The dislocation motion under an external stress field is greatly influenced by the internal stress field of the precipitate and dislocations. The spatial distribution of stress caused by internal stress origins is readily computed by Green’s function method based on micromechanics, which is well documented in textbook [
9]. First, total stress,
, is represented by the sum of external stress
and internal stress
as follows:
where
is external stress without the disturbance of internal stress,
is internal stress caused by dislocations,
, is internal stress caused by precipitates. Note that the internal stress field in Equation (1) varies depending on the geometry of the dislocation segments and precipitates. Since dislocation motion is predicted by the stress acted on the dislocation, the stress computation is performed only on the dislocation nodal points. This indicates that the elastic field of the entire body of material is not needed in the present simulation, so that dislocation motion can be predicted analytically without solving the boundary value problem, such as with FE analysis.
2.1. Stress Field of Precipitate
According to the Eshelby inclusion theory [
9,
10,
11,
12], the stress of the precipitate
at the field point
can be given through the Eshelby tensor as follows:
where
is Eshelby tensor,
is stiffness tensor,
is eigenstrain of region Ω. Supposing that the region Ω is subjected to uniform eigenstrain, the elastic field inside Ω becomes uniform when an ellipsoidal inclusion is assumed. The analytical solution of the Eshelby tensor for isotropic ellipsoidal inclusion is listed in textbook [
9], while the arbitrary shape of the inclusion should be computed numerically by Green’s function. Furthermore, the analytical solution for the stress outside Ω is limited for the isotropic case [
13], so that the exterior point stress should be computed also numerically. Assuming that the misfit strain is the major internal stress origin (inclusion problem), the superposition principle of stress holds even in the case of multiple inclusions. The general integral expression of inclusion stress is represented by the surface integral as follows:
where
Gij,k is the first derivative of Green’s function, n
k is outward vector normal to |Ω| for the interior point Eshelby tensor, while the inward direction is for exterior point Eshelby tensor. Note that Equation (3) gives the total strain at the region inside and outside, then eigenstrain should be subtracted as shown in Equation (2) when the evaluation point is taken at the interior point of Ω. In the case that the stiffness of inclusion is different from that of the matrix phase, the stress disturbance should be considered as an inhomogeneity problem. In such a case, the fictitious eigenstrain assumed in Ω is proportionally changed with the stress caused by other stress origins, such as dislocation, inhomogeneity, external stress, etc. Nevertheless, the eigenstrain formulation of the present study is still valid whenever the dislocation interacts with the elastic field of the misfit precipitate.
2.4. Energy Consideration of Precipitation Hardening
The evolution of dislocation microstructure under external stress is influenced by the dislocation motion among local internal stress fields of dislocations and precipitates, where the total plastic strain by dislocation motion should be treated as the macroscopic average of strain. First, the amount of plastic work performed on the material is mentioned by
where
is plastic work performed on the material,
is variation in plastic strain,
is total stress, respectively. However, due to the fact that the internal stress is zero in average,
in the entire volume of material equals external stress on the loading surface,
.
Accounting that the increment of plastic strain is mentioned by the total area swept by dislocations in considered volume, the discrete dislocation slip can be translated into the averaged plastic strain. According to Mura [
9], the amount of plastic strain in considered volume,
, can be mentioned by the product of Burgers vector,
, and the area swept by dislocations,
S, as follows:
where small displacement of dislocation motion
is assumed in line integral. By substituting Equation (7) into Equation (6), plastic work performed on the material is mentioned by that mediated by total dislocation motions in material as follows:
In order to discuss the elastic energy and its interactions associated with the dislocation and precipitate, Gibbs free energy change is considered. the total Gibbs free energy associated with external stress, precipitates and dislocations is prescribed by elastic strain energy,
, and potential energy,
, as follows:
where
is elastic energy of external stress, dislocations, and precipitates without interactions between them (self-energy), which is reduced to the following:
where superscript
and
indicates stress associated with
p-th inclusion and m-th dislocation, respectively. Summation is taken with respect to
and
. Meanwhile
is interaction energy between precipitates and dislocations, as follows:
where the first term is the interaction of precipitates and the second term is interaction of dislocations. The third terms are dislocation–precipitate interactions, which are expressed by dislocation stress and precipitate strain. Potential energy is defined by tractions and displacement on the external loading surface,
, which is as shown below:
The first term is potential energy due to external loading and the second and third terms are the interaction of external stress and eigenstrain associated with precipitates and dislocations. Note that interaction between external stress and internal stress is absent in the elastic energy of Equation (11) (Collonetti’s theorem [
9]), while external stress interacts with internal stress in potential energy in Equation (12). Superscript
and
indicate n-th inclusion and dislocation segment, respectively. Since the elastic strain of the internal stress origin should be zero at the traction-free surface,
and
are regarded as the displacement caused by average eigenstrain in the whole volume. Note that although the energy associated with the precipitate and dislocation in Equation (10)–(12) is computed by the volume integral of the entire volume, the integral region is eventually reduced to the region of eigenstrain due to the property of internal stress. By translating volume integrals associated with dislocation motion in Equation (10)–(12) into surface integrals, the energy associated with the dislocation motion can be obtained as follows:
For simplicity, the energy difference due to the dislocation motion is considered by assuming the small displacement of dislocation segments in Equation (13)–(15). Variation in Gibbs free energy associated with dislocation motion is mentioned by
as follows:
Therefore, by using the translation property of surface and line elements,
, variation form of energy components become the following;
where
,
,
is variation in dislocation self-energy, interaction-energy and potential energy, respectively. Note that, due to the singularity of the self-stress in the dislocation core, the self-energy of dislocation is often alternated by line tension approximation
. By taking the sum of energy differences of external and internal origins, variation in Gibbs free energy becomes the following;
Eventually, the stress mentioned in Equation (20) is the total stress acted on dislocation segments. Note that the numerical sum of
accounts for all the dislocation segments, where interaction stress (s ≠ t) is computed by Equation (4), while self-stress (s = t) can be alternated by line tension approximation. By comparison of Equation (20) to the macroscopic external plastic work in Equations (6) and (8), the second and third terms in Equation (20) are regarded as the change in interaction energy. Although the average internal stress should be zero in the entire volume of material (
), the dislocation motion is influenced by the internal stress field of dislocations and precipitates because the total sum of interaction stress at the dislocation nodal point is locally not zero. Recalling the fact that the dislocation glide motion and total force vectors should be the same directions,
is always decreased whenever dislocation motion takes place such as in the following:
Accordingly, depending on the signs and the magnitude of interaction forces, external and internal work is classified into the following three cases: (I) hardening behavior, (II) softening behavior, (III) annihilation behavior. In case (I), the direction of interaction force and external force is opposite , then the dislocation motion is retarded and the amount of becomes smaller since the amount of plastic displacement is decreased. In case (II), in contrast to the above, the total force acted on the dislocation is increased by internal stress ; therefore, the dislocation motion is promoted and the amount of is negatively increased with the increase in plastic displacement. In case (III), annihilation of dislocations takes place due to the negative direction of strong interaction forces since the displacement vector turns into negative direction. This situation may be made possible by the dislocation motion driven by the strong internal stress of the precipitate. In all the cases, is negatively increased by the dislocation motion, but the amount of is greatly influenced by the internal stress field, hence the interaction force acted on the dislocation is important to be clarified in a later section.