Elasticity/Antiplane shear

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Antiplane shear (or antiplane strain) is the state of strain that is obtained when the displacement field is of the form

$u_1=u_2=0,u_3=u_3(x_1,x_2)$

There is only an out of plane displacement.

Since the strains are given by

${\epsilon }_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i})=\frac{1}{2}(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i})$

we have

$\begin{array}{cc}{\epsilon }_{11}& =\frac{\partial u_1}{\partial x_1}=0\\ {\epsilon }_{22}& =\frac{\partial u_2}{\partial x_2}=0\\ {\epsilon }_{33}& =\frac{\partial u_3}{\partial x_3}=0\\ {\epsilon }_{23}& =\frac{1}{2}(\frac{\partial u_2}{\partial x_3}+\frac{\partial u_3}{\partial x_2})=\frac{1}{2}{u}_{3,2}\\ {\epsilon }_{31}& =\frac{1}{2}(\frac{\partial u_3}{\partial x_1}+\frac{\partial u_1}{\partial x_3})=\frac{1}{2}{u}_{3,1}\\ {\epsilon }_{12}& =\frac{1}{2}(\frac{\partial u_1}{\partial x_2}+\frac{\partial u_2}{\partial x_1})=0\end{array}$

Therefore, for antiplane shear, the only nonzero strains are the out-of-plane shear strains

${\epsilon }_{23}=\frac{1}{2}\frac{\partial u_3}{\partial x_2};{\epsilon }_{13}=\frac{1}{2}\frac{\partial u_3}{\partial x_1}$