Yield (engineering)
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Fernando Estel
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{{Mechanical failure modes}}
The '''yield strength''' or '''yield point''' of a [[material]] is defined in [[engineering]] and [[materials science]] as the [[Stress (physics)|stress]] at which a material begins to [[Plasticity (physics)|deform plastically]]. Prior to the yield point the material will deform [[Elasticity (physics)|elastically]] and will return to its original shape when the applied stress is removed. Once the yield point is passed some fraction of the deformation will be permanent and non-reversible.
In the three-dimensional space of the principal stresses (<math> \sigma_1, \sigma_2 , \sigma_3</math>), an infinite number of yield points form together a [[yield surface]].
Knowledge of the yield point is vital when designing a component since it generally represents an upper limit to the load that can be applied. It is also important for the control of many materials production techniques such as [[forging]], [[Rolling (metalworking)|rolling]], or [[Machine press|pressing]]. In structural engineering, this is a soft failure mode which does not normally cause [[catastrophic failure]] or [[ultimate failure]] unless it accelerates [[buckling]].
==Definition==
[[Image:Metal yield.svg|thumb|right|300px|Typical yield behavior for non-ferrous alloys.<br>
1: True elastic limit<br>
2: Proportionality limit<br>
3: Elastic limit <br>
4: Offset yield strength]]
It is often difficult to precisely define yielding due to the wide variety of [[stress–strain curve]]s exhibited by real materials. In addition, there are several possible ways to define yielding<ref name=dieter>G. Dieter, ''Mechanical Metallurgy'', McGraw-Hill, 1986</ref>:
; True elastic limit: The lowest stress at which [[dislocation]]s move. This definition is rarely used, since dislocations move at very low stresses, and detecting such movement is very difficult.
; Proportionality limit : Up to this amount of stress, stress is proportional to strain ([[Hooke's law]]), so the stress-strain graph is a straight line, and the gradient will be equal to the [[elastic modulus]] of the material.
; Elastic limit (yield strength): Beyond the elastic limit, permanent deformation will occur. The lowest stress at which permanent deformation can be measured. This requires a manual load-unload procedure, and the accuracy is critically dependent on equipment and operator skill. For [[elastomer]]s, such as [[rubber]], the elastic limit is much larger than the proportionality limit. Also, precise strain measurements have shown that plastic strain begins at low stresses. <ref>{{cite book |title= Engineering Materials and their Applications|last= Flinn|first= Richard A.|coauthors= Trojan, Paul K.|year= 1975|publisher= Houghton Mifflin Company|location= Boston|isbn= 0-395-18916-0|pages= 61}} </ref><ref>{{cite journal
| last = Kumagai
| first = Naoichi
| coauthors = Sadao Sasajima, Hidebumi Ito
| title = Long-term Creep of Rocks: Results with Large Specimens Obtained in about 20 Years and Those with Small Specimens in about 3 Years
| journal = Journal of the Society of Materials Science (Japan)
| volume = 27
| issue = 293
| pages = 157–161
| publisher = Japan Energy Society
| url = http://translate.google.com/translate?hl=en&sl=ja&u=http://ci.nii.ac.jp/naid/110002299397/&sa=X&oi=translate&resnum=4&ct=result&prev=/search%3Fq%3DIto%2BHidebumi%26hl%3Den
| date = 15 February 1978
| accessdate = 06-16-2008}}</ref>
; Offset yield point (proof stress) : This is the most widely used strength measure of metals, and is found from the stress-strain curve as shown in the figure to the right. A plastic strain of 0.2% is usually used to define the offset yield stress, although other values may be used depending on the material and the application. The offset value is given as a subscript, e.g. R<sub>p0.2</sub>=310 MPa. In some materials there is essentially no linear region and so a certain value of strain is defined instead. Although somewhat arbitrary, this method does allow for a consistent comparison of materials.
; Upper yield point and lower yield point: Some metals, such as mild steel, reach an upper yield point before dropping rapidly to a lower yield point. The material response is linear up until the upper yield point, but the lower yield point is used in structural engineering as a conservative value.
