**B. Converting the polar coordinate of the displacement field to the cartesian coordinate**

## **B.1 Plane strain solution**

Recall the plane strain solution for the hole with uniaxial far-field stress given by Eqs. (5) and (6)

$$u\_r = \sigma\_{\text{xx}-\text{os}} \left( \frac{\mathbf{1} + v}{2Er} \right) \left\{ r^2 (\mathbf{1} - 2v + \cos 2\theta) + a^2 \left[ \mathbf{1} + \left( 4(\mathbf{1} - v) - \frac{a^2}{r^2} \right) \cos 2\theta \right] \right\} \tag{\text{B1}}$$

$$u\_{\theta} = -\sigma\_{\text{xx}-\infty} \left( \frac{\mathbf{1} + v}{2Er} \right) \left[ r^2 + a^2 \left( 2(\mathbf{1} - 2v) + \frac{a^2}{r^2} \right) \right] \sin 2\theta \tag{\text{B2}}$$

$$u\_{\mathbf{x}} = u\_r \cos \theta - u\_\theta \sin \theta, \quad u\_\mathbf{y} = u\_r \sin \theta + u\_\theta \cos \theta$$

$$u\_x = \left[\sigma\_{\rm xx-so} \left(\frac{1+v}{2Er}\right) \left\{r^2 (1 - 2v + \cos 2\theta)\right\}\right] + a^2 \left[1 + \left(4(1-v) - \frac{a^2}{r^2}\right) \cos 2\theta\right] \cos \theta$$

$$- \left[-\sigma\_{\rm xx-so} \left(\frac{1+v}{2Er}\right) \left[r^2 + a^2 \left(2(1-2v) + \frac{a^2}{r^2}\right)\right] \sin 2\theta\right] \sin \theta \tag{B3}$$

$$u\_y = \left[\sigma\_{\rm xx-so} \left(\frac{1+v}{2Er}\right) \left\{r^2 (1 - 2v + \cos 2\theta)\right\}\right]$$

$$+ a^2 \left[1 + \left(4(1-v) - \frac{a^2}{r^2}\right) \cos 2\theta\right] \sin \theta \tag{B4}$$

$$+ \left[-\sigma\_{\rm xx-so} \left(\frac{1+v}{2Er}\right) \left[r^2 + a^2 \left(2(1-2v) + \frac{a^2}{r^2}\right)\right] \sin 2\theta\right] \cos \theta$$

$$\infty = r \cos \theta, \quad \text{y} = r \sin \theta, \quad r^2 = \mathfrak{x}^2 + \mathfrak{y}^2$$

$$u\_{\mathbf{x}} = \sigma\_{\mathbf{x}\mathbf{x}\rightarrow\mathbf{e}} \left(\frac{\mathbf{1}+\boldsymbol{v}}{2Er}\right) \left\{ \left[ r^2 \left( \mathbf{1} - 2\boldsymbol{v} + \frac{\mathbf{x}^2 - \boldsymbol{y}^2}{r^2} \right) + a^2 \left[ \mathbf{1} + \left( 4(\mathbf{1} - \boldsymbol{v}) - \frac{a^2}{r^2} \right) \frac{\mathbf{x}^2 - \mathbf{y}^2}{r^2} \right] \right] \frac{\mathbf{x}}{r} \right\}$$

$$\quad + \left[ \left[ r^2 + a^2 \left( 2(\mathbf{1} - 2\boldsymbol{v}) + \frac{a^2}{r^2} \right) \right] \frac{2\mathbf{x}\mathbf{y}}{r^2} \right] \frac{\mathbf{y}}{r}$$

$$\quad = \sigma\_{\mathbf{x}\mathbf{x}-\mathbf{a}} \left( \frac{\mathbf{1} + \boldsymbol{v}}{2E} \right) \left\{ \left[ r^2 (\mathbf{1} - 2\boldsymbol{v}) + \mathbf{x}^2 - \mathbf{y}^2 + a^2 + 4a^2 (\mathbf{1} - \boldsymbol{v}) \left( \frac{\mathbf{x}^2 - \mathbf{y}^2}{r^2} \right) \right. \tag{5.2}$$

