Unsheared triads and extended polar decompositions of the deformation gradient

In this paper, the concept of unsheared triads of material line elements at a point X is introduced. We find that there is an infinity of unsheared tr iads. More precisely, it is shown that, in general, for any given unsheared pair at X, a unique third material line element at X may be found such tha t the three material line elements form an unsheared triad. Special cases a re analyzed in detail. A link between unsheared triads and new decompositio ns of the deformation gradient, is exhibited. These decompositions generali ze the classical polar decomposition F = RU = VR of the deformation gradien t F, in which R is a proper orthogonal tensor and U, V are positive-definit e symmetric. Associated with any unsheared (oblique) triad is a new decompo sition F = QG = HQ, in which Q is a proper orthogonal tensor, but G and H a re no longer symmetric, but have three positive eigenvalues and three linea rly independent right eigenvectors. Because there is an infinity of unshear ed triads, there is an infinity of such decompositions. We call them "exten ded polar decompositions". Several examples of unsheared triads and extende d polar decompositions are presented. (C) 2000 Elsevier Science Ltd. All ri ghts reserved.