Convexity conditions and existence theorems in nonlinear elasticity
01 Dec 1976-Archive for Rational Mechanics and Analysis (Springer-Verlag)-Vol. 63, Iss: 4, pp 337-403
About: This article is published in Archive for Rational Mechanics and Analysis.The article was published on 1976-12-01. It has received 2329 citations till now. The article focuses on the topics: Convexity & Linear elasticity.
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TL;DR: A new numerical method based on a combination of the classical shape derivative and of the level-set method for front propagation, which can easily handle topology changes and is strongly dependent on the initial guess.
Abstract: In the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level-set method for front propagation. We implement this method in two and three space dimensions for a model of linear or nonlinear elasticity. We consider various objective functions with weight and perimeter constraints. The shape derivative is computed by an adjoint method. The cost of our numerical algorithm is moderate since the shape is captured on a fixed Eulerian mesh. Although this method is not specifically designed for topology optimization, it can easily handle topology changes. However, the resulting optimal shape is strongly dependent on the initial guess.
2,176 citations
Cites background from "Convexity conditions and existence ..."
...The minimization problem min v IðvÞ has a solution if W satisfies some convexity, growth and regularity conditions [8], but the question of existence of solutions to the boundary value problem (28) is still open....
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Book•
01 Jan 1994
TL;DR: This book discusses the theory and applications of Bifurcation Theory and its applications to Elasticity, as well as problems in Nonlinear Elasticity and Dynamical Problems.
Abstract: Preface* Chapter 1. Background* Chapter 2. The Equations of Motion for Extensible Strings* Chapter 3. Elementary Problems for Elastic Strings* Chapter 4. Planar Steady-State Problems for Elastic Rods* Chapter 5. Introduction to Bifurcation Theory and it's Applications to Elasticity* Chapter 6. Global Bifurcation Problems for Strings and Rods* Chapter 7. Variational Methods* Chapter 8. Theory of Rods Deforming in Space* Chapter 9. Spatial Problems for Rods* Chapter 10. Axisymmetric Equilibria of Shells* Chapter 11. Tensors* Chapter 12. 3-Dimensional Continuum* Chapter 13. 3-Dimensional Theory of Nonlinear Elasticity* Chapter 14. Problems in Nonlinear Elasticity* Chapter 15. Large-Strain Plasticity* Chapter 16. General Theories of Rods* Chapter 17. General Theories of Shells* Chapter 18. Dynamical Problems* Chapter 19. Appendix: Topics in Linear Analysis* Chapter 20. Appendix: Local Nonlinear Analysis* Chapter 21. Appendix: Degree Theory and it's Applications* References* Index
1,888 citations
TL;DR: In this article, the authors explore a theoretical approach to these fine phase mixtures based on the minimization of free energy and show that the α-phase breaks up into triangular domains called Dauphine twins which become finer and finer in the direction of increasing temperature.
Abstract: Solid-solid phase transformations often lead to certain characteristic microstructural features involving fine mixtures of the phases. In martensitic transformations one such feature is a plane interface which separates one homogeneous phase, austenite, from a very fine mixture of twins of the other phase, martensite. In quartz crystals held in a temperature gradient near the α-β transformation temperature, the α-phase breaks up into triangular domains called Dauphine twins which become finer and finer in the direction of increasing temperature. In this paper we explore a theoretical approach to these fine phase mixtures based on the minimization of free energy.
1,488 citations
TL;DR: In this article, the authors studied the large-data Cauchy problem for Boltzmann equations with general collision kernels and proved that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in L' to a solution.
Abstract: We study the large-data Cauchy problem for Boltzmann equations with general collision kernels. We prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in L' to a solution. From this stability result we deduce global existence of a solution to the Cauchy problem. Our method relies upon recent compactness results for velocity averages, a new formulation of the Boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions. It allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations.
1,155 citations
TL;DR: The energy functional of nonlinear plate theory is a curvature functional for surfaces rst proposed on physical grounds by G. Kirchhoff in 1850 as mentioned in this paper, and it arises as a 0-limit of three-dimensional nonlinear elasticity theory as the thickness of a plate goes to zero.
Abstract: The energy functional of nonlinear plate theory is a curvature functional for surfaces rst proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a 0-limit of three-dimensional nonlinear elasticity theory as the thickness of a plate goes to zero. A key ingredient in the proof is a sharp rigidity estimate for maps v V U ! R n , U R n . We show that the L 2 -distance of rv from a single rotation matrix is bounded by a multiple of the L 2 -distance from
748 citations
Cites background from "Convexity conditions and existence ..."
...Since we have assumed no particular convexity properties of W (such as those in [4]) away from SO(3), the infimum of the total energy at nonzero h may not be attained....
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References
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Book•
01 Jan 1949
TL;DR: In this paper, the Elasticity of Long-Chain Molecules (LCHs) and Elasticity in a Molecular Network (MNNs) is investigated. But the authors focus on the elasticity of the long chain Molecules.
Abstract: 1. General Physical Properties of Rubber 2. Internal Energy and Entropy Changes on Deformation 3. The Elasticity of Long-Chain Molecules 4. The Elasticity of a Molecular Network 5Ex5 Experimental Examination of the Statistical Theory 6. Non-Gaussian Chain Statistics and Network Theory 7. Swelling Phenomena 8. Cross-linking and Modulus 9. Photoelastic Properties of Rubbers 10. The General Strain: Phenomenological Theory 11. Alternative Forms of Strain-Energy Function 12. Large-Deformation Theory: Shear and Torsion 13. Thermodynamic Analysis of Gaussian Network
4,242 citations
Book•
01 Jan 1966
TL;DR: The merite as discussed by the authors is a date marque une date dans le progres des mathematiques and de la physique en levant l'ambiguite que constituait le succes des methodes de calcul symbolique aupres des physiciens and l'inacceptabilite de leurs formules au regard de la rigueur mathematiques.
Abstract: Ce traite a marque une date dans le progres des mathematiques et de la physique en levant l’ambiguite que constituait le succes des methodes de calcul symbolique aupres des physiciens et l’inacceptabilite de leurs formules au regard de la rigueur mathematiques Le merite revient a Laurent Schwartz d’avoir englobe dans une theorie qui est a la fois une synthese et une simplifications, des procedes heterogenes et souvent incorrects utilises dans des domaines tres divers
Une definition correcte et une etude systematique de ces etres nouveaux, les distributions, leur ont donne droit de cite dans l’usage courant Leur utilisation extensive dans de nombreuses branches des mathematiques pures et appliquees, de la physique et des sciences de l’ingenieur fait de ce livre un classique des mathematiques modernes
4,197 citations
01 Jan 2008
TL;DR: In this paper, a variational method in the theory of harmonic integrals has been proposed to solve the -Neumann problem on strongly pseudo-convex manifolds and parametric Integrals two-dimensional problems.
Abstract: Semi-classical results.- The spaces Hmp and Hmp0.- Existence theorems.- Differentiability of weak solutions.- Regularity theorems for the solutions of general elliptic systems and boundary value problems.- A variational method in the theory of harmonic integrals.- The -Neumann problem on strongly pseudo-convex manifolds.- to parametric Integrals two dimensional problems.- The higher dimensional plateau problems.
3,190 citations