International Journal of Fatigue 104 (2017) 231–242

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International Journal of Fatigue

journal homepage: www.elsevier .com/locate / i j fa t igue

Strategies for rapid parametric assessment of microstructure-sensitivefatigue for HCP polycrystals

http://dx.doi.org/10.1016/j.ijfatigue.2017.07.0150142-1123/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: mwpriddy@me.msstate.edu (M.W. Priddy), nhpnp3@gatech.

edu (N.H. Paulson), surya.kalidindi@me.gatech.edu (S.R. Kalidindi), david.mcdo-well@me.gatech.edu (D.L. McDowell).

Matthew W. Priddy a,⇑, Noah H. Paulson b, Surya R. Kalidindi b,c, David L. McDowell b,d

aDepartment of Mechanical Engineering, Mississippi State University, Mississippi State, MS 39762, United StatesbGeorge W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, United Statesc School of Computational Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332, United Statesd School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0245, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 April 2017Received in revised form 13 July 2017Accepted 15 July 2017Available online 18 July 2017

Keywords:Crystal plasticityHigh cycle fatigueExtreme value statisticsTitanium alloysHigh throughput

Traditionally, crystal plasticity finite element method (CPFEM) simulations have been used to capture thevariability in the microstructure-scale response of polycrystalline metals. However, these types of simu-lations are computationally expensive and require significant resources. To explore the large space ofmicrostructures (reflecting a variety of grain shape, size, and orientation distributions) within the prac-tical constraints of computational resources, a more efficient strategy is required. The purpose of thiswork is to explore the viability of leveraging the recently established, high-throughput MaterialsKnowledge System (MKS) for fast evaluation of high cycle fatigue (HCF) performance of candidatemicrostructures. More specifically, we explore the feasibility of estimating the mesoscale strain fieldsin hexagonal close packed (HCP) a-titanium polycrystals during HCF loading conditions using the com-putationally low-cost MKS approach, and subsequently estimating the slip system activities via decou-pled numerical integration of the relevant crystal plasticity (CP) constitutive relations. The computedslip activities are then used to arrive at extreme value distributions (EVDs) of fatigue indicator parame-ters (FIPs). As critical validation of this reduced-cost computational strategy, it is shown that the FIP dis-tributions in the HCF regime estimated using this novel strategy are in reasonable agreement with thosecomputed directly using the conventional CPFEM approach. Additionally, the computational advantagesof the MKS and decoupled numerical integration approach over the traditional, computationally-expensive, CPFEM approach are presented and discussed.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Titanium and its alloys have attractive strength-to-weightratios and corrosion resistance, both of which are vital for the aero-space, automotive, and biomedical industries. The a-phase of tita-nium has a hexagonal close packed (HCP) crystal structure whilethe b-phase has a body centered cubic (BCC) crystal structure. Ingeneral, titanium alloys can be categorized as a or near a;aþ b,or b alloys [1]. Titanium alloys exhibit an enormous diversity ofmicrostructure arrangements resulting from the combined effectof composition and thermo-mechanical processing routes [2]. Fur-thermore, these microstructures are characterized by elastic andinelastic anisotropy at multiple length-scales [3–6]. This variationin microstructure and resultant mechanical properties can lead

to competing objectives in the material selection or optimizationprocess.

Fatigue resistance is an important performance characteristicfor titanium alloys because of their applications of use. Experimen-tal evaluation of the fatigue resistance of materials is costly andlabor intensive. In addition, the information extracted is often lim-ited to simple metrics such as the number of cycles to failure.Unfortunately, it can be difficult to relate this data to microstruc-ture variables (e.g., grain size distribution and crystallographic tex-ture) because this requires parametric arrays of experiments andmaterial characterization. Computational frameworks such as thecrystal plasticity finite element method (CPFEM) have been usedrecently for relating microstructure features to macroscopic mate-rial properties [7–9]. CPFEM allows for the estimation of local plas-tic deformation and the associated fatigue indicator parameters(FIPs); however, the practical viability of employing CPFEM formicrostructure-sensitive design of polycrystalline microstructures

232 M.W. Priddy et al. / International Journal of Fatigue 104 (2017) 231–242

for improved high cycle fatigue (HCF) performance is rather lim-ited due to the high demands placed on computational resources.

To circumvent the difficulties mentioned above, an alternativeapproach is proposed in this work that leverages modern machinelearning techniques to significantly reduce the computational costof evaluating the HCF performance of polycrystalline materials.Specifically, this approach takes advantage of a convenient traitof the HCF regime; the magnitudes of the cyclic plastic strainsare generally much lower than the magnitudes of the cyclic elasticstrains. This feature enables the use of the recently developedMaterials Knowledge System (MKS) to efficiently predict the elas-tic strain fields in the polycrystalline microstructures [6,10–14].Stresses may be directly calculated from the elastic strains usingHooke’s Law and resolved onto each slip system. Plastic strainsmay then be estimated by numerically integrating a crystal plastic-ity flow rule over a specified number of cycles. As with traditionalcomputational approaches, these plastic quantities enable thecomputation of FIPs and the rank-ordering of the HCF resistanceof candidate microstructures. In this work, these novel protocolsare demonstrated for the analysis of the HCF resistance of a diverseset of a-titanium microstructures subjected to uniaxial loadingconditions. This represents the first use of the MKS framework toaccelerate fatigue analyses.

