Fracture mechanics
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'''Fracture mechanics''' is the field of [[mechanics]] concerned with the study of the formation 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 materials and components. It applies the [[physics]] of [[stress (physics)|stress]] and [[Strain (materials science)|strain]], in particular the theories of [[Elasticity (physics)|elasticity]] and [[plasticity (physics)|plasticity]], to the microscopic [[crystallographic defect]]s 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==
[[Image:Tay1.jpg|thumb|right|Tay Bridge Disaster (1879)]]
In many cases, failure of engineering structures through fracture can be fatal; one example is that of the [[Tay Rail Bridge]] disaster (right). Often disasters occur because engineering structures contain cracks—arising either during production or during service (e.g. from [[Fatigue (material)|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 (engineering)|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.
==History==
===Griffith's energy relation===
[[Image:EdgeCrack2D.png|thumb|right|An edge crack (flaw) of length <math>a</math>in a material.]]
Fracture Mechanics was invented during World War I by English aeronautical engineer, [[Alan Arnold Griffith|A.A.Griffith]], to explain the failure of brittle materials.<ref>[http://gallica.bnf.fr/ark:/12148/bpt6k560264/| Griffith, A.A. 1920. The phenomena of rupture and flow in solids. Phil.Trans.Roy.Soc.Lond. A221, pp. 163–198.]</ref> Griffith's work was motivated by a couple of facts:
* The [[Stress (physics)|stress]] needed to [[fracture]] bulk [[glass]] is around 100 MPa.
* The theoretical stress needed for breaking atomic bonds is approximately 10,000 MPa.
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 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 (<math>a</math>) and the stress at fracture (<math>\sigma_f</math>) was nearly constant, i.e.,
:<math>
\sigma_f\sqrt{a} \approx C ~.
</math>
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 <math>C</math> 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 an 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. The next step was to
* 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
:<math>
C = \sqrt{\cfrac{2E\gamma}{\pi}}
</math>
where <math>E</math> is the Young's modulus of the material and <math>\gamma</math> is the [[surface energy]] density of the material. Assuming <math>E = 62</math> GPa and <math>\gamma=1</math> J/m<sup>2</sup> gives excellent agreement of Griffith's predicted fracture stress with experimental results for glass.
===Irwin's modification of Griffith's energy relation===
[[Image:PlasticZone2D.png|400px|thumb|right|The plastic zone around a crack tip in a ductile material.]]
<blockquote>
''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.'' '''F. Erdogan (2000)'''<ref name=Erdogan00>E. Erdogan (2000) ''Fracture Mechanics'', International Journal of Solids and Structures, 27, pp. 171–183.</ref>
</blockquote>
Griffith's theory provides excellent agreement with experimental data for [[brittle]] materials such as [[glass]]. For [[ductile]] materials such as [[steel]], though the relation <math> \sigma_y\sqrt{a} = C </math> still holds, the [[surface energy]] (<math>\gamma</math>) predicted by Griffith's theory is usually unrealistically high. A group working under [[G. R. Irwin]]<ref name=Irwin57>Irwin G (1957), ''Analysis of stresses and strains near the end of a crack traversing a plate'', Journal of Applied Mechanics 24, 361–364.</ref> at the U.S. Naval Research Laboratory (NRL) during World War II realized that [[plasticity (physics)|plasticity]] must play a significant role in the fracture of [[ductile]] materials.
In [[ductile]] materials (and even in materials that appear to be [[brittle]]<ref>Orowan, E., 1948. ''Fracture and strength of solids''. Reports on Progress in Physics XII, 185–232.</ref>), 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 <math>G = 2\gamma + G_p</math> where <math>\gamma</math> is the surface energy and <math>G_p</math> 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
:<math>
\sigma_f\sqrt{a} = \sqrt{\cfrac{E~G}{\pi}} ~.
</math>
For [[brittle]] materials such as glass, the surface energy term dominates and <math>G \approx \gamma = 1</math> J/m<sup>2</sup>. For ductile materials such as steel, the plastic dissipation term dominates and <math>G \approx G_p = 1000</math> J/m<sup>2</sup>. For [[polymers]] close to the [[glass transition]] temperature, we have intermediate values of <math>G \approx 1-1000</math> J/m<sup>2</sup>.
