On uniqueness for the traction problem in finite elasticity

In many problems of interest the (Cauchy) surface traction is given as a function of position on the deformed surface. A class of loadings sufficiently general to include these problems is considered, and within the context of the fraction problem in finite elasticity, a number of uniqueness results are established. This work extends results obtained for the mixed problem by Gurtin and Spector.

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Resumé Dans plusieurs proble\`mes interessants la traction surfacique de Cauchy est donnée comme une fonction de la position dans la surface deformée. Une telle classe des charges, suffisamment géerale, est considerée et un nombre des résultats d'unicité est établit dans le cadre du probleme de traction de l'elasticité non linéaire. Cet ouure prolonge des résultats pour le proble\`me mixte obtenus par Gurtin et Spector.

     

A spatial dynamical problem of the theory of elasticity for a multiply-connected domain

International Applied Mechanics -     

On the uniqueness of solutions to the displacement problem in linear elastodynamics

In recent separate investigations, Kirchhoff's classical uniqueness theorem of elastodynamics has been extended in two ways. Gurtin and Toupin generalized the theorem to encompass bodies possessing an elasticity tensor that obeys the semi-strong ellipticity condition, whereas the author has extended the theorem to unbounded regions. These two results are brought together in the present paper.

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Resume Dans deux études récentes, le champ d'application du théoréme classique de Kirchhoff sur l'unicité de l'élastodynamique a été élargi de deux manières différentes. Gurtin et Toupin l'ont généralisé pour inclure les corps possédant un tenseur d'élasticité obéissant à la condition d'ellipticité demi-forte, alors que le présent auteur l'a étendu aux régions non finies. Ces deux résultats ont été combinés dans la présente communication.

     

A uniqueness theorem for the displacement problem in finite elastodynamics

Archive for Rational Mechanics and Analysis -     

Phragmen–Lindelof alternative for the displacement boundary value problem in a theory of non-linear micropolar elasticity

In this note we investigate the spatial behavior of the solutions for the displacement boundary value problem in a theory of non-linear micropolar elasticity. Under suitable conditions on the non-linear terms we obtain estimates for the solutions. The main tool is the energy method.     

On the (u, p) problem in the theory of elasticity

A problem of the theory of elasticity is considered for a body with vectors of displacements u and loads p simultaneously defined on one part of the body and with undefined conditions on the remaining part of the body. For a doubly connected domain, where the vectors u and p are set on one of its boundaries (inner or outer), an iterative method based on reduction of the initial problem to a sequence of mixed problems is justified. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 100–103, May–June, 2006.     

On uniqueness in finite elasticity with general loading

In many problems of interest the (Cauchy) surface traction is given as a function of position on the deformed surface. A class of loadings sufficiently general to include these problems is considered and within the context of finite elasticity a number of uniqueness results are established. A key ingredient is the result of Gurtin and Spector that uniqueness holds in any convex, stable set of deformations.

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Résumé Dans plusieurs problémes interessants la traction surfacique de Cauchy est donnée comme une fonction de la position dans la surface deformée. Une telle classe des charges, suffisamment générale, est considerée et un nombre des résultats d'unicité et établit dans le cadre d'élasticité non linéaire. Le résultat de Gurtin et Spector (unicité est valable sur un ensemble stable et convexe queleonque des déformations) est utilisé.

     

Axisymmetric problem for a half-space in the micropolar theory of elasticity

The general solution of the equilibrium equations is obtained for a half-space subjected to arbitrary normal pressure. Four particular cases, including a concentrated force, are considered in detail. The singular parts of the displacements, rotations and stresses are obtained for the case of a concentrated force. The corresponding classical results have been derived. The graphs for various physical and geometrical quantities have been drawn to illustrate the micropolar effects.

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Zusammenfassung Die ellgemeine Lösung für Gleichgewicht eines Halbraums unter beliebigem Normaldruck wird gefunden. Vier Spezialfälle darunter eine konzentrieste Kraft werden genauer untersucht. Für diesen Fall werden die Verschiebungen, Drehungen und Spannungen gefunden. Die entsprechenden klassischen Lösungen wurden erhalten. Zur Illustration der Mikropolareffekte wurden Diagramme der verschiedenen physikalischen und geometrischen Grössen gezeichnet.

     

On displacement methods in planar anisotropic elasticity

Two displacement formulation methods are presented for problems of planar anisotropic elasticity. The first displacement method is based on solving the two governing partial differential equations simultaneously/ This method is a recapitulation of the orignal work of Eshelby, Read and Shockley [7] on generalized plane deformations of anisotropic elastic materials in the context of planar anisotropic elasticity.The second displacement method is based on solving the two governing equations separately. This formulation introduces a displacement function, which satisfies a fourth-order partial differential equation that is identical in the form to the one given by Lekhnitskii [6] for monoclinic materials using a stress function. Moreover, this method parallels the traditional Airy stress function method and thus the Lekhnitskii method for pure plane problems. Both the new approach and the Airy stress function method start with the equilibrium equations and use the same extended version of Green's theorem (Chou and Pagano [13], p. 114; Gao [11]) to derive the expressions for stress or displacement components in terms of a potential (stress or displacement) function (see also Gao [10, 11]). It is therefore anticipated that the displacement function involved in this new method could also be evaluated from measured data, as was done by Lin and Rowlands [17] to determine the Airy stress function experimentally.The two different displacement methods lead to two general solutions for problems of planar anisotropic elasticity. Although the two solutions differ in expressions, both of the depend on the complex roots of the same characteristic equation. Furthermore, this characteristic equation is identical to that obtained by Lekhnitskii [6] using a stress formulation. It is therefore concluded that the two displacement methods and Lekhnitskii's stress method are all equivalent for problems of planar anisotropic elasticity (see Gao and Rowlands [8] for detailed discussions).     

Bone remodeling III: uniqueness and stability in adaptive elasticity theory

A uniqueness theorem for infinitesimal adaptive elasticity is proved. Two theorems establishing sufficient conditions for stability are demonstrated.

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Zusammenfassung Ein Eingigkeitsatz für unendich klein adaptives Elastizitätstheoric ist gebeweisen, Zwei Sätze aussagen hinreichende Bedingingen für Stabilität werden demonstrieren.