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

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Railway Bridge Collapse (1887)

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: Strain Energy Release Rate

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, as shown in the Appendix to this article.

Irwin's Modified Griffith Crack Theory: Fracture Toughness, Ksub_Ic

The S.S.Schenectady Split Apart by Brittle Fracture while in Harbor (1944)

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. Irwin and his colleagues developed a modified form of Griffith's approach; they reformulated it in terms of stress, rather than energy. Their work resulted in a new materials property, fracture toughness, which is denoted K

Ic, and is now universally accepted as the defining property of fracture mechanics (see Appendix for equations).

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 violating the underlyuing assumption of the theory. Linear-elastic fracture mechanics is limitied practical use for structural steels for two other reasons:

(1) Fracture toughness testing is very expensive and sufficient information can be obtained from the simpler and cheaper Charpy Impact Test, which is described on the UMIST web page.

(2) If a part's reasponse to load is sufficiently close to linear-elastic that K_{Ic can be measured, there is little plastic relaxation at the crack tip and the steel will be brittle. Structural steels, in particular, can be prone to brittle fracture, which has led to a number of catastrophic failures.}

Elastic-Plastic Fracture Mechanics

File:Aircraft Crash.jpg

Vertical Stabilizer, which Separated from the Aircraft Leading to a Fatal Crash(2001)

In the mid-1960's J.R.Rice (then at Brown University) developed a new toughness measure to describe the case where there is sufficient crack-tip deformation that the part no longer obeys the linear-elastic approximation. Rice's analysis, which assumes non-linear elastic deformation ahead of the crack tip, is designated the J integral. The complex mathematics employed in this approach is summarized on the Brown University Engineering Web Page. This analysis is limited to situations where plastic deformation at the crack tip does no extend to the furthest edge of the loaded part. It also demands that the assumed non-linear elastic behavior of the material is a reasonable approximation in shape and magnitude to the real

material's load response. If this is done, the J integral approach reduces to the Griffith theory for linear-elastic behavior.

Because engineers became accustomed to using K_{Ic to characterize fracture toughness, a relation has been used to reduce JIc to it:}

K_{Ic = {EJIc/(1 - ν^{2}1/2 (6)}}

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. Fracture mechanics as a subject for critical study has barely been around for a century and thus is relatively new. There is a high demand for engineers with fracture mechanics expertise - particularly in this day and age where engineering failure is considered 'shocking' amongst the general public.

APPENDIX: Mathematical Relations

Griffith's Energy Relation

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), σf 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.}

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Irwin's Modification of Griffith

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, Kc the fracture toughness, and ν is 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. 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 (3) will be different for geometries other than the center cracked plate, as discussed in the article on stress intensity.}

See Also

• Fatigue fracture

References

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