Magnetoelectric Coupling and Overall Properties of a Class of Multiferroic Composites

Abstract

In this chapter we present a widely useful composite model for the calculation of magnetoelectric coupling and all other properties of a two-phase multiferroic composite consisting of aligned piezomagnetic (or piezoelectric) spheroidal inclusions in a piezoelectric (or piezomagnetic) matrix. Both perfect and imperfect interface conditions are considered. Among the many features of the properties reported is the intriguing magnetoelectric coupling that signifies the $``0+0 \to 1"$ product effect of the multiferroic composite. It is also reported that, due to the piezoelectric-piezomagnetic interaction, the elastic C ₄₄ of the composite can be substantially higher than that of either of the two phases. We have used the theory to calculate the 17 independent material constants: 5 elastic, 3 piezoelectric, 3 piezomagnetic, 2 dielectric, 2 magnetic, and 2 magnetoelectric coefficients of a transversely isotropic BaTiO₃-CoFe₂O₄ composite, and show how these magneto-electro-elastic constants depend on the volume concentration, aspect ratio of inclusions, and the interface condition. We conclude by pointing out that a weak interface model is often required to capture the experimentally measured data of a bulk multiferroic composite.

Appendices

Appendix 1: The Eight Variants of the Coupled Constitutive Equations

Eight different types of thermodynamic potentials can be developed based on different choices of the (σ, ε),, (D, E), and (B, H) pairs. This leads to eight different variants of the magneto-electro-elastic constitutive equations. Following Soh and Liu (2005), we write

Type	Independent variables	Constitutive equations
1	ε, E, H	$\begin{array}{l}\sigma ={\mathbf{C}}_{E,H}\varepsilon -{\mathbf{e}}_H^{\mathrm{T}}E-{\mathbf{q}}_E^{\mathrm{T}}H\\ {}D={\mathbf{e}}_H\varepsilon +{k}_{\varepsilon, H}E+{\alpha}_{\varepsilon }H\\ {}B={\mathbf{q}}_E\varepsilon +{\boldsymbol{\upalpha}}_{\varepsilon }E+{\boldsymbol{\upmu}}_{\varepsilon, E}H\end{array}$
2	σ, D, B	$\begin{array}{l}\varepsilon ={\mathbf{S}}_{D,B}\sigma +{\mathbf{g}}_B^{\mathrm{T}}D+{\mathbf{m}}_D^{\mathrm{T}}B\\ {}E=-{\mathbf{g}}_B\sigma +{\beta}_{\sigma, B}D-{\lambda}_{\sigma }B\\ {}H=-{\mathbf{m}}_D\sigma -{\boldsymbol{\uplambda}}_{\sigma }D+{\boldsymbol{\upupsilon}}_{\sigma, D}B\end{array}$
3	ε, D, H	$\begin{array}{l}\sigma ={\mathbf{C}}_{D,H}\varepsilon -{\mathbf{h}}_H^{\mathrm{T}}D-{\mathbf{q}}_D^{\mathrm{T}}H\\ {}E=-{\mathbf{h}}_H\varepsilon +{\boldsymbol{\upbeta}}_{\varepsilon, H}D-{\boldsymbol{\upvarsigma}}_{\varepsilon }H\\ {}B={\mathbf{q}}_D\varepsilon +{\boldsymbol{\upvarsigma}}_{\varepsilon }D+{\boldsymbol{\upmu}}_{\varepsilon, D}H\end{array}$
4	σ, E, B	$\begin{array}{l}\varepsilon ={\mathbf{S}}_{E,B}\sigma +{\mathbf{d}}_B^{\mathrm{T}}E+{\mathbf{m}}_E^{\mathrm{T}}B\\ {}D={\mathbf{d}}_B\sigma +{k}_{\sigma, B}E+{\eta}_{\sigma }B\\ {}H=-{\mathbf{m}}_E\sigma -{\boldsymbol{\upeta}}_{\sigma }E+{\boldsymbol{\upupsilon}}_{\sigma, E}B\end{array}$
5	ε, E, B	$\begin{array}{l}\sigma ={\mathbf{C}}_{E,B}\varepsilon -{\mathbf{e}}_B^{\mathrm{T}}E-{\mathbf{n}}_E^{\mathrm{T}}B\\ {}D={\mathbf{e}}_B\varepsilon +{k}_{\varepsilon, B}E+{\eta}_{\varepsilon }B\\ {}H=-{\mathbf{n}}_E\varepsilon -{\boldsymbol{\upeta}}_{\varepsilon }E+{\boldsymbol{\upupsilon}}_{\varepsilon, E}B\end{array}$
6	σ, D, H	$\begin{array}{l}\varepsilon ={\mathbf{S}}_{D,H}\sigma +{\mathbf{g}}_H^{\mathrm{T}}D+{\mathbf{p}}_D^{\mathrm{T}}H\\ {}E=-{\mathbf{g}}_H\sigma +{\beta}_{\sigma, H}D-{\boldsymbol{\upvarsigma}}_{\sigma }H\\ {}B={\mathbf{p}}_D\sigma +{\boldsymbol{\upvarsigma}}_{\sigma }D+{\boldsymbol{\upmu}}_{\sigma, D}H\end{array}$
7	ε, D, B	$\begin{array}{l}\sigma ={\mathbf{C}}_{D,B}\varepsilon -{\mathbf{h}}_B^{\mathrm{T}}D-{\mathbf{n}}_D^{\mathrm{T}}B\\ {}E=-{\mathbf{h}}_B\varepsilon +{\boldsymbol{\upbeta}}_{\varepsilon, B}D-{\lambda}_{\varepsilon }B\\ {}H=-{\mathbf{n}}_D\varepsilon -{\boldsymbol{\uplambda}}_{\varepsilon }D+{\boldsymbol{\upupsilon}}_{\varepsilon, D}B\end{array}$
8	σ, E, H	$\begin{array}{l}\varepsilon ={\mathbf{S}}_{E,H}\sigma +{\mathbf{d}}_H^{\mathrm{T}}E+{\mathbf{p}}_E^{\mathrm{T}}H\\ {}D={\mathbf{d}}_H\sigma +{\kappa}_{\sigma, H}E+{\alpha}_{\sigma }H\\ {}B={\mathbf{p}}_E\sigma +{\boldsymbol{\upalpha}}_{\sigma }E+{\boldsymbol{\upmu}}_{\sigma, E}H\end{array}$

In the above constitutive equations, each subscript denotes that the corresponding tensor is measured under which kind of constant field. For instance, C _E,H means this elastic stiffness tensor is measured under constant electric and magnetic field, so it is different from C _E,B . However the coefficients of one pair are dependent upon the coefficients of another. From any given pair we can derive the rest seven pairs. They can be converted from each other, as below:

• 1. When all the independent variables are to be reversed, a direct inversion of the matrix is sufficient. For instance, given the matrix for independent variables (ε, E, H), we can find that for (σ, D, B) through

  $$ \left[\begin{array}{ccc}\hfill {\mathbf{S}}_{D,B}\hfill & \hfill {\mathbf{g}}_B^T\hfill & \hfill {\mathbf{m}}_D^T\hfill \\ {}\hfill -{\mathbf{g}}_B\hfill & \hfill {\beta}_{\sigma, B}\hfill & \hfill -{\lambda}_{\sigma}\hfill \\ {}\hfill -{\mathbf{m}}_D\hfill & \hfill -{\lambda}_{\sigma}\hfill & \hfill {\upsilon}_{\sigma, D}\hfill \end{array}\right]={\left[\begin{array}{ccc}\hfill {\mathbf{C}}_{E,H}\hfill & \hfill -{\mathbf{e}}_H^T\hfill & \hfill -{\mathbf{q}}_E^T\hfill \\ {}\hfill {\mathbf{e}}_H\hfill & \hfill {\kappa}_{\varepsilon, H}\hfill & \hfill {\alpha}_{\varepsilon}\hfill \\ {}\hfill {\mathbf{q}}_E\hfill & \hfill {\alpha}_{\varepsilon}\hfill & \hfill {\mu}_{\varepsilon, E}\hfill \end{array}\right]}^{-1}. $$

  (8.13)

