= Solution
<Plane strain> sets all <strains> involving $z$ to zero, including $e_{zz}=0$, while $e_{xx}=u_x$, $e_{yy}=v_y$ and $e_{xy}=(u_y+v_x)/2$. Thus
$$
\boxed{\tau=\begin{pmatrix}
(\lambda+2\mu)u_x+\lambda v_y&\mu(u_y+v_x)&0\\
\mu(u_y+v_x)&\lambda u_x+(\lambda+2\mu)v_y&0\\
0&0&\lambda(u_x+v_y)
\end{pmatrix}.}
$$
In particular, $\tau_{zz}$ generally does not vanish: the suppressed out-of-<plane strain> requires a reaction <stress>. Setting that entry to zero would instead be a plane-<stress> assumption.
Back to article page
