Editing Piezoelectricity (section)

===Mathematical description===
Linear piezoelectricity is the combined effect of
* The linear electrical behavior of the material:
:: <math>\mathbf{D} = \boldsymbol{\varepsilon}\,\mathbf{E} \quad \implies</math> [[Summation|<math>\quad D_i = \sum_j \varepsilon_{ij}\,E_j \;</math>]]
: where '''D''' is the electric flux density<ref>IEC 80000-6, item 6-12</ref><ref>{{Cite web|url=http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=121-11-40|title=IEC 60050 – International Electrotechnical Vocabulary – Details for IEV number 121-11-40: "electric flux density"|website=www.electropedia.org}}</ref> ([[electric displacement]]), '''ε''' is the [[permittivity]] (free-body dielectric constant), '''E''' is the [[electric field strength]], and [[divergence|<math> \nabla\cdot\mathbf{D} = 0 </math>]] , [[Curl (mathematics)|<math>\nabla \times \mathbf{E} = \mathbf{0} </math>]].
* [[Hooke's law]] for linear elastic materials:
::<math>\boldsymbol{S}=\mathsf{s}\,\boldsymbol{T} \quad \implies \quad S_{ij} = \sum_{k,\ell} s_{ijk\ell} \,T_{k\ell} \;</math>
: where '''S''' is the linearized [[Strain (materials science)|strain]], '''s''' is [[Compliance (mechanics)|compliance]] under short-circuit conditions, '''T''' is [[Stress (physics)|stress]], and
::<math> \nabla \cdot \boldsymbol{T} = \mathbf{0} \,\,,\, \boldsymbol{S} = \frac{\nabla \mathbf{u} + \mathbf{u} \nabla}{2}, </math>
: where '''u''' is the ''displacement vector''.

These may be combined into so-called ''coupled equations'', of which the '''strain-charge form''' is:<ref name=ikeda>{{cite book|title=Fundamentals of piezoelectricity|last=Ikeda|first=T.|year=1996|publisher=Oxford University Press}}{{ISBN missing}}</ref>
:<math>
 \begin{align}
 \boldsymbol{S} &= \mathsf{s}\,\boldsymbol{T} + \mathfrak{d}^t\,\mathbf{E}\ && \implies \quad
 S_{ij} = \sum_{k,\ell} s_{ijk\ell} \,T_{k\ell} + \sum_k d^t_{ijk} \,E_k, \\[6pt]
 \mathbf{D} &= \mathfrak{d}\,\boldsymbol{T} + \boldsymbol{\varepsilon}\,\mathbf{E} && \implies \quad
 D_i = \sum_{j, k} d_{ijk} \,T_{jk} + \sum_j \varepsilon_{ij}\,E_j,
 \end{align}
</math>
where <math>\mathfrak{d}</math> is the piezoelectric tensor and the superscript t stands for its transpose. Due to the symmetry of <math>\mathfrak{d}</math>, <math>d^t_{ijk}=d_{kji}=d_{kij}</math>.

In matrix form,
:<math>
 \begin{align}
 \{S\} &= \left [s^E \right ]\{T\}+[d^\mathrm{t}]\{E\}, \\[6pt]
 \{D\} &= [d]\{T\}+\left [ \varepsilon^T \right ] \{E\},
 \end{align}
</math>
where [''d''] is the matrix for the direct piezoelectric effect and [''d''{{i sup|t}}] is the matrix for the converse piezoelectric effect. The superscript ''E'' indicates a zero, or constant, electric field; the superscript ''T'' indicates a zero, or constant, stress field; and the superscript t stands for [[Transpose|transposition]] of a [[Matrix (mathematics)|matrix]].

Notice that the third order tensor <math>\mathfrak{d}</math> maps vectors into symmetric matrices. There are no non-trivial rotation-invariant tensors that have this property, which is why there are no isotropic piezoelectric materials.

The strain-charge for a material of the [[Tetragonal crystal system|4mm]] (C<sub>4v</sub>) [[crystal system|crystal class]] (such as a poled piezoelectric ceramic such as tetragonal PZT or BaTiO<sub>3</sub>) as well as the [[Hexagonal crystal system#Hexagonal crystal system|6mm]] crystal class may also be written as (ANSI IEEE 176):
:<math>
\begin{align}
& \begin{bmatrix} S_1 \\ S_2 \\ S_3 \\ S_4 \\ S_5 \\ S_6 \end{bmatrix}
= \begin{bmatrix} s_{11}^E & s_{12}^E & s_{13}^E & 0 & 0 & 0 \\
s_{21}^E & s_{22}^E & s_{23}^E & 0 & 0 & 0 \\
s_{31}^E & s_{32}^E & s_{33}^E & 0 & 0 & 0 \\
0 & 0 & 0 & s_{44}^E & 0 & 0 \\
0 & 0 & 0 & 0 & s_{55}^E & 0 \\
0 & 0 & 0 & 0 & 0 & s_{66}^E=2\left(s_{11}^E-s_{12}^E\right) \end{bmatrix}
\begin{bmatrix} T_1 \\ T_2 \\ T_3 \\ T_4 \\ T_5 \\ T_6 \end{bmatrix}
+
\begin{bmatrix} 0 & 0 & d_{31} \\
0 & 0 & d_{32} \\
0 & 0 & d_{33} \\
0 & d_{24} & 0 \\
d_{15} & 0 & 0 \\
0 & 0 & 0 \end{bmatrix}
\begin{bmatrix} E_1 \\ E_2 \\ E_3 \end{bmatrix} \\[8pt]
& \begin{bmatrix} D_1 \\ D_2 \\ D_3 \end{bmatrix}
=
\begin{bmatrix} 0 & 0 & 0 & 0 & d_{15} & 0 \\
0 & 0 & 0 & d_{24} & 0 & 0 \\
d_{31} & d_{32} & d_{33} & 0 & 0 & 0 \end{bmatrix}
\begin{bmatrix} T_1 \\ T_2 \\ T_3 \\ T_4 \\ T_5 \\ T_6 \end{bmatrix}
+
\begin{bmatrix} {\varepsilon}_{11} & 0 & 0 \\
0 & {\varepsilon}_{22} & 0 \\
0 & 0 & {\varepsilon}_{33} \end{bmatrix}
\begin{bmatrix} E_1 \\ E_2 \\ E_3 \end{bmatrix}
\end{align}
</math>
where the first equation represents the relationship for the converse piezoelectric effect and the latter for the direct piezoelectric effect.<ref name="DD1998">{{cite journal |last=Damjanovic |first=Dragan |year=1998 |title=Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics |journal=Reports on Progress in Physics |volume=61 |pages=1267–1324 |doi=10.1088/0034-4885/61/9/002|bibcode = 1998RPPh...61.1267D |issue=9 }}</ref>

