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Multi-scale simulation of nanoindentation on cast Inconel 718 and NbC precipitate for mechanical properties prediction
Materials Science & Engineering A 662 (2016) 385–394
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Multi-scale simulation of nanoindentation on cast Inconel 718 and NbC precipitate for mechanical properties prediction Chenghui Ye a,1, Jieshi Chen a,c,1, Mengjia Xu a, Xiao Wei a, Hao Lu a,b,n a
School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China Key Lab of Shanghai Laser Manufacturing and Materials Modification, Shanghai Jiao Tong University, Shanghai 200240, PR China c School of Materials Engineering, Shanghai University of Engineering Science, Shanghai 201620, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 13 October 2015 Received in revised form 16 February 2016 Accepted 17 March 2016 Available online 18 March 2016
Multi-scale simulation of nanoindentation on cast Inconel 718 superalloy and its NbC precipitate were performed with combined first principle study and crystal plasticity finite element method (CPFEM). The concerned parameters were calibrated through a representative volume element (RVE) model compared with the stress–strain curves obtained from tensile tests. Nanoindentation was carried out on the matrix. First principle calculations were applied to estimate the mechanical properties of precipitate NbC, including elastic modulus and hardness. The simulated force–displacement curves match well with the experimental results. The simulated results indicate that the local pile-up pattern in the indentation zone depends significantly on the crystallographic orientations. In addition, large precipitate NbC inserted in the matrix was also indented and simulated. The elastic modulus calculated by first principle is quite accurate while the yield stress is determined using inversion calculations. It appears that the proposed CPFE analysis approach combined with first principle calculation do help estimate the mechanical behavior of large precipitates on the Ni-based superalloy. & 2016 Elsevier B.V. All rights reserved.
Keywords: Finite element method Nanoindentation Nickel based superalloys Plasticity
1. Introduction Inconel 718 is a nickel based superalloy with superior mechanical properties and structural stability at elevated temperature resulting from precipitation hardening. The matrix is γ phase (FCC). The most common precipitates in cast Inconel 718 are MC (FCC), laves phases (HCP) and δ-Ni3Nb (D0a, orthorhombic), apart from γ′ (L12, cubic) and γ″(D022, tetragonal) strengthening phase in the matrix.[1–3]. To investigate local properties of Inconel 718, nanoindentation is a good choice that can measure mechanical properties such as hardness and elastic modulus even at micro-nano scales. The indentation depth ranges from tens of nanometers to several microns. This testing method was first introduced in mid-1970s [4–7] and it is now widely used to characterize micro-mechanical behavior. However, it is difficult to interpret the experimental data as it involves a very complex deformation field. With the help of finite element method (FEM), more information can be obtained. In addition, traditional plastic theory cannot illustrate the anisotropy and the evolution of orientation during the deformation. n Corresponding author at: School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China. E-mail address: [email protected] (H. Lu). 1 Co-first Authors.
http://dx.doi.org/10.1016/j.msea.2016.03.081 0921-5093/& 2016 Elsevier B.V. All rights reserved.
Therefore, the crystal plasticity (CP) constitutive models based on dislocation evolution in crystal have been developed and applied on nanoindentation simulation. The CP theory was first introduced by Taylor [8,9], afterwards, it was incorporated with FEM. CPFEM adapts well with complex boundaries and enables various of plastic flow and hardening laws. Liu et al. [10] carried out 3D CPFE simulations of nanoindentation on copper single crystals, combined with experimental force–displacement curves to predict the stress–strain curve at the mesoscale. Li et al. [11] simulated nanoindentation on aluminum alloy polycrystals to study the effect of orientation, grain tessellation and grain boundaries. Li et al. [12] presented a numerical simulation of variable amplitude loading effects on cyclic plasticity and microcrack initiation in 304 L steel. Nie et al. analyzed the size effect of precipitates on the creep properties of nickel based superalloys using creep based CPFE model [13]. In spite of these recent studies, the application of CPFEM in nanoindentation remains to be improved. There is seldom CPFEM simulation of nanoindentation on nickel based superalloy, mainly due to the fact that nickel based superalloys are filled with various precipitates. In addition, as far as the author's knowledge, there was no study on CPFE simulation of nanoindentation on materials with precipitates. How to estimate mechanical properties of precipitates in nickel based superalloys remains an issue. As a solution to this issue, first principle study provides a
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Table 1 Composition of Inconel 718 high temperature alloy (wt%). C
Cr
Ni
Mo
Al
Ti
(Nb þTa)
Fe
0.02–0.08
17.00–21.00
50.00–55.00
2.80–3.30
0.40–0.60
0.90–1.15
5.00–5.50
Rest
means of calculating mechanical behaviors of crystals based on stress-strain approach [14]. Therefore, multi-scale simulations provide a solution to simulate nanoindentation on materials with precipitates. In this study, CPFE simulation of nanoindentation combined with first principle study is applied to estimate the mechanical properties of NbC precipitate in Inconel 718. To ensure accurate simulation results, CP parameters are calibrated with representative volume element (RVE) model, and simulation results of force-displacement curve are compared with experimental data. The sections of the paper are arranged as follows: The theory of CP and the calibration of the CPRVE model are presented in Section 2. The calibration of Berkovich indenter and nanoindentation experimental results are reported in Section 3. Multi-scale simulation of Inconel 718 and NbC precipitate under nanoindentation compared with the experimental data is discussed in Section 4. Finally, conclusions are given in the last section.
effect of the mobile dislocation density. The critical shear stress τcα is thus expressed as a function of the forest dislocation ρF, such that:
τcα = Gb
12
∑β = 1 Aαβ ρFβ
=
∑β hαβ γ β
(5)
where A is the interaction matrix that describes the hindering behavior between different slip systems. G is the shear modulus, b is the burger's vector, h is the hardening strength and γ β stands for the shear strain. The interaction matrix is composed of four dislocation interaction types from a0 to a3. The evolution of dislocation density can be decomposed into a term of dislocation multiplication and a term of annihilation for the slip system α, as follows:
ρḞ α =
⎛ ⎞ β 1 ⎜ ∑α ≠ β ρF α⎟ α − ρ 2 y ⎟⎟ γ ̇ c b ⎜⎜ K ⎝ ⎠
(6)
where yc represents the critical annihilation length, which is related to the dynamic recovery. K is a constant.
2. Crystal plasticity theory and calibration 2.1. Crystal plasticity formulations
2.2. Calibration with RVE model The plastic deformation of a single crystal is regarded as the cause of crystalline slip. The deformation gradient F is the product of a lattice deformation gradient F* and a plastic deformation gradient Fp, assuming that the material flows depends on dislocation motion, and then the combination of elastic deformation and rigid body rotation [15].
