Extension of the concept of material objectivity to mixed Eulerian-Lagrangian reference systems

Summary.

The material objectivity of stress and strain measures forms one of the fundamental physical principles, which have to be taken into account in the formulation of constitutive laws. In the present paper we investigate its form in a mixed Eulerian-Lagrangian (E-L) reference system and show that it covers the well-known versions in the Eulerian (E) and Lagrangian (L) levels of description. The transformation rules for vector and tensor quantities are derived, and the concept of the Lie-derivative for time-dependent quantities is reviewed and generalized. Depending on the underlying formulation, different objective rate tensors for stress and strain, such as the Oldroyd and Truesdell rate, can be derived in a systematic way in the intermediary (E-L) system. This concept is then applied to elasto-viscoplastic and hypoelastic constitutive laws.