• 2. But when only one or two independent variables are to be changed, a sequential conversion is needed. For example, given the matrix for (ε, E, H), we can find that for (σ, E, H) by

  $$ \left[\begin{array}{ccc}\hfill {\mathbf{S}}_{E,H}\hfill & \hfill {\mathbf{d}}_H^T\hfill & \hfill {\mathbf{p}}_E^T\hfill \\ {}\hfill {\mathbf{d}}_H\hfill & \hfill {\kappa}_{\sigma, H}\hfill & \hfill {\alpha}_{\sigma}\hfill \\ {}\hfill {\mathbf{p}}_E\hfill & \hfill {\alpha}_{\sigma}\hfill & \hfill {\mu}_{\sigma, E}\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill {\mathbf{S}}_{D,B}\hfill & \hfill {\mathbf{g}}_B^T\hfill & \hfill {\mathbf{m}}_D^T\hfill \\ {}\hfill \mathbf{0}\hfill & \hfill \mathbf{I}\hfill & \hfill \mathbf{0}\hfill \\ {}\hfill \mathbf{0}\hfill & \hfill \mathbf{0}\hfill & \hfill \mathbf{I}\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill {\mathbf{C}}_{E,H}\hfill & \hfill -{\mathbf{e}}_H^T\hfill & \hfill -{\mathbf{q}}_E^T\hfill \\ {}\hfill {\mathbf{e}}_H\hfill & \hfill {\kappa}_{\varepsilon, H}\hfill & \hfill {\alpha}_{\varepsilon}\hfill \\ {}\hfill {\mathbf{q}}_E\hfill & \hfill {\alpha}_{\varepsilon}\hfill & \hfill {\mu}_{\varepsilon, E}\hfill \end{array}\right]{\left[\begin{array}{ccc}\hfill {\mathbf{C}}_{E,H}\hfill & \hfill -{\mathbf{e}}_H^T\hfill & \hfill -{\mathbf{q}}_E^T\hfill \\ {}\hfill \mathbf{0}\hfill & \hfill \mathbf{I}\hfill & \hfill \mathbf{0}\hfill \\ {}\hfill \mathbf{0}\hfill & \hfill \mathbf{0}\hfill & \hfill \mathbf{I}\hfill \end{array}\right]}^{-1}. $$

  (8.14)

Appendix 2: The Determination of the Magneto-Electro-Elastic S-Tensor

The magneto-electro-elastic S-tensor has been studied by Li and Dunn (1998a, b), Huang et al. (1998), and several others. Here we briefly summarize the method for calculating this S-tensor with the notations used here.

First, we define a material constant “tensor” (which is not a real tensor by rigorous definition) L _iJMn for the matrix phase of multiferroic composites, with subscript $i,n=1\sim 3$ and $J,M=1\sim 5$:

$$ {\mathbf{L}}_{iJMn}=\left\{\begin{array}{l}{\mathbf{C}}_{iJMn}, J,M=1,2,3,\\ {}{\mathbf{e}}_{niJ}, J=1,2,3,M=4,\\ {}{\mathbf{q}}_{niJ}, J=1,2,3,M=5,\\ {}{\mathbf{e}}_{iMn}, J=4,M=1,2,3,\\ {}{\mathbf{q}}_{iMn}, J=5,M=1,2,3,\\ {}-{\kappa}_{in}, J=4, M=4,\\ {}-{\alpha}_{in}, J=4, M=5 \& J=5, M=4,\\ {}-{\mu}_{in}, J=5, M=5.\end{array}\right. $$

(8.15)

With L _iJMn , we introduce a $5\times 5$ matrix K _MJ ,

$$ {\mathbf{K}}_{MJ}={\mathbf{L}}_{iJMn}{x}_i{x}_n, $$

(8.16)

where ${x}_i={\left[{x}_1, {x}_2, {x}_3\right]}^{\mathrm{T}}$. Then we define another pseudo tensor J _inMJ ,

$$ {\mathbf{J}}_{inMJ}\left({x}_1, {x}_2, {x}_3\right)={x}_i{x}_n{\mathbf{K}}_{MJ}^{-1}, $$

(8.17)

so that it is a function of x ₁, x ₂, and x ₃. Next we integrate J _inMJ over the volume of an ellipsoid inclusion $\Omega : {x}_1^2/{a}_1^2+{x}_2^2/{a}_2^2+{x}_3^2/{a}_3^2\le 1$. When this inclusion is spheroidal and symmetric about three-direction, it satisfies ${a}_1={a}_2$ and $\alpha ={\alpha}_3/{a}_1$, where α is the aspect ratio of inclusions used in the text. Hence the volume integral of J _inMJ can be written as

$$ \begin{array}{c}{\mathbf{H}}_{inMJ}={\displaystyle {\int}_{\Omega}{\mathbf{J}}_{inMJ}\left({x}_1/{a}_1, {x}_2/{a}_2, {x}_3/{a}_3\right)}\mathrm{d}V={\displaystyle {\int}_{\Omega}{\mathbf{J}}_{inMJ}\left({x}_1, {x}_2, {x}_3/\alpha \right)}\mathrm{d}V,\\ {}={\displaystyle \underset{-1}{\overset{1}{\int }}\mathrm{d}\tau {\displaystyle \underset{0}{\overset{2\pi }{\int }}{\mathbf{J}}_{inMJ}\left({y}_1, {y}_2,\kern0.5em {y}_3/\alpha \right)\mathrm{d}\theta },}\end{array} $$

(8.18)

where the second equality is based on the fact that J _inMJ is a homogeneous function of order zero; thus multiplying all the variables by a ₁ will not affect the integration. The third equality is given by applying a change of variables from x ₁, x ₂, and x ₃ to

$$ {y}_1=\sqrt{1-{\tau}^2} \cos \theta, {y}_1=\sqrt{1-{\tau}^2} \sin \theta, {y}_3=\tau, $$

(8.19)

with $\tau \in \left[-1, 1\right]$ and $\theta \in \left[0, 2\pi \right]$. Finally the S-tensor is determined by

$$ {\mathbf{S}}_{MnAb}=\left\{\begin{array}{l}\frac{1}{8\pi }{\mathbf{L}}_{iJAb}\left({\mathbf{H}}_{in MJ}+{\mathbf{H}}_{iMnJ}\right), M=1\sim 3,\\ {}\frac{1}{4\pi }{\mathbf{L}}_{iJAb}{\mathbf{H}}_{in4J}, M=4,\\ {}\frac{1}{4\pi }{\mathbf{L}}_{iJAb}{\mathbf{H}}_{in5J}, M=5.\end{array}\right. $$

(8.20)

Still S _MnAb is a pseudo tensor. For the convenience of calculation we now convert it into a $12\times 12$ matrix S by the Voigt-Nye contract notations.

For a general spheroid, evaluation of the integral in Eq. (8.18) can be carried out by the Gaussian quadrature method. It turns the definite integral into a weighted sum of function values at specified points within the domain of integration. Then it can be rewritten as

$$ {\mathbf{H}}_{inMJ}={\displaystyle \underset{-1}{\overset{1}{\int }}\mathrm{d}\tau {\displaystyle \underset{0}{\overset{2\pi }{\int }}{\mathbf{J}}_{inMJ}\left({y}_1, {y}_2, {y}_3/\alpha \right)\mathrm{d}\theta }}\approx {\displaystyle \sum_{i=1}^n{w}_if\left({\tau}_i\right)}, $$

(8.21)

where $f\left(\tau \right)={\displaystyle \underset{0}{\overset{2\pi }{\int }}{\mathbf{J}}_{inMJ}\left({y}_1, {y}_2, {y}_3/\alpha \right)\mathrm{d}\theta }$, n is the total number of specified points (usually $n=20$ or more will provide enough accuracy), and w _i is the weight at each specified point which can be constructed by different kinds of weight function. Up to this point the calculation of S-tensor is completed. In general, S-tensor is not symmetric, while $\mathbf{S}{\mathbf{L}}_0^{-1}$ must always be symmetric, as L ₀ (the material constant for matrix phase) written in Eq. (8.8) is symmetric. This can be used as a criterion to check if the calculated components of S-tensor are correct.

