A Screw Dislocation in a Monoclinic Tri-Material

Abstract

Employing primarily analytic continuation, we derive analytical solutions to the anti-plane problem associated with a screw dislocation located anywhere inside an anisotropic tri-material composed of an intermediate anisotropic elastic layer of finite thickness sandwiched between two semi-infinite anisotropic elastic media. All three phases of the tri-material are monoclinic with the symmetry plane at $x_{3}=0$ in a Cartesian coordinate system. We obtain explicit expressions for each one of the three analytic functions defined in its respective anisotropic elastic phase. This allows for the complete determination of the stresses and displacement in the tri-material. In addition, we present simple yet concise expressions for the image force acting on the screw dislocation. An interfacial Zener-Stroh crack in the tri-material is discussed using the analytical solution developed for an interfacial screw dislocation.

Appendices

Appendix A

Equation (7b) can be rewritten in the following form

$$ \frac{2f_{2}(z_{2})}{\varGamma _{2} + 1} = f_{3}(z_{3}) + K_{2}\bar{f}_{3}(\bar{z}_{3}),\quad x_{1} + \mathrm{i}x_{2} \in L_{2}. $$

(A.1)

In view of Eq. (9), Eq. (A.1) can be rewritten as

$$ \frac{2f_{2}(z_{2})}{\varGamma _{2} + 1} = g(z_{3}) + K_{2}\bar{g}( \bar{z}_{3}),\quad x_{1} + \mathrm{i}x_{2} \in L_{2}. $$

(A.2)

By substituting the expressions for $z_{3}$ and $\bar{z}_{3}$ in Eq. (13a) into Eq. (A.2), we arrive at Eq. (10).

In addition, Eq. (7a) can be written in the form

$$ \frac{2f_{1}(z_{1})}{\varGamma _{1} + 1} = f_{2}(z_{2}) + K_{1}\bar{f}_{2}(\bar{z}_{2}),\quad x_{1} + \mathrm{i}x_{2} \in L_{1}. $$

(A.3)

By substituting the expression for $f_{2}(z_{2})$ in Eq. (10) into Eq. (A.3) and making use of the expressions of $z_{2}$ and $\bar{z}_{2}$ in Eq. (13b), we finally obtain Eq. (11).

Appendix B

Equation (12) can be written in the form

$$\begin{aligned} \frac{4f_{1}(z_{1})}{(\varGamma _{1} + 1)(\varGamma _{2} + 1)} =& g(z) + K_{1}K_{2}g \biggl( z - h\frac{p''_{3} - \mathrm{i}p'_{3}}{p''_{2} - \mathrm{i}p'_{2}} - h\frac{p''_{3} - \mathrm{i}p'_{3}}{p''_{2} + \mathrm{i}p'_{2}} \biggr) \\ &{}+ K_{2}\bar{g} \biggl[ - \frac{p''_{3} + \mathrm{i}p'_{3}}{p''_{1} - \mathrm{i}p'_{1}}z_{1} - h\frac{p''_{2} + p''_{3} + \mathrm{i}(p'_{3} - p'_{2})}{p''_{2} - \mathrm{i}p'_{2}} \biggr] \\ &{}+ K_{1}\bar{g} \biggl[ - \frac{p''_{3} + \mathrm{i}p'_{3}}{p''_{1} - \mathrm{i}p'_{1}}z_{1} + h\frac{p''_{3} - p''_{2} + \mathrm{i}(p'_{3} - p'_{2})}{p''_{2} + \mathrm{i}p'_{2}} \biggr], \end{aligned}$$

(B.1)

where

$$ z = \frac{p''_{3} - \mathrm{i}p'_{3}}{p''_{1} - \mathrm{i}p'_{1}}z_{1} + h\frac{p''_{3} - p''_{2} - \mathrm{i}(p'_{3} - p'_{2})}{p''_{2} - \mathrm{i}p'_{2}}. $$

(B.2)

The expression for $g(z)$ in Eq. (14) is obtained by examining Eq. (B.1) and observing that $f_{1}(z_{1})$ exhibits the following logarithmic singular behavior at the location of the screw dislocation

$$ f_{1}(z_{1}) \cong \frac{b}{2\pi } \ln (z_{1} - d),\quad \mbox{as}\ z_{1} \to d, $$

(B.3)

but remains regular at all points away from the location of the screw dislocation.