A simple two-dimensional model of the earthquake cycle (preearthquake strain accumulation, coseismic strain release, and postseismic readjustment) has been constructed from the Nur-Mavko solution for a screw dislocation in an elastic plate (lithosphere) overlying a viscoelastic substrate (asthenosphere). The deformation at the free surface is calculated for an earthquake cycle imposed by prescribed slip on a transform fault. This deformation is compared to that produced by a similar cycle in an elastic half space so that the effects of viscoelastic relaxation in the asthenosphere may be isolated. The following conclusions are drawn: (1) The surface deformation produced by viscoelastic relaxation in the asthenosphere can be duplicated identically by a reasonable distribution of slip at depth on a vertical fault in an elastic half space. Thus differentiation of two possible modes of postearthquake readjustment will be difficult. (2) The effect of asthenosphere relaxation is important only if the depth of the seismic zone is comparable to the thickness of the lithosphere. If the seismic zone is 15 km deep and the lithosphere is 75 km thick, as commonly estimated for the San Andreas fault zone, asthenosphere relaxation is not particularly significant in determining surface deformation. (3) In a periodic sequence of earthquakes the principal observable effects of viscoelasticity in the asthenosphere are to produce a rapid postearthquake deformation and to concentrate strain accumulation and relaxation even closer to the fault than in the elastic half-space model.

Asthenosphere readjustment and the earthquake cycle

This paper outlines a new method, the “split node technique” for introducing fault displacements into finite element numerical computations. The value of the displacement at a single node point shared between two elements depends upon which element it is referred to, thus introducing a displacement discontinuity between the two elements. We show that the modification induced by this splitting can be contained in the load vector, so that the stiffness matrix is not altered. The number of degrees of freedom is not increased by splitting. This method can be implemented entirely on the local element level, and we show rigorously that no net forces or moments are induced on the finite element grid when isoparametric elements are used. This method is thus of great utility in many geological and engineering applications.

A simple and efficient method for introducing faults into finite element computations

Fully plastic crack problems: The center-cracked strip under plane strain

A New method to compute perturbed bifurcations: Application to the buckling of imperfect elastic structures

This article is concerned with the nonlinear analysis of the stability of thick elastic bodies subjected to finite elastic deformations. The analysis is based on the theory of small elastic deformations superimposed on a finite elastic deformation. Attention is drawn to methods developed in the stability analysis of fluids and of thin shells and plates which are readily applicable to the present circumstances. The state of development of the nonlinear stability analysis of thick elastic bodies is summarized in order to provide a basis for subsequent studies, and some new results relating to the stability of an elastic plate subjected to a pre-stress associated either with uniaxial thrust or with simple shear in the presence of all-round pressure are discussed. Near-critical modes in the neighbourhood of so-called critical configurations are considered to depend on, for example, a slow time variable, and nonlinear evolution equations for the mode amplitudes are derived both in the case of a monochromatic mode and for a resonant triad of modes. The crucial role of the ‘nonlinear coefficient’ in such an equation in the analysis of stability, imperfection sensitivity and localization is highlighted. An efficient (virtual work) method for the determination of this coefficient is described together with an alternative method based on the calculation of the total energy of a monochromatic near-critical mode. The influence of the boundary conditions and of the form of the pre-stress is examined and explicit calculation of the nonlinear coefficient is provided for the two representative pre-stress conditions mentioned in the above paragraph. It is shown, in particular, that a resonant triad of modes has an effect similar to that generated by the presence of a geometrical imperfection. The Appendices gather together for reference certain expressions which are used in the body of the article. These include expressions, not given previously in the literature, for the components of the tensor of third-order instantaneous elastic moduli in terms of the principal stretches of the deformation in respect of a general form of incompressible isotropic elastic strain-energy function.

Nonlinear stability analysis of pre-stressed elastic bodies

Asymptotic analyses of the buckling of imperfect columns on nonlinear elastic foundations

Fully plastic crack in an infinite body under anti-plane shear

The interaction of fault slip and lithospheric creep is studied on the basis of an idealized viscoelastic model of the lithosphere. Under a constant average driving stress the model displays a periodic slip-creep-slip limit cycle. A simple formula is found that relates effective lithospheric viscosity to fault depth, long-time average slip rate, and dynamic driving stress τAD = τA - τD, where τA is the average shear stress in the lithosphere and τD is the dynamic shear resistance to fault slip. An estimate is made of lithospheric viscosity in California.

Interaction of fault slip and lithospheric creep

Fully plastic center-cracked strip under anti-plane shear☆

On the J integral for an internally pressurized cylindrical shell with longitudinal crack

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