Orientation-preserving condition and polyconvexity on a surface: Application to nonlinear shell theory

We propose in this paper a definition of a “polyconvex function on a surface”, inspired by the definitions set forth in other contexts by J. Ball (1977) [3] and by J. Ball, J.C. Currie, and P.J. Olver (1981) [5]. When the surface is thought of as the middle surface of a nonlinearly elastic shell and the function as its stored energy function, we show that it is possible to assume in addition that this function is coercive for appropriate Sobolev norms and that it satisfies specific growth conditions that prevent the vectors of the covariant bases along the deformed middle surface to become linearly dependent, a condition that is the “surface analogue” of the orientation-preserving condition of J. Ball. We then show that a functional with such a polyconvex integrand is weakly lower semi-continuous, a property which eventually allows to establish the existence of minimizers. We also indicate how this new approach compares with the classical nonlinear shell theories, such as those of W.T. Koiter and P.M. Naghdi.     

The notion of a production function

Summary The mathematical notion of the production function is developed to relate input and output rates for unconstrained technological possibilities. It is shown that the structure of production may be expressed interchangeably, either by postulating the existence of a production function (X) satisfying certain properties or by a family of production possibility setsL (U), whenU is output rate,X is a vector of input rates,L (U) is the set of input rate vectors yielding at leastU, and . A class of production functions called homothetic is defined. This class is particularly useful in economic studies.

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Zusammenfassung In der vorliegenden Arbeit wird ein mathematischer Begriff der Produktionsfunktion entwickelt, um im Falle unbeschränkter technologischer Möglichkeiten Faktor- und Produktquantitäten zucinander in Beziehung zu setzen. Es wird gezeigt, daß die Produktionsstruktur wechselweise ausgedrückt werden kann, entweder durch Postulierung der Existenz einer Produktionsfunktion (X) mit gewissen Eigenschaften oder mit Hilfe einer Familie von MengenL (U) von Produktionsmöglichkeiten, wobeiU eine Produktquantität,X ein Vektor von Faktorquantitäten,L (U) die Menge der Faktorvektoren, die mindestensU erzeugen, und ist. Ferner wird eine Klasse sogenannter homothetischer Produktionsfunktionen definiert, die bei ökonomischen Untersuchungen besonders nützlich ist.

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This research was supported in part by the National Science Foundation Grant GP-4593 with the University of California, Berkeley; and the Logistics Research Project at Princeton University sponsored by the Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the United States Government.

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Vorgel. v.:W. Wittmann.     

A nonlinear Korn inequality on a surface

Let ω be a domain in ${\mathbb{R}}^2$ and let $\mathit{\theta }:\overline{\omega }\to {\mathbb{R}}^3$ be a smooth immersion. The main purpose of this paper is to establish a “nonlinear Korn inequality on the surface $\mathit{\theta }(\overline{\omega })$”, asserting that, under ad hoc assumptions, the ${\mathit{H}}^1(\omega )$-distance between the surface $\mathit{\theta }(\overline{\omega })$ and a deformed surface is “controlled” by the ${\mathit{L}}^1(\omega )$-distance between their fundamental forms. Naturally, the ${\mathit{H}}^1(\omega )$-distance between the two surfaces is only measured up to proper isometries of ${\mathbb{R}}^3$.This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let ${\mathit{\theta }}^k:\omega \to {\mathbb{R}}^3$, $k⩾1$, be mappings with the following properties: They belong to the space ${\mathit{H}}^1(\omega )$; the vector fields normal to the surfaces ${\mathit{\theta }}^k(\omega )$, $k⩾1$, are well defined a.e. in ω and they also belong to the space ${\mathit{H}}^1(\omega )$; the principal radii of curvature of the surfaces ${\mathit{\theta }}^k(\omega )$, $k⩾1$, stay uniformly away from zero; and finally, the fundamental forms of the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{L}}^1(\omega )$ toward the fundamental forms of the surface $\mathit{\theta }(\overline{\omega })$ as $k\to \infty$. Then, up to proper isometries of ${\mathbb{R}}^3$, the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{H}}^1(\omega )$ toward the surface $\mathit{\theta }(\overline{\omega })$ as $k\to \infty$.Such results have potential applications to nonlinear shell theory, the surface $\mathit{\theta }(\overline{\omega })$ being then the middle surface of the reference configuration of a nonlinearly elastic shell.     

A nonlinear extensible 4-node shell element based on continuum theory and assumed strain interpolations

Summary A quadrilateral continuum-basedC ⁰ shell element is presented, which relies on extensible director kinematics and incorporates unmodified three-dimensional constitutive models. The shell element is developed from the nonlinear enhanced assumed strain (EAS) method advocated by Sino & Armero [1] and formulated in curvilinear coordinates. Here, the EAS-expansion of the material displacement gradient leads to the local interpretation of enhanced covariant base vectors that are superposed on the compatible covariant base vectors. Two expansions of the enhanced covariant base vectors are given: first an extension of the underlying single extensible shell kinematic and second an improvement of the membrane part of the bilinear element. Furthermore, two assumed strain modifications of the compatible covariant strains are introduced such that the element performs well even in the case of very thin shells. This paper is dedicated to the memory of Juan C. Simo In honour of Professor Juan Simo who had significant collaboration with our institute and contributed important insights to our research work. This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan Simo.     

