M-SERVE and P-SERVE
85
s
α
cross =
s α
cross,oct =s α
cross,oct (τ α
pe , τ α
se , τ α
cb , T , , 111 , , 100 ) on octahedral slip systems
s α
cross,cube =s α
cross,cube (T )
on cube slip systems
(24)
The cross-slip resistance on the octahedral planes accounts for the non-Schmid shear
stresses and anisotropy in the antiphase boundary (APB) energy function and is
expressed as:
s
α
cross,oct = ξ 0 exp
A
T − T c
μ
ρ 0 exp
−
H α
k B T
(25)
where H
α
= c H
⎧
⎨
⎩
h + k 1 (t
α
pe − k 2 t
α
se ) +
1
√
3
−
010
111 + |t α
cb |
b
B
⎫
⎬
⎭
and ξ 0 , A, h, k 1 , k 2 are material constants, T c is the critical temperature, H α is
the enthalpy for the cross slip event, and ρ 0 is the initial cross-slipped dislocation
density. Furthermore,
t
α
xx =
τ α
xx
111 /b
, B =
μb 2
2ππ 111 , c H =
μb 3
4π
(26)
The cross-slip resistance in cube slip system is a function of temperature and is
defined as:
s
α
cross,cube =
s cc1
for T ≤ 915K
s cc1 − s cc2 (T − 915) for T > 915K
(27)
where s cc1 and s cc2 are material properties. It should be noted that the cube slip
systems are only activated when the temperature rises. Details of this model are
given in [23].
3.4.1 P-SERVE Convergence Studies with the Crystal Plasticity Model
The size of the P-SERVE is estimated through convergence studies of material properties and response functions. CPFE simulations are conducted for a large number of
simulated SERVEs of different sizes with variations in the microstructural statistics
using the calibrated material parameters. For each size, at least five realizations
are generated for simulations. For smaller sizes, where high scatter in material
properties is expected, larger number of SERVEs are simulated. For example, up
to 15 samples are simulated with a size of 25 μm.
Results of the CPFE simulations are used to determine the volume-averaged,
macroscopic strain energy, yield strength, and the hardening rate. The yield strength
is measured as 0.2% flow stress in the stress-strain plot and the hardening rate
is derived as average slope of the stress-strain plot in the post-yield plastic
deformation. The values of these CPFEM simulated volume-averaged quantities are
85
s
α
cross =
s α
cross,oct =s α
cross,oct (τ α
pe , τ α
se , τ α
cb , T , , 111 , , 100 ) on octahedral slip systems
s α
cross,cube =s α
cross,cube (T )
on cube slip systems
(24)
The cross-slip resistance on the octahedral planes accounts for the non-Schmid shear
stresses and anisotropy in the antiphase boundary (APB) energy function and is
expressed as:
s
α
cross,oct = ξ 0 exp
A
T − T c
μ
ρ 0 exp
−
H α
k B T
(25)
where H
α
= c H
⎧
⎨
⎩
h + k 1 (t
α
pe − k 2 t
α
se ) +
1
√
3
−
010
111 + |t α
cb |
b
B
⎫
⎬
⎭
and ξ 0 , A, h, k 1 , k 2 are material constants, T c is the critical temperature, H α is
the enthalpy for the cross slip event, and ρ 0 is the initial cross-slipped dislocation
density. Furthermore,
t
α
xx =
τ α
xx
111 /b
, B =
μb 2
2ππ 111 , c H =
μb 3
4π
(26)
The cross-slip resistance in cube slip system is a function of temperature and is
defined as:
s
α
cross,cube =
s cc1
for T ≤ 915K
s cc1 − s cc2 (T − 915) for T > 915K
(27)
where s cc1 and s cc2 are material properties. It should be noted that the cube slip
systems are only activated when the temperature rises. Details of this model are
given in [23].
3.4.1 P-SERVE Convergence Studies with the Crystal Plasticity Model
The size of the P-SERVE is estimated through convergence studies of material properties and response functions. CPFE simulations are conducted for a large number of
simulated SERVEs of different sizes with variations in the microstructural statistics
using the calibrated material parameters. For each size, at least five realizations
are generated for simulations. For smaller sizes, where high scatter in material
properties is expected, larger number of SERVEs are simulated. For example, up
to 15 samples are simulated with a size of 25 μm.
Results of the CPFE simulations are used to determine the volume-averaged,
macroscopic strain energy, yield strength, and the hardening rate. The yield strength
is measured as 0.2% flow stress in the stress-strain plot and the hardening rate
is derived as average slope of the stress-strain plot in the post-yield plastic
deformation. The values of these CPFEM simulated volume-averaged quantities are