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The 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 nonreversible.
In the threedimensional space of the principal stresses (), 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.
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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]}
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 , , and .
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.
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.
Maximum Principal Strain Theory  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:
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 exceeds the shear yield strength :
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 3dimensional stress state this is given by:
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. This is generally referred to as the Von Mises yield criterion and is expressed as:
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.
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:
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
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:
where α and A are constants and Z is the temperaturecompensated strainrate  often described by the ZenerHollomon parameter:
where Q_{HW} is the activation energy for hot deformation and T is the absolute temperature.
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
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:
where is the yield stress, G is the shear elastic modulus, b is the magnitude of the Burgers vector, and is the dislocation density.
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:
where 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 is the strain induced in the lattice due to adding the impurity.
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:
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.
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 HallPetch strengthening, this type of strengthening is governed by the formula:
where
Yield strength testing involves taking a small sample with a fixed crosssection 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.
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 failsafe failure mode. No safety factor is therefore needed when comparing limit loads (the highest loads expected during normal operation) to yield criteria.^{[citation needed]}
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)  2633  37  0.95  2.8 
Polypropylene  1243  19.780  0.91  1.3 
Stainless steel AISI 302  Coldrolled  520  860  
Cast iron 4.5% C, ASTM A48^{[11]}  *  172  7.20  2.4 
Titanium alloy (6% Al, 4% V)  830  900  4.51  18.8 
Aluminium alloy 2014T6  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]}  2400  0.97  400  
Bone (limb)  104121  130  3  
Nylon, type 6/6  45  75  2  
^{*}Grey cast iron does not have a well defined yield strength because the stressstrain relationship is atypical. The yield strength can vary from 65 to 80% of the tensile strength.^{[14]} 
Young's modulus (GPa)  Proof or yield stress (MPa)  Ultimate Tensile Strength (MPa)  

Aluminium  70  1520  4050 
Copper  130  33  210 
Iron  211  80100  350 
Nickel  170  1435  140195 
Silicon  107  50009000  
Tantalum  186  180  200 
Tin  47  914  15200 
Titanium  120  100225  240370 
Tungsten  411  550  550620 