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Modeling and Simulation of Microstructurally Small Crack Formation and Growth in Notched Nickel-base Superalloy Component
Available online at ScienceDirect
ScienceDirect J. Mater. Sci. Technol., 2014, 30(3), 203e212
Modeling and Simulation of Microstructurally Small Crack Formation and Growth in Notched Nickel-base Superalloy Component G.M. Owolabi*, H.A. Whitworth Department of Mechanical Engineering, Howard University, Washington, DC 20059, USA [Manuscript received January 29, 2013, in revised form May 14, 2013, Available online 25 September 2013]
Studies on microstructurally small fatigue cracks have illustrated that heterogeneous microstructural features such as inclusions, pores, grain size distribution as well as precipitate size distribution and volume fraction create stochasticity in their behavior under cyclic loads. Therefore, to enhance safe-life and damage-tolerance approaches, accurate modeling of the influence of these heterogeneous microstructural features on microstructurally small crack formation and growth from stress raisers is necessary. In this work, computational micromechanics was used to predict the high cycle fatigue of microstructurally small crack formation and growth in notched polycrystalline nickel-base superalloys and to quantify the variability in the driving force for formation and growth of microstructurally small crack from notch root in the matrix with non-metallic inclusions. The framework involves computational modeling to obtain three-dimensional perspectives of microstructural features influencing fatigue crack growth in notched nickel-base superalloys, which accounts for the effects of nonlocal notch root plasticity, loading, microstructural variability, and extrinsic defects on local cyclic plasticity at the microstructure-scale level. This approach can be used to explore sensitivity of minimum fatigue lifetime to microstructures. The simulation results obtained from this framework were calibrated to existing experimental results for polycrystalline nickel-base superalloys. KEY WORDS: Microstructurally small crack; Crystal plasticity; Nickel-base superalloy; Fatigue life
1. Introduction Current fatigue life prediction methods mainly consider two stages: crack initiation and crack propagation. Crack initiation is further decomposed into crack nucleation, microstructurally small crack (MSC) growth, and physically small crack (PSC) growth[1,2]. Crack nucleation is the locally complex process of crack formation on the microstructural scale, which is characterized by smooth fracture surfaces at angles inclined to the loading direction[3]. MSCs have a size on the order of microstructural features in a material, which ranges from micrometers to hundreds of micrometers[4]. The length of PSCs is on the order of 5e10 times of the microstructural scale. Microstructurally and physically small cracks can be identified depending on the reasons for their lack of similitude and their dissimilarities with long cracks[5]. Crack propagation is long crack growth to final * Corresponding author. Ph.D.; Tel.: þ1 202 8066594; Fax: þ1 202 4831396; E-mail address: [email protected] (G.M. Owolabi). 1005-0302/$ e see front matter Copyright Ó 2013, The editorial office of Journal of Materials Science & Technology. Published by Elsevier Limited. All rights reserved. http://dx.doi.org/10.1016/j.jmst.2013.09.011
fracture. The length of long cracks is more than 10e20 times of the microstructural dimension. Long crack propagation to final failure is the stage of damage accumulation that is well characterized using the linear elastic fracture mechanics (LEFM) or the elasticeplastic fracture mechanics (EPFM). The total fatigue life can be determined by adding the loading cycles for crack nucleation, MSC growth, PSC growth, and crack propagation. However, fatigue life prediction tools are currently limited in their ability to incorporate MSC growth[6] especially for materials with complex microstructure such as nickel-base superalloys. Nickel-base superalloys are a high performance material subject to severe operating conditions. They are widely used in aero engine components and power-plants because of their high strength and good creep, fatigue, and corrosion resistance at high temperatures[7,8]. Fatigue failure is of great importance for these components for the catastrophic consequence caused by cyclic stresses and strains. Fatigue failure of nickel-base superalloys is a limiting factor for their reliable use in many engineering applications. The total fatigue life of nickel-base superalloys is the addition of numbers of cycles for crack nucleation and growth through the MSC, PSC, and long crack growth regimes[9]. Studies[7e10] have shown that the crack nucleation and MSC growth of nickel-base superalloys is different from that in
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homogeneous base materials due to the high level of residual stresses, complex microstructure, complicated loading, and internal defects. Experimental studies on polycrystalline nickelbase superalloys indicate that the crack nucleation and MSC regimes are very important stages since the majority of the fatigue life in nickel-base superalloys used as turbine disc is spent in these fatigue regimes during service[11]. Prior research studies on the fatigue behavior of nickel-base superalloys are focused on higher length-scale phenomena and on direct experimental observation of fatigue damage processes[12]. Under high cycle fatigue (HCF) situation, crack nucleation and MSC growth is highly heterogeneous at the grain scale and the influence of microstructure on the small crack formation and growth has shown to be significant[13]. Microstructure features such as inclusions, pores, grain size distribution, grain boundaries as well as precipitate size distribution and volume fraction often have main influences on determining the crack formation and growth in both low cycle fatigue (LCF) and HCF in nickel-base superalloys[9]. Since polycrystalline nickel-base superalloys such as IN 100 is prepared by powder metallurgy technique, the presence of inclusions and pores in the microstructure always provide crack nucleation sites. Therefore, among these features, inclusions such as carbides within a grain or near grain boundaries and pores in the polycrystalline material can significantly reduce the number of cycles to form a fatigue crack because the incompatibility of deformations between the inclusions and the surrounding materials leads to local plasticity[14]. Over the past several decades, many attempts have been made to determine the critical value of inclusion size, below which the effect of inclusions on fatigue life could be ignored, while above which, the influences of inclusions should be examined and taken into consideration carefully. Uhrus[15] showed that for ball bearings, only oxide inclusions, which are more than 30 mm in diameter, should be taken into consideration for assessment of the fatigue life. Nishijima et al.[16] studied standard Japanese tempered martensitic steels, which showed that the critical size of inclusions is around 45 mm. The inclusions with an irregular shape and sharp edges could lead to larger stress concentrations compared to inclusions with smooth shape and edge, which further make it easier for crack initiation. In real microstructure, inclusions have irregular shape[17]. In this work, computational micromechanics was used to predict the high cycle fatigue of MSC formation and growth in notched polycrystalline nickelbase superalloys IN 100 and to quantify the variability in the driving force for MSC growth in the matrix with non-metallic inclusions. The simulation results were used to characterize the fatigue strength reduction effects of notched components with non-metallic inclusions at a given probability of failure. The approach presented in this study can also be used to support expensive and time consuming experiments as well as assist in the design of better fatigue resistant safety-critical notched components with defects such as non-metallic inclusions and pores.
Fig. 1 Heat treatment process for IN 100[18].
between 500 and 800 C[17]. Polycrystalline IN 100 was prepared using powder metallurgy technique. Powder process is able to produce the alloys with fine-grained microstructures with high strength and structure homogeneity due to its capability in reduction of individual phase segregation. In order to obtain the desired final microstructure, attention should be paid to heat treatment process, which determines the end product form and application. Optimum heat treatments should be designed for yielding the desired microstructure and phases. Typically, a heat treatment process for IN 100 has a three-step thermal cycle: solution treatment at high temperature, stabilization/stress relief at an intermediate temperature, and low temperature age as shown in Fig. 1[18]. The first cycle stands for a subsolvus or supersolvus heat treatment with temperature at 1143 or 1205 C for 2 h, the second cycle is stabilization process at 982 C for 1 h and the last step is aging process at 732 C for 8 h[18]. After heat treatment, two major phases are apparent: disordered g and ordered g0 as shown in Fig. 2[19]. Three different size distributions of g0 precipitates can be founded in this image. These size distributions are the primary g0 precipitate, the secondary g0 and the tertiary g0 , respectively, which are distinct through the diameters. The largest g0 precipitates are the primary g0 with a diameter around 1 mm formed during the first step of heat treatment, whose size and distribution is set by the solution treatment temperature; the secondary precipitate with diameter around 0.1 mm is the product of cooling process from solution temperature and is modified by the stabilization temperature; and tertiary precipitates with diameter around 0.01 mm form during subsequent aging process, which are coherent with the g matrix. Two types of carbides are seen in the microstructure of IN 100:
2. Material System The material used in this work was the polycrystalline IN 100 superalloy, which is commonly used for hot section applications in aircraft engines and power generation turbines due to its capability of retaining strength in excess of 1 GPa at high temperature. The yield strength of this nickel-base superalloy decreases with increase in temperature, with an anomalous behavior
Fig. 2 Schematic diagram of the microstructure of IN 100 after heat treatment[19].
G.M. Owolabi and H.A. Whitworth: J. Mater. Sci. Technol., 2014, 30(3), 203e212
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Fig. 4 (a) Octahedral slip system and (b) cube slip system. Fig. 3 Deformation process in kinematics formulation[17].
