Abstract Investigators have viewed the stress rate in two different ways: the material (body-fixed) point of view and the Eulerian point of view. We discuss the Zaremba–Jaumann rate and Oldroyd’s rate from the material viewpoint and apply them to the material formulation of a theory of plasticity for materials undergoing anisotropic plastic deformation. Significant advantages of the material formulation are that the derivation of equations is straight forward, the distortion of yield surface can be easily accounted for, and the issue of self-consistent elastic equation does not arise. A convected (body-fixed) coordinate system that is imbedded in the material element is used to define a convected material element. Constitutive equations in a theory of plasticity are written with respect to this element referred to the body-fixed coordinate system. A material type yield function is a function of stress components and it is invariant under either a superimposed rigid rotation of the body or an arbitrary change of observer. The kinematic hardening evolves by the co-rotational rate and it is disassociated from the general kinematic of deformation. The latter evolves by the convected rate.