$$\quad - a^4 \left( \frac{\mathbf{x}^2 - \mathbf{y}^2}{r^4} \right) \right] \frac{\mathbf{x}}{r} + \left[ 2\mathbf{x}\mathbf{y} + (\mathbf{1} - 2\boldsymbol{v}) \left( \frac{4a^2 \mathbf{x}\mathbf{y}}{r^2} \right) + \frac{2a^4 \mathbf{x}\mathbf{y}}{r^4} \right] \frac{\mathbf{y}}{r^2}$$

$$\begin{split} u\_{\eta} &= \sigma\_{\text{xx}-\text{so}} \left( \frac{\mathbf{1} + \boldsymbol{v}}{2E\boldsymbol{r}} \right) \left\{ \left[ r^2 \left( \mathbf{1} - 2\boldsymbol{v} + \frac{\mathbf{x}^2 - \boldsymbol{y}^2}{r^2} \right) + a^2 \left[ \mathbf{1} + \left( 4(\mathbf{1} - \boldsymbol{v}) - \frac{a^2}{r^2} \right) \frac{\mathbf{x}^2 - \boldsymbol{y}^2}{r^2} \right] \right] \frac{\mathbf{y}}{r} \right\} \\ &\quad - \left[ \left[ r^2 + a^2 \left( 2(\mathbf{1} - 2\boldsymbol{v}) + \frac{a^2}{r^2} \right) \right] \frac{2\mathbf{x}\mathbf{y}}{r^2} \right] \frac{\mathbf{x}}{r} \right\} \\ &= \sigma\_{\text{xx}-\text{so}} \left( \frac{\mathbf{1} + \boldsymbol{v}}{2E} \right) \left\{ \left[ r^2 (\mathbf{1} - 2\boldsymbol{v}) + \mathbf{x}^2 - \mathbf{y}^2 + a^2 + 4a^2 (\mathbf{1} - \boldsymbol{v}) \left( \frac{\mathbf{x}^2 - \mathbf{y}^2}{r^2} \right) \right. \\ &\left. - a^4 \left( \frac{\mathbf{x}^2 - \mathbf{y}^2}{r^4} \right) \right] \frac{\mathbf{y}}{r^2} - \left[ 2\mathbf{x}\mathbf{y} + (\mathbf{1} - 2\boldsymbol{v}) \left( \frac{4a^2 \mathbf{x}\mathbf{y}}{r^2} \right) + \frac{2\mathbf{a}^4 \mathbf{x}\mathbf{y}}{r^4} \right] \frac{\mathbf{x}}{r^2} \right\} \end{split} \tag{B6}$$

$$u\_{\mathbf{x}} = \sigma\_{\mathbf{xx} \to \mathbf{so}} \left( \frac{\mathbf{1} + \nu}{2E} \right) \left\{ \left[ \left( \mathbf{x}^2 + \mathbf{y}^2 \right) (\mathbf{1} - 2\nu) + \mathbf{x}^2 - \mathbf{y}^2 + a^2 + 4a^2 (\mathbf{1} - \nu) \left( \frac{\mathbf{x}^2 - \mathbf{y}^2}{\mathbf{x}^2 + \mathbf{y}^2} \right) \right. \\ \left. \left. \left( \frac{\mathbf{x}}{\mathbf{x}^2 + \mathbf{y}^2} \right) \right] \left( \frac{\mathbf{x}}{\mathbf{x}^2 + \mathbf{y}^2} \right) + \left[ 2\mathbf{x}\mathbf{y} + (\mathbf{1} - 2\nu) \left( \frac{4a^2 \mathbf{x}\mathbf{y}}{\mathbf{x}^2 + \mathbf{y}^2} \right) + \frac{2a^4 \mathbf{x}\mathbf{y}}{(\mathbf{x}^2 + \mathbf{y}^2)^2} \right] \left( \frac{\mathbf{y}}{\mathbf{x}^2 + \mathbf{y}^2} \right) \right\} \tag{\text{B7}}$$

