# A.2 Simple Shear

An alternative, and often simpler to implement, shear problem is that of simple shear. With such a deformation field, only one shear stress component is non-zero (like the pure shear case). The difference arises in that given a simple shear loading the diagonal stresses are not necessarily zero. This state may be produced by a motion, \(\chi(X_i,t)\) of the form \(\chi(X_i,t)=X_i+\gamma(t) X_2\delta_{i1}\). The resultant deformation gradient, \(F_{ij}\), takes the form,

\[F_{ij}=\delta_{ij}+\gamma\left(t\right)\delta_{i1}\delta_{j2}\]

and it is noted that this deformation is volume preserving (\(J=\det F_{ij}=1\)). Numerically, such a deformation field results from applying a displacement in the \(x\) direction along the \(y\) surface.