=== Stress intensity factor ===
Another significant achievement of [[G. R. Irwin|Irwin]] and his colleagues was to 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.<ref name="Irwin57" /> This asymptotic expression for the stress field around a crack tip is
:<math>
\sigma_{ij} \approx \left(\cfrac{K}{\sqrt{2\pi r}}\right)~f_{ij}(\theta)
</math>
where <math>\sigma_{ij}</math> are the Cauchy stresses, <math>r</math> is the distance from the crack tip, <math>\theta</math> is the angle with respect to the plane of the crack, and <math>f_{ij}</math> are functions that are independent of the crack geometry and loading conditions. Irwin called the quantity <math>K</math> the '''[[stress intensity factor]]'''. Since the quantity <math>f_{ij}</math> is dimensionless, the stress intensity factor can be expressed in units of <math>\text{Pa-}\sqrt{\text{m}}</math>.
=== Strain energy release rate ===
[[G. R. Irwin|Irwin]] was the first to observe that if the size of the plastic zone around a 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.<ref name="Erdogan00" /> 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 the change in elastic strain energy per unit area of crack growth, i.e.,
:<math>
G := -\left[\cfrac{\partial U}{\partial a}\right]_P = -\left[\cfrac{\partial U}{\partial a}\right]_u
</math>
where <math>U</math> is the elastic energy of the system and <math>a</math> is the crack length. Either the load <math>P</math> or the displacement <math>u</math> can be kept fixed while evaluating the above expressions.
Irwin showed that for a [[fracture#Crack_Separation_Modes|mode I crack]] the strain energy release rate and the [[stress intensity factor]] are related by
:<math>
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}
</math>
where <math>E</math> is the [[Young's modulus]], <math>\nu</math> is the [[Poisson's ratio]], and <math>K_I</math> is the [[stress intensity factor]] in [[fracture#Crack_Separation_Modes|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 [[fracture#Crack_Separation_Modes|mode I]], [[fracture#Crack_Separation_Modes|mode II]], and [[fracture#Crack_Separation_Modes|mode III]] 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 <math>G_{Ic}</math>. '''Today, it is the related quantity ''K<sub>Ic</sub>'' which is called the [[fracture toughness]] and is now universally accepted as the defining material property in linear elastic fracture mechanics.'''
=== Limitations of linear elastic fracture mechanics ===
[[Image:TankerSchenectady.jpg|thumb|right|The [[S.S. Schenectady|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 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 for another more practical reason. Fracture toughness testing is very expensive and engineers believe that sufficient information for selection of steels can be obtained from the simpler and cheaper [[Charpy impact test]]{{Fact|date=May 2008}}.
==Elastic-plastic fracture mechanics==
[[Image:Aircraft Crash.jpg|thumb|right|[[Vertical stabilizer]], which separated from the aircraft leading to a fatal crash(2001)]]
Most engineering materials show some inelastic behavior under operating conditions that involve large loads {{Fact|date=June 2008}}. In such materials the assumptions of linear elastic fracture mechanics may not hold, that is,
* the plastic zone at a crack tip may have a size of the same order of magnitude as the crack size
* the size and shape of the plastic zone may change as the applied load is increased and also as the crack length increases.
Therefore a more general theory of crack growth is needed for elastic-plastic materials that can account for:
* the local conditions for initial crack growth which include the nucleation, growth, and coalescence of voids or decohesion at a crack tip.
* a global energy balance criterion for further crack growth and unstable fracture.
=== R-curve ===
An early attempt in the direction of elastic-plastic fracture mechanics was [[G. R. Irwin|Irwin's]] '''crack extension resistance curve''' or '''R-curve'''. This curve acknowledges the fact that the resistance to fracture increases with growing crack size in elastic-plastic materials. The R-curve is a plot of the total energy dissipation rate as a function of the crack size and can be used to examine the processes of slow stable crack growth and unstable fracture. However, the R-curve was not widely used in applications until the early 1970s. The main reasons appear to be that the R-curve depends on the geometry of the specimen and the crack driving force may be difficult to calculate.<ref name="Erdogan00" />
=== J-integral ===
In the mid-1960s [[James R. Rice|J. R. Rice]] (then at [[Brown University]]) and G. P. Cherepanov independently 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]] (or monotonic [[deformation-theory]] [[plastic]]) deformation ahead of the crack tip, is designated the [[J integral]].<ref>[http://esag.harvard.edu/rice/015_Rice_PathIndepInt_JAM68.pdf Rice,J.R. 1968. A path independent integral and the approximate analysis of strain concentration by notches and cracks. Trans. ASME: J. Appl. Mech. 35, 379–386 ]</ref> 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 J<sub>Ic</sub> and is conventionally converted to K<sub>Ic</sub> 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==
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 (K<sub>Ic</sub>) 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 [[structural failure|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:
:<math>G = \frac{\pi \sigma^2 a}{E}\,</math> (1.1)
where <math>G</math> is the strain energy release rate, <math>\sigma</math> is the applied stress, <math>a</math> is half the crack length, and <math>E</math> is the [[Elastic modulus|Young’s modulus]]. The strain energy release rate can otherwise be understood as: <i>the rate at which energy is absorbed by growth of the crack<i>.
