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The primary focus of this research is to evaluate the time-dependent crack growth in rocks using lab tests and numerical modeling and its application to geologic hazard problems. This research utilized Coconino sandstone and Columbia granite as the study materials and produced the subcritical crack growth parameters in both mode I and II loadings using the rock materials. The mode I loading test employs three different types of fracture mechanics tests: the Double Torsion (DT), the Wedge Splitting (WS), and the Double Cantilever Beam (DCB) test. Each test measured the mode I crack velocity. The DT test indirectly measured the crack velocity using the load relaxation method. The WS and DCB tests directly measured the crack velocity by monitoring using a video recording. The different mode I subcritical crack growth parameters obtained from the three tests are discussed. For the mode II loading test, this study developed a new shear fracture toughness test called the modified Punch-Through Shear (MPTS). The MPTS test conducted at different loading rates produced the mode II subcritical crack growth parameters. These fracture mechanics tests were calibrated and simulated using the distinct element method (DEM) and the finite element method (FEM). DEM analysis employed the particle flow code (PFC) to simulate the mixed mode crack growth and to match with the failure strength envelop of the triaxial compressive tests. FEM analysis employed the Phase2 program to analyze the crack tip stress distribution and the FRANC2D program to calculate the modes I and II stress intensity factors. The fracture mechanics tests and numerical modeling showed well the dependency of the mode II subcritical crack growth parameters according to confining pressure, loading rate, and the mode II fracture toughness. Finally, the UDEC modeling based on DEM is utilized in this study to forecast the long-term stability of the Coconino rock slope, as one of geologic hazards. The fracture mechanics approach is implemented in the program using the modes I and II subcritical crack growth parameters obtained from the lab tests and numerical modeling. Considering the progressive failure of rock bridges due to subcritical crack growth, the UDEC results predicted the stable condition of the Coconino rock cliff over 10,000 years. This result was validated by comparing it with the previous planar failure case.

In classic GARCH models for financial returns the innovations are usually assumed to be normally distributed. However, it is generally accepted that a non-normal innovation distribution is needed in order to account for the heavier tails often encountered in financial returns. Since the structure of the normal inverse Gaussian (NIG) distribution makes it an attractive alternative innovation distribution for this purpose, we extend the normal GARCH model by assuming that the innovations are NIG-distributed. We use the normal variance mixture interpretation of the NIG distribution to show that a NIG innovation may be interpreted as a normal innovation coupled with a multiplicative random impact factor adjustment of the ordinary GARCH volatility. We relate this new volatility estimate to realised volatility and suggest that the random impact factors are due to a news noise process influencing the underlying returns process. This GARCH model with NIG-distributed innovations leads to more accurate parameter estimates than the normal GARCH model. In order to obtain even more accurate parameter estimates, and since we expect an information gain if we use more data, we further extend the model to cater for high, low and close data, as well as full intraday data, instead of only daily returns. This is achieved by introducing the Brownian inverse Gaussian (BIG) process, which follows naturally from the unit inverse Gaussian distribution and standard Brownian motion. Fitting these models to empirical data, we find that the accuracy of the model fit increases as we move from the models assuming normally distributed innovations and allowing for only daily data to those assuming underlying BIG processes and allowing for full intraday data. However, we do encounter one problematic result, namely that there is empirical evidence of time dependence in the random impact factors. This means that the news noise processes, which we assumed to be independent over time, are indeed time dependent, as can actually be expected. In order to cater for this time dependence, we extend the model still further by allowing for autocorrelation in the random impact factors. The increased complexity that this extension introduces means that we can no longer rely on standard Maximum Likelihood methods, but have to turn to Simulated Maximum Likelihood methods, in conjunction with Efficient Importance Sampling and the Control Variate variance reduction technique, in order to obtain an approximation to the likelihood function and the parameter estimates. We find that this time dependent model assuming an underlying BIG process and catering for full intraday data fits generated data and empirical data very well, as long as enough intraday data is available. / Thesis (Ph.D. (Risk Analysis))--North-West University, Potchefstroom Campus, 2006.

In classic GARCH models for financial returns the innovations are usually assumed to be normally distributed. However, it is generally accepted that a non-normal innovation distribution is needed in order to account for the heavier tails often encountered in financial returns. Since the structure of the normal inverse Gaussian (NIG) distribution makes it an attractive alternative innovation distribution for this purpose, we extend the normal GARCH model by assuming that the innovations are NIG-distributed. We use the normal variance mixture interpretation of the NIG distribution to show that a NIG innovation may be interpreted as a normal innovation coupled with a multiplicative random impact factor adjustment of the ordinary GARCH volatility. We relate this new volatility estimate to realised volatility and suggest that the random impact factors are due to a news noise process influencing the underlying returns process. This GARCH model with NIG-distributed innovations leads to more accurate parameter estimates than the normal GARCH model. In order to obtain even more accurate parameter estimates, and since we expect an information gain if we use more data, we further extend the model to cater for high, low and close data, as well as full intraday data, instead of only daily returns. This is achieved by introducing the Brownian inverse Gaussian (BIG) process, which follows naturally from the unit inverse Gaussian distribution and standard Brownian motion. Fitting these models to empirical data, we find that the accuracy of the model fit increases as we move from the models assuming normally distributed innovations and allowing for only daily data to those assuming underlying BIG processes and allowing for full intraday data. However, we do encounter one problematic result, namely that there is empirical evidence of time dependence in the random impact factors. This means that the news noise processes, which we assumed to be independent over time, are indeed time dependent, as can actually be expected. In order to cater for this time dependence, we extend the model still further by allowing for autocorrelation in the random impact factors. The increased complexity that this extension introduces means that we can no longer rely on standard Maximum Likelihood methods, but have to turn to Simulated Maximum Likelihood methods, in conjunction with Efficient Importance Sampling and the Control Variate variance reduction technique, in order to obtain an approximation to the likelihood function and the parameter estimates. We find that this time dependent model assuming an underlying BIG process and catering for full intraday data fits generated data and empirical data very well, as long as enough intraday data is available. / Thesis (Ph.D. (Risk Analysis))--North-West University, Potchefstroom Campus, 2006.

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