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Geometric Dislocation tensor in finite plasticity

Amit Acharya's picture

 in B. D. Reddy (ed.). IUTAM Symposium on Theoretical, Modelling, and Computational Aspects of Inelastic Media, 99-105. Springer Science, 2008.

The criteria of Cermelli and Gurtin (2001, J. Mech. Phys. Solids) for choosing a geometric dislocation tensor in finite plasticity are reconsidered. It is shown that physically reasonable alternate criteria could just as well be put forward to select other measures; overall, the emphasis should be on the connections between various physically meaningful measures as is customary in continuum mechanics and geometry, rather than on criteria to select one or another specific measure. A more important question is how the geometric dislocation tensor should enter a continuum theory and it is shown that the inclusion of the dislocation density tensor in the specific free energy function in addition to the elastic distortion tensor is not consistent with the free energy content of a body as predicated by classical dislocation theory. Even in he case when the specific free energy function is meant to represent some spatial average of the actual microscopic free energy content of the body, a dependence on the average dislocation density tensor cannot be adequate.

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Comments

Arash_Yavari's picture

Dear Amit:

Thanks for posting this interesting manuscript. I don't follow everything but have a few comments.

First, I should say I very much like your work on dislocations as you
have been solving concrete problems. This is something that has been
missing in defect mechanics and most arguments are more or less
abstract.

I read Cermelli and Gurtin's paper again without understanding all the
details. What I can say is that their argument on the existence of a
"unique" measure of dislocation density tensor is not strong and I
agree with you that there are other possibilities.

I have been trying to understand the works of Kondo, Bilby, Kroner and
others on connecting defect mechanics with the geometry of
Riemann-Cartan manifolds. I feel some of the arguments are handwaving
and this literature is not completely consistent in my opinion.

I have the following comments on Cermelli and Gurtin's paper and would like to know your opinion.

1) I don't see any mention of the fact that the decomposition F=F^e F^p
is not unique. I understand that in crystal plasticity one can define a
unique F^p but in general F^e F^p is unique only up to local rotations.
I'm guessing that this could affect their G tensor?

2) Page 1549: In the first line they mention that "The operation of
integration is physically meaningless on L(X) as integration is not
local,...". This is similar to saying that integration of vector fields
is meaningless on a manifold with curvature. The good thing here is
that they realize that in the intermediate (microstructural)
configuration one should be careful with integration. But in the
following, in my opinion, they contradict themselves.

3) Page 1551: In Eq.(4.1) Burger's vector is defined as an integral on
a referential curve but if you look at what they do I think integration
is done on the intermediate configuration as F^p dX is an element of
the tangent space (at the appropriate point) of the intermediate
configuration (this can be clearly seen if you look at the component
form). So, I think something is still missing here and their Burger's
vector is not well defined.

In your paper, on page 3, footnote 2, you mention objective stress
rates. You may want to mention the following too. Geometrically, all
the objective stress rates are the Lie derivatives (with respect to the
spatial velocity field) of different representations of Cauchy stress
(this is clearly explained in the elasticity book by Marsden and Hughes
on page 99). Cauchy stress is a second-order tensor and has covariant,
contravariant and two different mixed representations. To have a
feeling of what a Lie derivative is, note that conservation of mass is
equivalent to Lie derivative of mass density with respect to the
spatial velocity field being zero (in other words mass density is
constant along the flow of velocity field). Saying that one stress rate
is more objective would of course be meaningless as you emphasize in
your paper.

Regarding the "appropriate" kinetic field(s) that should enter free
energy, I'm not sure. I think there might be more than one possibility
depending on how you define your "geometry", though I'm not sure.

My last comment is on the linearized theory and making general
conclusions using it. I don't mean to criticize the linearized theory
but one should be very careful with any linearized theory as
linearization can be misleading and can obscure a lot of geometric
information. Of course, this doesn't mean the linear theory of
dislocation mechanics is not useful; as a matter of fact it is
practically very useful.

Regards,
Arash

Amit Acharya's picture

Hi Arash,

 Thank you for your comments. The manuscript is accepted for publication in the Proceedings of a recently held IUTAM symposium in Cape Town.

My Responses:

 0) On the Kondo, Bilby, Kroner genre - In Kroner's work there are key ideas that when combined with continuum mechanics and homogenization (of dynamical, nonlinear PDE systems!) can result in much progress for saying something about microstructure evolution and its macroscopic effects. Kondo's program is not yet entirely clear to me (although I have read a fair bit of his RAAG memoirs), and as I am 'narrowly' interested in what progress can be made in plasticity with these ideas, Kondo's stuff is too general. I find Bilby et al. too kinematics oriented on this particular topic and, while this is necessary, it is not sufficient for my goals.

I am not a stickler for 'rigor,' so as long as there are good ideas, it suffices for me - but again, this is a matter of taste.

