A micromechanical model for the effective properties of two-phase magnetoelectric composites

Abstract

A micromechanical model based on the bridging theory is proposed for predicting the effective coefficients of two-phase magnetoelectric composites. The 17 different coefficients, including the effective elastic, piezoelectric, piezomagnetic, dielectric, magnetic permeability and magnetoelectric coupling coefficients, are divided into two groups based on the analysis of existing results of the classical micromechanical models. The relationship between the strain field, electric field and the magnetic field of two constituents is presented. The results are well matched with the finite element results. The classical models for predicting the effective coefficients of magnetoelectric composites can be covered flexibly by the presented unified micromechanical model.

Appendices

Appendix A

The expressions B₁ appearing in Eq. (79) and B₂ appearing in Eqs. (70), (73) and (76) take the following form:

$$B_{1} = V_{f}^{2} + \left( {A_{44} + A_{77} } \right)V_{f} V_{m} + \left( { - A_{48} A_{75} + A_{44} A_{77} } \right)V_{m}^{2} ,$$

(94)

$$B_{2} = V_{f}^{2} + \left( {A_{1010} + A_{77} } \right)V_{f} V_{m} + \left( { - A_{107} A_{710} + A_{1010} A_{77} } \right)V_{m}^{2} .$$

(95)

The expressions, $C_{2}$ and $C_{3}$ appearing in Eqs. (70), (73), (76), (77), (79) and (81) take the following form:

$$C_{1} = A_{1010} + A_{44} + A_{77} ,$$

(96)

$$C_{2} = - A_{107} A_{710} - A_{48} A_{75} + A_{44} A_{77} + A_{1010} \left( {A_{44} + A_{77} } \right),$$

(97)

$$C_{3} = - A_{107} A_{44} A_{710} + A_{105} A_{48} A_{710} - A_{1010} A_{48} A_{75} + A_{1010} A_{44} A_{77} .$$

(98)

The expressions $D_{1}$, $D_{7}$ appearing in Eq. (70), $D_{{2}}$ and $D_{{3}}$ appearing in Eq. (76), $D_{{4}}$ and $D_{{5}}$ appearing in Eq. (77) and $D_{{6}}$ appearing in Eq. (79) take the following form:

$$D_{1} = A_{107} q_{15}^{m} V_{f} V_{m} + e_{15}^{f} V_{f} E_{3} - A_{48} C_{44}^{m} V_{m} E_{3} ,$$

(99)

$$D_{2} = A_{107} \left( { - \mu_{11}^{f} + \mu_{11}^{m} } \right)V_{f} + A_{48} q_{15}^{m} E_{3} ,$$

(100)

$$D_{3} = A_{48} A_{710} q_{15}^{m} V_{m}^{2} - \mu_{11}^{f} V_{f} E_{2} + \mu_{11}^{m} V_{f} E_{2} ,$$

(101)

$$D_{{4}} = - A_{107} A_{710} V_{m} + A_{77} E_{3} ,$$

(102)

$$D_{{5}} = A_{75} V_{f} - A_{105} A_{710} V_{m} + A_{1010} A_{75} V_{m} ,$$

(103)

$$D_{{6}} = A_{107} \mu_{11}^{m} E_{4} + A_{48} \left( {q_{15}^{m} V_{f} - A_{105} \mu_{11}^{m} V_{m} } \right),$$

(104)

$$D_{{7}} = q_{15}^{m} V_{f} E_{2} + A_{710} V_{m} \left( {e_{15}^{f} V_{f} - A_{48} C_{44}^{m} V_{m} } \right).$$

(105)

The expressions $E_{1}$ appearing in Eq. (35), $E_{2}$ appearing in Eq. (A.8) and (105), $E_{3}$ appearing in Eqs. (73), (77), (99), (100) and (102), $E_{4}$ appearing in Eqs. (77), (81) and (104) and $E_{{5}}$ appearing in Eqs. (66) and (67) take the following form:

$$E_{1} = \left( {v_{f} + \left( {A_{11} + A_{21} } \right)v_{m} } \right), \, E_{2} = V_{f} + A_{77} V_{m} ,$$

(106)

$$E_{3} = V_{f} + A_{1010} V_{m} , \, E_{4} = V_{f} + A_{44} V_{m} , \, E_{5} = V_{f} + A_{11} V_{m} .$$

(107)

Appendix B

In order to help readers to be clearer about the expression of the bridging element, we again give the nonzero elements of the bridging element in Appendix B for readers to browse. The non-elements of the bridging matrix are given by the order in which they appear in the article.

Equation (108) appearing in Eq. (34), and Eqs. (109), (110), (111) appearing in Eqs. (39) take the following form:

$$A_{33} = 1,A_{66} = 1,A_{99} = 1,A_{1212} = 1,$$

(108)

$$A_{{13}} = \frac{{( - 1 + A_{{11}} + A_{{21}} )(C_{{13}}^{f} - C_{{13}}^{m} )}}{{C_{{11}}^{f} + C_{{12}}^{f} - C_{{11}}^{m} C_{{12}}^{m} }},$$

(109)

$$A_{{19}} = \frac{{( - 1 + A_{{11}} + A_{{21}} )C_{{29}}^{f} }}{{C_{{11}}^{f} + C_{{12}}^{f} - C_{{11}}^{m} - C_{{12}}^{m} }}$$

