120
J. Liu et al.
Fig. 5.6 The model configuration of a crystal under a shear strain
deformation field in the amplitude order parameter in the APFC models [27, 28].
The “penalty term” approach is the most intuitive and simplest, which has the form
similar to Eq. (5.32). Hence in this section, we will focus on the “penalty term”
approach to apply the strain field in the PFC models.
The “penalty term” applied in the PFC models serves as traction boundary conditions. It is the square of the difference of the density field n and the imposed density
field n s [17, 18, 29, 30], i.e.,
F ext =
drM (r)(n − n s )
2
∂n
∂t
= ∇
2 δF
δn
= ∇
2
δF
δn
+
δF ext
δn
(5.33)
Note that the “penalty term” is modulated by a function M (r), i.e., M (r) = 0
outside the region where the traction is being applied. For the application of “penalty
term” for polycrystalline, n s needs to be adjusted to accommodate the various grain
orientations, which has the similar form of Eq. (5.17). Different grain orientations of
n is given by the rotation of the coordinates (x, y, z) to a new coordinates (x
, y
, z
),
i.e., (x
, y
, z
)
T
= R m (x, y, z)
T [31]. R m is the rotation matrix and T denotes the
transposition of a matrix. Figure 5.6 illustrates a PFC model with the application of
a shear strain.