45-degree rafting in Ni-based superalloys: A combined phase-field and strain gradient crystal plasticity study

Journal Pre-proof 45-degree rafting in Ni-based superalloys: A combined phase-field and strain gradient crystal plasticity study Muhammad Adil Ali, Waseem Amin, Oleg Shchyglo, Ingo Steinbach

PII: DOI: Reference:

S0749-6419(19)30491-7 https://doi.org/10.1016/j.ijplas.2020.102659 INTPLA 102659

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International Journal of Plasticity

Received date : 3 July 2019 Revised date : 19 December 2019 Please cite this article as: M.A. Ali, W. Amin, O. Shchyglo et al., 45-degree rafting in Ni-based superalloys: A combined phase-field and strain gradient crystal plasticity study. International Journal of Plasticity (2020), doi: https://doi.org/10.1016/j.ijplas.2020.102659. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.

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45-degree rafting in Ni-based superalloys: a combined phase-field and strain gradient crystal plasticity study Muhammad Adil Ali1a , Waseem Amina,b , Oleg Shchygloa , Ingo Steinbacha Interdisciplinary Center for Advanced Material Simulations, Ruhr Universit¨ at Bochum, Universit¨ atsstr. 150, 44801, Bochum, Germany b Department of Metallurgy and Materials Engineering, University of Engineering and Technology Taxila, 47050, Taxila, Pakistan

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Abstract

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45◦ rafting of Ni-based superalloys has been investigated with the help of creep test simulations applying a strain gradient crystal plasticity model coupled to the multi-phase field method. This mode of rafting lies in between P- and N-type rafting modes. The model parameters are calibrated against experimental data for N-type rafting under high temperature and low stress creep condition. By increasing the stress level, the mixed-mode rafting of precipitates with a clear tendency towards formation of 45◦ rafts is observed. We show that the key factor for the occurrence of this type of rafting is the generation of highly localized creep strain of more than 10% due to non-homogeneous creep deformation in the form of slip bands. We have successfully captured the evolution of microstructure under high stress leading to production of localized shear bands.

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Keywords: Superalloys, Dislocations, Rafting, Directional Coarsening, Phase Field, Crystal Plasticity 1. Introduction

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Ni-based single crystal superalloys containing γ 0 precipitates produce an outstanding resistance ing temperatures 650o C < T < 1100o C under the magnitudes [1]. Such a high thermo-mechanical

high volume fractions of to creep at higher workinfluence of higher stress stability is an attractive

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Corresponding author. E-mail address: [email protected] (Muhammad Adil Ali) Preprint submitted to International Journal of Plasticity

December 19, 2019

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property that makes superalloys perfect not only for the parts experiencing higher temperatures in the airplane or rocket engines, but also for the power plant gas turbines. To date numerous studies have addressed the basic mechanisms that control the creep deformation in single crystal superalloys including the evolution of dislocation network around γ 0 precipitates [2, 3, 4, 5], the role of lattice misfit stresses [6, 7, 8], dislocation-precipitate interactions [9, 10, 11, 12], accumulation of dislocations in γ channels [13, 14, 15]. Nonetheless, even using the most sophisticated experimental techniques it is difficult to capture all dynamic microstructural processes at in-service conditions. In contrast, creep simulations allow to capture all necessary information during material deformation process at the mesoscopic scale, using physics based micro-mechanics models where a higher level of abstraction along with a higher level of resolution can be achieved [16]. Another advantage of such simulation techniques as compared to the experiments is that the effect of microstructure evolution, the mechanisms controlling the dislocation activities and the interplay between them can be discriminated at all stages of the process. To date, a number of numerical models to predict mechanical response of superalloys have been published [17, 18, 19, 20, 21, 22, 23] but most of these models have been applied to study low temperature creep. In a typical superalloy microstructure, the long-range ordered L12 γ 0 precipitates with a volume fraction of about 70% and cuboidal morphology are coherently embedded into solid solution γ matrix which has face centered cubic (FCC) crystal structure [1, 24]. The cuboidal morphology forms due to the anisotropy of elastic moduli and negative lattice misfit between the two phases. In Ni-based superalloys γ 0 phase has a smaller lattice parameter value (a0γ ) as compared to the lattice parameter (aγ ) of γ matrix phase. The lattice misfit between γ 0 and γ phases reads

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1 (1) δ = (a0γ − aγ )/(a0γ + aγ ). 2 The microstructure consisting of regularly aligned cuboidal γ 0 precipitates significantly increases the creep strength of single crystal superalloys as compared to the smaller irregularly shaped precipitates [25]. Creep deformation of superalloys is typically divided into three well defined stages: primary, secondary and tertiary creep. The degree of primary creep deformation is the most significant design criterion because most of the total allowable creep deformation occurs during primary creep stage [26]. In particular, the initial cuboidal microstructure changes its topology during 2

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primary creep at high temperatures, called rafting [9, 27]. During the primary stage of creep, density of dislocations starts to increase and accumulates in γ channels [14]. Low stress-high temperature creep triggers dislocations in such a way that they enter the γ channels with a mixed character (with leading screw dislocation segments) along (111) planes and get deposited at an angle of 60o in roughly two third of the γ channels (see Fig.1) while the remaining channel space contains pure screw type dislocations close to γ/γ 0 interface [28].

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Figure 1: Schematic illustration of (a) (111) planes in γ 0 precipitates (triangles) and (b) cross section along (111) plane through the whole specimen showing the intersection of (111) plane with the γ/γ 0 interface.

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Upon uniaxial loading along [001] direction, γ channels normal to the loading direction experience higher effective stresses than the channels parallel to the loading direction due to negative lattice misfit in Ni-based superalloys [8, 29]. This enhances gliding of dislocations in γ channels perpendicular to the loading direction which relaxes internal stresses in these channels [15, 30] and thus increases the stability of the microstructure. This results in coalescence of γ 0 precipitates along the direction perpendicular to the loading direction leading to raft formation at 90o to the tensile loading direction, which is called N-type rafting. The rafting reverts its direction under compressive stress, i.e. it occurs along the direction parallel to the loading direction, and the resulting rafted morphology is called P-type rafting [31]. The initial cube like morphology of γ 0 phase transforms into plate like 3

