On finite shear

If a pair of material line elements, passing through a typical particle P i n a body, subtend an angle Theta before deformation, and Theta + gamma afte r deformation, the pair of material elements is said to be sheared by the a mount gamma. Here all pairs of material elements at P are considered for ar bitrary deformations. Two main problems are addressed and solved. The first is the determination of all pairs of material line elements at P which are unsheared. The second is the determination of that pair of material line e lements at P which suffers the maximum shear. All unsheared pairs of material elements in a given plane pi(S) with normal S passing through P are considered. Provided pi(S) is not a plane of centr al circular section of the C-ellipsoid at P (where C is the right Cauchy-Gr een strain tensor), it is seen that corresponding to any material element i n pi(S) there is, in general, one companion material element in pi(S) such that the element and its companion are unsheared. There are, however, two elements in pi(S) which have no companions. We call their corresponding directions limiting directions. Equally inclined to th e direction of least stretch in the plane pi(S), the limiting directions pl ay a central role. It is seen that, in a given plane pi(S), the pair of mat erial line elements which suffer the maximum shear lie along the limiting d irections in pi(S). If Theta(L) is the acute angle subtended by the limitin g directions in pi(S) before deformation, then this angle is sheared into i ts supplement n - Theta(L) so that the maximum shear gamma* in pi(S) is gam ma* = pi - 2 Theta(L). If S is given and C is known, then Theta(L) may be d etermined immediately. Its calculation does not involve knowing the eigenve ctors or eigenvalues of C. When all possible planes through P are considered, it is seen that the glob al maximum shear gamma(G)* occurs for material elements lying along the lim iting directions in the plane spanned by the eigenvectors of C correspondin g to the greatest principal stretch (lambda(3)) and the least (lambda(1)). The limiting directions in this principal plane of C subtend the angle 2 ta n(-1){(lambda(1)/lambda(3))(1/2)} and gamma(G)* = pi - 4 tan(-1) {(lambda(1 )/lambda(3))(1/2)}. Generally the maximum shear does not occur for a pair o f material elements which are originally orthogonal. For a given material element along the unit vector N, there is, in general, in each plane pi(S) passing through N at P, a companion vector M such that material elements along N and M are unsheared. A formula, originally due t o Joly (1905), is presented for M in terms of N and S. Given an unsheared pair in pi(S), the limiting directions in pi(S) are seen to be easily determined, either analytically or geometrically. Planar shear, the change in the angle between the normals of a pair of mate rial planar elements at X, is also considered. The theory of planar shear r uns parallel to the theory of shear of material line elements. Correspondin g results are presented. Finally, another concept of shear used in the geology literature, and appar ently due to Jaeger, is considered. The connection is shown between Cauchy shear, the change in the angle of a pair of material elements, and the Jaeg er shear, the change in the angle between the normal N to a planar element and a material element along the normal N. Although Jaeger's shear is descr ibed in terms of one direction N, it is seen to implicitly include a second material line element orthogonal to N.