Multiscale modeling of the mechanical behavior of IN718 superalloy based on micropillar compression and computational homogenization

Acta Materialia 98 (2015) 242–253

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Multiscale modeling of the mechanical behavior of IN718 superalloy based on micropillar compression and computational homogenization A. Cruzado a,1, B. Gan a,1, M. Jiménez a, D. Barba a, K. Ostolaza b, A. Linaza b, J.M. Molina-Aldareguia a, J. Llorca a,c,⇑, J. Segurado a,c a b c

IMDEA Materials Institute, C/Eric Kandel 2, 28906 Getafe, Madrid, Spain Industria de Turbo Propulsores, 48170 Zamudio, Bizkaia, Spain Department of Materials Science, Polytechnic University of Madrid, E.T.S. de Ingenieros de Caminos, 28040 Madrid, Spain

a r t i c l e

i n f o

Article history: Received 8 June 2015 Revised 2 July 2015 Accepted 2 July 2015

Keywords: Multiscale modeling Ni-based superalloys Crystal plasticity Micropillar Computational homogenization

a b s t r a c t A multiscale modeling strategy is presented to determine the effective mechanical properties of polycrystalline Ni-based superalloys. They are obtained by computational homogenization of a representative volume element of the microstructure which was built from the grain size, shape and orientation distributions of the material. The mechanical behavior of each grain was simulated by means of a crystal plasticity model, and the model parameters that dictate the evolution of the critical resolved shear stress in each slip system (including viscoplastic effects as well as self and latent hardening) were obtained from compression tests in micropillars milled from grains of the polycrystal in different orientations suited for single, double (coplanar and non coplanar) and multiple slip. The multiscale model predictions of the compressive strength of wrought IN718 were in good agreement with the experimental results. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

1. Introduction Polycrystalline IN718 is a Ni–Fe based superalloy strengthened with c0 and c00 precipitates, which is widely used for structural applications up to 650–700 °C because of its good castability and weldability, high mechanical properties and corrosion resistance [1]. The microstructure of IN718 is made up by a c phase (Ni FCC solid solution) which contains a dispersion of nm-sized c0 -Ni3(Al,Ti) and c00 -Ni3Nb coherent precipitates within the grains together with lm-sized metal carbides and d phase (Ni3Nb) particles at grain boundaries [2–4]. The c0 phase presents an ordered cubic (FCC) L12 structure, while the c00 phase shows a body centered tetragonal (BCT) D022 structure [5] and forms thin ellipsoidal disc-shape precipitates. The c00 phase (which provides most of the strengthening [6]) is a metastable form of Ni3Nb, which tends to stabilize to the orthorhombic d phase with the D0a structure preferentially at grain boundaries [7]. The volume fractions of c0 and c00 phases are in the range 3–5% and 10–20% respectively, depending on the bulk alloy composition, the heat-treatment and the degree of element segregation. In general the weight fraction of (c00 + c0 + d) is around 20% [8].

⇑ Corresponding author at: IMDEA Materials Institute, Spain. 1

E-mail address: [email protected] (J. Llorca). These authors contributed equally to this work.

http://dx.doi.org/10.1016/j.actamat.2015.07.006 1359-6454/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Polycrystalline IN718 presents a strength well above 1000 MPa, large ductility (>30%), a marked tension-compression strength differential [9], softening under cyclic loading conditions [10] and significant losses of strength above 650 °C. These properties are controlled by the microstructural features of the material (grain size, precipitate size and volume fraction, etc.) but physically-based models capable of establishing a quantitative relationship between the microstructure and the mechanical properties are scarce. This is important from the fundamental viewpoint as well as from perspective of including the microstructural features into the design of structural components. Most of the models for the mechanical behavior of polycrystalline IN718 are phenomenological [9,10]. More recently, Fisk and Lundbäckl [11] proposed a physically-based flow stress model to evaluate the residual stresses obtained during welding and heat treatment of IN718. The parameters of the dislocation-based model were obtained from the stress–strain curves of the polycrystalline material at different temperatures, taking into account the strengthening effects of precipitates. Following this work, Fisk et al. [12] developed a model that includes the evolution (nucleation, growth and coarsening) of precipitate size and volume fraction in order to simulate the aging process of IN718. These models [11,12] were a first attempt to link the microstructure with the macroscopic mechanical but the connection of the single crystal deformation with the macroscopic flow was carried out through

