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Why rate equations in Nonlinear FE?

ramdas chennamsetti's picture

Hi all!

I have a very fundamental question as follwing.

In Nonlinear FE formulations, we use rate equations (virtual work), but, in linear FE we don't use rate equations. Why???

Is it because Nonlinear solution is iterative solution (time may be virtual time).

I request those who have an idea to give some explanations.

Thanks in advance,


- Ramdas



Rui Huang's picture

I had the same question in mind some time ago. I believe you are right that the rate equation is needed for solving the nonlinear equation iteratively, for example, by the conjugate-gradient method. I am now facing another question about the rate equation. There are definitions of several different rates (Lie rate, Jaumann rate, etc.). Which one shall we use? Why?



arash_yavari's picture

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.


For more on stress rates see my notes posted on Wikiversity at

Which rate you use is not particularly important.  What is important is that your material parameters should be such that the rate that you use reflects reality.  That is, the parameters should be determined keeping a particular rate in mind and not just be the same as those used in small strain elasticity (for instance).   And you have to keep in mind that first-order hypoelasticity is not history-independent.

The Jaumann rate is popular because it's easy to implement. 

Rong Tian's picture

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

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

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

Please see for
the relationship among the three rates.

Rong Tian

ramdas chennamsetti's picture

Thank you!!! I downloaded your notes.

Here, I want to understand why the rate equation came into picture (not which rate equation it is)? What is the origin for this?

Why virtual work has to be expressed for per unit time? why can't we formulate non-linear FE frmulations without rate, using virtaul work without rate?

Kindly explain.

Thanks in advance,


- Ramdas 

Siva P V Nadimpalli's picture

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.




Hello Siva,

What you mentioned in your post is about the constitutive behavior of materials.

However we should distinguish that the rate form of constitutive equation is not a same concept as rate dependent.

In case of constitutive equations of nonlinear plasticity, the rate of loading also doesnt affect the plastic behavior.

But the rate form is still widely used.

That is because using the rate form we can get the full stress-strain path from incremental description, which is much more convinient.

Suppose the materials undergo cyclic loading or ratchetting, in such a case  using rate form can easily describe the cyclic stress-strain behavior, especially the plastic loading-elastic unloading transition. But using total form it may be a big problem because at the same strain you may have more than one stress value. You have to make a lot of effort on programming that situation.

The rate dependent (viscoplasticity/viscoelasticity) thing is totally another topic. Maybe we can start another thread to talk about it.


Xiaoteng Wang

typical ratchetting s-s curve,

Siva P V Nadimpalli's picture

Xiaoteng Wang,

Thanks for your explanation, I guess I didn't get the question properly. Being an experimental person when the rate was mentioned I was looking it from the constitutive behavior point of view. You are absolutely right, now I get what the question is mainly about.



Thank you for your site. I have found here much useful information...

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 

Rui Huang's picture

It appears to me that even for the total form hyperelasticity the rate equation is needed to define the tangent modulus tensor in the process of solving the nonlinear equilibrium equation. This is the question I have since the definition of tangent modulus depends on which stress rate you use. For example, in ABAQUS UMAT, a symmetric tangent modulus tensor is assumed, but it is not clear to me how the tangent modulus is defined and how it is used to solve the nonlinear equation. 


The tangent modulus (or Jacobian) defined by user is called by ABAQUS to formulate the initial stiffness matrix K.

Then ABAQUS solves the nonlinear equation f=Ku  using itterative method.

A symmetric tangent modulus tensor gives better itterative stability.

To formulate Jacobian, you have to write the relationship between stress increment tensor (ds) and total strain increment tensor (de) first. This relationship is not so easy to formulate but from the constitutive equations we can derive it finally.

Then the first order derivative d(ds)/d(de) is the material Jacobian J. To achieve that goal you need a bit tensor analysis.

Luckily we dont need a super exact J, we just need a J approximately close to the true value to help the solver solving the equation without so many itterations.

Once you defined your own constitutive model, you can always formulate the J. Since the relationship between plastic strain increment, total strain increment and stress increment is already defined by yourself.




1) The tangent modulus is dsigma/depsilon.  If you have sigma(epsilon) in closed form then the tangent can be calculated without resorting to a rate form. 

2) You're right that a rate equation is often used while deriving a consistent tangent modulus.  However, that rate equation can be generated by taking the time derivative of a hyperelastic relation in the material  frame and then pushing it forward to the spatial frame if needed.  That is equivalent to taking a Lie derivative but consistent with elastic material behavior (without history dependence).  You can check out my writeup at

3) You can get an exact relation for the consistent tangent modulus (unless the elastic moduli depend on pressure) after a bit of algebraic trauma.

That's not the case for hypoelastic models - no matter what objective stress rate you choose.

4) The tangent modulus is often unsymmetric for complicated material models, e.g., soil plasticity with non-associated flow rules.   I hope ABAQUS allows for unsymmetric tangent moduli but don't know enough about that software.

Prof. Govindjee is an expert on some of these issues as is Prof. Acharya.  Let's hope they can find the time to chime in and give their inputs.  I've found Simo's 1988 papers in CMAME particularly useful (though very dense). 

-- Biswajit 

 For what is worth, since you asked.  Here are my two bits:

1) There is nothing fundamental about using rate forms.  One can equally
well use non-rate forms (for the cases being considered).  Sometimes, however, people find it easier to think in terms of rates.

