Modeling of Brittle Failure and Comparisons

A numerical multiple crack interaction model was developed to simulate the failure process in brittle solids containing significant populations of flaws. The model, which is 2D, allows for the growth of microcracks on a regular array of potential crack sites. Individual cracks may be oriented vertically, horizontally or at 450 to the sample axes. Quasi-static equilibrium equations are expressed in terms of finite difference approximations, and are solved by applying a Renormalization Group Theory approach. More than 5,000 potential crack sites are included in the current version of the model. We have successfully duplicated a variety of brittle fracture phenomena observed in laboratory rock mechanics studies by employing a limited number of parameters and relations in the model. Included in the model are 1) Lam6 constants X, 9t for intact matrix material 2) coefficient of friction, f, for friction on cracks 3) an algorithm to allow for crack closure in response to normal stress 4) an initial crack population and 5) for time-dependent modeling, a power law sub-critical crack growth rule. A fracture mechanics approach is used to determine crack growth. Approximate stress intensity factors are computed for all cracks and when critical values are exceeded, cracks are allowed to grow in either mode I (tension) or mode II (in-plane shear). Simulations are performed by specifying a combination of stress and strain boundary conditions. The model is capable of duplicating experimentally observed features such as elastic moduli, dilatancy, acoustic velocities, peak strength, Mohr-Coulomb failure envelope and, to a limited degree, crack coalescence. A number of supplemental laboratory experiments have also been performed, focusing primarily on low-temperature creep. In one set of experiments, cylindrical samples of granite were deformed at 260 C, constant confining pressure (600 bars) and constant pore pressure (200 bars). Axial and volumetric strain were determined from changes in the output of resistance foil strain gauges bonded to the rock surface. In addition, DC electrical resistivity was measured parallel to the sample axis. During these experiments (typically lasting from one to two weeks), the deviatoric stress 0 d applied to the sample was cycled between 70% and 90% of the short-term failure strength. The majority of the experiments were conducted in the secondary or 'steady-state' creep regime. Inelastic volumetric strain rate was fo nd to obey the law logl0(iv) = A + B od + C log0(Ev) where B = 7.4 ± 0.2 kb1 and C -4. The C-coefficient represents a strain-hardening-like term. The stress-dependence is of the same form as the stress-dependence measured for mode I crack growth in double cantilever beam experiments. The observed creep behavior is analyzed in terms of stress corrosion and crack growth models. A second set of experiments was designed to measure the development of electrical resistivity anisotropy during deformation of granite. In this series, two triaxial deformation experiments were conducted on brine-saturated (0.1 and 0.01 Maq KC1) Westerly granite at effective pressures of 100 and 400 bars. Deformation histories included constant strain rate, constant stress and stress relaxation sequences. Complex resistivity, over the frequency range from 10-3 to 105 Hz, was measured in both axial and transverse directions during the experiments. Low-frequency-limit resistivity pDC increased with initial loading and then decreased steadily after the onset of dilatancy. The initial increase in pDC was greatest in the transverse direction, in one experiment reaching a peak contrast PDCI/PDCI = 1.4 at approximately 35% peak strength. With continued loading and the resultant opening of microcracks, the resistivity contrast decreased to approximately 0.8 by failure. In terms of the frequency dependence of resistivity, the general form of the real part of the resitivity varied little during deformation other than a uniform change in magnitude proportional to the changes in PDCUnder initial hydrostatic stress, the phase angle between current and voltage ranged from 10 to 30 mrad over the frequency range 0.001 to 100 Hz. With the onset of dilatancy, notable changes occured in the phase spectra in the 0.001 to 10 Hz region. The changes differed in the axial and transverse directions and appeared to be the result of changing pore structure