PARAMETRIZATION OF COSSERAT EQUATIONS

Jean-François Pommaret

hal-00363807, version 1

PARAMETRIZATION OF COSSERAT EQUATIONS

Jean-François Pommaret () ¹

(2009-02)

Abstract: As a matter of fact, the solution space of many systems of ordinary or partial differential equations in engineering or mathematical physics "can/cannot" be parametrized by a certain number of arbitrary functions behaving like potentials. For example, such a parametrization exists for a control system if and only if it is controllable and may be of high order. The first set of Maxwell equations admits a first order parametrization by the 4-potential. However, Einstein equations in vacuum do not admit any parametrization. Finally, the stress equations in continuum mechanics admit a second order parametrization by means of n^2(n^2-1)/12 arbitrary functions, the case n=2 being the Airy function. The purpose of this paper is to use unexpected deep results of homological algebra and algebraic analysis in order to prove for the first time that the stress/couple-stress Cosserat equations admit a first order parametrization by mens of n^2(n^2-1)/4 arbitrary functions

1: Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS)
Ecole des Ponts ParisTech

hal-00363807, version 1
http://hal.archives-ouvertes.fr/hal-00363807
oai:hal.archives-ouvertes.fr:hal-00363807
From: Jean-François Pommaret <>
Submitted on: Tuesday, 24 February 2009 14:54:11
Updated on: Wednesday, 15 December 2010 13:38:02

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%% hal-00363807, version 1
%% http://hal.archives-ouvertes.fr/hal-00363807
@unpublished{pommaret:hal-00363807,
    hal_id = {hal-00363807},
    url = {http://hal.archives-ouvertes.fr/hal-00363807},
    title = {{PARAMETRIZATION OF COSSERAT EQUATIONS}},
    author = {Pommaret, Jean-Fran{\c c}ois},
    abstract = {{As a matter of fact, the solution space of many systems of ordinary or partial differential equations in engineering or mathematical physics "can/cannot" be parametrized by a certain number of arbitrary functions behaving like potentials. For example, such a parametrization exists for a control system if and only if it is controllable and may be of high order. The first set of Maxwell equations admits a first order parametrization by the 4-potential. However, Einstein equations in vacuum do not admit any parametrization. Finally, the stress equations in continuum mechanics admit a second order parametrization by means of n^2(n^2-1)/12 arbitrary functions, the case n=2 being the Airy function. The purpose of this paper is to use unexpected deep results of homological algebra and algebraic analysis in order to prove for the first time that the stress/couple-stress Cosserat equations admit a first order parametrization by mens of n^2(n^2-1)/4 arbitrary functions}},
    keywords = {Adjoint operator, continuum mechanics, controllability, Cosserat, stress equations, couple-stress, homological algebra, parametrization},
    language = {English},
    affiliation = {Centre d'Enseignement et de Recherche en Math{\'e}matiques et Calcul Scientifique - CERMICS},
    note = {13 pp, To appear as CERMICS/ENPC preprint february 2009 submitted for the International Conference 15-17 july 2009 http://cosserat2009.enpc.fr Now published with modifications and additional material in the Springer journal Acta Mechanica. The online version (11/04/2010) can be obtained through the DOI: http://dx.doi.org/10.1007/s00707-010-0292-y The reference of the printed journal version is: Acta Mech 215, 43-55 (2010) },
    year = {2009-02},
    pdf = {http://hal.archives-ouvertes.fr/hal-00363807/PDF/COSSERAT.pdf},
}