Explicit forms of the S-tensor are available for 1-3 fibrous composite $\left(\alpha \to \infty \right)$ and 2-2 multilayered structure $\left(\alpha \to 0\right)$. These two connectivities represent the most widely used microstructures and are frequently adopted in experiments. Due to the transverse isotropy of the phase, its derivation is quite involved. Keeping in mind that direction 3 is symmetric and plane 1-2 isotropic, its components can be summarized as below:

$$ \begin{array}{ccc}\hfill \mathrm{Type}\hfill & \hfill \mathrm{Piezoelectric}\ \mathrm{matrix}\hfill & \hfill \mathrm{Piezomagnetic}\ \mathrm{matrix}\hfill \\ {}\hfill \hfill & \hfill {S}_{11}={S}_{22}=\frac{5{C}_{11}+{C}_{12}}{8{C}_{11}},\hfill & \hfill {S}_{11}={S}_{22}=\frac{5{C}_{11}+{C}_{12}}{8{C}_{11}},\hfill \\ {}\hfill \hfill & \hfill {S}_{12}={S}_{21}=-\frac{1}{8}+\frac{3{C}_{12}}{8{C}_{11}},\hfill & \hfill {S}_{12}={S}_{21}=-\frac{1}{8}+\frac{3{C}_{12}}{8{C}_{11}},\hfill \\ {}\hfill 1\hbox{-} 3\hfill & \hfill {S}_{13}={S}_{23}=\frac{C_{13}}{2{C}_{11}}, {S}_{19}={S}_{29}=\frac{e_{31}}{2{C}_{11}},\hfill & \hfill {S}_{13}={S}_{23}=\frac{C_{13}}{2{C}_{11}}, {S}_{1,12}={S}_{2,12}=\frac{q_{31}}{2{C}_{11}},\hfill \\ {}\hfill \hfill & \hfill {S}_{66}=\frac{3}{4}-\frac{C_{12}}{4{C}_{11}},\hfill & \hfill {S}_{66}=\frac{3}{4}-\frac{C_{12}}{4{C}_{11}},\hfill \\ {}\hfill \hfill & \hfill {S}_{44}={S}_{55}={S}_{77}={S}_{88}={S}_{10,10}={S}_{11,11}=\frac{1}{2},\hfill & \hfill {S}_{44}={S}_{55}={S}_{77}={S}_{88}={S}_{10,10}={S}_{11,11}=\frac{1}{2},\hfill \\ {}\hfill \hfill & \hfill \mathrm{all} \mathrm{the} \mathrm{other} \mathrm{components}\kern0.85em \mathrm{are}\kern0.85em \mathrm{zero}.\hfill & \hfill \mathrm{all}\kern0.85em \mathrm{the}\kern0.85em \mathrm{other}\kern0.85em \mathrm{components}\kern0.85em \mathrm{are}\kern0.85em \mathrm{zero}.\hfill \\ {}\hfill \hfill & \hfill {S}_{31}={S}_{32}=\frac{e_{31}{e}_{33}+{C}_{13}{\kappa}_{33}}{{e_{33}}^2+{C}_{33}{\kappa}_{33}},\hfill & \hfill {S}_{31}={S}_{32}=\frac{q_{31}{q}_{33}+{C}_{13}{\mu}_{33}}{{q_{33}}^2+{C}_{33}{\mu}_{33}},\hfill \\ {}\hfill 2\hbox{-} 2\hfill & \hfill {S}_{57}={S}_{48}=\frac{e_{15}}{C_{44}},\hfill & \hfill {S}_{5,10}={S}_{4,11}=\frac{q_{15}}{C_{44}},\hfill \\ {}\hfill \hfill & \hfill {S}_{91}={S}_{92}=\frac{-{C}_{33}{e}_{31}+{C}_{13}{e}_{33}}{{e_{33}}^2+{C}_{33}{\kappa}_{33}},\hfill & \hfill {S}_{12,1}={S}_{12,2}=\frac{-{C}_{33}{q}_{31}+{C}_{13}{q}_{33}}{{q_{33}}^2+{C}_{33}{\mu}_{33}},\hfill \\ {}\hfill \hfill & \hfill {S}_{33}={S}_{44}={S}_{55}={S}_{99}={S}_{12,12}=1,\hfill & \hfill {S}_{33}={S}_{44}={S}_{55}={S}_{99}={S}_{12,12}=1,\hfill \\ {}\hfill \hfill & \hfill \mathrm{all}\kern0.85em \mathrm{the}\kern0.85em \mathrm{other}\kern0.85em \mathrm{components}\kern0.85em \mathrm{are}\kern0.85em \mathrm{zero}.\hfill & \hfill \mathrm{all}\kern0.85em \mathrm{the}\kern0.85em \mathrm{other}\kern0.85em \mathrm{components}\kern0.85em \mathrm{are}\kern0.85em \mathrm{zero}.\hfill \end{array} $$

(8.22)

For the S-tensor of the interphase, which is isotropic, its components are

$$ {S}_{11}^{\mathrm{int}}={S}_{22}^{\mathrm{int}}=\frac{3}{8\left(1-{\nu}_0\right)}\frac{\alpha^2}{\alpha^2-1}+\frac{1}{4\left(1-{\nu}_0\right)}\left[1-2{\nu}_0-\frac{9}{4\left({\alpha}^2-1\right)}\right]g\left(\mathrm{g}\upalpha \right), $$

$$ {S}_{33}^{\mathrm{int}}=\frac{1}{2\left(1-{\nu}_0\right)}\left\{1-2{\nu}_0+\frac{3{\alpha}^2-1}{\alpha^2-1}-\left[1-2{\nu}_0+\frac{3{\alpha}^2}{\alpha^2-1}\right]g\left(\alpha \right)\right\}, $$

$$ {S}_{12}^{\mathrm{int}}={S}_{21}^{\mathrm{int}}=\frac{1}{4\left(1-{\nu}_0\right)}\left\{\frac{\alpha^2}{2\left({\alpha}^2-1\right)}-\left[1-2{\nu}_0+\frac{3}{4\left({\alpha}^2-1\right)}\right]g\left(\alpha \right)\right\}, $$

$$ {S}_{13}^{\mathrm{int}}={S}_{23}^{\mathrm{int}}=-\frac{1}{2\left(1-{\nu}_0\right)}\frac{\alpha^2}{\alpha^2-1}+\frac{1}{4\left(1-{\nu}_0\right)}\left\{\frac{3{\alpha}^2}{\alpha^2-1}-\left(1-2{\nu}_0\right)\right\}g\left(\alpha \right), $$

$$ {S}_{31}^{\mathrm{int}}={S}_{32}^{\mathrm{int}}=-\frac{1}{2\left(1-{\nu}_0\right)}\left[1-2{\nu}_0+\frac{1}{\alpha^2-1}\right]+\frac{1}{2\left(1-{\nu}_0\right)}\left[1-2{\nu}_0+\frac{3}{2\left({\alpha}^2-1\right)}\right]g\left(\alpha \right), $$

(8.23)

$$ {S}_{44}^{\mathrm{int}}={S}_{55}^{\mathrm{int}}=\frac{1}{2\left(1-{\nu}_0\right)}\left\{1-2{\nu}_0-\frac{\alpha^2+1}{\alpha^2-1}-\frac{1}{2}\left[1-2{\nu}_0-\frac{3\left({\alpha}^2+1\right)}{\alpha^2-1}\right]g\left(\alpha \right)\right\}, $$

$$ {S}_{66}^{\mathrm{int}}=\frac{1}{2\left(1-{\nu}_0\right)}\left\{\frac{\alpha^2}{2\left({\alpha}^2-1\right)}+\left[1-2{\nu}_0-\frac{3}{4\left({\alpha}^2-1\right)}\right]g\left(\alpha \right)\right\}, $$

$$ {S}_{77}^{\mathrm{int}}={S}_{88}^{\mathrm{int}}={S}_{10,10}^{\mathrm{int}}={S}_{11,11}^{\mathrm{int}}=\frac{1}{2}g\left(\alpha \right), $$

$$ {S}_{99}^{\mathrm{int}}={S}_{12,12}^{\mathrm{int}}=1-g\left(\alpha \right), $$

and all the other components are zero. Here ${\nu}_0=\left({C}_{11}^{\mathrm{int}}-2{C}_{44}^{\mathrm{int}}\right)/2\left({C}_{11}^{\mathrm{int}}-{C}_{44}^{\mathrm{int}}\right)$ is Poisson’s ratio of the interphase, and function g(α) depends on the aspect ratio, as

$$ g\left(\alpha \right)=\left\{\begin{array}{l}\frac{\alpha }{{\left(1-{\alpha}^2\right)}^{\frac{3}{2}}}\left[{ \cos}^{-1}\alpha -\alpha {\left(1-{\alpha}^2\right)}^{\frac{1}{2}}\right],\kern1.5em \alpha <1,\kern0.5em \\ {}\frac{\alpha }{{\left({\alpha}^2-1\right)}^{\frac{3}{2}}}\left[\alpha {\left({\alpha}^2-1\right)}^{\frac{1}{2}}-{ \cosh}^{-1}\alpha \right],\kern1.5em \alpha \ge 1.\end{array}\right. $$

(8.24)

This set of S-tensor can reduce to the commonly used S-tensor for the uncoupled elastic, dielectric, magnetic, electrical, or thermal conduction problem.