==Yield criterion==
A yield criterion, often expressed as [[yield surface]], or yield locus, is an hypothesis concerning the limit of elasticity under any combination of stresses. There are two interpretations of yield criterion: one is purely mathematical in taking a statistical approach while other models attempt to provide a justification based on established physical principles. Since stress and strain are [[tensor]] qualities they can be described on the basis of three principal directions, in the case of stress these are denoted by <math> \sigma_1 \,\!</math>, <math> \sigma_2 \,\!</math> and <math> \sigma_3 \,\!</math>.
The following represent the most common yield criterion as applied to an isotropic material (uniform properties in all directions). Other equations have been proposed or are used in specialist situations.
=== Isotropic yield criteria ===
'''Maximum Principal Stress Theory''' - Yield occurs when the largest principal stress exceeds the uniaxial tensile yield strength. Although this criterion allows for a quick and easy comparison with experimental data it is rarely suitable for design purposes.
: <math>\ \sigma_1 \le \sigma_y \,\!</math>
''' Maximum Principal Strain Theory''' - Yield occurs when the maximum principal [[strain (materials science)|strain]] reaches the strain corresponding to the yield point during a simple tensile test. In terms of the principal stresses this is determined by the equation:
: <math>\ \sigma_1 - \nu(\sigma_2 + \sigma_3) \le \sigma_y. \,\!</math>
'''Maximum Shear Stress Theory''' - Also known as the [[Tresca yield criterion]], after the French scientist [[Henri Tresca]]. This assumes that yield occurs when the shear stress <math>\tau\!</math> exceeds the shear yield strength <math>\tau_y\!</math>:
: <math>\ \tau = \frac{\sigma_1-\sigma_3}{2} \le \tau_{ys}. \,\!</math>
'''Total Strain Energy Theory''' - This theory assumes that the stored energy associated with elastic deformation at the point of yield is independent of the specific stress tensor. Thus yield occurs when the strain energy per unit volume is greater than the strain energy at the elastic limit in simple tension. For a 3-dimensional stress state this is given by:
: <math>\ \sigma_{1}^2 + \sigma_{2}^2 + \sigma_{3}^2 - 2 \nu (\sigma_1 \sigma_2 + \sigma_2 \sigma_3 + \sigma_1 \sigma_3) \le \sigma_y^2. \,\! </math>
'''Distortion Energy Theory''' - This theory proposes that the total strain energy can be separated into two components: the ''volumetric'' ([[hydrostatic]]) strain energy and the ''shape'' (distortion or [[Shearing (physics)|shear]]) strain energy. It is proposed that yield occurs when the distortion component exceeds that at the yield point for a simple tensile test. This is generally referred to as the [[Von Mises yield criterion]] and is expressed as:
: <math>\ \frac{1}{2} \Big[ (\sigma_1 - \sigma_2)^2 + (\sigma_2 - \sigma_3)^2 + (\sigma_3 - \sigma_1)^2 \Big] \le \ \sigma_y^2. \,\! </math>
Based on a different theoretical underpinning this expression is also referred to as '''octahedral shear stress theory'''.
Other commonly used isotropic yield criteria are the
* [[Mohr-Coulomb theory|Mohr-Coulomb yield criterion]]
* [[Drucker Prager|Drucker-Prager yield criterion]]
* [[Bresler Pister yield criterion|Bresler-Pister yield criterion]]
=== Anisotropic yield criteria ===
When a metal is subjected to large plastic deformations the grain sizes and orientations change in the direction of deformation. As a result the plastic yield behavior of the material shows directional dependency. Under such circumstances, the isotropic yield criteria such as the von Mises yield criterion are unable to predict the yield behavior accurately. Several anisotropic yield criteria have been developed to deal with such situations.
Some of the more popular anisotropic yield criteria are:
* [[Hill yield criteria|Hill's quadratic yield criterion]].
* [[Hill yield criteria|Generalized Hill yield criterion]].
