Piezoelectricity

Definition

Consider a linear dielectric elastic material with stress $T_{kl}$ and infinitesimal strain $S_{ij}$. Let $E_k$ be the electric field and $D_i$ the electric displacement, with indices from $1$ to $3$ and summation over repeated indices. Its linear constitutive law in strain–charge form is the following. $$S_{ij} = s^E_{ijkl} T_{kl} + d_{kij} E_k, \qquad D_i = d_{ikl} T_{kl} + \epsilon^T_{ik} E_k$$ Here $s^E$ is the elastic compliance at constant field, $\epsilon^T$ is the permittivity at constant stress, and $d_{ikl} = d_{ilk}$ is the tensor of piezoelectric coefficients. The material is piezoelectric if $d \neq 0$. The term $d_{ikl} T_{kl}$ is the direct piezoelectric effect and the term $d_{kij} E_k$ is the converse effect.