 & -y_{225} & 0 & 0 \\ 0 & 0 & 0 & -y_{225} - 2y_{246} & 0 & 0 \\ 0 & 0 & 0 & -y_{235} & 0 & 0 \\ -y_{225} & -y_{225} - 2y_{246} & -y_{235} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -y_{246} \\ 0 & 0 & 0 & 0 & -y_{246} & 0 \end{pmatrix}, \quad (7.33) $$

$$ \Xi_{(622)}^{(2)} = \begin{pmatrix} 0 & 0 & 0 & 0 & y_{225} + 2y_{246} & 0 \\ 0 & 0 & 0 & 0 & y_{225} & 0 \\ 0 & 0 & 0 & 0 & y_{235} & 0 \\ 0 & 0 & 0 & 0 & 0 & y_{246} \\ y_{225} + 2y_{246} & y_{225} & y_{235} & 0 & 0 & 0 \\ 0 & 0 & 0 & y_{246} & 0 & 0 \end{pmatrix}, \quad (7.34) $$

with $\Xi_{(622)}^{(3)} \equiv 0$. Hence,

$$ N_{\xi^{(5)}}^{(622)} = 3. \quad (7.35) $$

### 7.7.3 Classes 6, 6/m, and 6/mmm

For class 6 (rotation only), the independent count is

$$ N_{\xi^{(5)}}^{(6)} = 11, \quad (7.36) $$
with a structure obtained by enforcing (7.26) for $a_6$ alone.

For centrosymmetric classes 6/m and 6/mmm,

$$ \xi_{kIJ} \equiv 0, \quad N_{\xi^{(5)}}^{(6/m)} = N_{\xi^{(5)}}^{(6/mmm)} = 0. \qquad (7.37) $$

## 7.8 Dielectric Constants

### 7.8.1 Second-Order Dielectric Tensor

Sixfold rotational invariance implies transverse isotropy about $x_3$, so

$$ [\epsilon_{ij}]_{(\text{hex})} = \begin{pmatrix} \epsilon_{11} & 0 & 0 \\ 0 & \epsilon_{11} & 0 \\ 0 & 0 & \epsilon_{33} \end{pmatrix}, \quad N_{\epsilon^{(2)}}^{(\text{hex})} = 2, \qquad (7.38) $$

valid for all hexagonal classes.

### 7.8.2 Third-Order Dielectric Tensor (quadratic electric-field response)

In this book, the third-order dielectric response is represented by the polar rank-3 tensor $\epsilon_{kij}^{(3)}$ symmetric in the last two indices, written in Voigt form as $\epsilon_{kI}^{(3)}$. Its invariance condition has the same structure as for linear piezoelectricity:

$$ (\epsilon^{(3)})'_{kI} = a_{kp} K_{IJ} \epsilon_{pJ}^{(3)}, \quad (\epsilon^{(3)})'_{kI} = \epsilon_{kI}^{(3)}. \qquad (7.39) $$

Therefore, the hexagonal third-order dielectric patterns and counts mirror those of the linear piezoelectric tensor for the same point group:

$$ N_{\epsilon^{(3)}}^{(6)} = 4, \quad N_{\epsilon^{(3)}}^{(622)} = 1, \quad N_{\epsilon^{(3)}}^{(6mm)} = 3, \quad N_{\epsilon^{(3)}}^{(6/m)} = N_{\epsilon^{(3)}}^{(6/mmm)} = 0. \qquad (7.40) $$

## 7.9 Electrostrictive Tensor

Electrostriction is represented by a fourth-rank tensor $\gamma_{ijkl}$ symmetric within each index pair,

$$ \gamma_{ijkl} = \gamma_{jikl} = \gamma_{ijkl}, \qquad (7.41) $$

with no intrinsic symmetry assumed under interchange of the pairs $(ij) \leftrightarrow (kl)$. In Voigt form this corresponds to a general matrix $\Gamma_{IJ}$.

Sixfold invariance yields a transversely isotropic block structure:

$$ [\Gamma_{IJ}]_{(\text{hex})} = \begin{pmatrix} \Gamma_{11} & \Gamma_{12} & \Gamma_{13} & 0 & 0 & 0 \\ \Gamma_{12} & \Gamma_{11} & \Gamma_{13} & 0 & 0 & 0 \\ \Gamma_{31} & \Gamma_{31} & \Gamma_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & \Gamma_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & \Gamma_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & \Gamma_{66} \end{pmatrix}, \quad \Gamma_{66} = \frac{\Gamma_{11} - \Gamma_{12}}{2}, \qquad (7.42) $$
with

$$N_{\gamma}^{(\text{hex})} = 7 \qquad (7.43)$$

independent electrostrictive constants.

## 7.10 Summary and Physical Interpretation

Hexagonal symmetry enforces transverse isotropy about the $x_3$-axis. This produces characteristic degeneracies:

* all in-plane directions are equivalent, implying $C_{11} = C_{22}$, $C_{13} = C_{23}$, and $\varepsilon_{11} = \varepsilon_{22}$;

* the two shear moduli involving $x_3$ are equal, $C_{44} = C_{55}$, while the in-plane shear modulus is constrained by $C_{66} = (C_{11} - C_{12})/2$;

* trigonal couplings permitted by threefold symmetry (such as $C_{14}$ in 3m) are eliminated by sixfold rotational invariance.

<table><caption>Table 7.1: Independent constants for representative hexagonal classes (sixfold axis $x_3$).</caption><thead><tr><th>Property tensor</th><th>6</th><th>622</th><th>6mm</th><th>6/m</th><th>6/mmm</th></tr></thead><tbody><tr><td>Second-order elasticity C<sub>IJ</sub></td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td></tr><tr><td>Third-order elasticity C<sub>ijk</sub></td><td>12</td><td>12</td><td>12</td><td>12</td><td>12</td></tr><tr><td>Linear piezoelectricity &xi;<sub>kI</sub></td><td>4</td><td>1</td><td>3</td><td>0</td><td>0</td></tr><tr><td>Fifth-order piezoelectricity &xi;<sub>kIJ</sub></td><td>11</td><td>3</td><td>8</td><td>0</td><td>0</td></tr><tr><td>Dielectric &epsilon;<sub>ij</sub></td><td>2</td><td>2</td><td>2</td><td>2</td><td>2</td></tr><tr><td>Third-order dielectric &epsilon;<sub>kI</sub><sup>(3)</sup></td><td>4</td><td>1</td><td>3</td><td>0</td><td>0</td></tr><tr><td>Electrostriction &Gamma;<sub>IJ</sub></td><td>7</td><td>7</td><td>7</td><td>7</td><td>7</td></tr></tbody></table>

## 7.11 Practical relevance & typical material examples

* **Where it appears:** Sixfold rotational symmetry; common in crystals used for high-frequency piezoelectric and acoustic devices.

* **Why it matters:** Hexagonal symmetry imposes stronger transverse constraints than trigonal and frequently yields compact constitutive forms used directly in device modeling and homogenization.

* **Typical examples:** Wurtzite ZnO (hexagonal), AlN (wurtzite, hexagonal), many hexagonal close-packed metals (elastic anisotropy context).
# Chapter 8

## Cubic Symmetry

### 8.1 Cubic Point Groups and Generator Choice

Cubic crystals possess the highest symmetry among the crystallographic point groups. Representative cubic point groups are

$$ 23, \quad m\bar{3}, \quad 432, \quad \bar{4}3m, \quad m\bar{3}m. \qquad (8.1) $$

All cubic classes share the defining property that the three Cartesian axes are symmetry-equivalent, and that the symmetry group contains both fourfold-type and threefold-type proper rotations.

A convenient generator set for the proper cubic rotation group (from which all cubic classes are built by adding inversion and/or mirrors) is:

* a 90° rotation about the $x_3$-axis,

$$ \mathbf{a}_{4z} = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \qquad (8.2) $$

* a 120° rotation about the [111] axis,

$$ \mathbf{a}_{3(111)} = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}, \qquad (8.3) $$

which cyclically permutes the axes $x_1 \to x_2 \to x_3 \to x_1$.

The matrices (8.2) and (8.3) generate the proper rotational symmetry of cubic crystals. Adding inversion symmetry (when present) eliminates all odd-rank polar tensors (e.g. piezoelectricity and third-order dielectric response in the sense adopted in this book).
## 8.2 Strong Invariance Constraints and Axis Equivalence

Cubic symmetry imposes invariance under rotations that interchange the coordinate axes. As a consequence:

* all diagonal components associated with the axes become equal by invariance under $a_{3(111)}$;

* all off-diagonal patterns related by axis permutations are identified;

* mixed normal-shear couplings vanish under quarter-turn invariance $a_{4z}$;

* for fully symmetric higher-order tensors, only a small number of invariants survive.

This produces exceptionally compact constitutive forms.

