Predicting fatigue crack initiation from coupled microstructure and corrosion morphology effects

Engineering Fracture Mechanics 220 (2019) 106661

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Predicting fatigue crack initiation from coupled microstructure and corrosion morphology effects Andrea Nicolasa, Noelle Easter C. Cob, James T. Burnsb, Michael D. Sangida, a b

T

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School of Aeronautics and Astronautics, Purdue University, 701 W. Stadium Ave, West Lafayette, IN 47907, USA Department of Materials Science and Engineering, University of Virginia, 395 McCormick Road, Charlottesville, VA 22904, USA

A R T IC LE I N F O

ABS TRA CT

Keywords: Fatigue crack initiation Corrosion morphology Crystal plasticity modeling X-ray computer tomography Microstructure

The onset of fatigue crack initiation is driven by the microstructure and the corrosion morphology; however, these mechanisms are rarely studied synergistically. In this work, characterizations of the microstructural features and the corrosion morphology resulting from two different environmental exposures are instantiated into crystal plasticity models. Fatigue indicator parameters (FIPs) are calculated from the micromechanical fields to quantify failure. The FIPs compare favorably to predict the experimentally observed location of fatigue crack initiation. These results show the potential behind analyzing environmentally-assisted fatigue crack initiation from a multivariable perspective.

1. Introduction Corrosion in aging aircraft and infrastructure is currently a 20-billion-dollar problem [1], due to the increased maintenance hours needed to preserve flight safety in fleets exposed to harsh environmental conditions. Recent aircraft teardown analysis demonstrated that corrosion features initiate roughly 80% of the observed fatigue cracks [2]. The current maintenance and repair protocols are largely focused on the outright removal of corrosion upon discovery [3], since there is not a clear understanding of the mechanisms driving the transition from corrosion damage to crack initiation. Such an approach can be inefficient with respect to cost and airframe availability; furthermore in some instances, it may actually be detrimental to the overall service life of components [2,3], due to the higher stresses arising from generalized thinning in the material. In the near-term, these shortcomings motivate the extension of linear elastic fracture mechanics (LEFM) based damage tolerant approaches to safely forecast the remaining fatigue life of corroded components [4–18]. However, long term goals included structural life management concepts such as production and performance digital twins that show great potential of reducing costs given their tailored approach towards the maintenance and repair needs of each individual component or system [19]. Rigorous predictions require accurate inputs as to the as-built part geometry, environmental history, loading history, and the material pedigree. In this study, microstructure-based modeling is employed for specimens with representative corrosion morphologies, in order to ascertain the model’s ability to predict fatigue crack initiation. Moreover, the model is used to help decouple the mechanisms driving crack initiation in corroded materials. The mechanisms behind fatigue crack initiation in pristine materials has been thoroughly studied by the community both from an experimental and a computational perspective; for a detailed review, please refer to [20]. As early as 1903, Ewing and Humfrey concluded that fatigue crack initiation is mediated from a slip-based damage mechanism [21]. Experimental studies have shown that microstructural variables, such as the grain sizes [22] or types of grain boundaries [23,24], affect the dislocation activity and

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Corresponding author. E-mail address: [email protected] (M.D. Sangid).

https://doi.org/10.1016/j.engfracmech.2019.106661 Received 19 May 2019; Received in revised form 31 August 2019; Accepted 3 September 2019 Available online 07 September 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Fracture Mechanics 220 (2019) 106661

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Nomenclature

height at point i slip system shear strain rate of slip system α reference shear strain rate accumulated plastic shear strain Lagrangian total, elastic, and plastic strain plastic strain rate reference and asymptotic hardening rate Poisson ratio Cauchy stress tensor normal stress acting on slip plane p yield stress shear stress and critical resolved shear stress on slip system α τo , τ1 reference and asymptotic shear stress DP discrete pitting EBSD electron backscatter diffraction EVP-FFT Elasto Viscoplastic Fast Fourier Transform FIS fissure corrosion FIP fatigue indicator parameter GND geometrically necessary dislocations HEDM high energy x-ray diffraction microscopy IPF inverse pole figure L long rolling direction LT long transverse SEM scanning electron microscopy TL transverse long TS transverse short WLI white light interferometry XCT X-ray computer tomography

zi α γ̇α γ̇0 Γ ε ,ε el ,ε pl ε ̇pl θ0 , θ1 ν σ p σnor σY α τ α , τCRSS

AS

total accumulated shear strain across all slip systems b rate sensitivity exponent C stiffness tensor D number of slip systems height difference at point i dz i ¯ dz average height difference Ė macroscopic strain rate I pixel intensity I¯ average pixel intensity i point at cracked surface Fatemi-Socie scaling factor for normal stress k M symmetric Schmid tensor m slip direction vector n slip normal vector OP slip plane with maximum accumulated shear strain amongst planes subject to normal stresses ¯ OPED average opening plane energy density p slip plane positive integer q SI ,Sdz intensity and height difference standard deviation SOPED opening plane energy density standard deviation SP slip plane with maximum accumulated shear strain SS slip system with maximum accumulated plastic shear strain SSED, SPED, ASED, OPED energetic equivalents of SS, SP, AS, OP, respectively, in terms of energy density t , Δt time, time increment x spatial point in 3D reconstructions Y Young’s modulus

