Yossef H. Hatzor

The influence of grain size and porosity on crack initiation stress and critical flaw length in dolomites

Abstract The influence of rock texture on crack initiation stress ( σ i ) and critical flaw length ( L i ) is studied by a series of triaxial tests performed on monomineralic dolomites. The critical flaw length, as predicted by analytical models, is shown to be larger than the measured mean grain size ( d m ) by two–three orders of magnitude. This discrepancy is explained by rock texture variations, which influence the fracture propagation mode and consequently fracture initiation stress. The quantification of rock texture is accomplished using porosity. Fracture initiation stress is shown to be inversely related to both porosity and mean grain size. When porosity is low, the sensitivity of σ i to mean grain size is high. This effect is reduced with higher porosity values. A model for initial flaw length is developed by a synthesis of Griffith initiation criteria with our empirical model for fracture initiation stress. Initial flaw length is found to be directly proportional to the elastic modulus, mean grain size and porosity of the rock. When porosity and mean grain size decrease simultaneously, the initial flaw length rapidly decreases and approaches the mean grain size value. Therefore, the classical assumption that grain size scales initial flaw size is shown to be valid only in the very restricted case of low porosity-low grain size rocks. In such textures, where void space is minimal, available crystal faces function as truly initial flaws, and variations in mean grain size influence crack initiation stress significantly. In more porous textures, however, the initial flaw length is shown to be up to two orders of magnitude higher than the mean grain size in the rock, depending upon the porosity and mean grain size values. In such textures crack initiation stress is much less sensitive to variations in mean grain size, indicating that the role of individual grains is less significant.

Crack damage stress as a composite function of porosity and elastic matrix stiffness in dolomites and limestones

Abstract Twenty-five uniaxial compression tests were performed to determine stress at onset of dilation, referred to herein as “the crack damage stress,” in heterogeneous dolomites and limestones. A simplified model for crack damage stress (σcd) is developed here using porosity, elastic modulus, Poissons ratio and three empirical coefficients. The model shows that when porosity decreases and elastic modulus increases, σcd rapidly increases and approaches its maximum value. On the other hand, when porosity increases and elastic modulus decreases, σcd rapidly decreases and approaches its minimum value. The proposed model is validated for six heterogeneous limestone and dolomite formations which are widely distributed in Israel.

Constraints on the strength of the upper crust from stress inversion of fault slip data

The coefficient of friction of small faults in the field are estimated here by stress inversion of fault slip data. The small faults that were measured in Israel and the Grand Canyon, Arizona, are considered as representing natural friction experiments. The stresses associated with the faulting are determined by a stress inversion method which incorporates the Coulomb failure criterion [Reches, 1987]. The coefficients of friction determined for 27 fault clusters in limestone, sandstone, and basalt range from 0.0 to 1.3 with mean value of 0.58 ± 0.37. These values are in general agreement with the friction of 0.6–0.85 determined from laboratory experiments. The magnitudes of the calculated principal stresses are compared with in situ stress measurements in similar tectonic environments.

The validity of dynamic block displacement prediction using DDA

In 1965, Newmark [1] published his classic Rankine lecture on ‘‘effects of earthquakes on dams and embankments’’. In his paper, Newmark presented solutions for displacement of a mass along circular or planar sliding surface under earthquake loading. Newmark made the assumption that the mass moves as a single rigid body with resistance mobilized along the sliding surface. Newmark further considered only a single pulse of magnitude Ag lasting for a time interval t0, arguing that introduction of a sinusoidal pulse would complicate the expressions unnecessarily. Thus, the socalled ‘‘Newmark method’’ provided an estimate for the amount of mass displacement to be expected under ground acceleration of constant magnitude and given duration. Newmark admitted that his approach would generally overestimate the actual displacement because it ignores the earthquake pulse in the opposite direction. Goodman and Seed [2] studied experimentally the shear resistance of sand to cyclic loading and suggested an expression for shear strength degradation as a function of displacement. They used numerical integration to find the velocity and displacement of a block on an incline subjected to a sinusoidal acceleration function of the form

The elastic deformability and strength of a high porosity, anisotropic chalk

The submission explores the mechanical behavior of a very porous chalk formation, in which a system of ancient caverns was excavated. Incidents of general and localized failure of these ancient caverns initiated a comprehensive laboratory testing program aimed at investigating the anisotropic nature of the stress–strain response and strength of the material. It was felt that these aspects could be of profound importance in the stability of the cavern systems. The effect of water content over a broad range from 1.5% to saturation, on the compressive and tensile strength was also studied. Testing was based on the hollow cylinder methodology and was supplemented with uniaxial compression of solid cylinders and diametric compression of Brazilian disks. Use of the hollow cylinder methodology was extended to failure conditions. Test results illustrate the anisotropic nature of the stress–strain response of the chalk. The material clearly displays transverse isotropy, with horizontal bedding planes corresponding to the plane of material symmetry. The modulus of deformation within the plane of material symmetry is significantly higher than that perpendicular to bedding planes. Torsional shear of hollow cylinder specimens was employed to measure the shear modulus of the chalk. The testing carried out up to failure illustrated the anisotropy of the chalk strength. The compressive strength was found to be 50% higher in compression parallel to bedding than perpendicular to bedding. Increasing water content was found to have a consistent detrimental effect on compressive strength, tensile strength and material stiffness. The most drastic effect was found due to relatively small increases in water content, at initial water contents of less than 5%. Anisotropy of the chalk strength was found to persist over the entire range of water contents considered.

