Mathematical Analysis of Two Unequal Collinear Cracks in a Piezo-Electro-Magnetic Media

Abstract

In this chapter, we begin our work of studying two unequal collinear semi-permeable cracks in a magneto-electro-elastic media. We employ the Stroh’s formalism and complex variable technique to solve the physical problem. We derive the closed form analytic solutions for various fracture parameters, and study the effect of volume fraction and inter-crack distance on these parameters.

Appendices

Appendix (A)

$X_{1}(z)=\sqrt{(z-a)(z-b)(z-c)(z-d)},$         $P_{1}(z)=C_{0}z^{2}+C_{1}(z)+C_{2};$

$\triangle =\Lambda _{22}(\Lambda _{44}\Lambda _{55}-\Lambda _{45}\Lambda _{54}) - \Lambda _{24}(\Lambda _{42}\Lambda _{55}-\Lambda _{45}\Lambda _{52}) + \Lambda _{25}(\Lambda _{42}\Lambda _{54}-\Lambda _{44}\Lambda _{52});$

$\triangle _{1}=-\sigma _{22}^{\infty }(\Lambda _{44}\Lambda _{55}-\Lambda _{45}\Lambda _{54}) - (D^{c}-D_{2}^{\infty })(\Lambda _{24}\Lambda _{55}-\Lambda _{25}\Lambda _{54}) + (B^{c}-B_{2}^{\infty })(\Lambda _{25}\Lambda _{44}-\Lambda _{24}\Lambda _{45});$

$\triangle _{2}=\sigma _{22}^{\infty }(\Lambda _{42}\Lambda _{55}-\Lambda _{45}\Lambda _{52}) + (D^{c}-D_{2}^{\infty })(\Lambda _{22}\Lambda _{55}-\Lambda _{25}\Lambda _{52}) + (B^{c}-B_{2}^{\infty })(\Lambda _{25}\Lambda _{42}-\Lambda _{22}\Lambda _{45});$

$\triangle _{3}=\sigma _{22}^{\infty }(\Lambda _{44}\Lambda _{52}-\Lambda _{42}\Lambda _{54}) + (D^{c}-D_{2}^{\infty })(\Lambda _{24}\Lambda _{52}-\Lambda _{22}\Lambda _{54}) + (B^{c}-B_{2}^{\infty })(\Lambda _{22}\Lambda _{44}-\Lambda _{24}\Lambda _{42});$

$C_{0}=a_{11}a_{22}-a_{12}a_{21},$    $C_{1}=a_{20}a_{12}-a_{10}a_{22},$    $C_{2}=a_{21}a_{10}-a_{11}a_{20},$    $k^{2}=\dfrac{(a-b)(c-d)}{(a-c)(b-d)};$

$g=\dfrac{2}{\sqrt{(a-c)(b-d)}},$    $\alpha ^{2}=\dfrac{d-c}{a-c},$    $\beta ^{2}=\dfrac{a-b}{a-c},$    $a_{11}=g[aF(k)+(d-a){{\Pi }}(\alpha ^{2},k)];$

$a_{12}=gF(k),$         $a_{21}=g[cF(k)+(b-c){{\Pi }}(\beta ^{2},k)],$         $a_{22}=gF(k);$

$a_{10}=g\left[ a^{2}F(k)+2a(d-a){{\Pi }}(\alpha ^{2},k)+(d-a)^{2}V_{2}\right] ;$

$a_{20}=g\left[ c^{2}F(k)+2c(b-c){{\Pi }}(\beta ^{2},k)+(b-c)^{2}V_{3}\right] ;$

$V_{2}=\dfrac{1}{2(\alpha ^{2}-1)(k^{2}-\alpha ^{2})}\left\{ \alpha ^{2}E(k)+(k^{2}-\alpha ^{2})F(k)+(2\alpha ^{2}k^{2}+2\alpha ^{2}-\alpha ^{4}-3k^{2}){{\Pi }}(\alpha ^{2},k)\right\} ;$

$V_{3}=\dfrac{1}{2(\beta ^{2}-1)(k^{2}-\beta ^{2})}\left\{ \beta ^{2}E(k)+(k^{2}-\beta ^{2})F(k)+(2\beta ^{2}k^{2}+2\beta ^{2}-\beta ^{4}-3k^{2}){{\Pi }}(\beta ^{2},k)\right\} ;$

where F(k),  E(K) and ${{\Pi }}(\alpha ^{2},k)$ are the complete elliptic integrals of the first, second and third kind, respectively.