2. Background

2.1. Crystal plasticity framework

The anisotropic deformation response of polycrystalline materi-als can be simulated using crystal plasticity [9,15,16] to success-fully rank-order the HCF resistance of different a-titaniummicrostructures. Specifically, the crystal plasticity framework con-sidered in this work was initially developed as a CPFEM model for2D analysis of duplex Ti-6Al-4V [17], was extended to 3-D [18],and then further modified by various authors [19–23]. Mostrecently, Smith et al. [23] calibrated the model to three distincttitanium alloy microstructures via uniaxial tension and fully-reversed tension-compression experimental data (employing peri-odic boundary conditions for uniaxial stress in simulation). In theremainder of this section, the features of this crystal plasticityframework relevant to the present modeling of a-titanium are dis-cussed in detail (features relevant to the modeling of a-b titaniumcolony grains are not discussed).

HCP crystal structures have the following slip systems: (i) basalslip 0001f g 11 �20

� �, (ii) prismatic slip 10 �10

� �11 �20� �

, (iii) ah ipyramidal slip 10 �11

� �11 �20� �

, (iv) first-order cþ ah i pyramidal

slip 10 �11� �

11 �23� �

, and (v) second-order cþ ah i pyramidal slip

11 �22� �

11 �23� �

. These slip planes and their associated slip direc-tions are shown in Fig. 1.

In the a-phase of Ti-6Al-4V, prismatic slip has the smallest crit-ical resolved shear stress (CRSS), followed by basal slip, while thepyramidal families exhibit much higher slip-resistance values.Deformation twinning can also be present in titanium alloys, butis severely diminished with an Al content above 6% in Ti-6Al-4V[24]. Accordingly, we will only consider slip and will neglect twin-ning in this study.

The crystal plasticity model employed in this work is based onthe two-term multiplicative decomposition of the deformationgradient into elastic and plastic parts (i.e. F ¼ Fe � Fp). The plasticvelocity gradient is determined in the intermediate configuration[15], which is both isoclinic and lattice invariant. The symmetricsecond Piola-Kirchhoff stress, rPK2, is obtained by application oflinear elasticity in the intermediate isoclinic configuration, i.e.,

rPK2 ¼ C0 : Ee ð1Þ

where C0 is the fourth-rank elasticity tensor in the intermediateconfiguration. The elastic Green strain is defined by

Ee ¼ 12

Feð ÞT � Fe � Ih i

: ð2Þ

The Cauchy stress ðrÞ can be found by mapping the secondPiola-Kirchhoff stress to the current configuration, i.e.,

r ¼ 1det Feð Þ Fe � rPK2 � Feð ÞT

h i: ð3Þ

Finally, the resolved shear stress on slip system nð Þ is given by

s nð Þ ¼ rPK2 : s nð Þ0 � n nð Þ

0

� �: ð4Þ

where s nð Þ0 and n nð Þ

0 are the slip direction and slip plane normal,respectively, in the intermediate (and reference) configuration.

The isothermal slip system shearing rate, _c nð Þ is defined accord-ing to a power-law flow rule of the form

_c nð Þ ¼ _c0s nð Þ � v nð Þ�� ��� j nð Þ

D nð Þ

M

sgn s nð Þ � v nð Þ� �; ð5Þ

where _c0 is the reference shearing rate, v nð Þ is the back stress, j nð Þ isthe threshold stress, D nð Þ is the drag stress, and M is the inversestrain-rate sensitivity exponent. The threshold stress is defined asthe sum of a Hall-Petch strength term and a softening term dueto breakdown of short-range order, i.e.,

j nð Þ ¼ jyffiffiffid

p þ j nð Þs : ð6Þ

In Eq. (6), jy is the Hall-Petch slope, d is the mean slip distance in

the a-phase, and j nð Þs is a softening parameter. The evolution of

the threshold stress is governed solely by the softening term, whichfollows a dynamic recovery law and takes the form

_j nð Þ ¼ _jsnð Þ ¼ �ljs _c nð Þ�� �� ð7Þ

where l is the softening rate coefficient. The drag stress is the dif-ference between the CRSS (sCRSS) and the initial threshold stress[25], i.e.

D nð Þ ¼ s nð ÞCRSS � j nð Þ��

t¼0: ð8Þ

The drag stress does not evolve (i.e., _D nð Þ ¼ 0), while the backstress is initially set to zero and evolves according to a direct hard-ening/dynamic recovery relation of the form

_v nð Þ ¼ h _c nð Þ � hDv nð Þ _c nð Þ�� ��; ð9Þwhere h is the direct hardening coefficient and hD is the dynamicrecovery coefficient. The parameter values for the crystal plasticityframework described in this section are included below in Table 1[23].

2.2. Materials knowledge system

In recent years, a computationally efficient localization frame-work for hierarchical material microstructure called the MaterialsKnowledge System (MKS) has been developed [6,10–14,26]. TheMKS is an algebraic series capable of predicting response fieldson the mesoscale given the corresponding macroscale averagedloading or boundary conditions. The details of the MKS frameworkrelevant to its application in the present study are briefly describednext.

The response of hierarchical materials systems has beenaddressed using generalized composite theories [27–34], whereina localization tensor relates the material response at the mesoscaleto the macroscale averaged values. In the case of linear-elastic

Fig. 1. Schematic of (a) basal and prismatic slip planes and associated slip directions along with (b) ah i pyramidal and cþ ah i pyramidal slip planes and associated slipdirections.

