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| {{Mechanical failure modes}}
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| A '''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.
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| <br>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]].
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| 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]].
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| ==Definition==
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| [[File:Metal yield.svg|thumb|right|300px|Typical yield behavior for non-ferrous alloys.<br>
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| 1: True elastic limit<br>
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| 2: Proportionality limit<br>
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| 3: Elastic limit <br>
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| 4: Offset yield strength]]
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| 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>
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| ; True elastic limit
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| :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.
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| ; Proportionality limit
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| :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.
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| ; Elastic limit (yield strength)
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| :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|page= 61}}</ref><ref>{{Cite journal
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| | last = Kumagai
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| | first = Naoichi
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| | coauthors = Sadao Sasajima, Hidebumi Ito
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| | title = Long-term Creep of Rocks: Results with Large Specimens Obtained in about 20 Years and Those with Small Specimens in about 3 Years
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| | journal = Journal of the Society of Materials Science (Japan)
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| | volume = 27
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| | issue = 293
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| | pages = 157–161
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| | publisher = Japan Energy Society
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| | 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
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| | date = 15 February 1978
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| | accessdate = 2008-06-16
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| | ref = harv}}</ref>
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| ;Yield point : The point in the stress-strain curve at which the curve levels off and plastic deformation begins to occur.<ref>{{harvnb|Ross|1999|p=56}}.</ref>
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| ; Offset yield point ({{visible anchor|proof stress}}) : When a yield point is not easily defined based on the shape of the stress-strain curve an ''offset yield point'' is arbitrarily defined. The value for this is commonly set at 0.1 or 0.2% of the strain.<ref name="ross59">{{harvnb|Ross|1999|p=59}}.</ref> The offset value is given as a subscript, e.g., R<sub>p0.2</sub>=310 MPa.{{Citation needed|date=October 2010}} High strength steel and aluminum alloys do not exhibit a yield point, so this offset yield point is used on these materials.<ref name="ross59"/>
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| ; Upper yield point and lower yield point
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| :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. If a metal is only stressed to the upper yield point, and beyond, [[Lüders band]]s can develop.<ref>Degarmo, p. 377.</ref>
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| ==Yield criterion==
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| {{unreferenced section|date=June 2013}}
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| A yield criterion, often expressed as yield surface, or yield locus, is a 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>.
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| 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.
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| ===Isotropic yield criteria===
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| '''Maximum Principal Stress Theory''' – by W.J.M Rankine(1850). 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. This theory gives good predictions for brittle materials.
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| : <math>\ \sigma_1 \le \sigma_y \,\!</math>
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| ''' Maximum Principal Strain Theory''' – by St.Venant. 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:
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| : <math>\ \sigma_1 - \nu(\sigma_2 + \sigma_3) \le \sigma_y. \,\!</math>
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| '''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>:
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| : <math>\ \tau = \frac{\sigma_1-\sigma_3}{2} \le \tau_y. \,\!</math>
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| '''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:
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| : <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>
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| '''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.
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| Based on a different theoretical underpinning this expression is also referred to as '''octahedral shear stress theory'''.{{Citation needed|date=September 2011}}
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| Other commonly used isotropic yield criteria are the
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| * [[Mohr-Coulomb theory|Mohr-Coulomb yield criterion]]
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| * [[Drucker Prager|Drucker-Prager yield criterion]]
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| * [[Bresler Pister yield criterion|Bresler-Pister yield criterion]]
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| * [[Willam-Warnke yield criterion]]
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| The [[yield surface]]s corresponding to these criteria have a range of forms. However, most isotropic yield criteria correspond to [[convex polytope|convex]] yield surfaces.
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| ===Anisotropic yield criteria===
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| 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.
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| Some of the more popular anisotropic yield criteria are:
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| * [[Hill yield criteria|Hill's quadratic yield criterion]].
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| * [[Hill yield criteria|Generalized Hill yield criterion]].
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| * [[Hosford yield criterion]].
