1. Michele Cooke Department of Geosciences 1. Work Budget 2. Boundary Element Method 3. GROW

2. 2 Work min = limit analysis ?  Civil structures Attention to the most efficient mode of failure Efficient = least load at failure = min max load  Geologic structures Is the Earth lazy? Most efficient fault grows… or doesn’t Photo by Mike Gross

3. 3

4. 4 Fault Evolution: San Gorgonio Knot Modified from Matti et al, 1992 Up to ~500 ky Mission Creek Strand 500 ky -> ~120 ky Mill Creek Strand Reactivate San Gorgonio 120 ky -> Present Day San Bernardino Strand Garnet Hill Fault Reactivate Banning

5. 5 Work min = limit analysis ?  Civil structures Attention to the most efficient mode of failure Efficient = least load at failure = min max load  Geologic structures Is the Earth lazy? Most efficient fault grows… or doesn’t Photo by Mike Gross

6. 6 Ways to understand fault growth  Field Evidence: Secondary fractures reveal fault history  Empirical Criterion: Laboratory tests on intact rock  Theory: Linear Elastic Fracture Mechanics Corona fault, San Francisco

7. 7 Ways to understand fault growth  Field Evidence: Secondary fractures reveal fault history  Empirical Criterion: Laboratory tests on intact rock  Theory: Linear Elastic Fracture Mechanics Valley of Fire, NV Myers and Aydin, 2004, JSG Normal faults in Moab, UT

8. 8 Ways to understand fault growth  Field Evidence: Secondary fractures reveal fault history  Empirical Criterion: Laboratory tests on intact rock  Theory: Linear Elastic Fracture Mechanics Measure strength at different confining pressures -> Mohr-Coulomb Criterion  = c +  Image from EP solutions

9. 9 Ways to understand fault growth  Field Evidence: Secondary fractures reveal fault history  Empirical Criterion: Laboratory tests on intact rock  Theory: Linear Elastic Fracture Mechanics Faults grow by coalescence of cracks For faults G c not well-constrained Micromechanics Seismologic Failure when G >= G c

10. 10 How do faults grow and evolve? Is the Earth Lazy? whatever Active faults of southern California ( from Southern California Earthquake Center )  Minimization of work considers the behavior of the entire fault system

11. 11 How does the Earth know that it is lazy?  A ball rolling downhill doesn’t know that it is lazy but still follows the path of least resistance.

12. 12 Evidence of Work Minimization Geometry of spreading centers [Sleep, 1979] and mudcracks reflects work minimization accommodate shrinkage with minimum new fracture surface Faults become more smooth with greater slip Strike-slip traces [e.g. Wesnousky, 1988], extensional fault traces [Gupta et al., 1998], and lab [Scholz, 1990]. Rymer, 2000

13. 13 Applications of Work Minimization: Normal fault arrays Antithetic faults are favored over synthetic faults [Melosh & Williams, 1989] Photo by Marli Miller Antithetic Synthetic

14. 14 Applications of Minimum Work: fabric evolution  Code Elle uses minimization of average local work rate to simulate the evolution of microstructures during deformation and metamorphism [ e.g. Lebensohn et al., 2008, Griera et al, 2011] Griera et al., 2011

15. 15 Applications of Minimum Work: fold and thrust belts  Growth of critical tapered wedges [e.g. Masek and Duncan, 1998], duplexes [Mitra and Boyer, 1986] and folds [Ismat, 2009]  Burbidge and Braun [2002]: use work analysis to explain the accretion-underthrust cycle  Work minimization to predict fault evolution [Maillot & Leroy, 2003; Souloumiac et al., 2008; Cubas et al, 2008] from Dahlen, et al., 1984 From Cubas et al., 2008

16. 16 Mechanical work: Force * Distance  Deformation – stored work ½ stress * strain  Potential Energy weight * distance  Frictional Heat Shear stress * slip  Acoustic/Seismic Energy Shear stress drop * slip  Fracture energy Gibb’s free energy * surface area reversible irreversible Cooke & Murphy, 2004

17. 17 Work Budget: W int + W grav + W fric + W seis + W prop = W ext Cooke & Murphy, 2004 tectonic