M23C6 around grain boundary and random distributed MC with different shapes. Microstructure features, such as inclusions, pores, secondary phase, at different scales play a dominant role in limiting the fatigue life of nickel-base superalloys because these features tend to act as sources for crack formation. The study of Leverant and Gell[20] shows that the pre-cracked carbides or micropores provide crack initiation sites at lower temperature in nickel-base superalloys under both HCF and LCF situations (<870 C). At higher temperature, oxidation plays a dominant role in limiting fatigue life, particularly, when dealing with failure due to cyclic loading in nickel-base superalloys. The notched component used in this work is under 650 C, so the role of carbides is more important than oxidation. 3. Crystal Plasticity Framework Most of the prior simulations of small cracks make the assumption that a material can be modeled using the classical isotropic plasticity theory based on the Von Mises yield criterion. However, the crack-tip plastic zone in small crack regime is typically small and confined into one grain or between two adjacent grains, which makes the previous assumption invalid[21]. Therefore, the theory of crystal plasticity is used due to its capability of simulating the inhomogeneous nature of plastic deformation in a single grain and enables prediction of the amount of plastic straining that occurs along discrete planes of the crystallographic lattice under loading. This theory lays a foundation for the development of anisotropic constitutive equations governing the plastic deformation in a single grain. The crystal plasticity theory had been used to model the small features of polycrystalline materials, where the features of interest are of the same length scale as the average grain size[22]. Rice et al.[23] were the first to use crystal plasticity theory to model fatigue crack growth. Gall et al.[24] also used the crystal
g_
ðaÞ
study because it relates the grain scale stress to crystallographic slip response[25]. The 3D computational crystal plasticity model is a rate-dependent, microstructure-sensitive model that follows that of Przybyla and McDowell[26] and Shenoy[17]. The model is used to capture the influences of microstructure features on stressestrain response. The crystal plasticity model used in this work is a constitutive model, which has no explicit distinction between the g matrix and g0 precipitate but contains inclusions or pores inside of the whole matrix. The kinematic of crystal plasticity[27] theory is based on both continuum mechanics and materials science, which are employed using the multiplicative decomposition of the deformation gradient as shown in Fig. 3. The total deformation gradient tensor F consists of two parts: (1) Fe; the elastic deformation gradient representing elastic lattice stretch and rigid rotation and (2) Fp; the plastic deformation gradient describing the collective effect of dislocation glide along crystallographic planes relative to the fixed lattice in the reference configuration[28]. Fig. 3 shows the crystal structure movement, which begins with pure plastic deformation, then rotation and pure elastic stretching. The grids in Fig. 3 represent the crystal lattice. p The macroscopic plastic velocity gradient b L , is the summation of all the slip systems in the intermediate relaxed configuration given by p b bp þ W bp ¼ L ¼ F p $F p1 ¼ D
(1)
a
where mao is the unit normal vector of the close packed planes in the reference configuration for each of the a slip system, sao is the b p and W b p are the unit vector of slip directions of these plane, D intermediate configuration plastic rate of deformation and plastic spin separately given by the symmetric and anti-symmetric parts of the associated plastic velocity[11]. The inelastic slip rate along each of the active slip system (a), g_ a , depends on the resolved shear stress on the slip systems and is assumed to follow the form below using a two term potential given by
ðaÞ n ðaÞ
s xðaÞ kðaÞ n1 s xðaÞ 2 ðaÞ ðaÞ _ g s sgn ¼ g_ 1 þ x 2 DðaÞ DðaÞ
plasticity theory to compute the crack tip plastic zone for MSC in a single grain. In order to model heterogeneity and interaction across grains in the notch root field, a fully three-dimensional (3D) computational crystal plasticity model is adopted in this
Nslip X g_ a sao 5mao
(2)
where g_ 1 and g_ 2 are constants, n1 and n2 are flow exponents, k(a) is the threshold stress, D(a) is an average drag resistance, s(a) is the resolved shear stress on the slip system a and x(a) is the back stress (used to capture the Bauschinger effect) on slip system
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Fig. 5 A schematic showing an elliptical inclusion in the matrix (a ¼ 15 mm and b ¼ 5 mm).
a[17]. The first part of Eq. (2) is to characterize the dominant cyclic behavior, while the second part represents the effects of thermally activated flow. There are two types of dislocation slip systems in the nickelbase superalloys with fcc crystal structure: one is the twelve octahedral <110>{111} slip systems observed in g austenitic phase at elevated temperature, and the other is <110>{100} six cube slip systems observed in g0 phase, which are active at high homologous temperature and high resolved shear stress as shown in Fig. 4. The role of the cube slip systems is less well understood and characterized due to the challenging identification. Two sources of cube slip have been reported: the manifestation of the zigezag octahedral slip at the geg0 interface in the g matrix and the actual cube slip in g0 -precipitates at the higher temperatures[29]. Both sources of cube slip will be lumped into the present model used for this paper. Both the octahedral and cube slip systems are assumed to be active in this model at all times. The model used in this paper contains two internal state variables: the dislocation density, ra and the slip system back stress, ca, which are used to embed microstructure dependence. The hardening law defines the overall threshold slip resistance, which is used to model the initial crystallographic yield/flow behavior of two phase nickel-base superalloys under the given conditions as[17] qffiffiffiffiffi
kal ¼ kao;l þ ammin b ral
(3)
where l is either the octahedral or the cube slip system and mmin stands for the volume fraction averaged shear modulus. The hardening behavior of the flow stress is connected with the nucleation, the multiplication and the interactions of dislocations under applied loading. In this model, it is a function of dislocation storage, dynamic recovery, and the size and spacing of g0 precipitates. Dynamic recovery is caused by dislocation annihilation and rearrangement. The rate of dynamic recovery is proportional to the dislocation density given by[17] n
qffiffiffiffiffi
o
r_ al ¼ h0 Z0 þ k1 ral k2 ral jg_ a j
(4)
where k1, k2, and k3 are constants, h0 is hardening coefficients for octahedral, h0 ¼ 4.8, and for cube slip system, h0 ¼ 2.4. ral is the dislocation density. The equations described above consist of the main part of the crystal plasticity constitutive equations used in this work that are implemented as a user material subroutine (UMAT) in ABAQUS for 3D finite element analyses.
Fig. 6 3D finite element model with an elliptical inclusion generated using ABAQUS.
4. Finite Element Modeling 3D finite element model is built to perform the sequence of computation for cyclic plastic deformation, which further sheds light on the failure mechanism using ABAQUS 6.10. The modeled material consists essentially of a matrix and inclusions. In this 3D model, the matrix is built first in ABAQUS CAE and ellipses are cut from the matrix. Then elliptical parts with the same dimension as the cutting parts are drawn and subsequently tied to the matrix. The carbide inclusions were considered to be linear elastic in the finite element model with elastic modulus E ¼ 405 GPa and Poisson’s ratio n ¼ 0.14[1]. The matrix is a double-edge notch with a height of 20 mm and a width of 10 mm. Due to symmetry, only half of the double-edge notched specimen was modeled to render the simulations less computationally intensive. The inclusions are assumed to be elliptical in shape with a representative aspect ratio, Ar ¼ a/b ¼ 3, where a and b are semi-major and semi-minor dimensions of the ellipse, respectively as illustrated in Fig. 5 for a notched component with a single inclusion. The 3D finite element model with an ellipsoidal inclusion is illustrated in Fig. 6. A coarse mesh size is used in the far field zone to enhance computational efficiency, while near the notch and the inclusion regions, the mesh size is finer. 3D 10-node quadratic tetrahedral element (C3D10) is used due to the complex geometry. The matrix and the inclusion are intact with each other. Six steps are subsequently created for three loading cycles and boundary conditions are applied to the whole model. Remotely applied strain is added via displacement boundary conditions on the upper with the absolute value of DL ¼ L 0.5 3 y (3 y ¼ 0.60% is the macroscopic proportional limit of the polycrystalline IN 100 superalloy obtained through experimental stressestrain data[17]). The amplitudes of imposed completely reversed nominal axial cyclic strain (R3 ¼ 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 IN 100
G.M. Owolabi and H.A. Whitworth: J. Mater. Sci. Technol., 2014, 30(3), 203e212
polycrystalline materials. In other words, only a relatively small number density of grains yield in the notch root and inclusion region after the initial loading cycle. 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 and inclusion regions. The use of crystal plasticity is also 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 HCF regime. Using the constitutive model in Section 3, computational crystal plasticity finite element simulations with appropriate microstructure length scale dependence were performed for the notched components with non-metallic inclusions with notch root radii varying from 0.2 mm to 1 mm. For each loading condition, a set of different realizations of grains are implemented with randomly assigned orientation distribution to obtain an initially isotropic effective medium while gathering information regarding variability among instantiations. Random grain orientations were assigned since there was lack of information regarding orientation and misorientation distributions functions for the IN 100 microstructure used in this study. Two stages of fatigue life are modeled: crack incubation and MSC growth because they account for a large percentage of fatigue life under HCF and very high cycle fatigue (VHCF). Crack incubation is the event marked by the first appearance of new surface area, which can be seen as the formation of a crack on the order of the grain size. Inclusions, flaws, voids, and notches act as stress raisers to provide sites for crack incubation. The driving force for fatigue crack nucleation is the maximum range of cyclic effective plastic shear strain. Shenoy et al.[9] made an extension of Tanaka and Mura’s[30] crack incubation for estimation of crack nucleation life in IN 100. The number of cycles required for the formation of a grain-size crack is obtained using the relation[9] Ninc
Dgpmax
2
2 ¼ avg
ag dgrain
(5)
is the maximum range of the plastic shear strain where Dg over a fatigue cycle, ag ¼ 0.056 mm-cycle, and dgrain ¼ 34 mm is the average grain size of coarse grain in IN 100. This equation shows that the maximum range of the cyclic plastic shear strain averaged over the grain region is used as the driving force for crack incubation. The MSC growth begins after the formation of the crack on the order of the grain size. Previous work by Fatemi and Kurath[31] has demonstrated a solid correlation of the fatigue crack propagation life under various stress states with and without mean stress. In this work, the grain scale averaged FatemieSocie (FS) parameter[32] is used as the driving force to describe the growth of the MSC on the order of three times of the grain size. A modified form of the MSC growth law, which is put forward by Shenoy et al.[9] is given as p max
