iMechanica - opinion - Comments

linear connection and dislocation densitry

Amit Acharya — Sun, 11 May 2008 17:31:45 -0400

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

intermediate configuration

Arash_Yavari — Sun, 11 May 2008 16:41:23 -0400

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

RE to Anurag continuum defect mechanics

Amit Acharya — Sun, 11 May 2008 00:53:31 -0400

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.

Re to Arash, continuum defect mechanics, convected derivative...

Amit Acharya — Sun, 11 May 2008 00:17:18 -0400

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

Some comments

Anurag Gupta — Sat, 10 May 2008 17:29:16 -0400

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

interesting manuscript

Arash_Yavari — Fri, 09 May 2008 15:19:02 -0400

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

Re: Modeling R-Curve behaviour using CZM

Zhigang Suo — Wed, 30 Apr 2008 14:41:00 -0400

Here is one such paper:

Tvergaard, V., Hutchinson, J.W.,"
The relation between crack growth resistance and fracture process parameters in elastic-plastic solids." J. Mech. Phys. Solids 40, 1377-1397(1992).

Modeling R-Curve behaviour using CZM

Siva P V Nadimpalli — Wed, 30 Apr 2008 12:57:45 -0400

Hi All,

Thanks for your valuable comments and explanations about this topic. I am relatively new to this (Cohesive Zone Model) field. From a limited amount of reading and discussions with collegues I understood this topic to certain extent. From what I understand, CZM requires a traction separation law for modeling, which is obtained based on fracture energy.

My question is, because fracture energy is not a contstant value i.e., in case of ductile materials we have phenomenon called "R-Curve" behaviour (fracture energy changes with crack length initially), is it justifiable to use only one value of fracture energy (i.e., steady state value) to derive the traction separation law?

Please point me towards any article(s) if exist about modeling "R-Curve" behaviour using CZM.

Thanks a lot for your valuable time and suggestions.

--Siva

The convergence of cohesive models

Dong Kong — Mon, 21 Jan 2008 03:26:35 -0500

Thanks very much,

Using Abaqus default cohesive element, I model the peeling test (the height of the adhesive is 150 um and the length is very long). The interaction between the adhesive and the soft (E=3MPa) is Van De Waels force. The length scale of Van De Waels force is very short (10 nm), compared with the adhesive and substrate. The compution cann't be finished even for some increments. But if the length scale of cohesive stress (200 um) becomes very large,it works well.

Crack tip definition

Fred Nilsson — Tue, 08 Jan 2008 04:23:43 -0500

Dong,

In a cohesive zone model there is no clearly defined crack tip. Sometimes the front edge is used as reference, sometimes the trailing edge. Thus you can choose any definition that you prefer. I suppose that with the definition you suggest the crack tip will be at or near the front end.

Fred

How to define the crack tip

Dong Kong — Tue, 08 Jan 2008 02:21:49 -0500

Dear colleagues,

Many thanks for these very useful comments. Using cohesive law in Abaqus, I'm doing a peeling test simulation. Within the cohesive layer, the position of crack tip is defined as the element with the maximum S22 (Normal stresses). Is that reasonable?

Thanks,

Dong

Element size and cohesive zones

Fred Nilsson — Mon, 07 Jan 2008 07:59:53 -0500

Rahul,

What I meant with my last comment is that the cohesive zone must be at least of the size of several elements to accurately resolve it. If the parameters of the cohesive law are such that the zone size for a specific problem is only one or a few elements, there is a risk that a length parameter is introduced that bears no relation to the physical problem. Thus, the physical problem may demand (through the appropriate cohesive law) elements that are so small that the computations may be very expensive or even impossible to perform. It may then happen that the cohesive parameters (bearing in mind that in general very little is known about the appropriate values of the parameters) are adjusted so that a solution with an artificial length scale is obtained.

Fred

Length of cohesive zone

Dhruv Bhate — Fri, 04 Jan 2008 09:39:08 -0500

Rahul,

I know of one paper that may help you appreciate this a bit better (Fred, do correct me if I have misunderstood your statement)

A. Turon, C.G. Da´vila, P.P. Camanho, J. Costa, "An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models," Engineering Fracture Mechanics, 74, 2007, pp. 1665-1682

The paper is relatively easy to grasp, and collects all the various interpretations of length of cohesive zone...the process zone is essentially a highly nonlinear damaging region and several elements are needed to adequately describe the behavior in that zone.

I hope this helps.

Dhruv

cohesive zone will be the size of an element near the crack tip

rahul — Thu, 03 Jan 2008 15:23:50 -0500

Fred,

Thank you very much for your interesting comments. I am a newbie in this area and couldn't completely understood the last comment "Some caution must be observed
when using cohesive models in conjunction with numerical models such as
FEM. The minimum size of the cohesive zone will be the size of an
element near the crack tip. Should this be larger than the size of the
process zone of the physical problem, a length scale has been
introduced that does not exist in the physical problem. This is
frequently the case when using cohesive zone models for analysis of
fatigue crack growth." I would really appreciate if you can elaborate on this or point me to some article where this is not taken into account or has been discussed.

Some remarks about cohesive zone modelling

Fred Nilsson — Thu, 03 Jan 2008 13:02:00 -0500

I have followed the discussion on cohesive modelling and would like to give a few additional remarks,

1) The cohesive zone is basically a model concept that can be useful in certain cases. It can be used for instance when the fracture process zone is too large and a point-sized crack tip model is not adequate. It can also be used when modelling the initiation of a crack from a medium without cracks. The cohesive zone may or may not be a simulacrum of an actual physical process. It can be used to model different types of separation processes such as void growth and coalescence, fibre bridging, atomic separation, separation of adhesive layers such as glue etc. Once the cohesive law has been set the problem formulation is complete and no other fracture criterion is necessary.

2) There are different ways to derive a cohesive law.

a) By experimental measurements on special types of specimens (cf. T. Andersson and U. Stigh, (2004), Int. J. of Solids and Structures, 41, 413-434, B. F. Sørensen and E. K. Jacobsen, (1998), Composites Part A, 29A, 1442-1451. and several others).

b) By modelling (numerical or analytical) of the process that is to be replaced by the cohesive zone model.

c) By using a predefined functional assumption for the cohesive law, for instance as predefined in a numerical code (cf. ABAQUS). The parameters are estimated from experiments or by reasonable guesswork. This is probably the most common way.

3) It is often stated that a cohcsive zone model is equivalent to assuming that crack growth is governed by constant fracture energy. This is not true in general. Here are some situations when the fracture energy is non-constant and problem dependent.

a) A crack tip that is extending under conditions that are not steady-state (with or without inertia effects) (cf. L. B. Freund (1990), Dynamic Fracture Mechanics, pp. 237-238).

b) When large deformation effects are significant, depending on how the cohesive zone law is formulated (cf. F. Nilsson (2005), Int. J. Fract., 136, 133-142).

c) When the cohesive law depends on other quantities from the problem such as constraint, displacement rate etc.

When, as often is the case, setting up a cohesive law using fracture energy measured from experiments, it is thus important that this is a problem independent quantity.

4) Some caution must be observed when using cohesive models in conjunction with numerical models such as FEM. The minimum size of the cohesive zone will be the size of an element near the crack tip. Should this be larger than the size of the process zone of the physical problem, a length scale has been introduced that does not exist in the physical problem. This is frequently the case when using cohesive zone models for analysis of fatigue crack growth.