Direct measurements of slip irreversibility in a nickel-based superalloy using high resolution digital image correlation

Direct Measurements of Slip Irreversibility in a Nickel-Based Superalloy using High Resolution Digital Image Correlation

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Direct Measurements of Slip Irreversibility in a Nickel-Based Superalloy using High Resolution Digital Image Correlation J.C. Stinville, P.G. Callahan, M.A. Charpagne, M.P. Echlin, V. Valle, T.M. Pollock PII: DOI: Reference:

S1359-6454(19)30838-9 https://doi.org/10.1016/j.actamat.2019.12.009 AM 15705

To appear in:

Acta Materialia

Received date: Revised date: Accepted date:

5 August 2019 3 December 2019 3 December 2019

Please cite this article as: J.C. Stinville, P.G. Callahan, M.A. Charpagne, M.P. Echlin, V. Valle, T.M. Pollock, Direct Measurements of Slip Irreversibility in a Nickel-Based Superalloy using High Resolution Digital Image Correlation, Acta Materialia (2019), doi: https://doi.org/10.1016/j.actamat.2019.12.009

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Direct Measurements of Slip Irreversibility in a Nickel-Based Superalloy using High Resolution Digital Image Correlation J.C. Stinville1,a , P.G. Callahan2,a , M.A. Charpagnea , M.P. Echlina , V. Valleb , T.M. Pollocka b Institut

a University of California Santa Barbara, Santa Barbara, USA PPRIME, Université de Poitiers, CNRS, ENSMA, UPR 3346, 86962 Chasseneuil Cedex, France

Abstract Fatigue crack nucleation in crystalline materials typically develops due to highly localized cyclic slip. During a fatigue cycle, reverse slip differs locally from slip in the forward direction particularly in precipitatecontaining materials such as superalloys. In this paper we report the first direct measurements of irreversibility at the scale of individual slip bands by high-resolution digital image correlation (DIC) in a polycrystalline nickel-based superalloy. Quantitative measurements of the slip irreversibility are challenging for regions of material that have a size that captures the microstructure and its variability. High spatial resolution at the nanometer scale during experimental measurements is needed to observe slip localization during deformation. Moreover, large fields are also needed to obtain the material response over statistically representative populations of microstructural configurations. Recently, high resolution scanning electron microscope (SEM) digital image correlation (DIC) has been extended for quantitative analysis of discontinuities induced by slip events using the Heaviside-DIC method. This novel method provides quantitative measurements of slip localization at the specimen surface. In this paper, the Heaviside-DIC method is used to measure slip irreversibility and plastic strain accumulation in a nickel-based superalloy. The method detects bands with high levels of irreversibility early in cycling that ultimately form fatigue cracks upon further cycling. The local microstructural configurations that induces large amounts of plasticity and slip irreversibility correlated to crack nucleation locations. Keywords: High Resolution Digital Image Correlation, Polycrystalline René 88DT Nickel-Based Superalloy, Scanning Electron Microscopy Digital Image Correlation, Strain Localization and Accumulation, Slip Irreversibility, Low Cycle Fatigue, Heaviside-Digital Image Correlation

1. Introduction Fatigue damage occurs during the mechanical loading of polycrystalline metals and involves grain scale plastic strain localization and subsequent strain accumulation, leading to crack nucleation [1–3]. Measure∗ corresponding

∗∗ Now

author: [email protected] at US Naval Research Lab, Washington, DC, USA

Preprint submitted to Acta Materialia

December 17, 2019

ments of the heterogeneous strain distribution have successfully identified the detrimental microstructural configurations that promote damage [4–11]. However, strain fields are usually measured at the macro or grain scale and therefore prevent the measurement of the cyclic plastic strain localization and resulting damage, which often occurs at the sub-grain scale through the formation of physical slip bands, persistent slip bands (PSBs) or sub-grain scale fatigue shear bands [1–3]. Fatigue crack initiation is enhanced by cyclic slip irreversibility, which results from the plastic strain accumulation caused by the motion of dislocations in the forward part of the fatigue cycle that are not completely recovered in the reverse part of the cycle [1, 12–17]. The density of plastic strain localization sites resulting from slip irreversibility can be significant [18] and originate from dislocation interactions with microstructure, environment, free surfaces or other dislocations [16, 19–21]. Quantification of cyclic slip irreversibility and plastic strain accumulation as a function of the microstructure and the environment is therefore crucial for predicting fatigue crack initiation and propagation in polycrystalline metals [16–19, 22]. The cyclic slip irreversibility obtained from macroscopic strain measurements, defined as the fraction of the local irreversible plastic strain in regard to the total plastic strain, is observed to be an efficient and relevant life-determining fatigue damage parameter [1, 17, 23]. However, this approach does not differentiate the variability of the cyclic irreversibility at the scale of local microstructural features. Nanometer scale spatial resolution is required to experimentally observe slip events during loading [6]. Conversely, large millimeter-scale fields are also required to capture the material response over statistically representative populations of microstructural configurations (grains, macro-texture regions) [15]. Atomic force microscopy (AFM) has been used to measure slip band emergence and persistent slip markings associated with PSBs and consequently, the degree of irreversibility during cyclic loading at local microstructure features [3, 23–26]. However, conventional AFM measurements are usually time consuming and difficult to scale to large fields of view. The use of high-speed atomic force microscopy [27] is necessary to scale investigation to large fields of view. In addition, the registration of each slip event is difficult between the different cycles, which prevents the quantitative measurement of the evolution of slip irreversibility for individual slip bands. Recently, high resolution digital image correlation (HR-DIC) under scanning electron microscopy (SEMDIC) has proven to be an efficient experimental tool for quantification of strain fields at the microstructure scale [10, 15, 28–35]. In the present paper, the HR-DIC technique is used for quantitatively measuring slip irreversibility and plastic strain accumulation over millimeter fields of view. Performing accurate measurements is still complex due to image distortions and electron beam defects associated with scanning electron microscopy (SEM) [7, 36, 37]. However, these phenomena have been well characterized and their effects minimized or corrected [7, 36–40]. Therefore, these DIC developments enable new insights on strain localization produced by slip during plastic deformation of metallic materials [7, 32, 36, 41, 42]. In the past 10 years, increases in spatial resolution of DIC by the use of SEM imaging and speckle patterning techniques has 2

enabled the observation of discrete slip events at the specimen surface - with reference to the microstructure - during monotonic loading [7, 36, 41–43] and cyclic loading [7, 32, 44]. The quantitative measurement of the plastic strain localization remains challenging using conventional DIC methods/code. Conventional DIC, that derives strains from continuous displacement fields is not adapted to describe kinematical discontinuities that exist along discrete slip bands during plastic deformation [45]. It has been observed that the strain measurements and displacement values obtained at slip bands are dependent on DIC parameters, such as step and subset size [7, 45, 46], preventing the quantitative measurement of slip localization. SEM-DIC has been recently been extended for quantitative analysis of discontinuities produced by slip using the discontinuity-tolerant Heaviside-DIC method [45]. The Heaviside-DIC method is part of the new DIC codes with discontinuity implementations that have been applied in fracture mechanics and plasticity [45, 47–52]. These new DIC codes provide identification and characterization of discontinuous displacements that occur at cracks, shear bands, grain boundary sliding or slip events. The use of the discontinuity tolerant Heaviside-DIC method provides physically based full-field measurements of the magnitude of plastic strain localization at slip bands [45]. Furthermore, it provides a measurement of the magnitude of the physical inplane displacement (shearing and sliding) along slip planes [45], opening new opportunities for the systematic measurement of slip irreversibility and plastic strain accumulation. In the present paper, the discontinuity-tolerant DIC code is used to solve the problem of kinematical discontinuities and to measure physically based strains at sites of slip localization. The detection of slip events in the displacement field has allowed for the measurement of the slip irreversibility and plastic strain accumulation over mm-scaled fields of view and will be used to characterize the cyclic deformation of a nickel-based superalloy at room temperature in the low cycle fatigue regime. Emphasis is placed on the first cycle of the fatigue tests to provide information about this cycle on the local mechanical behavior of the material. 2. Experimental Materials 2.1. Material In this study, nickel-based superalloys with a coherent two-phase γ - γ0 fcc structure were studied [53]. Specifically, the powder metal processed alloy René 88DT was investigated having a composition: 13%Co, 16%Cr, 4%Mo, 4%W, 2.1%Al, 3.7%Ti, 0.7%Nb, 0.03%C, 0.015%B (wt%), as detailed previously [54]. There exists 2 populations of γ0 precipitates, tertiary and secondary, which exist at different length-scales and form during material processing. The tertiary precipitates are roughly 10’s of nm in size and the secondary precipitates are 100’s of nm [54]. The powder processing method yields a controlled grain size distribution centered around 26 µm with little appreciable texture [55]. The René 88DT does have a large population 3

