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A Micromechanics-Based Fatigue Damage Process Zone
Procedia Engineering
Available online at www.sciencedirect.com
Procedia Engineering 00 (2011) 000–000 Procedia Engineering 10 (2011) 496–505
www.elsevier.com/locate/procedia
ICM11
A Micromechanics-Based Fatigue Damage Process Zone G.M.Owolabia*, L. Shia, H.A. Whitwortha a
Department of Mechanical Engineering, Howard University, Washington, DC, USA, 20059
Abstract
A primary issue in high cycle fatigue life prediction is the appropriate definition of damage process zone (i.e. volume influencing fatigue crack initiation) for notched components. Several definitions of the process zone have been proposed based on stress distributions obtained using homogeneous elastic or elastic-plastic finite element analysis. It is generally accepted to be the region, adjacent to the surface of the specimen, where peak stress is highest and fatigue crack initiation occurs; assumed as perhaps several mean grain diameters in spatial extent. However most of these existing techniques have not yet been related to fatigue failure mechanisms via computational micromechanics studies. Therefore they do not address the role of microstructure explicitly in fatigue life prediction. In this study, computational micromechanics is used to clarify and distinguish process zone for crack formation relative to scale of notch root radius and spatial extent of stress concentration at the notch. A new nonlocal criterion for fatigue damage process zone based on the distribution of a shear-based fatigue indicator parameter is proposed and used along with a probabilistic mesomechanics approach to obtain a new microstructuresensitive fatigue notch sensitivity index, thereby extending notch sensitivity to explicitly incorporate microstructure sensitivity and attendant size effects via probabilistic arguments. The probabilistic approach presented in this study predicts the general trends of notch sensitivity obtained from experimental results in literature.
© 2011 Published by Elsevier Ltd. Selection and peer-review under responsibility of ICM11 Keywords: Crystal plasticity, Fatigue damage process zone, micromechanics, crack formation and growth, fatigue notch factor, notch sensitivity.
1. Introduction The development of simulation methods for calculating notch root parameters for purposes of estimating fatigue life is a critical aspect of designing against fatigue failures in notched components. Notches such as holes, keyways, threads, fillets, and weldments occur in most components. However, traditional local “hot-spot” methods of notch root analysis [1-8] based on maximum stress and strain components at the notch root are often inadequate for characterizing the fatigue life of notched components since the stress gradients at the notch root also contribute to fatigue failures [9] but are not accounted in the traditional hot spot methods. Conventional approaches for accounting for the combined effects of stress gradients and notch root plasticity is through the concept of notch root strength reduction, also known as fatigue notch factor, Kf.
* Corresponding author. Tel.: 1-202-806-6594; fax: 1-202-483-1396. E-mail address: [email protected] 1877–7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.04.084
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Motivated by microscopic observations that fatigue failure occurs by damage accumulation in a finite notch root damage process zone, which can encompass several grains, several research studies have explored estimation of the fatigue notch factor and the fatigue strength of notched components as a function of an average stress over the fatigue damage process zone [10-23]. These methods consider the average stress value over a significant length scale at the notch root when analyzing fatigue potency of a notched component. One of these methods; the the point method uses a specific distance from a critical distance from the notch-root which may correspond to a grain size [12] or plastic zone size [13] and its corresponding average stress level to predict the fatigue life of notched components. While a single parameter involving a critical distance may be a viable approach, size effects arising from deformation and damage phenomena may render an elementary simple single parameter approach like critical distance inapplicable [20]. Other methods [21-23] use the stress field intensity to define the fatigue notch factor and to propose a fatigue failure criterion. The stress field intensity is the average stress obtained by integrating the stress distribution obtained from finite element analyses over the damage process zone. The stress field intensity incorporates the effects of stress gradient. A weight function is also implicitly assumed to account for the contribution of stresses in the damage process zone and is incorporated into fatigue notch factor and failure criterion. This weight function depends on notch geometry, loading type, boundary conditions and material properties [19]. Although these techniques offer improvement over the local hot-spot methods and perhaps represent an important step towards a more reliable prediction of notch sensitivity, most of the models are based on assumptions necessary to simplify the complex nature of the problem since the size of the damage zone has been quantitatively determined [17]. While physical volume has been recognized as very vital in the development of more reliable fatigue notch factor estimation models and to fatigue damage (nucleation plus growth), there is not yet a suitable definition of fatigue damage process zone in literature based on underlying physical phenomenon at the microscale level that leads to the formation and growth of small cracks in real materials. These conventional methods for accounting for stress gradients are also deterministic and do not account for the role of microstructure explicitly. Therefore it is difficult to gain direct insight into the controlling physical phenomenon at the microstructure scale that gives more predictive explanation of the material dependence of the notch sensitivity. In HCF, the cyclic plastic strain is quite heterogeneously distributed within the damage process zone, an aspect that plays a strong role in the coupling of notch size and acuity with probability of fatigue crack formation and small crack growth from the notch root. As a consequence, it is unclear that volume averaged stress or stress field intensity is an appropriate indicator of the damage process zone driving force, based on macroscopic constitutive relations. In this study, computational micromechanics is used to clarify and distinguish fatigue damage process zone for crack formation relative to scale of notch root radius and spatial extent of stress concentration at the notch. A new nonlocal criterion for fatigue damage process zone based on the distribution of a shear-based fatigue indicator parameter is proposed and used along with a probabilistic mesomechanics approach to obtain a new microstructuresensitive fatigue notch sensitivity index, thereby extending notch sensitivity to explicitly incorporate microstructure sensitivity and attendant size effects via probabilistic arguments. 2. Fatigue Damage Process Zone Since fatigue crack initiation in polycrystalline materials is usually predicted from the critical plane containing the most favorably oriented grain, one appropriate approach to defining the fatigue damage process zone is to consider the statistical distributions of a nonlocal fatigue indicator parameter, , around the notch root. From these statistical distributions, the fatigue damage process zone may be assumed to consist of a number of grains with nonlocal values equal to or greater than a specified microscopic threshold value, th . For any given instance, it is assumed that damage at the microscale, i.e., microcracks, does not initiate in any grain with a nonlocal value below the threshold, th . The threshold fatigue indicator parameter, th , can be associated with the microscopic resolved shear stress threshold for yielding, y. y may be estimated from the macroscopic yield via the Taylor relation y/M, where M is the Taylor factor. To estimate th , y is used in the modified form of a microstructurally small crack growth law given by [24], i.e.,
da AFS y a b CTD CTDth dN
(1)
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for the nucleation and growth of microstructurally small cracks on the order of 3-10 grain size. In equation (1) AFS and are constants, “a” is the crack length, and b is the Burgers vector. The cyclic crack tip displacement range is given by CTD , and the threshold value, CTDth , is directly related to b . Assuming that cracks will nucleate if cyclic microplasticity exists in a given grain, i.e., > 0, the threshold value is given by da/dN = 0, resulting in:
