advances have been made in the areas of fatigue, creep, friction, and wear, but this broader range of topics is outside the scope of this discussion.
In 1939 Burgers10 developed a vector field theory for dislocations, including an expression for the displacement field of a dislocation loop in terms of line integrals over its length and an area integral over its enclosed area. In addition, Peach and Koehler11 presented an expression for the virtual thermodynamic force on a dislocation segment. Yet, in 1960 these concepts had been developed extensively only for straight dislocations. The theory of curved dislocations appeared during the past 25 years.
Mura12 transformed Burgers’s equation into a line integral for displacement gradients in terms of an integrand containing the elastic Green function tensor. This led to the Brown formula13–15 for the stress field at an arbitrary point, produced by a line segment of dislocation, in terms of elastic energy coefficients for an infinite straight dislocation pair passing through the point in question and the ends of the segment. The result applies for isotropic or anisotropic elasticity. Blin16 also presented an expression, based on Burgers’s equation, for the interaction energy of two dislocation loops. Manipulations of this expression and the Peach-Koehler result led to isotropic elastic expressions for the self- and interaction energies of dislocation segments,17,18 the interaction force between segments,19 and the displacement field of a segment.20 By approximating an arbitrary curved dislocation as a connected set of straight segments, one can use this set of relations to determine the elastic fields of complex dislocation arrays. Examples include stacking-fault tetrahedra, loops, double kinks, and double jogs.21
The elastic interaction of an arbitrarily inclined dislocation and a free surface was expressed by Lothe22 as an image interaction analogous to that in electrostatics. This interaction was also extended to the case of dislocation segments, with numerous subsequent applications.23,24
As an alternative to the eigenvalue sextic solution of Eshelby and co-workers,25 Stroh developed an explicit solution for the anisotropic elastic field of a straight dislocation.26 Another alternative formulation using Fourier analysis was presented by Willis.27 The Stroh theory was elaborated as an integral theory, facilitating numerical calculations, by Barnett and Lothe.28 Advanced anisotropic elastic calculations for complex dislocation configurations have now been initiated.29
With the advent of computers, atomic calculations have been used to estimate the Peierls stress and energy, the variation of energy with position