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structures normal to the applied stress [24] and the thickness of these plates increases with prolonged creep periods due to ripening [32]. The kinetics of the rafting process is controlled by several factor including plasticity; resulting from the dislocation motion in the γ channels [13], dislocation storage along the γ/γ 0 interfaces [33] and diffusion controlled microstructural evolution. This is also affected by the loss of interface coherency between γ and γ 0 phase, and by reduction of elastic misfit strains [34]. Dislocation density coupled phase-field simulations of creep show that plasticity in γ channels dominates the rafting process in Ni-based superalloys [35, 36]. Cottura et al. [23] coupled strain gradient crystal plasticity with phase-field method to simulate rafting in Ni-based superalloys by incorporating dislocation interactions and GND led back stresses. N-type, P-type and mix mode, i.e. 45◦ rafting in V-notched Ni-based superalloy specimens are found to be governed by the state of stress in the microstructure [37]. Specific specimen geometries can favor the production of high creep strains locally, resulting into formation of rafts at 45◦ [31]. The resulting microstructure of mix mode rafting is caused by the local creep strain of more than 10% and independent of the state of stress whether it is compressive or tensile. Touratier et al. [31] showed through scanning electron micrographs (SEM) that one region of the test specimen is under tension, whereas, other region is under compression. Tension produces Ntype rafting, whereas, compression zone produces P-type rafting and a shift of rafting toward 45◦ is observed with a mixed stress state. Some experimental findings show that mixed mode rafting is independent of the type of stress, whether it is tensile or compressive and this type of rafting is also observed under pure shear stress state [38]. Feller-Kniepmeier and Link [4] have investigated creep and found mixed mode rafting microstructure near the rupture zone and its absence at the regions away from the rupture zone. Epishin et al. [39] found that Normal N-type rafting is produced due to lower creep strain whereas mixed mode of rafting results from higher creep strain. le Graverend et al. [18] performed creep test on bi-notched sample and showed the orientation of rafts with respect to loading axis. The microstructural zone with highly non-uniform stress field showed large angle of alignment of rafts whereas it decreased while traversing away from this zone. The maximum angle between initial configuration of precipitates and final morphological alignment of rafts is around 18o which gradually reduces to zero at a distance of 3 mm from this zone in the microstructure. Moreover, it is found that traversing from the notch 4

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towards the center of the sample, this angle increases, which corresponds to a non-uniformity of the stress field. Epishin et al. [40] analyzed the SEM micrographs of ruptured specimen and found waviness in the raft morphology. Shi et al. [41] performed the creep test on Ni-based superalloys to study the rafting morphology near the rupture zone. They observed that the rafting direction tends to align along the axis of applied stress. The angle of rafting direction with the axis of applied stress is also measured at a region away from the rupture zone. It is reported that 45◦ rafts are found near the rupture zone and this angle decreases while traversing into the material away from the rupture zone. Moreover, he also observed most of the dislocations in the γ phase and along the γ/γ 0 interface whereas only few dislocations are found in γ 0 precipitates. Similar results are found by Guo et al. [42] and Nathal [43], where 45◦ observed near the rupture zone, whereas, Tian et al. [44] and Zhao et al. [45] found the waviness of rafts near the rupture zone. Yang et al. [46] simulated creep in Ni-based superalloys using phase-field method under tensile and shear loads and found 45◦ rafts only in the case of shear load. Other works [47, 38] showed that shear strains can also lead to an accumulation of high creep strain to produce rafts along 45◦ . There are only a handful of experimental studies which focus on mixed mode rafting and no theoretical studies to the best of our knowledge. Until now it is not clear whether the mixed mode of rafting is due to highly localized shear bands or because of mixed stress state. Therefore, in this work, we have addressed this problem by coupling non-local crystal plasticity with phase-field method. Doing so, we have successfully captured the evolution of microstructure under high stresses leading to production of localized shear bands. The structure of the paper is as follows: An introduction to the phasefield model and non-local crystal plasticity model is given in the following section. Then, the simulation setup is described along with simulation input parameters. Next, the results of rafting simulation are presented and discussed including normal N-type and mixed mode rafting under high tensile and compressive stresses.

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Methods

In this study a multi-phase field model coupled with continuum dislocation density based strain gradient crystal plasticity model developed by [48] has been employed. It serves the purpose to understand the temporal evolu5

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tion of superalloy microstructure. A detailed investigation is carried out to describe dislocation density evolution leading to a specific mode of rafting in Ni-based superalloys at high temperatures and low stresses.

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The phase-field model The multi-phase field (MPF) model employed in this work is based on [49]. It is an efficient method for simulating microstructure evolution with single/multi-phase and/or multi-component systems. It lies in its flexible nature enabling one to study the multi-physics problems ranging from thermodynamics to mechanics, electrics, magnetism and their coupled effects. The model is based on total free energy functional Z F = fch + fel + fint , (2)

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where fch , fel and fint are chemical, elastic and interfacial free energy densities of the system, respectively. The interfacial energy density in MPF model reads  2  N X N X η 8σαβ − 2 ∇φα · ∇φβ + φα φβ , = η π α=1 β>α

(3)

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where σαβ is the interfacial energy of α/β interfaces and individual phases are represented by phase fields indicators i.e. φα and φβ , for α and β phases respectively. η is the thickness of the diffuse interface. The chemical free energy density has the form " # N N X X fch = φα fα (cα ) + µ c − (φα cα ) , (4) α=1

α=1

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where c represents the spatial concentration of the system, cα is the spatial concentration of the α phase and fα (cα ) is the bulk free energy density of an arbitrary phase α. The evolution of chemical concentration field depends on the alloy composition (cα ) is given in [49]. µ is a Lagrange multiplier and has the form of generalized chemical potential ensuring local mass conservation. Finally, the elastic energy density reads N

 ijkl  kl  1 X  ij (p)ij (p)kl φα α − ∗ij Cα α − ∗kl . fel = α − α α − α 2 α=1 6

(5)

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1 φ˙ = − N

N X

µαβ

α=16=β



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The elastic energy density is formulated as a function of the total strains α , eigenstrains ∗α resulting from the lattice misfit between the γ and γ 0 phases defined in equation (1) and plastic strains pα of phase α. The resulting phase-field evolution equation has the following general form  δF δF − , δφα δφβ

(6)

N 1 X ˙ φα = µαβ N β=16=α

"

N X

(σβγ

γ=16=β

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where N represents the total number of active phase fields and µαβ is the interface mobility. Substituting the free energy functional, Eq. 2, into Eq. 6 we get the following form of the phase-field evolution equation #   π2 πp 2 ch el − σαγ ) ∇ φγ + 2 φγ + φα φβ (∆G + ∆G ) , η η

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(7) where ∆G and ∆G represent the chemical and elastic driving forces given below for two arbitrary phases α and β, ch

