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Hypoelastic-plasticity with logarithmic spin

In small strain elastoplasticity we start off with an additive decomposition of the total strain into elastic and plastic parts. In terms of strain rates we write

$\displaystyle \dot{\ensuremath{\boldsymbol{\varepsilon}}} = \dot{\ensuremath{\boldsymbol{\varepsilon}}_e} + \dot{\ensuremath{\boldsymbol{\varepsilon}}_p}.<br />
$

Prior to 1990 most large deformation plasticity algorithms extended this idea by postulating an additive decomposition of the Eulerian stretching tensor (rate of deformation):

$\displaystyle \ensuremath{\boldsymbol{d}}= \ensuremath{\boldsymbol{d}}_e + \ensuremath{\boldsymbol{d}}_p .<br />
$ 

Typically, a hypoelastic constitutive model was used to compute the Kirchhoff stress. For isotropic materials such a model is of the form

$\displaystyle \ensuremath{\overset{\circ}{\ensuremath{\boldsymbol{\tau}}}} = \l...<br />
...ht)} \ensuremath{\boldsymbol{\mathit{1}}}+ 2 \mu \ensuremath{\boldsymbol{d}}_e<br />
$ 

A question that occupied many stalwarts of mechanics for a while was how to choose the correct objective rate of the Kirchhoff stress from an infinite number of possibilities. For various reasons, the Zaremba-Jaumann rate became popular. However, it was soon discovered that there was a problem with this rate. An oscillating shear stress response was predicted for a monotonically applied shear deformation when the Zaremba-Jaumann rate was used. The shear oscillation problem was ameliorated to some extent by using the corotational Green-Naghdi rate but other issues remained.

It was found that not only did the hypoelastic model deviate considerably from Hooke's law for large deformations, the model was dissipative and path-dependent, and could not be derived from a potential. Therefore the hypoelastic material considered above was not really elastic in the usual (hyperelastic) sense of the word. Over time people forgot that purely elastic materials could be obtained as a subclass of hypoelastic materials under some circumstances. The use of hypoelastic material models became taboo in the finite plasticity community.

As the plasticity community moved to problems that involved initial elastic anisotropy they also discovered that classical hypoelastic material models were only applicable to initially isotropic materials. This put the final nail in the hypoelastic/additive decomposition coffin. Some researchers (including me) continue to pursue the classical path saying that most metals undergo only small elastic strains. Also simulations show that for the errors introduced by the hypoelastic assumption are smaller than the error in the experimental data that are used to fit various models used in elastoplasticity. However, these justifications are not really satisfactory.

Researchers tried to find a way out of the mess and decided to change tracks. They started using a multiplicative decomposition of the deformation gradient and hyperelastic material models to represent the elastic response. This approach has been quite successful though some fundamental questions regarding uniqueness of the decomposition remain. The literature is huge and often hard to square with experimental data.

New life was breathed into the additive decomposition approach by the discovery by a number of researchers in the late 1990s that an Eulerian corotational logarithmic strain rate was power conjugate to the Kirchhoff stress [1].

The Eulerian Hencky (logarithmic) strain is defined as

$\displaystyle \ensuremath{\boldsymbol{h}}= \ln\ensuremath{\boldsymbol{V}}= \sum...<br />
...ath{\ensuremath{\mathbf{n}}_i\boldsymbol{\otimes}\ensuremath{\mathbf{n}}_i} .<br />
$ 

An isotropic elastic model in terms of the Hencky strain can be written as

$\displaystyle \ensuremath{\boldsymbol{\tau}}= \lambda \ensuremath{\text{tr}\lef...<br />
...ht)} \ensuremath{\boldsymbol{\mathit{1}}}+ 2 \mu \ensuremath{\boldsymbol{h}} .<br />
$ 

The newly found relationship between the Hencky strain and the rate of deformation was

$\displaystyle \ensuremath{\boldsymbol{d}}= \ensuremath{\overset{\circ}{\ensurem...<br />
...<br />
\ensuremath{\boldsymbol{\Omega}}^{\text{log}}\cdot\ensuremath{\boldsymbol{h}}<br />
$ 

where the logarithmic spin is given by

$\displaystyle \ensuremath{\boldsymbol{\Omega}}^{\text{log}} = \ensuremath{\bold...<br />
...suremath{\boldsymbol{m}}_i\boldsymbol{\otimes}\ensuremath{\boldsymbol{m}}_i})<br />
$

 
and

$\displaystyle \ensuremath{\boldsymbol{b}}= \ensuremath{\boldsymbol{F}}\cdot\ens...<br />
...uremath{\boldsymbol{m}}_i\boldsymbol{\otimes}\ensuremath{\boldsymbol{m}}_i} .<br />
$

 
It was shown that purely elastic behavior without path dependence or dissipation could be obtained if the objective rate in the relation

$\displaystyle \ensuremath{\overset{\circ}{\ensuremath{\boldsymbol{\tau}}}} = \l...<br />
...ht)} \ensuremath{\boldsymbol{\mathit{1}}}+ 2 \mu \ensuremath{\boldsymbol{d}}_e<br />
$

 
was given by

$\displaystyle \ensuremath{\overset{\circ}{\ensuremath{\boldsymbol{\tau}}}} = \d...<br />
...uremath{\boldsymbol{\Omega}}^{\text{log}}\cdot\ensuremath{\boldsymbol{\tau}} .<br />
$

It was also shown that this objective rate is the only one that is allowable in a consistent Eulerian rate theory of elastoplasticity.