Although the above equations are the most used form in literature, some comments about the notation are necessary. Generally, ''D'' and ''E'' are [[Vector (geometric)|vectors]], that is, [[Cartesian tensor]]s of rank 1; and permittivity ''ε'' is a Cartesian tensor of rank 2. Strain and stress are, in principle, also rank-2 [[tensors]]. But conventionally, because strain and stress are all symmetric tensors, the subscript of strain and stress can be relabeled in the following fashion: 11&nbsp;→&nbsp;1; 22&nbsp;→&nbsp;2; 33&nbsp;→&nbsp;3; 23&nbsp;→&nbsp;4; 13&nbsp;→&nbsp;5; 12&nbsp;→&nbsp;6. (Different conventions may be used by different authors in literature. For example, some use 12&nbsp;→&nbsp;4; 23&nbsp;→&nbsp;5; 31&nbsp;→&nbsp;6 instead.) That is why ''S'' and ''T'' appear to have the "vector form" of six components. Consequently, ''s'' appears to be a 6-by-6 matrix instead of a rank-3 tensor. Such a relabeled notation is often called [[Voigt notation]]. Whether the shear strain components ''S''<sub>4</sub>, ''S''<sub>5</sub>, ''S''<sub>6</sub> are tensor components or engineering strains is another question. In the equation above, they must be engineering strains for the 6,6 coefficient of the compliance matrix to be written as shown, i.e., 2(''s''{{su|b=11|p=''E''}}&nbsp;−&nbsp;''s''{{su|b=12|p=''E''}}). Engineering shear strains are double the value of the corresponding tensor shear, such as ''S''<sub>6</sub>&nbsp;=&nbsp;2''S''<sub>12</sub> and so on. This also means that ''s''<sub>66</sub>&nbsp;=&nbsp;{{sfrac|1|''G''<sub>12</sub>}}, where ''G''<sub>12</sub> is the [[shear modulus]].

In total, there are four piezoelectric coefficients, ''d<sub>ij</sub>'', ''e<sub>ij</sub>'', ''g<sub>ij</sub>'', and ''h<sub>ij</sub>'' defined as follows:

:<math>\begin{align}
d_{ij} &= \phantom{+} \left ( \frac{\partial D_i}{\partial T_j} \right )^E
&&= \phantom{+} \left ( \frac{\partial S_j}{\partial E_i} \right )^T \\[6pt]
e_{ij} &= \phantom{+} \left ( \frac{\partial D_i}{\partial S_j} \right )^E
&&= -\left ( \frac{\partial T_j}{\partial E_i} \right )^S \\[6pt]
g_{ij} &= -\left ( \frac{\partial E_i}{\partial T_j} \right )^D
&&= \phantom{+} \left ( \frac{\partial S_j}{\partial D_i} \right )^T \\[6pt]
h_{ij} &= -\left ( \frac{\partial E_i}{\partial S_j} \right )^D
&&= -\left ( \frac{\partial T_j}{\partial D_i} \right )^S
\end{align}</math>
where the first set of four terms corresponds to the direct piezoelectric effect and the second set of four terms corresponds to the converse piezoelectric effect. The equality between the direct piezoelectric tensor and the transpose of the converse piezoelectric tensor originates from the [[Maxwell relations]] of thermodynamics.<ref>{{cite journal |last=Kochervinskii |first=V. |year=2003 |title=Piezoelectricity in Crystallizing Ferroelectric Polymers |journal=[[Crystallography Reports]] |volume=48 |issue= 4 |pages=649–675|doi=10.1134/1.1595194|bibcode = 2003CryRp..48..649K }}</ref> For those piezoelectric crystals for which the polarization is of the crystal-field induced type, a formalism has been worked out that allows for the calculation of piezoelectrical coefficients ''d<sub>ij</sub>'' from electrostatic lattice constants or higher-order [[Madelung constant]]s.<ref name="ZPB1995a"/>