Imechanica Journal Club: “The Ruga Mechanics”
Prepared by Mazen Diab, Ruike Zhao and Kyung-Suk Kim
Brown University, Providence, RI
Introduction
The ‘ruga’, a Latin term, means a single state of various corrugated material configurations, which form diverse 2D-patterns on solid surfaces, interfaces and in thin films. Typical ruga configurations include large-amplitude wrinkles, creases, folds, ridges, wrinklons, crinkles and crumples. While rugae (plural of ruga) in Latin originally meant variety of corrugated configurations, it has been narrowly used in anatomy (Wikipedia). In a broad sense, ruga structures are widely observed in Nature. They are on human, animal and plant skins, in the wings of insects, the surface of our brains, even the crust of the earth, the moon and the planets (Figure 1).
Ruga configurations have distinct geometrical characteristics, and each of them stands for an energetically monomorphic equilibrium configuration representing a single phase of ruga. Over the past half century important ruga phases have been identified – Biot for wrinkle [1], Gent and Cho for crease [2], Brau, et al. for period multiplication [3, 4], Sun, et al. & Pocivavsek, et al. for fold [5, 6], Cao and Hutchinson for ridge [7], Bowden, et al. for 2D wrinkles [8], Efimenko, et al. for hierarchical wrinkles [9], Ahmed, et al. for 2D ruga tessellation [10]. Diab, et al. recently constructed a comprehensive ruga phase diagram of a stiff surface boundary layer in a neo-Hoookean solid (figure 2B) [11, 12]. They showed that, similar to conventional material phase transitions for which a thermodynamic potential typically depicts the behavior, ruga phase transitions are naturally described by bifurcation analysis with a mechanical (structural) potential. Construction of iso-periodic [11, 12] and self-selective ruga phase diagrams [13] will not only enable analysis for coexistence, scaling or localization of multiple ruga phases, etc., but will also provide understanding on meso-scale self-organizing mechanisms that can be controlled by large-scale field parameters.
Recently, importance of ruga mechanics in science and technology appealed to public – NPR and Discover Magazine. In an application point of view, there has been an increasing interest in developing new classes of multi-functional soft composite materials at various length scales. For example, since nano-science has advanced to produce various 2D materials beyond graphene [14-18], we expect that nano ruga-structures in such 2D materials can create unprecedented functional properties. For examples of relatively large-scale engineering applications, the ability to control ruga patterns makes soft composites useful in a wide range of applications such as soft robotics, flexible electronics, constructing artificial skins, and morphing aircrafts and marine structures [19-34]. In addition, principles of ruga phase formation will uncover important self-organizing phenomena in brain science [35-37], medical science [38, 39] and geo science [40-43].
Figure 1: Ruga phases in nature and studies: a) scheme of ruga phases in 1D structure; b) 50nm gold film folds: blue-red: 0-2 MPa Mises stress; c) 2D ruga pattern on 10 nm ion beam implanted DLC on PDMS; d) graphene folds, on PDMS (top & side views); e) graphene nano sacks with “cargos”; f) 2D ruga patterns on an apple; g) brain cortex structure; h) 3D mapping of earth; i) 2D ruga tessellation and localization on the moon; j) carbon disk (top left image) and free-standing hollow carbon nanocones; k) paper folding; l) nanoribbons for high Memory Storage Density; m) folded sedimentary layers of Sullivan River, British Columbia.
