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1.
Okada (1992) provided expressions for the displacement and strain fields due to a finite rectangular source in an elastic, homogeneous and isotropic half-space. Starting with these results, we applied the correspondence principle of linear viscoelasticity to derive the quasi-static displacement, strain and stress fields in a viscoelastic, homogeneous and isotropic half-space. We assume that the medium deforms viscoelastically with respect to both the shear and the normal stresses but keeps a constant bulk modulus; in particular, the shear modulus relaxes as Maxwell fluid. We presented the viscoelastic effect on displacement, displacement gradient and stress fields, for a choice of parameter values. The viscoelastic effect due to the sudden dislocation reaches a limit value after about 10 times the Maxwell time. The expressions obtained here provide tools for the study of viscoelastic relaxation of lithosphere associated with seismic and volcanic phenomena.  相似文献   

2.
The degree-one deformation of the Earth (and the induced discrepancy between the figure centre and the mass centre of the Earth) is computed using a theoretical approach (Love numbers formalism) at short timescales (where the Earth has an elastic behaviour) as well as at long timescales (where the Earth has a viscoelastic or quasi-fluid behaviour). For a Maxwell model of rheology, the degree-one relaxation modes associated with the viscoelastic Love numbers have been investigated: the Mo mode does not exist and there is only one transition mode (instead of two) generated by a viscosity discontinuity.
The translations at each interface of the incompressible layers of the earth model [surface, 670 km depth discontinuity, core-mantle boundary (CMB) and inner-core boundary (ICB)] are computed. They are elastic with an order of magnitude of about 1 mm when the excitation source is the atmospheric continental loading or a magnetic pressure acting at the CMB. They are viscoelastic when the earth is submitted to Pleistocene deglaciation, with an order of magnitude of about 1 m. In a quasi-fluid approximation (Newtonian fluid) because of the mantle density heterogeneity their order of magnitude is about 100 m (except for the ICB, which is in quasi-hydrostatic equilibrium at this timescale).  相似文献   

3.
In this study we apply the correspondence principle for free vibrations of a homogeneous viscoelastic solid derived by Fisher & Leitman to obtain the torsional modes of a homogeneous viscoelastic rod. We also extend the correspondence principle, showing that it may be used to find the frequencies of Love waves in a stratified viscoelastic medium. Finally, we apply the correspondence principle to four viscoelastic materials, the Kelvin-Voigt solid, the Maxwell solid, the standard linear solid, and the Achenbach-Chao solid. We show that in each of these cases some care must be used in applying the correspondence principle because of the presence of multiple solutions. We also examine measures of dissipation of the free vibrations, and we determine the conditions under which the logarithmic decrement may be approximated by the process-independent l/ Q of O'Connell & Budiansky.  相似文献   

4.
Summary. We give the analytical formulation for calculating the transient displacement of fields produced by earthquakes in a stratified, selfgravitating, incompressible, viscoelastic earth. We have evaluated the potential of viscous creep in the asthenosphere in exciting the Chandler wobble by a four-layer model consisting of an elastic lithosphere, a two-layer Maxwell viscoelastic mantle, and an inviscid core. The seismic source is modelled as an inhomogeneous boundary condition, which involves a jump condition of the displacement fields across the fault in the lithosphere. The response fields are derived from the solution of a two-point boundary value problem, using analytical propagator matrices in the Laplace-transformed domain. Transient flows produced by post-seismic rebound are found to be confined within the asthenosphere for local viscosity values less than 1020P. The viscosity of the mantle below the low-viscosity channel is kept at 1022P. For low-viscosity zones with widths greater than about 100 km and asthenospheric viscosities less than 1018P, we find that viscoelasticity can amplify the perturbations in the moment of inertia by a factor of 4–5 above the elastic contribution within the time span of the wobble period. We have carried out a comparative study on the changes of the inertia tensor from forcings due to surface loading and to faulting. In general the global responses from faulting are found to be much more sensitive to the viscosity structure of the asthenosphere than those produced from surface loading.  相似文献   