th
b AFS y a
(2)
It is left to the characterization of the factors AFS and to quantify this threshold for each material of interest. Thus, for a notched specimen the process zone is defined as the volume around the notch root made of a number of grains with nonlocal fatigue indicator parameter, i , values above the threshold th (see Fig. 1 for illustration, the red color squares represent grains that satisfy this condition). The damage process zone can be mathematically represented as:
volume bounded by grains G (r ) at the notch root N r Vd ( N r ) satisfying i th , where i nonlocal FIP
(3)
Vd
i th Fig. 1: A sketch of the fatigue damage process zone (V d) in the notch root region showing heterogeneity in plastic strain and distribution of fatigue indicator parameter Note that any suitable parameter such as nonlocal hysteresis work density may be used as a driving force parameter in equations (2) and (3), but parameters which best reflect change in crack tip displacement, CTD , for crystallographic Stage I growth are preferred. The Fatemi Socie [25] parameter given by: p* max 2
nmax* 1 y
(4)
is used in this study as the fatigue indicator parameter, , since it can be used to approximate the mixed mode character of CTD in Eq. (1). In equation (4) is a coefficient that moderates the effect of normal stress to the maximum plastic shear strain range plane. The nonlocal peak stress normal, nmax* , to the plane of maximum shear strain amplitude is normalized by the polycrystalline cyclic yield strength, y. 3. Microstructure-Dependent Fatigue Notch Factor The statistical distribution of the nonlocal fatigue indicator parameter within in grains the well-defined damage process zone (illustrated by Fig. 1) discussed in the previous section is used in the development of a microstructure-
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dependent fatigue notch factor. This is an extension of the earlier work of Owolabi’ et al. [26] that combines computational crystal plasticity with a probabilistic framework to obtain a novel microscopic fatigue notch factor and associated notch sensitivity index. Here, 3D computational crystal plasticity was performed on OFHC copper to assess the degree of heterogeneity of cyclic plastic deformation as a function of notch size and notch root acuity for a range of strain amplitudes below the macroscopic yield strain and different realizations of aggregates of grains with random orientations at the notch root. Statistical information regarding the distributions of the shear-based fatigue indicator parameter is obtained providing useful insight into the dependence of fatigue notch factor and associated notch sensitivity index on the heterogeneity inherent in the actual microstructures. Results from simulations within the well-defined fatigue damage process zone are used in the development of new microstructure-dependent fatigue notch factor and associated sensitivity index. The method developed incorporates the effects of notch size, stress/strain gradients, and microstructural variability that have not been accounted for in previous traditional approaches. Using the Weibull’s weakest link theory and assuming that the nonlocal is a random variable, the microstructure-dependent fatigue notch factor is given in [26] as:
1 K f V d
th V net ( ave)
b dV
1
b
Vd Vo
1
b
,
(5)
where net is the net section nonlocal for the notched component, computed as the average of in the net section region outside the damage process zone, V. b and Vo are the shape parameter and reference volume respectively. An alternative statistical approach that considers the extreme value portion of the distribution in the fatigue damage process zone rather than the Weibull distribution used in [26] is used here to develop a new relation for the microstructure-dependent fatigue notch factor. Based on statistical distribution of , the differences between the extreme-values of in the tail section of the statistical distribution and the threshold th are fitted by an appropriate statistical distribution function. Fitting a parametric distribution to data sometimes leads to a model that agrees well with the data in high density regions, but poor accuracy in the low density regions due to sparse data [27]. In HCF applications in which few grains have nonlocal values greater than the threshold, fitting the data in the tail to an appropriate statistical distribution is associated with substantial challenges. Thus, the generalized Pareto distribution [28-29] that has been shown to model the tails of a wide variety of statistical distributions is used in this work. The distribution of the extreme values of nonlocal that exceed the threshold can be reasonably modeled using the generalized Pareto distribution with a distribution function, , of the form:
e th 1 1 Vo 0
1
,
(6)
where is equivalent to the shape parameter in equation (5), 0 is the scale parameter and e is the extreme value of at the tail end of the distribution function. The generalized Pareto distribution function in (6) is more appropriate than the Weibull’s distribution function used in [26] since in most engineering materials and components fatigue is manifested by extremal of microstructure attributes at various length scale that promote slip intensification. The extreme-value distribution function in equation (6) has three parameters: (i) the threshold or location, th , (ii) the shape, and (iii) the scale, 0 . Note that the shape parameter is a dimensionless quantity that depends on the material microstructure. Moreover, for the Fatemi-Socie nonlocal values the scale parameter is also dimensionless. Although the distribution functions have three parameters, the threshold value, th , which corresponds to the location parameter, is obtained using equation 2 as 6.14 x10-5. As a result, the number of unknown parameters reduces to two: i.e., and 0 . Since in HCF only a small number of grains of a smooth specimen will have non-zero microplasticity, the scale, o , and shape, , parameters are estimated (using the modified moment parameter estimation method [30]) respectively as 6.95 x10-6 and 0.85 the mean and variance of
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the plastic strain obtained experimentally for 25 notched specimens. Finite element simulation results are used to obtain Vd for different notch root radii using the condition stated in equation 3 at the notch root region. Using equation 6 above and following the probabilistic framework proposed in [26], a new relation is obtained for the microstructure-dependent fatigue notch factor of the form:
V K f Kt d , V0
(7)
where K t , is the microscopic concentration factor given by: 0 e th Kt dV . net ( ave ) Vd 1
(8)
The associated notch sensitivity index, q, is given as: q
K f 1 K t 1
(9)
The microscopic fatigue notch factor in equation (7) follows the spirit of the traditional fatigue notch factor i.e., the ratio of unnotched to notched values of driving force for a given HCF life and same probability of failure, but it is rooted in probabilistic arguments. The new probabilistic model for microstructure-dependent fatigue notch factor incorporates the strength of the notch root stress field gradient and distributions of microslip relative to underlying microstructure within a well-defined fatigue damage process zone. Equations (7) and (8) show that the degree of accuracy of the fatigue notch factor depends strongly on reliable prediction of the statistical distribution of the extreme-values of the shear-based fatigue indicator parameter. Thus, the fatigue damage process zone, Vd plays a major role in the determination of the microstructure-dependent fatigue notch factor. 4. Crystal Plasticity and Finite Element Implementation Since crystal plasticity models relate the grain scale stress to crystallographic slip response, they are suitable for studying heterogeneity and interaction across grains in the notch root field. The kinetics of dislocation glide are formulated by the relationships between the resolved shear stresses and the shearing rate along the slip systems. The shearing rate, , on the th slip system is described by the power law flow rule:
0
x g
m
sgn x ,
(10)
where 0 is the reference shearing rate, m is the inverse strain-rate sensitivity exponent, g is the isotropic drag strength, x is the back stress on the th slip system, and is the resolved shear stress, given by:
: s n .
