A note on some insights from decoupling the time derivative of an objective tensor

There are some standard procedures in Mechanics that lead to a final equation that presents an apparent objectivity unbalancing. Generally, this apparent unbalancing has its origin on the material time derivative of an objective quantity. In the present work we analyze three examples: the material time derivative of a scalar-valued function of a second order tensor, the relaxation of a simple viscoelastic material and the Cattaneo equation for the heat flux where this situation occurs. We show that, decoupling the material time derivative of the tensor into a part that is associated to the time derivative of its eigenvalues and another part that is associated to the time derivative of its eigenvectors, only the first part is significant. This conclusion can bring insights for different forms of constitutive assumptions and interpretation of general results.