WIKI STRESS|February 25, 2013 1:33 pm

Uniaxial stress

If two of the dimensions of the object are very large or very small compared to the others, the object may be modelled as one-dimensional. In this case the stress tensor has only one component and is indistinguishable from a scalar. One-dimensional objects include a piece of wire loaded at the ends and a metal sheet loaded on the face and viewed up close and through the cross section.

When a structural element is subjected to tension or compression its length will tend to elongate or shorten, and its cross-sectional area changes by an amount that depends on the Poisson’s ratio of the material. In engineering applications, structural members experience small deformations and the reduction in cross-sectional area is very small and can be neglected, i.e., the cross-sectional area is assumed constant during deformation. For this case, the stress is called engineering stress or nominal stress. In some other cases, e.g., elastomers and plastic materials, the change in cross-sectional area is significant, and the stress must be calculated assuming the current cross-sectional area instead of the initial cross-sectional area. This is termed true stress and is expressed as

\sigma_\mathrm{true} = (1 + \varepsilon_\mathrm e)(\sigma_\mathrm e)\,\!,

where

\varepsilon_\mathrm e\,\! is the nominal (engineering) strain, and
\sigma_\mathrm e\,\! is nominal (engineering) stress.

The relationship between true strain and engineering strain is given by

\varepsilon_\mathrm{true} = \ln(1 + \varepsilon_\mathrm e)\,\!.

In uniaxial tension, true stress is then greater than nominal stress. The converse holds in compression.

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