Fracture mechanics


Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. 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.
Theoretically, the stress ahead of a sharp crack tip becomes infinite and cannot be used to describe the state around a crack. Fracture mechanics is used to characterise the loads on a crack, typically using a single parameter to describe the complete loading state at the crack tip. A number of different parameters have been developed. When the plastic zone at the tip of the crack is small relative to the crack length the stress state at the crack tip is the result of elastic forces within the material and is termed linear elastic fracture mechanics and can be characterised using the stress intensity factor. Although the load on a crack can be arbitrary, in 1957 G. Irwin found any state could be reduced to a combination of three independent stress intensity factors:
  • Mode I – Opening mode,
  • Mode II – Sliding mode, and
  • Mode III – Tearing mode.
When the size of the plastic zone at the crack tip is too large, elastic-plastic fracture mechanics can be used with parameters such as the J-integral or the crack tip opening displacement.
The characterising parameter describes the state of the crack tip which can then be related to experimental conditions to ensure similitude. Crack growth occurs when the parameters typically exceed certain critical values. Corrosion may cause a crack to slowly grow when the stress corrosion stress intensity threshold is exceeded. Similarly, small flaws may result in crack growth when subjected to cyclic loading. Known as fatigue, it was found that for long cracks, the rate of growth is largely governed by the range of the stress intensity experienced by the crack due to the applied loading. Fast fracture will occur when the stress intensity exceeds the fracture toughness of the material. The prediction of crack growth is at the heart of the damage tolerance mechanical design discipline.

Motivation

The processes of material manufacture, processing, machining, and forming may introduce flaws in a finished mechanical component. 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 and those that are liable to propagate as cracks and so cause failure of the flawed structure. Despite these inherent flaws, it is possible to achieve through damage tolerance analysis the safe operation of a structure. Fracture mechanics as a subject for critical study has barely been around for a century and thus is relatively new.
Fracture mechanics should attempt to provide quantitative answers to the following questions:
  1. What is the strength of the component as a function of crack size?
  2. What crack size can be tolerated under service loading, i.e. what is the maximum permissible crack size?
  3. How long does it take for a crack to grow from a certain initial size, for example the minimum detectable crack size, to the maximum permissible crack size?
  4. What is the service life of a structure when a certain pre-existing flaw size is assumed to exist?
  5. During the period available for crack detection how often should the structure be inspected for cracks?

    Linear elastic fracture mechanics

Griffith's criterion

Fracture mechanics was developed during World War I by English aeronautical engineer A. A. Griffith – thus the term Griffith crack – to explain the failure of brittle materials. Griffith's work was motivated by two contradictory facts:
  • The stress needed to fracture bulk glass is around.
  • The theoretical stress needed for breaking atomic bonds of glass is approximately.
A theory was needed to reconcile these conflicting observations. Also, experiments on glass fibers that Griffith himself conducted suggested that the fracture stress increases as the fiber diameter decreases. Hence the uniaxial tensile strength, which had been used extensively to predict material failure before Griffith, could not be a specimen-independent material property. Griffith suggested that the low fracture strength observed in experiments, as well as the size-dependence of strength, was due to the presence of microscopic flaws in the bulk material.
To verify the flaw hypothesis, Griffith introduced an artificial flaw in his experimental glass specimens. The artificial flaw was in the form of a surface crack which was much larger than other flaws in a specimen. The experiments showed that the product of the square root of the flaw length and the stress at fracture was nearly constant, which is expressed by the equation:
An explanation of this relation in terms of linear elasticity theory is problematic. Linear elasticity theory predicts that stress at the tip of a sharp flaw in a linear elastic material is infinite. To avoid that problem, Griffith developed a thermodynamic approach to explain the relation that he observed.
The growth of a crack, the extension of the surfaces on either side of the crack, requires an increase in the surface energy. Griffith found an expression for the constant in terms of the surface energy of the crack by solving the elasticity problem of a finite crack in an elastic plate. Briefly, the approach was:
  • Compute the potential energy stored in a perfect specimen under a uniaxial tensile load.
  • Fix the boundary so that the applied load does no work and then introduce a crack into the specimen. The crack relaxes the stress and hence reduces the elastic energy near the crack faces. On the other hand, the crack increases the total surface energy of the specimen.
  • Compute the change in the free energy as a function of the crack length. Failure occurs when the free energy attains a peak value at a critical crack length, beyond which the free energy decreases as the crack length increases, i.e. by causing fracture. Using this procedure, Griffith found that
where is the Young's modulus of the material and is the surface energy density of the material. Assuming and gives excellent agreement of Griffith's predicted fracture stress with experimental results for glass.
For the simple case of a thin rectangular plate with a crack perpendicular to the load, the energy release rate,, becomes:
where is the applied stress, is half the crack length, and is the Young's modulus, which for the case of plane strain should be divided by the plate stiffness factor. The strain energy release rate can physically be understood as: the rate at which energy is absorbed by growth of the crack.
However, we also have that:
If ≥, this is the criterion for which the crack will begin to propagate.
For materials highly deformed before crack propagation, the linear elastic fracture mechanics formulation is no longer applicable and an adapted model is necessary to describe the stress and displacement field close to crack tip, such as on fracture of soft materials.

