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Covariant and contravariant stress tensors: difference between the Cauchy stress and the 2nd Piola-Kirrchoff stress

I am currently reading a book about "Material Inhomogeneities in Elasticity" written by Gérard Maugin.

He brings up the covariancy and contravariancy of some well-known stress tensors and in particular that of the Cauchy stress tensor. I was not very familiar with the concept of covariance, so i read up on this very specific tensorial notion. 

I have been able to demonstrate easily that the deformation gradient F is a mixed tensor (covariant and contravariant).

But i still don't understand why the Cauchy stress tensor σ is said to be covariant.  I know that covariancy and contravariancy has to do with how the components of the tensor are trasformed along with a change of coordinates. But it is not clear to me why the Cauchy stress tensor σ is covariant while the 2nd Piola Kirchoff stress tensor S is contravariant. 

Since  σ is defined as force per unit surface in the deformed configuration while S corresponds to a pull-back onto the undeformed configuration, i believe that the covariancy or the contravariancy has to do with the configuration accounted for by the diffent stress tensors introduced above.

I am looking for a demonstration or an explanation showing and proving why the Cauchy stress tensor is covariant whereas the 2nd Piola-Kirrchoff stress tensor is contravariant.

 

Let me know what you know or think about that.

 

Regards,

 

Malik Ait-Bachir

Hi,
As far as I know Cauchy stress is contravariant.
I try to explain this whole issue to you as I understand it.

Define a covariant triad consisting of covariant base vectors

for I=1,2,3.
Use this base for the construction of a contravariant triad consisting of contravariant base vectors:



If an orthogonal Cartesian coordinate system is used (this can be considered as standard, other systems will be cumbersome) covariant and contravariant bases are the same.
I think that this is the reason why many books on continuum mechanics skip a detailed explanation on that, because it is not crucial in the beginning.

Remind:

If something has to be obtained, which cannot be described locally through a scalar (as temperature for instance), it is described via tensor analysis (with all the rules of covariant and contravariant transformations etc.).
However, this tensorial item described by components and base vectors can be constructed as a pure covariant, mixed covariant-contravariant or pure contravariant tensor.
The described item stays the same in each description (as seen on invariants for example), but each tensor description looks differently and relies on mentioned base vectors.

A point in reference configuration is described by vector
.

Hence, through covariant component and contravariant base vector.
Normally, just components are written, therefore in indicial notation vector X is called covariant.
A point in current configuration is described by vector
.
Hence, through contravariant component and covariant base vector.
Why that was done in this particular way and not differently is, in my opinion (maybe I am wrong), arbitrary.
Nevertheless, there is a consensus on that.

The deformation gradient results out of that as:


Right Cauchy-Green deformation tensor as


and at last Green-Lagrange strain


Components of Green-Lagrange strain are covariant.
Remind Einstein notation: if an index appears twice, once covariant and once contravariant, they are summed up.
Green-Lagrange strain E and Second-Piola-Kirchhoff stress S are work-conjugated.
To obtain (scalar) work W components have to be written as (using double contraction):
.

The components of Second Piola Kirchhoff stress are contravariant, however, base vectors covariant:


Cauchy stress can be evaluated as "push-forward":


Finally:
.

Cauchy stress has contravariant components. 

Arash_Yavari's picture

Dear Malik:

A tensor field is a multi-linear map from a finite number of Cartesian products of tangent and cotangent spaces of your manifold (let's say M) to real numbers.
    
Given a manifold (something that locally looks like an Euclidean space but is globally curved), you can define a tangent space at each point (you can have a sphere in mind and the tangent plane at each point). A tangent space is a finite-dimensional linear space of the same dimension as that of the manifold. Tangent space being a finite dimensional linear space, you can pick a basis {E^A} for it. Physically you can think of a tangent space as the space of "velocities" at that point. There is always a dual space to tangent space called cotangent space. This is the space of linear functionals on the tangent space, i.e. given any vector in the tangent space a covector associates a scalar to it. Physically you can think of cotangent space as the space of "forces". Given a "velocity", "force" as a dual acts on it and gives you power (a scalar).

Having the basis {E^A} for the tangent space you can pick a basis {E_A} or {dX_A} for the cotangent space at the same point. One thing to note is that the the notion of dual does not require any metric structure.

By definition, any index that appears as a superscript is called a contravariant and any index that appears as a subscript is called covariant.
    
A two-point tensor is a tensor field that has two legs, i.e. it is defined with respect to two manifold. Deformation gradient is a two-point tensor. In the representation F^a_A, "a" is contravariant and "A" is covariant.
    