(11)
In equation (11), is the Cauchy stress tensor. A direct hardening-dynamic recovery format relation is employed for evolution of g [31], i.e.,
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N
1
1
g H q R g ,
501
(12)
where q are components of the hardening matrix and H and R are the direct hardening and dynamic recovery coefficients, respectively, for the isotropic hardening relation. In the present work, q = 1 such that isotropic hardening is the same on all slip systems. The back stress on each slip system evolves according to a nonlinear kinematic hardening rule of self hardening type, i.e., x h hD x , where h and hD are the direct hardening and dynamic recovery coefficients, respectively. The material used in this study is polycrystalline OFHC copper (Cu) with mean grain size of 62 m and consisting of grains with random crystallographic orientation distribution. For the polycrystalline OFHC Cu, the constants in the flow rule at room temperature are 0 = 0.001 s-1, m = 50, H =
225 MPa, R = 2.05 with no kinematic hardening (h = hD = 0) and initial values g 0 = 13 MPa and x 0 = 0 MPa. The cubic single crystal elastic constants for Cu at room temperature are C11 = 150 GPa, C12 = 75 GPa, and C44 = 37.5 GPa. Using the constitutive models above for OFHC Cu, three-dimensional (3D) finite element calculations are conducted for representative microstructures as a function of notch size and notch root acuity for a strain amplitude below the macroscopic yield strain (i.e. high cycle fatigue region). This yield strain is defined as that strain associated with the macroscopic yield point based on the proportional limit. Here we use a double edge-notched plate with notch root radii ranging from 200 m to 1000 m. In this study we focus on the effects of notch size and acuity as well as the orientation distribution of grains. For each loading condition, a set of 40 different realizations of grains within the notch root region are implemented with randomly assigned orientation distribution to obtain an initially isotropic effective medium while gathering information regarding variability among instantiations. Remote applied strain was imposed via displacement boundary conditions on the upper and lower surfaces, with traction-free lateral surfaces. Four-node 3D solid tetrahedron elements were used for meshing in all regions, with a resolution of four elements per grain. The amplitudes of imposed completely reversed nominal axial cyclic strain (R = -1) are selected to ensure that the macroscopic strain at the notch root is in the vicinity of or below the yield strain of the polycrystalline materials. In other words, only a relatively small number density of grains yields in the notch root region after the initial loading cycle. To obtain the nonlocal fatigue indicator parameter, we consider a set of planes at each integration point within the nonlocal averaging region. At each integration point within this averaging volume, the plastic shear strain p on the maximum plane is calculated. The nonlocal cyclic plastic shear strain range for each plane, averaged over volume is then calculated. The maximum of the range of p amongst all planes is taken to be the nonlocal plastic shear strain range used in equation 4, i.e., pmax max p
(13)
5. Results and Discussions Fatigue Damage Process zone The distributions of microplasticity enable assessment of statistical distributions of the shear-based nonlocal fatigue indicator parameter given by equation (4) which are used along with equations (2) and (3) to obtain the fatigue process zone volume. Five notch root radii in the range of 200 m to 1000 m were considered. The fatigue damage process zones obtained for forty different realizations of grains with randomly assigned orientation distribution are shown in Fig. 2. Figure 3 shows the average fatigue damage process zone obtained for different notch root radii. The results in Fig. 2 also show that for notch root radii less than 200 m, randomness of grain orientations has minimal effect on the distribution of fatigue damage process zone based on the distribution of microplasticity, but significant effects are observed for notch root radii above 400 m. From Fig. 3, it is observed that the average Vd increases with increasing notch root radius. Figure 3 indicates that notches with higher notch root radii have larger volume of fatigue damage process zone than those for smaller notch root radii. Therefore, larger notches have higher driving force for crack nucleation and microstructurally small crack growth, with
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corresponding higher notch sensitivity. The results also show that as the notch root radii decrease, the detrimental effect of the notch reduces.
Fig. 2: Fatigue damage process zone for 40 random realizations of grains at the same strain amplitude of (y = 0.027%) and R = -1 for notch root radii ranging from 200m to 1000 m.