5.
Geological studies show evidence for temporal clustering of large earthquakes on individual fault systems. Since post-seismic deformation due to the inelastic rheology of the lithosphere may result in a variable loading rate on a fault throughout the interseismic period, it is reasonable to expect that the rheology of the non-seismogenic lower crust and mantle lithosphere may play a role in controlling earthquake recurrence times. We study this phenomenon using a 2-D, finite element method continuum model of the lithosphere containing a single strike-slip fault. This model builds on a previous study using a 1-D spring-dashpot-slider analogue of a single fault system to study the role of Maxwell viscoelastic relaxation in producing non-periodic earthquakes. In our 2-D model, the seismogenic portion of the fault slips when a predetermined yield stress is exceeded; stress accumulated on the seismogenic fault is shed to the viscoelastic layers below and recycled back to the seismogenic fault through viscoelastic relaxation. We find that random variation of the fault yield stress from one earthquake to the next can cause the earthquake sequence to be clustered; the amount of clustering depends on a non-dimensional number, W , called the Wallace number defined as the standard deviation of the randomly varied fault yield stress divided by the effective viscosity of the system times the tectonic loading rate. A new clustering metric based on the bimodal distribution of interseismic intervals allows us to investigate clustering behaviour of systems over a wide range of model parameters and those with multiple viscoelastic layers. For models with   W ≥ 1  clustering increases with increasing W , while those with   W ≤ 1  are unclustered, or quasi-periodic.  相似文献   

6.
Summary. A microplate is modelled as an elastic plate with two long strike-slip boundaries, lying over a Maxwell-type viscoelastic asthenosphere. The microplate is subjected to a constant and uniform shear strain rate by the opposite motions of two adjoining larger plates. After the occurrence of an earthquake at one of the microplate boundaries, the time evolution of shear stress at the other boundary is studied. It is found that stress build-up at the second boundary is delayed due to stress diffusion governed by the asthenosphere relaxation. Earthquake occurrence at this latter boundary would be delayed depending upon both the microplate width and the ratio between the Maxwell relaxation time of the asthenosphere and a characteristic time required for tectonic strain to recover rupture conditions. It turns out that the parameters which determine the occurrence of seismic activity along the microplate boundaries are more strictly constrained in the presence of a viscoelastic asthenosphere than in the case of an elastic half-pace model.  相似文献   

7.
The dynamical origin of subduction zone topography   总被引：1，自引：0，他引：1  
Summary. Subduction zones are expressed topographically by long linear oceanic trenches flanked by a low outer rise on the seaward side and an island arc on the landward side. This topographic structure is reflected in free air gravity anomalies, suggesting that much of the topography originates from dynamical forces applied at the base of the crust. We have successfully reproduced the general topographic features of subduction zones by supposing that the stresses generated by the bending of the viscous lower lithosphere as it subducts are transmitted through the thin elastic upper portion of the lithosphere. The trench is due to a zone of extensional flow (associated with low pressure) in the upper part of the viscous lithosphere.
The stresses in the subducting slab are computed using a finite element technique, assuming a Maxwell viscoelastic constitutive relation. Various dips (10 to 90°) were investigated, as well as depth dependent and non-Newtonian (power law, n = 3) viscosities. Observed subduction zone dimensions are well reproduced by these models. The effective viscosity required at mid-depth in the lithosphere is about 6 × 1022 P. This low value is probably due to the stress dependence of the effective viscosity. However, these models also show that the topography of the subduction zone depends primarily upon the geometry of the subducting slab (dip, radius of curvature of the bend) rather than upon its rheology. Shear stresses beneath the trench reach maxima of approximately 50 MPa. An interesting feature of some solutions is a dynamically supported bench or platform between the trench and island arc.  相似文献   

8.
Summary. The space-time elastic wave motion generated by an impulsive line source in a homogeneous anisotropic medium is calculated with the aid of the Cagniard-de Hoop method. Two types of sources are considered in detail, viz. a line source of expansion (model for an explosive source) and a line force (model for a mechanical vibrator). Numerical results are presented for the radiated particle velocity in the medium including those regions of space where shear-wave triplication occurs. There is a marked difference in the time response observed for the two types of sources and for the different positions of the receiver with respect to the source position. These waveform differences are important when the radiated wave is used to determine experimentally the elastic properties of the medium. As compared with the traditional Fourier-integral transform method to handle this problem, the computation time with the present method is considerably less.  相似文献   