of annealing twins which have been characterized extensively using EBSD with both 2D cross-section [56] and 3D tomography [55]. The investigated René 88DT displayed 0.2% yield strength of 1080 MPa and a Young’s modulus of 217 GPa. 2.2. Mechanical Testing Cyclic tests were performed ex-situ at room temperature in air in the low cycle fatigue regime using the conventional symmetric, uniaxial, push-pull mode on an electromechanical machine. Tests were carried out in stress control mode with a R-ratios of -1 and a frequency of 1 Hz at two maximum applied stresses of 1080 MPa and 1140 MPa that correspond to 0.2% and 1.2% plastic deformation (0.7% and 1.8% total deformation) respectively. A Mechanical extensometer with 10 mm gauge was used to measure macroscopic strain during cycling. The evolution of the maximum strain during fatigue is reported in Figure 1(a) for the two loading conditions. Typical stress-strain hysteresis curves for specimens tested at maximum applied stresses of 1080 MPa and 1140 MPa are displayed in Figure 1(b and c) respectively. After the first cycle, an increase in strain amplitude is observed during subsequent cycling followed by a saturation regime. More details on the low cycle fatigue behavior of the investigated alloy are presented elsewhere [6, 8, 57, 58]. 2.3. Sample Preparation and Microscopy HR-DIC measurements were performed in a FEI Versa3D field emission gun system (FIB-SEM) on cylindrical dog-bone fatigue specimens with diameter of 6 mm. The geometry of the dog-bone specimens can be found elsewhere [59]. Two flat areas, 2.5 mm in width and 8 mm in length, were machined on the gauge of selected fatigue specimens. The two flats were positioned on opposite sides of the specimen in the gauge section. The geometry of the flat areas was designed to limit potential stress concentrations by removing the sharp edges produced during machining the flat areas and by careful mechanical polishing. Crack densities and lifetimes were indistinguishable in specimens with and without flat areas [6]. Preparation of the flat surfaces consisted of mechanical polishing with SiC papers up to 1200 grit, followed by polishing with a 6 µm diamond suspension and then chemo-mechanically polishing with 0.05 µm colloidal silica for 12 hours. The speckle pattern used for HR-DIC was formed by preferentially etching the intrinsic microstructural features of the René 88DT alloy by the previously mentioned chemo-mechanically polishing and using the procedure described in [7]. Prior to deformation, reference scanning electron microscopy images and Electron Backscatter Diffraction (EBSD) maps were acquired. EBSD measurements were performed with an EDAX OIM-Hikari XM4 EBSD detector using a step size of 0.8 µm. Diffraction patterns were acquired using an accelerating voltage of 20 kV, a 4 × 4 binning and a beam current of 0.2 nA. The chemical-mechanical polishing procedure reveals the microstructure, including the grain structure and twin boundary locations,

allowing sub-pixel spatial registration between the SEM imaging steps, the HR-DIC maps and the EBSD maps. Data registration is done using pairs of control points as described in ([60]). 4

2.4. Scanning Electron Microscopy Imaging Conditions and HR-DIC HR-DIC measurements were performed after interruption during the first cycle after tensile and compressive load for both loading conditions and after 400 and 5000 cycles for the specimen tested at a maximum applied stress of 1140 MPa and 2000 and 4000 cycles for the specimen tested at a maximum applied stress of 1080 MPa. SEM images for conventional surface investigation were also performed after 5000 cycles (about 95% of the fatigue life) for the specimen tested at a maximum applied stress of 1140 MPa and after 4000 cycles (about 45% of fatigue life) for the specimen tested at a maximum applied stress of 1080 MPa. These later number of cycles where selected to observe the damage induced by cycling that occurs at the surface of the specimens. SEM image sets for HR-DIC were acquired in the unloaded state during interruptions in the cycling loading steps for the polycrystalline René 88DT nickel-based superalloy. In order to minimize the distortion errors associated with SEM imaging [29, 37], SEM parameters were chosen following the recommendations of Kammers and Daly [29, 37], Stinville et al. [7] and Mello et al.[36]. A customized National InstrumentT M NI-DAQ scan controller was deployed to control beam scanning in the FEI microscope in order to minimize SEM beam artifacts [7, 39]. High magnification images were taken at horizontal field widths (HFWs) of 138 µm. Regions of 1 mm ×

1 mm were investigated during cyclic loading of the polycrystalline René 88DT nickel-based superalloy. Subset sizes of 25 × 25 pixels (825 nm × 825 nm) with a step size of 3 pixels (101 nm) were used for HR-

DIC measurements. Digital image correlation was performed using the Heaviside-DIC method [45, 47] for detection of discontinuous displacements within the displacement fields with discontinuity detection resolution between 0.2 and 0.3 pixels (7 nm and 10 nm respectively). The H-DIC method is described in detail elsewhere [45, 47] and briefly summarized in the following section. 2.5. Heaviside-DIC for Plasticity Measurements The Heaviside-DIC method improves upon the conventional DIC method by offering the ability to take into account the presence of a kinematical discontinuity in each subset. The shape function X ∗ is modified to have the jump/step vector multiplied by a Heaviside function, as is described here: Rigid body

X∗ = X + |

z}|{ U

First gradient

z }| { Jump function z}|{ Heaviside z }| { 0 ∂U + (X − X0 ) + U × H(X − X0 ) ∂X | {z } {z } Heaviside-DIC method

(1)

