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Besides being earthquake sources, active faults can slip aseismically. Aseismic slip plays an important role in releasing stress along plate boundaries and should be considered in the interpretation of geodetic measurements in tectonically active areas. An asperity model is presented showing that, due to the heterogeneity of fault surfaces, aseismic slip occurs through advancing dislocation fronts. The ground deformation pattern produced by such propagating dislocations is studied by calculating tilt as a function of time produced by a dislocation front moving at constant velocity in an elastic half-space model.

Crack models of faulting are presented in a layered medium. A free surface bounds a layer of low rigidity, in welded contact with a higher rigidity half-space, cut by an antiplane vertical shear crack reaching the interface between the two media. For comparison, the case is also considered of a homogeneous half-space. Solutions are found following an efficient numerical method which employes the integral properties of Jacobi and Chebyshev polynomials respectively in the two cases. Solutions for the slip amplitude on the crack surface and for the displacement field at the free surface show the importance of taking properly into account the presence of soft sedimentary layers when geodetic data are inverted for fault parameters after an earthquake. The major differences regard the maximum value of the displacement and its distance from the fault strike.

We suggest that the regular, long-period perturbations recorded by the Marussi pendulums at the Trieste Earth Tide Station over a period of three years preceding the 1976 Friuli earthquake, were due to silent earthquakes which occurred below the focal zone of the impending 1976 earthquake. The source of silent earthquakes is modelled in terms of slow slippage episodes on a buried fault with viscoelastic properties. The computed radiation emitted by such a source reproduces at least the order of magnitude of the dominant frequency, the mean duration and amplitude of the events observed at Trieste. Owing to the very long period and the low amplitudes involved, the wave emission is found to be detectable in the near field only, with appropriate instrumentation.

On the basis of observational evidence and theoretical models, we assess a possible mechanism for earthquake occurrence on plate margins. Since it is largely accepted that a condition of aseismic sliding may prevail at depth below active faults, we propose that the sequence: aseismic slip at depth rt arrow stress concentration on the upper brittle zone rt arrow seismic rupture may be a common mechanism on plate margins. In particular, we suggest that aseismic slip at depth could be responsible for the so-called 'silent earthquakes', those observed wavetrains which display a dominant low-frequency content, and propose a simple model which accounts for very low slip velocities on deep fault sections in terms of a viscous drag force.

SUMMARY There is evidence that magma chambers are not fed by a continuous flow, but in the form of discrete magma batches. The possibility is considered that the chamber is fed by a magma bubble ascending through the region underneath, driven by buoyancy force. Due to the high ambient temperatures, it is assumed that the bubble moves through a viscoelastic medium with temperature-dependent viscosity. The motion of a spherical magma bubble and its inflow into the chamber are studied. The bubble volume is assumed to be at least equal to magma volumes in typical effusive eruptions on Mount Etna, corresponding to bubble radii of a few hundred metres. Under some simplifying assumptions, the problem is solved analytically. The bubble velocity is directly proportional to the square of its radius and inversely to the viscosity of surrounding rocks, but it is independent of magma viscosity. Velocity can reach values of the order of tens of metres per year in the proximity of the chamber. Since the characteristic time for heat diffusion is several hundred years, the bubble can cover several kilometres with only moderate cooling. During ascent, forced convection takes place in the bubble. Equations for convection streamlines are obtained and traveltimes of magma are calculated, giving a mixing time of the order of hundred years below the chamber. Inflow of the bubble in the magma chamber produces a pulse in flow rate. Under reasonable assumptions, pulse shape and duration are calculated analytically. Pulse duration can be several tens of years and can give rise to a sequence of eruptions, depending on the size of the bubble and the critical overpressure for eruption.

Effusion rate in basaltic eruptions typically depends on time: there is an initial, relatively fast increase followed by a much slower decrease until the eruption vanishes; in addition, changes are observed in the effusion rate having durations much shorter than the total duration of the eruption. For an effusive eruption, we calculate the deformation of the volcanic conduit due to short-term pressure oscillations. The model considers an elliptical conduit embedded in a viscoelastic medium, described by a Maxwell body. As a consequence of pressure oscillations, the semi axes of the conduit are quasi periodic functions of time with the same period as pressure. For volcanic fissures the viscoelastic rheology entails a remarkable increase in oscillation amplitude of flow rate with respect to the elastic case and a time delay in flow rate oscillation with respect to overpressure oscillation. For a given value of overpressure amplitude, this effect is controlled by the conduit eccentricity and the ratio between overpressure period and Maxwell time; for larger values of this ratio and/or for eccentricity values closer to unity, flow rate oscillates around a value larger than its initial value and can vary from 5% to 30% with respect to it. The model can approximate the in-situ observations of short-time fluctuations of flow rate during the 2018 eruption of Kı̄lauea Volcano.

