YIELD (ENGINEERING)

(Redirected from Elastic limit)

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 plastically deform. 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 unless it accelerates buckling.

Contents

Definition

Yield criterion

Factors influencing yield stress

Implications for structural engineering

See also

References

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 yieldingG. Dieter, ''Mechanical Metallurgy'', McGraw-Hill, 1986:
; 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 : The point at which the stress–strain curve deviates from Hooke's law, i.e., becomes nonlinear.
; Elastic limit : 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. R_p0.2=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, reaches an upper yield point before it drops 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.

Yield criterion

A yield criterion, often expressed as yield surface, 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 $sigma\_3\; ,!$.
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:
: $sigma\_1\; -$
u(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 $au!$ exceeds the shear yield strength $au\_y!$:
:
'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:
: $sigma\_\{1\}^2\; +\; sigma\_\{2\}^2\; +\; sigma\_\{3\}^2\; -\; 2$
u (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:
:
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:
:
ight ]^{(1/n)} ,!
where α and A are constants and Z is the temperature-compensated strain-rate - often described by the Zener-Hollomon parameter:
:
ight ) ,!
where Q_HW is the activation energy for hot deformation and T is the absolute temperature.

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.

See also

★ Piola-Kirchhoff stress tensor

★ Strain tensor

★ Stress-energy tensor

★ Stress concentration

★ 3-D elasticity

★ Proof stress

★ Tensile strength

★ Elastic modulus

References

★ Mark's Standard Handbook for Mechanical Engineers, Avallone, Eugene A.; & Baumeister III, Theodore, , , McGraw-Hill, 1996, ISBN 0-07-004997-1

★ Roark's Formulas for Stress and Strain, 7th edition, Young, Warren C.; & Budynas, Richard G., , , McGraw-Hill, 2002, ISBN 0-07-072542-X

★ Engineer's Handbook

★ 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). ''Mechnical Engineering Design'', 5th edition. McGraw Hill. ISBN 0-07-056899-5