Extended Polar Decompositions for Plane Strain

In a finite deformation at a particle of a continuous body, a triad of infinitesimal material line elements is said to be “unsheared” when the angles between the three pairs of line elements of the triad suffer no change. In a previous paper, it has been shown that there is an infinity of unsheared oblique triads. With each oblique unsheared triad may be associated an “extended polar decomposition” F = QG = HQ of the deformation gradient F, in which Q is a rotation tensor, and G, H are not symmetric. Both G and H have the same real eigenvalues which are the stretches of the elements of the triad. In this paper, a detailed analysis of extended polar decompositions is presented in the case when the finite deformation is that of plane strain. Then, we may deal with a 2 × 2 deformation gradient F′ = Q′G′ = H′Q′ instead of the full 3 × 3 tensor F. In this case, the extended polar decompositions are associated with “unsheared pairs,” i.e., pairs of infinitesimal material line elements in the plane of strain which suffer no change in angle in the deformation. If one arm of an unsheared pair is chosen in the plane of strain, then, in general, its companion in the plane is determined. It follows that all possible extended polar decompositions may then be described in terms of a single parameter, the angle that the chosen arm makes with a coordinate axis in the plane. Explicit expressions for G′ and H′ are obtained, and various special cases are discussed. In particular, we note that the expressions for G′ and H′ remain valid even when the chosen arm is along a “limiting direction,” that is the direction of a line element which has no companion element in the plane forming an unsheared pair with it. The results are illustrated by considering the cases of simple shear and of pure shear.Dedicated to Professor Piero Villaggio as a symbol of our friendship and esteem.