Thus some of the major objections to using a hypoelastic model and an additive decomposition of the rate of deformation tensor have been removed, at least for moderate strains. It remains to be seen what modifications are needed to the theory to allow for initially anisotropic materials. I plan to try out the model in the next few days. If you have already done so please comment on what you found.

 

Bibliography

1   H. Xiao, O. T. Bruhns, and A. Meyers.
Elastoplasticity beyond small deformations.
Acta Mechanica, 182:31-111, 2006.

Comments

Konstantin Volokh's picture

Dear Biswajit,

Thanks for drawing our attention to this old and unresolved issue of the large-strain plasticity. The paper you quote is interesting though I am missing a point that you can probably clarify. It seems to me that what you and Xiao et al call Eulerian approach is not perfectly Eulerian because both Kirchhoff's stress and Hencky's strain require a reference configuration. A purely Eulerian approach should not consider any reference configuration, i.e. it should involve only Cauchy's stress and the rate of deformation tensor. The purely Eulerian hypoelasticity lacks physical grounds as it is well-known. The pseudo-Eulerian hypoelasticity that you and Xiao et al consider is not fundamentally different (except formulation) from the Naghdi or Lee approaches, which also rely upon the notion of a reference state.

-Kosta

Dear Kosta,

You are right that a reference state is needed in Xiao's formulation (I wouldn't claim it to be particularly knowledgable in their formulation considering that I found out about it only a few days ago following Andy's mention :) 

Indeed, any definition of strain requires a change of length.  Any change requires the knowledge of two states separated in time.  Therefore a strain, by definition, cannot be a purely Eulerian quantity.  Xiao's formulation uses the left stretch and therefore cannot be purely Eulerian.  The contortions provided by the logarithmic spin are needed to connect the left stretch to the purely Eulerian rate of deformation.  However, the Kirchhoff stress can be replaced with the Cauchy stress without changing any of their conclusions.   

My interest in Xiao et al.'s work was piqued by their claim that a linear rate constitutive equation using the corotational logarithmic rate of the Kirchhoff stress was the only such equation that could be derived from a potential.   Their approach appears to address some of the issues are paraded as severe weakness of the additive decomposition of the rate of deformation.  However, the complications involved in an incrementally objective numerical algorithm for Xiao et al.'s approach lessen its attractiveness.  Not that any of the other approaches are much better :)

Albert Meyers's picture

Dear Biswajit,

I am glad to see your elaboration  upon the role of the  logarithmic rate in elastoplasticity. I  am wondering that the use of the Kirchhoff stress is not generally accepted. Let me just comment on this and some other points.

  1. Of course, a  truly Eulerian constitutive law should be formulated in respect of proper Eulerian measures. The Cauchy stress tensor is not a proper tensor, the Kirchhoff stress tensor, however, is. There was an ardent discussion about that matter at the beginning of last century. Brillouin gave a review in [1]; see also the literature cited in his paper. By the way, this was the motivation for Hencky to revisit his  earlier elasticity law [2] and to exchange the Cauchy stress by the Kirchhoff stress [3].
  2. It is well known that a closed hyperelastic deformation path is without dissipation. It may be shown that a closed hypoelastic deformation path can only be without dissipation if the logarithmic stress rate is used (see [4] and related references therein). While the use of the logarithmic stress rate is necessary, it is not sufficient. Additionaly, the law must be transformable into the hyperelastic form. Hypoelasticity of grade zero is such a law. Thus, upon the logarithmic stress rate, at times it is possible to bring hypo- and hyperelastic formulations together.
  3. The elastoplasticity of Xiao&al. is free of the strain notion. It is a rate formulation in proper Eulerian measures, i.e. the Kirchhoff stress and the deformation rate, making the notion of strain obsolete.
  4. At some places the additive decomposition of the deformation rate is presented as being suitable for small deformation elastoplaticity only. The decomposition is based on the decomposition of the specific energy τ:D into a recoverable part τ:De and a dissipated part τ:Dp. Wherefrom does the limitation to small deformations emerge?