Mechanics of major ruga phases and their transitions:
Wrinkle:
Over the past several decades, mechanics research revealed that wrinkle has the following primary characteristics: 1) critical wavelength and critical strain, 2) distinct material property dependence, 3) weak imperfection sensitivity and stable amplitude growth. The research provided relatively simple relation among the critical wavelength, stiffness ratio and thickness. However, dependence of the relationship on material properties of the film and the substrate, and loading conditions including pre-stretch and finite deformation has been steadily and extensively investigated [1, 42, 44-64]. While the onset criticality of the wrinkling was analyzed by first-order perturbation, growth rate of the wrinkle amplitude and imperfection sensitivity of the criticality required higher-order-perturbation post-buckling analysis typically with the Koiter method [11, 12, 65-69]. The wrinkle characteristics of weak imperfection sensitivity and stable amplitude growth have been attractive features of wrinkle for experimental measurement of the property of the film relative to that of the substrate or vice versa [70, 71]. In addition, hierarchical wrinkles with their frequencies clustered in broad scales have been observed, and cause of the clustering has been debated [4, 9].
Crease:
In contrast to wrinkle, crease has three distinct characteristics: localization of deformation, snap buckling with strong imperfection sensitivity and global irreversibility of deformation. For the first aspect, Gent & Cho [2] discovered the criticality and the mode of the crease localization. Later, Diab et al., [11, 12] uncovered that the crease localization is caused by cascade instability either from a fundamental state (instantaneous crease) or from a wrinkle state (setback crease). For the second aspect, a series of investigations [72-76] revealed that snap buckling of creasing is a consequence of bifurcation onto a branch of solutions extended from a subcritical state to a supercritical state. For the third aspect, the global irreversibility exhibits hysteresis in loading and unloading despite that the local deformation is elastic (reversible) everywhere [12, 74]. In addition, it is found that creasing is sensitive to imperfections [69].
Fold:
When wrinkle amplitude grows large, the wrinkles often forms folds with high aspect ratio amplitude for self-contact, period multiplication of the wrinkle to set the fold periodicity, and subsequently leads to fold localization. The folds have attractive features for nano-manufacturing of surface nanostructures such as subsurface nano channels. Diverse characteristics of folding processes have been observed and investigated by several research groups. Those include folding of an elastic sheet floating on water [5], period multiplication of a wrinkle on a bilayer system, formation of periodic folds [6], folding of a graphene sheet on a PDMS [77], etc. The role of substrate nonlinearly in the post-buckling evolution of wrinkle has been revealed numerically by Sun, et al. [6] for fold, and further investigated by Hutchinson (?) [68] and Zhao, et al. [13]. However, a quantitative characterization of the critical conditions for the large amplitude wrinkle evolution is still elusive due to lack of analytical expressions of the wrinkle configuration with nonlinear finite deformation at the onset of period multiplication. Once the wrinkle surfaces make self-contacts for folding, some selective fold tips begin to advance into the substrate with a series of unfolding of nearby folds, leading to fold localization.
Ridge:
Ridge is a ruga phase of localized bulge that exhibits mechanical characteristics similar to those of crease for localization and irreversible mode of hysteresis behavior. It is typically observed in a surface layer compressed by releasing a large pre-stretch strain. For example, depending on the level of pre-stretches of an incompressible neo-Hookean substrate, the wrinkle generally bifurcates to a single isolated ridge followed by development of periodic ridges through frequency and period multiplications. Ridges are formed through snap buckling and the configurational evolution is irreversible in general. Ridges often grow into ridge-folds at large compression of the film. ([13, 32, 69, 78, 79])
Delamination ruga:
Delamination ruga develops when a thin surface-mounted film with is laterally compressed to be debonded. Blistering and telephone cord buckling of a thin film mounted on a stiff substrate was extensively studied in 1990s. In contrast, when a stiff film is delaminated through wrinkling on a soft substrate, the delamination process develops localized delamination ridges. The delamination ridges often transforms into delamination folds under large strain mismatch, typically observed in CVD grown films of graphene caused by large thermal strain mismatch. Isolation of bending localization on the debonded part of the film has attracted attention for potential use of delamination ruga in flexible electronics applications. Recently a delamination ruga phase diagram was proposed. ([80-86])
Two-dimensional Ruga:
Experimental observations of various 2D ruga phases such as periodic checkerboard, herringbone, labyrinth, crease and fold have been reported [8, 87-93]. Recently Ahmed et al., [10] showed that for a strain larger than 30 % the folds evolve in random orientations to create asymmetric disordered tessellation. Wrinkle instability of hyper-elastic halfspace under general biaxial compression wasstudied by Nowinski [94], and Usmani and Beatty [95]. However, it was not until recently that the community started to investigate the formation of the various 2D ruga phases of bilayer systems using numerical and/or analytical approaches [55, 96-105]. Most all of these studies haven’t considered the full sequential bifurcations of the various patterns at large strains. Many issues are still unexplained such as the disordered tessellation observed in [10], the labyrinthine pattern, mode jumping, coexistent of states, and the role of imperfections on the formation of the ruga patterns. More efforts are needed to develop analytical and numerical tools that are capable of handling the formation of the complex 2D ruga phases and transitions among them.
