Displacement field due to glide and climb of rectilinear dislocations in gradient elasticity

ABSTRACT

The aim of this article is to provide the displacement field of a straight dislocation in gradient elasticity that can be implemented in dislocation dynamics simulations. The displacement and plastic strain fields of a dislocation depend on the history of the dislocation motion. The cut surface (or the branch cut) represents this history. When an edge dislocation glides, the branch cut must be parallel to the Burgers vector, and when a dislocation moves by a combination of glide and climb, extra terms need to be added to the displacement field to capture the correct discontinuity in the displacement field. The existing formulas for the dislocation displacement in gradient elasticity lack such considerations, which make them unsuitable for simulations. In addition, the previous formulas both in classical and gradient elasticity have contained some inaccuracies and miscalculations. In this paper, the displacement field of a dislocation and the corresponding plastic strain are derived in detail and a distinction between the dislocation due to glide and that due to climb has been made. Moreover, we discuss the shortcomings of Mura's [Micromechanics of Defects in Solids, 2nd ed., Martinus Nijhoff Publishers, 1987.] and Lazar and Maugin's [Dislocations in gradient elasticity revisited, Proc. R. Soc. Math. Phys. Engin. Sci. 462(2075) (2006), pp. 3465–3480.] calculations. We illustrate that Orowan's law is a direct result of the correct displacement and plastic strain fields.

KEYWORDS:

• Gradient elasticity
• Dislocation
• Displacement
• Plastic Strain
• Glide
• Climb

Acknowledgments

The author wishes to thank Harvard University for providing research facilities during his PhD and Postdoctoral studies.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Aifantis and his collaborators use two different parameters for Equations (3(3) $(1-c^2{\mathrm{\nabla }}^2){ϵ}_{ij}^{\left(p\right)}={ϵ}_{ij}^{0\left(p\right)}$(3) ) and (4(4) $(1-c^2{\mathrm{\nabla }}^2){\sigma }_{ij}={\sigma }_{ij}^0$(4) ).

2 There are two inaccuracies in Mura's calculations: (1) He ignores the second term of the right hand side of Equation (10(10) ${\widehat{ϵ}}_{zy}^{\left(p\right)}=\mathfrak{F}\{{ϵ}_{zy};x\to {\omega }_1,y\to {\omega }_2,z\to {\omega }_3\}=\frac{ib\delta \left({\omega }_3\right)}{2\sqrt{2\pi }{\omega }_1}+\frac{b}{2}\sqrt{\frac{\pi }{2}}\delta \left({\omega }_1\right)\delta \left({\omega }_3\right)$(10) ). (2) The integral that he finds is equal to $u_z=-b/\left(2\pi \right){\tan }^{-1}\left(x/y\right)$ not $b/\left(2\pi \right){\tan }^{-1}\left(y/x\right)$.

3 Because ${\tan }^{-1}\left(\frac{y}{x}\right)=-{\tan }^{-1}\left(\frac{x}{y}\right)+\frac{\pi }{2}\mathrm{s}\mathrm{g}\mathrm{n}\left(xy\right),$Equation (12(12) $u_z\left(\mathbf{x}\right)=-\frac{b}{2\pi }{\tan }^{-1}\left(\frac{x}{y}\right)+\frac{b}{4}\mathrm{s}\mathrm{g}\mathrm{n}\left(y\right).$(12) ) and Equation (7(7) $u_z(r,\theta )=\frac{b}{2\pi }\theta .-\pi <\theta \le \pi$(7) ) are the same.

4 De Wit [25] originally wrote the extra expression as $\pi H(-x)\left[H\right(y)-H(-y\left)\right]$