84
S. Ghosh et al.
the polycrystalline SERVEs. The parameters of the AE-CP constitutive model are
calibrated from experimental data on polycrystalline superalloy Rene88-DT. Details
of the constitutive model and crystal plasticity parameter calibration are given in
[29].
The plastic slip-rate ˙
γ α in this model is governed by the Orowan’s equation,
reflecting a thermal activation relationship. For a given slip system,
˙
γ
α
=
⎧
⎨
⎩
0
f o r τ α
eff ≤ 0
˙
γ α
0 exp
−
Q
k B T
1 −
τ α
eff
s α
∗,tot
p 1 p 2
sign(τ α ) for 0 < τ α
eff ≤ s α
∗,tot
(21)
where ˙
γ α
0 is the reference slip rate, Q is the activation energy, k B is the Boltzmann
constant, T is the temperature, and p 1 and p 2 are material constants. The effective
shear stress in any slip system, τ α
eff = |τ α | − s α
a , is defined as the difference of
resolved shear stress τ α and the athermal obstacle resistance s α
a due to parallel
dislocations. The total thermal slip resistance is comprised of two parts, i.e. s α
∗,tot =
s α
∗ + s α
cross . The thermal slip resistance s α
∗ provides an impeding effect of obstacles
that can be overcome by thermally activated processes such as forest dislocations.
The cross-slip resistance s α
cross develops due to sessile dislocation segments creating
pinning points by formation of Kear-Wilsdorf (KW) configurations. The initial
values of athermal and thermal resistances, i.e. s α
a0 and s α
∗0 due to the presence
of statistically stored dislocations (SSDs) are determined through the experimental
calibration process.
The time evolution of the athermal and thermal slip resistances is a function of
the plastic slip-rate, in and out of the slip plane, corresponding to the effects of
parallel and forest dislocations respectively. The slip resistances evolve as:
˙
s
α
a =
N
β=1
h
αβ
a | ˙
γ
β sin(n
α , t
β )| and ˙
s
α
∗ =
N
β=1
h
αβ
∗ | ˙
γ
β cos(n
α , t
β )|
(22)
where N is the number of slip systems, m α is the slip direction, n α is the slip plane
normal, t α = m α × n α is the transverse direction. The interaction coefficients are
taken to be the same for both the athermal and thermal resistances, i.e. h αβ = h
αβ
a =
h
αβ
∗ . The hardening coefficients for self and latent hardening are expressed as:
h
αβ
= q
αβ h
β , where h
β
= h 0
1 −
s β
s
β
sat
r
sign
1 −
s β
s
β
sat
(23)
where h 0 is a material constant, s α is the shear resistance, s α
sat is the saturation slip
resistance, r is an exponent controlling the rate of saturation, and q αβ = q + (1 −
q)δ αβ and δ αβ is the Kronecker delta. Parameter q = 1.4 in this study. s α
cross is
another term contributing to the total thermal slip system resistance. This resistance
is associated with the accumulation of pinned screw dislocations after cross slip.
The evolution of the cross-slip resistance is computed for both the octahedral and
cube slip systems as:
S. Ghosh et al.
the polycrystalline SERVEs. The parameters of the AE-CP constitutive model are
calibrated from experimental data on polycrystalline superalloy Rene88-DT. Details
of the constitutive model and crystal plasticity parameter calibration are given in
[29].
The plastic slip-rate ˙
γ α in this model is governed by the Orowan’s equation,
reflecting a thermal activation relationship. For a given slip system,
˙
γ
α
=
⎧
⎨
⎩
0
f o r τ α
eff ≤ 0
˙
γ α
0 exp
−
Q
k B T
1 −
τ α
eff
s α
∗,tot
p 1 p 2
sign(τ α ) for 0 < τ α
eff ≤ s α
∗,tot
(21)
where ˙
γ α
0 is the reference slip rate, Q is the activation energy, k B is the Boltzmann
constant, T is the temperature, and p 1 and p 2 are material constants. The effective
shear stress in any slip system, τ α
eff = |τ α | − s α
a , is defined as the difference of
resolved shear stress τ α and the athermal obstacle resistance s α
a due to parallel
dislocations. The total thermal slip resistance is comprised of two parts, i.e. s α
∗,tot =
s α
∗ + s α
cross . The thermal slip resistance s α
∗ provides an impeding effect of obstacles
that can be overcome by thermally activated processes such as forest dislocations.
The cross-slip resistance s α
cross develops due to sessile dislocation segments creating
pinning points by formation of Kear-Wilsdorf (KW) configurations. The initial
values of athermal and thermal resistances, i.e. s α
a0 and s α
∗0 due to the presence
of statistically stored dislocations (SSDs) are determined through the experimental
calibration process.
The time evolution of the athermal and thermal slip resistances is a function of
the plastic slip-rate, in and out of the slip plane, corresponding to the effects of
parallel and forest dislocations respectively. The slip resistances evolve as:
˙
s
α
a =
N
β=1
h
αβ
a | ˙
γ
β sin(n
α , t
β )| and ˙
s
α
∗ =
N
β=1
h
αβ
∗ | ˙
γ
β cos(n
α , t
β )|
(22)
where N is the number of slip systems, m α is the slip direction, n α is the slip plane
normal, t α = m α × n α is the transverse direction. The interaction coefficients are
taken to be the same for both the athermal and thermal resistances, i.e. h αβ = h
αβ
a =
h
αβ
∗ . The hardening coefficients for self and latent hardening are expressed as:
h
αβ
= q
αβ h
β , where h
β
= h 0
1 −
s β
s
β
sat
r
sign
1 −
s β
s
β
sat
(23)
where h 0 is a material constant, s α is the shear resistance, s α
sat is the saturation slip
resistance, r is an exponent controlling the rate of saturation, and q αβ = q + (1 −
q)δ αβ and δ αβ is the Kronecker delta. Parameter q = 1.4 in this study. s α
cross is
another term contributing to the total thermal slip system resistance. This resistance
is associated with the accumulation of pinned screw dislocations after cross slip.
The evolution of the cross-slip resistance is computed for both the octahedral and
cube slip systems as: