Spectral Analysis of Piezoelectric Tensors

A third order real tensor is called a piezoelectric-type tensor if it is partially symmetric with respect to its last two indices. The piezoelectric tensor is a piezoelectric-type tensor of dimension three. We introduce C-eigenvalues and C-eigenvectors for piezoelectric-type tensors. Here, "C&q...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Chen, Yannan, Antal Jákli, Qi, Liqun
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 23.03.2017
Subjects
Eigenvalues
Eigenvectors
Electric fields
Electricity
Mathematical analysis
Piezoelectric crystals
Piezoelectricity
Tensors
Online AccessGet full text

Cover

More Information
Summary:A third order real tensor is called a piezoelectric-type tensor if it is partially symmetric with respect to its last two indices. The piezoelectric tensor is a piezoelectric-type tensor of dimension three. We introduce C-eigenvalues and C-eigenvectors for piezoelectric-type tensors. Here, "C" names after Curie brothers, who first discovered the piezoelectric effect. We show that C-eigenvalues always exist, they are invariant under orthonormal transformations, and for a piezoelectric-type tensor, the largest C-eigenvalue and its C-eigenvectors form the best rank-one piezoelectric-type approximation of that tensor. This means that for the piezoelectric tensor, its largest C-eigenvalue determines the highest piezoelectric coupling constant. We further show that for the piezoelectric tensor, the largest C-eigenvalue corresponds to the electric displacement vector with the largest \(2\)-norm in the piezoelectric effect under unit uniaxial stress, and the strain tensor with the largest \(2\)-norm in the converse piezoelectric effect under unit electric field vector. Thus, C-eigenvalues and C-eigenvectors have concrete physical meanings in piezoelectric effect and converse piezoelectric effect. Finally, we apply C-eigenvalues and associated C-eigenvectors for various piezoelectric crystals with different symmetries.
ISSN:2331-8422