9.
The 2004 M = 9.2 Sumatra–Andaman earthquake profoundly altered the state of stress in a large volume surrounding the ∼1400 km long rupture. Induced mantle flow fields and coupled surface deformation are sensitive to the 3-D rheology structure. To predict the post-seismic motions from this earthquake, relaxation of a 3-D spherical viscoelastic earth model is simulated using the theory of coupled normal modes. The quasi-static deformation basis set and solution on the 3-D model is constructed using: a spherically stratified viscoelastic earth model with a linear stress–strain relation; an aspherical perturbation in viscoelastic structure; a 'static' mode basis set consisting of Earth's spheroidal and toroidal free oscillations; a "viscoelastic" mode basis set; and interaction kernels that describe the coupling among viscoelastic and static modes. Application to the 2004 Sumatra–Andaman earthquake illustrates the profound modification of the post-seismic flow field at depth by a slab structure and similarly large effects on the near-field post-seismic deformation field at Earth's surface. Comparison with post-seismic GPS observations illustrates the extent to which viscoelastic relaxation contributes to the regional post-seismic deformation.  相似文献   

10.
The vibration of a fluid-filled crack is considered to be one of the most plausible source mechanisms for the long-period events and volcanic tremors occurring around volcanoes. As a tool for the quantitative interpretation of source process of such volcanic seismic signals, we propose a method to numerically simulate the dynamic response of a fluid-filled crack. In this method, we formulate the motions of the fluid inside and the elastic solid outside of the crack, using boundary integrals in the frequency domain and solve the dynamic interactions between the fluid and the elastic solid using the point collocation method. The present method is more efficient compared with the time-domain finite difference method, which has been used in simulations of a fluid-filled crack and enables us to study the dynamics of a fluid-filled crack over a wide range of physical parameters. The method also allows us direct calculation of the attenuation quality factor of the crack resonance, which is an indispensable parameter for estimating the properties of the fluid inside the crack. The method is also designed to be flexible to many applications, which may be encountered in volcano seismology, and thus, extensions of the method to more complicated problems are promising.  相似文献   

11.
The post-seismic response of a viscoelastic Earth to a seismic dislocation can be computed analytically within the framework of normal-modes, based on the application of propagator methods. This technique, widely documented in the literature, suffers from several shortcomings; the main drawback is related to the numerical solution of the secular equation, whose degree increases linearly with the number of viscoelastic layers so that only coarse-layered models are practically solvable. Recently, a viable alternative to the standard normal-mode approach, based on the Post–Widder Laplace inversion formula, has been proposed in the realm of postglacial rebound models. The main advantage of this method is to bypass the explicit solution of the secular equation, while retaining the analytical structure of the propagator formalism. At the same time, the numerical computation is much simplified so that additional features such as linear non-Maxwell rheologies can be simply implemented. In this work, for the first time, we apply the Post–Widder Laplace inversion formula to a post-seismic rebound model. We test the method against the standard normal-mode solution and we perform various benchmarks aimed to tune the algorithm and to optimize computation performance while ensuring the stability of the solution. As an application, we address the issue of finding the minimum number of layers with distinct elastic properties needed to accurately describe the post-seismic relaxation of a realistic Earth model. Finally, we demonstrate the potentialities of our code by modelling the post-seismic relaxation after the 2004 Sumatra–Andaman earthquake comparing results based upon Maxwell and Burgers rheologies.  相似文献   

12.
Analytical solutions for the piezomagnetic potentials are derived for strike-slip, dip-slip and tensile-opening fault motions with arbitrary dip and strike angles, so as to be applicable in various types of earthquakes. These solutions are expressed as the composition of elementary functions which are identical to the magnetic potentials produced by magnetic dipoles, quadrupoles and octupoles distributed on the fault plane and other planes. Therefore, the geomagnetic field changes due to the piezomagnetic effect are expressed by the superposition of the fields produced by these equivalent sources.
Examples of calculated results show characteristic features for various types of fault motions as follows: (1) the pattern of the geomagnetic field changes becomes significantly different depending on the strike direction, although the maximum amplitude is almost the same for all directions; (2) the geomagnetic field change reaches a maximum at a dip angle of 90° for strike-slip and tensile-opening fault motions and at 45° for dip-slip fault motion.  相似文献   