We assume that a magma pulse enters a magma chamber from the plumbing system, producing an overpressure triggering an effusive volcanic eruption. The chamber is modeled as a spherical cavity in an elastic half-space, connected to the Earth's surface by a vertical conduit, and the magma as an incompressible Newtonian liquid. Overpressure in the chamber is calculated as a function of time during the eruption and has different behaviors depending on the characteristics of the magma pulse and of the volcanic system. The time history of effusion rate is controlled by the pulse duration and the geometrical and rheological parameters of the model. The calculated effusion rates are compared with different classes of effusion rates extrapolated from historical data of Mount Etna, which are asymmetric functions of time. For appropriate values of the model parameters, the calculated time histories are found to fit well those derived from observation.

The acceleration of gravity in Earth’s interior is determined by the density distribution in Earth. A remarkable result is that the acceleration is approximately constant all over the mantle, which amounts to about 84% of Earth’s volume. This result can be explained by a simple two-layer model of Earth, showing that the constancy of the acceleration in the mantle is a consequence of the particular size and density of Earth’s core with respect to the size and density of the whole Earth. In other planets, with different mass distributions, the dependence of acceleration on depth could be very different.

A fault containing two asperities with different strengths is considered. The fault is embedded in a viscoelastic shear zone, subject to a constant strain rate by the motions of adjacent tectonic plates. The fault is modelled as a discrete dynamical system where the average values of stress, friction and slip on each asperity are considered. The state of the fault is described by three variables: the slip deficits of the asperities and the viscoelastic deformation. The system has four dynamic modes, for which the analytical solutions are calculated. The relationship between the state of the fault before a seismic event and the sequence of slipping modes in the event is enlightened. Since the moment rate depends on the number and sequence of slipping modes, the knowledge of the source function of an earthquake constrains the orbit of the system in the phase space. If the source functions of a larger number of consecutive earthquakes were known, the orbit could be constrained more and more and its evolution could be predicted with a smaller uncertainty. The model is applied to the 1964 Alaska earthquake, which was the effect of the failure of two asperities and for which a remarkable postseismic relaxation has been observed in the subsequent decades. The evolution of the system after the 1964 event depends on the state from which the event was originated, that is constrained by the observed moment rate. The possible durations of the interseismic interval and the possible moment rates of the next earthquake are calculated as functions of the initial state.

We study the dynamics of a fault containing two asperities having different strengths and subject to a constant strain rate. The fault is modeled by a discrete dynamical system with two degrees of freedom that are the slip deficits of the asperities. The radiation of elastic waves during the slipping modes is taken into account by introducing a term proportional to slip rate in the equations of motion. We give a complete analytical solution of the four dynamic modes of the system. Any seismic event can be expressed as a sequence of modes, for which the moment rate, the spectrum and the total seismic moment can be calculated. We consider the energy budget of the event and calculate the seismic efficiency. The presence of radiation may remarkably change the evolution of the system from a given state, since it moves the boundaries between the different subsets of the sticking region. In addition, the slip amplitude in a seismic event is smaller, while the slip duration is longer in the presence of radiation. The shape of the moment rate function depends on the seismic efficiency and the seismic moment decreases with increasing efficiency, at constant radiated energy. As an application of the model, we consider the 1964 Great Alaska earthquake and calculate the mode durations, the slip distribution, the moment rate function and the seismic moment of the earthquake, obtaining values that are consistent with observations.

We propose a model describing the thermal effects of a lava tube. The tube is a circular cylinder, embedded in a solid medium and filled with a Newtonian liquid flowing under the gravity force. Steady-state conditions are considered. The velocity field in the tube, evaluated by the Navier-Stokes equation, is introduced into the heat equation taking into account the viscous dissipation. The temperature distribution is evaluated both inside the tube and in the surrounding solid medium. Under the assumption that the lava tube is embedded in a solid half-space, the surface heat flow due to the presence of the tube is calculated. It is shown that heat flow measurements at the Earth's surface can give information on the depth, size and temperature of the buried lava tube. © 2002 Elsevier Science B.V. All rights reserved.

