We hope this content on epidemiology, disease modeling, pandemics and vaccines will help in the rapid fight against this global problem. Click on title above or here to access this collection. We report on several numerical experiments where the rank-one convexification of an energy density is computed. The explicit examples cover a whole spectrum of typical situations one may encounter. One of those is especially relevant for the computation of microstructures in crystalline solids. Sign in Help View Cart.

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Csr2 tempest 1 tier 5Notify Me! E-mail Alerts. RSS Feeds. SIAM J. Related Databases. Web of Science You must be logged in with an active subscription to view this. Keywords laminatesmaximum number of levelsrank-one convex hull. Publication Data.

Publisher: Society for Industrial and Applied Mathematics. Ernesto Aranda and Pablo Pedregal. Archive for Rational Mechanics and Analysis :3, Applications in continuum mechanics and physics of solids. Rate-Independent Systems, Continuum Mechanics and Thermodynamics 23 :6, BIT Numerical Mathematics 47 :3, Applied Mathematics and Computation :1, Computational Materials Science 32 Computer Methods in Applied Mechanics and Engineering Numerical Functional Analysis and Optimization 23 Banner art adapted from a figure by Hinke M.Unable to display preview.

Download preview PDF. Skip to main content. This service is more advanced with JavaScript available. Advertisement Hide. Authors Authors and affiliations M. Conference paper.

This is a preview of subscription content, log in to check access. Ball, J. Convexity conditions and existence theorems in nonlinear elasticity. Rational Mech. Fine phase mixtures as minimizers of energy. Proposed experimental tests of a theory of fine microstructure and the two-well problem.

Royal Soc. ADS Google Scholar. Bladon, P. Transitions and instabilities in liquidcrystal elastomers. E 47R—R Chipot, M. Equilibrium configurations of crystals. Dacorogna, B. Direct methods in the calculus of variationsSpringer, Berlin. Google Scholar. DeSimone, A. Macroscopic response of nematic elastomers via relaxation of a class of SO 3 -invariant energies. Preprint No.

Kohn, R. Pure Appl.An elliptope is that convex Euclidean body formed from elements that are vectorized matrices. Each matrix constituting an elliptope has 1 in each entry along the main diagonal and is positive semidefinite. These matrices are also known as the correlation matrices. The elliptope is important because some optimization problems involving positive semidefinite matrices can be restricted to the elliptope without loss of generality.

This constraint can simplify problems. The cone of Euclidean distance matrices can be formed by manipulating an elliptope. But the more conventional convex definition of vertex describes it as any point that can be isolated by a supporting hyperplane. Although the elliptope's relative boundary looks smooth, is is actually rough because of all the vertices.

Above is that pillow-shaped elliptope formed from all positive semidefinite 3x3 matrices having 1 along the main diagonal.

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In the example illustrated above, the elliptope is that line segment interior to the positive semidefinite cone of 2x2 matrices. Fantope is our nomenclature named after mathematician Ky Fan.

A Fantope is the convex hull of that set comprising outer product of all orthonormal matrices of particular dimension. In the example illustrated, the circular Fantope represents outer product of all 2x2 rank-1 orthonormal matrices. Home Elliptope and Fantope Elliptope and Fantope An elliptope is that convex Euclidean body formed from elements that are vectorized matrices.

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Data result lisboa 4dComplementarity problem. Compressive Sampling. Compressed Sensing.A necessary and sufficient condition for the rank 1 convexity of f in terms of f is given. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Aubert, Necessary and sufficient conditions for isotropic rank-one convex functions in dimension 2.

Elasticity 39 31— Aubert and R. Rational Mech. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Ball, Differentiability properties of symmetric and isotropic functions. Duke Math. Ball and R. James, Fine phase mixtures as minimizers of energy. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem.

London — Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Dacorogna, Direct methods in the calculus of variations. Springer, Berlin Google Scholar. Dacorogna, Necessary and sufficient conditions for strong ellipticity of isotropic functions in any dimension. Discrete Contin. B2 — Dacorogna and H. Koshigoe, On the different notions of convexity for rotationally invariant functions.

Toulouse II — Dacorogna and P. Marcellini, Implicit Partial Differential Equations. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain. Kohn and G. Pure Appl. Springer, New York Springer, Berlin pp. Pedregal, Parametrized Measures and Variational Principles. Rosakis, Characterization of convex isotropic functions. Elasticity 49 —To browse Academia. Skip to main content. Log In Sign Up. Miroslav Silhavy. Journal of Elasticity —, Printed in the Netherlands.

