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The mixed-mode analysis of a functionally graded orthotropic half-plane weakened by multiple curved cracks

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Abstract

The problem of functionally graded orthotropic half-plane with climb and glide edge dislocations is solved. Dislocations are used as the building blocks of defects to model cracks of modes I and II. Following a dislocation-based approach, the problem is reduced to a system of singular integral equations for dislocation density functions on the surfaces of smooth cracks. These integral equations enforce the crack-face boundary conditions and are solved numerically for the dislocation density. The numerical results include the stress intensity factors for several different cases of crack configurations and arrangements.

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Authors and Affiliations

  1. Faculty of Engineering, University of Zanjan, P. O. Box 45195-313, Zanjan, Iran

    M. M. Monfared & M. Ayatollahi

  2. Department of Civil and Structural Engineering, Aalto University, PO Box 12100, 00076, Espoo, Finland

    S. M. Mousavi

Authors
  1. M. M. Monfared
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  2. M. Ayatollahi
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  3. S. M. Mousavi
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Correspondence to M. Ayatollahi.

Appendices

Appendix A

The integrands of Eq. (18) are given as

$$\begin{aligned} H_{xx1} (x,y,\omega )= & {} \frac{\beta C_{660} (\alpha _2 +2\alpha _3 )}{\pi }\frac{1}{(r_1 -r_2 )}\left\{ b_x \left[ \frac{r_4 r_2^2 \mathrm{e}^{r_2 y\beta }}{(r_2 -r_3 )(r_2 -r_4 )}-\frac{r_3 r_2^2 \mathrm{e}^{r_1 y\beta }}{(r_1 -r_3 )(r_1 -r_4 )}\right. \right. \nonumber \\&+\,\frac{r_3^3 (r_2 -r_3 )\mathrm{e}^{(r_1 -r_3 )\beta h+r_3 y\beta }-r_4 r_3^2 (r_1 -r_3 )\mathrm{e}^{(r_2 -r_3 )\beta h+r_3 y\beta }}{(r_1 -r_3 )(r_2 -r_3 )(r_3 -r_4 )}\nonumber \\&\left. +\,\frac{r_3 r_4^2 (r_2 -r_4 )\mathrm{e}^{(r_1 -r_4 )\beta h+r_4 y\beta }-r_4^3 (r_1 -r_4 )\mathrm{e}^{(r_2 -r_4 )\beta h+r_4 y\beta }}{(r_4 -r_3 )(r_2 -r_4 )(r_1 -r_4 )}\right] \cos (\beta \omega x)\nonumber \\&+\,\omega b_y \left[ \frac{r_1^2 \mathrm{e}^{r_1 y\beta }}{(r_1 -r_3 )(r_1 -r_4 )}-\frac{r_2^2 \mathrm{e}^{r_2 y\beta }}{(r_2 -r_3 )(r_2 -r_4 )}\right. \nonumber \\&+\,\frac{r_3^2 (r_1 -r_3 )\mathrm{e}^{(r_2 -r_3 )\beta h+r_3 y\beta }-r_3^2 (r_2 -r_3 )\mathrm{e}^{(r_1 -r_3 )\beta h+r_3 y\beta }}{(r_1 -r_3 )(r_2 -r_3 )(r_3 -r_4 )}\nonumber \\&+\,\left. \left. \frac{r_4^2 (r_1 -r_4 )\mathrm{e}^{(r_2 -r_4 )\beta h+r_4 y\beta }-r_4^2 (r_2 -r_4 )\mathrm{e}^{(r_1 -r_4 )\beta h+r_4 y\beta }}{(r_4 -r_3 )(r_2 -r_4 )(r_1 -r_4 )}\right] \sin (\beta \omega x) \right\} , \end{aligned}$$
(34)
$$\begin{aligned} H_{yy1} (x,y,\omega )= & {} -\frac{\beta C_{660} (\alpha _2 +2\alpha _3 )}{\pi }\frac{\omega ^{2}}{(r_1 -r_2 )}\left\{ b_x \left[ \frac{r_4 \mathrm{e}^{r_2 y\beta }}{(r_2 -r_3 )(r_2 -r_4 )}-\frac{r_3 \mathrm{e}^{r_1 y\beta }}{(r_1 -r_3 )(r_1 -r_4 )}\right. \right. \nonumber \\&+\,\frac{r_3 (r_2 -r_3 )\mathrm{e}^{(r_1 -r_3 )\beta h+r_3 y\beta }-r_4 (r_1 -r_3 )\mathrm{e}^{(r_2 -r_3 )\beta h+r_3 y\beta }}{(r_1 -r_3 )(r_2 -r_3 )(r_3 -r_4 )} \nonumber \\&\left. +\,\frac{r_3 (r_2 -r_4 )\mathrm{e}^{(r_1 -r_4 )\beta h+r_4 y\beta }-r_4 (r_1 -r_4 )\mathrm{e}^{(r_2 -r_4 )\beta h+r_4 y\beta }}{(r_4 -r_3 )(r_2 -r_4 )(r_1 -r_4 )}\right] \cos (\beta \omega x) \nonumber \\&+\,\omega b_y \left[ \frac{\mathrm{e}^{r_1 y\beta }}{(r_1 -r_3 )(r_1 -r_4 )}-\frac{\mathrm{e}^{r_2 y\beta }}{(r_2 -r_3 )(r_2 -r_4 )}\right. \nonumber \\&+\,\frac{(r_1 -r_3 )\mathrm{e}^{(r_2 -r_3 )\beta h+r_3 y\beta }-(r_2 -r_3 )\mathrm{e}^{(r_1 -r_3 )\beta h+r_3 y\beta }}{(r_1 -r_3 )(r_2 -r_3 )(r_3 -r_4 )}\nonumber \\&+\,\left. \left. \frac{(r_1 -r_4 )\mathrm{e}^{(r_2 -r_4 )\beta h+r_4 y\beta }-(r_2 -r_4 )\mathrm{e}^{(r_1 -r_4 )\beta h+r_4 y\beta }}{(r_4 -r_3 )(r_2 -r_4 )(r_1 -r_4 )}\right] \sin (\beta \omega x) \right\} , \end{aligned}$$
(35)
