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The Baker-Ericksen inequalities

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 lauo
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(@lauo)
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Hello everybody

By researching about what restrictions should be applied to the strain energy density function of a hyperelastic model, I found some very interesting papers about the Baker-Ericksen inequalities. Truesdell in 1956 had already set some restrictions about the positivity of the derivatives of the strain energy function but some authors later showed that they are overly restrictive.

My question is if have someone here already applied the BE inequalities to any model and how have you done this, because Im getting confuse about how can I apply them to my equation.

Thank you all,

regards

Felipe Stumpf, Brazil

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(@jorgen)
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The conditions that you mention sound somewhat familiar, athough I dont recall the details at this time 🙁

I dont recall either seeing a paper that discusses the use of those inequalities for hyperelaticity. What people tend to use is the second law of thermodynamics in the form of the Clausius Duhem inequality. I have used that approach on different models.

-Jorgen

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 lauo
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(@lauo)
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Actually, theres a lot of authors applying the Baker-Ericksen inequalities for different hyperelasticity models, such as Marzano, Bilgili (who applied them to the Yeoh model), Criscione and Wilber, Hartmann, etc.

What Baker-Ericksen inequalities try to ensure is that the principal stress is in the same direction of the principal stretch, garanteeing physical meaning to the deformation process.

As far as I understood, Professor, the thermodynamical Clausius-Duhen inequality is applied when there is some kind of cyclic process involving loss of energy, such hysteresis effect. So, if my model does not catch these effects, the Clausius-Duhen inequalities would not be applicable, is that correct?

Another interesting restrictions to the strain energy function I have read about are those suggested by Truesdell (1956), which intends to garantee the positivity of the derivatives of W.

Ive read also about the mathematical restriction of the policonvexity of W, but some authors showed this is too restrictive.

So, is someone aware of this restrictions to W or have already applied them to any model?

Regards,

Felipe Stumpf, Brazil

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(@jorgen)
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Very Cool - thanks for the info.
Can you provide one or two main references. I would like to learn more, as I am sure other readers here...

Well, yes, the Clausius-Duhem inequality is mainly of interest for materials that dissipate energy. For hyperelasticity its a non-issue.

Your questions are very good indeed. I have not studied the polyconvexity issue in detail, but as you know, there are certainly books, papers, etc, about that.

To be honest, I have more or less stopped my hyperelastic research efforts. In my opinion (for now), hyperelasticity theory is very interesting, but the need for refined hyperelasticity models is (probably) of limited engineering significance. I could be wrong, but I find the viscoelasticity and viscoplasticity theories more interesting from a practical engineering viewpoint. That said, I encourage you to pursue theoretical developments in hyperelasticity if that interests you. There is certainly an interest in the academic community.

-Jorgen

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Posts: 32
 lauo
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(@lauo)
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The main reference about the Baker-Ericksen inequalities is:

Baker, M. and Ericksen, J.L. (1954). Inequalities Restricting the Form of the

StressDeformation Relations for Isotropic Elastic Solids and ReinerRivlin

Fluids, J. Wash. Acad. Sci., 44: 3335.

Another reference about restrictions to the energy function was written by Truesdell in 1956, which imposes the positivity of the derivatives of this funtion.

Truesdell, C.A. (1956). Das ungelste Hauptproblem der endlichen Elastizittstheorie, ZAMP, 36: 97103.

- Felipe Stumpf

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(@jorgen)
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Thanks for the references!

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