Probabilistic framework for a microstructure-sensitive fatigue notch factor

International Journal of Fatigue 32 (2010) 1378–1388

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Probabilistic framework for a microstructure-sensitive fatigue notch factor G.M. Owolabi a,b, R. Prasannavenkatesan c, D.L. McDowell b,* a

Department of Mechanical Engineering, Howard University, Washington, DC 20059, USA George W Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA c QuesTek Innovations LLC, Evanston, IL 60201, USA b

a r t i c l e

i n f o

Article history: Received 14 April 2009 Received in revised form 31 December 2009 Accepted 10 February 2010 Available online 17 February 2010 Keywords: Probabilistic mesomechanics Weakest link Microstructure-sensitive Fatigue notch factor Fatigue indicator parameters Fatigue crack formation

a b s t r a c t To advance fatigue life prediction algorithms, more comprehensive knowledge of the intensity and distribution of slip within microstructure at the notch root is desired to quantify notch size effects. This study utilizes 3D computational crystal plasticity to assess the degree of heterogeneity of cyclic plastic deformation as a function of notch size and notch root acuity for notch root strain amplitudes near and below the macroscopic yield strain (high cycle fatigue) and for several realizations of aggregates of grains with random orientation distribution at the notch root for polycrystalline OFHC Cu. By using different notch root radii and microstructure realizations of aggregates of grains, statistical information regarding the distributions of stress/strain gradients and fatigue indicator parameters provides useful insight into the microstructure dependence of driving forces for fatigue crack formation at the scale of mean grain size. Results from simulations within a quantitatively defined notch root damage process zone are used along with a probabilistic mesomechanics approach to quantify notch size effects by defining a new microstructure-sensitive fatigue notch factor that considers the probability distribution of the high cycle fatigue strength. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Use of notch root parameters for purposes of estimating fatigue life is a critical aspect of designing against fatigue failure in notched components. Although the bulk of the component may undergo macroscopic elastic deformation, the highly stressed region in the vicinity of the notch root may experience significant cyclic plastic deformation. Component high cycle fatigue (HCF) failure is generally due to the initiation of cracks at such notches. In many HCF situations, the peak notch root stress is below the macroscopic yield strength, but favorably oriented grains at the notch root undergo cyclic plasticity, crack formation and subsequent growth. In some cases, the local driving force is enhanced by the presence of pores or nonmetallic inclusions. State-of-the-art simplified methods of notch root analysis to evaluate driving forces for crack initiation are based on a global–local argument that assumes that stress and strain components at the notch root can be used to assess the fatigue life of engineering components using stress- or strain-life methods [1–6] that map onto smooth specimen experimental results. Some of these models have shown promising results for two-dimensional geometries un-

* Corresponding author. Tel.: +1 404 894 5128; fax: +1 404 894 0186. E-mail addresses: [email protected] (G.M. Owolabi), rprasanna@ questek.com (R. Prasannavenkatesan), [email protected] (D.L. McDowell). 0142-1123/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2010.02.003

der in-phase multiaxial loading. However, they lack explicit consideration of intrinsic material length scales (e.g., grain size) manifest notch size effects and typically do not consider stress gradients which have been shown to influence the fatigue life of complex notched components [7,8]. The fatigue notch factor, Kf, has commonly been employed to account for both notch root stress concentration and notch size effects; Kf is defined as the ratio of unnotched to notched fatigue strengths at a given number of completely reversed cycles (typically 106 or 107) to crack initiation. The classical Peterson equation [9] relating Kf to the theoretical elastic notch stress concentration factor, Kt, is a first order attempt to incorporate a length scale parameter in describing notch sensitivity as a function of notch size, when used in conjunction with a notch root analysis such as Neuber’s rule [1] or Moski and Glinka’s [2] equivalent strain energy method. Motivated by microscopic/macroscopic observations that fatigue failure occurs by damage accumulation within a finite notch root process zone that can encompass a number of grains, several research studies have explored estimation of Kf as a function of an average stress over a damage process zone [10–21]. One class of models that considers subsurface damage, critical distance theory, has been classified according to point, line and 2D/3D methods [21]. The point method employs a specific distance from the notch root, which may correspond to a grain size [10] or plastic zone size [11], and its corresponding average stress level over the notch root region to obtain Kf and related notch sensitivity

