On Infinitesimal Shear

Abstract

The setting for this note is the theory of infinitesimal strain in the context of classical linearized elasticity. As a body is subjected to a deformation the angle between a pair of material line elements through a typical point P is changed. The decrease in angle is called the shear of this pair of elements.

Here, we determine all pairs of material line elements at P which are unsheared in a deformation. It is seen, in general, that corresponding to any given material line element in a given plane through P, there is one corresponding “companion” material line element such that the given element and its conjugate are unsheared in the deformation. There are two exceptions. If the plane through P is a plane of central circular section of the strain ellipsoid, then every material line element through P in this plane has an infinity of companion elements in this plane — all pairs of material line elements in the plane(s) of central circular section of the strain ellipsoid are unsheared. If the plane through P is not a plane of central circular section of the strain ellipsoid, then there are two exceptional material line elements through P such that neither of them has a companion material line element forming an unsheared pair with it. The directions of these exceptional elements in the plane are called “limiting directions”. It is seen that it is the pair of elements along the limiting directions in a plane which suffer the maximum shear in that plane.

A geometrical construction is presented for the determination of the extensional strains along the pairs of elements which are unsheared. Also, it is shown that knowing one unsheared pair in a plane and their extensions is sufficient to determine the principal extensions and the principal axes in this plane. Expressions for all unsheared pairs in a given plane are given in terms of the normals to the planes of central circular sections of the strain ellipsoid.

Finally, for a given material line element, a formula is derived for the determination of all other material line elements which form an unsheared pair with the given element.