F = F *⋅Fp
(1)
The velocity gradient L can be defined as:
L = FḞ −1 = F *̇ F * − 1 + F *Fṗ F p−1F * − 1 = L* + F *L pF * − 1
(2)
The plastic velocity tensor Lp can be expressed by the sum of the shearing rate γ α̇ for all 12 slip systems of a face centered cubic (FCC) crystal structure (α = 1, 2, ⋯ , 12). The slip system α is speα α cified by the slip direction m and slip plane normal n in the global coordinate as the following equation: 12
Lp =
∑ γ α̇ mα ⊗ nα
(3)
α=1 α
The shearing rate γ ̇ for each slip system can be approximated by a power law:
⎛ τ α ⎞n γ α̇ = γ0⎜ α ⎟ si gn(τ α ) ⎝ τc ⎠
(4) α
where the rate exponent n represents the strain rate sensitivity. τ is the critical resolved shear stress for the slip system α, which stands for the projection of the nominal stress onto the slip plane, and it is the driving force of the plastic deformation. The strength τcα for the slip system α is the resistance of the plastic deformation, namely the stress needed for attaining reference velocity in the slip system α. Based on the hardening law [16], it is assumed that the dislocation cutting force is the major obstacle in plastic deformation. As a result, the plastic shear rate is related to the mean
To use the CP formulations listed above, the calibration of parameters is necessary. The composition of cast Inconel 718 used in this study is listed in Table 1. As the sample has undergone high temperature for a long time, the grains have grown coarser and the mechanical properties are not as good as those in standards. The samples are carefully polished and electrolytic etched with 10% chromic acid solution as electrolyte under a voltage of 45 V for 10 s for SEM observation. Also, EDS analysis was conducted to identify the precipitates. As is shown in Fig.1, significant dendritic segregation is observed in this alloy. Precipitates are strongly segregated and form dendrites. When the segregation spot is zoomed to 20,000 , both large precipitates of several microns and tiny needle-like precipitates of hundreds of nanometers are observed. According to Wen Sun [17], large precipitates in cast Inconel 718 alloys are mainly MC and tiny precipitates are Laves phase or δ phase. Cast Inconel 718 alloy is composed of polycrystals with a mean grain size around 300 mm. It is time-consuming to simulate all the grains in the tensile sample. CPRVE model is thus implemented in micro-scale simulations to effectively represent the mechanical behavior of the macro-scale specimen with the minimum number of grains. [18,19] The CP parameters were calibrated using standard tension tests with a CPRVE model. The CPRVE model used in this study is 4.4 4.4 1.1 mm3, containing 1000 equiaxed grains with different random orientations as shown in Fig. 2. Meanwhile, tensile tests of Inconel 718 bar samples are conducted on Zwick/ Roell Z100 with a constant strain rate of 1 mm/min. The tensile bar samples are 6 mm in diameter and 40 mm in gage length. The comparison between tensile test and CPRVE models is shown in Fig. 3. After times of parameter testing, good agreements are found with the CP parameters listed in Table 2. With these parameters, the mechanical properties of Inconel 718 superalloy can be numerically simulated with CPFEM method.
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Fig.1. SEM of cast Inconel 718 at (a) 500 (b) 2000 (c) 20,000 .
Table 2 Paremeters used in CPFE model. C11 (MPa)
C12 (MPa)
C44 (MPa)
τ0 (MPa)
γ0̇ (s 1)
n
a0
a1
203,030
149,870
134,980
32
0.001
2.703
0.045
0.625
a2
a3
ρ0 (m 2)
b (nm)
yc (nm)
K
0.255
1.43
64
0.137
0.122
10
12
3. Nanoindentation test
Fig.2. RVE model of Inconel 718 polycrystals tensile test.
Before nanoindentation, the sample of Inconel 718 was mechanically polished and then vibrant polished for 2 hours. It was not etched so that the structure of precipitates on the matrix is totally kept. Nanoindentation test of Inconel 718 was conducted on Hysitron TI 950 TriboIndenter which combines nanoindentation with in-situ scanning probe microscopy (SPM). The indenter was a Berkovich indenter made of diamond with an apex angle of 142.35° and a radius of 150 nm. To have a better comparison with the simulation results, displacement control method was adopted so that the indenter stopped and unloaded once the displacement had reached the designated indentation depth. Loading is a complex process which involves plastic deformation (material nonlinearity) and an increasing contact area (geometric nonlinearity), while unloading is purely elastic. To ensure accurate measurement, the shape of the indenter was also calibrated. According to Oliver and Pharr [20], a practical indenter is not as perfect as an ideal one, and its shape can be fitted with the following equation:
Ac = 24.56hc2 + C1hc1 + C2hc1/2 + C3hc1/4 + ⋯ + C8hc1/128
Fig.3. The comparison of Stress-Strain curve between tensile test and RVE model.
(7)
where Ac is the contact area, hc is the contact depth, C1 to C8 are constants, the first term 24.56hc2 represents the shape of an ideal Berkovich indenter and the rests are terms to modify its shape due to bluntness and defects. The actual shape of the Berkovich indenter was fitted as shown in Fig. 4. In order to clarify the hardening effect of NbC precipitate on Nibased superalloy, nanoindentation test is carried out on both matrix and NbC precipitate. It is worth noting that, in this paper, the matrix stands for Inconel 718 superalloy without visible precipitates, which has taken into account some tiny precipitates (distributed underneath the surface in the segregation spot). Since these tiny precipitates have a uniform distribution in the segregation spot [17,21], the matrix and tiny precipitates can be treated as one. As is shown in back-scattered Scanning Electron Microscopy (SEM) Fig. 5, four nanoindentation tests were conducted on
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Fig.4. (a) Sketch of a real Berkovich indenter, α ¼62.35° (b) The calibration relation between contact area and contact depth of the Berkovich indenter.
the matrix. The indentation was controlled by indentation depth, so that the indenter started unloading at an indentation depth of 250 nm. As for NbC precipitate, due to its extremely high hardness, the indentation depth was set to 200 nm, where the maximum load of indenter had been reached at 12833 μN. As to indent on NbC precipitate, we do not want to damage the matrix of Ni, as a result, only slight etching was allowed during vibrant polishing. Thus only precipitates at the surface of the sample can be observed. Also, limited by the resolution of SPM, only large precipitates at the surface of the sample can be identified, which are typically carbides such as NbC. Then a nanoindentation on NbC precipitate was also conducted as shown in Fig. 6a. The indentation was totally on the precipitate so that we can assume this structure as a NbC layer inserted to the surface of the matrix. As we can see in Fig. 6b, the Energy Dispersive Spectrometer (EDS) analysis of the composition of this precipitate proves that the precipitate is mainly NbC, since it is rich in Nb and C.
