Thanks to Marino, I have found the reason for the difference in our bending stiffness calculation. The original discussion is here:
http://imechanica.org/node/4029

The reason why we have a higher bending stiffness is due to the dihedral term. This dihedral term does have a significant contribution to the bending stiffness. However, in Ref. [26], apparently, this dihedral term was ignored.
I have written a short document showing the contribution of the dihedral term to the bending stiffness. Please take a look at the attachment.
I received great help from Dr. Huang and Marino. Thank you very much.

Qiang Lu and Rui Huang

Department of Aerospace Engineering and Engineering mechanics, University of Texas, Austin,
TX 78712, USA

In this paper we analyze evolution of wrinkle patterns of anisotropic crystal films on viscoelastic substrates. The effects of the residual stress state in the film and the anisotropic elastic property are emphasized. Analytical solutions for the initial growth kinetics and the equilibrium states are presented along with numerical simulations based on nonlinear evolution equations. Compared to wrinkling of isotropic elastic films, more ordered wrinkle patterns are predicted, including orthogonal, zigzag, parallel, and checkerboard patterns. Tranistion of the wrinkle patterns under various stress states is elucidated. Some related experimental works are referred to, but quantitative comparisons between the model the experiments await further studies.

This paper aims to illustrate how anisotropic elastic properties of the crystal substrate affect epitaxial surface evolution and pattern formation. Specifically, for Ge and SiGe films on silicon substrates of various surface orietations, it is shown that the elastic anisotropy plays an important role. However, it must be pointed out that the evolution dynamics of epitaxial surfaces can be much more complicated, due to the combination/competition of various anisotropic properties (e.g., surface energy, surface diffusivity, etc.). Furthermore, for some surface orietations. e.g., Si(111) and Si(113), discrete surface steps play critical roles in the nucleation and growth of epitaxial islands and other surface structures.

Abstract: 

H. Mei, Y. Pang, and R. Huang, International Journal of Fracture 148, 331-342 (2007).

Following a previous effort published in MRS Proceedings, we wrote a journal article of the same title, with more numerical results. While the main conclusions stay the same, a few subtle points are noted in this paper.

First, instead of using the approximate formula by Ye, Suo and Evans (1992), we calculate the energy release rate of interfacial delamination emanating from the channel crack exclusively by the finite element method. We found that the approximate formula is not accurate in several cases.

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)

Attached slides were presented at the 2007 ASME Congress at Seattle.

This paper has been published in Journal of the Mechanics and Physics of Solids 56 (2008), pp. 1609-1623 (doi:10.1016/j.jmps.2007.07.013).

Abstract

Channeling cracks in low-k dielectrics have been observed to be a key reliability issue for advanced interconnects. The constraint effect of surrounding materials including stacked buffer layers has been studied. This paper analyzes the effect of interfacial delamination on the fracture condition of brittle thin films on elastic substrates. It is found that stable delamination along with the growth of a channel crack is possible only for a specific range of elastic mismatch and interface toughness. An effective energy release rate is defined to account for the influence of interfacial delamination on both the driving force and the fracture resistance, which can be significantly higher than the case assuming no delamination.

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.

Y. Pang and R. Huang, J. Applied Physics 101, 023519 (2007).

Surface instability of epitaxial thin films leads to a variety of surface patterns. Anisotropy in surface and bulk properties has profound effects on the dynamics of pattern formation. In this study, we theoretically predict that, under anisotropic mismatch stresses, a bifurcation of surface pattern occurs in addition to generic symmetry breaking from isotropic systems. Numerical simulations based on a nonlinear evolution equation demonstrate pattern selection at an early stage and nontrivial patterns for long-time evolution.

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).

Y. Pang and R. Huang, Physical Review B 74, 075413 (2006).

An epitaxial thin film can undergo surface instability and break up into discrete islands. The stress field and the interface interaction have profound effects on the dynamics of surface evolution. In this work, we develop a nonlinear evolution equation with a second-order approximation for the stress field and a nonlinear wetting potential for the interface. The equation is solved numerically in both two-dimensional (2D) and three-dimensional (3D) configurations using a spectral method. The effects of stress and wetting are examined. It is found that the nonlinear stress field alone induces "blow-up" instability, leading to crack-like grooving in 2D and circular pit-like morphology in 3D. For thin films, the blow-up is suppressed by the wetting effect, leading to a thin wetting layer and an array of discrete islands. The dynamics of island formation and coarsening over a large area is well captured by the interplay of the nonlinear stress field and the wetting effect.