$$\begin{aligned} H_{xy1} (x,y,\omega )= & {} \frac{\beta C_{660} (\alpha _2 +2\alpha _3 )}{\pi }\frac{\omega }{(r_1 -r_2 )}\left\{ b_x \left[ \frac{r_2 r_4 \mathrm{e}^{r_2 y\beta }}{(r_2 -r_3 )(r_2 -r_4 )}-\frac{r_1 r_3 \mathrm{e}^{r_1 y\beta }}{(r_1 -r_3 )(r_1 -r_4 )}\right. \right. \nonumber \\&+\,\frac{r_3^2 (r_2 -r_3 )\mathrm{e}^{(r_1 -r_3 )\beta h+r_3 y\beta }-r_3 r_4 (r_1 -r_3 )\mathrm{e}^{(r_2 -r_3 )\beta h+r_3 y\beta }}{(r_1 -r_3 )(r_2 -r_3 )(r_3 -r_4 )}\nonumber \\&\left. +\,\frac{r_3 r_4 (r_2 -r_4 )\mathrm{e}^{(r_1 -r_4 )\beta h+r_4 y\beta }-r_4^2 (r_1 -r_4 )\mathrm{e}^{(r_2 -r_4 )\beta h+r_4 y\beta }}{(r_4 -r_3 )(r_2 -r_4 )(r_1 -r_4 )}\right] \sin (\beta \omega x)\nonumber \\&-\,\omega b_y \left[ \frac{r_1 \mathrm{e}^{r_1 y\beta }}{(r_1 -r_3 )(r_1 -r_4 )}-\frac{r_2 \mathrm{e}^{r_2 y\beta }}{(r_2 -r_3 )(r_2 -r_4 )}\right. \nonumber \\&+\,\frac{r_3 (r_1 -r_3 )\mathrm{e}^{(r_2 -r_3 )\beta h+r_3 y\beta }-r_3 (r_2 -r_3 )\mathrm{e}^{(r_1 -r_3 )\beta h+r_3 y\beta }}{(r_1 -r_3 )(r_2 -r_3 )(r_3 -r_4 )}\nonumber \\&+\,\left. \left. \frac{r_4 (r_1 -r_4 )\mathrm{e}^{(r_2 -r_4 )\beta h+r_4 y\beta }-r_4 (r_2 -r_4 )\mathrm{e}^{(r_1 -r_4 )\beta h+r_4 y\beta }}{(r_4 -r_3 )(r_2 -r_4 )(r_1 -r_4 )}\right] \cos (\beta \omega x) \right\} , \end{aligned}$$
(36)
$$\begin{aligned} H_{xx2} (x,y,\omega )= & {} \frac{\beta C_{660} (\alpha _2 +2\alpha _3 )}{\pi }\frac{1}{(r_1 -r_2 )(r_4 -r_3 )}\left\{ b_x \left[ \frac{r_3^3 (r_2 -r_3 )(1-\mathrm{e}^{(r_1 -r_3 )\beta h})-r_4 r_3^2 (r_1 -r_3 )(1-\mathrm{e}^{(r_2 -r_3 )\beta h})}{(r_2 -r_3 )(r_1 -r_3 )}\mathrm{e}^{r_3 y\beta } \right. \right. \nonumber \\&\left. +\,\frac{r_4^3 (r_1 -r_4 )(1-\mathrm{e}^{(r_2 -r_4 )\beta h})-r_3 r_4^2 (r_2 -r_4 )(1-\mathrm{e}^{(r_1 -r_4 )\beta h})}{(r_2 -r_4 )(r_1 -r_4 )}\mathrm{e}^{r_4 y\beta }\right] \cos (\beta \omega x)\nonumber \\&+\,\omega b_y \left[ \frac{r_3^2 (r_1 -r_3 )(1-\mathrm{e}^{(r_2 -r_3 )\beta h})-r_3^2 (r_2 -r_3 )(1-\mathrm{e}^{(r_1 -r_3 )\beta h})}{(r_2 -r_3 )(r_1 -r_3 )}\mathrm{e}^{r_3 y\beta }\right. \nonumber \\&\left. +\,\left. \frac{r_4^2 (r_2 -r_4 )(1-\mathrm{e}^{(r_1 -r_4 )\beta h})-r_4^2 (r_1 -r_4 )(1-\mathrm{e}^{(r_2 -r_4 )\beta h})}{(r_2 -r_4 )(r_1 -r_4 )}\mathrm{e}^{r_4 y\beta }\right] \sin (\beta \omega x) \right\} , \end{aligned}$$
