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Riemann-Cartan Geometry of Nonlinear Disclination Mechanics

arash_yavari's picture

In the continuous theory of defects in nonlinear elastic solids, it is known that a distribution of disclinations leads, in general, to a non-trivial residual stress field. To study this problem we consider the particular case of determining the residual stress field of a cylindrically-symmetric distribution of parallel wedge disclinations. We first use the tools of differential geometry to construct a Riemaniann material manifold in which the body is stress-free. This manifold is metric compatible, has zero torsion, but has non-vanishing curvature. The problem then reduces to embed this manifold in Euclidean 3-space following the procedure of a classical nonlinear elastic problem. We show that this embedding can be elegantly accomplished by using Cartan's method of moving frames and compute explicitly the residual stress field for various distributions in the case of a neo-Hookean material.

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Dear Arash,

Thanks for sharing the elegant work. I have a few comments.

  1. A connection is a mapping from smooth sections of a vector bundle to smooth sections of the product bundle, so at the end of page 1 in the definition of a connection, the arrow is pointing the opposite way.
  2. Method of moving frames is parrallel to generalized linear groups in Lie group theory. So I believe your work can be formulated in the Lie group context, with the Lie group being defined as transformations with respect to a reference configuration. The Maurer-Cartan form and the associated structure equations should direct to equations of motion.
  3. After all, I more believe geometry is the language for mechanics.

Xiaobo

arash_yavari's picture

Dear Xiaobo:

Thank you for pointing out the typo. I'll make sure to fix it.

I know that there are similarities between this formalism and Lie group theory. The physically important thing here is the Riemannian manifold in which the disclinated body is stress free. I think it would be worthwhile exploring this similarity in case Lie groups can help construct the material manifold more efficiently.

Another important thing to note is that this geometric formalism is not just about elegance; one can calculate the residual stress field of solids with distributed disclinations. I'm sure there are other examples that can be solved in this framework.

Regards,
Arash

Amit Acharya's picture

Hi Arash,

Very nice!

A question: do you have any sense for what the uniqueness situation might be for the Neo-Hookean material? The reason I ask is the following (and the immediate intent is not related to non-monotone stress consititutive assumption related issues). Suppose you choose a curvature field (disclination density) that does not interfere with it being the curvature of a connection that has zero torsion. Then in the language of continuum mechanics, your problem becomes that of solving for a stress distribution in equilibrium for a Neo-Hookean material whose F^e (elastic distortion) field is such that the finite strain incompatibility expression formed from its C^e = F^eT F^e field is equal to the specified curvature field.

Now, if you take the small strain analog of this problem, then this amounts to looking for a small elastic strain tensor field whose stress satisfies the equilibrium equations and whose st.venant incompatibility expression is set equal to the specified Kroner's incompatibility field (eta). Given traction free b.c.s on a known geometry, it is easy to prove uniqueness for the stress field here. From the problem definition, it is immediately clear (without solving anything), that this problem definition does not nail down the elastic distortion - i.e. the skew-symmetric part of the elastic distortion can be assigned at will. As an aside, note that when solving the dislocation problem, given a specified dislocation density (alpha), the stress may be determined by Kroner's method by forming eta from alpha, but this does not use all the information, as given the dislocation density the whole distortion can be determined uniquely (an observation implicit in Willis's formulation of dislocation mechanics).

So coming back to the disclination problem and the finite deformation case - now there is a difference. Even though the incompatibility eqn. is posed purely in terms of the metric (C^e), the equilibrium equations (essentially because of frame indifference) necessarily involve F^e, which is unlike the small strain case. So one cannot say immediately that this does or does not determine the full elastic distortion uniquely. But it would surely be good to know the answer, and if there is non-uniqueness in the elastic distortion, it would be good to have some sort of characterization of it.

Any thoughts?

- Amit

arash_yavari's picture

Hi Amit,

Sorry for the late reply. Let me first tell you what I do here and see if I understand your question correctly.

I directly work with the material manifold (where the disclinated body is stress-free) and try to embed it into the Euclidean 3-space to find the residual stresses. Deformation gradient is the tangent map of this embedding and is purely elastic (all the anelastic effects are buried in the material manifold). Now given a disclination distribution, as you correctly pointed out, is equivalent to having the curvature tensor (or curvature 1-forms) of the (Riemannian) material manifold.

Regarding uniqueness there are two issues here:

1) If I know what the material manifold is, its embedding into the Euclidean 3-space may not be unique. Here, I assume a symmetric solution (r,\phi,z)=(r(R ),\Phi,Z) and in this class of embeddings solution is unique.

2) You may ask the following question. Given the curvature tensor (disclination density tensor) is the material manifold unique? This is equivalent to asking whether the metric corresponding to the Riemann curvature tensor is unique. First, note that uniqueness or lack of uniqueness is independent of constitutive equations and is purely kinematical. Second, looking at definition of Riemann curvature tensor you see that you have a system of nonlinear PDEs for the unknown metric. So, my answer would be, no the metric is not unique, in general.
    For the problem of a cylindrically-symmetric distribution of wedge disclinations I use a semi-inverse method. I start with an orthonormal coframe field, which is very similar to the classical cylindrical coframe field for the flat Euclidean 3-space but with one unknown function f=f(R ). Now given a coframe field and knowing that material connection is torsion-free and metric-compatible we know that the connection 1-forms are uniquely determined (as a consequence of Cartan's Lemma). Having the connection 1-forms you can then calculate the curvature 1-forms (using Cartan's second structural equations) and see if they can be equal to the given curvature 1-forms (or disclination tensor). It turns out that for the assumed orthonormal coframe field this can happen and these equations give a differential equation for the unknown function f(R ). So, for the assumed orthonormal coframe field the metric (and consequently the Riemannain material manifld) is uniquely determined. But, this does not mean that the material manifold is unique. Uniqueness of the material manifold is a very interesting question that deserves a careful analysis. This remains to be done in the future.

Regards,
Arash

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