Finally the matrix form for the isotropic magneto-electro-elastic tensor of the interphase L _int is

$$ {\mathbf{L}}_{\mathrm{int}}=\left[\begin{array}{cccccccccccc}\hfill {C}_{11}^{\mathrm{int}}\hfill & \hfill {C}_{12}^{\mathrm{int}}\hfill & \hfill {C}_{12}^{\mathrm{int}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {C}_{12}^{\mathrm{int}}\hfill & \hfill {C}_{11}^{\mathrm{int}}\hfill & \hfill {C}_{12}^{\mathrm{int}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {C}_{12}^{\mathrm{int}}\hfill & \hfill {C}_{12}^{\mathrm{int}}\hfill & \hfill {C}_{11}^{\mathrm{int}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {C}_{44}^{\mathrm{int}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {C}_{44}^{\mathrm{int}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {C}_{44}^{\mathrm{int}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\kappa}^{\mathrm{int}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\kappa}^{\mathrm{int}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\kappa}^{\mathrm{int}}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mu}^{\mathrm{int}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mu}^{\mathrm{int}}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{\mu}^{\mathrm{int}}\hfill \end{array}\right], $$

(8.25)

where ${C}_{12}^{\mathrm{int}}={C}_{11}^{\mathrm{int}}-2{C}_{44}^{\mathrm{int}}$.

Appendix 3: Explicit Results for α ₃₃ and α ₁₁ of the 1-3 and 2-2 Multiferroic Composites with a Perfect and an Imperfect Interface

For fibrous composites and multilayers, explicit formulae for the components of S-tensor are listed in the Appendix 2. With it and the theory given in Eq. (8.8) for the perfect interface, and Eq. (8.9) together with Eq. (8.8) for the imperfect interface, we have obtained the explicit expressions for the magnetoelectric coupling coefficients, α ₃₃ and α ₁₁. As the 1-3 and 2-2 composites are widely used, we give the results here for ready reference.

In reading the following expressions, care must be exercised that superscript “e” always refers to the properties of BaTiO₃ regardless whether BTO exists as the matrix or inclusions, and superscript “m” always refers to the properties of CoFe₂O₄, also regardless whether it exists as inclusion or matrix. Superscript “int” on the other hand refers to the properties of the interface. In addition, for 1-3 composites, c ₁ and c ₀ denote the volume concentrations of the inclusions and matrix, respectively, which could be CFO or BTO. c _int denotes the volume concentration of interface in the thinly coated inclusion. While for 2-2 composites, c ₀, c ₁, and c _int denote the volume concentrations of BTO, CFO, and interface in the whole composite, so that ${c}_0+{c}_1+{c}_{\mathrm{int}}=1$.

8.3.1 The 1-3 Multiferroic Fibrous Composites with a Perfect Interface

With CoFe₂O₄ as inclusions and BaTiO₃ as the matrix, we find

$$ {\alpha}_{33}=-\frac{2{c}_0{c}_1{e}_{31}^{(e)}{q}_{31}^{(m)}}{C_{11}^{(e)}+{C}_{11}^{(m)}-{c}_0\left({C}_{12}^{(e)}-{C}_{12}^{(m)}\right)+{c}_1\left({C}_{11}^{(e)}-{C}_{11}^{(m)}\right)}, $$

(8.26)

$$ {\alpha}_{11}=-\frac{4{c}_0{c}_1{e}_{12}^{(e)}{q}_{15}^{(m)}{\kappa}_{11}^{(m)}{\mu}_{11}^{(e)}}{D_{\alpha 11}^{\mathrm{BTO}}}, $$

(8.27)

where denominator D ^BTO _α11 is

$$ \begin{array}{l}{D}_{\alpha 11}^{\mathrm{BTO}}={c}_0^2{q}_{15}^{(m)}{}^2\left[{\kappa}_{11}^{(e)}+{\kappa}_{11}^{(m)}+{c}_1\left({\kappa}_{11}^{(e)}-{\kappa}_{11}^{(m)}\right)\right]+\left\{{\left(1+{c}_1\right)}^2{e}_{15}^{(e)}2\right.\\ {}\kern3.4em +\left[{C}_{44}^{(e)}+{C}_{44}^{(m)}+{c}_1\left({C}_{44}^{(e)}-{C}_{44}^{(m)}\right)\right]\\ {}\kern3.5em \left.\left[{\kappa}_{11}^{(e)}+{\kappa}_{11}^{(m)}+{c}_1\left({\kappa}_{11}^{(e)}-{\kappa}_{11}^{(m)}\right)\right]\right\}\left[{\mu}_{11}^{(e)}+{\mu}_{11}^{(m)}+{c}_1\left({\mu}_{11}^{(e)}-{\mu}_{11}^{(m)}\right)\right].\end{array} $$

(8.28)

On the other hand with BaTiO₃ as inclusion phase, we have

$$ {\alpha}_{33}=-\frac{2{c}_0{c}_1{e}_{31}^{(e)}{q}_{31}^{(m)}}{C_{11}^{(e)}+{C}_{11}^{(m)}+{c}_0\left({C}_{12}^{(e)}-{C}_{12}^{(m)}\right)-{c}_1\left({C}_{11}^{(e)}-{C}_{11}^{(m)}\right)}, $$

(8.29)

$$ {\alpha}_{11}=\frac{4{c}_0{c}_1{e}_{15}^{(e)}{q}_{15}^{(m)}{\kappa}_{11}^{(m)}{\mu}_{11}^{(e)}}{D_{\alpha 11}^{\mathrm{CFO}}}, $$

(8.30)

where

$$ \begin{array}{l}{D}_{\alpha 11}^{\mathrm{CFO}}={c}_0^2{e}_{15}^{(e)}{}^2\left[{\mu}_{11}^{(m)}+{\mu}_{11}^{(e)}+{c}_1\left({\mu}_{11}^{(m)}-{\mu}_{11}^{(e)}\right)\right]+\left\{1,{\left(+,{c}_1\right)}^2\right.{q}_{15}^{(m)}2\\ {}\kern3.2em +\left[{C}_{44}^{(m)}+{C}_{44}^{(e)}+{c}_1\left({C}_{44}^{(m)}-{C}_{44}^{(e)}\right)\right]\\ {}\kern3.2em \left.\left[{\mu}_{11}^{(m)}+{\mu}_{11}^{(e)}+{c}_1\left({\kappa}_{11}^{(m)}-{\kappa}_{11}^{(e)}\right)\right]\right\}\left[{\kappa}_{11}^{(m)}+{\kappa}_{11}^{(e)}+{c}_1\left({\kappa}_{11}^{(m)}-{\kappa}_{11}^{(e)}\right)\right].\end{array} $$

(8.31)

8.3.2 The 1-3 Multiferroic Fibrous Composites with an Imperfect Interface

In order to obtain the effective magnetoelectric coefficients of 1-3 multiferroic composites with imperfect interface, we first obtain the relevant properties of coated inclusion from Eq. (8.9). This in turn can be used for the properties of the inclusion phase in Eq. (8.8) for the overall composite.

