This article is about the deformation of materials. For other meanings, see strain.

In any branch of science dealing with materials and their behaviour, strain is the geometrical expression of deformation caused by the action of stress on a physical body. Strain therefore expresses itself as a change in size and/or shape. In the case of geological action of the earth, if the release of stress through strain in rocks is sufficiently large, earthquakes may occur.

If strain is equal over all parts of a body, it is referred to as homogeneous strain; otherwise, it is inhomogeneous strain. In its most general form, the strain is a symmetric tensor.

Quantifying strain

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Given that strain results in the deformation of a body, it can be measured by calculating the change in length of a line or by the change in angle between two lines (where these lines are theoretical constructs within the deformed body). The change in length of a line is termed the stretch, absolute strain, or extension, and may be written as $\backslash delta\; \backslash ell$. Then the (relative) strain, $\backslash epsilon\; \backslash ;$, is given by

$\backslash epsilon\; =\; \backslash frac\; \{\backslash delta\; \backslash ell\}\{\backslash ell\_0\}$

where $\backslash ell\_o$ is the original length of the material. The extension ($\backslash delta\; \backslash ell$) is positive if the material has gained length (in tension) and negative if it has reduced length (in compression). Because $\backslash ell\_o$ is always positive, the sign of the strain is always the same as the sign of the extension.

Strain has no units of measure because in the formula the units of length are cancelled. Dimensions of metres/metre or inches/inch are sometimes used for convenience, but generally units are left off and the strain sometimes is given as a percentage.

Engineering strain vs. true strain

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The above definition (known technically as engineering strain) is not linear, in that strains cannot be totalled. Imagine that a body is deformed twice, first by $\backslash delta\; \backslash ell\_1$ and then by $\backslash delta\; \backslash ell\_2$ (cumulative deformation). The final strain

$\backslash epsilon\; =\; \backslash frac\{\backslash delta\; \backslash ell\_1\; +\; \backslash delta\; \backslash ell\_2\}\{\backslash ell\_0\}$

is slightly different from the sum of the strains:

$\backslash epsilon\_1\; =\; \backslash frac\{\backslash delta\; \backslash ell\_1\}\{\backslash ell\_0\}$

and

$\backslash epsilon\_2\; =\; \backslash frac\{\backslash delta\; \backslash ell\_2\}\{\backslash ell\_0\; +\; \backslash delta\; \backslash ell\_1\}$

As long as $\backslash delta\; \backslash ell\_1\; \backslash ll\; \backslash ell\_0$, it is possible to write:

$\backslash epsilon\_2\; \backslash simeq\; \backslash frac\{\backslash delta\; \backslash ell\_2\}\{\backslash ell\_0\}$

and thus

$\backslash epsilon\; \backslash simeq\; \backslash epsilon\_1\; \backslash ;\; +\; \backslash epsilon\_2\; \backslash ;$

True strain, however, can be totalled. This is defined by:

$\backslash exp(\backslash epsilon)\; =\; \backslash frac\{d\backslash ell\}\{\backslash ell\}$

and thus

$\backslash epsilon\; =\; \backslash ln\; \backslash left\; (\backslash frac\{\backslash ell\}\{\backslash ell\_0\}\; \backslash right\; )$

The engineering strain formula is the series expansion of the true strain formula.

See also

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• engineering
• geology
• Hooke's law
• Poisson's ratio
• plate tectonics
• strain gauge
• strain tensor
• Stress-strain curve

Physical quantity | Continuum mechanics

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