# Fracture mechanics

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The three fracture modes.

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.

In modern materials science, fracture mechanics is an important tool in improving the mechanical performance of mechanical 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 prediction of crack growth is at the heart of the damage tolerance discipline.

There are three ways of applying a force to enable a crack to propagate:

• Mode I fracture – Opening mode (a tensile stress normal to the plane of the crack),
• Mode II fracture – Sliding mode (a shear stress acting parallel to the plane of the crack and perpendicular to the crack front), and
• Mode III fracture – Tearing mode (a shear stress acting parallel to the plane of the crack and parallel to the crack front).

## Linear elastic fracture mechanics

### Griffith's criterion

An edge crack (flaw) of length ${\displaystyle a}$ in a material

Fracture mechanics was developed during World War I by English aeronautical engineer, A. A. Griffith, to explain the failure of brittle materials.[1] Griffith's work was motivated by two contradictory facts:

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 (a) and the stress at fracture (σf) was nearly constant, which is expressed by the equation:

${\displaystyle \sigma _{f}{\sqrt {a}}\approx C}$

An explanation of this relation in terms of linear elasticity theory is problematic. Linear elasticity theory predicts that stress (and hence the strain) 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 requires the creation of two new surfaces and hence an increase in the surface energy. Griffith found an expression for the constant C 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 (surface energy − elastic 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 by increasing the crack length, i.e. by causing fracture. Using this procedure, Griffith found that
${\displaystyle C={\sqrt {\cfrac {2E\gamma }{\pi }}}}$

where E is the Young's modulus of the material and γ is the surface energy density of the material. Assuming E = 62 GPa and γ = 1 J/m2 gives excellent agreement of Griffith's predicted fracture stress with experimental results for glass.

### Irwin's modification

The plastic zone around a crack tip in a ductile material

Griffith's work was largely ignored by the engineering community until the early 1950s. The reasons for this appear to be (a) in the actual structural materials the level of energy needed to cause fracture is orders of magnitude higher than the corresponding surface energy, and (b) 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. [2]

Griffith's theory provides excellent agreement with experimental data for brittle materials such as glass. For ductile materials such as steel, though the relation ${\displaystyle \sigma _{y}{\sqrt {a}}=C}$ still holds, the surface energy (γ) predicted by Griffith's theory is usually unrealistically high. A group working under G. R. Irwin[3] at the U.S. Naval Research Laboratory (NRL) during World War II realized that plasticity must play a significant role in the fracture of ductile materials.

In ductile materials (and even in materials that appear to be brittle[4]), 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 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 when 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 (and any other dissipative forces that may be at work). The dissipated energy provides the thermodynamic resistance to fracture. Then the total energy dissipated is
${\displaystyle G=2\gamma +G_{p}}$

where γ is the surface energy and Gp is the plastic dissipation (and dissipation from other sources) per unit area of crack growth.

The modified version of Griffith's energy criterion can then be written as

${\displaystyle \sigma _{f}{\sqrt {a}}={\sqrt {\cfrac {E~G}{\pi }}}.}$

For brittle materials such as glass, the surface energy term dominates and ${\displaystyle G\approx 2\gamma =2\,\,J/m^{2}}$. For ductile materials such as steel, the plastic dissipation term dominates and ${\displaystyle G\approx G_{p}=1000\,\,J/m^{2}}$. For polymers close to the glass transition temperature, we have intermediate values of ${\displaystyle G\approx 2-1000\,\,J/m^{2}}$.

### Stress intensity factor

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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.[3] This asymptotic expression for the stress field around a crack tip is

${\displaystyle \sigma _{ij}\approx \left({\cfrac {K}{\sqrt {2\pi r}}}\right)~f_{ij}(\theta )}$

where σij are the Cauchy stresses, r is the distance from the crack tip, θ is the angle with respect to the plane of the crack, and fij are functions that depend on the crack geometry and loading conditions. Irwin called the quantity K the stress intensity factor. Since the quantity fij is dimensionless, the stress intensity factor can be expressed in units of ${\displaystyle {\text{MPa-}}{\sqrt {\text{m}}}}$.

When a rigid line inclusion is considered, a similar asymptotic expression for the stress fields is obtained.