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index for notched components. Assuming that the stress near the notch root decreases linearly, Neuber [10] and Peterson [11] established empirical relations for the notch sensitivity factor as a function of the notch root radius; their relations invoked a material constant, assumed to be related to some microstructure attribute size. For a given material, the value of the fatigue notch factor (and associated notch sensitivity index) are commonly determined from costly, time consuming experiments on notched and smooth specimens at long fatigue lives. This empirical basis inhibits design projections for microstructures that have not yet been processed and tested. The empirical Kf approach reflects notch size effects without linking directly to microstructure and notch root geometry. Moreover, there is substantial uncertainty when estimating Kf and notch sensitivity index using empirical relations [13] if complete data are not available. The line [14,15] and the 2D/3D [16–18] methods use the socalled stress field intensity to define a fatigue notch factor and the failure criterion, defined as the average stress obtained by integrating the notch root stress distribution over the critical distance using macroscopic (continuum elastic–plastic) FE analyses. A weight function is implicitly assumed that accounts for the contribution of stresses in the damage zone and is incorporated into the definition of the notch factor and associated failure criterion; this weight function depends on notch geometry, loading type, boundary conditions and material properties. Although these techniques offer improvement over conventional local approaches, it is still not clear that the size of the damage zone has been adequately defined [19]. Some efforts [19–21] have been made to quantify the size of the damage zone using the concept of effective distance to define Kf and the failure criterion as a function of the stress field intensity. The effective distance, which is regarded as the boundary of the onset of fatigue damage processes, is obtained using a plot of the logarithm of stress distribution at the notch, based on FE analysis, versus the logarithm of distance from the notch root. Although this technique conceptually represents an important step towards prediction of notch sensitivity, most of the models are based on assumptions necessary to simplify the complex nature of the problem. In addition, while a single parameter involving a critical distance may be a viable approach, size effects associated with deformation and damage phenomena may render an elementary single parameter approach such as critical distance too approximate [22]. In HCF, the cyclic plastic strain is quite heterogeneously distributed within the damage process zone, an aspect that plays a strong role in the coupling of notch size and acuity with probability of fatigue crack formation and small crack growth from the notch. This heterogeneity has a close linkage with material microstructure. As a consequence, it is unclear that volume averaged stress or stress field intensity is an appropriate indicator of the damage process zone driving force, whether based on macroscopic constitutive relations or polycrystal plasticity. The heterogeneous distribution of cyclic plastic strain in the microstructure contributes to scatter in the HCF life. To account for this scatter, various probabilistic approaches have been developed [23–26], most of which are based on the weakest link theory. Some of these approaches are based on the distribution of stress [23,24] or strain [25,26] obtained from experimental results or macroscopic finite element analysis. However, typically no physical hypotheses justify this choice. An approach motivated by statistical/probabilistic mesomechanics treatment of the distribution of slip among grains seems closer to physical reality. In [27], the distribution of microscopic yield stress was used to derive a two-scale micromechanical model based on a localization relation between global and grain scale fields that accounts for scatter in HCF life based on a dissipated energy criterion and also for the thermal effects during cyclic loading in a unified framework. The model was based on the hypothesis that HCF damage is localized at the micro-

scopic scale, with characteristic dimension smaller than that of the mesoscopic representative volume element [27]. A similar framework was used in [28] to model multiaxial high cycle fatigue tests, again using a micromechanical approach rather than finite element simulations of fields. In [27], microplasticity was modeled at the scale of the slip bands; it was assumed that plasticity is described by Schmid’s law but only one slip system is active per grain. Treatment of notch sensitivity coupled with intrinsic length scales of grains or other material attributes is ultimately necessary to form a predictive basis for notch size effects in forming and growing small fatigue cracks in real materials. In the present work, we pursue use of more realistic 3D microstructure-sensitive models to identify and classify the damage process zone in terms of the distribution of cyclic plastic strain and its relation to notch root stress concentration and stress/strain gradients in HCF. A new concept is presented for a microstructure-sensitive fatigue notch factor that incorporates distributions of microstructure-scale slip information as a function of notch root acuity and grain size. It is amenable to computational assessment and is benchmarked by comparing to experimental data trends for notch sensitivity in polycrystalline OFHC Cu. 2. Computational micromechanics 2.1. Crystal plasticity Since crystal plasticity models relate the grain scale stress to crystallographic slip response, they are suitable for studying heterogeneity and interaction across grains in the notch root field [29– 31]. The use of crystal plasticity is relevant for accurate determination of the variation of the stress and plastic strain fields at the notch root within the microstructure to estimate the threshold of cyclic plasticity within grains for forming cracks in the high cycle fatigue regime. The crystal plasticity algorithm used here follows earlier developments [32–34]. The kinematics [35] of dislocation glide are employed using the multiplicative decomposition

F ¼ Fe  Fp; 



ð1Þ



where F is the total deformation gradient, F e is the elastic deforma  tion gradient representing elastic stretch and rotation (including rip gid rotation of the lattice), and F is the plastic deformation gradient  that describes the collective effect of dislocation glide along crystallographic planes relative to the fixed lattice in the reference configuration. In Fig. 1, the grids represent the crystal lattice; sa0 and na0 are unit vectors in the slip direction and the slip plane normal direction, respectively, for the ath slip system in the undeformed configuration, while, Sa and na are the corresponding stretched and rotated   vectors in the current configuration. The kinetics of dislocation glide are formulated by the relationships between the resolved shear stresses and the shearing rate along the slip systems. The shearing rate, c_ a , on the ath slip system is described by the power law flow rule





sa  xa m c_ a ¼ c_ 0  a  sgnðsa  xa Þ; g

ð2Þ

where c_ a0 is the reference shearing rate, m is the inverse strain rate sensitivity exponent, is the isotropic drag strength, xa is the back stress on the ath slip system, and sa is the resolved shear stress, given by

sa ¼ r : ðsa  n Þa :

ð3Þ

In Eq. (3), r is the Cauchy stress tensor. A direct hardening-dynamic  recovery format relation is employed for evolution of ga [32], i.e.,

g_ a ¼ H

N X b¼1

qab jc_ b j  Rg a

N X b¼1

jc_ b j;

