# Plane Strain in Anisotropic Materials

I'm looking into plane strain and I'm a little confused about the concept. If we assume the material is much longer in the $z$-direction, the typical assumptions I see are \begin{equation*} |\varepsilon_{12}|, |\varepsilon_{13}|, |\varepsilon_{33}| \ll 1 \end{equation*} where $\boldsymbol{\varepsilon}$ is the strain tensor. Even if we suppose those terms are zero, it seems to yield the following stresses are nonzero: \begin{equation*} \begin{split} \sigma_{33} =& C_{1133} \varepsilon_{11} + C_{2233} \varepsilon_{22} + C_{3312} \varepsilon_{12} \\ \sigma_{23} =& C_{1123} \varepsilon_{11} + C_{2223} \varepsilon_{22} + C_{2312} \varepsilon_{12} \\ \sigma_{13} =& C_{1113} \varepsilon_{11} + C_{2213} \varepsilon_{22} + C_{1312} \varepsilon_{12} \\ \end{split} \end{equation*} In practice are the $\sigma_{i3}$ typically ignored or assumed to be small? This formulation seems to imply we have $\sigma_{33}$ is not necessarily small even in the case of isotropy, for instance $C_{1133}$ or $C_{2233}$ $\gg 1$. As we already have three (hopefully independent) equations for $\sigma_{11},\sigma_{12},\sigma_{22}$ in terms of $\varepsilon_{11}, \varepsilon_{12},\varepsilon_{22}$ we could I suppose rewrite $\sigma_{33}, \sigma_{23},\sigma_{13}$ in terms of $\sigma_{11},\sigma_{12},\sigma_{22}$, but I wonder if I am perhaps missing a core assumption of plane strain. For instance we assume $\frac{\partial u_1}{\partial z} \approx -\frac{\partial u_3}{\partial x}$ since $|\varepsilon_{13}| \ll 1$, but would it be wrong to strengthen this and assume $\frac{\partial u_1}{\partial z} = \frac{\partial u_3}{\partial x} = 0$, where $u$ is the displacement?

• The plane strain assumption does not in any way imply the out of plane stress is negligible. Oct 24 '17 at 15:38
• Plane strain has nothing to do with the dimension of the material in the $z$ direction. What matters is the constraints on the structure, which force the three strain components you mention in your OP to be zero. For a general anisotropic material, assuming three strain components are zero might not make any sense, but it does make sense in some special cases, e.g. an orthotropic material where the material anisotropy is in the same direction as the $z$ axis. Oct 24 '17 at 20:03
• You might get some help if you look up LEFM in Wikipedia. Of the many calculations I have seen , I never saw any for anisotropic material. Nov 24 '17 at 21:34