Strain rates and material spins

The material time rate of Lagrangean strain measures, objective corotationa l rates of Eulerian strain measures and their defining spin tensors are inv estigated from a general point of view. First, a direct and rigorous method is used to derive a simple formula for the gradient of the tenser-valued f unction defining a general class of strain measures. By means of this formu la and the chain rule as well as Sylvester's formula for eigenprojections, explicit basis-free expressions for the material time rate of an arbitrary Lagrangean strain measure can be derived in terms of the right Cauchy-Green tensor and the material time rate of any chosen Lagrangean strain measure, e.g. Hencky's logarithmic strain measure. These results provide a new deri vation of Carlson-Hoger's general gradient formula for an arbitrary general ized strain measure and supply a new, rigorous proof for Carlson-Hoger's co njecture concerning the n-dimensional case. Next, by virtue of the aforemen tioned gradient formula, a general fact for objective corotational rates an d their defining spin tensors is disclosed: Let Omega = Y (B, D, W) be any spin tensor that is continuous with respect to B, where B, D and W are the left Cauchy-Green tensor, the stretching tensor and the vorticity tensor. T hen the corotational rate of an Eulerian strain measure defined by B is obj ective if and only if Omega = W + (Y) over tilde(B, D), where (Y) over tild e is isotropic. By means of this fact and certain necessary or reasonable r equirements, it is further found that a single antisymmetric function of tw o positive real variables can be introduced to characterize a general class of spin tensors defining objective corotational rates. A general basis-fre e expression for all such spin tensors and accordingly a general basis-free expression for a general class of objective corotational rates of an arbit rary Eulerian strain measure are established in terms of the left Cauchy-Gr een tensor B and the stretching tensor D as well as the introduced antisymm etric function. By choosing several particular forms of the latter, all com monly-known spin tensors and corresponding corotational rates are shown to be incorporated into these general formulas in a natural way. In particular , with the aid of these general formulae it is proved that an objective cor otational rate of the Eulerian logarithmic strain measure In V is identical with the stretching tensor D and moreover that in all possible strain tens or measures only In V enjoys this property.