9.6.4 Restrictions imposed by invariance under superposed rigid body motions and conservation of angular momentum

The constitutive equations (9.74) for the deformation-temperature-electric field-magnetic field formulation (or any other formulation for that matter, e.g., (9.64) or (9.70)), obtained from restrictions imposed by the second law, must also satisfy invariance under superposed rigid body motions. In particular, we must have [30]

${\left(E^{F\text{Θ}\mathit{eh}}\right)}^{+}=E^{F\text{Θ}\mathit{eh}}$

when

${\mathbf{F}}^{+}=\mathbf{QF},{\text{Θ}}^{+}=\text{Θ},{\left({\mathbf{e}}^{*}\right)}^{+}=\mathbf{Q}{\mathbf{e}}^{*},{\left({\mathbf{h}}^{*}\right)}^{+}=\mathbf{Q}{\mathbf{h}}^{*}$

for all proper orthogonal tensors Q(t). Equations (9.75a) and (9.75b) are referred to as invariance requirements. It can be shown that these invariance requirements demand that

$E^{F\text{Θ}\mathit{eh}}\left(\mathbf{F},\text{Θ},{\mathbf{e}}^{*},{\mathbf{h}}^{*}\right)={\tilde{E}}^{F\text{Θ}\mathit{eh}}\left(\mathbf{C},\text{Θ},{\mathbf{e}}_{\text{R}},{\mathbf{h}}_{\text{R}}\right),$

where

$\mathbf{C}={\mathbf{F}}^{\text{T}}\mathbf{F},{\mathbf{e}}_{\text{R}}={\mathbf{F}}^{\text{T}}{\mathbf{e}}^{*},{\mathbf{h}}_{\text{R}}={\mathbf{F}}^{\mathbf{T}}{\mathbf{h}}^{*}$

are the right Cauchy-Green deformation tensor, referential electric field, and referential magnetic field, respectively. We can then verify, through a change of independent variable from F to C, e* to e_R, and h* to h_R, that the constitutive equations (9.74) become

$\mathbf{T}=2\rho \mathbf{F}\frac{\partial {\tilde{E}}^{F\text{Θ}\mathit{eh}}}{\partial \mathbf{C}}{\mathbf{F}}^{\text{T}}-{\mathbf{e}}^{*}\otimes {\mathbf{p}}^{*}-{\mu }_{\text{o}}{\mathbf{h}}^{*}\otimes {\mathbf{m}}^{*},\eta =-\frac{\partial {\tilde{E}}^{F\text{Θ}\mathit{eh}}}{\partial \text{Θ}},{\mathbf{p}}^{*}=-\rho \mathbf{F}\frac{\partial {\tilde{E}}^{F\text{Θ}\mathit{eh}}}{\partial {\mathbf{e}}_{\text{R}}},{\mathbf{m}}^{*}=-\frac{\rho }{{\mu }_{\text{o}}}\mathbf{F}\frac{\partial {\tilde{E}}^{F\text{Θ}\mathit{eh}}}{\partial {\mathbf{h}}_{\text{R}}}.$

These are the invariant forms of the constitutive equations. We emphasize that ${\tilde{E}}^{F\text{Θ}\mathit{eh}}$ denotes the free energy function with C, Θ, e_R, and h_R as independent variables. A useful result follows from a change of the mechanical independent variable from the right Cauchy-Green deformation C to the Lagrangian strain E, where E = 1/2 (C − I), in which case (9.77) becomes

$\mathbf{T}=\rho \mathbf{F}\frac{\partial {\breve{E}}^{F\text{Θ}\mathit{eh}}}{\partial \mathbf{E}}{\mathbf{F}}^{\text{T}}-{\mathbf{e}}^{*}\otimes {\mathbf{p}}^{*}-{\mu }_{\text{o}}{\mathbf{h}}^{*}\otimes {\mathbf{m}}^{*},\eta =-\frac{\partial {\breve{E}}^{F\text{Θ}\mathit{eh}}}{\partial \text{Θ}},{\mathbf{p}}^{*}=-\rho \mathbf{F}\frac{\partial {\tilde{E}}^{F\text{Θ}\mathit{eh}}}{\partial {\mathbf{e}}_{\text{R}}},{\mathbf{m}}^{*}=-\frac{\rho }{{\mu }_{\text{o}}}\mathbf{F}\frac{\partial {\breve{E}}^{F\text{Θ}\mathit{eh}}}{\partial {\mathbf{h}}_{\text{R}}},$

where ${\breve{E}}^{F\text{Θ}\mathit{eh}}$ denotes the free energy function with E, Θ, e_R, and h_R as independent variables. It can be verified that (9.78) satisfies conservation of angular momentum. In the following section, we use (9.78) as a starting point for developing constitutive equations for materials with linear, reversible TEME response.