Figure 2: A) FEM results of 1D ruga phases; B) ruga phase diagram of graded material; C) various ruga phases delineated by the surfaces of nonlinear bifurcations; D) Hysteresis loops of creasing.
Applications of ruga mechanics
Typically ruga structures are formed with large configuational change which requires high compliance of the constituents at least at a reduced dimension. Such compliance is commonly encountered in a very high aspect ratio thin film structures, soft materials or very large-scale slender structures. Therefore, ruga mechanics is expected play important roles in nano, bio and geo science and technology.
Advanced materials research and development:
Ruga structures provide periodic symmetry-breaking sites where diverse quantum states can be generated in 2D materials. In turn, collective behavior of the quantum states offers new functional properties of the material, such as properties of electron transport and/or molecular-level reactivity. Understanding and controlling of such material properties will make quantum leaps in advancing nano science and technology for diverse applications like multifunctional electronics including nanoscale spintronic and flexible nano-electronics, optoelectronics of meta-materials.([16-18, 21, 22, 28, 31, 84, 106-114])
Bio science research and development:
Among many potential applications of ruga mechanics in bio science and technology, we have many examples of application, such as hierarchical ruga for bio adhesion control with super hydrophobic/hydrophilic surfaces including anti-biofouling, wrapping ruga of 2D materials for drug delivery, general ruga phases for bio-inspired technology, reversibility of sulcus ruga deformations for studying TBI, growth ruga for evolution of brain structures, and crinkle ruga for bio-informatics with dry sequencing of DNA or protein structures. ([29, 30, 38, 39, 113, 115-119])
Geo science research and development:
Ruga mechanics will play significant roles in studying geological wrinkles, folds and ridges, including microbial shape analysis of sedimentary wrinkle structures, ruga localization of folding for evolution of thrust faults, interface ruga developed by gravity in the earth, the moon and the planets, multi-layer ruga for evolution of mineral deposition, ruga structures in volcanic lava flow, etc. Recently, the GRAIL project of sensing gravitational field gradients revealed many interesting ruga structures in the earth, the moon and the planets. ([40-43])
Computational science research and development:
Much of recent progress in ruga mechanics research was possible thanks to advancement of computational mechanics, in particular, finite element method and hybrid computational techniques for multi-scale analysis. However, we still have lots of challenge in developing reliable numerical methods for critical bifurcation processes with multiple quasi- convexities in the potential, for dynamics of ruga deformations and for clustered wrinkling in broad scales, for scaling analysis of co-existing ruga distribution, for motion analysis of wrinklon rugae [120-122].
Experimental method research and development:
Experimental frequency control of ruga structures over an appreciable area is still a great challenge. In particular, nano-imprinting, nano-lithography, wetting and adhesion control with ruga structures are important experimental technology to be developed further. Material property characterization with ruga mechanics is a good experimental technique to be developed for materials in extreme dimensions or under extreme conditions. Experimental ruga dynamics is another challenging area of research.
In this journal club of imechanica of this month, we wish that further issues in ruga mechanics can be discussed actively, including more mathematical issues.
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