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. Solid mechanics is the branch of physics and mathematics that concern the behavior of solid matter under external actions (e. ... Solid mechanics is the branch of physics and mathematics that concern the behavior of solid matter under external actions (e. ...

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. The Materials Science Tetrahedron, which often also includes Characterization at the center Materials science is an interdisciplinary field involving the properties of matter and its applications to various areas of science and engineering. ... The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... Figure 1  Stress tensor A mature tree trunk may support a greater force than a fine steel wire but intuitively we feel that steel is stronger than wood. ... Look up strain in Wiktionary, the free dictionary. ... Elasticity has meanings in two different fields: In physics and mechanical engineering, the theory of elasticity describes how a solid object moves and deforms in response to external stress. ... For other uses, see Plasticity. ... Crystalline solids have a very regular atomic structure: that is, the local positions of atoms with respect to each other are repeated at the atomic scale. ...

The need for fracture mechanics

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 (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. Image File history File links Tay1. ... Image File history File links Tay1. ... Original Tay Bridge from the south) The Tay Bridge (sometimes unofficially the Tay Rail Bridge) is a railway bridge approximately two and a quarter miles (three and a half kilometres) long[1] that spans the Firth of Tay in Scotland, between the city of Dundee and the suburb of Wormit... In materials science, fatigue is the progressive, localized, and permanent structural damage that occurs when a material is subjected to cyclic or fluctuating strains at nominal stresses that have maximum values less than (often much less than) the static yield strength of the material. ... Yield strength, or the yield point, is defined in engineering and materials science as the stress at which a material begins to plastically deform. ...

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, as shown in the Appendix to this article. Alan Arnold Griffith (b 13 June 1893 - 13 Oct 1963) was a British engineer, who, among many other contributions, is best known for his work on stress and fracture in metals that is now known as metal fatigue, as well as being one of the first to develop a strong...

Irwin's modification of Griffith's energy relation

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 (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 K_Ic, and is now universally accepted as the defining property of fracture mechanics (see Appendix for equations). Image File history File links Download high resolution version (886x694, 295 KB) Summary The Design and Methods of Constructionof Welded Steel Merchant Vessels U.S. GPO (1947) Licensing File links The following pages link to this file: Fracture mechanics ... Image File history File links Download high resolution version (886x694, 295 KB) Summary The Design and Methods of Constructionof Welded Steel Merchant Vessels U.S. GPO (1947) Licensing File links The following pages link to this file: Fracture mechanics ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

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: In physics and materials science, plasticity is a property of a material to undergo a non-reversible change of shape in response to an applied force. ...

(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 The Charpy impact test is a standardized test, which determines the amount of energy absorbed by a material during fracture, which is a measure of a given materials toughness. ...

(2) If a part's response 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. A material is brittle if it is subject to fracture when subjected to stress i. ...

Elastic-plastic fracture mechanics

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

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 J_Ic and is conventionally converted to K_Ic 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. Image File history File links Download high resolution version (1000x660, 220 KB) Summary NTSB Report AAR-04-04 Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Download high resolution version (1000x660, 220 KB) Summary NTSB Report AAR-04-04 Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... (disputed—see talk page) In fracture mechanics, the J integral is a path integral around a crack tip representing the energy released per unit area. ...

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:

• 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 pessimistic 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_Ic) 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. Structural failure refers to loss of the load-carying capacity of a component or member within the structure or of the structure itself. ...

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:

                 (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. An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substances tendency to be deformed when a force is applied to it. ...

However, we also have that:

                 (1.2)

where G_{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.

Irwin's modified Griffith crack theory: fracture toughness

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: To meet Wikipedias quality standards, this article or section may require cleanup. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

                 (2.1)

and

(for plane stress)                 (2.2)

(for plane strain)                 (2.3)

where K_I is the stress intensity, K_c the fracture toughness, and ν is Poisson’s ratio. It is important to recognise the fact that fracture parameter K_c has different values when measured under plane stress and plane strain In real engineering components, stress (and strain) are 3-D tensors, however when one of the dimensions of the material is much smaller than the other two, it can be neglected and the resulting state of stress becomes bidimensional[1]. This state is known as plane stress because the normal... In real engineering components, stress (and strain) are 3-D tensors but in prismatic structures such as a long metal billet, the length of the structure is much greater than the other two dimensions. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... When a sample of material is stretched in one direction, it tends to get thinner in the other two directions. ...

Fracture occurs when K_I ≥ K_c. For the special case of plane strain deformation, K_c becomes 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 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 K_I 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: Tensile stress (or tension) is the stress state leading to expansion; that is, the length of a material tends to increase in the tensile direction. ... Shear stress is a stress state where the stress is parallel to a face of the material, as opposed to normal stress when the stress is perpendicular to the face. ... Shear stress is a stress state where the stress is parallel to a face of the material, as opposed to normal stress when the stress is perpendicular to the face. ... Image File history File links Fracture_modes. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In dimensional analysis, a dimensionless number (or more precisely, a number with the dimensions of 1) is a pure number without any physical units. ...

                 (2.4)

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

                 (2.5)

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

                 (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_Ic to characterise fracture toughness, a relation has been used to reduce J_Ic to it:

                 (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.

The University of Oxford, located in the city of Oxford, England, is the oldest university in the English-speaking world. ...

See also

• Fatigue fracture
• Stress corrosion cracking

In materials science, fatigue is the progressive, localized, and permanent structural damage that occurs when a material is subjected to cyclic or fluctuating strains at nominal stresses that have maximum values less than (often much less than) the static yield strength of the material. ... Stress corrosion cracking (SCC) is the unexpected sudden failure of normally ductile metals subjected to a constant tensile stress in a corrosive environment, especially at elevated temperature. ...

External links

• eFunda - Fracture Mechanics
• UMIST - Charpy Impact Test
• Brown University Engineering - Mathematical Relations
• Fracture Mechanics Notes from Cornell University