Antonio Bobet

Coalescence of multiple flaws in a rock-model material in uniaxial compression

A number of specimens made of gypsum with three and 16 flaws have been prepared and tested in compression. Results from these experiments are compared with observations from similar specimens with two flaws. The comparisons indicate that the cracking pattern observed in specimens with multiple flaws is analogous to the pattern obtained in specimens with two flaws. Two types of cracks initiate from the tips of the flaws: wing cracks and secondary cracks. Wing cracks are tensile cracks that initiate at an angle with the flaw and propagate in a stable manner towards the direction of maximum compression. Secondary cracks are shear cracks that initially propagate along their own plane in a stable manner. Two types of secondary cracks are possible: coplanar or quasi-coplanar, and oblique. As the load is increased, wing cracks propagate in a stable manner and secondary cracks may propagate in an unstable manner and produce coalescence, which occurs when two flaws are linked together. Nine types of coalescence have been observed, and each type is characteristic of a particular flaw geometry. The stresses at which wing and secondary cracks initiate and coalescence occurs strongly depend on the geometry of the flaws and on the number of the flaws; as the flaw inclination angle increases, the spacing increases, or the number of flaws decreases, initiation and coalescence stresses increase.

The initiation of secondary cracks in compression

Abstract In rocks and rock-model materials, two types of cracks are observed: wing cracks, and secondary cracks. Wing cracks are tensile cracks that initiate at the tips of pre-existing cracks (flaws) and propagate in a stable manner towards the direction of the maximum compressive stress. Secondary cracks initiate also from the tips of the flaws, propagate in a stable manner, and have been recognized by many researchers as shear cracks. Two initiation directions are possible: one coplanar or quasi-coplanar to the flaw, and the other one parallel to the wing cracks but in the opposite direction. Shear cracks quasi-coplanar to the flaw are observed in most of the experiments; shear cracks parallel to the wing cracks only in few cases. This indicates that the second direction may be material dependent. Secondary cracks play a major role in the cracking process of rocks in compression. Crack coalescence is caused in many instances by secondary cracks. In biaxial compression and for high confinement, cracking is only produced through secondary cracks. Conventional initiation criteria are suitable for predictions concerning tensile cracks, but are inadequate to predict initiation of secondary cracks. An extension of the maximum tangential stress criterion is proposed in which shear crack initiation is analogous to tensile crack initiation, except that the direction and stress level of initiation are determined by the direction and magnitude of the maximum shear stress. This criterion allows for the initiation of more than one kink from pre-existing cracks. This is in agreement with experiments, and with the fact that the stress singularity at the tip of a flaw does not disappear with the initiation of a kink. A limited number of comparisons between experiments and this model show promising results. Stress initiation and angle of initiation for both wing and secondary cracks can be determined within reasonable errors.

Numerical modeling of fracture coalescence in a model rock material

The crack pattern, as well as crack initiation, -propagation and -coalescence observed in experiments on gypsum specimens with pre-existing fractures in uniaxial, biaxial, and tensile loading are satisfactorily predicted with the numerical model presented in this paper. This was achieved with a new stress-based crack initiation criterion which is incorporated in FROCK, a Hybridized Indirect Boundary Element method first developed by Chan et al. (1990). The basic formulation of FROCK is described, and the code verified for both open and closed pre-existing fractures either with only friction or with friction and cohesion. The new initiation criterion requires only three material properties: σcrit, the critical strength of the material in tension; τcrit, the critical strength of the material in shear; r0, the size of the plastic zone. The three parameters can be determined with the results from only one test. Predictions using this model are compared with experiments on gypsum specimens with pre-existing fractures loaded in uniaxial and biaxial compression performed by the authors. Specifically, wing crack and shear crack initiation, crack propagation, coalescence stress and -type as well as the crack pattern up to coalescence can be modeled. The model can also duplicate experimental results in compression and tension obtained by other researchers. These results show that stress-based criteria can be effectively used in modeling crack initiation and crack coalescence.

Predictions of ground deformations in shallow tunnels in clay

Twenty-eight tunnels are used to evaluate predictions from an analytical solution for shallow tunnels in saturated ground. The solution assumes plane strain conditions at any cross-section perpendicular to the tunnel axis and poroelastic behavior of the ground and elastic behavior of the liner. Stresses and deformations are obtained with this method for short- and long-term conditions anywhere in the continuum. Of particular interest to this study are the short-term surface settlements at the ground surface. Comparisons between predictions and observations from actual tunnels show good agreement, generally within 15% difference. The analyses show that: (1) most of the ground movements are caused by the gap parameter, which is a measure of the three-dimensional deformations at the tunnel face, the physical gap between the liner and the perimeter of the excavation, and of the workmanship; (2) most of the ground deformations take place within a distance of three to four radii around the tunnel; (3) the bottom boundary of zero vertical deformation should be placed at a distance of two tunnel diameters below the tunnel center line, or at the location of a stiff soil layer, whichever comes first; (4) the horizontal movements around the tunnel are relatively smaller than the vertical movements; and (5) the analytical solution tends to underpredict the maximum soil deformations and overestimate the settlement trough; however, with an appropriate estimation of the gap parameter and small soil yield, the differences between predictions and observations are small.

