2025 - Sustainable Industrial Processing Summit
SIPS2025 Volume 14. Intl. Symp on Multiscale, Modelling, Nanotechnology and Modelling Materials
| Editors: | F. Kongoli, D. Bammann, R. Das, J.B. Jordon, R. Prabhu, A. Rajendran, P. Trovalusci, M. de Campos |
| Publisher: | Flogen Star OUTREACH |
| Publication Year: | 2025 |
| Pages: | 214 pages |
| ISBN: | 978-1-998384-64-8 (CD) |
| ISSN: | 2291-1227 (Metals and Materials Processing in a Clean Environment Series) |
CD shopping page
SOME THOUGHTS ON THE FORMULATION OF CONSERVATION LAWS FOR GENERALIZED CONTINUUM
Lalaonirina Rakotomanana1;1UNIVERSITY OF RENNES, Rennes, France;
Type of Paper: Regular
Id Paper: 292
Topic: 1
Abstract:
It is well sound that the rigorous formulation of equations governing complex materials, where multi-scale and multi-physics phenomenon are present, still remains a big challenge in continuum mechanics e.g. [1]. Distribution of line and surface defects as dislocations and disclinations in the material implies such difficulties. Indeed, the classical divergence of a stress tensor as originally developed by Cauchy requires some subtil conditions which might sometimes underestimated. Such is the case when distribution of defects (i.e. elastoplasticity) occurs within an otherwise virgin material.
The goal of this work is to derive the generic shape of conservation laws, specially equilibrium equations of such complex material in mechanics.
First, any mathematical equations governing physics models should be invariant under passive diffeomorphisms (i.e. change of coordinate system). We then propose a mathematical model of generalized continuum described by Lagrangian depending on metric, torsion and curvature of a Riemann-Cartan manifold [2].
Second, generic conservation laws of Noll first-gradient continuum (deduced rigorously by Lie derivative of metric) are extended to such generalized continuum by using active diffeomorphisms deduced by Lie derivatives of metric, torsion and curvature. The method thus uses the so-called Principle of General Covariance, based on these derivatives along a non-uniform vector field e.g. [3].
The present work although devoted to conservation laws lies heavily on the geometric approach of generalized continuum. The main concern is then the derivation of equilibrium equations in various situations, and with some applications to defected materials or even other physics as gravitation. If time is allowed some application in coupled physics as electromagnetism will be sketched. Some hints for the use of non-smooth elastioplasticity may be presented.