Geometrical Field Representation of Solid, Fluid, and Gas as Continuum in Rational Mechanics

TL;DR

This paper introduces a geometrical field representation in continuum mechanics that characterizes solids, fluids, and gases through intrinsic stretching and rotation, providing a unified framework for different material states.

Contribution

It proposes a novel continuum model based on points-set transformation, expressing classical strain as intrinsic stretching and rotation, and establishes constitutive equations for multiple material states.

Findings

Solid state has zero intrinsic stretching and local rotation.

Fluid state exhibits intrinsic plane stretching without changing gauge tensor.

Gas state shows intrinsic stretching along the rotation direction.

Abstract

Based on the points-set transformation concept about the motion transformation in continuum, the macro classical strain is expressed by the additive addition of the intrinsic stretching of material element and its intrinsic local rotation. For zero classical strain (no macro deformation observed on its configuration surface, suitable container is required for liquid and gas to make up macro invariant configuration), the results show that: (1) For solid, the local rotation angular is zero. The material element has no intrinsic stretching. (2) For liquid, the local rotation will not change the basic gauge tensor. The material element has intrinsic plane stretching on the rotation plane. (3) For gas state, the intrinsic local rotation will amplify the basic gauge tensor. The material element has intrinsic stretching along the rotation direction. Hence, under the condition of no macro classical strain be observed, the material element has three different physical states: solid (no intrinsic stretching), fluid (plane intrinsic stretching), and gas (directional intrinsic stretching). Furthermore, for the three states, the free conditions are defined by zero intrinsic stretching. Referring to this free condition, the constitutive equations for the materials at multiple states are established.