On non–corotational rates of Oldroyd's type and relevant issues in rate constitutive formulations

Abstract

The lower and upper Oldroyd rates are two well–known objective non–corotational rates. A remarkable property of these rates is that they establish direct relationships between finite strain measures and the Eulerian strain rate (stretching) D. Namely, the Oldroyd rates of the Almansi strain and the Finger strain, respectively, are exactly D. Here, we would like to study non–corotational rates of Oldroyd's type from a general point of view. We demonstrate that, for any given Hill's strain measure, there actually exist infinitely many definitions of non–corotational rates such that the rate of this strain evaluated according to each of these definitions is identical to D. A general explicit expression for the asymmetric second–order tensor defining all such non–corotational rates is derived in terms of the left Cauchy–Green deformation tensor and the velocity gradient. It is shown that a Hill–type non–corotational rate of any given Doyle–Ericksen or Seth–Hill strain can identically equal D. This markedly contrasts with a corresponding fact related to corotational rates. In a recent work, it has been established that, if a corotational rate of a strain is identical to D, there is only one choice under the doubly free choices of strain and spin, namely, Hencky's logarithmic strain and the logarithmic spin. The essence of this sharp contrast will be investigated from standpoints of both kinematics and rate constitutive formulations. In particular, relevant issues in traditional Eulerian rate formulations of finite elastoplasticity will be clarified by virtue of two consistency criteria that are essential to reasonable Eulerian rate formulations.