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Journal Club Theme of October 2012: Instabilities of Structures

Davide Bigoni's picture

The Journal Club Theme of February 2012 "Elastic Instabilities for Form and Function" (node/11812), coordinated by Doug Holmes (, has provided an excellent review of elastic instability as related to pattern formation in soft materials.
The October 2012's Journal Club is on a closely related subject, namely, "instabilities of structures". This is a mature research field, but still strongly vital. In particular, we would like to draw the attention on recent results on how to exploit the post-critial behaviour of an elastic structure to obtain flexible mechanisms with special behaviours: (i.) a spherical shell shrinking towards its center; (ii.) a one-degree-of-freedom elastic structure buckling in tension and compression and providing a constant force ("neutral") post-critical behaviour; (iii.) dynamical instabilities explain wrapping of a liquid drop by an elastic strip.

(i.) A spherical shell shrinking towards its center

A spherical shell patterned with a regular array of circular engraves is shown to buckle, when subject to external pressure, in a sort of torsional-mode, giving rise to a shrink of the shell toward its center (Shima et al. 2012), Fig. 1.  
This folding mechanism is induced by a mechanical instability in an elastic structure and therefore is fully reversible.

Fig. 1 Sequence of progressively deformed shapes of the buckliball (from Shima et al. 2012)

A video on the buckliball can be found at:

(ii.) A one-degree-of-freedom elastic structure buckling in tension and compression and providing a constant force ("neutral") post-critical behaviour

Zaccaria et al. (2011) have shown that elastic structures (in which each element is governed by the Euler's elastica) can be designed to buckle under purely tensile dead loading and that the shape of the buckled elastica in tension corresponds to the shape of a water meniscus in a capillary channel (Fig. 2).

Fig. 2 Analogy between an elastic rod buckled under tensile force (left) and a water meniscus in a capillary channel (right, superimposed to the solution of the elastica, marked in red): the deflection of the rod and the surface of the liquid have the same shape (from Zaccaria et al. 2011)

These concepts have been developed by Bigoni et al. (2012) to show that the profile of the constraint where an end of an elastic structure has to slide can be designed to obtain buckling in tension or compression, and two buckling loads for a one-degree-of-freedom structure. Moreover, the profile can be designed to obtain a "neutral" postcritical response, in which the displacement increases at constant load (Fig. 3).

Fig. 3 A one-degrees-of-freedom elastic structure with two (one tensile and one compressive) buckling loads and neutral postcritical behaviour. Right: experimental set-up. Left: theoretical predictions versus experimental results (from Bigoni et al. 2012)

A video on multiple bifurcations and neutral postcritical response can be found at:

(iii.) Dynamical instabilities explain wrapping of a liquid drop by an elastic strip

Experiments by Antkowiak et al. (2011) show that a liquid drop can be wrapped by an elastic thin film by a drop impact on millimetric and centimetric scales (Fig. 4). This capillary phenomenon, is competing with the weight of the elastic film and can be explained (Rivetti and Neukirch, 2012; Hure and Audoly, 2012) in terms of elastic instability.

Fig. 4 Instant capillary origami, obtained with a water droplet impacting a thin triangular polymer sheet with thickness (from Antkowiak et al. 2011).

A video on the wrapping of liquid drops can be found at:

In the previously-quoted works, the theoretical predictions are substantiated with experiments and a close agreement is shown, so that we may think that it will be possible in the future to realize flexible structures with designed pre- and post- critical behaviours. Results presented in (i.) and (iii.) can find applications in the industrial process of encapsulation, while results presented in (ii.) can be employed to design of shock-absobers for tensile forces.  


A. Antkowiak, B. Audoly, C. Josserand, S. Neukirch, and M. Rivetti (2011) "Instant fabrication and selection of folded structures using drop impact". PNAS 26, 108, 10400–10404.

D. Bigoni, D. Misseroni, G. Noselli and D. Zaccaria (2012) Effects of the constraint’s curvature on structural instability: tensile buckling and multiple bifurcations. Proc. R. Soc. A, 468, 2191-2209.

J. Hure, B. Audoly (2012) Capillary buckling of a thin film adhering to a sphere. J. Mech. Phys. Solids, DOI:

M. Rivetti and S. Neukirch (2012) "Instabilities in a drop-strip system: a simplified model." Proc. R. Soc. A doi:10.1098/rspa.2011.0589

J. Shima, C. Perdigoub, E.R. Chen, K. Bertoldi, and P.M. Reis (2012) "Buckling-induced encapsulation of structured elastic shells under pressure", PNAS 109, 16, pp. 5978-5983.

D. Zaccaria, D. Bigoni, G. Noselli and D. Misseroni (2011) Structures buckling under tensile dead load. Proc. R. Soc. A, 467, 1686-1700.


katia bertoldi's picture

Nice post on an exciting topic.

Quick question regarding elastic structures buckling in tension: have you found such behavior in a natural system?

Davide Bigoni's picture

I am sure that buckling in tension exists in nature. Although I am teaching a course named "Theoretical biomechanics" I am not expert enough to know a natural system buckling in tension. However, almost all enginnering structures find a counterpart in nature (see for instance the correspondence between truss structures and the vertebrate skeletons, or between a masonry and the microstructure of nacre, or between a dome and a natural shell, or between a vassoir arch and the plantar arch), so I feel confident that the mechanism of buckling in tension must exist in nature. Maybe some of the iMechanica's readers, or a biologist, could help me, and it will become a cool article ...


May be this post is not relavent to this topic, but I have a small doubt and would like to get solutions from experts in this forum. 

I have modelled 1D beam-column problem with axial and lateral loads by variational methods. There are 4 elements with midnodes and two degrees of freedom at each node.  The governing equation is as given below.


Where [Ke] = elastic stiffness matrix

P = buckling load

[Kg] = Geometric stiffness matrix

(Δ) = Global DOF matrix, and

(F) = lateral loads at each node.

I want to calculate buckling load and corresponding eigen vector. I am unable to solve this system of equations (I lack knowledge in solving this sytem of equations). Can anybody help me in modelling it as an incremental solution.

Best regards,

Brahmendra S Dasaka. 


Fan Xu's picture

Roughly speaking, to solve these kinds of nonlinear system of equations, there are two main ways as far as I know:


1. classic Newton-Raphson method or modified...

2. Asymptotic numerical method (Méthode Asymptotique Numérique).


The 2nd method is more powerful compared with the classic Newton-Raphson method but a little complicated, which is proposed by Pr. Michel Potier-Ferry. You can find a book named 'Méthode asymptotique numérique' published by Hermès 2007, or go to Sciencedirect to see many papers related to this method. 

zichen's picture

Thanks for coordinating discussions on this interesting topic! I'd like to add a few recent studies on mechanical instabilities in thin structures as follows.

1. S. Armon, et al. Geometry and mechanics in the opening of chiral seed pods. Science 333, 1726-1730 (2011).

2. Z. Chen, et al. Nonlinear Geometric Effects in Bistable Morphing Structures. Phys. Rev. Lett. 109, 114302 (2012).

3. J. Huang, et al. Spontaneous and deterministic three-dimensional curling of pre-strained elastomeric bi-strips. Soft Matter 8, 6291 (2012).

4. W. Shan, et al. Attenuated short wavelength buckling and force propagation in a biopolymer-reinforced rod, Soft Matter, Advance Article (DOI:10.1039/C2SM26974K).

5. M. Gigliotti, et al. Predicting loss of bifurcation behaviour of 0/90 unsymmetric composite plates subjected to environmental loads. Composite Structures 94, 2793–2808 (2012).

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