## 8.3 Second-Order Elastic Constants

### 8.3.1 Cubic reduction of $C_{IJ}$

Let $c_{ijkl}$ be the elastic stiffness tensor with intrinsic symmetries, represented by the symmetric Voigt matrix $C_{IJ}$. Imposing invariance under $a_{4z}$ and $a_{3(111)}$ yields the standard cubic stiffness form:

$$ [C_{IJ}]_{\text{(cubic)}} = \begin{pmatrix} C_{11} & C_{12} & C_{12} & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{12} & 0 & 0 & 0 \\ C_{12} & C_{12} & C_{11} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{44} \end{pmatrix}. \qquad (8.4) $$

Thus cubic symmetry reduces linear elasticity to exactly

$$ N_{C^{(2)}}^{(\text{cubic})} = 3 \qquad (8.5) $$

independent constants: $C_{11}$, $C_{12}$, $C_{44}$. (Equivalently, $C_{66} = (C_{11} - C_{12})/2$ is not an independent constant in cubic symmetry.)

## 8.4 Third-Order Elastic Constants

### 8.4.1 Independent set in generalized Voigt form

Third-order elasticity is represented by a fully symmetric generalized-Voigt tensor $C_{ijk}$. Cubic invariance under (8.2) and (8.3) yields the classical cubic third-order reduction with

$$ N_{C^{(3)}}^{(\text{cubic})} = 6. \qquad (8.6) $$
A standard independent parameter set in generalized Voigt notation is:

$$C_{111}, C_{112}, C_{123}, C_{144}, C_{155}, C_{456}. \qquad (8.7)$$

## 8.4.2 Complete cubic identifications

The nonzero third-order constants and their identifications are:

$$C_{111} = C_{222} = C_{333}, \qquad (8.8)$$

$$C_{112} = C_{113} = C_{221} = C_{223} = C_{331} = C_{332}, \qquad (8.9)$$

$$C_{123} \text{ is invariant under any index permutation of } (1, 2, 3), \qquad (8.10)$$

$$C_{144} = C_{244} = C_{344}, \qquad (8.11)$$

$$C_{155} = C_{166} = C_{255} = C_{266} = C_{355} = C_{366}, \qquad (8.12)$$

$$C_{456} \text{ is invariant under cubic symmetry (modulo full symmetry of } C_{IJK}). \qquad (8.13)$$

All other $C_{IJK}$ components that are not generated from (8.7) through (8.8)–(8.13) and full symmetry vanish identically by cubic invariance.

## 8.5 Linear Piezoelectric Constants

The linear piezoelectric tensor $\xi_{kij}$ (symmetric in $ij$) is represented in Voigt form as $\xi_{kl}$. Cubic classes split into two categories:

* centrosymmetric cubic classes (e.g. $m\bar{3}, m\bar{3}m$) forbid piezoelectricity;

* among the non-centrosymmetric cubic classes, piezoelectricity is admitted in 23 and $43m$, but not in 432.

For the piezoelectric cubic classes 23 and $\bar{4}3m$ (in the present Cartesian setting), invariance yields exactly one independent constant, conventionally denoted $\xi_{14}$:

$$[\xi_{kI}]_{(23)} = [\xi_{kI}]_{(43m)} = \begin{pmatrix} 0 & 0 & 0 & \xi_{14} & 0 & 0 \\ 0 & 0 & 0 & 0 & \xi_{14} & 0 \\ 0 & 0 & 0 & 0 & 0 & \xi_{14} \end{pmatrix}, \qquad N_{\xi}^{(23)} = N_{\xi}^{(43m)} = 1. \qquad (8.14)$$

For 432 and for centrosymmetric cubic classes,

$$\xi_{kI} \equiv 0. \qquad (8.15)$$
## 8.6 Second- and Third-Order Dielectric Constants

### 8.6.1 Second-order dielectric tensor

The dielectric tensor $\epsilon_{ij} = \epsilon_{ji}$ reduces under cubic symmetry to a scalar multiple of the identity:

$$[\epsilon_{ij}]_{\text{(cubic)}} = \begin{pmatrix} \epsilon & 0 & 0 \\ 0 & \epsilon & 0 \\ 0 & 0 & \epsilon \end{pmatrix}, \quad N_{\epsilon}^{(\text{cubic})} = 1. \qquad (8.16)$$

### 8.6.2 Third-order dielectric tensor (quadratic electric-field response)

In this book, the third-order dielectric response is represented by the polar rank-3 tensor $\epsilon_{kij}^{(3)}$ symmetric in the last two indices, written in Voigt form as $\epsilon_{kI}^{(3)}$. Its invariance condition has the same structure as for linear piezoelectricity:

$$(\epsilon^{(3)})'_{kI} = a_{kp} K_{IJ} \epsilon_{pJ}^{(3)}, \quad (\epsilon^{(3)})'_{kI} = \epsilon_{kI}^{(3)}. \qquad (8.17)$$

Therefore, the cubic third-order dielectric patterns and counts mirror those of the linear piezoelectric tensor for the same point group.

Hence:

* for the non-centrosymmetric piezoelectric cubic classes 23 and $\overline{4}3m$,

$$[\epsilon_{kI}^{(3)}]_{(23)} = [\epsilon_{kI}^{(3)}]_{(43m)} = \begin{pmatrix} 0 & 0 & 0 & \epsilon_{14}^{(3)} & 0 & 0 \\ 0 & 0 & 0 & 0 & \epsilon_{14}^{(3)} & 0 \\ 0 & 0 & 0 & 0 & 0 & \epsilon_{14}^{(3)} \end{pmatrix}, \qquad (8.18)$$
$$N_{\epsilon}^{(23)} = N_{\epsilon}^{(\overline{4}3m)} = 1.$$

* for 432 and for centrosymmetric cubic classes ($m\bar{3}$, $m\overline{3}m$),

$$\epsilon_{kI}^{(3)} = 0, \quad N_{\epsilon}^{(432)} = N_{\epsilon}^{(m\bar{3})} = N_{\epsilon}^{(m\overline{3}m)} = 0. \qquad (8.19)$$

## 8.7 Electrostrictive Tensor

Electrostriction is represented by a fourth-rank tensor $\gamma_{ijkl}$ symmetric within each index pair,

$$\gamma_{ijkl} = \gamma_{jikl} = \gamma_{ijkl}, \qquad (8.20)$$

with no intrinsic symmetry assumed under interchange of the pairs $(ij) \leftrightarrow (kl)$. In Voigt form this corresponds to a general 6 × 6 matrix $\Gamma_{I,J}$ transforming as

$$\Gamma' = K \Gamma K^T, \qquad (8.21)$$
with the same induced **K** acting on both the left and right symmetric-pair indices. Cubic invariance yields the standard cubic reduction with three independent constants:

$$ [\Gamma_{IJ}]_{\text{(cubic)}} = \begin{pmatrix} \Gamma_{11} & \Gamma_{12} & \Gamma_{12} & 0 & 0 & 0 \\ \Gamma_{12} & \Gamma_{11} & \Gamma_{12} & 0 & 0 & 0 \\ \Gamma_{12} & \Gamma_{12} & \Gamma_{11} & 0 & 0 & 0 \\ 0 & 0 & 0 & \Gamma_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & \Gamma_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & \Gamma_{44} \end{pmatrix}, \quad N_{\gamma}^{(\text{cubic})} = 3. \quad (8.22) $$

## 8.8 Comparison with the Isotropic Limit

Isotropic media are invariant under all rotations in SO(3). Cubic symmetry is the closest crystallographic analogue: it yields exceptionally compact tensor structures and, for second-order tensors, forces full isotropy already (e.g. $\epsilon_{ij} = \epsilon \delta_{ij}$). However:

* for linear elasticity, cubic symmetry retains three independent constants ($C_{11}$, $C_{12}$, $C_{44}$), whereas isotropy reduces this further to two independent Lamé-type constants;

* for third-order elasticity, cubic symmetry admits six independent constants (8.7), whereas isotropy yields a smaller invariant set (treated in Chapter 9).

Thus, cubic symmetry provides a natural bridge between general anisotropy and full isotropy and serves as the maximally symmetric crystallographic benchmark.

<table><caption>Table 8.1: Independent constants for representative cubic classes.</caption><thead><tr><th>Property tensor</th><th>23</th><th>m&#773;3</th><th>432</th><th>43m</th><th>m&#773;3m</th></tr></thead><tbody><tr><td>Second-order elasticity C<sub>IJ</sub></td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td></tr><tr><td>Third-order elasticity C<sub>IJK</sub></td><td>6</td><td>6</td><td>6</td><td>6</td><td>6</td></tr><tr><td>Linear piezoelectricity &xi;<sub>kI</sub></td><td>1</td><td>0</td><td>0</td><td>1</td><td>0</td></tr><tr><td>Dielectric &epsilon;<sub>ij</sub></td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>Third-order dielectric &epsilon;<sup>(3)</sup><sub>kI</sub></td><td>1</td><td>0</td><td>0</td><td>1</td><td>0</td></tr><tr><td>Electrostriction &Gamma;<sub>IJ</sub></td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td></tr></tbody></table>

## 8.9 Practical relevance & typical material examples

* **Where it appears:** Highest crystallographic symmetry; ubiquitous in semiconductors and many structural crystals.

* **Why it matters:** Cubic symmetry gives maximal crystallographic reduction and is the natural bridge to isotropy; it is also a standard reference case for validating tensor-reduction implementations.