4. Multi-scale simulation Multi-scale simulation is carried out including first principle study and CPFEM simulation of nanoindentation. At first, first principle study was conducted to predict the mechanical properties of NbC precipitate, and these properties would be used in CPFEM simulation. Then CPFEM models were built to simulate the process of nanoindentation of the matrix and NbC precipitate. 4.1. First principle study on mechanical properties of NbC To estimate the mechanical properties of precipitate NbC, first principle study was conducted. All calculations presented in this work were performed using the CASTEP plane-wave code [22] based on density-functional theory (DFT), and the exchange-correlation energy was calculated using the Perdew-Burke-Ernzerhof (PBE) general gradient approximation (GGA) [23]. Brillouin-zone integrations were performed using Monkhorst and Pack k-point
Fig.5. (a) SEM image of 4 nanoindentations on the matrix (b) SPM image of the same nanoindentations.
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Fig.6. (a) SEM image of nanoindentation on NbC precipitate. (b) EDS spectrum of the precipitate. Table 3 Structural properties obtained by GGA calculations, in comparison with other experimental works. Lattice parameters a (Å), b (Å), c (Å); Cell volume V (Å3); Information in square brackets shows the final setting values for k-point meshes and their corresponding k-point numbers. Phase Structure Space group NbC
a
Cub.
Method
Lattice parameters (Å)
GGA[6 6 6] a ¼ b¼ c ¼4.478 Exp. by others a ¼ b¼ c ¼4.468a
FM-3M
89.76 89.19a
569.2 (557.3 )
NbC a
C12 a
C44 a
172.8 (162.4 )
148.5 ( 146.5a)
meshes [24]. The final settings for these calculations are shown in Table 3. Besides, the elastic constants were calculated by the stress-strain approach [25]. The convergence tolerance were selected as follows: minimum energy less than 5.0 10–6 eV/atom, maximum force less than 0.01 eV Å 1, maximum stress less than 0.02 GPa, and maximum displacement less than 5 10–4 Å. The calculated elastic constants are listed in Table 4. Usually, on the basis of the elastic constants, the bulk modulus (B) and shear modulus (G) are determined by the Voigt-Reuss-Hill (VRH) averaging scheme [27]. The Voigt bounds on the effective bulk and shear modulus of cubic polycrystals are
GVoigt = (c11 − c12 + 3c44 )/5
GV R
G
+
BV BR
−6
(13)
U
A is identically set to zero for locally isotropy single crystals, and the departure of AU from zero indicates the extent of elastic anisotropy. The calculated value of AU is collected in Table 5, which indicates that NbC has slight elastic anisotropy. A three-dimensional (3D) curved surface, representing the dependence of elastic properties on crystallographic directions, can indicate the elastic anisotropy of crystal structure as shown in Fig. 7. The directional dependence of Young's modulus (E) for cubic crystals can be defined as [29]:
Cubic system:
Ref [14].
BVoigt = (c11 + 2c12)/3
(12)
The elastic anisotropy of the crystals can also be measured by the universal anisotropy index (AU) [28]:
AU = 5
Table 4 Elastic constants (GPa) of NbC. C11
9BG 3B − 2G ,ν= 3B + G 2(3B + G)
V(Å3)
Ref [26].
Phase
E=
1/E = S11 − 2(S11 − S12 − S44/2)(l12l22 + l22l32 + l32l12)
where Sij are the elastic compliance constants, namely the reverse of elastic constants, and l1, l2 are the directional cosines to the X, Y axes, respectively. According to Gao's theory [30], the theoretical hardness of crystal can be expressed as follow:
Hv = A 0 Pvb−5/3
(15)
where A0 is a proportional coefficient, set to 740, P is the Mulliken overlap population, and Vb is the bond volume. The following equation is used to calculate Vb: [31]
Vb = (d u)3/∑ [(d v )3Nbv ]
(16)
v
(8) (9)
(14)
Calculated results are listed in Table 6. These results are close to those calculated by YangZhen Liu. As empirical equation suggests that the hardness is about 3 times of yield stress, the yield stress
While for the Reuss bounds, the equations are
BReuss = (c11 + 2c12)/3 G Reuss =
5(c11 − c12)c44 4c44 + 3(c11 − c12)
(10)
(11)
Then the Young's modulus (E) and the averaged Poisson's ratio (ν) can be obtained:
Table 5 Calculated elastic properties of NbC. Bulk modulus B (GPa); Shear modulus G (GPa); Young's modulus E (GPa); Poisson's ratio ν; Universal anisotropy index (AU). Phase
Bhill
Ghill
ν
E
B/G
AU
NbC
304.9
166.7
0.27
423.0 (483.9a)
1.82
0.084
a
Ref [14].
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4.2.2. Nanoindentation on NbC precipitate inserted in the matrix For that the NbC precipitate is large and that the indentation was entirely on the precipitate (see Fig. 6a), it is assumed as an NbC layer on the top of the matrix as shown in Fig. 8c. Here the thickness of NbC layer is large enough that the deformation is concentrated in this region. This model with an NbC layer has the same mesh and boundary conditions with the previous one. However, differences lie in material properties. Here elements in green represent for the matrix, whose properties are given by CP laws. As NbC precipitate is extremely hard and quite brittle, here we only consider an ideal EP property without hardening. Due to the fact that NbC has slight anisotropy and that orientation has little effect on force-displacement curve, simple EP properties is enough to estimate mechanical properties of NbC. The elements in red represent for NbC whose mechanical properties have been calculated in the Section 4.1 with first principle. 4.3. Simulation results and discussion
Fig.7. The directional dependence of Young's Modulus (E) for NbC. Table 6 Calculated bond parameters Vickers hardness for NbC. The units for hardness are in GPa. Phase
bond
d (Å)
Nbv
P
vb (Å3)
Hv (Gpa)
NbC
Nb-C
2.248
12
0.75
7.48
19.38 (19.5a)
a
Ref [14]
used in the simulation is estimated around 6.46 GPa. And these properties can be used as reference for mechanical properties for NbC in FE simulations. 4.2. CPFE simulation procedure 4.2.1. Nanoindentation on the matrix The boundary effect has great influence on the simulation result as nanoindentation only induces local deformation underneath the indenter. To find a balance between boundary effect and computational efficiency, a core CP model is combined with an elasto-plastic (EP) model. As is shown in Fig. 8, the dimension of the model is 20 20 10 mm3, and it consists of 125800 hexahedral elements (C3D8R). The elements are finer in the middle and range from 100 nm to 760 nm. The Berkovich indenter is discrete rigid and consist of 53778 triangular elements (R3D3) to provide detailed shape at the tip of indenter and to reduce the mismatch of nodes on the indenter and the sample. CPFEM is applied in ABAQUS standard solver by using a UMAT subroutine. Since the force– displacement curve is independent from the coefficient of friction [10], the contact surface is then assumed to be frictionless in the tangential direction and “hard contact” in the normal direction. The bottom surface of the EP model is fixed in the Z direction and the lateral surfaces are fixed in the X or Y direction.
