'''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 (physics)|stress]] and [[strain]], in particular the theories of [[elasticity]] and [[plasticity (physics)|plasticity]], to the microscopic [[crystallographic defect]]s 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. In many cases, failure of engineering structures through fracture can be fatal. For instance, failure of pressure vessels due to crack propagation could cause an fatal explosion. If failure were ever to happen, we would rather it were by [[yield (engineering)|yield]] or by leak before break. As a consequence, a material property, 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.

==Griffith's Crack Theory: Stain Energy Release Rate, G ==

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 thin rectangular plate with a crack perpendicular to the load Griffith’s theory becomes:

:<math>G = \frac{\pi \sigma^2 a}{E}\,</math>                 (1)

where G is the strain energy release rate, σ is the applied stress, a is half the crack length, and E is the [[Elastic modulus|Young’s modulus]]. The strain energy release rate can otherwise be understood as: the rate at which energy is absorbed by growth of the crack.

However, we also have that:

:<math>G_c = \frac{\pi \sigma_f^2 a}{E}\,</math>                 (2)

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 [[Elastic modulus|Young’s modulus]]. This [[fracture energy]] can otherwise be understood as: the rate of strain energy release by growth of the crack.

If G ≥ Gc, this is the criterion for which the crack will begin to propagate.

== Griffith's Crack Theory: Stress Intensity Factor, K ==

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>                 (3)

and

:<math>K_c = \sqrt{E G_c}\,</math> (Plane stress)                 (4)
:<math>K_c = \sqrt{\frac{E G_c}{1 - \nu^2}}\,</math> (Plane strain)                 (5)

where KI is the [[stress intensity]], Kc the [[fracture toughness]], and ν is [[Poisson ratio|Poisson’s ratio]]. Fracture occurs when KI ≥ Kc. For the special case of plane strain deformation, Kc becomes KIc and is considered a material property. The subscript I arises because of the different ways of loading a material to enable a crack to propagate:

There are three ways of applying a force to enable a crack to propagate: 
Mode I: Opening mode (a tensile stress normal to the plane of the crack) 
Mode II: Sliding mode (a shear stress acting parallel to the plane of the crack and perpendicular to the crack front) 
Mode III: Tearing mode (a shear stress acting parallel to the plane of the crack and parallel to the crack front)

Note that the expression for KI in Eq (4) and (5) will be different for geometries other than the center cracked plate, as discussed in the article on [[stress intensity]]. To get around this problem, it is possible to introduce a non-dimensional unit Y, thus giving:

:<math>K_I = Y \sigma \sqrt{\pi a}\,</math>                 (6)

where Y is given by:

:<math>Y = \sqrt{sec(\frac{\pi a}{W})}\,</math>                 (7)

for a sheet of finite width W containing a through-thickness crack of length 2a or:

:<math>Y = 1.12 - \frac{0.41}{\sqrt{\pi}}\frac{a}{W} + \frac{18.7}{\sqrt{\pi}}(\frac{a}{W})^2\,</math>                 (8)

== Short Summary ==
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 [[structural failure|failure]] of the flawed structure.

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