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Extension of a Probabilistic Mesomechanics based Model for Fatigue Notch Factor to Titanium Alloy Components
Available online at www.sciencedirect.com
ScienceDirect Procedia Materials Science 3 (2014) 1860 – 1865
20th European Conference on Fracture (ECF20)
Extension of a Probabilistic Mesomechanics Based Model for Fatigue Notch Factor to Titanium Alloy Components Gbadebo Owolabi*, Oluwamayowa Okeyoyin, Oluwakayode Bamiduro, Horace Whitworth Department of Mechanical Mechanical Engineering, Howard University, Washington, DC, USA, 20059
Abstract This paper extends a recently developed probabilistic mesomechaniccs based model for fatigue notch factor to titanium alloy components. The notch size effects and notch root and inelastic behaviour are combined with probability distributions of microscale stress-strain gradient and small crack initiation to inform minimum life design methods. The fatigue notch factors predicted using the new model are in good agreements with the experimental results for the notched titanium alloy specimens. © 2014 Elsevier Ltd. Open access under CC BY-NC-ND license. © 2014 The Authors. - Published by Elsevier Ltd. Selection and peer-review under responsibility of the University of Science and Technology (NTNU), (NTNU), Department Selection and peer review under responsibility ofNorwegian the Norwegian University of Science and Technology Department of Structural Engineering. of Structural Engineering
Keywords: weakest link, fatigue notch factor, titanium alloy, fatigue damage process zone
1. Introduction Titanium is widely known for its good resistance to corrosion and high strength to weight ratio. When alloyed with other metals and heat treated, it can achieve a wide range of attractive properties both at low and high temperatures. Thus, it is widely used in engineering applications from airframe components and fans to compressor blades of jet engines. Ingestion of debris into the engine of aircraft during takeoff and landing causes nicks and dents to form on the leading and trailing edge of turbine blades as shown in Fig. 1. These dents and nicks can be treated as
* Corresponding author. Tel.: 202-806-6594; fax: 202-483-1396. E-mail address: [email protected]
2211-8128 © 2014 Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Structural Engineering doi:10.1016/j.mspro.2014.06.300
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Gbadebo Owolabi et al. / Procedia Materials Science 3 (2014) 1860 – 1865
small notches and thus serve as stress raiser and favorable zones for crack initiation therefore reducing the fatigue strength of the material. This phenomenon is referred to as foreign object damage (FOD) and can be modeled as small notches with notch root radius and notch depth [see Haritos et. al. and Yamashita et. al.].
Fig. 1. Image of FOD damage; (a) Fan blade schematics (b) FOD damage example on edges of airfoil [from Yamashita et. al.].
The fatigue notch factor, kf, is used in the estimation of fatigue life and strength of notched structural components. Different expressions based on different assumptions have been developed for kf in the past. The Neuber, Kuhn et al., and Peterson expressions are all based on average stress assumptions. One of the drawbacks of these models is that they do not incorporate explicit sensitivity to the combined effects of microstructure and strength of the notch root stress field gradient. Recent approaches have been developed to incorporate the stress gradient field at notches [see Adib-Ramezeani et. al; Ren and Nicholas], but are deterministic. It is therefore very difficult to link the kf obtained using these methods to realistic microstructure of the material such as grain size and orientation. Owolabi et al. recently developed a probabilistic model for fatigue notch factor where elements of crystal plasticity were combined with new probabilistic methods for notch sensitivity based on computed slip at the notch root for homogenous oxygen free high conductivity copper. The purpose of this paper is to extend this model to heterogeneous multiphase titanium alloy components. 2. Material and Crystal Plasticity Modeling of Notched Components The HCP structure of titanium, unlike for FCC structured materials, has several planes which are favorable for occurrence of slip or twinning. In most HCP materials, basal (0001) and prismatic ^1 010 ` have been identified as the primary slip planes with a closed packed direction 1 12 0 for the slip vector. The crystal plasticity model used in this work follows the existing work of Mayeur and McDowell and Zhang et.al. The relationship between the slip system shearing rate and the resolved shear stress of the D slip system is described by the power law flow rule given as: D
J
W
J0
D
D
F
N
D
D
D
M
s g n W
D
F
D
(1)
.
where, J o is the reference shearing rate, M is the inverse strain-rate sensitivity exponent which controls the rate sensitivity of flow, W D is the resolved shear stress, F D is the back stress, N D is the length scale-dependent threshold stress and D D is the drag stress. In Zhang et al., the drag stress is taken as a non-evolving constant, i.e. D D 0 , while the back stress evolves according to an Armstrong-Frederick direct hardening/dynamic recovery type of equation, i.e., F
D
hJ
D
hD F
D
J
D
(2)
The threshold stress is expressed as N
D
N
y
d
D
N
D s
(3)
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Gbadebo Owolabi et al. / Procedia Materials Science 3 (2014) 1860 – 1865
3. Three Dimensional Finite Element Implementation Procedure for Notched Titanium Alloy Component The crystal plasticity model in Section 2 is coded into ABAQUS 2006 UMAT. The notched geometry modeled in this work is a v-notched cylindrical component shown in Fig. 2. Finite element simulations were performed on three different geometries, meshed using 3D stress four-node linear tetrahedron element type (C3D4) to estimate the stress distribution and possible plastic straining that occur in the notched specimens. The dimensions of the specimens used and the different test cases are as given in Table 1.
