Computational Micromechanics Modeling of Polycrystalline Superalloys. . .
139
Three key ingredients are the basis of a successful simulation of a polycrystal
using computational homogenization: (1) the numerical approach to solve the
boundary value problem, (2) the representation of the microstructure, and (3) the
constitutive description of the single crystal behavior. The particular homogenization framework used for the simulation of the mechanical behavior of the alloy
Inconel 718 will be reviewed below, attending to these three aspects.
4.1 Boundary Value Problem and Boundary Conditions
Two approaches are normally used to solve the boundary problem in a homogenization problem, the finite element (FE) method or methods based on the fast
Fourier transform (FFT). FE is still the most common approach for polycrystalline
homogenization and has been profusely used to simulate the behavior of any type
of metals [57, 58]. FFT-based homogenization, proposed in the 1990s by [43],
soon attracted the attention of the homogenization community for its computational
efficiency and because meshing of the RVE is avoided. The use of FFT methods
for polycrystalline microstructures was proposed in [33], and the method has been
used since then mainly to simulate the response of polycrystalline alloys under
monotonic loading [19, 35]. Only very recently these methods have been applied
to obtain the cyclic and fatigue response of superalloys like Inconel100 [59] or
Inconel 718 [37].
In this work, the micromechanical simulations of the behavior of the alloy
Inconel 718 [14–17] have been performed using the FE method. The FE framework
for polycrystalline simulations shares many aspects with standard macroscopic
plasticity simulations, as the use of the same type of elements, linearization
schemes, or linear solvers, and relies very often in the use of commercial FE
packages as ABAQUS or MARC. The two main differences with standard plasticity
simulations are the constitutive equation used and the boundary conditions. Under
the framework of computational homogenization of polycrystals, each material
point is modeled using crystal plasticity, and the load history corresponds to the
macroscopic stress or strain that is introduced using special boundary conditions as
uniform displacement, uniform stress, or periodic boundary conditions.
Periodic boundary conditions are used for the micromechanical simulation
of Inconel 718. These boundary conditions fulfill the Hill-Mandel principle of
macro-homogeneity [30], and, among the boundary conditions compatible with this
principle, the results obtained under these conditions (even in the case of nonperiodic microstructures) show the fastest convergence toward the actual effective
response with increasing the RVE size [25]. Periodic boundary conditions assume
that the RVE deforms as a jigsaw puzzle. A cubic periodicity of the RVE is assumed
(Fig. 8a). Let l i = l i e i be the three orthogonal vectors defining the cubic periodicity,
and let e i be the corresponding unit vectors defining the basis. If x 1 , x 2 , x 3 are the
coordinates of a point in the RVE in the system defined by e i , the periodic boundary
139
Three key ingredients are the basis of a successful simulation of a polycrystal
using computational homogenization: (1) the numerical approach to solve the
boundary value problem, (2) the representation of the microstructure, and (3) the
constitutive description of the single crystal behavior. The particular homogenization framework used for the simulation of the mechanical behavior of the alloy
Inconel 718 will be reviewed below, attending to these three aspects.
4.1 Boundary Value Problem and Boundary Conditions
Two approaches are normally used to solve the boundary problem in a homogenization problem, the finite element (FE) method or methods based on the fast
Fourier transform (FFT). FE is still the most common approach for polycrystalline
homogenization and has been profusely used to simulate the behavior of any type
of metals [57, 58]. FFT-based homogenization, proposed in the 1990s by [43],
soon attracted the attention of the homogenization community for its computational
efficiency and because meshing of the RVE is avoided. The use of FFT methods
for polycrystalline microstructures was proposed in [33], and the method has been
used since then mainly to simulate the response of polycrystalline alloys under
monotonic loading [19, 35]. Only very recently these methods have been applied
to obtain the cyclic and fatigue response of superalloys like Inconel100 [59] or
Inconel 718 [37].
In this work, the micromechanical simulations of the behavior of the alloy
Inconel 718 [14–17] have been performed using the FE method. The FE framework
for polycrystalline simulations shares many aspects with standard macroscopic
plasticity simulations, as the use of the same type of elements, linearization
schemes, or linear solvers, and relies very often in the use of commercial FE
packages as ABAQUS or MARC. The two main differences with standard plasticity
simulations are the constitutive equation used and the boundary conditions. Under
the framework of computational homogenization of polycrystals, each material
point is modeled using crystal plasticity, and the load history corresponds to the
macroscopic stress or strain that is introduced using special boundary conditions as
uniform displacement, uniform stress, or periodic boundary conditions.
Periodic boundary conditions are used for the micromechanical simulation
of Inconel 718. These boundary conditions fulfill the Hill-Mandel principle of
macro-homogeneity [30], and, among the boundary conditions compatible with this
principle, the results obtained under these conditions (even in the case of nonperiodic microstructures) show the fastest convergence toward the actual effective
response with increasing the RVE size [25]. Periodic boundary conditions assume
that the RVE deforms as a jigsaw puzzle. A cubic periodicity of the RVE is assumed
(Fig. 8a). Let l i = l i e i be the three orthogonal vectors defining the cubic periodicity,
and let e i be the corresponding unit vectors defining the basis. If x 1 , x 2 , x 3 are the
coordinates of a point in the RVE in the system defined by e i , the periodic boundary