∆Gch αβ = −

∂ ∂ − )fch , ∂φα ∂φβ

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and ∆Gel αβ = −

∂ ∂ − )fel . ∂φα ∂φβ

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The continuum dislocation density based plasticity model To capture the plastic behavior of the system, a physically based nonlocal strain gradient crystal plasticity model [50] is implemented. Plastic deformation takes place only in γ phase under the conditions applied in this work, therefore the plasticity model is only active in the γ phase whereas γ 0 phase behaves elastically. Internal length scale parameter, e.g. distance between two nearest dislocations in the system, acts as the state parameter in this model. It governs the flow and hardening of the material based on the evolution of statistically stored dislocations (SSD) resulting from the random motion and entanglement of dislocations on different slip systems. Whereas geometrically necessary dislocation (GND) density evolves from the spatial 7

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γ˙s = ρs bν s ,

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gradients of plastic strains and describes the heterogeneous deformation due to lattice compatibility constraints. In such a model, Orowan law [51] is the underlying flow rule which describes dislocation slip rate in a crystal undergoing plastic deformation. This slip is controlled by the interactions among dislocations lying on different slip systems. The dislocation slip rate on a given slip system s reads (10)

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where ν is the dislocation velocity and b is the length of the Burgers vector. Dislocation interaction, due to slip on different slip systems, leads to the multiplication or annihilation of dislocations. The dislocation velocity ν [50], evolving on slip system s is given by τs 1 )n , (11) τcs where ν0 is the initial or reference dislocation velocity and n is a measure of the strain rate sensitivity of material. The quantitative measure of the slip resistance is given by the Taylor’s hardening law, which is used to describe hardening produced due to plastic deformation:

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ν s = ν0 (

√ τcs = τ0 + aGb ρs ,

τcs

(12)

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where is the critical resolved shear stress on slip system s, τ0 is the lattice friction stress [52], a is a geometrical factor [53], G is the shear modulus of the material and ρs = ρsSSD + ρsGN D

(13)

ρ is the total dislocation density defined as a sum [54] of statistically stored dislocation density ρsSSD within the bulk of phases and geometrically necessary dislocation density ρsGN D mainly at the interfacial regions. In turn, the evolution of ρsSSD is calculated by a modified version of Kocks-Mecking law [55] where GND density is also taken into consideration:

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ρ˙ sSSD = (k1

p s ρSSD + ρsGN D − k2 ρsSSD )γ˙s ,

(14)

where k1 describes the dislocation generation due to interactions of dislocations on the same or on different slip systems and k2 controls dislocation annihilation. 8

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The term ρsGN D in equation 14 depends on Nye’s tensor which is basically the curl of the plastic part of deformation gradient or gradient of the plastic strain in a way as proposed in [50]. In other words it is a measure of the lattice curvature produced during deformation to geometrically accommodate the effect of non-homogeneous deformation. The evolution of Nye’s tensor reads ˙ = (−ejkl ˙p )T ei ⊗ej , Λ il,k

(15)

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˙ is the Nye’s dislocation density tensor [56], ejkl represents a third where Λ order permutation tensor with component value 1 when indices are permuted in even order, 1 with odd order of permutation and otherwise it is equal to 0, ˙pil,k is the spatial gradient of the plastic strain rate and it involves the internal length scale of material. GND density is composed of edge ρsGN D(e) and screw ρsGN D(s) parts [23, 50, 57, 58, 59, 60] as given below ρ˙ sGN D = ρsGN D(e) + ρsGN D(s) ,

(16)

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which is illustrated schematically in Fig. 2.These components of GND density are computed by above mentioned Nye’s tensor as follows

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1 ˙ s | + |ds Λd ˙ s |), (17) ρ˙ sGN D = (|ds Λl b where ds is the slip direction vector, ls is the dislocation line vector on the slip plane s. The evolution of SSD and GND densities have been integrated using explicit scheme at the end of each increment.

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Figure 2: Schematic illustration of mixed geometrically necessary dislocation on slip system s [59].

We have coupled phase-field model with crystal plasticity in a similar way as done by [23, 46, 61, 62]. We employed our model to study the special type 9

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of rafting; therefore, some of the physical aspects have not been considered. Firstly, we have not considered the back stresses [23, 61, 63] induced by dislocation pile-ups along the interfaces. Such pile-ups account for short range interactions among dislocations and mimic Orowan hardening of the γ channels. They also reveal the impact of varying size of γ 0 precipitates on plasticity. Secondly, eigenstrains associated with dislocations [35, 36, 64] have been neglected for the sake of simplification which enabled us to analyze the accumulation of high local plastic strain. Eigenstrains play an important role during the creep and may produce interfacial ledges in highly strained superalloys as observed in experiments Shi et al. [41] and show long range interactions among dislocations. Lastly, since we have limited our analysis to the secondary creep stage whereas damage [46, 65] and shearing of the γ 0 precipitates normally occur during the tertiary stage of creep, therefore damage analysis is also not included. Simulation setup

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The simulations are performed on a 3D representative volume element (RVE) of 3.84 µm x 3.84 µm x 3.84 µm size and discretized using regular cubic grid with 30 nm grid point spacing. The initial 3D microstructure is obtained by nucleating the γ 0 precipitates quasi-randomly in the saturated γ matrix and allowed them to grow until chemical equilibrium is reached as shown in Fig. 3(a). Thus the initial microstructure consists of about 500 cuboidal inter-metallic γ 0 precipitates with a stoichiometric composition of N i3 Al in a solid solution matrix of γ phase. The precipitates are located at nearly equidistant positions in three dimensional space separated by the γ phase channels. The average diameter of initial γ 0 precipitates is 450 nm. Khachaturyan’s [66] elasticity homogenization scheme is applied to deal with the elastic properties within interfacial regions. Periodic boundary conditions are imposed on all of the physical quantities. A constant tensile stress of 350 MPa is applied to the system along [001] cubic axis. Since the microstructure is heterogeneous and it is very difficult to measure the exact dislocation density content of the material even with the most sophisticated experimental techniques, we have assigned constant initial dislocation density per slip system to the channels while assuming the γ 0 precipitates dislocation free. Coalescence of γ 0 precipitates is restricted in the absence of external stresses, due to the elastic stabilization force provided by the lattice misfit between the γ and the γ 0 phases [67, 68]. On the other hand, during the 10

