A generalized theory of tensor strain and stress measures in the classical continuum mechanics

A generalized theory of tensor measures of strains and stresses in classical continuum mechanics is discussed: the main axioms of the theory are proposed, the general formulas of new tensor measures are derived, the theorem of energy conjugation is established to separate the complete Lagrangean class of the measures. As a subclass, the simple Lagrangean class of energy conjugated measures of stresses and finite strains is constructed in which the families of holonomic and corotational measures are distinguished. By comparison of measures of the simple Lagrangean class with one another and by matching them with logarithmic measures, the characteristics of holonomic and corotational measures are studied. For the simple Lagrangean class and its families, their completeness and closure are established relative to any choice of a generating pair of energy conjugated measures. The applications of the new tensor measures in modeling the properties of plasticity, viscoelasticity, and shape memory are mentioned.