* **Typical examples:** Silicon (cubic), diamond (cubic), NaCl (cubic), many cubic perovskites (phase-dependent).
# Chapter 9

## Isotropic Media as a Limiting Case

### 9.1 Isotropy as Full Rotational Invariance

A material is isotropic if its constitutive tensors are invariant under all proper rotations:

$$ \mathbf{a} \in \text{SO}(3). \qquad (9.1) $$

Equivalently, for every tensor $T_{i_1 \cdots i_n}$ describing the material response,

$$ T_{i_1 \cdots i_n} = a_{i_1 j_1} \cdots a_{i_n j_n} T_{j_1 \cdots j_n} \quad \forall \mathbf{a} \in \text{SO}(3). \qquad (9.2) $$

This is a strictly stronger requirement than any crystallographic point group. In particular, isotropy can be viewed as the limiting case of cubic symmetry obtained by enlarging the discrete cubic rotation group to the full continuous group SO(3).

A central consequence is that isotropic tensors can be expressed entirely in terms of the Kronecker delta $\delta_{ij}$ (and, for pseudo-tensors, the Levi-Civita tensor $\epsilon_{ijk}$). For the polar tensors treated in this book, and with the intrinsic symmetries used throughout (symmetry within index pairs and, when applicable, energy-based major symmetries), the invariant building block needed is $\delta_{ij}$.

### 9.2 Recovery of Lamé Form for Linear Elasticity

#### 9.2.1 Fourth-rank isotropic stiffness tensor

The linear elastic stiffness tensor $c_{ijkl}$ has the intrinsic symmetries

$$ c_{ijkl} = c_{jikl} = c_{ijkl} = c_{klij}. \qquad (9.3) $$

The most general fourth-rank tensor compatible with SO(3)-invariance and these symmetries is a linear combination of the two independent isotropic invariants:

$$ c_{ijkl} = \lambda \delta_{ij} \delta_{kl} + \mu (\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk}), \qquad (9.4) $$
where $\lambda$ and $\mu$ are the Lamé constants.

### 9.2.2 Voigt form and isotropic condition as a cubic limit

In Voigt notation, (9.4) corresponds to

$$C_{11} = \lambda + 2\mu, \quad C_{12} = \lambda, \quad C_{44} = \mu, \qquad (9.5)$$

with all other entries fixed by symmetry.

Comparing with the cubic stiffness form, isotropy is obtained from cubic symmetry by imposing the additional constraint

$$C_{44} = \frac{C_{11} - C_{12}}{2}. \qquad (9.6)$$

Thus, isotropy reduces the cubic set {$C_{11}, C_{12}, C_{44}$} to two independent parameters {$\lambda, \mu$}.

## 9.3 Isotropic Higher-Order Elasticity

### 9.3.1 Energy-based characterization and invariant basis

For isotropic nonlinear elasticity, it is standard to define higher-order material constants through an isotropic strain-energy density. Let $\mathbf{E}$ denote the (symmetric) Green-Lagrange strain tensor. A common third-order (cubic-in-strain) isotropic energy expansion is the Murnaghan form:

$$W(\mathbf{E}) = \frac{\lambda}{2}(\text{tr } \mathbf{E})^2 + \mu \text{tr}(\mathbf{E}^2) + \frac{l}{3}(\text{tr } \mathbf{E})^3 + m(\text{tr } \mathbf{E})\text{tr}(\mathbf{E}^2) + n \text{tr}(\mathbf{E}^3), \qquad (9.7)$$

where $(\lambda, \mu)$ are the Lamé constants and $(l, m, n)$ are the three independent third-order isotropic elastic constants (Murnaghan constants). This invariant representation is equivalent (up to notational changes) to alternative isotropic parameterizations (e.g. Landau-type constants).

### 9.3.2 Sixth-rank third-order stiffness and independent count

The third-order elastic tensor $c_{ijklmn}$ (second-order nonlinearity in strain) is obtained from the third derivative of the energy with respect to strain components (evaluated at the reference configuration) and inherits the intrinsic symmetries implied by an energy density. Isotropy and these symmetries imply that the corresponding sixth-rank tensor has exactly

$$N_{C^{(3)}}^{(\text{iso})} = 3 \qquad (9.8)$$

independent constants.

This should be compared with the cubic third-order set, which has 6 independent constants. Isotropy therefore corresponds to additional constraints among the cubic
third-order constants, reflecting the enlargement of the symmetry group from discrete cubic rotations to full SO(3).

### 9.3.3 Practical representation

For practical work, Eq. (9.7) is often the cleanest starting point because it expresses isotropic third-order elasticity directly in terms of rotational invariants of E:

$$I_1 = \operatorname{tr} \mathbf{E}, \quad I_2 = \operatorname{tr}(\mathbf{E}^2), \quad I_3 = \operatorname{tr}(\mathbf{E}^3),$$

and guarantees by construction that the resulting $c_{ijklmn}$ is fully SO(3)-invariant and satisfies all required intrinsic symmetries.

## 9.4 Isotropic Dielectric Response & Odd-Rank Polar Tensors

### 9.4.1 Second-order dielectric tensor

For isotropic media, the dielectric tensor reduces to a scalar:

$$\varepsilon_{ij} = \varepsilon \delta_{ij}, \quad N_{\varepsilon(2)}^{(\text{iso})} = 1. \qquad (9.9)$$

### 9.4.2 Third-order dielectric tensor (quadratic electric-field response)

In this book, the quadratic dielectric response is represented by a polar rank-3 tensor $\varepsilon_{kij}^{(3)}$ symmetric in the last two indices,

$$\varepsilon_{kij}^{(3)} = \varepsilon_{kji}^{(3)}, \qquad (9.10)$$

or, equivalently, by its Voigt form $\varepsilon_{kI}^{(3)}$. There exists no nonzero SO(3)-invariant polar tensor of rank 3 constructed from $\delta_{ij}$. Consequently,

$$\varepsilon_{kij}^{(3)} \equiv 0 \quad \text{for isotropic media,} \qquad (9.11)$$

which is consistent with viewing isotropy as the limiting case of increasing rotational symmetry.

## 9.5 Isotropic Piezoelectricity

Piezoelectricity is described (in the present notation) by a polar rank-3 tensor $\xi_{kij}$ symmetric in $(ij)$. As with $\varepsilon_{kij}^{(3)}$, there exists no nonzero SO(3)-invariant polar tensor of this rank constructed from $\delta_{ij}$. Therefore,

$$\xi_{kij} \equiv 0 \quad \text{for isotropic media.} \qquad (9.12)$$
## 9.6 Isotropic Electrostriction

Electrostriction is represented by a fourth-rank tensor $\gamma_{ijkl}$ symmetric within each index pair,

$$ \gamma_{ijkl} = \gamma_{jikl} = \gamma_{ijkl}. \qquad (9.13) $$

(As throughout this book, no additional symmetry under interchange of the pairs $(ij) \leftrightarrow (kl)$ is assumed unless explicitly stated by a specific constitutive convention.)

The most general isotropic fourth-rank tensor satisfying these intrinsic minor symmetries is again a two-invariant form:

$$ \gamma_{ijkl} = \Gamma_1 \delta_{ij} \delta_{kl} + \Gamma_2 (\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk}), \qquad (9.14) $$

with

$$ N_{\gamma}^{(\text{iso})} = 2. \qquad (9.15) $$

In Voigt form, (9.14) implies the isotropic relations

$$ \Gamma_{11} = \Gamma_1 + 2\Gamma_2, \quad \Gamma_{12} = \Gamma_1, \quad \Gamma_{44} = \Gamma_2, \quad \Gamma_{44} = \frac{\Gamma_{11} - \Gamma_{12}}{2}, \qquad (9.16) $$

together with the usual isotropic identifications $\Gamma_{22} = \Gamma_{33} = \Gamma_{11}$, $\Gamma_{13} = \Gamma_{23} = \Gamma_{12}$, and $\Gamma_{55} = \Gamma_{66} = \Gamma_{44}$, and with all normal-shear couplings equal to zero.

## 9.7 Connection to Nonlinear Continuum Mechanics

The isotropic forms derived here are consistent with the standard invariant-theory treatment of constitutive laws in continuum mechanics:

* Linear isotropic elasticity is characterized by two constants $\lambda, \mu$ and the Lamé stiffness form (9.4).

* Third-order isotropic elasticity admits exactly three independent constants, conveniently represented by the Murnaghan energy expansion (9.7).

* Isotropy forbids all odd-rank polar material tensors such as $\xi_{kij}$ and $\epsilon_{kij}^{(3)}$, yielding (9.12) and (9.11).

In the logic of this book, isotropy is not an ad hoc idealization but the limiting case of increasing symmetry: cubic symmetry provides the maximally symmetric crystallographic baseline, and isotropy is obtained by extending cubic rotational invariance to the full continuous group SO(3), which enforces additional constraints such as (9.6) and reduces higher-order tensors to their minimal invariant parameter sets.
# Chapter 10

# Summary Tables for All Symmetry Classes

## 10.1 Purpose and Scope

This chapter provides a compact and definitive reference summarizing the results derived in Chapters 2–9. For each symmetry class and tensor type, it lists the *exact number of independent material constants*, assuming invariance under the stated point group, standard intrinsic permutation symmetries, polar tensor character unless explicitly stated otherwise, Cartesian tensor definitions consistent with Chapters 1–9.