ð4Þ

G.M. Owolabi et al. / International Journal of Fatigue 32 (2010) 1378–1388

m (α ) = m o(α ) ⋅ (F e ) −1

III. Current, deformed configuration s (α ) = F e ⋅ s (oα )

F = Fe ⋅ F p Fe

mo

mo

so

so

I. Reference, undeformed configuration

II. Intermediate configuration Fp

domly assigned orientation distribution to obtain an initially isotropic effective medium while gathering information regarding variability among instantiations. To reduce computational time, the notched specimen geometries are decomposed into three regions: an outermost region, far from the notch root, where isotropic linear elasticity is used (for the random orientation distribution of grains), an intermediate transition region employing macroscopic J2 cyclic plasticity theory along with isotropic linear elasticity to minimize effects of discontinuity between the exterior field and the notch root domain which employs 3D crystal plasticity simulations, each grain having anisotropic (cubic) elastic behavior. The domain decomposition is shown in Fig. 2 for a double edgenotched plate with notch root radius of 1 mm. The actual dimensions of the crystal plasticity region were chosen in a way that ensures that the distribution of microslip at the notch region is fully captured. Notch root plasticity is small scale in such cases. Remote applied strain e(t) was imposed via displacement boundary conditions on the upper and lower surfaces, with traction-free lateral surfaces. Four-node 3D solid tetrahedron elements were used for meshing in all regions, with a resolution of four elements per grain. It should be noted that the 4-node tetrahedron element suffers volumetric locking in simulating J2 plasticity under large strain, fully

Fig. 1. Kinematics of elastic–plastic deformation of crystalline solid deforming by crystallographic slip (I: undeformed, II: intermediate, and III: deformed configuration).

where qab are components of the hardening matrix and H and R are the direct hardening and dynamic recovery coefficients, respectively, for the isotropic hardening relation. In the present work, qab = 1 such that isotropic hardening is the same on all slip systems. The back stress on each slip system evolves according to a nonlinear kinematic hardening rule of self hardening type, i.e., x_ a ¼ hc_ a  hD xa jc_ a j, where h and hD are the direct hardening and dynamic recovery coefficients, respectively. Even without slip system kinematic hardening, a Bauschinger effect is manifested for the polycrystalline ensemble due to intergranular interactions as grains yield differentially with applied stress. It is noted that many alternative forms have been proposed and developed for the isotropic and kinematic hardening relations. The present forms serve as examples for illustration.

ε(t)

Linear Elastic

J2-plasticity

ρ = 1 mm

Crystalplasticity

20 mm

1380

2.2. 3D finite element analysis Using the constitutive models outlined in the previous section, three-dimensional (3D) finite element calculations are conducted for representative microstructures as a function of notch size and notch root acuity for a range of strain amplitudes below the macroscopic yield strain (high cycle fatigue region). This yield strain ey is defined on the basis of the macroscopic proportional limit. The material used in this study is polycrystalline OFHC copper (Cu) consisting of grains with random crystallographic orientation distribution. For the annealed polycrystalline OFHC Cu with mean grain size of 62 lm, the constants in the flow rule at room temperature are c_ 0 ¼ 0:001 s1 , m = 50, H = 225 MPa, R = 2.05 with no kinematic hardening (h = hD = 0) and initial values ga (0) = 13 MPa and xa (0) = 0 MPa. The cubic single crystal elastic constants for Cu at room temperature are C11 = 150 GPa, C12 = 75 GPa, and C44 = 37.5 GPa. For this material in this condition, ey = 0.0123%. These parameters were optimized by McGinty [32] for large deformation rather than small strain cyclic behavior, but they suffice to demonstrate the concepts involved. In this study we focus on the effects of notch size and acuity as well as the orientation distribution of grains. For each loading condition, a set of 40 different realizations of grains within the notch root region are implemented with ran-

2.5 mm 3.0 mm 5.0 mm Fig. 2. Domain decomposition of half of a double edge-notched plate with semicircular edge notches (the plate is divided into three different regions that employ different elastic and elastic–plastic models). The right boundary is a plane of symmetry.

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developed plasticity. However, this is not an issue in the present study because the local (integration point) plastic strain is very small. The constants for the elastic and elastic–plastic relations used in the outermost linear elastic and intermediate J2 plasticity regions are extracted from the uniaxial tension–compression hysteresis loops obtained using the 3D crystal plasticity models on a cubic volume with an appropriate aggregate of grains subjected to periodic boundary conditions. The crystal plasticity algorithm outlined in Section 2.1 is coded into an ABAQUS [36] User MATerial subroutine (UMAT) [32]. For a given deformation history, the UMAT is called for each load increment and provides updated stress components and state variables at the end of each time step, given the increment of the deformation gradient. The amplitudes of imposed completely reversed nominal axial cyclic strain (Re = 1) are selected to ensure that the macroscopic strain at the notch root is in the vicinity of or below the yield strain of the polycrystalline materials. In other words, only a relatively small number density of grains yield in the notch root region after the initial loading cycle. Three loading cycles are applied to the specimen in the axial direction. A strain rate of 104 s1 under room temperature conditions is used in all the simulations. 3. Fatigue indicator parameters The distributions of microplasticity (slip) among grains at the notch root enable assessment of statistical distributions of Fatigue Indicator Parameters (FIPs) in the fatigue process zone. FIPs are employed as a means of comparing fatigue susceptibility of various microstructures and as a basis for estimating failure probabilities over a polycrystalline ensemble of grains. For our purposes, several candidate FIPs could be used. One such FIP that is useful for fatigue in shear-dominated crack formation is the Fatemi–Socie (FS) critical plane parameter [37], given by

   Dcpmax rmax 1þj n ¼ DC : 2 ry

ð5Þ

Here, j is a coefficient that moderates the effect of normal stress to the plane of maximum plastic shear strain range. The FS parameter has been applied to multiaxial fatigue life prediction for polycrystals [38,39] that exhibit shear-dominated fatigue crack initiation, including microstructure-scale applications [32,40–42]. In this 

