Why rate equations in Nonlinear FE?

Dear Ramdas and Rui:

I'm not an expert in this topic but have a couple of comments that may be helpful.

1) As has already been mentioned by Biswajit, the existing objective time derivatives in continuum mechanics are all Lie derivatives (with respect to the spatial velocity field). As a matter of fact, the different objective derivatives are nothing but different representations of the same Lie derivative (using the metric tensor for raising and lowering indices). As is already known for a long time, there is no "more" objective stress rate.

One should note that the class of objective time derivatives is larger that just Lie derivatives. The following two papers explain this in some details.

i) F. Bampi and A. Morro, Objectivity and objective time derivatives in continuum physics, Foundations of Physics 10 (1980), 905-920.
ii)Jean-Luc Thiffeault, Covariant time derivative for dynamical systems, Journal of Physics A 34 (2001), 5875–5885.

2)The class of non-dissipative solids is much larger than hyperelastic solids. The following recent paper discusses some of these subtle points.

K.R. Rajagopal and A.R. Srinivasa, On the response of non-dissipative solids, Proceedings of the Royal Society A 463(2007), 357-367.

Regards,
Arash

I posted a question and just found this thread of
discussions. I post my question again.

http://imechanica.org/node/4725

In the above post, we showed that Truesdell rate can by
simplified to Green-Naghdi rate by assuming F .=. R and can be
further simplified to Jaumann rate by assuming W .=. R(.)R(T),
where .=. means approximately equal

In a stretch dominant deformation, the three rates give different
stress rate. This is usually explained that we need a different
tangential modulus for different objective rate. However, it is
hard to understand why we need to change "material" modulus when we use a
different "mathematical" form of objective rate as they are all supposed to be
"equivalent."

So a simple explanation may be that the Jaumann and Green-Naghdi rates
are "inaccurate" when stretch deformation dominants. As we know that
in a shear deformation, Jaumann rate gives a "sin" varying shear stress, while
Truesdell gives the exact answer (linear variation) (See Ted Belytschko, Wing Kam Liu, and Brian
Moran. Nonlinear Finite Elements for Continua and Structures.)  In other words,
the three rate forms might not be equivalent, espeically when stretch
deformation dominants

So a quick question is why are we often using Jaumann rate, instead of
Truesdell rate in large deformation FE analysis? From the viewpoint of
implementation, there should be no too much difference whichever rate we
choose.

Please see http://imechanica.org/files/stress%20rate.pdf for
the relationship among the three rates.

Rong Tian

Hello Ramdas,

I had the same questions some time back, but I found some helpful information about it.

I have not worked extensively on (formulation) coding etc, hence I dont know any reasons from the computational point of view for the above point. However, if we see this from experimental poit of view, it makes some sense.  

I guess we all know that when subjected to different loading rates, most of the engineering materials show significant changes in material response in the nonlinear portion (i.e., plastic deformation range). The elastic behavior of the material is not effected due to rate of loading (unless viscoelastic). This may be due to some microscopic mechanisms.

Hence,  the rate equatios may not be required in the elastic region to model the material response.

I hope this info is of some help to you.

Regards,

Siva 

There are two issues here:

1) Rate form virtual work (or the principle of virtual power)

2) Rate form constitutive equations

Neither of these is necessary for nonlinear finite element analysis, though rate forms are widely used (particularly in explicit codes).

"Computational Inelasticity" by Simo and Pister takes the approach of not using the rate form of the principle of virtual work.  That book discusses the problems with rate form constitutive equations for elasticity and ultimately proposes that total form hyperelasticity is the best way to go.  Rate forms are used for history dependent behavior for various reasons (some cited by by another commentor).

If you like the rate form of the principle of virtual work "Nonlinear Finite Elements for Continua And Structures" by Belytschko, Liu, Moran gives a ton of examples. 

Rate form constitutive equations are usually easier to contruct than total forms (and probably easier to express in incrementally linear form). 

I would like experts on constitutive relations to provide us with some insight into deeper reasons for the use of rate constitutive equations.

-- Biswajit 

    It seems that the "Rate" we are talking about has two meanings" "Rate" with respect to REAL time and "Rate" with respec to some parameters (for example, to the load increment).

   If the material behavior is rate-dependent or there is some energy dissipation in the system which is dependent on the velocity (for example velocity dependent friction) of the objects, then time related "Rate form" virtual work principle (virtual power principle) should be used. In this case, the test field is often named as "virtual velocity" and the basic unknow variable is the TRUE VELOCITY field in the structure.

   For the analysis of hyperelastic material, since this is a conservative system, the equilibrium equation can be expressed in "Total form". But this "Total form" equation(s) is (are) nonlinear in terms of state variables, such as displacement field, etc. Therefore lineariztion and iteration are necessary for the solution of it. At every step of iteration, an "Incrementally linear form" of virtual work principle should be used. In this case, the test field is often named as "virtual diplacement increment " and the basic unknow variable is the TRUE (in the sense of linearization) incremental diplacement  field in the structure assocaited with the current load step. The tangent modulus is also obtained in this linearization process.

  For the palsticity analysis of time rate-independent material, since it is always path-dependent, so "Increment form" must be used. In this case, At every step of iteration, an "Incrementally linear form" of virtual work principle should also be used.

  From my point of view, for RAEL-time independent problems, the term "Incrementally linear form" is more appropriate than "rate form" Of course, you can also think that "rate" is the rate with respec to the load increment...

   Hope this helps.
 
   best regards
    X.Guo

1) Rate equations are not needed even if the constitutive law is rate dependent.

e.g. S = A E + B dE/dt

is not a true "rate equation" even though dE/dt appears on the right.  We can just consider dE/dt to be an independent variable.

2) A  rate constitutive relation is often of the form

dS/dt = A(E) dE/dt

Such a relation can have various uses as discussed in earlier comments.

3) You don't "need" rate equations unless rates appear in some part of your formulation.  This is especially true of quasistatic problems.  Once again see the discussion with Rui.

4) The easiest way to find out is to generate your own contact formulation and see what constitutive model is needed.  Belytschko's book uses a virtual power approach.  That is not a must.  You can replaced the variation of the velocity with a variation of the displacement and work from there.

-- Biswajit