Preferred orientation in experimentally deformed dolomite

Fine grained dolomite has been deformed in over twenty compression experiments in a Griggs-type piston-cylinder apparatus at various P and T conditions. Preferred orientation determined quantitatively using X-ray techniques and spherical harmonic analysis of the data is presented in inverse pole-figures of — 2/m symmetry. In most cases specimens display strong preferred orientation which varies mainly as a function of temperature. At all conditions it is very different from calcitic limestone. Although there is no significant grain growth even at 1000 °C the simple c-axes maximum fabric above 700 °C might be the result of recrystallization or translation on c. Below 700 °C, the preferred orientation is much weaker and complex. The primary maximum in the inverse pole-figure is near e, a secondary maximum near a high angle positive rhomb, principal minima are at c and f. This inverse pole-figure is consistent with f-twinning and translation gliding on r (t = a ?), two mechanisms which counteract each other. The latter is a new deformation mechanism for dolomite which we propose in order to explain the pattern of preferred orientation. The minimum at c is less pronounced below 100 ° C suggesting that c-translation may be active, but in these fine-grained aggregates it appears to be less important than is expected from single crystal experiments (Higgs and Handin, 1959), at least at low temperatures.     

Deformation und Gefügeregelung von Steinsalz im Temperaturbereich 20–200° c

Cubic specimens of fine-grained artifical polycristalline rock salt were deformed at different principal stresses in the range of 20–200° C. The data from different types of tests are compared by plotting ₁–₃ against ₁, ₂, ₃. The deformation behavior is highly influenced by the intermediate principal stress ₂. An increase of the intermediate stress from compression (₁>₂=₃) to extension (₁=₂>₃) lowers the rate of deformation. An increase of temperature always gives rise to an increase of deformation rate.Translation gliding on (110) is assumed to be the dominant flow mechanism. Textural changes and development of preferred orientation were investigated by means of the X-ray diffractometer. Intensities were measured for (110), (111), and (100). Compression leads to girdle occupations about the unique principle compressive axis, extension to girdles about the unique principle axis of extension. When all principle stresses are different the polfigures tend to a position intermediate between the two occupations.     

Dreiaxiale verformungen an Solnhofener Kalkstein im Temperaturbereich von 20–650° C. Röntgenographische Gefügeuntersuchungen mit dem Texturgoniometer

Axial-symmetric and real triaxial deformation tests on Solnhofen limestones were made in the range of 20 to 650° C. Stress, strain and time observations as functions of temperature and the ratios of the external stresses are plotted as stress-strain and strain-time curves. Mechanical twinning on e (01¯12) and syntectonic recrystallization (at temperatures above 500° C) is assumed to be the dominant flow mechanism. On samples of high permanent strain the textural changes and development of preferred orientations were investigated by means of the X-ray diffractometer. In all tests the symmetry of the polefigures is exactly identical with the symmetry of the external stress-field.In tests with ₁>₂=₃ the poles of the planes investigated (10¯11; 10¯12; 11¯20) occupy girdles about the unique principle compressive axis. When all principle stresses are different (₁>₂>₃) the poles of the planes investigated tend to maxima (in the case of plane strain) ore to girdle-occupations about the principle axis of maximum extension. The deduced c-diagrams show a fairly similar tendency of the spacial orientations.The pole figures (as well as the calculated orientation of the c-axis) must be explained by the superposition of reflected intensities on planes in twinned and untwinned grains. In every diagram partial occupations—possibly representing the orientations of twinned grains-are in fairly good agreement with the orientation predicted by the theory of Brace-MacDonald for the stable orientation of anisotropic minerals in a non-hydrostatic stress-field.

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Herrn Prof. Dr. F. Karl danke ich sehr herzlich für zahlreiche Diskussionen sowie für die kritische Durchsicht des Manuskripts. Mein Dank gilt auch der Deutschen Forschungsgemeinschaft, welche die Untersuchung durch eine Sachbeihilfe unterstützte.     