Irwin's modification


Griffith's work was largely ignored by the engineering community until the early 1950s. The reasons for this appear to be in the actual structural materials the level of energy needed to cause fracture is orders of magnitude higher than the corresponding surface energy, and in structural materials there are always some inelastic deformations around the crack front that would make the assumption of linear elastic medium with infinite stresses at the crack tip highly unrealistic.

Griffith's theory provides excellent agreement with experimental data for brittle materials such as glass. For ductile materials such as steel, although the relation still holds, the surface energy predicted by Griffith's theory is usually unrealistically high. A group working under G. R. Irwin at the U.S. Naval Research Laboratory during World War II realized that plasticity must play a significant role in the fracture of ductile materials.
In ductile materials, a plastic zone develops at the tip of the crack. As the applied load increases, the plastic zone increases in size until the crack grows and the elastically strained material behind the crack tip unloads. The plastic loading and unloading cycle near the crack tip leads to the dissipation of energy as heat. Hence, a dissipative term has to be added to the energy balance relation devised by Griffith for brittle materials. In physical terms, additional energy is needed for crack growth in ductile materials as compared to brittle materials.
Irwin's strategy was to partition the energy into two parts:
  • the stored elastic strain energy which is released as a crack grows. This is the thermodynamic driving force for fracture.
  • the dissipated energy which includes plastic dissipation and the surface energy. The dissipated energy provides the thermodynamic resistance to fracture.
Then the total energy is:
where is the surface energy and is the plastic dissipation per unit area of crack growth.
The modified version of Griffith's energy criterion can then be written as
For brittle materials such as glass, the surface energy term dominates and. For ductile materials such as steel, the plastic dissipation term dominates and. For polymers close to the glass transition temperature, we have intermediate values of between 2 and 1000.

Stress intensity factor

Another significant achievement of Irwin and his colleagues was to find a method of calculating the amount of energy available for fracture in terms of the asymptotic stress and displacement fields around a crack front in a linear elastic solid. This asymptotic expression for the stress field in mode I loading is related to the stress intensity factor following:
where are the Cauchy stresses, is the distance from the crack tip, is the angle with respect to the plane of the crack, and are functions that depend on the crack geometry and loading conditions. Irwin called the quantity the stress intensity factor. Since the quantity is dimensionless, the stress intensity factor can be expressed in units of.
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:
and
where is the mode stress intensity, the fracture toughness, and is Poisson's ratio.
Fracture occurs when. For the special case of plane strain deformation, becomes and is considered a material property. The subscript arises because of the different ways of loading a material to enable a crack to propagate. It refers to so-called "mode " loading as opposed to mode or :
The expression for will be different for geometries other than the center-cracked infinite plate, as discussed in the article on the stress intensity factor. Consequently, it is necessary to introduce a dimensionless correction factor,, in order to characterize the geometry. This correction factor, also often referred to as the geometric shape factor, is given by empirically determined series and accounts for the type and geometry of the crack or notch. We thus have:
where is a function of the crack length and width of sheet given, for a sheet of finite width containing a through-thickness crack of length, by: