Abstract
In the Newtonian approach to mechanics, the concepts of objective tensors of various ranks and types are introduced. The tough classification of objective tensors is given, including tensors of material and spatial types. The diagrams are constructed for non-degenerate (“analogous”) relations between tensors of one and the same (any) rank, and of various types of objectivity. Mappings expressing dependence between objective tensor processes of various ranks and types are considered. The fundamental concept of frame-independence of such mappings is introduced as being inherent to constitutive relations of various physical and mechanical properties in the Newtonian approach. The criteria are established for such frame-independence. The mathematical restrictions imposed on the frame-independent mappings by the objectivity types of connected tensors are simultaneously revealed. The absence of such restrictions is established exclusively for mappings and equations linking tensors of material types. Using this, a generalizing concept of objective differentiation of tensor processes in time, and a new concept of objective integration, are introduced. The axiomatic construction of the generalized theory of stress and strain tensors in continuum mechanics is given, which leads to the emergence of continuum classes and families of new tensor measures. The axioms are proposed and a variant of the general theory of constitutive relations of mechanical properties of continuous media is constructed, generalizing the known approaches by Ilyushin and Noll, taking into account the possible presence of internal kinematic constraints and internal body-forces in the body. The concepts of the process image and the properties of the five-dimensional Ilyushin’s isotropy are generalized on the range of finite strains.
Highlights
The courses in continuum mechanics [1–9] are based on the fundamental works of the classics of science [10–13], which provided the foundations of the modern general theory of the constitutive relations of deformable media and contributed to the solution of the sixth Hilbert problem [14]
The presented material contributes to the creation and development of new approaches in various fields of Newtonian mechanics, in the branches of classical continuum mechanics
Related mechanical characteristics can be expressed by objective tensors of the same rank and of different types: vectors u(0), u(1), second rank tensors L(00), L(10), L(01), L(11), which are related pairwise by identities in the form of maps that make up commutative diagrams in the form
Summary
The courses in continuum mechanics [1–9] are based on the fundamental works of the classics of science [10–13], which provided the foundations of the modern general theory of the constitutive relations of deformable media and contributed to the solution of the sixth Hilbert problem [14]. The presented material contributes to the creation and development of new approaches in various fields of Newtonian mechanics, in the branches of classical continuum mechanics These new approaches are based on fundamental achievements of the science of mechanics [15–51] and mathematics [52–64]. They were carried out by the author [65–73] and were realized by his colleagues in different branches: In elasto-plasticity at finite strains [74,75], in constructing models of Cosserat type [75–79], in poromechanics [79,80], in the theory of shape-memory materials [81], in the generalization of the theory of elastic-plastic processes to finite deformations [82], in numerical methods for elastic-plastic problems at finite strains [83], in analytical research in hypo-elasticity [84], and in constructing new models of visco-elasticity at finite strains by using new methods, including the application of different objective derivatives [85–87]
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