Fig. 3: Average fatigue damage process zone volume at the same strain amplitude of (y = 0.027 %) and R = -1 for notch root radii ranging from 200 m to 1000 m. Microstructure-Dependent Fatigue Notch Factor The variation of the microstructure-dependent notch factor, K f , for forty different realizations of grains is shown in Fig. 4. The results show that the fatigue notch factor at the microscale level varies with the microstructure for a given notch root radius. The results also indicate that the fatigue notch factor in the notch root region dependent on the notch root radius but also on the materials microstructures. The average K fave and the theoretical elastic stress concentration factors, Kt, for each notch root radius are also plotted in Fig. 4 for comparison. For the notch root radius of 1000 m, K f ranges from 1.38 to 2.61 with an average of 1.78. Therefore there is a statistical chance that the driving force for crack formation may even exceed that indicated by the theoretical stress concentration factor, Kt = 2.41 for = 1000 m, although the probability is low. This is due to the additional enhancement of microstructure-induced strain concentration at the level of individual grains. It is important to state that the microscopic fatigue notch factor presented here is based on the distribution of microplasticity whereas the
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macroscopic one in literature is related to the ratio of average remote applied stress amplitudes corresponding to unnotched and notched specimens at a given HCF life, requiring the designer to apply a safety factor to account for uncertainty associated with material condition or microstructure variability. The microscopic fatigue notch factor K f is therefore useful in determining the effect of the notch on reduction of fatigue resistance in a way that directly incorporates microstructure. The trend in the notch sensitivity index, q obtained using equation (9) based on the average K fave results obtained here are compared to the trend in notch sensitivity index based on the experimentally measured K f values [32], as shown in Fig. 5. This figure shows that the probabilistic model adequately predicts the trends observed in the experimental results for the average values of the notch sensitivity index, q, for notch root radii 200 m to 800 m. The results also show that the value of q obtained using the extreme-value distribution function for each notch root radius is closer to the experimental results than the one obtained using the Weibull distribution function in [26] when compared with the experimental results. It is, however, important to state that the microscopic notch factor presented here is not applicable in the low cycle fatigue and transition fatigue regimes, where a portion of the total fatigue life is spent on the growth of microstructually small crack across multiple grains. K f K fave
Kf, Kt
Kt
Fig. 4: Distribution of K f at the same strain amplitude of 0.027% for notch root radii of 200 m, 400 m, 600 m, 800 m and 1000 m for 40 random realizations of grains. The average K f (ave) concentration factor Kt are also plotted for comparison.
and the elastic stress
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Fig. 5: The dependence of notch sensitivity index q on notch root radius comparing measures of q based on K f and experimentally determined conventional K f for OFHC Cu [32], based on the ratio of unnotched to notched fatigue strengths at long lives. 6. Conclusions In this paper, a new criterion was established to determine the volume of the fatigue damage process zone using a micromechanics-based approach. Simulation results obtained show that higher notch root radius produces high probability for crack nucleation and microstructurally small crack growth and corresponding higher notch sensitivity. Statistical information regarding the distribution of the extreme-value of a shear based fatigue indicator parameter was also obtained that was used in the development of a microstructure-dependent fatigue notch factor and associated notch sensitivity as a function of the heterogeneity inherent in the actual microstructures. The probabilistic model presented adequately predicts the trends observed in the experimental results for the average values of the notch sensitivity index as a function of notch root radius. The microstructure dependent fatigue notch factor and associated notch sensitivity are computable for a given microstructure, and its predictive capabilities can be further assessed by validation with experiments on specific numerous materials with different microstructures and the same notches. This approach has certain advantages relative to the conventional approach since it can account for the variation in the microstructure of a given material and can also be used for qualitative comparison of notch sensitivities of various notch geometries for a range of microstructures of both current and future advanced materials for numerous fatigue applications. Acknowledgements The authors are grateful for the financial support provided by the Mechanical Engineering Department, Howard University. References [1] Neuber H. Theory of stress concentration in shear strained prismatic bodies with arbitrary nonlinear stress law. Journal of Applied Mechanics 1961;28:544-550. [2] Moski K, Glinka G. A method of elastic-plastic stress-strain calculations at the notch-root. Material Science and Engineering 1981;50:93– 100. [3] Moftakhar A, Bucznski A, Glinka G. Calculation of elastic-plastic strains and stress in notches under multiaxial loading. International Journal of Fracture 1995;70:357-373.
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[4] Barkey ME, Socie DF, and Hsai KJ. A yield surface approach to the estimation of notch strains for proportional and non-proportional cyclic loading. Journal of Engineering Materials and Technology 1994;116:173-179. [5] Singh MNK, Glinka G, Dubey RN. Elastic-plastic stress-strain calculation in notched bodies subjected to non-proportional loading. International Journal of Fracture 1996;76:39-60. [6] Owolabi GM, Singh MNK. A comparison between two analytical models that approximate notch-root elastic-plastic stress-strain components in two-phase particle-reinforced metal matrix composites under multiaxial cyclic loading theory. International Journal of Fatigue 2006;28:910917. [7] Moftakhar A, Bucznski A, Glinka G. Calculation of elastic-plastic strains and stress in notches under multiaxial loading. International Journal of Fracture, 1995;70:357-373. [8] Gu R.J., Lee Y. A new method for estimating non-proportional notch root stresses and strains. J Eng Mater Technology, 1997, 119: 40–45. [9] Yamasita Y, Ueda Y, Kuroti H, Shinozaki M. Fatigue life of small notched Ti-6Al-4V specimens using critical distance. Engineering Fracture Mechanics 2010; 77:1439–1453. [10] Ranganathan RM, Gu RJ, Lee YL. An improved automated finite element analysis for fatigue life predictions of notched components. International Journal of Materials and Product Technology 2004; 21:539–554. [11] Ren W, Nicholas T. Notch size effects on high cycle fatigue limit stress of Udimet 720. Materials Science and Engineering A 2003; 357:141–152. [12] Neuber H. Theory of notch stresses. 