A set of collinear electrically charged interfacial cracks in magnetoelectroelastic bimaterial

Abstract

A bimaterial space composed of two semi-infinite magnetoelectroelastic spaces with a finite set of cracks along the material’s interface is considered. The cracks can have arbitrary lengths and location and their faces are covered with the electrodes having different electric net charge and zero magnetic net induction. The bimaterial is loaded by remote mixed mode mechanical loading, electric and magnetic fields, which do not change along the coordinate codirected with the crack fronts. The problems of linear relationship are formulated and solved analytically by using the presentations of electro-magneto-mechanical quantities via sectionally analytic functions. Using this solution all required mechanical, electric and magnetic components along the interface are presented in the closed form. Because the obtained solution has an oscillating singularity at the crack tips, the energy release rate is the most appropriate fracture parameter in this case. It was found analytically for all crack tips using the asymptotic presentations of all fields at the crack tips and the crack closure integral. Numerical results are presented in graph and table forms for different loading, crack locations, their number and lengths.

Appendices

Appendix 1

Solution of the system (26):

$$\begin{aligned} D_{r} = & \det \left( {\begin{array}{*{20}c} {r_{11} } &\quad {r_{14} } &\quad {r_{15} } \\ {r_{41} } &\quad {r_{44} } &\quad {r_{45} } \\ {r_{51} } &\quad {r_{54} } &\quad {r_{55} } \\ \end{array} } \right), \\ \sigma_{13}^{\left( 1 \right)} \left( {x_{1} ,0} \right) = & \det \left( {\begin{array}{*{20}c} { - {\text{Im}} \left[ {\Gamma_{1} \left( {x_{1} } \right)} \right]} &\quad {r_{14} } &\quad {r_{15} } \\ { - {\text{Im}} \left[ {\Gamma_{4} \left( {x_{1} } \right)} \right]} &\quad {r_{44} } &\quad {r_{45} } \\ { - {\text{Im}} \left[ {\Gamma_{5} \left( {x_{1} } \right)} \right]} &\quad {r_{54} } &\quad {r_{55} } \\ \end{array} } \right)/D_{r} , \\ E_{1}^{\left( 1 \right)} \left( {x_{1} ,0} \right) = & \det \left( {\begin{array}{*{20}c} {r_{11} } &\quad { - {\text{Im}} \left[ {\Gamma_{1} \left( {x_{1} } \right)} \right]} &\quad {r_{15} } \\ {r_{41} } &\quad { - {\text{Im}} \left[ {\Gamma_{4} \left( {x_{1} } \right)} \right]} &\quad {r_{45} } \\ {r_{51} } &\quad { - {\text{Im}} \left[ {\Gamma_{5} \left( {x_{1} } \right)} \right]} &\quad {r_{55} } \\ \end{array} } \right)/D_{r} , \\ H_{1}^{\left( 1 \right)} \left( {x_{1} ,0} \right) = & \det \left( {\begin{array}{*{20}c} {r_{11} } &\quad {r_{14} } &\quad { - {\text{Im}} \left[ {\Gamma_{1} \left( {x_{1} } \right)} \right]} \\ {r_{41} } &\quad {r_{44} } &\quad { - {\text{Im}} \left[ {\Gamma_{4} \left( {x_{1} } \right)} \right]} \\ {r_{51} } &\quad {r_{54} } &\quad { - {\text{Im}} \left[ {\Gamma_{5} \left( {x_{1} } \right)} \right]} \\ \end{array} } \right)/D_{r} . \\ \end{aligned}$$

(50)

The formulas (50) can be presented in the form:

$$\begin{gathered} \sigma_{13}^{\left( 1 \right)} \left( {x_{1} ,0} \right) = \rho_{\sigma 1} {\text{Im}} \left[ {\Gamma_{1} \left( {x_{1} } \right)} \right] + \rho_{\sigma 4} {\text{Im}} \left[ {\Gamma_{4} \left( {x_{1} } \right)} \right] + \rho_{\sigma 5} {\text{Im}} \left[ {\Gamma_{5} \left( {x_{1} } \right)} \right], \hfill \\ E_{1}^{\left( 1 \right)} \left( {x_{1} ,0} \right) = \rho_{E1} {\text{Im}} \left[ {\Gamma_{1} \left( {x_{1} } \right)} \right] + \rho_{E4} {\text{Im}} \left[ {\Gamma_{4} \left( {x_{1} } \right)} \right] + \rho_{E5} {\text{Im}} \left[ {\Gamma_{5} \left( {x_{1} } \right)} \right], \hfill \\ H_{1}^{\left( 1 \right)} \left( {x_{1} ,0} \right) = \rho_{H1} {\text{Im}} \left[ {\Gamma_{1} \left( {x_{1} } \right)} \right] + \rho_{H4} {\text{Im}} \left[ {\Gamma_{4} \left( {x_{1} } \right)} \right] + \rho_{H5} {\text{Im}} \left[ {\Gamma_{5} \left( {x_{1} } \right)} \right], \hfill \\ \end{gathered}$$

