Symmetry Classes and Matrix Representations of the 2D Flexoelectric Law
Abstract
:1. Introduction
- What types of anisotropies can flexoelectric tensors model?
- How does one characterize flexoelectric materials independently of their spatial orientation?
- How does one parametrize the different characteristics (anisotropy, coupling, etc.) of flexoelectric materials in an intrinsic way?
2. Notation
- the group of invertible matrices over ;
- the group of matrices satisfying and , where and stand for the inverse and the transpose of the matrix . This group is called the orthogonal group;
- the subgroup of of matrices of determinant 1, called the special orthogonal group.
- the identity group;
- the cyclic group with k elements, generated by . For , we have ;
- the dihedral group with elements, generated by and . For , the group will be denoted by .
- Centrosymmetric (denoted by ) if its contains the inversion , and non-centrosymmetric () otherwise;
- Chiral (denoted by ) if its does not contain reflection or mirror , and achiral () otherwise.
- ⊗ denotes the usual tensor product of two tensors or vector spaces;
- denotes the n-th power of the tensor product, e.g., (n copies);
- denotes the symmetrized tensor product;
- indicates the twisted tensor product defined by:
- is the space of n-th order tensors with no index symmetry on ;
- is the space of completely symmetric (by completely symmetric we mean symmetric with respect to all permutations of indices) n-th order tensors on ;
- is the space of n-th order harmonic tensors (i.e., completely symmetric and traceless tensors), with
- is the space of scalars;
- is the space of pseudo-scalars (i.e., scalars which change sign under improper transformations).
- is the second-order identity tensor with components , where is the Kronecker delta;
- is the fourth-order identity tensor on ;
- denotes the 2D Levi–Civita tensor defined by:
- is the space of dimensional square matrices;
- is the space of dimensional symmetric square matrices;
- is the space of rectangular matrices.
- ⊕ direct sum;
- ≃ denotes hereafter an isomorphism.
3. Flexoelectricity Law
3.1. Constitutive Equations
- is a fourth-order elasticity tensor;
- is a fifth-order elasticity tensor;
- is the fifth-order elasticity tensor defined as the transpose of in the following sense ;
- is the sixth-order elasticity tensor.
- is the dielectric susceptibility tensor;
- is the piezoelectric tensor;
- is the transpose of the piezoelectric tensor . The transposition is defined as ;
- is the flexoelectric tensor;
- is the transpose of the flexoelectric tensor . The transposition is defined as .
- State tensors: , , , , , and ;
- Constitutive tensors: , , , , , and .
3.2. Notions of Symmetry Group and Symmetry Class
3.3. Symmetry Classes of the Complete Flexoelectric Law
3.4. Isotypic Decomposition of the Space of Flexoelectric Tensors
⊗ | |||
× | × | ||
× | × | ||
× | × |
4. Matrix Representations
4.1. Matrix Representations of the Flexoelectric Tensor
4.1.1. -Class
4.1.2. -Class
4.1.3. -Class
4.1.4. -Class
4.1.5. -Class
4.1.6. -Class
4.2. Matrix Representations of the Complete Flexoelectric Law
- Any odd-order tensor vanishes for the centrosymmetric symmetry classes: , ;
- An even-order tensor can not see an odd-order material invariance (in 2D), the invariance seen will be twice the order of the former (see [36] (Theorem 5)).
5. Harmonic Parametrization of the Flexoelectric Tensor
5.1. Decomposition of the State Tensor Space
5.2. The Harmonic Basis
5.3. Explicit Harmonic Decomposition of the Flexoelectric Tensor
5.4. Associated Matrix Representations
6. Discussion
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Name | Digonal | Orthotropic | Tetrachiral | Tetragonal | Hemitropic | Isotropic |
---|---|---|---|---|---|---|
#indep () | 6 | 3 | 4 | 2 |
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Abdoul-Anziz, H.; Auffray, N.; Desmorat, B. Symmetry Classes and Matrix Representations of the 2D Flexoelectric Law. Symmetry 2020, 12, 674. https://doi.org/10.3390/sym12040674
Abdoul-Anziz H, Auffray N, Desmorat B. Symmetry Classes and Matrix Representations of the 2D Flexoelectric Law. Symmetry. 2020; 12(4):674. https://doi.org/10.3390/sym12040674
Chicago/Turabian StyleAbdoul-Anziz, Houssam, Nicolas Auffray, and Boris Desmorat. 2020. "Symmetry Classes and Matrix Representations of the 2D Flexoelectric Law" Symmetry 12, no. 4: 674. https://doi.org/10.3390/sym12040674