If you add more structure to your manifold, e.g. introduce a metric tensor g, then there will be a natural identification of tangent and cotanegt spaces: given a vector v^a, g_{ab}v^b is the associated covector. In this sense and in the case of Riemannian manifolds, one can have different equivalent representations for the same tensor field. If you write \sigma^{ab} for Cauchy stress, it means t^a=\sigma^{ab}n_b, where "n" is a unit normal vector and "t" is traction. Having the vector "n" you use the metric to make it a one-form, i.e. n_a=g_{ab}n^b and think of traction "t" as a vector by index raising using the metric tensor again. So, in this sense there is nothing deep behind "covariant" and "contravarant". The important thing is to know where the tensor lives, acts on what and gives what.

In the case of Cauchy stress, it acts on a unit normal of a deformed surface and gives a force in the deformed configuration. Cauchy stress being a tensor is a consequence of Cauchy's theorem. You can have different associated tensors that are contravariant or mixed but at the same time describe the same physical quantity.

For solids, it is desirable to formulate everything in terms of a reference configuration. First Piola-Kirchhoff stress acts on the unit normal to an undeformed surface and gives you the real force in the deformed configuration, while the second Piola-Kirchhoff stress acts on the same unit normal but gives the pulled-back force.

The book by Maugin should be read with caution when it comes to geometric ideas.

I hope this helps.

Regards,
Arash

Both of you explanations are pretty clear to me in the sense that they sum up some key points of the theory. However i have a few more questions...

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Dear Manfred, in your comment you  consider Riemann manifolds with a metric structure and then you make use of the metric tensor which clearly illustrates your remarks.  However, starting out from the beginning of your reasoning, you could also have defined a  contravariant triad consisting of contravariant vectors and then make use of the opposite description to describe a point in the reference configuration and a point in the current configuration:

Χ=Xi gi and x=xI GI

This yields the following results:

Deformation gradient: F=FIi GIgi

Right-Cauchy Green Tensor:

C=Cij gi gj

Green-Lagrange strain: 

E=Eij gi gj

 

As you said, the fact that is was done your way but not this way (by "this" i mean in my comment)  is, in your opinion, arbitrary. Frankly, i don't know how to solve this specific problem but it makes me think that the Cauchy stress can be arbitrarily covariant, contravariant or mixed covariant -contravariant.

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Dear Arash, 

Thanks for your precious help. Yours comment are very helpful in the sense that they remind me of the general theory with metric structures as a particular case. It is now much clearer to me what the different theoretical notions mean.

However i have a few more questions...

When you wrote about the deformation gradient F, you considered a mixed coraviant-contravariant representation. Is it an arbitrary choice?

Indeed you said that in a Riemann manifold with a metric structure,   getting a covariant or contravariant or even mixed representation is just a matter of mathematical transformation. This latter remark makes me think and believe that the Cauchy stress tensor σ and even the deformation gradient F can take on any representation i want. So speaking of contravariancy or covariancy for any of the stress tensors would not make any sens ...???

 

Actually i am missing some good points of your comment ine the last two paragraphs.

 

Indeed you say: "The important thing is to know where the tensor lives, acts on what and gives what."  I got the first speech but i didn't get the meaning that lies behind this relevant remark. Do you mean that due to its properties, the Cauchy stress tensor σ is more likely to be used in its covariant representation? Cause my question is more about the statement by Maugin: "The Cauchy stress is covariant". My question is simply why ???

If in a Riemann manifold with a metric structure any representation can be used to describe a tensor then why Maugin picked the covariant one for the Cauchy stress?

Finally you consider a contravariant basis for the Tangent space and thus a covariant basis for the Cotangent space. Would it make sense to consider the opposite: a covariant basis E_{A} for the tangent space and an contravariant basis E^{A} for the cotangent space.

 

I look forward to hearing from you soon.

Once again, i want to thank you both for your comments.

Malik

Arash_Yavari's picture

Dear Malik:

1) "When you wrote about the deformation gradient F, you considered a mixed coraviant-contravariant representation. Is it an arbitrary choice? "

---> F acts on a vector in the reference configuration and gives you another vector in the deformed configuration. In this sense F is a vector-valued one-form. The representation F^a_A really means that F=F^a_A e_a \otimes E^A. If you like to define F in terms of what it does on the corresponding one-forms you will get other representations, which are all related through the metrics G and g.

