1. Griffith Cracks Flaws Make the World Beautiful! Were it not for the flaws, rocks and mountains would have been perfectly boring

2. Griffith Cracks Rocks have mechanical inhomogeneities/discontinuities e.g.: Flaws, fossils, inclusions, cavities, grain boundaries, and microcracks These inhomogeneities have different elastic properties compared to the surrounding rock Their presence perturbs the otherwise homogeneous, mechanically- or thermally-induced remote stress field This leads to an inhomogeneous stress field that initiates joints when the concentrated local tensile stress exceeds the tensile strength of the rock

3. Micro-Flaws Micro-flaws are the main factor in structural failure in man-made structures (e.g., ship, bridge, dam), by producing stress concentration Facts: 1. Failures in high strength material commonly occur under low stress 2. Brittle solid materials are much weaker (i.e., have lower fracture strength) under tension than under compression

4. Micro-Flaws - Facts… 3. The fracture strength, which is an inherent property for an ideal, continuous brittle solid, and represents the critical stress needed to fracture, is not highly reproducible Testing methods, dimensions of test specimens, environmental conditions, and intrinsic structural characteristics are but a few factors influencing the variation of the fracture strength of brittle solid material

5. Strength Resistance of a rock to fracture Is a critical value of stress at which fracture occurs and rock fails The theoretical tensile strength, which is the stress needed to break atomic bonds of an ideal brittle material, is about one tenth of its Young's modulus (E/10). Recall that:  = E e The Young's modulus, E, for most rocks is commonly of the order of 10 5 or 10 6 bars, implying great strengths for these rocks (i.e., 10 4 or 10 5 bar or 10-100 kbar!)

6. Strength However, for real brittle material, the measured tensile strength is 1 to 2 orders of magnitude less than the theoretical tensile strength (i.e., E/1000-E/100) i.e., rock strength is in the order of: 10 2 -10 3 bars This indicates that the fracture strength in such solids is not an intrinsic material property The discrepancy between the molecular cohesive forces and the observed tensile strength of real material solids has been attributed by Griffith (1920) to the presence of planar defects or cracks that are since referred to as the Griffith cracks

7. Micro-Flaws … Structural Definition of Rock: Polycrystalline aggregate that commonly has a random population of mostly inhomogeneous and anisotropic pre-existing or mechanically-induced micro-flaws These flaws, that include micro-cracks, grain boundaries, and pores, control the mechanical behavior of imperfect rocks

8. Micro-Flaws … Problems dealing with crack initiation are concerned with how and where cracks start, whereas those dealing with propagation study the path that the cracks take, and the extent to which they grow Fracture mechanics established a relationship between fracture strength and micro-crack geometry and fracture toughness

9. Inglis (1913) Recognized the destructive influence of cracks in brittle material Determined stresses around an elliptical stress-free hole and an extreme case of a fine straight crack He examined a brittle, homogeneous, isotropic, plate under tension using a mathematical approach

10. Inglis (1913) Showed that a pull applied to the ends of an elastic plate would produce tensile stresses at the tip of a crack, that may exceed the elastic limit of the material and lead to the propagation of the crack Showed that the increase in the length of the crack exaggerates the stress even more, such that the crack would continue to spread

11. Experiment Assume an elliptical crack with semi-major, c, and semi- minor axis, b, with an aspect ratio of c/b Load the crack with a far remote tensile stress within the plate (  r ), normal to c The local tensile stress perpendicular to the c axis is magnified several times to  C ( stress concentration ) Inglis showed that stress,  C, at the tip of the crack, varies with the length and radius of curvature (  b 2 /c ) at the apices of the crack, and is proportional to the square root of length (c), and inversely proportional to the  of the crack