9.7.1 Small-deformation kinematics, kinetics, electromagnetic fields, and fundamental laws

The small-deformation theory (or, equivalently, the small-strain or infinitesimal theory) is customarily obtained by assuming that the displacements u are small, and expanding u in a power series with respect to a small parameter.¹² As a consequence of this approximation, (3.7), (3.27), (3.36), (3.54), and (3.55) become, to leading order,^13,14

$\mathbf{x}=\mathbf{X},\mathbf{F}=\mathbf{I},J=1,\mathbf{E}=\mathbf{e}=\frac{1}{2}\left[\text{grad}\mathbf{u}+{\left(\text{grad}\mathbf{u}\right)}^{\text{T}}\right].$

Hence, in the small-deformation theory, the reference and present configurations of the body are indistinguishable, the referential (Lagrangian) and spatial (Eulerian) descriptions of a given quantity are one and the same, and the Eulerian and Lagrangian strains are identical. In the small-deformation theory, we also have

$\stackrel{.}{⌽}={⌽}^{\prime },\text{Grad}⌽=\text{grad}⌽,$

where ø is a scalar-valued (or vector-valued or tensor-valued) function of position and time; refer to (3.20) and (3.21). It then follows from (9.41) and (9.79) that, to leading order,

${\mathbf{e}}_{\text{R}}={\mathbf{e}}^{*},{\mathbf{p}}_{\text{R}}={\mathbf{p}}^{*},{\mathbf{d}}_{\text{R}}={\mathbf{d}}^{*},{\mathbf{h}}_{\text{R}}={\mathbf{h}}^{*},{\mathbf{m}}_{\text{R}}={\mathbf{m}}^{*},{\mathbf{b}}_{\text{R}}={\mathbf{b}}^{*},{\sigma }_{\text{R}}={\sigma }^{*},{\mathbf{j}}_{\text{R}}={\mathbf{j}}^{*},\mathbf{P}=\mathbf{T},{\mathbf{q}}_{\text{R}}=\mathbf{q}.$

We assume that the electromagnetic fields are small, so their contributions to balance of linear momentum and angular momentum emerge as higher-order terms. In other words, the electromagnetic body force and body couple vanish at leading order, i.e.,

${\mathbf{f}}^{\text{em}}=\mathbf{0},{\text{Γ}}^{\text{em}}=\mathbf{0}.$

An additional implication of the small-field assumption is that

${\mathbf{e}}^{*}=\mathbf{e},{\mathbf{p}}^{*}=\mathbf{p},{\mathbf{d}}^{*}=\mathbf{d},{\mathbf{h}}^{*}=\mathbf{h},{\mathbf{m}}^{*}=\mathbf{m},{\mathbf{b}}^{*}=\mathbf{b},{\sigma }^{*}=\sigma ,{\mathbf{j}}^{*}=\mathbf{j}.$

That is, to leading order, the effective fields collapse to the corresponding standard fields, and all standard fields are equivalent (the v → 0 limit of (9.51)–(9.54)).

As a consequence of (9.79)–(9.83), we hereafter omit adjectives such as “referential,” “spatial,” “effective,” and “standard” that are no longer needed in the small-deformation/small-field theory, and instead refer to E as the infinitesimal strain, T the stress, e the electric field, p the electric polarization, d the electric displacement, and so on. Also, (9.79)–(9.83) imply that the fundamental laws (9.49a)–(9.49j), when expressed in Cartesian component notation, become, to leading order,

$\rho ={\rho }_{\text{R}},$

$\rho u_{i,\mathit{tt}}=\rho f_i^m+T_{\mathit{ij},j},$

$T_{\mathit{ij}}=T_{\mathit{ji}},$

$\rho {\epsilon }_{,t}=T_{\mathit{ij}}{E}_{\mathit{ij},t}+e_i{p}_{i,t}+{\mu }_{\text{o}}{h}_i{m}_{i,t}+\rho r^t+j_i{e}_i-q_{i,i},$

$\rho {\eta }_{,t}\ge \rho \frac{r^{\text{t}}}{\text{Θ}}-{\left(\frac{q_i}{\text{Θ}}\right)}_{,i},$

