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Formation of creases on the surfaces of elastomers and gels
When a block of an elastomer is bent, the compressed surface may form a crease. This paper analyzes the critical condition for creasing by comparing the elastic energy in a creased body and that in a smooth body. This difference in energy is expressed by a scaling relation. Critical conditions for creasing are determined for elastomers subject to general loads and gels swelling under constraint. The theoretical results are compared with existing experimental observations.
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questions on formation of creases
Dear Wei,
I read your manuscript with interest. I remember Zhigang talking about this idea more than a year ago and was waiting anxiously for this to come out. Anyway, I like the work in general but have a few questions.
First, in the discussion of Fig. 2, we know that the length L changes from a to b in a homogeneous deformation. Does the length L change from a to c? This would depend on how you specify the displacements in the finite element calculations.
Second, in Fig. 3 (Page 11), you plotted the energy versus strain, but the curve stopped before it reaches zero (the critical condition). Do you have any results for the energy to be negative, as you discussed in the text?
Finally, for the generalized plane-strain problem (Fig. 4), I agree with your discussion for the elastomers but have concerns about the same for swelling gels. Following your previous works on hydrogels, the swelling ratios (volume) are very different for a gel without constraint (stress-free state in a) and with constraint (b). In other words, under the same chemical potential, the concentrations of the solvent (water) in the gels are different, and thus the volumes are different too. So the relation between a and b would be different from that for an incompressible elastomor. Furthermore, from the intermediate state (c) to the final state (b), the stress state changes and the volume of the gel changes too. You may argue that the change in solvent concentration (or chemical potential) does not necessarily alter the critical condition for creasing, but I am not so sure. In fact, one of my students has been doing some simulations of creasing in swelling gels (using essentially the same model as in your previous works), and I have seen some promising results that hopefully will be ready for sharing soon.
Best regards,
RH
Crease length and gel crease
Dear Rui,
Thank you for your interest in our work, and sorry that it took so long.
1) The length is also free to change from a to c. I only constrained the horizontal displacement, with the vertical direction left free.
2) Very good observation. We did actually tried further. However, if you rethink about it, the (equilibrium) state corresponding to a negative energy does not exist. -- It is simply imposible to have a short contact length as prescribed beyond the critical strain. -- The material will fold back into itself if you allow penatration, or it will make a longer contact on top of A/A'. Since our Fig.3 describes the energy of a state in equilibrium with a prescribed (arbitrary but constant) crease length, the curve physically stops at the critical point. We can not get the exact critical point simply because of numerical reason.
3) Another very insightful point. What you pointed out is a equilibrium state, or long term response of a gel. What we described in the paper is a case in the other limit - the instantaneous response. What I have in mind is that the formation of a crease, especially for the formation of a very tiny one, can be very fast (the time scales with the size^2 under the model described in my previous paper). If we only look at the initiation of an infinitesimal crease, we should look at the instantaneous response of a gel - when time is short, the mobile molecules do not have time to relax or migrate, the gel is just like a piece of rubber. Ofcourse, in reality whenever a crease form, it has a finite size due to various reasons, and takes a finite time period. The real situition is somewhere between the two limiting cases and will need a proper dynamic model. It will be great if you can share your paper with us some time.
Wei
Re: formation of creases
Rui: Thank you for remarkably perceptive questions. Indeed, we struggled with these issues ourselves. The format of the paper is too short to expand these discussions.
Wei: Concerning your response 3), I'd like to add three sub-points:
(3a) The model does agree with several sets of experimental data within factor 2. The model may be incorrect for many reasons, but no model can ever be really correct, just as no painting can be real enough to be the real thing.
(3b) That said, however, some models are better than others. Regarding our long conversations about equilibrium vs. instantaneous models before you left Harvard last year, I now have second thoughts. If we are looking at creases formed duing the swelling of a gel, then the fact that gel swells seems to mean that time is enough for the surface layer to equilibrate with the external solvant. So perhaps equilibrium theory should be more appropriate.
(3c) I recall that you and Xuanhe had some FEM calculations using the equilibrium theory, and told me that results were not too different from the instataneous theory. We should reexamine these calculations, given that the instantaneous theory only predicts a single critical value, but experiments reported a range. Perhas the thermodynamic properties of the gel (e.g., the Flory parameter and Nv) will allow a range of critical values, sufficiant to explain the experimental observations.
crease on a gel
Yes, Zhigang you are right. It has been too long and I could not remember.
The instantaneous response we had in the paper does not need any calculation, it is a direct extension from the elastomer result.
We also did some simulation with our program for the equilibrium state. Now I don't remember whether it is close to that of an instantaneous response.
I could slightly recall that I told you the equilibrium critical condition is higher than that of the instantaneous one. But if it could form instantaneously, the crease will form (if there is no energy barrier), and may disappear as time evolves.