### Strain energy release

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Irwin was the first to observe that if the size of the plastic zone around a crack is small compared to the size of the crack, the energy required to grow the crack will not be critically dependent on the state of stress at the crack tip.[2] In other words, a purely elastic solution may be used to calculate the amount of energy available for fracture.

The energy release rate for crack growth or strain energy release rate may then be calculated as the change in elastic strain energy per unit area of crack growth, i.e.,

${\displaystyle G:=\left[{\cfrac {\partial U}{\partial a}}\right]_{P}=-\left[{\cfrac {\partial U}{\partial a}}\right]_{u}}$

where U is the elastic energy of the system and a is the crack length. Either the load P or the displacement u can be kept fixed while evaluating the above expressions.

Irwin showed that for a mode I crack (opening mode) the strain energy release rate and the stress intensity factor are related by:

${\displaystyle G=G_{I}={\begin{cases}{\cfrac {K_{I}^{2}}{E}}&{\text{plane stress}}\\{\cfrac {(1-\nu ^{2})K_{I}^{2}}{E}}&{\text{plane strain}}\end{cases}}}$

where E is the Young's modulus, ν is Poisson's ratio, and KI is the stress intensity factor in mode I. Irwin also showed that the strain energy release rate of a planar crack in a linear elastic body can be expressed in terms of the mode I, mode II (sliding mode), and mode III (tearing mode) stress intensity factors for the most general loading conditions.

Next, Irwin adopted the additional assumption that the size and shape of the energy dissipation zone remains approximately constant during brittle fracture. This assumption suggests that the energy needed to create a unit fracture surface is a constant that depends only on the material. This new material property was given the name fracture toughness and designated GIc. Today, it is the critical stress intensity factor KIc, found in the plane strain condition, which is accepted as the defining property in linear elastic fracture mechanics.

### Small scale yielding

When a macroscopic loading is applied to a structure with a crack, the stress near the vicinity of the crack approaches ∞ as r→0, where r is the distance from the crack tip. As the magnitude of the stress in a material is bounded by the yield stress, there exists a region near the crack tip, of size rp, in which the material has deformed plastically. The maximum size of this plastic zone can be estimated from the stress intensity factor KC and the yield stress σY; and can be expressed as

${\displaystyle r_{p}={\frac {K_{C}^{2}}{\sigma _{Y}^{2}}}.}$

In this region, the equations of linear elasticity are not valid. Therefore, for linear elastic fracture mechanics to be applicable, 2.5 rp should be much smaller than the relevant dimensions, such as the length, thickness and width of the structure. This condition, in which the plastic deformation of the structure is confined to a very small region near the crack tip, is commonly referred to as small scale yielding.

### Fracture toughness tests

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### Limitations

The S.S. Schenectady split apart by brittle fracture while in harbor, 1944.

But a problem arose for the NRL researchers because naval materials, e.g., ship-plate steel, are not perfectly elastic but undergo significant plastic deformation at the tip of a crack. One basic assumption in Irwin's linear elastic fracture mechanics is small scale yielding, the condition that the size of the plastic zone is small compared to the crack length. However, this assumption is quite restrictive for certain types of failure in structural steels though such steels can be prone to brittle fracture, which has led to a number of catastrophic failures.

Linear-elastic fracture mechanics is of limited practical use for structural steels and Fracture toughness testing can be expensive.

## Elastic–plastic fracture mechanics

Vertical stabilizer, which separated from American Airlines Flight 587, leading to a fatal crash

## Cracktip constraint under large scale yielding

{{ safesubst:#invoke:Unsubst||$N=Unreferenced section |date=__DATE__ |$B= {{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} }} Under small-scale yielding conditions, a single parameter (e.g., K, J, or CTOD) characterizes cracktip conditions and can be used as a geometry-independent fracture criterion. Single-parameter fracture mechanics breaks down in the presence of excessive plasticity, and the fracture toughness depends on the size and geometry of the test specimen. The theories used for large scale yielding is not very standardized. The following theories/ approaches are commonly used among researchers in this field

### J-Q Theory

By using FEM, one can establish a parameter Q to modify the stress field for a better solution when the plastic zone is growing. The new stress field is: σ_ij=(σ_ij)+Q*δ_ij*σ_yield where δ_ij=1 for i=j and 0 if not. Q usually takes values from -3 to +2. A negative value greatly changes the geometry of the plastic zone.

