Fracture Analysis in an Imperfect FGM Orthotropic Strip Bonded Between Two Magneto-Electro-Elastic Layers

R. Bagheri; A. M. Mirzaei

doi:10.1007/s40997-017-0129-6

Iranian Journal of Science and Technology, Transactions of Mechanical Engineering

pp 1–19 | Cite as

Fracture Analysis in an Imperfect FGM Orthotropic Strip Bonded Between Two Magneto-Electro-Elastic Layers

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R. BagheriEmail author
A. M. Mirzaei

Research Paper

First Online: 05 December 2017

Received: 25 November 2016

Accepted: 27 November 2017

26 Downloads

Abstract

In this study, a functionally graded orthotropic layer sandwiched between two magneto-electro-elastic layers under anti-plane time-harmonic mechanic and in-plane electromagnetic loadings is studied. This paper examines the modeling of cracks by distribution of strain nuclei along crack lines. During this investigation, the Volterra-type screw dislocation is employed in the FGM orthotropic layer and the magneto-electro-elastic coating. The material properties of the strip are considered to obey exponential variations. To solve the dislocation problem, the complex Fourier transform is applied. One merit of this technique is the possibility of determination of the stress intensity factors for multiple smooth cracks. The system of equations is derived by considering the distribution of line dislocation on the crack. These equations are of Cauchy singular type in the location of dislocation, which can be solved numerically to obtain the dislocation density on the faces of the cracks. Various examples are solved, and the dynamic stress intensity factors are obtained. The effect of the cracks configuration, bonding constant, angular frequency and material properties on the mode III stress intensity factor is studied, and the validity of the analysis is checked.

Keywords

Multiple smooth cracks Dynamic stress intensity factors Smart materials Dislocation technique

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Appendix A

$$ \begin{aligned} N_{1} = & - ((\sinh [(\eta + h_{1} )\alpha_{2} ]G_{y} {\kern 1pt} \Delta_{1} (g^{2} s^{2} - M_{2}^{2} ) \\ & + {\text{e}}^{{2\lambda h_{1} }} \sinh [h_{3} \alpha_{1} ]\tilde{c}_{44} k_{13} \alpha_{1} \,\Delta_{2} )( - \cosh [h_{2} \alpha_{1} ]G_{y} k_{12} \,\Delta_{3} + \sinh [h_{2} \alpha_{1} ]{\kern 1pt} \,\tilde{c}_{44} \alpha_{1} \,\Delta_{4} )) \\ \end{aligned} $$

$$ \begin{aligned} N_{2} = & (\alpha_{2} ( - {\text{e}}^{{2\lambda h_{1} }} \sinh [h_{1} \alpha_{2} ]\sinh [h_{2} \alpha_{1} ]\sinh [h_{3} \alpha_{1} ]\tilde{c}_{44}^{2} k_{12} k_{13} \alpha_{1}^{2} + \sinh [h_{1} \alpha_{2} ]G_{y}^{2} \,\Delta_{5} ( - \Delta_{1} )(g^{2} s^{2} - M_{2}^{2} ) \\ & + \tilde{c}_{44} G_{y} \alpha_{1} ( - {\text{e}}^{{2\lambda h_{1} }} \cosh [h_{2} \alpha_{1} ]\sinh [h_{3} \alpha_{1} ]k_{12} k_{13} \,\Delta_{6} \\ + \sinh [h_{2} \alpha_{1} ]({\text{e}}^{{2\lambda h_{1} }} \sinh [h_{3} \alpha_{1} ]\tilde{c}_{44} k_{13} \alpha_{1} \,\Delta_{6} - k_{12} \,\Delta_{1} \,\Delta_{7} )))) \\ \end{aligned} $$

$$ \begin{aligned} N_{3} = & - ((\sinh [(\eta + h_{1} )\alpha_{2} ]G_{y} \,\Delta_{1} (g^{2} s^{2} - M_{2}^{2} ) \\ & + {\text{e}}^{{2\lambda h_{1} }} \sinh [h_{3} \alpha_{1} ]\tilde{c}_{44} k_{13} \alpha_{1} \Delta_{2} )( - \cosh [h_{2} \alpha_{1} ]G_{y} k_{12} \sinh [\alpha_{2} y](g^{2} s^{2} - M_{2}^{2} ) + \sinh [h_{2} \alpha_{1} ]\tilde{c}_{44} \alpha_{1} \Delta_{8} )) \\ \end{aligned} $$

