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Constitutive models for rubber

Hi.

I found a very useful explanation about hyperelasticity here: http://en.wikipedia.org/wiki/Hyperelastic_material#cite_note-Ogden-1

 In the proof3, there is a good explanation (I can't found anything better than this one) of how we can express the Cauchy stresses in terms of the principal streches.

I am interested to incompressible material, so this is the equation:

eq1

 

But I have a problem here, because I found in many papers (for example here: http://www.mscsoftware.com/Submitted-Content/Resources/WP_Nonlinear_FEA-Elastomers.pdf  - p.11, last equation) that we can get the Cauchy stress in the direction 1 (because it is got from uniaxial tension test data), assuming a Neo-Hookean material, as:

 

eq2

 

that is different from the first equation, because in this one there isn't the stretch ratio to multiply the derivate of the energy W.

Why? Are they the same equations? 

And, finally, the last question is: why do we use the pressure p as "Lagrange multiplier"?

I hope in a reply, thank you very much.

Italo 

 

 

Matt Lewis's picture

HI Italo,

The first equation is correct for the Cauchy stress in an incompressible material.  The second equation is for the first Piola-Kirchoff stress, which some refer to as the nominal stress.  This all has to do with stress measures. 

The problem with the pressure is that in an incompressible material it is not solely a function of the deformation.  The stress calculated from the deformation for an incompressible material will only be a deviatoric stress.  The pressure field consistent with a deviatoric stress field must satisfy boundary conditions and equations of equilibrium (or of motion if the problem is dynamic).

To get a real understanding of this, I suggest you delve a little deeper into Ogden or into Holzapfel's Nonlinear Solid Mechanics. Cool

Matt Lewis
Los Alamos, New Mexico

Dear Matt,

 

thank you so much. You were able to clarify me this little doubts about these argouments.

Now I understand.

Italo 

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