4.3.1. Nanoindentation on the matrix Four indentations with different random crystallographic orientations are obtained from CPFE numerical simulations. The force-displacement curve obtained both from experiments and simulations are presented in Fig. 9. In experimental results, peak loads have a variation of 7%, which is resulted from misorientation and random distributed tiny precipitates. As for the simulation results, the peak loads only have a variation of 1.2%. Similar results have been found by Ling Li et al. [11], which show that the effect of orientation is limited in CPFE simulations. The accuracy of the CPFE prediction is evident as the model can capture the variations due to different crystallographic orientations and all the simulated force–displacement curves fall in the range of the experimental measurements regardless of the indentation depth. Pile-up patterns can reflect local deformation in the indentation zone. As is shown in Fig. 10, the pile-up patterns are asymmetric as the orientations are different from common specifically oriented single crystals. The simulated pile-up patterns are similar with SPM images in Fig. 5b. Besides, the patterns from different orientations are quite different from each other, which indicate that orientations will significantly affect the local deformation in the indentation zone. Therefore information on orientation can be obtained from pile-up patterns. Pattern (d) is close to the orientation of (100). Pattern (a) and pattern (d) are similar since their misorientation is small. While pattern (b) and pattern (c) have similar orientation, the difference lies in the stiffness. 4.3.2. Nanoindentation on NbC precipitate As is observed in Fig. 11, the experimental force-displacement curve reaches the maximum load of 12833 μN when the indentation depth is around 200 nm. This fact proves that NbC is much stronger and stiffer than the matrix. In the unloading period, the experimental and simulation results have similar slope, which illustrates that the elastic modulus calculated by first principle is comparable to the actual value. However, the yield stress derived from hardness is not accurate enough. The load-displacement curve with a yield stress of 6.46 GPa has a maximum load which is 14.7% larger than the experimental result. Therefore, inversion calculations are carried out to determine the optimized yield stress. As we can see, the force increases with the increasing yield stress non-linearly. The yield stress of 5.06 GPa matches well with the experimental curve. Then it can be estimated that the yield stress is around 5.06 GPa. The yield stress is lower than expected mainly due to crystalline defects that have not been considered in first principle calculations. During the growth of NbC precipitate, point defects and dislocations will inevitably appear in the
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Fig.8. (a) Core CP model combined with isotropic EP model with dimension; (b) CP core model with dimension; (c) Model of NbC precipitate inserted in the matrix of Ni, NbC in red and Ni in green.
The evolution of Von Mises Stress is illustrated in Fig. 12. During loading, stress accumulates under the indenter. As CP is not implemented, the stress field is quite isotropic. When the indenter reaches the maximum depth, the stress in the matrix is still below its yield stress. During unloading, maximum residual stress turns up right underneath the edges of Berkovich indenter, where NbC precipitate has undergone the largest deformation.
5. Conclusion
Fig.9. Force–displacement curves of Inconel 718 at indentation depth of 250 nm: comparison of the experimental results (Exp1–4) and CPFE simulations with four sets of crystallographic orientations (Sim-orient1-4).
precipitate and its hardness will be greatly reduced. Also, the empirical law between hardness and yield stress is not the case for every material.
In this study, 3D CPFE simulations combined with first principle calculations have been carried out to investigate the behavior of cast Inconel 718 under nanoindentation and to estimate the mechanical properties of NbC precipitate. CP parameters have been calibrated using standard tensile test through a 3D CPRVE model. Also, the indenter has been carefully calibrated. Indentation was carried out on both matrix and NbC precipitate with different indentation depths. The positions of indents have been observed by back-scattered SEM and the composition of the precipitate has been examined by EDS. Then multi-scale simulation including first principle study and CPFEM simulation have been performed. The main results are as follows: 1. First principle calculations have been carried out to estimate the
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Fig.10. Comparison of the vertical displacement contour at an indentation depth of 250 nm for four sets of grain orientations. Each orientation under the indenter is schematically shown at the corner of each contour. The displacements are in mm in the legend.
elastic modulus, Poisson's ratio and hardness of NbC precipitate. Good results were found and in agreement with previous studies. 2. The CPFE simulation of nanoindentation on the matrix has proved that calibrated parameters can reflect mechanical properties of Ni with all the simulated force-displacement curves within the range of experiments. 3. Orientation has limited effect on load-displacement curve, but it has great influence on pile-up patterns. The peak load only has a variation of 1.2%. 4. The simulation of nanoindentation on NbC precipitate has been done to estimate its mechanical properties. It turns out that elastic properties are quite accurate and the yield stress can be derived from inversion calculations.
Fig.11. Force–displacement curves of NbC precipitate inserted in the matrix at indentation depth of 200 nm: comparison of the experimental results and CPFE simulations with four sets of yield stress.
Consequently, the proposed CPFE simulation combined with first principle study shows a great potential to study polycrystalline materials and to estimate the mechanical properties of precipitates. Material behavior such as stress, strain and pile-up patterns which cannot be easily observed in experiments can be vividly obtained from simulations. Moreover, mechanical properties like Young's Modulus and yield stress of precipitates which cannot
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Fig.12. Evolution of Mises stress during nanoindentation on NbC precipitate at the displacement of (a) 100 nm loading, (b) 200 nm loading, (c) 190 nm unloading and (d) unloaded state. The unit of stress is in MPa.
be easily measured in experiments can be estimated with inversion methods.
Acknowledgement This work was supported by the National Natural Science Foundation of China (Grant Nos. 51575347, 51405297 and 51204107).
