In the attachment, we show that Truesdell rate can by simplified to Green-Naghdi rate by assuming F .=. R and can be further simplified to Jaumann rate by assuming W .=. R(.)R(T), where .=. means approximately equal
In a stretch dominant deformation, the three rates give different stress rate. This is usually explained by that we need a different tangential modulus for different objective rate. However, it is hard to understand why we need to change "material" modulus when we use a different "mathematical" form of objective rate as they are all supposed to be equivalent.