Editing Infinitesimal strain theory (section)

== Special cases ==

===Plane strain===
[[Image:Plane strain.svg|class=skin-invert-image|500px|right|thumb|Plane strain state in a continuum.]]
In real engineering components, [[Stress (physics)|stress]] (and strain) are 3-D [[tensor]]s but in prismatic structures such as a long metal billet, the length of the structure is much greater than the other two dimensions. The strains associated with length, i.e., the normal strain <math>\varepsilon_{33}</math> and the shear strains <math>\varepsilon_{13}</math> and <math>\varepsilon_{23}</math> (if the length is the 3-direction) are constrained by nearby material and are small compared to the ''cross-sectional strains''. Plane strain is then an acceptable approximation. The [[strain tensor]] for plane strain is written as:
<math display="block">\underline{\underline{\boldsymbol{\varepsilon}}} = \begin{bmatrix}
\varepsilon_{11} & \varepsilon_{12} & 0 \\
\varepsilon_{21} & \varepsilon_{22} & 0 \\
     0      &     0       & 0
\end{bmatrix}</math>
in which the double underline indicates a second order [[tensor]]. This strain state is called ''plane strain''. The corresponding stress tensor is:
<math display="block">\underline{\underline{\boldsymbol{\sigma}}} = \begin{bmatrix}
\sigma_{11} & \sigma_{12} & 0 \\
\sigma_{21} & \sigma_{22} & 0 \\
     0      &     0       & \sigma_{33}
\end{bmatrix}</math>
in which the non-zero <math>\sigma_{33}</math> is needed to maintain the constraint <math>\epsilon_{33} = 0</math>. This stress term can be temporarily removed from the analysis to leave only the in-plane terms, effectively reducing the 3-D problem to a much simpler 2-D problem.

=== Antiplane strain ===
{{main|Antiplane shear}}
Antiplane strain is another special state of strain that can occur in a body, for instance in a region close to a [[screw dislocation]]. The [[strain tensor]] for antiplane strain is given by
<math display="block">\underline{\underline{\boldsymbol{\varepsilon}}} = \begin{bmatrix}
0 & 0 & \varepsilon_{13} \\
0 & 0 & \varepsilon_{23}\\
\varepsilon_{13} & \varepsilon_{23} & 0
\end{bmatrix}</math>