subsequently the onset of crack initiation. Engineering alloys are tailored with complicated microstructures, in which a collection of attributes determine the crack initiation event. Computational mechanics models allow the analysis of competing mechanisms that would be difficult to isolate experimentally and have been used to address technical issues pertaining to crack initiation. These include developing a rank order of fatigue performance based on microstructure ensembles [25], such as-large-as grains in Ni-based superalloys [26], rogue grain combinations leading to dwell effects in Ti alloys [27], and critical pore size in additive manufacturing [28], to name a few. In each of these models, the slip-based deformation around the microstructural attributes results in heterogeneous strain accumulation and associated crack initiation event. Surrogate fatigue metrics have been proposed to identify the microstructure response to fatigue, which are known as Fatigue indicator parameters (FIPs) [29]. Various FIPs have been proposed and compared to assess their predictive nature [30,31]. The use of both computational mechanics models and crack initiation metrics have demonstrated the prognosis capabilities towards the concept of a digital twin. However, such efforts have not been performed to understand the fatigue crack initiation process when it is impacted by both microstructural attributes and localized corrosion damage. The fatigue crack initiation from localized corrosion damage has been of special interest in the 7xxx aluminum alloys (AA7xxx) series, which are commonly used alloys in aerospace applications due to their strength-to-weight ratio and their resistance to corrosion. In particular, AA7050-T7451 is of interest, due to its prevalence in aerospace stemming from its relative lack of quench sensitivity and resistance to corrosion / stress-corrosion cracking. This alloy contains several types of secondary phases: ~5 nm strengthening precipitates, such as MgZn2; ~20 nm recrystallization-controlling dispersoids, such as Al3Zr; and ~5 μm constituent particles, such as Al7Cu2Fe, Mg2Si, and Al2CuFe [32]. In the pristine condition (e.g. not corroded), AA7xxx alloys typically nucleate fatigue cracks on the surface at large constituent particles [33,15]. These brittle particles are stress concentrators [34], which is influenced by both its irregular geometry and the surrounding microstructure [35]. However, research has demonstrated that when present, corrosion damage will serve as preferential sites for fatigue crack nucleation and reduce the fatigue life due to the associated increase in local stress concentration [36]. Even small scale localized corrosion damage (≈25–50 μm) has been shown to lead to nearzero initiation lives, thus greatly reducing the total fatigue life of the material [37]. Generally, when exposed to a corrosive environment localized pitting will evolve at or proximate to the constituent particles, due to the fact that these particles are anodic or cathodic relative to the matrix [38,39]. In AA7050 the Al7Cu2Fe constituent particles are the main precursors to pitting due to their higher volume content (0.43%) [40], their larger size (~10 μm), and their cathodic behavior relative to the matrix [41]. It is necessary to understand the physics behind the pit-to-crack transition, in order to predict the location of damage nucleation 2

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and the remaining life of a corroded component, since the total life of corroded materials is governed by the crack initiation life. Generally, studies have concentrated on the geometry of the corrosion features as an indicator of fatigue life; specifically, the pit size is used as the crack size in fracture mechanics-based analyses. Initiation is considered to occur when the calculated stress intensity associated with the geometry, applied load, and flaw size (e.g. pit size) exceeds a threshold value [16]. There are several limitations to this approach: life-limiting pits can be as small as 20 μm [42], and at this length-scale, the material cannot be modeled as a homogeneous solid; a sharp 2D equivalent flaw is used to model a blunt 3D corrosion feature [43]; cracking is considered to immediately begin propagating about the periphery of the pit, which is not consistent with experimental observations [17,44]; and the mechanical effects from other microstructural features, such as the constituent particles, are usually neglected in the formulation of stress concentrations. To improve the understanding of the mechanisms behind the pit-to-crack transition, the 3D corrosion morphology, the constituent particle location, and the underlying microstructure need to be jointly evaluated. The study of the geometry alone is insufficient to capture the regions that may initiate cracking [32], since the impact of constituent particles on the local mechanics needs to be considered [35], as well as the role of local slip activity on the initiation process [45]. In other words, there is a need to study the pitto-crack transition via a multivariable approach, where the different mechanisms affecting the micromechanical fields are simultaneously taken into account [46,47]. To evaluate the mechanisms behind the pit-to-crack transition at the micro-scale, highresolution experimental characterizations of the corrosion morphology, the microstructure, and the constituent particles are necessary. Ideally, such analysis would be performed over a sufficiently large area such that a reasonable number of pits are properly captured. X-ray computer tomography (XCT) [48] is capable of capturing the full tortuosity of corrosion [44], as well as the shapes and location of the constituent particles [49], over regions spanning millimeters with submicron resolutions. The geometric characterizations from XCT can be directly used as input in computer models [50], with the current limitation being that the size of the reconstruction makes any XCT-informed crystal plasticity model computationally expensive. In this paper, our main goal is to investigate the driving mechanisms behind corrosion fatigue crack initiation and evaluate the predictive capability of crystal plasticity models in identifying the location of cracking. To do so, the joint effect that the corrosion topology, the constituent particles, and the microstructure has on the cracking of pre-corroded AA7050 is studied by instantiating a full 3D crystal plasticity model from the experimental characterizations of the corrosion morphology and the microstructure obtained via XCT and electron backscatter diffraction (EBSD). The resulting micromechanical field is then used to calculate several FIPs, which will be used to predict the onset of crack initiation in the material. These results will be compared with experimental results that identify the exact initiation location for the modeled microstructure. As a result, this work has five main goals: 1. Develop a procedure for building an equivalent computer model that can account for the multiple variables driving crack initiation and can ultimately be used as a precursor for highly descriptive digital twins. 2. Evaluate the different FIPs available in the literature and determine the metric that best represents the driving force for fatigue crack initiation from corrosion damage. 3. Evaluate the overall distribution of the FIPs in the reconstructed model, as well as their ability to predict the experimentally observed location of crack initiation. 4. Determine the level of significance of the different factors contributing towards the predictive capabilities of the FIP distributions (comparisons between microstructure, surface topology, and constituent particles). The completion of these goals will help understand how the different variables present in corroded materials affect crack initiation, as well as inform next generation fracture mechanics approaches to predict the remaining life of corroded components. 2. Materials and methodology 2.1. Material and corrosion protocols Four specimens were machined from a 50-mm thick rolled AA7050 plate parallel to the rolling direction (L) and centered at a through-thickness (t) location of t/8. That is, the specimens were machined 2.45 mm from the rolled surface of the plate to minimize residual stress effects. The specimens had a 7.60 mm thickness, uniform gage length of 20.96 mm, and reduced gage width of 7.60 mm. All specimens were polished down to a 600 SiC grit finish. A full description of the specimen geometry and preparation procedures can be found in [32]. For rolled AA7050, the average grain sizes are 20.38 μm in the thickness direction (S), 46.30 μm in the transverse direction (T), and ~1.5 mm in the longitudinal direction (L) [32], where variability of the grain sizes is expected within the anisotropic material [51]. To investigate the effect of the corrosion morphology on crack initiation, two different corrosion protocols were applied on the LS surface of the specimens. The first protocol aimed to achieve a discrete pitting (DP) morphology with isolated small pits, while the second protocol aimed to create a fissure (FIS) morphology with larger coalesced pits. A small area in the LS surface were exposed to specific electrochemical conditions and the rest of the specimens were masked off using a lacquer paint. To achieve the DP morphologies, two specimens, from here on referred to as D1 and D2, were held at −700 mVSCE using a potentiostat and were exposed to 0.5 M NaCl solution for 1.5 h, where the pH was adjusted to 8 by adding NaAlO2. To achieve the FIS morphologies, another two specimens, from here on referred to as F1 and F2, were exposed to a film of electrolyte with 1 M NaCl + 0.022 M AlCl3 + 0.05 M K2S2O8 solution for 168 h (7 days) while inside a relative humidity (RH) chamber. A stable electrolyte film was achieved on the FIS specimens by setting the RH to 96% and the temperature to 30 °C. Additional details of the electrolyte setup and 3