Dcpmax , 2

correspondstudy, the nonlocal plastic shear strain amplitude, ing to the plane of maximum plastic shear strain range, is averaged over a selected volume (e.g., grain size) and is computed as a function of distance from the notch root. The nonlocal peak stress nor mal to the plane of maximum plastic shear strain range, rmax , is n also computed and is normalized by the polycrystalline cyclic yield strength, ry, in the FS parameter. To obtain the nonlocal FS parameter, DC, we consider a set of planes at each integration point within the nonlocal averaging region, similar to the approach in Ref. [42]. At each integration point within this averaging volume, the plastic shear strain cph on the maximum plane is calculated by resolving the plastic strain tensor epij onto it at each integration point, i.e.,

cph ¼ ni epij tj h ¼ 1 . . . N;

ð6Þ

where ni is the unit normal vector on plane h, tj is a unit tangent vector in the considered direction along the plane, and N is the number of ‘‘bins” of discrete planes sampled. The nonlocal average plastic shear strain associated with the h plane is calculated by averaging the plastic shear strain on the h plane over the volume of the nonlocal region, i.e.,

cp h ¼

1 Vb

Z Vb

cph dV:

ð7Þ

The nonlocal cyclic plastic shear strain range for each plane, averaged over volume V b , is then calculated using Eq. (7) in the third cycle of the simulation. The maximum of the range of cp h amongst all planes is taken to be the nonlocal plastic shear strain range, i.e., p Dcp max ¼ maxðDch Þ:

ð8Þ

Note that this particular algorithm is suitable for proportional loading, as relevant to the present work. 4. Microstructure-sensitive probabilistic fatigue notch factor In this section, distributions of the nonlocal FIP obtained from simulations within a well-defined damage process zone (i.e., volume within which fatigue crack formation is most likely to occur) are used in the development of a new microstructure-sensitive fatigue notch factor, K lf . There is scatter in the nonlocal FIP values due to microstructure variability (see Fig. 5). Accordingly, a probabilistic mesomechanics approach, based on the statistical distribution of a nonlocal FIP in the damage process zone, is used to pursue l predictive relations for K f . The following probabilistic framework focuses on the formation of a fatigue crack on the scale of a single grain, and would therefore be applicable to the HCF and very high cycle fatigue (VHCF) regimes for which total life is strongly correlated to crack formation at this scale. Assume that slip drives crack nucleation and growth of small cracks to the scale of grains; a crack may form in any of the grains within the damage process zone, Vd, defined as the smallest volume corresponding to a locus of points enclosing grains with FIP levels above some threshold level. The damage process zone Vd in the notch root region is composed of a number of grains, each with volume dVi, considered as infinitesimal relative to Vd. Similar to the approach used in Refs. [23,24], the probability of fatigue crack formation in a grain within Vd is expressed as

dPf ;i ¼ ki ðDCi ðea or

ra ; R; rÞÞdV i :

ð9Þ

The probability of survival of the grain with an infinitesimal volume dVi is given by

dPs;i ¼ 1  dPf ;i ¼ 1  ki ðDCi ðea or

ra ; R; rÞÞdV i :

ð10Þ

In Eqs. (9) and (10), ki ðDCi Þ characterizes the failure probability of each infinitesimal volume dVi, and depends on DCi, which depends on the applied (remote) stress or strain amplitude (ea or ra), strain or stress ratio (R), and grain location (r) and orientation within Vd. Using weakest link theory, the probability of survival for the notched component with process zone Vd is obtained from the probability of survival of all ‘‘m” number of grains within Vd, i.e.,

Ps ¼

m Y

dPs;i ¼

i¼1

m Y

ð1  ki ðDCi ðea or

ra ; R; rÞÞdV i Þ:

ð11Þ

i¼1

Eq. (11) assumes independence of ki on the nonlocal DCi values in grains within the damage process zone. This is only reasonable when considering the formation of a fatigue crack(s) in HCF on the order of the grain size and not the propagation of small cracks away from the notch root across multiple grains. The latter would require treatment of conditional probabilities for further extension once a crack is formed. This may not be too limiting for HCF and VHCF regimes. The product of probabilities of survival given by Eq. (11) can be transformed to a sum by taking the natural logarithm of both sides of Eq. (11), resulting in

ln P s ¼

m X i¼1

ln ð1  ki ðDCi ðea or

ra ; R; rÞÞdV i Þ

ð12Þ

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Since we are concerned with low probability of failure in HCF in each infinitesimal volume dVi, the Taylor series expansion of the right hand side of Eq. (12) in terms of ki dV i gives

lnð1-ki ðDCi ðea or

ra ; R; rÞÞdV i Þ  ki ðDCi ðea or ra ; R; rÞÞdV i

ð13Þ

where only the linear term has been retained. Using Eq. (13) in Eq. (12), the probability of survival of the component (i.e., the probability of survival of Vd) can be obtained as

ln Ps ¼

m X

ðki ðDCi ðea or

ra ; R; rÞÞdV i Þ

ð14Þ

i¼1

Eq. (14) can be re-written in continuous form as

Ps ¼ exp 

!

