Fracture Overview

Fall 2011

Figure 6.1 Rectangular plate with hole subjected to axial load. (a) Plate with cross-sectional plane; (b) one-half of plate with stress distribution; (c) plate with elliptical hole subjected to axial load.

Rectangular Plate with Hole

• Thermal Stress:

• Stress = E.. T

Flaws Make the World Beautiful!

Were it not for the flaws, rocks and mountains would have been perfectly boring

Griffith (1920)

• Griffith (1920), realized the significance of microcracks 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

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

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

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

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

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

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

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., dWR<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 (dUs)

Griffith Energy-Balance Concept• As the crack propagates, the rock undergoes a change in strain

energy (dUE).

• The total change in energy for crack propagation is:

UT = Us - WR + UE

• Griffith energy balance concept:

–Propagation occurs without changing the total energy of the rock-crack system. – i.e., for an increment of crack extension (dc),

d UT /dc = 0

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