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Encyclopedia > Yield (engineering)
Mechanical failure modes
Buckling
Corrosion
Creep
Fatigue
Fracture
Melting
Rupture
Thermal shock
Wear
Yielding
This box: view  talk  edit  Typical yield behavior for non-ferrous alloys.
1: True elastic limit
2: Proportionality limit
3: Elastic limit
4: Offset yield strength

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: A stressâ€“strain curve is a graph derived from measuring load (stress â€“ Ïƒ) versus extension (strain â€“ Îµ) for a sample of a material. ...

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 Young's modulus of the material.
Elastic limit
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.
Offset yield point (yield strength or proof stress)
This is the most widely used strength measure of metals, and is found from the stress-strain curve as shown in the figure to the right. A plastic strain of 0.2% is usually used to define the offset yield stress, although other values may be used depending on the material and the application. The offset value is given as a subscript, e.g. Rp0.2=310 MPa. In some materials there is essentially no linear region and so a certain value of strain is defined instead. Although somewhat arbitrary, this method does allow for a consistent comparison of materials.
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.

In materials science, a dislocation is a crystallographic defect, or irregularity, within a crystal structure. ... In solid mechanics, Youngs modulus (E) is a measure of the stiffness of a given material. ... The term elastomer is often used interchangeably with the term rubber, and is preferred when referring to vulcanisates. ... This does not cite any references or sources. ...

## Yield criterion

A yield criterion, often expressed as yield surface, or yield locus, is an 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 $sigma_3 ,!$. Yield surface is described in three dimensional space of principal stresses (), and encompasses the elastic region of material behavior. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

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. $sigma_1 le sigma_y ,!$

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: This article is about the deformation of materials. ... $sigma_1 - nu(sigma_2 + sigma_3) le sigma_y. ,!$

Maximum Shear Stress Theory - Also known as the Tresca criterion, after the French scientist Henri Tresca. This assumes that yield occurs when the shear stress $tau!$ exceeds the shear yield strength $tau_y!$: Henri Tresca (1814â€“1884) was French Mechanical Engineer, professor of Conservatoire National des Arts et MÃ©tiers in Paris. ... $tau = frac{sigma_1-sigma_3}{2} le tau_{ys}. ,!$

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

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. This is generally referred to as the Von Mises criterion and is expressed as: Fluid pressure is the pressure on an object submerged in a fluid, such as water. ... Shearing in continuum mechanics refers to the occurrence of a shear strain, which is a deformation of a material substance in which parallel internal surfaces slide past one another. ... $frac{1}{2} Big[ (sigma_1 - sigma_2)^2 + (sigma_2 - sigma_3)^2 + (sigma_3 - sigma_1)^2 Big] le sigma_y^2. ,!$

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

## 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. 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 $sigma_y = C (dot{epsilon})^m ,!$

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

where α and A are constants and Z is the temperature-compensated strain-rate - often described by the Zener-Hollomon parameter: $Z = (dot{epsilon}) exp left ( frac{Q_{HW}}{RT} right ) ,!$

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:

1. Work Hardening - Where machining 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. Work hardening, or strain hardening, is an increase in mechanical strength due to plastic deformation. ... In materials science, a dislocation is a crystallographic defect, or irregularity, within a crystal structure. ...

The governing formula for this mechanism is: $Deltasigma_y = Gb sqrt{rho}$

where σy is the yield stress, G is the shear elastic modulus, b is the magnitude of the Burgers vector, and ρ is the dislocation density. In materials science, a dislocation is a linear crystallographic defect, or irregularity, within a crystal structure. ...

2. 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. Solid solution strengthening is a type of alloying that can be used to improve the strength of a pure metal. ... An alloy is a homogeneous hybrid of two or more elements, at least one of which is a metal, and where the resulting material has metallic properties. ...

The relationship of this mechanism goes as: $Deltatau = Gbsqrt{C_s}epsilon^{3/2}$

where τ 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 ε is the strain induced in the lattice due to adding the impurity. Shear stress is a stress state where the stress is parallel or tangential to a face of the material, as opposed to normal stress when the stress is perpendicular to the face. ...

3. 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. Precipitation hardening, also called age hardening or dispersion hardening, are heat treatment techniques used to strengthen malleable materials, especially non-ferrous alloys including most structural alloys of aluminium, magnesium and titanium. ...