(51)

where

$$\begin{aligned} \rho_{\sigma 1} = & \left( {r_{54} r_{45} - r_{44} r_{55} } \right)/D_{r} ,\;\rho_{\sigma 4} = \left( {r_{14} r_{55} - r_{54} r_{15} } \right)/D_{r} ,\;\rho_{\sigma 5} = \left( {r_{44} r_{15} - r_{14} r_{45} } \right)/D_{r} , \\ \rho_{E1} = & \left( {r_{41} r_{55} - r_{51} r_{45} } \right)/D_{r} ,\;\rho_{E4} = \left( {r_{51} r_{15} - r_{11} r_{55} } \right)/D_{r} ,\;\rho_{E5} = \left( {r_{11} r_{45} - r_{41} r_{15} } \right)/D_{r} , \\ \rho_{H1} = & \left( {r_{51} r_{44} - r_{41} r_{54} } \right)/D_{r} ,\;\rho_{H4} = \left( {r_{11} r_{54} - r_{51} r_{14} } \right)/D_{r} ,\;\rho_{H5} = \left( {r_{41} r_{14} - r_{11} r_{44} } \right)/D_{r} , \\ \end{aligned}$$

The expressions for $\varphi^{(1)} \left( {x_{1} ,0} \right)$ and $\psi^{(1)} \left( {x_{1} ,0} \right)$ can be found on the same formulas (51) as $E_{1}^{\left( 1 \right)} \left( {x_{1} ,0} \right)$ and $H_{1}^{\left( 1 \right)} \left( {x_{1} ,0} \right)$, respectively, provided $\left[ { - \hat{\Gamma }_{j} \left( {x_{1} } \right)} \right]$ instead $\Gamma_{j} \left( {x_{1} } \right)$ ($j = 1,\,4,\,5$) in these formulas are taken.

Solution of the system (29) is the following:

$$\begin{aligned} \left\langle {u^{\prime}_{1} \left( {x_{1} } \right)} \right\rangle = & \det \left( {\begin{array}{*{20}c} {{\text{Re}} \left[ {\theta_{1} \left( {x_{1} } \right)} \right]} & \quad {t_{14} } & \quad {t_{15} } \\ {{\text{Re}} \left[ {\theta_{4} \left( {x_{1} } \right)} \right]} & \quad {t_{44} } & \quad {t_{45} } \\ {{\text{Re}} \left[ {\theta_{5} \left( {x_{1} } \right)} \right]} & \quad {t_{54} } & \quad {t_{55} } \\ \end{array} } \right)/D_{t} , \\ \left\langle {D_{3} \left( {x_{1} } \right)} \right\rangle = & \det \left( {\begin{array}{*{20}c} {t_{11} } &\quad {{\text{Re}} \left[ {\theta_{1} \left( {x_{1} } \right)} \right]} &\quad {t_{15} } \\ {t_{41} } &\quad {{\text{Re}} \left[ {\theta_{4} \left( {x_{1} } \right)} \right]} &\quad {t_{45} } \\ {t_{51} } &\quad {{\text{Re}} \left[ {\theta_{5} \left( {x_{1} } \right)} \right]} &\quad {t_{55} } \\ \end{array} } \right)/D_{t} , \\ \left\langle {B_{3} \left( {x_{1} } \right)} \right\rangle = &\quad \det \left( {\begin{array}{*{20}c} {t_{11} } &\quad {t_{14} } &\quad {{\text{Re}} \left[ {\theta_{1} \left( {x_{1} } \right)} \right]} \\ {t_{41} } &\quad {t_{44} } &\quad {{\text{Re}} \left[ {\theta_{4} \left( {x_{1} } \right)} \right]} \\ {t_{51} } &\quad {t_{54} } &\quad {{\text{Re}} \left[ {\theta_{5} \left( {x_{1} } \right)} \right]} \\ \end{array} } \right)/D_{t} , \\ \end{aligned}$$

(52)

where

$$D_{t} = \det \left( {\begin{array}{*{20}c} {t_{11} } & {t_{14} } & {t_{15} } \\ {t_{41} } & {t_{44} } & {t_{45} } \\ {t_{51} } & {t_{54} } & {t_{55} } \\ \end{array} } \right).$$

