# Yield (engineering)

(Redirected from Elastic limit)
Stress–strain curve showing typical yield behavior for nonferrous alloys. (Stress${\displaystyle \sigma }$, shown as a function of strain${\displaystyle \epsilon }$.)

In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Prior to the yield point, a material will deform 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 and is known as plastic deformation.

The yield strength or yield stress is a material property and is the stress corresponding to the yield point at which the material begins to deform plastically. The yield strength is often used to determine the maximum allowable load in a mechanical component, since it represents the upper limit to forces that can be applied without producing permanent deformation. In some materials, such as aluminium, there is a gradual onset of non-linear behavior, making the precise yield point difficult to determine. In such a case, the offset yield point (or proof stress) is taken as the stress at which 0.2% plastic deformation occurs. Yielding is a gradual failure mode which is normally not catastrophic, unlike ultimate failure.

In solid mechanics, the yield point can be specified in terms of the three-dimensional principal stresses (${\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}}$) with a yield surface or a yield criterion. A variety of yield criteria have been developed for different materials.

## Definition

Material Yield strength
(MPa)
Ultimate strength
(MPa)
ASTM A36 steel 250 400
Steel, API 5L X65[1] 448 531
Steel, high strength alloy ASTM A514 690 760
Steel, prestressing strands 1650 1860
Piano wire   1740-3300[2]
Carbon fiber (CF, CFK) 5650[3]
High-density polyethylene (HDPE) 26–33 37
Polypropylene 12–43 19.7–80
Stainless steel AISI 302 – cold-rolled 520 860
Cast iron 4.5% C, ASTM A-48[4] 172
Titanium alloy (6% Al, 4% V) 830 900
Aluminium alloy 2014-T6 400 455
Copper 99.9% Cu 70 220
Cupronickel 10% Ni, 1.6% Fe, 1% Mn, balance Cu 130 350
Brass 200+ ~ 550
Spider silk 1150 (??) 1400
Silkworm silk 500
Aramid (Kevlar or Twaron) 3620 3757
UHMWPE[5][6] 20 35[7]
Bone (limb) 104–121 130
Nylon, type 6/6 45 75
Aluminium (annealed) 15–20 40–50[8]
Copper (annealed) 33 210
Iron (annealed) 80–100 350
Nickel (annealed) 14–35 140–195
Silicon (annealed) 5000–9000
Tantalum (annealed) 180 200
Tin (annealed) 9–14 15–200
Titanium (annealed) 100–225 240–370
Tungsten (annealed) 550 550–620

It is often difficult to precisely define yielding due to the wide variety of stress–strain curves exhibited by real materials. In addition, there are several possible ways to define yielding:[9]

True elastic limit
The lowest stress at which dislocations 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 elastic limit is, therefore, the lowest stress point at which permanent deformation can be measured. This requires a manual load-unload procedure, and the accuracy is critically dependent on the equipment used and operator skill. For elastomers, such as rubber, the elastic limit is much larger than the proportionality limit. Also, precise strain measurements have shown that plastic strain begins at very low stresses.[10][11]
Yield point
The point in the stress-strain curve at which the curve levels off and plastic deformation begins to occur.[12]
Offset yield point (proof stress)
When a yield point is not easily defined on the basis of 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% plastic strain.[13] The offset value is given as a subscript, e.g., ${\displaystyle R_{\text{p0.1}}=310}$ MPa or ${\displaystyle R_{\text{p0.2}}=350}$ MPa.[14] For most practical engineering uses, ${\displaystyle R_{\text{p0.2}}}$ is multiplied by a factor of safety to obtain a lower value of the offset yield point.[15] High strength steel and aluminum alloys do not exhibit a yield point, so this offset yield point is used on these materials.[13]
Upper and lower yield points
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 bands can develop.[16]

## Usage in 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.

## Testing

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. This is called a Tensile Test. Longitudinal and/or transverse strain is recorded using mechanical or optical extensometers.

Indentation hardness correlates roughly linearly with tensile strength for most steels, but measurements on one material cannot be used as a scale to measure strengths on another.[17] 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. However, for critical situations, tension testing is done to eliminate ambiguity.

## 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.

These mechanisms for crystalline materials include

### Work hardening

Where deforming the material will introduce 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:

${\displaystyle \Delta \sigma _{y}=Gb{\sqrt {\rho }}}$

where ${\displaystyle \sigma _{y}}$ is the yield stress, G is the shear elastic modulus, b is the magnitude of the Burgers vector, and ${\displaystyle \rho }$ is the dislocation density.

### Solid solution strengthening

By 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:

${\displaystyle \Delta \tau =Gb{\sqrt {C_{s}}}\epsilon ^{\frac {3}{2}}}$

where ${\displaystyle \tau }$ is the shear stress, related to the yield stress, ${\displaystyle G}$ and ${\displaystyle b}$ are the same as in the above example, ${\displaystyle C_{s}}$ is the concentration of solute and ${\displaystyle \epsilon }$ is the strain induced in the lattice due to adding the impurity.