—A model is proposed to study the modification of the stress field at a transcurrent plate boundary due to frictional heat production at depth. Two cases are considered a stable and a stretched lithosphere. The model is applied to those weak faults where the dynamic friction is small compared to a static one; if the deformation along the brittle portion of the fault is entirely accommodated by a series of seismic ruptures in a quasi-static state where the fault has been moving for millions of years, the long-term thermal field perturbation due to these ruptures results in only a few degrees and can be neglected. The boundary zone is considered as a viscoelastic body subject to a constant strain rate. The lower section of the boundary is assumed to slip aseismically along a vertical transcurrent fault and to completely accommodate the plate motion, while the upper section is locked. The slipping zone is divided into a semi-brittle zone, placed between the isothermal surfaces of 300°C and 450°C, and a ductile zone beneath. The frictional heat is calculated by assuming a linearly decreasing friction in the semi-brittle and a constant friction in the ductile zones. The heat modifies the temperature field, producing an upward movement of the semi-brittle and ductile fault sections. As a consequence, the thickness of the brittle fault section is reduced and friction at the base of this section is less. The stress field in the boundary zone is calculated as a function of time for different friction profiles and slip rates on the fault. Owing to heat production, a greater stress concentration is produced on the brittle fault section, while shear stress is lowered in regions occupied by the uplifted semi-brittle layer. These effects are found to be remarkable only in the case of a stable zone, with a standard unperturbed geotherm, while they are irrelevant in a stretched zone with a high geothermal gradient. In any case, the role of the semi-brittle layer appears to be more prominent in the case of boundaries with higher slip rates, due to the presence of higher stress values.

SUMMARY Volcanic rocks forming sills, dykes or lava flows may display a magnetic anisotropy derived from the viscous flow during their emplacement. We model a sill as a steadystate flow of a Bingham fluid, driven by a pressure gradient in a horizontal conduit. The magma velocity as a function of depth is calculated from the motion and constitutive equations. Vorticity and strain rate are determined for a reference system moving with the fluid. The angular velocity and the orientation of an ellipsoidal magnetic grain immersed in the fluid are calculated as functions of time or strain. Magnetic susceptibility is then calculated for a large number of grains with a uniform distribution of initial orientations. It is shown that the magnetic lineation oscillates in the vertical plane through the magma flow direction, and that the magnetic foliation plane changes periodically from horizontal to vertical. The results are compared with the magnetic fabric of Ferrar dolerite sills (Victoria Land, East Antarctica) derived from low-field susceptibility measurements.

The problem of post-seismic stress evolution at the lower dislocation edge of great shallow earthquakes is investigated by a model reproducing a transcurrent boundary zone subject to a constant strain rate. Dislocation models with frictional stress threshold for slip arrest must be considered in order to study post-seismic stress evolution in the proximity of dislocation edges. It is found that the boundary zone is vertically divided in two regions: an upper region, where stress increases with time and a lower region, where stress decreases. Dislocations which are nucleated in the seismogenic region can propagate downward into the aseismic region: in this case earthquakes are followed by relaxation of the stress concentration at the lower dislocation edge. This may have effects on the space-time distribution of aftershocks.

Tectonic boundary zones are characterized by a shallow, brittle region (seismogenic region) where frictional slip occurs on fault planes, and by a lower, ductile and aseismic region. The nucleation depth and the extent of slip planes in great interplate earthquakes are determined by the relationships between frictional resistance, applied stress and stress drop. The post-seismic stress evolution on a fault plane is investigated by a model which reproduces a transcurrent boundary zone subject to a constant strain rate. Dislocation models with frictional stress threshold for slip arrest must be considered in order to study post-seismic stress evolution in the proximity of dislocation edges. It is found that dislocations which are nucleated in the seismogenic region can propagate downward into the aseismic region: in this case earthquakes are followed by relaxation of the stress concentration at the lower dislocation edge. This has effects on the space-time distribution of aftershocks.

SUMMARY During 1997 several slow earthquakes have been recorded by a geodetic interferometer located beneath Gran Sasso, central Italy. The strain rise times of the events range from tens to thousands of seconds and strain amplitudes are of the order of 10−9. Amplitudes scale with the square root of the rise time and this suggests a diffusive behaviour of the slip propagation along the fault. In this work, we develop a model in which slip diffusion is the result of the presence of a gouge layer between fault faces, with a viscoplastic rheology. The fluid velocity field in the gouge layer diffuses in the directions of fault length and fault thickness, with different characteristic times. This model reproduces the relation between amplitude and rise time of measured strain signals. Synthetic straingrams, obtained for a horizontally layered, flat Earth and a source located a few kilometres from the instrument, are in agreement with observed signals.

