Fracture

Stress vs. strain curve typical of aluminum
1. Ultimate tensile strength
2. Yield strength
3. Proportional limit stress
4. Fracture
5. Offset strain (typically 0.2%)

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.

There are two types of fractures :

Brittle fracture

Brittle fracture in glass

Fracture of an aluminum crank arm of a bicycle, where Bright= brittle fracture, Dark= fatigue 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 (K_c), 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

Ductile failure of a specimen strained axially

Schematic representation of the steps in ductile fracture (in pure tension)

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. The extensive plasticity causes the crack to propagate slowly due to the absorption of a large amount of energy before fracture.[6][7]

Ductile fracture surface of 6061-T6 aluminum

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. Voids typically coalesce around precipitates, secondary phases, inclusions, and at grain boundaries in the material. Ductile fracture is typically transgranular and deformation due to dislocation slip can cause the shear lip characteristic of cup and cone fracture.[8]

There are three standard conventions for defining relative displacements in elastic materials in order to analyze crack propagation[3] as proposed by Irwin.[9] In addition fracture can involve uniform strain or a combination of these modes.[4]

Fracture crack separation modes

• 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 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.[10] 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 ${\textstyle \mathrm {K} _{\mathrm {c} }}$.[11]

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.

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.[11] 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.[11] 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.

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:

• Pressure vessels: Great Molasses Flood in 1919,[3] New Jersey molasses tank failure in 1973[4]
• Bridges: King Street Bridge span collapse in 1962, Silver Bridge collapse in 1967,[3] partial failure of the Hoan Bridge in 2000
• Ships: Titanic in 1912,[4] Liberty ships during World War II,[3] SS Schenectady in 1943[4]

• 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.

• Virtual museum of failed products at http://materials.open.ac.uk/mem/index.html
• Fracture and Reconstruction of a Clay Bowl
• Ductile fracture