Abstract
We determine the different symmetry classes of bi-dimensional flexoelectric tensors. Using the harmonic decomposition method, we show that there are six symmetry classes. We also provide the matrix representations of the flexoelectric tensor and of the complete flexoelectric law, for each symmetry class.
Highlights
In recent years, the study of electromechanical effects has attracted the attention of many researchers.The best known of these electromechanical effects are piezoelectricity [1,2,3] and flexoelectricity [4,5].Piezoelectricity is the appearance of a polarization in a dielectric material when it is subjected to an uniform mechanical strain, while flexoelectricity is the linear response of a polarization to a non-uniform strain or a strain-gradient
For each element H ∈ S, we distinguish two types of strata: The open stratum denoted Σ[ H ] which is the stratum of tensors whose symmetry class is exactly [ H ] and the closed stratum denoted Σ[ H ] which is the stratum of tensors whose symmetry class is at least [ H ] (in the set of all symmetry classes we introduce a partial ordering relation as follows: [ H1 ] [ H2 ] if there exists Q ∈ O(2) such that Q H1 QT ⊆ H2 )
In this work we solved the problem of the determination of the number and types of symmetry classes of the complete flexoelectricity law
Summary
The study of electromechanical effects has attracted the attention of many researchers. Some studies [21,22] provide the complete description of bi-dimensional anisotropic strain-gradient elasticity by determining the different symmetry classes of the constitutive tensors and their associated matrix forms. We will complete these studies in the case where strain-gradient effects are coupled with flexoelectric ones. To provide the matrix representation of the complete flexoelectric law we need to construct appropriate orthonormal bases in order to express the different constitutive tensors in a matrix form. We wish to emphasize that the explicit harmonic decomposition provided in Section 5 is a new result of practical importance for the computation of physically-based invariants of the flexoelectricity tensor
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