The Influence of Crystallographic Orientation and Grain Boundary on Nanoindentation Behavior of Inconel 718 Superalloy Based on Crystal Plasticity Theory

In this work, a crystal plasticity constitutive model is employed to represent the inelastic deformation of a polycrystalline material at the microscale. For the crystallographic model, the total deformation gradient can be multiplicatively decomposed into two parts based on large deformation theory [22], as follows [23]:where F_e denotes the elastic part caused by lattice stretching and rotation, and F_p denotes the plastic part representing the plastic deformation gradient of the crystal lattice along the slip direction.

For the plastic deformation of the αth slip system, the corresponding inelastic velocity gradient L_P depends on the plastic slip ratio \(\dot{{\gamma }^{{\alpha }}}\) [24], which can be defined as:where m^α and n^α are unit vectors used to represent the slip direction vector and the normal vector of the slip plane on the αth slip system, respectively. N delegates the gross amount of active slip systems, and the mathematical symbol ⊗ represents a dyadic product.

The crystallographic slip rate of the αth slip system \(\dot{{\gamma }^{\alpha }}\) is expressed as being correlated with the resolved shear stress τ^α, the back stress B^α, and the slip resistance S^α, following the activation flow criterion [25]:where \(\dot{{\gamma }_{0}}\), F₀, and τ₀ are components representing the reference strain rate, total free energy to overcome the lattice resistance, and lattice friction stress at 0 K, respectively; k donates the Boltzmann constant, T is the absolute temperature, and p and q are material parameters associated with the material properties. The mathematical symbol <·> is the MaCauley bracket, which represents the relationship that when x≥0 then < x > = x, and when x<0 then < x > = 0.

The resolved shear stress of the αth slip system, following the works of Li et al. [26, 27], can be written as:

The second Piola-Kirchoff stress, \(\varvec{T}^{*}\), is given by:where C is the fourth-order stiffness tensor, which is a function of three elastic stiffness constants for FCC crystals, and E^e is the elastic Green tensor [28].

The slip resistance S^α is described according to Ref. [29]:where S₀ is the initial slip resistance, S^β is the slip resistance on the βth slip system, S_sat is the saturated slip resistance, and h^αβ is the hardening matrix, given by:where h_s and δ_αβ are the hardening constant and Kronecker delta, respectively, and w refers to the latent hardening behavior of the material. The value of w in this work is assumed to be 1 [30].

The above-mentioned CPFEM model is written in a user-defined subroutine (UMAT) and integrated within the ABAQUS standard solver for finite element simulation.

In microscale simulation research, a representative volume element (RVE) model consisting of only a few grains to replace the mechanical behavior of large sized samples is typically used to greatly reduce the simulation time [31, 32]. In this work, a RVE model has been developed using the Voronoi tessellation (VT) method to calibrate the crystal plastic constitutive parameters of Inconel 718 superalloy. The RVE model is composed of 150 grains, which was proven to be enough to represent the mechanical behavior of large sized specimens [33]. All grains in the RVE model are randomly given different crystallographic orientations, as shown in Figure 3. The boundary conditions of the simulated model are as follows: the degrees of freedom in the x-direction of the left boundary are fully constrained but those of the y-direction are not constrained; the upper and top boundaries keep the same displacement in the y-direction by adopting multi-point constraints; and the target displacement load is applied on the right boundary.

The CPFEM results are compared with those of the macro-scale standard experiment [34], and the parameters of CPFEM model are calibrated using a trial-and-error technique. Figure 4 shows the comparison curves between the final stress-strain data obtained by CPFEM and the macro-scale standard experiment, indicating that after parameter calibration, the simulation results by CPFEM agree well with the experimental data. The CPFEM parameters of Inconel 718 are presented in Table 1. It should be noted that once the calibration of CPFEM parameters was successfully completed, the same set of CPFEM parameters was used in all subsequent nanoindentation simulations.

The nanoindentation model established in this study is illustrated in Figure 5. To simplify the finite element simulations, the indenter used in nanoindentation simulation can be assumed to be rigid because the indenter is usually made of hard materials such as diamond whose elastic modulus is much larger than that of the tested material. The diameter of the indenter’s spherical end is 1.5 μm, and the dimensions of the model are 10 μm × 10 μm × 5 μm (x × y × z). The finite element model is meshed with 3D hexahedric linear integration elements (C3D8). Because nanoindentation is a typical contact problem and the main focus area is underneath the indenter, for the purpose of improving the astringency and astringent rate of the simulation, the mesh in the contact area is refined while a lager grid structure is applied to the remaining area. The friction coefficients between the indenter and the tested sample of all simulations in this work are set to be zero, meaning that the contact conditions are assumed to be frictionless. The bottom surface is fixed in three normal directions and the four surrounding surfaces are constrained along the x or y direction for the entire numerical simulation process. The indenter can move freely only along the z direction, and the degrees of freedom in other directions are fully constrained. The displacement control method is adopted in this simulation, and the indenter moves along the z-axis at a fixed displacement rate. The thickness of the sample should be more than ten times the impressed depth to minimize the effect of external factors such as the specimen dimensions and boundaries on the simulation results. The maximum indentation depth of all the simulations performed in this work are set to be 500 nm, which meets the requirements mentioned above. Figure 6 shows the force-displacement curves under three different impressed depths (500 nm, 800 nm, and 1000 nm). It is apparent that the initial unloading stage under the three impressed depths have almost the same slope. Therefore, the plastic zone is reached at an indentation depth of 500 nm, which means the selected indentation depth of 500 nm meets the simulation requirements.