The J-Q-M theory includes another parameter, the mismatch parameter, which is used for welds to make up for the change in toughness of the weld metal (WM), base metal (BM) and heat affected zone (HAZ). This value is interpeted to the formula in a similar way as the Q-parameter, and the two are usually assumed to be independent of each other.

### T-modification

As an alternative to J-Q theory, a parameter T can be used. This only changes the normal stress in the x-direction (and the z-direction in the case of plane strain). T does not require the use of FEM, but is derived from constraint. It can be argued that T is limited to LEFM, but as the plastic zone change due to T never reaches the actual crack surface (except on the tip), its validity holds true not only under small scale yielding.

## Engineering applications

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

• Applied load
• 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. Ensuring safe operation of structure despite these inherent flaws is achieved through damage tolerance analysis. 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 criterion

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

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

where ${\displaystyle G}$ is the strain energy release rate, ${\displaystyle \sigma }$ is the applied stress, ${\displaystyle a}$ is half the crack length, and ${\displaystyle E}$ is the Young’s modulus, which for the case of plane strain should be divided by the plate stiffness factor (1-ν^2). 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:

${\displaystyle G_{c}={\frac {\pi \sigma _{f}^{2}a}{E}}\,}$                 (1.2)

If ${\displaystyle G}$${\displaystyle G_{c}}$, this is the criterion for which the crack will begin to propagate.

### Irwin's modifications

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:

${\displaystyle K_{I}=\sigma {\sqrt {\pi a}}\,}$                 (2.1)

and

${\displaystyle K_{c}={\sqrt {EG_{c}}}\,}$ (for plane stress)                 (2.2)
${\displaystyle K_{c}={\sqrt {\frac {EG_{c}}{1-\nu ^{2}}}}\,}$ (for plane strain)                 (2.3)

where KI is the stress intensity, Kc the fracture toughness, and ${\displaystyle \nu }$ is Poisson’s ratio. It is important to recognize the fact that fracture parameter Kc has different values when measured under plane stress and plane strain

Fracture occurs when ${\displaystyle K_{I}\geq K_{c}}$. For the special case of plane strain deformation, ${\displaystyle K_{c}}$ becomes ${\displaystyle K_{Ic}}$ 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 so-called "mode I" loading as opposed to mode II or III:

We must note that the expression for ${\displaystyle K_{I}}$ in equation 2.1 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, Y, in order to characterize the geometry. We thus have:

${\displaystyle K_{I}=Y\sigma {\sqrt {\pi a}}\,}$                 (2.4)

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

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

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

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

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

### Elasticity and plasticity

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

${\displaystyle K_{Ic}={\sqrt {E^{*}J_{Ic}}}\,}$          where ${\displaystyle E^{*}=E}$ for plane stress and ${\displaystyle E^{*}={\frac {E}{1-\nu ^{2}}}}$ for plane strain          (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.

## Applications of fracture mechanics

{{ safesubst:#invoke:Unsubst||$N=Unreferenced section |date=__DATE__ |$B= {{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} }} The design process for a component consists of choosing the appropriate geometry, the necessary material strength as per the loading conditions (either cyclic or constant loading), the temperature of usage and structural analysis (Testing and FEM analysis), so that it does not fail under load. The methodologies followed in design criteria traditionally pick up the conventional materials based on standard data and as per the loading conditions proportioning the geometry of the components on basis of analysis. This method is not applicable for some new innovation like usage of new material in design. Another method followed is that as per the loading conditions, static analysis is done for the structure taking into account the forces acting on each component, material strength and geometry. The material strength is chosen keeping in mind the factor of safety, i.e. the ultimate stress (where it fails) is much higher than maximum stress in the component. The general assumptions in the design criteria are: lack of discontinuities, no defects or cracks in the material, and even in the presence of discontinuities the material is assumed to have sufficient ductility to yield locally so that redistribution of stress at discontinuities can occur. Investigations of failed components proved that crack growth started because of such discontinuities.

