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Do Eshelby forces exist in elastic structures?

Davide Bigoni's picture

Do Eshelby forces exist in elastic structures?

We provide a positive answer to this question, see



More information about my research activity can be found in
More information about our experiments can be found in


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PDF icon eshelbylikeforcesMoM2013.pdf2.01 MB


Zhigang Suo's picture

Dear Davide:  Such a clever demonstration!  The vidoe of lecture is also terrific.  Thank you for making this concept so clear.  

The mechanics of "making beam longer" reminds me of an old paper in fracture mechanics:

Obreimoff, The splitting strength of mica, Proc. R. Soc Lond. A127, 290-297 (1930).

In Obreimoff's setup, the elastic energy release rate is balanced by surface tension.  In your setup, the energy release rate (i.e., the Eshelby force) causes the motion.  I'll use your video in my fracture mechanics course in the coming semester. 

Davide Bigoni's picture

Dear Zhigang,
Thanks for the compliments and the suggestion of the Obreimhoff paper,
which I will read with pleasure. In fact we have found an example of a
structure releasing elastic energy, in other words, the secret of Eshelby
Best wishes!

Thank you Davide for the paper and your excellent presentation of the key concepts.  This paper is timely for me in the sense that I have a better explanation for the need for frictional clamps in a double cantilever beam experiment under moment loading.  On the other hand, the effect of the Eshelby force on the mode mixity during crack propagation (so far ignored) now needs to be explained (or ignored again?).

-- Biswajit

Zhigang Suo's picture

Dear Biswajit:  Not sure if I understand your question correctly, but you might be interested in mixed mode fracture of beams.  You can find many examples in the following article:

Hutchinson and Suo, Mixed-mode cracking in layered materials.  Advances in Applied Mechanics 29, 63-191 (1992). 

In particular, take a look at Figs 20 and 21. 


Was away from a desktop for a while hence the delayed response.

I looked at your paper again and noticed that you had made a brief mention of configurational forces.  I haven't gone through all the results in your paper to see whether configurational forces have been used in their derivation.  However, I assume that is the case.

To clarify my earlier comment, consider the situation where the beam that Davide shows in his video has a crack (say midway along the thickness and through the width).  Suppose only the bottom is loaded by a dead load and that additional loads are added in the form of grains of sand, one grain at a time.  Suppose also that the additional load provides enough enegry that the crack can grow in a stable manner. 

Under these circumstances will the beam slide out (or in) first (because of the change in curvature) or will the crack grow first  (because of the change in configuration due to crack growth)?

In our experiments we ran into a situation where, unless one end of the beam was clamped instead of being constrained inside rollers, we obserrved large stick-slip motions even though the time rate of application of moments was small.  Of cource, our rollers were not frictionless.

-- Biswajit

Pu Zhang's picture

This work is amazing and thanks for your experimental proof.

Mathematically this is a variational problem with a moving boundary (Gurtin ME, Configurational forces as basic concepts of continuum physics). Then the 'Eshelby force' or configuration force comes up due to the equilibrium of the moving boundary.

Actually there is another example similar to this problem. The adhesion boundary of an elastica or plate also exhibits this configuration force, which can be found in (Majidi C. Remarks on formulating an adhesion problem using Euler's elastica, Mech Res Commun 2007, 34:85-90).  

Davide Bigoni's picture

Dear Pu,

In fact we have quoted both Gurtin and Majidi in our article, please see:

Best wishes,




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