The study of surface deformation due to seismic activity is often made using dislocations with uniform slip and simple geometries. A better modeling of coseismic and postseismic surface displacements can be obtained by using dislocations with variable slip and nonregular shapes. This is consistent with the asperity model of fault surfaces, assuming a friction distribution on faults made of locked zones with much higher friction than surrounding zones. In this paper we consider the 1997–1998 Colfiorito seismic sequence. The coseismic surface displacements in the Colfiorito zone are used in order to infer the slip distribution on the fault surface at different stages of the sequence. The displacement field has been modeled varying the slip distribution on the fault, and comparing the deformation observed by SAR and GPS techniques with model results. The slip distribution is calculated by Monte Carlo simulations on a normal fault with the dip angle equal to 40°. A good approximation is obtained by using square asperity units of 1.5 times 1.5 km2. In the first stage, we employed a simplified model with uniform slip, in which each asperity unit is allowed to slip a constant amount or not to slip at all, and in the second stage, we evaluate the slip distribution in the dislocation area determined by the Monte Carlo inversion: in this case we allow unit cells to undergo different values of slip in order to refine the initial dislocation model. The results show that the 1997 seismic events of the sequence can be modeled by irregular dislocations, obtaining a good fit to the DInSAR and GPS observations. The model also confirms the results of previous studies by a different methodology, defining the distribution of asperities on the fault plane using the fault geometry, the geodetic data and the seismic moment of the 1997–1998 Colfiorito seismic sequence. Furthermore, the analysis of 1997 aftershocks in the seismogenic region shows a strong correlation between most events and the asperity distribution, which can be considered as an independent test of the validity of the model.

The 1964 Alaska earthquake was the second largest seismic events in the 20th century. The aim of this work is the use of surface deformation data to determine asperity and slip distributions on the fault plane of the Alaska earthquake: these distributions are calculated by a Monte Carlo method. To this aim, we decompose the fault plane in a large number of small square asperity units with a side of 25 km; this allows us to obtain plane surfaces with an irregular shape. In the first stage, each asperity unit is allowed to slip a constant amount or not to slip at all, providing the geometry of the dislocation surface that best reproduces the observed displacements. To this purpose, a large number of slip distributions have been tried by the use of the Monte Carlo method. The slip amplitude is the same for all the asperities and is equal to the average fault slip inferred from the seismic moment. In the second stage, we evaluate the slip distribution in the dislocation area determined by the Monte Carlo inversion: in this case, we allow unit cells to undergo different values of slip in order to refine the initial dislocation model. The results confirm the previous finding that the slip distribution of the great Alaska earthquake was essentially made of two dislocation areas with a higher slip, the Prince William Sound and the Kodiak asperities. Analysis of the post-1964 seismicity in the rupture region shows a strong correlation between the larger earthquakes (Mw≥6) and the distribution of locked asperities following the 1964 event, which can be considered as an independent test of the validity of the model. We do not find slip values higher than 25 m for any of the patches, and we determine two separate high-slip zones: one correspondent to the Prince William Sound asperity, and one (∼18 m slip) to the Kodiak asperity. The slip distribution connected with the 1964 shock appears to be consistent with the following seismicity in the region.

We introduce a 3D model for near-vent channelized lava flows. We assume the lava to be an isothermal Newtonian liquid flowing in a rectangular channel down a constant slope. The flow velocity is calculated with an analytical steady-state solution of the Navier-Stokes equation. The surface velocity and the flow rate are calculated as functions of the flow thickness for different flow widths, and the results are compared with those of a 2D model. For typical Etna lava flow parameters, the influence of levees on the flow dynamics is significant when the flow width is less than 25 m. The model predicts the volume flow rate corresponding to the surface velocity, taking into account that both depend on flow thickness. The effusion rate is a critical parameter to evaluate lava flow hazard. We propose a model to calculate the effusion rate given the lava flow width, the topographic slope, the lava density, the surface flow velocity, and either the lava viscosity or the flow thickness.

A power-law, viscoelastic constitutive equation for lithospheric rocks, is considered. The equation is a nonlinear generalization of the Maxwell constitutive equation, in which the viscous deformation depends on the n-th power of deviatoric stress, and describes a medium which is elastic with respect to normal stress, but relaxes deviatoric stress. Power-law exponents equal to 2 and 3, which are most often found in laboratory experiments, are considered. The equation is solved by a perturbative method for a viscoelastic layer subjected to a constant, extensional or compressional, strain rate and yields stress as a function of time, temperature and rock composition. The solution is applied to an ideal extensional boundary zone and shows that the base of the crustal seismogenic layer may be deeper than predicted by a linear rheology.