Albert

References
[1] Léon Brillouin. Les lois de l’é́lasticité́ sous forme tensorielle valable pour des coordonnées quelconques. Ann. de Phys., 10e sé́rie, t. III:251–298, 1925.
[2] H. Hencky. Über die Form des Elastizitä̈tsgesetzes bei ideal elastischen Stoffen. Z. Techn. Phys., 9:215–223, 1928.
[3] H. Hencky. The law of elasticity for isotropic and quasi-isotropic substances by finite deformations. J. Rheol., 2:169–176, 1931.
[4] H. Xiao, O. T. Bruhns, and A. Meyers. Elastoplasticity beyond small deformations. Acta Mechanica, 182(1–2):31–111, 2005.

Dear Albert,

Thanks for the references that you've provided.  Do you have an English translation of the Brillouin paper?

Some more questions:

1) You say that ".. The Cauchy stress tensor is not a proper tensor ...".  Could you elaborate?

2) You also say that the notion of strain is made obsolete by your formulation.  This is an interesting point of view.  Does that mean that rate-independent formulations are now passe?

Regarding your point about the limitation of the additive decomposition to small deformations, I believe that what people mean is that the definitions of de and dp should not be ad-hoc but based on finite deformation kinematics. 

-- Biswajit 

 

Albert Meyers's picture

Dear Biswajit,

Thank you very much for your remarks. Let me shortly respond to your questions:

The Brillouin paper is in French; I didn't encounter an English translation. Brillouin distinguishes proper tensors (independent of arbitrary frame changes) from pseudo-tensors, tensorial densities and tensorial capacities. He refers to Hermann Weyl's "Raum-Zeit-Materie" (space-time-material, a book in German containing lectures on relativistic mechanics) where details seem to be explained. I didn't read that book.  

There is no unique deformation counterpart to De and Dp. Why are people basing the decomposition of D on kinematics then? I prefer to relate it to the decomposition of the specific energy. This decomposition is not ad-hoc. A part of the envolved energy is dissipated, another is recoverable.

 D, De, Dp are time dependent quantities, i.e. the elastoplasticity formulation is rate independent. To see this: the total D is equal to the log rate of the total Hencky strain. Details are in our paper cited in the earlier answer. It may be interesting to note that the relation between the stretching and the strain rate is unique if corotational time derivatives are considered, i.e. only the logarithmic rate of the Hencky strain is equal to D. No other rate and no other strain may be related in this way.

Albert

An interesting paper on hypoelasticity has been published by Romano and Barrettaa: Covariant hypo-elasticity, 2011, European Journal of Mechanics - A/Solidsdoi:10.1016/j.euromechsol.2011.05.005 .  Any comments?-- Biswajit

Abstract:  The theory of constitutive behavior, with explicit reference to
hypo-elastic materials, is addressed with a geometric approach which,
following physical arguments, leads to a covariant formulation. The
essential role played by a careful distinction between spatial vectors,
material-based spatial-vectors and material vectors is emphasized.
Definite answers to debated issues, such as the proper definition of
stress rate, the formulation of integrability conditions, the fulfilment
of material frame-indifference, and the task of evaluating the stress
state evolution, are given.Simple shear and extension of a specimen of a
hypo-elastic material are investigated as applications of the
theory.Improper statements and unsound physical responses of
hypo-elastic materials are overcome by the covariant theory, thus
restoring the proper role to this constitutive model, for both
theoretical and computational purposes.

Giovanni Romano's picture

Dear Dr. Biswajit Banerjee

I have really appreciated your interest in the paper by me and Dr. Barretta (a former PhD student of mine) on "Covariant hypo-elasticity". I have read some of your comments on issues in iMechanica and I would be interested in your comments on the new ideas and results there exposed. The investigation described in that paper was motivated by a deep feeling of unsatisfaction for the common treatment of time-rates. It became readily manifest that a proper physical-geometrical framework should be designed to provide a clear picture of Continuum Mechanics and of rate formulations of constitutive relations.

The effort has now progressed to provide a treatment in the four dimensional events space which is certainly the most satisfactory from the geometric point of view. It is my conviction that geometric methods are not to be considered just as fashionable presentations of known results, but rather should be deemed as needed tools to get a proper formulation of basic theoretical and computational issues in Non-Linear Continuum Mechanics (NLCM). Unfortunately the still lasting lack of familiarity with the fundamentals of Differential Geometry has, up to the present time, prevented many clever scholars (including the writer) to avoid misformulations and sterile debates about ill-defined concepts. A suitable training into this mathematical discipline is required to get a scenic and effective view of the geometric construction underlying NLCM, but the effort is certainly worthwhile, and in any case unavoidable.

A comprehensive presentation of the geometric approach to Continuum Mechanics will be presented in a new paper by me and coworkers and I will submit it to you as soon as available.

Best regards

Giovanni Romano

P.S. please visit my home-page at the URL

http://wpage.unina.it/romano

where the slides of a recent General Lecture by me (September 13, 2011)

on the issue is available in the list of presentations.

 

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