Complexities of rock fracture and rock friction from deformation of Westerly granite

A series of rock friction experiments has been carried out to study the complexities in rock fracture and rock friction. Intact Westerly granite samples were loaded to shear failure in a laboratory polyaxial loading apparatus. The resultant fractured samples were reloaded to cause frictional sliding. Both polyaxial loading (₁ > ₂ > ₃ > 0) and equal confining condition (₁ > ₂ = ₃ > 0) were used. The deformation processes were monitored by macroscopic axial stress-strain, optical holography, and ultrasonic velocity measurements.Intense localized deformation along the fracture occurred very early in the loading of fractured samples. Contacts on the fracture surfaces continuously broke during loading. No acoustic velocity anomaly was observed for the fractured sample, in contrast to a 25% drop in the velocity before the failure of the corresponding intact sample. The current study and previous research suggest that the deformation localization is an important process in governing the instability of rock friction. Instability analysis of rock friction needs to include not only the deformation processes along the sliding surfaces, but also those adjacent to the fractures such as the localized deformation along the fractures observed in the current study. The instability analysis of rock friction with rate- and state-dependent friction laws does not specifically include the deformation localization adjacent to the faults and thus ignores an important class of instability as described byRudnicki (1977).A dependence of frictional strengths on the stress components normal to the sliding and in the plane of the fracture surface was observed. This dependence can be understood by considering the loading of the irregular fracture surface under polyaxial loading conditions. This observation requires the friction laws in the macroscopic scale to be modified for those cases where the three principal stresses (₁, ₂, and ₃) are significantly different.     

Sealing law of far-field spectra based on observed parameters of the specific barrier model

A set of acceleration source spectra is constructed using the observed parameters of the specific barrier model of Papageorgiou and Aki. The spectra show a significant departure from the ²-model at the high frequency range. Specifically, the high frequency spectral amplitudes of seismic excitation are higher as compared to the level predicted by the ²-model. This is also supported by other observational evidence. The high frequency amplitudes of acceleration scale proportionally to the square root of the rupture areaS, to the rupture spreading velocityv, and to the local strain drop (/) (=strain drop in between barriers). The local strain drop in between barriers is not related in a simple way to the global strain drop, which is the strain drop estimated by assuming that it is uniform over the entire rupture area. Consequently, the similarity law does not apply. Using the source spectra which we constructed, we derive expressions for high frequency amplitudes of acceleration such asa _rms anda _max. Close to the fault both are independent of fault dimensions and scale as (/µ)(f)^1/2, while away from the fault plane they scale asW ^1/2(/µ)(f)^1/2, whereW is the width of the fault and f is the effective bandwidth of the spectra.     

Tensional crack development in physical models with inhomogeneities under load

Summary The paper deals with a study of the mean stress field (_M) and its gradient (_M/x) in uniaxially loaded two-dimensional models of small thickness with respect to other dimensions, in which two inhomogeneities, e.g., a circular opening and a linear slit have been made. Particular attention was paid to the stress field development in the treated models durin gthe change of the mutual geometric configuration of both the inhomogeneities in question. The problem was analysed theoretically by means of Muskhelishvili's potentials of the two-dimensional theory of elasticity. To compare the results of calculations, the schlieren method was used to visualize and record the studied gradients of stress fields in physical models. The results of the calculations were calibrated by laboratory tests. In this way the individual model situations were classified from the point of view of their resistance to the applied load.     

