Bending Moment in Membrane Theory

Bouchitté, Guy; Fonseca, Irene; Mascarenhas, M. L.

doi:10.1023/b:elas.0000029996.20973.92

Cited by 30 publications

(56 citation statements)

References 15 publications

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“…In particular, our results extend results obtained by G. Friesecke, R. D. James and S. Müller in 2002 [16] for rigid thin films, and by G. Bouchitté, I. Fonseca and M. L. Mascarenhas in 2004 [5] for thin films with bending moment. Their proofs require other techniques and we will discuss these issues in our forthcoming papers.…”

supporting

confidence: 90%

Effective energy integral functionals for thin films in the Orlicz–Sobolev space setting

Laskowski

Nguyêñ

2013

Demonstratio Mathematica

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Abstract. We consider an elastic thin film as a bounded open subset ω of R 2 . First, the effective energy functional for the thin film ω is obtained, by Γ-convergence and 3D-2D dimension reduction techniques applied to the sequence of re-scaled total energy integral functionals of the elastic cylinders) as the thickness ε goes to 0. Then we prove the existence of minimizers of the film energy functional. These results are proved in the case when the energy density function for the elastic cylinders has the growth prescribed by an Orlicz convex function M . Here M is assumed to be non-power-growth-type and to satisfy the conditions ∆2 and ∇2 (that is equivalent to the reflexivity of Orlicz and Orlicz-Sobolev spaces generated by M ). These results extend results of H. Le Dret and A. Raoult for the case M (t) = |t| p for some p ∈ (1, ∞).

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supporting

confidence: 90%

Effective energy integral functionals for thin films in the Orlicz–Sobolev space setting

Laskowski

Nguyêñ

2013

Demonstratio Mathematica

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“…Their proof can be found in [3] in the homogeneous case and the heterogeneity does not create any additional difficulty. Proposition 3.6.…”

Section: Lemma 35 For Anymentioning

confidence: 87%

“…In view of Lemma 2.2 in [3] (the proof in our context is identical to that of the homogeneous case), there exists a subsequence of {ε n } (not relabelled) and a sequence {v …”

Section: Lemma 35 For Anymentioning

confidence: 95%

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Spatial heterogeneity in 3D-2D dimensional reduction

Babadjian¹,

Francfort²

2004

ESAIM: COCV

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Abstract.A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret (1995). Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitté et al. (2002), the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as a problem of Γ-convergence of the elastic energy, as the thickness tends to zero.Mathematics Subject Classification. 49J45, 74B20, 74G65, 74K15, 74K35.

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“…In a different direction, the paper [5] investigates both mid-surface and cross-sectional behavior in the absence of interfacial energy, but this analysis only accounts for the bending moment -the average of the Cosserat vector through the cross section -and it is conjectured that the behavior becomes nonlocal if the actual Cosserat vector is kept in the formulation in lieu of its average.…”

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confidence: 99%

Thin elastic films: The impact of higher order perturbations

2006

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Abstract. The asymptotic behavior of an elastic thin film penalized by a van der Wals type interfacial energy is investigated when both its thickness and the magnitude of the additional energy vanish in the limit. Keeping track of both mid-plane and outof-plane deformations (through the introduction of the Cosserat vector), the resulting behavior strongly depends upon the ratio between thickness and interfacial energy.

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Bending Moment in Membrane Theory

Cited by 30 publications

References 15 publications

Effective energy integral functionals for thin films in the Orlicz–Sobolev space setting

Effective energy integral functionals for thin films in the Orlicz–Sobolev space setting

Spatial heterogeneity in 3D-2D dimensional reduction

Thin elastic films: The impact of higher order perturbations

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