A notion of rank in set theory without choice

 Starting from the definition of `amorphous set' in set theory without the axiom of choice, we propose a notion of rank (which will only make sense for, at most, the class of Dedekind finite sets), which is intended to be an analogue in this situation of Morley rank in model theory. Received: 22 September 2000 / Revised version: 14 May 2002 Published online: 19 December 2002 The research of the first author was supported by the SERC. Mathematics Subject Classification (2000): 03E25 Key words or phrases: Rank – Degree – Amorphous     

A problem on the nonlinear oscillations of a shallow shell in quasistatic formulation

Translated from Matematicheskie Zametki, Vol. 47, No. 4, pp. 128–137, April, 1990.     

Partial regularity for certain classes of polyconvex functionals related to nonlinear elasticity

Consider the variational integral where Ω⊂ℝ ⁿ andp≥n≥2. H: (0, ∞)→[0, ∞) is a smooth convex function such that . We approximateJ by a sequence of regularized functionalsJ _δ whose minimizers converge strongly to anJ-minimizing function and prove partial regularity results forJ _δ-minimizers.     

An example of a C1,1 polyconvex function with no differentiable convex representative

We construct a C^1,1 polyconvex function W such that there exists a fixed 2×2 matrix Y with the property that all convex representatives of W have at least two distinct subgradients (and are hence not differentiable) at the point (Y,detY), showing in particular that a polyconvex function can be smoother than any of its convex representatives. To cite this article: J. Bevan, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A finite-variation method in nonlinear shell mechanics

A numerical algorithm is proposed for calculating coefficients of first-and second-order variations of strain energy in a nonlinear finite-element model of a shell, which are necessary to define equilibrium states of the shell and investigate stability of the states. Several numerical schemes are considered based on various finite-difference approximations. For these schemes, the accuracy, convergence, and computation time are analyzed using popular geometrically nonlinear problems of deformation of elastic plates and shells.     

Shift invariant manifolds and nonlinear analytic function theory

The ultimate goal of our campaign is to generalize a substantial collection of results in classical complex variables to highly nonlinear situations. In [BH1] and subsequent works (c. f. [BGR],[H]) it was shown how an extension of the classical Beurling-Lax-Halmos theorem to Hilbert spasces with a signed bilinear form (indefinite metric) could be regarded as the key to many theorems in complex analysis. Thus our approach to the nonlinear case is to first extend our indefinite metric Beurling-Lax-Halmos theory to nonlinear situations that is the subject of this article.Supported in part by the Air Force Office of Scientific Research, the National Science Foundation and the Office of Naval Research.     

Another approach to linear shell theory and a new proof of Korn's inequality on a surface

We propose a new approach to the quadratic minimization problems arising in Koiter's 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. This approach also provides a new proof of Korn's inequality on a surface. To cite this article: P.G. Ciarlet, L. Gratie, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

A solution to the equations of shell theory

In this paper the Vekua-Dikmen formulation of the equations of shell theory as a heirarchy are presented as a hypercomplex system. Using the function theory associated with this algebra the torsion problem is solved as an integral representation.     

A comparison between John's refined interior shell equations and classical shell theory

Zusammenfassung Die linearisierte Form der verfeinten Feldgleichungen vonJohn für dünne elastische Schalen wird verglichen mit der klassischen Schalentheorie in der Form der Gleichungen vonSanders undKoiter. Im Falle der dehnungslosen axialsymmetrischen Biegung einer Schale in der Form einer Schraubenfläche stimmen beide Theorien überein in ihren Aussagen. Ein wesentlicher Unterschied zeigt sich jedoch für Schalen mit einem kovariant konstanten zweiten Haupttensor (Kugelschale oder Kreiszylinderschale). Der Vergleich mit einer exakten Lösung für zylindrische Biegung einer Kreiszylinderschale bestätigt die (annähernde) Gültigkeit der klassischen Gleichungen. Dieser Unterschied, und die Übereinstimmung im Falle der Schraubenfläche, lenken die Aufmerksamkeit auf vier numerische Koeffizienten in den äusserst komplizierten Gleichungen vonJohn, die einer Nachprüfung auf Schreibfehler bedürfen.

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Dedicated to ProfessorHenry Görtler on the occasion of his 60th anniversary     

Estimation of cardinal utility based on a nonlinear theory

Empirical studies have demonstrated that cardinal utility functions assessed via gamble-based methods are often incoherent because of the probability and certainty effects. These effects are caused by apparent risk attitudes different from those admissible within the linear (expected) utility theory. The incoherences can also be accentuated by the effects of chaining and serial positioning of responses. To filter out these effects and obtain an unbiased measurement of the strength of preference, and a simultaneous measurement of risk attitude, we devised the independent-gamble, nonlinear-inference (IGNI) method: the utility function of outcomes and the risk function of probabilities are estimated jointly from assessed certainty equivalents of independent gambles by using a nonlinear utility theory for inference. The method contrasts with all popular utility assessment techniques in that it estimates a cardinal function in the two-dimensional space of outcomes and probabilities. Hence, it allows us to obtain novel insights into the nature of utility functions and the probability effect. Both are illustrated by empirical results for fifty-four subjects.     

A nonlinear shell model of Koiter's type

We define a new two-dimensional nonlinear shell model “of Koiter's type” that can be used for the modeling of any type of shell and boundary conditions and for which we establish an existence theorem. The model uses a specific three-dimensional stored energy function of Ogden's type that satisfies all the assumptions of John Ball's fundamental existence theorem in three-dimensional nonlinear elasticity and that is adapted here to the modeling of thin nonlinearly elastic shells by means of specific deformations that are quadratic with respect to the transverse variable.