207
vectors. The nonlocal FS fatigue indicator parameter, DG, is given by
Dgpmax
DG ¼
2
smax 1þk n
sy
(7)
where k ¼ 1.0 is a material parameter that is used to take into consideration the effect of normal stress in crack formation and early growth, sy is the macroscopic cyclic yield strength, and smax is the maximum normal stress acting on the defined n plane with grain scale averaged value. To obtain the nonlocal DG from the finite element simulation, we consider a set of planes at each integration point lying within the nonlocal averaging region. For consistency in comparison of the current results with previous work in literature[34], Dgpmax and 3 max sn are averaged over a volume of (4 mm ) and calculated over the third fatigue cycle for all the simulation instantiations since this volume is smaller than the statistically small grain size, capable of detecting gradient in plastic shear within the grains, and is sufficiently large to enable the convergence of the response DG[34]. At each integration point, the plastic shear strain gpq on the q plane is calculated by resolving the plastic strain tensor 3 pij onto it at each integration point, i.e.,
gpq ¼ ni 3 pij tj ;
q ¼ 1.N
(8)
where ni is the unit normal vector on plane q, tj is a unit tangent vector in the considered direction along the plane, and N is the number of discrete planes sampled. Discrete set of tangent vectors are screened on any given q plane to estimate the plastic shear strain. The nonlocal average plastic shear strain associated with the q plane is then calculated by averaging the plastic shear strain on the q plane over the volume Vb of the nonlocal region around the notch and the inclusions, i.e.,
gpq ¼
1 Vb
Z
gpq dV
(9)
Vb
The maximum of the range of gpq amongst all planes is taken to be the nonlocal plastic shear strain range, i.e.,
Dgpmax ¼ max Dgpq
(10)
Once the critical plane and the nonlocal maximum plastic shear strain range Dgpmax are determined, DG is calculated using Eq. (7) at every integration point on the critical plane (deter mined by Dgpmax ) and a volume average is performed in the nonlocal region similar to Eq. (9); this effectively results in a nonlocal DG. 5. Probabilistic Framework 5.1. Probability of failure for crack formation
da ¼ AFS sy DGa hb dN
(6)
where AFS ¼ 8.1 104 (MPa)1 for IN 100 is a constant calibrated through experiments, sy is the critical resolved shear stress given as sy =M with M ¼ 3.06, which is the Taylor factor for a texture free polycrystalline aggregate[9,17], h is a constant with the value close to 1 and b is the magnitude of the Burger’s
The objective of this probabilistic approach is to characterize the effect of inclusions and microstructurally-sensitive notch root effect on the HCF life of the polycrystalline IN 100 superalloy, which considers the probability of crack formation within a grain size. The FS fatigue indicator parameter distribution function is determined through the computational crystal plasticity model described before. The prior work by Owolabi
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et al.[33] lays foundation for this work. The probability of formation of a crack is given as[33] 8 9 < 1 Z DG DG bG = th Pf ¼ 1 exp dV : V0 ; DG0
(11)
Vd
In Eq. (11), DG0 is the scale parameter, bG is the shape parameter used to measure the scatter in the DG distribution among grains within the damage process zone at the notch root, and DGth is the threshold fatigue indicator parameter, below which no microdamage will occur at a given number of cycles in the HCF region, Vd is defined as the smallest volume corresponding to a locus of points enclosing grains with fatigue indicator parameter values above the threshold value and V0 is the volume of a reference smooth specimen. 5.2. Probability of failure for MSC growth In order to calculate the probability of forming and growing a crack to a transition crack length, l, the fatigue indicator parameter-based transition crack length approach developed by Musinski and McDowell[34] is adopted. It considers the intensity of the fatigue indicator parameter over a characteristic length to assess the probability of forming and propagating of a crack from the notch root to a transition crack length. In this approach, the l is the distance, in which a crack propagates in a certain length, which has no influence on the notch root stress concentration. For semi-circular notch, the l is equal to 0.13r according to the work done by Smith and Miller[35]. For the notches with size less than 0.5 mm, l is only 1 or 2 grain diameters. The transition cracks for this work are 0.1 mm for notch radius 0.2 mm, 0.78 mm for notch with radius 0.6 mm, and 0.13 mm for notch with radius 1.0 mm. The probability for crack propagation to the transition length is affected by the applied stress or strain amplitude, the notch root size, and the mechanisms of MSC growth, which is dependent on the size of preexisting crack at a given number of cycles. The cumulative distribution function (CDF) is given as h a i n CDF ¼ 1 exp h l
(12)
where h is equal to ln(0.5) used to provide normalization to the case for CDF ¼ 0.5 when crack propagates at the length of l[34]. The crack length an can be found by integrating the crack growth rate in Eq. (6) and substituting the resulting expression in Eq. (12) to obtain 2 6 CDF ¼ 1 exp6 4h
Z
3 AFS sy DGmax adN 7 7 5 l
(13)
6. Results and Discussion 6.1. Fatigue indicator parameter distribution vs distance from notch root The magnitude and distribution of the effective plastic strain among grains for different notch root radii is an important factor in determining the crack nucleation and the early growth of
Fig. 7 Maximum fatigue indicator parameter vs distance from notch root for r ¼ 0.2 mm (a), r ¼ 0.6 mm (b), and r ¼ 1.0 mm (c) at strain amplitudes of 0.53 y.
microstructurally small fatigue cracks at notch root, which is further used to assess the statistical distributions of the fatigue indicator parameters and to obtain the volume of the fatigue damage process zone. In this work, the maximum fatigue indicator parameter distribution is utilized to define the fatigue driving force for the formation and growth of small crack at the notch root. Fig. 7 shows the distribution of the maximum fatigue indictor parameter, DGmax with respect to the distance in x-direction from the notch root for different notch radii (0.2 mm, 0.6 mm, and 1.0 mm). As can be seen from the figure, there is a rapid decrease in DGmax values at the beginning of the plot, and it follows by an embossment at the position around 1 mm for 0.2 mm notch, 0.6 mm position for 0.6 mm, and 0.3 mm position for 1.0 mm notch. These positions are also where the inclusions are added for the notched component. So this phenomenon shows that the inclusion acts as DG raiser. Also, there is a significant difference in DG intensity among three different notch
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Fig. 9 MSC growth fatigue life vs notch radius.
Fig. 8 Incubation life vs notch radius.
radii. The larger the notch root radius, the higher the maximum fatigue indicator parameter intensity. It means that the driving force for crack incubation and MSC growth in larger notch radius component is higher than that for smaller ones.
The fatigue crack initiation life is the sum of the number of cycles required for an MSC to form and grow outside the influence of the notch root plastic zone. Eq. (5) is used in this work to estimate the incubation life, Ninc for the notched IN 100 component with inclusions. Rearranging Eq. (5), it gives p 2 Dgmax = dgrain 2 avg ag
(14)
The crack incubation lives for different random orientation of grains at different notch radii are calculated using Eq. (11). Note that the grain with the smallest computed incubation life is assumed to be the grain in which crack incubation will occur. Fig. 8 shows the incubation life distribution for different n. It is very obvious that there is a decrease trend for crack incubation life at different notch sizes. For notch size 1.0 mm, the crack incubation life is the lowest. 6.3. MSC growth life MSC growth begins once a grain-size crack is formed. In this study, the MSC life, NMSC is defined as the number of cycles for the incubated crack to grow to 3 grains diameter since it has been noted in literature[34] that the crack growth curves obtained from experiments show that small crack and long crack curves typically merge at crack length in the range of 3e10 grain diameters. After integrating and rearranging Eq. (6), the number of cycles to propagate a crack from aj1 to aj is given as[34] 0 NMSC aj1 /aj ¼
trend as the incubation life; that is, the larger the notch radius, the higher the probability of propagation of small crack and the smaller the MSC growth life. 6.4. Calibration to experiments and notched IN 100 without inclusions
6.2. Crack incubation life and MSC growth life
Ninc ¼
209
1
mb B aj AFS sy DGave C 1 C lnB A b AFS sy DGave @aj1 A smDG FS y ave
(15)
where DGave is the average of the maximum FS parameter over the increment of crack length from aj to aj1. Fig. 9 shows the average MSC growth life vs the notch radius. It has the same
Under HCF and VHCF regimes, the total cycles for crack incubation and MSC growth (Ninc þ NMSC) take up to nearly 90% or more of a component’s fatigue lifetime. So the total fatigue life is close to the sum of crack incubation fatigue cycles and MSC growth fatigue cycles. In order to assess the influence of notches and inclusions on the fatigue life of the notched component using the computational results obtained previously, the experimental results for smooth specimen from the work by Cowles et al.[36] and the work by Bathias and Paris[37] are used to illustrate the notch effect on the fatigue life, while the simulation results from the work by Musinski and McDowell[34] for notched components without inclusions is employed to assess the effect of inclusions only on the fatigue life. Cowles et al.[36] performed an extensive study on five different nickel-base alloys in order to evaluate their total crack incubation and crack propagation fatigue life. All the experiments were carried out under fully reversed strain-controlled situation and under both cyclic and cyclic-hold condition at 650 C. In this work, the test results for HIP Astroloy nickel-base superalloy were used. The work done by Bathias and Paris[37] was in the regimes of HCF (w106) and VHCF (w109). In this regime, crack usually nucleated at microscopic flaws and defects. So the study of a nickel-based alloy (N18) had been chosen to compare the effect of inclusions on IN 100. The work done by Musinski and McDowell[34] was mainly on the smooth and notched IN 100 components with different notch radii without inclusions. It is important to state that the previous work by Musinski and McDowell[34] also used the experimental results from Cowles et al.[36] for HIP Astroloy and the experimental results from Bathais and Paris[37] to calibrate the simulation results for the smooth IN 100 superalloys without non-metallic inclusions. The justification for using these experimental results have been detailed in Section 3 of reference[34], but a summary is provided here along with the limitations. The Astroloy nickel-base superalloy experimental data were used because its grain size, which is in the range of 30 mme70 mm, is similar to the grain size of the IN 100 with an average mean grain size of 34 mm. Furthermore, since there was no detailed experimental data
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Fig. 10 Comparison among results from experiments, notched with inclusion component, notched without inclusion components: (a) r ¼ 0.2 mm, (b) r ¼ 0.6 mm, (c) r ¼ 1.0 mm, (d) comparison of all notch root radii.
