# Toughness

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This article is about toughness of physical objects. For the mathematical concept in graph theory, see graph toughness.

In materials science and metallurgy, toughness is the ability of a material to absorb energy and plastically deform without fracturing.[1] One definition of material toughness is the amount of energy per volume that a material can absorb before rupturing. It is also defined as the resistance to fracture of a material when stressed.

Toughness requires a balance of strength and ductility.[1]

## Mathematical definition

It can be determined by integrating the stress-strain curve[1] (i.e., finding the area underneath) and its energy of mechanical deformation per unit volume prior to fracture. The explicit mathematical description is:

$\frac{\mbox{energy}}{\mbox{volume}} = \int_{0}^{\epsilon_f} \sigma\, d\epsilon$

Where

• $\epsilon_{}$ is strain
• $\epsilon_f$ is the strain upon failure
• $\sigma$ is stress

Another definition is the ability to absorb mechanical (or kinetic) energy up to failure. The area covered under stress strain curve is called toughness.

If the upper limits of integration up to the yield point is restricted, then the energy absorbed per unit volume is known as the modulus of resilience. Mathematically, the modulus of resilience can be expressed by the product of the square of the yield stress divided by two times the Young's modulus.

## Toughness tests

Tests can be done by using a pendulum and basic physics to measure how much energy it will hold when released from a particular height. By having a sample at the bottom of its swing, a measure of toughness can be found, as in the Charpy and Izod impact tests.

## Unit of toughness

Tensile toughness (or, deformation energy, UT) is measured in units of joule per cubic metre (J·m–3) in the SI system and inch-pound-force per cubic inch (in·lbf·in–3) in US customary units; i.e.. deformation energy per volume of test specimen (merely for gage-length part). 1.00 N·m/m−30.000145 in·lbF·in−3 and 1.00 in·lbF·in−3 ≃ 6.89 kN·m/m−3.

In terms of SI system, the unit of tensile toughness can be easily calculated by using area underneath the stress–strain (σε) curve, which gives tensile toughness value, as given below:[2]

UT = Area underneath the stress–strain (σε) curve = σ × ε
UT = MPa × % = (N·m−2·106)·(m·m−1·10−2)
UT = N·m·m−3·104
UT = J·m−3·104

## Toughness and strength

Toughness is also defined as area of stress-strain diagram. Toughness is related to the area under the stress-strain curve. In order to be tough, a material must be both strong and ductile. For example, brittle materials (like ceramics) that are strong but with limited ductility are not tough; conversely, very ductile materials with low strengths are also not tough. To be tough, a material should withstand both high stresses and high strains. Generally speaking, strength indicates how much force the material can support, while toughness indicates how much energy a material can absorb before rupturing.

## References

1. ^ a b c "Toughness", NDT Education Resource Center, Brian Larson, Editor, 2001-2011, The Collaboration for NDT Education, Iowa State University
2. ^ O.Balkan and H.Demirer (2010). Polym. Compos. 31. p. 1285. ISSN 1548-0569.