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Collecting phenomena of large elastic deformation

Zhigang Suo's picture

It might be fun for us to work together to collect phenomena of large elastic deformation.  These phenomena will enliven teaching and motivate research.  As inspiration, here are two albums of fluid motion

In teaching large elastic deformation (notes on special cases), I describe several phenomena of large elastic deformation.

Necking or not.  When pulled, a metal bar forms a neck, but a rubber band does not.

Inflation of a spherical balloon.  When a balloon is inflated, the pressure-volume curve is not monotonic.  Thus, when two identical balloons are connected, one balloon expands more than the other.

Inflation of a cylindrical balloon.  When a cylindrical balloon is inflated, one segment of the balloon inflates first, and the remaining part of the balloon deforms slighly.  As more air is pumped in, the more part of the balloon becomes inflated.

Cavitation.  When a block of rubber is subject to triaxial tension, a cavity forms when the tension reaches a critical value.

Crease.  When a block of rubber is compressed, the surface forms creases when the compression reaches a critical level.

In the notes on special cases, I also gave references.

If you think of a phenomenon of large elastic deformation, please leave a comment below.

Comments

Rui Huang's picture

This is a great idea. Many of us are working along this line. I would add one here:

Stretch-induced wrinkling of thin membranes: when pulling a thin membrane (such as plastic food wrap), parallel wrinkles are ofen observed. A model problem is studied in the following paper, and a second paper with experimental measurements is coming soon.

1. V. Nayyar, K. Ravi-Chandar, R. Huang, Stretch-induced stress patterns and wrinkles in hyperelastic thin sheets. International Journal of Solids and Structures 48, 3471-3483 (2011). DOI: 10.1016/j.ijsolstr.2011.09.004.

2. V. Nayyar, K. Ravi-Chandar, R. Huang, Stretch-induced wrinkling of polyethylene thin sheets: experiments and modeling. Submitted. 

RH

Xuanhe Zhao's picture

This is a topic of great intersts to us. Mechanics of film-substrate systems have been studied over decades. Recently, a number of groups discovered that large deformation and instabilities in film-substrate systems can lead to intriguing phenomena as well as transformative applications. Some of our recent works are listed here:

1. Phase diagrams that can quantitatively predict various modes of instabilities, including wrinkle, crease, delaminated buckle, period double, localized ridge, fold, and coexistence of states.

2. Harnessing delaminated-buckle instability in graphene for reversible crumpling and unfolding graphene.

3. Harnessing localized-ridge instability in nanofilms for extremely high-aspect-ratio surface patterns that give stretchable superhydrophobic coatings and dynamic control of cell alignment. 

Wei Hong's picture


Nice initiative on this interesting collection for mechanicians.


As suggested by Zhigang in his original post, crease is really a unique type of
instability.  Unlike in wrinkles, ridges,
or other types of surface instabilities on solids, the large deformation (or
geometric nonlinearity to be exact) in crease is intrinsic.  The self-contact of a (free) surface
distinguishes crease from regular wrinkles. 
While wrinkles start from small amplitude and small strain, a crease is
born with finite (or infinite) strain.  In
the idealized case of a crease on a semi-infinite solid, a crease is
self-similar and thus the strain profile is independent of the size of the
crease (contact area).  If one neglects the
effect of surface energy, this idealized scenario is indeed the onset of crease
on any surface.


For an incompressible material, the strain at the crease tip
is finite.  For compressible solids, the
strain is singular.  The stress, on the
other hand, is always singular.


An ideal crease has 0 radius of curvature at the tip.  In reality, due to the presence of surface
energy, the tip exhibits finite radius of curvature, and the onset of instability
is also dependent on the surface energy. 
For materials with significant surface energy (or a stiff skin layer),
the crease instability is delayed and may take place after wrinkling.  With large radius of curvature (a tunnel below
the contact area), such creases are more often referred to as folding.  This is basically the mechanism of iron-free
/ wrinkle-free fabrics – maybe we should call them “crease-free fabric”?


Although the ideal case (no surface energy) is not possible physically,
it is appreciated by its mathematical beauty. 
To date, there is no effective way of rigorously predicting the onset of
the instability.  The mathematical
problem behind it is clear but challenging. 
It is basically a nonlinear eigenvalue problem, with the difference
between the homogeneous solution and the bifurcation nonlinear (and singular).  Because of the intrinsic geometric
nonlinearity, linearization of any kind is doomed to fail.  A singular solution like the crack-tip field
may be sought, but is very difficult because of the nonlinearity (due to self-contact).