For the coated CoFe₂O₄ inclusion—with a prime added to superscript m—we obtain from Eq. (8.9)

$$ \begin{array}{l}{C}_{11}^{\left({m}^{\prime}\right)}=\frac{C_{11}^{\left(\operatorname{int}\right)}\left({C}_{11}^{(m)}+{C}_{12}^{(m)}+2{C}_{44}^{\left(\operatorname{int}\right)}\right)}{2{C}_{11}^{\left(\operatorname{int}\right)}+{c}_{\mathrm{int}}\left({C}_{11}^{(m)}+{C}_{12}^{(m)}-2{C}_{11}^{\left(\operatorname{int}\right)}+2{C}_{44}^{\left(\operatorname{int}\right)}\right)}\\ {}+\frac{2\left(1-{c}_{\mathrm{int}}\right){C}_{11}^{\left(\operatorname{int}\right)}{C}_{44}^{\left(\operatorname{int}\right)}\left({C}_{11}^{(m)}-{C}_{12}^{(m)}-2{C}_{44}^{\left(\operatorname{int}\right)}\right)}{4{C}_{11}^{\left(\operatorname{int}\right)}{C}_{44}^{\left(\operatorname{int}\right)}+{c}_{\mathrm{int}}\left({C}_{11}^{\left(\operatorname{int}\right)}+{C}_{44}^{\left(\operatorname{int}\right)}\right)\;\left({C}_{11}^{(m)}-{C}_{12}^{(m)}-2{C}_{44}^{\left(\operatorname{int}\right)}\right)},\end{array} $$

(8.32)

$$ \begin{array}{l}{C}_{12}^{\left({m}^{\prime}\right)}=\frac{C_{11}^{\left(\operatorname{int}\right)}\left({C}_{11}^{(m)}+{C}_{12}^{(m)}+2{C}_{44}^{\left(\operatorname{int}\right)}\right)}{2{C}_{11}^{\left(\operatorname{int}\right)}+{c}_{\mathrm{int}}\left({C}_{11}^{(m)}+{C}_{12}^{(m)}-2{C}_{11}^{\left(\operatorname{int}\right)}+2{C}_{44}^{\left(\operatorname{int}\right)}\right)}\\ {}-\frac{2{C}_{44}^{\left(\operatorname{int}\right)}\left[{C}_{11}^{\left(\operatorname{int}\right)}\left({C}_{11}^{(m)}-{C}_{12}^{(m)}+2{C}_{44}^{\left(\operatorname{int}\right)}\right)+{c}_{\mathrm{int}}{C}_{44}^{\left(\operatorname{int}\right)}\left({C}_{11}^{(m)}-{C}_{12}^{(m)}-2{C}_{44}^{\left(\operatorname{int}\right)}\right)\right]}{4{C}_{11}^{\left(\operatorname{int}\right)}{C}_{44}^{\left(\operatorname{int}\right)}+{c}_{\mathrm{int}}\left({C}_{11}^{\left(\operatorname{int}\right)}+{C}_{44}^{\left(\operatorname{int}\right)}\right)\kern0.5em \left({C}_{11}^{(m)}-{C}_{12}^{(m)}-2{C}_{44}^{\left(\operatorname{int}\right)}\right)},\end{array} $$

(8.33)

$$ \begin{array}{l}{C}_{44}^{\left({m}^{\prime}\right)}={C}_{44}^{\left(\operatorname{int}\right)}\times \\ {}\frac{4{C}_{44}^{(m)}{\mu}^{\left(\operatorname{int}\right)}+2{c}_{\mathrm{int}}\left[{q}_{15}^{(m)}{}^2+{C}_{44}^{(m)}\left({\mu}_{11}^{(m)}-{\mu}^{\left(\operatorname{int}\right)}\right)-{\mu}^{\mathrm{int}}\left({C}_{44}^{(m)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\right]-{c_{\mathrm{int}}}^2\left[{q}_{15}^{(m)}{}^2+\left({C}_{44}^{(m)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\left({\mu}_{11}^{(m)}-{\mu}^{\left(\operatorname{int}\right)}\right)\right]}{4{C}_{44}^{\left(\operatorname{int}\right)}{\mu}^{\left(\operatorname{int}\right)}+2{c}_{\mathrm{int}}\left[{C}_{44}^{\left(\operatorname{int}\right)}\left({\mu}_{11}^{(m)}-{\mu}^{\left(\operatorname{int}\right)}\right)+{\mu}^{\left(\operatorname{int}\right)}\left({C}_{44}^{(m)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\right]+{c_{\mathrm{int}}}^2\left[{q}_{15}^{(m)}{}^2+\left({C}_{44}^{(m)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\left({\mu}_{11}^{(m)}-{\mu}^{\left(\operatorname{int}\right)}\right)\right]},\end{array} $$

(8.34)

$$ {q}_{31}^{\left({m}^{\prime}\right)}={\kappa}^{\left(\operatorname{int}\right)}\frac{2{\kappa}_{11}^{(m)}-{c}_{\mathrm{int}}\left({\kappa}_{11}^{(m)}-{\kappa}^{\left(\operatorname{int}\right)}\right)}{2{\kappa}^{\left(\operatorname{int}\right)}+{c}_{\mathrm{int}}\left({\kappa}_{11}^{(m)}-{\kappa}^{\left(\operatorname{int}\right)}\right)}, $$

(8.35)

$$ {q}_{15}^{\left({m}^{\prime}\right)}=\frac{4\left(1-{c}_{\mathrm{int}}\right){q}_{15}^{(m)}{C}_{44}^{\left(\operatorname{int}\right)}{\mu}^{\left(\operatorname{int}\right)}}{4{C}_{44}^{\left(\operatorname{int}\right)}{\mu}^{\left(\operatorname{int}\right)}+2{c}_{\mathrm{int}}\left[{C}_{44}^{\left(\operatorname{int}\right)}\left({\mu}_{11}^{(m)}-{\mu}^{\left(\operatorname{int}\right)}\right)+{\mu}^{\left(\operatorname{int}\right)}\left({C}_{44}^{(m)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\right]+{c_{\mathrm{int}}}^2\left[{q}_{15}^{(m)}{}^2+\left({C}_{44}^{(m)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\left({\mu}_{11}^{(m)}-{\mu}^{\left(\operatorname{int}\right)}\right)\right]}, $$

(8.36)

$$ {\kappa}_{11}^{\left({m}^{\prime}\right)}={\kappa}^{\left(\operatorname{int}\right)}\frac{2{\kappa}_{11}^{(m)}-{c}_{\mathrm{int}}\left({\kappa}_{11}^{(m)}-{\kappa}^{\left(\operatorname{int}\right)}\right)}{2{\kappa}^{\left(\operatorname{int}\right)}+{c}_{\mathrm{int}}\left({\kappa}_{11}^{(m)}-{\kappa}^{\left(\operatorname{int}\right)}\right)}, $$

(8.37)

$$ \begin{array}{l}{\mu}_{11}^{\left({m}^{\prime}\right)}={\mu}^{\left(\operatorname{int}\right)}\times \\ {}\frac{4{\mu}_{11}^{(m)}{C}_{44}^{\left(\operatorname{int}\right)}+2{c}_{\mathrm{int}}\left[{q}_{15}^{(m)}{}^2+{\mu}_{11}^{(m)}\left({C}_{44}^{(m)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)-{C}_{44}^{\left(\operatorname{int}\right)}\left({\mu}_{11}^{(m)}-{\mu}^{\left(\operatorname{int}\right)}\right)\right]-{c_{\mathrm{int}}}^2\left[{q}_{15}^{(m)}{}^2+\left({C}_{44}^{(m)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\left({\mu}_{11}^{(m)}-{\mu}^{\left(\operatorname{int}\right)}\right)\right]}{4{\mu}^{\left(\operatorname{int}\right)}{C}_{44}^{\left(\operatorname{int}\right)}+2{c}_{\mathrm{int}}\left[{\mu}^{\left(\operatorname{int}\right)}\left({C}_{44}^{(m)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)+{C}_{44}^{\left(\operatorname{int}\right)}\left({\mu}_{11}^{(m)}-{\mu}^{\left(\operatorname{int}\right)}\right)\right]+{c_{\mathrm{int}}}^2\left[{q}_{15}^{(m)}{}^2+\left({C}_{44}^{(m)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\left({\mu}_{11}^{(m)}-{\mu}^{\left(\operatorname{int}\right)}\right)\right]}.\end{array} $$

(8.38)

This set can be used to replace the properties of CFO in Eqs. (8.25)–(8.27) to obtain α ₃₃ and α ₁₁ of the CFO-in-BTO composite with an imperfect interface.