No ranges, benchmarks, or conditional counts appear in this chapter. Each entry corresponds to a uniquely defined invariant tensor space.

## 10.2 Symmetry Classes Considered

The symmetry classes included are:

* Triclinic: 1

* Monoclinic: 2, *m*

* Orthorhombic: 222, *mm*2, *mmm*

* Tetragonal: 4, 422, 4*m*mm, 4/*m*/*m*

* Trigonal: 3, 32, 3*m*, 3*m*

* Hexagonal: 6, 622, 6*m*mm, 6/*m*/*m*

* Cubic: 23, *m*3, 432, *d*3*m*, *m*3*m*

* Isotropic: full SO(3)

Classes sharing identical reductions are grouped.
## 10.3 Second-Order Elasticity $c_{ijkl}$

<table><thead><tr><th>Symmetry</th><th>1</th><th>2/m</th><th>Ortho</th><th>4</th><th>4mm</th><th>3m</th><th>6mm</th><th>Cubic</th><th>Iso</th></tr></thead><tbody><tr><th>$N_C^{(2)}$</th><td>21</td><td>13</td><td>9</td><td>7</td><td>6</td><td>6</td><td>5</td><td>3</td><td>2</td></tr></tbody></table>

## 10.4 Third-Order Elasticity $c_{ijklmn}$

<table><thead><tr><th>Symmetry</th><th>1</th><th>2/m</th><th>Ortho</th><th>4</th><th>4mm</th><th>3m</th><th>6mm</th><th>Cubic</th><th>Iso</th></tr></thead><tbody><tr><th>$N_C^{(3)}$</th><td>56</td><td>32</td><td>20</td><td>16</td><td>12</td><td>14</td><td>12</td><td>6</td><td>3</td></tr></tbody></table>

## 10.5 Linear Piezoelectricity $\xi_{kij}$

<table><thead><tr><th>Symmetry</th><th>1</th><th>2</th><th>m</th><th>222</th><th>mm2</th><th>4mm</th><th>6mm</th><th>43m</th></tr></thead><tbody><tr><th>$N_\xi^{(2)}$</th><td>18</td><td>8</td><td>10</td><td>3</td><td>5</td><td>3</td><td>3</td><td>1</td></tr></tbody></table>

Centrosymmetric classes admit no linear piezoelectricity.

## 10.6 Higher-Order Piezoelectricity $\xi_{kijmn}$

<table><thead><tr><th>Symmetry</th><th>1</th><th>2</th><th>m</th><th>Ortho</th><th>4mm</th><th>3m</th><th>6mm</th><th>Cubic</th></tr></thead><tbody><tr><th>$N_\xi^{(5)}$</th><td>63</td><td>29</td><td>34</td><td>13</td><td>10</td><td>12</td><td>8</td><td>0</td></tr></tbody></table>

Isotropic media admit no odd-rank polar tensors.

## 10.7 Dielectric Tensors

### Second order $\epsilon_{ij}$

<table><thead><tr><th>Symmetry</th><th>1</th><th>2/m</th><th>Ortho</th><th>Tetra</th><th>Trig/Hex</th><th>Cubic</th><th>Iso</th></tr></thead><tbody><tr><th>$N_\epsilon^{(2)}$</th><td>6</td><td>4</td><td>3</td><td>2</td><td>2</td><td>1</td><td>1</td></tr></tbody></table>

### Third order $\epsilon_{ijk}$

<table><thead><tr><th>Symmetry</th><th>1</th><th>2/m</th><th>Ortho</th><th>Tetra</th><th>Trig</th><th>Hex</th><th>Cubic/Iso</th></tr></thead><tbody><tr><th>$N_\epsilon^{(3)}$</th><td>10</td><td>6</td><td>3</td><td>2</td><td>1</td><td>1</td><td>0</td></tr></tbody></table>

## 10.8 Independent Constants: Electrostriction

Electrostriction is described here by a fourth-rank tensor $\gamma_{ijkl}$ that is symmetric within each index pair but *not* assumed to possess major symmetry:

$$ \gamma_{ijkl} = \gamma_{jikl} = \gamma_{ijkl}, \quad (\text{no assumption that } \gamma_{ijkl} = \gamma_{klij}). \quad (10.1) $$
Accordingly, in Voigt form $\Gamma_{IJ}$ is a general 6 × 6 matrix (not necessarily symmetric), and the triclinic baseline is 36 independent constants.

<table><caption>Table 10.1: Independent electrostrictive constants $\gamma_{ijkl}$ (minor symmetries only, no major symmetry $\gamma_{ijkl} \neq \gamma_{klij}$).</caption><thead><tr><th>Symmetry</th><th>1</th><th>2 / m</th><th>Orthorhombic</th><th>4</th><th>422 / 4mm</th><th>4/mmm</th><th>3m</th><th>Hexagonal</th><th>Cubic</th><th>Isotropic</th></tr></thead><tbody><tr><th>N<sub>γ</sub></th><td>36</td><td>20</td><td>12</td><td>10</td><td>7</td><td>8</td><td>7</td><td>3</td><td>2</td></tr></tbody></table>

## 10.9 Concluding Remarks

All entries in this chapter correspond to uniquely defined invariant tensor spaces derived from exact tensor transformation laws, intrinsic permutation symmetries, minimal generator sets of the corresponding point groups.

This chapter therefore serves as a definitive lookup reference consistent with the full derivations presented earlier in the book.
# Chapter 11

# Conclusions and Outlook

## 11.1 Summary of the Invariance-Based Framework

This book has developed a unified, constructive framework for determining symmetry relations among material tensors of arbitrary rank in crystalline and isotropic media. The guiding axiom throughout is simple, but powerful:

*A material tensor is admissible for a given symmetry class if and only if it is invariant under every point-group operation of that class, in addition to satisfying its intrinsic permutation symmetries.*

Mathematically, for a material tensor $T_{i_1...i_n}$ and any point-group operation $a \in G$, admissibility requires

$$T_{i_1...i_n} = a_{i_1j_1} \cdots a_{i_nj_n} T_{j_1...j_n}, \quad \forall a \in G, \qquad (11.1)$$

together with the intrinsic index symmetries implied by the constitutive framework, such as symmetry of stress and strain, existence of an energy density, or minor and major symmetries for elastic-type tensors.

In practice, invariance is enforced on a generator set of the point group. The consequence is always a finite system of homogeneous linear equations whose solution forces certain tensor components to vanish, and imposes exact linear relations among the remaining components.

Within fixed coordinate conventions and constitutive assumptions, this procedure yields a complete and canonical reduced tensor form for each symmetry class and tensor type treated in this book.

## 11.2 Achievements Across Tensor Rank and Symmetry Class

The central achievement of this work is the demonstration that symmetry reduction—independent of tensor rank, physical coupling, or crystal class—can be treated as a purely algebraic problem governed by linear invariance equations.
This methodology has been applied exhaustively to:

* second- and third-order elastic tensors,

* linear and higher-order piezoelectric tensors,

* linear and nonlinear dielectric response tensors,

* electrostrictive and mixed electromechanical couplings,

* all crystallographic point groups, from triclinic symmetry to the isotropic limit.

Each symmetry class is treated in a uniform and systematic manner. The triclinic case establishes the unconstrained baseline, while successive chapters demonstrate how additional symmetry generators progressively collapse the space of admissible material constants. This controlled reduction reveals clear structural patterns in how symmetry governs anisotropy and coupling behavior.

The summary tables collected in Chapter 10 provide a compact, internally consistent reference that extends well beyond what is currently available in the literature, particularly for higher-order tensors.

## 11.3 Robustness, Transparency, and Reproducibility

Symmetry reductions of higher-rank tensors are often viewed as error-prone due to the rapid growth of component count and the difficulty of auditing index manipulations. The workflow developed here addresses these issues directly:

* **Generator-based enforcement.** Only a minimal generator set is required to guarantee full point-group invariance.

* **Exact transformation laws.** Every reduction starts from the universal tensor transformation rule, avoiding heuristic pattern recognition.

* **Controlled representational reduction.** Voigt and generalized Voigt notation are used strictly as bookkeeping devices, without altering tensorial meaning.

* **Linear-algebra closure.** Invariance conditions close to finite linear systems, enabling internal cross-checks and independent verification.

* **Rank independence.** The same logic applies uniformly to second-, third-, and higher-order tensors and to different coupled physical fields.

For this reason, the results should be read not merely as static tables, but as outputs of a reproducible and verifiable procedure.
## 11.4 Position Within the Literature

Classical treatments of crystal physics and elasticity, notably those by Nye, Musgrave, and their contemporaries, established the foundational role of symmetry in continuum theory. However, those works were necessarily limited in their treatment of higher-order constitutive behavior.

Subsequent literature addressing higher-order tensors has remained fragmented, often restricted to selected symmetry classes, based on compact derivations that are difficult to audit componentwise, inconsistent in notation and convention, incomplete due to combinatorial complexity at higher rank.

