ABSTRACT:Crack nucleation, first spall generation, and spall growth in roll-ing contact fatigue共RCF兲 are known to be highly sensitive to the heteroge-neity of the microstructure. Yet the current state-of-the-art in the design of high performance bearing materials and microstructures is highly empirical requiring substantial lengthy experimental testing to validate the reliability and performance of these new materials and processes. We have laid the groundwork necessary to determine the influence of microstructure in RCF to aid in the development and processing of bearing steels. Microstructure at-tributes that may control the fatigue behavior are explicitly modeled in a 41xxx steel. The methodology is demonstrated by studying the role of an aluminum oxide inclusion embedded in a matrix of tempered martensite and retained austenite. The matrix is represented by crystal plasticity, which pro-vides more realistic accumulations of localized plastic strains with cycling compare to homogenized J2plasticity. As a demonstration of the approach, the relative influence of the volume fraction of retained austenite on RCF is evaluated.
KEYWORDS: bearing steel, rolling contact fatigue, crystal plasticity, non-metallic inclusion, martensite, retained austenite, transformation
Introduction
Crack nucleation, first spall generation, and spall growth in rolling contact fatigue共RCF兲 are known to be highly sensitive to the heterogeneity of the
mi-Manuscript received June 19, 2009; accepted for publication October 23, 2009; published online December 2009.
1The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Tech-nology, Atlanta, GA 30332-0405.
2Timken Research Center, Canton, OH 44706.
3The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Tech-nology, Atlanta, GA 30332-0405,共Corresponding author兲 e-mail: [email protected] Cite as: Alley, E. S., Sawamiphakdi, K., Anderson, P. I. and Neu, R. W., ‘‘Modeling the Influence of Microstructure in Rolling Contact Fatigue,’’ J. ASTM Intl., Vol. 7, No. 2.
doi:10.1520/JAI102629.
Copyright © 2010 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959.
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crostructure at the microscopic level. Features in the microstructure include non-metallic inclusions, either hard共e.g., Al2O3and TiN兲 or soft 共e.g., MnS兲, the size, morphology, and volume fraction of the different phases共e.g., martensite, retained austenite, bainite patches兲, and the crystallographic orientation of each of the phases relative to each other关1兴. Furthermore, the elastic and pltic properties at the crystalline scale are highly anisotropic. The typical as-sumption of homogeneous, isotropic material, even with consideration of em-bedded inclusions, is often highly idealistic. Because of the difficulty in predicting the influence of microstructural heterogeneity on RCF life, the cur-rent state-of-the-art in the design of high performance bearing materials and microstructures continues to be highly empirical requiring substantial lengthy bench testing to validate the reliability and performance of new materials and processes.
It is well known that the empirical parameters correlated to bench tests are sensitive to the microstructual attributes of the material关2兴. Based on fracture mechanics arguments, the fatigue life is known to be inversely proportional to the square-root of the size of the non-metallic inclusions 关3兴. Many other at-tributes of the microstructure can influence RCF. These include inclusion com-position, size, orientation, their distribution and volume fraction, interfacial/
particle strength, grain size, morphology and distributions, microtexture, compositional variations共e.g., banding兲, microhardness, percentage of retained austenite, and residual stress. Experimental work to evaluate the influence of all of these attributes on bearing fatigue can be costly and can take long amounts of time to complete, and thus it desirable to have a numerical model for predicting the influence of different microstructural attributes on RCF. With a numerical model, many different microstructural realizations can potentially be evaluated more efficiently to help reduce the number of costly and time-consuming bench tests and hence to more quickly hone in on optimum pro-cessing conditions.
In RCF, the crack often initiates at a subsurface location where the cyclic shear stresses are greatest. Hence, the heterogeneity of the microstructure plays a more important role compared to typical fatigue scenarios where cracks form on the surface. There are key microstructural attributes known to be associated with RCF. For example, the formation of “butterfly wings,” typically around hardAl2O3inclusions, as seen in Fig. 1, is driven by the accumulated deforma-tion accentuated by the difference in the mechanical properties of the inclu-sion, matrix, and interface 关1,4兴. The location and orientation of the altered microstructure adjacent to the inclusion is not random but is linked to the directions of the local cyclic deformation in the vicinity of the inclusion关1,5,6兴.