Table 1Elastic and inelastic crystal plasticity model parameters for a-titanium.

Parameter Value (GPa) Parameter Value (MPa) Parameter Value

c11 172.8 sbasalCRSS350 h 50 MPa

c12 97.9 sprismCRSS275 hD 50 MPa

c13 73.4 spyrhaiCRSS470 d 146 lm

c33 192.3 spyrhaþciCRSS

570 l 2

c44 49.7 js 50

M.W. Priddy et al. / International Journal of Fatigue 104 (2017) 231–242 233

response, the fourth-order localization tensor AðxÞ relates the elas-tic strain at a location in the microstructure, x, to the averagemacroscopic strain imposed on the microstructure:

� xð Þ ¼ A xð Þ : h� xð Þi ð10aÞA xð Þ ¼ I� hC x;x0ð Þ : C 0 x0ð Þi þ hC x;x0ð Þ : C 0 x0ð Þ : C x0;x00ð Þ : C 0 x00ð Þi � . . .

ð10bÞIn Eq. (10b), I is the fourth-rank identity tensor, C 0 xð Þ is the devia-tion in elastic stiffness from some arbitrary reference medium ata location x;C is a symmetrized derivative of the Greens functiondefined with the elastic properties of the reference medium andhf i signifies the ensemble average of a variable f over all spatiallocations in the microstructure.

Eq. (10b) can be transformed to a more convenient formthrough the introduction of the microstructure function, m x;hð Þ[35], which captures the probability density of finding local stateh at the spatial location x. The local state descriptors are selectedin such a manner that allows one to define the local mesoscaleproperties at the spatial location x (these may include phase iden-tifiers, crystal orientation, etc.). Through the introduction ofm x;hð Þ, substitution of r ¼ x� x0 and invocation of the ergodichypothesis, Eqs. (10a) and (10b) can be rewritten as

� xð Þ¼ I�ZR

ZHa r;hð Þm xþr;hð Þdhdr

�

þZR

ZR

ZH

ZH

~a r;r0;h;h0� �m xþr;hð Þm xþrþr0;h0� �

dhdh0drdr0 � . . .

�: h� xð Þi

ð11Þ

where a r;hð Þ and ~a r; r0;h;h0� �are the first- and second-order influ-

ence functions [11], respectively, H is the set of all possible distinct

local states h 2 Hð Þ and R is the set of all vectors r 2 Rð Þ. The first-order influence functions a r;hð Þ quantify the contribution to thelocal response in the current spatial location due to the presenceof local state h at a vector r away. Note that the influence functionsare fourth-rank tensors and satisfy the mapping established in Eq.(10). The influence functions are computationally advantageous asthey are completely independent of microstructure. Eq. (11) is aninfinite series where each successive term captures the influenceof the local topology for higher levels of interactions between thelocal states [14]. It is worth noting that when the variation of localproperties (or contrast) throughout the range of H is low, the seriesin both Eqs. (10b) and (11) can be truncated to the first-order termswith minimal loss of accuracy [6,10,11,14].

Unfortunately, C rð Þ has a singularity as r approaches zero, andthe convergence of the series is highly sensitive to the selectionof the reference medium. The MKS avoids these computationalissues through a calibration of the influence functions using resultsfrom numerical simulations (e.g., based on finite element simula-tions) that include a variety of microstructures and their localresponse fields. Once the influence functions are calibrated, theresulting linkages can be used to predict the response field ofany new microstructure in the materials system at a far lowercomputational cost than using existing numerical frameworks.

Next, a generalized MKS framework is presented which extendsEq. (11) to complex microstructures (e.g., polycrystallinemicrostructures studied in this work). First, m x;hð Þ and a x;hð Þare expressed as Fourier series using products of orthonormal basisover both the local state space and the spatial domain of themicrostructure [6,14]:

m x;hð Þ ¼XL

Xs

MLsQL hð Þvs xð Þ; ð12Þ

Fig. 2. Illustration of the two parameters computed to assess the Fatemi-SocieFatigue Indicator Parameter FIPFSð Þ.

234 M.W. Priddy et al. / International Journal of Fatigue 104 (2017) 231–242

a r;hð Þ ¼XL

Xt

ALtQL hð Þvt rð Þ; ð13Þ

where QL hð Þ and vs xð Þ denote the basis indexed by L and s, respec-tively. vs xð Þ are referred as the indicator basis and return one for allvalues of x within the spatial bin indexed by s, and zero elsewhere.This uniform discretization of the spatial domain allows for theapplication of the FFT algorithm in the evaluation of Eqs. (12) and(13). If the same indicator basis is chosen for QL hð Þ, the local statespace will be binned discretely (e.g., a two phase composite mate-rial). If the desired local state space is crystal lattice orientation, thisbinning strategy proves computationally inefficient as the numberof bins required to accurately represent the orientation space is verylarge (1,944,000 bins for 1� spacing in each of the three Bunge-Euler[36] angles, /1;U and /2, used to describe the lattice orientation in ahexagonal crystal). It is far more efficient in this case to use the gen-eralized spherical harmonics (GSH) [36]; a continuous, periodic, andorthonormal basis for functions on the orientation space definedusing the Bunge-Euler angles.