* [[Hosford yield criterion]].
==Factors influencing yield stress==
The stress at which yield occurs is dependent on both the rate of deformation (strain rate) and, more significantly, the temperature at which the deformation occurs. Early work by Alder and Philips in 1954 found that the relationship between yield stress and strain rate (at constant temperature) was best described by a power law relationship of the form
: <math> \sigma_y = C (\dot{\epsilon})^m \,\! </math>
where C is a constant and m is the strain rate sensitivity. The latter generally increases with temperature, and materials where m reaches a value greater than ~0.5 tend to exhibit [[super plastic behaviour]].
Later, more complex equations were proposed that simultaneously dealt with both temperature and strain rate:
: <math> \sigma_y = \frac{1}{\alpha} \sinh^{-1} \left [ \frac{Z}{A} \right ]^{(1/n)} \,\! </math>
where α and A are constants and Z is the temperature-compensated strain-rate - often described by the Zener-Hollomon parameter:
: <math> Z = (\dot{\epsilon}) \exp \left ( \frac{Q_{HW}}{RT} \right ) \,\! </math>
where Q<sub>HW</sub> is the activation energy for hot deformation and T is the absolute temperature.
====Strengthening Mechanisms====
There are several ways in which crystalline and amorphous materials can be engineered to increase their yield strength. By altering dislocation density, impurity levels, grain size (in crystalline materials), the yield strength of the material can be fine tuned. This occurs typically by introducing defects such as impurities dislocations in the material. To move this defect (plastically deforming or yielding the material), a larger stress must be applied. This thus causes a higher yield stress in the material. While many material properties depend only on the composition of the bulk material, yield strength is extremely sensitive to the materials processing as well for this reason.
These mechanisms for crystalline materials include:
1. [[Work hardening|Work Hardening]] - Where machining the material will introduce [[Dislocation|dislocations]], which increases their density in the material. This increases the yield strength of the material, since now more stress must be applied to move these dislocations through a crystal lattice. Dislocations can also interact with each other, becoming entangled.
The governing formula for this mechanism is:
: <math> \Delta\sigma_y = Gb \sqrt{\rho} </math>
where <math> \sigma_y </math> is the yield stress, G is the shear elastic modulus, b is the magnitude of the [[Burgers vector]], and <math> \rho </math> is the dislocation density.
2. [[Solid solution strengthening|Solid Solution Strengthening]] - By [[Alloy|alloying]] the material, impurity atoms in low concentrations will occupy a lattice position directly below a dislocation, such as directly below an extra half plane defect. This relieves a tensile strain directly below the dislocation by filling that empty lattice space with the impurity atom.
The relationship of this mechanism goes as:
: <math> \Delta\tau = Gb\sqrt{C_s}\epsilon^{3/2} </math>
where <math> \tau </math> is the [[Shear stress|shear stress]], related to the yield stress, G and b are the same as in the above example, C_s is the concentration of solute and <math> \epsilon </math> is the strain induced in the lattice due to adding the impurity.
3. [[Precipitation strengthening| Particle/Precipitate Strengthening]] - Where the presence of a secondary phase will increase yield strength by blocking the motion of dislocations within the crystal. A line defect that, while moving through the matrix, will be forced against a small particle or precipitate of the material. Dislocations can move through this particle either by shearing the particle, or by a process known as bowing or ringing, in which a new ring of dislocations is created around the particle.
The shearing formula goes as:
<math> \Delta\tau = \cfrac{r_{particle}}{l_{interparticle}} \gamma_{particle-matrix} </math>
and the bowing/ringing formula:
<math> \Delta\tau = \cfrac{Gb}{l_{interparticle}-2r_{particle}} </math>
In these formulas, <math> r_{particle} </math> is the particle radius, <math> \gamma_{particle-matrix} </math> is the surface tension between the matrix and the particle, <math> l_{interparticle} </math> is the distance between the particles.