Key words: rank 1 convex functions, rotational invariance. In memory of Clifford Truesdell 1. If the problem has a solution, i. Subsequently it was found that the existence of the solution, and its further properties, are directly related to the semiconvexity properties i. For finite- valued functions, quasiconvexity implies rank 1 convexity. If f is not quasiconvex, the material exhibits microstructure and phase transformation [5—8, 13, 15].

The effective energy is given by the relaxation of Ii. One defines the rank 1 convex hull Rf similarly. For example, stored energies of isotropic solids have this property. The reader is referred to [2, 10], and [9] for additional information. In view of its global nature, Theorem 6, and the results of [20—22, 25], can be used to define iterative procedures for evaluating the rank 1 convex hull of an in- variant function [26, 23]. Theorem 6, and its proof, has two parts.

One part item i of the theorem is the monotonicity of invariant rank 1 convex functions as estab- lished in [24], which is closely related to the Baker—Ericksen inequalities in the differentiable case.

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In the special case of O n invariant functions, a similar result has been established in [11]; however, the result does not apply to SO n invariant functions treated here. Namely, the rank 1 perturbations described in Proposition 1 have certain minimum properties stated in Lemma 4. Combining these with items iii and the use of some continuity and density arguments Lemmas 2 and 5 then completes the proof.

Apart from the notation and the bilat- eral interlacing inequalities, Section 2 is not needed for the statement of Theorem 6; it only gathers a material for the proof, and can be used as reference as needed. It would be desirable to integrate the two conditions of Theorem 6 into a single condition.

One connection between iii is established in Lemma 4, which is the main technical improvement with respect to the previous papers of the author. The lemma shows that the rank 1 perturbations underlying condition ii are local minimizers of the partial products of signed singular values occurring in condition i.

This establishes the special positions of these particular rank 1 perturbations.One of these conditions allows us to understand better the gap between the rank-one convexity and the quasiconvexity. This is a preview of subscription content, log in to check access.

Rent this article via DeepDyve. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Alibert and B. Dacorogna, An example of a quasiconvex function not polyconvex in dimension two.

Numanovic stoliceModel and Num. Google Scholar. Aubert and R. Ball, Existence theorems in non linear elasticity; Arch. Dacorogna, Direct methods in the calculus of variations; Springer Verlag, Berlin Knowles and E.

Sternberg, On the failure of ellipticity of the equations of finite elasticity. Davies, A simple derivation of necessary and sufficient conditions for strong ellipticity of isotropic hyperelastic materials in plane strain. Elasticity 26 Morrey, Multiple integrals in the calculus of variations; Springer Verlag, Berlin Rosakis and H.

Simpson, On the relation between polyconvexity and rank-one convexity in nonlinear elasticity. Elasticity 37 — Simpson and S. Spector, On compositive matrices and strong ellipticity for isotropic material. Sverak, Rank-one convexity does not imply quasiconvexity. Download references. Reprints and Permissions.

The companies in bondeno (ferrara). local firms and products.Aubert, G. Necessary and sufficient conditions for isotropic rank-one convex functions in dimension 2. J Elasticity 39, 31—46 Download citation.The microstructures accompanying phase transitions in solids are governed by the minimum energy principle.

The occurrence of microstructures is connected with the incompatibility of individual solid phases and with the nonexistence of minimizers of energy. In their presence, one is concerned with effective properties.

The procedure is the relaxation. Unable to display preview. Download preview PDF. Skip to main content. This service is more advanced with JavaScript available. Advertisement Hide.

Authors Authors and affiliations M. This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access.

## Necessary and sufficient conditions for isotropic rank-one convex functions in dimension 2

Acerbi and N. Semicontinuity problems in the calculus of variations. Rational Mech. Alibert and B. An example of a quasiconvex function that is not polyconvex in two dimensions. Anderson, D. Carlson and E. A continuum mechanical theory for nematic elastomers. Elasticity33—58, EdinburghA: —, Necessary and sufficient conditions for isotropic rank-one convex functions in dimension 2. Elasticity31—46, Aubert and R. Paris—, Convexity conditions and existence theorems in nonlinear elasticity.

Ball and R. Fine phase mixtures as minimizers of energy. Proposed experimental tests of a theory of fine microstructure and the two-well problem. Bladon, E. Terentjev and M. Transitions and instabilities in liquid—crystal elastomers. Google Scholar. Buttazzo, B. Dacorogna and W.

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