Objective Tensors and Their Mappings in Classical Continuum Mechanics

— The fundamentals of the theory of objective tensor mechanical characteristics are presented: the general concept of objectivity, types of objectivity, simple tensor analogs of mechanical characteristics and simple commutative diagrams connecting them. For objective tensors of the second rank, a general classification of simple diagrams is given, examples of diagrams are given. The mappings connecting objective mechanical tensor processes of various ranks and types of objectivity are considered. The concept of mappings independent of the frame of reference is introduced, which form the basis of the theory of constitutive relations of media, and a theorem on the necessary and sufficient conditions for such independence is established. Examples are given. For two arbitrary diagrams of objective tensor processes (generally speaking, of different ranks), a complete set (package) of related mappings (conductors) of all possible analogs of one diagram into all possible analogs of another diagram is considered—a package of conductors, or a mapping of diagrams. For coinciding diagrams (of the same rank with the same intertwining operators), a generalizing concept of objective derivatives is introduced that make up a subpackage of conductors connecting tensors of the same (all possible) types of objectivity. General equations for the objective derivatives of scalars, vectors, and second-rank tensors are presented, presented in Lagrangian and Eulerian forms. It is shown that the well-known derivatives of Zaremba–Jaumann, Oldroyd, Cotter–Rivlin, Truesdell, Hill, Sedov, Dienes, Gordon–Schowalter are covered by these equations. The objective integration operator is constructed as the inverse operator to the objective derivative. Examples of the use of objective derivatives and objective integrals in the construction of the constitutive relations of media at finite deformations are given.