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Research Papers

# Wrinkling Phenomena in Neo-Hookean Film/Substrate Bilayers

[+] Author and Article Information
Yanping Cao

AML, Department of Engineering Mechanics,  Tsinghua University, 100084, Beijing, People’s Republic of China

John W. Hutchinson1

School of Engineering and Applied Sciences,  Harvard University, Cambridge, MA 02138hutchinson@husm.harvard.edu

In passing it is worth mentioning that surface wrinkling strain, $ɛW=0.456$, is also the critical strain for wrinkling localized at the bonded interface between two semi-infinite half-spaces of neo-Hookean materials with different ground state moduli. This result, due to Biot, can be readily appreciated by noting that each half-space undergoes traction-free surface wrinkling at the same strain with arbitrary sinusoidal wavelength. Thus, the two wrinkled half-spaces can be “fit together” satisfying continuity of displacements and tractions.

1

Corresponding author.

J. Appl. Mech 79(3), 031019 (Apr 05, 2012) (9 pages) doi:10.1115/1.4005960 History: Received August 20, 2011; Revised November 28, 2011; Posted February 13, 2012; Published April 04, 2012; Online April 05, 2012

## Abstract

Wrinkling modes are determined for a two-layer system comprised of a neo-Hookean film bonded to an infinitely deep neo-Hookean substrate with the entire bilayer undergoing compression. The full range of the film/substrate modulus ratio is considered from the limit of a traction-free homogeneous substrate to very stiff films on compliant substrates. The role of substrate prestretch is considered wherein an unstretched film is bonded to a prestretched substrate with wrinkling arising as the stretch in the substrate is relaxed. An exact bifurcation analysis reveals the critical strain in the film at the onset of wrinkling. Numerical simulations carried out within a finite element framework uncover advanced post-bifurcation modes including period-doubling, folding and a newly identified mountain ridge mode.

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## Figures

Figure 1

Geometry of the film/substrate system and illustrations of wrinkling modes involving interaction between the film and substrate and a shallow surface mode. The cases considered in this paper all correspond to incremental plane strain bifurcations in the (x1,x2) plane.

Figure 2

Compressive strain in the film at wrinkling, ɛW, as a function of the film/substrate modulus ratio of the two neo-Hookean materials for plane strain compression with no substrate prestretch. The prediction, ɛ0, of the simple formula, Eq. 5, for wrinkling of a stiff linear elastic film on a compliant linear substrate is also shown.

Figure 3

Compressive strain in the film at wrinkling, ɛW, as a function of the film/substrate modulus ratio of the two neo-Hookean materials for plane strain compression showing the influence of plane strain substrate prestretch, λ1s0. The prediction, ɛ0, of the simple formula, Eq. 5, for wrinkling of a stiff linear elastic film on a compliant prestretched substrate is also shown. The range 1≤μf/μs≤5 is magnified in Fig. 4.

Figure 4

Compressive strain in the film at wrinkling, ɛW, as a function of the film/substrate modulus ratio of the two neo-Hookean materials for plane strain compression showing the influence of plane strain substrate prestretch, λ1s0, in the range in which the ratio of the film to substrate modulus is not large

Figure 5

Compressive strain in the film at wrinkling, ɛW, for plane strain compression showing the influence of plane strain substrate prestretch, λ1s0, for neo-Hookean films and substrates that have the same ground state modulus (μf/μs=1). In the range of prestretch, 1<λ1s0<5, the critical mode is not the short wavelength surface mode but rather a mode with wavelength that is long compared to film thickness.

Figure 14

Evolution of wrinkling mode under plane strain compression with a plane strain substrate prestretch λ1s0=2 for μf/μS=836. The sinusoidal wrinkling mode associated with bifurcation at ɛW≅0.01 is stable to ɛ=0.036 but at ɛ=0.0367 a mountain ridge has formed at the right end of the model and the amplitude of the undulations near the ridge have been reduced. By ɛ=0.054 a second mountain ridge is clearly forming near the left end, and by ɛ=0.099 this ridge is fully developed with a third ridge beginning to emerge near the center. At ɛ=0.116 three fully developed mountain ridges have formed and have relaxed the undulation amplitudes between the ridges. The ridges are a form of localization under compression.