Through the utilization of the orthogonality of the bases,expressions for ML

s and ALt can be derived from Eqs. (12) and (13)

[14]. Introducing these definitions, the MKS can be expressed as

�s ¼XL

Xt

DNL

ALtM

Lsþt þ

XL

XL0

Xt

Xt0

D2

NLNL0ALL0

t MLsþtM

L0sþtþt0 þ . . .

!

: h�i:ð14Þ

where D is the volume of a spatial bin and NL is a constant that maydepend on L.

2.3. Fatigue indicator parameters

The driving force for growth of long cracks in metals is ade-quately described by the stress intensity factor in the context ofLEFM [37]. In particular, a variety of stress-, strain-, and energy-based relations have been employed to predict fatigue crackgrowth at the macroscopic length scale [38]. The driving force forsmall fatigue crack formation and early growth, however, isstrongly dependent on the local driving force, which in turn isdirectly linked to the microstructure of the material and local cyc-lic slip conditions [39]. Fatigue indicator parameters (FIPs) havebeen used in macroscale and mesoscale data analysis to serve assurrogate measures of the driving force for fatigue crack nucleationand early growth [40,41].

The HCF regime typically requires more than 100,000 cycles tofailure and is characterized by heterogeneous plastic deformationamong grains, with the majority of grains undergoing elastic defor-mation throughout the specimen. The majority of cycles to failurefor HCF involves processes of crack nucleation and early growththrough the first few grains. Additionally, the fatigue crack forma-tion process is strongly affected by the spatial variation ofmicrostructure features (e.g., grain size, orientation, disorientation)that give rise to heterogeneous stress and plastic strain states.These local elevated states of cyclic plastic shear strain can leadto localized damage and, ultimately, the formation of a single dom-inant crack that eventually propagates to failure [42]. For example,for Ti-6Al-4V tested at room temperature with R = 0.1, more than85% of the high-cycle fatigue life (strain amplitude not reported)was spent initiating and propagating small cracks up to 0.5 mmin length [43].

Distinct FIPs were introduced [44] to explicitly account for cyc-lic (reversed) slip per cycle as well as cumulative directional slipand have since been used in conjunction with macroscopic exper-imental [45–50] and computational [51] data of titanium alloys.FIPs have also been combined with finite element simulations to

make life estimates [52,53] and compare the relative fatigue resis-tance of multiple microstructures or materials [54]. In these cases,the selection of a specific FIP is important to reflect the deforma-tion mechanisms that contribute to fatigue crack formation andearly growth. For example, the Fatemi-Socie [40,55] shear-basedFIP has been shown to correlate well in the LCF and HCF regimesfor multiaxial fatigue crack initiation [56] and is defined by [54]

FIPFS ¼ D�cpmax

21þ k

�rnmax

ry

� �; ð15Þ

where Dcpmax is the maximum cyclic plastic shear strain range, rnmax

is the maximum stress normal to the maximum cyclic plastic shearstrain, ry is the macroscopic yield strength of the material and k is aconstant with typical values between 0.5 and 1. The overbarD�cpmax; �rn

max

� �indicates that volumetric averaging should be per-

formed [54]. In this work, the volumetric averaging is performedover 2 � 2 � 2 voxel regions to better account for the stress andplastic strain in a region and not values from a single material point.The Fatemi-Socie FIP correlates well with fatigue crack formationand early growth for metals that exhibit planar slip [25]; a graphicalrepresentation is shown in Fig. 2. Additionally, FIP values can beextended to assist in the calculation of life estimates [52,53,57],which could in turn be used to make a probabilistic prediction ofHCF life and capture the scatter associated with these life measure-ments. The FIPFS in Eq. (15) serves as an effective grain-level surro-gate measure for the cyclic crack tip displacement range of smallcrystallographic fatigue cracks [52]. More recent work has exploredlength – scale dependent FIPs [58], but these models differ from theone presented here because of the inclusion of strain gradienteffects and geometrically-necessary dislocations in the modelformulation.

2.4. Extreme value statistics

Microstructure locations with the lowest resistance to fatiguecrack formation are associated with the largest FIP values identi-fied through Eq. (15). A single FIP value, however, is not sufficientto evaluate a microstructure’s resistance to HCF. Instead, the distri-bution of the most extreme FIP values for a sufficient volume ofmaterial gives some indication of the relative presence of HCF sus-ceptible locations in the microstructure. In previous work [54,59], astatistical approach was taken where multiple instantiations ofeach microstructure were simulated using CPFEM and the maxi-mum FIP values in each were used to perform an EVD analysis.The FIP distributions were fit to a Gumbel distribution [60], i.e.,

FYn ynð Þ ¼ exp �ean yn�unð Þ� �; ð16Þ

M.W. Priddy et al. / International Journal of Fatigue 104 (2017) 231–242 235

where FYn ynð Þ is the probability that Yn will be equal to or less thanyn;un is the characteristic largest value of the sampled population,and an is an inverse measure of dispersion of the largest values ofthe population. The Gumbel distribution is unbounded and theshape of the probability density function is the same regardless ofthe fitting parameters. The maximum FIP values are arranged inincreasing magnitude and their probability estimated by

FYjyj� � ¼ j� 0:3

nþ 0:4; ð17Þ

where n is the total number of simulations and j is the rank-order ofeach maximum FIP value [61]. For plotting and regression purposes,the Gumbel distribution is re-formulated as a linear function of ywith expressions for slope (an) and y-intercept (anun), i.e.,

ln1

ln FYn ynð Þ½ �� �

¼ anyn � anun: ð18Þ

These resultant distributions are the basis for comparing fatigueresistance among different microstructures in this work.