Likewise for the coated BaTiO₃ inclusion—also with a prime added to e—we have

$$ \begin{array}{l}{C}_{11}^{\left({e}^{\prime}\right)}=\frac{c_{11}^{\left(\operatorname{int}\right)}\left({c}_{11}^{(e)}+{c}_{12}^{(e)}+2{C}_{44}^{\left(\operatorname{int}\right)}\right)}{2{C}_{11}^{\left(\operatorname{int}\right)}+{c}_{\mathrm{int}}\left({C}_{11}^{(e)}+{C}_{12}^{(e)}-2{C}_{11}^{\left(\operatorname{int}\right)}+2{C}_{44}^{\left(\operatorname{int}\right)}\right)}\\ {}\kern3.2em +\frac{2\left(1-{c}_{\mathrm{int}}\right){C}_{11}^{\left(\operatorname{int}\right)}{C}_{44}^{\left(\operatorname{int}\right)}\left({C}_{11}^{(e)}-{C}_{12}^{(e)}-2{C}_{44}^{\left(\operatorname{int}\right)}\right)}{4{C}_{11}^{\left(\operatorname{int}\right)}\;{C}_{44}^{\left(\operatorname{int}\right)}+{c}_{\mathrm{int}}\left({C}_{11}^{\left(\operatorname{int}\right)}+{C}_{44}^{\left(\operatorname{int}\right)}\right)\left({C}_{11}^{(e)}-{C}_{12}^{(e)}-2{C}_{44}^{\left(\operatorname{int}\right)}\right)},\end{array} $$

(8.39)

$$ \begin{array}{l}{C}_{12}^{\left({e}^{\prime}\right)}=\frac{c_{11}^{\left(\operatorname{int}\right)}\left({c}_{11}^{(e)}+{c}_{12}^{(e)}+2{C}_{44}^{\left(\operatorname{int}\right)}\right)}{2{C}_{11}^{\left(\operatorname{int}\right)}+{c}_{\mathrm{int}}\left({C}_{11}^{(e)}+{C}_{12}^{(e)}-2{C}_{11}^{\left(\operatorname{int}\right)}+2{C}_{44}^{\left(\operatorname{int}\right)}\right)}\\ {}-\frac{2{C}_{44}^{\left(\operatorname{int}\right)}\left[{C}_{11}^{\left(\operatorname{int}\right)}\left({C}_{11}^{(e)}-{C}_{12}^{(e)}+2{C}_{44}^{\left(\operatorname{int}\right)}\right)+{c}_{\mathrm{int}}{C}_{44}^{\left(\operatorname{int}\right)}\left({C}_{11}^{(e)}-{C}_{12}^{(e)}-2{C}_{44}^{\left(\operatorname{int}\right)}\right)\right]}{4{C}_{11}^{\left(\operatorname{int}\right)}\;{C}_{44}^{\left(\operatorname{int}\right)}+{c}_{\mathrm{int}}\left({C}_{11}^{\left(\operatorname{int}\right)}+{C}_{44}^{\left(\operatorname{int}\right)}\right)\left({C}_{11}^{(e)}-{C}_{12}^{(e)}-2{C}_{44}^{\left(\operatorname{int}\right)}\right)},\end{array} $$

(8.40)

$$ \begin{array}{l}{C}_{44}^{\left({e}^{\prime}\right)}={C}_{44}^{\left(\operatorname{int}\right)}\\ {}\kern3em \times \frac{4{C}_{44}^{(e)}{\kappa}^{\left(\operatorname{int}\right)}+2{c}_{\mathrm{int}}\left[{e}_{15}^{(e)}{}^2+{C}_{44}^{(e)}\left({\kappa}_{11}^{(e)}-{\kappa}^{\left(\operatorname{int}\right)}\right)-{\kappa}^{\mathrm{int}}\left({C}_{44}^{(e)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\right]-{c_{\mathrm{int}}}^2\left[{e}_{15}^{(e)}{}^2+\left({C}_{44}^{(e)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\left({\kappa}_{11}^{(e)}-{\kappa}^{\left(\operatorname{int}\right)}\right)\right]}{4{C}_{44}^{\left(\operatorname{int}\right)}{\kappa}^{\left(\operatorname{int}\right)}+2{c}_{\mathrm{int}}\left[{C}_{44}^{\left(\operatorname{int}\right)}\left({\kappa}_{11}^{(e)}-{\kappa}^{\left(\operatorname{int}\right)}\right)+{\kappa}^{\left(\operatorname{int}\right)}\left({C}_{44}^{(e)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\right]+{c_{\mathrm{int}}}^2\left[{e}_{15}^{(e)}{}^2+\left({C}_{44}^{(e)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\left({\kappa}_{11}^{(e)}-{\kappa}^{\left(\operatorname{int}\right)}\right)\right]},\end{array} $$

(8.41)

$$ {e}_{31}^{\left({e}^{\prime}\right)}=\frac{2\left(1-{c}_{\mathrm{int}}\right){e}_{31}^{(e)}{C}_{11}^{\left(\operatorname{int}\right)}}{c_{\mathrm{int}}\left({C}_{11}^{(e)}+{C}_{12}^{(e)}-2{C}_{11}^{\left(\operatorname{int}\right)}+2{C}_{44}^{\left(\operatorname{int}\right)}\right)+2{C}_{11}^{\left(\operatorname{int}\right)}}, $$

(8.42)

$$ \begin{array}{l}{e}_{15}^{\left({e}^{\prime}\right)}\\ {}\kern1em =\frac{4\left(1-{c}_{\mathrm{int}}\right){e}_{15}^{(e)}{C}_{44}^{\left(\operatorname{int}\right)}{\kappa}^{\left(\operatorname{int}\right)}}{4{C}_{44}^{\left(\operatorname{int}\right)}{\kappa}^{\left(\operatorname{int}\right)}+2{c}_{\mathrm{int}}\left[{C}_{44}^{\left(\operatorname{int}\right)}\left({\kappa}_{11}^{(e)}-{\kappa}^{\left(\operatorname{int}\right)}\right)+{\kappa}^{\left(\operatorname{int}\right)}\left({C}_{44}^{(e)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\right]+{c_{\mathrm{int}}}^2\left[{e}_{15}^{(e)}{}^2+\left({C}_{44}^{(e)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\left({\kappa}_{11}^{(e)}-{\kappa}^{\left(\operatorname{int}\right)}\right)\right]},\end{array} $$

(8.43)

$$ \begin{array}{l}{\kappa}_{11}^{\left({e}^{\prime}\right)}={\kappa}^{\left(\operatorname{int}\right)}\\ {}\kern3em \times \frac{4{\kappa}_{11}^{(e)}{C}_{44}^{\left(\operatorname{int}\right)}+2{c}_{\mathrm{int}}\left[{e}_{15}^{(e)}{}^2+{\kappa}_{11}^{(e)}\left({C}_{44}^{(e)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)-{C}_{44}^{\left(\operatorname{int}\right)}\left({\kappa}_{11}^{(e)}-{\kappa}^{\left(\operatorname{int}\right)}\right)\right]-{c_{\mathrm{int}}}^2\left[{e}_{15}^{(e)}{}^2+\left({C}_{44}^{(e)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\left({\kappa}_{11}^{(e)}-{\kappa}^{\left(\operatorname{int}\right)}\right)\right]}{4{\kappa}^{\left(\operatorname{int}\right)}{C}_{44}^{\left(\operatorname{int}\right)}+2{c}_{\mathrm{int}}\left[{\kappa}^{\left(\operatorname{int}\right)}\left({C}_{44}^{(e)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)+{C}_{44}^{\left(\operatorname{int}\right)}\left({\kappa}_{11}^{(e)}-{\kappa}^{\left(\operatorname{int}\right)}\right)\right]+{c_{\mathrm{int}}}^2\left[{e}_{15}^{(e)}{}^2+\left({C}_{44}^{(e)}-{C}_{44}^{\left(\operatorname{int}\right)}\right)\left({\kappa}_{11}^{(e)}-{\kappa}^{\left(\operatorname{int}\right)}\right)\right]},\end{array} $$

(8.44)

$$ {\mu}_{11}^{\left({e}^{\prime}\right)}={\mu}^{\left(\operatorname{int}\right)}\frac{2{\mu}_{11}^{(e)}-{c}_{\mathrm{int}}\left({\mu}_{11}^{(e)}-{\mu}^{\left(\operatorname{int}\right)}\right)}{2{\mu}^{\left(\operatorname{int}\right)}+{c}_{\mathrm{int}}\left({\mu}_{11}^{(e)}-{\mu}^{\left(\operatorname{int}\right)}\right)}. $$

(8.45)

This set can be used to replace the properties of BTO in Eqs. (8.28)–(8.30) to obtain α ₃₃ and α ₁₁ of the BTO-in-CFO composite with an imperfect interface.