EFFECT OF PORE WATER PRESSURE ON TUNNEL SUPPORT DURING STATIC AND SEISMIC LOADING

Abstract The support of underground structures must be designed to withstand static overburden loads as well as seismic loads. New analytical solutions for a deep tunnel in a saturated poroelastic ground have been obtained for static and seismic loading. The static solution accounts for drainage and no-drainage conditions at the ground–liner interface. Linear elasticity of the liner and ground, and plane strain conditions at any cross-section of the tunnel are assumed. For tunnels in which ground stresses and pore pressures are applied far from the tunnel center, the drainage conditions at the ground–liner interface do not affect the stresses in the liner. The analytical solution shows that the stresses in the liner are exactly the same whether there is drainage or not at the ground–liner interface. Hence, if the drainage conditions in the tunnel are changed from full drainage to no-drainage or vice versa the stresses in the liner are not affected. However, the stresses and displacements in the ground change significantly from drainage to no-drainage conditions. For seismic loading a new analytical formulation is presented which provides the complete solution for the ground and the liner system for both dry and saturated ground conditions. The formulation is based on quasi-static seismic loading and elastic ground response; for a saturated ground, undrained conditions are assumed which indicate that the excess pore pressures generated during the seismic event do not dissipate. The results show that the racking deformations of a liner in dry or saturated ground are highly dependent on the flexibility of the liner.

Numerical Models in Discontinuous Media: Review of Advances for Rock Mechanics Applications

The paper presents a description of the methods used to model rock as discontinuous media. The objective of the work is to bring to the geomechanics community recent advances in numerical modeling in the field of rock mechanics. The following methods are included: (1) the distinct element method; (2) the discontinuous deformation analysis method; and (3) the bonded particle method. A brief description of the fundamental algorithms that apply to each method is included, as well as a simple case to illustrate their use.

Drucker–Prager Criterion

List of Symbols k Drucker–Prager material constant j Drucker–Prager material constant J2 Second invariant of the stress deviator tensor I1 First invariant of the effective stress tensor r1 Major principal effective stress r2 Intermediate principal effective stress r3 Minor principal effective stress soct Octahedral shear stress roct Octahedral effective normal stress C0 Uniaxial compressive strength T0 Uniaxial tensile strength h Lode angle b MSDPu parameter that defines the shape of the criterion in the p-plane (usually, b % 0.75) a1 MSDPu parameter a2 MSDPu parameter / Angle of internal friction c Cohesion

A practical iterative procedure to estimate seismic-induced deformations of shallow rectangular structures

An iterative procedure is proposed to estimate seismic-induced distortions of cut-and-cover rectangular structures. The procedure is based on an existing analytical solution for deep rectangular structures subjected to far-field shear stress which assumes elastic behavior of the soil and structure, tied contact at the soil–structure interface, and static loading. The new proposed procedure builds on the analytical solution and approximates dynamic response with a pseudo-static analysis and incorporates soil-stiffness degradation through an iterative scheme where the soil shear modulus is changed in each iteration based on the shear strain of the soil obtained in the previous iteration. The presence of the ground surface and slip at the soil–structure interface are neglected in the method proposed, but their effects are shown to be small and have compensating results when soil nonlinearity is introduced. Predictions obtained from the analytical solution have been verified by a series of numerical tests, wh...

Liquefaction Mitigation Using Bentonite Suspensions

AbstractOttawa sand specimens premixed with 0, 3, and 5% bentonite by dry mass of sand were tested under undrained static and cyclic loading to investigate the effects of bentonite on the static and cyclic shear strength of the sand. The results show that allowing the bentonite to hydrate within the sand pore space increases the cyclic resistance of the sand. For the same skeleton relative density and cyclic stress ratio, cyclic tests on specimens with sufficient hydration times showed a significant increase in the number of cycles required for liquefaction compared with clean sand. When the specimens were allowed an extended postconsolidation aging period, the cyclic resistance increased further. Resonant column and cyclic triaxial tests showed that this is a result of the delay in the generation of excess pore pressure in the presence of the bentonite suspension in the pore space. The improvement in cyclic behavior does not occur at the expense of the static resistance of the soil under working loads be...

Classification of Organic Soils

The presence of organics in soils is generally associated with high compressibility, significant secondary compression, often unsatisfactory strength characteristics, and low unit weight. As a result of the above, many state departments of transportation (DOT) in the United States have strict limits on the maximum value of the organic content (2-7%) that can be present in soils to be used as subgrades and backfills. The loss on ignition test is the most widely used technique for measuring organic content. However, especially for low organic content soils, this method can lead to significantly overestimating the true organic content. As a result, certain soils may be incorrectly classified and erroneously considered unviable for certain applications; in other cases unnecessary costly treatments may be requested, even if not required. These are the issues motivating the research presented in this report, which addressed the classification of organic soils and the quantification of organic matter in soils. The research reviewed existing classification systems for organic soils, the effects of organic matter on the geotechnical properties of soils, and the methods for determination of organic content. In addition to the review of the existing literature, this research also included experimental work conducted on natural soils with varying organic content, as well as on laboratory prepared (“artificial”) organic soils. The experiments performed included loss on ignition tests (LOI), Atterberg limits, colorimetric tests, dry combustion tests, thermal analyses, and X-ray diffraction analyses. This work led to proposing a system for classifying organic soils which is based on the percentage of organic matter present: soils with organic content 3% and < 15%, soils are classified as mineral soils with organics; when the organic content exceeds 15% but is <30%, the term organic soil is employed. Finally, soils with organic content higher than 30% are termed highly organic soils or peats. Given the potential errors associated with measuring organic content using the LOI method, this research proposes an approach based on the combined use of the LOI test, the liquid limit test, and the the colorimetric test.