In MEMS fields, a need arises in engineering practice to predict accurately the nonlinear response of slender post-buckling beams, especially the nonlinear transverse stiffness. The bistability of the post-buckling beams is excellent in reducing power consumption of micro-devices or micro-systems. However, the major difficulty in analyzing the post-buckling and snap-through response is the intractability of the geometric nonlinear control equations of large deflection beams.

I. Arias and M. Arroyo, Size-Dependent Nonlinear Elastic Scaling of Multiwalled Carbon Nanotubes, Phys. Rev. Lett. 100, 085503 (2008).

Size matters for the mechanics of multiwalled carbon nanotubes (MWCNTs). It has been known for some time that MWCNTs often wrinkle under deformation exhibiting the so-called rippling deformation pattern, which makes MWCNTs much softer. Through large-scale multiscale simulations we have characterized with a power law the softer wrinkled response, and showed that the transition strain between the super-stiff behavior attributed to MWCNTs and this softer regime scales as the inverse of the tube diameter. Thus, the tera Pascal Young’s modulus can be fully exploited in devices and materials only for moderately sized tubes. Similarly, in interpreting experiments or designing devices, the classical Euler-Bernouilli beam theory can only be applied to such tubes. The elasticity of thicker tubes is nonlinear, typically display mixtures of wrinkled and unwrinkled sections, and often exhibit hysteretic mechanical behavior.

See http://imechanica.org/node/2395 for a related post.

Many studies on the thin film/substrate structure and its failure mechanism were reported in recent years. The direct experimental results of thin film/substrate structure by scanning electron microscopy (SEM) presents an intriguing problem:there exists a buckling failure mechanism at the lateral edge of metal film under pure bending. The qualitative theoretical analysis has been done on such buckling failure of thin film/substrate structure. The experimental results and theoretical analysis are helpful to understand the extrinsic stresses or deformations that are induced by external physical effects. Accepted by Appl. Phys. Lett.

B. Li, M. K. Kang, K. Lu, R. Huang, P. S. Ho, R. A. Allen, and M. W. Cresswell, Nano Letters 8, 92 -98 (2008). (Web Release Date: 07-Dec-2007; DOI: 10.1021/nl072144i)

Can we derive governing differential equations for bukling of plate and/or shell using variational principles ?

H. Mei, J.Y. Chung, H.-H. Yu, C.M. Stafford, and R. Huang, Buckling modes of elastic thin films on elastic substrates. Applied Physics Letters 90, 151902 (2007).

Two modes of thin film buckling are commonly observed, one with interface delamination (e.g., telephone cord blisters) and the other with no delamination (i.e., wrinkling). Which one would occur for your film?

This Letter gives a quantitative criterion for the selection of the buckling modes. An experiment with a polystyrene film on a PDMS substrate was described showing a transition of the buckling modes.

Appl Phys Lett, 90, 201910, 2007

(J Appl Phys, 100, 114327, 2006) 

R. Huang, Applied Physics Letters 87, 151911 (2005).

The stability of a conductive thin film on a dielectric substrate subjected to a transverse electric field and a residual strain is analyzed. Under a uniform electric field, an equilibrium state exists with a constant thickness reduction of the substrate. The equilibrium state however can be unstable, depending on the intensity of the electric field, the stiffness and Poisson’s ratio of the substrate, and the residual strain in the film. Based on a linear perturbation analysis, the critical condition is determined, beyond which wrinkling of the film is predicted.

Rui Huang and Se Hyuk Im, Physical Review E 74, 026214 (2006).

A stressed thin film on a soft substrate can develop complex wrinkle patterns. The onset of wrinkling and initial growth is well described by a linear perturbation analysis, and the equilibrium wrinkles can be analyzed using an energy approach. In between, the wrinkle pattern undergoes a coarsening process with a peculiar dynamics. By using a proper scaling and two-dimensional numerical simulations, this paper develops a quantitative understanding of the wrinkling dynamics from initial growth through coarsening till equilibrium. It is found that, during the initial growth, a stress-dependent wavelength is selected and the wrinkle amplitude grows exponentially over time. During coarsening, both the wrinkle wavelength and amplitude increases, following a simple scaling law under uniaxial compression. Slightly different dynamics is observed under equi-biaxial stresses, which starts with a faster coarsening rate before asymptotically approaching the same scaling under uniaxial stresses. At equilibrium, a parallel stripe pattern is obtained under uniaxial stresses and a labyrinth pattern under equi-biaxial stresses. Both have the same wavelength, independent of the initial stress. On the other hand, the wrinkle amplitude depends on the initial stress state, which is higher under an equi-biaxial stress than that under a uniaxial stress of the same magnitude.

A recent discussion here about the effect of surface stress on vibrations of microcantilever has gained some interest from our members. A few years ago, Zhigang and I looked at surface effect on buckling of a thin elastic film on a viscous layer (Huang and Suo, Thin Solid Films 429, 273-281, 2003). Although the physical phenomena (buckling vs vibrations) are different, the conclusion is quite consistent with Wei Hong and Pradeep's comments toward the end of the discussion. That is, surface stress only contributes as a residual stress and thus does not affect the buckling wavelength (frequency in space in analogy to frequency in time for vibrations).

A possible explanation of the variety of fingerprints comes from the consideration of the mechanics of tissue growth. Formation of fingerprints can be a result of the surface buckling of the growing skin. Remarkably, the surface bifurcation enjoys infinite multiplicity. The latter can be a reason for the variety of fingerprints. Tissue morphogenesis with the surface buckling mechanism and the growth theory underlying this mechanism are presented in the attached notes.