数式メモ:DD

Dislocation dynamics (DD) simulation is now widely used for the
studies of work hardening, BDTT etc.
The Discrete Dislocation dynamics(DDD)

In order to precicely reproduce the behaviour of the dislocation pinning,
one has to model the dislocation lines with reasonable parameters.


imprement the criteria for the
different dislocation lines with different Burger's vector

As far as the pinning process of single dislocation is concerned,
the parameter we have to fix is the dislocation mobility
and

The "classical" critical resolved stress (Gottstein, Physical Foundations of Materials Science) is as follows:
\tau_0 = \frac{1}{l-2r} f_e /b
where f_e denotes line tension of an edge dislocation.
In ref. \cite{gottstein}, f_e is set to G b^2 where G = E/2(1+\nu) is the shear modulus. We take the orientation dependence of the line tension into account and introduce cutoff length:
f(\theta)= \frac{f_0(\theta) G b^2}{4\pi} \log R_{max}
The line tension of a curved dislocation segment of curvatur radius $R$ is then (f(\theta)+f^{\prime\prime}(\theta))/R per unit length.

以下捨て

The energy increase by the presence of a hard precipitate of radius d is proportional to d^3 \epsilon^2, where $\epsilon$ is the strain at the precipitate sphere induced by a dislocation segment.
The PK strain is
\epsilon_{xz}=\frac{b_x x}{R(R+y)}
\epsilon_{yz}=\frac{-b_x \nu}{R}+\frac{b_y x(1-\nu)}{R(R+y)}
and the interaction force is
\epsilon_{xz} \partial \epsilon_{xz}/\partial x+\epsilon_{yz} \partial \epsilon_{yz}/\partial x

\frac{\partial}{\partial x}\frac{1}{R} = \frac{x}{R}
\frac{\partial}{\partial x}\frac{x}{R(R+y)} = \frac{1}{R(R+y)} -\frac{x^2}{R^3(R+y)} -\frac{x(x+Ry)}{R^2(R+y)^2}

The interaction force between a hard precipitate and a dislocation segment can be calculated from the interaction bewteen the dislocation and its "mirror image" inside the precipitate. In the MD simulation, atomic displacement on the surface of the precipitate is fixed to zero. Note that the relation between the dislocation configuration and the strain field is covariant under the inversion transformation \vec{R}\rightarrow R_0 \vec{R}/\vec{R}^2 where R_0 denotes the radius of the precipitate, and the displacement field induced by the inverted dislocation exactly cancels out the outside one on the surface of the precipitate.

To avoid singular behavior of the Peach-Koehler force, interactions between neighboring segments are calculated separately. We assume that the dislocation line is a smooth curve whose local curvature \kappa is equal to \delta \theta /0.5(l_i+l_j) where \delta \theta is the angle between the neighboring segment and l_i denotes the length of the i'th segment. If a local coorinate system x'y' is introduced so that x' and y' is parallel and perpendiculat to the tangent vector of the dislocation, the dislocation curve is approximated by y'=\kappa x'^2. Integration of PK force acting on the segment junction gives the following line tension force:
\kappa f(\theta) (\log l_i/b+log l_j/b)
where b is the short-range cutoff length.

Assume that we have a protion of curved dislocation line of Burger's vector (bx,by), expressed by a curve x=\kappa y^2. The PK force from the curve acting on the point (0,0), integrated over a length r is G/4\pi 2\kappa\log( r/b) (b_x^2/(1-\nu)+b_y^2).

FCC Metal Al Ni Cu Ag Au
\gamma_{SF}(mJ/m^2) 200 80 40 20 33
b_p=b/2 (Angstrom) 1.43 1.24 1.28 1.45 1.44
\tau_{UB}=\gamma_{SF}/b_p(MPa) 1400 640 300 140 230

Strength of shear stress at which split partials become unbounded.

The anglar dependence of deislocation energy is measured by molecular dynamics simulation, using the well-developed embedded atom method potential. We choose several kinds of simulation cell configurations which assume various angles to the Burger's vector. Table x summarizes these configurations. Periodic boundary condition is imposed in the direction parallel to the dislocation line, and free-boundary condition is used in the other two directions. We measured the energy increase around each split partial, summed over atoms within a cylinder of radius $r$, whose center is located at the center of each split partials. The energy increase can be attributed to the elastic energy, core energy, and the stacking fault energy: E(r)=E_{core}(\theta)+ \gamma r + Gb^2/4\pi f(\theta) \log(r/r_{core}), where \gamma is the stacking fault energy, G is the shear modulus, and r_core is the core cutoff length.
We measured E(r) for the two cases $r=3b$ and $r=4b$.
the difference betweeen them is well fitted by the known parameters $\gamma$ and $G$. By subtracting the stacking fault energy and elastic energy and setting $r_{core}=3b$, we have estimated the core energy $E_core(\theta)$.

Assumption
E = Vh p
Vh=
T-sites in BCC

Only mixed 110-100
p=(C11 Uxx)/3
P.B.C for XYZ

D=D0 exp(-Em/kT)
D0=
Em=
Experimental/first-principle
Phys.Rev.B 70, 064102(2004)

Assumption
Barrier-height
(maybe lower)
T-T
E > -0.3eV

kMC