$$ N_{4} = ((\sinh [\eta \alpha_{2} ]G_{y} \Delta_{5} (g^{2} s^{2} - M_{2}^{2} ) - \sinh [h_{2} \alpha_{1} ]\,\tilde{c}_{44} k_{12} \alpha_{1} \,\Delta_{9} )(\cosh [h_{3} \alpha_{1} ]G_{y} k_{13} \Delta_{10} - \sinh [h_{3} \alpha_{1} ]\;\tilde{c}_{44} \alpha_{1} \Delta_{11} )) $$

$$ \begin{aligned} N_{5} = & ((\sinh [\eta \alpha_{2} ]G_{y} \Delta_{5} (g^{2} s^{2} - M_{2}^{2} ) \\ & - \sinh [h_{2} \alpha_{1} ]\tilde{c}_{44} k_{12} \alpha_{1} \Delta_{9} )(\cosh [h_{3} \alpha_{1} ]G_{y} k_{13} \sinh [\alpha_{2} (y + h_{1} )](g^{2} s^{2} - M_{2}^{2} ) - \sinh [h_{3} \alpha_{1} ]\;\tilde{c}_{44} \alpha_{1} \Delta_{12} )) \\ \end{aligned} $$

$$ \Delta_{1} = (\cosh [h_{3} \alpha_{1} ]k_{13} - \sinh [h_{3} \alpha_{1} ]\tilde{c}_{44} \alpha_{1} )$$

$$\Delta_{2} = (\alpha_{2} \cosh [(\eta + h_{1} )\alpha_{2} ] - \lambda \sinh [(\eta + h_{1} )\alpha_{2} ]) $$

$$ \Delta_{3} = (\lambda \sinh [\alpha_{2} y] + \alpha_{2} \cosh [\alpha_{2} y])$$

$$\Delta_{4} = ((k_{12} + G_{y} \lambda )\sinh [\alpha_{2} y] + G_{y} \alpha_{2} \cosh [\alpha_{2} y])) $$

$$ \Delta_{5} = (\cosh [h_{2} \alpha_{1} ]k_{12} - \sinh [h_{2} \alpha_{1} ]\tilde{c}_{44} \alpha_{1} )$$

$$\Delta_{6} = (\alpha_{2} \cosh [\alpha_{2} h_{1} ] - \lambda \sinh [\alpha_{2} h_{1} ]) $$

$$ \Delta_{7} = (\alpha_{2} \cosh [\alpha_{2} h_{1} ] + \lambda \sinh [\alpha_{2} h_{1} ]) $$

$$ \Delta_{8} = (( - \lambda {\kern 1pt} k_{12} + G_{y} (g^{2} s^{2} - M_{2}^{2} ))\sinh [\alpha_{2} y] + k_{12} \alpha_{2} \cosh [\alpha_{2} y]) $$

$$ \Delta_{9} = (\alpha_{2} \cosh [\eta \alpha_{2} ] - \lambda \sinh [\eta \alpha_{2} ]) $$

$$ \Delta_{10} = (\lambda \sinh [\alpha_{2} (y + h_{1} )] + \alpha_{2} \cosh [\alpha_{2} (y + h_{1} )]) $$

$$ \Delta_{11} = (( - {\text{e}}^{{2\lambda h_{1} }} k_{13} + G_{y} \lambda )\sinh [\alpha_{2} (y + h_{1} )] + G_{y} \alpha_{2} \cosh [\alpha_{2} (y + h_{1} )]) $$

$$ \Delta_{12} = (({\text{e}}^{{2\lambda h_{1} }} \lambda {\kern 1pt} k_{13} + G_{y} (g^{2} s^{2} - M_{2}^{2} ))\sinh [\alpha_{2} (y + h_{1} )] - {\text{e}}^{{2\lambda h_{1} }} \alpha_{2} k_{13} \cosh [\alpha_{2} (y + h_{1} )]) $$

$$ \alpha_{1} = \sqrt {s^{2} - M_{1}^{2} } $$

$$ \alpha_{2} = \sqrt {\lambda^{2} + g^{2} s^{2} - M_{2}^{2} } $$

Appendix B

$$ \begin{aligned} C_{1} = & \tanh [\alpha_{1} (\eta - h_{2} )]\{ \,\varOmega_{8} \cosh (\alpha_{1} \eta ){\kern 1pt} {\kern 1pt} + G_{y}^{2} \,\varOmega_{5} \varOmega_{1} \sinh (\alpha_{2} h_{1} )(g^{2} s^{2} - M_{2}^{2} ) \\ + \tilde{c}_{44} G_{y} \alpha_{1} [{\text{e}}^{{2\lambda h_{1} }} k_{13} \,\varOmega_{5} \varOmega_{3} \sinh (\alpha_{1} h_{3} ) + k_{12} \,\varOmega_{1} \,\varOmega_{4} \cosh (\alpha_{1} \eta )]\} \\ \end{aligned} $$