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| ==Factors influencing yield stress==
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| {{unreferenced section|date=June 2013}}
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| 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. In general, the yield strength increases with strain rate and decreases with temperature. When the latter is not the case, the material is said to exhibit [[yield strength anomaly]], which is typical for [[superalloys]] and leads to their use in applications requiring high strength at high temperatures.
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| 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
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| : <math> \sigma_y = C (\dot{\epsilon})^m \,\! </math>
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| 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]].
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| Later, more complex equations were proposed that simultaneously dealt with both temperature and strain rate:
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| : <math> \sigma_y = \frac{1}{\alpha} \sinh^{-1} \left [ \frac{Z}{A} \right ]^{(1/n)} \,\! </math>
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| where α and A are constants and Z is the temperature-compensated strain-rate – often described by the [[Zener-Hollomon parameter]]:
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| : <math> Z = (\dot{\epsilon}) \exp \left ( \frac{Q_{HW}}{RT} \right ) \,\! </math>
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| where Q<sub>HW</sub> is the activation energy for hot deformation and T is the absolute temperature.
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| ===Strengthening mechanisms===
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| 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.
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| These mechanisms for crystalline materials include
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| * [[Work hardening]]
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| * [[Solid solution strengthening]]
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| * [[Precipitation strengthening]]
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| * [[Grain boundary strengthening]]
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| ====Work hardening====
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| Where deforming the material will introduce [[dislocation]]s, 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.
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| The governing formula for this mechanism is:
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| : <math> \Delta\sigma_y = Gb \sqrt{\rho} </math>
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| 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.
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| ====Solid solution strengthening====
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| By [[alloy]]ing 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.
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| The relationship of this mechanism goes as:
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| : <math> \Delta\tau = Gb\sqrt{C_s}\epsilon^{3/2} </math>
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| where <math> \tau </math> is the [[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.
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| ====Particle/Precipitate strengthening====
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| 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.
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| The shearing formula goes as:
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| <math> \Delta\tau = \cfrac{r_{\rm{particle}}}{l_{\rm{interparticle}}} \gamma_{\rm{particle-matrix}} </math>
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| and the bowing/ringing formula:
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| <math> \Delta\tau = \cfrac{Gb}{l_{\rm{interparticle}}-2r_{\rm{particle}}} </math>
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| In these formulas, <math> r_{\rm{particle}}\, </math> is the particle radius, <math> \gamma_{\rm{particle-matrix}} \,</math> is the surface tension between the matrix and the particle, <math> l_{\rm{interparticle}} \,</math> is the distance between the particles.
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| ====Grain boundary strengthening====
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| 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 strengthening, this type of strengthening is governed by the formula:
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| :<math> \sigma_y = \sigma_0 + kd^{-1/2} \,</math>
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| where
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| :<math> \sigma_0 </math> is the stress required to move dislocations,
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| :k is a material constant, and
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| :d is the grain size.
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| ==Testing==
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| Yield strength testing involves taking a small sample with a fixed cross-section area, and then pulling it with a controlled, gradually increasing force until the sample changes shape or breaks. Longitudinal and/or transverse strain is recorded using mechanical or optical extensometers.
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| [[Indentation hardness]] correlates linearly with tensile strength for most steels.<ref>[http://www.springerlink.com/content/q86642448t84g267/ Correlation of Yield Strength and Tensile Strength with Hardness for Steels , E.J. Pavlina and C.J. Van Tyne, Journal of Materials Engineering and Performance, Volume 17, Number 6 / December, 2008]</ref> Hardness testing can therefore be an economical substitute for tensile testing, as well as providing local variations in yield strength due to e.g. welding or forming operations.
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| ==Implications for structural engineering==
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| 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.{{Citation needed|date=September 2007}}
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| ==Typical yield and ultimate strengths==
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| Note: many of the values depend on manufacturing process and purity/composition.
