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.

     

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.     

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.     

A new dissipation model based on memory mechanism

Summary The model of dissipation based on memory introduced by Caputo is generalized and checked with experimental dissipation curves of various materials.List of symbols unidimensional stress - unidimensional strain - Q ^–1 specific dissipation function - c(t) creep compliance - m(t) relaxation modulus - c ₀ instantaneous compliance - m equilibrium modulus - (t) creep function - relaxation function - () spectral distribution of retardation times - spectral distribution of relaxation times - c ^*() complex compliance - m ^*() complex modulus - tang loss-tangent     

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.     

Changes of electric conductivity in the neighbourhood of the curie temperature of basalts

Summary In investigating the electric conductivity of rocks as a function of the temperature it was found that rocks containing ferrimagnetic minerals display a change in the slope of the functionlg =f(1/T) in the neighbourhood of the Curie temperature . In order to explain these changes the curveslg =f(1/T) and the Curie temperatures , obtained from the temperature dependence of the saturated magnetization J_s=f(T), were compared. Eight samples of basalts, 2 samples of haematite and magnetite were used to demonstrate the relation between and the changes in the pattern of the electric conductivity, caused by the variations of the exchangeable energy of ferrimagnetic minerals.     

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.     

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.     

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.     

On the Upper Bound of the Liquid Core Viscosity

By using Molodensky and Sasao (1995a) and Molodensky and Groten (1998) an approach of expansion in powers of the small parameter =(+)/ (where + and are the frequencies of nutational motion in space and in a mantle-fixed reference frame, respectively, is tidal frequency) a theory of diurnal Earth tides and nutation for a realistic model of the Earth with an inhomogeneous, viscous liquid core and an anelastic mantle is constructed. It is shown that our approach is self-consistent for semi-annual, annual, and principal nutational components (when ||1/180). By comparing the results of modern VLBI-nutational data and the results of our calculations, we have found the region of possible values of the parameters which describe the anelastic properties of the Earth's lower mantle and the viscosity of the liquid core. It is shown that modern VLBI-data are about six orders of magnitude more sensitive to the liquid core viscosity than modern seismic data or Earth free oscillations data.     

Radioactivity and geochemistry of granulites and some related rocks from the saxonian granulite complex

Summary The distribution of radioactive(Th, U, K), major and selected trace(Rb, Sr, Ba, Y, Zr, V, Cr, Ni) elements of granulites from the Saxonian Granulite Complex was studied. Similarly to the South Bohemian granulites, the Saxonian granulites can be divided according to the contents of their major and trace elements into two main groups, groupA containing mostly acid and subacid granulites (K ₂ O>2.5%, SiO ₂ >68%), and groupB containing mostly intermediate and basic granulites (K ₂ O<2.5%, SiO ₂ <68%). Statistically significant differences between groupsA andB were found for all major oxides and several trace elements(Rb, V, Cr, Ni). The Saxonian granulites follow the same calc-alkaline trend as the South Bohemian, granulitesA being placed mostly in the rhyolite field and granulitesB mostly in the dacite, andesite and basalt fields of this trend. The investigated granulites are characterized by a considerable scatter ofTh andU contents accompanied by very variableTh/U ratios; theTh andU concentrations of granulitesA are substantially lower than is usual for rocks of corresponding acidity.

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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.     

The inverse gravimetric problem: Existence,uniqueness and stability of the solution

Summary The paper is concerned with the mathematical properties of the density distribution within the Earth obtained by inverting of the external gravity field, provided an Earth's reference density model, used as the initial guess, is available. The method of regularization, which proves the existence, the uniqueness and the stability of the solution, which is nearest the initial guess in the L ² norm, is described.

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Topographic characteristics of laboratory induced shear fractures

The topography of laboratory induced shear fracture surfaces of Westerly granite was studied. Three types of fracture surfaces were examined: (1) a fresh fracture from the shear failure of an intact sample under polyaxial loading (₂ = 40 MPa > ₃ = 15 MPa); (2) a shear fracture subjected to frictional sliding of 100 m under polyaxial loading; (3) a shear fracture subjected to frictional sliding of 800 m under conventional triaxial loading (₁ > ₂ = ₃ = 40 MPa). Both sliding distances are within the range of the grain size of Westerly granite. The results are represented by a power spectral method.Similar to the power spectra from natural rock surfaces, the power spectra of the induced shear fracture surfaces fall off about 2 orders of magnitude per decade increase in spatial frequency. No corner frequency exists in the power spectra over a spatial frequency range from that corresponding to the profile length to the Nyquist frequency. A slope break in the power spectrum was identified, however. It separates a steeper low frequency segment from a less steep high frequency segment. The spatial frequency at the slope break corresponds to a wavelength of several hundred microns which is on the scale of the microcracking and contact breaking on the fractures. Upon re-examining power spectra of natural fault traces and fault surfaces obtained in previous studies, we noted similar slope breaks. We suggest that this slope break may have significant implications in the scaling problem. Both the induced fracture surfaces and natural faults exhibit topographic characteristics different from those of sawcut surfaces, which have been widely used in laboratory rock friction experiments. In the present study, we observed that even a small amount of sliding (less than a grain size) already results in significant mismatches between the paired sliding surfaces in the direction normal to sliding.     

Luni-solar precession constant and accuracy of the Earth's principal moments of inertia

¶rt;am aum ua mm, m um aua m uuu u u amu n¶rt;u. aamuam m unau au au ma ¶rt; u nmm ¶rt; u. ¶rt;a u auu m uuu ¶rt;a, amuu u      

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.     

The derivatives of the lunar disturbing potential harmonics with respect to euler's angles

Summary The derivatives of the harmonicsP _n ^(k) (sin _O)cos kT_O andP _n ^(k) (sin _O)sin kT_O, occurring in the development of the lunar disturbing potential, are derived upto n=4 and for k== 0, 1, ..., n. The equatorial co-ordinates _OT_O are referred to the Moon's mass centre; the procedure for the solar disturbing potential is formally identical.     

Die Seismizität der Türkei

Summary Using the fromulae given byGutenberg andRichter, the writer has computed the magnitude and energy of 1804 earthquakes which occurred in Turkey during the period 1850–1960. For drawing the Isenerget, the formula =log₁₀ S has been used in accordance with the definitions given byToperczer andTrapp, whereS=e _i/F·p represents the energy in erg/m² h corresponding to the surface element of 0.5° Lat. x 0.5° Long. Also the relationship between the seismicity and the tectonics of Turkey has been studied by drawing the maps of the epicenters, the focus-depths and the frequences of the earthquakes with various intensities.     

Source parameters of some large earthquakes in the eastern Mediterranean region based on an itertive maximum entropy technique

TeleseismicP-waves of some large earthquakes that occurred in the eastern Mediterranean region have been analysed by using an iterative maximum entropy technique in order to obtain the independent spectral parameters, the long-period spectral level ₀ and the corner frequencyf ₀ of the far-field displacement spectra.Based on these parameters, the seismic source parameters seismic momentM ₀, source dimensionr, fault lengthl, average displacement u, shear stress drop , radiated energyE _s and apparent stressn are calculated for the considered earthquakes by using Brune's and Madariaga's models.The striking feature of the source parameters obtained in this study is the low stress drop value which varies between 5 and 15 bars. If Madariaga's model had been used, higher stress drop values would have been obtained.The low stress drop earthquakes in the eastern Mediterranean region might be interpreted either by the possible presence of low strength material near the source or by the partial stress drop model.     

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.