Strain rates and material spins

The material time rate of Lagrangean strain measures, objective corotational rates of Eulerian strain measures and their defining spin tensors are investigated 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 function defining a general class of strain measures. By means of this formula 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 derivation of Carlson-Hoger's general gradient formula for an arbitrary generalized strain measure and supply a new, rigorous proof for Carlson-Hoger's conjecture concerning the n-dimensional case. Next, by virtue of the aforementioned gradient formula, a general fact for objective corotational rates and 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. Then the corotational rate of an Eulerian strain measure defined by B is objective if and only if Omega = W + (Y) over tilde(B, D), where (Y) over tilde is isotropic. By means of this fact and certain necessary or reasonable requirements, it is further found that a single antisymmetric function of two positive real variables can be introduced to characterize a general class of spin tensors defining objective corotational rates. A general basis-free expression for all such spin tensors and accordingly a general basis-free expression for a general class of objective corotational rates of an arbitrary Eulerian strain measure are established in terms of the left Cauchy-Green tensor B and the stretching tensor D as well as the introduced antisymmetric function. By choosing several particular forms of the latter, all commonly-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 corotational rate of the Eulerian logarithmic strain measure In V is identical with the stretching tensor D and moreover that in all possible strain tensor measures only In V enjoys this property.