8.3.3 The 2-2 Multiferroic Multilayers with a Perfect Interface

For the 2-2 multiferroic composites with a perfect interface, we find

$$ {\alpha}_{33}=-\frac{c_0{c}_1{e}_{33}^{(e)}{q}_{33}^{(m)}{\kappa}_{33}^{(m)}{\mu}_{33}^{(e)}}{D_{\alpha 33}}, $$

(8.46)

$$ {\alpha}_{11}=-\frac{c_0{c}_1{e}_{15}^{(e)}{q}_{15}^{(m)}}{c_0{C}_{44}^{(m)}+{c}_1{C}_{44}^{(e)}}, $$

(8.47)

where the denominator D _α33 is

$$ \begin{array}{l}{D}_{\alpha 33}={c_0}^2\left({c}_0{\kappa}_{33}^{(m)}+{c}_1{\kappa}_{33}^{(e)}\right)\left({q}_{33}^{(m)}{}^2+{C}_{33}^{(m)}{\mu}_{33}^{(m)}\right)+{c}_0{c}_1\left[\left({c}_0{\kappa}_{33}^{(m)}+{c}_1{\kappa}_{33}^{(e)}\right){C}_{33}^{(m)}{\mu}_{33}^{(e)}\right.\\ {}\kern3em \left.+\left({c}_0{\mu}_{33}^{(m)}+{c}_1{\mu}_{33}^{(e)}\right){C}_{33}^{(e)}{\kappa}_{33}^{(m)}\right]+{c_1}^2\left({c}_0{\mu}_{33}^{(m)}+{c}_1{\mu}_{33}^{(e)}\right)\left({e}_{33}^{(e)}{}^2+{C}_{33}^{(e)}{\kappa}_{33}^{(e)}\right),\end{array} $$

(8.48)

where c ₀ is for BTO and c ₁ for CFO.

8.3.4 The 2-2 Multiferroic Multilayers with an Imperfect Interface

With an imperfect interface, we have

$$ {\alpha}_{33}=-\frac{c_0{c}_1{e}_{33}^{(e)}{q}_{33}^{(m)}{\kappa}_{33}^{(m)}{\mu}_{33}^{(e)}{C}_{11}^{\left(\operatorname{int}\right)}{\kappa}^{\mathrm{int}}{\mu}^{\mathrm{int}}}{D_{\alpha 33}}, $$

(8.49)

$$ {\alpha}_{11}=-\frac{c_0{c}_1{e}_{15}^{(e)}{q}_{15}^{(m)}{C}_{44}^{\left(\operatorname{int}\right)}}{c_0{C}_{44}^{(m)}{C}_{44}^{\left(\operatorname{int}\right)}+{c}_1{C}_{44}^{(e)}{C}_{44}^{\left(\operatorname{int}\right)}+{c}_{\mathrm{int}}{C}_{44}^{(e)}{C}_{44}^{(m)}}, $$

(8.50)

where

$$ \begin{array}{l}{D}_{\alpha 33}={\lambda}_1{c_0}^3+{\lambda}_2{c_1}^3+{\lambda}_3{c_{\mathrm{int}}}^3+{\lambda}_4{c_0}^2{c}_1+{\lambda}_5{c_0}^2{c}_{\mathrm{int}}+{\lambda}_6{c_1}^2{c}_0+{\lambda}_7{c_1}^2{c}_{\mathrm{int}}+{\lambda}_8{c_{\mathrm{int}}}^2{c}_0\\ {}\kern2.8em +{\lambda}_9{c_{\mathrm{int}}}^2{c}_1+{\lambda}_{10}{c}_0{c}_1{c}_{\mathrm{int}}.\end{array} $$

(8.51)

The coefficients λ ₁ to λ ₁₀ are given by

$$ {\lambda}_1={\kappa}_{33}^{(m)}\left({q}_{33}^{(m)}{}^2+{C}_{33}^{(m)}{\mu}_{33}^{(m)}\right){C}_{11}^{\left(\operatorname{int}\right)}{\kappa}^{\mathrm{int}}{\mu}^{\mathrm{int}}, $$

$$ {\lambda}_2={\mu}_{33}^{(e)}\left({e}_{33}^{(e)}{}^2+{C}_{33}^{(e)}{\kappa}_{33}^{(e)}\right){C}_{11}^{\left(\operatorname{int}\right)}{\kappa}^{\mathrm{int}}{\mu}^{\mathrm{int}}, $$

$$ {\lambda}_3={\mu}_{33}^{(e)}\left({e}_{33}^{(e)2}+{C}_{33}^{(e)}{\kappa}_{33}^{(e)}\right){C_{11}}^{\left(\operatorname{int}\right)}{\kappa}^{\mathrm{int}}{\mu}^{\mathrm{int}}, $$

$$ {\lambda}_4={C}_{11}^{\left(\operatorname{int}\right)}{\kappa}^{\mathrm{int}}{\mu}^{\mathrm{int}}\left[{\kappa}_{33}^{(e)}\left({q_{33}^{(m)}}^2+{C}_{33}^{(m)}{\mu}_{33}^{(m)}\right)+{\kappa}_{33}^{(m)}\left({C}_{33}^{(e)}{\mu}_{33}^{(m)}+{C}_{33}^{(m)}{\mu}_{33}^{(e)}\right)\right], $$

$$ {\lambda}_5={\kappa}_{33}^{(m)}\left({q}_{33}^{(m)}{}^2+{C}_{33}^{(m)}{\mu}_{33}^{(m)}\right)\left[{C}_{33}^{(e)}{\kappa}^{\mathrm{int}}{\mu}^{\mathrm{int}}+{C}_{11}^{\left(\operatorname{int}\right)}\left({\kappa}^{\mathrm{int}}{\mu}_{33}^{(e)}+{\kappa}_{33}^{(e)}{\mu}^{\mathrm{int}}\right)\right], $$

$$ {\lambda}_6={C}_{11}^{\left(\operatorname{int}\right)}{\kappa}^{\mathrm{int}}{\mu}^{\mathrm{int}}\left[{\mu}_{33}^{(m)}\left({e}_{33}^{(e)}{}^2+{C}_{33}^{(e)}{\kappa}_{33}^{(e)}\right)+{\mu}_{33}^{(e)}\left({C}_{33}^{(e)}{\kappa}_{33}^{(m)}+{C}_{33}^{(m)}{\kappa}_{33}^{(e)}\right)\right], $$

$$ {\lambda}_7={\mu}_{33}^{(e)}\left({e}_{33}^{(e)}{}^2+{C}_{33}^{(e)}{\kappa}_{33}^{(e)}\right)\left[{C}_{33}^{(m)}{\kappa}^{\mathrm{int}}{\mu}^{\mathrm{int}}+{C}_{11}^{\left(\operatorname{int}\right)}\left({\kappa}^{\mathrm{int}}{\mu}_{33}^{(m)}+{\kappa}_{33}^{(m)}{\mu}^{\mathrm{int}}\right)\right], $$

$$ {\lambda}_8={\kappa}_{33}^{(m)}\left({q}_{33}^{(m)}{}^2+{C}_{33}^{(m)}{\mu}_{33}^{(m)}\right)\left[{\mu}^{\mathrm{int}}\left({e}_{33}^{(e)}{}^2+{C}_{33}^{(e)}{\kappa}_{33}^{(e)}\right)+{\mu}_{33}^{(e)}\left({C}_{33}^{(e)}{\kappa}^{\mathrm{int}}+{C}_{11}^{\left(\operatorname{int}\right)}{\kappa}_{33}^{(e)}\right)\right], $$