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[20] W.C. Oliver, G.M. Pharr, An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments, J. Mater. Res. 7 (06) (1992) 1564–1583. [21] N. Zhou, D.C. Lv, H.L. Zhang, et al., Computer simulation of phase transformation and plastic deformation in IN718 superalloy: microstructural evolution during precipitation, Acta Mater. 65 (6) (2014) 270–286. [22] S.J. Clark, M.D. Segall, C.J. Pickard, P.J. Hasnip, M.I.J. Probert, K. Refson, M. C. Payne, First principles methods using CASTEP, Z. Krist. 220 (2005) 567. [23] (a) J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1997) 3865; (b) J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 78 (1997) 1396. [24] G. Kresse, J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6 (1) (1996) 15–50. [25] B.B. Karki, L. Stixrude, S.J. Clark, et al., Structure and elasticity of MgO at high pressure, Am. Mineral. 82 (1) (1997) 51–60. [26] Georg Will, Rolf Platzbecker., Crystal structure and electron density distribution in niobium carbide, Z. Anorg. Allg. Chem. 627 (9) (2001) 2207–2210. [27] R. Hill, The elastic behaviour of a crystalline aggregate, Proc. Phys. Soc. Sect. A 65 (5) (1952) 349. [28] S.I. Ranganathan, M. Ostoja-Starzewski, Universal elastic anisotropy index, Phys. Rev. Lett. 101 (5) (2008) 055504. [29] J.F. Nye, Physical properties of crystals: their representation by tensors and matrices, Oxford University Press, 1985. [30] F. Gao, Theoretical model of intrinsic hardness, Phys. Rev. B 73 (13) (2006) 132104. [31] J.J. Gilman, J.H. Westbrook, H. Conrad, The Science of Hardness Testing and its Research Applications, American Society for Metals, Metals Park, Ohio 1973, p. 54.
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Multi-scale simulation of nanoindentation on cast Inconel 718 and NbC precipitate for mechanical properties prediction Chenghui Ye a,1, Jieshi Chen a,c,1, Mengjia Xu a, Xiao Wei a, Hao Lu a,b,n a
School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China Key Lab of Shanghai Laser Manufacturing and Materials Modification, Shanghai Jiao Tong University, Shanghai 200240, PR China c School of Materials Engineering, Shanghai University of Engineering Science, Shanghai 201620, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 13 October 2015 Received in revised form 16 February 2016 Accepted 17 March 2016 Available online 18 March 2016
Multi-scale simulation of nanoindentation on cast Inconel 718 superalloy and its NbC precipitate were performed with combined first principle study and crystal plasticity finite element method (CPFEM). The concerned parameters were calibrated through a representative volume element (RVE) model compared with the stress–strain curves obtained from tensile tests. Nanoindentation was carried out on the matrix. First principle calculations were applied to estimate the mechanical properties of precipitate NbC, including elastic modulus and hardness. The simulated force–displacement curves match well with the experimental results. The simulated results indicate that the local pile-up pattern in the indentation zone depends significantly on the crystallographic orientations. In addition, large precipitate NbC inserted in the matrix was also indented and simulated. The elastic modulus calculated by first principle is quite accurate while the yield stress is determined using inversion calculations. It appears that the proposed CPFE analysis approach combined with first principle calculation do help estimate the mechanical behavior of large precipitates on the Ni-based superalloy. & 2016 Elsevier B.V. All rights reserved.
Keywords: Finite element method Nanoindentation Nickel based superalloys Plasticity
1. Introduction Inconel 718 is a nickel based superalloy with superior mechanical properties and structural stability at elevated temperature resulting from precipitation hardening. The matrix is γ phase (FCC). The most common precipitates in cast Inconel 718 are MC (FCC), laves phases (HCP) and δ-Ni3Nb (D0a, orthorhombic), apart from γ′ (L12, cubic) and γ″(D022, tetragonal) strengthening phase in the matrix.[1–3]. To investigate local properties of Inconel 718, nanoindentation is a good choice that can measure mechanical properties such as hardness and elastic modulus even at micro-nano scales. The indentation depth ranges from tens of nanometers to several microns. This testing method was first introduced in mid-1970s [4–7] and it is now widely used to characterize micro-mechanical behavior. However, it is difficult to interpret the experimental data as it involves a very complex deformation field. With the help of finite element method (FEM), more information can be obtained. In addition, traditional plastic theory cannot illustrate the anisotropy and the evolution of orientation during the deformation. n Corresponding author at: School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China. E-mail address: [email protected] (H. Lu). 1 Co-first Authors.
http://dx.doi.org/10.1016/j.msea.2016.03.081 0921-5093/& 2016 Elsevier B.V. All rights reserved.
Therefore, the crystal plasticity (CP) constitutive models based on dislocation evolution in crystal have been developed and applied on nanoindentation simulation. The CP theory was first introduced by Taylor [8,9], afterwards, it was incorporated with FEM. CPFEM adapts well with complex boundaries and enables various of plastic flow and hardening laws. Liu et al. [10] carried out 3D CPFE simulations of nanoindentation on copper single crystals, combined with experimental force–displacement curves to predict the stress–strain curve at the mesoscale. Li et al. [11] simulated nanoindentation on aluminum alloy polycrystals to study the effect of orientation, grain tessellation and grain boundaries. Li et al. [12] presented a numerical simulation of variable amplitude loading effects on cyclic plasticity and microcrack initiation in 304 L steel. Nie et al. analyzed the size effect of precipitates on the creep properties of nickel based superalloys using creep based CPFE model [13]. In spite of these recent studies, the application of CPFEM in nanoindentation remains to be improved. There is seldom CPFEM simulation of nanoindentation on nickel based superalloy, mainly due to the fact that nickel based superalloys are filled with various precipitates. In addition, as far as the author's knowledge, there was no study on CPFE simulation of nanoindentation on materials with precipitates. How to estimate mechanical properties of precipitates in nickel based superalloys remains an issue. As a solution to this issue, first principle study provides a
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Table 1 Composition of Inconel 718 high temperature alloy (wt%). C
Cr
Ni
Mo
Al
Ti
(Nb þTa)
Fe
0.02–0.08
17.00–21.00
50.00–55.00
2.80–3.30
0.40–0.60
0.90–1.15
5.00–5.50
Rest
means of calculating mechanical behaviors of crystals based on stress-strain approach [14]. Therefore, multi-scale simulations provide a solution to simulate nanoindentation on materials with precipitates. In this study, CPFE simulation of nanoindentation combined with first principle study is applied to estimate the mechanical properties of NbC precipitate in Inconel 718. To ensure accurate simulation results, CP parameters are calibrated with representative volume element (RVE) model, and simulation results of force-displacement curve are compared with experimental data. The sections of the paper are arranged as follows: The theory of CP and the calibration of the CPRVE model are presented in Section 2. The calibration of Berkovich indenter and nanoindentation experimental results are reported in Section 3. Multi-scale simulation of Inconel 718 and NbC precipitate under nanoindentation compared with the experimental data is discussed in Section 4. Finally, conclusions are given in the last section.
effect of the mobile dislocation density. The critical shear stress τcα is thus expressed as a function of the forest dislocation ρF, such that:
τcα = Gb
12
∑β = 1 Aαβ ρFβ
=
∑β hαβ γ β
(5)
where A is the interaction matrix that describes the hindering behavior between different slip systems. G is the shear modulus, b is the burger's vector, h is the hardening strength and γ β stands for the shear strain. The interaction matrix is composed of four dislocation interaction types from a0 to a3. The evolution of dislocation density can be decomposed into a term of dislocation multiplication and a term of annihilation for the slip system α, as follows:
ρḞ α =
⎛ ⎞ β 1 ⎜ ∑α ≠ β ρF α⎟ α − ρ 2 y ⎟⎟ γ ̇ c b ⎜⎜ K ⎝ ⎠
(6)
where yc represents the critical annihilation length, which is related to the dynamic recovery. K is a constant.