Analysis of the variability of unconfined compression tests of rock

Summary The laboratory testing program reported herein describes the statistical correlations of unconfined compressive strength ( _c ), Young's modulus (E _t50), and modulus ratio (MR) to their coefficients of variation (CV( _c ), CV(E _t50), and CV(MR)). Negative correlation of both _c andE _t50 to CV( _c ) indicates that data scatter tends to decrease with increasing strength and stiffness. A positive correlation between MR and CV( _c ) indicates that stiffer rocks are more variable for a given compressive strength.     

Deformationsmechanik eines offenen Konglomeratgefüges—Ein Beitrag zur statischen Tektonik

Zusammenfassung Offene Faltung in proterozoischen Tilliten der Huronian Supergroup führte zu geringen Differentialverschiebungen zwischen Matrix und Grobkomponenten innerhalb von konglomeratischen Schichtgliedern. Einmessung der aus der Bewegung resultierenden Striemungen und feiner Risse erlaubt die Bestimmung der regionalen Spannungsachsen (₁ > ₂ > ₃).

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Deformation mechanics of an open conglomerate fabric—A contribution to static tectonics

Summary Open folding of Proterozoic tillites (Huronian Supergroup) caused minor differential displacements between matrix and clasts within conglomeratic units. Detailed measurements of striations and ruptures along the clast surface permitted the deduction of regional stress axes (₁ > ₂ > ₃).

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Mit 6 Abbildungen

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Herrn Prof. Dr. Dr. h. c.B. Sander zum 90. Geburtstag gewidmet.     

Modeling soil variability as a random field

The observed variability in the spatial distribution of soil properties suggests that it is natural to describe such distribution as a random field. One of the ways to study engineering problems in such a stochastic setting is by the use of the Monte-Carlo simulation procedure. Application of this technique requires the capability to generate a large number of realizations of a given random field. A numerical procedure for the generation of such realizations in two-dimensional space is introduced as a finite difference approximation of a stochastic differential equation. The equation used was that suggested by Heine (1955). The resulting procedure is essentially similar to other autoregressive procedures used for the same purpose (Whittle, 1954; Smith and Freeze, 1979). However, contrary to these procedures, the present one is defined in terms of physically significant parameters:r ₀, the autocorrelation distance;, the discretization size; and the standard deviation, . Formulating the simulation procedure in terms of the physically significant parameters (r ₀,, ) greatly simplifies the task of generating realizations that are compatable with a given soil deposit.     

Ground failure around buried tubes

Summary The hoop forces which develop in circular tubes buried in elastic-plastic ground are investigated. A closed form solution is used to determine the hoop forces which develop when the field stresses in the elastic-plastic ground are initially uniform. The finite element method is used to solve the problem for biaxial field stress. A parametric study is undertaken to assess the influence of tube stiffness and ground strength on the hoop forces, and use is made of elastic stress contours to predict the likely extent of material failure around tubes buried in ground with biaxial prestress.Notation a tube radius - c cohesion - D flexural stiffness of the structure - E _i Young's modulus of structure - E _s Young's modulus of ground - F hoop force (compression positive) - G _s shear modulus of ground - H hoop stiffness of the structure - K coefficient of lateral pressure - N tan²(45+/2) - q ( ₁– ₁)/2 - S _f relative flexural stiffness of the structure - S _h relative hoop stiffness of the structure - t structural thickness - v circumferential displacement - w radial displacement - v _l Poisson's ratio of structure - v _s Poisson's ratio of ground - normal traction acting on the structure - _d deviatoric component of field stress - _h horizontal field stress - _m uniform component of field stress - _v vertical field stress - ₁ major principal stress - ₃ minor principal stress - tangential traction acting on the structure - angle of internal friction of the ground - angle of dilation of the ground     

An internal variable model for the creep of rocksalt

Summary The creep strain rate of rocksalt, like that of other ductile crystalline materials, can be described by a power law equation of the type ( ) ⁿ , where the active stress is the difference between the total deviatoric applied stress and an internal stress _i . In this paper, the origin and the nature of this internal stress, which develops during inelastic deformation of the material, are discussed. It is shown that this internal stress can serve as an internal (or state) variable in the constitutive model of rocksalt, which reflects the microstructure evolution of the material under the competitive action of hardening and recovery mechanisms.An analysis of experimental data, both our own and those taken from the literature, demonstrates that such a law is able to correctly reproduce rocksalt creep test results in the steady-state domain. The proposed model is in accordance with the macroscopic and microscopic behavior of salt, and with direct measurements of the internal stresses made by others on this material.     