Conventional DIC method

where X is the subset position in the reference image, (X − X0 ) is the position in the subset and U is an

in-plane translation and

∂U ∂X

is the first gradient of the transformation corresponding to the kinematical 0

description in the conventional DIC method. The jump/step function U describes the amplitude of the kinematical discontinuity, dx and dy , along the horizontal and vertical directions as shown in Figure 2(a), 5

respectively. The Heaviside function provide the exact location and angle of the discontinuity in regard to the subset center (r∗ , θ∗ ) as displayed in Figure 2(a) All parameters are retrieved in a process of minimization using Newton’s algorithm. Subsets containing discontinuities are identified as those with non-zero jump values, along with the location and orientation of the discontinuity (r∗ ,θ∗ ) and the displacement between the two sides of the discontinuity (dx , dy ) as shown in Figure 2(a). The Heaviside-DIC method maintains a discontinuity spatial detection of nearly one pixel, independently of the subset size. The Heaviside-DIC method greatly enhances levels of spatial resolution on the strain and displacement fields in presence of kinematical discontinuities compared to conventional DIC methods [45, 47]. During mechanical loading, metals such as nickel-based superalloys undergo non-reversible plasticity in the form of slip bands. During plastic deformation, dislocations emerge at the surface along crystallographic planes lead to the formation of slip traces at the free surfaces of specimens [1]. The in-plane displacements occur locally in the material between both sides of the slip trace and can be described by a vector named in-plane slip ~τ as depicted in green in Figure 2(b). The in-plane slip vector represents the physical in-plane displacement, produced by a slip event, between the material on the "left" and "right" side of the slip trace. The Heaviside-DIC method systematically provides the jump/step function referenced to the slip trace. Therefore, the full in-plane description of the slip displacements (in-plane slip vector) is obtained from the Heaviside-DIC method at every point in the HR-DIC map. The amplitude (norm) in nanometers of the slip vector k~τ k for each measured point is given in Figure 2(c) for a nickel-based superalloy after deformation at 1.83%. The intensity of the slip is obtained for each single slip trace at the surface of the specimen with exceptional spatial (less than 33 nm) and amplitude resolution (less than 10 nm). Amplitudes lower than 10 nm are not detected, however the average strain amplitude is well captured [45]. If multiple discontinuities are occurring within a spacing of 100 to 200 nanometers, the Heaviside-DIC will display a single discontinuity that has the cumulative amplitude of all discontinuities. A sensitive study of the method in regards to band spacing is details in appendix A. In addition, the direction of the slip vector is given using the angle γ ∗ , which is the angle between the discontinuity (slip trace / L in Figure 2(b)) and the in-plane slip vector. An angle of 0◦ indicates pure in-plane shearing displacement along the slip trace, while an angle of 90◦ is pure in-plane sliding displacement along the direction transverse to the slip trace, i.e., no in-plane shearing displacement along the direction longitudinal to the slip trace. The in-plane slip angle γ ∗ for each subset is given in Figure 2(d) for a nickel-based superalloy after being deformed to 1.83%. It is worth noting that the value of the γ ∗ angle is constant for a given slip band. This is due to a unique active slip direction along the active slip plane [45]. This information is also useful for direct identification of distinct slip systems (slip plane and slip direction), Figure 2(d).

6

3. Results 3.1. Slip Recovery and Cyclic Slip Accumulation Measurements Fatigue tests were performed ex-situ in the low cycle fatigue regime with HR-DIC measurements over one millimeter fields of view during the first cycle after the tensile and compressive loads and after subsequent cycles. An example of HR-DIC dataset is presented in the supplementary material for the specimen tested at a maximum applied stress of 1080 MPa and during the first cycle after the tensile load. Conventional HR-DIC and Heaviside-DIC measurements are reported in the supplementary material. An associated EBSD inverse pole figure map along the loading direction is also reported for the investigated region. At first, average strain along the loading direction obtained from HR-DIC were extracted from the investigated one millimeter fields of view for both loading conditions. The average strains from HR-DIC are reported in Figure 1(a and b) and Table 1 during the first cycle after tensile and compressive load and after 400 and 5000 cycles for the specimen tested at a maximum applied stress of 1140 MPa, and after 2000 and 4000 cycles for the specimen tested at a maximum applied stress of 1080 MPa. The measurements were performed after unloading from the tensile part of the cycles. Very good agreement (4% average difference) is observed between the strain data obtained from mechanical extensometer and HR-DIC measurements. HR-DIC measurements are registered to the initial undeformed state, which allows for an overlap with subpixel accuracy [45, 47, 48] of the investigated region between the different loading states. Pixel-to-pixel comparisons were made between the different HR-DIC maps obtained during the first cycle after tensile and compressive loads and after 400 cycles. For the sake of visualization, only small regions of interest will be displayed in the present paper. However, statistical data were extracted from regions of 1 mm × 1 mm. The norm k~τ k and angle γ ∗ of the in-plane slip from HR-DIC measurements with Heaviside-DIC method is displayed in Figure 3 for a reduced

region of interest after tensile and the following compressive loads during the first cycle. The HR-DIC maps display bands of high in-plane slip values that correspond to the occurrence of slip bands that form during plastic deformation at room temperature. The highly planar character of slip in nickel-based superalloys [1, 17], due to the presence of γ0 precipitates, induces large amplitude surface extrusions that result in large in-plane slip amplitude values as high as hundreds of nanometers (Figure 3). After compressive loading, inplane slip values significantly decrease along existing slip events, as observed in Figure 3(b). This indicates that there is recovery of slip localization from the tensile to the compressive loading accommodated by the motion of dislocations in a reverse direction compared to the dislocation motion during tensile loading. The recovery of slip localization at existing slip bands is reflected in reduced positive slip step height, i.e. by reduction of in-plane slip k~τ k, and is due to both the motion of new dislocations that nucleate during

compressive loading and the motion of existing dislocations that remained in the bulk and did not emerge at the surface after the tensile loading [18, 61]. While many of the slip bands fully recover during the first cycle after compressive loading, a few slip bands only partially recover. The local amplitude of the recovery 7

is measured at each point using the metric R defined as :

R=

st

1 cycle 1st cycle

~τtension − ~τcompression 1st cycle

(2)

k~τ ktension

1 cycle 1 cycle where ~τtension and ~τcompression is the in-plane slip during the first cycle after tensile and compressive st

st

loads, respectively. Using the present definition of recovery R, it is observed that values above 1 can be obtained. This occurs when slip bands fully recover during the compressive loading, plus an increment of slip occurs in the backward direction with regard to slip that occurs during the tensile loading, as shown for the slip band at the white arrow in Figure 3(a and b). This slip band, which initially developed during the tensile loading, has the opposite slip direction after tensile and compressive loading as observed in the in-plane slip angle γ ∗ maps in Figure 3(c and d). To aid in visualization, the average direction of the in-plane slip vector for the slip band of interest after tensile and compressive loads is displayed with a white arrows in Figure 3(d and e). The root of the arrow is positioned on the corresponding slip band. Schematics of the displacement of the slip band of interest are given after tensile and compressive loading in Figure 3(f and g). Negative values (average value of -8◦ ) of the in-plane slip angle γ ∗ are reported along this band after the compressive load, indicating a slip step with "negative" height as shown in Figure 3(f). The slip band that developed after the tensile load has positive values (average value of 172◦ ) of the in-plane slip angle γ ∗ , indicating a slip step with positive height in Figure 3(e). The denomination "negative" associated with slip step height indicates that there is a physical intrusion of the material from one side of the slip in regard to the other side (Figure 3(f)). For all investigated slip events, one of four slip band configurations were observed at the end of the first fatigue cycle: 1. Slip bands showing full recovery during compressive loading (R=1) - their in-plane slip amplitude k~τ k is null (under the detection threshold) after compressive load;

2. Slip bands with partial or no recovery (0≤R<1) - their in-plane slip amplitude k~τ k is positive and they display positive and identical in-plane slip angle γ ∗ after tensile and compressive load;