2) ".... This latter remark makes me think and believe that the Cauchy stress tensor σ and even the deformation gradient F can take on any representation i want. So speaking of contravariancy or covariancy for any of the stress tensors would not make any sens ...??? "

---> It does make sense to say a given tensor is covariant or contravariant or mixed, however all these can, in general, represent the same physical quantity.

3) "Do you mean that due to its properties, the Cauchy stress tensor σ is more likely to be used in its covariant representation? Cause my question is more about the statement by Maugin: "The Cauchy stress is covariant". My question is simply why ??? "

---> One way of looking at this is the following. You choose your deformation measure and then you will have to work with the corresponding energy conjugate force measure. If you pick F^a_A then your force measure is the first Piola-Kirchhoff stress P_a^A. A statement like "The Cauchy stress is covariant." is fine as long as you understand in what context it is used.

4) "If in a Riemann manifold with a metric structure any representation can be used to describe a tensor then why Maugin picked the covariant one for the Cauchy stress? "

---> I would say convenience or simply a choice.

5) "Finally you consider a contravariant basis for the Tangent space and thus a covariant basis for the Cotangent space. Would it make sense to consider the opposite: a covariant basis E_{A} for the tangent space and an contravariant basis E^{A} for the cotangent space. "

---? You wouldn't say I pick a "contravariant basis" for a tangent space. You simply pick a basis for the tangent space and then the corresponding components of a tensor with respect to this basis are called contravariant components.

Regards,
Arash

Dear Arash,

 

1) Thanks for making things much clearer to me. The way you introduce the F_a^A representation for the deformation gradient by explaining what it acts on and what it gives makes it very obvious. Indeed, as you noticed, in this sense F is a vector-valued one-form and the representation suddenly makes sense.

 

3) Moreover, as you wrote it down, on writting the balance of Mechanical Energy in the material description it turns out that if you pick F_a^A for your deformation measure then  the corresponding energy conjugate force measure is the first Piola-Kirchhoff stress. Hence the Cauchy stress σ is covariant in this context.

 

Besides that i need your opinion. Indeed i started reading a book by Marcelo Epstein about "Material Inhomogeneities and their Material Evolution". I don't know if you had the chance to read it but i would like to know if it worth spending some time reading it regarding the topic of Configurational Mechanics and  its application to cracks...It's based on a pure Geometric Approcach..

 

Finally i would like to read up  more on Tensor Theory. If you know about any book published that is worth it, please let me know.

 

Regards,

 

Malik Ait-Bachir

Arash_Yavari's picture

Dear Malik:

By the way, although some people may disagree, I would not take the so-called "configurational mechanics" too seriously. There have been many works since what Eshelby did but to this date it's not completely clear what "configurational forces" are. There are some old and new controversies and many of the discussions are philosophic more than anything else. The idea of working with an evolving reference configuration is interesting and useful but many of the so-called "material balance laws" are vague from a fundamental point of view. It is wonderful to try to build a theory that can unify all the known examples of defects (including cracks), but in my opinion, such a general framework does not exist yet.

Regards,
Arash

Arash_Yavari's picture

Dear Malik:

I have read most parts of the recent book by Marcelo Epstein. Marcelo is an exceptionally sharp person and I think this is a nice book. However, I'm not sure if this whole idea of "material uniformity" can be used for anything real. This was introduced by Noll and Wang in the late fifties. I think it's time for some fresh ideas.

There are many books on tensor analysis. If I were you I would start with a book on tensor analysis on Euclidean spaces first. Although not completely rigorous, I found the following book very useful.

A. I. Borisenko and I. E. Tarapov, Vector and Tensor Analysis, Dover, 1979.

For more rigorous and geometric treatments you can look at the following two books:

1) Abraham, R., J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Springer-Verlag, New York, 1988.

2) Nakahara, M., Geometry, Topology and Physics, Taylor & Francis, New York, 2003.

Regards,
Arash

Dear Arash,

 

I understand that you are quite skeptical about Configurational Mechanics in the sense that you think and believe there is no way to build a theory that can unify all the known examples of defectfs. And as you say, most of the discussions about this topic are more philosophic than anything else. Frankly i do only partially agree with you on the latter point.

 

However, no one claims that it is a new theory. It's just a projection of the balance laws of continuum mechanics onto the material manifold (see Maugin).  In this context it is basically an extension of "Continuum Mechanics"... For instance, the Eshelby tensor  is expressed through quantities that are defined in the framework of "classical continuum mechanics"...

