324
M. R. Karamooz-Ravari et al.
Table 17.4 Utilized material parameters for simulation of cyclic loading in shape memory regime
(Karamooz-Ravari et al. 2018)
M f ( ◦ C) σ cr
s0 (MPa) C M0 (MPa/ ◦ C) C Af0 (MPa/ ◦ C) E M0 (MPa) σ M0
ye (MPa) ε ∗
0
26
100
NA
NA
80,000
600
0.0315
M s ( ◦ C) σ cr
se (MPa) C Me (MPa/ ◦ C) C Afe (MPa/ ◦ C) E Me (MPa) σ A0
ye (MPa) ε ∗
e
54.1
5
NA
NA
62,000
NA
0.015
A s ( ◦ C)
σ cr
f 0 (MPa)
C As0 (MPa/ ◦ C)
E A0 (MPa)
σ M0
y0 (MPa)
h(MPa)
μ
59
380
NA
NA
350
30,000
6.5
A f ( ◦ C)
σ cr
fe (MPa)
C Ase (MPa/ ◦ C)
E Ae (MPa)
σ A0
y0 (MPa)
ε
(max)
p
m
84.5
600
NA
NA
NA
0.054
2.8
Fig. 17.10 Comparison of the predicted cyclic stress–strain responses in shape memory regime
with experimental ones (Karamooz-Ravari et al. 2018)
17.3 Conclusions
This chapter was allotted to 3-D constitutive modeling of shape memory alloys using
microplane approach. The basic concepts of this method and its different formulations were presented. The potential directional bias in conventional microplane theory
was eliminated by developing modified formulations based on one shear component
on each plane. 1-D constitutive equations of SMAs were generalized for multiaxial
loadings using microplane theory, and an efficient numerical implementation technique for statically constrained formulation was proposed. By introducing different
transformation responses in tension and compression, the model was then modified in such a way that tension–compression asymmetry was taken into account. The
obtained results were compared against experimental findings in uniaxial tension and
compression tests as well as four-point bending, and good agreements were observed.
Finally, considering plastic strains as the source of residual strains upon unloading,
M. R. Karamooz-Ravari et al.
Table 17.4 Utilized material parameters for simulation of cyclic loading in shape memory regime
(Karamooz-Ravari et al. 2018)
M f ( ◦ C) σ cr
s0 (MPa) C M0 (MPa/ ◦ C) C Af0 (MPa/ ◦ C) E M0 (MPa) σ M0
ye (MPa) ε ∗
0
26
100
NA
NA
80,000
600
0.0315
M s ( ◦ C) σ cr
se (MPa) C Me (MPa/ ◦ C) C Afe (MPa/ ◦ C) E Me (MPa) σ A0
ye (MPa) ε ∗
e
54.1
5
NA
NA
62,000
NA
0.015
A s ( ◦ C)
σ cr
f 0 (MPa)
C As0 (MPa/ ◦ C)
E A0 (MPa)
σ M0
y0 (MPa)
h(MPa)
μ
59
380
NA
NA
350
30,000
6.5
A f ( ◦ C)
σ cr
fe (MPa)
C Ase (MPa/ ◦ C)
E Ae (MPa)
σ A0
y0 (MPa)
ε
(max)
p
m
84.5
600
NA
NA
NA
0.054
2.8
Fig. 17.10 Comparison of the predicted cyclic stress–strain responses in shape memory regime
with experimental ones (Karamooz-Ravari et al. 2018)
17.3 Conclusions
This chapter was allotted to 3-D constitutive modeling of shape memory alloys using
microplane approach. The basic concepts of this method and its different formulations were presented. The potential directional bias in conventional microplane theory
was eliminated by developing modified formulations based on one shear component
on each plane. 1-D constitutive equations of SMAs were generalized for multiaxial
loadings using microplane theory, and an efficient numerical implementation technique for statically constrained formulation was proposed. By introducing different
transformation responses in tension and compression, the model was then modified in such a way that tension–compression asymmetry was taken into account. The
obtained results were compared against experimental findings in uniaxial tension and
compression tests as well as four-point bending, and good agreements were observed.
Finally, considering plastic strains as the source of residual strains upon unloading,