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phase-field simulations, an artificially wide diffuse interface allows γ 0 precipitates to merge by overlapping of their phase-field regions, if the width of the γ channels becomes narrower than the typical interface width used in our phase-field simulations. To restrict the merging of γ 0 precipitates due to the numerical effects, the wetting condition is imposed by increasing the interface energy of σγ 0 −γ 0 interfaces by a factor of three as compared to the σγ 0 −γ interface energies [69]. It is known for Ni-based superalloys that the dislocations form networks around the γ 0 precipitates during the deformation process at sufficient stress levels. Such networks decrease the negative lattice misfit and interface coherency between the γ and γ 0 phases and thus enables the coalescence of the γ 0 precipitates [68]. These phenomena are taken into account numerically by altering the artificial wetting condition which is used to control the coalescence of γ 0 precipitates in this study. This constraint is removed to allow the coalescence of γ 0 precipitates when their phase fields overlap and the total dislocation density ρ in the γ channels reaches a critical value allowing the formation of the dislocation networks at the interfaces. In addition, coalescence of neighboring γ 0 precipitates is enabled only if they belong to the same anti-phase domain as explained below. γ 0 precipitates form by an atomic ordering on the FCC lattice with L12 crystal structure, thus an anti-phase boundary (APB) is generated by neighbouring precipitates due to random phase shift along the three different h110i directions between the formed ordered structures. This leads to the formation of four different L12 variants which form APBs. This prevents the coalescence of γ 0 precipitates due to high energy associated with the APB formation in Ni-based superalloys [70, 71]. In this study the formation of anti-phase domains is accounted for by assigning an index corresponding to every γ 0 precipitate variant generated randomly and using it to prevent the coalescence of anti-phase related domains during the simulations.

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Value 1x1010 2x103 1x1011 1x10−13 215.5 162 77.6 222.7 164.2 85.6 -0.003 10 0.2 1283 30 1.83x10−17 4dx 0.05 0.15 2.65x10−17 950

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Symbol k1 k2 ρ νo C11 C12 C44 C11 C12 C44 ε∗11 = ε∗22 = ε∗33 τo n Ω dx D η σγ−γ 0 σγ 0 −γ 0 µ T

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Parameters Dislocation production parameter Dislocation annihilation parameter Total initial dislocation density in γ Initial dislocation velocity Anisotropic elasticity constant γ [72] Anisotropic elasticity constant γ Anisotropic elasticity constant γ Anisotropic elasticity constant γ 0 Anisotropic elasticity constant γ 0 Anisotropic elasticity constant γ 0 Eigenstrain γ 0 [6] Lattice friction stress γ Strain rate sensitivity exponent [73] System size Space discretization Diffusivity constant Interface width Interface energy[69, 68] Interface energy Interface mobility System temperature

Unit m−2 ms−1 GPa GPa GPa GPa GPa GPa MPa grid points nm m2 s−1 grid points Jm−2 Jm−2 m4 J−1 s−1 o C

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Table 1: Simulation parameters used in this study

Results and Discussion

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Starting from the initial microstructure as described above, the external stress of 350 MPa is applied along [001] direction at a temperature of 950o C. The external applied stress disturbs the stress distribution in the γ channels, which facilitates the directional coarsening of γ 0 precipitates by minimizing the mechanical free energy of the system.

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Figure 3: Simulation results. (a) Initial microstructure where external stress is applied and final microstructure after 1.0 % of creep strain.

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Plastic activity in the channels normal to the loading direction i.e. [100] and [010], experience higher stresses, which further enhances the kinetics of the microstructure evolution. The neighbouring precipitates along the coarsening direction eventually touch each other and coalescence takes place on the basis of the merging criteria described above, leading to the formation of N-type rafts. After merging, bigger precipitates start to coarsen at the expense of smaller precipitates to minimize the interfacial energy of the system by creating a completely rafted microstructure shown in Fig. 3(b). The 13

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resulting average creep strain in the system is around 1%.

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Figure 4: 2D cross sections of 3D simulation results showing the evolution of microstructure during N-type rafting under tensile stress with increasing creep strain from 0% to 1%.

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Fig. 4 shows 2D cross sections of 3D simulation domain at different plastic strains up to secondary creep regime i.e. 1%. Kinetics of microstructural evolution is comparatively fast in the primary creep regime i.e. up to 0.4% creep strain where precipitate merging takes place and gradually slows down in the secondary creep regime. It enforces precipitates to align themselves along [010] direction whereas channels get wider along [001] direction. Narrowing channels along [010] crystallographic direction help merging of γ 0 precipitates during primary creep regime. From 0.4 % to 1.0 % plastic strain, i.e. secondary creep regime, only directional coarsening of microstructure is observed but at a very low rate. Hence these results are qualitatively consistent with the findings presented in [23, 74]. 14

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Figure 5: Simulation results. (a) GND density and (b) local creep strain distribution for 1.0 % average creep strain.

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Fig. 5 shows the distribution of local creep strain and geometrically necessary dislocation (GND) density at 1.0% average creep strain. During creep, the creep strain evolves in the horizontal channels and accommodates the high stresses present in these channels. The change in local stresses lowers the creep strain increment and hence, decreases the creep rate. Moreover, the internal stresses are relaxed not only by the plastic activity but also by the microstructural evolution of the γ 0 phase. Since the local creep strain is relatively low, only a few slip bands can be observed in the γ channels. The local GND density evolves by the spatial gradient of the creep strain increment and surrounds the γ 0 precipitates. Since the applied stress is lower than the yield strength of the γ 0 phase, it does not allow the dislocations to enter into γ 0 phase. To account for this phenomenon, the model has been modified such that the growing γ 0 precipitates push away the dislocations along their slip systems into the γ phase. The mechanism of pushing dislocations is outlined in the equation 18, where m corresponds to the current state of the material and m + 1 represents the new state. The x describes current spatial position and x + 1 corresponds to the neighbouring point along a given slip direction. ρ represents the total dislocation density which includes ρGN D and ρSSD density, 15

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ρxm+1 = 0,

m ρm x ∗ φx . φm+1 x+1

(18)

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m+1 ρx+1 = ρm x+1 +

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and φ is phase fraction of the corresponding phase. Dislocations move away from the interface in the slip direction only if the local grid point corresponds to γ phase and in next time step, the grid point will become a part of the γ 0 phase as demonstrated in Fig. 6.

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Figure 6: Schematic illustration of the mechanism used to push the dislocations away from γ 0 -phase. The ργ is the dislocation density which belongs to γ phase and is overgrown by γ 0 phase, so ργ moves outward into the γ phase along its slip direction.