Appendix (B)

$\alpha _{1}^{2}=\dfrac{a}{d}\alpha ^{2},$    $\beta _{1}^{2}=\dfrac{c}{b}\beta ^{2},$    $\nu =sin^{-1}\sqrt{\dfrac{(a-c)(y-d)}{(d-c)(a-y)}},$    $\psi =sin^{-1}\sqrt{\dfrac{(a-c)(y-b)}{(a-b)(y-c)}};$

$S_{1}=\alpha ^{2}E(\nu ,k)+(k^{2}-\alpha ^{2})F(\nu ,k)+(2\alpha ^{2}k^{2}+2\alpha ^{2}-\alpha ^{4}-3k^{2}){{\Pi }}(\nu ,\alpha ^{2},k) - \dfrac{\alpha ^{4}\text {sn}{u}\text {cn}{u}\text {dn}{u}}{1-\alpha ^{2}\text {sn}^{2}{u}};$

where snu, cnu and dnu are the Jacobian elliptic functions.

$S_{2}=\beta ^{2}E(\psi ,k)+(k^{2}-\beta ^{2})F(\psi ,k)+(2\beta ^{2}k^{2}+2\beta ^{2}-\beta ^{4}-3k^{2}){{\Pi }}(\psi ,\beta ^{2},k) - \dfrac{\beta ^{4}\text {sn}{u}\text {cn}{u}\text {dn}{u}}{1-\beta ^{2}\text {sn}^{2}{u}};$

$S_{3}=d^{2}g\dfrac{\alpha _{1}^{4}}{\alpha ^{4}}\left\{ F(\nu ,k)+\dfrac{2(\alpha ^{2}-\alpha _{1}^{2})}{\alpha _{1}^{2}}{{\Pi }}(\nu ,\alpha ^{2},k)+ \dfrac{(\alpha ^{2}-\alpha _{1}^{2})^{2}}{2\alpha _{1}^{4}(\alpha ^{2}-1)(k^2-\alpha ^{2})}S_{1}\right\} ;$

$S_{4}=dg\dfrac{\alpha _{1}^{2}}{\alpha ^{2}}\left\{ F(\nu ,k)+\dfrac{\alpha ^{2}-\alpha _{1}^{2}}{\alpha _{1}^{2}}{{\Pi }}(\nu ,\alpha ^{2},k)\right\} ,$         $S_{5}=gF(\nu ,k);$

$S_{6}=b^{2}g\dfrac{\beta _{1}^{4}}{\beta ^{4}}\left\{ F(\psi ,k)+\dfrac{2(\beta ^{2}-\beta _{1}^{2})}{\beta _{1}^{2}}{{\Pi }}(\psi ,\beta ^{2},k)+ \dfrac{(\beta ^{2}-\beta _{1}^{2})^{2}}{2\beta _{1}^{4}(\beta ^{2}-1)(k^2-\beta ^{2})}S_{2}\right\} ;$

$S_{7}=bg\dfrac{\beta _{1}^{2}}{\beta ^{2}}\left\{ F(\psi ,k)+\dfrac{\beta ^{2}-\beta _{1}^{2}}{\beta _{1}^{2}}{{\Pi }}(\psi ,\beta ^{2},k)\right\} ,$         $S_{8}=gF(\psi ,k);$

where F(, k),  E(, k) and ${{\Pi }}(,\,\,k)$ are the incomplete elliptic integrals of first, second and third kinds, respectively.