Application of nonlocal elasticity to the energetics for solute atoms in body-centered cubic transition metals with disl
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I.
II. E D G E D I S L O C A T I O N WITH SOLUTE ATOM
INTRODUCTION
THE interaction energy between dislocations and solute atoms and the elastic energy due to interstitial atoms are the important physical quantities to understanding and calculating properties of metals and alloys. The pinning down of solute atoms to edge and screw dislocations is the basic mechanism of yield phenomenon. However, the precise calculation of the interaction between dislocations and solute atoms on the atomic scale has not been solved in a satisfactory manner, tm] Theoretical predictions that the interstitial atom occupies the tetrahedral or octahedral sites in body-centered cubic (bcc) metals vary with the author. I3] Just recently, we applied nonlocal linear elasticity theory to calculate the interaction energy between dislocations and point defects, t1'21 The results are in very good agreement with the experiments, which stimulates us to do further work. The purpose of the present article is to formulate the self-energy of interstitial atoms, the interaction energies between edge or screw dislocations, and the interstitial atoms at the tetrahedral or octahedral positions in bcc transition metals by using nonlocal elastic theory. All quantities calculated, e . g . , t h e distances, sites, and volume changes, are unambiguous. The results show that the energies can be calculated within an error of 0.1 eV by applying nonlocal linear elasticity. The basic idea of nonlocal elasticity is that the theory takes into account long-range (nonlocal) interatomic interactions in the determination of the elastic stresses originating from a displacement field, eliminating the stress singularities which appear in classical (local) elasticity. The present treatment will be restricted to linear, isotropic, elastic medium with classical elastic modulus tensor. The theory is linear in strain, like the classical (local) continuum theory of elasticity. Other contributions ( e . g . , chemical or electronic) to the binding of solute are not taken into account for the present.
By considering the interaction due to the size effect of a solute atom as being due to the effect of a dilatation center, in a linear, isotropic, nonlocal elastic medium, we obtained the elastic interaction energy between an edge dislocation and a solute atom in nonlocal elasticity as [I ]
~b(1 + v) U(r) -
3zr(1 - v)
c
sin0[ -
1 - exp
/~b(1 + v )
sin0 c --
37r(1 - v)
METALLURGICAL TRANSACTIONS A
[1]
where /x and v are the L a m e ' s constant and Poisson's ratio; b is the Burgers vector of the dislocation; and c = v - v0 is the volume change due to an impurity defect in the host, which can be obtained from atomistic theodes of the region surrounding the defect or from experimental data. For present isotropic assumption, the attenuation factor k = 1.65, which is determined from the dispersion curve of one-dimensional plane waves based on nonlocal elasticity coinciding with that based on the Born-K~m~in model of atomic lattice dynamics in the entire Brillouin zone with an
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