About fracture mechanics
Template:Cleanup-date Fracture mechanics is a method for predicting failure of a structure containing a crack. It uses methods of analytical Solid Mechanics to calculate the driving force on a crack and those of experimental Solid Mechanics to characterize the material's resistance to fracture.
In modern Materials Science, Fracture Mechanics is an important tool in improving the mechanical performance of materials and components. It applies the physics of stress and strain, in particular the theories of elasticity and plasticity, to the microscopic crystallographic defects found in real materials in order to predict the macroscopic mechanical failure of bodies.
An excellent introduction to Fracture Mechanics is: Adrian Demaid Fail Safe Open University (2004).
The Need for Fracture Mechanics
Engineering structures often contain cracks - arising either during production or during serevice (e.g. from Fatigue (material)). These cracks can lower the strength of the structure beyond that due to loss of load-bearing area. As a consequence, a material propety, above and beyond conventional strength, is needed to describe the fracture resistance of engineering materials. this is the reason for the need for fracture mechanics - the evaluation of the strength of cracked structures.
Fracture Mechanics
Fracture Mechanics was invented during World War I by English aeronautical engineer, A.A.Griffith, to explain the failure of brittle materials. Griffith was faced with the problem that theoretical calculations showed that the stress at the tip of a sharp crack approaches infinity. Accordingly, any structure containing a crack should fail, no matter how small the crack or how light the load. To solve this dilemma, Griffith developed a thermodynamic approach. He assumed that growth of a crack requires creation of surface energy, which is supplied by the loss of strain energy accompanying the relaxation of local stresses as the crack advances. Failure occurs when the loss of strain energy is sufficient to provide the increase in surface energy. For the simple case of a rectangular plate with a crack perpendicular to the load Griffith’s theory becomes:
- <math>G_c = \frac{ \pi \sigma_f ^2 a}{ E }\,</math> (1)
where Gc the critical strain energy release rate (also fracture energy), σf is the brittle fracture stress, a is half the crack length, and E is the Young’s modulus.
Griffith’s work was ignored for over twenty years until a group under G. R. Irwin at the U.S. Naval Research Laboratory took it up during World War II. Their problem was that naval materials, e.g. ship-plate steel, are not perfectly elastic but undergo plastic deformation at the tip of a crack. Eventually a modification of Griffith’s theory emerged from this work; a term called stress intensity replaced strain energy release rate and a term called fracture toughness replaced surface energy. Both of these terms are simply related to the energy terms that Griffith used:
- <math>K_I = \sigma \sqrt{ \pi a }\,</math> (2)
and:
[EGc/(1 - ν2)]0.5 = Kc (3)
where K is the stress intensity, Kc the fracture toughness, and ν is Poisson’s ratio. Fracture occurs when K = Kc. Note that the expression for K in Eq. (2) will be different for geometries other than the center cracked plate, as discussed in the article on Stress Intensity. For the special case of plane strain deformation, Kc becomes KIc and is considered a material property.
Arising from the manufacturing process, interior and surface flaws are found in all metal structures. Not all such flaws are unstable under service conditions. Fracture mechanics is the analysis of flaws to discover those that are safe (that is, do not grow) and those that are liable to propagate as cracks and so cause failure of the flawed structure.