A model is proposed describing the mechanical evolution of a shear zone along compressional and extensional plate boundaries, subject to constant strain rate. The shear zones are assumed as viscoelastic with Maxwell rheology and with elastic and rheological parameters depending on temperature and petrology. Stress and strain are computed as functions of time and depth. For both kinds of boundaries the model reproduces the existence of a shallow seismogenic zone, characterized by a stress concentration. The thickness of the seismogenic layer is evaluated considering the variations of shear stress and frictional strength on faults embedded in the shear zone. Assuming that a fault dislocation takes place, the brittle-ductile transition is assumed to occur at the depth at which the time derivative of total shear stress changes from positive to negative values. The effects of different strain rates and geothermal gradients on the depth of the brittle-ductile transition are studied. The model predictions are consistent with values inferred from seismicity data of different boundary zones. © 1993 Birkhäuser Verlag.

The process of earthquake nucleation is studied assuming that faults are rupture surfaces on which sliding is controlled by friction. Earthquakes are assumed to arise through an instability of frictional sliding. Empirical slip laws indicate that, under constant ambient conditions, friction depends on time, slip rate and slip history. Regular stick-slip behaviour is induced by velocity weakening, a negative dependence of friction on slip rate. Velocity weakening is introduced into a model for a propagating Somigliana dislocation under slowly increasing ambient shear stress. The instability occurs when the rate at which friction decreases becomes greater than the rate at which the applied stress must increase to produce an advance of fault slip. The possibility that this condition is fulfilled depends on the velocity dependence and on the spatial distribution of friction on the fault. A critical nucleation width of the dislocation is associated with the instability and is controlled by the friction distribution, which determines the size of the initial slipping patch. Depending on the stress drop and the characteristic slip distance, the critical nucleation width may be greater for small earthquakes than for large earthquakes, with respect to the initial slipping patch.

A 2-D model is presented for the propagation of a Somigliana dislocation along a fault with nonuniform friction. Fault slip is driven by a uniform ambient shear stress, slowly increasing with time. The dislocation is nucleated in the lowest-friction region of the fault plane and is confined by the surrounding higher-friction regions (asperities). The case studied involves asperities which have smooth borders, characterized by a constant friction gradient. For values of ambient shear stress near to the weak-zone friction, the propagation of dislocation is slowed down by the presence of asperities. Only when it goes beyond the border does the dislocation front move at increasing velocity. The model shows that, at a given value of ambient shear stress, the slip amplitude is larger in the case of finite and constant friction gradient than in the case of asperities with sharp borders. Unlike the propagation velocity, slip rate is ever increasing during the dislocation process. The model shows to what extent a dislocation is influenced by the distribution of friction on the fault. A detailed knowledge of slip rate and slip history is needed to understand the mechanism of frictional instability on faults.

A model is proposed for studying the mechanical behaviour of faults during their interseismic periods. The model considers a plane fault surface in an elastic medium, subject to a uniform shear stress which increases slowly with time. A1-D friction distribution is assumed on the fault, characterized by asperities and a weaker zone. The traction vector on the fault plane has an arbitrary orientation: in particular, it can be nonperpendicular to the asperity borders. Aseismic fault slip takes place when the applied stress exceeds the frictional resistance: slip starts in weak zones and is confined by asperities, where it propagates at increasing velocity. Propagation into asperities is characterized by a dislocation front, advancing perpendicularly to the asperity border. Fault slip does not take prate in the direction of traction, except when traction is perpendicular or parallel to the asperity border. The propagation of such aseismic dislocations produces a stress redistribution along the fault and can play a key role in determining the conditions which give rise to earthquakes.

SUMMARY A 2-D model which represents a slipping fault with non-uniform Coulomb friction is studied. The fault plane is subject to a uniform ambient shear stress, slowly increasing with time. Aseismic fault creep is assumed to start in a weak zone, when the ambient stress reaches a strength threshold. The solution for the resulting dislocation is worked out analytically using a technique based on Chebyshev polynomials. It is found that the dislocation partially propagates into the adjacent asperities, concentrating stress onto them and preparing the conditions which will produce the asperity failure and the accompanying earthquake. Propagation is not self-similar and occurs at increasing velocity. A non-linear slip hardening effect is reproduced. The nearness to earthquake instability is measured by a dimensionless parameter which depends on Coulomb friction and ambient shear stress and decreases to zero with time. An upper boundary to the critical value of this parameter, at which instability may occur, is estimated and is found to depend on the ratio between the sizes of the asperity and the weak zone.