2. Crystal plasticity theory and calibration 2.1. Crystal plasticity formulations
2.2. Calibration with RVE model The plastic deformation of a single crystal is regarded as the cause of crystalline slip. The deformation gradient F is the product of a lattice deformation gradient F* and a plastic deformation gradient Fp, assuming that the material flows depends on dislocation motion, and then the combination of elastic deformation and rigid body rotation [15].
F = F *⋅Fp
(1)
The velocity gradient L can be defined as:
L = FḞ −1 = F *̇ F * − 1 + F *Fṗ F p−1F * − 1 = L* + F *L pF * − 1
(2)
The plastic velocity tensor Lp can be expressed by the sum of the shearing rate γ α̇ for all 12 slip systems of a face centered cubic (FCC) crystal structure (α = 1, 2, ⋯ , 12). The slip system α is speα α cified by the slip direction m and slip plane normal n in the global coordinate as the following equation: 12
Lp =
∑ γ α̇ mα ⊗ nα
(3)
α=1 α
The shearing rate γ ̇ for each slip system can be approximated by a power law:
⎛ τ α ⎞n γ α̇ = γ0⎜ α ⎟ si gn(τ α ) ⎝ τc ⎠
(4) α
where the rate exponent n represents the strain rate sensitivity. τ is the critical resolved shear stress for the slip system α, which stands for the projection of the nominal stress onto the slip plane, and it is the driving force of the plastic deformation. The strength τcα for the slip system α is the resistance of the plastic deformation, namely the stress needed for attaining reference velocity in the slip system α. Based on the hardening law [16], it is assumed that the dislocation cutting force is the major obstacle in plastic deformation. As a result, the plastic shear rate is related to the mean
To use the CP formulations listed above, the calibration of parameters is necessary. The composition of cast Inconel 718 used in this study is listed in Table 1. As the sample has undergone high temperature for a long time, the grains have grown coarser and the mechanical properties are not as good as those in standards. The samples are carefully polished and electrolytic etched with 10% chromic acid solution as electrolyte under a voltage of 45 V for 10 s for SEM observation. Also, EDS analysis was conducted to identify the precipitates. As is shown in Fig.1, significant dendritic segregation is observed in this alloy. Precipitates are strongly segregated and form dendrites. When the segregation spot is zoomed to 20,000 , both large precipitates of several microns and tiny needle-like precipitates of hundreds of nanometers are observed. According to Wen Sun [17], large precipitates in cast Inconel 718 alloys are mainly MC and tiny precipitates are Laves phase or δ phase. Cast Inconel 718 alloy is composed of polycrystals with a mean grain size around 300 mm. It is time-consuming to simulate all the grains in the tensile sample. CPRVE model is thus implemented in micro-scale simulations to effectively represent the mechanical behavior of the macro-scale specimen with the minimum number of grains. [18,19] The CP parameters were calibrated using standard tension tests with a CPRVE model. The CPRVE model used in this study is 4.4 4.4 1.1 mm3, containing 1000 equiaxed grains with different random orientations as shown in Fig. 2. Meanwhile, tensile tests of Inconel 718 bar samples are conducted on Zwick/ Roell Z100 with a constant strain rate of 1 mm/min. The tensile bar samples are 6 mm in diameter and 40 mm in gage length. The comparison between tensile test and CPRVE models is shown in Fig. 3. After times of parameter testing, good agreements are found with the CP parameters listed in Table 2. With these parameters, the mechanical properties of Inconel 718 superalloy can be numerically simulated with CPFEM method.
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Fig.1. SEM of cast Inconel 718 at (a) 500 (b) 2000 (c) 20,000 .
Table 2 Paremeters used in CPFE model. C11 (MPa)
C12 (MPa)
C44 (MPa)
τ0 (MPa)
γ0̇ (s 1)
n
a0
a1
203,030
149,870
134,980
32
0.001
2.703
0.045
0.625
a2
a3
ρ0 (m 2)
b (nm)
yc (nm)
K
0.255
1.43
64
0.137
0.122
10
12
3. Nanoindentation test
Fig.2. RVE model of Inconel 718 polycrystals tensile test.
Before nanoindentation, the sample of Inconel 718 was mechanically polished and then vibrant polished for 2 hours. It was not etched so that the structure of precipitates on the matrix is totally kept. Nanoindentation test of Inconel 718 was conducted on Hysitron TI 950 TriboIndenter which combines nanoindentation with in-situ scanning probe microscopy (SPM). The indenter was a Berkovich indenter made of diamond with an apex angle of 142.35° and a radius of 150 nm. To have a better comparison with the simulation results, displacement control method was adopted so that the indenter stopped and unloaded once the displacement had reached the designated indentation depth. Loading is a complex process which involves plastic deformation (material nonlinearity) and an increasing contact area (geometric nonlinearity), while unloading is purely elastic. To ensure accurate measurement, the shape of the indenter was also calibrated. According to Oliver and Pharr [20], a practical indenter is not as perfect as an ideal one, and its shape can be fitted with the following equation:
Ac = 24.56hc2 + C1hc1 + C2hc1/2 + C3hc1/4 + ⋯ + C8hc1/128
Fig.3. The comparison of Stress-Strain curve between tensile test and RVE model.
(7)
where Ac is the contact area, hc is the contact depth, C1 to C8 are constants, the first term 24.56hc2 represents the shape of an ideal Berkovich indenter and the rests are terms to modify its shape due to bluntness and defects. The actual shape of the Berkovich indenter was fitted as shown in Fig. 4. In order to clarify the hardening effect of NbC precipitate on Nibased superalloy, nanoindentation test is carried out on both matrix and NbC precipitate. It is worth noting that, in this paper, the matrix stands for Inconel 718 superalloy without visible precipitates, which has taken into account some tiny precipitates (distributed underneath the surface in the segregation spot). Since these tiny precipitates have a uniform distribution in the segregation spot [17,21], the matrix and tiny precipitates can be treated as one. As is shown in back-scattered Scanning Electron Microscopy (SEM) Fig. 5, four nanoindentation tests were conducted on
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Fig.4. (a) Sketch of a real Berkovich indenter, α ¼62.35° (b) The calibration relation between contact area and contact depth of the Berkovich indenter.
the matrix. The indentation was controlled by indentation depth, so that the indenter started unloading at an indentation depth of 250 nm. As for NbC precipitate, due to its extremely high hardness, the indentation depth was set to 200 nm, where the maximum load of indenter had been reached at 12833 μN. As to indent on NbC precipitate, we do not want to damage the matrix of Ni, as a result, only slight etching was allowed during vibrant polishing. Thus only precipitates at the surface of the sample can be observed. Also, limited by the resolution of SPM, only large precipitates at the surface of the sample can be identified, which are typically carbides such as NbC. Then a nanoindentation on NbC precipitate was also conducted as shown in Fig. 6a. The indentation was totally on the precipitate so that we can assume this structure as a NbC layer inserted to the surface of the matrix. As we can see in Fig. 6b, the Energy Dispersive Spectrometer (EDS) analysis of the composition of this precipitate proves that the precipitate is mainly NbC, since it is rich in Nb and C.