Requirements for accurate estimation of fractal parameters for self-affine roughness profiles using the line scaling method

Summary A new concept of feature size range of a roughness profile is introduced in the paper. It is shown that this feature size range plays an important role in estimating the fractal dimension,D, accurately using the divider method. Discussions are given to indicate the difficulty of using both the divider and the box methods in estimatingD accurately for self-affine profiles. The line scaling method's capability in quantifying roughness of natural rock joint profiles, which may be self-affine, is explored. Fractional Brownian profiles (self-affine profiles) with and without global trends were generated using known values ofD, input standard deviation, , and global trend angles. For different values of the input parameter of the line scaling method (step sizea ₀),D and another associated fractal parameterC were calculated for the aforementioned profiles. Suitable ranges fora ₀ were estimated to obtain computedD within ±10% of theD used for the generation. Minimum and maximum feature sizes of the profiles were defined and calculated. The feature size range was found to increase with increasingD and , in addition to being dependent on the total horizontal length of the profile and the total number of data points in the profile. The suitable range fora ₀ was found to depend on bothD and , and then, in turn, on the feature size range, indicating the importance of calculating feature size range for roughness profiles to obtain accurate estimates for the fractal parameters. Procedures are given to estimate the suitablea ₀ range for a given natural rock joint profile to use with the line scaling method in estimating fractal parameters within ±10% error. Results indicate the importance of removal of global trends of roughness profiles to obtain accurate estimates for the fractal parameters. The parametersC andD are recommended to use with the line scaling method in quantifying stationary roughness. In addition, one or more parameters should be used to quantify the non-stationary part of roughness, if it exists. The estimatedC was found to depend on bothD and and seems to have potential to capture the scale effect of roughness profiles.     

Hydrofracture mechanisms in rock during pressure grouting

Summary Hydrofracture Mechanisms in Rock During Pressure Grouting. The paper examines the basic meachnisms controlling the initiation of fractures in rocks and layered soils during pressure grouting, and their subsequent propagation into the ground mass. Previous analyses of fracture initiation have tended to concentrate on simplified models in which the ground is treated as an impervious elastic or Mohr-Coulomb continuum. The present method allows for the porous or fissured nature of the ground by considering the effect of seepage forces induced by the pore pressure gradient. The effect is quantified by use of a parameterN such that the ratio of fluid force used in expanding the injection hole, to that used in forcing fluid through void spaces, isN to (1—N).Analysis of hydrofracture propagation is based on stress analysis of a borehole in an elastic continuum, the propagating fracture zone around the borehole being represented as a non-elastic material governed by the Mohr-Coulomb failure criterion. This is supplemented by an energy approach which equates the energy supplied to the ground from the injection pump, with the energy stored in the ground and the energy necessary to fracture it.

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Zusammenfassung Brucherscheinungen im Fels bei Verprearbeiten. In der Arbeit wird der grundlegende Mechanismus untersucht, welcher bei Verpreßarbeiten in Fels und geschichtetem Boden zur Einleitung und Ausbreitung von Brüchen führt. Frühere Untersuchungen des Bruchbeginns stützten sich im wesentlichen auf vereinfachte Modelle, in denen der Untergrund als undurchlässiges Kontinuum angesehen wird, das entweder elastisch ist oder der Mohr-Coulombschen Bruchbedingung genügt. Die neue Methode berücksichtigt dagegen eine Porosität oder Klüftung des Untergrundes durch Ansatz der vom strömenden Medium auf das Gebirge ausgeübten Belastung. Diese Belastung wird aufgeteilt in einen Druckverlust an der Bohrlochwand (gleich ParameterN mal Verpreßdruck) und die entsprechende, über den gesamten durchströmten Bereich verteilte Belastung.Die Untersuchung der Bruchausbreitung geht von der Spannungsermittlung um ein Bohrloch in einem elastischen Kontinuum aus, wobei in der sich ausbreitenden Bruchzone um das Bohrloch herum nichtelastisches Material angenommen wird, das dem Mohr-Coulombschen Bruchkriterium genügt. Zur Ergänzung dient eine Energie-Betrachtung, bei der die von der Injektionspumpe abgegebene Energie gleichgesetzt wird der im Untergrund gespeicherten Energie und der aufgewendeten Brucharbeit.