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 constagt 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 halfspace model.

A crack model in antiplane shear configuration is shown representing creep processes interpreted in terms of ‘viscous’ deformation of a narrow plastic layer, characterized by inhomogeneous rheological properties, embedded within a homogeneous elastic medium. The evolution in time of slip and stress over the crack plane is studied through a truncated expansion in Chebyshev polynomials, and convergence is proved to be fast in the simple examples considered. Finite‐stress solutions are found which are compatible with constitutive relations of elasto‐plastic materials and furthermore these allow us to simulate creep propagation and stress transfer between locked and unlocked fault segments. This model provides a simple interpretation of the shallow depth of the seismogenic layer observed in several areas of the world and lends itself to modelling creep processes during either post‐seismic rebound or pre‐seismic stress buildup. Stress transfer is accomplished mostly by the slow extension of the creeping section. During a seismic cycle it is envisaged that different regimes dominate over deep, intermediate and shallow sections of faults: (i) slow pre‐seismic stress build‐up accompanied by creep and stress migration toward intermediate depths; (ii) brittle fracture over shallow and intermediate sections of faults; (iii) post‐seismic rebound over intermediate and deep sections of faults. The present crack model, while providing finite‐stress solutions, allows a better understanding of how stress may accommodate at different depths over a fault plane during a seismic cycle.

A Maxwell viscoelastic model is proposed for the lithosphere in the proximity of a long, vertical strike slip fault. In this framework, formulas for the recurrence time of great earthquakes are derived by imposing boundary conditions of (1) uniform strain rate in the fault region or (2) steady, aseismic sliding at depth. In both cases a minimum effective viscosity of the fault region is found in order that a critical stress for earthquake occurrence can be reached; at higher viscosities the recurrence time approaches a minimum, limiting value. Application of the model to the northern ‘locked’ segment of the San Andreas fault would indicate a recurrence time significantly longer than the often quoted 100-year period. These results differ considerably from those obtained from similar models employing boundary conditions of assigned stress at large distance from the fault.

Theoretical arguments and petrological observation indicate that magma chambers are surrounded by a thermal metamorphic shell. Owing to their higher temperature, the rocks of this shell have mechanical behaviour different from that of the cooler, more distant crustal rocks. This situation is represented by a model in which a spherical magma chamber is surrounded by a viscoelastic shell. The shell is elastic dilatational and Maxwell deviatoric; outside, the medium is elastic. Displacement and stress fields outside the magma chamber are calculated analytically, assuming that the chamber is internally pressurized, with different source functions. The model illustrates some time-dependent effects which are a result of the presence of the viscoelastic shell: amplification of displacement in the surrounding medium, relaxation of shear stress within the shell and stress concentration outside it. These phenomena may be relevant to the interpretation of some aspects of the ground deformation and induced seismicity, which were observed in the Campi Flegrei volcanic area, Italy, during the period 1982–1984.

Summary. We consider a long strike-slip fault in a lithosphere modelled as an elastic slab. To the base of the slab a shear stress distribution is applied which simulates the viscous drag exerted by the asthenosphere. The resulant stress on the fault plane may directly fracture the lithosphere in its brittle upper portion; alternatively it may give rise at first to a stable aseismic sliding in the lower portion. In the latter case, stress concentration due to the deep aseismic slip is the relevant feature of the pre-seismic stress acting on the upper section of the lithosphere. The two cases are examined by use of dislocation theory and their observable effects compared. Different depths of the aseismic slip zone and the presence or absence of a uniform friction on the seismic fault are allowed for. If the model is applied to the San Andreas fault region, where a steady sliding condition actually seems to be present at shallow depth, it turns out that the slip amplitudes commonly associated with large earthquakes are consistent with average basal stress values which can be substantially lower than a few bars, a value often quoted as the steady state basal stress due to a velocity gradient in the upper asthenosphere.

We study the change in the moment of inertia due to shallow faulting in a planet composed of an elastic lithosphere and a viscoelastic mantle. The rheology of the mantle is that of a Burgers body, which is characterized by two viscosity values, η 1 and η 2 , and by two rigidity values, μ 1 and μ 2 . Here η 1 is the long-term viscosity, while μ 1 is the instantaneous rigidity of the mantle. It is found that an earthquake produces an instantaneous elastic change in the moment of inertia, followed by a long-term viscoelastic change. The viscoelastic change occurs through two jumps with timescales controlled, respectively, by the two viscosity values of the mantle. The magnitudes of the two jumps depend instead upon the ratio μ 2 / μ 1 between the two rigidity values of the mantle. For a thin-lithosphere planet (such as the Earth) the total viscoelastic change in the moment of inertia is four times larger than the elastic change. In this case, it is found that the earlier jump in the moment of inertia would be such as to excite significantly the Chandler wobble only if η 2 is at least four orders of magnitude less than η 1 and the rigidity μ 2 is smaller than the instantaneous rigidity μ 1 . However, such viscosity values are not compatible with observations of the Chandler wobble period.

