1. Brittle Creep: How it works and its role in fracture 02/05/2014 Stephen Perry

2. Confining pressure :  c =  3 =  2. Differential stress :   =  1 -  3 The axial stress generally exceeds the confining pressure, the axial pressure =  and the confining pressure =  =  3; System of stress in a conventional ‘triaxial’ test Sample before and after triaxial test

3. Normal Failure due to Coalescense of Cracks Stage IStage IIIStage IV From Scholz, 2002 -Most tests relate to a stress failure criterion where the sample is loaded with increasing stress -Failure occurs when become large enough to move past the failure criterion

4. Static Load Experimental Setup - Static load experiments show that samples can fail below their short term failure strength – this is called static fatigue (Brantut, in press)

5. Measurements in Real Rocks (Lockner) -16 constant strain rate experiments conducted on room-dry samples of Westerly granite -Axial shortening rates of 1 um/s and 0.01 um/s were used (0.01 um/s shown on right) -Failure criterion do slightly depend on loading rate -> shows some time dependence in the system (Lockner, 1998)

6. - 3 characteristic creep phases are shown: -1. primary/transient creep (strain rate α 1/t) -2. constant strain rate -3. accelerating creep culminating in failure - The sample fails by Static Fatigue after sub-critical crack growth - Rock strength is time dependent (Lockner, 1998) Creep Experiment Simulation

7. Mechanism for Brittle Creep? Propagation of sub- critical cracks in a statically loaded sample (below critical stress intensity factor) Cracks propagate through stress corrosion – chemical bonds are broken over time at the crack tips (Brantut, 2012)

8. 3 Phases of Brittle Creep Primary – formation of microcracks Secondary – cracks don’t interact; strain levels off Tertiary – interaction between cracks causes increase in strain and failure

9. Formulation of peak strength vs. strain rate Stress logarithmically depends on strain rate Create a curve based on loading phase, beginning of failure, and an inferred complete failure curve (dotted line) Differential Stress Peak Strength (Lockner, 1998) Inelastic strain Rate Characteristic Peak Strain Rate

10. Brantut Brittle Deformation Experiments Constant strain rate and constant stress triaxial deformation tests on 3 types of sandstone (Brantut, in press)

11. Axial Strain becomes the important factor Measured evolution of P wave speeds is similar for 2 different types of tests even with differences in strain rate of several orders of magnitude Thus wave speed is dependent on axial inelastic strain and not necessarily the process Link between microstructural state and inelastic axial strain (Brantut, in press)

12. Brantut Axial Strain Rate Relation (Brantut, in press) Differential Stress Characteristic “activation” stress Instantaneous Brittle Creep Rate Reference strain rate (applied during constant strain rate test)

13. Rate and State (General): Axial Strain Rate Relation: Brantut’s Formulations vs. R+S Friction

14. Lockner’s Formulation vs. R+S friction Rate and State: Peak Strength vs. Strain Rate -Both have the same form except there is no evolution term in the intact rock relation -Possible to recover (1) by integrating (2) over a population of contacts? (1) (2)

15. A Tale of Two State Variables Lockner – absense of an explicit state variable – This is ok because state variable relates to asperity contact area evolution which doesn’t have an analog in bulk material Brantut – state variable relates to inelastic axial strain – Inelastic axial strain corresponds to the density of cracks which is related to the stiffness of the material

16. Relation to Other Mechanisms – Pressure Solution Creep - Brittle Creep is primarily a high stress / low temperature effect

17. Additional Factors and Conclusions Moisture content in samples can affect propagation rate of microcracks – it is somewhat a chemical process R+S friction could be brittle creep on the asperity level - both experiments see a rate dependency State variable is less clear, but may be a consequence of R+S being on a planar surface

19. How to cracks normally propagate? Crack has very high stress near the tip (near infinite) The bigger the crack gets the faster it propagates (it accelerates up to Rayleigh wave speed) Talk about Stress Intensity factor? Put in diagram from Nadia’s class

20. Consequences of Brittle Creep -Collapse in mining structures years after they are built -Potential cause of roof collapse in the “Big Dig” in Boston in 2006

21. Static Fatigue Simulations using data from experiments -3 characteristic creep phases are shown: -1. primary/transient creep (ε. α 1/t) -2. constant strain rate -3. accelerating creep culminating in failure -Can be used to compute failure time for granite loaded to some fraction of its short term failure strength