The present work unifies these disparate strands by maintaining a single, invariant-based framework across all chapters, with explicit component-level results and consistent conventions. Where overlap with established results exists, agreement follows naturally from invariance. Where results extend beyond the classical literature, they do so in a fully documented and reproducible manner.

## 11.5 Relevance to Modern Research and Applications

Higher-order material constants are increasingly important in contemporary science and engineering. Nonlinear elastic and electromechanical effects play a decisive role in nonlinear acoustics and finite-amplitude wave propagation, high-field and high-power piezoelectric devices, bias-stressed and prestressed sensing elements, energy harvesting and electromechanical transduction, sonar, medical ultrasound, and space-borne instrumentation, engineered materials and metamaterials with tailored nonlinear response.

In all these contexts, reliable symmetry information is essential. Incorrect or incomplete symmetry reductions can introduce spurious couplings and lead to flawed physical interpretation. By providing a comprehensive and verified symmetry framework, this book aims to serve as a dependable foundation for both theoretical modeling and experimental analysis.

## 11.6 Outlook and Future Directions

The framework developed here admits several natural extensions:

* inclusion of magnetic and magneto-electric tensors, requiring explicit treatment of axial character and time-reversal symmetry,

* incorporation of dissipative and non-conservative couplings,

* extension to even higher tensor orders relevant to extreme-field applications,

* algorithmic automation of the invariance procedure using symbolic or computational tools.
Because symmetry reduction is reduced to explicit linear constraints, the method is particularly well suited for computational implementation. The structure presented in this book may therefore serve as a blueprint for future symmetry engines capable of generating constitutive laws automatically, without loss of rigor or transparency.

## 11.7 Final Perspective

Symmetry constraints on material constants are not matters of convention; they are strict consequences of invariance. By treating invariance as the unifying axiom and applying it systematically across tensor ranks and symmetry classes, this work provides a stable and reproducible foundation for higher-order constitutive modeling.

It is hoped that this book will serve both as a practical reference and as a methodological guide, enabling researchers to extend higher-order tensor frameworks confidently, correctly, and transparently across physics, engineering, and materials science.
# Appendix A

## Tensor Transformation Rules

### A.1 Purpose and Scope of This Appendix

This appendix collects, in a single self-contained place, the tensor transformation rules used throughout the book. These rules constitute the mathematical backbone of the invariance-based procedure developed in Chapter 1 and applied systematically in Chapters 2–10.

All transformation laws stated here are exact consequences of classical tensor analysis and apply independently of any material model. They are valid for linear and nonlinear constitutive theories, static and dynamic regimes, and arbitrary tensor rank.

### A.2 Orthogonal Transformations and Notation

Let $\mathbf{a}$ be an orthogonal transformation representing a point-group operation:

$$ \mathbf{a} \in O(3), \quad a_{ip}a_{jp} = \delta_{ij}, \quad a_{pi}a_{pj} = \delta_{ij}. \qquad (A.1) $$

Proper rotations satisfy det $\mathbf{a} = +1$; improper rotations (including reflections and inversion) satisfy det $\mathbf{a} = -1$.

Throughout this book:

* Latin indices $i, j, k, \dots$ run over $\{1, 2, 3\}$.

* Einstein summation convention is used.

* All tensors are expressed in a fixed Cartesian basis unless explicitly stated otherwise.

### A.3 Transformation of Vectors and Covectors

A polar vector $v_i$ transforms under $\mathbf{a}$ as

$$ v_i' = a_{ij}v_j. \qquad (A.2) $$
For orthogonal transformations, covariant and contravariant components transform identically, and index position is not distinguished.

An axial (pseudo-)vector $w_i$ transforms as

$$w_i' = (\det \mathbf{a}) a_{ij} w_j, \qquad (A.3)$$

which differs from (A.2) by the parity factor $\det \mathbf{a}$.

## A.4 General Transformation Law for Cartesian Tensors

A Cartesian tensor of rank $n$,

$$T_{i_1 i_2 \dots i_n}, \qquad (A.4)$$

transforms under $\mathbf{a}$ according to

$$T'_{i_1 i_2 \dots i_n} = a_{i_1 j_1} a_{i_2 j_2} \dots a_{i_n j_n} T_{j_1 j_2 \dots j_n}, \qquad (A.5)$$

for polar tensors. For axial tensors, an additional factor ($\det \mathbf{a}$) appears for each axial index.

Equation (A.5) is the starting point of every invariance derivation in this book.

## A.5 Invariance Condition

A tensor $T$ is invariant under a symmetry operation $\mathbf{a}$ if and only if

$$T'_{i_1 i_2 \dots i_n} = T_{i_1 i_2 \dots i_n}. \qquad (A.6)$$

Combining (A.5) and (A.6) yields a system of linear equations for the tensor components:

$$T_{i_1 \dots i_n} = a_{i_1 j_1} \dots a_{i_n j_n} T_{j_1 \dots j_n}. \qquad (A.7)$$

All symmetry relations derived in the main text follow from enforcing (A.7) for a generator set of the point group.

## A.6 Transformation of Second-Rank Tensors

For a second-rank tensor $T_{ij}$,

$$T'_{ij} = a_{ip} a_{jq} T_{pq}. \qquad (A.8)$$

### A.6.1 Symmetric and Antisymmetric Parts

The symmetric and antisymmetric parts transform independently:

$$T_{ij}^{(s)} = \frac{1}{2}(T_{ij} + T_{ji}), \qquad (A.9)$$

$$T_{ij}^{(a)} = \frac{1}{2}(T_{ij} - T_{ji}), \qquad (A.10)$$
with each part obeying (A.8). This justifies treating symmetric tensors (e.g. strain, stress, dielectric tensors) independently of antisymmetric ones.

## A.7 Transformation of Fourth-Rank Tensors

A fourth-rank tensor $T_{ijkl}$ transforms as

$$ T'_{ijkl} = a_{ip}a_{jq}a_{kr}a_{ls}T_{pqrs}. \qquad (A.11) $$

For material tensors derived from an energy density (e.g. elastic stiffness, electrostriction), intrinsic symmetries such as

$$ T_{ijkl} = T_{jikl} = T_{ijkl} = T_{klij} \qquad (A.12) $$

are imposed before enforcing invariance under (A.11). This separation is essential for clarity and error control.

## A.8 Transformation of Higher-Rank Tensors

For higher-rank tensors (e.g. sixth-rank third-order elasticity tensors or fifth-rank piezoelectric tensors), the transformation law follows directly from (A.5). For example:

### A.8.1 Sixth-Rank Tensors

$$ T'_{ijklmn} = a_{ip}a_{jq}a_{kr}a_{ls}a_{mt}a_{nu}T_{pqrstu}. \qquad (A.13) $$

### A.8.2 Fifth-Rank Tensors

$$ T'_{ijklm} = a_{ip}a_{jq}a_{kr}a_{ls}a_{mt}T_{pqrst}. \qquad (A.14) $$

In all cases, intrinsic permutation symmetries (if present) are enforced prior to applying invariance conditions.

## A.9 Voigt and Generalized Voigt Transformation Rules

### A.9.1 Voigt Mapping for Symmetric Index Pairs

For symmetric second-rank tensors $S_{ij} = S_{ji}$, the Voigt mapping

$$ (11, 22, 33, 23, 13, 12) \leftrightarrow (1, 2, 3, 4, 5, 6) \qquad (A.15) $$

is used. Under an orthogonal transformation $a$, the corresponding Voigt components transform as

$$ S'_{i} = K_{ij}(a) S_{j}, \qquad (A.16) $$

where the 6 × 6 matrix $K(a)$ is uniquely determined by $a$.
For diagonal generators $\mathbf{a} = \text{diag}(s_1, s_2, s_3)$,

$$ \mathbf{K}(\mathbf{a}) = \text{diag}(1, 1, 1, s_2s_3, s_1s_3, s_1s_2). \qquad (A.17) $$

### A.9.2 Generalized Voigt Notation

For higher-rank tensors built from multiple symmetric index pairs, each pair is mapped independently using the same Voigt convention. For example:

* $c_{ijkl} \to C_{IJ}$,

* $c_{ijklmn} \to C_{IJK}$,

* $\xi_{kijmn} \to \xi_{kIJ}$.

Under a transformation $\mathbf{a}$,

$$ C'_{IJK} = K_{IP}(\mathbf{a}) K_{JQ}(\mathbf{a}) K_{KR}(\mathbf{a}) C_{PQR}, \qquad (A.18) $$

with analogous expressions for mixed tensors involving both Voigt indices and Cartesian indices.

## A.10 Mixed Polar–Tensor Transformations

For tensors carrying both vector indices and symmetric index pairs (e.g. piezoelectric tensors $\xi_{kij}$), the transformation law combines Cartesian and Voigt actions:

$$ \xi'_{kI} = a_{kp} K_{IJ}(\mathbf{a}) \xi_{pJ}. \qquad (A.19) $$

This mixed form is used systematically in Chapters 3–8 to derive piezoelectric and higher-order electromechanical symmetry relations.

## A.11 Summary

All symmetry reductions in this book follow from:

1. the general tensor transformation law (A.5),

2. the invariance condition (A.6),

3. intrinsic permutation symmetries imposed by the constitutive framework,

4. and the Voigt or generalized Voigt representations as purely notational devices.

