Static elastic deformation in an orthotropic half-space with rigid boundary model due to non-uniform long strike slip fault

Abstract

The solution of static elastic deformation of a homogeneous, orthotropic elastic uniform half-space with rigid boundary due to a non-uniform slip along a vertical strike-slip fault of infinite length and finite width has been studied. The results obtained here are the generalisation of the results for an isotropic medium having rigid boundary in the sense that medium of the present work is orthotropic with rigid boundary which is more realistic than isotropic and the results for an isotropic case can be derived from our results. The variations of displacement with distance from the fault due to various slip profiles have been studied to examine the effect of anisotropy on the deformation. Numerically it has been found that for parabolic slip profile, the displacement in magnitude for isotropic elastic medium is greater than that for an orthotropic elastic half-space, but, in case of linear slip, the displacements in magnitude for an orthotropic medium is greater than that for the isotropic medium.

Appendices

Appendix 1

For the single couple (12)

$$\begin{aligned} A_0 =\frac{F_{12}}{2\pi c\alpha }, \quad B_0 =0 . \end{aligned}$$

For the single couple (13)

$$\begin{aligned} A_0 =0, \quad B_0 =\pm \frac{F_{13}}{2\pi c}, \end{aligned}$$

where the upper sign is for $x_3 >y_3$ and the lower sign is for $x_3 <y_3$.

Appendix 2

Stresses in case of uniform slip

$$\begin{aligned} P_{12}= & {} \frac{c\alpha ^{3}b_0}{2\pi } \left[ {\frac{L-x_3}{A^{2}}-\frac{L+x_3}{B^{2}}+\frac{2x_3}{C^{2}}} \right] ,\\ P_{13}= & {} \frac{c\alpha b_0}{2\pi } \left[ {\frac{x_2}{A^{2}}+\frac{x_2}{B^{2}}-\frac{2x_3}{C^{2}}} \right] . \end{aligned}$$

Stresses in case of parabolic slip

$$\begin{aligned} P_{12}= & {} -\frac{c\alpha b_0}{2\pi L}\left[ \frac{2x_2}{\alpha L}\left\{ \tan ^{-1}\frac{\alpha \left( {L-x_3} \right) }{x_2}\right. \right. \\&\left. \left. -\tan ^{-1}\frac{\alpha \left( {L+x_3} \right) }{x_2} +2\tan ^{-1}\frac{\alpha x_3}{x_2} \right\} \right. \\&\left. -\alpha \left( {L-\frac{x_3^{2}}{L}+\frac{x_2^{2}}{\alpha ^{2}L}} \right) \left\{ \frac{\alpha (L-x_{3})}{A^{2}}\right. \right. \\&\left. \left. -\frac{\alpha (L+x_{3})}{B^{2}}\right. \right. \left. \left. +\frac{2\alpha x_{3}}{C^{2}} \right\} -\frac{2x_3}{L}\mathrm{log}\left( {\frac{AB}{C^{2}}} \right) \right. \\&\left. -\frac{2x_2^{2} x_3}{L}\left\{ {\frac{1}{A^{2}}+\frac{1}{B^{2}}-\frac{2}{C^{2}}} \right\} \right] \\ P_{13}= & {} \frac{cb_0}{2\pi \alpha L}\left[ \frac{2\alpha x_3}{L}\left\{ \tan ^{-1}\frac{\alpha \left( {L-x_3} \right) }{x_2}\right. \right. \\&\left. \left. -\tan ^{-1}\frac{\alpha \left( {L+x_3} \right) }{x_2}\right. \right. \left. \left. +2\tan ^{-1}\frac{\alpha x_3}{x_2} \right\} \right. \\&\left. +\alpha ^{2}x_2 \left( {L-\frac{x_3^{2}}{L}+\frac{x_2^{2}}{\alpha ^{2}L}} \right) \left\{ {\frac{1}{A^{2}}+\frac{1}{B^{2}}-\frac{2}{C^{2}}} \right\} \right. \\&\left. +\frac{2x_2}{L}\mathrm{log}\left( {\frac{AB}{C^{2}}} \right) \right. -\frac{2\alpha ^{2}x_2 x_3}{L}\\&\left. \times \left\{ {\frac{L-x_3}{A^{2}}-\frac{L+x_3}{B^{2}}+\frac{2x_3}{C^{2}}} \right\} \right] \end{aligned}$$