13.
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.  相似文献   

14.
We combine Global Positioning System (GPS) measurements with forward modelling of viscoelastic relaxation and after-slip to study the post-seismic deformation of the 1997 Umbria-Marche (Central Apennines) moderate shallow earthquake sequence. Campaign GPS measurements spanning the time period 1999–2003 are depicting a clear post-seismic deformation signal. Our results favour a normal faulting rupture model where most of the slip is located in the lower part of the seismogenic upper crust, consistent with the rupture models obtained from the inversion of strong motion data. The preferred rheological model, obtained from viscoelastic relaxation modelling, consists of an elastic upper crust, underlain by a transition zone with a viscosity of 1018 Pa s, while the rheology of deeper layers is not relevant for the observed time-span. Shallow fault creep and after-slip at the base of the seismogenic upper crust are the first order processes behind the observed post-seismic deformation. The deep after-slip, below the fault zone at about 8 km depth, acting as a basal shear through localized time-dependent deformation, identifies a rheological discontinuity decoupling the seismogenic upper crust from the low-viscosity transition zone.  相似文献   

15.
Non-linear elastic behaviour of damaged rocks   总被引：1，自引：0，他引：1  
The pervasive damage of rocks by microcracks and voids strongly affects their macroscopic elastic properties. To evaluate the damage effects, we derive here the macroscopic stress-strain relations for a 3-D elastic solid with non-interacting cracks embedded inside a homogeneous matrix. The cracks considered are oriented either perpendicular to the maximum tension axis, or perpendicular to the maximum compression axis. In the first case they dilate during loading and in the second case they contract during loading. We derive a solution for the elastic energy of this rock following the self-consistent scheme of Budiansky & O'Connell (1976). The solution describes the stress-strain relations in terms of Λd and μd, which are the modified Lame constants, and an additional parameter Λ. The latter accounts for the non-linear behaviour of the solid and is related to crack density. The solution predicts a non-linear elastic rheology even for an infinitesimal strain of ɛ < 0.001, and abrupt change in the elastic moduli when the loading reverses from uniaxial compression to uniaxial tension.
We use the derived solution to analyse rock-mechanics experiments in which beams of Indiana limestone were deformed under four-point loading. This configuration provides simultaneously the apparent tensile and compressive moduli for small strains. The apparent moduli fit the effective elastic moduli calculated with the present damage model well, including the differences between tensile and compressive moduli.  相似文献   

16.
Previous studies of the wander of the rotation pole associated with the Late Pleistocene glacial cycles indicate that the predicted polar wander speed is sensitive to the density jump at the 670 km discontinuity, the thickness of the elastic lithosphere, and the lower mantle viscosity. In particular, the M1 mode related to the density jump at 670 km depth has been shown to contribute a dominant portion of predicted polar wander speed for sufficiently small lower mantle viscosities. In this study, we examine the sensitivity of polar wander to variations in the viscosity of the viscoelastic lithosphere using simplified compressible Maxwell viscoelastic earth models. Model calculations for earth models with a viscoelastic lithosphere of finite viscosity indicate that the contribution of the M1 mode is similar to those associated with the density discontinuity at the core–mantle boundary (C0 mode) and the lithosphere (L0 mode). We speculate that this is due to the interaction between the M1 mode and the transient mode associated with the viscoelastic lithosphere, which reduces the magnitude of polar wander rates. Therefore, the M1 mode does not contribute a dominant portion of the predicted polar wander speed for earth models with a viscoelastic lithosphere of finite viscosity. In this case, predictions of polar wander speed as a function of lower mantle viscosity exhibit the qualitative form of an 'inverted parabola', as predicted for the J ˙2 curve. We caution, however, that these results are obtained for simplified earth models, and the results for seismological earth models such as PREM may be complicated by the interaction between the M1 mode and the large set of transient modes.  相似文献   