Fig. 2. Gage section of the cylindrical specimen with a circumferential V-notch. Table 1. Notch Root Geometries and Load Test Cases.
Test Case 1 2 3 4 5 6 7
Kt 2.78 2.78 2.78 2.78 2.78 2.78 2.78
Notch radius, ૉ (mm) 0.330 0.330 0.330 0.203 0.203 0.127 0.127
Notch depth, h (mm) 0.729 0.729 0.729 0.254 0.254 0.127 0.127
R-ratio -1 0.1 0.5 0.1 0.5 0.1 0.5
Average alternating HCF 6 strength at 10 cycles (MPa) 173.6 158.9 104.6 167.2 105.2 144.7 111.0
4. Probabilistic Framework Following the framework presented in Owolabi et.al., the probability of survival of a smooth component having a fatigue damage process zone of volume V, divided into small volume elements, dV is given as: Ps
§ 1 exp ¨ ¨ V0 ©
³ Vd
§§V V ¨¨ ¨ ©© V0
b
th
·· ¸ ¸¸ d V ¹¹
· ¸ ¸ ¹
(4)
where, b, Vth and σ0 represent 3-parameter Weibull shape, location and scale parameters respectively. V is the stress distribution. The cumulative probability of HCF failure of the component, can be obtained from Equation (5) as:
Pf
§ 1 1 exp ¨ ¨ V0 ©
³ Vd
§§V V ¨¨ ¨ ©© V0
b
th
·· ¸ ¸¸ d V ¹¹
· ¸ ¸ ¹
(5)
Gbadebo Owolabi et al. / Procedia Materials Science 3 (2014) 1860 – 1865
1863
To facilitate development of the expression for the fatigue notch factor from Equation (5), the concept of stress homogeneity factor is introduced here. Thus, Equation (5) could be re-written as, b
Pf
§ k V d § V m ax · · 1 exp ¨ ³ ¨ ¸ ¸ ¨ V0 © V 0 ¹ ¸ © Vd ¹
(6)
where
k
1 § §V V ¨ ¨ ³ Vd ¨ V © V 0 © d
th
· ¸ ¹
b
· ¸ dv ¸ ¹
(7)
is regarded as the stress homogeneity factor. Conventionally, the fatigue notch factor is the ratio of unnotched to notched fatigue strength at the same probability of failure (usually 50%). Using Equation (6), the probability of failure of unotched specimen and a notched specimen will be the same when ° k s V s § V m a x , s · °½ exp ® ¨ ¸¾ °¯ V 0 © V 0 ¹ °¿
° k n V n § V m a x , n · °½ exp ® ¨ ¸¾ °¯ V 0 © V 0 ¹ °¿
(8)
where the subscripts n and s represent the respective value of the variable for notched and smooth (unmatched) specimens. The ratio of the smooth to notch fatigue driving force parameters (i.e., the stress amplitude) is used to define a new fatigue notch factor given as: kf
V
m ax, s
V
m ax, n
§ kn · ¨ ¸ © ks ¹
1
b
§ Vn · ¨ ¸ © Vs ¹
1
(9)
b
For smooth specimen that is loaded at a very low stress or strain amplitude in the HCF regime, the number of critically stressed grains (or elements) is very small. Thus for the life limiting case in which only one grain or element is critically stressed above the threshold, Vs = Ve (i.e. volume of element or grain) and k s = 1; thus Equation (9) becomes
k
f
ª 1 º « » ³ ¬ V d ¼ Vd
§§V V th ¨¨ ¨¨ V n m a x , ©©
· ¸ dv ¸ ¹
· ¸ ¸ ¹
1
b
§ Vn · ¨ ¸ © Ve ¹
1
b
(10)
The yield stress Vth for titanium alloy it is 990 MPa, b is determined to be 7.7, while the scale parameter V 0 is determined to 3205.03 MPa. From the geometry of the specimens, the reference volume V0 is calculated to be 16.96 mm3. 5. Results and Discussion The stress distribution obtained from the finite element analysis was used in the developed model to determine the probability of failure for each test case. Also, the average fatigue notch factor is computed for each test case using the developed probabilistic framework. Figure 3 shows probability of failure plotted against the notch root radii for ten different grain orientations. Figure 3 shows that the probability of failure increases with increasing notch root radius. It is interesting to note that for titanium alloy specimen with 0.127 mm notch root radius, some of the probabilities of failure for some of the grain orientations are greater than for the remaining higher notch root radii of 0.203 mm and 0.33 mm. This is an indication that grain orientation around the notch root region also plays
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Gbadebo Owolabi et al. / Procedia Materials Science 3 (2014) 1860 – 1865
an equally important role in determining the occurrence of fatigue failure in the notched titanium alloy specimen. The fatigue notch factor determined using the developed method is compared to experimentally obtained values from Naik et al., close from solution, and other existing conventional method such as Neuber as shown in Figure 4 and Table 2. It can be noted from Table 2 that the percentage variance for Weibull fatigue notch is lower compared to Neuber’s predictions. Thus, the developed probabilistic model is more accurate in predicting fatigue notch factor of titanium alloy compared to the Neuber method.
