A screw dislocation interacting with a semi-infinite interface crack and two semi-infinite cracks perpendicular to the bimaterial interface

Abstract

Using conformal mapping and the image method, we derive an analytic solution to the problem of an isotropic elastic bimaterial involving a screw dislocation interacting with a semi-infinite interface crack, a semi-infinite crack in the upper half-plane perpendicular to the interface, and another semi-infinite crack in the lower half-plane also perpendicular to the interface. Closed-form expressions of the image force acting on the screw dislocation and the mode III stress intensity factors at the three crack tips are derived.

Appendices

Appendix A

When $M=-1$, the entire $x_{1}$-axis is traction-free. In this case, we can also introduce the following conformal mapping function [14]:

$$\begin{aligned} z=\omega (\xi )=\frac{1}{2}\left( {\xi -\frac{a^{2}}{\xi }} \right) ,\, \, \xi =\omega ^{-1}(z)=z+\sqrt{z^{2}+a^{2}} , \end{aligned}$$

(A1)

which maps the upper half-plane containing the semi-infinite vertical crack onto the right and upper quarter plane: $\left\{ {\text {Re}\left\{ \xi \right\} \ge 0,\, \text {Im}\left\{ \xi \right\} \ge 0} \right\}$; maps the two surfaces of the crack onto the vertical semi-infinite line $\left\{ {\text {Re}\left\{ \xi \right\} =0^{+},\, 0\le {\text {Im}}\left\{ \xi \right\} <+\infty } \right\}$; maps the $x_{1}$-axis onto the horizontal semi-infinite line $\left\{ {0\le {\text {Re}}\left\{ \xi \right\} <+\infty ,\, \text {Im}\left\{ \xi \right\} =0^{+}} \right\}$; and maps the location of the screw dislocation at $z=z_{0}$ onto the point $\xi =\xi _{0} =\omega ^{-1}(z_{0} )$. In this case, also by using the image method [10], the analytic function $f_{1} (\xi )$ is simply given by

$$\begin{aligned} f_{1} (\xi )=\frac{b}{2\pi }\ln \frac{(\xi -\xi _{0} )(\xi +\xi _{0} )}{(\xi -\bar{{\xi }}_{0} )(\xi +\bar{{\xi }}_{0} )},\, \, \text {Re}\left\{ \xi \right\} \ge 0\, \text{ and }\, \text {Im}\left\{ \xi \right\} \ge 0. \end{aligned}$$

(A2)

The image force acting on the screw dislocation can be finally derived as

$$\begin{aligned} F_{1} -\text{ i }F_{2} =\frac{\mu _{1} b^{2}}{\pi }\frac{\frac{3}{2\xi _{0} }-\frac{\xi _{0} }{\xi _{0}^{2} +a^{2}}-\frac{1}{\xi _{0} -\bar{{\xi }}_{0} }-\frac{1}{\xi _{0} +\bar{{\xi }}_{0} }}{1+\frac{a^{2}}{\xi _{0}^{2} }}. \end{aligned}$$

(A3)

When choosing $\xi _{0} =a\text{ e }^{\mathrm{i}\theta },\, \, 0<\theta <\tfrac{\pi }{2}$, the screw dislocation is located just on the positive $x_{2}$-axis below the vertical crack with $z_{0} =-\bar{{z}}_{0} =\text{ i }y_{0} ,\, 0<y_{0} <a$. In this case, the image force in Eq. (A3) becomes

$$\begin{aligned} F_{1} =0,\, \, F_{2} =\frac{\mu _{1} b^{2}(1-3\cos 2\theta )}{4\pi a\sin 2\theta \text{ cos }\theta }=\frac{\mu _{1} b^{2}(3y_{0}^{2} -a^{2})}{4\pi y_{0} (a^{2}-y_{0}^{2} )}, \end{aligned}$$

(A4)

which is found in agreement with the results in Figs. 4 and 5 for $M=-1$. It is seen from Eq. (A4) that there is an unstable equilibrium position for the screw dislocation at $y_{0} =a / {\sqrt{3} }=0.5774a$, which can also be observed from Fig. 5. When the coordinate $y_{0}$ is above this value, the screw dislocation is attracted to the upper semi-infinite crack; when $y_{0}$ is below this value, the screw dislocation is attracted to the traction-free $x_{1}$-axis.

When the upper elastic half-plane weakened by the semi-infinite crack $L_{1}$ is perfectly bonded to the lower elastic half-plane weakened by the semi-infinite crack $L_{2}$, and the screw dislocation is located on the positive $x_{2}$-axis below the vertical crack, the image force acting on the screw dislocation can be derived as

$$\begin{aligned} F_{1} =0,\, \, F_{2} =\frac{\mu _{1} b^{2}\left[ {(2-M)y_{0}^{2} +Ma^{2}} \right] }{4\pi y_{0} (a^{2}-y_{0}^{2} )}. \end{aligned}$$

(A5)

It is seen from Eq. (A5) that when $-1\le M\le 0$, there is an unstable equilibrium position for the screw dislocation at

$$\begin{aligned} y_{0} =a\sqrt{\frac{-M}{2-M}} ,\, \, \, \, -1\le M\le 0. \end{aligned}$$

(A6)

The unstable equilibrium position determined by Eq. (A6) as a function of M is illustrated in Fig. 8.

