Multiple integrals under differential constraints: two-scale convergence and homogenization

Fonseca, Irene; Krömer, Stefan

doi:10.1512/iumj.2010.59.4249

Cited by 28 publications

(48 citation statements)

References 10 publications

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“…Homogenization in the context of variational problems restricted to A-free fields was first studied by Braides, Fonseca & Leoni in [9]. Later, this result was enhanced in [15], where the use of two-scale techniques allowed for weaker assumptions on the integrand.…”

Section: Collection Of Results On Homogenizationmentioning

confidence: 99%

Heterogeneous Thin Films: Combining Homogenization and Dimension Reduction with Directors

Kreisbeck¹,

Krömer²

2016

SIAM J. Math. Anal.

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Abstract. We analyze the asymptotic behavior of a multiscale problem given by a sequence of integral functionals subject to differential constraints conveyed by a constant-rank operator with two characteristic length scales, namely the film thickness and the period of oscillating microstructures, by means of Γ-convergence. On a technical level, this requires a subtle merging of homogenization tools, such as multiscale convergence methods, with dimension reduction techniques for functionals subject to differential constraints. One observes that the results depend critically on the relative magnitude between the two scales. Interestingly, this even regards the fundamental question of locality of the limit model, and, in particular, leads to new findings also in the gradient case.MSC (2010): 49J45 (primary); 35E99, 74K15, 74Q05

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Section: Collection Of Results On Homogenizationmentioning

confidence: 99%

Heterogeneous Thin Films: Combining Homogenization and Dimension Reduction with Directors

Kreisbeck¹,

Krömer²

2016

SIAM J. Math. Anal.

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“…Due to the non-convexity of the sets M s and SO (2), it does not fall within the scope of works on gradient-constraint problems like [7,8,12], either. For a study of homogenization problems that involve constraints imposed by special linear first order partial differential equations we refer to [5,18,21].…”

Section: Introductionmentioning

confidence: 99%

Homogenization of layered materials with rigid components in single-slip finite crystal plasticity

Christowiak

Kreisbeck

2017

Calc. Var.

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We determine the effective behavior of a class of composites in finite-strain crystal plasticity, based on a variational model for materials made of fine parallel layers of two types. While one component is completely rigid in the sense that it admits only local rotations, the other one is softer featuring a single active slip system with linear self-hardening. As a main result, we obtain explicit homogenization formulas by means of -convergence. Due to the anisotropic nature of the problem, the findings depend critically on the orientation of the slip direction relative to the layers, leading to three qualitatively different regimes that involve macroscopic shearing and blocking effects. The technical difficulties in the proofs are rooted in the intrinsic rigidity of the model, which translates into a non-standard variational problem constraint by non-convex partial differential inclusions. The proof of the lower bound requires a careful analysis of the admissible microstructures and a new asymptotic rigidity result, whereas the construction of recovery sequences relies on nested laminates. Mathematics Subject Classification 49J45 (primary) · 74Q05, 74C15

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“…Let us emphasize that variational problems with differential constraints naturally appear in hyperelasticity, electromagnetism, or in micromagnetics [7,25,26] and are closely related to the theory of compensated compactness [24,28,29]. The concept of A-quasiconvexity goes back to [5] and has been proved to be useful as a unified approach to variational problems with differential constraints, including results on homogenization [4,11], dimension reduction [19] and characterization of generalized Young measures [2] in the A-free setting. Moreover, first results on A-quasiaffine functions and weak continuity appeared in [16].…”

Section: Introductionmentioning

confidence: 99%

𝒜$\mathcal{A}$-quasiconvexity at the boundary and weak lower semicontinuity of integral functionals

Kramer

Krömer

Kružík

et al. 2017

Advances in Calculus of Variations

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We state necessary and su cient conditions for weak lower semicontinuity of integral functionals of the form u → ∫ Ω h (x, u(x)) dx, where h is continuous and possesses a positively p-homogeneous recession function, p > , and u ∈ L p (Ω; ℝ m ) lives in the kernel of a constant-rank rst-order di erential operator A which admits an extension property. In the special case A = curl, apart from the quasiconvexity of the integrand, the recession function's quasiconvexity at the boundary in the sense of Ball and Marsden is known to play a crucial role. Our newly de ned notions of A-quasiconvexity at the boundary, generalize this result. Moreover, we give an equivalent condition for the weak lower semicontinuity of the above functional along sequences weakly converging in L p (Ω; ℝ m ) and approaching the kernel of A even if A does not have the extension property.

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Multiple integrals under differential constraints: two-scale convergence and homogenization

Abstract: Two-scale techniques are developed for sequences of mapssatisfying a linear dierential constraint Au k = 0. These, together with Γ-convergence arguments and using the unfolding operator, provide a homogenization result for energies of the type

Cited by 28 publications

References 10 publications

Heterogeneous Thin Films: Combining Homogenization and Dimension Reduction with Directors

Heterogeneous Thin Films: Combining Homogenization and Dimension Reduction with Directors

Homogenization of layered materials with rigid components in single-slip finite crystal plasticity

𝒜$\mathcal{A}$-quasiconvexity at the boundary and weak lower semicontinuity of integral functionals

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