Tension field theory and a far-from-threshold expansion of Föppl–von Kármán equations: Quantitative approach to pattern formation in stressed sheets
Benny Davidovitch (UMass Amherst)
SES Medal Symposium in honor of D.J. Steigmann
Mon 10:45 - 12:15
MacMillan 115
When confined, thin solid sheets are strongly deformed. Tension field theory, rigorously formulated by Pipkin and Steigmann in the 1980-90’s, posits that the deformations of very thin sheets, often described as “folds” or “wrinkles”, reflect a compression-free state approached at the singular limit of vanishing bending modulus. Hence, standard post-buckling theory, which consists of expansion around the compressed state of the sheet, cannot describe the deformed state of very thin sheets. Recently, we have developed a far-from-threshold (FT) theory for the buckling of very thin sheets [1,2]. In contrast to post-buckling theory, the FT theory is a singular expansion of Föppl–von Kármán equations around a compression-free state. The small parameter here is not the deflection amplitude (as in post-buckling theory), but rather a dimensionless parameter that is proportional to the bending modulus and inversely proportional to the characteristic tensile load. Tension field theory, that captures the macro-scale features of the deformed state (such as the extent of a wrinkled zone), is recovered as the leading order of the FT expansion. The fine features of the pattern (such as the number of wrinkles [1,3], the emergence of wrinkle cascades [4] and of localized, stress-focusing zones [3,5]) are described by the sub-leading order of the FT expansion. We applied the FT theory for experiments in which capillary forces are exerted on floating, ultrathin polymer films [1,3,4]. This analysis shows that the thinner the sheet is, the smaller is the compressive load above which the FT analysis is required, emphasizing its relevance for nanomechanics applications. [1] B. Davidovitch et al., Proc. Nat. Acad. Sci. 108, 18227 (2011) [2] B. Davidovitch et al., Phys. Rev. E, 85 066115 (2012) [3] H. King et al., Proc. Nat. Acad. Sci. 109, 9716 (2012) [4] J. Huang et al. Phys. Rev. Lett. 105, 038302 (2010) [5] R.D. Schroll et al, Phys. Rev. Lett. 106, 074301 (2011)