No additional assumptions or heuristic rules are required. The transformation laws summarized in this appendix therefore provide the complete mathematical foundation for the invariant-based derivations presented throughout the book.
# Appendix B

## Voigt and Generalized Voigt Notation

### B.1 Purpose of This Appendix

This appendix provides a precise and unambiguous definition of Voigt and generalized Voigt notation as used throughout the book. The purpose is strictly representational: Voigt notation is introduced as a compact indexing scheme for tensors possessing intrinsic symmetries, without altering their tensorial meaning or transformation properties.

Because errors and ambiguities in Voigt notation are a common source of sign mistakes and incorrect symmetry reductions in the literature, this appendix explicitly states:

* the index mappings employed,

* the transformation behavior under orthogonal operations,

* the distinction between tensorial objects and their Voigt representations.

All derivations in the main text rely on the conventions fixed here.

### B.2 Symmetric Second-Rank Tensors

#### B.2.1 Intrinsic Symmetry

Let $S_{ij}$ be a symmetric second-rank tensor:

$$S_{ij} = S_{ji}. \qquad (B.1)$$

Such tensors arise naturally in continuum physics (e.g. strain, stress, dielectric tensors). Because of the symmetry, only six independent components exist.
## B.2.2 Voigt Index Mapping

We adopt the standard tensorial Voigt mapping

$$ (11, 22, 33, 23, 13, 12) \leftrightarrow (1, 2, 3, 4, 5, 6), \qquad (B.2) $$

so that

$$ \begin{aligned} S_1 &= S_{11}, \\ S_2 &= S_{22}, \\ S_3 &= S_{33}, \\ S_4 &= S_{23} = S_{32}, \\ S_5 &= S_{13} = S_{31}, \\ S_6 &= S_{12} = S_{21}. \end{aligned} \qquad (B.3) $$

**Important remark.** No numerical prefactors (such as factors of 2) are absorbed into the Voigt components. All components retain their tensorial magnitudes. This is essential for preserving exact transformation laws and avoiding hidden convention-dependent factors.

## B.3 Transformation Behavior in Voigt Form

Under an orthogonal transformation **a**, the tensor $S_{ij}$ transforms as

$$ S'_{ij} = a_{ip}a_{jq}S_{pq}. \qquad (B.4) $$

In Voigt form, this induces a linear transformation

$$ S'_{I} = K_{I,J}(\mathbf{a}) S_{J}, \qquad (B.5) $$

where the 6 × 6 matrix $K(\mathbf{a})$ is uniquely determined by **a** and the mapping (B.2).

### B.3.1 Diagonal Orthogonal Transformations

For diagonal transformations

$$ \mathbf{a} = \mathrm{diag}(s_1, s_2, s_3), \quad s_i = \pm 1, \qquad (B.6) $$

the induced Voigt transformation is diagonal:

$$ K(\mathbf{a}) = \mathrm{diag}(1, 1, 1, s_2 s_3, s_1 s_3, s_1 s_2). \qquad (B.7) $$

This form is used extensively for monoclinic, orthorhombic, and related symmetry classes.
## B.3.2 Non-Diagonal Rotations

For non-diagonal rotations (e.g. 90°, 120°, or 60° rotations), the matrix $K(\mathbf{a})$ is generally non-diagonal and mixes Voigt components. Explicit examples for tetragonal, trigonal, hexagonal, and cubic generators are given in the corresponding chapters.

# B.4 Fourth-Rank Tensors in Voigt Form

## B.4.1 Intrinsic Pair Symmetries

Let $C_{ijkl}$ be a fourth-rank tensor satisfying the intrinsic symmetries

$$C_{ijkl} = C_{jikl} = C_{ijkl} = C_{klij}. \qquad (B.8)$$

Such tensors arise from quadratic energy densities (e.g. elastic stiffness, electrostatic).

## B.4.2 Voigt Representation

Using the mapping (B.2) independently on each symmetric index pair,

$$C_{ijkl} \rightarrow C_{IJ}, \qquad (B.9)$$

where $C_{IJ} = C_{JI}$ is a symmetric 6 × 6 matrix.

Under an orthogonal transformation,

$$C'_{IJ} = K_{IP}(\mathbf{a}) K_{JQ}(\mathbf{a}) C_{PQ}, \qquad (B.10)$$

which is the Voigt-space equivalent of $C'_{ijkl} = a_{ip}a_{jq}a_{kr}a_{ls}C_{pqrs}$.

# B.5 Generalized Voigt Notation

## B.5.1 Motivation

Higher-order material tensors often involve multiple symmetric index pairs. Generalized Voigt notation provides a systematic way to compress such tensors while preserving intrinsic permutation symmetries, exact transformation behavior, and linear structure of invariance equations.

## B.5.2 Third-Order Elasticity

For the third-order elastic tensor $c_{ijklmn}$, intrinsic symmetries imply symmetry in each index pair $(ij)$, $(kl)$, $(mn)$, as well as full symmetry under permutation of the three pairs. Using the Voigt mapping on each pair,

$$c_{ijklmn} \rightarrow C_{IJK}, \qquad (B.11)$$
where $C_{IJK}$ is fully symmetric in $(I, J, K)$.

Under an orthogonal transformation,

$$C'_{IJK} = K_{IP}(\mathbf{a}) K_{JQ}(\mathbf{a}) K_{KR}(\mathbf{a}) C_{PQR}. \qquad (B.12)$$

### B.5.3 Mixed Cartesian–Voigt Tensors

For tensors involving both vector indices and symmetric index pairs, generalized Voigt notation is applied only to the symmetric pairs.

**Example: linear piezoelectric tensor.** The piezoelectric tensor $\xi_{kij} = \xi_{kji}$ is represented as $\xi_{kI}$. Its transformation law is

$$\xi'_{kI} = a_{kp} K_{IJ}(\mathbf{a}) \xi_{pJ}. \qquad (B.13)$$

**Example: higher-order piezoelectric tensor.** For $\xi_{kijmn} = \xi_{kjinm}$, the representation $\xi_{kIJ} = \xi_{kJI}$ is used, with

$$\xi'_{kIJ} = a_{kp} K_{IP}(\mathbf{a}) K_{JQ}(\mathbf{a}) \xi_{pPQ}. \qquad (B.14)$$

## B.6 Distinction Between Tensor and Representation

It is essential to emphasize that:

* Voigt and generalized Voigt notation do not define new tensors.

* They are linear representations of tensors with intrinsic symmetries.

* All physical meaning resides in the underlying Cartesian tensor.

Symmetry relations derived in Voigt form are therefore exact translations of the Cartesian invariance conditions.

## B.7 Common Pitfalls and How They Are Avoided Here

The conventions adopted in this book avoid several common sources of confusion:

* No engineering-strain factors (e.g. factors of 2 in shear components) are introduced.

* No mixed conventions are used across chapters.

* All transformation matrices $K(\mathbf{a})$ are derived explicitly from Cartesian transformations.

This guarantees consistency across tensor ranks and symmetry classes.
## B.8 Summary

Voigt and generalized Voigt notation provide a compact and exact framework for handling intrinsically symmetric material tensors. When used with explicitly stated mappings and transformation rules, they enable systematic, error-controlled derivations of symmetry relations without obscuring the underlying tensorial structure.

All subsequent appendices and derivations rely exclusively on the conventions fixed in this appendix.
# Appendix C

## Point-Group Generators

### C.1 Purpose of This Appendix

This appendix specifies the explicit point-group generators used throughout the book. The choice of generators is a crucial practical step in the invariance-based procedure: invariance under a *minimal generator set* is both necessary and sufficient to guarantee invariance under the full point group.

All generators listed here are orthogonal 3 × 3 matrices acting on a fixed Cartesian basis. They are chosen to be as simple as possible (diagonal where feasible), align with standard crystallographic conventions, minimize algebraic complexity in invariance equations, allow direct cross-checking with the literature.

Once these generators are fixed, all symmetry relations in the main text follow uniquely.

### C.2 General Principles for Generator Selection

For a given point group $G$:

* A *generator set* {$a_α$} is a subset of $G$ such that every element of $G$ can be written as a finite product of generators.

* Invariance under all generators implies invariance under the entire group:

$$ T \text{ invariant under all } a_α \iff T \text{ invariant under } G. \quad (C.1) $$

* When possible, diagonal generators (reflections or twofold rotations) are preferred, because they induce diagonal Voigt transformations and lead to transparent sign constraints.

* Non-diagonal generators (e.g. 90°, 120°, or 60° rotations) are introduced only when required by symmetry.
## C.3 Triclinic Class

### C.3.1 Class 1

The triclinic class 1 contains only the identity:

$$ \mathbf{I} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}. \qquad (C.2) $$

No nontrivial generators exist. All tensors are unconstrained beyond intrinsic permutation symmetries.

## C.4 Monoclinic Classes

Throughout this book, the monoclinic unique axis is chosen along $x_2$.

### C.4.1 Class 2

A 180° rotation about $x_2$:

$$ \mathbf{a}_{2y} = \mathrm{diag}(-1, +1, -1). \qquad (C.3) $$

### C.4.2 Class m

A mirror plane normal to $x_2$:

$$ \mathbf{m}_y = \mathrm{diag}(+1, -1, +1). \qquad (C.4) $$

## C.5 Orthorhombic Classes

The orthorhombic axes coincide with ($x_1, x_2, x_3$).