3. Proposed methodology

The main purpose of this work is to explore a novel,computationally-efficient method for rank-ordering the HCF resis-tance of polycrystalline microstructures. The conventionalapproach, shown in the bottom row of Fig. 3, adheres to the follow-ing steps: (i) digital microstructures are obtained, (ii) CPFEM isused to determine the local cyclic plastic strain tensors, (iii) FIPfields are computed for each microstructure, and (iv) extremevalue distributions of FIPs are constructed to rank-order themicrostructures in terms of their HCF resistance. The newapproach explored in this study (also displayed in Fig. 3) utilizesthe MKS method to determine the local total strain fields and cal-culate the local plastic strain tensors through a decoupled integra-tion scheme that employs the relevant constitutive relationsdescribed in Section 2.1. The remainder of this section describesthis new approach in detail.

First, a diverse set of microstructures of interest are identifiedfor a given composition as a function of thermo-mechanical pro-cess path. Traditionally, a representative volume element (RVE) isestablished to capture the microstructure for subsequent computa-tional evaluations. RVEs are intended to approximate the responseof the overall material microstructure by encompassing a volume

Fig. 3. Flowchart for the insertion of MKS into the traditional workflow fo

that is large enough to contain a sufficient number ofmicrostructure-specific features for statistical homogeneity [62].Additionally, RVEs must have measurable properties (e.g., elasticmodulus, thermal conductivity) that are in agreement with theproperties of an extremely large volume of the true materialmicrostructure. Unfortunately, this typically requires performingsimulations on prohibitively large volumes of material, especiallyto describe extreme value phenomena such as HCF. Instead, themicrostructures may be represented by an ensemble of smallerstatistical volume elements (SVEs), which are constructed suchthat their size is sufficient to sample microstructure-specific fea-tures (e.g., grains), but whose individual responses might differfrom that of an RVE. In this framework, a large enough set of SVEsis selected for each microstructure to demonstrate the convergenceof a representative property.

Next, if they have not been previously computed for the specificmaterial system being studied, the MKS influence functions are cal-ibrated with an ensemble of calibration SVEs and their responsesas computed using linear-elastic FE simulations. These simulationsare performed using appropriate periodic boundary conditions forthe applied loading of interest and the elastic model parametersgiven in Table 1. The use of an MKS calibrated with linear-elasticsimulations is justified as the cyclic plastic strains in the HCFregime are expected to be orders of magnitude smaller than theelastic strains; therefore the total strains are approximately equalto the elastic strains. Once the Fourier coefficients of the influencefunction (herein called influence coefficients) are calibrated foreach set of boundary conditions and for all unique componentsof the strain tensor, Eq. (14) is employed to efficiently predict theelastic strain fields in each SVE at the minimum and maximumapplied strains in strain-controlled cyclic loading. The stress tensoris calculated from the strain tensor in each voxel for the localfourth-order elastic stiffness tensor (see Eq. (1)). Assuming a linearrelationship between the initial and final stress tensors, each load-ing segment is discretized into a number of increments. The stresstensor is then used to determine the resolved shear stress in eachslip system (s nð Þ) for each time increment, according to

s nð Þ ¼ r : s nð Þ � n nð Þ� �; ð19Þ

where s nð Þ and n nð Þ are the slip direction and slip plane normal,respectively. The resolved shear stress is then used to solve forthe slip system shearing rate (Eq. (5)) and evolution equations

r producing extreme value distributions of the extreme FIP responses.

236 M.W. Priddy et al. / International Journal of Fatigue 104 (2017) 231–242

(Eqs. (7) and (9)) in each time increment. A forward Euler routine,commonly used in explicit CPFEM simulations [63], is employedto obtain the cumulative plastic strains on the slip systems. Finally,the plastic strain tensor at a time indexed by imay be approximatedas

�pi ¼

Xn

c nð Þi s nð Þ � n nð Þ� �

sym: ð20Þ

from the associated cumulative plastic shear strains. This methodaccounts for the spatial variance of the local fatigue response inthe digital microstructures due to local crystal orientations. As such,the local stress/strain quantities and derivative quantities accountfor the geometric imperfections present in the digital microstruc-ture, which are primarily the grain boundaries for this study.

A graphical representation of this procedure is shown in Fig. 4.The back stress and plastic shear strain are initialized at zero foreach slip system. The plastic strain quantities are orders of magni-tude smaller than the total strain quantities in the HCF regime, andtherefore their impact on redistribution of the stress and totalstrain tensors relative to the elastic solution are minimal. Accord-ingly, phenomena of cyclic stress redistribution and relaxation areneglected in this work, as well as local lattice rotation.

Finally, FIP fields are computed in all SVEs using Eq. (15), andthe extreme value distributions are extracted following the proce-

Fig. 4. Flowchart for the forward Euler integration scheme used to estimate thelocal plastic strain tensor �pð Þ from the local total strain tensor �ð Þ provided by MKS.

dures of Section 2.4. Through the FIP EVDs, microstructures may berank-ordered by their resistance to HCF.

