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Stress–strain curve showing typical yield behavior for nonferrous alloys. (Stress, present as a function of strain.)

In content 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. Below the yield point, a contentwill 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 contentproperty and is the stress corresponding to the yield point at which the contentlaunch to deform plastically. The yield strength is often utilize to determine the maximum allowable load in a mechanical component, since it represents the upper limit to forces that shouldbe applied without producing permanent deformation. In some content, 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 shouldbe specified in rulesof the three-dimensional principal stresses () with a yield surface or a yield criterion. A variety of yield criteria have been developed for different content.

## Definition

Material Yield strength
(MPa)
Ultimate strength
(MPa)
ASTM A36 steel 250 400
Steel, API 5L X65 448 531
Steel, high strength alloy ASTM A514 690 760
Steel, prestressing strands 1650 1860
Piano wire   1740–3300
Carbon fiber (CF, CFK) 5650
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-48 172
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 20 35
Bone (limb) 104–121 130
Nylon, kind6/6 45 75
Aluminium (annealed) 15–20 40–50
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 content. In addition, there are several possible method to define yielding:

True elastic limit
The lowest stress at which dislocations move. This definition is rarely utilize 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 shouldbe measured. This requires a manual load-unload procedure, and the accuracy is critically dependent on the equipment utilize and operator skill. For elastomers, such as rubber, the elastic limit is much huge than the proportionality limit. Also, precise strain measurements have present that plastic strain launch at very low stresses.
Yield point
The point in the stress-strain curve at which the curve levels off and plastic deformation launch to occur.
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. The offset value is given as a subscript, e.g., MPa or MPa. For most practical engineering utilize, is multiplied by a factor of securityto obtain a lower value of the offset yield point. High strength steel and aluminum alloys do not exhibit a yield point, so this offset yield point is utilize on these content.
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 contentresponse is linear up until the upper yield point, but the lower yield point is utilize in structural engineering as a conservative value. If a metal is only stressed to the upper yield point, and beyond, Lüders bands shoulddevelop.

## 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 tinysample with a fixed cross-section locationand then pulling it with a controlled, gradually increasing force until the sample modify 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 contentcannot be utilize as a scale to measure strengths on another. Hardness testing shouldtherefore 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 method in which crystalline content shouldbe engineered to increase their yield strength. By altering dislocation density, impurity levels, grain size (in crystalline content), the yield strength of the contentshouldbe 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 huge stress must be applied. This thus causes a higher yield stress in the material. While many contentproperties depend only on the composition of the bulk material, yield strength is extremely sensitive to the content processing as well.

These mechanisms for crystalline content include

### Work hardening

Where deforming the contentwill introduce dislocations, which increases their density in the material. This increases the yield strength of the contentsince now more stress must be applied to move these dislocations through a crystal lattice. Dislocations shouldalso interact with each other, becoming entangled.

The governing formula for this mechanism is:

where is the yield stress, G is the shear elastic modulus, b is the magnitude of the Burgers vector, and 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 zonewith the impurity atom.

The relationship of this mechanism goes as:

where is the shear stress, associatedto the yield stress, and are the same as in the above example, is the concentration of solute and 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 versusa tinyparticle or precipitate of the material. Dislocations shouldmove through this particle either by shearing the particle or by a process known as bowing or ringing, in which a freshring of dislocations is madearound the particle.

The shearing formula goes as:

and the bowing/ringing formula:

In these formulas, is the particle radius, is the surface tension between the matrix and the particle, 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 locationto 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 kindof strengthening is governed by the formula:

where

is the stress neededto move dislocations,
is a contentconstant, and
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 excellentcrystal is much higher than the observed stress at the initiation of plastic flow.

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

The theoretical yield strength shouldbe estimated by considering the process of yield at the atomic level. In a excellentcrystal, shearing effect 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 freshlattice site. The applied stress to overcome the resistance of a excellentlattice 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.

where is the interatomic separation distance. Since τ = G γ and dτ/dγ = G at tinystrains (i.e. Single atomic distance displacements), this equation becomes:

For tinydisplacement of γ=x/a, where a is the spacing of atoms on the slip plane, this shouldbe rewritten as:

Giving a value of τmax equal to:

The theoretical yield strength shouldbe approximated as .

### Bibliography

• Avallone, Eugene A. & Baumeister III, Theodore (1996). Mark's Standard Handbook for Mechanical Engineers (8th ed.). FreshYork: McGraw-Hill. ISBN 978-0-07-004997-0.
• Avallone, Eugene A.; Baumeister, Theodore; Sadegh, Ali; Marks, Lionel Simeon (2006). (11th, Illustrated ed.). McGraw-Hill Professional. ISBN 978-0-07-142867-5..
• Beer, Ferdinand P.; Johnston, E. Russell; Dewolf, John T. (2001). (3rd ed.). McGraw-Hill. ISBN 978-0-07-365935-0..
• Boresi, A. P., Schmidt, R. J., and Sidebottom, O. M. (1993). Advanced Mechanics of Content, 5th edition John Wiley & Sons. ISBN 0-471-55157-0
• Degarmo, E. Paul; Black, J T.; Kohser, Ronald A. (2003). Content and Processes in Manufacturing (9th ed.). Wiley. ISBN 978-0-471-65653-1..
• Oberg, E., Jones, F. D., and Horton, H. L. (1984). Machinery's Handbook, 22nd edition. Industrial Press. ISBN 0-8311-1155-0
• Ross, C. (1999). . City: Albion/Horwood Pub. ISBN 978-1-898563-67-9.
• Shigley, J. E., and Mischke, C. R. (1989). Mechanical Engineering Design, 5th edition. McGraw Hill. ISBN 0-07-056899-5
• Young, Warren C. & Budynas, Richard G. (2002). Roark's Formulas for Stress and Strain, 7th edition. FreshYork: McGraw-Hill. ISBN 978-0-07-072542-3.

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