Fracture Mechanics in Rock Breaking

1. Fracture Mechanics in Rock Breaking John Kemeny

2. What is Fracture Mechanics? Traditionally, material failure based on the concept of strength (i.e., stress>strength) Cannot take into account presence of flaws or propagation of single fractures => airplane wing! In the 20s, Griffith and later Irwin pioneered energy concept for crack growth LEFM - linear elastic fracture mechanics Applications in rocks => blasting, mechanical excavation, drilling, earthquake rupture, etc.

3. Some General Concepts Crack growth from pre-existing flaws (microcracks, pores, grains, etc.) Crack growth in the direction of the maximum compressive principal stress (perpendicular to maximum tensile stress)

4. Crack Growth in Rock Breaking Blasting => internal pressure creates compressive radial stress and tensile hoop stress

5. Crack Growth in Rock Breaking Drilling and mechanical excavation => indenter creates chips which allows tool to advance

6. Crack Growth in Rock Breaking Drilling and mechanical excavation => indenter creates chips which allows tool to advance

7. Crack Growth in Rock Breaking Caving => compression and shearing due to the draw results in crack growth

8. Stress Intensity Factors 3 Modes (KI, KII, KIII), we will only consider Mode I today Actually rock tends to fracture in Mode I no matter how you load it

9. Stress Intensity Factors KI relates the stress field around a crack tip to far field boundary conditions Stress becomes infinite at the crack tip! Square root singularity KI has the unit of stress vdistance, and it remains finite

10. Stress Intensity Factors Stress intensity factor solution for a single crack under tension Analytic, numerical solutions for KI

11. Crack Growth Criteria Clever idea => use stress intensity factor as criteria for crack growth KI = KIC => crack growth KI < KIC => no crack growth KIC is the fracture toughness, a material property that can be measured in the lab Indiana limestone => KIC = 0.99 MPavm Tennessee sandstone => KIC = 0.54 MPavm

12. Crack Growth Example 1 Estimate the tensile strength of Indiana limestone Based on grain size (why?), assume the largest cracks in the limestone are about 4 mm (why the largest?) The crack keeps growing across the sample, because KI keeps increasing with increasing crack length

13. Crack Growth Example 2 Use the �star crack� KI solution to estimate how long blasting cracks will grow and the optimal number of cracks around the hole

14. Crack Growth Example 2 For R=1.5 inches, P=100 MPa, E=50 GPa, KIC=1.25MPavm, set KI=KIC and solve for a to get stable crack length For n=2 => a=11.5 meters For n=3 => a=9.1 meters For n=4 => a=7.0 meters For n=5 => a=5.6 meters For n=6 => a=4.7 meters

15. Crack Growth and Energy It takes energy to create crack surface area Crack growth will occur when a critical energy is supplied G = GC => crack growth G is the energy release rate, ?Ue/?a GC is the critical value of G, a material property Energy required = GC * surface area

16. Crack Growth and Energy G and KI are related by the relationship: From KI solution, can calculate energy to create cracks Optimal configuration is the one that maximizes the strain energy (uses up as much strain energy as possible as the crack grows)

17. Crack Growth and Energy Back to the star crack example, can integrate to get strain energy for n=1, n=2, etc. : For n=2 => Ue=3842 Joules For n=3 => Ue=4404 Joules For n=4 => Ue=4291 Joules For n=5 => Ue=4136 Joules => optimal configuration for n=3! (depends on properties)

21. Energy Example Surface area => 48 m2/m3 Assume: 80% of fractures pre-existing, 18% tensile, 2% compressive Gc values: Pre-existing fractures: 10 J/m2 New tensile fractures: 100 J/m2 New compressive fractures: 5000 J/m2 Energy per m3: 6,050 J/m3 Energy supplied:1.5 MJ/m3 (1,500,000) Difference of 250. Where does the rest of the energy go?

22. Things To Think About Create as many microcracks in blasting as possible, it helps in crushing and grinding Grinding takes the more energy than blasting or crushing, why? Other ways to create cracks => thermal, laser, water, etc.

23. Energy Problems Electricity Environment => global warming, acid rainfall, nuclear waste Supply and demand => Calif. (and US) crisis Transportation Traffic congestion Safety Supply and demand => gas prices

24. Energy solutions Electricity Alternatives to fossil fuels Wind, solar, PV cells Conservation, efficiency Transportation Conservation, efficiency => incentives Alternatives to the auto Alternative fuels for autos => hydrogen

25. Basic sciences in energy Thermodynamics Conversion losses Energy units Heat transfer Electricity and Magnetism Relationship between E and M Generating electricity

26. The Scientific Method Careful observation Formulation of rules of empirical laws Develop a hypothesis Test the hypothesis

27. Scientific Method Example Observation => low pH in lakes More observations => associated with acid rainfall, areas where no soil buffering

28. Scientific Method Example Empirical law => always occurs in areas downwind of power plants that burn fossil fuels, or a major city Hypothesis => SO2 emissions from power plant and/or Nox (power plants and smog) main cause of acid lakes

29. Scientific Method Example Test => reduce SO2, NOx emissions

30. Other Similar Examples Grand Canyon haze Global warming High cancer rates in certain neighborhoods in certain cities

31. For Wednesday Read first chapter in book Print out first activity and bring to class