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replacement of the electrolyte film can be found in [32]. After the specimens were corroded, their surfaces were cleaned with nitric acid (HNO3) for a maximum of 3 min to remove any corrosion product, followed by step-wise ultrasonic cleaning using deionized water, then acetone, and finally methanol, in order to remove the corrosion product buildup. The final corrosion morphology in each specimen was characterized via optical microscopy, white light interferometry (WLI) at 5× and 20×, and scanning electron microscopy (SEM). After the surface characterizations, all specimens were subjected to fatigue cycling in a controlled environment until failure. The RH was kept at 90% during the entire duration of the fatigue test. The maximum stress applied was 200 MPa with a frequency of 20 Hz and a fatigue loading ratio R of 0.5. All fatigue tests were performed in a high RH of 90% up to the final failure event. A programmed loading sequence was used to create distinct fatigue markings (e.g. markerbands) at the fracture surface that enable precise identification of the initiation location and quantification of the initiation life. Further details of the markerband protocol and experimental set-up are available in [32,18].

2.2. Experimental characterization To characterize the location of crack initiation, fractography was performed on all specimens using a FEI Quanta 650 SEM with a working distance of 10–15 mm, accelerating voltage of 10 kV, spot size of 4, and a magnification of 100×–250×. The first marker band, generated after ~5000 cycles, was located < 10 μm away from the crack initiation point. This initial marker band, alongside with fractography, helped locate the crack initiation point in each specimen. After fatigue testing, both the lower and upper sections of the fractured specimens were characterized using XCT to identify the 3D corrosion geometry and the constituent particle distribution. An Xradia MicroXCT-200 microscope was used with the following parameters: 80 kV source voltage, 8 source power, 100 μA current, 25–30 mm source-sample distance, and 17–25 mm detectorsample distance. Each section was rotated 180° around the L-direction and imaged for 20–30 s per image for a total of 500 images per characterization. Based on the size of the region of interest (ROI) desired for each specimen, imaging was performed using a either 2× magnification with 2 μm pixel resolution or a 10× magnification with 0.7 μm pixel resolution. The grayscale image stacks were post-processed to remove noise arising from center shift and beam hardening using the TXRM reconstruction program. The image stacks were then sectioned to display a ROI around the known crack initiation points enclosing the most significant corrosion features, and reconstructed into the final 3D geometry using the Avizo software [52]. The final solid 3D volumes had a voxel size of 1.5 μm for specimen D2, 1.6 μm for specimens D1 and F2, and 3 μm for specimen F1. To properly characterize the constituent particles, the sectioned grayscale image stack was post-processed with an algorithm developed in-house. This algorithm first blurs the 3D image stack sufficiently to remove all particles and preserve the grayscale background characterizing the matrix using a pixel-wise adaptive low-pass Wiener filter [53]. Afterwards it subtracts the blurred 3D stack from the original 3D stack, thus leaving only pixels with intensity outliers. To ensure capturing of the brighter pixels, only the pixels with intensity I higher than 1 standard deviation I ≥ I¯ + 1SI are retained, where I¯ is the average intensity of the pixels and SI is the standard deviation of the pixel intensities. Finally, an erosion-dilation procedure is performed on the 3D segmented particles to remove unrealistic single-pixel particles and improve the connectivity inside each particle. This algorithm also enables the verification of the matrix segmentation performed by Avizo by capturing the darker pixels characterizing the gas phase in the blurred 3D stack where I ≤ I¯ − 1SI . Once both the AA7050 matrix and the constituent particles were segmented, the lower and the upper sections of each specimen were aligned, such that the overall distance between the cracked surfaces was minimized, as described in Eq. (1):

Fig. 1. Geometry reconstruction from the post-mortem XCT specimen scans. (a) The specimen halves were aligned to minimize their vertical separation and according to the corrosion front, after which (b) the middle section was filled and the crack plane was calculated. (c) The middle section was further populated with cathodic particles statistically reconstructed from the experimental characterization. 4

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∑ (ziupper − zilower ) → 0,

∀ z iupper > z ilower

(1)

where z is the height of each point i within the cracked surface studied. This technique automated the alignment process between the post-mortem halves. Since the cracked surfaces experienced plasticity, elastic recovery, and other degradation during the cracking and final failure process [54], the cracked surfaces did not have a perfect fit with each other. Therefore, even after performing the distance minimization algorithm between the cracked surfaces, gaps were present between the reconstructed sections, which needed to be filled appropriately with a solid phase, as seen in Fig. 1a. To ensure that the corrosion front was preserved and not inadequately filled by a solid phase, only the points in the upper surface that had a close neighbor with a point in the lower surface were allowed to be connected by a matrix phase, which is described in Eq. (2) as:

¯ + 3Sdz dz i = z iupper − z ilower ≤ dz

(2)

¯ is the average of the height differences dz i , and Sdz is the respective standard deviation. This allowed the sections to be where dz properly connected by a solid phase, while preserving the corrosion topography, as seen in Fig. 1b. The geometry of the corrosion topography was further verified with the optical and WLI surface corrosion images that were obtained prior to fatigue loading, as described in Section 2.1. Afterwards, an equivalent crack plane was calculated by averaging the lower and upper cracked surfaces, such that the points in the crack plane were always in the middle of the cracked surfaces, such that, as seen in Eq. (3): z iCrackPlane = (z iupper + z ilower )/2

(3)

During loading the mid-section of the material experienced particle fallout. Therefore, to ensure a proper representation of the material, the filled mid-section was populated with statistically reconstructed particles, as seen in Fig. 1c, where the statistics necessary to reconstruct these particles, such as the particle sizes and volume fractions, were directly obtained from the experimentally obtained XCT particles in the lower and upper sections of the specimen. Each statistically reconstructed particle was randomly placed within the mid-section of each reconstruction. The stochastic placement of the particles effect on fatigue crack initiation will be further evaluated in Section 3. Finally, to ensure that the smaller tortuosity details of the corrosion morphology were preserved in the mid-section, the corrosion front profile observed via fractography was enforced on the crack plane, as seen in Fig. 2a. This ensured the minimization of geometric uncertainty for the crack morphology in this critical plane of interest. To characterize the microstructure in each specimen, EBSD was performed on the plane parallel and directly adjacent to the fracture surface. To obtain this data the cracked surface of the lower section was polished until a flat surface was achieved, in all instances this was within 100 μm of the cracked surface. To obtain the surface finish necessary for EBSD, the surface was polished down to a 1200 grit, followed by sequential fine polish using diamond suspension with 3 μm, 1 μm, and 0.25 μm sizes. Care was taken to obtain the grain structure underneath, but as close as possible, to the fracture surface. Afterwards the specimen was sonicated with acetone and methanol. The polished specimens were subjected to flat ion milling prior to EBSD characterization. An FEI Quanta 650 FE-SEM was used with the following acquisition settings: 20 kV accelerating voltage, 10–15 mm working distance, 250× magnification, spot size 4, step size 1 μm, and 70° tilt. Only specimen F1 was scanned using a 100× magnification given its larger ROI. Any noise present in the EBSD scan was removed with standard filters available in the HKL Channel 5 Tango software. The final Inverse Pole Figure (IPF) map of the material in the TS plane is shown in Fig. 2b. A critical analysis assumption is that the grain structure present on the fracture surface is accurately reflected by the EBSD data gathered < 100 μm from the fracture surface. Such an

Fig. 2. Post-mortem characterizations of (a) the corrosion front profile, and (b) the grain orientations at the crack plane. Both (a) and (b) are needed to create (c) the final 3D reconstruction containing both the corrosion topography and the local grain structure. The experimentally observed crack initiation point is signaled by a white arrow. 5

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assumption is reasonable given the highly anisotropic nature of the grains in rolled materials. Specifically, the average grain size in the L-direction is ~1.5 mm; as such, it is reasonable to assume that the EBSD data gathered < 100 μm below the surface provides a first-order estimate of the grain structure on the fracture surface. Thus during modeling, the grains were assumed to be perfectly elongated through the thickness of the material for all specimen reconstructions. Therefore, the microstructural data inside each 3D reconstruction consisted of grain orientations extruded in the L direction, as seen in Fig. 2c. The final IPF maps for all specimens can be observed in Fig. 3, for which the average grain sizes in the TS plane were as follows: 35.73 μm for D1, 36.61 μm for D2, 52.04 μm for F1, and 55.4 μm for F2. All specimens exhibited an aspect ratio of 2–2.5. 2.3. Models and simulation The 3D reconstructions from XCT and EBSD were input into one single data structure that contained both the corrosion geometry and the microstructure of the material, as seen in Fig. 4. The resolution of all structures was limited by the XCT reconstructions. The reconstructed structure for each specimen was used as input within an Elasto Viscoplastic Fast Fourier Transform (EVP-FFT) framework [55], which is a spectral solving method [56] capable of converting the convolution integral of crystal plasticity modeling into a tensor multiplication, therefore reducing the computational expense by several orders of magnitude. The effect of corrosion induced compositional changes (that if present, would be localized within < 1 μm), precharged H from the corrosion process (which would outgas during the 2–4 week time between the corroding process and the fatigue testing), and the localized H uptake during the cyclic loading in the moist air environment are all reasonably assumed to have a secondary impact on the grain scale plasticity. As such, these effects are not included in the EVP-FFT mechanical simulations. The primary goal is to predict the location of crack initiation, for this reason, the loading was exaggerated to develop more plasticity within the material, in order to increase the signal to noise ratio. Each simulation was loaded up to 3% strain (530 MPa), past the macroscopic yield point of the material. Full details of the formulation can be found in [55] and [56]. For this model, only the macroscopic strain rate, E ̇ , along the L-direction was enforced, while the other strain rate components were allowed to freely adjust to achieve both compatibility and equilibrium. The EVP-FFT formulation calculates both a compatible strain field and an equilibrated stress field at every spatial point, x , in the model. The elastic relation, in the crystal frame, at time t + Δt is described in Eq. (4) as:

σ (x ) = C (x ): ε el (x ) = C (x ): (ε (x ) − ε pl (x )) = C (x ): (ε (x ) − ε pl, t (x ) − ε ̇pl, t Δt )

(4)

where C (x ) is the fourth-order stiffness tensor, ε pl, t is the plastic strain at time t , and the plastic strain rate is described in Eq. (5) as: D

ε ̇pl (x ) =

D

b

|M α (x ): σ (x )| ⎞ α ⎟ sgn (M (x ): σ (x )) α ⎝ τCRSS (x ) ⎠

∑ M α (x ) γ ̇α (x ) = γ0̇ ∑ M α (x ) ⎛ ⎜

α=1

α=1

(5)

α (x ) and M α (x ) are the critical where D is the number of slip systems, γ̇α is the resolved shear strain rate on a given slip system α , τCRSS resolved shear stress (CRSS) and the symmetric Schmid tensor for the slip system α and spatial point x , respectively, and where ε ̇pl (x ) and σ (x ) are the strain rate and the stress tensors at each spatial point x , respectively. Additionally, γ̇0 is a shear strain rate normalization factor and b is the rate sensitivity exponent. This equation shows that the plastic flow on a given slip system is governed by the resolved shear stress. The resolved shear stress on a slip system α can be determined from the stress tensor σ (x ) , as described in

Fig. 3. EBSD orientation scans for discrete pitting specimens (a) D1 and (b) D2, and for fissure corrosion specimens (c) F1 and (d) F2. 6

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Fig. 4. Full 3D reconstructions and mean crack planes for discrete pitting (D1, D2) and fissure corrosion (F1, F2) specimens. Unique grain ID colors are plotted on each crack plane.

Eq. (6):

τ α (x ) = M α (x ): σ (x ) =

1 α (m (x ) ⊗ nα (x ) + nα (x ) ⊗ mα (x )): σ (x ) 2

nα

(6)

mα

is the slip direction for a slip system α . The AA7050 matrix was modeled as an elastowhere is the normal of the slip plane and viscoplastic medium, in which the hardening behavior is modeled by a generalized Voce law. This hardening law was selected given the reduced amount of variables involved in the calculation, where only four parameters need to be fit, which in turn minimizes the parametric uncertainty of the model. It should be noted that this hardening law is isotropic where each slip system for a given material point point, x , hardens at the same rate. The Voce law is described in Eq. (7) as:

τ (Γ) = τo + (τ1 + θ1 Γ) ⎡1 − e− ⎣

Γθ 0 τ1 ⎤

(7)

⎦

where τ0 and θ0 pertain to the initial yield stress and hardening rate, respectively, and τ1 and θ1 are the asymptotic parameters. The term Γ is the accumulated shear strain. These hardening parameters were obtained from previous studies of the AA7050 matrix across various textures [51], where the hardening parameters were found to be τo = 135.8 MPa, τ1 = 8.5 MPa, θ0 = 3061.6 MPa, and θ1 = 107.6 MPa. For the elastic behavior of the material, the AA7050 matrix was modeled as anisotropic with the components of the stiffness matrix C being obtained from [57] where C11 = 111.2 GPa, C12 = 57.4 GPa, and C44 = 57.4 GPa. The constituent particles, given their brittle nature, were modeled as purely elastic and isotropic, with a Young modulus Y = 160.2 GPa and a Poisson ratio ν = 0.33 [58]. Due to the requirements of the FFT formulation, the material was padded in the L direction, to ensure a continuous load path, until a size of 2q was reached, where q is a positive integer. To prevent both any artificial stress concentrations from the periodic boundary condition and corrosion features from acting as artificial corner notches, each reconstruction was padded with a solid phase on all sides, except for the corroded surface, which was padded with a dummy gas phase. A sensitivity analysis indicated that a minimum padding of 32 voxels was necessary to ensure no cross-talk was present in the micromechanical fields across the periodic domains. The final sizes of each EVP-FFT model were as follows: 340 × 240 × 640 voxels for specimen D1 (1 voxel = 1.6 μm), 368 × 256 × 640 voxels for specimen D2 (1 voxel = 1.5 μm), 392 × 176 × 256 voxels for specimen F1 (1 voxel = 3 μm), and 368 × 368 × 640 voxels for specimen F2 (1 voxel = 1.6 μm). All of the voxel resolutions are a direct result of the pixel sizes in the cleaned grayscale image stacks from XCT imaging. Given that each reconstruction contained over 107 voxels, a parallelization method of the EVP-FFT analysis was used to achieve convergence under reasonable computational times [59]. Each specimen roughly required 7–10 h of runtime using 6 HP compute nodes with two 10-core Intel Xeon-E5 processors (20 cores per node) and 64 GB of memory. 2.4. Crack nucleation metrics In an effort to identify the most likely location for the nucleation event, eight different FIPs were calculated from the micromechanical fields calculated from the simulations. The first four parameters studied, Slip System (SS), Slip Plane (SP), Accumulated Slip (AS), and Opening Plane (OP), evaluate the maximum accumulated shear strain per slip system, slip plane, or material point [30]. The SS metric assumes failure will occur at the slip system with the highest amount of plastic shear, the SP metric assumes failure will occur on the slip plane with the highest accumulated shear over all slip directions. The AS metric quantifies the entire accumulated shear at a material point, x , by summing the shear strains across all slip systems, whereas the OP metric studies the slip plane with the highest combination of accumulated shear and opening stress (i.e. stress normal to the slip plane). The OP metric is 7

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based on the Fatemi-Socie metric for multi-axial damage [60] and applied to the micro-scale analysis [61]. These four slip-based FIPs are described in units of strain (mm/mm). The last four parameters, Slip System Energy Density (SSED), Slip Plane Energy Density (SPED), Accumulated Slip Energy Density (ASED) and Opening Plane Energy Density (OPED), are the energetic counterparts [62] for the first four parameters that represent the dissipated energy over the respective slip system, slip plane, or material point. The energy density is calculated by multiplying the shear strains (mm/mm) with their respective shear stresses (MPa), therefore yielding units of MJ/m3. These FIPs are described in Eq. (8) to Eq. (15) as:

SS = max |Γ α|

(8)

α D

SP = max p

∑ |Γαp|

(9)

α=1

D

AS =

∑ |Γα|

(10)

α=1

Dp

OP = max p

∑ |Γαp| ⎛1 + k ⎜

⎝

α=1

SSED = max

p 〈σnor 〉⎞ σY ⎠ ⎟

(11)

|τ α Γ α|

(12)

α D

SPED = max p

∑ |τpα Γαp|

(13)

α=1

D

ASED =

∑ |τ α Γα|

(14)

α=1 Dp

OPED = max p

∑ |τpα Γαp| ⎛1 + k ⎜

α=1

⎝

p 〈σnor 〉⎞ σY ⎠ ⎟

(15)

where τ is the resolved shear stress, Γ is the accumulated plastic shear strain, α is the slip system number, p is the index of each slip plane, Dp is the number of slip systems associated with a particular slip plane p , D is the total number of slip systems, k is a scaling p 〉 is the opening stress normal to the slip plane p , σY is the yield stress (460 MPa for this material), factor set to 0.5 as seen in [60], 〈σnor and 〈∙〉 are the Macaulay brackets where 〈x 〉 = x if x ≥ 0 , and 〈x 〉 = 0 if x < 0 . As seen in Fig. 5, there are no significant differences between spatial fields for the FIPs analyzed. Asides from some increased heterogeneity observed for the Opening Plane parameters (OP and OPED), which includes additional variables (i.e. the stress normal to the slip plane of interest), all FIP distributions are similar in nature. In fact, the FIP distributions along the corrosion front of the

Fig. 5. Comparison of the eight different Fatigue Indicator Parameters (FIP) in the crack plane of discrete pitting specimen D1. The (d) opening plane energy density FIP is used in further evaluations as a predictor of crack initiation. The experimentally observed crack initiation point is signaled by a white arrow. 8