Rui Huang's picture

Dear Wei,

Interesting thoughts on crease instability. I agree that crease is different from wrinkling, but I would argue that there is an intrinsic connection between the two. The paper by Cao and Hutchinson (Proc. R. Soc. A, vol. 468, 94-115, 2012) offers some idea. From my own experience, by linear perturbation analysis, if there exists a finite wavelength at the onset of wrinkling instability, there would be no crease (to begin with). On the other hand, if the linear perturbation analysis predicts a critical condition at an infinitely short wavelength (or independent of wrinkle wavelength such as in Biot's analysis), crease would most likely preceed wrinkling. An example problem is considered in the paper below for a bilayer hydrogel.

Z. Wu, N. Bouklas, R. Huang, Swell-induced surface instability of hydrogel layers with material properties varying in thickness direction. Int. J. Solids and Structures 50, 578-587 (2013). DOI: 10.1016/j.ijsolstr.2012.10.022.

RH

Douglas P Holmes's picture

Hi Zhigang,

Snapping plates and shells 

The first example I'd like to add is in reference to the snap-buckling instability of plates and shells. Unlike classical Euler buckling, snap-through leads to a global loss of structural stability - once the critical threshold is reached, a large jump in deformation occurs. Recently, work we've done in collaboration with Dominic Vella and Derek Moulton at Oxford has discussed the mechanics and dynamics of snapping plates and shells. Additionally, we done work on creating surfaces of shells that undergo large, elastic deformations on-command - either snapping or buckling.

 

  1. A. Pandey, D. Moulton, D. Vella, and D.P. Holmes, "Dynamics of snapping beams and jumping poppers," arXiv:1310.3703 [link]
  2. D.P. Holmes and A.J. Crosby, "Snapping Surfaces," Advanced Materials, 19, 3589, (2007). [link]
  3. D.P. Holmes, M. Ursiny, and A.J. Crosby. "Crumpled Surface Structures," Soft Matter, 4, 82, (2008). [link]

Wrinkles, creases, and folds

Wrinkles and creases have already been mentioned, but I'd like to quickly add a few comments.  The non-uniform swelling of elastomers can lead to either surface deformation (creasing) or structural deformation (bending and/or buckling). These large deformations are entirely reversible, and can lead to strange behavior, such as a travelling buckle wave around a circular disc.

  1. A. Pandey and D.P. Holmes, "Swelling-Induced Deformations: A Materials-Defined Transition from Macroscopic to Microscopic Deformations," Soft Matter, 9, 5524, (2013). [link]
  2. D.P. Holmes, M. Roche, T. Sinha, and H.A. Stone, "Bending and Twisting of Soft Materials by Non-homogenous Swelling," Soft Matter, 7, 5188, (2011). [link

In addition to wrinkles, the wrinkle-to-fold transition we studied in axisymmetric thin plates is also reversible. These wrinkles grow in amplitude and then localize into sharp folds. With elastic materials, such as silicone rubber, the sharp folds will disappear once the load is removed.

  1. D.P. Holmes and A.J. Crosby, "Draping Films: A Wrinkle to Fold Transition," Physical Review Letters, 105, 038303, (2010). [link] 

 

 Cheers,

-- Douglas Holmes

Zhigang Suo's picture

Dear Xuanhe and Doug:  Thank you both for the examples from your recent work.  The two jClub themes led by you highlight one reason for the recent interest in elastic instability:  link instabilities to functions.  For people who missed the two themes, here they are

Cai Shengqiang's picture

Dear Zhigang, thank you for initiating this discussion. 

One problem we are working on is probably interesting to some of you. Large deformation in the problem is very critical. 

The system is following: chemical reacations can happen inside gels. The swelling degree of gels are influenced by the concentration of reactants (and/or products). There are two revelant time scales in the system: the time for reaction and the time for diffusion of different solutes. Because the fime scale for the diffusion are dertermined by the size of gels, large defromation in gels can change diffusion time scale dramatically.  It is observed that for certain initial conditions, chemical reacations can induce oscillatory deformations in the gel. (The chemical reacation itself is not oscialltory at all) 

 This idea is first proposed by Boissonade in the paper below.

http://prl.aps.org/abstract/PRL/v90/i18/e188302 

Zhigang Suo's picture

Thank you Shengqiang for this interesting paper.  Your jClub theme on elastocapillarity also provides a set of excellent examples of large elastic deformation.

