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{{about|toughness of physical objects|the mathematical concept in graph theory|graph toughness}}
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{{refimprove|date=January 2011}}
 
In [[materials science]] and [[metallurgy]], '''toughness''' is the ability of a material to absorb energy and plastically deform without fracturing.<ref name=NDT>[http://www.ndt-ed.org/EducationResources/CommunityCollege/Materials/Mechanical/Toughness.htm "Toughness"], [http://www.ndt-ed.org/index_flash.htm NDT Education Resource Center], Brian Larson, Editor, 2001-2011, The Collaboration for NDT Education, [[Iowa State University]]</ref>  
One definition of material toughness is the amount of [[energy]] per volume that a material can absorb before [[rupture (engineering)|rupturing]]. It is also defined as the resistance to [[fracture]] of a material when [[stress (physics)|stressed]].
 
Toughness requires a balance of [[Strength of materials|strength]] and [[ductility]].<ref name=NDT/>
 
==Mathematical definition==
Toughness can be determined by measuring the area (i.e., by taking the integral) underneath the [[stress-strain curve]]<ref name=NDT/> and its energy of mechanical deformation per unit volume prior to fracture. The explicit mathematical description is:
 
: <math> \frac{\mbox{energy}}{\mbox{volume}} = \int_{0}^{\epsilon_f} \sigma\, d\epsilon </math>
 
Where
* <math> \epsilon_{} </math> is strain
* <math> \epsilon_f </math> is the strain upon failure
* <math> \sigma </math> 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 limit of integration up to the yield point is restricted, then the energy absorbed per unit volume is known as the [[resilience|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 impact test|Charpy]] and [[Izod impact strength test|Izod impact tests]].
 
==Unit of toughness==
Tensile toughness (aka ''deformation energy'', ''U''<sub>T</sub>) is measured in units of [[joule]] per [[cube metre]] (J·m<sup>–3</sup>) in the [[SI]] system and inch-[[pound-force]] per [[cube inch]] (in·lbf·in<sup>–3</sup>) in [[US customary units]], ''i.e.'' deformation energy per volume of test specimen (merely for gage-length part).
 
In terms of [[SI]] system, the unit of tensile toughness can be easily calculated by using area underneath the stress–strain (''σ''–''ε'') curve, which gives tensile toughness value, as given below:<ref>{{cite journal|author=O.Balkan and H.Demirer|title=Polym. Compos.|volume=31|page=1285|year=2010|ISSN=1548-0569}}</ref>
 
''U''<sub>T</sub> = Area underneath the stress–strain (''σ''–''ε'') curve = ''σ'' × ''ε''
 
''U''<sub>T</sub> = MPa × % = (N·m<sup>−2</sup>·10<sup>6</sup>)·(m·m<sup>–1</sup>·10<sup>–2</sup>)
 
''U''<sub>T</sub> = N·m·m<sup>–3</sup>·10<sup>4</sup>
 
'''''U''<sub>T</sub> = J·m<sup>–3</sup>·10<sup>4</sup>'''
 
==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.
 
==See also==
* [[Fracture toughness]]
* [[Graph toughness]]
* [[Hardness]]
* [[Impact (mechanics)]]
* [[Resilience]]
* [[Shock (mechanics)]]
* [[Tablet hardness testing]]
 
==References==
{{reflist}}
 
[[Category:Continuum mechanics]]
[[Category:Materials science]]
[[Category:Physical quantities]]

Latest revision as of 18:22, 18 June 2014

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