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the Taylor factor and this simplification limited the scope of the approach as microscopic stress and strain fields within each grain were not considered. Polycrystalline homogenization [13] offers an alternative approach for developing models of the mechanical behavior based on the microstructure. Within this framework, the grain structure is considered through the definition of a representative volume element (RVE), while the single-crystal behavior is accounted for by a crystal plasticity model. In particular, computational homogenization models [14,15] (based on the resolution of a boundary volume problem on an RVE using either finite elements or fast Fourier transform) can account for the influence of the grain size, shape and orientation distributions on the mechanical properties. The accuracy of these models is strongly dependent on the single-crystal behavior, which can be represented by a physically-based constitutive model [16–19] or by simpler phenomenological crystal plasticity (CP) model [20–22]. In any case, the determination of the single-crystal behavior still remains the main issue in polycrystalline homogenization. Different approaches can be used to obtain the single crystal properties: from a multiscale bottom-up approach ranging from the heterogeneous subgrain microstructure to the grain mesoscale [18], to an inverse analysis strategy in which the single crystal behavior is chosen to reproduce the results of a set of mechanical tests in polycrystals [19–23] or of a set of indentation tests in single crystals [24,25]. Neither strategy is fully convincing as multiscale modeling strategies are not yet mature enough and the extrapolation of the parameters obtained by inverse analysis strategies to other scenarios (different from the one used in the inverse analysis) can be questioned. In this investigation, an alternative approach is proposed to obtain the single-crystal behavior based on the use of micropillar compression tests within individual grains from the polycrystalline microstructure [26–28]. Micropillar compression allows the direct determination of the single-crystal stress–strain behavior without the need of complex inverse analysis, as in the case of inverse analysis strategies based on indentation. However, it is well known that the mechanical behavior of micropillars might be affected by their size [29], and this approach would only be reliable if the micropillar

(a)

behavior is size independent and representative of the single crystal behavior within the polycrystalline material. This hypothesis seems reasonable in the case of IN718 because the internal length scale controlling the strength, i.e., the c00 + c0 precipitate spacing, is much smaller than the micropillar dimensions. The multiscale modeling approach based on micropillar compression tests and computational homogenization is developed and validated on a precipitation-hardened coarse-grain IN718 wrought alloy. The room temperature single-crystal behavior was experimentally measured by compressing single-crystal micropillars machined by focused ion beam (FIB) out of individual grains with different crystallographic orientations. The results of these tests were used to calibrate a CP model of the c-Ni phase that accounts for the effect of the c00 + c0 precipitation hardening. Finally, the polycrystal behavior was obtained by computational homogenization of an RVE of the microstructure that accounted for the actual grain size, shape and orientation distributions. The multiscale model predictions were in good agreement with the macroscopic response of the polycrystal.

2. Experimental methodology 2.1. Microstructural characterization The microstructure of the IN718 wrought alloy was characterized by optical microscopy, scanning electron microscopy (SEM), using a FEI Helios Nanolab 600i instrument, and electron backscatter diffraction (EBSD) in an Oxford AZTEC system. The microstructure was mainly composed of c phase grains, with negligible amounts of dispersed metal carbides (<1 vol.%), Fig. 1a. The average grain size was 95 lm and the grain orientations presented a random crystallographic texture, as shown in the inverse pole figure (IPF) map of Fig. 1b. Negligible amounts of d phase particles (<1 vol.%) were found, all decorating the grain boundaries (Fig. 1c). Transmission electron microscopy analysis, performed using a JEOL 3000F instrument, revealed the presence of profuse amounts of c0 and c00 precipitates in the interior of the grains with sizes below 50 nm (Fig. 1d).

(b)

Carbides 250 μm

(c)

125 μm

(d)

γ’’ phases δ phases 20 μm

γ’ phases 20 nm

Fig. 1. Microstructure of wrought IN718 superalloy. (a) Optical image showing the homogeneous distribution of metal carbides throughout the sample. (b) Inverse pole figure showing the grain orientation distribution. (c) SEM micrograph showing the decoration of grain boundary by d phase. (d) TEM micrograph showing the distribution of c0 and c00 precipitates within the grain.

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Based on the microstructural observations, the metal carbide and d phase content was rendered negligible and the material was modeled as a polycrystalline aggregate of c grains, containing a fine dispersion of c0 and c00 precipitates. Thus, the micropillar compression tests in single crystals provided the mechanical response of the c0 and c00 strengthened c-Ni phase. 2.2. Mechanical characterization The compressive stress–strain behavior of the wrought IN718 alloys was determined using a universal testing machine. Tests were conducted at room temperature and a constant strain rate of 5.0  104 s1 on cylindrical specimens 6 mm in diameter and 9 mm in length.

2.3. Micropillar compression tests The CP model of the single crystal was calibrated from the results of micropillar compression tests machined out of individual grains within the polycrystal. The micropillars were milled in the center of selected grains (Fig. 2a) by means of FIB (FEI Helios Nanolab 600i) following an annular milling strategy, with a final polishing step using a current of 230 pA to minimize FIB surface damage. Special care was taken to ensure that the resulting pillars were completely embedded in individual grains to avoid grain boundary effects. The crystallographic orientation of the pillars was determined by EBSD (Fig. 2b). A wide range of crystallographic orientations were tested, and they are shown in the reduced inverse pole figure (IPF) of Fig. 2b. Plastic deformation in FCC c-Ni phase occurs on the octahedral {1 1 1} h1 1 0i slip systems and the micropillar orientations were selected to promote single slip, double (coplanar and non-coplanar) and multiple slip.

