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# Journal Club Theme of September 2009: Elasticity and Failure

Let us consider interaction of two atoms/molecules/particles. The reference distance between them corresponds to zero interaction force and zero stored energy. The interaction passes three stages with the increase of the distance. At the first stage the force increases proportionally to the increasing distance: the linear stage. At the second stage the force-distance relationship deviates from the linear proportionality: the nonlinear stage. At the third stage the force drops with the increasing distance: the separation or failure stage.

In the case of solids composed of many particles two first stages of the particle interaction are described by the linear and nonlinear theories of elasticity correspondingly where the changing distance between particles is averaged by a continuum strain measure and the energy of the particle interaction is averaged by a strain energy function. Surprisingly, the third, failure, stage of the particle interaction is beyond the scope of elasticity theories. However, the failure description can still be introduced in elasticity by analogy with the failure description in the particle interaction. Indeed, the force of the pair interaction decreases with the increase of the interaction distance because the energy that can be stored during separation is limited by the constant of the bond energy. If the energy limiter exists for the pair interaction then it should exist in the multiple interactions. The latter means that we should limit the magnitude of the strain energy in order to describe material failure within the framework of elasticity.

This simple qualitative reasoning encouraged my search of the strain energy functions that include energy limiters for a description of material failure [1]. Such functions can be material- and/or problem- dependent because the failure modes can be different for different loading conditions. I examined various theories and applications of elasticity with energy limiters including the problems of the crack initiation in Hookean [2] and neo-Hookean [3] materials; cavitation in Hookean [4] and soft [5] materials; failure of arteries [6] and abdominal aortic aneurysm [7]; dynamic failure propagation [8]. Extending the idea of energy limiters to liquids I examined a new scenario of the transition to turbulence through the viscosity failure [9]. Lastly, I proposed a universal or ‘try-first’ formula for enhancing a material description with failure [10].

I emphasize two limitations of the developed approach. Firstly, failure of materials exhibiting sound inelastic deformations is beyond the scope of elasticity with energy limiters, though in some cases the material evolution can be accounted for. Secondly, elasticity with energy limiters is a local continuum theory, which is not directly applicable to problems where failure tends to localize in material volumes smaller than the representative ones, e.g. cracks, and regularization techniques must be used to model sharp localizations of failure.

Finally, I should say that the introduction of energy limiters in elasticity is a guiding idea rather than an accomplished theory and iMechanicians can contribute much better models than those considered in the references.

References (PDFs can also be found on my website):

- Volokh KY (2004) Nonlinear elasticity for modeling fracture of isotropic brittle solids . J. Appl. Mech. 71:141-143
- Volokh KY, Trapper P (2008) Fracture toughness from the standpoint of softening hyperelasticity . J. Mech. Phys. Solids 56:2459-2472
- Trapper P, Volokh KY (2008) Cracks in rubber . Int. J. Solids Struct. 45:6034-6044
- Volokh KY (2007) Softening hyperelasticity for modeling material failure: analysis of cavitation in hydrostatic tension . Int. J. Solids Struct. 44:5043-5055
- Volokh KY (2007) Hyperelasticity with softening for modeling materials failure . J. Mech. Phys. Solids 55:2237-2264
- Volokh KY (2008) Prediction of arterial failure based on a microstructural bi-layer fiber-matrix model with softening . J. Biomech. 41:447-453
- Volokh KY, Vorp DA (2008) A model of growth and rupture of abdominal aortic aneurysm . J. Biomech. 41:1015-1021
- Trapper P, Volokh KY (2009) Elasticity with energy limiters for modeling dynamic failure propagation, submitted (and attached)
- Volokh KY (2009) An investigation into the stability of a shear thinning fluid . Int. J. Eng. Science 47:740-743
- Volokh KY (2009) On modeling failure of rubber-like materials, submitted (and attached)

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## Comments

## Nice concepts

Dear Kosta,

I read your JMPS papers recently and found the idea of energy limiters very interesting and conceptually appealing. One of the problems we are looking at in our research group is the description of damage or so-called plasticization in polymers due to diluent ingress. I am trying to figure out if we could cast the problem of elastic softening (one aspect of plasticization, the other being the glass transition temperature) into the framework that you describe. The core idea would be to identify right physics for the softening function within the context of the problem. We are still cranking our way through the chemical and physical changes to a polymer (e.g. epoxy) that manifest in modulus degradation.

It would be great to discuss the problem with you to pick your brain on this.

Thanks,

~Shailendra

## re

Dear Shailendra,

Good to hear from you . I do not know the physics behind the phenomenon you describe. However, my feeling is that you probably need the damage description of the sort that they use for a description of the Mullins effect in rubber. Your problem looks more like a problem of the material evolution rather than the complete bond rupture where energy limiters could be good.

-Kosta

## Question

Dear Prof Volokh,

I have some questions regarding your paper”

On modeling failure of rubber-like materials”

Figure 6 shows critical failure stretches based on hyperelasticity with energy limiters and experiment. As you have mentioned in the paper you have varied n (biaxiality ratio) to create different load conditions so each of the points on this plot corresponds to different biaxiality ratio. My question is why experimental data distribution is different compare to theoretical data. Thanks

Azadeh

## re: question

Dear Azadeh,

There are two possible answers:

1. The experiments have to be done more carefully. ( I like this answer)

2. The theoretical calibration should be done for the whole range of the biaxial data while it was done for the uniaxial failure only.

And, finally, I do not think that the experimental data is THAT different from the theory .

Sincerely,

Kosta