Invariant Form of Linear Twin-Property Constitutive Equations and Its Application to Piezoelectricity

Acoustic properties of piezoelectric cubic crystals

“…Because of their importance in applications, we give complete prescriptions for their determination in general in Sections 3.4 and 3.5, which are specialized for cubics. Applied to cubics, and with the definition $${k^2} = e_{14}^2/( {c_{44}^E \cdot \varepsilon _{11}^S} )$$ (the square of Baerwald's invariant coupling factor 249 ), one has the following identities: $$1 \equiv ( {s_{11}^D - s_{12}^D} ) \cdot ( {c_{11}^E - c_{12}^E} ) = ( {s_{11}^D + 2 \cdot s_{12}^D} ) \cdot ( {c_{11}^E + 2 \cdot c_{12}^E} ) = ( {s_{44}^D \cdot c_{44}^E} ) \cdot ( {1 + {k^2}} ) = ( {{g_{14}} \cdot {e_{14}}} ) \cdot ( {1 + {k^2}} )/{k^2} = ( {\beta _{11}^T \cdot \varepsilon _{11}^S} ) \cdot ( {1 + {k^2}} )$$. Nonlinear piezo‐elastic constitutive sets are considered by McMahon 250 .…”

“…Additional references pertinent to axial transformations of elastic, piezoelectric, and dielectric coefficients, constitutive equations, and piezoelectric coupling coefficients are given in Refs. [33, 181, 187, 235, 237, 238–244, 249–251, 323–325, 329, 379–400]. In the previous discussion, couplings to thermal and magnetic fields have been neglected, but the formalism is easily extended to take these variables into account.…”

“…It is found that the 6 × 6 matrix is positive‐definite; its determinant factors to $${( {{c_{44}} \cdot {\varepsilon _{11}} + e_{14}^2} )^3} \cdot {( {{c_{11}} - {c_{12}}} )^2} \cdot ( {{c_{11}} + 2 \cdot {c_{12}}} ) > 0$$. This result follows as well from the more recondite and completely general group‐theoretical considerations given by Baerwald 251,508 . His representation leads to at most three distinct invariant coupling factors for a given piezoelectric point group; the two cubic classes have only one invariant coupling factor, and this is triply degenerate.…”