2015_Kindrachuk

Here x : = ( x+ x)/2
denotes the McCauley brackets and K g and
n g are slip system dependent model
parameters. In the following, slip system dependent model parameters will be assumed to take different values
1,12
g Π13,18 slip systems. The overstress
for octahedral ( g Π[ ])
and cubic ( [ ])
f = t -x -R - r
(4)
g g g 0g g
is a function of the resolved shear stress t = s:
m , the back stress x g and the isotropic hardening contribution
g
g
r g to the yield stress R 0g . Equation f g = 0 defines the boundary of elasticity domain. The evolution of
kinematic hardening follows from the Armstrong and Frederick equation (Armstrong and Frederick, 2007). The
growth of the back-stress x g is compensated by the dynamic ( DR g ) and static ( SR g ) recovery terms
x
= c a ,
g g g
vp
g = g - g ⋅DRg -SRg
a& g& w&
.
(5)
Here
a g is a strain-like kinematic internal variable associated to the back stress;
g
vp
g
w& = g& denotes the
accumulated shear strain rate, c g is a material parameter. Both recovery terms are power law functions of
like in the flow rule (3), i.e.
a g ,
DR
g
Êag
ˆ
= DR ( a ) =Á ,
ÁD
˜
Ë g ¯
g g g
m
d
g
g
Ê x ˆ Ê
g m
a ˆ
g
g
SRg = sign( xg) ⋅ Á ˜ = cg ⋅sign( ag) ⋅Á ˜ .
ÁMg
˜ ÁMg
˜
Ë ¯ Ë ¯
m
(6)
The Cailletaud model considers an interaction between slip systems at the level of the isotropic hardening. The
interaction is expressed by the interaction matrix H (influence of the slip system „i“ on the system “g”). Thus
the isotropic internal variable accounting for the interaction is given as
gi
 ( 1 exp( ))
. (7)
r = Q ⋅ H ⋅ - -bw
g g gi i i
iŒG
In case of self-hardening, the interaction matrix reduces to the identity matrix.
3.2 Extension of the Model of Cailletaud for the Rate-Independent Behavior
Viscoplastic constitutive models can properly simulate the high-temperature behavior of metals and alloys.
However, using viscoplasticity for low or even moderate temperatures can lead to numerical difficulties. This is
caused by high Norton’s exponents needed in a nearly rate-independent regime. Therefore, a numerically robust
model valid over the whole temperature range should incorporate both viscoplastic and plastic behaviors. With
this in mind, we assume that the inelastic shear strain rate is the sum of the viscoplastic and plastic contributions
where
Ê
n g
f ˆ
g
p
g& g = Á + l&
˜
g ⋅sign( tg - xg) = w& g ⋅sign( tg -xg),
(8)
Á K ˜
g
Ë
¯
w&
g denotes further the inelastic accumulated shear strain rate
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