regarding the crack length measured in literature[36] at a 5% stiffness drop, Musinski and McDowell[34] assumed that this level of stiffness drop corresponds to the crack initiation and the MSC growth through three adjacent grain sizes. Musinski and McDowell[34] also indicated that the ultrasonic fatigue testing of the IN18 at 450 C can provide sufficient comparison to the fatigue testing of the polycrystalline IN 100 at 650 C both in the HCF and VHCF regimes. It is interesting to note that the simulation results obtained for the smooth specimens in literature[34] for the IN 100 superalloy predict the general trends observed in the experimental results for the HIP Astroloy nickelbase superalloy[36] and the N18 nickel-base superalloy[37] very well at both the LCF and the HCF regimes and are very close to the strain life curve in the HCF and VHCF regimes. Thus, these experimental results for the smooth specimens in literature[36,37] are sufficient for the calibration of the simulation results for the notched specimen with and without non-metallic inclusions used in this study. For this study, the experimental results are also adequate to characterize the fatigue strength reduction effects of the notch and non-metallic inclusions at a given probability of failure. Fig. 10(aec) are the results for notch radii (0.2 mm, 0.6 mm, 1.0 mm) and Fig. 10(d) is the comparison among all notch radii. The existence of notch leads to a drastic drop in component’s life. The comparison between the notched component with inclusions and the notched component without inclusion also shows that the presence of inclusion in component causes a significant decrease in fatigue life. Another interesting observation is that a crack may not initiate in the smooth specimen at the strain amplitude (3 a ¼ 0.53 y) used even in the regime of VHCF,
while for notched components with inclusions, a crack initiates at the HCF regime. This observation agrees well with the initial observations in literature[34] for notched components without non-metallic inclusions. It is noted here that the simulation results could have been more efficient if the results have been compared with experimental results from fatigue data obtained using the actual polycrystalline IN 100 superalloys notched components under the same loading conditions. Additionally more robust simulation results could have been obtained: i) if the actual size and volume distributions of the non-metallic inclusions in the specimens are obtained using characterization tools such as computed tomography and the information embedded in the current model, ii) by accounting for the grain size distributions in the IN 100 since larger grains in the materials microstructure have higher probability for fatigue crack initiation than smaller grains and iii) by accounting for the initial residual stresses due to manufacturing processes[34]. These limitations will be addressed in the future studies for various materials, notch root geometries and acuity as well as different loading conditions. 6.5. Probability of failure for crack formation and growth The following probabilistic approach is used to determine the probability of formation and growth of a fatigue crack on the scale of a single grain, which is developed by Owolabi et al.[33] for HCF and VHCF fatigue regimes and given in Eq. (11). The threshold fatigue indicator parameter in Eq. (11) is estimated through Eq. (6). That is, when the crack growth rate is zero, the threshold value is obtained as
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211
Fig. 11 Probability of failure vs notch root radius for notched component with inclusion and notched component without inclusion.
DGth ¼
hb AFS sy a
(16)
where a is considered to be the grain size, and sy ¼ 241.83 MPa. The threshold value for polycrystalline IN 100 is 3.04 104. The definition of failure in probabilistic framework[33] is the formation of a crack on the order of a grain size. The probability of failure, Pf vs notch size for both notched components with and without inclusions is shown in Fig. 11. As seen from the plot, the existence of inclusion increases the probability of formation of MSC in a notched component. The probabilistic approach for crack growth to a transition length is the extension of work done by Musinski and McDowell[34]. In order to calculate the CDF, the maximum value of the fatigue indicator parameter should be obtained first through simulation. The relationship between DGmax and the distance, x from the notch root up to 5 times the grain size is shown in Fig. 12. For notched components with non-metallic inclusions, there is no linear relationship between DGmax and the distance from notch root unlike the linear relation between the maximum fatigue indicator parameter and the distance from notch root obtained in Musinski and McDowell[34], which does not account for the presence of inclusions in the notched IN 100 component. Applying the fatigue indicator parameter-based transition length crack probabilistic framework for the notched IN 100 superalloy component with inclusions, the cumulative distribution values are shown in Fig. 13(aec). It can be seen that
Fig. 13 Cumulative distribution function for IN 100 with different notch radius at 3 ¼ 0.53 y and R3 ¼ 1: (a) r ¼ 0.2 mm, (b) r ¼ 0.6 mm, (c) r ¼ 1.0 mm.
cumulative distribution function can be calculated for any number of cycles and any probability of failure. This approach also shows that the presence of inclusions in notched component has higher failure probability than notched component without inclusions and the larger the notch radius, the higher the probability of formation and growth of MSC. 7. Conclusion
Fig. 12 Maximum fatigue indicator parameter values vs distance from the notch root for r ¼ 1.0 mm within five grains length.
A computational framework was presented to determine the influence of non-metallic inclusions and notches on the formation and growth of MSC in polycrystalline IN 100 superalloy and to assess the effects of the non-metallic inclusions on the fatigue life and the probability of fatigue failure of notched polycrystalline IN 100 components. The distribution of the effective plastic strains around the notch and the inclusions is used to
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estimate the driving forces for crack formation and MSC growth in the notched IN 100 with non-metallic inclusions. The finite element simulation results show that the driving force for notched component with inclusions is higher than that for the notched component without inclusions. The results also show that the larger the notch radius, the higher the probability of formation and growth of MSC. Comparison between fatigue lives for notched component with inclusions and experimental results for smooth component shows that the presence of notch and inclusions leads to a drastic drop in the fatigue life. Acknowledgments The authors are grateful for the funding provided by the Department of Defense (DoD) through the research and educational program for HBCU/MI (Contract No. # W911NF-11-1014) monitored by Dr. Larry Russell (Program Manager, ARO) and Dr. David Stargel (Program Manager, AFOSR). REFERENCES [1] D.L. McDowell, K. Gall, M.F. Horstemeyer, J. Fan, Eng. Fract. Mech. 70 (2007) 49e80. [2] D.L. McDowell, Int. J. Fatigue 19 (1997) 127e135. [3] R.G. Tryon, NASA Contractor Report 202342, 1997. [4] R.O. Ritchie, J. Lankford (Eds.), Small Fatigue Cracks, Metall. Soc. Inc., Pennsylvania 15086, 1986, pp. 1e5. [5] B. Lin, L.G. Zhao, J. Tong, Eng. Fract. Mech. 78 (2011) 2174e2192. [6] J.D. Hochhalter, D.J. Littlewood, M.G. Veilleux, J.E. Bozek, A.M. Maniatty, A.D. Rollett, A.R. Ingraffea, Model. Simul. Mater. Sci. Eng. 19 (2011) 035008. [7] F. Tancret, Processing for China, Sterling Publications, London, 2000, pp. 56e58. [8] S.A. Padula, A. Shyam, R.O. Ritchie, W.W. Milligan, Int. J. Fatigue 21 (1999) 725e731. [9] M. Shenoy, J. Zhang, D.L. McDowell, Fatigue Fract. Eng. Mater. Struct. 30 (2007) 889e904. [10] C. Mercer, A.B.O. Soboyejo, W.O. Soboyejo, Mater. Sci. Eng. A 270 (1999) 308e322. [11] M. Goto, D.M. Knowles, Eng. Fract. Mech. 60 (1998) 223e232. [12] C.P. Przybyla, R. Prasannavenkatesan, N. Salajegheh, D.L. McDowell, Int. J. Fatigue 32 (2010) 512e525. [13] C.P. Przybyla, D.L. McDowell, Procedia Eng. 2 (2010) 1045e 1052.