2rd ed. Berlin: Springer;1961. [13] Beremin FM. A local criterion for cleavage fracture of a nuclear pressure vessel steels. Metallurgical Transactions 1983;11:2277–2287 [14] Weixing Y. On the notched strength of composite laminates. Composite Science and Technology 1992;45:105–110 [15] Doudard C, Hild F. A probabilistic model for multiaxial high cycle fatigue. Fatigue and Fracture of Engineering Materials and Structures, 2007, 30: 107-114 [16] Delahay T, Palin-Luc T. Estimation of the fatigue strength in high cycle multiaxial fatigue taking into account the stress-strain gradient effect. International Journal of Fatigue, 2006, 28: 474-484. [17] Qylafku G, Azari Z, Kadi N, Gjonaj M, Pluvinage G. Application of a new model proposal for fatigue life prediction on notches and keyseats. International Journal of Fatigue 1999;21:753–760. [18] Qylafku G, Azari Z, Gjonaj M, Pluvinage G. On the fatigue failure and life prediction for notched specimens. Matericals Science 1998;34:604–618. [19] Adib-Ramezani H, Jeong J. Advanced volumetric method for fatigue life prediction using stress gradient effects at notch root. Computational Materials Science 2007; 39:649–663. [20] Lanning DB, Nicholas T, Haritos GK. On the use of critical distance theories for the prediction of the high cycle fatigue limit stress in notched Ti-6Al-4V. International Journal of Fatigue 2005;27:45–57 [21] Weixing Y. Stress field intensity approach for predicting fatigue life. International Journal of Fatigue 1993;15:243–245 [22] Weixing Y, Kaiquan X, Yi G. On the fatigue notch factor, Kf. International Journal of Fatigue 1995;17:245–251 [23] Bomas H, Linkewitz T, Mayr P. Application of a weakest-link concept to the fatigue limit of the bearing steel SAE 52100 in a bainitic condition. Fatigue & Fracture of Engineering Materials & Structures 1999;22:733–741 [24] Shenoy M, Zhang J, McDowell DL. Estimating fatigue sensitivity to polycrystalline Ni-base superalloy microstructures using a computational approach. Fatigue and Fracture of Engineering Materials and Structures 2007;30:889–904. [25] Fatemi A. Socie D. A critical plane approach to multiaxial fatigue damage including out of phase loading. Fatigue and Fracture of Engineering Materials and Structure 1988;11:145–165. [26] Owolabi, GM, Prasannavenkatesan R, McDowell DL. Probabilistic framework for a microstructure-sensitive fatigue notch factor. International Journal of Fatigue 2010;32:1378–1388. [27] The Mathsworks. “Fitting a Generalized Pareto distribution to tail data” http://www.mathworks.com [28] Davidson AC, Smith RL. Modeling for exceedances over high threshold. Journal of the Royal Statistical Society Series 1990;52:393–442. [29] Coles S. An introduction to statistical modeling of extreme values. London: Springer Series in Statistics;2001. [30] Cohen AC, Whitten BJ. Parameter estimation in reliability and life span models. 3rd ed. New York; Marcel Dekker:1988. [31] McGinty RD. Multiscale Representation of Polycrystalline Inelasticity. Ph.D. thesis, Georgia Institute of Technology, 2001. [32] Lukas P, Kunz L, and Svoboda M, Fatigue notch sensitivity of ultrafine-grained copper. Materials Science and Engineering A 2005;39: 337– 341.
Available online at www.sciencedirect.com
Procedia Engineering 00 (2011) 000–000 Procedia Engineering 10 (2011) 496–505
www.elsevier.com/locate/procedia
ICM11
A Micromechanics-Based Fatigue Damage Process Zone G.M.Owolabia*, L. Shia, H.A. Whitwortha a
Department of Mechanical Engineering, Howard University, Washington, DC, USA, 20059
Abstract
A primary issue in high cycle fatigue life prediction is the appropriate definition of damage process zone (i.e. volume influencing fatigue crack initiation) for notched components. Several definitions of the process zone have been proposed based on stress distributions obtained using homogeneous elastic or elastic-plastic finite element analysis. It is generally accepted to be the region, adjacent to the surface of the specimen, where peak stress is highest and fatigue crack initiation occurs; assumed as perhaps several mean grain diameters in spatial extent. However most of these existing techniques have not yet been related to fatigue failure mechanisms via computational micromechanics studies. Therefore they do not address the role of microstructure explicitly in fatigue life prediction. In this study, computational micromechanics is used to clarify and distinguish process zone for crack formation relative to scale of notch root radius and spatial extent of stress concentration at the notch. A new nonlocal criterion for fatigue damage process zone based on the distribution of a shear-based fatigue indicator parameter is proposed and used along with a probabilistic mesomechanics approach to obtain a new microstructuresensitive fatigue notch sensitivity index, thereby extending notch sensitivity to explicitly incorporate microstructure sensitivity and attendant size effects via probabilistic arguments. The probabilistic approach presented in this study predicts the general trends of notch sensitivity obtained from experimental results in literature.
© 2011 Published by Elsevier Ltd. Selection and peer-review under responsibility of ICM11 Keywords: Crystal plasticity, Fatigue damage process zone, micromechanics, crack formation and growth, fatigue notch factor, notch sensitivity.
1. Introduction The development of simulation methods for calculating notch root parameters for purposes of estimating fatigue life is a critical aspect of designing against fatigue failures in notched components. Notches such as holes, keyways, threads, fillets, and weldments occur in most components. However, traditional local “hot-spot” methods of notch root analysis [1-8] based on maximum stress and strain components at the notch root are often inadequate for characterizing the fatigue life of notched components since the stress gradients at the notch root also contribute to fatigue failures [9] but are not accounted in the traditional hot spot methods. Conventional approaches for accounting for the combined effects of stress gradients and notch root plasticity is through the concept of notch root strength reduction, also known as fatigue notch factor, Kf.
