Relations between solutions of the Zorawski condition and motions with constant stretch history

摘要

In the present work, we examine the relationship between some skew-symmetric tensors that appear in the field of Continuum Mechanics, namely Motions With Constant Stretch History (MWCSH), the Zorawski condition, and Rivlin-Ericksen (R-E) tensors. MWCSH are motions that are steady from the Lagrangian viewpoint. The Zorawski condition is met when there is a second reference frame for a given transient velocity field where this flow is steady from an Eulerian viewpoint. The R-E tensors are n-orders convected time derivatives of the identity tensor that can help describe a material's strain history. Our theoretical analysis has shown that the rate-of-rotation of the R-E tensors in a MWCSH is the same and is equal to the skew-symmetric tensor that rules this kind of motion, generalizing some analytical results of the literature concerning the conditions for a solution of the equations concerning MWCSH. In addition, we found a compact form for the Zorawski condition. These findings enable a theoretical analysis of cases where we can anticipate if the motion is intrinsically unsteady or not. In particular, we analyzed combinations of homogeneity/non-homogeneity and if a motion is or is not a MWCSH. In this regard, we discussed the existence of Zorawski solutions where the rate-of-rotation of the second observer is the rate-of-rotation of the eigenvectors of the R-E tensor. An unsteady non-homogeneous flow analyzed in the literature was shown to obey this condition.