3. Slip bands having recovery values larger than 1 such as the band in Figure 3 - These bands display positive and negative values of the in-plane slip angles γ ∗ ; i.e. opposite direction of the in-plane slip vector ~τ after tensile and compressive loads, respectively; 4. Slip bands that develop after the compressive load that were not present after tensile loading. This will be discussed further in Section 3.2. The slip recovery during the first fatigue cycle is shown in Figure 4(d) for a specimen tested in low cycle fatigue at a maximum applied stress of 1140 MPa. The in-plane slip k~τ k maps during the first cycle after

tensile and compressive loads are in Figure 4(a and c). Many slip bands display a high degree of recovery

with the exception of bands that develop near and parallel to particular twin boundaries, indicated by the 8

black arrow within the grain pair labeled "A" and "B" in Figure 4(b). These slip bands display slip recovery values as low as 3% indicating that motion of dislocations did not occur along these slip planes during the compressive load. Average values of the slip recovery over the millimeter field of view obtained by the Heaviside-DIC method for the two loading conditions and during the first cycle are reported in Table 1 and compared with macroscopic strain recovery defined as the the amplitude between the macroscopic strain after tensile and compressive loads divided by the strain after the tensile load. A good agreement is obtained between macroscopic strain recovery and average slip recovery indicating that the Heaviside-DIC method and HRDIC measurement capture the right amount of slip recovery at the scale of each slip. The in-plane slip k~τ k map after 400 fatigue cycles is displayed in Figure 4(e). The in-plane slip k~τ k

increases significantly after 400 fatigue cycles for slip bands near the twin boundaries in the grain pair labeled "A" and "B", when compared to the in-plane slip values obtained at the end of the first cycle. This indicates plastic strain accumulation during cycling as a consequence of cyclic slip irreversibility. Other slip bands that were present after the tensile load and display full recovery after the first cycle eventually do accumulate irreversible plasticity during cycling, such as the one in the grain labeled "G" in Figure 4(b). A slip accumulation metric is defined, at each pixel, as the difference between in-plane slip k~τ k after a given cycle number and the first cycle (after compressive load). The slip accumulation metric is reported in

Figure 5(e and f) after 400 cycles for two regions of interest tested in the low cycle regime at a maximum applied stress of 1140 MPa. The associated in-plane slip k~τ k maps after the tensile load during the first

cycle are given in Figure 5(c and d). Significant values (2 to 3 times the average) of slip accumulation are observed along locations of slip localization that occurs near and parallel to particular twin boundaries as indicated by arrows in Figure 5(a and b). In addition, new slip traces that developed after the tensile loading during the first cycle are observed on the slip accumulation map at the white arrow in Figure 5(f). This is evidence of the formation of a persistent slip band (PSB) [1, 2, 62], also known as a fatigue shear band or a deformation band [3, 8, 63, 64]. In nickel-based superalloys, PSBs are comprised of multiple highly intense planar slip bands along parallel {111} slip planes [1, 8, 17]. In the investigated nickel-based superalloy, twin boundaries that develop slip localization near and parallel to them are preferential crack nucleation sites [6, 65]. Therefore, a more focused investigation of these specific microstructure features follows. The slip bands that occur nearby (within 2 µm) and parallel to twin boundaries after the first cycle of tensile loading were systematically extracted by conventional image segmentation techniques and then analyzed. The average values of slip accumulation after 400 cycles and the average values of slip recovery during the first cycle were calculated for each slip band and are plotted in Figure 6(a and b) as a function of the maximum Schmid factor and the elastic modulus difference of the twin and parent grain pair that exists at each of the investigated twin boundaries. Sometimes, several slip bands were associated with a single twin boundary; therefore, the average values of slip accumulation and slip recovery for all of the slip bands 9

nearby that boundary are considered. The elastic modulus and Schmid factor calculations were obtained from the average crystallographic orientation of each grain with respect to the sample loading direction and the compliance matrix for the superalloy [6]. Only slip bands with lengths greater than 10 µm were considered. Slip bands with slip accumulation, after 400 cycles, that was lower than 30 nm are indicated by crosses in Figure 6(a and b). Slip bands with a high intensity of slip accumulation after 400 cycles and low slip recovery after the first cycle occurred near twin boundaries that maximize either the Schmid factor or the elastic modulus difference between the twin and parent grain pair. It is also observed that slip bands with high slip recovery after the first cycle do not have significant slip accumulation after 400 cycles. Maps of slip recovery R after the first cycle for three regions of interest are displayed in Figure 7(g-i) for a specimen tested in low cycle fatigue at a lower maximum applied stress of 1080 MPa. The associated microstructure and in-plane slip k~τ k after the tensile load during the first cycle are presented in Figure 7(a-c

and d-f). Specimens were cycled to 4500 cycles, and back scatter electron (BSE) images were collected to investigate damage nucleation at the surface of the specimen. The BSE images of three regions of interest in Figure 7(j-l) show that crack nucleation sites that are localized near and parallel to twin boundaries develop slip bands during the first cycle, with low levels of slip recovery. The slip bands near and parallel to twin boundaries with slip recovery lower than 50% are displayed in Figure 8, according to the maximum Schmid factor and the elastic modulus difference of the twin and parent grain associated with the twin boundary. As observed for the test performed at higher maximum applied stress, grain pairs with slip bands with low recovery during the first cycle have either a high Schmid factor or a high elastic modulus difference. The size of the symbols in Figure 8 scale with the occurrence of a crack nucleation site (or not) after 4500 cycles. Crack nucleation occurs near twin boundaries in grain pairs that have the lowest slip recovery during the first cycle. 3.2. Origin of low slip recovery and high cyclic slip accumulation near twin boundaries The in-plane slip k~τ k maps after the tensile and compressive loads during the first cycle, in Figure 9(b

and c), show a twin labeled "A" and associated parent grain labeled "B" with slip bands near and parallel

to twin boundaries having low slip recovery. The grain pair labeled "A" and "B" was previously described in Figure 4. Interestingly, a new slip band is formed during compressive loading with high in-plane slip values (slip band labeled "SB2" in Figure 9(b)). The existing slip bands that developed during tensile loading near the investigated twin boundaries still have high intensity in-plane slip after the compressive load cycle. The in-plane slip angle γ ∗ maps in Figure 9(c and d) after the first cycle of tensile and compressive loading show that the slip band that developed during the compressive load has negative in-plane slip angles γ ∗ that are different than those that develop near slip bands after tensile loading. From Figure 9, it is observed that the slip band denoted "SB1" developed during the first cycle of tensile loading, while the slip band denoted "SB2" developed after the first cycle of compressive loading. The white arrows in Figure 9(d) show the 10

direction of the average in-plane slip vector ~τ for the observed slip bands. The values of the average in-plane slip angles for the band labeled "SB1" and "SB2" are also reported in the Table 2. The slip band labeled "SB2" that developed during the first cycle of compressive loading has an in-plane slip vector direction that is in the opposite direction of the slip bands that developed during the tensile loading. Schematics of the morphology of the bands labeled "SB1" and "SB2" are given in Figure 3(f and g). A positive slip step height (Figure 3(f)) is observed during the first cycle after tensile loading along the slip band labeled "SB1", while a "negative" slip step height ((Figure 3(g)) is observed along the band labeled "SB2" during the first cycle after compressive loading. The denomination "negative" associated with slip step height indicates that there is a physical intrusion of the material from one side of the slip in regard to the other side (Figure 3(f)). Figure 9 also shows that several slip bands such as the slip band labeled "SB3" that developed after tensile loading near and parallel to the twin boundaries in grain "A" and "B" experience slip recovery higher than 1, as evidenced by the inversion of the direction of the in plane slip vector after compressive loading. While slip with positive slip step height (positive γ ∗ ) was observed along these bands after tensile loading, slip with "negative" slip step height (negative γ ∗ ) developed after compressive loading. Theoretical values of the in-plane slip angle γ were calculated based on the EBSD measurements from the slip band labelled "SB1", "SB2" and "SB3" and are provided for the 3 possible slip directions within the active slip plane in Table 2. The grain orientation measurement and the twelve possible {111}h011i slip systems present in an fcc crystal allow the calculation of the Burgers vector and the direction of the in-plane slip vector ~τ , i.e. in-plane slip angle, as outlined in more detail elsewhere [45]. The active {111} slip plane is identified using the trace of the slip band from HR-DIC, and the active h011i slip direction is

identified using the direction of the in-plane slip vector ~τ from Heaviside-DIC. The in-plane slip angle, γ ∗ , from HR-DIC corresponds to the theoretical in-plane slip system slip system with the highest Schmid factor for both investigated slip bands. A similar configuration was observed during fatigue testing at the lower maximum applied stress as shown in Figure 10. A set of investigated slip bands labeled "SB4" and "SB5" in Figure 10 have opposite direction of the in-plane slip vector from each other. The band labeled "SB5" developed during the first cycle after the compressive load, while the band labeled "SB4" developed after the tensile load. The in-plane slip angles γ ∗ are consistent with the flip in the Burgers vector direction as shown in Table 2. The band labeled "SB3" in Figure 9 has an inversion of the in-plane slip direction as shown in Table 2 during the compressive load, which is consistent with the theoretical inversion of the slip direction between the tensile and compressive load. The in-plane slip amplitude k~τ k and angle γ ∗ along the slip trace from the slip bands labeled "SB1",

"SB2" and "SB3" in Figure 9 are displayed in Figure 11(a-b and c-d) after the tensile and compressive load.

