Dear Jean-Paul,
Thank you so much for your timely and detailed response. If I repeat what I
understood is
following, in the given expression [image: J][image: [\widehat{p}\,
\mathbf{I}\otimes\mathbf{I}-2p\mathcal{I}]] with [image: \widehat{p} = p +
J\dfrac{dp}{dJ}] for the case of
three variable formulation [image: p] and [image: J] are independent
hence [image:
\widehat{p} = p] , as [image: \dfrac{dp}{dJ} = 0].
Regards,
Navneet R
On Fri, Mar 6, 2020 at 2:47 AM Jean-Paul Pelteret <[email protected]>
wrote:
> Dear Naveet,
>
> I think that I see where things are not explained completely clearly. The
> sections “Hyperelastic materials”, “Neo-Hookean materials”, and “Elasticity
> tensors” all describe the theory for compressible materials. This is
> specifically mentioned in the “Neo-Hookean materials” part, and was
> probably done to give context for the modelling of quasi-incompressible
> models (although I don’t remember this with 100% clarity). So the “p” and
> “\hat{p}” that are mentioned there are the pressure-like terms that can be
> computed directly from the constitutive law for compressible materials.
> These, however, no longer apply for the 3-field formulation.
>
> The strain energy function used in the section “Principle of stationary
> potential energy and the three-field formulation” (which derives the
> governing equations as implemented in the code) has the same form as that
> before, but different arguments. Specifically, the volumetric part now
> depends on the dilatation field “\tilde{J}" rather than the point-wise
> volumetric Jacobian “J". Similarly, the pressure response “p" is no longer
> computed from the energy density function, but is a field variable
> “\tilde{p}" itself. How these relate to the original volumetric/isochoric
> stresses and tangents is captured in the underpriced parts of the equations
> in “Principle of stationary potential energy and the three-field
> formulation”, namely the residual equation and its consistent linearisation.
>
> What you had written is indeed correct for one-field elasticity. There is
> a code-gallery example (one-field elasticity) that use this exact
> expression for the volumetric material tangent:
>
> https://github.com/dealii/code-gallery/blob/master/Quasi_static_Finite_strain_Compressible_Elasticity/cook_membrane.cc#L654-L666
> and another 3-field elasticity example that uses the definition as given
> in step-44:
>
> https://github.com/dealii/code-gallery/blob/master/Quasi_static_Finite_strain_Quasi_incompressible_ViscoElasticity/viscoelastic_strip_with_hole.cc#L561-L568
>
> So, in summary, the definitions of the volumetric stress and tangent
> differ to one-field elasticity due to the introduction of the additional
> fields. This follows directly from the expression of the stationary point
> and linearisation, rather than from the constitutive law in the code.
> However, one can interpret various terms in the residual and linearisation
> as the equivalent of the volumetric stress and tangent (even though they
> only indirectly governed by the constitutive law, since they yield the
> pressure response to the added primary field). That’s what we tried to
> capture in the code.
>
> I hope that this helps clarify any misunderstanding. Suggestions as to how
> this can be made more clear in the introductory text are always welcome, so
> please feel to open a pull request for them.
>
> Best,
> Jean-Paul
>
> On 05 Mar 2020, at 12:51, navneet roshan <[email protected]> wrote:
>
> Dear delii members
>
> While modifying the material model in the step-44.cc, implementation of
> the function *get_Jc_vol(). *I found the implementation in the code is
> different than the formula provided. The worse part is that the solution
> stops converging after correcting the implementation. I will be really
> grateful if some one can hint me as, I am stuck between convergence and
> divergence of material models from quite some time.
>
> The given implementation is:
> SymmetricTensor<4, dim>
> <https://www.dealii.org/current/doxygen/deal.II/classSymmetricTensor.html>
> get_Jc_vol() const
> {
> return p_tilde * det_F *
> (Physics::Elasticity::StandardTensors<dim>::IxI
> <https://www.dealii.org/current/doxygen/deal.II/classPhysics_1_1Elasticity_1_1StandardTensors.html>
> -
> (2.0 * Physics::Elasticity::StandardTensors<dim>::S
> <https://www.dealii.org/current/doxygen/deal.II/classPhysics_1_1Elasticity_1_1StandardTensors.html>
> ))
> }
>
> The implementation as per formula, should have been:
>
> SymmetricTensor<4, dim> get_Jc_vol() const
> {
> return det_F * ( (get_dPsi_vol_dJ() +
> det_F*get_d2Psi_vol_dJ2() )
> *Physics::Elasticity::StandardTensors<dim>::IxI
> - (2.0 * get_dPsi_vol_dJ() *
> Physics::Elasticity::StandardTensors<dim>::S) );
> }
>
> Thank you,
>
> Regards,
> Navneet R
>
>
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