The micropillar aspect ratio (length/diameter) was 2.4 to avoid buckling during compression [30] and the micropillar diameter was varied between 1 and 18 lm to determine the range of micropillar sizes that rendered a size independent response. One of the as-milled micropillars, with 5 lm in diameter and 12 lm in length, is depicted in Fig. 3a. The annular milling parameters used resulted in a minimum taper (<1.5°) of the pillars, as shown in the FIB cross-section of Fig. 3b. Most of the pillars were compressed using a circular diamond flat punch of 10 lm in diameter inside an instrumented nanoindentation system (Hysitron TI950). Tests were carried out in displacement control at three different strain rates (104, 103 and 102 s1). Only pillars with diameter <7 lm (which constitute the majority of the results included in this paper) could be compressed in this system because of the flat punch diameter. Larger pillars (up to 18 lm in diameter) were tested in a Micromaterials Nanotest system (maximum load of 500 mN) with a circular diamond flat punch of 30 lm in diameter. The experimental load–displacement curves were corrected for the extra compliance associated with the elastic deflection of the matrix at the base of the pillar using the Sneddon correction [31]. They were transformed in either engineering stress–strain curves or engineering resolved shear stress– strain curves from the initial length and upper diameter of the pillar and the initial Schmid factor calculated from the crystallographic orientation of the pillar. 3. Multiscale modeling strategy 3.1. Crystal plasticity model A detailed description of the CP constitutive model and its implementation is provided in [14]. It is briefly outlined here for completeness. The CP model is based on the multiplicative

Fig. 2. (a) SEM micrograph of polycrystalline IN718 with the micropillars machined out in the center of various grains. (b) IPF showing the orientation of the micropillars.

Fig. 3. (a) SEM micrograph of a micropillar machined from the IN718 polycrystalline specimen. (b) Cross-section of the micropillar showing the upper and lower diameter and the height.

A. Cruzado et al. / Acta Materialia 98 (2015) 242–253

decomposition of the deformation gradient, F, into its elastic Fe and plastic part Fp according to

F ¼ Fe Fp

ð1Þ

The plastic velocity gradient Lp in the intermediate (relaxed) configuration is defined as the sum of the shear rates, c_ a , on all the slip systems a according to 1 Lp ¼ F_ p Fp ¼

N slip X

c_ a ðsa0  ma0 Þ

ð2Þ

a¼1

where sa0 and ma0 are the unit vectors in the slip direction and the normal to the slip plane in the reference configuration respectively. The elastic strain is defined using the Green-Lagrange strain tensor, Ee, is expressed as

Ee ¼

1 eT e ðF F  IÞ 2

ð3Þ

where I stands for 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 ¼ CEe

ð4Þ

where C stands for the fourth order elastic stiffness tensor on the single crystal. Thus, the resolved shear stress sa on the slip plane a is obtained as the projection of the Kirchoff stress on the slip system according to

sa ¼ C

  1 eT e ðF F  IÞ : ðsa0  ma0 Þ 2

ð5Þ

The crystal is assumed to behave as an elasto-viscoplastic solid in which the plastic slip rate for a given slip plane system follows a power law dependency

c_ a ¼ c_ 0

 a 1=m js j

sac

sgnðsa Þ

ð6Þ

where c_ 0 stands for the reference strain rate, sac is the critical resolved shear stress (CRSS) of ath slip system at the reference strain rate and m the rate sensitivity parameter. The evolution of the CRSS of a given slip system a, sac , is expressed as,

s_ ac ¼

X hab jc_ ðbÞ j

ð7Þ

b

where b stands for any slip system. hab is the strain hardening modulus due to self and latent hardening, which can be expressed in a simplified manner according to

hab ¼ qab h

hardening modulus at large strains and ca is the accumulated shear strain in all slip systems, which is given by

ca ¼

Z tX jc_ a jdt 0

3.2. Single-crystal CP constitutive model The CP model was implemented in a user subroutine UMAT for the c0 and c00 strengthened c-Ni phase. Slip trace analysis in polycrystalline IN718 [33], as well as the micropillar compression tests that will be presented below, indicate that plastic deformation only took place in octahedral {1 1 1} h1 1 0i slip systems. This is consistent with the FCC structure of the c phase and it is presumably due to the low volume fraction (<20%) of c0 and c00 small precipitates, as opposed to the high volume fraction (>50%) of large c0 precipitates observed in other Ni-based superalloys where pseudo-cubic systems operate [34]. Therefore, only the 12 octahedral {1 1 1} h1 1 0i slip systems were implemented in the CP model. The single crystal elastic properties were obtained from literature [21] and can be found in Table 1. The parameters that control the plastic deformation of each slip system s0 , ss , h0 , hs and qab were obtained by comparison of the experimental stress-strain curves of micropillars of 5 lm in diameter in different orientations with the finite element simulations of the micropillar compression tests presented below. 3.3. Finite element model of micropillar compression The FE model used for the simulation of the micropillar compression tests is shown in Fig. 4. The model represents the geometry of the micropillars with 5 lm in diameter, including the radius of curvature of the fillet (approximately r = 1.5 lm) and the taper angle b  1.5°, as measured in Fig. 3b. Including the taper is crucial because it is has been shown that neglecting taper might result in an overestimation of the elastic modulus and of strain hardening and an increase of the apparent yield stress [35]. The micropillar and the supporting material were modeled as a single crystal. Eight node lineal brick elements (C3D8 with full integration in Abaqus [36]) were used for discretization. The total number of elements in the model was 15359 elements, following a mesh convergence analysis. The flat punch was modeled as a rigid body, with a lateral stiffness of 10 lN/nm, which approximately corresponds to the lateral stiffness of the indenter used in the experiment [37]. The contact between the flat punch and the micropillar head was modeled according to a Coulomb friction