17.
Elastic finite element models are applied to investigate the effects of topography and medium heterogeneities on the surface deformation and the gravity field produced by volcanic pressure sources. Changes in the gravity field cannot be interpreted only in terms of gain of mass disregarding the ground deformation of the rocks surrounding the source. Contributions to gravity changes depend also on surface and subsurface mass redistribution driven by dilation of the volcanic source. Both ground deformation and gravity changes were firstly evaluated by solving a coupled axisymmetric problem to estimate the effects of topography and medium heterogeneities. Numerical results show significant discrepancies in the ground deformation and gravity field compared to those predicted by analytical solutions, which disregard topography, elastic heterogeneities and density subsurface structures. With this in mind, we reviewed the expected gravity changes accompanying the 1993–1997 inflation phase on Mt Etna by setting up a fully 3-D finite element model in which we used the real topography, to include the geometry, and seismic tomography, to infer the crustal heterogeneities. The inflation phase was clearly detected by different geodetic techniques (EDM, GPS, SAR and levelling data) that showed a uniform expansion of the overall volcano edifice. When the gravity data are integrated with ground deformation data and a coupled FEM modelling was solved, a mass intrusion could have occurred at depth to justify both ground deformation and gravity observations.  相似文献   

18.
Summary. The tidal deformation of a homogeneous viscoelastic sphere due to the gravitational attraction of an external body is calculated. The sphere is modelled as an incompressible Kelvin-Voigt solid. An equation for the displacement field is obtained assuming that strains are small and inertia is negligible. This equation has a series solution in terms of Legendre polynomials. The resulting expression for the displacement field reduces to that for an elastic solid and a viscous fluid in the appropriate limits of the material constants. The first term in the viscoelastic solution is used to calculate the moments induced by tidal deformation assuming a circular orbit. In the absence of obliquity and precession, these moments reduced to a torque about the spin axis. This torque is compared to that predicted by a phase lag analysis. These two approaches are formally equivalent if the tidal dissipation function Q −1 depends in a specific way on the difference of the spin and orbital angular velocities.  相似文献   

19.
Analytical approach for the toroidal relaxation of viscoelastic earth   总被引：1，自引：0，他引：1  
This paper is concerned with post-seismic toroidal deformation in a spherically symmetric, non-rotating, linear-viscoelastic, isotropic Maxwell earth model. Analytical expressions for characteristic relaxation times and relaxation strengths are found for viscoelastic toroidal deformation, associated with surface tangential stress, when there are two to five layers between the core–mantle boundary and Earth's surface. The multilayered models can include lithosphere, asthenosphere, upper and lower mantles and even low-viscosity ductile layer in the lithosphere. The analytical approach is self-consistent in that the Heaviside isostatic solution agrees with fluid limit. The analytical solution can be used for high-precision simulation of the toroidal relaxation in five-layer earths and the results can also be considered as a benchmark for numerical methods. Analytical solution gives only stable decaying modes—unstable mode, conjugate complex mode and modes of relevant poles with orders larger than 1, are all excluded, and the total number of modes is found to be just the number of viscoelastic layers between the core–mantle boundary and Earth's surface—however, any elastic layer between two viscoelastic layers is also counted. This confirms previous finding where numerical method (i.e. propagator matrix method) is used. We have studied the relaxation times of a lot of models and found the propagator matrix method to agree very well with those from analytical results. In addition, the asthenosphere and lithospheric ductile layer are found to have large effects on the amplitude of post-seismic deformation. This also confirms the findings of previous works.  相似文献   

20.
A new model that accounts for the stress dependence of the phase velocity of elastodynamic waves propagating in a cracked solid under compression is presented. The phase velocities of longitudinal and shear waves are derived from the effective elastic properties of a cracked solid, which are evaluated within the framework of Kachanov's approach. Following Kachanov, the extra-compliance tensor of the cracked solid is related to the crack compliances, which display a marked non-linear behaviour when subjected to a compressive load. Such non-linear behaviour is shown to be derived from the elastic interaction between the contacting crack faces under compression. This work does not address the effect of mutual interaction among cracks and the generation of higher harmonics due to the medium non-linearity. Numerical examples are presented that illustrate the phase velocity changes occurring in a solid with a random distribution of parallel cracks as a function of an external compressive load. A distinctive feature of the acoustoelastic effect in solids with large parallel fractures and in solids with distributions of aligned microcracks is also illustrated.  相似文献   

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