(110)

$$A_{{112}} = \frac{{( - 1 + A_{{11}} + A_{{21}} )C_{{212}}^{m} }}{{C_{{11}}^{f} + C_{{12}}^{f} - C_{{11}}^{m} - C_{{12}}^{m} }},$$

(111)

where equations related to A₁₁ and A₂₁ are:

$$A_{{11}} = \frac{{3C_{{11}}^{f} - C_{{12}}^{f} + C_{{11}}^{m} + C_{{12}}^{m} }}{{2(C_{{11}}^{f} - C_{{12}}^{f} + C_{{11}}^{m} + C_{{12}}^{m} )}},\,A_{{21}} = \frac{{C_{{11}}^{f} + C_{{12}}^{f} - C_{{11}}^{m} - C_{{12}}^{m} }}{{2(C_{{11}}^{f} - C_{{12}}^{f} + C_{{11}}^{m} + C_{{12}}^{m} )}}$$

(112)

Equation (113)–(118) appearing in Eqs. (45)–(50) take the following form:

$$A_{1010} = \frac{{A_{1010}^{1} }}{A},$$

(113)

$$A_{75} = - \frac{{A_{75}^{1} }}{A},$$

(114)

$$A_{710} = \frac{{A_{710}^{1} }}{A},$$

(115)

$$A_{48} = - \frac{{A_{48}^{1} }}{A},$$

(116)

$$A_{105} = \frac{{A_{105}^{1} }}{A},$$

(117)

$$A_{{107}} = \frac{{A_{{107}}^{1} }}{A},$$

(118)

where equations related to A, $A_{1010}^{1} ,A_{75}^{1} ,A_{710}^{1} ,A_{48}^{1} ,A_{105}^{1}$ and $A_{107}^{1}$ are:

$$A = \left( {\kappa_{11}^{m} } \right)^{2} \mu_{11}^{m} C_{44}^{m} + \left( {\kappa_{11}^{f} } \right)^{2} \left( {q_{15}^{m} } \right)^{2} - C_{44}^{f} \kappa_{11}^{f} \mu_{11}^{m} \kappa_{11}^{m} - \mu_{11}^{f} \kappa_{11}^{f} C_{44}^{m} \kappa_{11}^{m} + \mu_{11}^{f} C_{44}^{f} \left( {\kappa_{11}^{m} } \right)^{2} ,$$

(119)

$$\begin{aligned} A_{{1010}}^{1} & = A_{{44}} \left( {\kappa _{{11}}^{f} } \right)^{2} \mu _{{11}}^{m} C_{{44}}^{m} + A_{{44}} \left( {\kappa _{{11}}^{f} } \right)^{2} \left( {q_{{15}}^{m} } \right)^{2} - A_{{44}} \left( {e_{{15}}^{f} } \right)^{2} \mu _{{11}}^{m} \kappa _{{11}}^{m} + A_{{77}} \left( {e_{{15}}^{f} } \right)^{2} \mu _{{11}}^{m} \kappa _{{11}}^{m} \\ & \quad - A_{{44}} C_{{44}}^{f} \kappa _{{11}}^{f} \mu _{{11}}^{m} \kappa _{{11}}^{m} - A_{{44}} \mu _{{11}}^{f} \kappa _{{11}}^{f} C_{{44}}^{m} \kappa _{{11}}^{m} + A_{{44}} \mu _{{11}}^{f} C_{{44}}^{f} \left( {\kappa _{{11}}^{m} } \right)^{2} , \\ \end{aligned}$$

(120)

$$A_{75}^{1} = \left( {A_{44} - A_{77} } \right)e_{15}^{f} \left[ {\kappa_{11}^{f} \mu_{11}^{m} C_{44}^{m} + \kappa_{11}^{f} \left( {q_{15}^{m} } \right)^{2} - \mu_{11}^{f} C_{44}^{m} \kappa_{11}^{m} } \right],$$

(121)

$$A_{710}^{1} = - A_{44} \mu_{11}^{f} e_{15}^{f} q_{15}^{m} \kappa_{11}^{m} + A_{77} \mu_{11}^{f} e_{15}^{f} q_{15}^{m} \kappa_{11}^{m} ,$$

(122)

$$A_{{48}}^{1} = (A_{{44}} - A_{{77}} )e_{{15}}^{f} \kappa _{{11}}^{m} ( - \kappa _{{11}}^{f} \mu _{{11}}^{m} + \mu _{{11}}^{f} \kappa _{{11}}^{m} )$$

(123)

$$A_{{105}}^{1} = - A_{{44}} (e_{{15}}^{f} )^{2} q_{{15}}^{m} \kappa _{{11}}^{m} + A_{{77}} (e_{{15}}^{f} )^{2} q_{{15}}^{m} \kappa _{{11}}^{m} ,$$

(124)

$$A_{107}^{1} = - A_{44} e_{15}^{f} \kappa_{11}^{f} q_{15}^{m} \kappa_{11}^{m} + A_{77} e_{15}^{f} \kappa_{11}^{f} q_{15}^{m} \kappa_{11}^{m} .$$

(125)

Equation (126)–(127) appearing in Eqs. (62)–(63) takes the following form:

$$A_{{44}} = 1 - \beta (1 - C_{{44}}^{f} /C_{{44}}^{m} ),$$

(126)

$$A_{{77}} = 1 - \gamma (1 - C_{{77}}^{f} /C_{{77}}^{m} ),$$

(127)

where the parameters $\beta$ and $\gamma$ are given as follows:

$$12 < \beta < 16.$$

(128)

$$0 < \gamma < 1.$$

(129)