The three slip band configurations observed during the first cycle near these particular twin boundaries are summarized by the following: 11

1. Identical in-plane slip angles are observed for the slip band labeled "SB1" after tensile and compressive loading. The in-plane slip amplitude values sightly decrease indicating partial slip recovery within the slip band; 2. The slip band "SB3" has an inversion in the direction of the in-plane slip vector and high in-plane slip amplitude values after compressive loading, therefore, slip recovery is found to be larger than 1 (average slip recovery R of 1.51). The motion of dislocations generated during the compressive loading portion of the cycle along the slip band labeled "SB3" drive the full recovery of the slip step developed along this band after tensile loading, plus an additional increment of slip that results in a "negative" slip step height of the slip band after the compressive load; 3. The slip band labeled "SB2" developed during the compressive load with a "negative" slip step height. The total slip increment at each pixel along slip bands induced by the first tensile and compressive loads

st

st

1 cycle

1 cycle 1st cycle during the first cycle can be obtained as ~τtension

and ~τcompression − ~τtension , respectively. The slip increment produces positive and "negative" slip step displacements during tensile and compressive loads

respectively. The slip increment for the slip bands "S1","S2" and "S3" that formed near and parallel to the twins boundaries in grain pairs labeled "A" and "B" in Figure 9 is plotted in Figure 12 during tensile and

compressive loading. The slip increment can be linked to the equivalent number of emerging dislocations that produced the slip event. The unit in-plane slip k~τ kunit corresponding to one dislocation emerging at

the surface is equal to the Burger vector component in the plane of the free surface:

a√ (3) 2(cos δ − cos β sin δ) 2 ~ β is the angle between the vector L ~ and the where δ is the angle between the Burgers vector ~b and L, k~τ kunit = kbk (cos δ − cos β sin δ) =

slip plane - as depicted in Figure 2, and a is the lattice parameter of the nickel-based superalloy. This approach is similar to the determination of the the number of dislocations by experimental measurement of slip step [61, 66]. This approach consider that the dislocations that emerge at the free surface along an active slip plane, producing a surface step, move along a direction equivalent to the active slip direction (Burgers vector). Near the twin boundaries in the grain pair labeled "A" and "B", three kinds of slip bands are observed: 1. Slip bands that partially recover slip such as the band labeled "SB1" that have low slip increment values during the compressive load indicating low levels of dislocation motion in the backward direction within the slip plane; 2. Slip bands that are fully recovered and display larger slip amplitude during the compressive load such as the band labeled "SB3". They have a stronger dislocation motion activity along their slip planes in the backward direction in comparison to the motion in the forward direction during tensile load, 12

resulting in a higher number of dislocations emerging at the surface during compressive loading along the slip planes. 3. Slip bands that develop during the compressive loading such as "SB2". The formation of slip bands that develop near specific twin boundaries during the first loading cycle are early indicators of the formation of PSBs and subsequent crack nucleation that will develop later during cycling, as shown in Figure 9(f) for the investigated twin boundaries in grain pair labeled "A" and "B". 3.3. Persistent Slip Bands In-plane slip k~τ k maps after the first loading cycle and after 400 cycles are shown in Figure 13(b-d) for

a specimen tested in low cycle fatigue at a maximum applied stress of 1140 MPa. The associated in-plane

slip angle γ ∗ maps are provided in Figure 13(e-g). The development of additional slip bands with negative in-plane slip angle γ ∗ values ("negative" slip step height) are observed after the compressive load during the first cycle near and parallel to twin boundaries, as previously described in 3.2. Interestingly, after 400 cycles the number of slip bands that developed parallel and near to twin boundaries increases significantly, resulting in a decrease of band spacing by approximately a factor of two. The alternating between positive and negative in-plane slip angle γ ∗ values are observed in slip bands within the PSB near twin boundaries and show the intrinsic intrusion-extrusion character of a PSB [12]. The distribution of the in-plane slip k~τ k for the bands that developed in the grain pair labeled "E" and

"F" that contain a PSB (Figure 13) are presented in Figure 14(a-c) during the first cycle after tensile and compressive loading and also after 400 cycles. The sign of the in-plane slip angle γ ∗ for every pixel along each slip band allows for the differentiation between slip bands with positive (γ ∗ > 0) and "negative" (γ ∗ < 0) slip step heights. It is logical to observe no distribution of in-plane slip with "negative" slip step height during the first cycle after tensile loading. After compressive loading, the slip bands that had high in-plane slip after tensile loading partially or fully recover from the plastic deformation. Interestingly, a significant

portion of slip bands display "negative" slip step height during the first cycle after compressive loading. After 400 cycles, slip accumulation occurs and result in an increasing number of bands with low in-plane slip intensities. The amplitudes in the distribution of in-plane slip increase significantly in the investigated grain pairs that have PSBs. The distribution of the in-plane slip increments and the equivalent number of dislocations emerging at the surface for the grain pair labeled "E" and"F" is reported in Figure 14(a and b), after the tensile and compressive load steps respectively. The ratio of the cumulative in-plane slip increment for the grain pair labeled "E" and "F", between the tensile and compressive loading can be calculated from the distributions in Figure 14 and is found to be 1.02 indicating that the magnitude of in-plane slip generated during the compressive loading is roughly equal to the in-plane slip generated during tensile load near this specific twin boundary. A value greater than 1 indicates that the compressive load did not fully recover the slip steps 13

