HI Michael. > To be honest, I have a hard time to understand the three-filed formulation in > Step 44.
That’s a fair comment, and a known deficit in our current tutorial list. I hope that by the end of the year we will have a tutorial that uses one-field elasticity, and would be a much better starting point for you and others (WIP: https://github.com/dealii/dealii/pull/10394 <https://github.com/dealii/dealii/pull/10394>). > However, I’m still interested in implementing the pure geometrical large > finite deformation model. I believe that will make things simpler and help me > understand more complex nonlinear models. Meanwhile, I try to compare the > results with a reference which only consider the geometrical nonlinearity. It sounds to me like you’re looking to implement finite strain elasticity with a St Venant-Kirchhoff constitutive law. That is, as far as I understand, the simplest constitutive law for finite strain elasticity. https://en.wikipedia.org/wiki/Hyperelastic_material#Saint_Venant%E2%80%93Kirchhoff_model <https://en.wikipedia.org/wiki/Hyperelastic_material#Saint_Venant%E2%80%93Kirchhoff_model> In this model, the elastic tangent looks like that for linear elasticity, but the strain tensor of interest is no longer the linear/small strain tensor, but rather the (nonlinear) Green-Lagrange strain tensor. (Nevertheless, even though the material law is linear, you still have the geometric tangent contribution that appears after linearisation of the residual, which comes from the linearisation of the now nonlinearly deformation-dependent test function “de” in \int de : \sigma dv .) If this is indeed what you want, then it is possible to swap out the Neo-Hookean constitutive law that is implemented in the code-gallery example with the St-Venant Kirchhoff constitutive law. You’d get “for free” the geometric stiffness contribution to the tangent, and would just have to worry about the material tangent and definition of the stress. I think that the only complication is the default parameterisation for the constitutive law — it was parameterised in terms of the left Cauchy-Green tensor as the whole problem is set up with the Kirchhoff stress (that’s the Cauchy stress, but per unit reference volume). Nevertheless, with the help of these “push-forward” transformations <https://www.dealii.org/current/doxygen/deal.II/namespacePhysics_1_1Transformations_1_1Contravariant.html> you would be able to use whatever parameterisation you’d like for the constitutive law (e.g. in terms of the Green-Lagrange strain tensor) and transform the resulting stress and material tangent to the correct (i.e. the spatial) configuration as is required for that implementation of the weak forms. The important thing to remember is that, despite all of the transformations involved, you’re still solving the same BVP with the same constitutive law irrespective of which choice of stress-strain (energetic) conjugate pairs you use, and irrespective of whether its expressed in the total or updated Lagrangian form. But anyway, this is just a suggestion. Irrespective of whether or not you choose to go that route, I wish you luck in implementing what you’re needing to. Best, Jean-Paul > On 25. Jun 2021, at 02:32, Michael Li <lianxi...@gmail.com> wrote: > > Andrew, thanks for confirming that. The missing 1/2 does not affect the > demonstration of functionalities of deal.II but it may change the results. > > Jean-Paul, thanks for commenting on my second question. I want to study the > pure geometrical nonlinear elasticity (large deformation with linear > material). It should be the simplest nonlinear model in elasticity. Step 18 > looks like a good start as an updated Lagrangian formulation; but it does not > include the nonlinear part of the finite strain. I spent some on Step 44 > because “ Quasi-Static Finite-Strain Compressible Elasticity > <https://dealii.org/current/doxygen/deal.II/code_gallery_Quasi_static_Finite_strain_Compressible_Elasticity.html>” > also mentions it is based on Step 44. To be honest, I have a hard time to > understand the three-filed formulation in Step 44. So maybe I should get into > “Quasi-Static … Elasticity” first which uses classical one-field formulation. > However, I’m still interested in implementing the pure geometrical large > finite deformation model. I believe that will make things simpler and help me > understand more complex nonlinear models. Meanwhile, I try to compare the > results with a reference which only consider the geometrical nonlinearity. > > -Michael > > > From: Andrew McBride <mailto:mcbride.and...@gmail.com> > Sent: Thursday, June 24, 2021 2:23 PM > To: deal.II User Group <mailto:dealii@googlegroups.com> > Subject: Re: [deal.II] Questions on Step-18 > > Hi both, > > Jean-Paul has addressed the second point nicely. On the first point, I think > there is a 1/2 missing. The curl of the velocity gradient is the vorticity > which is twice the angular velocity - hence I think you need a 1/2. Happy to > be corrected on this. > > Best, > Andrew > > > On 24 Jun 2021, at 21:03, Jean-Paul Pelteret <jppelte...@gmail.com > <mailto:jppelte...@gmail.com>> wrote: > > Hi Michael, > > I cannot comment on the first question, but might be able to assist a bit > with the second. But may I first ask, what precisely are you trying to > achieve with this extension? > > As interesting as it is, in the past I had found step-18 to deviate too > significantly from the “classical” approach to elasticity to be a natural > candidate extend towards finite strain elasticity, for example (this is > kind-of implied by the caveat that you partially quoted). I’ve been looking > at some of my textbooks (e.g. reviewing the topic of the updated Lagrangian > formulation in Holzapfel’s “Nonlinear solid mechanics” and Wrigger’s > “Nonlinear finite element methods”) to try to answer (2), but cannot > trivially reconcile the two approaches. I think that there’s a bit too much > going on to be able to correctly deduce by eye what the requisite changes > are. I think that you’d