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Controversy: Dynamic Peierls-Nabarro equations

In 2010,  Yves-Patrick Pellegrini published a paper in Physical Review B called

 Dynamic Peierls-Nabarro equations for elastically isotropic crystals.  with the abstract

 The dynamic generalization of the Peierls-Nabarro equation for dislocations cores in an isotropic elastic medium is derived for screw and edge dislocations of the “glide” and “climb” type, by means of Mura’s eigenstrains method. These equations are of the integrodifferential type and feature a nonlocal kernel in space and time. The equation for the screw differs by an instantaneous term from a previous attempt by Eshelby. Those for both types of edges involve in addition an unusual convolution with the second spatial derivative of the displacement jump. As a check, it is shown that these equations correctly reduce, in the stationary limit and for all three types of dislocations, to Weertman’s equations that extend the static Peierls-Nabarro model to finite constant velocities.

 Xanthippi Markenscoff  responded to the paper in 2011 saying that

 The paper by Pellegrini [Phys. Rev. B 81, 024101 (2010)]
introduces additional “distributional terms” to the displacement of the static field of a dislocation and claims that they are needed so that Weertman's equation for the steady-state motion of the Peierls-Nabarro dislocation be recovered. He also claims that the [Eshelby, Phys. Rev. 90, 248 (1953)] solution for a moving screw is wrong, a statement with which I disagree. The same [Eshelby, Phys. Rev. 90, 248 (1953)] solution is also obtained and used by the eminent dislocation scientists Al’shitz and Indenbom in Al’shitz et al. [Sov. Phys. JETP 33, 1240 (1971)] that the author ignores. A key reference in the formulation of the problem as a 3D inclusion with eigenstrain is Willis [J. Mech. Phys. Solids 13, 377 (1965)] who showed that, in the transient fields, the static Eshelby equivalence of dislocations to inclusions (with eigenstrain) does not hold, but only at long times when they tend to the static ones. In this Comment the author provides the fundamental physics of the behavior of a moving Volterra dislocation in nonuniform motion by showing how the
singular fields near the moving core are obtained from “first principles” (without solving for the full fields). The limit to the steady-state motion of a Peierls-Nabarro dislocation is also shown how to be obtained from first principles from the Volterra one by taking the appropriate limit, without the need of the additional distributional terms that Pellegrini introduces.

Pellegrini has responded saying

" The Comment by Markenscoff that criticizes a recent dynamic extension of the Peierls-Nabarro equation [Y.-P. Pellegrini, Phys. Rev. B 81, 024101 (2010)] is refuted by means of simple examples that illustrate the interest of using an approach based on generalized functions to compute dynamic stress fields."

Can someone who uderstands the subject throw some more light on the matter?



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