Figure 13

Numerical simulations of the post-bifurcation modes for plane strain compression with a plane strain prestretch of the substrate, λ1s0=2. The curve for the bifurcation strain, ɛW, at the onset of sinusoidal wrinkling is that from the theoretical calculation in Sec. 3. For this level of prestretch the secondary post-bifurcation mode is the mountain ridge mode (see Fig. 1) at all values of μf/μS plotted. The compressive strain at its onset is denoted by ɛMR. In the range where μf/μS is not large, mountain ridges form with relatively small additional compression after bifurcation. When μf/μS is large, mountain ridges form at compressive strains that are small and roughly twice ɛW.

Figure 7

Compressive strain in the film at wrinkling, ɛW, as a function of the film/substrate modulus ratio of the two neo-Hookean materials for uniaxial compression showing the influence of substrate prestretch under uniaxial tension. The prediction, ɛ0, of the simple formula, Eq. 6, for wrinkling of a stiff linear elastic film on a compliant prestretched substrate is also shown.

Figure 12

Evolution of wrinkling mode under plane strain compression with a plane strain substrate prestretch λ1s0=1.3 for μf/μS=186. The sinusoidal wrinkling mode associated with bifurcation at ɛW=0.019 is stable to much larger strains (e.g., ɛ=0.278 above). The onset of period-doubling is evident at ɛ=0.314 and is fully developed at ɛ=0.367.

Figure 11

Numerical simulations of the post-bifurcation modes for plane strain compression with a relatively small plane strain prestretch of the substrate, λ1s0=1.3. The curve for the bifurcation strain, ɛW, at the onset of sinusoidal wrinkling is that from the theoretical calculation in Sec. 3. The post-bifurcation behavior with small prestretch is qualitatively similar to the case with no prestretch, although the secondary modes are delayed to larger overall compressive strains. Period-doubling occurs when μf/μs≥20 (see Fig. 1) with folding at smaller values of the modulus ratio.

Figure 6

Dimensionless wave number of the wrinkling mode as a function of the film/substrate modulus ratio of the two neo-Hookean materials for plane strain compression including the influence of plane strain substrate prestretch. The prediction, k0h, from the simple formula, Eq. 5, for wrinkling of a stiff linear elastic film on a compliant prestretched substrate is also shown in the range of large stiffness ratio. The normal deflection of the top surface of the film has the form u2∝cos(kx1) where x1 identifies material point locations in the undeformed film. Other than the limit for μf/μs=1 with λ1s0=1 which is not plotted, the critical mode has a wavelength that is large compared to the film thickness.

Figure 10

Plane strain compression with μf/μs=5 and no substrate prestretch. Bifurcation into the sinusoidal wrinkling mode occurs at ɛ≅0.175. Already at a strain slightly above the onset of bifurcation (ɛ=0.182) the deflection is showing signs that it is evolving away from the sinusoidal mode. At ɛ=0.195 the film deflection has localized into two side-by-side incipient folds. With a slight additional increase in strain (ɛ=0.1954) the fold on the left becomes dominant and a crease has begun to form in the film at the point of maximum local compression. This behavior is representative of bilayers with μf/μs<10 and no prestretch.

Figure 9

Plane strain compression with μf/μs=30 and no substrate prestretch. Bifurcation first occurs as a sinusoidal wrinkling mode (ɛ≅0.055). The sinusiodal mode is stable to much larger strains (e.g., ɛ=0.18). At a compressive strain of approximately ɛ=0.2 a secondary bifurcation occurs corresponding to the onset of period doubling. With a slight further increase of compression to ɛ=0.226 the period-doubling mode is firmly established. This behavior is representative of bilayers with μf/μs>10 if there is no prestretch.

Figure 8

Post-bifurcation behavior in the neo-Hookean bilayer under plane strain compression with no substrate prestretch computed using the finite element model. The critical compressive strain at the onset of wrinkling, ɛW, is the theoretical result of Sec. 3. The compressive strain at the onset of two distinct post-bifurcation modes is also indicated. For μf/μs≥10, period-doubling (see Fig. 9) is the first post-bifurcation mode encountered at strain ɛPD. For μf/μs<10, the post-bifurcation mode is a fold that occurs at a strain denoted by ɛF that is only slightly greater than ɛW. It subsequently develops a local crease (see Fig. 1).

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