*3.6.3 Cartesian coordinates of delta between plane strain and plane stress solutions (uniaxial far-field stress)*

Using the same transformation as in Section 3.6.1, Eqs. (11) and (12) are converted to Cartesian coordinate (see Appendix B for details):

$$u\_{\mathbf{x}} = -\sigma\_{\mathbf{x}\mathbf{x}-\mathbf{e}\bullet} \left(\frac{v^2}{E(\mathbf{x}^2+\mathbf{y}^2)}\right) \left[ \left(\mathbf{x}^2+\mathbf{y}^2+2a^2\frac{\mathbf{x}^2-\mathbf{y}^2}{\mathbf{x}^2+\mathbf{y}^2}\right)\mathbf{x} + \left(4a^2\frac{\mathbf{x}\mathbf{y}}{\mathbf{x}^2+\mathbf{y}^2}\right)\mathbf{y} \right] \tag{25}$$

$$u\_y = -\sigma\_{\text{xx}-\text{ss}} \left( \frac{v^2}{E(\mathbf{x}^2 + \mathbf{y}^2)} \right) \left[ \left( \mathbf{x}^2 + \mathbf{y}^2 + 2a^2 \frac{\mathbf{x}^2 - \mathbf{y}^2}{\mathbf{x}^2 + \mathbf{y}^2} \right) \mathbf{y} - \left( 4a^2 \frac{\mathbf{x}\mathbf{y}}{\mathbf{x}^2 + \mathbf{y}^2} \right) \mathbf{x} \right] \tag{26}$$

One can also obtain Eqs. (25) and (26) by subtracting Eqs. (17) and (18) from Eqs. (15) and (16) (see Appendix B).