where qab are the interaction coefficients that represent the influence of the hardening between different slip systems, and h corresponds to the self hardening modulus. The evolution of the self hardening was described according to the Voce hardening model proposed in [32],

ð9Þ

where h0 is the initial hardening modulus, s0 the initial yield shear stress, ss the saturation yields shear stress, hs the saturation

Table 1 Elastic constants of the cubic IN718 single crystal [21]. C11 (GPa)

C12 (GPa)

C44 (GPa)

259.6

179

109.6

ð10Þ

a

ð8Þ

  h0 hs ca exph0 ca =ðss s0 Þ hðca Þ ¼ hs þ h0  hs þ ss  s0

245

Fig. 4. Finite element model of the micropillar compression tests.

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Fig. 5. Log-normal grain size distribution of polycrystalline IN718 (solid line) and the corresponding grain size distribution in the RVE of the microstructure.

model, with a friction coefficient of 0.1 [37]. The base of the supporting material was fully constrained, while a flat punch was moved in the vertical direction at a constant speed. 3.4. Computational homogenization framework The effective behavior of polycrystalline IN718 was obtained by computational homogenization of an RVE of the microstructure. The grain size distribution in the polycrystal was measured from a cross-section that included approximately 300 grains. The experimental 2D grain size distribution was transformed to a 3D distribution assuming spherical grains. The open source software StripStar [38] was used to this purpose. The 3D grain radii distribution was then approximated by a lognormal function (r = 0.70862, l = 3.65) by means of the Levenberg–Marquadt algorithm (correlation coefficient of 0.93) and is shown in Fig. 5. The mechanical behavior of each crystal in the RVE was given by the CP model presented in Sections 3.1 and 3.2. The cubic RVEs were generated from the Voronoi tessellation of a set of points, which divided the initial volume in a number of polyhedral [39]. A Monte Carlo algorithm was developed to generate the position of the set of points used

for the tessellation, so the grain size distribution in the RVE followed the experimental grain size distribution (Fig. 5). The details of the algorithm developed can be found in Appendix A. The Voronoi tessellation was carried out with Neper [40] from an extended cloud of points obtained by a periodic copy in the three directions of space of the point distribution obtained by the Monte-Carlo method in order to preserve the periodicity of the microstructure in the RVE. The final shape of the RVE was not cubic because grains intersecting the cube faces were not cut and copied into the opposite face but were maintained in their positions. This strategy avoided meshing problems related with the development of very small grains near the cube surfaces. The periodic RVE was finally meshed with the open source program Gmsh [41]. 10-node quadratic tetrahedral elements (C3D10 with full integration in Abaqus [36]) were used for the discretization and the mesh quality was very high (only 0.26% of distorted elements). Because the texture in the material was negligible, the grains orientations were randomly generated in the rotation group SO3. Finite element models of RVEs containing from 20 to 472 grains and 615 to 26007 elements per grain were created to analyze the influence of element size and number of grains in the numerical results. One of them, which includes 210 grains, is depicted in Fig. 6. The generated RVEs are periodic along the axes of the original cube of length L although they do not have a cubic shape because grains are not limited by the faces of the original cube to avoid distorted elements. Nodes A and B in the same location of opposite surfaces can be grouped in pairs of periodic nodes and the initial distance between them is given by xB  xA ¼ LAB , where LAB = (L,0,0), (0,L,0) or (0,0,L) depending of the surface orientation. Uniaxial compression of the RVE was simulated by applying periodic boundary conditions to each node pair according to

  IÞLAB uB  uA ¼ ð F

ð11Þ

 stands for prescribed macroscopic deformation gradient where F and uA and uB the displacement vectors of the paired nodes. If some components of the far-field deformation gradient are not known a priori (mixed boundary conditions, as in uniaxial compression), the corresponding components of the effective stresses are applied instead following the strategy presented in [23]. 4. Results 4.1. Effect of pillar size The effect of pillar size on the experimental stress–strain response was assessed by performing tests on pillars with

Fig. 6. Finite element model with 130000 elements of an RVE containing 210 grains following the lognormal distribution in Fig. 5. A cross-section of the RVE parallel to one of the cube faces is shown to the right, showing the periodicity of the microstructure.