$$ {\lambda}_9={\mu}_{33}^{(m)}\left({q}_{33}^{(m)2}+{C}_{33}^{(m)}{\mu}_{33}^{(m)}\right)\left[{\kappa}^{\mathrm{int}}\left({q}_{33}^{(m)2}+{C}_{33}^{(m)}{\mu}_{33}^{(m)}\right)+{\mu}_{33}^{(m)}\left({C}_{33}^{(m)}{\mu}^{\mathrm{int}}+{C}_{11}^{\left(\operatorname{int}\right)}{\kappa}_{33}^{(m)}\right)\right], $$

$$ \begin{array}{l}{\lambda}_{10}={C}_{11}^{\left(\operatorname{int}\right)}\left[{\kappa}_{33}^{(m)}{\mu}_{33}^{(m)}\mu \left({e}_{33}^{(e)2}+{C}_{33}^{(e)}\;{\kappa}_{33}^{(e)}\right)+{\kappa}_{33}^{(e)}\;{\mu}_{33}^{(e)}{\kappa}^{\mathrm{int}}\left({q}_{33}^{(m)2}+{C}_{33}^{(m)}{\mu}_{33}^{(m)}\right)\right.\\ {}\kern3.75em \left.+\kern0.15em {\kappa}_{33}^{(m)}{\mu}_{33}^{(e)}\left({C}_{33}^{(e)}{\mu}_{33}^{(m)}{\kappa}^{\mathrm{int}}+{C}_{33}^{(m)}{\kappa}_{33}^{(e)}{\mu}^{\mathrm{int}}\right)\right]+{\kappa}^{\mathrm{int}}{\mu}^{\mathrm{int}}\left({e}_{33}^{(e)2}+{C}_{33}^{(e)}{\kappa}_{33}^{(e)}\right)\left({q}_{33}^{(m)2}+{C}_{33}^{(m)}{\mu}_{33}^{(m)}\right).\end{array} $$

(8.52)

Most of Appendix 2 and 3 can also be found in Wang et al. (2015).

Appendix 4: The Elastic C ₄₄ of the Fibrous Multiferroic Composite and the Purely Elastic Composite

We have seen the extraordinary value of C ₄₄ of the multiferroic composite in Fig. 8.19, and that its value can be higher than the individual value of the constituent phases. But this is not the case with the purely elastic composite shown in Fig. 8.20. This extraordinary value is an outcome of the piezoelectric and piezomagnetic interactions. To make such a difference more apparent, we give the explicit form of C ₄₄ with 1-3 connectivity.

Here we use the superscript (e) to denote the piezoelectric phase such as BTO, and the superscript (m) to denote the piezomagnetic phase such as CFO. We further use C ^E₄₄ to denote the C ₄₄ value of the purely elastic composite, and C ^M₄₄ to denote the C ₄₄ of the multiferroic composite. After making use of the explicit form of S-tensor for a circular cylinder, we have used Eq. (8.8) to derive the effective C ₄₄ in both cases.

First with the piezoelectric BTO as the matrix, we have found that, for the elastic composite,

$$ {C}_{44}^{\mathrm{E}}=\frac{C_{44}^{(e)}\left[\left({C}_{44}^{(e)}+{C}_{44}^{(m)}\right)-{c}_1\left({C}_{44}^{(e)}-{C}_{44}^{(m)}\right)\right]}{\left({C}_{44}^{(e)}+{C}_{44}^{(m)}\right)+{c}_1\left({C}_{44}^{(e)}-{C}_{44}^{(m)}\right)}, $$

(8.53)

and for the multiferroic composite, we have

$$ {C}_{44}^{\mathrm{M}}={C}_{44}^{\mathrm{E}}+\frac{4{c}_1\left(1-{c}_1\right)\left({C}_{44}^{(e)}{}^2{q}_{15}^{(m)}2{\tilde{\kappa}}_{11}^{(e)}+{C}_{44}^{(e)}{}^2{e}_{15}^{(e)}{}^2{\tilde{\mu}}_{11}^{(e)}+{e}_{15}^{(e)}{}^2{q}_{15}^{(e)}{}^2{\tilde{C}}_{44}^{(e)}\right)}{{\tilde{C}}_{44}^{(e)}\left[{\tilde{C}}_{44}^{(e)}{\tilde{\kappa}}_{11}^{(e)}{\tilde{\mu}}_{11}^{(e)}+{\left(1+{c}_1\right)}^2{e}_{15}^{(e)}{}^2{\tilde{\mu}}_{11}^{(e)}+{\left(1-{c}_1\right)}^2{q}_{15}^{(e)}{}^2{\tilde{\mu}}_{11}^{(e)}\right]}, $$

(8.54)

where

$$ {\tilde{C}}_{44}^{(e)}={C}_{44}^{(e)}+{C}_{44}^{(m)}+{c}_1\left({C}_{44}^{(e)}-{C}_{44}^{(m)}\right), $$

$$ {\tilde{\kappa}}_{11}^{(e)}={\kappa}_{11}^{(e)}+{\kappa}_{11}^{(m)}+{c}_1\left({\kappa}_{11}^{(e)}-{\kappa}_{11}^{(m)}\right), $$

(8.55)

$$ {\tilde{\mu}}_{11}^{(e)}={\mu}_{11}^{(e)}+{\mu}_{11}^{(m)}+{c}_1\left({\mu}_{11}^{(e)}-{\mu}_{11}^{(m)}\right). $$

On the other hand with the piezomagnetic CFO as the matrix, we have

$$ {C}_{44}^{\mathrm{E}}=\frac{C_{44}^{(m)}\left[\left({C}_{44}^{(m)}+{C}_{44}^{(e)}\right)-{c}_1\left({C}_{44}^{(m)}-{C}_{44}^{(e)}\right)\right]}{\left({C}_{44}^{(m)}+{C}_{44}^{(e)}\right)+{c}_1\left({C}_{44}^{(m)}-{C}_{44}^{(e)}\right)}, $$

(8.56)

and for the multiferroic composite, we find

$$ {C}_{44}^{\mathrm{M}}={C}_{44}^E+\frac{4{c}_1\left(1-{c}_1\right)\left({C}_{44}^{(e)}{}^2{q}_{15}^{(e)}{}^2{\tilde{\kappa}}_{11}^{(m)}+{C}_{44}^{(e)}{}^2{e}_{15}^{(e)}{}^2{\tilde{\mu}}_{11}^{(m)}+{e}_{15}^{(e)}{}^2{q}_{15}^{(m)}{}^2{\tilde{C}}_{44}^{(m)}\right)}{{\tilde{C}}_{44}^{(m)}\left[{\tilde{C}}_{44}^{(m)}\;{\tilde{\kappa}}_{11}^{(m)}\;{\tilde{\mu}}_{11}^{(m)}+{\left(1+{c}_1\right)}^2{q}_{15}^{(m)}2{\tilde{\kappa}}_{11}^{(m)}+{\left(1-{c}_1\right)}^2{e}_{15}^{(e)}2{\tilde{\mu}}_{11}^{(m)}\right]}, $$

(8.57)

where

$$ {\tilde{C}}_{44}^{(m)}={C}_{44}^{(m)}+{C}_{44}^{(e)}+{c}_1\left({C}_{44}^{(m)}-{C}_{44}^{(e)}\right), $$

$$ {\tilde{\kappa}}_{11}^{(m)}={\kappa}_{11}^{(m)}+{\kappa}_{11}^{(e)}+{c}_1\left({\kappa}_{11}^{(m)}-{\kappa}_{11}^{(e)}\right), $$

(8.58)

$$ {\tilde{\mu}}_{11}^{(m)}={\mu}_{11}^{(m)}+{\mu}_{11}^{(e)}+{c}_1\left({\mu}_{11}^{(m)}-{\mu}_{11}^{(e)}\right). $$

This set of results also gives rise to the outcome of ${C}_{44}^{\mathrm{M}}\ge {C}_{44}^{\mathrm{E}}$, regardless of the particular phase serving as the matrix.