Dimension Reduction for Thin Films with Transversally Varying Prestrain: Oscillatory and Nonoscillatory Cases

Abstract

We study the non-Euclidean (incompatible) elastic energy functionals in the description of prestressed thin films, at their singular limits (Γ-limits) as h → 0 in the film's thickness h. Firstly, we extend the prior results [39,12,40] to arbitrary incompatibility metrics that depend on both the midplate and the transversal variables (the "non-oscillatory" case). Secondly, we analyze a more general class of incompatibilities, where the transversal dependence of the lower order terms is not necessarily linear (the "oscillatory" case), extending the results of [3,47] to arbitrary metrics and higher order scalings. We exhibit connections between the two cases via projections of appropriate curvature forms on the polynomial tensor spaces. We also show the effective energy quantisation in terms of scalings as a power of h and discuss the scaling regimes h 2 (Kirchhoff), h 4 (von Kármán) in the general case, as well as all possible (even powers) regimes for conformal metrics, thus paving the way to the subsequent complete analysis of the non-oscillatory setting in [34]. Thirdly, we prove the coercivity inequalities for the singular limits at h 2 -and h 4 -scaling orders, while disproving the full coercivity of the classical von Kármán energy functional at scaling h 4 .

Quantitative Immersability of Riemann Metrics and the Infinite Hierarchy of Prestrained Shell Models

“…We further prove compactness and the lower bound, at any of the new viable scaling levels inf E h ∼ h 2(n+1) , completing thus the analysis done for n = 0 in [28,6] and for n = 1 in [29,30]:…”

“…In this paper, we determine the possible energy scalings: inf E h ∼ h β in the limit of vanishing thickness h → 0, and the corresponding variational limits (Γ-limits) I β of h −β E h , in the regime β > 4 that has not been analyzed before. We thus complete the discussion of weakly prestrained films, started in our previous works [28,6,29,30] that covered the range β ∈ [2,4]. The singular limits I β are typically given by energies of the form I = T ensor(y) 2 E defined on the appropriate set of limiting deformations/displacements y of the midplate ω.…”

Dimension reduction through Gamma convergence for general prestrained thin elastic sheets

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We study thin films with residual strain by analyzing the Γ−limit of non-Euclidean elastic energy functionals as the material's thickness tends to 0. We begin by extending prior results (Bhattacharya, Lewicka, & Schäffner, 2016) (Agostiniani, Lucantonio, & Lucic, 2019) (Lewicka & Lucic, 2018) (Schmidt, 2007, to a wider class of films, whose prestrain depends on both the midplate and the transversal variables. The ansatz for our Γ−convergence result uses a specific type of wrinkling, which is built on exotic solutions to the Monge-Ampere equation, constructed via convex integration (Lewicka & Pakzad, 2017).

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“…Thin elastic sheets that deform because of residual strain have recently been the focus of numerous mathematical (Agostiniani et al, 2019) (Bhattacharya et al, 2016) (Lewicka & Lucic, 2018) (Schmidt, 2007) (Lewicka, Mahadevan, & Pakzad, 2010) (Lewicka, Mora, & Pakzad, 2010) (Lewicka, Mora, & Pakzad, 2011) and engineering/physics (Gladman, Matsumoto, Nuzzo, Mahadevan, & Lewis, 2016) (Miskin et al, 2018) (Kim, Hanna, Byun, Santangelo, & Hayward, 2012) (Klein, Efrati, & Sharon, 2007) (Pezzulla, Shillig, Nardinocchi, & Holmes, 2015) (Aharoni, Xia, Zhang, Kamien, & Yang, 2018) (Sharon, Roman, Marder, Shin, & Swinney, 2002) studies. Mathematically, this represents the challenge of generalizing the seminal work of (Friesecke, James, & Müller, 2002) to noneuclidean geometries, while from the point of view of applications, careful design of prestrain in thin sheets can be used to 3D print surfaces.…”

Derivation of a Homogenized Bending–Torsion Theory for Rods with Micro-Heterogeneous Prestrain

“…In general, we expect that the type of limiting model does not only depend on α but also on other properties of the prestrain. Indeed, in a situation without homogenization it is shown in [8,36] that bending and von Karman type plate models arise in the case α = 0 depending on the geometry of the prestrain (see also [37] for related results in the case of rods and [31,32] for recent results for plates and shells beyond the von Karman regime). It is an interesting question if these results are stable with respect to (small) rapidly oscillating perturbations.…”