Yield Point
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In materials science and
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, the yield point is the point on a stress-strain curve that indicates the limit of
elastic Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects togeth ...
behavior and the beginning of
plastic Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adaptab ...
behavior. Below the yield point, a material will deform elastically and will return to its original shape when the applied
stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
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 In engineering, deformation refers to the change in size or shape of an object. ''Displacements'' are the ''absolute'' change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strain ...
. The yield strength or yield stress is a
material property A materials property is an intensive property of a material, i.e., a physical property that does not depend on the amount of the material. These quantitative properties may be used as a metric by which the benefits of one material versus another c ...
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 Aluminium (aluminum in American and Canadian English) is a chemical element with the symbol Al and atomic number 13. Aluminium has a density lower than those of other common metals, at approximately one third that of steel. I ...
, 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 Failure causes are defects in design, process, quality, or part application, which are the underlying cause of a failure or which initiate a process which leads to failure. Where failure depends on the user of the product or process, then human er ...
which is normally not catastrophic, unlike
ultimate failure In mechanical engineering, ultimate failure describes the breaking of a material. In general there are two types of failure: fracture and buckling. Fracture of a material occurs when either an internal or external crack elongates the width or len ...
. In
solid mechanics Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ot ...
, the yield point can be specified in terms of the three-dimensional principal stresses (\sigma_1, \sigma_2 , \sigma_3) with a
yield surface A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of ''inside'' the yield surface is elastic. When the stress state lies on the surface the materi ...
or a yield criterion. A variety of yield criteria have been developed for different materials.


Definition

It is often difficult to precisely define yielding due to the wide variety of
stress–strain curve In engineering and materials science, a stress–strain curve for a material gives the relationship between stress and strain. It is obtained by gradually applying load to a test coupon and measuring the deformation, from which the stress and ...
s exhibited by real materials. In addition, there are several possible ways to define yielding:G. Dieter, ''Mechanical Metallurgy'', McGraw-Hill, 1986 ; True elastic limit: The lowest stress at which
dislocation In materials science, a dislocation or Taylor's dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allow atoms to sl ...
s 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 In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
), so the stress-strain graph is a straight line, and the gradient will be equal to the
elastic modulus An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
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
elastomer An elastomer is a polymer with viscoelasticity (i.e. both viscosity and elasticity) and with weak intermolecular forces, generally low Young's modulus and high failure strain compared with other materials. The term, a portmanteau of ''elastic p ...
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 very low stresses. ;Yield point: The point in the stress-strain curve at which the curve levels off and plastic deformation begins to occur. ; Offset yield point (): 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., R_\text = 310 MPa or R_\text= 350 MPa. For most practical engineering uses, R_\text is multiplied by a factor of safety to 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 used on these materials. ; Upper and lower yield points: Some metals, such as
mild steel Carbon steel is a steel with carbon content from about 0.05 up to 2.1 percent by weight. The definition of carbon steel from the American Iron and Steel Institute (AISI) states: * no minimum content is specified or required for chromium, cobalt ...
, 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 Lüders bands, is type of slip bands in metals or stretcher-strain marks which are formed due to localized bands of plastic deformation in metals experiencing tensile stresses, common to low-carbon steels and certain Al-Mg alloys. First reported b ...
s can develop.


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 Indentation hardness tests are used in mechanical engineering to determine the hardness of a material to deformation. Several such tests exist, wherein the examined material is indented until an impression is formed; these tests can be performed on ...
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. 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. For critical situations, tension testing is often done to eliminate ambiguity. However, it is possible to obtain stress-strain curves from indentation-based procedures, provided certain conditions are met. These procedures are grouped under the term
Indentation plastometry Indentation plastometry is the idea of using an indentation-based procedure to obtain (bulk) mechanical properties (of metals) in the form of stress-strain relationships in the plastic regime (as opposed to hardness testing, which gives numbers ...
.


