{{Short description|Material ability to absorb energy and plastically deform without fracturing}} {{About|toughness of physical objects|the mathematical concept in graph theory|Graph toughness}} {{Use dmy dates|date=November 2017}} {{more citations needed|date=January 2011}} [[File:Toughness area under curve.svg|thumb|Toughness as defined by the area under the stress–strain curve for one unit volume of the material.]] 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.nde-ed.org/EducationResources/CommunityCollege/Materials/Mechanical/Toughness.htm "Toughness"], [http://www.nde-ed.org/index_flash.htm NDT Education Resource Center], Brian Larson, editor, 2001–2011, The Collaboration for NDT Education, [[Iowa State University]]</ref> Toughness is the strength with which the material opposes rupture. One definition of material toughness is the amount of energy per unit volume that a material can absorb before [[rupture (engineering)|rupturing]]. This measure of toughness is different from that used for [[fracture toughness]], which describes the capacity of materials to resist fracture.<ref>{{Cite book|last1=Askeland|first1=Donald R.|title=The science and engineering of materials|last2=Wright|first2= Wendelin J.|author2-link=Wendelin Wright|date=January 2015 |isbn=978-1-305-07676-1|edition= Seventh|location=Boston, MA|pages=208|oclc=903959750}}</ref> Toughness requires a balance of [[Strength of materials|strength]] and [[ductility]].<ref name=NDT/>
==Toughness and strength== 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, [[brittleness|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.
==Mathematical definition== Toughness can be determined by [[Integration (mathematics)|integrating]] the stress-strain curve.<ref name=NDT/> It is the energy of mechanical deformation per unit volume prior to fracture. The explicit mathematical description is:<ref>{{Cite book|last=Soboyejo|first=W. O.|title=Mechanical properties of engineered materials|date=2003|publisher=Marcel Dekker|isbn=0-8247-8900-8|chapter=12.3 Toughness and Fracture Process Zone|oclc=300921090}}</ref>
<math display="block"> \tfrac{\mbox{energy}}{\mbox{volume}} = \int_{0}^{\varepsilon_f} \sigma\, d\varepsilon </math>
where * <math> \varepsilon </math> is strain * <math> \varepsilon_f </math> is the strain upon failure * <math> \sigma </math> is stress
If the upper limit of integration up to the yield point is restricted, the energy absorbed per unit volume is known as the [[resilience (materials science)|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 of elasticity. That is,
{{block indent|1=Modulus of resilience = {{sfrac|Yield stress<sup>2</sup>|2 (Young's modulus)}}}}
==Toughness tests== The toughness of a material can be measured using a small specimen of that material. A typical testing machine uses a pendulum to deform a notched specimen of defined cross-section. The height from which the pendulum fell, minus the height to which it rose after deforming the specimen, multiplied by the weight of the pendulum, is a measure of the energy absorbed by the specimen as it was deformed during the [[impact (mechanics)|impact]] with the pendulum. The [[Charpy impact test|Charpy]] and [[Izod impact strength test|Izod]] notched impact strength tests are typical [[ASTM]] tests used to determine toughness.
==Unit of toughness== Tensile toughness (or ''deformation energy'', ''U''<sub>T</sub>) is measured in units of [[joule]] per cubic metre (J·m<sup>−3</sup>), or equivalently newton-metre per cubic metre (N·m·m<sup>−3</sup>), in the [[SI]] system and inch-[[pound-force]] per cubic inch (in·lbf·in<sup>−3</sup>) in [[US customary units]]: *1.00 N·m·m<sup>−3</sup> ≃ {{gaps|0.000|145}} in·lbf·in<sup>−3</sup> * 1.00 in·lbf·in<sup>−3</sup> ≃ 6.89 kN·m·m<sup>−3</sup>.
In the [[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|first1=O. |last1=Balkan |first2=H. |last2=Demirer|title=Mechanical properties of glass bead- and wollastonite-filled isotactic-polypropylene composites modified with thermoplastic elastomers|url=https://4spepublications.onlinelibrary.wiley.com/doi/10.1002/pc.20953|journal=Polymer Composites |volume=31|pages=1285–1308|year=2010|issue=7 |issn=1548-0569|doi=10.1002/pc.20953|url-access=subscription}}</ref>
*''U''<sub>T</sub> = Area underneath the stress–strain (''σ''–''ε'') curve = ''σ'' × ''ε'' *''U''<sub>T</sub> [=] F/A × ΔL/L = (N·m<sup>−2</sup>)·(unitless) *''U''<sub>T</sub> [=] N·m·m<sup>−3</sup> *''U''<sub>T</sub> [=] J·m<sup>−3</sup>
== Toughest material == An alloy made of almost equal amounts of [[chromium]], [[cobalt]], and [[nickel]] (CrCoNi) is the toughest material discovered thus far.<ref>{{cite web |last1=Kovner |first1=Aliyah |title=Say Hello to the Toughest Material on Earth |url=https://newscenter.lbl.gov/2022/12/08/say-hello-to-the-toughest-material-on-earth/ |website=News From Berkeley Lab |publisher=Berkeley Lab |access-date=4 September 2024 |date=8 December 2022}}</ref> It resists fracturing even at incredibly cold temperatures close to absolute zero. It is being considered as a material to be used in building spacecraft.<ref>{{Cite web |last=Sparkes |first=Matthew |date=14 December 2022 |title=Toughest material ever is an alloy of chromium, cobalt and nickel |url=https://www.newscientist.com/article/2350789-toughest-material-ever-is-an-alloy-of-chromium-cobalt-and-nickel/ |access-date=2023-03-18 |website=New Scientist |language=en-US}}</ref>
==See also== * [[Hardness]] * [[Rubber toughening]] * [[Shock (mechanics)]] * [[Tablet hardness testing]]
==References== {{Reflist}}
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[[Category:Continuum mechanics]] [[Category:Materials science]]