# Fracture

A fracture is the separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity surfaces within the solid. If a displacement develops perpendicular to the surface of displacement, it is called a normal tensile crack or simply a crack; if a displacement develops tangentially to the surface of displacement, it is called a shear crack, slip band, or dislocation.[1]

Brittle fractures occur with no apparent deformation before fracture; ductile fractures occur when visible deformation does occur before separation. Fracture strength or breaking strength is the stress when a specimen fails or fractures. A detailed understanding of how fracture occurs in materials may be assisted by the study of fracture mechanics.

## Fracture strength

Fracture strength, also known as breaking strength, is the stress at which a specimen fails via fracture.[2] This is usually determined for a given specimen by a tensile test, which charts the stress–strain curve (see image). The final recorded point is the fracture strength.

Ductile materials have a fracture strength lower than the ultimate tensile strength (UTS), whereas in brittle materials the fracture strength is equivalent to the UTS.[2] If a ductile material reaches its ultimate tensile strength in a load-controlled situation,[Note 1] it will continue to deform, with no additional load application, until it ruptures. However, if the loading is displacement-controlled,[Note 2] the deformation of the material may relieve the load, preventing rupture.

## Types

There are two types of fractures :

### Brittle fracture

In brittle fracture, no apparent plastic deformation takes place before fracture. Brittle fracture typically involves little energy absorption and occurs at high speeds—up to 2133.6 m/s (7000 ft/s) in steel.[3] In most cases brittle fracture will continue even when loading is discontinued.[4]

In brittle crystalline materials, fracture can occur by cleavage as the result of tensile stress acting normal to crystallographic planes with low bonding (cleavage planes). In amorphous solids, by contrast, the lack of a crystalline structure results in a conchoidal fracture, with cracks proceeding normal to the applied tension.

The theoretical strength of a crystalline material is (roughly)

${\displaystyle \sigma _{\mathrm {theoretical} }={\sqrt {\frac {E\gamma }{r_{o}}}}}$

where: –

${\displaystyle E}$ is the Young's modulus of the material,
${\displaystyle \gamma }$ is the surface energy, and
${\displaystyle r_{o}}$ is the equilibrium distance between atomic centers.

On the other hand, a crack introduces a stress concentration modeled by

${\displaystyle \sigma _{\mathrm {elliptical\ crack} }=\sigma _{\mathrm {applied} }\left(1+2{\sqrt {\frac {a}{\rho }}}\right)=2\sigma _{\mathrm {applied} }{\sqrt {\frac {a}{\rho }}}}$ (For sharp cracks)

where: –

${\displaystyle \sigma _{\mathrm {applied} }}$ is the loading stress,
${\displaystyle a}$ is half the length of the crack, and
${\displaystyle \rho }$ is the radius of curvature at the crack tip.

Putting these two equations together, we get

${\displaystyle \sigma _{\mathrm {fracture} }={\sqrt {\frac {E\gamma \rho }{4ar_{o}}}}.}$

Looking closely, we can see that sharp cracks (small ${\displaystyle \rho }$) and large defects (large ${\displaystyle a}$) both lower the fracture strength of the material.

Recently, scientists have discovered supersonic fracture, the phenomenon of crack propagation faster than the speed of sound in a material.[5] This phenomenon was recently also verified by experiment of fracture in rubber-like materials.

The basic sequence in a typical brittle fracture is: introduction of a flaw either before or after the material is put in service, slow and stable crack propagation under recurring loading, and sudden rapid failure when the crack reaches critical crack length based on the conditions defined by fracture mechanics.[4] Brittle fracture may be avoided by controlling three primary factors: material fracture toughness (Kc), nominal stress level (σ), and introduced flaw size (a).[3] Residual stresses, temperature, loading rate, and stress concentrations also contribute to brittle fracture by influencing the three primary factors.[3]

Under certain conditions, ductile materials can exhibit brittle behavior. Rapid loading, low temperature, and triaxial stress constraint conditions may cause ductile materials to fail without prior deformation.[3]