induced by the tensile load. The amplitude of in-plane slip at the surface of the specimen at the end of the first cycle and 400 cycles is reported in Figure 14(b). A large number of low amplitude in-plane slip events developed after 400 cycles, which is evidence of plastic accumulation. The ratio of cumulative in-plane slip for the grain pair labeled "E" and "F" between 400 cycles and the first cycle can be calculated from the distributions in Figure 15(b) and are found to be 1.43 indicating that the amount of in-plane slip generated by the grain pair is increasing with cycling. 4. Discussion It is important to note that the tests conducted here were in stress control mode; further experiments in strain control mode would give further insight to the processes that occur in turbine components that are subject to this type of cycling. Of most importance, stress control test usually lead to most of the plastic strain occuring in the first cycle, which is not consistent with what would happen in components. Nevertheless, the aim of the paper is to demonstrate the possibility of performing direct measurement of slip irreversibility using advanced HR-DIC measurements coupled with discontinuity-tolerant code analysis. The presented experimental method measures the accumulation of slip reversibility/irreversibility during fatigue loading that is critical for calculating cyclic slip irreversibility. Since the measurement can be performed for each single slip band during cycling within the sensitivity of the technique, the slip reversibility can be directly described by the recovery of slip during the compressive load during a single cycle, and cyclic slip irreversibility can be described by the residual plastic strain localization increment after a complete cycle. HR-DIC measurements combined with the discontinuity-tolerant Heaviside-DIC method provide new information on individual slip bands over millimeter fields of view. For each subset, the local in-plane slip ~τ of the discontinuity as shown in Figure 2(b) provides a physically based value of the plastic strain localization. Of most interest for fatigue, this local approach allows for the analysis of the in-plane character of all slip bands, including measurements of positive and "negative" slip step heights that are physical evidence of PSBs [12, 17]. The local cyclic slip irreversibility is critical for developing and validating models for crack nucleation [16– 19, 22]. In the literature, different definitions of slip and cyclic slip irreversibility exist. They are obtained either from bulk measurements [1] or from local measurements [1, 18]. The local definitions are mainly based on the measurements of slip steps/extrusion forming at the surface using profilometry techniques such as AFM. Local measurements provide an efficient means to investigate the effect of the microstructure on surface crack nucleation [1, 67] in comparison to bulk measurements that consider plasticity in the bulk of the specimen. The use of AFM for the measurement of slip irreversibility provided opportunities to develop physics based models that accurately predict crack initiation processes [1, 23, 67]. However, the measurements of slip irreversibility by AFM are limited to small regions of interest and are relatively 14

slow. Furthermore, the registration of single slip bands/traces from one load step to the next is difficult to perform with AFM data due to the lack of unique tracking features. Therefore, the measurement of the slip irreversibility/reversibility for a single slip band is hard to scale to fields of view that are large enough to characterize the distribution of crack nucleation events. Our new HR-DIC measurements provide sub-pixel registration of each loading step to the undeformed state. With HR-DIC, the local slip displacements at a given pixel can be tracked throughout fatigue cycling, guaranteeing the investigation of the same exact pixels at each loading step. 4.1. Twin Boundaries While many grains experience full slip recovery after the first cycle of compressive loading in low cycle fatigue, grain pairs with particular twin boundaries have complex slip redistribution during compressive loading, which results in residual slip localization at the surface. In addition, these grain pairs experience elevated levels of slip accumulation during cycling, indicating high levels of cyclic slip irreversibility at these locations. The interaction of dislocation with twin boundaries is of most importance for incipient plastic strain localization and the formation of PSBs in nickel-based superalloys or fcc materials [1, 12, 31, 56, 68–73]. Early plastic strain localization is favored in the case of fcc materials containing coherent twins in the parallel slip configuration [7], which occurs when an activated {111}h110i slip plane is parallel to the twin plane. Fatigue crack nucleation was first reported nearby twin boundaries by Heinz and Neumann [68] in stainless steels. It is reported that stress concentrations due to the high elastic anisotropy enhance dislocations gliding adjacent to twin boundaries. It has been recently observed using experimental and computational tools that twin boundaries display higher plastic strain localization parallel to their boundary than a general boundary with oblique activated slip planes [31, 68, 74–76]. Moreover, this phenomenon is strongly intensified by the presence of the γ0 precipitates [1]. It is interesting to note that the in-plane slip displacements that occurs along the most active slip bands in the first cycle are of the order of 100 - 200nm, of the same order as the diameter of the secondary γ0 precipitates. Thus, these intense shearing events completely destruct the precipitates, as has occasionally been observed by TEM studies previously [58]. The PSBs observed in precipitate containing material are structures with multiple high intensity planar slip bands along parallel {111} slip planes [1, 8, 17, 77, 78] as observed in the present paper. Previous studies indicated that preferential crack nucleation sites for the polycrystalline René 88DT nickel-based superalloy are twin boundaries that display a parallel slip configuration [31, 56]. In addition, the grains on either side of the twin boundary must either have a high elastic modulus difference or a high Schmid factor in order to nucleate cracks [6]. In the present paper, it is also observed that these particular twin boundaries localize strain during the first cycle after tensile loading. It appears that slip recovery during the first cycle near these specific twin boundaries for certain slip bands is very low. Indeed, the slip 15

accumulation that follows during cycling is observed to nucleate cracks at sites with low slip recovery during the first cycle, and suggest that the first cycle in low cycle fatigue is the most important for predicting fatigue. 4.2. Slip Redistribution The present paper focuses on the early formation of PSBs in a nickel-based alloy and especially on slip recovery during the first cycle. The structure of PSBs in nickel-based alloys are different with regards to PSBs in single phase alloys such as copper or austenitic stainless steel. The structure of the PSBs in nickel-based superalloys displays closely spaced parallel slip bands with significant slip step height for each of the slip band [8, 58, 79]. At the origin of the structure, during the first cycle, different types of slip reversibility/irreversibility behavior were observed according to the location of the slip band with respect to the microstructure. Many of the slip bands have close to full reversibility during the first cycle of compressive loading. The slip steps at the surface are near to fully recovered by the compressive load, indicating that the increment of plastic strain due to the compressive load provides dislocation motion along the same slip planes that were activated during tensile load, as shown in the grain labelled "G" in Figure 4. The increment of plasticity during the compressive load is accommodated by the activation of new dislocation sources or by dislocations that already existed on the planes activated during tensile loading. Unusual slip reversibility/irreversibility behavior is observed along slip bands that are located near and parallel to the particular twin boundaries displayed in Figure 9. Two additional slip reversibility/irreversibility behaviors are observed: (i) partial or no recovery of slip during the compressive load, indicated by high remaining in-plane slip values along the slip band after the compressive load and (ii) total slip recovery plus an additional slip in the backward direction. The latter reversibility/irreversibility behavior was previously observed in [80] during cyclic loading and is associated with the formation of PSBs and damage during cyclic loading [66, 80]. A quantitative analysis of these phenomenon can now be provided by HR-DIC measurements with the Heaviside-DIC method. In addition to slip reversibility/irreversibility from existing slip bands after tensile loading, new slip bands are observed to develop during the compressive load with a "negative" slip step height as shown in Figure 9. The two previously mentioned slip reversibility/irreversibility behaviors, in addition to the occurrence of new slip events during the first cycle and nearby particular twin boundaries, are the source of PSB formations as shown in Figure 13 and Figure 6 and the damage nucleation in Figure 7. The cumulative in-plane slip events generated by the increment of plasticity from the first compressive step in the grains that generate PSBs later in fatigue life are observed to be identical to the cumulative in-plane slip generated during tensile loading as shown in Figure 15. This indicates that a similar level of dislocations are emerging from the sample surface during tension and compression, as observed for the grains that do not display PSBs. However, the distribution of slip within the grains that have these par16