Zhengjin WANG 王正锦's picture

Dear Zhigang, thank your for this interesting topic.

Fingering instability has been researched for several decades. It has been widely reported about the fingering instability at the interface of liquids or that between soft solids and rigid substrate. Shull et al. [1] used a semi-sphere probe to contact with the soft gel and then pulled up at a constant velocity. They observed that the fingers formed and propagated within the bulk of the gel. This kind of instability can be driven by injecting air or pulling the confinements, in which conditions the interfaces are glued well. According to the results of Mahadevan et al. [2] and Saintyves et al. [3], the bulk fingering instability appears at a critical strain, independent of any material parameters. Moreover, the elastic instability is reversible and rate-independent. However, Foyart et al. [4] recently found a discontinuous transition between fingering instability and fracture, which is rate-dependent and governed by the flow properties of the gel ahead of the finger tip.

[1]. K. R. Shull, C. M. Flanigan, and A. J. Crosby, Physical Review Letters 84, 3057 (2000).
[2]. J. S. Biggins, B. Saintyves, Z. Wei, E. Bouchaud, and L. Mahadevan, Proceedings of the National Academy of Sciences (2013).
[3]. B. Saintyves, O. Dauchot, and E. Bouchaud, Physical Review Letters 111, 047801 (2013).
[4]. G. Foyart, L. Ramos, S. Mora, and C. Ligoure, Soft Matter 9, 7775 (2013).

Oscar Lopez-Pamies's picture

Thanks Zhigang for initiating this!

A problem that we have been working on for a while now is that of cavitation in rubber.

Under certain conditions, large enclosed cavities may suddenly ``appear'' in the interior of rubber. This phenomenon has come to be popularly known as cavitation. It corresponds, in essence, to nothing more than to the growth of defects inherent in rubber (Lopez-Pamies et al., 2011). Such defects can be of various natures (e.g., weak regions of the polymer network, actual holes, particles of dust) and of various geometries ranging from submicron to supramicron in length scale (Gent, 1991).

We are currently studying the roles that, both, elastic and fracture properties have on cavitation (Lefèvre et al., 2014). The movie below shows a simulation of one of the poker-chip tests of Gent and Lindley (1959) illustrating that - rather unexpectedly - the elastic properties of rubber do play a significant role.


Poker-Chip Simulation

Figure

 

1. Lopez-Pamies, O., Idiart, M.I., Nakamura, T., 2011. Cavitation in elastomeric solids: I - A defect-growth theory. Journal of the Mechanics and Physics of Solids 59, 1464-1487. [link]

2. Gent, A.N., 1991. Cavitation in rubber: A cautionary tale. Rubber Chemistry and Technology 63, G49-G53. [link]

3. Lefèvre, V., Ravi-Chandar, R., Lopez-Pamies, O., 2014. Cavitation in rubber: An elastic instability or a fracture criterion?. In preparation.

4. Gent, A.N., Lindley, P.B., 1959. Internal rupture of bonded rubber cylinders in tension. Proc. R. Soc. London A 249, 195-205. [link]

5. Bayraktar, E., Bessri, K., Bathias, C., 2008. Deformation behaviour of elastomeric matrix composites under static loading conditions. Eng. Fract. Mech. 75, 2695-2706. [link]

 

Zhigang Suo's picture

Dear Oscar:  Thank you very much for an update of your work on cavitation.  Your jClub discussion on cavitation is an exellent place to begin to learn about the subject.

katia bertoldi's picture

Hi Zhigang

I would add another one:

Instabilities in porous structures

By carefully designing the microstructure we can trigger short wavelength modes

      Mullin T, Deschanel S, Bertoldi K, Boyce MC. Pattern Transformation Triggered by Deformation.       Physical Review Letters. 2007;99:084301


      Michel, J.C., Lopez-Pamies, O., Ponte Castañeda, P., Triantafyllidis, N. 2007. Microscopic and       macroscopic instabilities in finitely strained porous elastomers. Journal of the Mechanics and       Physics of Solids 55, 900–938

that can be exploited to

  • form complex patterns

         Zhang, Y. Matsumoto, E. A., Peter, A., Lin, P., Kamien, R. D.*, and Yang, S.*, “One-step          nanoscale assembly of complex structures via harnessing of an elastic instability”, Nano Lett.          2008 , 8 (4), 1192-1196

