Conservation laws of linear elasticity in stress formulations

2005 ◽  
Vol 461 (2053) ◽  
pp. 99-116 ◽  
Author(s):  
Shaofan Li ◽  
Anurag Gupta ◽  
Xanthippi Markenscoff
Keyword(s):  
Conservation Laws ◽  
Linear Elasticity ◽  
Three Dimensional ◽  
Cauchy Stress ◽  
General Boundary Conditions ◽  
Cauchy Stress Tensor ◽  
Equilibrium Equations ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Three Dimensional Elasticity

In this paper, we present new conservation laws of linear elasticity which have been discovered. These newly discovered conservation laws are expressed solely in terms of the Cauchy stress tensor, and they are genuine, non–trivial conservation laws that are intrinsically different from the displacement conservation laws previously known. They represent the variational symmetry conditions of combined Beltrami–Michell compatibility equations and the equilibrium equations. To derive these conservation laws, Noether's theorem is extended to partial differential equations of a tensorial field with general boundary conditions. By applying the tensorial version of Noether's theorem to Pobedrja's stress formulation of three–dimensional elasticity, a class of new conservation laws in terms of stresses has been obtained.

Introduction

2017 ◽  
Author(s):  
Laurent Baulieu ◽  
John Iliopoulos ◽  
Roland Sénéor
Keyword(s):  
Conservation Laws ◽  
Lagrangian Mechanics ◽  
General Introduction ◽  
Noether’S Theorem ◽  
Noether's Theorem

General introduction with a review of the principles of Hamiltonian and Lagrangian mechanics. The connection between symmetries and conservation laws, with a presentation of Noether’s theorem, is included.

scholarly journals Conservation Laws in Physics – Emmy Noether’s Theorem

2014 ◽  
Vol 29 ◽  
pp. 55-61
Author(s):  
Daniela Manolea
Keyword(s):  
Conservation Laws ◽  
Celestial Body ◽  
Practical Importance ◽  
Microscopic Level ◽  
Human Knowledge ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
The World ◽  
Physical Laws ◽  
Practical Impact

The study is explanatory-interpretative and argues the practical character of Physics. It starts from premise that formation of a correct conception of the world begins with the understanding of physics. It is one of the earliest chapters of human knowledge, studying the material world from the microscopic level of the particles to the macroscopic level of the celestial body. As an example for the practical importance of applying the laws of physics take the set of physical laws of conservation, in particular, it explains the practical impact of Emmy Noether's Theorem.

Conservation Laws in Systems With Variable Mass

1993 ◽  
Vol 60 (4) ◽  
pp. 954-958 ◽  
Author(s):  
L. Cveticanin
Keyword(s):  
Variational Principle ◽  
Dynamic System ◽  
Conservation Laws ◽  
Dynamic Systems ◽  
Conservation Law ◽  
Variable Mass ◽  
Mass Variation ◽  
Noether’S Theorem ◽  
Noether's Theorem

In this paper, a method for obtaining conservation laws of dynamic systems with variable mass is developed. It is based on Noether’s theorem to the existence of conservation laws and D’Alembert’s variational principle. In the general case, a dynamic system with variable mass is purely nonconservative. Noether’s identity for such a case is expanded by the terms that describe the mass variation. If Noether’s identity if satisfied, a conservation law exists. Two groups of systems with variable mass are considered: a nonlinear vibrating machine and a rotor with variable mass. For these systems, conservation laws are obtained using the procedure developed in this paper.

Uniqueness and regularity of flows of non-Newtonian fluids with critical power-law growth

2019 ◽  
Vol 29 (06) ◽  
pp. 1207-1225 ◽  
Author(s):  
Miroslav Bulíček ◽  
Petr Kaplický ◽  
Dalibor Pražák
Keyword(s):  
Power Law ◽  
Three Dimensional ◽  
Uniqueness Result ◽  
Newtonian Fluids ◽  
Index Formula ◽  
Cauchy Stress ◽  
Cauchy Stress Tensor ◽  
Energy Equality ◽  
Regularity Properties ◽  
Uniqueness Of The Solution

We deal with flows of non-Newtonian fluids in three-dimensional setting subjected to the homogeneous Dirichlet boundary condition. Under the natural monotonicity, coercivity and growth condition on the Cauchy stress tensor expressed by a power index [Formula: see text], we establish regularity properties of a solution with respect to time variable. Consequently, we can use this better information for showing the uniqueness of the solution provided that the initial data are good enough for all power–law indices [Formula: see text]. Such a result was available for [Formula: see text] and therefore the paper fills the gap and extends the uniqueness result to the whole range of [Formula: see text]’s for which the energy equality holds.

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