Analysis of State of Stress During The 1997 Earthquake Swarm In Western Bohemia

Focal mechanisms of 70 events of the January 1997 earthquake swarm were analysed using the Gephart and Forsyth method to determine the state of stress in the West Bohemia/Vogtland region and to reveal possible stress variations during the swarm activity. The method was applied to the whole set of 70 focal mechanisms and to several subsets distinguishing magnitudes of events and the spatial and temporal distribution of the swarm. The three representative stress models A ( ₁ ,A: 0/45 (az/dip), ₂ A: 134/35, ₃ A: 243/25), B ( ₁ ,B: 162/38, ₂ B: 263/14, ₃ B: 10/49), and C ( ₁ C: 135/30, ₂ C: 1/50, ₃ C: 239/23) which could characterise the state of stress in the region were found. Model C can be considered to be the most probable stress model because of its consistency with the European stress field. The results of the stress analysis applied to the individual subsets of focal mechanisms indicate that the state of stress was more uniform during the first phase of the swarm. To distinguish between the fault plane and the auxiliary plane of fault plane solutions a statistical approach was used. The results showed that the fault planes striking NW-SE and dipping 45° to NE were activated during the swarm.     

Explanation of the internal air pressure effect on the worden gravity meter dial constant

Summary A model, explaining the effect of air pressure changes in the space of the measuring system on the fine dial counter constant of the Worden and similar gravity meters, is presented. It is based on the changes of the bellows volume with compensating the gravity changes. It is shown that the resetting the gravity meter in the whole range has practically no influence on the fine dial counter constant.

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u¶rt;a ¶rt;, na m uu ¶rt;au ¶rt;a nmam uum um a nm a umaauma ¶rt; u nuaum. ¶rt; aa a uu a ua umu uma nu nauu uu u mmu. aa, m nma ¶rt;uanaa um namuu um a uu nm a uma.

     

Zur Analyse der Genauigkeit von mehrmals mit einer Gravimetergruppe gemessenen gravimetrischen Verbindungen

Summary In order to evaluate the accuracy of measurements repeated by a set of gravimeters the semi-systematic errors ₂ and ₃ were introduced besides the random error ₁ in[1]. It is shown that Eqs.(10) and(11), given in[1], should not be used to determine these errors, but Eqs.(8) and(9).     

Anisotropic magnetic susceptibility of rocks under mechanical stresses

Summary The magnetic susceptibility of a rock under a uniaxial compression () decreases along the axis of compression and increases along the direction perpendicular to the axis, with an increase of . Thus, the magnetic susceptibility of a compressed rock becomes anisotropic.The decrease of longitudinal susceptibility,K (), and the increase of transverse susceptibility,K (), are theoretically derived from a model of rock which assumes the uniaxial anisotropy and the isotropic magnetostriction of magnetic minerals in rocks and a random orientation of the minerals. Results show thatK () decreases toward zero whereasK () increases and approaches a finite asymptotic value with an increase of , and –(/)K () is twice as large as /K () for small values of . These results are in good agreement with experimental data.

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Zusammenfassung Die magnetische Suszeptibilität eines Steines unter zunehmender uniachsigen Druckspannung () nimmt ab längs der Achse der Druckspannung und nimmt zu längs der Richtung senkrecht der Achse. Somit wird die magnetische Suszeptibilität des gedrückten Steines anisotrop.Die Abnahme der longitudinalen Suszeptibilität,K (), und die Zunahme der transversalen Suszeptibilität,K (), werden theoretisch von einem Modell eines Steines hergeleitet, das die uniachsige Anisotropie, die isotrope Magnetostriktion, und eine nichtbevorzugte Orientierung der magnetischen Minerals im Stein annimmt. Die Ergebnisse zeigen, dass mit einer Zunahme des ,K () gegen Null abnimmt, währendK () zunimmt und sich einem begrenzten asymtotitschen Wert nähert und, dass für kleine Werte des , –(/)K () zweimal so gross wie /K () ist. Diese Ergebnisse stimmen gut mit den Versuchangaben überein.

     

The optimal shape design method applied to modelling thermal thinning of the oceanic lithosphere near hot spots

Summary Using the optimal shape design method, which is generally described, and von Herzen's et al. measurements of the heat flow, the shape of the lithosphere and its thermal field is computed in the vertical plane parallel to the hot spot source versus the plate velocity at a distance of about 250 km from the axis of the Hawaiian Island chain. The results are compared with the computations based on Crough's idea of thermal rejuvenation of the oceanic lithosphere above a hot spot source. If we assume that the lateral cross-section of the lithospheric bottom is described by the Gaussian curve h=h₀ exp (–y²/2²), we obtain h₀35 km and 130 km, where h is the value of lithospheric thinning and y the lateral coordinate. We thus obtain the lower limit of the lateral dimension of the Hawaiian anomaly.