[14] D. Raabe, F. Roters, F. Barlat, L.Q. Chen, Continuum Scale Simulation of Engineering Materials: FundamentalseMicrostructureseProcess Applications, first ed., Wiley-VCH, Weinheim, 2004, p. 500. [15] L.O. Uhrus, Iron Steel Inst. 77 (1963) 104e109. [16] S. Nishijima, K. Tanaka, H. Sumiyoshi, Fracture 84, in: Proc. 6th Int. Conf. Fracture (ICF6), Adv. Fracture Res., New Delhi, India, December 4e10, second ed. vol. 3, 1984, pp. 1719e1726. [17] M.M. Shenoy, Constitutive Modeling and Life Predictions in Nibase Superalloy (Ph.D. thesis), Georgia Institute of Technology, 2006. [18] R.W. Kozar, A. Suzuki, W.W. Milligan, J.J. Schirra, M.F. Savage, T.M. Pollock, Metall. Mater. Trans. A 40 (2009) 1588e1603. [19] A.M. Wusatowska-Sarnek, M.J. Blackburn, M. Aindow, Mater. Sci. Eng. A 360 (2003) 390e395. [20] G. Leverant, M. Gell, Trans. Metall. Soc. AIME 245 (1969) 1167e 1173. [21] G.P. Potirniche, S.R. Daniewicz, J.C. Newman Jr., Fatigue Fract. Eng. Mater. Struct. 27 (2004) 59e71. [22] G.P. Potirniche, S.R. Daniewicz, Eng. Fract. Mech. 70 (2003) 1623e1643. [23] J.R. Rice, D.E. Hawk, R.J. Asaro, Int. J. Fatigue 42 (1990) 301e 321. [24] K. Gall, H. Sehitoglu, Y. Kadioglu, Metall. Mater. Trans. A 27 (1996) 3491e3502. [25] P.R. Dawson, Int. J. Solids Struct. 37 (2000) 115e130. [26] C.P. Przybyla, D.L. McDowell, Int. J. Plast. 26 (2010) 372e394. [27] R.J. Asaro, Adv. Appl. Mech. 23 (1983) 1e115. [28] R. Prasannavenkatesan, Microstructure-sensitive Fatigue Modeling of Heat Treated and Shot Peened Martensitic Gear Steels (Ph.D. thesis), Georgia Institute of Technology, 2009. [29] Y.Q. Sun, P.M. Hazzledeng, Dislocat. Solids 10 (1996) 27e68. [30] K. Tanaka, T. Mura, J. Appl. Mech. 48 (1981) 97e103. [31] A. Fatemi, P. Kurath, J. Eng. Mater. Technol. Trans. ASME 110 (1988) 380e388. [32] A. Fatemi, D.F. Socie, Fatigue Fract. Eng. Mater. Struct. 11 (1988) 149e165. [33] G.M. Owolabi, R. Prasannavenkatesan, D.L. McDowell, Int. J. Fatigue 32 (2010) 1378e1388. [34] W.D. Muskinski, D.L. McDowell, Int. J. Fatigue 37 (2012) 41e53. [35] R.A. Smith, K.J. Miller, Int. J. Mech. Sci. 19 (1977) 11e22. [36] B.A. Cowles, J.R. Warren, F.K. Haake, Evaluation of the Cyclic Behavior of Aircraft Turbine Disk Alloys, Part II, NASA CR165123, Pratt & Whitney Aircraft, 1980. [37] C. Bathias, P.C. Paris, Int. J. Fatigue 32 (2010) 894e897.
ScienceDirect J. Mater. Sci. Technol., 2014, 30(3), 203e212
Modeling and Simulation of Microstructurally Small Crack Formation and Growth in Notched Nickel-base Superalloy Component G.M. Owolabi*, H.A. Whitworth Department of Mechanical Engineering, Howard University, Washington, DC 20059, USA [Manuscript received January 29, 2013, in revised form May 14, 2013, Available online 25 September 2013]
Studies on microstructurally small fatigue cracks have illustrated that heterogeneous microstructural features such as inclusions, pores, grain size distribution as well as precipitate size distribution and volume fraction create stochasticity in their behavior under cyclic loads. Therefore, to enhance safe-life and damage-tolerance approaches, accurate modeling of the influence of these heterogeneous microstructural features on microstructurally small crack formation and growth from stress raisers is necessary. In this work, computational micromechanics was used to predict the high cycle fatigue of microstructurally small crack formation and growth in notched polycrystalline nickel-base superalloys and to quantify the variability in the driving force for formation and growth of microstructurally small crack from notch root in the matrix with non-metallic inclusions. The framework involves computational modeling to obtain three-dimensional perspectives of microstructural features influencing fatigue crack growth in notched nickel-base superalloys, which accounts for the effects of nonlocal notch root plasticity, loading, microstructural variability, and extrinsic defects on local cyclic plasticity at the microstructure-scale level. This approach can be used to explore sensitivity of minimum fatigue lifetime to microstructures. The simulation results obtained from this framework were calibrated to existing experimental results for polycrystalline nickel-base superalloys. KEY WORDS: Microstructurally small crack; Crystal plasticity; Nickel-base superalloy; Fatigue life
1. Introduction Current fatigue life prediction methods mainly consider two stages: crack initiation and crack propagation. Crack initiation is further decomposed into crack nucleation, microstructurally small crack (MSC) growth, and physically small crack (PSC) growth[1,2]. Crack nucleation is the locally complex process of crack formation on the microstructural scale, which is characterized by smooth fracture surfaces at angles inclined to the loading direction[3]. MSCs have a size on the order of microstructural features in a material, which ranges from micrometers to hundreds of micrometers[4]. The length of PSCs is on the order of 5e10 times of the microstructural scale. Microstructurally and physically small cracks can be identified depending on the reasons for their lack of similitude and their dissimilarities with long cracks[5]. Crack propagation is long crack growth to final * Corresponding author. Ph.D.; Tel.: þ1 202 8066594; Fax: þ1 202 4831396; E-mail address: [email protected] (G.M. Owolabi). 1005-0302/$ e see front matter Copyright Ó 2013, The editorial office of Journal of Materials Science & Technology. Published by Elsevier Limited. All rights reserved. http://dx.doi.org/10.1016/j.jmst.2013.09.011
fracture. The length of long cracks is more than 10e20 times of the microstructural dimension. Long crack propagation to final failure is the stage of damage accumulation that is well characterized using the linear elastic fracture mechanics (LEFM) or the elasticeplastic fracture mechanics (EPFM). The total fatigue life can be determined by adding the loading cycles for crack nucleation, MSC growth, PSC growth, and crack propagation. However, fatigue life prediction tools are currently limited in their ability to incorporate MSC growth[6] especially for materials with complex microstructure such as nickel-base superalloys. Nickel-base superalloys are a high performance material subject to severe operating conditions. They are widely used in aero engine components and power-plants because of their high strength and good creep, fatigue, and corrosion resistance at high temperatures[7,8]. Fatigue failure is of great importance for these components for the catastrophic consequence caused by cyclic stresses and strains. Fatigue failure of nickel-base superalloys is a limiting factor for their reliable use in many engineering applications. The total fatigue life of nickel-base superalloys is the addition of numbers of cycles for crack nucleation and growth through the MSC, PSC, and long crack growth regimes[9]. Studies[7e10] have shown that the crack nucleation and MSC growth of nickel-base superalloys is different from that in
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homogeneous base materials due to the high level of residual stresses, complex microstructure, complicated loading, and internal defects. Experimental studies on polycrystalline nickelbase superalloys indicate that the crack nucleation and MSC regimes are very important stages since the majority of the fatigue life in nickel-base superalloys used as turbine disc is spent in these fatigue regimes during service[11]. Prior research studies on the fatigue behavior of nickel-base superalloys are focused on higher length-scale phenomena and on direct experimental observation of fatigue damage processes[12]. Under high cycle fatigue (HCF) situation, crack nucleation and MSC growth is highly heterogeneous at the grain scale and the influence of microstructure on the small crack formation and growth has shown to be significant[13]. Microstructure features such as inclusions, pores, grain size distribution, grain boundaries as well as precipitate size distribution and volume fraction often have main influences on determining the crack formation and growth in both low cycle fatigue (LCF) and HCF in nickel-base superalloys[9]. Since polycrystalline nickel-base superalloys such as IN 100 is prepared by powder metallurgy technique, the presence of inclusions and pores in the microstructure always provide crack nucleation sites. Therefore, among these features, inclusions such as carbides within a grain or near grain boundaries and pores in the polycrystalline material can significantly reduce the number of cycles to form a fatigue crack because the incompatibility of deformations between the inclusions and the surrounding materials leads to local plasticity[14]. Over the past several decades, many attempts have been made to determine the critical value of inclusion size, below which the effect of inclusions on fatigue life could be ignored, while above which, the influences of inclusions should be examined and taken into consideration carefully. Uhrus[15] showed that for ball bearings, only oxide inclusions, which are more than 30 mm in diameter, should be taken into consideration for assessment of the fatigue life. Nishijima et al.[16] studied standard Japanese tempered martensitic steels, which showed that the critical size of inclusions is around 45 mm. The inclusions with an irregular shape and sharp edges could lead to larger stress concentrations compared to inclusions with smooth shape and edge, which further make it easier for crack initiation. In real microstructure, inclusions have irregular shape[17]. In this work, computational micromechanics was used to predict the high cycle fatigue of MSC formation and growth in notched polycrystalline nickelbase superalloys IN 100 and to quantify the variability in the driving force for MSC growth in the matrix with non-metallic inclusions. The simulation results were used to characterize the fatigue strength reduction effects of notched components with non-metallic inclusions at a given probability of failure. The approach presented in this study can also be used to support expensive and time consuming experiments as well as assist in the design of better fatigue resistant safety-critical notched components with defects such as non-metallic inclusions and pores.
Fig. 1 Heat treatment process for IN 100[18].
between 500 and 800 C[17]. Polycrystalline IN 100 was prepared using powder metallurgy technique. Powder process is able to produce the alloys with fine-grained microstructures with high strength and structure homogeneity due to its capability in reduction of individual phase segregation. In order to obtain the desired final microstructure, attention should be paid to heat treatment process, which determines the end product form and application. Optimum heat treatments should be designed for yielding the desired microstructure and phases. Typically, a heat treatment process for IN 100 has a three-step thermal cycle: solution treatment at high temperature, stabilization/stress relief at an intermediate temperature, and low temperature age as shown in Fig. 1[18]. The first cycle stands for a subsolvus or supersolvus heat treatment with temperature at 1143 or 1205 C for 2 h, the second cycle is stabilization process at 982 C for 1 h and the last step is aging process at 732 C for 8 h[18]. After heat treatment, two major phases are apparent: disordered g and ordered g0 as shown in Fig. 2[19]. Three different size distributions of g0 precipitates can be founded in this image. These size distributions are the primary g0 precipitate, the secondary g0 and the tertiary g0 , respectively, which are distinct through the diameters. The largest g0 precipitates are the primary g0 with a diameter around 1 mm formed during the first step of heat treatment, whose size and distribution is set by the solution treatment temperature; the secondary precipitate with diameter around 0.1 mm is the product of cooling process from solution temperature and is modified by the stabilization temperature; and tertiary precipitates with diameter around 0.01 mm form during subsequent aging process, which are coherent with the g matrix. Two types of carbides are seen in the microstructure of IN 100:
2. Material System The material used in this work was the polycrystalline IN 100 superalloy, which is commonly used for hot section applications in aircraft engines and power generation turbines due to its capability of retaining strength in excess of 1 GPa at high temperature. The yield strength of this nickel-base superalloy decreases with increase in temperature, with an anomalous behavior
Fig. 2 Schematic diagram of the microstructure of IN 100 after heat treatment[19].