On your comments on C&G:

1) I don't think there is a problem.

Assume it is a good day, and you have a unique motion with all kinds of smoothness - so the deformation gradient at a point is unique. Now if F^p is unique and invertible (as would happen from a crystal plasticity specification of L^p, assuming everything is going well with the ibvp) then F^e is.

2 + 3) I think this is OK too. I will discuss the issue you raise in a minute.

As I understand it, what they are referring to is the fact that the intermediate 'configuration' is really no coherent configuration but a 'collection of pieces' if you will, one piece corresponding to each material point. So you cannot have a nicely defined continuous curve on such a beast for doing a line integral, because to do so you need to define a tangent to the curve etc.

What you are referring to is how legitimate is it to add vectors on different tangent spaces of the target manifold. Basically, one makes the identification that each tangent space is the same vector space represented by the translation space of three dimensional Euclidean space, so one can add vectors on these different tangent spaces without ambiguity.

From the continuum mechanics point of view this is OK, but you might object to it from the geometric point of view - it is a matter of taste. In my opinion, this is fine, as I entirely buy physical notions like statics means 'traction integrated over the surface of a body is zero' and balance of linear momentum and these statements all rely on the abovementioned identification.

4) On Objective rates as Lie derivatives: Actually, there is a little technicality here, if you want to split hairs. Not all objective rates are Lie derivatives w.r.t to the spatial velocity field or any flow for that matter.

The basic point here is as follows: An objective rate (convected derivative in the language of Hill, 1978) at a material point is defined through the existence of a time dependent invertible tensor function of time that allows one to pull back, do a time derivative on the pullback, and then a push forward - it is a purely pointwise operation. It may turn out that the field of invertible tensors over a local patch of material points may not be compatible so that it cannot be written as the two point gradient tensor of a flow. In that case you cannot define an objective rate as a strict Lie derivative w.r.t some flow.

(just definitions, basically).

However, the motivation behind the definition of something as weird as a convected derivative of some sort of (volumetric, areal, linear) 'density' field is physically best understood, in my opinion, in the context when it is a Lie derivative w.r.t. some flow, coupled with the question of determining the time derivative of the integral of the density field over a time-dependent volume/area/curve, respectively.

Basically Reynolds transport theorems for volumes, areas, curves...

And, yes, of course, the covariant, contravariant, and mixed convected derivatives of the same tensor when convected by the deformation gradient are not all the same tensors (but precisely related), and if you took the same tensor and now convected it with an invertible tensor other than the deformation gradient, you would generate many more objective rates. Hill's Invariance in Solid Mechanics (1978) article gets through an awful lot of high powered stuff in a just a few pages in a very simple but beautiful way. I think this treatment is ideal for the student of continuum mechanics who may not have had any exposure to differential geometry. My students and I go through these in their plasticity class.

5) Regarding what should enter the free energy - The main point there is, we can of course start to put whatever we want but in this case there is a good successful standard to go with if we want to talk about dislocations and their energy, which is the classical theory. I do not think this is a matter of geometry or mathematics - ultimately it is a matter of material behavior. I think a gradient can be important to represent core effects but it will be a small contribution to the energy. One might see a dependence arising due to averaging too (but this requires that one does the required averaging), and in the last paragraph of my paper I have recorded some elementary ideas on what I think of this matter.

6) I have worked out the general nonlinear theory in my 2004 JMPS paper, and even there I work only with a dependence of the free energy on the elastic distortion tensor and no gradients. Things work out quite well in the general theory too.

all the best,

Amit

Arash_Yavari's picture

Dear Amit:

Thanks for your detailed and informative response.

Just one comment.

I don't think one can identify tangent spaces in the intermediate
configuration the way you're describing it. This is fine as long as
you're dealing with Euclidean spaces. In the case of non-Euclidean
spaces (and as far as I can understand it the intermediate
configuration is not Euclidean) you cannot make such an identification
of tangent spaces; you would need a "connection" to be able to parallel
transport vectors (or tensors in general). Maybe the way Burger's
vector is defined traditionally works for practical problems but as far
as I can tell the definition in its present form is not mathematically
consistent. I don't think this is a matter of taste; if one realizes
that a given space is not Euclidean and defines it only locally (and
calls it an "anholonomic space", a "collection of pieces", etc.) then
integration of a vector field would be meaningless (this integration is
meaningless even for a smooth curve in the intermediate configuration).
You're correct that we integrate tractions in continuum mechanics but
that's because continuum mechanics is formulated on Euclidean spaces.
And of course, this is reasonable as Euclidean space is where we
observe the deformed body and is where our deformed bodies live. But as
soon as you define something like an intermediate configuration, things
are completely different. I agree that "rigor" is not always necessary
in engineering problems and many important problems can be solved by
engineering/physical intuition, but I think in the works that argue a
geometric (or mathematical) treatment of defect mechanics, Burger's
vector is ill-defined and intermediate configuration seems to be more
or less like a mysterious object.