The shearing formula goes as: $Deltatau = cfrac{r_{particle}}{l_{interparticle}} gamma_{particle-matrix}$

and the bowing/ringing formula: $Deltatau = cfrac{Gb}{l_{interparticle}-2r_{particle}}$

In these formulas, rparticle is the particle radius, γparticlematrix is the surface tension between the matrix and the particle, linterparticle is the distance between the particles.

4. 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 strenghthening, this type of strengthening is governed by the formula: Figure 1: Hall-Petch Strengthening is limited by the size of dislocations. ... $sigma_y = sigma_0 + kd^{-1/2} ,$

where

σ0 is the stress required to move dislocations,
k is a material constant, and
d is the grain size.

## 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.[citation needed]

## Typical yield strength

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

Material Yield strength
(MPa)
Ultimate strength
(MPa)
Density
(g/cm³)
Structural steel ASTM A36 steel 250 400 7.8
Steel, API 5L X65 (Fikret Mert Veral) 448 531 7.8
Steel, high strength alloy ASTM A514 690 760 7.8
Steel, prestressing strands 1650 1860 7.8
Steel Wire     7.8
Steel (AISI 1060 0.6% carbon) Piano wire 2200-2482 MPa   7.8
High density polyethylene (HDPE) 26-33 37 0.95
Polypropylene 12-43 19.7-80 0.91
Stainless steel AISI 302 - Cold-rolled 520 860
Cast iron 4.5% C, ASTM A-48 276 (??) 200
Titanium alloy (6% Al, 4% V) 830 900 4.51
Aluminium alloy 2014-T6 400 455 2.7
Copper 99.9% Cu 70 220 8.92
Cupronickel 10% Ni, 1.6% Fe, 1% Mn, balance Cu 130 350 8.94
Brass approx. 200+ 550 5.3
Tungsten   1510 19.25
Glass   50 (in compression) 2.53
E-Glass N/A 3450 2.57
S-Glass N/A 4710 2.48
Basalt fiber N/A 4840 2.7
Marble N/A 15
Concrete N/A 3
Carbon Fiber N/A 5650 1.75
Spider silk 1150 (??) 1200
Silkworm silk 500
Aramid (Kevlar or Twaron) 3620   1.44
UHMWPE 23 46 0.97
UHMWPE fibers (Dyneema or Spectra) 2300-3500 0.97
Vectran   2850-3340
Pine Wood (parallel to grain)   40
Bone (limb) 104-121 130
Nylon, type 6/6 45 75
Rubber - 15
Boron N/A 3100 2.46
Silicon, monocrystalline (m-Si) N/A 7000 2.33
Silicon carbide (SiC) N/A 3440
Sapphire (Al2O3) N/A 1900 3.9-4.1
Carbon nanotube (see note above) N/A 62000 1.34
Elements in the annealed state Young's Modulus
(GPa)
Proof or yield stress
(MPa)
Ultimate strength
(MPa)
Aluminium 70 15-20 40-50
Copper 130 33 210
Gold 79   100
Iron 211 80-100 350
Nickel 170 14-35 140-195
Silicon 107 5000-9000
Silver 83   170
Tantalum 186 180 200
Tin 47 9-14 15-200
Titanium 120 100-225 240-370
Tungsten 411 550 550-620
Zinc (wrought) 105   110-200

This article needs to be cleaned up to conform to a higher standard of quality. ... The strain tensor, Îµ, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation: the diagonal coefficients Îµii are the relative change in length in the direction of the i direction (along the xi-axis) ; the other terms Îµij = 1/2 Î³ij (i... This article is in need of attention from an expert on the subject. ... A stress concentration is a phenomenon encounterered in mechanical engineering where an object under load has higher than average local stresses due to its shape. ... // Linear elasticity The linear theory of elasticity models the macroscopic mechanical properties of solids assuming small deformations. ... The proof stress (also called offset yield strength) is the stress in a material that causes a small, specified amount of (permanent) plastic deformation in a test piece. ... Tensile strength isthe measures the force required to pull something such as rope, wire, or a structural beam to the point where it breaks. ... An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substances tendency to be deformed when a force is applied to it. ...

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