Line tension of a dislocation moving through an anisotropic crystal

ABSTRACT

Plastic deformation, at all strain rates, is accommodated by the collective motion of crystalline defects known as dislocations. Here, we extend an analysis for the energetic stability of a straight dislocation, the so-called line tension (Γ), to steady-state moving dislocations within elastically anisotropic media. Upon simplification to isotropy, our model reduces to an explicit analytical form yielding insight into the behaviour of Γ with increasing velocity. We find that at the first shear wave speed within an isotropic solid, the screw dislocation line tension diverges positively indicating infinite stability. The edge dislocation line tension, on the other hand, changes sign at approximately of the first shear wave speed, and subsequently diverges negatively indicating that the straight configuration is energetically unstable. In anisotropic crystals, the dependence of Γ on the dislocation velocity is significantly more complex; at velocities approaching the first shear wave speed within the plane of the crystal defined by the dislocation line, Γ tends to diverge, with the sign of the divergence strongly dependent on both the elastic properties of the crystal and the orientation of the dislocation line. We interpret our analyses within the context of recent molecular dynamics simulations of the motion of dislocations near the first shear wave speed. Both the simulations and our analyses are indicative of instabilities of nominally edge dislocations within fcc crystals approaching the first shear wave speed. We apply our analyses towards predicting the behaviour of dislocations within bcc crystals in the vicinity of the first shear wave speed.

KEYWORDS:

• Moving dislocations
• line tension
• dynamic loading
• anisotropy

Acknowledgments

We thank D.J. Luscher for enlightening discussions. B.A.S. thanks Joshua Crone for fruitful discussions. We also thank the anonymous referee for valuable comments. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the US Government. The US Government is authorised to reproduce and distribute reprints for government purposes notwithstanding any copyright notation herein.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. One exception is [8], as the authors discussed only screw dislocations and naturally no jump was found there.

2. What that new shape is exactly will be more difficult to determine and may be pursued in future studies.

3. Since we are considering Cartesian coordinates we do not worry about co-/contra-variant indices.

4. We follow [15, pp. 467–478] below.

5. Analytic results for , are much more difficult to obtain and are known only in the simplest of cases, namely within the limit of isotropy [14], and for dislocations oriented along within an fcc lattice [15].

6. By considering differences in energy between two similar dislocation configurations, within a sufficiently large crystal the difference in linear elastic strain and kinetic energy will always dominate because it scales logarithmically with the crystal radius, R. We hence neglect effects associated with the dislocation core, not only for simplicity but also assuming that the cores remain similar across two configurations and their effects on the line tension is therefore subleading. For additional details on the energetics of the dislocation core, we refer to [35–38].

7. A second suite of computations of Equation (16(16) ) was performed within Mathematica for validation purposes. Within this implementation, differentiation w.r.t. ϑ was performed symbolically and only the integrations over ϕ were completed numerically.

8. Note that our definition of γ matches the one known from special relativity and is related to Weertman's [13] notation via , .

Additional information

Funding

This work was performed in part under the auspices of the U.S. Department of Energy under contract DE-AC52-06NA25396. In particular, the authors are grateful for the support of the Advanced Simulation and Computing, Physics and Engineering Models Program. In addition, research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-17-2-0224.