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Dynamics of wrinkle growth and coarsening in stressed thin films
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.
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Kinematics
Hi Sehyuk,
Very interesting paper! I have a question regarding kinematics. You consider nonlinear strain-displacement relationship for the film and the linear one for the viscoelastic layer (if I am not missing something). Is that compatible? I mean that the particles on the boundary between the film and the layer should deform similarly on both sides of the boundary.
Regards,
Kosta
compatibility or continuity is the key
Dear Kosta:
Thanks for your comment on our work. You are right that the kinematics must be compatible across the boundary (interface) between the film and the viscoelastic layer. In fact, this is the key to form the coupled equations. Both the displacement and traction must be continuous across the interface, but they are related in different ways for the film and the substrate. We use a simple model of linear viscoelasticity to relate the surface displacement of the substrate to the surface tractions (normal and shear), with the help of a thin layer approximation. For the film, we use the nonlinear von Karman plate equation, which is only geometrically nonlinear for moderately large deflections while the material deformation remains in the regime of linear elasticity. The two sets of equations are then coupled together through the continuity condition at the interface.
The following two papers contain more details about derivation of the equations.
[1] S.H. Im and R. Huang, Evolution of wrinkles in elastic-viscoelastic bilayer thin films. J. Applied Mechanics 72, 955-961 (2005).
[2] R. Huang, Kinetic wrinkling of an elastic film on a viscoelastic substrate. J. Mech. Phys. Solids 53, 63-89 (2005).
By the way, I also enjoyed reading your works on fingerprints, which share some similar mechanics with synthetic wrinkles in thin films.
RH
Thank you Rui! My question
Thank you Rui!
My question was not clearly formulated. Of course, you meet the continuity condition. I meant something else. If the boundary particle on the film side undergoes, say, large rotation then its adjacent twin on the layer side should also undergo large rotation. The latter, however is ruled out by the linear kinematics...
Kosta
continuous rotation of particles?
Kosta:
Good point! I start to see what could be a flaw in the analysis. We included the contribution of rotation (second order term) in the stretch of the plate, but did not do the same for the surface of the underlayer! If this is what you meant, I need to think more about it. For a quick defense, I may point out that the in-plane deformation of the substrate surface plays a secondary role in the whole problem. In other words, the compatibility of the lateral deflection more or less controls the dynamics, as we did in the scaling analysis (in-plane deformation ignored completely). On the other hand, the in-plane kinematics may be important for long term evolution and even the equilibrium state.
Thank you for the insightful comments!
RH
Rui,Yes, you are right. This
Rui,Yes, you are right. This may be a problem. May not... The wrinkles' amplitudes are important as compared to, say, the film thickness. Of course, pure bifurcation analysis should not be strongly affected by this simplification because the bifurcation modes are not fully defined numerically. If you want to see a 'real' distribution of wrinkles, however, nonlinear kinematics of the substrate can matter...
Kosta