Z

kðDCðea or

ra ; R; rÞÞdV

ð15Þ

Vd

 1 DCðea or ra ; R; rÞ  DCth ðea or V0 DC0 ðea or ra ; R; rÞ

ra ; R; rÞ

bC :

ð16Þ

This represents an implicit parameterization in terms of remote applied stress, with the DC explicitly computed using computational crystal plasticity. Eq. (16) is motivated by the form of the Weibull stress function [43] that has been used in a number of statistical approaches in HCF based on experimental results or FE analyses. However, it is emphasized that it is based on the nonlocal FIP DC rather than stress or strain in each element. In Eq. (16), DC0 and bC are scale and shape parameters, respectively. The shape parameter, bC, is a measure of the scatter in the DC distribution among grains within the damage process zone at the notch root; DCth is the threshold FIP (location parameter) below which no microdamage will occur at a given number of cycles in the HCF regime, and V0 is a reference volume. It is estimated in this study as the threshold condition for arrest of a crack with length on the order of the grain size, based on a micromechanical growth law as DCth = 2b/(AFSsydgr) from the work of Shenoy et al. [44], where b is the magnitude of the Burgers vector, dgr is the mean grain size, AFS is a constant, and sy is the critical resolved shear stress. The failure function in Eq. (16) employs a statistical distribution of DC, thus enabling the conversion of deterministic multiaxial failure criteria into a more realistic probabilistic approach that addresses the significant degree of scatter inherent in HCF. Substituting (16) into (15) yields,

(

1 Ps ¼ exp  V0

)  Z  DCðea or ra ; R; rÞ  DCth ðea or ra ;R;rÞ bC dV DC0 ðea or ra ;R;rÞ Vd

ð17Þ The cumulative probability of HCF failure of the notched component, specifically defined here as the nucleation and growth of a crack to the order of the grain size, can be obtained from Eq. (17) as

Pf ¼ 1  Ps ¼ 1 (

1  exp  V0

(   DCnetðaveÞ ðea or ra ; R; rÞ bC V d 1 Ps ¼ exp  DC0 ðea or ra ; R; rÞ V0 Vd )  Z  DCðea or ra ; R; rÞ  DCth ðea or ra ; R; rÞ bC  dV DCnetðaveÞ ðea or ra ; R; rÞ V

)  Z  DCðea or ra ;R;rÞ  DCth ðea or ra ; R; rÞ bC dV DC0 ðea or ra ; R; rÞ Vd ð18Þ

At this point it is desired to introduce the notion of a net section average value of DC to facilitate expression of results in terms of a relative measure of stress levels for notched and unnotched cases

ð19Þ

In Eq. (19), DCnet(ave) is the net section average nonlocal FIP for the notched component, computed either as: (i) the average of DC in the net section region outside the damage process zone, Vd, having non zero DC or (ii) the average of DC over the entire net section. We next introduce a new microscopic concentration factor l K C defined by

(

l

KC ¼

The failure probability parameter k is prescribed in terms of a constitutive law dependence on a driving force which depends on applied remote stress and location within the notch root field. Here, we assume that it depends on the FIP, DC, in each respective subvolume element (e.g., grain) by adopting a power law of the form

k¼

for the same probability of survival. The probability of survival of the notched component in Eq. (17) can be re-written as

1 Vd

)1=bC   DCðea or ra ; R; rÞ  DCth ðea or ra ; R; rÞ bC dV DCnetðaveÞ ðea or ra ; R; rÞ Vd

Z

ð20Þ The DCnet(ave) based on definition (i) above is used to determine the microscopic concentration factor in Eq. (20). It is noted that the average value of DC over the notch root damage process zone Vd could alternatively have been used as the scale parameter in definl ing K C in Eq. (20), replacing DCnet(ave) in both places in Eq. (19) in that case. Only the scaling is changed, in effect. Substituting Eq. (20) into Eq. (19) yields,

( bC ) Vd l DCnetðaveÞ ðea or ra ; R; rÞ Ps ¼ exp K C DC0 ðea or ra ; R; rÞ V0

ð21Þ

Using Eq. (21), the net section average nonlocal FIP, DCnet(ave), for a notched specimen (i.e., DCnotch(net ave)) with a damage process zone Vd at the same number of cycles to crack formation, Ni, and probability of survival as that of a reference smooth specimen of volume V0 with diameter or width corresponding to the net section width of the notched specimen and having an average nonlocal FIP, DCreference(ave), is given as

DCnotchðnet

e or ra ; R; rÞNi ¼

aveÞ ð a

DCreferenceðaveÞ ðea or ra ; R; rÞNi  1=bC l Vd KC V0 ð22Þ

Note that in Eq. (21), for a smooth specimen K lC = 1 and Vd = V0. Moreover, for the smooth specimen, the entire volume is considered as having sub volumes with DC at or above the threshold value DCth. Eq. (22) is obtained from Eq. (21) using Ps(notch) = Ps(reference), meaning that the probability of survival is the same for notched specimen with a damage process zone Vd at the same number of cycles to crack formation as that of a reference smooth specimen of volume V0 with diameter or width corresponding to the net section width of the notched specimen. Thus, a new quantitative definition of the fatigue notch factor for the formation and growth of crack on the order of the grain size in HCF is obtained from Eq. (22) as l

Kf ¼

 1=bC DCreferenceðaveÞ ðea or ra ; R; rÞNi l Vd ¼ KC DCnotchðnet aveÞ ðea or ra ; R; rÞNi V0 l

ð23Þ

In principle, the fatigue notch factor K f follows the spirit of the traditional definition for Kf, i.e., the ratio of unnotched to notched values of assumed driving force for a given HCF life, but is rooted in probabilistic arguments, based on the distribution of DC. It is interesting to note that the average reference smooth specimen val lue DCreference(ave) is not required for the computation of K C since it does not appear on the extreme right hand relation in Eq. (23). Only the reference smooth specimen volume, V0, is needed along with the distribution of DC within the damage process zone, Vd, as well as the value of DC averaged over the entire net section.