Inclusions can appear singly, as shown in Fig. 1, or grouped together in long chains known as stringers. When grouped in close proximity, these are often treated as a single, elongated inclusion. The orientation of such stringers can have a significant effect on fatigue performance.
The amount of retained austenite 共RA兲 affects the fatigue performance of steel based on the loading conditions, i.e., differently for RCF as opposed to bending fatigue. For bending fatigue, higher amounts of RA are detrimental to high cycle fatigue共HCF兲, while RA can be beneficial for low cycle, high strain fatigue关7兴, which suggests that RA should be detrimental for RCF. However, for
some RCF conditions, the fatigue life has been found to increase as RA in-creases 关8兴. One beneficial role of RA in RCF is preventing fatigue cracking from debris dents, most likely due to the compressive residual stress field that forms around the debris dents due to transformation of austenite to martensite 关1兴. Since increased amounts of RA reduce the surface hardness, it is important to balance the appropriate amount of RA with the anticipated operating condi-tions. This balance is critical, as the benefit of increased life in debris contain-ing environments can be negated by transformation-induced distortion when RA transforms to untempered martensite during RCF conditions关1兴. When RA transforms to martensite, there is a volumetric transformation strain that needs to be modeled since it clearly alters the local residual stresses, which in turn affect the fatigue drivers.
Under long-term RCF conditions, changes in the microstructure are often observed in the subsurface regions undergoing the greatest loading. One change is the formation of microtexture, which is a preferred orientation of the crystallographic grains关9,10兴. Generation of microtexture depends on loading, temperature, and number of cycles. Bearings that run cool at extreme loads tend to form deformation textures, whereas bearings that run warm at nominal loads tend to promote recrystallization textures. Texture has been shown to influence the character of the spall that forms under RCF关10兴.
Bearing steels comprise a complex hierarchical heterogeneous microstruc-ture of mostly lath martensite in austenite. Plastic deformation is driven by the magnitude of resolved shear stress on individual slip systems. The resolved shear stress depends on the crystal structure and orientation with respect to the
FIG. 1—White etching near Al2O3inclusion关5兴.
loading. In polycrystalline and multiphase systems, the resolved shear stress also depends on the intergranular constraints and anisotropic properties of the matrix and inclusions. Hence, a crystal plasticity formation is needed to cap-ture the highly non-uniform cyclic strain field developed due to the heteroge-neous microstructure. In HCF problems, crystal plasticity has been shown to be a much more realistic model to predict the localized cyclic deformation 关11–13兴. In fact, when using a conventional J2plasticity formulation to model HCF and RCF, the analysis almost always predicts the response remains elastic, yet it is well-understood that fatigue is driven by the accumulation of plastic strains locally in the microstructure关14兴. Crystal plasticity captures these local plastic strains under conditions when HCF life is known to be finite. Because of the fine microstructure of steels that is comprised of multiple phases and struc-tures, they are more challenging to model using crystal plasticity than other classes of metals. To reduce the complexity, polycrystal plasticity models typi-cally assume that steel can be described by a single phase共i.e., all of the crys-talline grains have the same crystal structure and properties兲. Single-phase crystal plasticity have been used to model low carbon steels关15兴, HSLA steels 关16兴, pressure vessel steels 关17兴, austenitic stainless steel 关15兴, and martensitic gear steels关18兴. Simulations using crystal plasticity have been used to study the influence of texture on fatigue of a HSLA steel 关19兴 and the influence of inter-granular constraints in both low carbon steel and austenitic stainless steel关15兴.
Both studies clearly showed the importance of the local ratcheting as a driver for fatigue damage.
In this paper, a new two-phase crystal plasticity model for martensitic bear-ing steels containbear-ing RA is described. Prior crystal plasticity models of steels only considered a single phase. This material model is coded as a User MATerial 共UMAT兲 subroutine for the general-purpose finite element code ABAQUS 关20兴.
Using a volume element approach described in a companion paper关14兴, a fully three-dimensional finite element simulation of RCF is performed to capture end effects of elongated aluminum oxide inclusions not possible with two-dimensional plane-strain models, providing for a more realistic assessment of inclusion morphology and arbitrary orientations. The scaling of the finite ele-ment model is optimized to capture the cyclic microplasticity around the inclu-sion accurately and efficiently. The links between the microstructural attributes and RCF are evaluated using fatigue indicator parameters. Using the new ma-terial model, the influence of the volume fraction of RA on RCF is examined.