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Simulated and experimental creep curves are displayed in Fig. 7(a). The simulated creep curve is fitted against the experimental data to get the right set of parameters for precise creep simulations. Trial and error method is used to determine appropriate plasticity parameters. During this process, initial dislocation density ρ, strain rate sensitivity parameter n and lattice friction stress τ0 are kept constant whereas dislocation storage parameter k1 , dislocation annihilation parameter k2 and initial dislocation velocity are varied to determine the simulated creep curve by matching it to the experimental data. The creep curve is obtained by taking the creep strain averaged over the entire simulation domain. In this study, we only focus on the primary and secondary creep i.e. up to 1% global creep strain, while the tertiary creep regime can be investigated by incorporating an appropriate damage model. The evolution of the averaged total dislocation density, which is the sum of the statistically stored dislocation density and the geometrically necessary 16

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dislocation density, along with the evolution of GND densities are displayed in Fig. 7(b). The averaged dislocation density is calculated by taking the average of the dislocation densities on all slip systems over the entire simulation domain. Dislocation density is increased with a factor of 10 up to 1% creep strain, which is consistent with the findings of Cottura et al. [23]. Fig. 7 also shows that high level of stress in the simulation of 45◦ rafting results in high local creep strain during secondary creep regime and thus storage of high amount of average dislocation density in the system.

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Figure 7: Experimental and simulated creep curves. (a) Simulated creep curve is fitted against experimental data to get the appropriate model parameters and creep curve for 45◦ rafted microstructure. (b) Evolution of total global dislocation density and geometrically necessary dislocation density with N-type and 45◦ rafting.

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In the next step, we calibrated our model for low stress (N-type rafting) condition and then investigated high stress (700 MPa) creep at same temperature of 950 o C. Tensile and compressive stresses of 700 MPa are applied along [001] direction. Under low applied stress, tensile stress produces N-type rafting and compressive stress produces P-type rafting. But under higher stress condition, the rafting direction is changed toward 45◦ . High external stress produces high local creep strains in the simulation. The formation of shear bands due to dislocation glide in γ channels along the most favorable slip systems is shown in Fig. 13(b). As the local creep strain increases in the γ channels, it restricts the normal coarsening of the γ 0 particles and shifts the coarsening of channels towards 45◦ . In our simulations mix mode rafting is observed when high creep strain is accumulated locally (> 10%), as shown 17

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in Fig. 13(b) and Fig. 14(b). These results are consistent with experimental and simulations results reported in [31]. In the former case higher heterogeneity of strain results in finer rafts as shown in Fig. 8 and in the latter case, the lower heterogeneity of strain leads to coarser rafts as shown in Fig. 9.

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Figure 8: Mixed type of rafting in Ni-based superalloys: (a) γ 0 precipitates and (b) γ channels under tensile stress.

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Figure 9: Mixed type of rafting in Ni-based superalloys: (a) γ 0 precipitates and (b) γ channels under compressive stress.

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The cross sectional views of the cubic simulation domain are shown in Fig. 10, which show the angles between the orientation of γ channels and (001) plane. It can be seen that γ channels are tilted towards [001] direction, which is the axis of applied stress. The observed angles range from 19◦ to 37◦ in our simulation results, which indicate clearly a tendency of rafting towards 45◦ . At low level of local creep strain (< 10%), rafting microstructure shows N-type character and rafts are orthogonal to the applied stress. But as soon as the localized creep strain increases beyond 10%, the orientation of the rafts starts to change and the resulting local shear bands in the γ matrix force the γ 0 precipitates to align themselves at a certain angle to (001) plane. Similar observations can be made in Fig.11 but with the coarser rafts and wider γ channels.

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Figure 10: (100) and (010) side views of the rafted microstructure. The orientation of the channels is measured with respect to the direction which is normal to the applied loading direction. Tensile stress is applied along < 001 > direction.

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Figure 11: (100) and (010) side views of the rafted microstructure. The orientation of the channels is measured with respect to the (001) plane which is normal to the loading direction. Compressive stress is applied along < 001 > direction.

Fig. 10 indicates that in our simulations of mixed mode rafting, rafts do not grow along 45◦ orientation as it is observed experimentally, but show 20

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a clear tendency towards 45◦ orientation. The inability to obtain a perfect 45◦ rafting can be explained on the basis of periodic boundary conditions imposed by the use of spectral elasticity solver in our simulations. This reduces the degree of freedom significantly for the evolution of γ 0 precipitates and prevents the formation of continuous 45◦ rafts due to the interaction of <011>- and <101>-type rafting directions across the boundaries of the simulation domain. Fig. 12 shows a two dimensional simulation result at a high external stress of 700 MPa. It can be seen that the reduced dimensions, and thus the complexity of the slip system interaction, result in a more pronounced mix mode rafting with rafts angle approaching 45◦ . At the same time, competition between formation of slip bands, and thus the rafts, along the < 101 > and < ¯101 > directions is clearly visible.

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Figure 12: 2D results of mix mode rafting in Ni-based superalloys. Spatial distribution of creep strain is shown in the γ channels.

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Fig. 13 shows distribution of GND density and creep strain in rafted microstructure under tensile stress. It is evident that GND density is highly localized in specific regions in the γ matrix due to higher gradients of local creep strain. It is also observable that local shear bands have creep strains of more than 10%. It can be further deduced that the high plastic activity due to the mobile dislocations facilitates the generation of the creep strain p > 10% and renders the microstructure less stable due to 45◦ rafting. Such a rafted microstructure facilitates the movement of the dislocations along <011> direction and makes the material prone to rupture. This phenomenon 21

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is self accelerating since high creep strain produces shear bands which further enforce the tendency towards 45◦ rafting. Whereas the mode of rafting under compressive stress changes as shown in Fig. 14. It can be observed that in this case concentration of dislocations and plastic strain is not as strong as it is observed under tensile stress. It leads to merging of precipitates to rafts with coarser morphology as compared to the former case.

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Figure 13: Spatial distribution of (a) geometrically necessary dislocation density and (b) creep strain under tensile applied stress.

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Figure 14: Spatial distribution of (a) geometrically necessary dislocation density and (b) creep strain under compressive applied stress.

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Figure 15: Rafted microstructure of Ni-based superalloy with different initial dislocation densities (a) 0.5x1011 m−2 , (b) 1x1011 m−2 and (c) 5x1011 m−2 under 700 MPa tensile stress.

Rafting of γ 0 precipitates is very sensitive to plastic strain [34, 75, 22] and dislocation density of material stored in γ matrix prior to creep [76]. Fig. 15 shows three microstructures with different initial dislocation densities with a value of (a) 0.5x1011 m−2 , (b) 1x1011 m−2 and (c) 5x1011 m−2 , which 23

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correspond to different pre-straining, subjected to a high tensile stress of 700 MPa. Fig. 15a having lowest initial dislocation density as compared to other specimens produces highest local creep strain. It also shows higher tendency towards 45 degree rafting compared to others. The increased initial dislocation content in γ matrix results into rafting with a tendency towards N- type rafts. It can also be observed from Fig. 16, where (a) creep curves and (b) evolution of dislocation density with different initial dislocation density are shown. Higher dislocation density restricts plastic activity by increasing the hardness of the sample [48, 77].

