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# 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. Fractography is widely used with Fracture Mechanics to understand the causes of failures and also verify the theoretical failure predictions with real life failures.

## The need for fracture mechanics

In many cases, failure of engineering structures through fracture can be fatal; one example is that of the Tay Rail Bridge disaster (left). Often disasters occur because engineering structures contain cracks - arising either during production or during service (e.g. from fatigue). For instance, growth of cracks in pressure vessels due to crack propagation could cause a fatal explosion. If failure were ever to happen, we would rather it were by yield or by leak before break.

Since cracks can lower the strength of the structure beyond that due to loss of load-bearing area 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.

## The history of fracture mechanics

### Griffith's energy relation

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.

### Irwin's modification of Griffith's energy relation

Griffith’s work was ignored for over twenty years until a group under G.R. Irwin at the U.S. Naval Research Laboratory (NRL) 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 KIc, and is now universally accepted as the defining property of fracture mechanics.

But a problem arose for the NRL researchers because naval materials, e.g. ship-plate steel, are not perfectly elastic but undergo plastic deformation at the tip of a crack violating the underlying assumption of the theory. Linear-elastic fracture mechanics is of limited practical use for structural steels for two other reasons:

(1) Fracture toughness testing is very expensive and sufficient information for selection of steels can be obtained from the simpler and cheaper Charpy impact test

(2) If a part's response to load is sufficiently close to linear-elastic that KIc 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

In the mid-1960s 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. This analysis is limited to situations where plastic deformation at the crack tip does not 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. The elastic-plastic failure parameter is designated JIc and is conventionally converted to KIc using Equation (3.1) of the Appendix to this article. Also note that the J integral approach reduces to the Griffith theory for linear-elastic behavior.

## Fully plastic fracture mechanics

If the alloy is so tough that the yielded region ahead of the crack extends to the far edge of the specimen before fracture, the crack is no longer an effective stress concentrator. Instead, the presence of the crack merely serves to reduce the load-bearing area. In this regime the failure stress is conventionally assumed to be the average of the yield and ultimate strengths of the alloy.

## Engineering applications of fracture mechanics

The following information is needed for a fracture mechanics prediction of failure:

• Residual stress
• Size and shape of the part
• Size, shape, location, and orientation of the crack

Usually not all of this information is available and conservative assumptions have to be made.

Occasionally post-mortem fracture-mechanics analyses are carried out. In the absence of an extreme overload, the causes are either insufficient toughness (KIc) or an excessively large crack that was not detected during routine inspection.

## 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 crack theory: strain energy release rate

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

$G = \frac{\pi \sigma^2 a}{E}\,$                 (1.1)

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:

$G_c = \frac{\pi \sigma_f^2 a}{E}\,$                 (1.2)

If GGc, this is the criterion for which the crack will begin to propagate.

### Irwin's modified Griffith crack theory: fracture toughness

Eventually a modification of Griffith’s solids theory emerged from this work; a term called stress intensity replaced strain energy release rate and a term called fracture toughness replaced surface weakness energy. Both of these terms are simply related to the energy terms that Griffith used:

$K_I = \sigma \sqrt{\pi a}\,$                 (2.1)

and

$K_c = \sqrt{E G_c}\,$ (for plane stress)                 (2.2)
$K_c = \sqrt{\frac{E G_c}{1 - \nu^2}}\,$ (for plane strain)                 (2.3)

where KI is the stress intensity, Kc the fracture toughness, and ν is Poisson’s ratio. It is important to recognise the fact that fracture parameter Kc has different values when measured under plane stress and plane strain

Fracture occurs when KIKc. 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 butt 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 crack - Opening mode (a tensile stress normal to the plane of the crack)
• Mode II crack - Sliding mode (a shear stress acting parallel to the plane of the crack and perpendicular to the crack front)
• Mode III crack - Tearing mode (a shear stress acting parallel to the plane of the crack and parallel to the crack front)

We must note that the expression for KI in Eq (2.1) will be different for geometries other than the center cracked plate, as discussed in the article on stress intensity. Consequently, it is necessary to introduce a dimensionless correction factor, Y, in order to characterise the geometry. We thus have:

$K_I = Y \sigma \sqrt{\pi a}\,$                 (2.4)

where Y is a function of the crack length and width of sheet given by:

$Y \left ( \frac{a}{W} \right ) = \sqrt{\sec\left ( \frac{\pi a}{W} \right )}\,$                 (2.5)

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

$Y \left ( \frac{a}{W} \right ) = 1.12 - \frac{0.41}{\sqrt \pi} \frac{a}{W} + \frac{18.7}{\sqrt \pi} \left ( \frac{a}{W} \right )^2 - \cdots\,$                 (2.6)

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

### Elastic-plastic fracture mechanics theory

Since engineers became accustomed to using KIc to characterise fracture toughness, a relation has been used to reduce JIc to it:

$K_{Ic} = \sqrt{\frac{E J_{Ic}}{1 - \nu^2}}\,$                 (3.1)

The remainder of the mathematics employed in this approach is interesting, but is probably better summarised in external pages due to its complex nature (refer to the Useful Websites section).

## References

• C. P. Buckley, "Material Failure", Lecture Notes (2005), University of Oxford
• T. L. Anderson, "Fracture Mechanics: Fundamentals and Applications" (1995) CRC Press.