4. [[Grain boundary strengthening]] - Where a buildup of dislocations at a grain boundary causes a repulsive force between dislocations. As grain size decreases, the surface area to volume ratio of the grain increases, allowing more buildup of dislocations at the grain edge. Since it requires a lot of energy to move dislocations to another grain, these dislocations build up along the boundary, and increase the yield stress of the material. Also known as Hall-Petch strenghthening, this type of strengthening is governed by the formula:
:<math> \sigma_y = \sigma_0 + kd^{-1/2} \,</math>
where
:<math> \sigma_0 </math> is the stress required to move dislocations,
:k is a material constant, and
:d is the grain size.
==Implications for structural engineering==
Yielded structures have a lower stiffness, leading to increased deflections and decreased buckling strength. The structure will be permanently deformed when the load is removed, and may have residual stresses. Engineering metals display strain hardening, which implies that the yield stress is increased after unloading from a yield state. Highly optimized structures, such as airplane beams and components, rely on yielding as a fail-safe failure mode. No safety factor is therefore needed when comparing limit loads (the highest loads expected during normal operation) to yield criteria.{{Fact|date=September 2007}}
==Typical yield strength==
Note: many of the values depend on manufacturing process and purity/composition.
{| border="1" cellpadding="2" cellspacing="0"
|-
! Material !! Yield strength<br> ([[Mega|M]][[Pascal (unit)|Pa]]) !! Ultimate strength<br> (MPa) !! Density<br> (g/cm³)
|-
| Structural [[steel]] ASTM [[A36 steel]] || 250 || 400 || 7.8
|-
| Steel, API 5L X65 (Fikret Mert Veral) || 448 || 531 || 7.8
|-
| Steel, high strength alloy ASTM [[A514]] || 690 || 760 || 7.8
|-
| [[High tensile steel|Steel, prestressing strands]] || 1650 || 1860 || 7.8
|-
| Steel Wire || || || 7.8
|-
| Steel (AISI 1060 0.6% carbon) [[Piano wire]] || 2200-2482 MPa<ref>[http://www.djaerotech.com/dj_askjd/dj_questions/musicwire.html Don Stackhouse @ DJ Aerotech]</ref> || || 7.8
|-
| [[High density polyethylene]] (HDPE) || 26-33 || 37 || 0.95
|-
| [[Polypropylene]] || 12-43 || 19.7-80 || 0.91
|-
| [[Stainless steel]] AISI 302 - Cold-rolled || 520 || 860 ||
|-
| [[Cast iron]] 4.5% C, ASTM A-48 || 276 (??) || 200 ||
|-
| [[Titanium alloy]] (6% Al, 4% V) || 830 || 900 || 4.51
|-
| [[Aluminium alloy]] 2014-T6 || 400 || 455 || 2.7
|-
| [[Copper]] 99.9% Cu || 70 || 220 || 8.92
|-
| [[Cupronickel]] 10% Ni, 1.6% Fe, 1% Mn, balance Cu || 130 || 350 || 8.94
|-
| [[Brass]] || approx. 200+|| 550 || 5.3
|-
| [[Tungsten]] || || 1510 || 19.25
|-
| [[Glass]] || || 50 (in compression) || 2.53
|-
| [[Fibreglass|E-Glass]] || N/A || 3450 || 2.57
|-
| [[Fibreglass|S-Glass]] || N/A || 4710 || 2.48
|-
| [[Basalt fiber]] || N/A || 4840 || 2.7
|-
| [[Marble]] || N/A || 15 ||
|-
| [[Concrete]] || N/A || 3 ||
|-
| [[Carbon Fiber]] || N/A || 5650 || 1.75
|-
| [[Spider silk]] || 1150 (??) || 1200 ||
|-
