156
A. Cruzado et al.
volume elements [13]) depending on the author. Therefore, the phenomenological
expression for life predictions will be applied to all the RVE ensemble, resulting in
a statistical distribution of fatigue lives.
The simulation of the cyclic response using computational micromechanics
was covered in the previous section, and the microscopic fields resulting of these
simulations are used here to obtain the FIP distributions. Among the different FIPs
proposed in the literature for the study of the fatigue life [7, 8, 39, 44, 53, 64, 69–
71, 74, 75], the local crystallographic strain energy dissipated per cycle W b
cyc was
used in this study to obtain the fatigue life on Inconel 718 alloy. This FIP is
expressed as
W
α
cyc (x) =
cyc
τ α (x) ˙
γ α (x)dt
(27)
where τ α and ˙
γ α are the resolved shear stress and the shear strain rate on the slip
system α, respectively, for a given location x in the polycrystal. The local value of
the FIP, as described in Eq. 27, is calculated in the centroid of each element in the
RVE. However, this local value is not representative of a fatigue damage region,
as the formation of persistent slip bands which leads into crack incubation affects
a particular finite volume of the material. To overcome this issue, the local FIPs
are averaged along narrow bands parallel to the slip planes, similar to the approach
proposed by Castelluccio et al. [9]. Under these assumptions, the FIP representative
of an RVE subjected to cyclic deformation, W b
cyc , is obtained as the maximum of
the band-averaged local FIP throughout the RVE, according to
W
b
cyc = max
i=1,nb
max
β i
1
V i
V i
W
β i
cyc (x)dV i
(28)
where β i (= 1, 2, 3) corresponds to the three different slips systems contained in
the slip plane parallel to the band i, V i is the volume of that band, and nb is the
total number of bands considered, which is four times the number of elements in the
RVE. This volume-averaging approach also presents the advantage of mitigating
spurious stress concentrations and mesh size effects.
The fatigue crack initiation is then related to the stabilized value of the cyclic FIP
W b
cyc . Cruzado et al. [17] showed that a linear relation between the number of cycles
for crack nucleation and the FIP value – approach followed in many microstructurebased models [39, 71, 74] – did not work for Inconel 718. The deficient predicting
capacity of a linear law in Inconel 718 was attributed to the dual slope CoffinManson behavior, described in Sect. 3.2.2, which is a consequence of a change in the
deformation mechanism that controls the nucleation of fatigue cracks from localized
deformation in a few grain at small cyclic strain amplitudes to homogeneous plastic
deformation at large cyclic strain amplitudes. Under these conditions, a power law
relation of the fatigue life N i with W b
cyc was proposed
A. Cruzado et al.
volume elements [13]) depending on the author. Therefore, the phenomenological
expression for life predictions will be applied to all the RVE ensemble, resulting in
a statistical distribution of fatigue lives.
The simulation of the cyclic response using computational micromechanics
was covered in the previous section, and the microscopic fields resulting of these
simulations are used here to obtain the FIP distributions. Among the different FIPs
proposed in the literature for the study of the fatigue life [7, 8, 39, 44, 53, 64, 69–
71, 74, 75], the local crystallographic strain energy dissipated per cycle W b
cyc was
used in this study to obtain the fatigue life on Inconel 718 alloy. This FIP is
expressed as
W
α
cyc (x) =
cyc
τ α (x) ˙
γ α (x)dt
(27)
where τ α and ˙
γ α are the resolved shear stress and the shear strain rate on the slip
system α, respectively, for a given location x in the polycrystal. The local value of
the FIP, as described in Eq. 27, is calculated in the centroid of each element in the
RVE. However, this local value is not representative of a fatigue damage region,
as the formation of persistent slip bands which leads into crack incubation affects
a particular finite volume of the material. To overcome this issue, the local FIPs
are averaged along narrow bands parallel to the slip planes, similar to the approach
proposed by Castelluccio et al. [9]. Under these assumptions, the FIP representative
of an RVE subjected to cyclic deformation, W b
cyc , is obtained as the maximum of
the band-averaged local FIP throughout the RVE, according to
W
b
cyc = max
i=1,nb
max
β i
1
V i
V i
W
β i
cyc (x)dV i
(28)
where β i (= 1, 2, 3) corresponds to the three different slips systems contained in
the slip plane parallel to the band i, V i is the volume of that band, and nb is the
total number of bands considered, which is four times the number of elements in the
RVE. This volume-averaging approach also presents the advantage of mitigating
spurious stress concentrations and mesh size effects.
The fatigue crack initiation is then related to the stabilized value of the cyclic FIP
W b
cyc . Cruzado et al. [17] showed that a linear relation between the number of cycles
for crack nucleation and the FIP value – approach followed in many microstructurebased models [39, 71, 74] – did not work for Inconel 718. The deficient predicting
capacity of a linear law in Inconel 718 was attributed to the dual slope CoffinManson behavior, described in Sect. 3.2.2, which is a consequence of a change in the
deformation mechanism that controls the nucleation of fatigue cracks from localized
deformation in a few grain at small cyclic strain amplitudes to homogeneous plastic
deformation at large cyclic strain amplitudes. Under these conditions, a power law
relation of the fatigue life N i with W b
cyc was proposed