Regards,
Arash

Amit Acharya's picture

17:10:0917:10:09

Dear Arash,

 If one does not want to think about the intermediate configuration, that is fine - it is a conceptual crutch, for those that find it useful to visualize things.

Given the field F^e-1 on the current configration, one can define a linear connection and then a torsion tensor (you know about all this) - all kosher geometrically - and of course what is defined as the two-point dislocation density  in terms of curl Fe^-1 on the current configuration can be precisely related to the torsion.

In my paper with John Bassani in 2000, we work out these relations between the geometric and continuum mechanics points of view. I'll upload it to my original post  just in case some people are interested.

best,

Amit 

Anurag Gupta's picture

Prof. Acharya. Many thanks for bringing up this interesting topic. In the following I would like to mention my view point on the issues you raised in your manuscript. 

 1) True dislocation density (Cermelli & Gurtin): While it is correct to say that, there are other equally valid measures of dislocation content in a body, the importance of the dislocation density tensor mentioned by Cermelli & Gurtin (it appears earlier in the works of Kondo and Noll) is highlightened by the fact that it appears naturally in various constitutive functions. Following the work of Davini, Parry, Cermelli/ Gurtin and Epstein, one can show that for a constitutive function to be invariant with respect to arbitrary (compatible) changes in the reference configuration, the dependence on plastic distortion $F^p$, can only be through a rate term (of the type ${F^p}^{-1} \dot{F^p}$) or through the true dilocation density. Requirement for such an invariance comes from the fact that our choice of a reference configuration is purely a convinience and should not affect the material response.

2) Dislocation density as a state variable: I am not sure if I follow your arguments correctly, but it seems to me that you are using a global argument (boundary value problem) to conclude a local result (constitutive assumption is a local argument).  

3) This is in response to the first question raised by Prof. Yavari: Under suitable constitutive assumptions and the assumption of material homogeneity, one can show that the non-uniqueness in obtaining an intermediate relaxed configuration is via a (uniform) rotation and (uniform) translation. Uniformity of the rotation, however, implies that the true dislocation density remains invariant under such changes in the relaxed configuration. See for example, Noll, Cohen/Epstein and Gupta et al.

 

References:

P. Cermelli and M. E. Gurtin. On the characterization of geometrically necessary islocations in finite plasticity. Journal of Mechanics and Physics of Solids, 49:1539, 2001.

W. Noll. Materially uniform simple bodies with inhomogeneities. Archive of Rational echanics and Analysis, 27:1-32, 1967.

H. Cohen and M. Epstein. Remarks on uniformity in hyperelastic materials. Internaional Journal of Solids and Structures, 20(3):233-243, 1984.

G. P. Parry. Generalized elastic-plastic decompositions in defective crystals. In. Capriz and P. M. Mariano, editors, Advances in Multifield Theories for Continua with Substructure, pages 33-50. BirkhÄauser, 2004

M. Epstein. Toward a complete second order evolution law. Mathematics and Mechanics of Solids, 4(2):251-266, 1999

A. Gupta, D. J. Steigmann, and J. Stolken. On the evolution of plasticity and incompatibility. Mathematics and Mechanics of Solids, 12:583-610, 2007

Amit Acharya's picture

Dear Anurag,

 Thank you for your comments. My responses, following your numbering scheme:

1) As you might have noted, I do not imply that the C&G measure is not important, especially kinematically. My partial concern in the paper is purely with its appearance in the free energy function and I provide a concrete argument for saying so. Please also see my response to Arash in this regard.

One can also formulate a dislocation mechanics and plasticity without any physical need whatsoever for a reference configuration, without involving the dislocation tensor in the free energy and, for that matter, without involving the tensor F^p altogether - if you are interested, you can see my JMPS 2004 paper for the situation where the scale of resolution is such that there are no 'statistical dislocations' and the last section of another JMPS paper in 2006 with Roy for the mesoscale situation.

2)  I don't understand your point here and whether you have an objection or not. I suggest a local constitutive equation for the free energy and the stress (not including the dislocation tensor) and show that with such constitutive equations, one recovers the stress field prediction of classical dislocation theory related to individual or collections of discrete dislocations with non-singular cores. Moreover, the argument also shows that if you stuck in a dependence on the dislocation tensor as well, you would overestimate the strain energy of the dislocation distribution.

Interestingly the argument also says that if in conventional plasticity theory one puts in a plastic distortion field whose curl corresponds to a given dislocation density field (including a single discrete dislocation), and then one solves the equlibrium equation, classical plasticty would deliver the stress field of that dislocation field. As an aside, to pull this off numerically the equations have to be solved somewhat differently and a JMPS paper paper with Roy in 2005 shows this.

I don't think this point is appreciated widely enough. 

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