Yield (engineering)

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Yield strength, or the yield point, 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. 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, yield is the permanent plastic deformation of a structural member under stress. This is a soft failure mode which does not normally cause catastrophic failure unless it accelerates buckling.

In three dimensional space of principal stresses (σ1,σ2,σ3), an infinite number of yield points form together a yield surface.

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[edit] Definition

It is often difficult to precisely define yield due to the wide variety of stress-strain behaviours exhibited by real materials. In addition there are several possible ways to define the yield point in a given material:

  • The point at which dislocations first begin to move. Given that dislocations begin to move at very low stresses, and the difficulty in detecting such movement, this definition is rarely used.
  • Elastic Limit - The lowest stress at which permanent deformation can be measured. This requires a complex iterative load-unload procedure and is critically dependent on the accuracy of the equipment and the skill of the operator.
  • Proportional Limit - The point at which the stress-strain curve becomes non-linear. In most metallic materials the elastic limit and proportional limit are essentially the same.
  • Offset Yield Point (proof stress) - Due to the lack of a clear border between the elastic and plastic regions in many materials, the yield point is often defined as the stress at some arbitrary plastic strain (typically 0.2% [1]). This is determined by the intersection of a line offset from the linear region by the required strain. In some materials there is essentially no linear region and so a certain value of plastic strain is defined instead. Although somewhat arbitrary this method does allow for a consistent comparison of materials and is the most common.

[edit] 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 - \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\!:

\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:

\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.

[edit] 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.

[edit] Implications for structural engineering

Yielded structures have a lower and less constant modulus of elasticity, so deflections increase and buckling strength decreases, and both become more difficult to predict. When load is removed, the structure will remain permanently bent, and may have residual pre-stress. If buckling is avoided, structures have a tendency to adapt a more efficient shape that will be better able to sustain (or avoid) the loads that bent it. Because of this, highly engineered structures rely on yielding as a graceful failure mode which allows fail-safe operation. In aerospace engineering, for example, no safety factor is needed when comparing limit loads (the highest loads expected during normal operation) to yield criteria. Safety factors are only required when comparing limit loads to ultimate failure criteria, (buckling or rupture.) In other words, a plane which undergoes extraordinary loading beyond its operational envelope may bend a wing slightly, but this is considered to be a fail-safe failure mode which will not prevent it from making an emergency landing.

[edit] See also

[edit] References

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

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

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