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material remain identical regardless of the FIPs analyzed; it is only the FIP magnitude that changes. One of the reasons behind the similarity in the FIP fields is the flow rule in Eq. (5), which relates the shear stresses on a particular slip system to the shear strains. This relationship makes the slip-based FIPs similar in nature to the energy density-based FIPs. Moreover, there is usually a single slip system that is significantly more active than the others, which explains the similarity in the FIP fields examining the representative shear strains or energy density over the maximum slip system (SS, SSED), the maximum slip plane (SP, SPED), or at a material point (AS, ASED). In a recent study, the predictive nature of these FIPs were assessed through a Bayesian network when compared to experimental data and it was found that each FIP provided a similar quantified reliability and precision, yet the OPED metric provided the highest degree of mutual information between the predictions and experiments, since it included more variables [31]. Therefore, all subsequent crack initiation studies in this paper were performed using the energy density-based OPED parameter, as it reasonably represents the mechanisms behind the onset of failure, where both the shear along a slip plane and the corresponding opening stress facilitates the opening of a crack. Finally, to properly capture the length-scale of the micromechanical fields driving crack initiation and to remove any artificially high values from the Gibbs effect either at the corrosion front or at grain boundaries as discussed and quantified in [59], several nonlocal regularization schemes were performed. Moreover, the crystal plasticity formulation used is length-scale independent, and hence, the regularization scheme provides an ad hoc means to include a characteristic length scale for crack initiation [63]. The different non-local averaging schemes aim to regularize the FIP values, while preserving the intrinsic microstructure variability, over three types of domains: grains, slip bands, and localized regions within the slip bands. As seen in the schematics in Fig. 6, these domains are captured with the three averaging metrics used: averaging around each point (Fig. 6a), averaging per slip band (Fig. 6b) and averaging per grain (Fig. 6c). The latter two, averaging per slip band and averaging per grain, have been used previously in literature [64] to perform microstructurally sensitive averaging on equiaxed grains. The more refined metric, averaging around each point, is known to adequately smooth out extreme values while at the same time preserving the heterogeneity within the slip bands [28]. When analyzing each of the averaging schemes for all the FIP metrics, it was observed that the regularization around each point was the most suitable for preserving the heterogeneity of the micromechanical field. A comparison of Fig. 7a with Fig. 7b shows that the overall FIP distributions are preserved when using the averaging around each point metric, with the yellow FIP hotspots at the corrosion front still being discernible in Fig. 7b. For the regularization of the FIP metrics over a slip band shown in Fig. 7c, overaveraging has caused some of the yellow FIP hotspots to disappear. This can be explained by the fact that the slip-based averaging metric is performed along a slip band that crosses the entire grain, for which a bigger grain size translates into a larger averaging volume size. Given the high level of morphological anisotropy present in rolled AA7050, where the average grain size is 50 μm but where grains as-large-as 300 μm are present in the microstructure in the TS plane (Fig. 3), there is a high probability of overaveraging for slip planes belonging to larger grains. A similar grain size dependency is observed in the averaging per grain metric in Fig. 7d, where only the small grains preserve the heterogeneity observed in the yellow FIP hotspots. As expected, micromechanical fields, such as strain accumulation, have been experimentally observed to localize within a grain or a given slip system [23], due to the directionality of slip and pile-up of dislocations at grain boundaries. Fig. 7e confirms these observations, with the averaging around each point metric (yellow line) being capable to fully retain the heterogeneity of the voxel values (black line), albeit with lower magnitudes due to the removal of artificially high values; followed by the averaging metrics per slip band (blue line), which partially preserves the FIP heterogeneity across the topology; and with the averaging metrics per grain (grey line) showing significant over-averaging of the FIP distribution across the experimental corrosion topology. Therefore, while all three averaging metrics may be more suitable for microstructures with small equiaxed grains, where the micromechanical fields are likely to span the entire grain, anisotropic-shaped microstructures benefit from the averaging around each point metric, since it preserves the length-scale of the micromechanical fields within the grains. Sensitivity analyses were performed on each specimen to identify the optimal non-local averaging volume necessary when averaging around each point. The results for the sensitivity analysis of the non-local averaging volume (length × width × thickness) for each specimen are as follows: 16 × 16 × 2 voxels for D1 (1 voxel = 1.6 μm), 8 × 8 × 2 voxels for D2 (1 voxel = 1.5 μm), 20 × 20 × 2 voxels for F1 (1 voxel = 3.0 μm), and 8 × 8 × 2 voxels for F2 (1 voxel = 1.6 μm). Therefore, the non-local averaging length-scale of the

Fig. 6. Schematic of different averaging metrics: (a) averaging around each point with the volume determined via a sensitiviy analysis, (b) averaging per slip band, and (c) averaging per grain. 9

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Fig. 7. Comparison of the Opening Plane Energy Density distributions, before and after averaging, in the crack plane of discrete pitting specimen D1. The distributions of (a) the voxel value and the various regularization schemes: (b) around each point, (c) per slip band, and (d) per grain. (e) shows a quantitative comparison of the distributions (a, b, c, d) along the experimental corrosion topology A–A′.

micromechanical fields across all specimens was ~25 μm. 3. Results and discussion The highest non-local averaged FIP values were used to evaluate the locations with the highest likelihood to nucleate a crack, in both the 3D corrosion geometry and the 2D crack plane. Fig. 8 shows that, for the 3D corrosion geometry analysis, the location of crack initiation predicted by the absolute highest FIP values do not necessarily coincide with the location of crack intiation that resulted in a dominate crack and final fracture, as experimentally observed via fractography. In fact, for each specimen reconstruction, the maximum FIP value identifies the most probable point for crack initiation at regions away from the crack plane. However, it is important to note that each simulation does not represent a single value prediction for the location of crack initiation, each simulation provides a set of > 107 calculated values, each of which provides a relative probability of the location of crack initiation. When analyzing other regions with relatively high FIPs, it was found that all crack initiation points are located within the four highest FIP values in each reconstruction. For the crack initiation points in the discrete pitting specimens, D1 was predicted by

Fig. 8. Crack initiation predictions in the entire 3D morphology based on the highest value of the opening plane energy density (averaged around each point), for discrete pitting specimens (a) D1 and (b) D2, and for fissure corrosion specimens (c) F1 and (d) F2. The experimentally observed crack initiation point is signaled by a white arrow. 10