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Résumé Les mécanismes de la fracturation hydraulique dans les roches pendant les injections sous pression. Le mémoire examine les mécanismes fondamentaux qui gouvernent l'initiation des ruptures dans les roches et les sols stratifiés, au cours des injections et leur propagation dans les massifs. Les analyses antérieures de l'initiation de la rupture, se sont concentrées sur des modèles simplifiés où l'on considérait la roche comme un milieu élastique et imperméable, ou comme un milieu de Mohr-Coulomb. La présente méthode admet que le massif est poreux ou fissuré, en considérant l'action des forces de percolation engendrées par le gradient de pression interstitielle. Cette action est quantifiée par un paramètreN, tel que le rapport de la force du liquide employée à dilater le forage d'injection, à celle employée pour forcer le coulis à travers les vides soitN/(1—N). L'analyse de la propagation des ruptures se base sur l'analyse des contraintes autour d'un forage dans un milieu élastique, alors que la zone de la rupture qui se propage autour du forage est représentée par un milieu non-élastique admettant le critère de rupture de Mohr-Coulomb. Cette analyse est complétée par une approche énergétique, où l'énergie qui est fournie au massif par la pompe d'injection est égalée à l'énergie emmagasinée dans la roche et à l'énergie de rupture.

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Nomenclature A area of new cracks created per unit volume of time t - E total work done by injection fluid - E _i irrecoverable component of energy - E _{i c} work done in fracturing rock or soil - E _{i l} work done to overcome various frictional forces in grouting system - E _{i p} work done to cause plastic deformation of fractured zone - E _{i s} work done to overcome shear strength of fluid during flow - E _{i v} work done to overcome frictional drag between fluid and rock in soil surfaces during flow - E _r recoverable component of energy - E _{r f} elastic strain energy stored in fluid - E _{r s} elastic strain energy stored in rock or soil - h height of overburden - i j 1, 2, 3 - K ₀ coefficient of horizontal earth pressure - k permeability of ground to grout - L length of cylindrical grout source - n rock or soil porosity - p average fluid pressure between timet and (t + t) - p ₀ injection pressure - R radius of grout front - r radial distance from borehole axis - r ₀ radius of borehole - r ₁ radius of fractured zone - S specific surface area of rock or soil - S _T tensile strength of rock or soil - t time - u grout seepage velocity - V volume of grout injected - v volumetric strain - specific surface energy of rock - bulk density of rock or soil - _{i j} ^e elastic strain increment tensor - _{i j} plastic strain increment tensor - v Poisson's ratio - _{i j} average stress tensor in the ground during timet and (t + t) - _R , _T, _Z radial, tangential and vertical stresses induced by grouting - _r , _t , _z radial, tangential and vertical stress around borehole before grouting - grout shear strength - angle of internal friction of rock or soil With 7 Figures     

Petrographic and geochemical evidence for hydrothermal evolution of the North Deposit,Mt Tom Price,Western Australia