Fracture mechanics follows one of two design principles: either fail-safe or safe-life. In fail safe mode, even if a component fails, the entire structure is not at risk (failure of redundant members). According to the safe life principle throughout the life, no component of the structure may fail. Fracture mechanics estimated the maximum crack that a material can withstand before it fails through analysis taking into consideration the overall dimensions of the structure, the stress value where crack initiation takes place, notch toughness value (ability of a material to absorb energy in the presence of a crack for crack propagation), the behavior of materials under the action of stresses by finding out the stress intensity factor (K), fatigue crack growth and stress corrosion crack growth. As in basic solid mechanics analysis, stresses in the component should be lower than the yield stress; application of the same principle is means that the stress intensity factor should be less than the critical stress intensity factor. Major applications of fracture mechanics design are material selection, effect of defects, failure analysis and control/monitoring of components. Fracture analysis includes the usage of mathematical models such as linear elastic fracture mechanics (LEFM), crack opening displacement (COD) and J-integral approaches by using finite element analysis (FEM).

The relationship used for estimating stress intensity factor is

${\displaystyle K=c\sigma {\sqrt {a}}}$

where K is the critical fracture toughness value, c a constant that depends on crack and specimen dimensions, σ the applied stress, and a the flaw size.

The above relation is very general and as per the shape of the crack, relations available in standard data books or course books are to be used, any general crack can be approximated to standard shapes used in writing the relations.

For a given material the value of K is dependent on stresses acting and flaw size. Flaw size decreases as the stress increases. Thus a design engineer can dictate the life of a component by choosing appropriate values of K, a and σ. Even there are other parameters that estimate the life of a component like working temperature, loading rate (fatigue), residual stress and stress concentration. The higher the K value, the higher is the resistance to crack growth, and the material can resist higher stresses. Designers try to decrease the defects in the component arising in casting or manufacturing processes by following good fabrication processes and inspection, and estimate notch-toughness values of materials using methods like charpy V-notch impact test, or drop weight tests. In many investigations it was proved that the material failed at a very much lower than the critical stress intensity factor because of defects in the material or micro cracks. Analysis proved that for any component there are two phases for crack development, i.e. crack initiation and second phase crack growth until failure. Of the two, the first phase covers a larger percentage of fatigue life, and under very large high cycle loading conditions second phase is instantaneous.

The factor (K/σ)² is used for estimating design of component because it estimates crack size, more the value better the resistance to the forces(Stress). But how large this factor has to be is decided by considering type of the structure, frequency of inspection, access to inspection, design life of the structure, consequences of failure, probability of over load, methods of fabrication, required quality, material cost in addition to the results obtained by fracture mechanics analysis.

## References

### Notes

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2. E. Erdogan (2000) Fracture Mechanics, International Journal of Solids and Structures, 37, pp. 171–183.
3. Irwin G (1957), Analysis of stresses and strains near the end of a crack traversing a plate, Journal of Applied Mechanics 24, 361–364.
4. Orowan, E., 1948. Fracture and strength of solids. Reports on Progress in Physics XII, 185–232.
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### Bibliography

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

## Further reading

• Davidge, R.W., Mechanical Behavior of Ceramics, Cambridge Solid State Science Series, (1979)
• Demaid, Adrian, Fail Safe, Open University (2004)
• Green, D., An Introduction to the Mechanical Properties of Ceramics, Cambridge Solid State Science Series, Eds. Clarke, D.R., Suresh, S., Ward, I.M. (1998)
• Lawn, B.R., Fracture of Brittle Solids, Cambridge Solid State Science Series, 2nd Edn. (1993)
• Farahmand, B., Bockrath, G., and Glassco, J. (1997) Fatigue and Fracture Mechanics of High-Risk Parts, Chapman & Hall.
• Chen, X., Mai, Y.-W., Fracture Mechanics of Electromagnetic Materials: Nonlinear Field Theory and Applications, Imperial College Press, (2012)
• Fracture mechanics fundamentals and applications by Anderson.
• Chapter 10 – Strength of Elastomers, A.N. Gent, W.V. Mars, In: James E. Mark, Burak Erman and Mike Roland, Editor(s), The Science and Technology of Rubber (Fourth Edition), Academic Press, Boston, 2013, Pages 473-516, ISBN 9780123945846, 10.1016/B978-0-12-394584-6.00010-8