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c.Ascompared with the geometrical nonlinearity, the constitutivemodelof the substrate plays a relatively minor role on the prestrain-dependent wavelength and amplitude. Other constitutive models(e.g., linear relation between the second Piola–Kirchhoff stress andGreen strain) that have the same linear elastic properties giveessentially the same prestrain dependence of the wavelength. Forthe prestrain pre 28.4%, the present model gives the wavelength12.9 m, which agrees satisfactorily with the experimentally mea-sured wavelength 12.1 0.4 m. Without accounting for thegeometrical nonlinearity, the wavelength would be 14.6 m, whichshows the dominant effect of geometrical nonlinearity at the largestrain.The amplitude, according to the finite deformation theory, isA A0 1 pre 1 1/3, [4]whereA0 is the amplitude in Eq. 2.As shown in Fig. 4, the amplitudegiven by the Eq. 4 expression agrees well with the experimentaldata, whereas the amplitude A0 clearly overestimates. Similar to Eq.3, an intuitive understanding of Eq. 4 is as follows: A0/ 1 prerepresents the change of amplitude expected based on simpleaccordion bellows mechanics; 1/(1 )1/3, which depends only onprestrain, results from the geometrical nonlinearity and nonlinearconstitutive model in the substrate.Membrane and Peak Strains in the Thin Film. The new theory and theaccuracy with which it reproduces experimental observation pro-vide opportunities to reexamine the nature of strains and displace-ments in buckled systems. For pre c ( 0.034%for the Si/PDMSsystem), relaxing the prestrain does not lead to buckling. Instead,the film supports small compressive strain (0) that is very close to pre, which we refer to as membrane strain.
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When pre
c, thefilm buckles to relieve some of the strain; the membrane strain, mem, as evaluated at the plane that lies at the midpoint of thethickness of the film, remains and has a magnitude almost equal to c. The peak strains peak in the film are equal to the sum ofmembrane strain mem and the strain induced by the buckledgeometry. In most cases of practical interest, the strain associatedwith the buckled geometry is much larger than mem, thus this peakstrain can be written peak 2 pre c 1 1 3 1 pre. [5]The magnitude of peak is typically much smaller than the overallstrain, pre mem, that the film accommodates by buckling. Forexample, in the case of pre 28%, peak is only 1.8%for the systemof Fig. 2. This mechanical advantage provides an effective level ofstretchability/compressibility in materials that are intrinsically brit-tle. As a result, peak determines the point at which fracture occursin the film. For Si, the fracture strain is in the range of fracture (foreither compression or tension). The maximum allowable prestrainis, therefore, approximately fracture24 c 1 4348 fracture24 c ,which, for the system examined here, is 37% or almost 20 timeslarger than fracture.Fig. 5A shows the peak and mem as a function of pre. Themembrane strain is negligible compared with the peak strain.Likewise, the peak strain is much smaller than the prestrain, suchthat the system can accommodate large strains. The measuredcontour length of the buckled Si film on PDMS substrate, shown inFig. 5B, is approximately constant and is independent of theprestrain. This result is consistent with a negligibly small value for mem.Postbuckling Behavior. When the buckled system is subjected to anapplied strain applied, the wavelength and amplitude become 0 1 applied 1 pre 1 applied 1/3,A h pre applied / c 1 1 pre 1 applied 1/3,[6]where 5( pre applied)(1 pre)/32. The amplitude A vanisheswhen the applied strain reaches the prestrain plus the critical strain c. In this situation, themembrane strain is equal to c. Additionalapplied strain relaxes the membrane strain and then, ultimately,appears as tensile strain in the silicon up to the point of fracture.The peak strain in the film is peak 2 pre applied c 1 applied 1/3 1 pre. [7]01230 10203040051015A λ Experiment Finite-Deform. Previous Model Accordion Model e d u t i l p m A A ( µ ) m h t g n e l e v a W λ ( µ ) mPrestrain εpre (%)Fig. 4. Wavelength and amplitude of buckled structures of Si (100 nmthickness) on PDMS as a function of the prestrain. The finite-deformationbuckling theory yieldswavelengths and amplitudes that both agreewellwithexperiments. Also shown are results from previous mechanics models (i.e.,small deformation limit) and the simple accordion model. Fig. 6 gives the experimentally measured and theoretically pre-dicted wavelength and amplitude A versus applied strain appliedfor a buckled Si thin-film/PDMS substrate formed with a prestrainof 16.2%, and other parameters the same as those of the examplesdescribed in the other sections. The constant wavelength and theamplitude predicted by the existingmechanicsmodels, given by Eq.2 with pre replaced by pre applied, are also shown.