Wrinkle to fold transition: influence of the substrate response

Abstract

Spatially confined rigid membranes reorganize their morphology in response to imposed constraints. Slight compression of a rigid membrane resting on a soft foundation creates a regular pattern of sinusoidal wrinkles with a broad spatial distribution of energy. For larger compression, the deformation energy is progressively localized in small regions which ultimately develop sharp folds. We review the influence of the substrate on this wrinkle to fold transition by considering two models based on purely viscous and purely elastic foundations. We analyze and contrast the physics and mathematics of both systems.

The role of substrate pre-stretch in post-wrinkling bifurcations

“…Over this range of strain, the energy U 3 remains above both U 2 and U 4 , and thus period tripling is energetically disfavored. These findings are in excellent agreement with our experiments at λ 0 = 1.0, as well as previous work, showing clean period doubling and period quadrupling bifurcations [32,[37][38][39] .…”

“…For an elastomeric foundation, wrinkles lose their initial periodicity through the emergence of sub-harmonic modes due to nonlinear contributions to the elastic response of the substrate [36,37] . Typically, a progression from wrinkles to a period-doubled state is observed, followed by a period-quadrupled state, and finally the formation of selfcontacting folds [32,[37][38][39] . The emergence of sub-harmonic spatial modes in this context is analogous to appearance of temporal sub-harmonics for a non-linear oscillator [38] .…”

Elastic membranes in confinement

“…We then calculate the membrane response ( p, b) to increased length S. The second assumes the membrane tension is a material parameter (fixed) and the system is 'open' to allow fluid transport through the 'intermembrane space'. It is clear from the comparison of our results that the second scenario most probably governs the mechanics of the inner membrane, whereas the first scenario is potentially relevant to other applications such as wrinkling [26] and folding [27,28] in elastic sheets.…”

Surface patterns in drying films of silica colloidal dispersions

“…Folds, described as a secondary instability arising from wrinkles 8 , present strong similarities such as the localization of the deformation, the subdivision of domains and the crossing patterns 11 . However, the observation of wrinkles is expected but it has never been observed in our systems.…”

“…Beyond a critical strain, it results in a periodic sinusoidal deformation of the interface [3][4][5][6] for which the periodicity has been derived theoretically 2,7 . When the strain is further increased, a secondary instability occurs leading to wrinkle-to-fold transition 8,9 ; this results in a localization of the deformation 10 . Under a biaxial compressive stress, it has been shown that a repetitive wrinkleto-fold transition produces a hierarchical network of folds 11,12 .…”