Representative Bearing Steel Microstructure and Properties
The 41XXX series steel was chosen as a representative bearing steel for this study. The microstructure consists of lath martensite formed in a prior auste-nite grain. Pockets of RA can remain between the laths depending on heat treatment. The response behavior of these steels can vary greatly by the initial volume fraction of RA, heat treatment, and distribution of primary carbides.
Two variations with different carbon content, 4145 and 41100, were used to calibrate the material model. The heat treatment of 41100 steel was varied to control the initial volume fraction of RA, as illustrated in Fig. 2. Three different
heat treatments were used. The direct quench resulted in the greatest amount of RA. It was 35 %, as measured by x-ray diffraction. A conventional heat treatment共CH兲 resulted in 22 % RA and a special heat treatment resulted in 13
% RA. The lower levels of RA were primarily due to the increasing amount of primary carbides that form during the re-austenitization step. The 4145 steel was processed using the CH.
Optical images of the microstructures are shown in Fig. 3. The 4145 mate-rial is nearly fully martensitic, as indicated by the densely packed martensitic laths in Fig. 3共a兲. Thus it can be reasonably treated as a single-phase martensite component and is used to calibrate the martensitic plasticity portion of the material model. The 41100 steel with 13 % RA, shown in Fig. 3共b兲. As the volume fraction of RA increases to 22 % 关Fig. 3共c兲兴, visible portions of RA 共ligher shade areas兲 are detectable and when RA is 35 % 关Fig. 3共d兲兴, significant RA is visible. As seen in Fig. 3共c兲, the RA distribution is also often inhomoge-neous, with some prior austenite grain regions containing more RA than oth-ers. This can be accounted for and studied using the newly developed model described herein by assigning individual elements or element regions different initial volume fractions of RA.
Uniaxial mechanical tests were conducted on cylindrical dog-bone speci-mens with a gage diameter of6.35 mm共0.25 in.兲 and gage length of 12.7 mm 共0.50 in.兲. Both the axial and diametral strains were measured simultaneously using two extensometers. The experiments were conducted at room tempera-ture and a strain rate of1.0⫻10−4s−1. The tensile response of the 4145 steel is shown in Fig. 4 and the responses of the three 41100 steels are shown in Fig. 5.
Note that the response of the fully martensitic 4145 steel is similar to the 41100 steel with lowest amount of RA. With increasing RA, the apparent yield strength decreases due primarily to the transformation of RA to martensite.
Since there is a volume change associated with the transformation, the FIG. 2—Heat treatments for 41100 steel.
amount and evolution of the transformation can be determined by tracking the volumetric transformation strain, as shown in Fig. 6. The volumetric transfor-mation strain was determined from the strain readings after subtracting the elastic strain, assuming plastic strains do not cause any volume change, follow-ing the procedure described in Neu and Sehitoglu 关21兴. The results shown in Fig. 6 confirm that the apparent yielding is due to transformation of RA to martensite.
The uniaxial compression response is shown in Fig. 7. The amount of transformation is reduced in compression compared to tension for the same amount of applied axial strain. This suggests that the transformation is prima-rily controlled by a stress-assisted mechanism and that hydrostatic stress, which is smaller under uniaxial compression than under uniaxial tension, plays a role in controlling the transformation. In addition, cyclic deformation experi-ments were conducted to study the evolution of transformation with cycling.
The rate of measurable transformation diminishes to negligible levels per cycle after the first couple of cycles under constant amplitude cycling.
Hybrid Transformation and Crystal Plasticity Model
To explore bearing performance in multi-phase alloys, the effects of RA must be considered. The austenite can promote ductility and can also induce residual stress fields due to volume expansion during the phase change. Thus, the two-phase model was developed to account for both stress-assisted transformation FIG. 3—Optical images of共a兲 4145 steel, 共b兲 41100 steel with 13 % RA, 共c兲 41100 steel with 22 % RA, and共d兲 41100 steel with 35 % RA.
and plasticity. The transformation model tracks the evolution of the volume fraction of RA that is transformed, as well as captures the volumetric strains produced by such activity.
