NDLI: Energy Bounds for a Compressed Elastic Film on a Substrate

Content Provider	Springer Nature Link
Author	Bourne, David P. Müller, Stefan Conti, Sergio
Copyright Year	2016
Abstract	We study pattern formation in a compressed elastic film which delaminates from a substrate. Our key tool is the determination of rigorous upper and lower bounds on the minimum value of a suitable energy functional. The energy consists of two parts, describing the two main physical effects. The first part represents the elastic energy of the film, which is approximated using the von Kármán plate theory. The second part represents the fracture or delamination energy, which is approximated using the Griffith model of fracture. A simpler model containing the first term alone was previously studied with similar methods by several authors, assuming that the delaminated region is fixed. We include the fracture term, transforming the elastic minimisation into a free boundary problem, and opening the way for patterns which result from the interplay of elasticity and delamination. After rescaling, the energy depends on only two parameters: the rescaled film thickness, $${\sigma }$$ , and a measure of the bonding strength between the film and substrate, $${\gamma }$$ . We prove upper bounds on the minimum energy of the form $${\sigma }^a {\gamma }^b$$ and find that there are four different parameter regimes corresponding to different values of a and b and to different folding patterns of the film. In some cases, the upper bounds are attained by self-similar folding patterns as observed in experiments. Moreover, for two of the four parameter regimes we prove matching, optimal lower bounds.
Ending Page	494
Page Count	42
Starting Page	453
File Format	PDF
ISSN	09388974
e-ISSN	14321467
Journal	Journal of Nonlinear Science
Issue Number	2
Volume Number	27
Language	English
Publisher	Springer US
Publisher Date	2016-10-17
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Branching Analysis of microstructure Pattern formation ApplicationMathematics/Computational Methods of Engineering Analysis Theoretical, Mathematical and Computational Physics Classical Mechanics Economic Theory/Quantitative Economics/Mathematical Methods Thin-film elasticity Variational methods including variational inequalities Energy scaling Plates
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Engineering Modeling and Simulation

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