The finite element mesh control plays a crucial role in the simulation results. A mesh-sensitivity study of the nanoindentation model is necessary to avoid the mesh effect on nanoindentation simulation analysis and to determine the best mesh scheme based on simulation time and accuracy. Four different mesh sizes were selected for analysis, and the minimum mesh sizes of the model were set to be 0.1, 0.2, 0.3, and 0.4 μm, respectively. From the macroscopic simulation results, a relatively large gap between the indentation force-displacement curves obtained by the coarsest mesh and the finest mesh can be observed, as shown in Figure 7, while the force-displacement curves obtained with mesh sizes values of 0.3 and 0.4 μm coincide quite well. Considering that the simulation time with the finest mesh value of 0.4 μm is over 10 times that with a mesh value of 0.3 μm and that the convergence of the indentation contact analysis will deteriorate when the mesh refinement is excessive, a mesh size value of 0.3 μm can be considered as the best meshing scheme for this simulation work.

To investigate the influence of crystallographic orientation on the nanoindentation mechanical behavior of Inconel 718, six sets of different crystallographic orientations (Euler angles) are selected and assigned to the same grain in the model (grain 1 in Figure 8(a)), while the crystallographic orientations of the remaining grains remain unchanged, as shown in Table 2. The indentation positions are selected at the center of the indented grain (grain 1) to reduce the impact of grain boundaries on the simulation results.

Figure 9 shows the force-displacement curves under different crystallographic orientation and demonstrates that the maximum variation of the peak load when the indentation depth reaches 500 nm is 3.55%. All six simulations with different crystallographic orientations have similar slopes during the unloading period. This indicates that the influence of crystallographic orientation on the force-displacement curve of nanoindentation is limited. Local deformation occurs when the indenter is applied to the grain, and this deformation is constrained by the indentation depth and surrounding materials. The indentation area shows a complex multi-directional stress state. The loading-unloading process of the indentation reflects the comprehensive mechanical response of the material in different directions. Although different crystallographic mechanical properties are set for each grain in the finite element simulation, the final reflection includes comprehensive material properties in all directions due to the action characteristics of the indentation test. Thus, there are no obvious differences in the force-displacement curves. However, the change of crystallographic orientation has a great influence on the stress distribution and pile-up patterns around the indentation. The stress distributions of different crystallographic orientations at the maximum indentation depth are shown in Figure 10. This demonstrates that the state of stress distribution depends on the crystallographic orientation to a certain degree. The stress is symmetrically distributed, and the axis of symmetry changes as the crystallographic orientation changes.

The evolution of the pile-up patterns in different crystallographic orientations is illustrated in Figure 11. The pile-up patterns are symmetrically distributed in different crystallographic orientations, which is consistent with the results of monocrystalline copper under different crystallographic orientations observed by Wang et al. [35]. Besides, the differences in indentation pile-up patterns are connected to the crystallographic misorientation. The cases OR1 and OR2 have the smallest misorientation, and the pile-up patterns of the two are the most similar, with four protrusions. The cases OR1 and OR4 have the largest misorientation, which leads to the obvious differences between their pile-up patterns. In other words, differences in the pile-up patterns increase as misorientation between the crystal grains increases. The crystallographic orientation dependence of the pile-up patterns is mainly caused by anisotropic crystallographic plastic deformation [17].

The comparison of the maximum Schmid factor of the grain (grain 1 in Figure 8) and the maximum plastic strain along different crystallographic orientations is shown in Figure 12. The Schmid factor is a typical parameter that can be used to describe the influence of crystallographic orientation [36]. A larger Schmid factor results in a greater degree of plastic slip [37]. It can be seen in Figure 12 that the variation tendency of the maximum local plastic strain under the indenter along different crystallographic orientations is consistent with the change of Schmid factor of the indented grain. The grain (grain 1) in case OR4 has the smallest Schmid factor, and its corresponding maximum plastic strain value is also the smallest.

The effect of grain boundary on the indentation behavior at the microscale is studied by changing the indentation position. As shown in Figure 8(b), it is indented at the grain center (center), boundary of two grains (GB2), and trigeminal grain boundary (GB3). Figure 13 shows the force-displacement curves obtained by nanoindentation simulation. It can be clearly seen that the maximum variation of the indentation peak load is only 2.94%, indicating that the existence of the grain boundary has little effect on the force-displacement curve of indentation. However, the grain boundary plays a critical part in changing the pile-up patterns and stress distributions. When the indentation position changes, the stress distribution and pile-up patterns exhibit significant differences, as illustrated in Figure 14.

For the sake of further revealing how the grain boundary affects the indentation behavior, the grains (grain 1 and grain 2) in Figure 8(a) on both sides of the grain boundary (GB2) are given different crystallographic orientations, as indicated in Table 3. Figure 15 shows the indentation force-displacement curves for different grain boundary angles. As shown in the figure, the influence on indentation force-displacement curves is still extremely limited, and the maximum peak load error is only 2.78%. Figure 16 shows the stress distribution at different grain boundary angles. This means that the resistance effect of grain boundaries on the continuity of the stress distribution becomes greater as the grain boundary angle increases (from case OM4 to case OM1).

With respect to the pile-up patterns, as shown in Figure 17, changes in the shape of the pile-up patterns are more obvious with an upward trend in grain boundary angle. The grain boundary angles of case OM3 and OM4 are low angle (13.2° and 2.2°, respectively), and the stress distribution and pile-up patterns of the two are almost the same as those of case OR1. On the contrary, the grain boundary angles of case OM1 and OM2 are large angle (53.6° and 38.2°, respectively), and the stress distribution at the grain boundary is obviously discontinuous with a significant change in the shape of the pile-up patterns.