The linear piezoelectric effect in a solid is described by a rank-three tensor through

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids

Invariance, Ch Index notation and Cartesian tensors

Problem

The linear piezoelectric effect in a solid is described by a rank-three tensor

d_{ijk}

through

P_i=d_{ijk}e_{jk}

, where

P_i

is the induced electric polarisation and

e_{jk}=e_{kj}

is the strain. Here

d_{ijk}

couples polarisation to strain; the

d

coefficients in data tables usually couple it to stress, but the argument is the same for either. Because the strain is symmetric, only the part of

d_{ijk}

symmetric in

j

and

k

affects

P_i

, and we choose that measurable representative,

d_{ijk}=d_{ikj}

. The only rank-three tensor unchanged by every proper rotation is a multiple of the Levi-Civita symbol (a result quoted without proof, as the rank-four one is), so for a solid isotropic under rotations,

d_{ijk}=c\,\epsilon_{ijk}

for some scalar

c

. ((a)) Using

d_{ijk}=d_{ikj}

together with

d_{ijk}=c\,\epsilon_{ijk}

, show that isotropy forces

c=0

. ((b)) State, in one sentence, what this means for the piezoelectric response of an isotropic solid such as glass or an untextured polycrystal.

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