Finite deformation mechanics in buckled thin filmson compliant supportsHanqing Jiang*, Dahl-Young Khang†, Jizhou Song‡, Yugang Sun§, Yonggang Huang¶ , and John A. Rogers†‡***Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287;†Department of Materials Science and Engineering,Beckman Institute, and Seitz Materials Research Laboratory, University of Illinois at Urbana–Champaign, Urbana, IL 61801;‡Department of MechanicalScience and Engineering, University of Illinois at Urbana–Champaign, Urbana, IL 61801;§Center for Nanoscale Materials, Argonne National Laboratory,9700 South Cass Avenue, Argonne, 43860
L 60439; and ¶Department of Civil and Environmental Engineering and Department of Mechanical Engineering,Northwestern University, Evanston, IL 60208Edited by John W. Hutchinson, Harvard University, Cambridge, MA, and approved August 3, 2007 (received for review March 29, 2007)We present detailed experimental and theoretical studies of themechanics of thin buckled films on compliant substrates. In par-ticular, accuratemeasurements of thewavelengths and amplitudesin structures that consist of thin, single-crystal ribbons of siliconcovalently bonded to elastomeric substrates of poly(dimethylsi-loxane) reveal responses that include wavelengths that change inan approximately linear fashion with strain in the substrate, for allvalues of strain above the critical strain for buckling. Theoreticalreexamination of this system yields analytical models that canexplain these and other experimental observations at a quantita-tive level.We showthat the resultingmechanics hasmany featuresin common with that of a simple accordion bellows. These resultshave relevance to the many emerging applications of controlledbuckling structures in stretchable electronics, microelectrome-chanical systems, thin-film metrology, optical devices, and others.buckling   stiff thin film   compliant substrate   stretchable electronicsNonlinear buckling of thin, high modulus plates on compliantsupports represents a classic problem in mechanics. Over thelast several decades, numerous theoretical and experimental studiesof this phenomenon have been performed (1–17). Although buck-ling has historically been viewed as a mechanism for structuralfailure, pioneering work in the late 1990s (18) showed that thisbehavior can be controlled in micro- and nanoscale systems togenerate interesting structures with well defined geometries anddimensions in the 100 nm to 100  m range. These observationscreated renewed interest in this area that persists today, with manyactive research groups currently exploring basic scientific aspects aswell as applications in stretchable electronics (10–14, 19), micro-and nanoelectromechanical systems (MEMS and NEMS) (20),tunable phase optics (1, 21), force spectroscopy in cells (22),biocompatible topographic matrices for cell alignment (23, 24),high-precisionmicro- and nanometrologymethods (15–17, 25), andpattern formation for micro/nanofabrication (18, 26–31). In thesesystems, controlled buckling is realized in thin films deposited,typically by vapor phase or physical transfer processes, onto pre-strained elastomeric substrates. The prestrain is usually generatedby one of two methods. The first involves thermally expandedelastomeric substrates, where the strains are on the order of a fewpercent (1, 10, 15–17).Depositing a filmonto a heated substrate andthen cooling the systemlead to compressive strains in the filmwhenits coefficient of thermal expansion is smaller than that of thesubstrate. Sufficiently large compression leads to buckling instabil-ities in the film that create ‘‘wavy’’ deformations (i.e., periodic,out-of-plane displacements of the film and surface region of thesubstrate). The second method uses mechanically stretched elas-tomeric substrates where the strain associated with the stretching(i.e., the prestrain) can range from a fraction of a percent to a fewtens (1, 19) and even a few hundreds of a percent (32). Depositinga film on this stretched substrate and then releasing the prestraincan create wavy structures. The designs of these systems can rangefrom simple layouts consisting of uniform films on f lat substrates(16, 18, 21) to complex lithographically patterned films on sub-strates with structures of relief embossed on their surfaces (18, 33,34). The persity of wavy geometries enabled by these strategiescreates considerable engineering f lexibility in the types of structuresthat can be formed. A fundamental understanding of the physics isimportant because it can provide a foundation for developing routesto exploit this behavior in fields ranging from biology to nanoscalemetrology to unusual electronics.This article focuses on a class of system whose fabrication isillustrated in Fig.