### Ductile fracture

In ductile fracture, extensive plastic deformation (necking) takes place before fracture. The terms rupture or ductile rupture describe the ultimate failure of ductile materials loaded in tension. Rather than cracking, the material "pulls apart," generally leaving a rough surface. In this case there is slow propagation and an absorption of a large amount of energy before fracture.[6]

Because ductile rupture involves a high degree of plastic deformation, the fracture behavior of a propagating crack as modelled above changes fundamentally. Some of the energy from stress concentrations at the crack tips is dissipated by plastic deformation ahead of the crack as it propagates.

The basic steps in ductile fracture are void formation, void coalescence (also known as crack formation), crack propagation, and failure, often resulting in a cup-and-cone shaped failure surface.

## Fracture modes and characteristics

There are three standard conventions for defining relative displacements in elastic materials in order to analyze crack propagation[3] as proposed by Irwin.[7] In addition fracture can involve uniform strain or a combination of these modes.[4]

• 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)

The manner in which a crack propagates through a material gives insight into the mode of fracture. With ductile fracture a crack moves slowly and is accompanied by a large amount of plastic deformation around the crack tip. A ductile crack will usually not propagate unless an increased stress is applied and generally cease propagating when loading is removed.[4] In a ductile material, a crack may progress to a section of the material where stresses are slightly lower and stop due to the blunting effect of plastic deformations at the crack tip. On the other hand, with brittle fracture, cracks spread very rapidly with little or no plastic deformation. The cracks that propagate in a brittle material will continue to grow once initiated.

Crack propagation is also categorized by the crack characteristics at the microscopic level. A crack that passes through the grains within the material is undergoing transgranular fracture. A crack that propagates along the grain boundaries is termed an intergranular fracture. Typically, the bonds between material grains are stronger at room temperature than the material itself, so transgranular fracture is more likely to occur. When temperatures increase enough to weaken the grain bonds, intergranular fracture is the more common fracture mode.[4]

## Fracture Testing

Fracture in materials is studied and quantified in multiple ways. Fracture is largely determined by the fracture toughness (${\textstyle \mathrm {K} _{\mathrm {c} }}$), so fracture testing is often done to determine this. The two most widely-used techniques for determining fracture toughness are the Three-point flexural test and the compact tension test.

By performing the compact tension and three-point flexural tests, one is able to determine the fracture toughness through the following equation:

${\displaystyle \mathrm {K_{c}} =\sigma _{\mathrm {F} }{\sqrt {\pi \mathrm {c} }}\mathrm {f\ (c/a)} }$

Where:-

${\displaystyle \mathrm {f\ (c/a)} }$ is an empirically-derived equation to capture the test sample geometry
${\displaystyle \sigma _{\mathrm {F} }}$ is the fracture stress, and
${\displaystyle \mathrm {c} }$ is the crack length.

To accurately attain ${\textstyle \mathrm {K} _{\mathrm {c} }}$, the value of ${\textstyle \mathrm {c} }$ must be precisely measured. This is done by taking the test piece with its fabricated notch of length ${\textstyle \mathrm {c\prime } }$ and sharpening this notch to better emulate a crack tip found in real-world materials [8]. Cyclical prestressing the sample can then induce a fatigue crack which extends the crack from the fabricated notch length of ${\textstyle \mathrm {c\prime } }$ to ${\textstyle \mathrm {c} }$. This value ${\textstyle \mathrm {c} }$ is used in the above equations for determining ${\textstyle \mathrm {K} _{\mathrm {c} }}$.[9]

Following this test, the sample can then be reoriented such that further loading of a load (F) will extend this crack and thus a load versus sample deflection curve can be obtained. With this curve, the slope of the linear portion, which is the inverse of the compliance of the material, can be obtained. This is then used to derive f(c/a) as defined above in the equation. With the knowledge of all these variables, ${\textstyle \mathrm {K} _{\mathrm {c} }}$ can then be calculated.

