| Keywords: | algebraic weight, pseudotensor, nanoscale, microscale, energy form, integer rational algebraic invariant, irreducible system of invariants, cubic approximation, hemitropic micropolar elastic solid |
| Abstracts: | This study deals with an algorithm for deriving a quintic-order approximation of the potential for force and moment stresses in a hemitropic micropolar elastic solid. The formulation accounts for corrections up to the fifth algebraic order—extending beyond the fundamental quadratic approximation through the systematic application of algebraic invariant theory.
To this end, the complete set of irreducible invariants for a system of two asymmetric secondrank tensors is discussed and represented in the form of invariant traces. Consequently, an initial set of 86 invariant traces is proposed, comprising 8 single invariants, 17 dual combinations, 44 invariant triples, and 17 invariant quadruples. This classification is based on the number of tensor literals involved, with a maximum of four literals. The maximum degree of the initial invariants is six.
From these 86 elements, 63 traces are subsequently selected according to the rule of increasing algebraic degree: 2 linear invariants, 6 quadratic, 12 cubic, 19 of the fourth degree, and 24 invariants of the fifth degree. A scheme for obtaining the fifth-degree invariants is introduced, partitioned for convenience into seven groups based on the following rules: products of linear invariants with each other, pairwise products of quadratic and cubic invariants, pairwise products of first- and fourthdegree invariants, products of linear and cubic invariants, products of linear invariants raised to the third power with quadratic invariants, products of linear invariants with squares of quadratic invariants, and the original fifth-degree invariants.
Thus, the hemitropic micropolar potential is characterized by 366 mechanical moduli. Constitutive equations for force and couple stresses are derived, incorporating second-, third-, and fourth-order algebraic corrections, and are formulated to be valid in an arbitrary curvilinear coordinate system.
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| The contact details of authors: | Evgenii V. Murashkin, Cand. Sci. Phys. & Math., MD, Senior Researcher, Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences; e-mail: evmurashkin@gmail.ru; https://orcid.org/0000-0002-3267-4742; AuthorID: 129570
Yuri N. Radayev, Dr. Sci. Phys. & Math., Prof., Leading Researcher, Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences; e-mail: radayev@ipmnet.ru; https://orcid.org/0000-0002-0866-2151; AuthorID: 103116
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