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Saint Venant compatibility equations in curvilinear coordinates

By Philippe G. Ciarlet, Cristinel Mardare and Ming Shen

Abstract

International audienceWe establish that the linearized strains in curvilinear coordinates associated with a given displacement field necessarily satisfy “Saint Venant equations in curvilinear coordinates”. Furthermore, we show that these equations are also sufficient, in the following sense: If a symmetric matrix field defined over a simply- connected open set satisfies the Saint Venant equations in curvilinear coordinates, then its coefficients are the linearized strains associated with a displacement field. In addition, our proof provides an explicit algorithm for recovering such a displacement field from its linear strains in curvilinear coordinates. This algorithm may be viewed as the linear counterpart of the reconstruction of an immersion from a given flat Riemannian metric

Topics: Differential geometry, Elasticity, [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG], [MATH] Mathematics [math], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph]
Publisher: World Scientific Publishing
Year: 2007
OAI identifier: oai:HAL:hal-00139199v1
Provided by: Hal-Diderot
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