Rigidity and gamma convergence for solid‐solid phase transitions with SO(2) invariance

Conti, Sergio; Schweizer, Ben

doi:10.1002/cpa.20115

Cited by 79 publications

(167 citation statements)

References 36 publications

(50 reference statements)

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“…With respect to the quantitative results of [4,8,11,14], we stress that, in spite of being also rigidity results for nonzero energy states, they are essentially different from the ones in the present paper. First, they differ in spirit, because the former ones rely on estimates involving the distances between ru and K, and ru and a particular energy well (estimates which we don't attempt).…”

Section: Introductioncontrasting

confidence: 71%

“…More precisely, he proved that if the perimeter of the transition in a ball, as controlled by kD 2 uk.B 1 /, is sufficiently small, there exist a certain power < 1 of the L 1 -norm of the distance between ru and the set K that controls the L 1 -norm of the distance of ru to one of the wells in some smaller ball. Lorent's result was later improved by Conti and Schweizer [8] to D 1 in the case of two wells with positive determinant and maps u 2 W 1;1 such that kD 2 uk. / is small compared with a quantity that depends on the geometry of .…”

Section: Introductionmentioning

confidence: 93%

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A rigidity result for a perturbation of the geometrically linear three‐well problem

Capella

Otto

2009

Comm Pure Appl Math

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We study a three-dimensional model for alloys that undergo a cubic-to-tetragonal phase transition in the martensitic phase. Any pair of the three martensitic variants can form a stress-free laminate. However, this laminate is only compatible on average with the remaining variant. The resulting local stresses favor a microstructure if all three variants are present (for instance, because of an externally imposed average strain).Next to the linearized elastic energy, the variational model features an interfacial energy between the three variants. This introduces a material length scale, which together with the sample size (which we mimic by periodic boundary conditions) gives rise to a nondimensional parameter Á.We rigorously establish the scaling of the minimal energy e per volume in case of externally imposed volume fractions of the martensitic variants in Á. More precisely, we show e Á 2=3 . The upper bound construction is achieved by a few patches of branched laminates; the lower bound relies on suitable interpolation inequalities. This is in the spirit of a celebrated work by Kohn and Müller and relies on techniques developed for domain branching in micromagnetics.We also prove a rigidity result in the sense that if the energy per volume e of a configuration is much smaller than Á 2=3 , the configuration is approximately a simple unbranched laminate. In particular, one of the three volume fractions has to be small. This is related to similar rigidity results by Dolzmann and Müller and by Kirchheim.

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Section: Introductioncontrasting

confidence: 71%

Section: Introductionmentioning

confidence: 93%

A rigidity result for a perturbation of the geometrically linear three‐well problem

Capella

Otto

2009

Comm Pure Appl Math

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“…Using methods of [9] (for the non-linear functional) Conti, Schweizer proved a strong generalisation of Theorem 1.1, their strategy was to use hypotheses (4) and (5) …”

Section: Specifically It Was Shown That For Eachmentioning

confidence: 99%

“…The naive guess that the optimal bound is given by cκ is false 1 , this follows from the construction of [7], see [9] for more details. The assumption that u is bilipschitz is a technical one, however it is used in an essential way many times in the proof.…”

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confidence: 99%

A Two Well Liouville Theorem

Lorent¹

2005

ESAIM: COCV

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Abstract. In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller.Let1 . There exists positive constants c1 < 1 and c2 > 1 depending only on σ, ζ1, ζ2 such that if ∈ (0, c1) and u satisfies the following inequalitiesthen there exists J ∈ {Id, H} and R ∈ SO (2) such thatMathematics Subject Classification. 74N15.

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“…In a BD setting one only obtains control of the longitudinal component of u. Therefore one needs to consider lines with many different orientations, making sure to choose them so that they do not intersect the jump set, a strategy that was used for proving density in [Cha04,Cha05] and for proving rigidity in one and multiwell settings for example in [Koh82,DM95,CS06]. More details are explained in the introduction to Section 2.…”

Section: Introductionmentioning

confidence: 99%

Korn-Poincare inequalities for functions with a small jump set

Chambolle¹,

Conti²,

Francfort³

2016

Indiana Univ. Math. J.

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Functions in SBDp arise naturally in the study of geometrically linear fracture models. They have a jump set of finite (n − 1)-dimensional measure and, away from the jump set, a symmetrized gradient e(u) = (∇u + ∇u T )/2 in L p , p ≥ 1. We show that if the measure of the jump set is sufficiently small with respect to the size of the domain, then the function u can be approximated by an affine function away from a small exceptional set, with an error which depends solely on e(u). We also derive a corresponding trace statement.

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Rigidity and gamma convergence for solid‐solid phase transitions with SO(2) invariance

Cited by 79 publications

References 36 publications

A rigidity result for a perturbation of the geometrically linear three‐well problem

A rigidity result for a perturbation of the geometrically linear three‐well problem

A Two Well Liouville Theorem

Korn-Poincare inequalities for functions with a small jump set

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