## Brittle Fracture of Ceramics and Inorganic Glasses

Ceramics and inorganic glasses have fracturing behavior that differ those of metallic materials. Ceramics have high strengths and perform well in high temperatures due to the material strength being independent of temperature. Ceramics have low toughness as determined by testing under a tensile load; often, ceramics have ${\textstyle \mathrm {K} _{\mathrm {c} }}$ values that are ~5% of that found in metals.[9] However, ceramics are usually loaded in compression in everyday use, so the compressive strength is often referred to as the strength; this strength can often exceed that of most metals. However, ceramics are brittle and thus most work done revolves around preventing brittle fracture. Due to how ceramics are manufactured and processed, there are often preexisting defects in the material introduce a high degree of variability in the Mode I brittle fracture.[9] Thus, there is a probabilistic nature to be accounted for in the design of ceramics. The Weibull distribution predicts the survival probability of a fraction of samples with a certain volume that survive a tensile stress sigma, and is often used to better assess the success of a ceramic in avoiding fracture.

## Notable fracture failures

Failures caused by brittle fracture have not been limited to any particular category of engineered structure.[3] Though brittle fracture is less common than other types of failure, the impacts to life and property can be more severe.[3] The following notable historic failures were attributed to brittle fracture:

## Notes

1. A simple load-controlled tensile situation would be to support a specimen from above, and hang a weight from the bottom end. The load on the specimen is then independent of its deformation.
2. A simple displacement-controlled tensile situation would be to attach a very stiff jack to the ends of a specimen. As the jack extends, it controls the displacement of the specimen; the load on the specimen is dependent on the deformation.

## References

1. Cherepanov, G.P., Mechanics of Brittle Fracture
2. Degarmo, E. Paul; Black, J T.; Kohser, Ronald A. (2003), Materials and Processes in Manufacturing (9th ed.), Wiley, p. 32, ISBN 0-471-65653-4.
3. Rolfe, John M. Barsom, Stanley T. (1999). Fracture and fatigue control in structures : applications of fracture mechanics (3. ed.). West Conshohocken, Pa.: ASTM. ISBN 0803120826.
4. Campbell, edited by F.C. (2012). Fatigue and fracture : understanding the basics. Materials Park, Ohio: ASM International. ISBN 1615039767.CS1 maint: extra text: authors list (link)
5. C. H. Chen; H. P. Zhang; J. Niemczura; K. Ravi-Chandar; M. Marder (November 2011). "Scaling of crack propagation in rubber sheets". Europhysics Letters. 96 (3): 36009. Bibcode:2011EL.....9636009C. doi:10.1209/0295-5075/96/36009.
6. Perez, Nestor (2016). Fracture Mechanics (2nd ed.). Springer. ISBN 3319249975.
7. Jin, C.T. Sun, Z.-H. (2012). Fracture mechanics. Waltham, MA: Academic Press. ISBN 9780123850010.
8. EFM - Stress concentration at notches a closer look
9. Courtney, Thomas H. (2000), Mechanical behavior of materials (3nd ed.), McGraw Hill, ISBN 1-57766-425-6.
• Alireza Bagher Shemirani, Haeri, H., Sarfarazi,V., Hedayat, A., A review paper about experimental investigations on failure behaviour of non-persistent joint.Geomechanics and Engineering, Vol. 13, No. 4, (2017), 535–570,
• Dieter, G. E. (1988) Mechanical Metallurgy ISBN 0-07-100406-8
• A. Garcimartin, A. Guarino, L. Bellon and S. Cilberto (1997) " Statistical Properties of Fracture Precursors ". Physical Review Letters, 79, 3202 (1997)
• Callister, Jr., William D. (2002) Materials Science and Engineering: An Introduction. ISBN 0-471-13576-3
• Peter Rhys Lewis, Colin Gagg, Ken Reynolds, CRC Press (2004), Forensic Materials Engineering: Case Studies.