OOF: Finite Element Analysis of Microstructures

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Plane Stress and Plane Strain

Plane strain is defined by the strain state $\epsilon_{iz} = 0$; it is the limiting condition in the center plane of a very thick specimen. An out-of-plane stress $\sigma_{iz}$ is required to maintain plane strain.

Plane stress is defined by the stress state $\sigma_{iz} = 0$; it holds at the surface of a thin specimen. The displacement field in the $z$ direction is not zero nor is it uniform in an heterogeneous grid.

Using plane stress or plane strain, the stiffness tensor $C_{ijkl}$ can be reduced from three to two dimensions, because the $z$ components of stress and strain have been eliminated from the problem. In the matrix representation, $C$ becomes a $3\times3$ matrix, with rows labeled 11, 22, and 12 in the original tensor indices.

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