| [[Silkworm]] silk || 500 || ||
|-
| [[Aramid]] ([[Kevlar]] or [[Twaron]]) || 3620 || || 1.44
|-
| [[Ultra high molecular weight polyethylene|UHMWPE]]|| 23 || 46 || 0.97
|-
| [[Ultra high molecular weight polyethylene|UHMWPE]] fibers<ref>[http://www-lgm2b.iut.u-bordeaux1.fr/publi/PE%20fibre.pdf Tensile and creep properties of ultra high molecular weight PE fibres]</ref><ref>[http://www.mse.mtu.edu/~drjohn/my4150/props.html Mechanical Properties Data<!-- Bot generated title -->]</ref> (Dyneema or Spectra) || || 2300-3500 || 0.97
|-
| [[Vectran]] || || 2850-3340 ||
|-
| [[Pine]] [[Wood]] (parallel to grain) || || 40 ||
|-
| [[Bone]] (limb) || 104-121 || 130 ||
|-
| [[Nylon]], type 6/6 || 45 || 75 ||
|-
| [[Rubber]] || - || 15 ||
|-
| [[Boron]] || N/A || 3100 || 2.46
|-
| [[Silicon]], monocrystalline (m-Si) || N/A || 7000 || 2.33
|-
| [[Silicon carbide]] (SiC) || N/A || 3440 ||
|-
| [[Sapphire]] (Al<sub>2</sub>O<sub>3</sub>) || N/A || 1900 || 3.9-4.1
|-
| [[Carbon nanotube]] (see note above) || N/A || 62000 || 1.34
|-}
{| border="1" cellpadding="2" cellspacing="0"
|-
! Elements in the [[Annealing (metallurgy)|annealed]] state !! [[Young's modulus|Young's Modulus]]<br> (GPa) !! Proof or [[yield stress]]<br> (MPa) !! Ultimate strength<br> (MPa)
|-
| [[Aluminium]] || 70 || 15-20 || 40-50
|-
| [[Copper]] || 130 || 33 || 210
|-
| [[Gold]] || 79 || || 100
|-
| [[Iron]] || 211 || 80-100 || 350
|-
| [[Lead]] || 16 || || 12
|-
| [[Nickel]] || 170 || 14-35 || 140-195
|-
| [[Silicon]] || 107 || 5000-9000 ||
|-
| [[Silver]] || 83 || || 170
|-
| [[Tantalum]] || 186 || 180 || 200
|-
| [[Tin]] || 47 || 9-14 || 15-200
|-
| [[Titanium]] || 120 || 100-225 || 240-370
|-
| [[Tungsten]] || 411 || 550 || 550-620
|-
| [[Zinc]] (wrought) || 105 || || 110-200
|-
|}
(Source: A.M. Howatson, P.G. Lund and J.D. Todd, "Engineering Tables and Data" p41)
==See also==
* [[Piola-Kirchhoff stress tensor]]
* [[Strain tensor]]
* [[Stress-energy tensor]]
* [[Stress concentration]]
* [[Linear elasticity]]
* [[Tensile strength]]
* [[Elastic modulus]]
* [[Yield surface]]
==References==
<references/>
*{{cite book | author=Avallone, Eugene A.; & Baumeister III, Theodore | title=Mark's Standard Handbook for Mechanical Engineers | location=New York | publisher=McGraw-Hill | year=1996 | editor= | id=ISBN 0-07-004997-1}}
*{{cite book | author=Young, Warren C.; & Budynas, Richard G. | title=Roark's Formulas for Stress and Strain, 7th edition | location=New York | publisher=McGraw-Hill | year=2002 | editor= | id=ISBN 0-07-072542-X}}
* [http://www.engineershandbook.com/Materials/mechanical.htm Engineer's Handbook]
* Boresi, A. P., Schmidt, R. J., and Sidebottom, O. M. (1993). ''Advanced Mechanics of Materials'', 5th edition John Wiley & Sons. ISBN 0-471-55157-0
*Oberg, E., Jones, F. D., and Horton, H. L. (1984). ''Machinery's Handbook'', 22nd edition. Industrial Press. ISBN 0-8311-1155-0
* Shigley, J. E., and Mischke, C. R. (1989). ''Mechnical Engineering Design'', 5th edition. McGraw Hill. ISBN 0-07-056899-5
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