13. Experiment… The highest tensile stress at the end of crack is  C =  r (1+ 2 c/b) The stress concentration is maximum at the crack tip (where  is minimum), and rapidly decreases within a short distance from the crack tip  C is approximated by:  C = 2  r (c/  )1/2 Note:  C depends on shape (i.e., aspect ratio) and not on the size of the elliptical cavity When b= 0 (i.e.,  = b 2 /c = 0), the stress concentration at the crack tip is infinity

14. Stress Concentration Factor The minimum value of the radius of curvature at the end of major axis of the elliptical cavity, , is at point C and is given by:  = b2/c In Fig. 1.1, note that  is zero at tip of sharp crack, where stress concentration is maximum, and is given by:  C =  r (1+2c/b)=  r [(1+2(c/  )1/2] The ratio of stress concentration,  C at point c, to the applied stress,  A, is called the elastic stress concentration factor, which for a thin and long ellipse (b < c) is given by:  C /  r ~ 2c/b = 2(c/  )1/2

15. Griffith (1920) Griffith (1920), realized the significance of micro-cracks in reducing the fracture strength Applied the mathematical work of Inglis (1913) for an elliptical hole, and developed a theoretical criterion of rupture based on the concept of minimum potential energy of classical mechanics and thermodynamics which seeks a minimum total free energy of a system

16. Griffith Theory In the Griffith theory, the theoretical strength is the microscopic fracture stress which is actually reached in a very small volume of the rock while the mean stress may remain very low Griffith's work, which has since been known as the Griffith energy balance approach, and has served as a foundation for fracture mechanics, deals with the equilibrium state of an elastic, solid body, deformed by specified surface forces

17. Griffith (1920) … Griffith extended the theorem of minimum energy by accounting for the increase of surface energy which occurs during formation of cracks He assumed that the equilibrium position is one in which rupture of the solid occurs if the system is allowed to pass from an unbroken to a broken state through a process involving continuous reduction of potential energy

18. Griffith (1920) … Griffith (1920) argued that brittle solids fail by incremental propagation of a multitude of randomly-oriented, small pre-existing cracks Griffith cracks are common in rocks and include intragranular and intergranular microcracks (grain boundaries) and larger transgranular cracks In a larger scale, the Griffith flaws include joints, faults, and bedding planes

19. Fracture Strength Brittle fracture strength depends largely on the elastic properties of the elastic rock and the length and sharpness of the flaws Stress concentrators such as low aspect ratio cavities, inclusions, material property mismatches, and fossils, give rise to tensile stresses that may fracture rocks even when applied stresses are compressive provided they are non-hydrostatic

20. Griffith (1920) The intensification of stress depends on the: Length and orientation of the crack with respect to the applied stress Radius of curvature at their tips, rendering certain cracks more vulnerable than others The propagation of these Griffith cracks involves the creation of two new incremental crack surfaces which is a process that absorbs energy

21. Griffith (1920) … The creation of these new surfaces in the interior of a solid (by crack propagation) leads to an increase in potential energy as work must be done against the cohesive forces of the molecules on either side of the crack The work is part of the potential surface energy of the system. Thus bounding surfaces posses a surface tension and a corresponding amount of potential energy

22. Griffith Energy-Balance Concept If we subject the outer boundary of a rock to tension (such that boundary moves out) This decreases the potential energy (i.e., dW R <0), of the loading device (Fig. 3.2 Engelder). R designates rock The work to propagate the crack is positive, and is defined as an increase in surface energy (dU s )

23. Griffith Energy-Balance Concept As the crack propagates, the rock undergoes a change in strain energy (dU E ). The total change in energy for crack propagation is:  U T =  U s -  W R +  U E Griffith energy balance concept: Propagation occurs without changing the total energy of the rock-crack system. i.e., for an increment of crack extension (d c ), d U T /d c = 0

24. Griffith Energy-Balance Concept … This means that the mechanical and surface energy terms within the rock-crack system must balance The motion of the crack walls represents a decrease in mechanical energy While work (as surface energy) must be done to remove the restraints across crack increment