Noncommutative deformation and essence of a postulate of isotropy Iliusin

TL;DR

This paper proposes a noncommutative deformation framework for elastic field theory, aiming to better understand isotropy and plasticity by extending classical models through symplectic structures and noncommutative geometry.

Contribution

It introduces a noncommutative approach to elastic theory, linking deformation space to cotangent bundles and symplectic structures, providing a new perspective on isotropy and plasticity.

Findings

Noncommutative deformation leads to a symplectic structure.

The approach generalizes classical elastic theory.

Distribution of transition probabilities to plasticity is derived.

Abstract

For construction of the general theory it is necessary, first of all, to refuse from traditional performance of the elastic theory of field, within the framework of which the static processes with commutative quantities are described only. The space of deformation should be considered as noncommutative \. Under action of any forces this space will proceed in cotangent bundle. When the action will achieve limiting value the status of deformation will be in strong fluctuation . Then we receive distribution of probability of transition to plasticity. Besides thanking noncommutativity we shall receive the symplectic structure. Only in this case it is possible to understand the essence of a postulate of isotropy Iliusin. And, may be, all other results of the classical theory can be received as special cases of our general model.