High-grade iron mineralisation (>65%Fe) in the North Deposit occurs as an E-W trending synclinal sheet within banded iron formation (BIF) of the Early Proterozoic Dales Gorge Member and consists of martite-microplaty hematite ore. Three hypogene alteration zones between unmineralised BIF and high-grade iron ore are observed: (1) distal magnetite-siderite-iron silicate, (2) intermediate hematite-ankerite-magnetite, and (3) proximal martite-microplaty hematite-apatite alteration zones. Fluid inclusions trapped in ankerite within ankerite-hematite veins in the hematite-ankerite-magnetite alteration zone revealed mostly H₂O–CaCl₂ pseudosecondary and secondary inclusions with salinities of 23.9±1.5 (1, n=38) and 24.4±1.5 (1, n=66) eq.wt.% CaCl₂, respectively. Pseudosecondary inclusions homogenised at 253±59.9°C (1, n=34) and secondary inclusions at 117±10.0°C (1, n=66). The decrepitation of pseudosecondary inclusions above 350°C suggests that their trapping temperatures are likely to be higher (i.e. 400°C). Hypogene siderite and ankerite from magnetite-siderite-iron silicate and hematite-ankerite-magnetite alteration zones have similar oxygen isotope compositions, but increasingly enriched carbon isotopes from magnetite-siderite-iron silicate alteration (–8.8±0.7, 1, n=17) to hematite-ankerite-magnetite alteration zones (–4.9±2.2, 1, n=17) when compared to the dolomite in the Wittenoom Formation (0.9±0.7, 1, n=15) that underlies the deposit. A two-stage hydrothermal-supergene model is proposed for the formation of the North Deposit. Early 1a hypogene alteration involved the upward movement of hydrothermal, CaCl₂-rich brines (150–250°C), likely from the carbonate-rich Wittenoom Formation (¹³C signature of 0.9±0.7, 1, n=15), within large-scale folds of the Dales Gorge Member. Fluid rock reactions transformed unmineralised BIF to magnetite siderite-iron silicate BIF, with subsequent desilicification of the chert bands. Stage 1b hypogene alteration is characterised by an increase in temperature (possibly to 400°C), depleted ¹³C signature of –4.9±2.2 (1, n=17), and the formation of hematite-ankerite-magnetite alteration and finally the crystallisation of microplaty hematite. Late Stage 1c hypogene alteration involved the interaction of low temperature (~120°C) basinal brines with the hematite-ankerite-magnetite hydrothermal assemblage leaving a porous martite-microplaty hematite-apatite mineral assemblage. Stage 2 supergene enrichment in the Tertiary resulted in the removal of residual ankerite and apatite and the weathering of the shale bands to clay.Editorial handling: B. Lehmann     

Statistical Properties of Ideal Rock Textures: Relationship Between Crystal Size Distribution and Spatial Correlation of Minerals

The concept of an ideal rock texture, in which crystals are distributed randomly in space, is proposed for use in general analysis of rock textures. The spatial correlation function for the ideal rock texture was examined and the function, a specific kind of spatial correlation function, is related to crystal size distribution and to some extent to crystal boundaries. The function is unity at distance zero, and monotonically decreases with increasing distance for the ideal texture. This behavior of the function is observed for any size distribution. In an ideal texture, the function is directly related to crystal size distribution and crystal shape. It is important in stereology because the crystal size distribution in three dimensions may be deduced from analyzing a function that is obtained from analyzing two-dimensional section images. Crystal shape is also related to the function. If crystals are concave in shape, or have inclusions of other phases, the function may show a hump or plateau when plotted against distance. However, the crystal shape effect cannot produce values smaller than zero. If values become negative, the texture is no longer considered ideal. The ideal textures for two model size distributions—step and delta functions—are considered. The rate of decrease of values is more strongly dependent on size distribution than on system dimension.     