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| {| class="wikitable sortable" border="1"
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| |-
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| ! Material !! data-sort-type="number" | Yield strength<br>(MPa) !! data-sort-type="number" | Ultimate strength<br>(MPa) !! data-sort-type="number" | Density<br>(g/cm³) !! data-sort-type="number" | [[Free breaking length|free breaking length<br>(km)]]
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| |-
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| | ASTM [[A36 steel]] || 250 || 400 || 7.85 || 3.2
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| |-
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| | Steel, API 5L X65<ref>[http://www.ussteel.com/corp/tubular/linepipe-seamless.asp ussteel.com]</ref> || 448 || 531 || 7.85 || 5.8
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| |-
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| | Steel, high strength alloy ASTM [[A514 steel|A514]] || 690 || 760 || 7.85 || 9.0
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| |-
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| | Steel, prestressing strands || 1650 || 1860 || 7.85 || 21.6
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| |-
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| | [[Piano wire]] || || 2200–2482 <ref>[http://www.djaerotech.com/dj_askjd/dj_questions/musicwire.html Don Stackhouse @ DJ Aerotech]</ref> || 7.8 || 28.7
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| |-
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| | [[Carbon Fiber]] (CF, CFK) || || 5650 <ref>[http://www.complore.com/properties-materials-tensile-strength complore.com]</ref> || 1.75 ||
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| |-
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| | [[High density polyethylene]] (HDPE) || 26–33 || 37 || 0.95 || 2.8
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| |-
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| | [[Polypropylene]] || 12–43 || 19.7–80 || 0.91 || 1.3
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| |-
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| | [[Stainless steel]] AISI 302 – Cold-rolled || 520 || 860 || ||
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| |-
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| | [[Cast iron]] 4.5% C, ASTM A-48<ref>{{harvnb|Beer|Johnston|Dewolf|2001|p=746}}.</ref> || * || 172 || 7.20 || 2.4
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| |-
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| | [[Titanium alloy]] (6% Al, 4% V) || 830 || 900 || 4.51 || 18.8
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| |-
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| | [[Aluminium alloy]] 2014-T6 || 400 || 455 || 2.7 || 15.1
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| |-
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| | [[Copper]] 99.9% Cu || 70 || 220 || 8.92 || 0.8
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| | [[Cupronickel]] 10% Ni, 1.6% Fe, 1% Mn, balance Cu || 130 || 350 || 8.94 || 1.4
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| |-
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| | [[Brass]] || approx. 200+|| 550 || 5.3 || 3.8
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| |-
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| | [[Spider silk]] || 1150 (??) || 1400 || 1.31 || 109
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| |-
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| | [[Silkworm]] silk || 500 || || || 25
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| |-
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| | [[Aramid]] ([[Kevlar]] or [[Twaron]]) || 3620 || || 1.44 || 256.3
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| |-
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| | [[Ultra high molecular weight polyethylene|UHMWPE]]<ref>[http://www.plastic-products.com/spec11.htm Technical Product Data Sheets UHMWPE]</ref><ref>[http://www.unitex-deutschland.eu/pdf/download/Dyneema-Version-web-db.pdf unitex-deutschland.eu]</ref> || 20 || 35<ref>[http://matweb.com/search/DataSheet.aspx?MatGUID=f9470672aa5549cb9c7b157677d02062&ckck=1 matweb.com]</ref> || 0.97 || 400
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| |-
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| | [[Bone]] (limb) || 104–121 || 130 || || 3
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| | [[Nylon]], type 6/6 || 45 || 75 || || 2
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| | colspan=5 | <sup>*</sup>Grey cast iron does not have a well defined yield strength because the stress-strain relationship is atypical. The yield strength can vary from 65 to 80% of the tensile strength.<ref>{{harvnb|Avallone|Baumeister|Sadegh|Marks|2006|p=6‐35<!-- Not a range -->}}.</ref>
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| |}
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| {| class="wikitable" border="1"
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| |+Elements in the annealed state<ref>A.M. Howatson, P.G. Lund and J.D. Todd, "Engineering Tables and Data", p. 41.</ref>
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| |-
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| ! !! Young's modulus<br> (GPa) !! Proof or yield stress<br> (MPa) !! Ultimate Tensile Strength<br> (MPa)
