Plane strain

From Wikipedia, the free encyclopedia

In real engineering components, stress (and strain) are 3-D tensors 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 $ε 33$ and the shear strains $ε 13$ and $ε 23$ (if the length is the 3-direction) are constrained by nearby material and are small compared to the cross-sectional strains. The strain tensor can then be approximated by:

$\underline{\underline{\epsilon}} = \begin{bmatrix} \epsilon_{11} & \epsilon_{12} & 0 \\ \epsilon_{21} & \epsilon_{22} & 0 \\ 0 & 0 & 0\end{bmatrix}$

in which the double underline indicates a second order tensor. This strain state is called plane strain. The corresponding stress tensor is:

$\underline{\underline{\sigma}} = \begin{bmatrix} \sigma_{11} & \sigma_{12} & 0 \\ \sigma_{21} & \sigma_{22} & 0 \\ 0 & 0 & \sigma_{33}\end{bmatrix}$

in which the non-zero $σ 33$ is needed to maintain the constraint $ε 33 = 0$ . 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.

Plane strain

From Wikipedia, the free encyclopedia

Views

Navigation

interaction

Search