* Corresponding author. Tel.: 1-202-806-6594; fax: 1-202-483-1396. E-mail address: [email protected] 1877–7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.04.084
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Motivated by microscopic observations that fatigue failure occurs by damage accumulation in a finite notch root damage process zone, which can encompass several grains, several research studies have explored estimation of the fatigue notch factor and the fatigue strength of notched components as a function of an average stress over the fatigue damage process zone [10-23]. These methods consider the average stress value over a significant length scale at the notch root when analyzing fatigue potency of a notched component. One of these methods; the the point method uses a specific distance from a critical distance from the notch-root which may correspond to a grain size [12] or plastic zone size [13] and its corresponding average stress level to predict the fatigue life of notched components. While a single parameter involving a critical distance may be a viable approach, size effects arising from deformation and damage phenomena may render an elementary simple single parameter approach like critical distance inapplicable [20]. Other methods [21-23] use the stress field intensity to define the fatigue notch factor and to propose a fatigue failure criterion. The stress field intensity is the average stress obtained by integrating the stress distribution obtained from finite element analyses over the damage process zone. The stress field intensity incorporates the effects of stress gradient. A weight function is also implicitly assumed to account for the contribution of stresses in the damage process zone and is incorporated into fatigue notch factor and failure criterion. This weight function depends on notch geometry, loading type, boundary conditions and material properties [19]. Although these techniques offer improvement over the local hot-spot methods and perhaps represent an important step towards a more reliable prediction of notch sensitivity, most of the models are based on assumptions necessary to simplify the complex nature of the problem since the size of the damage zone has been quantitatively determined [17]. While physical volume has been recognized as very vital in the development of more reliable fatigue notch factor estimation models and to fatigue damage (nucleation plus growth), there is not yet a suitable definition of fatigue damage process zone in literature based on underlying physical phenomenon at the microscale level that leads to the formation and growth of small cracks in real materials. These conventional methods for accounting for stress gradients are also deterministic and do not account for the role of microstructure explicitly. Therefore it is difficult to gain direct insight into the controlling physical phenomenon at the microstructure scale that gives more predictive explanation of the material dependence of the notch sensitivity. In HCF, the cyclic plastic strain is quite heterogeneously distributed within the damage process zone, an aspect that plays a strong role in the coupling of notch size and acuity with probability of fatigue crack formation and small crack growth from the notch root. As a consequence, it is unclear that volume averaged stress or stress field intensity is an appropriate indicator of the damage process zone driving force, based on macroscopic constitutive relations. In this study, computational micromechanics is used to clarify and distinguish fatigue damage process zone for crack formation relative to scale of notch root radius and spatial extent of stress concentration at the notch. A new nonlocal criterion for fatigue damage process zone based on the distribution of a shear-based fatigue indicator parameter is proposed and used along with a probabilistic mesomechanics approach to obtain a new microstructuresensitive fatigue notch sensitivity index, thereby extending notch sensitivity to explicitly incorporate microstructure sensitivity and attendant size effects via probabilistic arguments. 2. Fatigue Damage Process Zone Since fatigue crack initiation in polycrystalline materials is usually predicted from the critical plane containing the most favorably oriented grain, one appropriate approach to defining the fatigue damage process zone is to consider the statistical distributions of a nonlocal fatigue indicator parameter, , around the notch root. From these statistical distributions, the fatigue damage process zone may be assumed to consist of a number of grains with nonlocal values equal to or greater than a specified microscopic threshold value, th . For any given instance, it is assumed that damage at the microscale, i.e., microcracks, does not initiate in any grain with a nonlocal value below the threshold, th . The threshold fatigue indicator parameter, th , can be associated with the microscopic resolved shear stress threshold for yielding, y. y may be estimated from the macroscopic yield via the Taylor relation y/M, where M is the Taylor factor. To estimate th , y is used in the modified form of a microstructurally small crack growth law given by [24], i.e.,
da AFS y a b CTD CTDth dN
(1)
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for the nucleation and growth of microstructurally small cracks on the order of 3-10 grain size. In equation (1) AFS and are constants, “a” is the crack length, and b is the Burgers vector. The cyclic crack tip displacement range is given by CTD , and the threshold value, CTDth , is directly related to b . Assuming that cracks will nucleate if cyclic microplasticity exists in a given grain, i.e., > 0, the threshold value is given by da/dN = 0, resulting in:
th
b AFS y a
(2)
It is left to the characterization of the factors AFS and to quantify this threshold for each material of interest. Thus, for a notched specimen the process zone is defined as the volume around the notch root made of a number of grains with nonlocal fatigue indicator parameter, i , values above the threshold th (see Fig. 1 for illustration, the red color squares represent grains that satisfy this condition). The damage process zone can be mathematically represented as:
volume bounded by grains G (r ) at the notch root N r Vd ( N r ) satisfying i th , where i nonlocal FIP
(3)
Vd
i th Fig. 1: A sketch of the fatigue damage process zone (V d) in the notch root region showing heterogeneity in plastic strain and distribution of fatigue indicator parameter Note that any suitable parameter such as nonlocal hysteresis work density may be used as a driving force parameter in equations (2) and (3), but parameters which best reflect change in crack tip displacement, CTD , for crystallographic Stage I growth are preferred. The Fatemi Socie [25] parameter given by: p* max 2
nmax* 1 y
(4)
is used in this study as the fatigue indicator parameter, , since it can be used to approximate the mixed mode character of CTD in Eq. (1). In equation (4) is a coefficient that moderates the effect of normal stress to the maximum plastic shear strain range plane. The nonlocal peak stress normal, nmax* , to the plane of maximum shear strain amplitude is normalized by the polycrystalline cyclic yield strength, y. 3. Microstructure-Dependent Fatigue Notch Factor The statistical distribution of the nonlocal fatigue indicator parameter within in grains the well-defined damage process zone (illustrated by Fig. 1) discussed in the previous section is used in the development of a microstructure-
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dependent fatigue notch factor. This is an extension of the earlier work of Owolabi’ et al. [26] that combines computational crystal plasticity with a probabilistic framework to obtain a novel microscopic fatigue notch factor and associated notch sensitivity index. Here, 3D computational crystal plasticity was performed on OFHC copper to assess the degree of heterogeneity of cyclic plastic deformation as a function of notch size and notch root acuity for a range of strain amplitudes below the macroscopic yield strain and different realizations of aggregates of grains with random orientations at the notch root. Statistical information regarding the distributions of the shear-based fatigue indicator parameter is obtained providing useful insight into the dependence of fatigue notch factor and associated notch sensitivity index on the heterogeneity inherent in the actual microstructures. Results from simulations within the well-defined fatigue damage process zone are used in the development of new microstructure-dependent fatigue notch factor and associated sensitivity index. The method developed incorporates the effects of notch size, stress/strain gradients, and microstructural variability that have not been accounted for in previous traditional approaches. Using the Weibull’s weakest link theory and assuming that the nonlocal is a random variable, the microstructure-dependent fatigue notch factor is given in [26] as:
1 K f V d
th V net ( ave)
b dV
1
b
Vd Vo
1
b
,
(5)
