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JClub June 2010: Geometric Nonlinearities in Thin Elastic Objects
Welcome to this month’s (June) Journal club, where I propose a discussion on the elasticity of thin objects (rods, plates and shells). It is a field with a long history that is being revived and rapidly burgeoning in new contexts. It is bringing together seemingly separate communities ranging from structural mechanics, statistical physics, differential geometry and nanotechnology. The main source of the complexity in the predictive understanding of thin structures is that large displacements can give rise to non-negligible geometric nonlinearities during their deformation, even if its material properties remain linear. Moreover, coupling the elasticity of thin objects with phenomena such as fracture, flow, surface tension and adhesion at solid interfaces, to name a few, raises new fundamental problems arising frequently in nature and technology.
The elasticity of thin rods, sheets or shells is highly nontrivial both in terms of their static configurations and dynamics . Examples of thin membranes and filaments in biology are far more numerous than predictive explanations of their mechanical behavior. Also, it is thought that mechanical stresses play a significant role in some instances of morphogenesis in thin membranes of animal  and plant  tissue. Furthermore, in technology there is an increasing presence of thin elastic films undergoing large deformations. This upsurge of interest has been largely due to the development of novel micro-fabrication techniques to produce ultra-thin materials; the utmost examples being carbon nanotubes  and graphene  but there are many others. By way of illustration, knowledge of the detailed mechanics of thin materials is paramount in the emerging field of stretchable electronics [10,11], amongst many other applications.
T. A. Witten, "Stress focusing in elastic sheets" Rev. Mod. Phys. 79, 643 (2007).
E. Cerda and L. Mahadevan, "Conical Surfaces and Crescent Singularities in Crumpled Sheets", Phys. Rev. Lett. 80 2358 (1998).
One of the first problems regarded in this spirit was the description of a crumpled elastic sheet and its singularities . Crumpled paper has become a canonical example of the high degree of complexity that can arise in this class of systems. For example, a theoretical description that predicts the shape of regions of stress and strain localization along singular points (developable-cones) and lines (ridges) has only recently been attained [1, 2, 11]. A developable-cone (d-cones) is a conical singularity obtained when an initially flat plate is pushed with a point load into a cylindrical annulus. Due to the excessive cost of in-plane stretching energy (~Eh, where E is the Young’s modulus of the material and h is the sheet’s thickness) compared with bending energy (~Eh^3), the plate deforms out of plane. It becomes only partially in contact with the annulus in a nonaxisymmetric way. An informative schematic diagram of the process is provided in the seminal paper of Cerda and Mahadevan . Based on a coupling between elasticity (in-plane stresses and bending) and geometry (gaussian curvature) the authors provide an inextensible analytical solution for the shape of a d-cone that characterizes the singularity far from the tip. Finally, they provide a scaling for the size of the core where stretching, plasticity, and other nonlinear effects that may be material dependent become important. An extensive discussion of this and other type of singularities (e.g. linear ridges) is presented in a recent review article by Tom Witten .
Another system in which large deformation of thin sheets can give rise to localization occurs when a thin sheet is laid flat on a air-water interface and an uniaxial load is then applied at two of its sides . Upon compression, the sheet initially buckles in an extended wrinkling configuration with a wavelength that is set by a balance between bending energy of the sheet against the hydrostatic loading of the liquid foundation. This is the first-order linear response of the membrane. Under further compression, the wrinkles develop into a single localized fold which involves highly localized curvature and breaks up-down symmetry. Here, nonlinear geometric effects become non-negligible. Scaling laws for the length-scales for both the wrinkling and localized states are given in this paper. The authors suggest that this wrinkling to fold transition is important in the context of biological membranes such as lung surfactant and nanoparticle thin films.
In the last topic I would like to discuss two technological instances where the large deformation of thin films is important.
The first appears in the context of stretchable electronics . When depositing electronic circuitry on a flexible substrate, a major challenge is primarily mechanical and involves the loading of the conducting elements. If these are laid down flat on the substrate (as is common in standard PCB boards) the deformation will often induce fatigue or fracture of the wires due to large in-plane stresses. Instead, Sun et.al.  propose one strategy is to dispose the conducting elements in an array of out-of-plane delaminated ‘bumps’. The deformation will be accommodated through low energetic cost in bending of the thin elements, making the whole system much more compliant (hence, stretchable). Again, understanding the mechanics of these systems involves large deformation of thin plates.
Finally, a recent paper  has shown that the conical singularities discussed above are relevant for in the deformation of graphene sheets. Graphene, the thinest of thin sheets, consists of a single atomic layer of carbon atoms disposed in a hexagonal lattice and has recently been receiving much attention due to its novel electronic properties . Its mechanical properties are also gaining increasing focus. For example, graphene is know to have the the largest in-plane Young’s modulus of all materials . Pereira et. al.  explore the coupling between the geometry set by the thin-sheet mechanics with the electronic structure and transport properties. The authors suggest that the electronic properties of graphene sheets can be engineered by exploiting localized structures such as those mentioned above. Testing the degree of validity of continuum formulations at these small scales remains an open question.
The deformation of thin objects has a long history and has been addressed by many giant mechanicians including Euler, Föppl, von Kármán and Timoshenko. However, new technological advances in microfabrication and increasing interest in thin biological membranes make the revival and further development of this topic both relevant and a timely effort.
 E. Brouze ́s, W. Supatto, and E. Farge. "Is mechano-sensitive expression of twist involved In mesoderm formation?" Biology of the Cell 96, 471 (2004).
 M. J. Jaffe and S. Forbes "Thigmomorphogenesis: the effect of mechanical perturbation on plants” Plant Growth Regulation 12, 313 (1993).
 S. Iijima and T. Ichihashi, "Single-shell carbon nanotubes of 1-nm diameter" Nature 363, 603 (1993).
 K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, and A.A. Firsov, “Electric Field Effect in Atomically Thin Carbon Films” Science 306, 666 (2004).
 D.-Y. Khang, H. Jiang, Young Huang, J. A. Rogers, "A Stretchable Form of Single-Crystal Silicon for High-Performance Electronics on Rubber Substrates" Science 311, 208 (2006).
 J.A. Rogers, and Y. Huang, "A curvy, stretchy future for electronics" Proc. Natl. Acad. Sci. U.S.A. 106, 10875 (2009).
 E. Cerda, S. Chaieb, F. Melo and L. Mahadevan, "Conical dislocations in crumpling", Nature 401, 46 (1999).
 C. Lee, X. Wei, J.W. Kysar and J. Hone "Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene" Science 321, 385 (2008).