18. 18 Work Budget: W int + W grav + W fric + W seis + W prop = W ext deformation Cooke & Murphy, 2004 tectonic

19. 19 Work Budget: W int + W grav + W fric + W seis + W prop = W ext uplift against gravity deformation Cooke & Murphy, 2004 tectonic

20. 20 Work Budget: W int + W grav + W fric + W seis + W prop = W ext uplift against gravity deformation heat Cooke & Murphy, 2004 tectonic

21. 21 Work Budget: W int + W grav + W fric + W seis + W prop = W ext uplift against gravity ground shaking deformation heat Cooke & Murphy, 2004 tectonic

22. 22 Work terms associated with weakening  Seismologists divide as E F, G and E R Cooke & Murphy, 2004Savage & Cooke, 2010

23. 23 Work Budget: W int + W grav + W fric + W seis + W prop = W ext uplift against gravity ground shaking deformation new fault surfaces heat tectonic Lab:10-10 4 J/m 2 ( Wong, 1982, 1986; Cox & Scholz, 1988; Lockner et al., 1992 ). Field: 10 5 -10 6 J/m 2 ( Wilson et al 2005; Pittarello et al, 2008 ). Cooke & Murphy, 2004

24. 24 Fric2D  Two-dimensional Boundary Element Method code Continuum mechanics Discretize boundaries and faults into linear dislocation elements  Crack/fault propagation via addition of elements  Static friction along faults Non-linear behavior requires iterative convergence  Other features not presented here Growth of fault damage (e.g. Savage & Cooke, 2010)

25. 25 Analog models provide direct observation of fault growth from Ask & Morgan, 2010 from Adam et al., 2005 from Cubas et al., 2010

26. 26 New faults grow during accretion a)Accretion: new forethrust b)Underthrusting Wedge thickening c)Accretion: new forethrust

27. 27 Sandbox experiments  Particle Image Velocimetry (PIV) records the development of accreting forethrust with 2.2 cm of contraction Adam et al. 2005 Henry Cadell ~1880

28. 28 Model Set-Up  Boundary Element Method (Fric2d)  Simulate %0.5 cm of contraction  Frictional slip along faults  Medium sand E = 10 MPa;  = 1732 kg/m 3 Forethrust

29. 29 Thrust Sheet Growth  Total work increases during underthrusting  With addition of the forethrust, work decreases  Increased W int is offset by decreased W fric Del Castello and Cooke, 2007

30. 30 Energy of Fault Growth W int + W fric + W grav W prop + W seis Del Castello and Cooke, 2007

31. 31 Location and vergence of most efficient thrust Test a suite of locations and vergence 30˚ dipping forethrusts ahead of the wedge are more efficient than 40˚ dipping backthrusts The preferred location and dip match the sandbox Del Castello and Cooke, 2007

32. 32 Force drop with fault growth observed in sandbox From Cubas et al., 2008 Nieuwland et al, 2001

33. 33 Evolution of force during accretion sandbox experiment from Université de Cergy-Pontoise sandbox experiment at Stanford (Cruz et al, 2010)

34. 34 ½ ΔF Δd = ΔW ΔW = γΔS + W seis + W fric Cost of fault growth 80 mJ/m 2 We can use the observed change in work per unit fault area to predict fault growth Measuring W prop + W seis

35. 35 Calibration Stiff model approximates first 4 cm Soft model matches past 6 cm Basal friction 0.5 static 0.35 dynamic within range of Souloumiac et al. ( 2012, EGU and JSG)

36. 36 Timing of fault growth Work Minimization Analog Experiments Numerical Simulations Conclusions  Hypothesis: The development of faults is more productive at peak loading than prior to peak The addition of a fault to the stiffer sand produces greater change in work than the softer sand. Early compaction of the sand facilitates the development of faults.

37. 37 What does this mean for fault growth? Lazy?  Can we use the energy of fault growth to predict timing of fault development in the sandbox?  How much energy does it take to grow a fault in the crust? Lab:10-10 4 J/m2 (Wong, 1982, 1986; Cox & Scholz, 1988; Lockner et al., 1992). Field: 10 5 -10 6 J/m 2 (Wilson et al 2005; Pittarello et al, 2008). Need more constraints If W prop were negligible then faults would not be long- lived.