4. Case study

To demonstrate the protocols described in Section 3, four differ-ent a-titaniummicrostructures in three separate loading directionsare rank-ordered by their HCF resistance. Synthetic microstructurevolumes are first generated using the open-source DREAM.3D soft-ware. The MKS is then calibrated with linear-elastic FE simulationsand employed to predict elastic strain fields in the volumes. Stres-ses are calculated from the elastic strain fields and the integrationscheme is used to compute the cyclic plastic strains. These quanti-ties are used to calculate the FIP fields in each SVE. The microstruc-tures and loading directions are rank-ordered by their HCFresistance through a comparison of the FIP EVDs. The details of thiscase study are provided in the remainder of this section.

4.1. Digital microstructures

Four crystallographic textures are selected for analysis in thiswork: (a) random, (b) b-annealed, (c) transverse, and (d) basal/-transverse. The b-annealed texture is taken from previous work[23] while the other textures are extracted from the literature[2,64–66]. All three-dimensional digital microstructures are gener-ated using the open-source DREAM.3D software [67] with inputs ofdesired grain-size, orientation, and misorientation distributions.The grain-size distribution for all microstructures is modeled as alog-normal distribution with a mean and standard deviation of30 lm and 15 lm, respectively. The instantiated microstructurescontain 9,261 hexahedron elements, each with 8 integration points(i.e., C3D8-type in ABAQUS 6.10-1[68]) and an individual sidelength of 10 lm. Each digital SVE has a total side-length of210 lm in the x-, y-, and z-directions. A total of 500 SVEs areinstantiated for each of the four microstructures and their statisticsare compared to the desired statistics. Representative pole figuresfor each microstructure analyzed in this work are shown in Fig. 5.DREAM.3D is used to generate the fully periodic microstructuresemployed in this work.

To determine how many SVEs are adequate to approximate anRVE, the mean and standard deviation of a particular value of inter-est (elastic stiffness in this instance) are compared with tolerancevalues [69]. The details of these calculations are provided in theAppendix. As shown in Fig. 6, the mean of the elastic stiffness con-verges extremely quickly (in fewer than 50 simulations) for load-ing in each direction while the standard deviation requiresapproximately 250 simulations for each loading direction. Similarresults are found for the other textures.

4.2. MKS model calibration

The MKS framework described in Section 2.2 was successfullyemployed in prior work for the prediction of elastic strain fieldsin polycrstalline HCP microstructures with a wide range elasticconstants, resulting in a variety of contrasts between local con-stituents [6,14]. In this study, however, the influence coefficientsof Eq. (13) are re-calibrated for the set of elastic parameters inTable 1. Truncation to the first term of Eq. (14) is justified as a-titanium single crystals exhibit low contrast (e.g., the ratio of thehighest modulus to the lowest modulus is approximately 1.40);prior work has shown that this truncation provides excellentresults for composite systems with low to moderate contrasts[6,10–12,14]. Consequently, the influence coefficients calibratedin this work are expected to provide high prediction accuracy forthe textures under consideration. Furthermore, only fifteen GSH

Fig. 5. Representative pole figures for (a) b-annealed, (b) basal/transverse, (c) transverse, and (d) random texture inputs to DREAM.3D.

M.W. Priddy et al. / International Journal of Fatigue 104 (2017) 231–242 237

basis functions are employed in Eqs. (12) and (13), as it has beenpreviously demonstrated that this is sufficient for the predictionof the elastic response of hexagonal polycrystals [6]. The influencecoefficients are calibrated for all unique components of the straintensor for three different loading directions.

The influence coefficients are calibrated using the linear-elasticFE responses (using ABAQUS) of uniform-textured SVEs (generatedvia DREAM.3D). For each loading direction (x, y and z), periodic,displacement-controlled boundary-conditions are employed such

that the macroscopic strain tensor only has only one non-zerocomponent (in the direction of loading). It is noted that the MKSinfluence coefficients may be calibrated for any desired macro-scopic strain tensor; uniaxial strain is selected in this work becauseit has been successfully employed in many previous studies [6,10–12,14]. The number of SVEs used in calibration of the MKS is deter-mined through examining the mean and maximum error metricsin the strain field predictions for different sets of validation SVEs.From this analysis, it was seen that an ensemble of 400 SVEs is suf-

(a) (b)

Fig. 6. The (a) mean and (b) standard deviation of the effective elastic modulus values for N number of MKS simulations of the b-annealed microstructure and for each loadingdirection.

238 M.W. Priddy et al. / International Journal of Fatigue 104 (2017) 231–242

ficient to calibrate the influence coefficients. When elastic strainfields are compared for 100 SVEs not included in the calibrationset, the mean voxel-to-voxel difference between MKS and linear-elastic FEM responses is 0.28% and the maximum error permicrostructure averages to only 1.50% for the �11 component. Fur-thermore, the MKS evaluation requires only 0.7 s on a single pro-cessor compared to the 10 min on four processors required forthe linear-elastic FE simulation of a single SVE. The local stressfields are also predicted with high accuracy. The mean and averagemaximum error per microstructure are 1.00% and 2.37%, respec-tively, for the r11 fields from 100 instantiations of the b-annealedmicrostructure simulated with CPFEM (subject to the cyclic load-ing described in Section 4.3) and the MKS computed fields. Theseresults indicate that stress relaxation or redistribution are minimalin the CPFEM simulations, which is an expected result for the HCFregime. Furthermore, the accuracy of the prediction for the b-annealed microstructure demonstrates the predictive capabilitiesof the MKS framework for new microstructures.