$$ \begin{aligned} C_{2} = & \tilde{c}_{44} G_{y} \alpha_{1} [{\text{e}}^{{2\lambda h_{1} }} k_{13} \varOmega_{2} \,\varOmega_{3} \sinh (\alpha_{1} h_{3} ) + k_{12} \,\varOmega_{1} \,\varOmega_{4} \sinh (\alpha_{1} h_{2} )] + \varOmega_{8} \sinh (\alpha_{1} h_{2} )\, \\ & \quad + \sinh (\alpha_{2} h_{1} )G_{y}^{2} \,\varOmega_{2} \,\varOmega_{1} (g^{2} s^{2} - M_{2}^{2} ) \\ \end{aligned} $$

$$ \begin{aligned} C_{3} = & \sinh [\alpha_{1} (\eta - h_{2} )]\{ \varOmega_{8} \cosh (\alpha_{1} y) + G_{y}^{2} {\kern 1pt} \varOmega_{6} {\kern 1pt} \varOmega_{1} \sinh (\alpha_{2} h_{1} )(g^{2} s^{2} - M_{2}^{2} ) \\ \quad + \tilde{c}_{44} G_{y} \alpha_{1} [{\text{e}}^{{2\lambda h_{1} }} k_{13} \varOmega_{6} {\kern 1pt} \varOmega_{3} \sinh (\alpha_{1} h_{3} ) + k_{12} \varOmega_{1} {\kern 1pt} \varOmega_{4} \cosh (\alpha_{1} y)]\} \\ \end{aligned} $$

$$ \begin{aligned} C_{4} = & \alpha_{1} \sinh [\alpha_{1} (\eta - h_{2} )]\{ \varOmega_{8} \sinh (\alpha_{1} y)\, + G_{y}^{2} {\kern 1pt} \varOmega_{7} {\kern 1pt} \varOmega_{1} \sinh (\alpha_{2} h_{1} )(g^{2} s^{2} - M_{2}^{2} ) \\ + \tilde{c}_{44} G_{y} \alpha_{1} [{\text{e}}^{{2\lambda h_{1} }} k_{13} \varOmega_{7} \varOmega_{3} \sinh (\alpha_{1} h_{3} ) + k_{12} \varOmega_{1} \varOmega_{4} \sinh (\alpha_{1} y)]\} \\ \end{aligned} $$

$$ \varOmega_{1} = \tilde{c}_{44} \alpha_{1} \sinh (\alpha_{1} h_{3} ) - k_{13} \cosh (\alpha_{1} h_{3} ),$$

$$\varOmega_{2} = k_{12} \cosh (\alpha_{1} h_{2} ) - \tilde{c}_{44} \alpha_{1} \sinh (\alpha_{1} h_{2} ) $$

$$ \varOmega_{3} = \lambda \sinh (\alpha_{2} h_{1} ) - \alpha_{2} \cosh (\alpha_{2} h_{1} ),$$

$$\varOmega_{4} = \lambda \sinh (\alpha_{2} h_{1} ) + \alpha_{2} \cosh (\alpha_{2} h_{1} ) $$

$$ \varOmega_{5} = k_{12} \sinh (\alpha_{1} \eta ) - \tilde{c}_{44} \alpha_{1} \cosh (\alpha_{1} \eta ),$$

$$\varOmega_{6} = k_{12} \sinh (\alpha_{1} y) - \tilde{c}_{44} \alpha_{1} \cosh (\alpha_{1} y) $$

$$ \varOmega_{7} = k_{12} \cosh (\alpha_{1} y) - \tilde{c}_{44} \alpha_{1} \sinh (\alpha_{1} y),$$

$$\varOmega_{8} = - e^{{2\lambda h_{1} }} \tilde{c}_{44}^{2} k_{12} k_{13} \alpha_{1}^{2} \sinh (\alpha_{2} h_{1} )\sinh (\alpha_{1} h_{3} ) $$

