Editing Ductility (section)

==Quantification==
===Basic definitions===
The quantities commonly used to define ductility in a tension test are [[relative elongation]] (in percent, sometimes denoted as <math>\varepsilon_{\mathrm f}</math>) and reduction of area (sometimes denoted as <math>q</math>) at fracture.<ref name="dieter">Dieter, G. (1986) ''Mechanical Metallurgy'', McGraw-Hill, {{ISBN|978-0-07-016893-0}}</ref> Fracture strain is the [[Deformation (physics)#Engineering strain|engineering strain]] at which a test specimen fractures during a [[Index ellipsoid|uniaxial]] [[tensile test]]. Percent elongation, or engineering strain at fracture, can be written as:<ref name=DR>{{Cite web|title=Ductility Review - Strength Mechanics of Materials - Engineers Edge|url=https://www.engineersedge.com/material_science/ductility.htm|access-date=2020-07-14|website=www.engineersedge.com}}</ref><ref name=":1">{{Cite book|last=Askeland|first=Donald R.|url=https://www.worldcat.org/oclc/903959750|title=The science and engineering of materials|others=Wright, Wendelin J.|year=2016|isbn=978-1-305-07676-1|edition=Seventh|location=Boston, MA|pages=195|chapter=6-4 Properties Obtained from the Tensile Test|oclc=903959750}}</ref><ref name=":2">{{Cite book|last=Callister|first=William D. Jr.|url=https://www.worldcat.org/oclc/401168960|title=Materials science and engineering : an introduction.|others=Rethwisch, David G.|year=2010|isbn=978-0-470-41997-7|edition=8th|location=Hoboken, NJ|pages=166|chapter=6.6 Tensile Properties|oclc=401168960}}</ref>

:<math>\%\mathrm{EL} = \frac{\text{final gauge length - initial gauge length}}{\text{initial gauge length}} = \frac{l_{\mathrm f} - l_0}{l_0} \cdot 100</math>

Percent reduction in area can be written as:<ref name=":1" /><ref name=":2" />

:<math>\%\mathrm{RA} = \frac{\text{change in area}}{\text{original area}} = \frac{A_0 - A_{\mathrm f}}{A_0} \cdot 100</math>

where the area of concern is the cross-sectional area of the gauge of the specimen.

According to ''Shigley's Mechanical Engineering Design'',<ref name="Shigley2"/> 'significant' denotes about 5.0 percent elongation.

=== Effect of sample dimensions ===
An important point concerning the value of the ductility (nominal strain at failure) in a tensile test is that it commonly exhibits a dependence on sample dimensions.  However, a universal parameter should exhibit no such dependence (and, indeed, there is no dependence for properties such as stiffness, yield stress and ultimate tensile strength).  This occurs because the measured strain (displacement) at fracture commonly incorporates contributions from both the uniform deformation occurring up to the onset of necking and the subsequent deformation of the neck (during which there is little or no deformation in the rest of the sample).  The significance of the contribution from neck development depends on the "aspect ratio" (length / diameter) of the gauge length, being greater when the ratio is low. This is a simple geometric effect, which has been clearly identified.  There have been both experimental studies<ref name="Matic">{{cite journal |last1=Matic |first1=P |title=The Relation of Tensile Specimen Size and Geometry Effects to Unique Constitutive Parameters for Ductile Materials |journal= Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences|date=1988 |volume=417 |issue=1853 |pages=309–333 |doi=10.1098/rspa.1988.0063|bibcode=1988RSPSA.417..309M |s2cid=43033448 }}</ref> and theoretical explorations<ref name="Havner">{{cite journal |last1=Havner |first1=K |title=On the Onset of Necking in the Tensile Test |journal=International Journal of Plasticity |date=2004 |volume=20 |issue=4–5 |pages=965–978 |doi=10.1016/j.ijplas.2003.05.004}}</ref><ref name="Kim">{{cite journal |last1=Kim |first1=H |title=Finite Element Analysis of the Onset of Necking and the Post-Necking Behaviour During Uniaxial Tensile Testing |journal=Materials Transactions |date=2005 |volume=46 |issue=10 |pages=2159–2163 |doi=10.2320/matertrans.46.2159|doi-access=free }}</ref><ref name="Joun">{{cite journal |last1=Joun |first1=M |title=Finite Element Analysis of Tensile Testing with Emphasis on Necking |journal=Computational Materials Science |date=2007 |volume=41 |issue=1 |pages=63–69 |doi=10.1016/j.commatsci.2007.03.002}}</ref><ref name="Osovski">{{cite journal |last1=Osovski |first1=S |title=Dynamic Tensile Necking: Influence of Specimen Geometry and Boundary Conditions. |journal=Mechanics of Materials |date=2013 |volume=62 |pages=1–13 |doi=10.1016/j.mechmat.2013.03.002|bibcode=2013MechM..62....1O |hdl=10016/17020 |hdl-access=free }}</ref> of the effect, mostly based on [[Finite Element Method]] (FEM) modelling. Nevertheless, it is not universally appreciated and, since the range of sample dimensions in common use is quite wide, it can lead to highly significant variations (by factors of up to 2 or 3) in ductility values obtained for the same material in different tests.