4.3. Auxiliary decoupled estimation of plastic strains

In this study, cyclic plastic strain fields are computed from thestress fields in each SVE using the decoupled numerical integrationscheme described in Section 3. Three cycles of fully reversedR ¼ �1ð Þ cyclic loading are performed (see Fig. (7)) to ensure thatcyclic plastic strains saturate before FIPs are computed.Furthermore, loading is performed to 0.5% strain amplitude so thatj �p j�j �e j and stress redistribution is minimal. This roughly

Fig. 7. Schematic of the imposed strain versus time curve used in the work.

corresponds to 60–67% of the applied strain to yield, dependingon the specific microstructure and loading direction. As a conse-quence, the stress and elastic strain fields are identical at pointsA, C, and E for any selected microstructure. However, this is notthe case for the corresponding plastic strain fields; their evolutionis a result of the plasticity-related fields computed using the crystalplasticity framework presented earlier. In this decoupled numeri-cal integration algorithm, the stress tensor is discretized into 50increments for each loading segment (e.g., A-B, B-C, etc.).

To check the accuracy of the predicted plastic strain fields,CPFEM simulations are performed for 100 SVEs belonging to theb-annealed microstructure. The crystal plasticity formulationdescribed in Section 2.1 is employed in Abaqus/Standard [68]through a User MATerial subroutine (UMAT) [70]. Three cycles offully reversed loading are applied to the volumes using the samedisplacement-controlled, uniaxial-strain boundary-conditionsdescribed in Section 4.2. Fig. 8 displays the plastic strain fieldsobtained using the novel protocols and CPFEM for a slice of anexample SVE.

Fig. 8 demonstrates that the plastic strain fields computed usingthe two approaches are indeed very similar. Note that the magni-tudes of the plastic strains are several orders smaller than the elas-tic strain fields (the average �11 strain is 0.5%). In this example SVE,the mean and maximum relative errors (of the novel approach ver-sus CPFEM) do not exceed 0.39% and 0.84%, respectively, for anycomponent of the plastic strain tensor. This error is defined asthe local difference in plastic strains normalized by the maximumplastic strain obtained via CPFEM. This definition ensures that rel-ative error is well characterized for locations in the volumeexhibiting high levels of plastic strain. Later results will demon-strate that this level of agreement is sufficient to reliably rank-order the HCF performance of the microstructures studied in thiswork. The main benefit of the proposed approach continues to bethe dramatic savings in computational cost. The CPFEM simulationrequires approximately 45 min to complete on four processors(and a total of 8 ABAQUS licenses) per SVE, while the decouplednumerical integration scheme requires 70 s on a single processor.

4.4. High cycle fatigue analysis

Given the results of Section 4.3, FIP fields are computed for allSVEs in each loading direction using Eq. (15). FIP EVDs are then cal-culated by compiling the maximum FIP value in each SVE. At thisstage, it would be desirable to compare the FIP EVDs resulting from

Fig. 8. A two-dimensional cross-section comparison of the �p11 values from CPFEM (left) and MKS plus decoupled numerical integration (right).

M.W. Priddy et al. / International Journal of Fatigue 104 (2017) 231–242 239

the novel protocols described in this work and the traditionalCPFEM approach. Therefore, FIP EVDs are obtained from the CPFEMpredicted plastic strain fields computed in the previous section for100 b-annealed SVEs. In general, the FIP EVDs from the new proto-col closely match the CPFEM results, as demonstrated in Fig. 9. Thetwo approaches indicate identical rank-ordering of HCF resistancebased on a specified loading direction at a selected high probabilityof failure. At a low probability of failure, however, the rank-ordering is different between the two methods. This discrepancycan be traced back to the difference in the local plastic strain ten-sors, and that both methods approximate the local plastic straintensors via Eq. (20). The plastic strain quantities are an approxima-tion because they are a function of the time step. The time step canvary within CPFEM simulation depending on the integration crite-ria and the nonlinearity of the simultaneous equations, while theproposed methodology uses an explicit integration method withfixed time increments.

Due to the high computational costs of the conventionalapproach (which provide the main motivation for this work), com-parisons between the two approaches were not performed for allmicrostructures studied here. Instead, the FIP EVDs were con-structed using the novel protocols presented in this work for allfour microstructure classes, each with a full ensemble of 500 SVEs(note that comparison presented in Fig. 9 used only 100 SVEs).Fig. 10 compares the resultant FIP EVDs for all four microstructuresand three loading directions. As expected, the FIP EVDs of the ran-dom textured microstructure are nearly coincident for all loadingdirections. The basal/transverse and b-annealed textured

(a)

Fig. 9. Comparison of the b-annealed microstructure extreme value distribution Fatemintegration and (b) CPFEM.

microstructures exhibit similar FIP EVD responses, with the lowestfatigue resistance for loading in the z-direction.