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u m¶rt; nmua nua amu, m u ma nuam, u ¶rt;a mn nm, ua a um u mn n mua nmu, naa mu um mum umua mu u ¶rt;a nuuum 250 m uuuaa aunaa. mam a uuu, au a u¶rt;u aa (Crough), aauu mn mu au um a¶rt; umu mu. u n¶rt;num, m ama nn u umu ¶rt;a nuam u aa h=h₀ exp (–y²/2²), m num h₀ 35 u 130 ,¶rt; h—umu mu u —amaa ¶rt;uama. ¶rt;am, =130 m u n¶rt; ama aaaa aauu.

     

Velocity changes associated with generalized triaxial deformation of pyrophyllite

Compressional and shear-wave velocities (V _p andV _s ) were measured during the generalized triaxial deformation (i.e. _{1 2}=2 ₃) of pyrophyllite. Observed velocity changes could be ascribed to crack development during dilatancy. Velocity changes were very localized with respect to the ultimate failure plane. The orientation and development of the failure plane was continuously observed with laser holography. Velocity reverals, i.e. changes from a decreasing trend to an increasing trend, were documented in a wet sample in bothV _p andV _s . These changes in bothV _p andV _p are inconsistent with dialatancy-diffusion models. The reversals were interpreted as a reflection of local stress reorientation caused by a slowly propagating fault.     

Effect of the additional low-sensitivity channel on the amplitude response of the electromagnetic seismograph

u¶rt;m ¶rt; anum¶rt;-amm aamumu maum a aa nu mummu (). ma¶rt;am aamumu uu mau auma uu am auu a uu aaa aa.     

Coulomb constitutive laws for friction: Contrasts in frictional behavior for distributed and localized shear

We describe slip-rate dependent friction laws based on the Coulomb failure criteria. Frictional rate dependence is attributed to a rate dependence of cohesionc and friction angle . We show that differences in the stress states developed during sliding result in different Coulomb friction laws for distributed shear within a thick gouge layer versus localized shear within a narrow shear band or between bare rock surfaces. For shear within gouge, shear strength is given by =c cos + _n sin, whereas for shear between bare rock surfaces the shear strength is =c cos + _n tan, where and _n are shear and normal stress, respectively. In the context of rate-dependent Coulomb friction laws, these differences mean that for a given material and rate dependence of the Coulomb parameters, pervasive shear may exhibit velocity strengthening frictional behavior while localized shear exhibits velocity weakening behavior. We derive from experimental data the slip-rate dependence and evolution ofc and for distributed and localized shear. The data show a positive rate dependence for distributed shear and a negative rate dependence for localized shear, indicating that the rate dependence ofc and are not the same for distributed and localized shear, even after accounting for differences in stress state. Our analysis is consistent with the well-known association of instability with shear localization in simulated fault gouge and the observation that bare rock surfaces exhibit predominantly velocity weakening frictional behavior whereas simulated fault gouge exhibits velocity strengthening followed by a transition to velocity weakening with increasing displacement. Natural faults also exhibit displacement dependent frictional behavior and thus the results may prove useful in understanding the seismic evolution of faulting.     

Changes in the shape of the circumpolar vortex due to heating from above

Summary Based on model considerations it is shown that, under certain assumptions, zonalization of tropospheric circulation may be expected in the region of the auroral oval as a result of heat released at the time energetic electrons penetrate from the Sun into the lower stratosphere.

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a auu ¶rt; a¶rt;u naa, m nu m n¶rt;nu u¶rt;am auau mn uuu amu aa aa mam ¶rt;u mna nuu mumu m u a u mam.

