Ordinary differential equations of non-linear elasticity I: Foundations of the theories of non-linearly elastic rods and shells

Antman, Stuart S.

doi:10.1007/bf00250722

Cited by 157 publications

(88 citation statements)

References 27 publications

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“…This viewpoint is thus analogous to that of Antman [3], who, back in 1976, was the first to suggest that the metric tensor field of a deformed configuration in nonlinear threedimensional elasticity could be considered as the primary unknown on its own, instead of the position vector field as is customary. It was likewise, but more recently, recognized that the first and second fundamental form of a deformed middle surface in nonlinear shell theory could be considered as primary unknowns on their own, instead of the position vector field of the middle surface (for recent developments and references on such approaches see [14] and [9]).…”

Section: Introductionmentioning

confidence: 71%

A New Approach to the Fundamental Theorem of Surface Theory

Ciarlet

Gratie²,

Mardare³

2007

Arch Rational Mech Anal

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ABSTRACT. The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a αβ ) of order two and a field of symmetric matrices (b αβ ) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a connected and simply-connected open subset ω of R 2 , then there exists an immersion θ : ω → R 3 such that these fields are the first and second fundamental forms of the surface θ(ω) and this surface is unique up to proper isometries in R 3 .In this paper, we identify new compatibility conditions, expressed again in terms of the functions a αβ and b αβ , that likewise lead to a similar existence and uniqueness theorem. These conditions take the formwhere A 1 and A 2 are antisymmetric matrix fields of order three that are functions of the fields (a αβ ) and (b αβ ), the field (a αβ ) appearing in particular through its square root. The unknown immersion θ : ω → R 3 is found in the present approach in function spaces "with little regularity", viz.,Une nouvelle approche du théorème fondamental de la théorie des surfaces.RÉSUMÉ. Le théorème fondamental de la théorie des surfaces affirme classiquement que, si un champ de matrices (a αβ ) symétriques définies positives d'ordre deux et un champ de matrices (b αβ ) symétriques d'ordre deux satisfont ensemble leséquations de Gauss et Codazzi-Mainardi dans un ouvert ω ⊂ R 2 connexe et simplement connexe, alors il existe une immersion θ : ω → R 3 telle que ces deux champs soient les première et deuxième formes fondamentales de la surface θ(ω), et cette surface est unique aux isométries propres de R 3 près.Dans cet article, nous identifions de nouvelles conditions de compatibilité, exprimées a nouveauà l'aide des fonctions a αβ et b αβ , qui conduisent aussià un théorème analogue d'existence et d'unicité. Ces conditions sont de la formeoù A 1 et A 2 sont des champs de matrices antisymétriques d'ordre trois, qui sont des fonctions des champs (a αβ ) et (b αβ ), le champ (a αβ ) apparaissant en particulier par l'intermédiaire de sa racine carrée. L'immersion inconnue θ : ω → R 3 est trouvée dans cette approche dans des espaces fonctionnels "de faible régularité",à savoir W 2,p loc (ω; R 3 ), p > 2.

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Section: Introductionmentioning

confidence: 71%

A New Approach to the Fundamental Theorem of Surface Theory

Ciarlet

Gratie²,

Mardare³

2007

Arch Rational Mech Anal

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“…In 1864, A. J. C. B. de Saint Venant showed that, if functions e ij = e ji ∈ C 3 (Ω) satisfy in a simply-connected open subset Ω of R 3 ad hoc compatibility relations that since then bear his name, then there exists a vector field (v i ) ∈ C 4 (Ω) such that e ij = 1 2 (∂ j v i + ∂ i v j ) in Ω. Thanks to Theorem 3.1, it can be shown that the same St Venant compatibility relations are also sufficient conditions in the sense of distributions, according to the following result (again due to Ciarlet and Ciarlet, Jr. 10 ):…”

Section: Idea Of the Proofmentioning

confidence: 99%

A New Approach to Linear Shell Theory

Ciarlet

Gratie

2005

Math. Models Methods Appl. Sci.