$$ P_{1} = \text{csch} (sh_{2} )\sinh (s\eta ), $$

$$P_{2} = \text{csch} (sh_{2} )\sinh (s(\eta - h_{2} )) $$

Appendix C

$$ {E}_{1} = \text{csch} (h_{2} \alpha_{1} )\{ Q_{1} + \tilde{c}_{44} G_{y} \alpha_{1} [Q_{2} + k_{12} (Q_{3} + k_{13} (Q_{4} + Q_{5} \alpha_{2} ))]\} $$

$$ {E}_{2} = Q_{6} - Q_{7} + \tilde{c}_{44} \alpha_{1} G_{y} (Q_{8} + Q_{9} ) $$

$$ {E}_{3} = \text{csch} (h_{2} \alpha_{1} )[Q_{10} + Q_{11} + k_{13} (Q_{12} + Q_{13} \alpha_{2} )] $$

$$ {E}_{4} = Q_{14} + \tilde{c}_{44} G_{y} \alpha_{1} (Q_{15} + k_{12} (Q_{16} + Q_{17} )) $$

$$ \begin{aligned} {E}_{5} = & - 2{\text{e}}^{{h_{1} \alpha_{2} }} \sinh (h_{1} \alpha_{2} )G_{y}^{2} \left( {g^{2} s^{2} - M_{2}^{2} } \right)T_{5} T_{2} + 2{\text{e}}^{{h_{1} \alpha_{2} }} \tilde{c}_{44} G_{y} ( - {\text{e}}^{{2\lambda h_{1} }} \sinh (h_{3} \alpha_{1} )k_{13} T_{3} T_{2} + \sinh (h_{2} \alpha_{1} )k_{12} T_{5} T_{4} )\alpha_{1} \\ & - 2{\text{e}}^{{2\lambda h_{1} + h_{1} \alpha_{2} }} \sinh (h_{2} \alpha_{1} )\sinh (h_{1} \alpha_{2} )\sinh (h_{3} \alpha_{1} )\tilde{c}_{44}^{2} k_{12} k_{13} \alpha_{1}^{2} \\ \end{aligned} $$

$$ {E}_{6} = {\text{e}}^{{ - (3\lambda + \alpha_{2} )h_{1} }} [{\text{e}}^{{2\lambda h_{1} }} k_{13} + G_{y} ( - \lambda + \alpha_{2} )](Q_{18} + \tilde{c}_{44} G_{y} \alpha_{1} (Q_{19} + k_{12} \{ Q_{20} + k_{13} [Q_{21} + 2{\text{e}}^{{h_{1} \lambda }} Q_{22} \alpha_{2} ]\} ))(\beta_{11}^{2} - d_{11} \gamma_{11} )^{5} $$

$$ \begin{aligned} {E}_{7} = & \cosh (\alpha_{1} (h_{1} + h_{3} ))\{ Q_{23} + Q_{24} [Q_{25} + \sinh (h_{2} \alpha_{1} )\tilde{c}_{44} \alpha_{1} Q_{26} ]\} (\beta_{11}^{2} - d_{11} \gamma_{11} )^{5} \\ & + \sinh (\alpha_{1} (h_{1} + h_{3} ))\{ Q_{27} [Q_{28} + \sinh (h_{2} \alpha_{1} )\tilde{c}_{44} \alpha Q_{29} ](\beta_{11}^{2} - d_{11} \gamma_{11} )^{5} + Q_{30} \} \\ \end{aligned} $$

$$ {E}_{8} = \frac{{D_{0} \gamma_{11} - B_{0} \beta_{11} }}{{d_{11} \gamma_{11} - \beta_{11}^{2} }} $$

$$ {E}_{9} = \frac{{ - D_{0} \beta_{11} + B_{0} d_{11} }}{{d_{11} \gamma_{11} - \beta_{11}^{2} }} $$

$$ \begin{aligned} E_{10} = & s( - {\text{e}}^{{\frac{{\alpha_{2} h_{1} }}{2}}} G_{y} k_{12} T_{5} T_{16} + {\text{e}}^{{\lambda h_{1} }} k_{13} ((\sinh (h_{2} \alpha_{1} )\sinh (y\alpha_{2} ) \\ + {\text{e}}^{{\lambda h_{1} }} \sinh (h_{3} \alpha_{1} )\sinh (\alpha_{2} (y + h_{1} ))\tilde{c}_{44} k_{12} \alpha_{1} + G_{y} T_{2} T_{17} ))(\varGamma_{3} B_{0} + \varGamma_{2} D_{0} + \tau_{0} )) \\ \end{aligned} $$