ticular twin boundaries are different. The activation of dislocation motion along new slip planes is more preferred near these particular twin boundaries than the dislocation motion along existing slip planes. The amount of plasticity that is not recovered along certain bands near these particular twin boundaries is either contributing to new slip plane activation or to full recovery of slip plus an additional slip increment with "negative" slip step height. Therefore, a slip redistribution is observed rather than local additional slip increment during the compressive load near these particular twin boundaries. During subsequent cycling slip accumulation occurs in grains with particular twin boundaries, such as the grain pair labeled "A"-"B" in Figure 9, but also in grains without slip redistribution during the first cycle as observed in the grain labeled "G" in Figure 6(b,e). However, the grain pair labeled "A"-"B" forms PSBs and damage as shown in Figure 9, while the grain labeled "G" does not form PSBs or nucleate cracks. This observation suggests that the slip redistribution during the first cycle is important for the formation of PSBs and crack nucleation processes. It must be mentioned that fatigue tests in the present paper were performed in stress control which may not be representative of the fatigue behavior of actual structural parts. Plastic strain control tests should be performed to confirm the occurrence of the slip redistribution mechanism. While many models for crack nucleation [1, 16, 18] consider only cyclic slip irreversibility and strain accumulation, the additional use of the redistribution of slip may improve models for crack nucleation. The mechanisms that generate the slip redistribution near these particular twin boundaries that develop PSBs is not known. The slip planes with high dislocation activity during the tensile load are not necessarily those that display high dislocation activity during the compressive load. The dislocations that emerge from the surface during compressive loading are taking another pathway than those from tensile loading near these particular twin boundaries. This suggests that either the stress distribution is locally different between the tensile and compressive load, activating different dislocation sources, or that new intrinsic dislocation mechanisms are active during compression such as cross slip. These differences in tension and compression may be enhanced by the presence of the precipitates and their configurations near the dislocation sources. The twin boundary character is also certainly contributing to the slip redistribution by the complex effect of dislocations interaction with the coherent boundaries [81, 82]. The slip redistribution mechanism that is observed during cyclic loading is of most importance for the formation of PSBs in nickel-based superalloy. The consideration of the cyclic slip irreversibility by formation of new slip or incremental slip in the backward direction also seems important for the formation of PSBs. Local softening by the formation of slip during the tensile load is not required in this process; the formation of a PSB occurs to accommodate the reverse deformation during the compressive load. This provides important insights for formulation of non-local damage metrics such as fatigue indicator parameters for crack nucleation and growth [83–86].

17

5. Conclusions The Heaviside-DIC was used to solve the problem of kinematical discontinuities that result from slip localization. The discontinuity-tolerant code allows quantitative measurement of plasticity localization for individual slip events. The approach was applied to measure slip reversibility/irreversibility and slip accumulation during low cycle fatigue loading of a nickel-based superalloy. The present method provides full field characterization of the formation of slip events during fatigue in a nickel-based superalloy. Emphasis was placed on the first cycle of the fatigue tests to provide information on the earliest stages of cyclic strain localization.Bands of localized slip with recovery below 50% that form near twin boundaries in the first loading cycle are nuclei for cracks that form much later in cycling. It is observed during the first cycle after compressive load, a slip redistribution mechanism near particular twin boundaries in grain pairs that maximize either the Schmid factor or the elastic modulus difference across the twin boundary. For the present test conditions, a significant fraction of slip during reversed loading is accomplish by formation of new slip bands near twin boundaries. The slip redistribution is observed to be correlated with the formation of the persistent slip bands and ultimately crack nucleation. 6. Acknowledgements The authors appreciate useful discussions with J. Laflen, A. Loghin, J. Marte and M. Soare. General Electric is acknowledged for providing the material. The support of U.S. Department of Energy, Office of Basic Energy Sciences Grant DE-SC0018901 is gratefully acknowledged. Chris Torbet is acknowledged for his support of the experimental effort.

18

19 1.21%/ 1.20%

Applied stress of 1140 MPa

0.25%/ 0.24%

-0.10%/ -0.10%

macro./ strain DIC

macro./ DIC strain 0.20%/ 0.19%

1st cycle(compressive load)

1st cycle( tensile load)

Applied stress of 1080 MPa

recovery obtain from Heaviside-DIC.

79%/ 80%

150%/ 153%

macro./ DIC strain

strain recovery

85%

148%

Average slip recovery

Macroscopic strain recovery obtained during the first cycle due to compressive load calculated from the macroscopic and average DIC measurements. Average slip

Table 1: Macroscopic and average DIC strain obtained during the first cycle after the tensile and compressive loads for the two investigated loading conditions.

Table 2: A comparison between the theoretical angle γ from EBSD measurements for the three possible activated slip directions on the active {111} slip plane obtained from slip trace analysis and the in-plane slip vector angle γ ∗ obtained from HeavisideDIC for the slip bands labeled SB1 to SB4 from the regions in Figure 9 and Figure 10. The angles obtained from Heaviside-DIC are an average of all angle measurements along a single slip band. The in-plane components of displacement generated by slip events at the DIC subset scale are forming the in-plane slip vector as depicted in green in Figure 2(b). The angle γ ∗ represents the inclination of the in-plane slip vector, referenced to the slip trace i.e. the angle between the reference vector L and the in-plane slip vector as shown in Figure 2(b).

In-plane slip angle γ from EBSD

HDIC

(111)[011]

(111)[101]

(111)[110]

In-plane slip angle γ ∗

SB1

6.4◦

175.9◦

56.9◦

169.9◦ ± 5.3◦

SB2

-173.6◦

-4.1◦

-123.1◦

-5.8◦ ± 4.5◦

SB3 (after tension)

6.4◦

175.9◦

56.9◦

172◦ ± 3.9◦

SB3 (after compression)

-173.6◦

-4.1◦

-123.1◦

-6.3◦ ± 5.7◦

(111)[011]

(111)[101]

(111)[110]

SB4

4.8◦

54.0◦

169.7◦

172.2◦ ± 5.2◦

SB5

-175.2◦

-126.0◦

-10.3◦

-11.2◦ ± 6.2◦

(111)[011]

(111)[101]

(111)[110]

In-plane slip angle γ ∗

SB6

26.3◦

60.8◦

169.9◦

57.7◦ ± 5.1◦

SB7

-153.7◦

-119.2◦

-10.1◦

-123.2◦ ± 6.2◦

20

Figure 1:

(a) Evolution of the maximum strain during cycling for specimens tested in the low cycle fatigue regime at

maximum applied stresses of 1080 MPa and 1140 MPa for the polycrystalline nickel-based superalloy Rene 88DT. (b,c) Stressstrain hysteresis curves of specimens tested in the low cycle fatigue regime at applied stresses of (a) 1080 MPa and (b) 1140 MPa. HR-DIC measurement were performed during the first cycle after the tensile and compressive loads and after subsequent cycles over one millimeter field of view. Measurements were performed after unloading (after tensile load) and obtained average strain along the loading direction over the field of view are reported.

21

Figure 2: (a) Discontinuity description inside a single subset by the Heaviside-DIC method. Subsets that have discontinuities are detected and described by their location inside the subset (r∗ , θ∗ ) and the jump/step of the discontinuity (dx , dy ) along the horizontal and vertical directions. (b) During mechanical loading, polycrystalline metals such as the investigated nickel-based superalloy undergoes non-reversible plasticity. Consequently, slip traces at the free surfaces of specimens are observed, each associated with a local surface step. The in-plane sliding and shearing displacements are contained in the plane of the free surface and are the two components of the in-plane slip vector ~ τ in the slip trace reference (T,L). The in-plane slip vector ~ τ represents the physical in-plane displacement of the material on the "left" side of the slip trace in comparison to the material on the "right" side of the slip trace. The in-plane slip vector can be obtained systematically by solving the problem of kinematical discontinuities at each subset using the Heaviside-DIC method. (c) The amplitude of displacement induced by slip events

22norm, k~τ k) giving in nanometers for a region in a nickel-based obtained using the Heaviside-DIC method (in-plane slip vector

superalloy specimen deformed at 1.83% deformation. (d) The associated angle γ ∗ of the in-plane slip vector ~ τ.

Figure 3: Recovery of slip localization during low cycle fatigue loading of the nickel-based superalloy René 88DT at a maximum applied stress of 1140 MPa during the first cycle. (a,b) The norm of the in-plane slip k~ τ k that describe the amplitude of the

local displacement/step at the surface of the specimen induced by slip during the first cycle after tension and the following compressive load, respectively. (c,d) The angle of the in-plane slip γ ∗ that describe the direction of the local displacement/step at the surface of the specimen induced by slip during the first cycle after tensile and following compressive load, respectively. Inverse pole figure map of the associated microstructure from EBSD measurements is given in insert. (e,f) Schematics of the displacements at the slip band after tensile and compressive loading as indicated by the white arrow in (a) and (b) respectively. The average direction of the in-plane slip vector ~ τ along a slip band of interest is given by white arrow in (c) and (d).