4. Multi-scale simulation Multi-scale simulation is carried out including first principle study and CPFEM simulation of nanoindentation. At first, first principle study was conducted to predict the mechanical properties of NbC precipitate, and these properties would be used in CPFEM simulation. Then CPFEM models were built to simulate the process of nanoindentation of the matrix and NbC precipitate. 4.1. First principle study on mechanical properties of NbC To estimate the mechanical properties of precipitate NbC, first principle study was conducted. All calculations presented in this work were performed using the CASTEP plane-wave code [22] based on density-functional theory (DFT), and the exchange-correlation energy was calculated using the Perdew-Burke-Ernzerhof (PBE) general gradient approximation (GGA) [23]. Brillouin-zone integrations were performed using Monkhorst and Pack k-point
Fig.5. (a) SEM image of 4 nanoindentations on the matrix (b) SPM image of the same nanoindentations.
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Fig.6. (a) SEM image of nanoindentation on NbC precipitate. (b) EDS spectrum of the precipitate. Table 3 Structural properties obtained by GGA calculations, in comparison with other experimental works. Lattice parameters a (Å), b (Å), c (Å); Cell volume V (Å3); Information in square brackets shows the final setting values for k-point meshes and their corresponding k-point numbers. Phase Structure Space group NbC
a
Cub.
Method
Lattice parameters (Å)
GGA[6 6 6] a ¼ b¼ c ¼4.478 Exp. by others a ¼ b¼ c ¼4.468a
FM-3M
89.76 89.19a
569.2 (557.3 )
NbC a
C12 a
C44 a
172.8 (162.4 )
148.5 ( 146.5a)
meshes [24]. The final settings for these calculations are shown in Table 3. Besides, the elastic constants were calculated by the stress-strain approach [25]. The convergence tolerance were selected as follows: minimum energy less than 5.0 10–6 eV/atom, maximum force less than 0.01 eV Å 1, maximum stress less than 0.02 GPa, and maximum displacement less than 5 10–4 Å. The calculated elastic constants are listed in Table 4. Usually, on the basis of the elastic constants, the bulk modulus (B) and shear modulus (G) are determined by the Voigt-Reuss-Hill (VRH) averaging scheme [27]. The Voigt bounds on the effective bulk and shear modulus of cubic polycrystals are
GVoigt = (c11 − c12 + 3c44 )/5
GV R
G
+
BV BR
−6
(13)
U
A is identically set to zero for locally isotropy single crystals, and the departure of AU from zero indicates the extent of elastic anisotropy. The calculated value of AU is collected in Table 5, which indicates that NbC has slight elastic anisotropy. A three-dimensional (3D) curved surface, representing the dependence of elastic properties on crystallographic directions, can indicate the elastic anisotropy of crystal structure as shown in Fig. 7. The directional dependence of Young's modulus (E) for cubic crystals can be defined as [29]:
Cubic system:
Ref [14].
BVoigt = (c11 + 2c12)/3
(12)
The elastic anisotropy of the crystals can also be measured by the universal anisotropy index (AU) [28]:
AU = 5
Table 4 Elastic constants (GPa) of NbC. C11
9BG 3B − 2G ,ν= 3B + G 2(3B + G)
V(Å3)
Ref [26].
Phase
E=
1/E = S11 − 2(S11 − S12 − S44/2)(l12l22 + l22l32 + l32l12)
where Sij are the elastic compliance constants, namely the reverse of elastic constants, and l1, l2 are the directional cosines to the X, Y axes, respectively. According to Gao's theory [30], the theoretical hardness of crystal can be expressed as follow:
Hv = A 0 Pvb−5/3
(15)
where A0 is a proportional coefficient, set to 740, P is the Mulliken overlap population, and Vb is the bond volume. The following equation is used to calculate Vb: [31]
Vb = (d u)3/∑ [(d v )3Nbv ]
(16)
v
(8) (9)
(14)
Calculated results are listed in Table 6. These results are close to those calculated by YangZhen Liu. As empirical equation suggests that the hardness is about 3 times of yield stress, the yield stress
While for the Reuss bounds, the equations are
BReuss = (c11 + 2c12)/3 G Reuss =
5(c11 − c12)c44 4c44 + 3(c11 − c12)
(10)
(11)
Then the Young's modulus (E) and the averaged Poisson's ratio (ν) can be obtained:
Table 5 Calculated elastic properties of NbC. Bulk modulus B (GPa); Shear modulus G (GPa); Young's modulus E (GPa); Poisson's ratio ν; Universal anisotropy index (AU). Phase
Bhill
Ghill
ν
E
B/G
AU
NbC
304.9
166.7
0.27
423.0 (483.9a)
1.82
0.084
a
Ref [14].