${\sigma }_{,t}+j_{i,i}=0,$

$b_{i,i}=0,$

$b_{i,t}+{\epsilon }_{\mathit{ijk}}{e}_{k,j}=0,$

$d_{i,i}=\sigma ,$

$d_{i,t}+j_i={\epsilon }_{\mathit{ijk}}{h}_{k,j}.$

The strain-displacement relationship (9.79)₄, expressed in Cartesian component notation, becomes

$E_{\mathit{ij}}=\frac{1}{2}\left(u_{i,j}+u_{j,i}\right).$

Note that ()_,t denotes partial differentiation with respect to time, e.g.,

$b_{i,t}\equiv \frac{\partial b_i\left(x_1,x_2,x_3,t\right)}{\partial t}.$

Also note that conservation of angular momentum (9.84c) and definition (9.85) imply that the stress T and infinitesimal strain E are symmetric.

9.7.2 Linear constitutive equations

With use of relationships (9.79)–(9.83), the TEME constitutive equations (9.78) become, to leading order in the small-deformation/small-field theory,

$T_{\mathit{ij}}=\rho \frac{\partial \psi }{\partial E_{\mathit{ij}}},\eta =-\frac{\partial \psi }{\partial \text{Θ}},p_i=-\rho \frac{\partial \psi }{\partial e_i},m_i=-\frac{\rho }{{\mu }_{\text{o}}}\frac{\partial \psi }{\partial h_i},$

where we have introduced $\psi ={\breve{E}}^{F\text{Θ}\mathit{eh}}$ for notational brevity. In what follows, starting from (9.86), we derive linear TEME constitutive equations for ferroic materials within the near-equilibrium regime. To obtain a functional form of the free energy ψ that characterizes this linear, reversible TEME response, we perform a Taylor series expansion of ψ in terms of its independent variables E, Θ, e, and h in the neighborhood of a thermodynamic equilibrium state x_o = (E^o, Θ°, e^o, h^o):

${\breve{E}}^{F\text{Θ}\mathit{eh}}\equiv \psi \left(\mathbf{E},\text{Θ},\mathbf{e},\mathbf{h}\right)={\psi |}_{x_o}+\frac{1}{2}{\frac{{\partial }^2\psi }{\partial E_{\mathit{ij}}\partial E_{\mathit{kl}}}|}_{x_{\text{o}}}{E}_{\mathit{ij}}{E}_{\mathit{kl}}+\frac{1}{2}{\frac{{\partial }^2\psi }{\partial e_i\partial e_j}|}_{x_{\text{o}}}{e}_i{e}_j+\frac{1}{2}{\frac{{\partial }^2\psi }{\partial h_i\partial h_j}|}_{x_{\text{o}}}{h}_i{h}_j+\frac{1}{2}{\frac{{\partial }^2\psi }{\partial {\text{Θ}}^2}|}_{x_{\text{o}}}{\text{Θ}}^2+{\frac{{\partial }^2\psi }{\partial E_{\mathit{ij}}\partial e_k}|}_{x_{\text{o}}}{E}_{\mathit{ij}}{e}_k+{\frac{{\partial }^2\psi }{\partial E_{\mathit{ij}}\partial h_k}|}_{x_{\text{o}}}{E}_{\mathit{ij}}{h}_k+{\frac{{\partial }^2\psi }{\partial e_i\partial h_j}|}_{x_{\text{o}}}{e}_i{h}_j+{\frac{{\partial }^2\psi }{\partial \text{Θ}\partial E_{\mathit{ij}}}|}_{x_{\text{o}}}\text{Θ}{E}_{\mathit{ij}}+{\frac{{\partial }^2\psi }{\partial \text{Θ}\partial e_i}|}_{x_{\text{o}}}\text{Θ}{e}_i+{\frac{{\partial }^2\psi }{\partial \text{Θ}\partial h_i}|}_{x_{\text{o}}}\text{Θ}{h}_i+\text{higher oreder terms}.$

Note that the TEME quantities E_ij, Θ, e_i, and h_i in (9.87) are perturbed about the equilibrium statex_o. Disregarding the higher-order terms (i.e., truncating at second order), we substitute the free energy expansion (9.87) into (9.86) to obtain the linear form of the constitutive equations:

$T_{\mathit{ij}}={\frac{{\partial }^2\overline{\psi }}{\partial E_{\mathit{ij}}\partial E_{\mathit{kl}}}|}_{x_{\text{o}}}{E}_{\mathit{kl}}+{\frac{{\partial }^2\overline{\psi }}{\partial E_{\mathit{ij}}\partial e_k}|}_{x_{\text{o}}}{e}_k+{\frac{{\partial }^2\overline{\psi }}{\partial E_{\mathit{ij}}\partial h_k}|}_{x_{\text{o}}}{h}_k+{\frac{{\partial }^2\overline{\psi }}{\partial E_{\mathit{ij}}\partial \text{Θ}}|}_{x_{\text{o}}}\text{Θ},$

$p_i=-{\frac{{\partial }^2\overline{\psi }}{\partial E_{\mathit{jk}}\partial e_i}|}_{x_{\text{o}}}{E}_{\mathit{jk}}-{\frac{{\partial }^2\overline{\psi }}{\partial e_i\partial e_j}|}_{x_{\text{o}}}{e}_j-{\frac{{\partial }^2\overline{\psi }}{\partial e_i\partial h_j}|}_{x_{\text{o}}}{h}_j-{\frac{{\partial }^2\overline{\psi }}{\partial e_i\partial \text{Θ}}|}_{x_{\text{o}}}\text{Θ},$

${\mu }_{\text{o}}{m}_i=-{\frac{{\partial }^2\overline{\psi }}{\partial E_{\mathit{jk}}\partial h_i}|}_{x_{\text{o}}}{E}_{\mathit{jk}}-{\frac{{\partial }^2\overline{\psi }}{\partial e_j\partial h_i}|}_{x_{\text{o}}}{e}_j-{\frac{{\partial }^2\overline{\psi }}{\partial h_i\partial h_j}|}_{x_{\text{o}}}{h}_j-{\frac{{\partial }^2\overline{\psi }}{\partial h_i\partial \text{Θ}}|}_{x_{\text{o}}}\text{Θ},$

$\overline{\eta }=-{\frac{{\partial }^2\overline{\psi }}{\partial E_{\mathit{ij}}\partial \text{Θ}}|}_{x_{\text{o}}}{E}_{\mathit{ij}}-{\frac{{\partial }^2\overline{\psi }}{\partial e_i\partial \text{Θ}}|}_{x_{\text{o}}}{e}_i-{\frac{{\partial }^2\overline{\psi }}{\partial h_i\partial \text{Θ}}|}_{x_{\text{o}}}{h}_i-{\frac{{\partial }^2\overline{\psi }}{\partial {\text{Θ}}^2}|}_{x_{\text{o}}}\text{Θ},$

where $\overline{\psi }=\rho \psi$ and $\overline{\eta }=\rho \eta$ denote the free energy per unit volume and the free entropy per unit volume, respectively. The constant coefficients arising in the linearized constitutive equations (9.88a)–(9.88d) are characterized using experimental data specific to the material. For instance, ${{\partial }^2\overline{\psi }/\left(\partial E_{\mathit{ij}}\partial E_{\mathit{kl}}\right)|}_{x_{\text{o}}}$ represents the component of the elasticity tensor at equilibrium state x_o, which is specific to the material being characterized; refer to Table 9.7. The other coefficients can be described in a similar manner. With use of the nomenclature shown in Table 9.7, the free energy per unit volume (9.87) becomes

$\overline{\psi }=\frac{1}{2}{C}_{\mathit{ijkl}}{E}_{\mathit{ij}}{E}_{\mathit{kl}}-\frac{1}{2}{\chi }_{\mathit{ij}}^{\text{e}}{e}_i{e}_j-\frac{1}{2}{\chi }_{\mathit{ij}}^{\text{m}}{h}_i{h}_j-\frac{1}{2}c{\text{Θ}}^2-d_{\mathit{ijk}}^{\text{e}}{E}_{\mathit{ij}}{e}_k-d_{\mathit{ijk}}^{\text{m}}{E}_{\mathit{ij}}{h}_k-{\beta }_{\mathit{ij}}{E}_{\mathit{ij}}\text{Θ}-{\chi }_{\mathit{ij}}^{\text{em}}{e}_i{h}_j-L_i^{\text{e}}\text{Θ}{h}_i,$

and the linearized set of constitutive equations (9.88a)–(9.88d) simplify to

$T_{\mathit{ij}}=C_{\mathit{ijkl}}{E}_{\mathit{kl}}-d_{\mathit{ijk}}^{\text{e}}{e}_k-d_{\mathit{ijk}}^{\text{m}}{h}_k-{\beta }_{\mathit{ij}}\text{Θ},$