$$ \begin{aligned} E_{11} = - (e^{{2\lambda h_{1} }} \sinh (h_{1} \alpha_{2} )\sinh (h_{2} \alpha_{1} )\sinh (h_{3} \alpha_{1} )\tilde{c}_{44}^{2} k_{12} k_{13} \alpha_{1}^{2} + \sinh (h_{1} \alpha_{2} )G_{y}^{2} T_{2} \hfill \\ T_{5} (g^{2} s^{2} - M_{2}^{2} ) + \tilde{c}_{44} G_{y} \alpha_{1} (e^{{2\lambda h_{1} }} \sinh (h_{3} \alpha_{1} )k_{13} T_{2} T_{14} - \sinh (h_{2} \alpha_{1} )k_{12} - T_{5} T_{15} )) \hfill \\ \end{aligned} $$

$$ \begin{aligned} E_{12} = & ( - G_{y} ({\text{e}}^{{\lambda h_{1} }} T_{2} \sinh (\alpha_{2} y)k_{13} - {\text{e}}^{{\frac{{\alpha_{2} h_{1} }}{2}}} \sinh \left( {\alpha_{2} (y + \frac{{h_{1} }}{2}))k_{12} T_{5} } \right)(g^{2} s^{2} - M_{2}^{2} ) \\ \quad + {\text{e}}^{{\lambda h_{1} }} \tilde{c}_{44}^{2} k_{12} k_{13} \alpha_{1} (\sinh (h_{2} \alpha_{1} )(T_{12} ) + {\text{e}}^{{\lambda h_{1} }} \sinh (h_{3} \alpha_{1} )(T_{13} )))(\varGamma_{3} B_{0} + \varGamma_{2} D_{0} + \tau_{0} ) \\ \end{aligned} $$

$$ Q_{1} = 2{\text{e}}^{{2\lambda h_{1} + h_{1} \alpha_{2} }} \cosh (\alpha_{1} y)\sinh (h_{3} \alpha_{1} )\sinh (h_{1} \alpha_{2} )\tilde{c}_{44}^{2} k_{12} k_{13} \alpha_{1}^{2} - 2{\text{e}}^{{h_{1} \alpha_{2} }} \sinh (h_{1} \alpha_{2} )G_{y}^{2} (g^{2} s^{2} - M_{2}^{2} )T_{5} T_{10} $$

$$ Q_{2} = 2{\text{e}}^{{(2\lambda + \alpha_{2} )h_{1} }} \cosh (\alpha_{1} y)\sinh (h_{3} \alpha_{1} )\tilde{c}_{44} k_{13} \alpha_{1} T_{3} $$

$$ Q_{3} = - \cosh (\alpha_{1} y)\sinh (h_{3} \alpha_{1} )\tilde{c}_{44} \alpha_{1} T_{7} $$

$$ Q_{4} = 2{\text{e}}^{{h_{1} \alpha_{2} }} \lambda \sinh (h_{1} \alpha_{2} )[\cosh (\alpha_{1} y)\cosh (h_{3} \alpha_{1} ) - {\text{e}}^{{2\lambda h_{1} }} \sinh (\alpha_{1} y)\sinh (h_{3} \alpha_{1} )] $$

$$ Q_{5} = \{ - 2{\text{e}}^{{(\lambda + \alpha_{2} )h_{1} }} \cosh (\alpha_{1} (y - h_{2} )) + 2{\text{e}}^{{h_{1} \alpha_{2} }} \cosh (\alpha_{2} h_{1} )[\cosh (\alpha_{1} y)\cosh (h_{3} \alpha_{1} ) + {\text{e}}^{{2h_{1} \lambda }} \sinh (\alpha_{1} y)\sinh (h_{3} \alpha_{1} )]\} $$

$$ Q_{6} = 2{\text{e}}^{{2\lambda h_{1} + h_{1} \alpha_{2} }} \sinh (h_{1} \alpha_{2} )\sinh (h_{3} \alpha_{1} )\tilde{c}_{44}^{2} k_{12} k_{13} \alpha_{1}^{2} $$

$$ Q_{7} = \sinh (h_{1} \alpha_{2} )G_{y}^{2} 2{\text{e}}^{{h_{1} \alpha_{2} }} (g^{2} s^{2} - M_{2}^{2} )T_{5} [\coth (h_{2} \alpha_{1} )k_{12} - \tilde{c}_{44} \alpha_{1} ] $$