(37)
$$\begin{aligned} H_{yy2} (x,y,\omega )= & {} -\frac{\beta C_{660} (\alpha _2 +2\alpha _3 )}{\pi }\frac{\omega ^{2}}{(r_1 -r_2 )(r_4 -r_3 )}\left\{ b_x \left[ \frac{r_3 (r_2 -r_3 )(1-\mathrm{e}^{(r_1 -r_3 )\beta h})-r_4 (r_1 -r_3 )(1-\mathrm{e}^{(r_2 -r_3 )\beta h})}{(r_2 -r_3 )(r_1 -r_3 )}\mathrm{e}^{r_3 y\beta } \right. \right. \nonumber \\&\left. +\,\frac{r_4 (r_1 -r_4 )(1-\mathrm{e}^{(r_2 -r_4 )\beta h})-r_3 (r_2 -r_4 )(1-\mathrm{e}^{(r_1 -r_4 )\beta h})}{(r_2 -r_4 )(r_1 -r_4 )}\mathrm{e}^{r_4 y\beta }\right] \cos (\beta \omega x)\nonumber \\&+\,\omega b_y \left[ \frac{(r_1 -r_3 )(1-\mathrm{e}^{(r_2 -r_3 )\beta h})-(r_2 -r_3 )(1-\mathrm{e}^{(r_1 -r_3 )\beta h})}{(r_2 -r_3 )(r_1 -r_3 )}\mathrm{e}^{r_3 y\beta }\right. \nonumber \\&+\,\left. \left. \frac{(r_2 -r_4 )(1-\mathrm{e}^{(r_1 -r_4 )\beta h})-(r_1 -r_4 )(1-\mathrm{e}^{(r_2 -r_4 )\beta h})}{(r_2 -r_4 )(r_1 -r_4 )}\mathrm{e}^{r_4 y\beta } \right] \sin (\beta \omega x) \right\} , \end{aligned}$$
(38)
$$\begin{aligned} H_{xy2} (x,y,\omega )= & {} \frac{\beta C_{660} (\alpha _2 +2\alpha _3 )}{\pi }\frac{\omega }{(r_1 -r_2 )(r_4 -r_3 )}\left\{ b_x \left[ \frac{r_3^2 (r_2 -r_3 )(1-\mathrm{e}^{(r_1 -r_3 )\beta h})-r_3 r_4 (r_1 -r_3 )(1-\mathrm{e}^{(r_2 -r_3 )\beta h})}{(r_2 -r_3 )(r_1 -r_3 )}\mathrm{e}^{r_3 y\beta } \right. \right. \nonumber \\&\left. +\,\frac{r_4^2 (r_1 -r_4 )(1-\mathrm{e}^{(r_2 -r_4 )\beta h})-r_3 r_4 (r_2 -r_4 )(1-\mathrm{e}^{(r_1 -r_4 )\beta h})}{(r_2 -r_4 )(r_1 -r_4 )}\mathrm{e}^{r_4 y\beta }\right] \sin (\beta \omega x)\nonumber \\&-\,\omega b_y \left[ \frac{r_3 (r_1 -r_3 )(1-\mathrm{e}^{(r_2 -r_3 )\beta h})-r_3 (r_2 -r_3 )(1-\mathrm{e}^{(r_1 -r_3 )\beta h})}{(r_2 -r_3 )(r_1 -r_3 )}\mathrm{e}^{r_3 y\beta }\right. \nonumber \\&+\,\left. \left. \frac{r_4 (r_2 -r_4 )(1-\mathrm{e}^{(r_1 -r_4 )\beta h})-r_4 (r_1 -r_4 )(1-\mathrm{e}^{(r_2 -r_4 )\beta h})}{(r_2 -r_4 )(r_1 -r_4 )}\mathrm{e}^{r_4 y\beta }\right] \cos (\beta \omega x) \right\} . \end{aligned}$$
(39)