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l

5. Results and discussion 5.1. Fatigue indicator parameters The magnitude and distribution of cyclic plastic strain among grains (microplasticity) is a significant factor in nucleation and early growth of microstructurally small fatigue cracks at the notch root. Five notch root radii ranging from 200 lm to 1000 lm with the same plate length and width, shown in Fig. 2, were employed in this investigation. All notches were semicircular in shape; the stress concentration factor based on net section stress varies with notch root radius as shown in Table 1. As stated earlier, the distributions of microplasticity also enable assessment of statistical distributions of the nonlocal FIP for various notch root radii in the fatigue damage process zone. As an illustration, the maximum values of the nonlocal FIP in Eq. (5) as a function of the distance from the notch root for the notch root radii listed in Table 1 are shown in Fig. 3 for a remote strain amplitude of 0.5ey for a single realization of grains. The nonlocal FIP plot in Fig. 3 indicates that larger notches have increased probability of high FIP values in more grains and larger volume of the fatigue damage process zone than smaller notches, with corresponding higher notch sensitivity. As the distance from the notch root increases, as expected the nonlocal FIP values decrease for all the notch root radii. 5.1.1. Effect of strain amplitude Calculations were conducted at three different strain amplitudes (0.4ey, 0.5ey, 0.6ey) and Re = 1 for various notch root radii. The distributions of maximum nonlocal FIP computed over the third cycle are shown in Fig. 4 as a distance from the notch root for the three strain amplitudes and various notch root radii. At

Table 1 Stress concentration factor based on net section stress for a plate with double edge semicircular notch. Notch root radius, q (lm)

Stress concentration factor (Kt) based on net section

200 400 600 800 1000

2.93 2.79 2.66 2.54 2.41

1

ρ = 1000 μm ρ = 800 μm

0.75

ρ = 600 μm

4

Max. ΔΓ (x10 )

In addition, K C is not a constant but depends on microstructure. This approach considers the multiaxial nature of the stress distribution around the notch since any suitable multiaxial FIP can be used. In addition, it considers notch size and stress field gradients. It is suitable for simple or arbitrary notched geometries. With this approach, we discern regimes of notch-sensitive and notch-insensitive behavior in terms of quantitative measures based on nonlocal FIP distributions, thereby offering a microstructure-sensitive approach as an alternative to the conventional fatigue notch factor. The value of this concept is that it can be used to estimate the influence of microstructure on notch sensitivity for microstructure conditions that have not been processed, which is normally not possible. The tradeoff, of course, is that computational simulations demand an adequate treatment of the distribution of heterogeneous yielding among grains within the notch root region. Another important restriction on the present work is that it relates to formation of a crack on the order of grain size, a criterion which generally could not be compared directly to Kf values measured from experiments with failure defined as complete specimen fracture. However, in HCF the fraction of total life spent forming cracks on the order of grain size is high, so qualitative comparison may be possible.

ρ = 400 μ m 0.5

ρ = 200 μ m

0.25

0 0.03

0.09

0.15

0.21

0.27

Distance from notch root (mm) Fig. 3. Maximum values of nonlocal FIP (max DC) distributions after the third cycle for an applied remote strain amplitude of 0.5ey (ey = 0.0123%) and Re = 1 for notch root radii (q) ranging from 200 lm to 1000 lm.

the lowest strain amplitude of 0.4ey and for small notch root radii, the nonlocal FIP distributions are negligible in magnitude, e.g., maximum FIP = 9.38  107 for the notch root radius of 200 lm; this represents a fatigue limit based on the absence of microplasticity. 5.1.2. Influence of grain orientation distribution To determine the effects of microstructure on notch root microplasticity, four different realizations, each with random grain orientation distribution, were considered. For each of these simulations, the same applied strain amplitude of 0.5ey was imposed with Re = 1. The results in Fig. 5 show that in all cases there is variability of DCmax among realizations, indicating that the sampled notch root volume does not comprise a representative volume in terms of statistics of fatigue failure. A number of such realizations must be computed to develop such representative statistics. Of course, a limited number of laboratory specimens (say less than five) at a given condition do not provide representative statistics either, indicating the value of augmenting with a larger number of simulations. Significant variability is observed among realizations for notch root radii above 400 lm. 5.2. Microstructure-sensitive fatigue notch factor and associated failure probability As before, 5 notch root radii over the range of 200 lm to 1000 lm were considered. For each notch root radius, 40 different realizations of grain distributions were computed. The location, scale and shape parameters are respectively given by 6.14  105, 7.05  106 and 1.13. These parameters were obtained in the present work using the modified moment estimation (MME) [45,46] technique. This technique, which is easy to use and low in bias with respect to the mean and variance, uses the first two sample moments and a function of a first order statistic of the computed DC. The variation in microstructure-sensitive fatigue notch factor, l K f , for the 40 different realizations of grain distributions is shown in Fig. 6. The results show that the fatigue notch factor at the microscale level varies with the microstructure for a given notch root radius. The elastic stress concentration factor Kt is superimposed on this plot for each notch root radius for comparison. The

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Fig. 4. Maximum values of nonlocal FIP (max DC) as a function of distance from the notch root surface for different completely reversed applied strain amplitudes and notch root radii (q) ranging from 200 lm to 1000 lm (shown in inset box in each plot).

