Homogenization of integral energies under periodically oscillating differential constraints

Davoli, Elisa; Fonseca, Irene

doi:10.1007/s00526-016-0988-5

Cited by 12 publications

(16 citation statements)

References 55 publications

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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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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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“…where f : Ω×R N ×R d → [0, +∞) has standard p-growth, Ω ⊂ R N is a bounded open set, ε → 0, and the fields u ε : Ω → R d are subjected to x−dependent differential constraints of the type 1) or in divergence form We recently analyzed in [10] the limit case in which α = 0, β > 0, the energy density is independent of the first two variables, and the fields {u ε } are subjected to (1.2). We will consider here the case in which α > 0, β = 0 and (1.1), i.e., the energy density is oscillating but the differential constraint is fixed and in "nondivergence" form.…”

Section: Introductionmentioning

confidence: 99%

“…As in [10] and [11], the proof of this result is based on the unfolding operator, introduced in [6,7] (see also [21,22]). In contrast with [11, Theorem 1.1] (i.e.…”

Section: Introductionmentioning

confidence: 99%

“…In [10] the issue of defining a projection operator was tackled by imposing an additional invertibility assumption on A and by exploiting the divergence form of the differential constraint. We do not add this invertibility requirement here, instead we use the fact that in our framework the differential operator depends on the "macro" variable x but acts on the "micro" variable y (see (1.5)).…”

Section: Introductionmentioning

confidence: 99%

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Periodic homogenization of integral energies under space-dependent differential constraints

Davoli¹,

Fonseca²

2016

Port. Math.

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A homogenization result for a family of oscillating integral energies uε →ˆΩ f (x, x ε , uε(x)) dx, ε → 0 + is presented, where the fields uε are subjected to first order linear differential constraints depending on the space variable x. The work is based on the theory of A-quasiconvexity with variable coefficients and on two-scale convergence techniques, and generalizes the previously obtained results in the case in which the differential constraints are imposed by means of a linear first order differential operator with constant coefficients. The identification of the relaxed energy in the framework of A-quasiconvexity with variable coefficients is also recovered as a corollary of the homogenization result.

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“…The main objective is to study properties for ε → 0 of I ε (u ε ) := Ω f (x, x ε , ∇u ε (x)) dx, where f is (0, 1) nperiodic in the middle variable. Homogenization problems are not only restricted to gradients but new results in the context of A-quasiconvexity (even with non-constant coefficients) recently appeared, e.g., in [74]. Various other generalizations including stochastic features to describe randomness in the distribution of inhomogeneities can be found in the papers by Dal Maso and Modica [71], Messaoudi and Michaille [168] or the recent work of Gloria, Neukamm, and Otto [104].…”

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confidence: 99%

Weak Lower Semicontinuity of Integral Functionals and Applications

Benešová¹,

Kružík²

2017

SIAM Rev.

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Minimization is a reoccurring theme in many mathematical disciplines ranging from pure to applied ones. Of particular importance is the minimization of integral functionals that is studied within the calculus of variations. Proofs of the existence of minimizers usually rely on a fine property of the involved functional called weak lower semicontinuity. While early studies of lower semicontinuity go back to the beginning of the 20th century the milestones of the modern theory were set by C.B. Morrey Jr. [176] in 1952 and N.G. Meyers [169] in 1965. We recapitulate the development on this topic from then on. Special attention is paid to signed integrands and to applications in continuum mechanics of solids. In particular, we review the concept of polyconvexity and special properties of (sub)determinants with respect to weak lower semicontinuity. Besides, we emphasize some recent progress in lower semicontinuity of functionals along sequences satisfying differential and algebraic constraints which have applications in elasticity to ensure injectivity and orientation-preservation of deformations. Finally, we outline generalization of these results to more general first-order partial differential operators and make some suggestions for further reading.If we, however, consider a slight modification of Y by changing the boundary condition at x = 1, and consider Y 1 := {u ∈ W 1,∞ (0, 1); −1 ≤ u ≤ 1, u(0) = 0, u(1) = 1} then min Y1

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Homogenization of integral energies under periodically oscillating differential constraints

Cited by 12 publications

References 55 publications

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

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

Periodic homogenization of integral energies under space-dependent differential constraints

Weak Lower Semicontinuity of Integral Functionals and Applications

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