In this chapter, we deal with the theory of finite strain in the context of nonlinear elasticity. As a body is subjected to a finite deformation, the angle between a pair of material line elements through a typical point is changed. The change in angle is called the “shear” of this pair of material line elements. Here we consider the shear of all pairs of material line elements under arbitrary deformation. Two main problems are addressed and solved. The first is the determination of all “unsheared pairs”, that is all pairs of material line elements which are unsheared in a given deformation. The second is the determination of those pairs of material line elements which suffer the maximum shear.

Also, triads of material line elements are considered. It is seen that, for an arbitrary finite deformation, there is an infinity of oblique triads which are unsheared in this deformation and it is seen how they are constructed from unsheared pairs.

Finally, for the sake of completeness, angles between intersecting material surfaces are considered. They are also changed as a result of the deformation. This change in angle is called the “planar shear” of a pair of material planar elements. A duality between the results for shear and for planar shear is exhibited.

6.1 Introduction

At a typical particle P in a body, material line elements are generally translated, rotated and stretched as a result of a deformation, so that angles between intersecting material lines are generally changed.