Summary
A new basis-free expression is derived for the logarithmic spin tensor, which assumes a unified form for all the three cases of coalescence of the three eigenvalues of the left Cauchy-Green deformation tensorB. It is shown that this new expression is endowed with particular continuity properties, i.e., each of its coefficients either accompanies the vanishing of its associated term or remains valid whenever the eigenvalues ofB become repeated. These favorable properties, which are not enjoyed by the expressions derived earlier, result in the foregoing unified form of expression which could not be achieved by means of the previous expressions. At the same time, a fully explicit, basis-free expression is obtained for the spin tensor of the Eulerian triad, which is expressed straightforwardly in terms of the three basic invariants ofB and hence renders computations of the eigenvalues ofB unnecessary. Moreover, remarks are given towards explaining and clarifying some issues concerning the hypo-elastic equation of grade zero with the logarithmic rate, including the exact integrability and unstable stress responses associated with initial shear stresses at simple shear deformations, etc.
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Bruhns, O.T., Xiao, H. & Meyers, A. New results for the spin of the Eulerian triad and the logarithmic spin and rate. Acta Mechanica 155, 95–109 (2002). https://doi.org/10.1007/BF01170842
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DOI: https://doi.org/10.1007/BF01170842