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Fig. 4 (a) Octahedral slip system and (b) cube slip system. Fig. 3 Deformation process in kinematics formulation[17].
M23C6 around grain boundary and random distributed MC with different shapes. Microstructure features, such as inclusions, pores, secondary phase, at different scales play a dominant role in limiting the fatigue life of nickel-base superalloys because these features tend to act as sources for crack formation. The study of Leverant and Gell[20] shows that the pre-cracked carbides or micropores provide crack initiation sites at lower temperature in nickel-base superalloys under both HCF and LCF situations (<870 C). At higher temperature, oxidation plays a dominant role in limiting fatigue life, particularly, when dealing with failure due to cyclic loading in nickel-base superalloys. The notched component used in this work is under 650 C, so the role of carbides is more important than oxidation. 3. Crystal Plasticity Framework Most of the prior simulations of small cracks make the assumption that a material can be modeled using the classical isotropic plasticity theory based on the Von Mises yield criterion. However, the crack-tip plastic zone in small crack regime is typically small and confined into one grain or between two adjacent grains, which makes the previous assumption invalid[21]. Therefore, the theory of crystal plasticity is used due to its capability of simulating the inhomogeneous nature of plastic deformation in a single grain and enables prediction of the amount of plastic straining that occurs along discrete planes of the crystallographic lattice under loading. This theory lays a foundation for the development of anisotropic constitutive equations governing the plastic deformation in a single grain. The crystal plasticity theory had been used to model the small features of polycrystalline materials, where the features of interest are of the same length scale as the average grain size[22]. Rice et al.[23] were the first to use crystal plasticity theory to model fatigue crack growth. Gall et al.[24] also used the crystal
g_
ðaÞ
study because it relates the grain scale stress to crystallographic slip response[25]. The 3D computational crystal plasticity model is a rate-dependent, microstructure-sensitive model that follows that of Przybyla and McDowell[26] and Shenoy[17]. The model is used to capture the influences of microstructure features on stressestrain response. The crystal plasticity model used in this work is a constitutive model, which has no explicit distinction between the g matrix and g0 precipitate but contains inclusions or pores inside of the whole matrix. The kinematic of crystal plasticity[27] theory is based on both continuum mechanics and materials science, which are employed using the multiplicative decomposition of the deformation gradient as shown in Fig. 3. The total deformation gradient tensor F consists of two parts: (1) Fe; the elastic deformation gradient representing elastic lattice stretch and rigid rotation and (2) Fp; the plastic deformation gradient describing the collective effect of dislocation glide along crystallographic planes relative to the fixed lattice in the reference configuration[28]. Fig. 3 shows the crystal structure movement, which begins with pure plastic deformation, then rotation and pure elastic stretching. The grids in Fig. 3 represent the crystal lattice. p The macroscopic plastic velocity gradient b L , is the summation of all the slip systems in the intermediate relaxed configuration given by p b bp þ W bp ¼ L ¼ F p $F p1 ¼ D
(1)
a
where mao is the unit normal vector of the close packed planes in the reference configuration for each of the a slip system, sao is the b p and W b p are the unit vector of slip directions of these plane, D intermediate configuration plastic rate of deformation and plastic spin separately given by the symmetric and anti-symmetric parts of the associated plastic velocity[11]. The inelastic slip rate along each of the active slip system (a), g_ a , depends on the resolved shear stress on the slip systems and is assumed to follow the form below using a two term potential given by
ðaÞ n ðaÞ
s xðaÞ kðaÞ n1 s xðaÞ 2 ðaÞ ðaÞ _ g s sgn ¼ g_ 1 þ x 2 DðaÞ DðaÞ
plasticity theory to compute the crack tip plastic zone for MSC in a single grain. In order to model heterogeneity and interaction across grains in the notch root field, a fully three-dimensional (3D) computational crystal plasticity model is adopted in this
Nslip X g_ a sao 5mao
(2)
where g_ 1 and g_ 2 are constants, n1 and n2 are flow exponents, k(a) is the threshold stress, D(a) is an average drag resistance, s(a) is the resolved shear stress on the slip system a and x(a) is the back stress (used to capture the Bauschinger effect) on slip system
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Fig. 5 A schematic showing an elliptical inclusion in the matrix (a ¼ 15 mm and b ¼ 5 mm).
a[17]. The first part of Eq. (2) is to characterize the dominant cyclic behavior, while the second part represents the effects of thermally activated flow. There are two types of dislocation slip systems in the nickelbase superalloys with fcc crystal structure: one is the twelve octahedral <110>{111} slip systems observed in g austenitic phase at elevated temperature, and the other is <110>{100} six cube slip systems observed in g0 phase, which are active at high homologous temperature and high resolved shear stress as shown in Fig. 4. The role of the cube slip systems is less well understood and characterized due to the challenging identification. Two sources of cube slip have been reported: the manifestation of the zigezag octahedral slip at the geg0 interface in the g matrix and the actual cube slip in g0 -precipitates at the higher temperatures[29]. Both sources of cube slip will be lumped into the present model used for this paper. Both the octahedral and cube slip systems are assumed to be active in this model at all times. The model used in this paper contains two internal state variables: the dislocation density, ra and the slip system back stress, ca, which are used to embed microstructure dependence. The hardening law defines the overall threshold slip resistance, which is used to model the initial crystallographic yield/flow behavior of two phase nickel-base superalloys under the given conditions as[17] qffiffiffiffiffi
kal ¼ kao;l þ ammin b ral
(3)
where l is either the octahedral or the cube slip system and mmin stands for the volume fraction averaged shear modulus. The hardening behavior of the flow stress is connected with the nucleation, the multiplication and the interactions of dislocations under applied loading. In this model, it is a function of dislocation storage, dynamic recovery, and the size and spacing of g0 precipitates. Dynamic recovery is caused by dislocation annihilation and rearrangement. The rate of dynamic recovery is proportional to the dislocation density given by[17] n
qffiffiffiffiffi
o
r_ al ¼ h0 Z0 þ k1 ral k2 ral jg_ a j
(4)
where k1, k2, and k3 are constants, h0 is hardening coefficients for octahedral, h0 ¼ 4.8, and for cube slip system, h0 ¼ 2.4. ral is the dislocation density. The equations described above consist of the main part of the crystal plasticity constitutive equations used in this work that are implemented as a user material subroutine (UMAT) in ABAQUS for 3D finite element analyses.
Fig. 6 3D finite element model with an elliptical inclusion generated using ABAQUS.