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We proposed a new approach to the existence theory for quadratic minimization problems that arise in linear shell theory. The novelty consists in considering the linearized change of metric and change of curvature tensors as the new unknowns, instead of the displacement vector field as is customary.Such an approach naturally yields a constrained minimization problem, the constraints being ad hoc compatibility relations that these new unknowns must satisfy in order that they indeed correspond to a displacement vector field. Our major objective is thus to specify and justify such compatibility relations in appropriate function spaces. Interestingly, this result provides as a corollary a new proof of Korn's inequality on a surface. While the classical proof of this fundamental inequality essentially relies on a basic lemma of J. L. Lions, the keystone in the proposed approach is instead an appropriate weak version of a classical theorem of Poincaré.The existence of a solution to the above constrained minimization problem is then established, also providing as a simple corollary a new existence proof for the original quadratic minimization problem.

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“…There are numerous models of this sort in the literature, of course, such as [2], [3], [4], [5], [6], [7], [8], [10], [11], and [13]. In this respect, our primary "point of departure" is to assume that "in line" or "in plane" displacements are of the same order as the square of the transverse and/or shearing displacements generally admitted into linear models of the same structures.…”

Section: Introductionmentioning

confidence: 99%

An Elementary Nonlinear Beam Theory with Finite Buckling Deformation Properties

Russell¹,

White²

2002

SIAM J. Appl. Math.

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Abstract.A simple nonlinear beam model is derived from basic principles. The assumption upon which the derivation is based is that axial motions are of second order compared with transverse motions of the beam. The existence of solutions is established. Issues concerning the uniqueness and nonuniqueness of solutions are examined with regard to buckling behavior. The numerical treatment of problems with nonunique solutions is presented. The results of buckling calculations are presented. 1. Introduction. The objectives of this work are to (i) present in one place a physical derivation of a simple extension of the classic linear beam equation to a model which can capture nonlinear effects such as buckling, (ii) derive an associated variational principle that submits itself to analysis for existence, uniqueness, and nonuniquenss, and (iii) describe some of the numerical results in which nontrivial solutions are obtained. The study of elastic structures such as beams and plates is, to a large extent, dominated by linear models. Even the most familiar model used to predict the onset of buckling in elastic beams subject to longitudinal forces is a linear one. Nevertheless, it is well understood that incorporation of both "pre-buckling" and "post-buckling" phases into a single model requires the proposed model, at the least, to account for nonlinear geometric effects.There [13]. In this respect, our primary "point of departure" is to assume that "in line" or "in plane" displacements are of the same order as the square of the transverse and/or shearing displacements generally admitted into linear models of the same structures. The development of these models has been given new emphasis by recent developments in "formation theory," the deliberate modification of the configuration of an elastic body by means of attached or imbedded actuators [14]. We will not pursue such a study in this paper, leaving that work for another article. The objective here is to present a derivation of the model from first principles and to indicate how the model may be useful for the analysis of "buckling"-type bifurcation problems, not only with regard to prediction of "critical" loads initiating such bifurcation, already possible with the standard linear model, but also permitting studies of the progressive growth of the buckling phenomenon as the load is increased beyond the critical buckling load.In section 2, we derive a set of geometrically nonlinear equations for the elastic beam. These equations compare with those presented in [10]. Here, however, derivations, although ad hoc, are based on explicitly stated fundamental principles. Then, in section 3, we demonstrate existence and uniqueness of solutions from a variational

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Ordinary differential equations of non-linear elasticity I: Foundations of the theories of non-linearly elastic rods and shells

Cited by 157 publications

References 27 publications

A New Approach to the Fundamental Theorem of Surface Theory

A New Approach to the Fundamental Theorem of Surface Theory

A New Approach to Linear Shell Theory

An Elementary Nonlinear Beam Theory with Finite Buckling Deformation Properties

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