$$ Q_{8} = {\text{e}}^{{2\lambda h_{1} }} \sinh (h_{3} \alpha_{1} )k_{13} \tilde{c}_{44} \alpha_{1} [2{\text{e}}^{{h_{1} \alpha_{2} }} \lambda \sinh (\alpha_{2} h_{1} ) - \alpha_{2} \cosh (\alpha_{2} h_{1} )] $$

$$ Q_{9} = k_{12} [ - {\text{e}}^{{2\lambda h_{1} }} \coth (h_{2} \alpha_{1} )\sinh (h_{3} \alpha_{1} )k_{13} T_{3} - T_{5} T_{7} ] $$

$$ Q_{10} = 2{\text{e}}^{{2\lambda h_{1} + h_{1} \alpha_{2} }} \sinh (\alpha_{1} y)\sinh (h_{3} \alpha_{1} )\sinh (h_{1} \alpha_{2} )\tilde{c}_{44}^{2} k_{12} k_{13} \alpha_{1}^{2} + 2{\text{e}}^{{h_{1} \alpha_{2} }} \sinh (h_{1} \alpha_{2} )G_{y}^{2} (g^{2} s^{2} - M_{2}^{2} )T_{5} T_{11} $$

$$ Q_{11} = \tilde{c}_{44} G_{y} \alpha_{1} (2{\text{e}}^{{(2\lambda + \alpha_{2} )h_{1} }} \sinh (\alpha_{1} y)\sinh (h_{3} \alpha_{1} )\tilde{c}_{44} k_{13} \alpha_{1} T_{3} + k_{12} ( - \sinh (\alpha_{1} y)\sinh (h_{3} \alpha_{1} )\tilde{c}_{44} \alpha_{1} T_{7} $$

$$ Q_{12} = 2{\text{e}}^{{h_{1} \alpha_{2} }} \lambda \sinh (h_{1} \alpha_{2} )[\sinh (\alpha_{1} y)\cosh (h_{3} \alpha_{1} ) - {\text{e}}^{{2\lambda h_{1} }} \cosh (\alpha_{1} y)\sinh (h_{3} \alpha_{1} )] $$

$$ Q_{13} = - 2{\text{e}}^{{(\lambda + \alpha_{2} )h_{1} }} \sinh (\alpha_{1} (y - h_{2} )) + 2{\text{e}}^{{h_{1} \alpha_{2} }} \cosh (\alpha_{2} h_{1} )[\sinh (\alpha_{1} y)\cosh (h_{3} \alpha_{1} ) + {\text{e}}^{{2h_{1} \lambda }} \cosh (\alpha_{1} y)\sinh (h_{3} \alpha_{1} )] $$

$$ Q_{14} = - 2{\text{e}}^{{h_{1} \alpha_{2} }} \sinh (h_{1} \alpha_{2} )G_{y}^{2} (g^{2} s^{2} - M_{2}^{2} )T_{1} T_{2} - 2{\text{e}}^{{2\lambda h_{1} + h_{1} \alpha_{2} }} \cosh (\alpha_{1} h_{1} )\sinh (h_{2} \alpha_{1} )\sinh (h_{1} \alpha_{2} )\tilde{c}_{44}^{2} k_{12} k_{13} \alpha_{1}^{2} $$

$$ Q_{15} = - 2{\text{e}}^{{(2\lambda + \alpha_{2} )h_{1} }} \cosh (\alpha_{1} h_{1} )\sinh (h_{2} \alpha_{1} )\tilde{c}_{44} k_{13} \alpha_{1} T_{3} $$

$$ Q_{16} = 2{\text{e}}^{{h_{1} \alpha_{2} }} \cosh (\alpha_{1} h_{1} )\sinh (h_{2} \alpha_{1} )\tilde{c}_{44} \alpha_{1} T_{4} $$

$$ Q_{17} = k_{13} (2{\text{e}}^{{h_{1} \alpha_{2} }} \lambda \sinh (h_{1} \alpha_{2} )T_{8} + [2{\text{e}}^{{(\lambda + \alpha_{2} )h_{1} }} \cosh (\alpha_{1} (h_{1} + h_{3} )) - 2{\text{e}}^{{h_{1} \alpha_{2} }} \sinh (\alpha_{2} h_{1} )T_{9} ]\alpha_{2} ) $$