l

average K f for notch root radii in Fig. 6 are plotted in Fig. 7. Note the characteristic sigmoidal shape of the fatigue notch factor in Fig. 7; the point with maximum slope is about 10 times grain size.

l

For a notch root radius of 1000 lm, K f ranges from 1.42 to 2.69 with an average of 1.83. There is a statistical chance that the driving force for crack formation will even exceed that indicated by Kt,

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1

1

200 μm

400 μm 0.75

ran1

ran1

ran2

ran2

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Max ΔΓ (x104)

0.75

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Distance from notch root (mm)

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600 μm 0.75

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ran1

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Max ΔΓ (x104)

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1

1

ran3

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ran4

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Distance from notch root (mm) 1.25

1000 μm

1

Max ΔΓ (x104)

ran1 0.75

ran2 ran3 ran4

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0 0.03

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Distance from notch root (mm) Fig. 5. Maximum values of nonlocal FIP (max DC) as a function of distance from the notch root surface for four realizations with different random grain orientation distributions at the same strain amplitude of 0.5ey and Re = 1 for notch root radii ranging from 200 lm to 1000 lm (shown in inset box in each plot).

as observed in Fig. 6 for q = 1000 lm, although the probability is low. This is due to the additional enhancement of microstructure-induced strain concentration at the level of individual grains.

This approach enables us to gain direct insight into the controlling physical phenomena at the microstructure scale and offers improved identification of cause-effect relations for the material

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4

μ

Kf Kt

μ Kf, Kt

3

2

1

0 0

200

400

600

800

1000

1200

Notch root radius (μm) Fig. 6. Distribution of K lf at the same strain amplitude of 0.5ey and Re = 1 for notch root radii ranging from 200 lm to 1000 lm. Elastic stress concentration factor Kt is also plotted for comparison.

for scatter in fatigue life data due to microstructure variability and notch/component size effects. Thus, the relationship between strain amplitude and the number of cycles to fatigue failure should not be a single-valued function but must also take into consideration the trends or probability of failure to reflect scatter in fatigue life. The average probability of fatigue crack formation increases with the notch root radius, as shown in Fig. 9. Of course, the distribution of Pf obtained from a large number of realizations enables estimation of the envelope for maximum failure probability as a function of notch root radius, as shown by the solid line in Fig. 8. l Direct comparison between the K f results obtained using the probabilistic approach developed and the conventional macroscopic Kf obtained from experimental results is complicated for a number of reasons. First, K lf is a ‘‘microscopic” variable that considers the microslip distribution in the volume of material within the notch root field; the conventional Kf is based on the ratio of average remote applied stress amplitudes corresponding to unnotched and notched specimens at a given HCF life, requiring the designer to apply a safety factor to account for uncertainty associated with matel rial condition or microstructure variability. Second, K f is based on the concentration of the FIP (DC in this study) rather than a stress measure per se. For the Fatemi–Socie FIP used here, it can be seen that the normal stress is effectively weighted by the maximum 1

2.5

0.8

2

0.6

1.5

Pf

μ Kf

3

0.4

1

0.2 0.5

0 0

0 0

200

400

600

800

1000

200

1200

400

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1000

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Notch root radius (μm)

Notch root radius (μm) l

Fig. 7. Average K f at an applied strain amplitude of 0.5ey and Re = 1 for notch root radii ranging from 200 lm to 1000 lm.

Fig. 8. Probability of fatigue crack initiation for 40 random realizations of grains at the same strain amplitude of 0.5ey and Re = 1 for notch root radii ranging from 200 lm to 1000 lm, with the upper bound trend line superimposed.

1

0.8

0.6

Pfave

dependence of fatigue scatter, size effects, and stress field gradient l effects. In view of the association of K f with the formation of a crack on the order of grain size, and in view of the dominance of the first major barrier (e.g., grain boundary) in arresting growth l of microstructurally small cracks, we may regard K f in the long life limit as relating closely to fatigue limit behavior for notched components. In the LCF and transition fatigue regimes for which the cyclic plastic strain amplitude at the notch root substantially exceeds the elastic strain amplitude, it is likely that this particular bal sis for K f will have less utility than a formulation that considers small crack propagation through a network of grains at the notch root. The probability of fatigue crack initiation, obtained using Eq. (18), for the 40 realizations with random grain orientation distribution at a given strain amplitude of 0.5ey and Re = 1, is shown in Fig. 8 for notch root radii ranging from 200 lm to 1000 lm. Fig. 8 also shows that the notch sensitivity of the material depends not only on the notch root radius but also on the microstructure, which can evolve significantly with processing. It also accounts

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400

600

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Notch root radius (μm) Fig. 9. Average probability of fatigue crack initiation at the same strain amplitude of 0.5ey and Re = 1 for notch root radii ranging from 200 lm to 1000 lm.

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cyclic plastic shear strain range. Hence, grains with higher plastic shear strain range weight the corresponding stress more highly. In addition, the correlation length between grains with significant cyclic plastic shear strain range may be 5–10 times the mean grain size. It is interesting to consider if this may relate more to the sharp transition region of the characteristic sigmoidal function for notch root sensitivity shown in Fig. 7. Therefore, direct comparison with experimental data is not feasible since no comparable fatigue notch factor has been assessed experimentally (based on formation of a crack at the scale of a grain, with the ratio of the weighted stress fields instead of direct stress fields). Notch factor K lf is useful in determining the effect of the notch on reduction of fatigue resistance in a way that directly incorporates microstructure. In addition, adequate experimental results of scatter in fatigue life for the reference specimen(s) necessary to obtain more reliable parameters are not typically available in literature. Considerable cost is associated with obtaining such data in the laboratory. Furthermore, the notion of crack initiation as typically applied in the determination of fatigue notch factor Kf used in conventional stress and strain-life approaches is ambiguous since it entails both the formation and growth of small cracks. While the amplitude of applied loading has been shown to affect the size of nucleated cracks [47], a number of experiments show that the nucleation size is on the order of grain size [48,49] in the HCF regime. Crack formation at the scale of an individual grain size should be modeled distinctly using corresponding statistical models based on cyclic microplasticity, as is done in this paper. The incorporation of stages of subsequent microstructurally and physically small crack growth over a number of grains, a topic for future consideration, demands a framework that acknowledges the role of microstructure in influencing crack growth away from the site of formation. In crack growth across sets of grains, the nature of the grain boundary network and associated crystallographic orientation and disorientation distributions play a strong role in crack growth retardation l and/or arrest. Accordingly, different values of K f are assumed to apply to crack formation at the scale of a grain and to subsequent growth through a specific number of grains within the notch root field. l Kf  1 based on the The trend in the notch sensitivity index q ¼ l KC  1 l average K f results obtained here are comparable to the trend in

0.8

micro q- theoretical macro q - experimental 0.6

q

0.4

0.2

0 0

200

400

600

800

1000

Notch root radius (μm) Fig. 10. The dependence of notch sensitivity index q on notch root radius l comparing measures of q based on K f and experimentally determined conventional Kf for OFHC Cu [50], based on the ratio of unnotched to notched fatigue strengths at long lives.

notch sensitivity index q ¼

K f 1 K t 1

based on the experimentally mea-

sured Kf values [50], as shown in Fig. 10. This figure shows that the probabilistic model adequately predicts the trends observed in the experimental results for the average values of the notch sensitivity as a function of the notch root radius. It is noted that the average grain size in the experimental results was 50 lm, slightly less than the 62 lm average grain size of the material in the present study. 6. Conclusions A probabilistic mesomechanics approach that accounts for the cyclic microplastic strain distributions and related fatigue indicator parameters with a damage process zone at a notch root is introduced along with a new microstructure-sensitive fatigue notch factor. The probabilistic methodology presented can also be used to assess the effects of microstructure attributes on probability of fatigue failure at critical locations such as the notch root regions. This approach has certain advantages relative to the conventional approach since it can account for the variation in the microstructure of a given material. For most of the notch root radii considered, the microstructure-sensitive fatigue notch factor and associated probability of failure are observed to be random variables exhibiting significant scatter. Both are computable for a given microstructure and applied load. The predictive capabilities of the stochastic framework presented in this study can be validated using experimental results on specific materials with different microstructures and the same notches, but care must be taken in defining the crack formation event(s) to avoid exercising the model beyond its limits. Acknowledgements G.M. Owolabi is grateful for the financial support provided by the Natural Science and Engineering Research Council of Canada (NSERC). D.L. McDowell is grateful for the support of the Carter N. Paden, Jr. Distinguished Chair in Metals Processing. References [1] Neuber H. Theory of stress concentration in shear strained prismatic bodies with arbitrary non-linear stress law. J Appl Mech 1961;28:544–50. [2] Moski K, Glinka G. A Method of elastic-plastic stress-strain calculations at the notch-root. Mat Sci Eng 1981;50:93–100. [3] Hoffman M, Seeger T. A generalization method for estimating multiaxial elastic-plastic notch stresses and strains. Part 1 and part 2. J Eng Mater Technol 1985;107:250–60. [4] Barkey ME, Socie DF, Hsai KJ. A yield surface approach to the estimation of notch strains for proportional and non-proportional cyclic loading. J Eng Mater Technol 1994;116:173–9. [5] Moftakhar A, Bucznski A, Glinka G. Calculation of elastic-plastic strains and stress in notches under multiaxial loading. Int J Fract 1995;70:357–73. [6] Gu RJ, Lee Y. A new method for estimating non-proportional notch root stresses and strains. J Eng Mater Technol 1997;119:40–5. [7] Phillip CE, Heywood RB. The size effect in fatigue of plain and notched steel specimens under reversed direct stress. Proc Inst Mech Eng 1951;165:113–24. [8] Bellett D, Taylor D, Marco S, Mazzeo E, Guillois J, Pircher T. The fatigue behaviour of three-dimensional stress concentrations. Int J Fatigue 2005;27:207–21. [9] Peterson RE. Stress concentration factors. New York: Wiley; 1974. [10] Neuber H. Theory of notch stresses. Berlin: Springer; 1958. [11] Peterson RE. Notch sensitivity. In: Sines G, Waisman JL, editors. Metal Fatigue. McGraw-Hill: New York; 1959. p. 293–306. [12] Ranganathan RM, Gu RJ, Lee YL. An improved automated finite element analysis for fatigue life predictions of notched components. Int J Mater Prod Technol 2004;21(6):539–54. [13] Ren W, Nicholas T. Notch size effects on high cycle fatigue limit stress of Udimet 720. Mater Sci Eng, A 2003;357:141–52. [14] Buch A. Fatigue strength calculation, vol. 6. USA: Material Science Surveys; 1988. [15] Beremin FM. A local criterion for cleavage fracture of a nuclear pressure vessel steels. Metall Trans 1983;11:2277–87. 14A. [16] Weixing Y. On the notched strength of composite laminates. Compos Sci Technol 1992;45:105–10.

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