Abstract:
The present study is devoted to employing the theory of algebraic invariants for deriving an approximation of the potential of force and couple stresses of the fourth degree for a nonlinear hemitropic micropolar elastic solid. The complete set of irreducible invariants for a system of two asymmetric second-rank tensors in the form of invariant traces is studied using the theory of integer rational algebraic invariants (semi-invariants).
As a result, a set of 86 invariant traces is obtained. This set comprises 8 individual invariants, 17 doublets, 44 triplets, and 17 quadruplets. From these 86 elements, 39 invariants were selected according to the rule of increasing algebraic degrees: 2 linear invariants, 6 quadratic, 12 cubic, and 19 quartic. The 39 fourth-degree invariants are divided into four groups based on the following rules: products of linear invariants with each other, products of quadratic invariants with each other, products of linear and quadratic invariants, pairwise products of linear and cubic invariants, and proper fourth-degree invariants.
The potential of force and couple stresses of a hemitropic micropolar elastic solid is constructed, containing quadratic, cubic, and quartic algebraic terms. Thus, the micropolar potential contains a total of 124 constitutive modules. Formulas for calculating all 39 invariants in mixed tensor components are provided. As a result, 87 quartic corrections to the cubic potential of force and couple stresses of a nonlinear hemitropic micropolar elastic solid are obtained.
Keywords:nanoscale, microscale, energy form, integer rational algebraic invariant, irreducible system of invariants, cubic approximation, hemitropic micropolar elastic solid
Fulltext:
PDF file (897 kB)
The content is published under the terms of the Creative Commons Attribution 4.0 International License