However, we also have that:
:<math>G_c = \frac{\pi \sigma_f^2 a}{E}\,</math> (1.2)
If <math>G</math> ≥ <math>G_c</math>, 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:
:<math>K_I = \sigma \sqrt{\pi a}\,</math> (2.1)
and
:<math>K_c = \sqrt{E G_c}\,</math> (for [[plane stress]]) (2.2)
:<math>K_c = \sqrt{\frac{E G_c}{1 - \nu^2}}\,</math> (for [[plane strain]]) (2.3)
where ''K''<sub>I</sub> is the [[stress intensity]], ''K<sub>c</sub>'' the [[fracture toughness]], and <math>\nu</math> is [[Poisson ratio|Poisson’s ratio]]. It is important to recognize the fact that fracture parameter ''K''<sub>c</sub> has different values when measured under plane stress and plane strain
Fracture occurs when <math>K_I \geq K_c</math>. For the special case of plane strain deformation, <math>K_c</math> becomes <math>K_{Ic}</math> 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:
[[Image:Fracture modes v2.svg|thumb|The three fracture modes.]]
There are three ways of applying a force to enable a crack to propagate:<br>
*'''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 <math>K_I</math> in equation 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 number|dimensionless correction factor]], ''Y'', in order to characterize the geometry. We thus have:
:<math>K_I = Y \sigma \sqrt{\pi a}\,</math> (2.4)
where ''Y'' is a function of the crack length and width of sheet given by:
:<math>Y \left ( \frac{a}{W} \right ) = \sqrt{\sec\left ( \frac{\pi a}{W} \right )}\,</math> (2.5)
for a sheet of finite width ''W'' containing a through-thickness crack of length 2''a'', or
:<math>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\,</math> (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 ''K''<sub>Ic</sub> to characterise fracture toughness, a relation has been used to reduce ''J''<sub>Ic</sub> to it:
:<math>K_{Ic} = \sqrt{E^* J_{Ic}}\,</math> where <math>E^* = E</math> for plane strain and <math>E^* = \frac{E}{1 - \nu^2}</math> for plane stress (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==
{{reflist}}
*C. P. Buckley, "Material Failure", Lecture Notes (2005), [[University of Oxford]]
*T. L. Anderson, "Fracture Mechanics: Fundamentals and Applications" (1995) CRC Press.
==See also==
*[[Fracture]]
*[[Fracture toughness]]
*[[fatigue (material)|Fatigue]]
*[[Stress corrosion cracking]]
*[[Stress intensity factor]]
*[[Strain energy release rate]]
==External links==
*[http://www.efunda.com/formulae/solid_mechanics/fracture_mechanics/fm_intro.cfm eFunda – Fracture Mechanics]
*[http://www2.umist.ac.uk/material/research/intmic/features/charpy/notes.htm UMIST – Charpy Impact Test]
*[http://www.engin.brown.edu/courses/EN175/Notes/Failure_Plasfrac/Failure_Plasfrac.htm Brown University Engineering – Mathematical Relations]
*[http://hdl.handle.net/1813/3075 Fracture Mechanics Notes] by Prof. Alan Zehnder (from Cornell University)
*[http://imechanica.org/node/755 Nonlinear Fracture Mechanics Notes] by Prof. John Hutchinson (from Harvard University)
*[http://imechanica.org/node/903 Notes on Fracture of Thin Films and Multilayers] by Prof. John Hutchinson (from Harvard University)
*[http://www.seas.harvard.edu/suo/papers/17.pdf Mixed mode cracking in layered materials] by Profs. John Hutchinson and Zhigang Suo (from Harvard University)
*[http://www.mate.tue.nl/~piet/edu/frm/sht/bmsht.html Fracture Mechanics] by Prof. Piet Schreurs (from TU Eindhoven, Netherlands)
*[http://www.dsto.defence.gov.au/publications/1880/DSTO-GD-0103.pdf Introduction to Fracture Mechanics] by Dr. C. H. Wang (DSTO – Australia)
*[http://imechanica.org/node/2621 Fracture mechanics course notes] by Prof. Rui Huang (from Univ. of Texas at Austin)
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