2) Not all objective rates seen in continuum mechanics are
simple variations of the Lie derivative -- though many can be
expressed as reasonably simple variation of a Lie derivative
(simple being in the eye of the beholder).  What really counts is: Can you more
easily express the behavior of your material in terms of one rate or

3) The history of rate forms in mechanics and computational mechanics in
particular is more for convenience than anything else.  Written in rate
form the constitutive relations of finite deformation look a lot more like
those of the small strain formulation.  This made it easier for people
think about models.  Unfortunately when computation came along this also gave people the false impression that they knew how to implement them in finite deformation
simulation codes.  Note that I say think!  They actually did not do it
correctly from a purist point of view.  Even today many commercial code do
not do it correctly (elastic or inelastic); they leave out terms
which results in constitutive evaluations that violate the 2nd Law of Thermodynamics.
Simo and Pister have a nice paper on this.  Here too is reference to a small paper on some aspects of this issue:

Govindjee, S., ``Accuracy
and Stability for Integration of Jaumann Stress Rate Equations in Spinning
Bodies," Engng. Comp., 14, 14-30 (1997).

4) The rate forms are only helpful for expressing the tangent matrix needed
in finite element computations when the rate form is EXACTLY integrated in
the constitutive evaluation subroutine.  Otherwise what the global Newton-Raphson iteration needs is the algroithmic tangent matrix; i.e. the tangent matrix
associated with the algorithm used to integrate the rate form.



Prof. Dr. Sanjay Govindjee
University of California, Berkeley

WaiChing Sun's picture

Dear Professor Govindjee,

     I have heard comments which says that the non-assicoate plasticity model may  violates the second law of thermodynamcs even for small deformation case. Would you mind to give us some reference about this topic? 





I guess we all know that when subjected to different loading rates,
most of the engineering materials show diploma at home 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

high school ged | diploma for home school

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. Essay Writing Help

    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

Rui Huang's picture

My thanks to Xu, Xiaoteng, and Biswajit for helping me out with the tangent modulus in hyperelasticity. Biswajit's writeup at wikiversity is particularly helpful with detailed tensor algebra.

Starting from an arbitrary strain energy density function for hyperelasticity, I was able to reach the total-form nonlinear equation in the weak form for finite element. To solve this nonlinear equation by iteration, I have to linearize it in an incremental form. Here I ran into the question about the tangent modulus, because the increment of the Kirchhoff stress consists of two parts. One can be written as c:d where c is the fourth elasticity tensor in Biswajit's write up. In addition, the increment also depends on the true stress at the current state as well as the rotation tensor, which is unsymmetric in general. This second part disappears by using the Lie derivative. However I believe both parts are needed to solve the nonlinear equation. In ABAQUS UMAT, it seems that only the first part is needed. My guess is that the second part is taken care of automatically by the nonlinear solver, since it does not really depend on the material. But I am not certain. I would appreciate comments from any ABAQUS experts.


Hello RH,

In UMAT the default formulation is hypoelastic, not hyperelastic. If you activate the large deformation option, hypoelastic formulation is adopted and the rotation tensor R is automatically  calculated and passed by the solver. That means ABAQUS calculates and passes the state variables in a default hypoelastic structure, and automatically rotate the stress rate to the Jaumann rate. The second part which you mentioned before is calculated by ABAQUS but maybe it is incompatible with other formulation....In your case you write your constitutive model based on hyperelastic hypothesis, thus you have to be careful to define your own stress tensor and store it to the state variable matrix and you'd better not use the stress increment tensor given by ABAQUS. 



Xiaoteng Wang

without rate equation you can not express an exact path.

A virtual work  W=int(x(y)dy), where x is load, usually an implicit function of y, and dy is virtual displacement.

The equation above is an integration process.

There are thousands of possible paths to get a same W.

The rate equation describes the exact path implicitly.

In linear FE formulation, the relation between x and y is simply linear, so we can directly write the virtual work.

Also in nonlinear FE, if you can get the integrated function explicitly, you can formulate it without rate form.

ramdas chennamsetti's picture

Hi all!

Thank you for good discussion. I went through the thread.

 I understood the use of rate equations if constitutive law is rate dependent. I couldn't really understand why rate equations are required for rate independent constitutive law.

Say, I am working on geometric nonlinearities or contact analysis, do I need to use rate equations? Please explain in simple terms.

Thank you,


- Ramdas

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 

ramdas chennamsetti's picture


Thank you!!!

Right. I am following Belytschko's book. The approach is virtual power. Do you mean to say that, variation of spatial velocity gradient has to be replaced with variational strain. Then we need to modify constitutive law accordingly. Am I right?

Thank you,

With regards,

- Ramdas

ramdas chennamsetti's picture

Prof. Govindji Sir,

Thank you very much for your nice explanation. Thanks to Dr. Biswajitda and all others those who particpated in this thread.

With regards,

- Ramdas

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.

distinguish that the rate form of constitutive equation is not a same concept as rate dependent.
In case of constitutive equations of nonlinear plasticity, the rate of loading also doesnt affect the plastic behavior. But the rate form is still widely used.That is because using the rate form we can get the full stress-strain path from incremental description

Zhigang Suo's picture

I wrote up the notes for my lectures on frame indifference and postes them online.  Several people made comments.  Both the comments and the notes might contribute to this thread of discussion.  

Note a constructive introduction of the objective time derivatives based on the formulation of solid mechanics as a simple Lagrangian system in [1]. This enables to distinguish between deformation rates, which are in principle Lie derivatives, and stress rates, which are actually covariant derivatives along a curve representing a deformation process. Besides, the role of Daleckii-Krein formula in understanding the theory with generalized strains is highlighted, and a special attention is also paid to the logarithmic time derivative.

[1] Fiala, Z.: Objective time derivatives revised. ZAMP 71, Article number: 4 (2020)

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