A. Cruzado et al. / Acta Materialia 98 (2015) 242–253

247

Fig. 7. CRSS vs. strain curves of micropillars of different diameter oriented for single slip tested in compression. (a) Diameter range from 1 to 7.5 lm. (b) Diameters of 6 and 18 lm.

diameters in the range 1–18 lm at an average strain rate of 103 s1. To this end, micropillars with different size were milled in grains with the same or similar crystallographic orientation (close to either h2 4 5i or h2 3 5i) to rule out any plastic anisotropy effects. They were favorably oriented for single slip with Schmid factors (SF) of 0.452 and 0.445 for the orientations h2 4 5i and h2 3 5i respectively. The critical resolved shear stress (CRSS) vs. strain curves of 4 micropillars with diameters between 1 and 7.5 lm are plotted in Fig. 7a and the corresponding SEM micrographs of the deformed micropillars are shown in Fig. 8a–d, which confirmed that plastic deformation took place by single slip. The micropillar of 1 lm in diameter presented a stiffer initial response and a higher CRSS while the CRSS–strain behavior of the micropillars with diameters between 3 lm and 7.5 lm were practically identical, except for a slight reduction in the slope of the initial loading slope with pillar diameter, that will be discussed in more detail in the next section. In all cases, the CRSS–strain curves displayed unloading events as a result of dislocation bursts, upon which the high frequency feedback loop of the system reacts relaxing the load to maintain the prescribed displacement rate. In addition to these tests, two new pillars with 6 and 18 lm in diameter were milled in grains with h3 2 6i (SF = 0.467) orientation. The CRSS–strain curves are plotted in Fig. 7b and the SEM micrographs of the deformed micropillars are depicted in Fig. 8e and f, showing single slip. These micropillars were tested in a different nanoindentation system, as described in Section 2.2, due to the large load required for the 18 lm pillar. The dynamic response of this system was not fast enough to unload the pillar in reaction to the dislocation bursts, resulting in repeated pop-in events, instead of load relaxations. Except for this, the CRSS–strain curves measured in these micropillars were indistinguishable from those measured in micropillars with diameter between 3 and 7.5 lm. These results confirm the hypothesis that the micropillar response can be regarded as independent of the micropillar diameter, and representative of the single-crystal bulk behavior, at least for micropillars with diameters above 3 lm in the case of IN718. This behavior presumably occurs because the internal length scale controlling the strength, i.e. the c00 + c0 precipitate spacing, is much smaller (of the order of 50 nm) than the micropillar dimensions. However, the CRSS–strain curves in Fig. 7a also showed a clear size effect of the type ‘‘the smaller the stronger’’ in micropillars of 1 lm in diameter. The origin of this behavior is intriguing but is beyond the scope of this paper. Based on these results, the CP model

parameters were obtained from the behavior of micropillars of 5 lm in diameter. 4.2. Calibration of the CP model As shown in Fig. 7, the slope of the initial loading decreases with micropillar diameter, leading to a much more compliant response than that found upon unloading. While the unloading slope yields the correct elastic modulus, it is well known that the initial loading slope in a micropillar tests is strongly affected by asperities of the surface and/or incorrect alignment between the flat punch and the head of the pillar [37]. This misalignment induces a stress concentration at the initial contact point of the micropillar, which leads to early yielding and to the apparent reduction of the initial loading stiffness while the unloading slope is free for this artifact once full plastic contact has been established between the flat punch and the pillar. Unfortunately, this experimental uncertainty makes it difficult to establish the onset of plastic yielding from the stress– strain curves, and this information is necessary to calibrate the parameters of the CP model. This limitation can be overcome following the strategy proposed by Kupka et al. [26] to obtain the evolution of the stress with the plastic strain in micropillar compression tests. To this effect, the experimental strain eðrÞ was first corrected by subtracting the elastic strain, which was obtained for each stress level from the unloading slope Sunload according to

e1 ðrÞ ¼ eðrÞ 

r Sunload

ð12Þ

The original and the corrected stress-strain curves are shown in Fig. 9a for the case of a micropillar of 3 lm in diameter oriented for single slip in the [3 2 5] direction. After this correction, the unloading segment of the stress–strain curve appears as a vertical line. Afterwards, the initial yield point (ry , e1y ) was determined as the point at which the maximum slope of the loading curve decreased by 20% with respect to the initial slope, and the corrected plastic strain – free of the initial loading artifacts-was given by

ep ðrÞ ¼ e1 ðrÞ  ðe1y Þ

ð13Þ

and the corresponding stress–plastic strain curve can be found in Fig. 9b. The CRSS–plastic strain curves corresponding to the micropillar compression tests were obtained using this procedure and they were used to determine the parameters of the CP model as follows.

