We consider the principles of coordinate, rotational, and initial independence of the equations of state for a deformable material and the theorem on the existence of elasticity potential connected with them. We show that the well-known axiomatic substantiation and mathematical representation of these principles in “rational continuum mechanics” as well as the proof of the theorem are erroneous. A correct proof of the principles and theorem is presented for the most general case (a stressed anisotropic body under the action of an arbitrary tensor field) without applying any axioms. On this basis, we eliminated the dependence on an arbitrary initial state and the corresponding accumulated strain from the system of equations of state of a deformable material. The obtained forms of equations are convenient for constructing and analyzing the equations of local influence of initial stresses on physical fields of different nature. Finally, these equations represent governing equations for the problems of nondestructive testing of inhomogeneous three-dimensional stress fields and for theoretical-and-experimental investigation of the nonlinear equations of state.
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 52, No. 3, pp. 90–102, July–September, 2009.
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Prokopovych, I.B. Independence principles in the equations of state of a deformable material. J Math Sci 171, 534–547 (2010). https://doi.org/10.1007/s10958-010-0155-z
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DOI: https://doi.org/10.1007/s10958-010-0155-z