23

Figure 4: Slip localization during low cycle fatigue loading of René 88DT at a maximum applied stress of 1140 MPa during the first cycle after tension (a) and (b) compression as displayed in (f) and after 400 cycles (e). HR-DIC measurements were performed ex-situ after unloading the specimen. (a,c,d) The amplitude of the in-plane slip that describes the local displacement/step at the surface of the specimen induced by slip. (b) Inverse pole figure map of the associated microstructure given by EBSD measurements. (d) Percentage of the slip irreversibility, which presents for each single slip band the amount

24

of local in-plane displacement/step that are not recovered during the first cycle between the tensile and compressive loading. The slip irreversibility is calculated from the difference between the value of the amplitude of the in-plane slip after tensile and compressive load during of the first cycle. This difference is normalized according the value of the in-plane slip after tensile load of the first cycle.

Figure 5: Slip localization from the low cycle fatigue loading of René 88DT at a maximum applied stress of 1140 MPa during the first cycle after the tensile loading (c,d) for two regions of interest in (a) and (b) respectively. (a,b) Inverse pole figure maps along the loading direction of the two regions of interest given by EBSD measurements. (c,d) The amplitude of the in-plane slip that describes the local displacement/step at the surface of the specimen induced by slip. (e,f) Accumulation of slip observed at the surface of the specimen after 400 cycles for the region of interest (a) and (b), respectively. The slip accumulation is the

25

additional amount of local in-plane displacement that is observed between the end of the first cycle (tension-compression) and the 400th cycle.

Figure 6: (a-b) Average slip irreversibility and the accumulation of slip bands that develop from low cycle fatigue loading of René 88DT at a maximum applied stress of 1140 MPa. Only slip bands that are observed near twin boundaries when slip localization develop parallel and near to the twin boundaries are shown. The slip irreversibility and accumulation is given as a function of the maximum Schmid factor and the elastic modulus difference of the twin and parent grain where the slip band developed. Data displayed by crosses and circles in (a) and (b) represent slip bands that display slip accumulation less and higher than 30 nm respectively. The circle size scales with slip accumulation and irreversibility in (a) and (b) respectively. Dashed lines represent the theoretical limits of elastic modulus difference as a function of maximum Schmid factor.

26

Figure 7: Slip localization from low cycle fatigue loading of René 88DT at a maximum applied stress of 1080 MPa during the first cycle after the tensile (a) and (b) compressive loading for three regions of interest. (a-c) Inverse pole figure maps along

27

the loading direction of the three regions of interest given by EBSD measurements. (d-i) The amplitude of the in-plane slip that describes the local displacement/step at the surface of the specimen induced by slip during the first cycle after the tensile load (d-f) and the following compressive load (g-i). (j-l) Backscatter electron images of the three region of interest after 10000 cycles.

Figure 8: Average slip irreversibility of slip bands that develop from low cycle fatigue loading of René 88DT at a maximum applied stress of 1080 MPa. Slip bands near twin boundaries where slip localization develops parallel and near to the twin boundaries and with average slip irreversibilty higher than 50% are reported. The slip irreversibility is given according the maximum Schmid factor and the elastic modulus difference of the twin and parent grain where the slip band developed. The slip events that later developed a crack after 10000 cycles are indicated with large circles. No crack were observed near slip bands that are displayed by small circles. Dashed lines represent the theoretical limits of elastic modulus difference according to the maximum Schmid factor.

28

Figure 9: Characterization of the slip bands that develop near a twin boundary after tensile (a,c) and compressive (b,d) loading during the first cycle of low cycle fatigue in René 88DT at a maximum applied stress of 1140 MPa. The grain pair labeled "A"-"B" from Figure 4 is considered (a-b). The amplitude of the in-plane slip that describes the local displacement/step at the surface of the specimen induced by slip during the first cycle after the tensile (a) and compressive (b) load steps. (c-d) The associated angle γ ∗ of the in-plane slip vector. The average directions of the in-plane slip vector ~ τ along a slip band is given by white arrows. (e) EBSD map of the region of interest. (f) Secondary electron images at the surface of the specimen after 5000 cycles.

29

Figure 10: Occurrence of new slip events after compressive loading during the first low cycle fatigue loading of the René 88DT at a maximum applied stress of 1080 MPa. (a) Inverse pole figure maps along the loading direction of the region of interest given by EBSD measurements. (b-e) The amplitude (b-c) and angle γ ∗ (d-e) of the in-plane slip vector that describe the local displacement/step at the surface of the specimen induced by slip during the first cycle after the tensile load step (b,d) and the following compressive load step (c,e). The average directions of the in-plane slip vector ~ τ along a slip band are given by white arrows in (2).

30

Figure 11: Characterization of the slip bands that develop near a twin boundary during the first low cycle fatigue loading step of René 88DT at a maximum applied stress of 1140 MPa. (a-b) The amplitude of the in-plane slip along the slip bands labeled "SB1", "SB2" and "SB3" in Figure 9 during the first tensile cycle (a) and compressive load (b). The in-plane slip describes the local displacement/step at the surface of the specimen induced by slip. (c-d) The associated in-plane slip angle γ ∗ along the slip bands.

31

Figure 12: In-plane slip amplitude induced by the tensile and compressive loading during the first cycle along the slip bands labeled "SB1", "SB2" and "SB3" in Figure 9. The in-plane slip amplitude along a slip band is linked to the number of dislocation that emerged at the surface during loading. The unit in-plane slip k~ τ kunit corresponding to the emergence at the surface of one dislocation is equal to the Burgers vector component in the plane of the free surface as indicated in (3).

32

Figure 13: Formation of a persistent slip band during low cycle fatigue loading of René 88DT at a maximum applied stress of 1140 MPa. (a) Inverse pole figure maps along the loading direction of the region of interest given by EBSD measurements. (b-g) The amplitude (b-d) and angle γ ∗ (e-g) of the in-plane slip vector that describe the local displacement/step at the surface of the specimen induced by slip during the first cycle after the tensile load (b,e), after the following compressive load setp (c,f) and after 400 cycles (d,e) respectively.

33

Figure 14: The in-plane slip distribution at the early stages of the formation of a fatigue shear band during low cycle fatigue loading of René 88DT at a maximum applied stress of 1140 MPa. The in-plane slip amplitude distributions are reported for slip bands that develop in the grain pair labeled "E" and "F" in Figure 13 during the first cycle after tensile (a) and compressive (b) loading and after 400 cycles(c). Values of the amplitude of the in-plane slip that are associated with positive in-plane slip angle γ ∗ , indicating positive slip step height of the slip band, are reported in black. Values of the amplitude of the in-plane slip that are associated with negative in-plane slip angle γ ∗ , indicating "negative" slip step height of the slip band, are reported in blue.

34

Figure 15: (a) The in-plane slip amplitude distribution induced by the tensile and compressive load during the first cycle for slip bands that develop in grain pairs labeled "E" and "F" in Figure 13. The in-plane slip amplitude along a slip band is linked to the number of dislocation that emerged at the surface during loading. The unit in-plane slip k~ τ kunit corresponding to the

emergence at the surface of one dislocation is equal to the Burgers vector component in the plane of the free surface as indicated in (3). (b) In-plane slip amplitude distribution after the first cycle and 400 cycles for slip bands that develop in the grain pair labeled "E" and "F" in Figure 13.

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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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