### 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:

${\displaystyle \Delta \tau ={\frac {r_{\text{particle}}}{l_{\text{interparticle}}}}\gamma _{\text{particle-matrix}}}$

and the bowing/ringing formula:

${\displaystyle \Delta \tau ={\frac {Gb}{l_{\text{interparticle}}-2r_{\text{particle}}}}}$

In these formulas, ${\displaystyle r_{\text{particle}}\,}$ is the particle radius, ${\displaystyle \gamma _{\text{particle-matrix}}\,}$ is the surface tension between the matrix and the particle, ${\displaystyle l_{\text{interparticle}}\,}$ is the distance between the particles.

### 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 strengthening, this type of strengthening is governed by the formula:

${\displaystyle \sigma _{y}=\sigma _{0}+kd^{-{\frac {1}{2}}}\,}$

where

${\displaystyle \sigma _{0}}$ is the stress required to move dislocations,
${\displaystyle k}$ is a material constant, and
${\displaystyle d}$ is the grain size.

## Theoretical yield strength

Material Theoretical Shear Strength (GPa) Experimental Shear Strength (GPa)
Ag 1.0 0.37
Al 0.9 0.78
Cu 1.4 0.49
Ni 2.6 3.2
α-Fe 2.6 27.5

The theoretical yield strength of a perfect crystal is much higher than the observed stress at the initiation of plastic flow.[18]

That experimentally measured yield strength is significantly lower than the expected theoretical value can be explained by the presence of dislocations and defects in the materials. Indeed, whiskers with perfect single crystal structure and defect-free surfaces have been shown to demonstrate yield stress approaching the theoretical value. For example, nanowhiskers of copper were shown to undergo brittle fracture at 1 GPa,[19] a value much higher than the strength of bulk copper and approaching the theoretical value.

The theoretical yield strength can be estimated by considering the process of yield at the atomic level. In a perfect crystal, shearing results in the displacement of an entire plane of atoms by one interatomic separation distance, b, relative to the plane below. In order for the atoms to move, considerable force must be applied to overcome the lattice energy and move the atoms in the top plane over the lower atoms and into a new lattice site. The applied stress to overcome the resistance of a perfect lattice to shear is the theoretical yield strength, τmax.

The stress displacement curve of a plane of atoms varies sinusoidally as stress peaks when an atom is forced over the atom below and then falls as the atom slides into the next lattice point.[20]

${\displaystyle \tau =\tau _{\max }\sin \left({\frac {2\pi x}{b}}\right)}$

where ${\displaystyle b}$ is the interatomic separation distance. Since τ = G γ and dτ/dγ = G at small strains (ie. Single atomic distance displacements), this equation becomes:

${\displaystyle G={\frac {d\tau }{dx}}={\frac {2\pi }{b}}\tau _{\max }\cos \left({\frac {2\pi x}{b}}\right)={\frac {2\pi }{b}}\tau _{\max }}$

For small displacement of γ=x/a, where a is the spacing of atoms on the slip plane, this can be rewritten as:

${\displaystyle G={\frac {d\tau }{d\gamma }}={\frac {2\pi a}{b}}\tau _{\max }}$

Giving a value of ${\displaystyle \tau _{\max }}$τmax equal to:

${\displaystyle \tau _{\max }={\frac {Gb}{2\pi a}}}$

The theoretical yield strength can be approximated as ${\displaystyle \tau _{\max }=G/30}$.

## Yield criterion

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 ${\displaystyle \sigma _{1}\,\!}$, ${\displaystyle \sigma _{2}\,\!}$, and ${\displaystyle \sigma _{3}\,\!}$.

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 – 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.

${\displaystyle \sigma _{1}\leq \sigma _{y}\,\!}$

Maximum Principal Strain Theory – by St.Venant. Yield occurs when the maximum principal 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:

${\displaystyle \sigma _{1}-\nu \left(\sigma _{2}+\sigma _{3}\right)\leq \sigma _{y}.\,\!}$

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 ${\displaystyle \tau \!}$ exceeds the shear yield strength ${\displaystyle \tau _{y}\!}$:

${\displaystyle \tau ={\frac {\sigma _{1}-\sigma _{3}}{2}}\leq \tau _{y}.\,\!}$

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:

${\displaystyle \sigma _{1}^{2}+\sigma _{2}^{2}+\sigma _{3}^{2}-2\nu \left(\sigma _{1}\sigma _{2}+\sigma _{2}\sigma _{3}+\sigma _{1}\sigma _{3}\right)\leq \sigma _{y}^{2}.\,\!}$

Maximum Distortion Energy Theory (von Mises yield criterion) – This theory proposes that the total strain energy can be separated into two components: the volumetric (hydrostatic) strain energy and the shape (distortion or 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 theory is also known as the von Mises yield criterion.

Based on a different theoretical underpinning this expression is also referred to as octahedral shear stress theory.[citation needed]

Other commonly used isotropic yield criteria are the

The yield surfaces corresponding to these criteria have a range of forms. However, most isotropic yield criteria correspond to convex yield surfaces.

### 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:

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