The shear strength of rock joints in theory and practice

SummaryThe Shear Strength of Rock Joints in Theory and Practice The paper describes an empirical law of friction for rock joints which can be used both for extrapolating and predicting shear strength data. The equation is based on three index parameters; the joint roughness coefficientJRC, the joint wall compressive strengthJCS, and the residual friction angle _r . All these index values can be measured in the laboratory. They can also be measured in the field. Index tests and subsequent shear box tests on more than 100 joint samples have demonstrated that _r can be estimated to within ± 1° for any one of the eight rock types investigated. The mean value of the peak shear strength angle (arctan/ _n ) for the same 100 joints was estimated to within 1/2°. The exceptionally close prediction of peak strength is made possible by performing self-weight (low stress) sliding tests on blocks with throughgoing joints. The total friction angle (arctan/ _n ) at which sliding occurs provides an estimate of the joint roughness coefficientJRC. The latter is constant over a range of effective normal stress of at least four orders of magnitude. However, it is found that bothJRC andJCS reduce with increasing joint length. Increasing the length of joint therefore reduces not only the peak shear strength, but also the peak dilation angle and the peak shear stiffness. These important scale effects can be predicted at a fraction of the cost of performing large scale in situ direct shear tests.With 20 Figures     

Über Beanspruchung und Verformung von Gesteinen

7 coaxial (p ₁>p ₂=p ₃) and 6 real triaxial (p ₁>p ₂>p ₃) compression tests (0,5 to 6 hours duration) on cubic samples of homogeneous, statistically low anisotropic marble from Auerbach (Germany) were made at room temperature and different ratios of the main pressures and rates of loading. 3 samples of strong anisotropic marble of Drama (Greece) were deformed at p ₁p ₂p ₃=841 and constant rate of loading in the three orientations: parallel, at 45° and perpendicular to the foliation (12 hours duration). As can be deduced from the orientation diagrams, the deformation proceeds mainly by twin gliding. In most of the tests reported here the symmetry of the external loading (coaxial or orthorhombic) can be correlated with the symmetry of the orientation diagrams. Real triaxial deformation (P ₁Nep ₂Nep ₃) produces two sets of planes ± perpendicular to each other as B B. In the tests with the strong anisotropic material the strain was measured in the three main directions and plotted as strain-time curves. When the foliation is oriented perpendicularly or even obliquely to p ₁ compression dominates at the beginning in the direction of the intermediate loading p ₂ producing preferred orientation of the twin lamellae. Later on there is compression in p ₁ and extension in p ₂ and p ₃. This phase of deformation is characterized by brittle fracture and increase of volume.     

Paleomagnetic constraints on Zn–Pb ore genesis of the Pillara Mine, Lennard Shelf, Western Australia

The Pillara Zn–Pb deposit is the largest of several known Mississippi Valley-type (MVT) deposits in the Lennard Shelf of the Canning Basin. Paleomagnetic and rock magnetic measurements are reported for 294 specimens from 23 sites in mineralization and its carbonate host rocks from the deposit as well as on 15 artificial specimens of zinc and lead concentrate and of tailings. Pyrrhotite carries the characteristic remanent magnetization (ChRM) in nearly all specimens. The ChRM postdates most faulting as shown by breccia tests and most minor regional tilting as shown by the degraded fit on tilt correction. The mean ChRM direction for all sites is D=20.6°, I=–27.5° (N=23, ₉₅=5.3°, k=34.1), yielding an age of 358±5 Ma (2) that is similar to the comparable age of 354±8 Ma (2) for the Kapok MVT deposit. Host rock diagenesis with attendant secondary remagnetization yields an age of 361±5 Ma (1) and the MVT mineralization with a primary chemical remanent magnetization gives an age of 356±3 Ma (1), co-eval with a published Rb–Sr sphalerite age of 357±3 Ma. Interpretation of this temporal data suggests that the MVT deposits of the southeastern Lennard Shelf originated during extension, probably in response to rift-related topography-driven fluid flow.Editorial handling: C. Brauhart     

Effects of ageing and overconsolidation on the elastic stiffness of a remoulded clay

Summary Results of resonant column tests were used to determine values of low amplitude shear modulus (G ₀) of a remoulded kaolinite clay for different durations of ageing and for different values of consolidation stress (₀) and overconsolidation ratio (OCR). It was found that after completion of primary consolidation the values of G ₀ increased linearly with the logarithm of time and after a week of confinement the value of normalized rate of secondary increase of shear modulus (N _G ) could be reliably estimated. Values of N _G were found to derease linearly with the logarithm of OCR and with the logarithm of ageing duration. This similarity of behaviour provided a basis for establishing an equivalency between age and equivalent overconsolidation ratio (OCR)_eq. The effects of ₀ and OCR on the value of G ₀ were also established in a functional form that indicated a stronger influence compared to that predicted by the Hardin Equation.     