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the third highest value (129 MJ/m3) and D2 was predicted by the fourth highest value (55 MJ/m3). For the crack initiation points in the fissure corrosion specimens, F1 was predicted by the fourth highest value (89 MJ/m3) and F2 was predicted by the third highest value (153 MJ/m3). All of these values, while not being the absolute hotspot, are statistical extremes lying outside 3 standard ¯ = 22 MJ/m3, the standard deviations of the FIP distributions, where the average FIP value across all specimens was found to be OPED 3 deviation was SOPED = 10 MJ/m , and consequently the ± 3SOPED bounding values were between 0 and 52 MJ/m3 across all specimens. For D1, F1, and F2, the calculated FIP value (OPED) corresponding to the location of experimentally observed crack initiation were +6 standard deviations beyond the average value (6SOPED = 82 MJ/m3). The reader should be reminded that there are no negative values present in the FIP calculations, based on the construction of Eqs. (8)–(15). It can be surmised that the FIPs statistically capture the regions with a higher probability of failure. When analyzing the averaged FIPs at the 2D projection of the crack plane, Fig. 9 shows that the highest FIP values accurately predict the location of the experimentally observed crack initiation points for the fissured specimens F1 and F2, but not for the discrete pitting specimens D1 and D2. The discrepancy in crack initiation predictions between the full 3D geometry analysis and the 2D crack plane analysis is mainly due to the increased level of microstructural uncertainty of the spatial points away from the crack plane. This increased uncertainty arises from the assumption of perfectly elongated grains, which is only suitable for spatial points close to the crack plane [54]. In other words, the actual grain structure, in terms of morphologies and orientations, away from the crack plane are unknown, which in turn means that the calculated micromechanical response in the points further away from the middle section have a higher degree of error arising from uncertainty [65]. Conversely, the points at the crack plane have the lowest level of joint geometric and microstructural uncertainty, thus becoming the region with the best capability of predicting crack initiation from calculated FIP values. When analyzing the crack prediction results at the crack plane, the FIPs are capable of predicting crack initiation at the heavily corroded FIS specimens, but not at the less severely corroded DP specimens. This may be due to the micromechanical effect that the constituent particles may have on the less corroded DP specimens, for which pit coalescence and particle fallout has not significantly occurred, especially when compared to the fissured specimens, meaning that there may still be some particles present ahead of the corrosion front [18]. Therefore, to study the role of the constituent particles in the prediction of crack initiation, several representative particles were placed ahead of corrosion front on both a FIS specimen and a DP specimen. As seen in Fig. 1, several particles with an average diameter size of 10 μm, the median diameter of the constituent particles, were placed ahead of the corrosion front of specimens D1 and F2. The crack prediction results from the modeling of particles ahead of the corrosion front were then compared with the results from the modeling of randomly instantiated particles. It was observed in Fig. 10 that while the presence of the particles has a slight effect on the local micromechanics of the material (slightly increasing the values, as seen in Fig. 10c and d, or changing altogether the predicted location of crack initiation, as seen in Fig. 10a and b), the placement of particles at the corrosion front does not improve the ability of the models to predict crack initiation. The lesser effect of the particles is due to the smaller mechanical impact that the 10 μm constituent particles may have relative to larger features such as the grains (30–120 μm) and the corroded geometry (30–150 μm). Therefore, the particles are a secondary mechanism behind the mechanical and geometric heterogeneity in the material. It should be noted that other studies have shown a much larger influence of the particles on the mechanical effect in corroded materials, for which the sizes are comparable to other micromechanical features in the material [66]. Additionally, the non-local regularization schemes further reduce the mechanical

Fig. 9. Crack initiation predictions on the crack planes given the highest value of the opening plane energy density (averaged around each point), for the discrete pitting specimens (a) D1 and (b) D2, and the fissure corrosion specimens (c) F1 and (d) F2. The experimentally observed crack initiation point is signaled by a white arrow. 11

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Fig. 10. Crack initiation predictions given two types of particles locations: (a, c) random, and (b, d) ahead of the corrosion front, for (a, b) discrete pitting specimen D1 and (c, d) fissure corrosion specimen F2. The opening plane energy density distributions were averaged around each point. The experimentally observed crack initiation point is signaled by a white arrow.

heterogeneity produced around the particles, especially since the radius of influence around the particles (~10 μm = ~6 voxels) has a similar or smaller size than the averaging volumes used. Furthermore, the resolution of the tomography reconstructions may have an effect on the modeling ability to predict crack initiation. Whereas for the fissured specimens, the resolution is large enough to capture the tortuosity in the larger geometric features; this may not be the case for the less corroded discrete pitting specimens, for which the tomography resolution may need to be improved to properly capture the smaller geometric features present on the corroded surface. It should be noted that aside from the uncertainty that arises from the perfectly elongated microstructure assumption, there are additional sources of uncertainty that may affect the final modeling results in the material. One of the principal uncertainty sources is the model form error within the crystal plasticity modeling of the material, specifically the use of the simple hardening law, e.g. the generalized Voce law, which isotropically hardens each slip system equally at a material point. However, the fact that there are only four parameters in this relatively simple hardening law lowers the uncertainty that may arise from the calibration of these parameters [67]. Another source of uncertainty is the uniform boundary condition applied to all EVP-FFT models. While the specimens were exposed to a uniform load at the grips, the actual boundary conditions applied to the microstructure of the reconstructed regions are heterogeneous and are dictated by the unknown neighboring grains. Any improvement of the crystal plasticity models would require prior knowledge of the stresses that the neighboring grains apply on the studied grains [68]. However, while the addition of more realistic boundary conditions can improve the overall representation and accuracy of the crystal plasticity model, its effect is more pronounced on the individual voxel values near the applied boundary conditions [59]. The effect of the boundary conditions is minimized when the region of interest is located three grains away from the applied boundary conditions, meaning that if the material is sufficiently padded the results will remain unaffected [59,65]. Another limitation in crystal plasticity modeling is the lack of information regarding pre-existing stresses and the degree of recrystallization in the microstructure studied. For a full 3D description of the residual stresses in the material, which would be of interest when studying 3D corrosion morphologies, a slice-by-slice destruction of the specimen would be necessary for the complete 3D characterization of the microstructure, which could be then used to calculate the geometrically necessary dislocations (GND) [69] and thus the associated stresses. This type of characterization is time consuming and does not allow for any further evaluation in the specimen. Furthermore, as shown in [69], including a spatial description of the GND distributions in the backstress formulation has a minimal effect in the overall mechanical response of the material. Thus, in the present study, any pre-existing stresses are expected to provide a slight increase of the stress concentrations arising from the combined effect of the corrosion morphology and the microstructure, and therefore their additions are not expected to change the results presented in this document. Moreover, even though the uncertainty has been minimized for most of the variables affecting the modeling results in the reconstructed specimens, there will always be an innate amount of uncertainty that may have propagated through the various steps leading to the final micromechanical fields [67]. Therefore, this has an impact on the determination of a single critical FIP value that can be used as a threshold for determining the onset of crack initiation, as it relies heavily on the uncertainty present in the model. Furthermore, it is possible to predict the overall probability of failure in pristine materials, based on the microstructure, while performing uncertainty quantification and propagation analyses [70]. Further work will adapt this life prediction framework onto corroded material, using the critical FIP values with uncertainty quantification [67], as a metric to predict the crack initiation life (which has been experimentally determined) and crack growth in the material. Ideally, the combination of these techniques will 12