The expressions for $\left\langle {u_{1} \left( {x_{1} } \right)} \right\rangle$, $\left\langle {\hat{D}_{3} \left( {x_{1} } \right)} \right\rangle$ and $\left\langle {\hat{B}_{3} \left( {x_{1} } \right)} \right\rangle$ can be found on the same formulas (52) as $\left\langle {u^{\prime}_{1} \left( {x_{1} } \right)} \right\rangle ,\,\left\langle {D_{3} \left( {x_{1} } \right)} \right\rangle ,\,\left\langle {B_{3} \left( {x_{1} } \right)} \right\rangle$, respectively, provided $\hat{\theta }_{j} \left( {x_{1} } \right)$ instead $\theta_{j} \left( {x_{1} } \right)$ ($j = 1,\,4,\,5$) in these formulas are taken. For this case, the mentioned formulas can be presented in the form

$$\begin{aligned} \left\langle {u_{1} \left( {x_{1} } \right)} \right\rangle =\, & \eta_{u1} {\text{Re}} \left[ {\hat{\theta }_{1} \left( {x_{1} } \right)} \right] + \eta_{u4} {\text{Re}} \left[ {\hat{\theta }_{4} \left( {x_{1} } \right)} \right] + \eta_{u5} {\text{Re}} \left[ {\hat{\theta }_{5} \left( {x_{1} } \right)} \right], \\ \left\langle {\hat{D}_{3} \left( {x_{1} } \right)} \right\rangle =\, & \eta_{D1} {\text{Re}} \left[ {\hat{\theta }_{1} \left( {x_{1} } \right)} \right] + \eta_{D4} {\text{Re}} \left[ {\hat{\theta }_{4} \left( {x_{1} } \right)} \right] + \eta_{D5} {\text{Re}} \left[ {\hat{\theta }_{5} \left( {x_{1} } \right)} \right], \\ \left\langle {\hat{B}_{3} \left( {x_{1} } \right)} \right\rangle =\, & \eta_{B1} {\text{Re}} \left[ {\hat{\theta }_{1} \left( {x_{1} } \right)} \right] + \eta_{B4} {\text{Re}} \left[ {\hat{\theta }_{4} \left( {x_{1} } \right)} \right] + \eta_{B5} {\text{Re}} \left[ {\hat{\theta }_{5} \left( {x_{1} } \right)} \right], \\ \end{aligned}$$

(53)

where

$$\begin{gathered} \eta_{u1} = (t_{44} t_{55} - t_{54} t_{45} )/D_{t} ,\;\eta_{u4} = (t_{54} t_{15} - t_{14} t_{55} )/D_{t} ,\;\eta_{u5} = (t_{14} t_{45} - t_{44} t_{15} )/D_{t} , \hfill \\ \eta_{D1} = (t_{51} t_{45} - t_{41} t_{55} )/D_{t} ,\;\eta_{D4} = (t_{11} t_{55} - t_{51} t_{15} )/D_{t} ,\;\eta_{D5} = (t_{41} t_{15} - t_{11} t_{45} )/D_{t} , \hfill \\ \eta_{B1} = (t_{41} t_{54} - t_{51} t_{44} )/D_{t} ,\;\eta_{B4} = (t_{51} t_{14} - t_{11} t_{54} )/D_{t} ,\;\eta_{B5} = (t_{11} t_{44} - t_{41} t_{14} )/D_{t} . \hfill \\ \end{gathered}$$

Appendix 2

The expressions for $h_{1k}^{a}$, $h_{2k}^{a}$ and $h_{4k}^{a}$ from the formula (49) are the following:

$$h_{1k}^{a} = q_{1k}^{a} p_{2k}^{a} + q_{2k}^{a} p_{1k}^{a} ,\;h_{2k}^{a} = q_{2k}^{a} p_{2k}^{a} - q_{1k}^{a} p_{1k}^{a} ,\;h_{4k}^{a} = \rho_{\sigma 45}^{a} \eta_{u45}^{a} - \rho_{E45}^{a} \eta_{D45}^{a} - \rho_{H45}^{a} \eta_{B45}^{a} ,$$