$$ Q_{18} = - 2{\text{e}}^{{\lambda h_{1} }} \sinh (h_{1} \alpha_{2} )G_{y}^{2} ( - g^{2} s^{2} + M_{2}^{2} )T_{2} T_{6} - 2{\text{e}}^{{3\lambda h_{1} }} \sinh (h_{1} \alpha_{1} )\sinh (h_{2} \alpha_{1} )\sinh (h_{1} \alpha_{2} )\tilde{c}_{44}^{2} k_{12} k_{13} \alpha_{1}^{2} $$

$$ Q_{19} = - 2{\text{e}}^{{3\lambda h_{1} }} \sinh (\alpha_{1} h_{1} )\sinh (h_{2} \alpha_{1} )\tilde{c}_{44} k_{13} \alpha_{1} T_{3} $$

$$ Q_{20} = 2{\text{e}}^{{h_{1} \lambda }} \sinh (\alpha_{1} h_{1} )\sinh (h_{2} \alpha_{1} )\tilde{c}_{44} \alpha_{1} T_{4} $$

$$ Q_{21} = 2{\text{e}}^{{\lambda h_{1} }} \lambda \sinh (h_{1} \alpha_{2} )[{\text{e}}^{{2\lambda h_{1} }} \sinh (\alpha_{1} h_{1} )\cosh (h_{2} \alpha_{1} ) + \cosh (\alpha_{1} h_{1} )\sinh (h_{2} \alpha_{1} )] $$

$$ Q_{22} = {\text{e}}^{{2\lambda h_{1} }} \cosh (h_{2} \alpha_{1} )\cosh (\alpha_{2} h_{1} )\sinh (\alpha_{1} h_{1} ) + \cosh (\alpha_{1} h_{1} )\cosh (\alpha_{2} h_{1} )\sinh (\alpha_{1} h_{2} ) + {\text{e}}^{{h_{1} \lambda }} \sinh (\alpha_{1} (h_{1} + h_{3} )) $$

$$ Q_{23} = 2\cosh (\alpha_{1} h_{1} )k_{13}^{2} \alpha_{2} G_{y} \{ \sinh (h_{2} \alpha_{1} )\tilde{c}_{44} \alpha_{1} [k_{12} + G_{y} (\lambda - \alpha_{2} )] + \cosh (h_{2} \alpha_{1} )G_{y} k_{12} ( - \lambda + \alpha_{2} )\} $$

$$ Q_{24} = 2{\text{e}}^{{ - (2\lambda + \alpha_{2} )h_{1} }} \{ \sinh (\alpha_{1} h_{1} )\tilde{c}_{44} \alpha_{1} [ - {\text{e}}^{{2\lambda h_{1} }} k_{13} + G_{y} (\lambda - \alpha_{2} )] + \cosh (\alpha_{1} h_{1} )G_{y} k_{13} (\lambda - \alpha_{2} )\} $$

$$ Q_{25} = \cosh (h_{2} \alpha_{1} )G_{y} k_{12} [\sinh (h_{1} \alpha_{2} )G_{y} ( - g^{2} s^{2} + M_{2}^{2} ) - {\text{e}}^{{2\lambda h_{1} }} T_{3} k_{13} ] $$

$$ Q_{26} = \{ - \cosh (h_{1} \alpha_{2} )G_{y} (k_{12} + {\text{e}}^{{2\lambda h_{1} }} k_{13} )\alpha_{2} + \sinh (h_{1} \alpha_{2} )\{ {\text{e}}^{{2\lambda h_{1} }} (k_{12} + G_{y} \lambda )k_{13} + G_{y} [ - \lambda k_{12} + G_{y} (g^{2} s^{2} - M_{2}^{2} )]\} $$

$$ Q_{27} = - 2{\text{e}}^{{ - (2\lambda + \alpha_{2} )h_{1} }} \{ \cosh (\alpha_{1} h_{1} )\tilde{c}_{44} \alpha_{1} [ - {\text{e}}^{{2\lambda h_{1} }} k_{13} + G_{y} (\lambda - \alpha_{2} )] + \sinh (\alpha_{1} h_{1} )G_{y} k_{13} (\lambda - \alpha_{2} )\} $$

$$ Q_{28} = \cosh (h_{2} \alpha_{1} )G_{y} k_{12} [\sinh (h_{1} \alpha_{2} )G_{y} (g^{2} s^{2} - M_{2}^{2} ) + {\text{e}}^{{2\lambda h_{1} }} T_{3} k_{13} ] $$

