iMechanica - patterns - Comments

Hi Rui, Any bifurcation

Konstantin Volokh — Sun, 15 Oct 2006 02:40:57 -0400

Hi Rui,

Any bifurcation analysis is nonlinear including the one I considered (look at the Appendices of my paper).

Sometimes, however, people distinguish between the "linear" and "nonlinear" stability analyses. The former means that the material/structural behavior is linear prior to the bifurcation while the latter means that the material/structural behavior is nonlinear prior to the bifurcation. For example, the Euler beam and the example I considered are linear prior to the bifurcation.

The nature of the instability is similar in these two problems in the following sense. The beam buckles because its bending stiffness is much less than the axial one. The surface buckles because the half-space stiffness is much less in the direction perpendicular to the surface than in the direction parallel to the surface. The anisotropy is crucial! The isotropic half-space would not undergo the surface buckling. However, if you consider isotropy of the half-space with the nonlinear material behavior, i.e. "nonlinear" instability, then the bifurcation can occur.

Again, I recommend you to look at the pioneering work of Biot (see the exact reference in my paper). The only problem with this book is that it was written before Truesdell et al shaped the foundations of continuum mechanics and the Biot treatment of nonlinearities, as well as his implicit and explicit additional assumptions, is somewhat vague (at least to me).

Kosta

still need nonlinearity here

Rui Huang — Sat, 14 Oct 2006 23:09:03 -0400

Hi Teng:

Of course I am aware of this work, even before it was published in paper. For that phenomenon, I believe nonlinear material behavior (e.g., hyperelasticity) plays the key role in forming multiple wavelengths.

Another paper of interest you may find interesting.

Teng Li — Sat, 14 Oct 2006 22:21:45 -0400

The following paper reported interesting phenomena in model elastomeric artificial skins. When compressed, such artificial skins wrinkle in a hierarchical pattern consisting of self-similar buckles extending over five orders of magnitude in length scale, ranging from a few nanometers to several milimeters.

Nested self-similar wrinkling patterns in skins Nature Materials 4, 293–297 (2005)

Linear perturbation predicts when incipient instability occurs

Teng Li — Sat, 14 Oct 2006 22:03:12 -0400

Rui,

Take necking as an example. The bifurcation analysis in our paper is essentially a linear perturbation approach. Such an analysis yields the critical strain above which incipient nonlinear deformation occurs. Since linear perturbation analysis cannot predict the amplitude of the bifurcation deformation, "a meaningful analysis" considering geometric nonlinearity, that is, large amplitude nonlinear deformation, is needed to describe the necking development and the resulted rupture. For example, finite element method (FEM) can be used to simulate the necking process. The final rupture strain obtained from the FEM simulations can be substantially different from the critical strain predicted from a bifurcation analysis, depending on both the material behavior and the structure.

Buckling as a linear perturbation of a nonlinear problem

Zhigang Suo — Sat, 14 Oct 2006 19:04:37 -0400

Rui: The analysis of beam buckling has one aspect common to a nonlinear analysis: The moment balance equation is established in the deformed configuration. You may as well regard the analysis as a linear perturbation of a nonlinear problem. Indeed, you can fiirst establish the full nonlienar equations, and then gives a small perturbation.

nonlinear or not

Rui Huang — Sat, 14 Oct 2006 14:42:24 -0400

Kosta and Zhigang:

From your comments, I sense that both of you believe, without any nonlinearity in material law or geometry, the surface buckling will occur for a half plane under uniform compression. While Kosta brought up the effect of anisotropy, Zhigang implied even that is unnecessary as we see from the examples. I don't know how to argue against Euler buckling where a linear, isotropic analysis predicts bifurcation (maybe there is something in the approximations of the beam theory?). For necking, however, is the competition between material hardening and geometric softening a nonlinear behavior? In Teng Li's paper, a nonlinear power law is assumed for the material. If we take N = 1 (linear elastic), the critical strain for a free-standing film would be 100%, way too big for the linear elastic description, which means geometric nonlinearity must be taken into consideration for a meaningful analysis even the material remains linear elastic. Back to the half-plane problem, I don't see how the bifurcation would occur without either material or geometrical nonlinearity. Kosta: do you think anisotropy alone can make it happen?

Material nonlinearity may be unnecessary for surface instability

Zhigang Suo — Sat, 14 Oct 2006 14:01:26 -0400

While material nonlinearity affects the conditions for instability, it is unnecessary for instability to occur. The mother of this kind of problem is the buckling of a beam. Here buckling is mainly a geometric effect: it happens whether material is linear or not.

