Static Deformation of a Uniform Half-Space with Rigid Boundary Due To a Vertical Dip-Slip Line Source

Abstract

The Airy stress function for a vertical dip-slip line source buried in a homogeneous, isotropic, perfectly elastic half-space with rigid boundary is obtained. This Airy stress function is used to derive closed- form analytical expressions for the stresses and displacements at an arbitrary point of the half-space caused by vertical dip-slip line source. The variation of the displacements and stress fields with distance from the fault and depth from the fault is studied numerically. Keywords - Dip-slip faulting, Half-space, Rigid boundary, Static deformation I. Introduction Strains and stresses within the Earth constitute important precursors of earthquakes. Therefore, the determination of the static deformation of an Earth model around surface faults is important for any scheme for prediction of earthquakes. Static dislocation models are used to analyze the static deformation of the medium caused by earthquake faults. Steketee (1958a, b) applied the elasticity theory of dislocations in the field of seismology. For the sake of simplicity, Steketee ignored the curvature of the Earth, its gravity, anisotropy and non-homogeneity and dealt with a semi-infinite, non-gravitating, isotropic and homogeneous medium. Homogeneity means that the medium is uniform throughout, whereas isotropy specifies that the elastic properties of the medium are independent of direction. Maruyama (1964) calculated all the sets of Green's functions required for the displacement and stress fields around faults in a half-space. Jungels and Frazier (1973) described a finite element variational method applied to plain strain analysis. This technique presents a suitable tool for the analysis of permanent displacements, tilts and strains caused by seismic events. The accuracy of technique was demonstrated by comparing the numerical results for the static field due to long dislocation in a homogeneous half-space from closed form analytical solution with those obtained from the finite element method. Sato (1971) and Sato and Yamashita (1975) derived the expressions for the static surface deformations due to two-dimensional strike slip and dip-slip faults located along the dipping boundary between the two different media. Freund and Barnett (1976) gave a two-dimensional analysis of surface deformation due to dip- slip faulting in a uniform half-space, using the theory of analytic functions of a complex variable. Singh and Garg (1986) obtained the integral expressions for the Airy stress function in an unbounded medium due to various two- dimensional seismic sources. Singh et al. (1992) followed a similar procedure to obtain closed-form analytical expression for the displacements and stresses at any point of either of two homogeneous, isotropic, perfectly elastic half-spaces in welded contact due to two-dimensional sources. Singh and Rani (1991) obtained closed-form analytical expressions for the displacements and stresses at any point of a two-phase medium consisting of a homogeneous, isotropic, perfectly elastic half-space in welded contact with a homogeneous, orthotropic, perfectly elastic half-space caused by two-dimensional seismic sources located in the isotropic half-space. Bonafede and Rivalta (1999a) obtained analytical solutions for the elementary tensile dislocation problem in a layered elastic medium composed of two welded, semi-infinite half-spaces. A plain - strain configuration was considered and different rigidities and Poisson ratios were assumed for the two half- spaces. The elementary dislocation problem refers to a dislocation surface over which a jump discontinuity with constant amplitude (Burgers vector) is prescribed for the displacement field. Similar dislocation models in homogeneous half-spaces (e.g. Okada, 1992) are often employed to model dyke injection within the crust (e.g. Bonaccorso and Davis 1993), although a constant-displacement discontinuity, in general, is not the most realistic description of dyke opening. Bonafede and Rivalta (1999b) obtained the solutions for the displacement and stress fields produced by a vertical tensile crack, opening under the effect of an assigned overpressure within it, in the proximity of the welded boundary between two media characterized by different elastic parameters. Singh et al.(2011) obtained analytical expressions for stresses at an arbitrary point of homogeneous, isotropic, perfectly elastic half-space with rigid boundary caused by a long tensile fault of finite width.