Systems of Cosserat--Zhilin in Newtonian Mechanics

TL;DR

This paper introduces Cosserat--Zhilin systems in Newtonian mechanics using new vector calculus notions, deriving motion equations and defining stress measures for classical and polar continua in 2D and 3D.

Contribution

It presents a novel framework for modeling Cosserat--Zhilin systems with new vector calculus tools, deriving equations of motion and stress measures in classical and polar continua.

Findings

Derived differential equations of motion for Cosserat--Zhilin systems.

Defined multiplicative groups for stress measures as isotropic maps.

Extended the framework to 2D and 3D classical and polar continua.

Abstract

Mechanical systems of Cosserat--Zhilin are introduced as the main object of Newtonian (non--relativistic) mechanics on the base of new notions of vector calculus - sliders and screw measures (bi-measures). The differential equations of motion are derived for different types of Cosserat--Zhilin systems in the case where Stocks theorem is applicable. The paper defines multiplicative groups which represent the measure of stress as a (well-defined) linear isotropic map of {strain} tensor or tensor of {strain velocities} for classical and polar continua in 2- and 3-dimensional cases.