Structure of the set of equilibrated loads in nonlinear elasticity and applications to existence and nonexistence
type: Item
instance of: scholarly article
title: Structure of the set of equilibrated loads in nonlinear elasticity and applications to existence and nonexistence
zbMATH Open document ID: 0654.73029
DOI: 10.1007/BF00043020
author: Hervé Le Dret
published in: Journal of Elasticity
publication date: 1987-01-01T00:00:00Z
review text: The paper is divided into three sections. In Section I, we describe the geometrical structure of the set of equilibrated loads found in the traction problem of nonlinear elasticity. Using this structure, we present in Section II existence theorems for nonlinear traction problems, and in Section III nonexistence theorems for the linear traction problem. In the first Section, we consider Signorini's condition of compatibility. This condition asserts that a deformation \(\phi\) and a corresponding load (b,t), where b and t are force densities (which may or may not depend on \(\phi)\) on the reference configuration \(\Omega\) and its boundary \(\partial \Omega\), may be in equilibrium only if \[ (0)\quad \int_{\Omega}\phi \wedge b+\int_{\partial \Omega}\phi \wedge t=0, \] so that the total torque vanishes. In the dead load traction problem of nonlinear elasticity, relation (0) cannot be imposed in advance as a condition to be satisfied by the load (by contrast with the linear case) since it involves the unknown deformation \(\phi\). Signorini suggested that condition (0) be regarded as a `condition of compatibility' to be checked a posteriori once the problem is solved. On the contrary, we consider here that (0) is an a priori condition to be satisfied by the admissible triples (\(\phi\),b,t). More precisely, we show that condition (0) defines a subset M of the product space \(W^{m+2,p}(\Omega)^ 3\times W^{m,p}(\Omega)^ 3\times W^{m+1-1/p,p}(\partial \Omega)^ 3\), \(p>3\), a dense subset of which, \(M\setminus N\), has a \(C^{\infty}\)- manifold structure. A further subset, \(M\setminus N'\), is actually endowed with a natural vector bundle structure, and condition (0) simply says that (b,t) must belong to the fiber attached to \(\phi\). The space \[ {\mathcal L}_ e=\{(b,t),\quad \int_{\Omega}x\wedge b+\int_{\partial \Omega}x\wedge t=0\}, \] which appears in the classical linear theory and in Signorini's and Stoppelli's theorems [\textit{C. C. Wang} and \textit{C. Trues…
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 58B20
Mathematics Subject Classification ID: 57R22
Mathematics Subject Classification ID: 74G30
Mathematics Subject Classification ID: 74H25
zbMATH DE Number: 4068398
zbMATH Keywords: geometrical structure
zbMATH Keywords: set of equilibrated loads
zbMATH Keywords: Signorini's condition of compatibility
zbMATH Keywords: vector bundle structure
zbMATH Keywords: property of transversality
zbMATH Keywords: a priori nonexistence
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00043020
OpenAlex ID: W1974361665
description: scientific article; zbMATH DE number 4068398