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4.2.2. Nanoindentation on NbC precipitate inserted in the matrix For that the NbC precipitate is large and that the indentation was entirely on the precipitate (see Fig. 6a), it is assumed as an NbC layer on the top of the matrix as shown in Fig. 8c. Here the thickness of NbC layer is large enough that the deformation is concentrated in this region. This model with an NbC layer has the same mesh and boundary conditions with the previous one. However, differences lie in material properties. Here elements in green represent for the matrix, whose properties are given by CP laws. As NbC precipitate is extremely hard and quite brittle, here we only consider an ideal EP property without hardening. Due to the fact that NbC has slight anisotropy and that orientation has little effect on force-displacement curve, simple EP properties is enough to estimate mechanical properties of NbC. The elements in red represent for NbC whose mechanical properties have been calculated in the Section 4.1 with first principle. 4.3. Simulation results and discussion
Fig.7. The directional dependence of Young's Modulus (E) for NbC. Table 6 Calculated bond parameters Vickers hardness for NbC. The units for hardness are in GPa. Phase
bond
d (Å)
Nbv
P
vb (Å3)
Hv (Gpa)
NbC
Nb-C
2.248
12
0.75
7.48
19.38 (19.5a)
a
Ref [14]
used in the simulation is estimated around 6.46 GPa. And these properties can be used as reference for mechanical properties for NbC in FE simulations. 4.2. CPFE simulation procedure 4.2.1. Nanoindentation on the matrix The boundary effect has great influence on the simulation result as nanoindentation only induces local deformation underneath the indenter. To find a balance between boundary effect and computational efficiency, a core CP model is combined with an elasto-plastic (EP) model. As is shown in Fig. 8, the dimension of the model is 20 20 10 mm3, and it consists of 125800 hexahedral elements (C3D8R). The elements are finer in the middle and range from 100 nm to 760 nm. The Berkovich indenter is discrete rigid and consist of 53778 triangular elements (R3D3) to provide detailed shape at the tip of indenter and to reduce the mismatch of nodes on the indenter and the sample. CPFEM is applied in ABAQUS standard solver by using a UMAT subroutine. Since the force– displacement curve is independent from the coefficient of friction [10], the contact surface is then assumed to be frictionless in the tangential direction and “hard contact” in the normal direction. The bottom surface of the EP model is fixed in the Z direction and the lateral surfaces are fixed in the X or Y direction.
4.3.1. Nanoindentation on the matrix Four indentations with different random crystallographic orientations are obtained from CPFE numerical simulations. The force-displacement curve obtained both from experiments and simulations are presented in Fig. 9. In experimental results, peak loads have a variation of 7%, which is resulted from misorientation and random distributed tiny precipitates. As for the simulation results, the peak loads only have a variation of 1.2%. Similar results have been found by Ling Li et al. [11], which show that the effect of orientation is limited in CPFE simulations. The accuracy of the CPFE prediction is evident as the model can capture the variations due to different crystallographic orientations and all the simulated force–displacement curves fall in the range of the experimental measurements regardless of the indentation depth. Pile-up patterns can reflect local deformation in the indentation zone. As is shown in Fig. 10, the pile-up patterns are asymmetric as the orientations are different from common specifically oriented single crystals. The simulated pile-up patterns are similar with SPM images in Fig. 5b. Besides, the patterns from different orientations are quite different from each other, which indicate that orientations will significantly affect the local deformation in the indentation zone. Therefore information on orientation can be obtained from pile-up patterns. Pattern (d) is close to the orientation of (100). Pattern (a) and pattern (d) are similar since their misorientation is small. While pattern (b) and pattern (c) have similar orientation, the difference lies in the stiffness. 4.3.2. Nanoindentation on NbC precipitate As is observed in Fig. 11, the experimental force-displacement curve reaches the maximum load of 12833 μN when the indentation depth is around 200 nm. This fact proves that NbC is much stronger and stiffer than the matrix. In the unloading period, the experimental and simulation results have similar slope, which illustrates that the elastic modulus calculated by first principle is comparable to the actual value. However, the yield stress derived from hardness is not accurate enough. The load-displacement curve with a yield stress of 6.46 GPa has a maximum load which is 14.7% larger than the experimental result. Therefore, inversion calculations are carried out to determine the optimized yield stress. As we can see, the force increases with the increasing yield stress non-linearly. The yield stress of 5.06 GPa matches well with the experimental curve. Then it can be estimated that the yield stress is around 5.06 GPa. The yield stress is lower than expected mainly due to crystalline defects that have not been considered in first principle calculations. During the growth of NbC precipitate, point defects and dislocations will inevitably appear in the
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Fig.8. (a) Core CP model combined with isotropic EP model with dimension; (b) CP core model with dimension; (c) Model of NbC precipitate inserted in the matrix of Ni, NbC in red and Ni in green.
The evolution of Von Mises Stress is illustrated in Fig. 12. During loading, stress accumulates under the indenter. As CP is not implemented, the stress field is quite isotropic. When the indenter reaches the maximum depth, the stress in the matrix is still below its yield stress. During unloading, maximum residual stress turns up right underneath the edges of Berkovich indenter, where NbC precipitate has undergone the largest deformation.
5. Conclusion
Fig.9. Force–displacement curves of Inconel 718 at indentation depth of 250 nm: comparison of the experimental results (Exp1–4) and CPFE simulations with four sets of crystallographic orientations (Sim-orient1-4).
precipitate and its hardness will be greatly reduced. Also, the empirical law between hardness and yield stress is not the case for every material.
In this study, 3D CPFE simulations combined with first principle calculations have been carried out to investigate the behavior of cast Inconel 718 under nanoindentation and to estimate the mechanical properties of NbC precipitate. CP parameters have been calibrated using standard tensile test through a 3D CPRVE model. Also, the indenter has been carefully calibrated. Indentation was carried out on both matrix and NbC precipitate with different indentation depths. The positions of indents have been observed by back-scattered SEM and the composition of the precipitate has been examined by EDS. Then multi-scale simulation including first principle study and CPFEM simulation have been performed. The main results are as follows: 1. First principle calculations have been carried out to estimate the
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Fig.10. Comparison of the vertical displacement contour at an indentation depth of 250 nm for four sets of grain orientations. Each orientation under the indenter is schematically shown at the corner of each contour. The displacements are in mm in the legend.
elastic modulus, Poisson's ratio and hardness of NbC precipitate. Good results were found and in agreement with previous studies. 2. The CPFE simulation of nanoindentation on the matrix has proved that calibrated parameters can reflect mechanical properties of Ni with all the simulated force-displacement curves within the range of experiments. 3. Orientation has limited effect on load-displacement curve, but it has great influence on pile-up patterns. The peak load only has a variation of 1.2%. 4. The simulation of nanoindentation on NbC precipitate has been done to estimate its mechanical properties. It turns out that elastic properties are quite accurate and the yield stress can be derived from inversion calculations.
Fig.11. Force–displacement curves of NbC precipitate inserted in the matrix at indentation depth of 200 nm: comparison of the experimental results and CPFE simulations with four sets of yield stress.
Consequently, the proposed CPFE simulation combined with first principle study shows a great potential to study polycrystalline materials and to estimate the mechanical properties of precipitates. Material behavior such as stress, strain and pile-up patterns which cannot be easily observed in experiments can be vividly obtained from simulations. Moreover, mechanical properties like Young's Modulus and yield stress of precipitates which cannot
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Fig.12. Evolution of Mises stress during nanoindentation on NbC precipitate at the displacement of (a) 100 nm loading, (b) 200 nm loading, (c) 190 nm unloading and (d) unloaded state. The unit of stress is in MPa.
be easily measured in experiments can be estimated with inversion methods.
Acknowledgement This work was supported by the National Natural Science Foundation of China (Grant Nos. 51575347, 51405297 and 51204107).
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