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(a)

(425)

(b)

μm 1μ m

500 nm

(c)

(235)

(d)

1.5 μm

(e)

(326)

1.5 μm

(325)

(245)

2 μm

(f)

(326)

10 μm

Fig. 8. SEM micrographs of micropillars with different size deformed in the single slip condition. (a) 1 lm oriented in [4 2 5]. (b) 3 lm oriented in [3 2 5]. (c) 5 lm oriented in [2 3 5]. (d) 7.5 lm oriented in [2 4 5]. (e) 6 lm oriented in [3 2 6] and (f) 18 lm oriented in [3 2 6].

Fig. 9. Correction of the stress–strain curves of the micropillar compression tests to obtain the stress–plastic strain curve. (a) Original and corrected stress–strain curve according to Eq. (12). The yield point is indicated in the corrected curve. (b) Stress–plastic strain curve obtained from Eq. (13). The results in this figure correspond to a micropillar of 3 lm in diameter oriented for single slip in the [3 2 5] direction and tested at an average strain rate of 103 s1.

According to Eq. (6), the actual CRSS in a given slip system a, depends on the shear strain rate c_ a in the form

ln

 a c_ 1 sa ¼ ln j sa j m c_ a0 sc

sas ,

ð14Þ

where c_ a0 and sac stand for the reference strain rate and the CRSS in the system a at the reference strain rate. The rate sensitivity exponent m was obtained from the CRSS–plastic strain curves at average strain rate of 102 s1 in h1 2 3i micropillars and at 103 s1 and 104 s1 in h2 3 5i micropillars oriented for single slip (SF = 0.467

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249

Fig. 10. (a) CRSS vs. plastic strain of micropillar oriented for single slip tested at different strain rates at room temperature. (b) Strain rate sensitivity parameter m of IN718 at room temperature as obtained from micropillar compression tests.

Fig. 11. Experimental and simulated CRSS vs. plastic strain curves of compression tests of micropillars of 5 lm in diameter oriented for single slip. (a) Average strain rate 103 s1, micropillar orientation [2 3 5]. (b) Average strain rates 102 s1 and 103 s4, micropillar orientation [1 2 3] and [2 3 5].

Table 2 Self hardening and latent hardening parameters of the Voce law.

s0 (MPa)

ss (MPa)

h0 (GPa)

hs (GPa)

qab

465.5

598.5

6.0

0.3

1

and SF = 0.451 for h1 2 3i and h2 3 5i, respectively), Fig. 10a. The average strain rate in each slip system can be computed as the average strain rate of the micropillar divided by the Schmid factor and the corresponding CRSS was taken as the one at a plastic strain of 0.04. The choice of the plastic strain to determine the CRSS was not very critical as the CRSS–plastic strain curves were practically parallel for plastic strains >0.02. These results (normalized by the shear strain rate and the CRSS for the tests carried out at an average strain rate of 103 s1) are plotted in Fig. 10b in bilogarithmic coordinates. The strain rate sensitivity parameter m = 0.017 was obtained from the inverse of the slope, the straight line fitted by the least squares method to these experimental data, according to Eq. (14). This result indicates that strain rate sensitivity of IN718 at room temperature is very small and it is not surprising that linear

regression shows a relative poor correlation index with the experimental data. The second step to calibrate the crystal plasticity model was to obtain the parameters of the Voce law each slip system, namely h0 , s0 , ss and hs . To this end, the finite element model of the micropillar compression test, presented in Section 3.3, was used to simulate the stress–strain curve of h2 3 5i micropillars, oriented favorably for slip with a SF = 0.451 at an average strain rate of 103 s1. The experimental CRSS vs. plastic curve is plotted in Fig. 11a together with the results of the numerical simulation obtained with the parameters shown in Table 2. It is worth noting that the values of the initial and saturation CRSSs (s0 and ss ) obtained from the finite element simulation of the micropillar are slightly below the ones that would be directly obtained from the CRSS–plastic strain curve and the Schmid factor because the numerical model accounts for the geometrical strain hardening introduced by the tapering of the micropillar as well as for lattice rotation during the test. The numerical model can also take into account the slight influence of the strain rate on the mechanical response of the micropillars, as shown in Fig. 11b, in which the results of the numerical simulations are compared with the CRSS vs. plastic

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Fig. 12. (a) Simulated stress vs. plastic strain curves of micropillars oriented for single slip (h2 3 5i) and double slip (h4 1 4i) obtained with qab = 1 and 2. (b) Experimental and simulation results of micropillars oriented in h0 1 2i (non co-planar double slip) and in h4 1 4i (co-planar double slip). Simulations were carried out with qab = 1. (c) SEM of a micropillar deformed along h0 1 2i showing non coplanar double slip. (d) SEM of a micropillar deformed along h4 1 4i showing coplanar double slip. All simulations and experiments in this figure were carried out in micropillars of 5 lm in diameter at an average strain rate of 103 s1.

