Plastic deformation of solids
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Аннотация
We present a phase-field theory of dislocation dynamics
in binary alloys, in which the elastic energy
is a periodic function of the anisotropic strain components. We introduce a elongation elastic energy $\Phi$ and a shear elastic energy $\Psi$ as
eqnarray*
\Phi=\frac\mu_2(\psi)8\pi^2 \times \\
&&łeft3-\cos2\piłeft(e_2-e_33\right)-\cos2\piłeft(e_2+e_33\right)-\cosłeft(4e_33\right)\right,
eqnarray*
$$\Psi = \mu_3(\psi)4\pi^2łeft3-\cos(2e_4)-\cos(2e_5)-\cos(2e_6)\right, $$
where $\mu_2(\psi)$ and $\mu_3(\psi)$ are the composition dependent elastic modului. We also introduce local tetragonal strains $e_2$ and $e_3$, and local shear strains $e_4$, $e_5$ and $e_6$ defined by
$$e_2 = \nabla_x u_x-\nabla_y u_y,$$
$$e_3= 13(2\nabla_zu_z - \nabla_y u_y - \nabla_x u_x),$$
$$e_4 = \nabla_x u_y + \nabla_y u_x,$$
$$e_5=\nabla_yu_z+\nabla_z u_y,$$
$$e_6=\nabla_xu_z+\nabla_zu_x, $$
where we have introduced a displacement vector $u$.
The composition difference of two metals $\psi$ is twofold coupled to the elastic field
via the lattice misfit and via the composition-dependence
of the elastic moduli (elastic inhomogeneity). So the total energy density is defined by
$$f(\psi, u) = f_BW(\psi) + e_1+ f_el(u, \psi),$$
where the first term is the Bragg-Williams free energy density, and the second term represents the lattice misfit.
We numerically solve the dynamic equations
for the lattice displacement $u$ and the composition $\psi$
$$\partial\psit = \nablaL_0(\psi)\nabla\delta\mathcalF\delta\psi,$$
$$\rho_0vt = -\delta\mathcalFu +
\eta_0\nabla^2v +(\zeta_0+\eta_0/3)\nabla\nablav, $$
to describe
various dislocation processes
in three dimensions, where $F$ is the free energy functional, and $L_0(\psi)$ is the kinetic coefficient.
On stretching in one-phase states,
dislocations are proliferated to form a tangle.
They tend to be created near preexisting dislocations.
On stretching in two-phase states,
dislocations appear in the interface region
and glide into the soft region.
They are detached from the interface
to expand as closed loops, where
a hard precipitate is acting as a dislocation mill as shown below.
1) A.~Minami, A.~Onuki, Phys. Rev. B 70 (2004) 184114.\\
2) A.~Minami, A.~Onuki, Phys. Rev. B 72 (2005) 100101.\\
3) A.~Minami, A.~Onuki, Acta Mater. 55 (2007) 2375.