$$u\_{\gamma} = \sigma\_{\text{xx}-\text{ss}} \left(\frac{1+\nu}{2E}\right) \left\{ \left[ \left(\mathbf{x}^2 + \mathbf{y}^2\right) (\mathbf{1} - 2\nu) + \mathbf{x}^2 - \mathbf{y}^2 + a^2 + 4a^2 (\mathbf{1} - \nu) \left(\frac{\mathbf{x}^2 - \mathbf{y}^2}{\mathbf{x}^2 + \mathbf{y}^2}\right) \right. \\ \left. \left. \left(\frac{\mathbf{x}^2 - \mathbf{y}^2}{\mathbf{x}^2 + \mathbf{y}^2}\right) - \left[2\mathbf{x}\mathbf{y} + (\mathbf{1} - 2\nu) \left(\frac{4a^2 \mathbf{x}\mathbf{y}}{\mathbf{x}^2 + \mathbf{y}^2}\right) + \frac{2\mathbf{a}^4 \mathbf{x}\mathbf{y}}{(\mathbf{x}^2 + \mathbf{y}^2)^2} \right] \right\}\_{\text{(\mathbf{B}\mathbf{S}})} \right\} \tag{\text{B8}}$$

$$u\_{\mathbf{x}} = u\_r \cos \theta - u\_\theta \sin \theta, \quad u\_\mathbf{y} = u\_r \sin \theta + u\_\theta \cos \theta$$

$$\begin{split} u\_{\rm x} &= \left( -\sigma\_{\rm xx-\infty} \left( \frac{v^2}{Er} \right) (r^2 + 2a^2 \cos 2\theta) \right) \cos \theta - \left( \sigma\_{\rm xx-\infty} \left( \frac{v^2}{Er} \right) (2a^2 \sin 2\theta) \right) \sin \theta \\ &= -\sigma\_{\rm xx-\infty} \left( \frac{v^2}{Er} \right) [(r^2 + 2a^2 \cos 2\theta) \cos \theta + (2a^2 \sin 2\theta) \sin \theta] \end{split}$$
 
$$\begin{split} &= -\sigma\_{\rm xx-\infty} \left( \frac{v^2}{E(\mathbf{x}^2 + \mathbf{y}^2)} \right) \left[ \left( \mathbf{x}^2 + \mathbf{y}^2 + 2a^2 \frac{\mathbf{x}^2 - \mathbf{y}^2}{\mathbf{x}^2 + \mathbf{y}^2} \right) \mathbf{x} + \left( 4a^2 \frac{\mathbf{x}\mathbf{y}}{\mathbf{x}^2 + \mathbf{y}^2} \right) \mathbf{y} \right] \end{split} \tag{B9}$$

$$\begin{split} u\_{\mathcal{Y}} &= \left( -\sigma\_{\text{xx}-\text{os}} \left( \frac{v^2}{Er} \right) (r^2 + 2a^2 \cos 2\theta) \right) \sin \theta + \left( \sigma\_{\text{xx}-\text{os}} \left( \frac{v^2}{Er} \right) (2a^2 \sin 2\theta) \right) \cos \theta \\ &= -\sigma\_{\text{xx}-\text{os}} \left( \frac{v^2}{Er} \right) [(r^2 + 2a^2 \cos 2\theta) \sin \theta - (2a^2 \sin 2\theta) \cos \theta] \\ &= -\sigma\_{\text{xx}-\text{os}} \left( \frac{v^2}{E(\mathbf{x}^2 + \mathbf{y}^2)} \right) \left[ \left( \mathbf{x}^2 + \mathbf{y}^2 + 2a^2 \frac{\mathbf{x}^2 - \mathbf{y}^2}{\mathbf{x}^2 + \mathbf{y}^2} \right) \mathbf{y} - \left( 4a^2 \frac{\mathbf{x}\mathbf{y}}{\mathbf{x}^2 + \mathbf{y}^2} \right) \mathbf{x} \right] \tag{\text{B10}} \end{split}$$

*Asymptotic Solutions for Multi-Hole Problems: Plane Strain versus Plane Stress Boundary… DOI: http://dx.doi.org/10.5772/intechopen.105048*