strain curves measured at 102 and 104 s1 in micropillars oriented for single slip in orientations h1 2 3i and h2 3 5i, respectively. The previous simulations were carried out under the assumption that qab = 1 (Eq. 8). The choice of qab was irrelevant in the case of micropillars oriented for single slip, as shown in the simulations depicted in Fig. 12a for a micropillar of 5 lm in diameter oriented along h2 3 5i for single slip. Nevertheless, the contribution of latent hardening was not negligible in the case of double slip and the flow stress of micropillars oriented along h4 1 4i for double slip depended on qab (Fig. 12a). The actual value of qab was obtained from the mechanical response of micropillars oriented along h4 1 4i and h0 1 2i. h0 1 2i micropillars showed activity in two non coplanar slip systems, {1 1 1} h1 0 1i and {1 1 1}h1 0 1i, with a SF = 0.49 (Fig. 12c), while the micropillar oriented in h4 1 4i deformed in two coplanar slip systems, {1 –1 1}h0 1 1i and {1 –1 1}h1 1 0i, with SF = 0.43 (Fig. 12d). The simulations carried out with qab = 1 provided the best approximation to the experimental data (Fig. 12b) indicating that IN718 follows an isotropic hardening model. The CP model of IN718 was validated by comparing the results of the micropillar compression tests in different orientations with actual experimental results. They are shown in Fig. 13a for micropillars of 5 lm in diameter tested at an average strain rate of 103 s1 along h1 2 6i (SF = 0.488) and h4 5 6i (SF = 371) orientations. Numerical simulations were very close to the experimental results in h1 2 6i while the simulations in h4 5 6i orientation overestimated the experimental flow stress. This difference may be due to the proximity of the h4 5 6i direction to the h1 1 1i orientation,

which is the stiffest and strongest orientation of the crystal. Deformation along h1 1 1i leads to the formation of a shear band (Fig. 13c), presumably as a result of the activation of a secondary slip system during deformation, although the exact sequence of events leading to this behavior is not known and is currently under study. Finally, simulation of the deformation along h0 0 1i was very accurate (Fig. 13b) although this orientation led to multiple slip with the activation of 8 simultaneous slip systems (Fig. 13d). Finally, the predictive capacity of the CP model can be assessed by comparing the SEM of the deformed micropillar after compression along the h2 1 2i orientation (where both coplanar slip systems are active) (Fig. 14a), with the corresponding predictions of the finite element model (Fig. 14b). The model was able to reproduce accurately the shape of the deformed micropillar as well as the location of the slip band. 4.3. Computational homogenization The CP model calibrated in the previous section was able to reproduce the mechanical behavior of IN718 single crystals in different orientations, which include single, double (coplanar and non coplanar) and multiple slip. This model was used as the constitutive equation of the single crystals within a computational homogenization framework to obtain the effective properties of the polycrystal. A sensitivity analysis was first carried out to assess the influence the mesh size and of the number of grains in the RVE on the predictions of the effective response of the polycrystal.

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(c)

(111)

(d)

(001)

Fig. 13. (a) Experimental and simulation results of micropillars oriented in h5 4 6i and h2 1 6i orientations. (b) Experimental and simulation results of micropillars oriented in h0 0 1i orientation (c) SEM micrograph of a square micropillar deformed along h1 1 1i showing the formation of a shear band. (d) SEM micrograph of a micropillar deformed along h0 0 1i showing multiple slip. All simulations and experiments in this figure were carried out in micropillars of 5 lm in diameter at an average strain rate of 103 s1.

Fig. 14. (a) SEM micrograph of micropillar of 5 lm in diameter deformed up to 0.2 in h2 1 2i orientation at an average strain rate of 103 s1. (b) Contour plot of the accumulated plastic strain superposed to the deformed mesh obtained from the finite element model.

Numerical simulations were performed with three different mesh sizes with 615, 2466 and 26007 finite element per grain in RVEs containing 21 and 199 grains. The differences in the stress-strain curve between the coarse and the fine meshes were always below 1% up to an applied strain of 20%. Similarly, simulations were carried out with RVEs containing 20, 210 and 472 grains and 610 finite elements per grain. The differences in the stress–strain curves between the models were also very small (<1%). Thus, the model with 210 grains and 610 finite elements pre grain was selected to predict the mechanical behavior of the IN718 polycrystal. The grain size distribution followed the

log-normal distribution measured experimentally and numerical simulations were carried out with four different realizations of the random grain orientation distribution. The differences in the mechanical response calculated with different realizations were below 1.3%, which indicates the error in the effective response of the polycrystalline model for a given set of CP parameters. The effective response obtained by computational homogenization is compared with the experimental behavior in Fig. 15 in terms of the true stress–logarithmic strain curves. The agreement between them was fairly good: the maximum difference in the compressive flow stress was below 4% and the strain hardening

