About fracture mechanics

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. 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 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 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 G_{c_{the critical strain energy release rate (also fracture energy), σ_{f_{is the brittle fracture stress, a is half the crack length, and E is the Young’s modulus. This fracture energy can otherwise be understood as: the rate of strain energy release by growth of the crack.}}}}
If G ≥ G_{c_{, 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> (for plane stress) (4)
<math>K_c = \sqrt{\frac{E G_c}{1 - \nu^2}}\,</math> (for plane strain) (5)

where K_{I_{is the stress intensity, K_{c_{the fracture toughness, and ν is Poisson’s ratio. Fracture occurs when K_{I_{≥ K_{c_{. For the special case of plane strain deformation, K_{c_{becomes K_{Ic_{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. It refers to loading via Mode I - the most common form of loading:}}}}}}}}}}}}

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 K_{I_{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)

for a sheet of finite width W containing a through-thickness edge-crack of length a.

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 failure of the flawed structure.

References
C.P.Buckley, "Material Failure (notes)" (2005), University of Oxford

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