Some authors like Gerard Maugin and co. or Kienzler and Herrmann worked a lot on this problem. In his book Maugin shows how "Configurational Mechanics" is related to "Continuum Mechanics". On comparing with what has been done in "Continuum Mechanics", they introduced the concept of "Configurational Forces" which only makes sense as a consequence of the dimension of components of the Eshelby Tensor . Indeed, it has energetic components (energy density = energy/volume). On using the dimension equality Energy/Volume=Force/Surface and on comparing with the components of the Cauchy stress σ, they introduced the concept of "Configurational Forces". 

In the reference configuration, if you pick a material surface with normal N  then ∑·N is the "material force" to which the material surface is subjected. This notion is the same as the notion of traction vector σ ·n in continuum mechanics and exists on its own. It turns out that when you integrate the "material force" on a countour that surrounds the crack tip, you get exactly the Energy Release Rate. For the time being that's the most meticulous definition for "material forces".

The Eshelby Tensor was introduced first by Eshelby when he was tryng to calculate the energy necessary to shift a defect in the reference configuration by an amount δX given that the Boundary Conditions remain the same. Hence he used an energetic approach to introduce it. Moreover speaking of forces in the reference configuration differs form the way they are usualy thought of in Physics.

To me it s the reason why people feel uncomfortable with the notion of "Configurational Forces". They want to feel it and be able to touch it whereas "Configurational Forces" represent something that exists but that is not touchable.

The theory of "Configurational Mechanics" is already well-established. Its application to defects is still provoking debates but it has to be noticed that a few studies have been carried out that prove it to be a promising approach to study defects.

 

In my opinon, a lot of things remain to be done in the framework of "Configurational Mechanics" but the foundations have already been laid down. However, an equivalent of the laws of Thermodynamics has to be written in this "new" framework to help people comprehend the notion that lies behind. Anyway, the notion of "Configurational Forces" makes sense even though it differs from the classcal notion of forces.

I am not trying to convince  you to believe in "Configurational Mechanics". I admit that it looks like a pure theoretical approach but it definitely remains part of the framework of Physics except that it introduces new notions that might seem unneeded. No one pretends that it is a new theory. On the contrary, it is more an extension than anything else. But it turns out to be very helpfuland very promissing.

 

It summarizes my feelings and thoughts about the question that you raised.

 

Kind regards,

 

Malik

Arash_Yavari's picture

Dear Malik:

I didn't say it's impossible to have a unifying theory but I don't think that theory exists in Maugin's or Gurtin's works. Statements like "projection of the balance laws of continuum mechanics onto the material manifold " are superficial in my opinion. Yes, you can pull back balance of linear momentum back to the reference configuration (in the absence of defects) and it is trivially satisfied. But then what? Eshelby's seminal work is certainly very useful but even he was not sure what the material forces were. If you look at his work on liquid crystals you can see a discussion on if the defect force is a "real" force or not. Ericksen has a few papers discussing this too. Of course you can always define analogues of Cauchy stress and use complicated-looking expressions but that doesn't mean everything is clearly understood. It is also true that some well-known concepts like J-integral can be reinterpreted using Eshelby's tensor but that doesn't necessarily give you anything new. I think this is a fascinating field but new ideas other than "projection of the balance laws of continuum mechanics onto the material manifold" are needed.

Regards,
Arash 

I understand your point of view.

You are definetly right when you say that new ideas other than  "projection of the balance laws of continuum mechanics onto the
material manifold" are needed. Indeed we have to move on to a theory where things will be better understood than they currently are.

Anyway something important is missing in that framework: the laws of thermodynamics. The Eshelby tensor Σ was introduced by calculating the change of energy  due to the shifting of the defect by an amount δX while the boundary conditions remain the same. And that change of energy is thus not a physical quantity. What it is exactly and what it thermodynamically represents? Frankly i don't know....

I even wonder if it has to be positive or negative? I mean as for the second law of thermodynamics that predicts the direction of evelution of the physical system, there should be a kind of "material law of thermodynamics" that can predict the direction of evolution of the material system, i.e. of that change of energ when the defect is shifted.

 

Because up till now, other than equations (balance law projected onto the material manifold) i have never heard of a  kind of law of thermodynamics in the material manifold giving the direction of evolution between two reference configurations. I am really interested in knowing if someone tried to build up such a "material law of thermodynamics.

I am going to post a new topic to raise that question.

 

Take care,

 

Malik

Sia Nemat-Nasser's picture

a tensor is neither covariant nor contravariant, while it can be expressed by its covariant, contravariant, or mixed *components* with respect to any arbitrary coordinate system. See a short explanation.

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