Forces and the existence of stresses in invariant continuum mechanics

Segev, Reuven

doi:10.1063/1.527406

Cited by 59 publications

(67 citation statements)

References 9 publications

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“…Alternative approaches to the modeling of elastic bodies in a Riemannian manifold can be found elsewhere in the literature; see, for instance, [11,12,17,19,20,21,22] and the references therein. Our approach is akin to the one in Ciarlet [6], but is formulated in a Riemannian manifold instead of the three-dimensional Euclidean space.…”

Section: Introductionmentioning

confidence: 99%

The equations of elastostatics in a Riemannian manifold

Grubic¹,

LeFloch²,

Mardare³

2014

Journal de Mathématiques Pures et Appliquées

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To begin with, we identify the equations of elastostatics in a Riemannian manifold, which generalize those of classical elasticity in the three-dimensional Euclidean space. Our approach relies on the principle of least energy, which asserts that the deformation of the elastic body arising in response to given loads minimizes over a specific set of admissible deformations the total energy of the elastic body, defined as the difference between the strain energy and the potential of the loads. Assuming that the strain energy is a function of the metric tensor field induced by the deformation, we first derive the principle of virtual work and the associated nonlinear boundary value problem of nonlinear elasticity from the expression of the total energy of the elastic body. We then show that this boundary value problem possesses a solution if the loads are sufficiently small (in a sense we specify). RésuméDans un premier temps, nous identifions leséquations de l'élastostatique dans une variété riemannienne, qui généralisent celles de la théorie classique de l'élasticité dans l'espace euclidien tridimensionnel. Notre approche repose sur le principe de moindre action, qui affirme que la déformation du corpsélastique sous l'action des forces externes minimise sur l'ensemble des déformations admissibles l'énergie totale du corpsélastique, définie comme la différence entre l'énergie de déformation et le potentiel des forces externes. Sous l'hypothèse que l'énérgie de déformation est une fonction du champ de tenseurs métriques induit par la déformation, nous dérivons dans un premier temps le principe des travaux virtuels et le problème aux limites associéà partir de l'expression de l'énergie totale du corpsélastique. Nous démontrons ensuite que ce problème aux limites admet une solution si les forces externes sont suffisamment petites (en un sens que nous précisons).

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Section: Introductionmentioning

confidence: 99%

The equations of elastostatics in a Riemannian manifold

Grubic¹,

LeFloch²,

Mardare³

2014

Journal de Mathématiques Pures et Appliquées

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“…Following the ideas of Germain [1972], dell 'Isola et al [2015b;2011;2012] obtain from the theory of distributions by L. Schwartz [1951] a representation of the virtual work in terms of N -th order stresses which are defined as the quantities dual to the N -th gradients of the virtual displacement field. A similar representation of forces in a continuum has already been proposed by Segev [1986]. In this generalized theory, the classical continuum is embedded and obtained for N = 1.…”

Section: Introductionmentioning

confidence: 99%

“…The body Ꮾ, as suggested by [Noll 1959;Segev 1986], is a three-dimensional compact differentiable manifold. A point of the body manifold x ∈ Ꮾ is called a material point of Ꮾ.…”

Section: Kinematics Of the Continuummentioning

confidence: 99%

Untitled

2017

Math. Mech. Compl. Sys.

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The minimum properties that allow a dissipation functional to describe a behavior of viscoplastic type are analyzed. It is considered a material model based on an internal variable description of the irreversible processes and characterized by the existence of an elastic region. The dissipation functional derived includes the case of time-independent plasticity in the limit. The complementary dissipation functional and the flow rule are also stated. The model analyzed leads naturally to the fulfillment of the maximum dissipation postulate and thus to associative viscoplasticity. A particular class of models is analyzed, and similarities to and differences from diffused viscoplastic formats are given.

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“…In fact, the mere requirement that a force be an element of C 1 (B, R 3 ) * implies through a representation theorem for such functionals several features we expect of stress theory of continuum mechanics (see [13,14]) some of which we will see below.…”

Section: 2mentioning

confidence: 99%

“…In order to write an expression for it, we follow the method used by Federer [8, pp. 367-368] to represent Whitney's [15] flat norm of chains and we utilize the interpretation of the stress as a tensor valued measure representing a C 1 -functional as in [13,14].…”

Section: Introductionmentioning

confidence: 99%

On Norms of Force Functionals and Stress Representations

Segev

deBotton

2006

Mathematics and Mechanics of Solids

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ABSTRACT. Forces in continuum mechanics are analyzed as 0-currents of geometric measure theory. The representation of forces by stresses is discussed and the flat norm of a force is expressed in terms of stress fields. An analogous treatment expresses the Sobolev norm of a force in terms of stress fields. In both cases, one obtain bounds on the stress fields that are in equilibrium with a given force. The analysis is universal in the sense that it is independent of any constitutive relation.

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Forces and the existence of stresses in invariant continuum mechanics

Cited by 59 publications

References 9 publications

The equations of elastostatics in a Riemannian manifold

The equations of elastostatics in a Riemannian manifold

Untitled

On Norms of Force Functionals and Stress Representations

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