need to reformulate the balance laws and consider > their implications for the implemented weak forms — in particular, I think > that you’d be missing a contribution that (for finite deformation) looks like > the geometric stiffness, but there could be further differences that extend > from the overall approach taken to the problem. > > If you’re interested in an examples that are more closely aligned with what > you might see in the literature, then you can take a look at step-44 > <https://www.dealii.org/current/doxygen/deal.II/step_44.html> or the > “Quasi-Static Finite-Strain Compressible Elasticity > <https://dealii.org/current/doxygen/deal.II/code_gallery_Quasi_static_Finite_strain_Compressible_Elasticity.html>” > code-gallery example, which is effectively step-44 reduced to the one-field > total Lagrangian formulation that you’d find in many standard textbooks. It > would be easy enough to rework this to use the updated Lagrangian approach, > if that it what you desire. > > I hope that this helps you a little. > > Best, > Jean-Paul > > > > On 22. Jun 2021, at 15:24, Michael Lee <lianxi...@gmail.com > <mailto:lianxi...@gmail.com>> wrote: > > Hello, > I have two questions when studying Step-18. > > 1) Should there be a factor 1/2 when calculating the rotation matrix angle > (code in tutorial: angle > <https://www.dealii.org/current/doxygen/deal.II/grid__tools__nontemplates_8cc.html#a1b9d6e95246f7a6b7ecc7430631dd0b6> > = std::atan > <https://www.dealii.org/current/doxygen/deal.II/namespaceDifferentiation_1_1SD.html#a803fcea270d7a523a91e3b7c173059f9>(curl) > ) ? This is a mathematical problem. Can anyone give some material for the > formula to clear out my doubt? > > 2) In the introduction, it says "we will consider a model in which we produce > a small deformation, deform the physical coordinates of the body by this > deformation, and then consider the next loading step again as a linear > problem. This isn't consistent, since the assumption of linearity implies > that deformations are infinitesimal and so moving around the vertices of our > mesh by a finite amount before solving the next linear problem is an > inconsistent approach." My question is how to make it consistent. Can I just > add the nonlinear part of the strain tensor like the following (of course in > both get_strain functions)? I just want to consider the geometrical > nonlinearity. > > template <int dim> > inline SymmetricTensor<2, dim> > get_strain(const std::vector<Tensor<1, dim>> &grad) > { > Assert(grad.size() == dim, ExcInternalError()); > > SymmetricTensor<2, dim> strain; > for (unsigned int i = 0; i < dim; ++i) > strain[i][i] = grad[i][i]; > > for (unsigned int i = 0; i < dim; ++i) > for (unsigned int j = i + 1; j < dim; ++j) > strain[i][j] = (grad[i][j] + grad[j][i] + // linear part > grad[i][0] * grad[j][0] + //nonlinear part > grad[i][1] * grad[j][1] + > grad[i][2] * grad[j][2]) / 2; > > return strain; > } > > Thanks, > Michael > > -- > The deal.II project is located at http://www.dealii.org/ > <http://www.dealii.org/> > For mailing list/forum options, see > https://groups.google.com/d/forum/dealii?hl=en > <https://groups.google.com/d/forum/dealii?hl=en> > --- > You received this message because you are subscribed to the Google Groups > "deal.II User Group" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to dealii+unsubscr...@googlegroups.com > <mailto:dealii+unsubscr...@googlegroups.com>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/dealii/02f465df-3cab-4d84-bd48-92ca4a7981efn%40googlegroups.com > > <https://groups.google.com/d/msgid/dealii/02f465df-3cab-4d84-bd48-92ca4a7981efn%40googlegroups.com?utm_medium=email&utm_source=footer>. > > > -- > The deal.II project is located at http://www.dealii.org/ > <http://www.dealii.org/> > For mailing list/forum options, see > https://groups.google.com/d/forum/dealii?hl=en > <https://groups.google.com/d/forum/dealii?hl=en> > --- > You received this message because you are subscribed to the Google Groups > "deal.II User Group" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to dealii+unsubscr...@googlegroups.com > <mailto:dealii+unsubscr...@googlegroups.com>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/dealii/C2745EC6-0BED-4AFC-A21C-C003D851C869%40gmail.com > > <https://groups.google.com/d/msgid/dealii/C2745EC6-0BED-4AFC-A21C-C003D851C869%40gmail.com?utm_medium=email&utm_source=footer>. > > -- > The deal.II project is located at http://www.dealii.org/ > <http://www.dealii.org/> > For mailing list/forum options, see > https://groups.google.com/d/forum/dealii?hl=en > <https://groups.google.com/d/forum/dealii?hl=en> > --- > You received this message because you are subscribed to the Google Groups > "deal.II User Group" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to dealii+unsubscr...@googlegroups.com > <mailto:dealii+unsubscr...@googlegroups.com>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/dealii/D7696104-6C14-43F2-8B03-CF7AA0E89391%40gmail.com > > <https://groups.google.com/d/msgid/dealii/D7696104-6C14-43F2-8B03-CF7AA0E89391%40gmail.com?utm_medium=email&utm_source=footer>. > > > -- > The deal.II project is located at http://www.dealii.org/ > <http://www.dealii.org/> > For mailing list/forum options, see > https://groups.google.com/d/forum/dealii?hl=en > <https://groups.google.com/d/forum/dealii?hl=en> > --- > You received this message because you are subscribed to the Google Groups > "deal.II User Group" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to dealii+unsubscr...@googlegroups.com > <mailto:dealii+unsubscr...@googlegroups.com>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/dealii/5A41FB64-E568-44C2-82CA-2A7CAE923CD4%40hxcore.ol > > <https://groups.google.com/d/msgid/dealii/5A41FB64-E568-44C2-82CA-2A7CAE923CD4%40hxcore.ol?utm_medium=email&utm_source=footer>. -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en --- You received this message because you are subscribed to the Google Groups "deal.II User Group" group. 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