Conclusion

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Figure 16: (a) Creep curves and (b) homogenised global dislocation density resulting from microstructures with different initial dislocation density 0.5x1011 m−2 , 1x1011 m−2 and 5x1011 m−2 under high tensile stress of 700 MPa.

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We studied the microstructure evolution in single crystal Ni-based superalloys by combining a dislocation density based strain gradient crystal plasticity model with the multi-phase field method. To illustrate its capabilities, it is applied to study the formation of 45◦ rafts in Ni-based superalloy subjected to high temperature and low stress condition. The following conclusions can be drawn from this study:

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• Mix mode rafting in Ni-based superalloys can be reliably reproduced by using dislocation density based crystal plasticity coupled phase-field model.

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• Simulations with up to 1% average creep strain reveal that even at such a low strain, the local creep strain can be significantly higher and reach values up to 4%, which is also manifested by the highly localized distribution of the GND density. • High local stresses either tensile or compressive, leading to highly localized creep strains of more than 10%, result in appearance of 45◦ rafts of γ 0 precipitates.

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• Highly non-homogeneous deformation results in the production of slip bands in localized regions, which eventually favours formation of 45◦ rafts. It leads to self-amplifying phenomenon where the formation of 45◦ rafts further increases the creep deformation along slip bands.

Acknowledgements

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The above mentioned self amplifying phenomenon can explain the observation of mix mode rafting near the rupture zones in the experiments. It can also indicate that the mix mode rafting can be a precursor effect or even a trigger for the rupture.

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References

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The authors acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG) within the framework of the collaborative research center SFB / TR 103 on single crystal superalloys through project C5. Also the support from the Fundamental Research Program of Korea Institute of Materials Science (PNK6410) is highly acknowledged.

[1] R. C. Reed, The superalloys, foundamentals and applications, volume 53, 2006.

Jo

[2] D. F. Lahrman, R. D. Field, R. Darolia, H. L. Fraser, R. D. Field, U. Champaign, Investigation of Techniques for Measuring Lattice Mismatch in a Rhenium, Acta Metallurgica 36 (1988) 1309–1320. [3] R. R. Keller, H. J. Maier, H. Mughrabi, Characterization of interfacial dislocation networks in a creep-deformed nickel-base superalloy, Scripta Metallurgica et Materiala 28 (1993) 23–28. 25

Journal Pre-proof

ro o f

[4] M. Feller-Kniepmeier, T. Link, Correlation of microstructure and creep stages in the 100 oriented superalloy SRR 99 at 1253 K, Advances in Plasticity 20 (1989) 353–356. [5] B. Fedelish, A microstructural model for the monotonic and the cyclic mechanical behavior of single crystals of superalloys at high temperatures, International Journal of Plasticity 18 (2002) 1–49.

e-p

[6] R. V¨olkl, U. Glatzel, M. Feller-Kniepmeier, Measurement of the lattice misfit in the single crystal nickel based superalloys CMSX-4, SRR99 and SC16 by convergent beam electron diffraction, Acta Materialia 46 (1998) 4395–4404. [7] V. Sass, U. Glatzel, M. Feller-Kniepmeier, Anisotropic creep properties of the nickel-base superalloy CMSX-4, Acta Materialia 44 (1996) 1967– 1977.

Pr

[8] L. M¨ uller, U. Glatzel, M. Feller-Kniepmeier, Calculation of the internal stresses and strains in the microstructure of a single crystal nickel-base superalloy during creep, Acta Metallurgica et Materialia 41 (1993) 3401– 3411.

al

[9] A. Epishin, T. Link, Mechanisms of high-temperature creep of nickelbased superalloys under low applied stresses, Philosophical Magazine 84 (2004) 1979–2000.

urn

[10] A. Kostka, G. M¨alzer, G. Eggeler, A. Dlouhy, S. Reese, T. Mack, L12phase cutting during high temperature and low stress creep of a Recontaining Ni-base single crystal superalloy, Journal of Materials Science 42 (2007) 3951–3957. [11] P. M. Sarosi, Observations of a [010] dislocations during the hightemperature creep of Ni-based superalloy single crystals deformed along the [001] orientation, Acta Materialia 55 (2007) 2509–2518.

Jo

[12] T. Link, A. Epishin, M. Klaus, U. Br¨ uckner, A. Reznicek, [100] Dislocations in nickel-base superalloys: Formation and role in creep deformation, Materials Science and Engineering A 405 (2005) 254–265.

26

Journal Pre-proof

ro o f

[13] T. M. Pollock, A. S. Argon, Creep resistance of CMSX-3 nickel base superalloy single crystals, Acta Metallurgica et Materialia 40 (1992) 1–30. [14] C. Mayr, G. Eggeler, A. Dlouhy, Analysis of dislocation structures after double shear creep deformation of CMSX6-superalloy single crystals at temperatures above 1000 C, Materials Science and Engineering A 207 (1996) 51–63.

e-p

[15] C. Carry, J. L. Strudel, Apparent and effective creep parameters in single crystals of a nickel base superalloy-I Incubation period, Acta Metallurgica 25 (1977) 767–777. [16] H. J. Chang, M. C. Fivel, J.-l. Strudel, Micromechanics of primary creep in Ni base superalloys, International Journal of Plasticity 108 (2018) 21–39.

Pr

[17] Y. K. Kim, D. Kim, H. K. Kim, E. Y. Yoon, Y. Lee, C. S. Oh, B. J. Lee, A numerical model to predict mechanical properties of Ni-base disk superalloys, International Journal of Plasticity 110 (2018) 123–144.

al

[18] J.-B. le Graverend, J. Cormier, F. Gallerneau, P. Villechaise, S. K. J. Mendez, A microstructure-sensitive constitutive modeling of the inelastic behavior of single crystal nickel-based superalloys at very high temperature, International Journal of Plasticity 59 (2014) 55–83.

urn

[19] Y.-K. Kim, D. Kim, H.-K. Kim, C.-S. Oh, B.-J. Lee, An intermediate temperature creep model for Ni-based superalloys, International Journal of Plasticity 79 (2016) 153–175. [20] D. Barba, E. Alabort, D. Garcia-Gonzalez, J. Moverare, R. Reed, A. Jrusalem, A thermodynamically consistent constitutive model for diffusion-assisted plasticity in Ni-based superalloys, International Journal of Plasticity 105 (2018) 74–98.

Jo

[21] Y. Zhao, M. J. Mills, Y. Wang, S. R. Niezgoda, A homogenized primary creep model of nickel-base superalloys and its application to determining micro-mechanistic characteristics, International Journal of Plasticity 110 (2018) 202–219.