A more meaningful representation of ductility would be obtained by identifying the strain at the onset of necking, which should be independent of sample dimensions. This point can be difficult to identify on a (nominal) stress-strain curve, because the peak (representing the onset of necking) is often relatively flat. Moreover, some (brittle) materials fracture before the onset of necking, such that there is no peak.  In practice, for many purposes it is preferable to carry out a different kind of test, designed to evaluate the toughness (energy absorbed during fracture), rather than use ductility values obtained in tensile tests.

In an absolute sense, "ductility" values are therefore virtually meaningless.  The actual (true) strain in the neck at the point of fracture bears no direct relation to the raw number obtained from the nominal stress-strain curve; the true strain in the neck is often considerably higher.  Also, the true stress at the point of fracture is usually higher than the apparent value according to the plot.  The load often drops while the neck develops, but the sectional area in the neck is also dropping (more sharply), so the true stress there is rising.  There is no simple way of estimating this value, since it depends on the geometry of the neck. While the true strain at fracture is a genuine indicator of "ductility", it cannot readily be obtained from a conventional tensile test.

The Reduction in Area (RA) is defined as the decrease in sectional area at the neck (usually obtained by measurement of the diameter at one or both of the fractured ends), divided by the original sectional area.  It is sometimes stated that this is a more reliable indicator of the "ductility" than the elongation at failure (partly in recognition of the fact that the latter is dependent on the aspect ratio of the gauge length, although this dependence is far from being universally appreciated).  There is something in this argument, but the RA is still some way from being a genuinely meaningful parameter.  One objection is that it is not easy to measure accurately, particularly with samples that are not circular in section.  Rather more fundamentally, it is affected by both the uniform plastic deformation that took place before necking and by the development of the neck.  Furthermore, it is sensitive to exactly what happens in the latter stages of necking, when the true strain is often becoming very high and the behavior is of limited significance in terms of a meaningful definition of strength (or toughness).  There has again been extensive study of this issue.<ref name="Choung">{{cite journal |last1=Choung |first1=J |title=Study on True Stress Correction from Tensile Tests |journal=Journal of Mechanical Science and Technology |date=2008 |volume=22 |issue=6 |pages=1039–1051 |doi=10.1007/s12206-008-0302-3|s2cid=108776720 }}</ref><ref name="Ho">{{cite journal |last1=Ho |first1=H |title=Modelling Tensile Tests on High Strength S690 Steel Materials Undergoing Large Deformations |journal=Engineering Structures |date=2019 |volume=192 |pages=305–322 |doi=10.1016/j.engstruct.2019.04.057|bibcode=2019EngSt.192..305H |s2cid=182744244 |hdl=10397/101163 |hdl-access=free }}</ref><ref name="Samuel">{{cite journal |last1=Samuel |first1=E |title=Inter-Relation between True Stress at the Onset of Necking and True Uniform Strain in Steels - a Manifestation of Onset to Plastic Instability |journal=Materials Science and Engineering A-Structural Materials Properties Microstructure and Processing |date=2008 |volume=480 |issue=1–2 |pages=506–509 |doi=10.1016/j.msea.2007.07.074}}</ref>