### Elastic Limit

A yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to deform plastically. Prior to the yield point the 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.
In the three-dimensional space of the principal stresses ($\sigma_1, \sigma_2 , \sigma_3$), 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, or 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

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:[1]

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 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 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 low stresses.[2][3]
Yield point
The point in the stress-strain curve at which the curve levels off and plastic deformation begins to occur.[4]
Offset yield point (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.[5] The offset value is given as a subscript, e.g., Rp0.2=310 MPa. High strength steel and aluminum alloys do not exhibit a yield point, so this offset yield point is used on these materials.[5]
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. If a metal is only stressed to the upper yield point, and beyond, Lüders bands can develop.[6]

## 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 $\sigma_1 \,\!$, $\sigma_2 \,\!$, and $\sigma_3 \,\!$.

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.

$\ \sigma_1 \le \sigma_y \,\!$

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:

$\ \sigma_1 - \nu\left(\sigma_2 + \sigma_3\right) \le \sigma_y. \,\!$

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 $\tau\!$ exceeds the shear yield strength $\tau_y\!$:

$\ \tau = \frac\left\{\sigma_1-\sigma_3\right\}\left\{2\right\} \le \tau_y. \,\!$

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:

$\ \sigma_\left\{1\right\}^2 + \sigma_\left\{2\right\}^2 + \sigma_\left\{3\right\}^2 - 2 \nu \left(\sigma_1 \sigma_2 + \sigma_2 \sigma_3 + \sigma_1 \sigma_3\right) \le \sigma_y^2. \,\!$

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 shear) strain energy. It is proposed that yield occurs when the distortion component exceeds that at the yield point for a simple tensile test.

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

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:

## 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. 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.

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

$\sigma_y = C \left(\dot\left\{\epsilon\right\}\right)^m \,\!$

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:

$\sigma_y = \frac\left\{1\right\}\left\{\alpha\right\} \sinh^\left\{-1\right\} \left \left[ \frac\left\{Z\right\}\left\{A\right\} \right \right]^\left\{\left(1/n\right)\right\} \,\!$

where α and A are constants and Z is the temperature-compensated strain-rate – often described by the Zener-Hollomon parameter:

$Z = \left(\dot\left\{\epsilon\right\}\right) \exp \left \left( \frac\left\{Q_\left\{HW\right\}\right\}\left\{RT\right\} \right \right) \,\!$

where QHW 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

#### 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:

$\Delta\sigma_y = Gb \sqrt\left\{\rho\right\}$

where $\sigma_y$ is the yield stress, G is the shear elastic modulus, b is the magnitude of the Burgers vector, and $\rho$ 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:

$\Delta\tau = Gb\sqrt\left\{C_s\right\}\epsilon^\left\{3/2\right\}$

where $\tau$ 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 $\epsilon$ 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:

$\Delta\tau = \cfrac\left\{r_\left\{\rm\left\{particle\right\}\right\}\right\}\left\{l_\left\{\rm\left\{interparticle\right\}\right\}\right\} \gamma_\left\{\rm\left\{particle-matrix\right\}\right\}$

and the bowing/ringing formula:

$\Delta\tau = \cfrac\left\{Gb\right\}\left\{l_\left\{\rm\left\{interparticle\right\}\right\}-2r_\left\{\rm\left\{particle\right\}\right\}\right\}$

In these formulas, $r_\left\{\rm\left\{particle\right\}\right\}\,$ is the particle radius, $\gamma_\left\{\rm\left\{particle-matrix\right\}\right\} \,$ is the surface tension between the matrix and the particle, $l_\left\{\rm\left\{interparticle\right\}\right\} \,$ 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:

$\sigma_y = \sigma_0 + kd^\left\{-1/2\right\} \,$

where

$\sigma_0$ is the stress required to move dislocations,
k is a material constant, and
d is the grain size.

## 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. Longitudinal and/or transverse strain is recorded using mechanical or optical extensometers.

Indentation hardness correlates linearly with tensile strength for most steels.[7] 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.

## 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.

## Typical yield and ultimate strengths

Note: many of the values depend on manufacturing process and purity/composition.

Material Yield strength
(MPa)
Ultimate strength
(MPa)
Density
(g/cm³)
free breaking length
(km)
ASTM A36 steel 250 400 7.85 3.2
Steel, API 5L X65[8] 448 531 7.85 5.8
Steel, high strength alloy ASTM A514 690 760 7.85 9.0
Steel, prestressing strands 1650 1860 7.85 21.6
Piano wire   2200–2482 [9] 7.8 28.7
Carbon Fiber (CF, CFK) 5650 [10] 1.75
High density polyethylene (HDPE) 26–33 37 0.95 2.8
Polypropylene 12–43 19.7–80 0.91 1.3
Stainless steel AISI 302 – Cold-rolled 520 860
Cast iron 4.5% C, ASTM A-48[11] * 172 7.20 2.4
Titanium alloy (6% Al, 4% V) 830 900 4.51 18.8
Aluminium alloy 2014-T6 400 455 2.7 15.1
Copper 99.9% Cu 70 220 8.92 0.8
Cupronickel 10% Ni, 1.6% Fe, 1% Mn, balance Cu 130 350 8.94 1.4
Brass approx. 200+ 550 5.3 3.8
Spider silk 1150 (??) 1400 1.31 109
Silkworm silk 500     25
Aramid (Kevlar or Twaron) 3620   1.44 256.3
UHMWPE[12][13] 20 35[14] 0.97 400
Bone (limb) 104–121 130   3
Nylon, type 6/6 45 75   2
*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.[15]
Elements in the annealed state[16]
Young's modulus
(GPa)
Proof or yield stress
(MPa)
Ultimate Tensile Strength
(MPa)
Aluminium 70 15–20 40–50
Copper 130 33 210
Iron 211 80–100 350
Nickel 170 14–35 140–195
Silicon 107 5000–9000
Tantalum 186 180 200
Tin 47 9–14 15–200
Titanium 120 100–225 240–370
Tungsten 411 550 550–620

## References

### Bibliography

• .
• .
• 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). Mechanical Engineering Design, 5th edition. McGraw Hill. ISBN 0-07-056899-5
• Engineer's Handbook

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