Homogenization of free discontinuity problems

Braides, Andrea; Defranceschi, Anneliese; Vitali, Enrico

doi:10.1007/bf02198476

Cited by 116 publications

(238 citation statements)

References 28 publications

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“…Remark 1 Theorem 1 of Braides and Lukkassen [9,22] is a generalization of the case of quadratic forms given by Bensoussan, Lions and Papanicolaou in the book [7] where the concept of reiterated homogenization was introduced. Later on M• uller, Braides and Defranceschi generalized this result to standard non-convex Lagrangians (see [8]). The corresponding proofs are quite di erent.…”

Section: Theoremmentioning

confidence: 94%

On hierarchical structures and reiterated homogenization

Ćwikel¹,

Engliš²,

Kufner³

et al. 2002

Function Spaces, Interpolation Theory and Related Topics

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Section: Theoremmentioning

confidence: 94%

On hierarchical structures and reiterated homogenization

Ćwikel¹,

Engliš²,

Kufner³

et al. 2002

Function Spaces, Interpolation Theory and Related Topics

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show abstract

“…A large number of works devoted to homogenization have been published since the seminal articles [34,87,94] that were at the origin of the De Giorgi theory of Γ -convergence [20,21,31,33] and of the Tartar-Murat theory of compensated compactness [75,76,94]. Homogenization was applied to continuum mechanics in, for example, [5,12,27,30,51,53,71,76,85].…”

Section: Literaturementioning

confidence: 99%

“…(i) The functional Λ may be compared with the homogenized functional of the calculus of variations (see, for example, [20,21,31,66]). However, the latter concerns the stationary setting, whereas Λ applies to processes.…”

Section: And Only If (Denoting By B * the Transpose Of The Tensor B) mentioning

confidence: 99%

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Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl—Reuss model of elastoplasticity

Visintin

2008

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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This paper deals with processes in nonlinear inelastic materials whose constitutive behaviour is represented by the inclusionhere we denote by σ the stress tensor, by ε the linearized strain tensor, by B(x) the compliance tensor and by ∂ϕ(·, x) the subdifferential of a convex function ϕ (·, x). This relation accounts for elasto-viscoplasticity, including a nonlinear version of the classical Maxwell model of viscoelasticity and the Prandtl-Reuss model of elastoplasticity. The constitutive law is coupled with the equation of continuum dynamics, and well-posedness is proved for an initial-and boundary-value problem. The function ϕ and the tensor B are then assumed to oscillate periodically with respect to x and, as this period vanishes, a two-scale model of the asymptotic behaviour is derived via Nguetseng's notion of two-scale convergence. A fully homogenized single-scale model is also retrieved, and its equivalence with the two-scale problem is proved. This formulation is non-local in time and is at variance with that based on so-called analogical models that rest on a mean-field-type hypothesis.

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“…The notation F = (F|b) means that the first two columns and final column of the 3 × 3 matrix F are replaced byF and b, respectively. The exact definition of Qϕ 0 can be found in 6,8) and is not shown here as it requires advanced mathematical analysis which is beyond the purpose of this paper.…”

Section: Single Crystal Filmmentioning

confidence: 99%

Shape-Memory Micropumps

Shu

2002

Mater. Trans.

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Motivated by many experimental efforts to develop suitable shape-memory micropumps, we propose a multiscale framework to study the behavior of pressurized films. We use recoverable deflection as a measure to design large stroke micropumps and develop a model to estimate it. We show that the recoverable deflection of a polycrystalline shape-memory film depends on the transformation strain of the underlying martensitic transformation, the texture and especially on the size effects. We find that flat grains are preferable to long grains in columnar films concerning the purpose of large recoverable strain. We also show that common sputtering texture is not ideal for recoverable deflection in both Ti-Ni and Cu-based shape-memory films. It turns out that {100} Cu-based films may have better behavior than Ti-Ni films. We conclude with comparison with experiment.

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Homogenization of free discontinuity problems

Cited by 116 publications

References 28 publications

On hierarchical structures and reiterated homogenization

On hierarchical structures and reiterated homogenization

Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl—Reuss model of elastoplasticity

Shape-Memory Micropumps

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