where net is the net section nonlocal for the notched component, computed as the average of in the net section region outside the damage process zone, V. b and Vo are the shape parameter and reference volume respectively. An alternative statistical approach that considers the extreme value portion of the distribution in the fatigue damage process zone rather than the Weibull distribution used in [26] is used here to develop a new relation for the microstructure-dependent fatigue notch factor. Based on statistical distribution of , the differences between the extreme-values of in the tail section of the statistical distribution and the threshold th are fitted by an appropriate statistical distribution function. Fitting a parametric distribution to data sometimes leads to a model that agrees well with the data in high density regions, but poor accuracy in the low density regions due to sparse data [27]. In HCF applications in which few grains have nonlocal values greater than the threshold, fitting the data in the tail to an appropriate statistical distribution is associated with substantial challenges. Thus, the generalized Pareto distribution [28-29] that has been shown to model the tails of a wide variety of statistical distributions is used in this work. The distribution of the extreme values of nonlocal that exceed the threshold can be reasonably modeled using the generalized Pareto distribution with a distribution function, , of the form:
e th 1 1 Vo 0
1
,
(6)
where is equivalent to the shape parameter in equation (5), 0 is the scale parameter and e is the extreme value of at the tail end of the distribution function. The generalized Pareto distribution function in (6) is more appropriate than the Weibull’s distribution function used in [26] since in most engineering materials and components fatigue is manifested by extremal of microstructure attributes at various length scale that promote slip intensification. The extreme-value distribution function in equation (6) has three parameters: (i) the threshold or location, th , (ii) the shape, and (iii) the scale, 0 . Note that the shape parameter is a dimensionless quantity that depends on the material microstructure. Moreover, for the Fatemi-Socie nonlocal values the scale parameter is also dimensionless. Although the distribution functions have three parameters, the threshold value, th , which corresponds to the location parameter, is obtained using equation 2 as 6.14 x10-5. As a result, the number of unknown parameters reduces to two: i.e., and 0 . Since in HCF only a small number of grains of a smooth specimen will have non-zero microplasticity, the scale, o , and shape, , parameters are estimated (using the modified moment parameter estimation method [30]) respectively as 6.95 x10-6 and 0.85 the mean and variance of
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the plastic strain obtained experimentally for 25 notched specimens. Finite element simulation results are used to obtain Vd for different notch root radii using the condition stated in equation 3 at the notch root region. Using equation 6 above and following the probabilistic framework proposed in [26], a new relation is obtained for the microstructure-dependent fatigue notch factor of the form:
V K f Kt d , V0
(7)
where K t , is the microscopic concentration factor given by: 0 e th Kt dV . net ( ave ) Vd 1
(8)
The associated notch sensitivity index, q, is given as: q
K f 1 K t 1
(9)
The microscopic fatigue notch factor in equation (7) follows the spirit of the traditional fatigue notch factor i.e., the ratio of unnotched to notched values of driving force for a given HCF life and same probability of failure, but it is rooted in probabilistic arguments. The new probabilistic model for microstructure-dependent fatigue notch factor incorporates the strength of the notch root stress field gradient and distributions of microslip relative to underlying microstructure within a well-defined fatigue damage process zone. Equations (7) and (8) show that the degree of accuracy of the fatigue notch factor depends strongly on reliable prediction of the statistical distribution of the extreme-values of the shear-based fatigue indicator parameter. Thus, the fatigue damage process zone, Vd plays a major role in the determination of the microstructure-dependent fatigue notch factor. 4. Crystal Plasticity and Finite Element Implementation Since crystal plasticity models relate the grain scale stress to crystallographic slip response, they are suitable for studying heterogeneity and interaction across grains in the notch root field. The kinetics of dislocation glide are formulated by the relationships between the resolved shear stresses and the shearing rate along the slip systems. The shearing rate, , on the th slip system is described by the power law flow rule:
0
x g
m
sgn x ,
(10)
where 0 is the reference shearing rate, m is the inverse strain-rate sensitivity exponent, g is the isotropic drag strength, x is the back stress on the th slip system, and is the resolved shear stress, given by:
: s n .
(11)
In equation (11), is the Cauchy stress tensor. A direct hardening-dynamic recovery format relation is employed for evolution of g [31], i.e.,
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N
1
1
g H q R g ,
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(12)
where q are components of the hardening matrix and H and R are the direct hardening and dynamic recovery coefficients, respectively, for the isotropic hardening relation. In the present work, q = 1 such that isotropic hardening is the same on all slip systems. The back stress on each slip system evolves according to a nonlinear kinematic hardening rule of self hardening type, i.e., x h hD x , where h and hD are the direct hardening and dynamic recovery coefficients, respectively. The material used in this study is polycrystalline OFHC copper (Cu) with mean grain size of 62 m and consisting of grains with random crystallographic orientation distribution. For the polycrystalline OFHC Cu, the constants in the flow rule at room temperature are 0 = 0.001 s-1, m = 50, H =
225 MPa, R = 2.05 with no kinematic hardening (h = hD = 0) and initial values g 0 = 13 MPa and x 0 = 0 MPa. The cubic single crystal elastic constants for Cu at room temperature are C11 = 150 GPa, C12 = 75 GPa, and C44 = 37.5 GPa. Using the constitutive models above for OFHC Cu, three-dimensional (3D) finite element calculations are conducted for representative microstructures as a function of notch size and notch root acuity for a strain amplitude below the macroscopic yield strain (i.e. high cycle fatigue region). This yield strain is defined as that strain associated with the macroscopic yield point based on the proportional limit. Here we use a double edge-notched plate with notch root radii ranging from 200 m to 1000 m. In this study we focus on the effects of notch size and acuity as well as the orientation distribution of grains. For each loading condition, a set of 40 different realizations of grains within the notch root region are implemented with randomly assigned orientation distribution to obtain an initially isotropic effective medium while gathering information regarding variability among instantiations. Remote applied strain was imposed via displacement boundary conditions on the upper and lower surfaces, with traction-free lateral surfaces. Four-node 3D solid tetrahedron elements were used for meshing in all regions, with a resolution of four elements per grain. The amplitudes of imposed completely reversed nominal axial cyclic strain (R = -1) are selected to ensure that the macroscopic strain at the notch root is in the vicinity of or below the yield strain of the polycrystalline materials. In other words, only a relatively small number density of grains yields in the notch root region after the initial loading cycle. To obtain the nonlocal fatigue indicator parameter, we consider a set of planes at each integration point within the nonlocal averaging region. At each integration point within this averaging volume, the plastic shear strain p on the maximum plane is calculated. The nonlocal cyclic plastic shear strain range for each plane, averaged over volume is then calculated. The maximum of the range of p amongst all planes is taken to be the nonlocal plastic shear strain range used in equation 4, i.e., pmax max p
(13)
5. Results and Discussions Fatigue Damage Process zone The distributions of microplasticity enable assessment of statistical distributions of the shear-based nonlocal fatigue indicator parameter given by equation (4) which are used along with equations (2) and (3) to obtain the fatigue process zone volume. Five notch root radii in the range of 200 m to 1000 m were considered. The fatigue damage process zones obtained for forty different realizations of grains with randomly assigned orientation distribution are shown in Fig. 2. Figure 3 shows the average fatigue damage process zone obtained for different notch root radii. The results in Fig. 2 also show that for notch root radii less than 200 m, randomness of grain orientations has minimal effect on the distribution of fatigue damage process zone based on the distribution of microplasticity, but significant effects are observed for notch root radii above 400 m. From Fig. 3, it is observed that the average Vd increases with increasing notch root radius. Figure 3 indicates that notches with higher notch root radii have larger volume of fatigue damage process zone than those for smaller notch root radii. Therefore, larger notches have higher driving force for crack nucleation and microstructurally small crack growth, with
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corresponding higher notch sensitivity. The results also show that as the notch root radii decrease, the detrimental effect of the notch reduces.