27

Journal Pre-proof

ro o f

[22] C. Wang, M. Adil Ali, S. Gao, J. V. G¨orler, I. Steinbach, Combined phase-field crystal plasticity simulation of P- and N-type rafting in Cobased superalloys, Acta Materialia 175 (2019) 21–34. [23] M. Cottura, B. Appolaire, A. Finel, Y. Le Bouar, Coupling the Phase Field Method for diffusive transformations with dislocation densitybased crystal plasticity: Application to Ni-based superalloys, Journal of the Mechanics and Physics of Solids 94 (2016) 473–489.

e-p

[24] B. Ruttert, D. B¨ urger, L. M. Roncery, A. B. Parsa, P. Wollgramm, G. Eggeler, W. Theisen, Rejuvenation of creep resistance of a Ni-base single-crystal superalloy by hot isostatic pressing, Materials and Design 134 (2017) 418–425. [25] P. Caron, T. Khan, Improvement of creep strength in a nickel-based single-crystal superalloy by heat treatment, Materials Science and Engineering 61 (1983) 173–184.

Pr

[26] P. Zhao, Y. Wang, S. R. Niezgoda, Microstructural and micromechanical evolution during dynamic recrystallization, International Journal of Plasticity 100 (2018) 52–68.

al

[27] F. R. N. Nabarro, Rafting in superalloys, Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science 27 (1996) 513–530.

urn

[28] J. X. Zhang, T. Murakumo, Y. Koizumi, T. Kobayashi, H. Harada, Slip geometry of dislocations related to cutting of the gamma prime phase in a new generation single-crystal superalloy, Acta Materialia 51 (2003) 5073–5081. [29] A. Royer, A. Jacques, P. Bastie, M. Ve, The evolution of the lattice parameter mismatch of nickel based superalloy : an in situ study during creep deformation, Materials Science and Engineering A 321 (2001) 800–804.

Jo

[30] M. V´eron, Y. Brechet, F. Louchet, Strain induced directional coarsening in Ni based superalloys, Scripta Materialia 34 (1996) 1883–1886.

28

Journal Pre-proof

ro o f

[31] F. Touratier, E. Andrieu, D. Poquillon, B. Viguier, Rafting microstructure during creep of the MC2 nickel-based superalloy at very high temperature, Materials Science and Engineering A 510-511 (2009) 244–249. [32] M. Nathal, L. Ebert, Gamma prime shape changes during creep of a Nickel-based superalloy, Scripta Metallurgica 17 (1983) 1151–1154. [33] T. Gabb, S. Draper, D. Hull, R. Mackay, M. Nathal, The role of interfacial dislocation networks in high temperature creep of superalloys, Materials Science and Engineering: A 118 (1989) 59–69.

e-p

[34] M. Adil Ali, J. V. G¨orler, I. Steinbach, Role of coherency loss on rafting behavior of Ni-based superalloys, Computational Materials Science 171 (2020) 109279.

Pr

[35] N. Zhou, C. Shen, M. J. Mills, Y. Wang, Phase field modeling of channel dislocation activity and γ’ rafting in single crystal Ni-Al, Acta Materialia 55 (2007) 5369–5381. [36] N. Zhou, C. Shen, M. J. Mills, Y. Wang, Contributions from elastic inhomogeneity and from plasticity to γ 0 rafting in single-crystal Ni-Al, Acta Materialia 56 (2008) 6156–6173.

al

[37] V. Caccuri, J. Cormier, R. Desmorat, γ 0 -Rafting mechanisms under complex mechanical stress state in Ni-based single crystalline superalloys, Materials & Design 131 (2017) 487–497. [38] M. Kamaraj, Rafting in single crystal nickel-base superalloys - An overview, Sadhana 28 (2003) 115–128.

urn

[39] A. Epishin, T. Link, P. D. Portella, U. Bru, Evolution of the γ/γ 0 microstructure during high-temperature creep of a Ni-base superalloy, Acta Materialia 48 (2000) 4169–4177.

Jo

[40] A. Epishin, T. Link, U. Br¨ uckner, P. D. Portella, Kinetics of the topolog0 ical inversion of the γ/γ -microstructure during creep of a nickel-based superalloy, Acta Materialia 49 (2001) 4017–4023. [41] H. H. Shi, J. S. Hou, J. T. Guo, L. Z. Zhou, M. Maldini, G. Angella, R. Donnini, D. Ripamonti, Microstructure evolution and its influence on deformation mechanisms during tensile creep of DD417G alloy, Materials Research Innovations 18 (2014) 319–323. 29

Journal Pre-proof

ro o f

[42] X. Guo, H. Fu, J. Sun, K. Kusabiraki, Deformed microstructure of the single-crystal superalloy NASAIR 100 at 1050◦ C, Metallurgical and Materials Transactions A 30 (1999) 2843–2852. [43] L. J. Nathal, M. V.and Ebert, The influence of cobalt, tantalum, and tungsten on the microstructure of single crystal nickel-base superalloys, Metallurgical Transactions A 16 (1985) 1849–1862.

e-p

[44] S. Tian, Y. Su, B. Qian, X. Yu, F. Liang, A. Li, Creep behavior of a single crystal nickel-based superalloy containing 4.2% Re, Materials Design 37 (2012) 236–242. [45] G. Zhao, S. Tian, S. Zhang, N. Tian, L. Liu, Deformation and damage features of a Re/Ru-containing single crystal nickel base superalloy during creep at elevated temperature, Progress in Natural Science: Materials International 29 (2019) 210–216.

Pr

[46] M. Yang, J. Zhang, H. Wei, Y. Zhao, W. Gui, H. Su, T. Jin, L. Liu, Study of γ 0 rafting under different stress states a phase-field simulation considering viscoplasticity, Journal of Alloys and Compounds 769 (2018) 453–462.

al

[47] M. Kamaraj, C. Mayr, M. Kolbe, G. Eggeler, On the influene of stress state on rafting in the single crystal superalloy CMSX-6 under conditions of high temperature and low stress creep, Scripta Materialia 38 (1998) 589–594.

urn

[48] W. Amin, M. Adil Ali, N. Vajragupta, H. A., Studying grain boundary strengthening by dislocation-based strain gradient crystal plasticity coupled with a multi-phase-field model, Materials 12 (2019) 2977. [49] I. Steinbach, M. Apel, Multi phase field model for solid state transformation with elastic strain, Physica D 217 (2006) 153–160.