$$ Q_{29} = \sinh (h_{1} \alpha_{2} )[ - {\text{e}}^{{2\lambda h_{1} }} (k_{12} + G_{y} \lambda )k_{13} + G_{y} (\lambda k_{12} - G_{y} (g^{2} s^{2} - M_{2}^{2} ))] + \cosh (h_{1} \alpha_{2} )G_{y} (k_{12} + {\text{e}}^{{2\lambda h_{1} }} k_{13} )\alpha_{2} $$

$$ Q_{30} = - 2\sinh (\alpha_{1} h_{1} )k_{13}^{2} \alpha_{2} G_{y} [\sinh (h_{2} \alpha_{1} )\tilde{c}_{44} \alpha_{1} (k_{12} + G_{y} (\lambda - \alpha_{2} )) + \cosh (h_{2} \alpha_{1} )G_{y} k_{12} ( - \lambda + \alpha_{2} )](\beta_{11}^{2} - d_{11} \gamma_{11} )^{5} $$

$$ T_{1} = \sinh (\alpha_{1} h_{1} )k_{13} + \cosh (\alpha_{1} h_{1} )\tilde{c}_{44} \alpha_{1} $$

$$ T_{2} = \sinh (\alpha_{1} h_{2} )\tilde{c}_{44} \alpha_{1} - \cosh (\alpha_{1} h_{2} )k_{12} $$

$$ T_{3} = \sinh (\alpha_{1} h_{2} )\lambda - \cosh (\alpha_{1} h_{2} )\alpha_{2} $$

$$ T_{4} = \sinh (\alpha_{1} h_{2} )\lambda + \cosh (\alpha_{1} h_{2} )\alpha_{2} $$

$$ T_{5} = \sinh (\alpha_{1} h_{3} )\tilde{c}_{44} \alpha_{1} - \cosh (\alpha_{1} h_{3} )k_{13} $$

$$ T_{6} = - \sinh (\alpha_{1} h_{1} )\tilde{c}_{44} \alpha_{1} - \cosh (\alpha_{1} h_{1} )k_{13} $$

$$ T_{7} = 2{\text{e}}^{{\alpha_{2} h_{1} }} \sinh (\alpha_{2} h_{1} )\lambda + \cosh (\alpha_{2} h_{1} )\alpha_{2} $$

$$ T_{8} = \sinh (\alpha_{1} h_{1} )\sinh (\alpha_{1} h_{2} ) + {\text{e}}^{{2\lambda h_{1} }} \cosh (\alpha_{1} h_{1} )\cosh (\alpha_{1} h_{2} ) $$

$$ T_{9} = - \sinh (\alpha_{1} h_{1} )\sinh (\alpha_{1} h_{2} ) + {\text{e}}^{{2\lambda h_{1} }} \cosh (\alpha_{1} h_{1} )\cosh (\alpha_{1} h_{2} ) $$

$$ T_{10} = \sinh (\alpha_{1} y)k_{12} - \cosh (\alpha_{1} y)\tilde{c}_{44} \alpha_{1} $$

$$ T_{11} = \sinh (\alpha_{1} y)\tilde{c}_{44} \alpha_{1} - \cosh (\alpha_{1} y)k_{12} $$

$$ T_{12} = \lambda \sinh (\alpha_{2} y) - \alpha_{2} \cosh (\alpha_{2} y) $$

$$ T_{13} = \lambda \sinh (\alpha_{2} (y + h_{1} )) - \alpha_{2} \cosh (\alpha_{2} (y + h_{1} )) $$

$$ T_{14} = \lambda \sinh (\alpha_{2} h_{1} ) - \alpha_{2} \cosh (\alpha_{2} h_{1} ) $$

$$ T_{15} = \alpha_{2} \cosh (\alpha_{2} h_{1} ) + \lambda \sinh (\alpha_{2} h_{1} ) $$

$$ T_{16} = \alpha_{2} \cosh \left( {\alpha_{2} \left( {y + \frac{{h_{1} }}{2}} \right)) + \lambda \sinh (\alpha_{2} \left( {y + \frac{{h_{1} }}{2}} \right)} \right) $$

$$ T_{17} = \lambda \sinh (\alpha_{2} y) + \alpha_{2} \cosh (\alpha_{2} y) $$

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R. Bagheri
- 1
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A. M. Mirzaei
- 2

1.Department of Mechanical Engineering, Mechatronics Faculty, Karaj BranchIslamic Azad UniversityAlborzIran
2.Miyaneh Technology and Engineering FacultyUniversity of TabrizMiyaneh, TabrizIran

Fracture Analysis in an Imperfect FGM Orthotropic Strip Bonded Between Two Magneto-Electro-Elastic Layers

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