$p_i=d_{\mathit{jki}}^{\text{e}}{E}_{\mathit{jk}}+{\chi }_{\mathit{ij}}^{\text{e}}{e}_j+{\chi }_{\mathit{ij}}^{\text{em}}{h}_j+L_i^e\text{Θ},$

${\mu }_{\text{o}}{m}_i=d_{\mathit{jki}}^{\text{m}}{E}_{\mathit{jk}}+{\chi }_{\mathit{ji}}^{\text{em}}{e}_j+{\chi }_{\mathit{ij}}^{\text{m}}{h}_j+L_i^{\text{m}}\text{Θ},$

$\overline{\eta }={\beta }_{\mathit{ij}}{E}_{\mathit{ij}}+L_i^{\text{e}}{e}_i+L_i^{\text{m}}{h}_i+c\text{Θ}.$

Table 9.7

Material Constants and Their Representations for Linear Reversible Processes

Constant	Representation	Constant	Representation
Elasticity constant	$C_{\mathit{ijkl}}={\frac{{\partial }^2\overline{\psi }}{\partial E_{\mathit{ij}}\partial E_{\mathit{kl}}}|}_{x_{\text{o}}}$	Piezoelectric constant	$d_{\mathit{ijk}}^{\text{e}}={-\frac{{\partial }^2\overline{\psi }}{\partial E_{\mathit{ij}}\partial e_k}|}_{x_{\text{o}}}$
Piezomagnetic constant	$d_{\mathit{ijk}}^{\text{m}}={-\frac{{\partial }^2\overline{\psi }}{\partial E_{\mathit{ij}}\partial h_k}|}_{x_{\text{o}}}$	Coefficient of thermal stress	${\beta }_{\mathit{ij}}={-\frac{{\partial }^2\overline{\psi }}{\partial \text{Θ}\partial E_{\mathit{ij}}}|}_{x_{\text{o}}}$
Electric susceptibility	${\chi }_{\mathit{ij}}^{\text{e}}={-\frac{{\partial }^2\overline{\psi }}{\partial e_i\partial e_j}|}_{x_{\text{o}}}$	Magnetoelectric constant	${\chi }_{\mathit{ij}}^{\text{em}}={-\frac{{\partial }^2\overline{\psi }}{\partial e_i\partial h_j}|}_{x_{\text{o}}}$
Pyroelectric constant	$L_i^{\text{e}}={-\frac{{\partial }^2\overline{\psi }}{\partial e_i\partial \text{Θ}}|}_{x_{\text{o}}}$	Magnetic susceptibility	${\chi }_{\mathit{ij}}^{\text{m}}={-\frac{{\partial }^2\overline{\psi }}{\partial h_i\partial h_j}|}_{x_{\text{o}}}$
Pyromagnetic constant	$L_i^{\text{m}}={-\frac{{\partial }^2\overline{\psi }}{\partial h_i\partial \text{Θ}}|}_{x_{\text{o}}}$	Specific heat	$c={-\frac{{\partial }^2\overline{\psi }}{\partial {\text{Θ}}^2}|}_{x_{\text{o}}}$

9.7.3 Material symmetry

The symmetric infinitesimal strains and stresses, as well as the existence of a free energy function $\overline{\psi }$, lead to the following restrictions on the material constants in the constitutive equations (9.90a)–(9.90d):

$C_{\mathit{ijkl}}=C_{\mathit{jikl}}=C_{\mathit{ijlk}}=C_{\mathit{klij}},d_{\mathit{ijk}}^{\text{e}}=d_{\mathit{jik}}^{\text{e}},d_{\mathit{ijk}}^{\text{m}}=d_{\mathit{jik}}^{\text{m}},{\chi }_{\mathit{ij}}^{\text{e}}={\chi }_{\mathit{ji}}^e,{\chi }_{\mathit{ij}}^m={\chi }_{\mathit{ji}}^m,{\beta }_{\mathit{ij}}={\beta }_{\mathit{ji}}.$

These restrictions reduce the total number of unknown material constants to 169. The number of unknown material constants can be further reduced using crystal symmetry arguments. Materials that undergo one or more linear thermo-electro-mechanical processes (refer to Figure 9.2) can be classified into 32 crystallographic symmetry groups. These groups are based on rotation, reflection, and inversion symmetry of the crystal structure. A detailed description of each of these symmetry groups is provided in [63]. For magnetic materials, the concept of time inversion symmetry becomes an additional consideration, which increases the total number of possible symmetry groups from 32 to 122 (90 magnetic and 32 crystallographic symmetry groups).

