# Dictionary:Strain (εij)

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{{#category_index:S|strain (εij)}} The change of dimensions or shape produced by a stress. Strain is usually expressed in dimensionless units such as change of length per unit of length, angle of twist, change of volume per unit of volume. Rotation or translation without change of shape is not strain. See elastic constants and Sheriff and Geldart (1995, 36–37). If u, v, w are the stress-produced displacements in the x,y,z directions of a point in an anisotropic body, the strains are:

normal strains

${\displaystyle \varepsilon _{xx}={\frac {\partial u}{\partial x}}}$,

${\displaystyle \varepsilon _{yy}={\frac {\partial v}{\partial y}}}$,

${\displaystyle \varepsilon _{zz}={\frac {\partial w}{\partial z}}}$.

shearing strains

${\displaystyle \varepsilon _{xy}=\varepsilon _{yx}={\frac {\partial v}{\partial x}}+{\frac {\partial u}{\partial y}}}$,

${\displaystyle \varepsilon _{yz}=\varepsilon _{zy}={\frac {\partial w}{\partial y}}+{\frac {\partial v}{\partial z}}}$,

${\displaystyle \varepsilon _{zx}=\varepsilon _{xz}={\frac {\partial u}{\partial z}}+{\frac {\partial w}{\partial x}}}$.

The fractional change in volume (dilatation) ${\displaystyle \Delta }$ is

${\displaystyle \Delta =\varepsilon _{xx}+\varepsilon _{yy}+\varepsilon _{zz}}$.

Note: The "shearing strains" defined above are so-called "engineering" strains. They differ from the corresponding components of the strain tensor by a factor of 1/2. See also the Strain page.