Strengthening mechanisms

There are several ways in which crystalline 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 In materials science, work hardening, also known as strain hardening, is the strengthening of a metal or polymer by plastic deformation. Work hardening may be desirable, undesirable, or inconsequential, depending on the context. This strengt ...
*
Solid solution strengthening In metallurgy, solid solution strengthening is a type of alloying that can be used to improve the strength of a pure metal. The technique works by adding atoms of one element (the alloying element) to the crystalline lattice of another element ...
*
Precipitation strengthening Precipitation hardening, also called age hardening or particle hardening, is a heat treatment technique used to increase the yield strength of malleable materials, including most structural alloys of aluminium, magnesium, nickel, titanium, and so ...
*
Grain boundary strengthening In materials science, grain-boundary strengthening (or Hall–Petch strengthening) is a method of strengthening materials by changing their average crystallite (grain) size. It is based on the observation that grain boundaries are insurmountabl ...


Work hardening

Where deforming the material will introduce
dislocation In materials science, a dislocation or Taylor's dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allow atoms to sl ...
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. The governing formula for this mechanism is: : \Delta\sigma_y = Gb \sqrt where \sigma_y is the yield stress, G is the shear elastic modulus, b is the magnitude of the
Burgers vector In materials science, the Burgers vector, named after Dutch physicist Jan Burgers, is a vector, often denoted as , that represents the magnitude and direction of the lattice distortion resulting from a dislocation in a crystal lattice. The ve ...
, and \rho is the dislocation density.


Solid solution strengthening

By
alloy An alloy is a mixture of chemical elements of which at least one is a metal. Unlike chemical compounds with metallic bases, an alloy will retain all the properties of a metal in the resulting material, such as electrical conductivity, ductility, ...
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. The relationship of this mechanism goes as: : \Delta\tau = Gb\sqrt\epsilon^\frac where \tau is the
shear stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
, 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 = \frac \gamma_\text and the bowing/ringing formula: : \Delta\tau = \frac In these formulas, r_\text\, is the particle radius, \gamma_\text\, is the surface tension between the matrix and the particle, l_\text\, 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^\, where : \sigma_0 is the stress required to move dislocations, : k is a material constant, and : d is the grain size.


Theoretical yield strength

The theoretical yield strength of a perfect crystal 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 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, 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. : \tau = \tau_\max \sin\left( \frac \right) where b is the interatomic separation distance. Since τ = G γ and dτ/dγ = G at small strains (i.e. Single atomic distance displacements), this equation becomes: : G = \frac =\frac \tau_\max\cos\left ( \frac \right ) = \frac \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: : G = \frac = \frac \tau_\max Giving a value of \tau_\maxτmax equal to: : \tau_\max = \frac The theoretical yield strength can be approximated as \tau_\max = G/30.


See also

* Specified minimum yield strength *
Ultimate tensile strength Ultimate tensile strength (UTS), often shortened to tensile strength (TS), ultimate strength, or F_\text within equations, is the maximum stress that a material can withstand while being stretched or pulled before breaking. In brittle materials t ...
*
Yield curve (physics) In finance, the yield curve is a graph which depicts how the yields on debt instruments - such as bonds - vary as a function of their years remaining to maturity. Typically, the graph's horizontal or x-axis is a time line of months or y ...
*
Yield surface A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of ''inside'' the yield surface is elastic. When the stress state lies on the surface the materi ...


References


Bibliography

* * . *. * Boresi, A. P., Schmidt, R. J., and Sidebottom, O. M. (1993). ''Advanced Mechanics of Materials'', 5th edition John Wiley & Sons. * . * Oberg, E., Jones, F. D., and Horton, H. L. (1984). ''Machinery's Handbook'', 22nd edition. Industrial Press. * * Shigley, J. E., and Mischke, C. R. (1989). ''Mechanical Engineering Design'', 5th edition. McGraw Hill. *
Engineer's Handbook
{{DEFAULTSORT:Yield (Engineering) Elasticity (physics) Mechanics Plasticity (physics) Solid mechanics Deformation (mechanics) Structural analysis