Piezoelectricity

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Piezoelectricity is the property presented by certain materials that exhibit an electric polarization when submitted to an applied mechanical stress such as a uniaxial compression. Conversely, their shape changes when they are submitted to an external electric field; this is the converse piezoelectric effect. The piezoelectric effect and the converse effect are described by third-rank tensors:

For a small stress, represented by a second-rank tensor, T_ij, the resulting polarization, of components P_k , is given by:

P_k = d_kijT_ij

where d_kij is a third-rank tensor representing the direct piezoelectric effect.

For a small applied electric field, of components E_k, the resulting strain, represented by a second-rank tensor, S_ij, is given by:

S_ij = d_ijkE_k + Q_ijklE_kE_l

where the first-order term, d_ijk, represents the inverse piezoelectric effect and the second-order term, Q_ijkl, a symmetric fourth-rank tensor, the electrostriction effect. The sense of the strain due to the piezoelectric effect changes when the sign of the applied electric field changes , while that due to electrostriction, a quadratic effect, does not.

The matrices associated to the coefficients d_kij and d_kij of the direct and converse piezoelectric effects, respectively, are transpose of one another.