Appendix B

$$\begin{aligned} H_{xx1\infty } (x,y,\omega )= & {} \frac{\beta C_{660} (\alpha _2 +2\alpha _3 )\mathrm{e}^{\frac{y\beta }{2}}}{2\pi (r_{11} -r_{22} )(r_{11} +r_{22} )}\left[ {\left( {r_{11}^2 \mathrm{e}^{-r_{11} y\beta \omega }-r_{22}^2 \mathrm{e}^{-r_{22} y\beta \omega }} \right) \cos (\beta x\omega )b_x } \right. \nonumber \\&-\left. {\left( {r_{11} \mathrm{e}^{-r_{11} y\beta \omega }-r_{22} \mathrm{e}^{-r_{22} y\beta \omega }} \right) \sin (\beta x\omega )b_y } \right] , \end{aligned}$$
(40)
$$\begin{aligned} H_{yy1\infty } (x,y,\omega )= & {} -\frac{\beta C_{660} (\alpha _2 +2\alpha _3 )\mathrm{e}^{\frac{y\beta }{2}}}{2\pi (r_{11} -r_{22} )(r_{11} +r_{22} )}\left[ {\left( {\mathrm{e}^{-r_{11} y\beta \omega }-\mathrm{e}^{-r_{22} y\beta \omega }} \right) \cos (\beta x\omega )b_x } \right. \nonumber \\&-\left. {\left( {\frac{\mathrm{e}^{-r_{11} y\beta \omega }}{r_{11} }-\frac{\mathrm{e}^{-r_{22} y\beta \omega }}{r_{22} }} \right) \sin (\beta x\omega )b_y } \right] , \end{aligned}$$
(41)
$$\begin{aligned} H_{xy1\infty } (x,y,\omega )= & {} -\frac{\beta C_{660} (\alpha _2 +2\alpha _3 )\mathrm{e}^{\frac{y\beta }{2}}}{2\pi (r_{11} -r_{22} )(r_{11} +r_{22} )}\left[ {\left( {r_{11} \mathrm{e}^{-r_{11} y\beta \omega }-r_{22} \mathrm{e}^{-r_{22} y\beta \omega }} \right) \sin (\beta x\omega )b_x } \right. \nonumber \\&+\left. {\left( {\mathrm{e}^{-r_{11} y\beta \omega }-\mathrm{e}^{-r_{22} y\beta \omega }} \right) \cos (\beta x\omega )b_y } \right] , \end{aligned}$$
(42)
$$\begin{aligned} H_{xx2\infty } (x,y,\omega )= & {} -\frac{\beta C_{660} (\alpha _2 +2\alpha _3 )\mathrm{e}^{\frac{y\beta }{2}}}{2\pi (r_{11} -r_{22} )(r_{11} +r_{22} )}\left[ {\left( {r_{11}^2 \mathrm{e}^{r_{11} y\beta \omega }-r_{22}^2 \mathrm{e}^{r_{22} y\beta \omega }} \right) \cos (\beta x\omega )b_x } \right. \nonumber \\&+\left. {\left( {r_{11} \mathrm{e}^{r_{11} y\beta \omega }-r_{22} \mathrm{e}^{r_{22} y\beta \omega }} \right) \sin (\beta x\omega )b_y } \right] , \end{aligned}$$
(43)
$$\begin{aligned} H_{yy2\infty } (x,y,\omega )= & {} \frac{\beta C_{660} (\alpha _2 +2\alpha _3 )\mathrm{e}^{\frac{y\beta }{2}}}{2\pi (r_{11} -r_{22} )(r_{11} +r_{22} )}\left[ {\left( {\mathrm{e}^{r_{11} y\beta \omega }-\mathrm{e}^{r_{22} y\beta \omega }} \right) \cos (\beta x\omega )b_x } \right. \nonumber \\&+\left. {\left( {\frac{\mathrm{e}^{r_{11} y\beta \omega }}{r_{11} }-\frac{\mathrm{e}^{r_{22} y\beta \omega }}{r_{22} }} \right) \sin (\beta x\omega )b_y } \right] , \end{aligned}$$
(44)
$$\begin{aligned} H_{xy2\infty } (x,y,\omega )= & {} -\frac{\beta C_{660} (\alpha _2 +2\alpha _3 )\mathrm{e}^{\frac{y\beta }{2}}}{2\pi (r_{11} -r_{22} )(r_{11} +r_{22} )}\left[ {\left( {r_{11} \mathrm{e}^{r_{11} y\beta \omega }-r_{22} \mathrm{e}^{r_{22} y\beta \omega }} \right) \sin (\beta x\omega )b_x } \right. \nonumber \\&-\left. {\left( {\mathrm{e}^{r_{11} y\beta \omega }-\mathrm{e}^{r_{22} y\beta \omega }} \right) \cos (\beta x\omega )b_y } \right] . \end{aligned}$$
(45)

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Monfared, M.M., Ayatollahi, M. & Mousavi, S.M. The mixed-mode analysis of a functionally graded orthotropic half-plane weakened by multiple curved cracks. Arch Appl Mech 86, 713–728 (2016). https://doi.org/10.1007/s00419-015-1057-9

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