where

$$\begin{gathered} q_{1k}^{a} = {\text{Re}} \left( {\overline{R}_{1k}^{a} } \right),\;q_{2k} = - {\text{Im}} \left( {\overline{R}_{1k}^{a} } \right),\;q_{2k} = - {\text{Im}} \left( {\overline{R}_{1k}^{a} } \right),\;p_{2k}^{a} = - {\text{Im}} \left[ {L_{1k}^{a} /\left( {i\varepsilon_{1} + 0.5} \right)} \right], \hfill \\ \overline{R}_{1k}^{a} = \left( { - 1} \right)^{n} \left( {1 + \gamma_{1} } \right)\overline{P}_{n1} \left( {a_{k} } \right)\left( {b_{k} - a_{k} } \right)^{{ - 0.5 + i\varepsilon_{1} }} \prod\limits_{j = 1,j \ne k}^{n} {\left( {a_{j} - a_{k} } \right)^{{ - 0.5 - i\varepsilon_{1} }} \left( {b_{j} - a_{k} } \right)^{{ - 0.5 + i\varepsilon_{1} }} } \hfill \\ L_{1k}^{a} = - \frac{{\left( {1 + \gamma_{1} } \right)}}{{\gamma_{1} }}P_{n1} \left( {a_{k} } \right)\left( {a_{k} - b_{k} } \right)^{{ - 0.5 - i\varepsilon_{1} }} \prod\limits_{j = 1,j \ne k}^{n} {\left( {a_{k} - a_{j} } \right)^{{ - 0.5 + i\varepsilon_{1} }} \left( {a_{k} - b_{j} } \right)^{{ - 0.5 - i\varepsilon_{1} }} } \hfill \\ \rho_{\sigma 45}^{a} = \rho_{\sigma 4} {\text{Im}} \left( {\overline{R}_{4k}^{a} } \right) + \rho_{\sigma 5} {\text{Im}} \left( {\overline{R}_{5k}^{a} } \right),\;\rho_{E45}^{a} = \rho_{E4} {\text{Im}} \left( {\overline{R}_{4k}^{a} } \right) + \rho_{E5} {\text{Im}} \left( {\overline{R}_{5k}^{a} } \right) \hfill \\ \rho_{H45}^{a} = \rho_{H4} {\text{Im}} \left( {\overline{R}_{4k}^{a} } \right) + \rho_{H5} {\text{Im}} \left( {\overline{R}_{5k}^{a} } \right),\;\eta_{u45}^{a} = - 2\left[ {\eta_{u4} {\text{Re}} \left( {L_{4k}^{a} } \right) + \eta_{u5} {\text{Re}} \left( {L_{5k}^{a} } \right)} \right], \hfill \\ \eta_{D45}^{a} = - 2\left[ {\eta_{D4} {\text{Re}} \left( {L_{4k}^{a} } \right) + \eta_{D5} {\text{Re}} \left( {L_{5k}^{a} } \right)} \right],\;\eta_{B45}^{a} = - 2\left[ {\eta_{B4} {\text{Re}} \left( {L_{4k}^{a} } \right) + \eta_{B5} {\text{Re}} \left( {L_{5k}^{a} } \right)} \right] \hfill \\ L_{sk}^{a} = - 2P_{n\,s} \left( {a_{k} } \right)\frac{1}{{\sqrt {a_{k} - b_{k} } }}\prod\limits_{j = 1,j \ne k}^{n} {\frac{1}{{\sqrt {\left( {a_{k} - a_{j} } \right)\left( {a_{k} - b_{j} } \right)} }}} ,\;(s = 4,5). \hfill \\ \hfill \\ \end{gathered}$$

Appendix 3

Effective properties of BaTiO₃—CoFe₂O₄ composite for different volume fractions of BaTiO₃ [38, 39]

Properties	Vf = 0.1	Vf = 0.5
$c_{11}$(GPa)	274	226
$c_{33}$(GPa)	161	124
$c_{13}$(GPa)	259	216
$c_{44}$(GPa)	45	44
$e_{31}$(C/m2)	− 4.4	− 2.2
$e_{15}$(C/m2)	1.86	9.3
$e_{33}$(C/m2)	1.16	5.8
$\alpha_{11}$ $\left( { \times 10^{ - 10} {\text{C}}^{2} /{\text{Nm}}^{2} } \right)$	11.9	56.4
$\alpha_{33}$ $\left( { \times 10^{ - 10} {\text{C}}^{2} /{\text{Nm}}^{2} } \right)$	13.4	63.5
$h_{31}$ $\left( {{\text{N}}/{\text{Am}}} \right)$	522.3	290.2
$h_{33}$ $\left( {{\text{N}}/{\text{Am}}} \right)$	629.7	350.0
$h_{15}$ $\left( {{\text{N}}/{\text{Am}}} \right)$	495.0	275.0
$\mu_{11}$ $\left( { \times 10^{ - 6} {\text{Ns}}^{2} /{\text{C}}^{2} } \right)$	531.5	297.0
$\mu_{33}$ $\left( { \times 10^{ - 6} {\text{Ns}}^{2} /{\text{C}}^{2} } \right)$	142.3	83.5