1. Animations to illustrate the effect of uncertainty in crack orientation View these slides in SlideShow mode of PowerPoint (not OpenOffice). Press Shift-F5 to get into slideshow mode. Copyright Rebecca.Brannon@utah.edu

2. Single crack Probability of failure radius of Mohr’s circle TXE TXC Stress can extend BEYOND the failure line because... The sample won’t fail unless a crack exists in a “dangerous” orientation. PROBLEM: Crack orientations are unknown. SOLUTION: treat this problem probabilistically. TXE TXC A simple model: Crack grows when , minus shear carried by friction (-  n ), reaches a critical value. This is a straight line in Mohr diagram: Let’s compare Triaxial extension (TXE) with Triaxial compression (TXC) at the same pressure and equivalent shear. Probability of failure = probability the crack normal lies in the “danger zone” on the sphere = area fraction of danger zones Crack orientation is a unit normal. Unit normals are points on the unit sphere. Copyright Rebecca.Brannon@utah.edu

3. Failure probability (multiple cracks) Let g(a) = probability that a single crack of size “a” is safe. Upper bound shear stress 100 exponentially distributed cracks Single crack shear stress Implicitly a function of stress, X(  ). MEASURE IT (easily might not be Weibull). Copyright Rebecca.Brannon@utah.edu

4. Accounting for uncertainty in crack orientation is only qualitatively similar to Weibull theory Weibull theory is simple and consistent with our lower bound analysis. BrnnonCopyright Rebecca.Brannon@utah.edu

5. Does Weibull theory give the correct sample size dependence? smaller sample larger sample Baseline Copyright Rebecca.Brannon@utah.edu

6. SIMULATED AND MEASURED SPALL DATA SIMULATED AND MEASURED BRAZILIAN DATA Deviation of measured data is comparable to finite sampling errors for simulated data that are exactly Weibull distributed. measured best fit simulated measured best fit simulated Copyright Rebecca.Brannon@utah.edu

7. EVIDENCE OF SIZE EFFECTS Sample-to-Sample variability in strength at large scales. SIGNIFICANT size effects – not negligible. Copyright Rebecca.Brannon@utah.edu

8. specimen large specimen small pressure (GPa) strength (GPa) Initial state: Small elements are stronger on average, but also more variable. conventional damage model same model with uncertainty and size effects Spatial Variability & Length Scale Effects small elements LARGE elements Experiment Copyright Rebecca.Brannon@utah.edu