Modal and vectorial analysis for determination of stress axes associated with fault slip data

Based on Anderson's faulting theory, for a given set of fault slip data including the sense of slip on faults and fault planes, this paper provides two possible methods to reconstruct the principal stress axes using vectorial and modal analysis procedure. The vectorial analysis consists of computing eigenvectors of the orientation matrices defined by axes P, B, and T (P, B, and T being geometrical axes parallel to axes ₁, ₂, ₃ associated with a single striated fault) which are determined geometrically knowing the slip vector S, the normal to fault plane N, and the dihedral angle 2. A parameter R, related to the maximum eigenvalue of the orientation matrix and the size of data sample, is shown to be a good test value for the homogeneity of the data. A process of refinement of this parameter enables bad data (representing faults generated under different tectonic events) to be ignored in the final computation of the principal stress axes; results are thus significantly improved and the vectorial analysis procedure enables the orientation of stress to be numerically determined. On defining probability density regions around each P, B, and T axes that incorporate the variation of fault geometry, the modal analysis is carried out in two density functions established by a convolution process to locate modes which represent optimal locations of the principal stress axes in spherical space. Splitting of heterogeneous data samples into homogeneous subsets is achieved by a dynamical cluster procedure which enables the principal stress axes associated with each subset to be determined separately.     

Heterogeneous deformation resulting from stress gradients in calcite single crystals

The stress gradient calculated from an isotropic elastic approximation does not directly reflect the distribution of permanent deformation in a crystal of the same shape under the same conditions. However, with additional crystallographic constraints, it serves to predict locations where twinning and slip are first activated in a stressed crystal. In this study, thick-walled hollow cylinders were cored from single calcite crystals parallel to 0001 and . One cylinder of each orientation was loaded at room temperature under one of two conditions: internal pressure (P _p )=60 MPa, external pressure (P _c )=100 MPa, or P _p = 20 MPa, P _c = 50 MPa. These conditions would produce a radial stress gradient in an isotropic elastic cylinder. Mechanical twins in the deformed calcite samples had a hexagonal distribution in the 0001 oriented cylinder and an orthorhombic distribution in the oriented cylinder.Zones of dense r-slip dislocations were observed in the cylinder. Calculated resolved shear stresses for r-slip in either cylinder remained far below the published critical resolved shear stress (c.r.s.s.) value. Calculated contributions from twinning back stress and anisotropy do not account for the difference between the resolved shear stress and the c.r.s.s. These results underscore the necessity of considering dislocation activiation stresses rather than c.r.s.s. in quantitative analyses of heterogeneous of deformation in single crystals.     

Statistical Methods for Jointly Estimating the Decay Constant of 40K and the Age of a Dating Standard

To simultaneously evaluate the decay constant of ⁴⁰K () and the age of a standard (t _std) using isotopic data from geologic materials, we applied a series of statistical methods. The problem of estimating the most probable intercept of many nonlinear curves in and t _std space is formulated by an errors-in-variables nonlinear regression model. Then a maximum likelihood method is applied to the model for a point estimate, which is equivalent to the nonlinear least square method when measurement error distributions are Gaussian. Uncertainties and confidence regions of the estimates can be approximated using three methods: the asymptotic normal approximation, the parametric bootstrap method and Bonferroni confidence regions. Five pairs of published data for samples with ages from 2 ka to 4.5 Ga were used to estimate and the age of Fish Canyon sanidine (t _FCs). The statistical procedure yields most probable estimates of (5.4755 ± 0.0170 × 10^–10 (1)/year) and t _FCs (28.269 ± 0.0661 (1) Ma) which are in between previously published values. These results indicate the power of our approach to provide improved constraints on these parameters, although the preliminary nature of some of the input data require further review before the values can be adopted.