Symmetry of the stress, strain, and material constant tensors allows further simplification of the constitutive equations (9.90a)–(9.90d) using Voigt notation. Voigt notation is a standard mapping, typically used to reduce the order (or rank) of symmetric tensors. The indices are mapped as follows:

$11\to 1,22\to 2,33\to 3,23\to 4,13\to 5,12\to 6.$

For example, Voigt notation simplifies the customary three-by-three matrix representation of the symmetric second-order stress tensor (refer, for instance, to (2.34)) to a single column matrix with six independent components. Similar reductions are accomplished for matrix representations of the fourth-order elasticity tensor, third-order piezoelectric and piezomagnetic coupling tensors, and second-order strain tensor. With use of this shorthand notation, the constitutive equations (9.90a)–(9.90d), for the special case of a fully coupled TEME material with hexagonal crystal symmetry (i.e., the C_6v crystallographic symmetry group), can be presented in matrix form as

$\left[\begin{array}{c}{T}_1\\ T_2\\ T_3\\ T_4\\ T_5\\ T_6\\ p_1\\ p_2\\ p_3\\ {\mu }_{\text{o}}{m}_1\\ {\mu }_{\text{o}}{m}_2\\ {\mu }_{\text{o}}{m}_3\\ \overline{\eta }\end{array}\right]=\left[\begin{array}{ccccccccccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& 0& 0& 0& -d_{13}^{\text{e}}& 0& 0& -d_{13}^{\text{m}}& -{\beta }_{11}\\ C_{12}& C_{11}& C_{13}& 0& 0& 0& 0& 0& -d_{13}^{\text{e}}& 0& 0& -d_{13}^{\text{m}}& -{\beta }_{11}\\ C_{13}& C_{13}& C_{33}& 0& 0& 0& 0& 0& -d_{33}^{\text{e}}& 0& 0& -d_{33}^{\text{m}}& -{\beta }_{33}\\ 0& 0& 0& C_{44}& 0& 0& -d_{41}^{\text{e}}& -d_{51}^{\text{e}}& 0& 0& -d_{51}^{\text{m}}& 0& 0\\ 0& 0& 0& 0& C_{44}& 0& -d_{51}^{\text{e}}& d_{41}^{\text{e}}& 0& -d_{51}^{\text{m}}& 0& 0& 0\\ 0& 0& 0& 0& 0& C_{66}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& d_{41}^{\text{e}}& d_{51}^{\text{e}}& 0& {\chi }_{11}^{\text{e}}& 0& 0& {\chi }_{11}^{\text{em}}& 0& 0& 0\\ 0& 0& 0& d_{51}^{\text{e}}& -d_{41}^{\text{e}}& 0& 0& {\chi }_{11}^{\text{e}}& 0& 0& {\chi }_{11}^{\text{em}}& 0& 0\\ d_{13}^{\text{e}}& d_{13}^{\text{e}}& d_{33}^{\text{e}}& 0& 0& 0& 0& 0& {\chi }_{33}^{\text{e}}& 0& 0& {\chi }_{33}^{\text{em}}& L_3^{\text{e}}\\ 0& 0& 0& 0& d_{51}^{\text{m}}& 0& {\chi }_{11}^{\text{em}}& 0& 0& {\chi }_{11}^{\text{m}}& 0& 0& 0\\ 0& 0& 0& d_{51}^{\text{m}}& 0& 0& 0& {\chi }_{11}^{\text{em}}& 0& 0& {\chi }_{11}^{\text{m}}& 0& 0\\ d_{13}^{\text{m}}& d_{13}^{\text{m}}& d_{33}^{\text{m}}& 0& 0& 0& 0& 0& {\chi }_{33}^{\text{em}}& 0& 0& {\chi }_{33}^{\text{m}}& L_3^{\text{m}}\\ {\beta }_{11}& {\beta }_{11}& {\beta }_{33}& 0& 0& 0& 0& 0& L_3^{\text{e}}& 0& 0& L_3^{\text{m}}& c\end{array}\right]\left[\begin{array}{c}{E}_1\\ E_2\\ E_3\\ E_4\\ E_5\\ E_6\\ e_1\\ e_2\\ e_3\\ h_1\\ h_2\\ h_3\\ \text{Θ}\end{array}\right]$

where C₆₆ = 1/2(C₁₁ − C₁₂). Clearly, crystal symmetry considerations greatly reduce the number of unknown material constants, which in turn reduces the number of experiments needed to completely characterize a material.