Finally, the HCF resistance of the transverse texturedmicrostructure is distinctly different from the other microstruc-tures, exhibiting a significant positive correlation with the effectivemodulus in the direction of loading. This microstructure is distinctas compared to the other three due to a strong c-axis fiber texturecomponent parallel to the sample x-axis. This particular texturefeature essentially dominates this microstructure. As a result,when this sample is loaded along the x-axis, we should expect aminimum amount of interactions between neighboring grains,especially in the elastic regime due to the transverse isotropy ofthe elastic response in each crystal in the plane normal to the c-axis. It is therefore not surprising that x-axis loading producesthe best HCF response for this microstructure with most of theFIP values being lower compared to the loading in the other direc-tions. Note that this is the case despite the higher stiffness of thematerial along the x-axis (i.e., stresses are higher because the samestrain amplitude is imposed in all cases) as compared to the otherdirections. It is also interesting to note that there are a few high FIPvalues in the x-axis loaded microstructure that are comparable tothe values obtained in the other two loading directions on thesame microstructure. This is because the high FIP values wouldcorrespond to locations of high local plastic strains. In the case ofx-axis loading, once plastic deformation initiates, transverseisotropy in the local response of the c-axis grains is completely lost(note that only the elastic response is transversely isotropic in theHCP crystals) and the degree of interactions between neighboring

(b)

i-Socie FIP plot with 100 SVEs obtained via (a) MKS plus decoupled numerical

(a) (b)

(c) (d)

Fig. 10. The extreme value distribution Fatemi-Socie FIP plot generated from the MKS and the decoupled numerical integration scheme with 500 SVEs for microstructureswith the following textures: (a) basal/transverse, (b) transverse, (c) b-annealed and (d) random.

240 M.W. Priddy et al. / International Journal of Fatigue 104 (2017) 231–242

grains dramatically increases. As a result of these strong interac-tions, the local plastic strains can be high in this microstructure.Overall, it should be noted that this microstructure provides thebest HCF response out of all of the microstructures studied. Thisimproved performance is largely attributed to lower levels of inter-actions between the neighboring grains (in the HCF loading regimedominated by elasticity).

5. Conclusions

A newmethodology that employs MKS and a decoupled numer-ical integration scheme is presented to reliably predict cyclic plas-tic strain fields in hexagonal, a-titanium, polycrystallineaggregates under low-amplitude loading conditions. These quanti-ties inform the computation of FIP distributions which are used torank-order the HCF resistance of polycrystalline microstructures.This analysis is specifically designed to support rapid microstruc-ture selection and optimization, assuming the proper materialand application have been identified. These new protocols aredemonstrated on an ensemble comprising over 6000 individualSVEs; the same task would demandmajor computational resourceswhen using traditional CPFEM-based approaches. In fact, the lowcomputational cost allows for the consideration of such a largeensemble of microstructures in the present work. The approximatespeed-up of the new protocols is approximately 40� versus tradi-tional CPFEM. The study also identifies the critical role of grain

interactions in the HCF performance through a consideration oftwelve different conditions (four microstructures, each loaded inthree directions). While this study has established the feasibilityof the new protocols, much additional work is needed to extractnew insights into the HCF response of polycrystalline microstruc-tures. These will be targeted in future studies where the new pro-tocols developed in this work will be employed on a much largerensemble of microstructures. Additionally, the computational fati-gue response needs to be validated against experimental data, suchas damage measurements relevant to HCF.

Acknowledgments

This work was supported by the National Science Foundationunder Grant No. CMMI-1333083. Any opinions, findings, and con-clusions or recommendations expressed in this material are thoseof the authors and do not necessarily reflect the views of theNational Science Foundation. MWP would like to thank Mr. Ben-jamin Smith for experimental characterizations of the b-annealedmicrostructure.

Appendix A

In this work, the number of SVEs required to approximate theRVE for a microstructure of interest is selected by examining theconvergence of the mean and standard deviation of the effective

Table 2Mean directional effective elastic modulus ( std. dev.) for each microstructure.

Texture Ex (GPa) Ey (GPa) Ez (GPa)

Random 123.4 1.2 123.4 1.2 120.0 1.0b-annealed 116.5 0.8 118.6 1.0 135.0 1.7

Basal/transverse 122.8 2.0 114.1 0.2 135.6 1.8Transverse 144.8 0.6 112.7 0.2 112.8 0.2

M.W. Priddy et al. / International Journal of Fatigue 104 (2017) 231–242 241

elastic modulus in each loading direction. In this section, protocolsare set forth to calculate these moduli values for the boundary con-ditions employed in this work.

HCP crystal structures have a transversely isotropic elasticresponse, meaning that the elastic properties exhibit rotationalsymmetry around the HCP crystal c-axis. For a-titanium, the elasticmodulus can vary between 104 and 146 GPa, the shear modulusbetween 40 and 47 GPa, and Poissons ratio between 0.265 and0.337. This degree of anisotropy for the elastic modulus (40%)and the shear modulus (18%) is much higher than that of otherHCP metals [5] and can have a pronounced effect on the materialdeformation response.

Mechanical properties (such as elastic stiffness) can be deter-mined from datasets generated in this study. Since the boundaryconditions used in this work result in non-zero normal stress com-ponents in all normal directions, additional steps are taken todetermine directional effective elastic modulus values. The MKSdetermines local quantities, therefore the stress and strain compo-nents are volume-averaged to obtain the macroscopic quantities

(e.g., hriji ¼PN

k¼1rkij=N where N is the total number of centroidal

averaged values for uniform voxel size). The averaged stress andstrain quantities are then used to determine the effective stiffnesscomponents. The effective stiffness tensor can then be inverted torecover the effective compliance tensor. Subsequently, the effec-tive elastic modulus values for each loading direction are extractedfrom the diagonal components of the compliance tensor. The meanand standard deviation of the effective elastic modulus values foreach loading direction and microstructure are presented in Table 2.

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