     

Three-dimensional inverse problem for inhomogeneous transversely isotropic media

Summary The basic formula used in the presented paper gives the relation between the P wave travel-time perturbation and the perturbation of an inhomogeneous transversely isotropic medium, expressed by four perturbations of elastic parameters and by two angles of orientation of the axis of symmetry of transverse isotropy in space. The travel time perturbation is computed along the ray in the unperturbed inhomogeneous isotropic medium. Four elastic parameters and two angles are parametrized in the model under study and a system of equations for many rays is constructed. The equations are linear in the sought elastic parameters and nonlinear in the sought angles, and the iterative Levenberg-Marquardt algorithm is thus used to solve them. The theoretical 3-D inverse problem was solved in the presented numerical example. The data, simulating teleseismic data, were computed in the direct problem and then inverted. The results indicate the applicability and limitation of the presented algorithm in real problems.

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a a, unaa n¶rt;aa am, ¶rt;am mu ¶rt; uu u na u uu ¶rt;¶rt; nn umn ¶rt;, a m nuu naamau u ¶rt; au umauu u umuu nn umnuu nmam. u u na um ¶rt; a aa ¶rt;¶rt; umn ¶rt;. nu naam u ¶rt;a a naamuua ¶rt;u u nma uma au ¶rt; u . au u n um nu naama u u n um a umauu umuu, nm un m umamu aum a-aa¶rt;ma ¶rt; u u. am nu¶rt; ¶rt; m u nu. nu muu ¶rt;a aaa a na a¶rt;aa u am ¶rt;a a. mam naam auu u mu nuu nu¶rt;uma a a.

     

On the diffusion of the perturbing toroidal magnetic field from the core to the mantle

Zusammenfassung Es werden die Wurzeln einer charakteristischen Gleichung für die Randbedingung der Diffusion des toroidalen magnetischen Störungsfeldes berechnet, das in dünner Schicht unterhalb der Grenze Kern — Mantel, als eine Folge von Störung der Winkelgeschwindigkeit dieser Schicht in Anwesenheit des magnetischen Dipolfeldes induziert wird. Die Wurzeln werden für verschiedene Werte des Verhältnisses der Leitfähigkeit des Erdmantels und Erdkerns ausgerechnet. Es wird gezeigt, dass das aus dem Milieu mit der elektrischen Leitfähigkeit in das Milieu mit elektrischer Leitfähigkeit _m diffundierende toroidale magnetische Störungsfeld vom Werte _m/<10^–2 an von dem Verhältnis elektrischer Leitfähigkeiten beider Milieus unabhänging bleibt.     

On the estimation of the heat flow at the earth's crust lower boundary

Summary The paper deals with a comparison of two models used in determining the heat flow q_M at the lower boundary of two-dimensional geothermal models of the Earth's crust. Method I is based on a linear relationship between the component of the surface heat flow, which is not generated by heat sources inside the model, and the heat flow q_M. This method uses a regularization process, in which the variation of the sought function q_M is limited from above. Method II is the most frequently used iterative method, in which the (i + 1)-th approximation of q_M is determined from the surface heat flow, corresponding to the i-th approximation of q_M used as a boundary condition in solving the direct problem. The comparison of both methods has revealed that the solution obtained by method I satisfies the supposed reality better than the solution obtained by method II. Method II is attractive especially for its simplicity. To eliminate the local variations of the estimated heat flow q_M, which are due to automatic transmission from the surface heat flow, a combination of method II with some smoothing procedure could be applied.

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¶rt;am ¶rt;a m¶rt;a uu mn nma q_M a uau ¶rt; muu ¶rt; . m¶rt; unm u ¶rt; ma nm nma, ma uaa umuau mna mu ¶rt;u, u nm q_M. m m¶rt; nu n uauu, auuau auau u uu q_M . m m¶rt;, mu nm ¶rt; u n a , umau: (i + 1)-a annuau nma q_M um n auma nm nma, ma i- annuauu q_M. au ¶rt;m, m nm q_M, a¶rt; n m¶rt;, mam n¶rt;naa ¶rt;mummu , u, n m m¶rt;. m m¶rt; nuam nmm. ¶rt; mau a auau a¶rt; nma q_M, uu ¶rt;mu amamu na u auau nm nma, ¶rt;a uuam au-u¶rt; m¶rt; auau.

