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Injectivity and self-contact in second-gradient nonlinear elasticity

Injectivity and self-contact in second-gradient nonlinear elasticity We prove the existence of globally injective weak solutions in mixed boundary-value problems of second-gradient nonlinear elastostatics via energy minimization. This entails the treatment of self-contact. In accordance with the classical (first-gradient) theory, the model incorporates the unbounded growth of the potential energy density as the local volume ratio approaches zero. We work in a class of admissible vector-valued deformations that are injective on the interior of the domain. We first establish a rigorous Euler–Lagrange variational inequality at a minimizer. We then define a self-contact coincidence set for an admissible deformation in a natural way, which we demonstrate to be confined to a closed subset of the boundary of the domain. We then prove the existence of a non-negative (Radon) measure, vanishing outside of the coincidence set, which represents the normal contact-reaction force distribution. With this in hand, we obtain the weak form of the equilibrium equations at a minimizer. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

Injectivity and self-contact in second-gradient nonlinear elasticity

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References (15)

Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
DOI
10.1007/s00526-017-1212-y
Publisher site
See Article on Publisher Site

Abstract

We prove the existence of globally injective weak solutions in mixed boundary-value problems of second-gradient nonlinear elastostatics via energy minimization. This entails the treatment of self-contact. In accordance with the classical (first-gradient) theory, the model incorporates the unbounded growth of the potential energy density as the local volume ratio approaches zero. We work in a class of admissible vector-valued deformations that are injective on the interior of the domain. We first establish a rigorous Euler–Lagrange variational inequality at a minimizer. We then define a self-contact coincidence set for an admissible deformation in a natural way, which we demonstrate to be confined to a closed subset of the boundary of the domain. We then prove the existence of a non-negative (Radon) measure, vanishing outside of the coincidence set, which represents the normal contact-reaction force distribution. With this in hand, we obtain the weak form of the equilibrium equations at a minimizer.

Journal

Calculus of Variations and Partial Differential Equations – Springer Journals

Published: Jul 12, 2017

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