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New inroads in an old subject: plasticity, from around the atomic to the macroscopic scale
(in Journal of the Mechanics and Physics of Solids)
Nonsingular, stressed, dislocation (wall) profiles are shown to be 1-d equilibria of a non-equilibrium theory of Field Dislocation Mechanics (FDM). It is also shown that such equilibrium profiles corresponding to a given level of load cannot generally serve as a traveling wave profile of the governing equation for other values of nearby constant load; however, one case of soft loading with a special form of the dislocation velocity law is demonstrated to have no ‘Peierls barrier’ in this sense. The analysis is facilitated by the formulation of a 1-d, scalar, time-dependent, Hamilton-Jacobi equation as an exact special case of the full 3-d FDM theory accounting for non-convex elastic energy, small, Nye-tensor dependent core energy, and possibly an energy contribution based on incompatible slip. Relevant nonlinear stability questions, including that of nucleation, are formulated in a non-equilibrium setting. Elementary averaging ideas show a singular perturbation structure in the evolution of the (unsymmetric) macroscopic plastic distortion, thus pointing to the possibility of predicting generally rate-insensitive slow response constrained to a tensorial ‘yield’ surface, while allowing fast excursions off it, even though only simple kinetic assumptions are employed in the microscopic FDM theory. The emergent small viscosity on averaging that serves as the small parameter for the perturbation structure is a robust, almost-geometric consequence of large gradients of slip in the dislocation core and the persistent presence of a large number of dislocations in the averaging volume. In the simplest approximation, the macroscopic yield criterion displays anisotropy based on the microscopic dislocation line and Burgers vector distribution, a dependence on the Laplacian of the incompatible slip tensor and a nonlocal term related to a Stokes-Helmholtz-curl projection of an ‘internal stress’ derived from the incompatible slip energy.
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Arash: 1-d compatibility
Arash,
This issue of compatibility for 1-d manifolds is somewhat interesting. I say this for the following reason:
First consider the following:
It is obvious that there are distortion fields (i.e. a candidate 'deformation gradient' field if the distortion were to be compatible) of 3-d bodies which vary only in 1 direction and can be incompatible.
E.g. take a cylinder, say of uniform rectangular x-secn., with say rectangular Cartesian directions, 1, 2 in the x-secn. Define a distortion field with only non-zero components
F_11 = 1
F_22 = 1
F_33 = 1
F_12 (x_3)
which varies only in the direction of the axis of the cylinder, x_3. As you can see this distortion field is incompatible, while the deformation is really 1-d, in the sense described.
Actually, I consider such a situation to construct a 'simple' example of my theory in the paper posted above.
Since you are the resident geometer amongst us, my question to you is, how would this situation go:
Taka genuine 1-d manifold, say a rod but which has director dofs to represent shear deformations. Is there some sense in which the above situation related to incompatibility in the context of the 3-d body occur for the rod? Or is it that as soon as one adds the director field, the manifold does not remain 1-d anymore?
Thanks,
- Amit
Rod Compatibility
Dear Amit:
Your example is perfectly fine; you can have a metric that locally depends only on one coordinate but still has a nonzero curvature tensor.
Having a metric, you can locally measure distances and this is why metric explicitly shows up in elasticity.
In a 1D manifold, intrinsically (i.e. without any need for an embedding) you can only measure distances between points and their changes, i.e. stretch. As soon as you add a director field you will have a bundle with the 1D manifold as its base. So, one would in general lose the trivial flatness. I haven't seen any detailed discussions on compatibility equations for rods anywhere but the following paper has some discussions that may be relevant.
ANTMAN SS, MARLOW RS, MATERIAL CONSTRAINTS, LAGRANGE MULTIPLIERS, AND COMPATIBILITY - APPLICATIONS TO ROD AND SHELL THEORIES, ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS 116(3), 257-299, 1991
Regards,
Arash
Arash - Yes, this question
Arash - Yes, this question of compatibility of rod with director is curious.Thanks for the Marlowe Antman reference.
On compatibility in the dislocation case - I agree with most you say except the issue here is not restricted to just strain compatibility as posed in terms of the metric tensor. As we have discussed before on imechanica, one could have a compatible metric and yet have nontrivial incompatibility of the elastic distortion
- Amit