since they diminished upon removal of deviatoric stress. Although the variations in complex resistivity were subtle, their occurance in the Earth may be exploited as a means of identifying secular changes in stress or strain through induced polarization or magnetotelluric measurements. Finally, a series of experiments was conducted to measure changes in electrical conductivity during densification of water-saturated quartz powders at elevated temperatures. Starting material was ultra-fine quartz powder (5-10 pm-diameter). Confining pressure ranged from 2,000 to 3,700 bars and pore pressure from 300 to 2,000 bars. All runs were conducted at 7000C and were saturated with distilled water. Initial porosity in all experiments was in excess of 40%. Experiments lasted from 10 hr to 8 days, with ending porosities from 19% to as little as 8 + 1%. In all experiments, initial volumetric compaction rates were rapid (10-5 to 10-6 s-1), decreasing quickly to rates in the range 10-7 to 10-8 s-1 after approximately 1 day. Electrical conductivity as well as porosity decreased monotonically during the experiments. Conductivity ranged from 10-2 to 10 -4 mho/m. A model is presented in which decrease in conductivity is initially controlled by the loss of fluid filled pore volume, followed by a transition, at approximately 15% porosity, to a condition in which conductivity in controlled by constrictions in interconnecting channels. Thesis Supervisor: Dr. Theodore R. Madden Title: Professor of Geophysics Co-supervisor: Dr. Brian Evans Title: Professor of Geophysics BIOGRAPHY 1968 1973: Attended University of Rochester, N.Y., received A.B. and M.S. in Geology. 1973 1974: Attended Stanford University, Calif. (no degree). 1974 present: Research scientist in Office of Earthquake Studies, U.S. Geological Survey, Menlo Park, California. Publications: Bonham-Carter, G., Thomas, J., and Lockner, D. A., 1973, A numerical model of steady wind-driven currents in Lake Ontario and the Rochester Embayment based on shallow-lake theory, Internat. Field Year for the Great Lakes, Rochester Embayment Project, Dept. of Geol. Sci., Univ. of Rochester, New York. Lockner, D. A., 1973, Sensitivity of a numerical circulation model for Lake Ontario to changes in lake symmetry and friction depth, and to variable wind stress, Rochester Embayment Project, Dept. of Geol. Sci., Univ. of Rochester, New York. Byerlee, J. D. and Lockner, D. A., 1977, Acoustic emission during fluid injection in rock, Proceedings, First Conference on Acoustic Emission/Microseismic Activity in Geological Structures and Materials, Trans Tech Publications, Clausthal-Zellerfeld, W. Germany, p. 87-98. Lockner, D. A. and Byerlee, J. D., 1977, Acoustic emission and fault formation in rocks, Proceedings, First Conference on Acoustic Emission/Microseismic Activity in Geological Structures and Materials, Trans Tech Publications, Clausthal-Zellerfeld, W. Germany, p. 99-107. Lockner, D. A., Walsh, J. B., and Byerlee, J. D., 1977, Changes in seismic velocity and attenuation during deformation of granite, J. Geophys. Res., v. 82, n. 33, p. 5374-5378. Solberg, P. H., Lockner, D. A., and Byerlee, J. D., 1977, Shear and tension hydraulic fractures in low permeability rocks, Pure and Applied Geophys., v. 115, p. 191-198. Lockner, D. A. and Byerlee, J. D., 1977, Acoustic emission and creep in rock at high confining pressure and differential stress, Bull. of the Seismological Society of Amer., v. 67, no. 2, p. 247-258. Lockner, D. A. and Byerlee, J. D., 1977, Hydrofracture in Weber sandstone at high confining pressure and differential stress, J. Geophys. Res., v. 82, no. 14, p. 2018-2026. Lockner, D. A. and Byerlee, J. D., 1978, Velocity anomalies: an alternative explanation based on data from laboratory experiments, Pure and Applied Geophys., v. 116, p. 765-772. Weeks, J. D., Lockner, D. A., and Byerlee, J. D., 1978, Changes in b-value during movement on cut surfaces in granite, Bull. of the Seismological Society of Amer., v. 68, no. 2, p. 333-341. Lindh, A. G., Lockner, D. A., and Lee, W. H. K., 1978,Velocity anomalies: an alternative explanation, Bull. of the Seismological Society of Amer., v. 68, no. 3, p. 721-734. Solberg, P., Lockner, D. A., Summers, R., Weeks, J., and Byerlee, J. D., 1978, Experimental fault creep under constant differential stress and high confining pressure, 19th U. S. Symposium of Rock Mechanics, p. 118-120. i-~.-~~ -~----~ ^--"--~'-"'---^~"-~*~