Variational models for microstructure and phase transitions

Müller, Stefan

doi:10.1007/bfb0092670

Cited by 415 publications

(347 citation statements)

References 139 publications

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“…If φ n ∈ A is any sequence and φ n ⇀ φ * weakly in some The previous remark suggests considering the problem of minimizing the relaxed energyF p , namely the largest weakly lower semi-continuous function less than or equal to F p on M [38,39]. Let B denote the weak closure of A in some L p ′ (the precise value of p ′ is irrelevant as indicated in Remark 5.2).…”

Section: Lower Bounds For the Curvature Of Smooth Isometric Immersionmentioning

confidence: 99%

Shape selection in non-Euclidean plates

Gemmer

Venkataramani

2011

Physica D: Nonlinear Phenomena

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We investigate isometric immersions of disks with constant negative curvature into R 3 , and the minimizers for the bending energy, i.e. the L 2 norm of the principal curvatures over the class of W 2,2 isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in H 2 into R 3 . In elucidating the connection between these immersions and the non-existence/singularity results of Hilbert and Amsler, we obtain a lower bound for the L ∞ norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally W 2,2 , and numerically have lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grow approximately exponentially with the radius of the disc. We discuss the implications of our results on recent experiments on the mechanics of non-Euclidean plates.

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Section: Lower Bounds For the Curvature Of Smooth Isometric Immersionmentioning

confidence: 99%

Shape selection in non-Euclidean plates

Gemmer

Venkataramani

2011

Physica D: Nonlinear Phenomena

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“…A characterization of Young measures generated by gradients was completely given by Kinderlehrer and Pedregal [21,22], cf. also [32,33]. To our knowledge, the first attempt to characterize both oscillations and concentrations in sequences of gradients is due to Fonseca, Müller, and Pedregal [14].…”

Section: Introductionmentioning

confidence: 99%

Oscillations and concentrations in sequences of gradients

Kałamajska

Kružík

2007

ESAIM: COCV

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Abstract.We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, {∇u k }, bounded in L p (Ω; R m×n ) if p > 1 and Ω ⊂ R n is a bounded domain with the extension property in W 1,p . Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of Ω are required and links with lower semicontinuity results by Meyers and by Acerbi and Fusco are also discussed.Mathematics Subject Classification. 49J45, 35B05.

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“…Thus, we refer to (4.2) as the relaxed or effective energy of the system. It turns out that (4.2) is equal to Φ qc (F), the quasiconvex envelope of Φ (see, e.g., [12]). Moreover, if K is the zero level set of Φ, then the zero level set of Φ qc (F) is K qc , the quasiconvex hull of K. Thus K qc is the set of macroscopic deformations that can be imposed at zero energy cost, i.e., the soft deformations.…”

Section: Quasiconvexity: Motivationmentioning

confidence: 99%

“…There is a well-developed mathematical literature on variational approaches to phase transitions in crystalline solids (see, e.g., [2] or the lecture notes [12]). Many relevant questions on the mechanical response of materials described by multiwell energies can thus be given a precise mathematical formulation.…”

Section: Introductionmentioning

confidence: 99%

Material instabilities in nematic elastomers

DeSimone

Dolzmann

2000

Physica D: Nonlinear Phenomena

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Soft deformation paths and domain patterns in nematic elastomers are analyzed through the minimization of a nonconvex free-energy recently proposed in the literature. The free-energy density has multiple wells, and is not restricted to small deformations. The problems of calculating the quasiconvex hull of the energy wells and the quasiconvex envelope of the free-energy density are formulated and solved (the latter only in two spatial dimensions). This leads to a complete characterization of the set of soft deformations paths available to a given material, and of its effective macroscopic energy.

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Variational models for microstructure and phase transitions

Cited by 415 publications

References 139 publications

Shape selection in non-Euclidean plates

Shape selection in non-Euclidean plates

Oscillations and concentrations in sequences of gradients

Material instabilities in nematic elastomers

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