### C.5.1 Class 222

Generators: 180° rotations about $x_1$ and $x_2$:

$$ \mathbf{a}_{2x} = \mathrm{diag}(+1, -1, -1), \qquad (C.5) $$

$$ \mathbf{a}_{2y} = \mathrm{diag}(-1, +1, -1). \qquad (C.6) $$

Rotation about $x_3$ follows as their product.

### C.5.2 Class mm2

Generators: two orthogonal mirror planes and a twofold axis:

$$ \mathbf{m}_x = \mathrm{diag}(-1, +1, +1), \qquad (C.7) $$

$$ \mathbf{m}_y = \mathrm{diag}(+1, -1, +1). \qquad (C.8) $$
The twofold rotation about $x_3$ is $\mathbf{m}_x\mathbf{m}_y$.

### C.5.3 Class mmm

Generators: three mutually orthogonal mirror planes:

$$ \mathbf{m}_x = \mathrm{diag}(-1, +1, +1), \qquad (C.9) $$

$$ \mathbf{m}_y = \mathrm{diag}(+1, -1, +1), \qquad (C.10) $$

$$ \mathbf{m}_z = \mathrm{diag}(+1, +1, -1). \qquad (C.11) $$

This class is centrosymmetric.

## C.6 Tetragonal Classes

The fourfold axis is aligned with $x_3$.

### C.6.1 Fourfold Rotation

Common generator:

$$ \mathbf{a}_4 = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}. \qquad (C.12) $$

### C.6.2 Class 4

Generator: $\mathbf{a}_4$.

### C.6.3 Class 422

Generators:

$$ \mathbf{a}_4, \quad \mathbf{a}_{2x} = \mathrm{diag}(+1, -1, -1). \qquad (C.13) $$

### C.6.4 Class 4mm

Generators:

$$ \mathbf{a}_4, \quad \mathbf{m}_x = \mathrm{diag}(-1, +1, +1). \qquad (C.14) $$

### C.6.5 Class 4/mmm

Generators:

$$ \mathbf{a}_4, \quad \mathbf{m}_x, \quad \mathbf{m}_z. \qquad (C.15) $$

This class is centrosymmetric.

## C.7 Trigonal Classes

The threefold axis is aligned with $x_3$.
### C.7.1 Threefold Rotation

Generator:

$$ a_3 = \begin{pmatrix} -\frac{1}{2} & \frac{\sqrt{3}}{2} & 0 \\ -\frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \\ 0 & 0 & 1 \end{pmatrix}. \qquad (C.16) $$

### C.7.2 Class 3

Generator: $a_3$.

### C.7.3 Class 32

Generators:

$$ a_3, \quad a_{2x} = \operatorname{diag}(+1, -1, -1). \qquad (C.17) $$

### C.7.4 Class $3m$

Generators:

$$ a_3, \quad m_x = \operatorname{diag}(-1, +1, +1). \qquad (C.18) $$

### C.7.5 Class $\bar{3}m$

Generators:

$$ a_3, \quad m_x, \quad m_z. \qquad (C.19) $$

This class is centrosymmetric.

## C.8 Hexagonal Classes

The sixfold axis is aligned with $x_3$.

### C.8.1 Sixfold Rotation

Generator:

$$ a_6 = \begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} & 0 \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} & 0 \\ 0 & 0 & 1 \end{pmatrix}. \qquad (C.20) $$

### C.8.2 Class 6

Generator: $a_6$.

### C.8.3 Class 622

Generators:

$$ a_6, \quad a_{2x} = \operatorname{diag}(+1, -1, -1). \qquad (C.21) $$
### C.8.4 Class 6mm

Generators:

$$ a_6, \quad m_x = \text{diag}(-1, +1, +1). \qquad (C.22) $$

### C.8.5 Class 6/mmm

Generators:

$$ a_6, \quad m_x, \quad m_z. \qquad (C.23) $$

This class is centrosymmetric.

## C.9 Cubic Classes

### C.9.1 Proper Cubic Rotation Generators

A minimal generator set for the proper cubic rotation group consists of:

$$ a_{4z} = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \qquad (C.24) $$

$$ a_{3(111)} = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}. \qquad (C.25) $$

### C.9.2 Non-centrosymmetric cubic classes

Classes 23 and $\bar{4}3m$ are generated by (C.24) and (C.25), with or without additional improper operations.

### C.9.3 Centrosymmetric cubic classes

Classes $m\bar{3}$ and $m\bar{3}m$ additionally include inversion:

$$ i = -I. \qquad (C.26) $$

## C.10 Isotropic Case

The isotropic case is not a crystallographic point group. Instead, it corresponds to invariance under the full rotation group:

$$ \mathcal{G} = \text{SO}(3). \qquad (C.27) $$

In practice, isotropy may be enforced by requiring invariance under all rotations about a fixed axis, and all rotations that permute the coordinate axes.

This limit is treated explicitly in Chapter 9.
C.11 Summary

This appendix provides the explicit generator matrices used throughout the book. Combined with the tensor transformation rules of Appendix A and the Voigt conventions of Appendix B, these generators fully determine all invariance equations and symmetry reductions presented in the main text.
# Appendix D

## Worked Invariance Examples

### D.1 Purpose of This Appendix

This appendix provides fully worked, step-by-step examples of the invariance procedure developed in Chapter 1 and formalized in Appendices A–C. The goal is not to introduce new results, but to:

* demonstrate explicitly how invariance equations are constructed,

* show how zero components and linear relations emerge,

* illustrate how Voigt and generalized Voigt notation are used operationally,

* provide concrete templates that can be followed for arbitrary tensor rank and symmetry class.

Each example is chosen to highlight a distinct aspect of the methodology while remaining algebraically transparent.

### D.2 Example 1: Second-Order Elasticity in Monoclinic Class 2

#### D.2.1 Problem Setup

Consider the elastic stiffness tensor $c_{ijkl}$, with intrinsic symmetries

$$c_{ijkl} = c_{jikl} = c_{ijkl} = c_{klij}. \qquad (D.1)$$

In Voigt notation this corresponds to a symmetric 6×6 matrix $C_{I,J}$ with 21 independent triclinic components.

We determine the monoclinic class 2 reduction using the generator

$$\mathbf{a}_{2y} = \text{diag}(-1, +1, -1), \qquad (D.2)$$

representing a 180° rotation about the $x_2$-axis.
### D.2.2 Voigt Transformation Matrix

From Appendix B, the induced Voigt transformation is diagonal:

$$ \mathbf{K}(\mathbf{a}_{2y}) = \mathrm{diag}(1, 1, 1, -1, 1, -1). \qquad (\mathrm{D.3}) $$

### D.2.3 Invariance Equation

The Voigt-form invariance condition is

$$ C_{IJ} = K_I K_J C_{IJ}, \qquad (\mathrm{D.4}) $$

where $K_I$ denotes the diagonal entries of $\mathbf{K}$.

### D.2.4 Component Analysis

Equation (D.4) implies:

* if $K_I K_J = +1$, then $C_{IJ}$ is unconstrained,

* if $K_I K_J = -1$, then $C_{IJ} = 0$.

Applying this rule yields the monoclinic stiffness matrix

$$ [C_{IJ}]_{(2)} = \begin{pmatrix} C_{11} & C_{12} & C_{13} & 0 & C_{15} & 0 \\ C_{12} & C_{22} & C_{23} & 0 & C_{25} & 0 \\ C_{13} & C_{23} & C_{33} & 0 & C_{35} & 0 \\ 0 & 0 & 0 & C_{44} & 0 & C_{46} \\ C_{15} & C_{25} & C_{35} & 0 & C_{55} & 0 \\ 0 & 0 & 0 & C_{46} & 0 & C_{66} \end{pmatrix}, \qquad (\mathrm{D.5}) $$

with 13 independent constants.

## D.3 Example 2: Linear Piezoelectricity in mm2

### D.3.1 Problem Setup

Consider the linear piezoelectric tensor $\xi_{kij} = \xi_{kji}$, represented in Voigt form as $\xi_{kI}$. We determine its structure for orthorhombic class mm2, with generators

$$ \mathbf{m}_x = \mathrm{diag}(-1, +1, +1), \qquad \mathbf{m}_y = \mathrm{diag}(+1, -1, +1). \qquad (\mathrm{D.6}) $$

### D.3.2 Transformation Law

From Appendix B, the mixed Cartesian–Voigt transformation is

$$ \xi'_{kI} = a_{kp} K_{IJ}(\mathbf{a}) \xi_{pJ}. \qquad (\mathrm{D.7}) $$
### D.3.3 Invariance under $m_x$

For $m_x$, the Voigt signs are

$$ \mathbf{K}(\mathbf{m}_x) = \mathrm{diag}(1, 1, 1, 1, -1, -1). \qquad (\mathrm{D.8}) $$

The invariance condition

$$ \xi_{kI} = a_{kk} K_I \xi_{kI} \qquad (\mathrm{D.9}) $$

forces all components with $a_{kk}K_I = -1$ to vanish.

