A finite element analysis of tidal deformation of the entire earth with a discontinuous outer layer

Tidal deformation of the Earth is normally calculated using the analytical solution with some simplified assumptions, such as the Earth is a perfect sphere of continuous media. This paper proposes an alternative way, in which the Earth crust is discontinuous along its boundaries, to calculate the tidal deformation using a finite element method. An in-house finite element code is firstly introduced in brief and then extended here to calculate the tidal deformation. The tidal deformation of the Earth due to the Moon was calculated for an geophysical earth model with the discontinuous outer layer and compared with the continuous case. The preliminary results indicate that the discontinuity could have different effects on the tidal deformation in the local zone around the fault, but almost no effects on both the locations far from the fault and the global deformation amplitude of the Earth. The localized deformation amplitude seems to depend much on the relative orientation between the fault strike direction and the loading direction (i.e. the location of the Moon) and the physical property of the fault.     

Ground deformation in a viscoelastic medium composed of a layer overlying a half-space: a comparison between point and extended sources

We obtain and compare analytical and numerical solutions for ground displacement caused by an overpressurized magma chamber placed in a linear viscoelastic medium composed of a layer over a half-space. Different parameters such as size, depth and shape of the chamber, crustal rheology and topography are considered and discussed. Numerical solutions for an axisymmetric extended source are computed using a finite element method (FEM). Analytical solutions for a point source are obtained using the dislocation theory and the propagator matrix technique. In both cases, the elastic solutions are used together with the correspondence principle of linear viscoelasticity to obtain the solution in the Laplace transform domain. Viscoelastic solutions in the time domain are derived inverting the Laplace transform using the Prony series method. The differences between the results allow us to constrain the applicability of the point source and the flat surface hypothesis, which are usually implicitly assumed when analytical solutions are derived. The effect of the topography is also considered. The results obtained show that neglecting the topographic effects may, in some cases, introduce an error greater than that implicit in the point-source hypothesis. Therefore, for an adequate modelling and interpretation of the time-dependent displacements, topography must be considered.     

State of stress within a bending spherical shell and its implications for subducting lithosphere

The state of stress within a bending spherical shell has some special features that are caused by sphericity. While most lithospheres are more like spherical shells than flat plates, our ideas of the state of stress have been dominated by flat-plate models. As a consequence, we might be missing some important aspects of the state of stress within subducting lithospheres. In order to examine this problem, we analyse spherical-shell bending problems from basic equations. We present two approaches to solve spherical-shell bending problems: one by the variational approach, which is suitable for global-scale problems, and the other by the asymptotic equation, which is valid to first order in h/R , where h is the thickness of the lithosphere and R is its curvature radius (i.e. under the assumption of small curvature). The form of the equation for displacement shows that wavelengths of deformation are determined by the spherical (elastic) effect and the gravitational buoyancy effect, for which only the latter effect is included in the usual flat-plate formulations. In the case of the Earth, the buoyancy force is dominant and, consequently, spherical effects are suppressed to a large extent; this explains why flat-plate models have been successful for Earth's lithospheric problems. On the other hand, the state of stress shows interesting spherical effects: while bending (fibre) stress along the subduction zone is always important, bending stress along the trench-strike direction can also be important, in particular when the subduction zone arc is small. Numerical results also indicate that compressive normal stress along the trench-strike direction is important when a subduction zone arc is large. These two stresses, the bending stress and the compressive normal stress, both along the trench-strike direction, may have important implications for intraplate earthquakes at subduction zones.     

Displacements and stresses due to a single force in a half-space in welded contact with another half-space

Closed-form analytical expressions for the displacements and stresses induced by a single force of arbitrary orientation located in an elastic half-space in welded contact with another elastic half-space are obtained. These expressions are valid for arbitrary values of the Poisson's ratio and for arbitrary source and observer locations. The final results are given in a form that makes numerical computation straightforward and accurate.     

Deformation of a modern alluvial plain

The alluvial plain of the Waal River (The Netherlands), over 4000 m², showed large flow-lobe like structures after it had fallen dry following floodings. The upstream-directed parabola-shaped folds had mutual distances from < 1 m to about 10 m and amplitudes of up to a few decimeters. The deformation is ascribed to a rare coincidence of hydrological, topographic, and weather conditions. Local high-energy currents developed opposite to the main river current direction, dragging the top layer along over several meters. A drilled section indicates that no true detachment plane but rather a detachment zone is present. Similar structures formed under comparable conditions have not been described previously in the literature.     

Surface vertical displacements, potential perturbations andgravity changes of a viscoelastic earth model induced by internal point dislocations

Using the viscoelastic correspondence principle, we utilize the surface coseismic spheroidal deformation fields (i.e. vertical displacements, potential perturbations and gravity changes) of SNREI earth models caused by four typical types of point dislocation, derived by Sun & Okubo (1993 ), to deduce the fundamental formulas for spheroidal fields relevant to viscoelastic earth models. In computations, we employ a strike-slip dislocation on a vertical plane buried at the bottom of the lithosphere to estimate the maximal viscous relaxation responses to this kind of source that possibly exist on the surface of the earth. We take the seismic moment as 10²²  N  m, which is characteristic of an average large earthquake. The numerical results demonstrate that, if we take the viscosity as 10¹⁹  Pa  s in the asthenosphere, and 10²¹  Pa  s in the other mantle layers, the rates of surface vertical displacements and gravity changes within about 2.5° for the 10 postseismic years are respectively 1.5–8.1  cm  yr⁻¹ and 4.0–14.9  μgal  yr⁻¹ : the viscous relaxation for this mantle viscosity profile proceeds much faster than for a constant mantle viscosity of 10²¹  Pa  s.