Bifurcation and Deformation in Periodic Pattern Evolution on a Gel Film Bonded on a Soft Substrate

The present study investigates the bifurcation and deformation in periodic pattern evolution on a gel film bonded on a soft substrate. The inhomogeneous field theory for polymeric gels was used in 3D finite element analysis, while the buckling and postbuckling analyses are performed for the periodic pattern evolution from the occurrence of the hexagonal dimple mode at the first bifurcation. The second bifurcation consists of the rectangular checkerboard modes in three symmetric directions. The rectangular checkerboard modes are explained well using the sinusoidal wrinkle modes categorized by m and n (i.e., M=12, the duodecuple bifurcation). The evolution of the hexagonal dimple pattern on the first bifurcated path allows the three rectangular checkerboard modes to occur selectively. The resulting deformation patterns on the second bifurcated paths were in good agreements with experiments, which surprisingly elucidated an analogy with the in-plane buckling behavior of hexagonal honeycombs. The first bifurcation yields the hexagonal dimple structure, while the second bifurcation produces the uniaxial, biaxial and equibiaxial (flower-like) patterns, which not only consist of the periodic arrangements of distorted dimples but also are caused by the three representative combinations of the three rectangular checkerboard modes, Modes I, II and III, respectively. These patterns can appear almost equally. The third bifurcation on the second bifurcated path from Mode I results in the occurrence and evolution of the herringbone pattern. The third bifurcation generates a single bifurcation mode that plays a trigger to cause the coalescence of the distorted dimples alternatively arranged in the vertical direction. The characteristic dimensions of the herringbone pattern are uniquely estimated using the sinusoidal wrinkle modes categorized by m and n, i.e., the ratio of the width and breadth is just the ratio of the dominant wavelengths, 10 and 7.6, for the first and second bifurcations. In addition, the inclination angle is 49°, which is the characteristic angle between the two sinusoidal wrinkle modes needed to express the resulting rectangular checkerboard mode. These dimensional values are considerably consistent with the results obtained by Chen and Hutchinson (2004). They provided an open question in their paper and the present study answered that the herringbone pattern is not a bifurcation mode but just a deformation pattern caused by the three sequential bifurcations. The further bifurcations from Modes II and III lead to the occurrence and evolution of the periodical labyrinth pattern. The third and fourth bifurcations yield the zigzag grooves consisting of the coalescence of the four dimples. Modes II and III provide the different bifurcation modes, which reproduce the different coalescence processes but the resulting deformation pattern is identical as the periodical labyrinth pattern formed by periodic arrangements of the zigzag grooves. It revealed that the occurrence of the rectangular checkerboard modes in the three symmetric directions at the second bifurcation plays the critical role as the missing link in the pattern evolution from the hexagonal dimples pattern to the herringbone and labyrinth patterns.