Benjamin Davidovitch, University of Massachusetts
Thin solid sheets exhibit a rich variety of patterns under generic confinements and featureless distribution of forces. This morphological diversity reflects the coupling between geometry and mechanics in elastic sheets, a problem whose foundations had been set many years ago. This field has seen recently a surge of research activity, driven by studies that demonstrated its relevance for morphogenetic processes, and by developing material applications at ever decreasing scales that enable the mechanical control of tiny structures. A major theoretical challenge posed by this progress is the development of a formalism that explains the basic mechanisms for pattern formation in thin sheets under various loadings and confinements. The hurdle here is associated with the highly nonlinear nature and the geometric complexity of the problem when the sheet thickness becomes exceedingly small. This complexity requires the development of new concepts and methods, beyond traditionally used ones.
During 2011-2012, the ACS-PRF grant supported our theoretical studies in this field. In particular, we continued to develop the “far-from-threshold” formalism of wrinkle pattern, and applied it to studies of ultrathin polymer films on curved liquid interfaces. Furthermore, we developed a theory that “regularizes” the singular nature of wrinkle patterns, making it a quantitatively predictive tool.
1. Ultrathin sheet on a liquid drop:
The adhesion of a solid sheet onto a curved substrate underlies numerous biomechanical phenomena and a broad range of technologies, from the production of a hemispherical electronic eye to wear-resisting coating of joint implants. Similarly to the unavoidable distortion of distances in planar maps of earth, solid sheet must develop elastic stress when attached to a curved substrate. The inevitable stress may relax through several types of instabilities: Delamination (blistering) of the sheet may occur if the substrate is highly stiff, wrinkling may occur on a sufficiently soft substrate, whereas a crumpled shape that involves a substantial deformation of the spherical substrate may occur if the imposed curvarture is sufficiently large. Despite the rich phenomenology and technological importance of this problem, numerous basic questions have not been addressed yet: What are the general conditions under which blisters, wrinkles and crumples emerge? Can these (and possible other) modes of deformation coexist in the same pattern? Can delamination be avoided if the curved substrate is sufficiently soft? Can a crumpled shape appear also on a stiff, spherically shaped substrate?
In a recent work “Elastic sheet on a liquid drop reveals wrinkling and crumpling as distinct symmetry-breaking instabilities” King et al., PNAS 109 9716 (2012)), we commenced an attack of these questions. This work, which was also featured on PNAS cover, described an experimental-theoretical study of a circular ultrathin sheet placed in a liquid drop. Upon a gradual increase of the drop’s curvature (by increasing the Laplace’s pressure inside the drop), the sheet develops radial wrinkles, which give way to sharp folds (dubbed “crumples”). Our theoretical work revealed the dimensionless groups (called “bendability” and confinement”) that govern these morphological transitions of the system. To the best of our knowledge, this experiment provides the first conclusive evidence for the validity and relevance of the “far-from-threshold” theory for elasto-capillary phenomena in ultra-thin sheets. The theoretical development was the work of former postdoc R. Schroll (supported by ACS-PRF in Sep-Dec 2011).
Following up on this work we commenced a systematic program that addresses the interplay between wrinkling, crumpling and delamination through a simple-yet-nontrival model system. This work is in collaboration with postdoc E. Hohlfeld, and is using previous results by former postdoc P. Buchak (Buchak was supported by ACS-PRF in 2011, Hohlfeld was supported in 2012 by NSF, and will be supported in 2013 by ACS-PRF). The model consists of a thin sheet that adheres to a spherically-shaped, solid-like substrate characterized by local (so-called Wrinkler’s) response to deformations. In addition to bendability and confinement, this model is characterized by an additional dimensionless group (called “deformability”) that encapsulates the effects of the stiff substrate. A fascinating prediction of our preliminary studies is the suppression of delamination of ultarthin sheets from highly stiff substrates. This prediction, which points to applicative methods for the efficient suppression of delamniation, was obtained through asymptotic analysis of one “corner” of the deformability-bendability- confinement parameter space. We will continue to explore the rich phenomenology of this system through asymptotic analysis of other regimes of this parameters space.
2. Regularizing the singular wrinkle theory.
Our previous work on the ``proto-typical model for tensional wrinkling in thin sheets” (Davidovitch et al., PNAS 108 18227 (2011), partially supported by ACS-PRF) identified two parameter regimes that exhibit wrinkle patterns of markedly different features: These were called ``near-threshold” (NT) and “far-from-threshold” (FFT). The conceptual difference between these patterns, which underlie numerous morphological features, is related to the compression in the sheet: finite in the NT regime, but vanishes from the sheet as its thickness decreases (in the FFT regime).
The distinct nature of the NT and FFT expansions raises a natural question: How does a wrinkle pattern transform from NT behavior to FFT behavior (Eq. 5) when the confinement imposed on the sheet is smoothly increased? In a recent work we addressed this question through a non-perturbative wrinkling model that recovers both NT and FFT limits. This non-perturbative theory addresses the high bendability regime, and is markedly different from standard methods that describe the post-buckling state through coupling between several unstable modes. A central outcome of the analysis of the non-perturbative wrinkling model was the regularization of the singular FFT expansion. The original derivation of the FFT expansion assumed a “pointwise matching” between the compression-free (wrinkled) zone and the unwrinkled region (in which both stress components are tensile). This assumption was shown to give rise to spurious divergence in the wrinkle energy. Our analysis of the non-perturbative model resolves this problem, showing that the transition between the wrinkled and unwrinkled zones occurs through a “compressional annulus”. We expect that this regularization mechanism underlies wrinkle patterns of finite extent. This theoretical work was in collaboration with former postdoc R. Schroll (supported by ACS-PRF in Sep-Dec 2011).