4. Finite Element Modeling 3D finite element model is built to perform the sequence of computation for cyclic plastic deformation, which further sheds light on the failure mechanism using ABAQUS 6.10. The modeled material consists essentially of a matrix and inclusions. In this 3D model, the matrix is built first in ABAQUS CAE and ellipses are cut from the matrix. Then elliptical parts with the same dimension as the cutting parts are drawn and subsequently tied to the matrix. The carbide inclusions were considered to be linear elastic in the finite element model with elastic modulus E ¼ 405 GPa and Poisson’s ratio n ¼ 0.14[1]. The matrix is a double-edge notch with a height of 20 mm and a width of 10 mm. Due to symmetry, only half of the double-edge notched specimen was modeled to render the simulations less computationally intensive. The inclusions are assumed to be elliptical in shape with a representative aspect ratio, Ar ¼ a/b ¼ 3, where a and b are semi-major and semi-minor dimensions of the ellipse, respectively as illustrated in Fig. 5 for a notched component with a single inclusion. The 3D finite element model with an ellipsoidal inclusion is illustrated in Fig. 6. A coarse mesh size is used in the far field zone to enhance computational efficiency, while near the notch and the inclusion regions, the mesh size is finer. 3D 10-node quadratic tetrahedral element (C3D10) is used due to the complex geometry. The matrix and the inclusion are intact with each other. Six steps are subsequently created for three loading cycles and boundary conditions are applied to the whole model. Remotely applied strain is added via displacement boundary conditions on the upper with the absolute value of DL ¼ L 0.5 3 y (3 y ¼ 0.60% is the macroscopic proportional limit of the polycrystalline IN 100 superalloy obtained through experimental stressestrain data[17]). The amplitudes of imposed completely reversed nominal axial cyclic strain (R3 ¼ 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 IN 100
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polycrystalline materials. In other words, only a relatively small number density of grains yield in the notch root and inclusion region after the initial loading cycle. 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 and inclusion regions. The use of crystal plasticity is also 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 HCF regime. Using the constitutive model in Section 3, computational crystal plasticity finite element simulations with appropriate microstructure length scale dependence were performed for the notched components with non-metallic inclusions with notch root radii varying from 0.2 mm to 1 mm. For each loading condition, a set of different realizations of grains are implemented with randomly assigned orientation distribution to obtain an initially isotropic effective medium while gathering information regarding variability among instantiations. Random grain orientations were assigned since there was lack of information regarding orientation and misorientation distributions functions for the IN 100 microstructure used in this study. Two stages of fatigue life are modeled: crack incubation and MSC growth because they account for a large percentage of fatigue life under HCF and very high cycle fatigue (VHCF). Crack incubation is the event marked by the first appearance of new surface area, which can be seen as the formation of a crack on the order of the grain size. Inclusions, flaws, voids, and notches act as stress raisers to provide sites for crack incubation. The driving force for fatigue crack nucleation is the maximum range of cyclic effective plastic shear strain. Shenoy et al.[9] made an extension of Tanaka and Mura’s[30] crack incubation for estimation of crack nucleation life in IN 100. The number of cycles required for the formation of a grain-size crack is obtained using the relation[9] Ninc
Dgpmax
2
2 ¼ avg
ag dgrain
(5)
is the maximum range of the plastic shear strain where Dg over a fatigue cycle, ag ¼ 0.056 mm-cycle, and dgrain ¼ 34 mm is the average grain size of coarse grain in IN 100. This equation shows that the maximum range of the cyclic plastic shear strain averaged over the grain region is used as the driving force for crack incubation. The MSC growth begins after the formation of the crack on the order of the grain size. Previous work by Fatemi and Kurath[31] has demonstrated a solid correlation of the fatigue crack propagation life under various stress states with and without mean stress. In this work, the grain scale averaged FatemieSocie (FS) parameter[32] is used as the driving force to describe the growth of the MSC on the order of three times of the grain size. A modified form of the MSC growth law, which is put forward by Shenoy et al.[9] is given as p max
207
vectors. The nonlocal FS fatigue indicator parameter, DG, is given by
Dgpmax
DG ¼
2
smax 1þk n
sy
(7)
where k ¼ 1.0 is a material parameter that is used to take into consideration the effect of normal stress in crack formation and early growth, sy is the macroscopic cyclic yield strength, and smax is the maximum normal stress acting on the defined n plane with grain scale averaged value. To obtain the nonlocal DG from the finite element simulation, we consider a set of planes at each integration point lying within the nonlocal averaging region. For consistency in comparison of the current results with previous work in literature[34], Dgpmax and 3 max sn are averaged over a volume of (4 mm ) and calculated over the third fatigue cycle for all the simulation instantiations since this volume is smaller than the statistically small grain size, capable of detecting gradient in plastic shear within the grains, and is sufficiently large to enable the convergence of the response DG[34]. At each integration point, the plastic shear strain gpq on the q plane is calculated by resolving the plastic strain tensor 3 pij onto it at each integration point, i.e.,
gpq ¼ ni 3 pij tj ;
q ¼ 1.N
(8)
where ni is the unit normal vector on plane q, tj is a unit tangent vector in the considered direction along the plane, and N is the number of discrete planes sampled. Discrete set of tangent vectors are screened on any given q plane to estimate the plastic shear strain. The nonlocal average plastic shear strain associated with the q plane is then calculated by averaging the plastic shear strain on the q plane over the volume Vb of the nonlocal region around the notch and the inclusions, i.e.,
gpq ¼
1 Vb
Z
gpq dV
(9)
Vb
The maximum of the range of gpq amongst all planes is taken to be the nonlocal plastic shear strain range, i.e.,
Dgpmax ¼ max Dgpq
(10)
Once the critical plane and the nonlocal maximum plastic shear strain range Dgpmax are determined, DG is calculated using Eq. (7) at every integration point on the critical plane (deter mined by Dgpmax ) and a volume average is performed in the nonlocal region similar to Eq. (9); this effectively results in a nonlocal DG. 5. Probabilistic Framework 5.1. Probability of failure for crack formation
da ¼ AFS sy DGa hb dN
(6)
where AFS ¼ 8.1 104 (MPa)1 for IN 100 is a constant calibrated through experiments, sy is the critical resolved shear stress given as sy =M with M ¼ 3.06, which is the Taylor factor for a texture free polycrystalline aggregate[9,17], h is a constant with the value close to 1 and b is the magnitude of the Burger’s
The objective of this probabilistic approach is to characterize the effect of inclusions and microstructurally-sensitive notch root effect on the HCF life of the polycrystalline IN 100 superalloy, which considers the probability of crack formation within a grain size. The FS fatigue indicator parameter distribution function is determined through the computational crystal plasticity model described before. The prior work by Owolabi
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et al.[33] lays foundation for this work. The probability of formation of a crack is given as[33] 8 9 < 1 Z DG DG bG = th Pf ¼ 1 exp dV : V0 ; DG0
(11)
Vd
In Eq. (11), DG0 is the scale parameter, bG is the shape parameter used to measure the scatter in the DG distribution among grains within the damage process zone at the notch root, and DGth is the threshold fatigue indicator parameter, below which no microdamage will occur at a given number of cycles in the HCF region, Vd is defined as the smallest volume corresponding to a locus of points enclosing grains with fatigue indicator parameter values above the threshold value and V0 is the volume of a reference smooth specimen. 5.2. Probability of failure for MSC growth In order to calculate the probability of forming and growing a crack to a transition crack length, l, the fatigue indicator parameter-based transition crack length approach developed by Musinski and McDowell[34] is adopted. It considers the intensity of the fatigue indicator parameter over a characteristic length to assess the probability of forming and propagating of a crack from the notch root to a transition crack length. In this approach, the l is the distance, in which a crack propagates in a certain length, which has no influence on the notch root stress concentration. For semi-circular notch, the l is equal to 0.13r according to the work done by Smith and Miller[35]. For the notches with size less than 0.5 mm, l is only 1 or 2 grain diameters. The transition cracks for this work are 0.1 mm for notch radius 0.2 mm, 0.78 mm for notch with radius 0.6 mm, and 0.13 mm for notch with radius 1.0 mm. The probability for crack propagation to the transition length is affected by the applied stress or strain amplitude, the notch root size, and the mechanisms of MSC growth, which is dependent on the size of preexisting crack at a given number of cycles. The cumulative distribution function (CDF) is given as h a i n CDF ¼ 1 exp h l
(12)
where h is equal to ln(0.5) used to provide normalization to the case for CDF ¼ 0.5 when crack propagates at the length of l[34]. The crack length an can be found by integrating the crack growth rate in Eq. (6) and substituting the resulting expression in Eq. (12) to obtain 2 6 CDF ¼ 1 exp6 4h
Z
3 AFS sy DGmax adN 7 7 5 l
(13)
6. Results and Discussion 6.1. Fatigue indicator parameter distribution vs distance from notch root The magnitude and distribution of the effective plastic strain among grains for different notch root radii is an important factor in determining the crack nucleation and the early growth of
Fig. 7 Maximum fatigue indicator parameter vs distance from notch root for r ¼ 0.2 mm (a), r ¼ 0.6 mm (b), and r ¼ 1.0 mm (c) at strain amplitudes of 0.53 y.
microstructurally small fatigue cracks at notch root, which is further used to assess the statistical distributions of the fatigue indicator parameters and to obtain the volume of the fatigue damage process zone. In this work, the maximum fatigue indicator parameter distribution is utilized to define the fatigue driving force for the formation and growth of small crack at the notch root. Fig. 7 shows the distribution of the maximum fatigue indictor parameter, DGmax with respect to the distance in x-direction from the notch root for different notch radii (0.2 mm, 0.6 mm, and 1.0 mm). As can be seen from the figure, there is a rapid decrease in DGmax values at the beginning of the plot, and it follows by an embossment at the position around 1 mm for 0.2 mm notch, 0.6 mm position for 0.6 mm, and 0.3 mm position for 1.0 mm notch. These positions are also where the inclusions are added for the notched component. So this phenomenon shows that the inclusion acts as DG raiser. Also, there is a significant difference in DG intensity among three different notch
G.M. Owolabi and H.A. Whitworth: J. Mater. Sci. Technol., 2014, 30(3), 203e212
Fig. 9 MSC growth fatigue life vs notch radius.
Fig. 8 Incubation life vs notch radius.
radii. The larger the notch root radius, the higher the maximum fatigue indicator parameter intensity. It means that the driving force for crack incubation and MSC growth in larger notch radius component is higher than that for smaller ones.