     

Die Lösung der Strahlungs-Übertragung in der Erdatmosphäre anhand der Monte-Carlo-Methode

ua bmam umumu ¶rt;uauu mu ¶rt;u , n no mo¶rt; me a [3] u no mo¶rt; [5] ¶rt; u a mauu uu amum. na¶rt;u ¶rt; m a h₀=6°.     

A note on the stress-dilatancy relation for simulated fault gouge

Theoretical constraints on the stress-dilation relation for a deforming Coulomb material requirev ifC=0 andv sin^-1( _m / _m ) always, wherev is the dilation angle, is the friction angle,C is cohesion, _m is the maximum shear stress, and _m is the mean effective stress. Recent laboratory measurements of friction and dilatancy of simulated fault gouge show that small amplitude shear-load cycling causes compaction and consolidation. Comparison of the data with theory indicates that such load cycling produces: (1) increased coefficient of friction (or friction angle), (2) increased cohesion, and (3) increased dilatancy rate (or dilation angle). Under certain conditions of load cycling without significant plastic shear strain accumulation ( ^p <0.005) we find thatv exceeds both and, in contrast to theory, sin^-1( _m / _m ). This result is interpreted in terms of enhanced cohesion and overconsolidation, which lead to residual stresses within the gouge. An analogy is drawn between these special loading conditions and those extant on natural faults. In particular, our results imply that jostling and minor stress variations associated with microearthquakes may produce strengthening of fault gouge and changes in the fault zone's stress-dilatancy relation. Hence, compaction associated with microseismicity may lead to subsequent dilation of fault gouge, even for faults with large displacement rates and large net offsets (e.g., San Andreas). In regions where such dilation persists over sufficient displacements (on the order of the critical slip distance for seismic faulting) it may tend to inhibit unstable slip.     

Solution of stochastic groundwater flow by infinite series,and convergence of the one-dimensional expansion

This paper investigates analytical solutions of stochastic Darcy flow in randomly heterogeneous porous media. We focus on infinite series solutions of the steady-state equations in the case of continuous porous media whose saturated log-conductivity (lnK) is a gaussian random field. The standard deviation of lnK is denoted . The solution method is based on a Taylor series expansion in terms of parameter , around the value =0, of the hydraulic head (H) and gradient (J). The head solution H is expressed, for any spatial dimension, as an infinite hierarchy of Green's function integrals, and the hydraulic gradient J is given by a linear first-order recursion involving a stochastic integral operator. The convergence of the -expansion solution is not guaranteed a priori. In one dimension, however, we prove convergence by solving explicitly the hierarchical sequence of equations to all orders. An infinite-order stochastic solution is obtained in the form of a -power series that converges for any finite value of . It is pointed out that other expansion methods based on K rather than lnK yield divergent series. The infinite-order solution depends on the integration method and the boundary conditions imposed on individual order equations. The most flexible and general method is that based on Laplacian Green's functions and boundary integrals. Imposing zero head conditions for all orders greater than one yields meaningful far-field gradient conditions. The whole approach can serve as a basis for treatment of higher-dimensional problems.     

Optimum determination of Euler Angles

a m m n¶rt; uua ¶rt; au anauu a nmu n¶rt;nuu, m u ¶rt;a a ¶rt;u -um, ma u amu um ¶rt;uam. a namu unm ¶rt;a anau, nu a¶rt;u m m aua n-a. nuam n¶rt; n¶rt;u au — D-nmuamu — naa uu anau ¶rt; n¶rt;u a.     

A remark on the hydrostatic approximation

Summary For an autobarotropic atmosphere the exact linearized equations of motion can be solved for the pressure oscillation when =0 and =2 (=frequency of oscillation, =angular velocity of the earth's rotation). The pressure computed in this way is shown to be in close agreement to the pressure computed by using the hydrostatic approximation.