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The mechanical behavior in compression of the polycrystal predicted with this multiscale strategy was in good agreement with the experimental results and validated the whole strategy. The methodology presented in this paper, together with the recent advances in the characterization of the mechanical properties of lm-sized specimen milled from grains and grain boundaries in the polycrystal, presents a large potential to carry out virtual tests of polycrystalline Ni-based superalloys to predict the influence of the microstructure on key macroscopic features. This includes, among others, the strength differential, the effect of temperature on the strength, the creep and the fatigue resistance, etc. Acknowledgments This investigation was supported by the MICROMECH, funded by the European Union under the Clean Sky Joint Undertaking, 7th Framework Programme (CS-GA-2013-620078). TEM was carried out in the National Center for Electron Microscopy of the Complutense University of Madrid. Fig. 15. Experimental result and numerical simulation obtained by computational homogenization of an RVE of the true stress–strain curve in compression of IN718.

of the model and of the material was identical. This is remarkable taking into account the scatter associated with the micropillar compression tests was around 5%. Moreover, it should be highlighted the multiscale nature of the approach because all the ingredients in the polycrystalline homogenization strategy were obtained at lower length scales from the microstructure of the polycrystal and the mechanical properties of the single crystals within the polycrystal. An interesting consequence of the accuracy of this prediction is that grain boundaries do not contribute to the hardening of this coarse grained polycrystal. If this mechanism were relevant here, the simulated stress–strain curves (that do not account for slip gradients) should have underestimated the experimental flow stress. The physical reason of the negligible effect of grain boundaries is probably the small distance between precipitates that controls the plastic response of the material, and renders negligible the strengthening contribution of grain boundaries for this grain size. 5. Conclusions A multiscale modeling strategy has been developed to predict the mechanical behavior of polycrystalline Ni-based superalloys. The effective properties of the polycrystalline material were obtained by computational homogenization of an RVE of the microstructure that accounts for the grain size, shape and orientation distributions. The mechanical behavior of each grain is given by a crystal plasticity model and the model parameters that dictate the evolution of the CRSS in each slip system (including viscoplastic effects as well self and latent hardening) were obtained from compression tests in micropillars milled from grains of the polycrystal in different orientations suited for single, double (coplanar and non coplanar) and multiple slip. The ability of the crystal plasticity model to reproduce the mechanical behavior of micropillars was validated by the accurate prediction of micropillar compression tests in different orientations and at various strain rates. In addition, the model was able to capture the dominant deformation mechanisms in the micropillars, including the formation of the slip bands observed. It was demonstrated that mechanical behavior of micropillars was size independent (and, thus representative of the behavior of the material within the grains) for micropillars with a diameter above 3 lm because the flow stress in this material was controlled by the distance between nm-sized c0 and c00 precipitates dispersed in the solid solution c face.

Appendix A. An algorithm has been developed to generate the coordinates of a set of N points p = pi that can lead by Voronoi tessellation to an RVE whose grain size distribution follows a prescribed one. To this end, it is necessary to minimize an error function O(p) defined as

OðpÞ ¼

N X jPDF exp ðV j Þ  PDF p ðV j Þj DV

ðA1Þ

j¼1

where PDF exp and PDF p are the probability density functions of the experimental and the target grain size distributions, respectively, which indicate the probability for a grain to have a volume in the range V j and V j þ DV, where DV ¼ ðV max þ V min Þ=N and V max and V min stand for the maximum and minimum grain sizes, respectively. An iterative process based on the Monte Carlo method was used to minimize O(p). Let p(k) the set of points at iteration k, that leads to an error OðpðkÞ Þ higher than a prescribed value. The new set p(k+1) is obtained by moving a random point pi in a random direction a random distance Dx in the range between 0 and 3.5D, where D is the average grain size of the experimental distribution. A new Voronoi tessellation is carried for the set pðkþ1Þ , and the error function Oðpðkþ1Þ Þ is computed from the new PDF p . Whether or not the set pðkþ1Þ is chosen as the starting point for the next iteration depends on the parameter prob, which is given by

   Oðpðkþ1Þ Þ  OðpðkÞ Þ prob ¼ min 1; exp  ðkÞ A Oðp Þ

ðA2Þ

where A is a parameter, chosen equal to 0.02 in this case. If the error is reduced, Oðpðkþ1Þ Þ 6 OðpðkÞ Þ, prob = 1 and the set pðkþ1Þ is chosen as the starting point for the next iteration. On the contrary, prob 2 ð0; 1Þ if the error increases and a random number is generated, r 2 ð0; 1Þ. The set pðkþ1Þ is chosen as the starting point for the next iteration if r < prob; otherwise, the old set pðkÞ is again used as the starting point. This algorithm, that is able to accept distributions that slightly increase the error, leads to an improvement of the efficiency of the Monte Carlo method. The process is repeated until the error function OðpÞ is smaller than a prescribed tolerance. This algorithm has been programmed in Matlab. References [1] H.J. Wagner, A.M. Hall, Physical metallurgy of alloy 718. DMIC Report 217, Battelle Memorial Institute, Columbus, June 1, 1965. [2] Y.S. Song, M.R. Lee, J.T. Kim, Effect of grain size for the tensile strength and the low cycle fatigue at elevated temperature of alloy 718 cogged by open die

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