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| |-
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| | [[Aluminium]] || 70 || 15–20 || 40–50
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| |-
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| | [[Copper]] || 130 || 33 || 210
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| | [[Iron]] || 211 || 80–100 || 350
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| |-
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| | [[Nickel]] || 170 || 14–35 || 140–195
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| |-
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| | [[Silicon]] || 107 || 5000–9000 ||
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| |-
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| | [[Tantalum]] || 186 || 180 || 200
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| |-
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| | [[Tin]] || 47 || 9–14 || 15–200
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| |-
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| | [[Titanium]] || 120 || 100–225 || 240–370
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| |-
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| | [[Tungsten]] || 411 || 550 || 550–620
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| |-
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| |}
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| ==See also==
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| * [[Elastic modulus]]
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| * [[Linear elasticity]]
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| * [[Piola-Kirchhoff stress tensor]]
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| * [[Strain tensor]]
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| * [[Stress concentration]]
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| * [[Tensile strength]]
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| * [[Virial stress]]
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| * [[Yield curve (physics)]]
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| * [[Yield surface]]
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| ==References==
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| === Notes ===
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| {{Reflist}}
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| ===Bibliography===
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| * {{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 | edition=8th | isbn=0-07-004997-1}}
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| * {{Cite book| last = Avallone | first = Eugene A. | last2 = Baumeister | first2 = Theodore | last3 = Sadegh | first3 = Ali | last4 = Marks | first4 = Lionel Simeon | title = Mark's Standard Handbook for Mechanical Engineers | publisher = McGraw-Hill Professional | year = 2006 | edition = 11th, Illustrated | url = http://books.google.com/?id=oOKqwp3CIt8C | isbn = 978-0-07-142867-5 | ref = harv | postscript = <!--None-->}}.
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| *{{Cite book| last = Beer | first = Ferdinand P. | last3 = Dewolf | first3 = John T. | last2 = Johnston | first2 = E. Russell | title = Mechanics of Materials | publisher = McGraw-Hill | year = 2001 | edition = 3rd | url = http://books.google.com/?id=TSDcA2-N2_sC | isbn = 978-0-07-365935-0 | ref = harv | postscript = <!--None-->}}.
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| * 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
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| * {{Cite book| last = Degarmo | first = E. Paul | last2 = Black | first2 = J T. | last3 = Kohser | first3 = Ronald A. | title = Materials and Processes in Manufacturing | publisher = Wiley | year = 2003 | edition = 9th | isbn = 0-471-65653-4 | ref = harv | postscript = <!--None-->}}.
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| * Oberg, E., Jones, F. D., and Horton, H. L. (1984). ''Machinery's Handbook'', 22nd edition. Industrial Press. ISBN 0-8311-1155-0
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| *{{cite book | last = Ross | first = C. | title = Mechanics of Solids | publisher = Albion/Horwood Pub | location = City | year = 1999 | url = http://books.google.com/books?id=H_5zV2twBtwC | isbn = 978-1-898563-67-9 }}
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| * Shigley, J. E., and Mischke, C. R. (1989). ''Mechanical Engineering Design'', 5th edition. McGraw Hill. ISBN 0-07-056899-5
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| * {{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= | isbn=0-07-072542-X}}
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| * [http://www.engineershandbook.com/Materials/mechanical.htm Engineer's Handbook]
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| {{Use dmy dates|date=September 2010}}
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| {{DEFAULTSORT:Yield (Engineering)}}
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| [[Category:Elasticity (physics)]]
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| [[Category:Mechanics]]
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| [[Category:Plasticity]]
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| [[Category:Solid mechanics]]
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| [[Category:Deformation]]
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