         Kang SH, Shan S, Noorduin WL, Khan M, Aizenberg J, Bertoldi K. Buckling-Induced Reversible          Symmetry Breaking and Amplification of Chirality Using Supported Cellular Structures.          Advanced Materials. 2013;25(24):3380-3385

  • achieve color switching in patterned membranes

         Li J, Shim J, Deng J, Overvelde JTB, Zhu X, Bertoldi K, Yang S. Switching periodic membranes          via pattern transformation and shape memory effect. Soft Matter. 2012;8:10322-10328

  • manipulate the propagation of elastic waves

         Bertoldi K, Boyce MC. Mechanically-Triggered Transformations of Phononic Band Gaps in          Periodic Elastomeric Structures. Physical Review B. 2008;77:052105


         Jang JH, Koh CY, Bertoldi K, Boyce MC, Thomas EL. Combining Pattern Instability and Shape-          Memory Hysteresis for Phononic Switching. Nano Letters. 2009;9:2113-2119


         Wang P, Shim J, Bertoldi K. Effects of Geometric and Material Non-linearities on the Tunable          Response of Phononic Crystals. Physical Review B. 2013;88:014304


  • design foldable structures

         Shim J, Perdigou C, Chen ER, Bertoldi K, Reis PM. Buckling-induced encapsulation of          structured elastic shells under pressure. Proceedings of the National Academic of Science of          the United States of America. 2012;109:5978-5983


  • design materials with unusual mechancial properties

         Bertoldi K, Reis PM, Willshaw S, Mullin T. Negative Poisson's ratio behavior induced by an          elastic instability. Advanced Materials. 2010;22:361-366


         Babaee S, Shim J, Weaver JC, Patel N, Bertoldi K. 3D Soft Metamaterials with Negative          Poisson’s Ratio. Advanced Materials. 2013.

Zhigang Suo's picture

Thank you for this update of your wonderful work on holely sheet.  I also enjoyed reading the News Freature on your work in Nature.  The article captured the excitment of the young community of "extreme mechanics".   

Zhigang Suo's picture

Here is a recent paper published by the Chaudhury group.

Interaction between hard particles in a soft elastic gel.  When hard particles are dropped into an exceptionally soft gel (of modulus of a few pascals), the particles interact with each other over a long distance.  Several particles can dance together, forming intricate patterns.  Watch the movies by downloading the Supporting Information.  

Jing_Yang's picture

Hi, Prof. Suo

I read in your slides on "Finite Deformation" attached in imechanica(both general and special parts), but it seems that no expositions on nonlinear membrane theory are included.

On the other hand, the finite deformations of shell or membrane, which is made up dielectric elastomers, is unavoidable.


Would you recommend me some classical treatise on finite deformations of or shell membrane?

I'm also an applicant for your 2014 fall Ph.D. program.

Sincerely

Zhigang Suo's picture

Dear jarvisyang:

My notes on finite deformation are for segment of a course.  These notes now have four parts:

Hopefully this iMechanica thread will help to collect some phenomena that will supplement the last part of the notes.

I have not looked at current works on shells, and do not know what to recammend.  

Recently we have spent considerable amount of time on large deformation of elastomeric membranes.  Here is one paper, and it contains references to the literature.

Tiefeng Li, Christoph Keplinger, Richard Baumgartner, Siegfried Bauer, Wei Yang, Zhigang Suo. Giant voltage-induced deformation in dielectric elastomers near the verge of snap-through instability. Journal of the Mechanics and Physics of Solids 61, 611–628 (2013).

Hi all,

I want to learn the steps of the simulation program on the subject of Abacus about (split Hopkinson pressure bar compression)with and without effect temperature .This device gives high strain rate impact where it two long bars between it put the specimen and then strike with short bar. Our data collection system consists of a pair of strain gages midway along each bar. Data is recorded and presented as the initial impulse wave passes the incident bar strain gages, a reflected pulse from when the wave hit the end of the bar and reflected back to the incident bar strain gage and finally a transmitted pulse which is what remains of the wave after it has passed through a sample. In all cases the voltage reading from the strain gages should return to zero once the wave has passed. However with the new bar the signal lingers after the wave has passed a gage.
my version 6.12
Please anyone can help me
regards

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