Austenite-to-martensite transformation deformation parameters are for-mulated in a manner described by Gall and Sehitoglu关22兴, with transformation directions and habit plane normals relative to orientation of a single crystal depicted in Fig. 8. Based on the mechanical tests, the stress-assisted transfor-mation model proposed by Suiker and Turtletaub关23兴 was identified as the best choice for implementing transformation. Their formulation was chosen over other formulations for steels such as that proposed by Karaman et al. 关24兴 because a key feature of the model is its multiplicative decomposition, consis-tent with formulations of crystal plasticity models. A rate-dependent crystal plasticity model formulated by Asaro 关25兴 and McGinty 关26兴 was chosen to describe the dislocation slip behavior.
The new two-phase hybrid model is built around a triple multiplicative decomposition of the deformation gradientF>, given as
FIG. 4—Uniaxial tensile experimental response and calibration of the two-phase crystal plasticity model for 4145 steel.
F> = F>e· F>p· F>tr 共1兲 where F>tr accounts for the volumetric strain produced by the austenite-to-martensite phase transformation,F>paccounts for polycrystalline plasticity, and F>eaccounts for elastic deformations and rigid body translations. Plasticity and transformation occur along slip and transformation systems associated with the lattice structures of the martensite and austenite. The evolution of the crys-tal plasticity follows the formulation work of Asaro关25兴. The model uses the 48 body-centered-cubic 共bcc兲 slip systems to represent the behavior of the tem-pered martensite. This is a reasonable assumption since during the early stages of tempering, the body-centered-tetragonal 共bct兲 lattice of virgin martensite quickly relaxes to obtain a bcc symmetry. The transformation is controlled by the formulation based on Suiker and Turtletaub关23兴. This is calculated on the 24 face-centered-cubic 共fcc兲 transform systems for austenite-to-martensite transformation.
For the transformation model, the unit vectors bˆ>0共兲 andn>0共兲 correspond to the transformation direction, also referred to as the shape strain direction, and the habit plane normal, respectively, for the th transformation system in the FIG. 5—Uniaxial tensile experimental response and calibration of the two-phase crystal plasticity model for 41100 steel.
intermediate configuration. Transformation is said to occur when a critical driving stress on a given transformation system is reached. The driving stress ftr共兲 on a transformation system is related to the transformation and habit vectors, and the Cauchy stress by
ftr共兲=>:共␥T· bˆ>共兲丢n>共兲兲 共2兲 where the transformation direction and habit plane normal are rotated into the current configuration by
bˆ>共兲= F>e· bˆ>0共兲 共3兲 and
n>共兲=共F>e兲−T· n>0共兲 共4兲 and ␥T is the shape strain magnitude, a parameter which is uniform for all transformation systems. When the driving stress given by Eq 2 exceeds the FIG. 6—Measured volumetric transformation strain under uniaxial tensile loading and correlation to the two-phase crystal plasticity model for 41100 steel with different initial volume fractions of RA.
critical driving stressfcr共兲 for the transformation system, the rate of the volume fraction of transformation˙共兲on that system is given by
˙共兲=˙max· tanh
冉
1tr·具ftr共兲− fcr共兲典
fcr共兲
冊
共5兲where:
˙max ⫽ the maximum rate of transformation, as the hyperbolic tangent reaches a maximum value of one.
The transform viscosity parametertrcontrols the amount of rate dependence in the transformation. When all of the transformation systems are summed together,
FIG. 7—Uniaxial compression experimental response and prediction of the two-phase crystal plasticity model for 41100 steel.
V˙trans=N=1兺trans˙共兲 共6兲
they give the rate of change of the volume fraction transformed from austenite to martensite, which is then multiplied by the time step,⌬t, to obtain the total volume fraction transformed,
Vtrans= V˙trans·⌬t 共7兲
Limits placed on the system transform amounts 共兲 are based on the initial volume fraction of RA in the material, designated as RAinitial. The total transformation, as well as on any given system, cannot exceed the initial amount of austenite present, as governed by
0艋 Vtrans艋 RAinitial 共8兲
FIG. 8—Transformation direction and habit plane normal关22兴.
0艋共兲艋 RAinitial 共9兲 When activated, the transform rates˙共兲on the transformation systems drive the
0艋共兲艋 RAinitial 共9兲 When activated, the transform rates˙共兲on the transformation systems drive the