The principal purpose of this paper is to examine whether movement of matter in a stratified viscoelastic earth can enhance the excitation of the Chandler wobble. We have constructed analytically a two-layer model consisting of an elastic lithosphere overlying a Maxwell viscoelastic mantle in order to calculate explicitly the temporal evolution of the inertia tensor from earthquake faulting. It is found that the ability of the viscoelastic flows to produce an additional increment of the inertia tensor depends critically on R, the ratio of the inner radius to the outer radius. Only for R ≳ 0.7 is there a substantial increase. For the earth (R ≃ 0.98) an amplification of between four and six is obtained from the models considered. However, the time scales over which this enhancement of the moment of inertia takes place is about 10 Maxwell times, thus making this model unlikely to excite the Chandler wobble for values of the mantle viscosity derived from postglacial uplifts. This phenomenon of strain amplification occurs by virtue of ‘transient membrane mechanics,’ as the ratio between the horizontal wavelength of the deformation field is large compared with the elastic shell. By means of a similarity argument we have found that for angular orders less than 40 to 45 ‘transient membrane mechanics’ can be important in global postseismic deformation in the earth. This mode of transient deformation arises from the three-dimensional nature of the problem and hence cannot be modeled with conventional flat earth models.

Some earthquake models based on the elastic theory of dislocations are presented. Earthquake occurrence is modelled as the opening of a crack in an infinite elastic medium triggered by the action of localized stress distributions. The fracture is modelled as a continuum of infinitesimal dislocations. This approach allows us to have complete information about the stress field and the displacement field and to make a first step towards understanding the relation between source mechanism and stress distributions.

Summary. Most crustal earthquakes of the world are observed to occur within a seismogenic layer which extends from the Earth's surface to a depth of a few tens of kilometres at most. A model is proposed in which the shear zone along a transcurrent plate margin is represented as a viscoelastic medium with depth-dependent power-law rheology. A frictional resistance linearly increasing with depth is assumed on a vertical transcurrent fault within the shear zone. Such a model is able to reproduce a continuous transition from the brittle behaviour of the upper crust to the ductile behaviour at depth. Assuming that the shear zone is subjected to a constant strain rate from the opposite motions of the two adjacent plates, it is found that there exists a maximum depth H below which tectonic stress can never reach the frictional threshold: this may be identified as the maximum depth of earthquake nucleation. The value of H is consistent with observations for plausible values of the model parameters. The stress evolution in the shear zone is calculated in the linear approximation of the constitutive equation. A change in rigidity with depth, which is also introduced in the model, may reproduce the high vertical gradient of shear stress, which has been measured across the San Andreas fault, and the fact that most earthquakes are nucleated at some depth in the seismogenic layer. A crack which drops the ambient stress to the dynamic frictional level is then introduced in the model. To this aim, a crack solution is employed without a stress singularity at its edges, which is compatible with a frictional stress threshold criterion for fracture. A constraint on the vertical friction gradient is obtained if such cracks are assumed to be entirely confined within the seismogenic layer.

The behaviour of a lava flow is reproduced by a two-dimensional model of a Bingham liquid flowing down a uniform slope. Such a liquid is described by two rheological parameters, yield stress and viscosity, both of which are strongly temperature-dependent. Assuming a flow rate and an initial temperature of the liquid at the eruption vent, the temperature decrease due to heat radiation and the consequent change in the rheological parameters are computed along the flow. Both full thermal mixing and thermal unmixing are considered. The equations of motion are solved analytically in the approximation of a slow downslope change of the flow parameters. Flow height and velocity are obtained as functions of the distance from the eruption vent; the time required for a liquid element to reach a certain distance from the vent is also computed. The gross features of observed lava flows are reproduced by the model which allows us to estimate the sensitivity of flow dynamics to changes in the initial conditions, ground slope and rheological parameters. A pronounced increase in the rate of height increase and velocity decrease is found when the flow enters the Bingham regime. The results confirm the observation according to which lava flows show an initial rapid advance, followed by a marked deceleration, while the final length of a flow is such that the Graetz number is in the order of a few hundreds.

