2018/07/05 by Marta Lewicka, Lewicka, Marta, Danka Lučić +1 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Materials and Mechanics #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Coercivity #Condensed matter physics #Conformal map #Curvature #Dimension (graph theory) #Energy (signal processing) #Euclidean geometry #FOS: Mathematics #Geometry #Mathematical analysis #Mathematics #Order (exchange) #Physics #Polynomial #Pure mathematics #Scaling #Statistics #Structural Analysis and Optimization #Transversal (combinatorics) #math.AP
paper · pdf · doi:10.48550/arxiv.1807.02060
published in arXiv (Cornell University) (Cornell University) · Accepted in CPAM 2019
openalex publication_date 2018/07/05 · openalex created_date 2018/07/10 · arxiv created 2019/02/07 · arxiv updated 2019/02/08 · openalex updated_date 2026/07/28
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 [Lewicka-Pakzad, Bhattacharya-Lewicka-Schaffner, Lewicka-Raoult-Ricciotti] 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 [Agostiniani-Lucic-Lucantonio, Schmidt] 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 h2 (Kirchhoff), h4 (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 [Lewicka]. Thirdly, we prove the coercivity inequalities for the singular limits at h2- and h4- scaling orders, while disproving the full coercivity of the classical von Kármán energy functional at scaling h4.