144
A. Cruzado et al.
where s α and m α stand for the unit vectors in the slip direction and the normal to
the slip plane, respectively, in the reference configuration.
The Green-Lagrange measure of deformation is used to characterize the elastic
deformation, E e ,
E
e
=
1
2
F
eT F
e
− I
(9)
being I the second order identity tensor. The symmetric second Piola-Kirchhoff
stress tensor in the intermediate configuration, S, is related with the Green-Lagrange
strain tensor according to
S = C : E
e
(10)
where C is for the fourth-order elastic stiffness tensor of the single crystal. The
driving force for the plastic slip is the resolved shear stress τ α on the slip plane α,
and it is obtained as the projection of the second Piola-Kirchhoff stress on the slip
system according to
τ
α
= S : (s
α
⊗ m
α )
(11)
Finally, the Cauchy stress is obtained as
σ =
1
J e F
e SF
eT
(12)
where J e is the determinant of F e .
The laws defining the slip rate as function of the resolved shear stress and the
internal variables depend on the particular regime considered, monotonic or cyclic,
and are included in its respective sections of this chapter.
5 Monotonic Behavior
The mechanical response of Inconel 718 under uniaxial monotonic loading is simulated using the computational homogenization framework presented in the previous
section. The grain size distributions extracted from experimental microstructures are
used as input for the RVEs. The strategy followed to model the crystal behavior of
a coarse-grained Inconel 718 (ASTM 3) was using a simple crystal plasticity model
in which the crystal parameters were experimentally measured from micromechanical tests. To extend the model to other microstructures, the crystal plasticity
model was modified to include the effect of grain size in the crystal strength
combining in this case microscopic data with macroscopic results for different
microstructures.
A. Cruzado et al.
where s α and m α stand for the unit vectors in the slip direction and the normal to
the slip plane, respectively, in the reference configuration.
The Green-Lagrange measure of deformation is used to characterize the elastic
deformation, E e ,
E
e
=
1
2
F
eT F
e
− I
(9)
being I the second order identity tensor. The symmetric second Piola-Kirchhoff
stress tensor in the intermediate configuration, S, is related with the Green-Lagrange
strain tensor according to
S = C : E
e
(10)
where C is for the fourth-order elastic stiffness tensor of the single crystal. The
driving force for the plastic slip is the resolved shear stress τ α on the slip plane α,
and it is obtained as the projection of the second Piola-Kirchhoff stress on the slip
system according to
τ
α
= S : (s
α
⊗ m
α )
(11)
Finally, the Cauchy stress is obtained as
σ =
1
J e F
e SF
eT
(12)
where J e is the determinant of F e .
The laws defining the slip rate as function of the resolved shear stress and the
internal variables depend on the particular regime considered, monotonic or cyclic,
and are included in its respective sections of this chapter.
5 Monotonic Behavior
The mechanical response of Inconel 718 under uniaxial monotonic loading is simulated using the computational homogenization framework presented in the previous
section. The grain size distributions extracted from experimental microstructures are
used as input for the RVEs. The strategy followed to model the crystal behavior of
a coarse-grained Inconel 718 (ASTM 3) was using a simple crystal plasticity model
in which the crystal parameters were experimentally measured from micromechanical tests. To extend the model to other microstructures, the crystal plasticity
model was modified to include the effect of grain size in the crystal strength
combining in this case microscopic data with macroscopic results for different
microstructures.