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enable accurate life predictions in materials that have experienced corrosion. Lastly, tremendous potential exists for the development and implementation of performance digital twins, which act as digital representations of serializable components or systems. Their aim is to predict future performance based on current knowledge, including real-time sensor information of the mechanical loads and stored log data of the environment conditions. While great progress has been made in the incorporation of sensor data, mechanical loading conditions, and digital microstructures, the incorporation of environmental effects into the structural prognosis capabilities is lagging. Holistically incorporation of environmental effects should include (1) the prediction of the location and evolution of corrosion damage, (2) how such damage impacts the current and future structural integrity of a component, and (3) how environmental conditions during loading impact the rate of crack propagation. The current effort provides critical insights into (2) by investigating the efficacy of microstructural scale EVP-FFT modeling to predict the location of crack formation in a galvanically corroded material during fatigue loading. This work is a foundational first step demonstrating that microstructure-based modeling and the relevant macro- and micro-corrosion morphologies can be used to accurately predict the crack initiation. This knowledge enables future development and use of this approach to address component and application specific challenges. For example, performing such an analysis at various component locations via a statistically instantiated microstructure that has a generic (but statistically representative) corrosion morphology would identify the component locations most prone to corrosion enhanced fatigue initiation. This or similar analyses would enable refinement of the Digital Twin prognosis capability to reduce the uncertainty in the overall life predictions and inspection schedule.

4. Conclusions In this work, XCT and EBSD characterizations have been used as input for EVP-FFT simulations, in order to calculate the micromechanical fields at and near the corroded surface. From the resulting simulations, a series of FIPs were constructed to predict the most probable location of crack initiation in the corroded material. Two different types of corrosion morphologies were investigated, specifically discrete pitting that form over a relative short duration and fissure corrosion that form during a relative long duration due to coalesced pits. The significant conclusions from this study are as follows: 1. It is possible to generate highly detailed crystal plasticity models of corroded materials, which can produce microstructure-based predictions of the mechanisms that drive crack initiation. These simulations provide a definitive step towards prognosis of the remaining life of corroded materials subjected mechanical loading. 2. Four strain based and four energy density based FIP metrics were analyzed, and each metric provided similar qualitative fields of the calculated values. Further, there is no marked difference in the predicted hotspots between the various FIP metrics analyzed. Each of these FIP metrics benefited from the use of a non-local averaging schemes, to regularize spuriously high calculations at individual material points and provide a means of a characteristic length-scale for crack initiation. It was observed that non-local averaging around a point, preserved the heterogeneities in the micromechanical fields around the microstructural and corrosion features, as opposed to slip band or grain averaging schemes. 3. The probable location of crack initiation is properly captured by extreme values in the calculated FIP values within the full 3D distribution of the material. In fact, all experimentally observed sites for crack initiation were at locations where the FIP values were, in general, at least six standard deviations greater than the average value. When confining the model results to analyzing only the experimental crack plane, the highest FIP value in the distribution is able to predict the experimentally observed location of crack initiation for the fissure corrosion specimens. The reason for the improved prognosis behavior is due to the 2D microstructural characterization being performed directly under the crack plane, thus providing the lowest degree of uncertainty in the underlying microstructure. 4. The prediction of crack initiation based on the FIP values coincides with the experimentally observed crack initiation points in the material, with the best results arising from the fissure corrosion specimens. As the fissure corrosion specimen experienced coalesced pitting, thus the model instantiations were able to fully capture their pronounced corrosion tortuosity. Higher resolution tomography characterization is necessary to capture the corrosion topology for the discrete pitting samples, which is anticipated to result in higher reliability for the location of crack initiation. In toto, the current results suggest that the FIP values are a reasonable metric for prognosis activities for the site of crack initiation. 5. The microstructure and the corrosion topography play a vital role in the crack initiation of the material. Through multiple instantiations, the position of the constituent particles were varied within the simulations, although the placement of these particles did not influence the ability of the models to predict the location of crack initiation. It was therefore concluded that the constituent particles play a less significant role in the location of crack initiation, as compared to the underlying grain structure and corrosion topography. In summary, this work further emphasizes that crack initiation in corroded materials is driven by multiple variables, namely the underlying microstructure and the corrosion topology. These results justify further development of microstructure-based prediction methods to better understand the fatigue crack behavior on corroded components, such that ultimately, the uncertainty for life prognosis can be reduced, within a performance digital twin framework, by including the environmentally-assisted surface degradation and morphology in a component or system. Such methods will enhance the mechanistic understanding which will inform near-term modification to existing LEFM prognosis methods and next-generation methods prognosis efforts that aim to explicitly capture microstructural effects on fatigue damage evolution. 13

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Declaration of Competing Interest The authors declare no conflicts of interest in preparing this article. Acknowledgements The authors acknowledge funding from the Office of Naval Research, [N00014-14-1-0544] and [N00014-14-1-0012] under program manager Mr. Bill Nickerson. The authors would like to thank Dr. Andrea Rovinelli for his technical support with the parallelization of the EVP-FFT formulation. Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.engfracmech.2019.106661. References [1] Hertzberg EF, Guo S, Stroh RF, Chan TK, Morris A, Stevenson A. Estimated impact of corrosion on cost and availability of DoD weapon systems: FY2018 Update. Virginia: LMI; 2018. [2] Shoales GA, Fawaz SA, Walters MR. Compilation of damage findings from multiple recent teardown analysis programs. Bridg. Gap between Theory Oper. Pract. 25th ICAF Symp., 2009, p. 187–207. https://dx.doi.org/10.1007/978-90-481-2746-7_11. 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