### D.3.4 Invariance under $m_y$

Repeating the procedure for $m_y$ and intersecting the allowed sets yields the final form

$$ [\xi_{kI}]_{(mm2)} = \begin{pmatrix} 0 & 0 & 0 & 0 & \xi_{15} & 0 \\ 0 & 0 & 0 & \xi_{24} & 0 & 0 \\ \xi_{31} & \xi_{32} & \xi_{33} & 0 & 0 & 0 \end{pmatrix}, \qquad (\mathrm{D.10}) $$

with 5 independent constants.

## D.4 Example 3: 4-fold Rotational Invariance of a 2nd-Rank Tensor

### D.4.1 Problem Setup

Let $S_{ij} = S_{ji}$ be a symmetric second-rank tensor (e.g. dielectric tensor). We enforce tetragonal fourfold invariance about $x_3$ using

$$ \mathbf{a}_4 = \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}. \qquad (\mathrm{D.11}) $$

### D.4.2 Direct Cartesian Evaluation

Applying $S'_{ij} = a_{ip}a_{jq}S_{pq}$ gives

$$ S'_{11} = S_{22}, \quad S'_{22} = S_{11}, \quad S'_{12} = -S_{12}. \qquad (\mathrm{D.12}) $$

Invariance therefore requires

$$ S_{11} = S_{22}, \quad S_{12} = 0. \qquad (\mathrm{D.13}) $$
### D.4.3 Final Tetragonal Form

The invariant tensor is

$$ [S_{ij}]_{(4)} = \begin{pmatrix} S_{11} & 0 & 0 \\ 0 & S_{11} & 0 \\ 0 & 0 & S_{33} \end{pmatrix}, \qquad (D.14) $$

illustrating how rotational symmetry enforces transverse isotropy.

## D.5 Example 4: Third-Order Elasticity in Cubic Symmetry

### D.5.1 Problem Setup

Consider the generalized Voigt tensor $C_{IJK}$, fully symmetric in its indices. We impose invariance under:

* a 90° rotation about $x_3$,

* a 120° permutation of axes.

### D.5.2 Structural Consequences

Quarter-turn invariance eliminates all normal-shear couplings, while axis permutation invariance enforces equality among constants related by cyclic index permutations. Solving the resulting invariance equations yields exactly six independent constants:

$$ C_{111}, C_{112}, C_{123}, C_{144}, C_{155}, C_{456}. \qquad (D.15) $$

### D.5.3 Interpretation

This example illustrates how high symmetry collapses a large tensor space (56 triclinic constants) into a small invariant subspace, and how the same invariance logic applies at higher rank without modification.

## D.6 Example 5: Centrosymmetry & Odd-Rank Tensors

### D.6.1 Problem Setup

Let $\xi_{kij}$ be a polar rank-3 tensor. Consider inversion symmetry,

$$ i = -I. \qquad (D.16) $$

### D.6.2 Transformation and Invariance

Under inversion,

$$ \xi'_{kij} = (-1)^3 \xi_{kij} = -\xi_{kij}. \qquad (D.17) $$
The invariance condition $\xi'_{kij} = \xi_{kij}$ therefore implies

$$ \xi_{kij} = 0. \qquad (D.18) $$

### D.6.3 Conclusion

This simple argument explains, in a single line, the systematic disappearance of piezoelectricity and other odd-rank polar tensors in centrosymmetric classes.
# Acknowledgments

I am deeply grateful to my doctoral advisor, Distinguished Professor Dr. Mack A. Breazeale (1930–2009), whose scientific rigor and clarity of judgment shaped my early work on nonlinear acoustics and piezoelectric media. In 2006, shortly after I joined the Georgia Institute of Technology, I discussed with him my intention to develop a comprehensive reference on symmetry relations for higher-order material constants. He immediately recognized the value of such an effort, remarking that without a systematic and transparent framework researchers are repeatedly driven to ad hoc derivations and trial-and-error symmetry reductions. His encouragement at that formative stage remained a lasting motivation throughout the long development of this work.

I also wish to acknowledge my Belgian doctoral advisor, Professor Dr. H. C. Oswald J. Leroy (1936–2022), whose profound theoretical background in nonlinear mechanics and exceptional physical intuition had a decisive influence on my scientific formation. Although his own research did not focus directly on nonlinear acoustics, his way of reasoning about nonlinear phenomena, and his ability to elucidate complex theoretical structures with conceptual clarity, resonated strongly with my own approach to science. Our discussions on acoustics and acousto-optic interactions were marked by a shared intuitive understanding, and his influence played an important role in shaping my confidence and independence as a young researcher.

In a broader intellectual sense, I have always been aware that my scientific development was informed by two distinct but complementary traditions. Through Mack A. Breazeale, I was connected to the experimental—theoretical lineage originating from Nobel laureate James Franck (1882–1964), characterized by physical clarity, empirical grounding, and structural simplicity. Through Oswald J. Leroy, I was influenced by a tradition rooted in the mathematical and conceptual legacy of Henri Poincaré (1854–1912), emphasizing symmetry, structure, and the deep interplay between mathematics and physical law. The convergence of these two perspectives has profoundly shaped the manner in which I approach theoretical modeling, symmetry, and invariance.

The finalization of the present book was supported within the framework of the grant *Initiative d'Excellence Lorraine* (LUE), reference ANR-15-IDEX-04-LUE.
About the Author

Nico F. Declercq is Full Professor in the George W. Woodruff School of Mechatronic Engineering at the Georgia Institute of Technology, with duties at Georgia Tech Europe in Metz, France. He received his Ph.D. in Engineering Physics from Ghent University in 2005 (summa cum laude), following earlier degrees in Physics from the Katholieke Universiteit Leuven. His doctoral work constituted an early integration of solid mechanics, wave physics, and nonlinear acoustics.

Professor Declercq's research spans nonlinear acoustics, elastic and piezoelectric wave propagation, anisotropic and biased media, and symmetry-based constitutive modeling. He is particularly known for his work on finite-amplitude wave phenomena, higher-order elastic and electromechanical constants, and the interaction of waves with periodic, corrugated, and architected structures. His contributions combine rigorous theoretical analysis with experimental validation, and they have been widely published in leading journals in acoustics, applied physics, and solid mechanics. He also investigates fundamental physics, such as through his developed Trembling Spacetime Relativity Theory.

A recurring theme in his work is the systematic use of tensor invariance and symmetry principles to derive physically correct constitutive relations. This perspective was shaped by the complementary influences of his advisors: the experimental rigor and nonlinear acoustics tradition associated with Mack A. Breazeale, and the strong theoretical and continuum-mechanical foundations exemplified by Oswald J. Leroy. Through these influences, Professor Declercq developed an approach to nonlinear material behavior that emphasizes clarity of assumptions, explicit derivations, and internal consistency—principles that underpin the present book.

Professor Declercq has held academic appointments at Georgia Tech since 2006, progressing from Assistant Professor to Associate Professor and, in 2022, to Full Professor. He has also maintained strong international engagement through Georgia Tech Europe and extensive collaborations across Europe, Asia, and the Americas. His work has been supported by national and international research grants, and he has supervised numerous graduate students and postdoctoral researchers.

His scientific contributions have been recognized by several international honors, including the International Dennis Gabor Award (2006), the ICA Early Career Award of the International Commission for Acoustics (2007), and multiple invited plenary and keynote lectures at major international conferences. In addition to his research activities, he is deeply committed to education, having received repeated recognition
for excellence in teaching at Georgia Tech.

The present book reflects more than two decades of engagement with tensorial descriptions of material behavior in anisotropic and nonlinear media. It distills methods and insights developed across nonlinear acoustics, piezoelectricity, and elasticity into a unified, invariance-based framework. The author's intent is that the book serve not only as a reference for practitioners, but also as a durable methodological guide for researchers who require symmetry arguments that are explicit, reproducible, and free of heuristic ambiguity.
Bibliography

[1] Nico F. Declercq, J. Degrieck, R. Briers, and O. Leroy. Sound propagation in stressed, biased piezoelectric crystals. *Annalen der Physik*, 15(4):265-280, April 2006.

[2] J. F. Nye. *Physical Properties of Crystals: Their Representation by Tensors and Matrices*. Oxford University Press, Oxford, 2nd edition, 1985. First published in 1957.

[3] A. W. Joshi. *Matrices and Tensors in Physics*. New Age International (P) Ltd., New Delhi, 3rd edition, 1995. Originally published by Wiley Eastern Ltd.

[4] M. J. P. Musgrave. *Crystal Acoustics*. Introduction to Acoustics. Acoustical Society of America, Melville, NY, 2003. Reprint of the original 1970 Holden-Day edition.

[5] Gérard A. Maugin. *Nonlinear Waves in Elastic Crystals*. Oxford Science Publications. Oxford University Press, Oxford, 1999.

[6] Nico F. Declercq. *Interaction of Complex Harmonic Elastic Waves with Periodically Corrugated Surfaces and with Anisotropic Viscoelastic and/or Piezoelectric Layered Media*. Ph.d. thesis, Universiteit Gent, Ghent, Belgium, 2005.