132
A. Cruzado et al.
0
0.02
0.04
0.06
0.08
0
100
200
300
400
500
600
700
800
0
0.02
(a)
(b)
0.04
0.06
–4
–3
–2
–1
0
1
2
3
4
Linear fit
Fig. 3 Strain rate sensitivity analysis of micropillars of 5 μm in diameter oriented for single slip
at room temperature. (a) RSS vs. plastic strain for three different strain rates: 10 −2 , 10 −3 , and
10 −4 s −1 and (b) linear regression curve to obtain the strain rate sensitivity m parameter
given slip plane α can be expressed as ˙
γ α = ˙
γ 0 (|τ α |/τ α
c |) 1/m sgn(τ α ). Here ˙
γ 0 is the
reference strain rate, τ α
c the critical resolved shear stress at the reference strain rate,
and m the rate sensitivity exponent. Taking the natural logarithm of the viscoplastic
flow, the following equation is derived:
ln
˙
γ α
˙
γ α
0
=
1
m
ln
τ α
s
τ α
c
(1)
so that the rate sensitivity exponent m can be obtained as the slope of the linear
regression between ln
˙
γ α
˙
γ α
0
and ln
τ α
s
τ α
c
. The average strain rate ˙
γ α for each slip
system is accounted, and the average strain of the micropillar divided by SF and
the corresponding CRSS τ α
s was taken at a plastic strain of 0.04. These results are
normalized against the reference strain rate considered in this work as 10 −3 s −1 .
The linear regression performed in Fig. 3b renders a strain rate sensitivity exponent
of m = 0.017. This exponent indicates a very small rate sensitivity of the material at
room temperature.
The SEM micrographs of micropillars deformed in single and multiple slip
orientation are shown in Fig. 4, respectively. The micropillar oriented along 235
clearly deforms through slip in a single plane, while the micropillar 001 shows the
activation of multiple slip systems. The rest of the parameters involved in the crystal
plasticity model that will be described in Sect. 5 have to be calibrated comparing
the results obtained in tests performed on pillars with crystallographic orientations
deforming under single and double slip with the corresponding numerical simulations. In particular, this method allowed to identify the critical resolved shear stress
and the parameters defining the strain hardening of the crystal [14], accounted here
A. Cruzado et al.
0
0.02
0.04
0.06
0.08
0
100
200
300
400
500
600
700
800
0
0.02
(a)
(b)
0.04
0.06
–4
–3
–2
–1
0
1
2
3
4
Linear fit
Fig. 3 Strain rate sensitivity analysis of micropillars of 5 μm in diameter oriented for single slip
at room temperature. (a) RSS vs. plastic strain for three different strain rates: 10 −2 , 10 −3 , and
10 −4 s −1 and (b) linear regression curve to obtain the strain rate sensitivity m parameter
given slip plane α can be expressed as ˙
γ α = ˙
γ 0 (|τ α |/τ α
c |) 1/m sgn(τ α ). Here ˙
γ 0 is the
reference strain rate, τ α
c the critical resolved shear stress at the reference strain rate,
and m the rate sensitivity exponent. Taking the natural logarithm of the viscoplastic
flow, the following equation is derived:
ln
˙
γ α
˙
γ α
0
=
1
m
ln
τ α
s
τ α
c
(1)
so that the rate sensitivity exponent m can be obtained as the slope of the linear
regression between ln
˙
γ α
˙
γ α
0
and ln
τ α
s
τ α
c
. The average strain rate ˙
γ α for each slip
system is accounted, and the average strain of the micropillar divided by SF and
the corresponding CRSS τ α
s was taken at a plastic strain of 0.04. These results are
normalized against the reference strain rate considered in this work as 10 −3 s −1 .
The linear regression performed in Fig. 3b renders a strain rate sensitivity exponent
of m = 0.017. This exponent indicates a very small rate sensitivity of the material at
room temperature.
The SEM micrographs of micropillars deformed in single and multiple slip
orientation are shown in Fig. 4, respectively. The micropillar oriented along 235
clearly deforms through slip in a single plane, while the micropillar 001 shows the
activation of multiple slip systems. The rest of the parameters involved in the crystal
plasticity model that will be described in Sect. 5 have to be calibrated comparing
the results obtained in tests performed on pillars with crystallographic orientations
deforming under single and double slip with the corresponding numerical simulations. In particular, this method allowed to identify the critical resolved shear stress
and the parameters defining the strain hardening of the crystal [14], accounted here