The fatigue crack initiation life is the sum of the number of cycles required for an MSC to form and grow outside the influence of the notch root plastic zone. Eq. (5) is used in this work to estimate the incubation life, Ninc for the notched IN 100 component with inclusions. Rearranging Eq. (5), it gives p 2 Dgmax = dgrain 2 avg ag
(14)
The crack incubation lives for different random orientation of grains at different notch radii are calculated using Eq. (11). Note that the grain with the smallest computed incubation life is assumed to be the grain in which crack incubation will occur. Fig. 8 shows the incubation life distribution for different n. It is very obvious that there is a decrease trend for crack incubation life at different notch sizes. For notch size 1.0 mm, the crack incubation life is the lowest. 6.3. MSC growth life MSC growth begins once a grain-size crack is formed. In this study, the MSC life, NMSC is defined as the number of cycles for the incubated crack to grow to 3 grains diameter since it has been noted in literature[34] that the crack growth curves obtained from experiments show that small crack and long crack curves typically merge at crack length in the range of 3e10 grain diameters. After integrating and rearranging Eq. (6), the number of cycles to propagate a crack from aj1 to aj is given as[34] 0 NMSC aj1 /aj ¼
trend as the incubation life; that is, the larger the notch radius, the higher the probability of propagation of small crack and the smaller the MSC growth life. 6.4. Calibration to experiments and notched IN 100 without inclusions
6.2. Crack incubation life and MSC growth life
Ninc ¼
209
1
mb B aj AFS sy DGave C 1 C lnB A b AFS sy DGave @aj1 A smDG FS y ave
(15)
where DGave is the average of the maximum FS parameter over the increment of crack length from aj to aj1. Fig. 9 shows the average MSC growth life vs the notch radius. It has the same
Under HCF and VHCF regimes, the total cycles for crack incubation and MSC growth (Ninc þ NMSC) take up to nearly 90% or more of a component’s fatigue lifetime. So the total fatigue life is close to the sum of crack incubation fatigue cycles and MSC growth fatigue cycles. In order to assess the influence of notches and inclusions on the fatigue life of the notched component using the computational results obtained previously, the experimental results for smooth specimen from the work by Cowles et al.[36] and the work by Bathias and Paris[37] are used to illustrate the notch effect on the fatigue life, while the simulation results from the work by Musinski and McDowell[34] for notched components without inclusions is employed to assess the effect of inclusions only on the fatigue life. Cowles et al.[36] performed an extensive study on five different nickel-base alloys in order to evaluate their total crack incubation and crack propagation fatigue life. All the experiments were carried out under fully reversed strain-controlled situation and under both cyclic and cyclic-hold condition at 650 C. In this work, the test results for HIP Astroloy nickel-base superalloy were used. The work done by Bathias and Paris[37] was in the regimes of HCF (w106) and VHCF (w109). In this regime, crack usually nucleated at microscopic flaws and defects. So the study of a nickel-based alloy (N18) had been chosen to compare the effect of inclusions on IN 100. The work done by Musinski and McDowell[34] was mainly on the smooth and notched IN 100 components with different notch radii without inclusions. It is important to state that the previous work by Musinski and McDowell[34] also used the experimental results from Cowles et al.[36] for HIP Astroloy and the experimental results from Bathais and Paris[37] to calibrate the simulation results for the smooth IN 100 superalloys without non-metallic inclusions. The justification for using these experimental results have been detailed in Section 3 of reference[34], but a summary is provided here along with the limitations. The Astroloy nickel-base superalloy experimental data were used because its grain size, which is in the range of 30 mme70 mm, is similar to the grain size of the IN 100 with an average mean grain size of 34 mm. Furthermore, since there was no detailed experimental data
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Fig. 10 Comparison among results from experiments, notched with inclusion component, notched without inclusion components: (a) r ¼ 0.2 mm, (b) r ¼ 0.6 mm, (c) r ¼ 1.0 mm, (d) comparison of all notch root radii.
regarding the crack length measured in literature[36] at a 5% stiffness drop, Musinski and McDowell[34] assumed that this level of stiffness drop corresponds to the crack initiation and the MSC growth through three adjacent grain sizes. Musinski and McDowell[34] also indicated that the ultrasonic fatigue testing of the IN18 at 450 C can provide sufficient comparison to the fatigue testing of the polycrystalline IN 100 at 650 C both in the HCF and VHCF regimes. It is interesting to note that the simulation results obtained for the smooth specimens in literature[34] for the IN 100 superalloy predict the general trends observed in the experimental results for the HIP Astroloy nickelbase superalloy[36] and the N18 nickel-base superalloy[37] very well at both the LCF and the HCF regimes and are very close to the strain life curve in the HCF and VHCF regimes. Thus, these experimental results for the smooth specimens in literature[36,37] are sufficient for the calibration of the simulation results for the notched specimen with and without non-metallic inclusions used in this study. For this study, the experimental results are also adequate to characterize the fatigue strength reduction effects of the notch and non-metallic inclusions at a given probability of failure. Fig. 10(aec) are the results for notch radii (0.2 mm, 0.6 mm, 1.0 mm) and Fig. 10(d) is the comparison among all notch radii. The existence of notch leads to a drastic drop in component’s life. The comparison between the notched component with inclusions and the notched component without inclusion also shows that the presence of inclusion in component causes a significant decrease in fatigue life. Another interesting observation is that a crack may not initiate in the smooth specimen at the strain amplitude (3 a ¼ 0.53 y) used even in the regime of VHCF,
while for notched components with inclusions, a crack initiates at the HCF regime. This observation agrees well with the initial observations in literature[34] for notched components without non-metallic inclusions. It is noted here that the simulation results could have been more efficient if the results have been compared with experimental results from fatigue data obtained using the actual polycrystalline IN 100 superalloys notched components under the same loading conditions. Additionally more robust simulation results could have been obtained: i) if the actual size and volume distributions of the non-metallic inclusions in the specimens are obtained using characterization tools such as computed tomography and the information embedded in the current model, ii) by accounting for the grain size distributions in the IN 100 since larger grains in the materials microstructure have higher probability for fatigue crack initiation than smaller grains and iii) by accounting for the initial residual stresses due to manufacturing processes[34]. These limitations will be addressed in the future studies for various materials, notch root geometries and acuity as well as different loading conditions. 6.5. Probability of failure for crack formation and growth The following probabilistic approach is used to determine the probability of formation and growth of a fatigue crack on the scale of a single grain, which is developed by Owolabi et al.[33] for HCF and VHCF fatigue regimes and given in Eq. (11). The threshold fatigue indicator parameter in Eq. (11) is estimated through Eq. (6). That is, when the crack growth rate is zero, the threshold value is obtained as
G.M. Owolabi and H.A. Whitworth: J. Mater. Sci. Technol., 2014, 30(3), 203e212
211
Fig. 11 Probability of failure vs notch root radius for notched component with inclusion and notched component without inclusion.
DGth ¼
hb AFS sy a
(16)
where a is considered to be the grain size, and sy ¼ 241.83 MPa. The threshold value for polycrystalline IN 100 is 3.04 104. The definition of failure in probabilistic framework[33] is the formation of a crack on the order of a grain size. The probability of failure, Pf vs notch size for both notched components with and without inclusions is shown in Fig. 11. As seen from the plot, the existence of inclusion increases the probability of formation of MSC in a notched component. The probabilistic approach for crack growth to a transition length is the extension of work done by Musinski and McDowell[34]. In order to calculate the CDF, the maximum value of the fatigue indicator parameter should be obtained first through simulation. The relationship between DGmax and the distance, x from the notch root up to 5 times the grain size is shown in Fig. 12. For notched components with non-metallic inclusions, there is no linear relationship between DGmax and the distance from notch root unlike the linear relation between the maximum fatigue indicator parameter and the distance from notch root obtained in Musinski and McDowell[34], which does not account for the presence of inclusions in the notched IN 100 component. Applying the fatigue indicator parameter-based transition length crack probabilistic framework for the notched IN 100 superalloy component with inclusions, the cumulative distribution values are shown in Fig. 13(aec). It can be seen that
Fig. 13 Cumulative distribution function for IN 100 with different notch radius at 3 ¼ 0.53 y and R3 ¼ 1: (a) r ¼ 0.2 mm, (b) r ¼ 0.6 mm, (c) r ¼ 1.0 mm.
cumulative distribution function can be calculated for any number of cycles and any probability of failure. This approach also shows that the presence of inclusions in notched component has higher failure probability than notched component without inclusions and the larger the notch radius, the higher the probability of formation and growth of MSC. 7. Conclusion
Fig. 12 Maximum fatigue indicator parameter values vs distance from the notch root for r ¼ 1.0 mm within five grains length.
A computational framework was presented to determine the influence of non-metallic inclusions and notches on the formation and growth of MSC in polycrystalline IN 100 superalloy and to assess the effects of the non-metallic inclusions on the fatigue life and the probability of fatigue failure of notched polycrystalline IN 100 components. The distribution of the effective plastic strains around the notch and the inclusions is used to
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estimate the driving forces for crack formation and MSC growth in the notched IN 100 with non-metallic inclusions. The finite element simulation results show that the driving force for notched component with inclusions is higher than that for the notched component without inclusions. The results also show that the larger the notch radius, the higher the probability of formation and growth of MSC. Comparison between fatigue lives for notched component with inclusions and experimental results for smooth component shows that the presence of notch and inclusions leads to a drastic drop in the fatigue life. Acknowledgments The authors are grateful for the funding provided by the Department of Defense (DoD) through the research and educational program for HBCU/MI (Contract No. # W911NF-11-1014) monitored by Dr. Larry Russell (Program Manager, ARO) and Dr. David Stargel (Program Manager, AFOSR). REFERENCES [1] D.L. McDowell, K. Gall, M.F. Horstemeyer, J. Fan, Eng. Fract. Mech. 70 (2007) 49e80. [2] D.L. McDowell, Int. J. Fatigue 19 (1997) 127e135. [3] R.G. Tryon, NASA Contractor Report 202342, 1997. [4] R.O. Ritchie, J. Lankford (Eds.), Small Fatigue Cracks, Metall. Soc. Inc., Pennsylvania 15086, 1986, pp. 1e5. [5] B. Lin, L.G. Zhao, J. Tong, Eng. Fract. Mech. 78 (2011) 2174e2192. [6] J.D. Hochhalter, D.J. Littlewood, M.G. Veilleux, J.E. Bozek, A.M. Maniatty, A.D. Rollett, A.R. Ingraffea, Model. Simul. Mater. Sci. Eng. 19 (2011) 035008. [7] F. Tancret, Processing for China, Sterling Publications, London, 2000, pp. 56e58. [8] S.A. Padula, A. Shyam, R.O. Ritchie, W.W. Milligan, Int. J. Fatigue 21 (1999) 725e731. [9] M. Shenoy, J. Zhang, D.L. McDowell, Fatigue Fract. Eng. Mater. Struct. 30 (2007) 889e904. [10] C. Mercer, A.B.O. Soboyejo, W.O. Soboyejo, Mater. Sci. Eng. A 270 (1999) 308e322. [11] M. Goto, D.M. Knowles, Eng. Fract. Mech. 60 (1998) 223e232. [12] C.P. Przybyla, R. Prasannavenkatesan, N. Salajegheh, D.L. McDowell, Int. J. Fatigue 32 (2010) 512e525. [13] C.P. Przybyla, D.L. McDowell, Procedia Eng. 2 (2010) 1045e 1052.
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