Another example is the necking instability. A tensile bar will neck even if material is linearly elastic, so long as it does not fracture. In this context, necking is an outcome of a competition between material hardening and geometric softening. Necking can be prevented if a material hardens exponentially. This explains why a rubber band does not neck.

In recent years, the Biot-type analysis has been revived in several contexts. Teng Li and I have published a paper, entitled Deformability of thin metal films on elastomer substrates (International Journal of Solids and Structures 43, 2351-2363 (2006)). The paper contains a review of literature. We have also shown that the Biot-type linear stability analysis may give misleading impressions of some phenomena. Full nonlinear analysis is often needed.

Hi Rui, You are right

Konstantin Volokh — Sat, 14 Oct 2006 13:33:19 -0400

Hi Rui,

You are right and the nonlinear isotropic elasticity can be more relevant to the problem than the linear anisotropy. I was thinking about that but I did not have time to do calculations. Go ahead! It can be a better argument than the one I brought up.

You can get the surface bifurcation without anisotropy if you include the material nonlinearity. This circle of questions is addressed in Biot's book on incremental elastic deformations. He considers the surface instability of a nonlinear isotropic hyperelastic material in compression. I have to admit that I experience some difficulties in reading Biot's book, however, you might find it insightful.

Best regards,

Kosta

Inhomogeneous elastic deformation

Rui Huang — Sat, 14 Oct 2006 13:10:27 -0400

Konsta:

After quickly reading your paper on surface buckling mechanism, without going into the details of your theory and solution, it appears to me your result suggests an interesting generic problem in elasticity. Consider a homogeneous, anisotropic elastic half plane, subjected to a uniform compression (not necessarily due to tissue growth). According to your solution (if I understand correctly), the deformation of the elastic half plane bifurcates at a critical stress to form surface buckles. Consequently, the deformation in the elastic half plane becomes inhomogeneous! I am aware of some works by John Hutchinson on surface instability of elastic-plastic solids, where material nonlinearity plays an important role. It seems to me that your theory does not involve any nonlinearity in the constitutive equations, but may have geometric nonlinearity through the incremental equations. While you emphasized the role of elastic anisotropy, I would think some nonlinearity has to be involved to give an inhomogeneous solution in an otherwise homogeneous system. If this is true, compression may not be a necessary condition. As we know, elastic-plastic deformation bifurcates (necking) under uniaxial tension. Anything wrong with my understanding?

Thanks.

Fingerprints may have little relation to gene

Weixu Zhang — Sat, 14 Oct 2006 11:02:19 -0400

Thank you.

I think your work is worthful.

I observed carefully at my fingerprints. They all seems perpendicular to main tensile direction. They must fit the mechanical mechanism. So I think your theory is quite reasonable. Your theory indicates that perhaps a lot of things of body are not controlled by gene, they are controlled by physical rules.

With regards

I think fingerprints do not

Konstantin Volokh — Sat, 14 Oct 2006 08:06:02 -0400

I think fingerprints do not change after their formation, i.e. after the surface buckling. The question of the role of genetic and epigenetic factors is open. May be you can shed new light on it!?

very interesting! reply to"Why fingerprints are different"

Weixu Zhang — Sat, 14 Oct 2006 05:44:01 -0400

Very interesting!!!

More than two weeks ago (2006.9.26), Prof J. Hutchinson came to Xi'an Jiaotong University and gave a lecture to the studernt on surface buckling. Before that time I knew little about surface buckling. After his lecture, I saw my fingerprints, suddenly I thought that maybe the fingerprints has some relation to the surface buckling. Before I investigated the literatures, I saw your answer!!!!!! Well done!

Here is still another question, as you said 'the surface bifurcation enjoys infinite multiplicity' this maybe the reason different person has different fingerprints. But this theory also seems that after the fingerprints wearing off, they would grow up in different form. According to the life experience, it do grow up in the same form ( I am not sure about this). What is the reason? It is controlled by DNA?

Rui,Yes, you are right. This

Konstantin Volokh — Wed, 27 Sep 2006 13:47:26 -0400

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

continuous rotation of particles?

Rui Huang — Wed, 27 Sep 2006 12:40:27 -0400

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!

Thank you Rui! My question

Konstantin Volokh — Wed, 27 Sep 2006 11:32:55 -0400

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