Fig. 2: Fatigue damage process zone for 40 random realizations of grains at the same strain amplitude of (y = 0.027%) and R = -1 for notch root radii ranging from 200m to 1000 m.
Fig. 3: Average fatigue damage process zone volume at the same strain amplitude of (y = 0.027 %) and R = -1 for notch root radii ranging from 200 m to 1000 m. Microstructure-Dependent Fatigue Notch Factor The variation of the microstructure-dependent notch factor, K f , for forty different realizations of grains is shown in Fig. 4. The results show that the fatigue notch factor at the microscale level varies with the microstructure for a given notch root radius. The results also indicate that the fatigue notch factor in the notch root region dependent on the notch root radius but also on the materials microstructures. The average K fave and the theoretical elastic stress concentration factors, Kt, for each notch root radius are also plotted in Fig. 4 for comparison. For the notch root radius of 1000 m, K f ranges from 1.38 to 2.61 with an average of 1.78. Therefore there is a statistical chance that the driving force for crack formation may even exceed that indicated by the theoretical stress concentration factor, Kt = 2.41 for = 1000 m, although the probability is low. This is due to the additional enhancement of microstructure-induced strain concentration at the level of individual grains. It is important to state that the microscopic fatigue notch factor presented here is based on the distribution of microplasticity whereas the
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macroscopic one in literature is related to the ratio of average remote applied stress amplitudes corresponding to unnotched and notched specimens at a given HCF life, requiring the designer to apply a safety factor to account for uncertainty associated with material condition or microstructure variability. The microscopic fatigue notch factor K f is therefore useful in determining the effect of the notch on reduction of fatigue resistance in a way that directly incorporates microstructure. The trend in the notch sensitivity index, q obtained using equation (9) based on the average K fave results obtained here are compared to the trend in notch sensitivity index based on the experimentally measured K f values [32], as shown in Fig. 5. This figure shows that the probabilistic model adequately predicts the trends observed in the experimental results for the average values of the notch sensitivity index, q, for notch root radii 200 m to 800 m. The results also show that the value of q obtained using the extreme-value distribution function for each notch root radius is closer to the experimental results than the one obtained using the Weibull distribution function in [26] when compared with the experimental results. It is, however, important to state that the microscopic notch factor presented here is not applicable in the low cycle fatigue and transition fatigue regimes, where a portion of the total fatigue life is spent on the growth of microstructually small crack across multiple grains. K f K fave
Kf, Kt
Kt
Fig. 4: Distribution of K f at the same strain amplitude of 0.027% for notch root radii of 200 m, 400 m, 600 m, 800 m and 1000 m for 40 random realizations of grains. The average K f (ave) concentration factor Kt are also plotted for comparison.
and the elastic stress
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Fig. 5: The dependence of notch sensitivity index q on notch root radius comparing measures of q based on K f and experimentally determined conventional K f for OFHC Cu [32], based on the ratio of unnotched to notched fatigue strengths at long lives. 6. Conclusions In this paper, a new criterion was established to determine the volume of the fatigue damage process zone using a micromechanics-based approach. Simulation results obtained show that higher notch root radius produces high probability for crack nucleation and microstructurally small crack growth and corresponding higher notch sensitivity. Statistical information regarding the distribution of the extreme-value of a shear based fatigue indicator parameter was also obtained that was used in the development of a microstructure-dependent fatigue notch factor and associated notch sensitivity as a function of the heterogeneity inherent in the actual microstructures. The probabilistic model presented adequately predicts the trends observed in the experimental results for the average values of the notch sensitivity index as a function of notch root radius. The microstructure dependent fatigue notch factor and associated notch sensitivity are computable for a given microstructure, and its predictive capabilities can be further assessed by validation with experiments on specific numerous materials with different microstructures and the same notches. This approach has certain advantages relative to the conventional approach since it can account for the variation in the microstructure of a given material and can also be used for qualitative comparison of notch sensitivities of various notch geometries for a range of microstructures of both current and future advanced materials for numerous fatigue applications. Acknowledgements The authors are grateful for the financial support provided by the Mechanical Engineering Department, Howard University. References [1] Neuber H. Theory of stress concentration in shear strained prismatic bodies with arbitrary nonlinear stress law. Journal of Applied Mechanics 1961;28:544-550. [2] Moski K, Glinka G. A method of elastic-plastic stress-strain calculations at the notch-root. Material Science and Engineering 1981;50:93– 100. [3] Moftakhar A, Bucznski A, Glinka G. Calculation of elastic-plastic strains and stress in notches under multiaxial loading. International Journal of Fracture 1995;70:357-373.
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