Summary The Somigliana dislocation theory is applied to the study of strike-slip faulting in an isotropic, homogeneous, elastic half space in the presence of localized distributions of strain nuclei. Vanishing shear tractions on the dislocation surface are assumed and the slip amplitude as a function of depth is derived, together with the stress drop pattern and the energy release. The results show indications that the condition of continuous stable sliding prevailing at depth along the locked sections of faults may strongly affect the slip, the stress drop and the energy release of a seismic dislocation event. Moreover, fault widths derived from dislocation models driven from underneath are greater than those evaluated from models with a uniform initial stress field and thus tend to explain in simple terms the common finding in some regions that earthquake foci are deeper than evaluated widths of dislocation surfaces.

Summary. Two‐dimensional crack problems in elastic homogeneous isotropic media are considered which describe rupture over a fault surface characterized by non‐uniform stress drop. Solutions can be found in which the stress field is finite at the crack tips and the rupture surface is not assigned a priori, but is part of the solution. These crack models are found to be consistent with the frictional stress threshold criterion for slip arrest over pre‐existing fault surfaces. A crack is found to stop when its contribution to the stress field is opposite to the stress drop at the crack tips. The quasi‐static propagation of a crack up to the arrest configuration is studied in terms of the minimum energy principle. The crack spontaneously propagates in such a way as to make the value of the stress intensity factor at one tip equal to the value at the other tip. Furthermore a tip propagating in a region with higher friction is found to move more slowly than the other tip propagating in a region with lower friction. Simple criteria for fracture arrest are derived, in terms of a properly averaged stress drop. Piecewise constant stress drop profiles are explicitly considered yielding a variety of solutions which can be applied to modelling asperities or barriers over a fault plane. The evaluation of the amount of the energy released during the quasi‐static crack propagation shows that stopping phases cannot be efficiently radiated if the crack comes to rest in a low friction region.

A model is proposed for the opening of ephemeral vents in the solid front of a stationary lava flow. The front is considered to be a viscoelastic shell which deforms due to the pressure exerted on it by the inner fluid lava. The vent opens when the normal stress in the front overcomes the tensile strength of solid lava. Cooling of the front is considered and the increase in crust thickness, as the isothermal surface at the solidus temperature deepens into the lava body, is calculated. The thermal effect is negligible, if the timescale for the opening of ephemeral vents is on the order of days, as is often observed. Considering stationary flow lengths of a few hundred meters, this timescale corresponds to an average viscosity of the crust of 10 12 Pa s, assuming reasonable values of the crust strength and initial thickness. Observation of ephemeral vents on Mount Etna is found to be consistent with the model predictions.

The formation of lava tubes is a common phenomenon on some basaltic volcanoes, such as Etna. A model for tube formation by roofing of a channel is proposed and involves first describing lava as a Bingham liquid flowing down a slope. It is further assumed that lava flows in a channel with rectangular cross section: as a result of heat loss into the atmosphere, a crust is gradually formed on the upper surface of the flow and this crust eventually welds to the channel levees. We assume that a lava tube is formed when such a crust is sufficiently thick to resist the drag of the underlying flow and to sustain itself under its own weight. The minimum thickness of the crust satisfying such conditions depends on the tensile strength and shear strength of the crust itself. Assuming that the growth of the crust produces a downflow linear increase of the shear stress at the interface between flowing lava and the crust, the distance is evaluated between the eruption vent and the point where the tube is formed. The model predicts that if the flow rate is constant, the thickness of the flow increases as the crust fragments grow and weld to each other, and the velocity of the crust decreases to zero. Once the lava tube is formed, the initial flow rate can be achieved by a flow thickness smaller than the vertical size of the tube, with the same viscous dissipation: this may explain why under steady state conditions, the lava level inside a tube is frequently lower than the roof of the tube itself.

Observations of very long period elastic waves are attributed to 'silent' earthquakes, events which are presumed to occur at depth in a nonbrittle regime. The fault at depth is modeled as a thin viscoelastic gouge layer embedded in an elastic medium. The source of silent earthquakes is envisaged as a prolonged, slow slippage of the fault faces, which yields a radiation spectrum strongly depleted in the high-frequency components and dominated by the near-field term. The results of the model are compared with the dominant frequencies, mean duration, and amplitude of a long-period perturbation recorded at Trieste in the years preceding the 1976 Friuli (Italy) earthquake. The role that silent earthquakes may play in the overall earthquake mechanisms (in particular as precursory phenomena) is discussed.