Fig. 8

The unstable equilibrium position for a screw dislocation on the positive $x_{2}$-axis below the vertical crack as a function of M when the two elastic half-planes are perfectly bonded

Appendix B

In the case when for the geometry shown in Fig. 9 in which the parameter h describes the general position of the tip of the semi-infinite interface crack on the $x_{1}$-axis can assume positive as well as negative values, the modified conformal mapping function is given by:

$$\begin{aligned} z=\omega (\xi )=\frac{1}{2}\left( {\sqrt{\xi ^{2}+c^{2}} -\frac{a^{2}}{\sqrt{\xi ^{2}+c^{2}} }} \right) ,\, \, \xi =\omega ^{-1}(z),\, \, \, \, \text {Re}\left\{ \xi \right\} \ge 0, \end{aligned}$$

(B1)

where $c=h+\sqrt{h^{2}+a^{2}} >0$. It is seen that $c>a$ if $h>0$; $c<a$ if $h<0$; and $c=a$ if $h=0$.

Fig. 9

The geometry of the composite in which the tip of the semi-infinite interface crack is located at an arbitrary position on the $x_{1}$-axis

Fig. 10

The image $\xi$-plane

As shown in Fig. 10, using the mapping function in Eq. (B1), the upper half-plane containing the semi-infinite crack $L_{1}$ is mapped onto the right and upper quarter plane: $\left\{ {\text {Re}\left\{ \xi \right\} \ge 0,\, \text {Im}\left\{ \xi \right\} \ge 0} \right\}$; the lower half-plane containing the semi-infinite crack $L_{2}$ is mapped onto the right and lower quarter plane: $\left\{ {\text {Re}\left\{ \xi \right\} \ge 0,\, \text {Im}\left\{ \xi \right\} \le 0} \right\}$; the upper surface of the semi-infinite interface crack $L_{0} \big \{ -\infty <x_{1} \le h,\, x_{2} =0^{+} \big \}$ is mapped onto the segment $\left\{ {\text {Re}\left\{ \xi \right\} =0^{+},\, 0\le {\text {Im}}\left\{ \xi \right\} \le c} \right\}$; the lower surface of the semi-infinite interface crack $L_{0} \quad \left\{ {-\infty <x_{1} \le h,\, x_{2} =0^{-}} \right\}$ is mapped onto the segment $\left\{ {\text {Re}\left\{ \xi \right\} =0^{+},\, -c\le {\text {Im}}\left\{ \xi \right\} \le 0} \right\}$; the perfect interface: $\left\{ {h\le x_{1} <+\infty ,\, x_{2} =0} \right\}$ is mapped onto the semi-infinite line $\left\{ {0\le \text {Re}\left\{ \xi \right\} <+\infty ,{\text {Im}}\left\{ \xi \right\} =0} \right\}$; the right surface of the semi-infinite crack $L_{1}$$\left\{ {x_{1} =0^{+},\, a\le x_{2} <+\infty } \right\}$ is mapped onto the semi-infinite line $\left\{ {\text {Re}\left\{ \xi \right\} =0^{+},\, \sqrt{c^{2}+a^{2}} \le {\text {Im}}\left\{ \xi \right\} <+\infty } \right\}$; the left surface of the semi-infinite crack $L_{1} \quad \left\{ {x_{1} =0^{-},\, a\le x_{2} <+\infty } \right\}$ is mapped onto the segment $\left\{ {\text {Re}\left\{ \xi \right\} =0^{+},\, c\le {\text {Im}}\left\{ \xi \right\} \le \sqrt{c^{2}+a^{2}} } \right\}$; the right surface of the semi-infinite crack $L_{2} \big \{x_{1} =0^{+},\, -\infty <x_{2} \le -a \big \}$ is mapped onto the semi-infinite line $\big \{\text {Re}\left\{ \xi \right\} =0^{+},\, -\infty <{\text {Im}}\left\{ \xi \right\} \le -\sqrt{c^{2}+a^{2}} \big \}$; the left surface of the semi-infinite crack $L_{2} \quad \left\{ {x_{1} =0^{-},\, -\infty <x_{2} \le -a} \right\}$ is mapped onto the segment $\left\{ {\text {Re}\left\{ \xi \right\} =0^{+},\, -\sqrt{c^{2}+a^{2}} \le {\text {Im}}\left\{ \xi \right\} \le -c} \right\}$; and the location of the screw dislocation at $z=z_{0}$ is mapped onto the point $\xi =\xi _{0} =\omega ^{-1}(z_{0} )$.