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Unsheared triads and extended polar decompositions of the deformation gradient

par Boulanger, Philippe ;Hayes, Michael
Référence International journal of non-linear mechanics, 36, 3, page (399-420)
Publication Publié, 2001-05

Article révisé par les pairs

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Résumé :

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 triads. 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 that the three material line elements form an unsheared triad. Special cases are analyzed in detail. A link between unsheared triads and new decompositions of the deformation gradient, is exhibited. These decompositions generalize the classical polar decomposition F = RU = VR of the deformation gradient F, in which R is a proper orthogonal tensor and U, V are positive-definite symmetric. Associated with any unsheared (oblique) triad is a new decomposition F = QG = HQ, in which Q is a proper orthogonal tensor, but G and H are no longer symmetric, but have three positive eigenvalues and three linearly independent right eigenvectors. Because there is an infinity of unsheared triads, there is an infinity of such decompositions. We call them `extended polar decompositions'. Several examples of unsheared triads and extended polar decompositions are presented.