Pure shear is the deformation in the x axis resulting in no change of area. During pure shear the axes of the strain ellipsoid do not rotate and the incremental and finite strain ellipsoids are coaxial. Simple shear results when a body is subjected to a uniform shear, parallel to some direction, involving no change in area. It can be argued that as most geological deformations do involve area changes the pure and simple shear models are not realistic. However observations do show that deformation does closely relate to these two types of deformation.
The above diagram represents strain ellipsoids for pure shear. There is no rotation only compression and exstension.
This diagram shows that simple shear involves rotation about a point. The amount of rotation is the augular shear as descibed in 2.2.
Pure and simple shear are almost always found together. The above object has gone through rotation and compression / exstension. The order in which the two occurr is very important and produce completely different finite strain ellipsoids as can be seen below.