Continuous and discontinuous stability analysis of the bell-shaped caverns at Bet Guvrin, Israel

Abstract The stability of two systems of bell-shaped caverns excavated some 1000 years ago at Bet Guvrin National Park is investigated. The caverns were excavated in a weak, anisotropic, and moderately discontinuous chalk. The cavern stability is considered based on two separate and independent methods: a continuum model framework—FLAC, used for stress analysis, and a discontinuous approach—block theory, used for critical key block analysis. The numerical stress analysis reveals that in the case of very large span openings, tensile fracture of intact rock may be responsible for instabilities, which may lead to global failure. Evidence of tensile rupture at margins of failed caverns is abundant at the Park. The discontinuous block theory analysis reveals that the moderate joint set spacing at Bet Guvrin, up to 45% of the roof area may be comprised of removable blocks. The removable keyblocks in the roof remain in place due to arching stresses, which develop through the roof material. The chalk at the roof can sustain the maximum loads in existing caverns, as predicted by the numerical stress analysis. However, local failures due to exceedingly high compressive stresses at the abutments or by tensile fracture at the roof, may lead to relaxation of arching stresses followed by keyblock displacement. Such a “mixed failure mode” process could eventually lead, over time, to global collapse. Indications that “mixed failure mode” processes are presently active in the studied caverns are substantiated by in-situ measurement of keyblock displacements. It is suggested that in weak and discontinuous rock environments where “mixed failure mode” processes may be active, long term stability evaluation should be based on both continuous and discontinuous stability analyses.

Microstructure effects on microcracking and brittle failure of dolomites

Abstract In this paper the influence of microstructure on crack initiation stress and ultimate strength is investigated using results and analysis of 32 triaxial compression tests performed on cylindrical cores of dolomite samples which exhibit a wide range of grain sizes and mosaic textures. All tests were performed at a constant strain rate, under confining pressures between 0 to 40 MPa. The predictive capability of conventional criteria for ultimate strength which are based on empirical fitting parameters such as cohesion and internal friction angle, or mechanical properties such as unconfined compressive strength, is shown to be quite poor, due to the influence of microstructure. Microstructure controls ultimate strength to such a degree that an assumed mechanical property such as unconfined compressive strength may vary by more than a factor of two, where two different microstructure patterns are present. The validity of published analytical expressions which predict fracture initiation stress assuming the sliding crack model is tested using both mean and maximum grain size, and inserting the measured fracture initiation stress as the remote stress. It is shown that these approximate models fail to describe true behaviour because they ignore boundary conditions which exist at the tip of the leading crack at different mosaic textures. Early attempts to discuss the influence of microstructure on rock strength have shown that ultimate strength is inversely related to mean grain size. This study demonstrates that grain size alone can not be used in correlation with ultimate strength. Rather, the combination of both grain size and porosity dominate the mechanical response of the rock. Fracture initiation stress is found to be more sensitive to the influence of grain size than ultimate strength, possibly because the length of initial cracks controls the level of stress concentration at the tip of leading cracks. However, fracture initiation stress is shown to be inversely related to both porosity and mean grain size, thus the importance of porosity in the initiation process must be recognized. Ultimate strength is influenced primarily by porosity and mosaic texture, and is less sensitive to mean grain size, possibly because once fracture propagation is initiated, grain arrangement controls fracture interaction processes which lead to macroscopic failure.

A New Viscous Boundary Condition in the Two-Dimensional Discontinuous Deformation Analysis Method for Wave Propagation Problems

Viscous boundaries are widely used in numerical simulations of wave propagation problems in rock mechanics and rock engineering. By using such boundaries, reflected waves from artificial boundaries can be eliminated; therefore, an infinite domain can be modeled as a finite domain more effectively and with a much greater accuracy. Little progress has been made, thus far, with the implementation and verification of a viscous boundary in the numerical, discrete element, discontinuous deformation analysis (DDA) method. We present in this paper a new viscous boundary condition for DDA with a higher absorbing efficiency in comparison to previously published solutions. The theoretical derivation of the new viscous boundary condition for DDA is presented in detail, starting from first principles. The accuracy of the new boundary condition is verified using a series of numerical benchmark tests. We show that the new viscous boundary condition works well with both P waves as well as S waves.

The block failure likelihood: A contribution to rock engineering in blocky rock masses

Abstract This paper summarizes some results of an applied block theory research conducted at Berkeley over the last three years. The validity of the critical-key-block concept is tested here using block mould statistics sampled from side walls of two tunnels, the raw data of which are shown. The predictive capabilities of the three block-failure-likelihood parameters are tested by correlation with the block mould sample population, and a modification is introduced. A possible method to integrate a block motion criteria in empirical rock mass classification systems is proposed.

The shear strength of clay-filled bedding planes in limestones - back-analysis of a slope failure in a phosphate mine, Israel

SummaryThe failure of a slope in a phosphate mine by shear-sliding along a clay-filled bedding plane in limestone, and by separation across a tension crack at the back, is back-analysed. The failure cannot be explained using laboratory measured values of the shear strength parameters. In order to simulate field conditions better two ‘physical models’ of the bedding plane were prepared for testing under triaxial compression. Cylindrical cores with an inclined saw-cut discontinuity were filled with remoulded montmorillonite. It is shown that failure in the models initiates along the contacts between the clay infilling and the limestone boundaries, and not through the clay itself, as would be intuitively expected. Furthermore, it is argued that in the analysis of rock slope stability in general, and particularly in the case of clay-filled discontinuities, the influence of paleo-overburden stress on frictional resistance must be resolved before the appropriate constitutive law can be established for analysis.