Jo

[50] P. Engels, A. Ma, A. Hartmaier, Continuum simulation of the evolution of dislocation densities during nanoindentation, International Journal of Plasticity 38 (2012) 159–169. [51] E. Orowan, Problems of plastic gliding, Proceedings of the Physical Society 52 (1940) 8–22. 30

Journal Pre-proof

[52] D. Kuhlmann-Wilsdorf, Frictional stress acting on a moving dislocation in an otherwise perfect crystal, Physical Review 120 (1960) 773–781.

ro o f

[53] M. F. Ashby, The deformation of plastically non-homogeneous materials, The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics 21 (1970) 399–424. [54] N. A. Fleck, G. M. Muller, M. F. Ashby, J. W. Hutchinson, Strain gradient plasticity: Theory and experiment, Acta Metallurgica et Materialia 42 (1994) 475–487.

e-p

[55] U. K. H. Mecking, Kinetics of flow and strain-hardening, Acta Metallurgica, 29 (1981) 1865–1875. [56] J. F. Nye, Some geometrical relations in dislocated crystals, Acta Metallurgica 1 (1953) 153–162.

Pr

[57] R. Kametani, K. Kodera, D. Okumura, N. Ohno, Implicit iterative finite element scheme for strain gradient crystal plasticity model based on self energy of geomatrically necessary dislocations, Computational material science 53 (2012) 53–59.

al

[58] M. E. Gurtin, L. Anand, , P. L. Suvrat, Gradient single-crystal plasticity with free energy dependent on dislocation densities, Journal of the Mechanics and Physics of Solids 55 (2007) 1853–1878.

urn

[59] N. Ohno, D. Okumura, Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations, Journal of the Mechanics and Physics of Solids 55 (2007) 1879–1898. [60] M. Kuroda, V. Tvergaard, Effects of microscopic boundary conditions on plastic deformations of small-sized single crystals, International Journal of Solids and Structures 46 (2009) 4396–4408.

Jo

[61] R. Wu, M. Zaiser, S. Sandfeld, A continuum approach to combined γ/γ 0 evolution and dislocation plasticity in nickel-based superalloys, International Journal of Plasticity 95 (2017) 142–162. [62] Y. Tsukada, Y. Murata, T. Koyama, M. Morinaga, Phase-Field Simulation on the Formation and Collapse Processes of the Rafted Structure in Ni-Based Superalloys, Materials Transactions 49 (2008) 484–488. 31

Journal Pre-proof

ro o f

[63] C. J. Bayley, W. A. M. Brekelmans, M. G. D. Geers, A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity, International Journal of Solids and Structures 43 (2006) 7268– 7286. [64] N. Zhou, C. Shen, P. M. Sarosi, M. J. Mills, T. Pollock, Y. Wang, Y’ rafting in single crystal blade alloys: a simulation study, Materials Science and Technology 25 (2009) 205–212.

e-p

[65] R. Harikrishnan, J. B. le Graverend, A creep-damage phase-field model: Predicting topological inversion in Ni-based single crystal superalloys, Materials and Design 160 (2018) 405–416. [66] A. G. Khachaturyan, Theory of structural transformations in solids, John Wiley and Sons, New York (1983) 25–45.

Pr

[67] J. V. G¨orler, S. Brinckmann, O. Shchyglo, I. Steinbach, Gamma-channel stabilization mechanism in Ni-base superalloys, Philosophical Magazine Letters 95 (2015) 519–525. [68] J. V. G¨orler, I. L´opez-Galilea, L. Mujica Roncery, O. Shchyglo, W. Theisen, I. Steinbach, Topological phase inversion after long-term thermal exposure of nickel-base superalloys: Experiment and phase-field simulation, Acta Materialia 124 (2017) 151–158.

urn

al

[69] M. Kumar Rajendran, O. Shchyglo, I. Steinbach, Large scale 3-D phasefield simulation of coarsening in Ni-base superalloys, MATEC Web of Conferences 14 (2014) 11001. [70] Y. Wang, A. Khachaturyan, Effect of antiphase domains on shape and spatial arrangement of coherent ordered intermetallics., Scripta Metallurgica et Materialia 31 (1994) 1425–1430.

Jo

[71] Y. Wang, D. Banerjee, C. C. Su, A. G. Khachaturyan, Field kinetic model and computer simulation of precipitation of L12 ordered intermettalics from F.C.C. solid solution, Acta Materialia 46 (1998) 2983–3001. [72] J. Kundin, L. Mushongera, T. Goehler, H. Emmerich, Phase-field modeling of the γ-coarsening behavior in Ni-based superalloys, Acta Materialia 60 (2012) 3758–3772. 32

Journal Pre-proof

ro o f

[73] P. Wollgramm, H. Buck, K. Neuking, A. B. Parsa, S. Schuwalow, J. Rogal, R. Drautz, G. Eggeler, On the role of Re in the stress and temperature dependence of creep of Ni-base single crystal superalloys, Materials Science and Engineering A 628 (2015) 382–395. [74] A. Gaubert, Y. Le Bouar, A. Finel, Coupling phase field and viscoplasticity to study rafting in Ni-based superalloys, Philosophical Magazine 90 (2010) 375–404.

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[75] B. Ruttert, O. Horst, I. L´opez-Galilea, D. Langenk¨amper, a. Kostka, C. Somsen, J. V. G¨orler, M. Adil Ali, O. Shchyglo, I. Steinbach, G. Eggeler, W. Theisen, Rejuvenation of Single-Crystal Ni-Base Superalloy Turbine Blades: Unlimited Service Life?, Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science (2018) 1–12.

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[76] H. Long, S. Mao, Y. Liu, H. Wei, Q. Denga, Y. Chen, Z. Zhang, X. Han, Effect of pre-straining treatment on high temperature creep behavior of ni-based single crystal superalloys, Materials Design 167 (2019) 107633.

Jo

urn

al

[77] E. Ghassemali, M.-J. Tan, C. Beng Wah, S. C. V. Lim, A. E. W. Jarfors, Effect of cold-work on the hall-petch breakdown in copper based microcomponents, Mechanics of Materials 80 (2015) 124–135.

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Highlights •

Strain gradient crystal plasticity model is combined with the phase-field framework.

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3D creep simulation up to 1 % averaged creep strain is performed to study the Ntype rafting in single crystal Ni-based superalloys in the high temperature, low stress regime.

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High stresses (> 600 MPa) lead to localized creep strains and a change in the

Localized creep strain higher than 10% tends to tilt the rafting of microstructure

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toward 45° to the direction of applied stress.

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microstructure evolution to so-called mixed mode rafting.

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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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CRedit author statement

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Muhammad Adil Ali: Writing - Original Draft, Software, Data Curation, Visualization, Validation, Methodology, Waseem Amin: Software, Writing Review Editing, Methodology, Oleg Shchyglo: Software, Supervision, Writing - Review Editing, Conceptualization, Ingo Steinbach: Supervision, Writing Review Editing, Conceptualization.

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