Fig. 3. Probability of failure vs. notch radius for notched titanium alloy at load ratio R= 0.1.
Fig. 4. Fatigue notch factor as a function notch root radius and load ratio compared to experimental values.
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Gbadebo Owolabi et al. / Procedia Materials Science 3 (2014) 1860 – 1865 Table 2. Fatigue notch factor % variance from experimental value for Weibull, close form solution and Neuber methods.
Radius
0.127 0.203 0.33
Experiment kf R=0.1 1.98 1.71 1.80
R=0.5 1.65 1.74 1.75
Weibull@R=0.1
Weibull@R=0.5
Kf 2.05 1.86 1.89
kf 1.72 1.83 1.82
% Var -3.54 -8.77 -5.00
% Var -4.24 -5.17 -4.00
Neuber kf 1.79 1.89 2.00
% Var 9.60 -10.53 -11.11
Closed Form Sol. kf % Var 2.00 -1.01 1.80 -5.26 1.88 -4.44
6. Conclusions A recently developed probabilistic model for microstructure-sensitive fatigue notch factor was extended to titanium alloy components. The result shows that the probability of failure and the fatigue notch factor increase with increasing notch root radius. Also it is noted that the grain orientation around the notch root region plays an important role in determining or predicting the fatigue strength of the material; the probability of failure varies at the same notch root radius for different grain orientation. Acknowledgements The authors of this paper express their profound gratitude to the Department of Defense for the financial support provided through contract # W911NF-11-1-041 (Dr. L. Russell, Program Manager, Army Research Office and Dr. D. Stargel, Program Manager, Air Force Office of Scientific Research.) References Adib-Ramezani H. and Jeong J., 2007,"Advanced Volumetric method for fatigue life prediction using stress gradient effects at notch root," Computational Material Science , vol. 39, pp. 649 - 663. Haritos G K, Nicholas T, & Lanning D B, 1999, "Notch Size Effects in HCF Behavior of Ti-6Al-4V," Int. J. Fatigue, vol. 21, pp. 643 – 652. Kuhn P; Hardraht H.F,1952, "An Engineering Method for Estimating the Notch-Size Effect in Fatigue Tests on Steel," in NACA TN2805Langley Aeronautical Laboratory , Washington. Mayeur J. R., McDowell D.L, 2007 "A three-dimensional crystal plasticity model of duplex Ti-6Al-4V.," Int.Journal of Plasticity, vol. 23, pp. 1457-1485. Naik R.A., Lanning D.B., Nicholas T., Kallmeyer A.R.,2005, "A critical plane gradient approach for the prediction of notched HCF life.," Int. J. Fatigue, vol. 27, pp. 481-492. Neuber H.J., 1961 "Theory of Stress Concentration for Shear-Strained Prismatical Bodies with Arbitrary Nonlinear Stress-Strain Law.," J. Appl. Mech., vol. 28, no. 4, pp. 544 - 550, 1961. Owolabi G., Egboiyi B., Shi L., Withworth H., 2011 "Microstructure-Dependent Fatigue Damage Process Zone and Notch Sensitivity Index," International Journal of Fracture , pp. 159-173. Peterson R. E., 1959,"Notch Sensitivity," in Metal Fatigue, G. Sines and J. Waisman, Eds., New York, McGrawHill, pp. 293-306. Ren W., Nicholas T.,2003, "Notch Size Effects on High Cycle Fatigue Limit Stress of Udimet 720," Material Science and Engineering, vol. A357, pp. 141-152. Yamashita Y., Ueda Y., Kuroki H., Shinozaki M., 2010, "Fatigue Life Prediction of Small Notched Ti-6Al-4V Specimens Using Critical Distance," Engineering Fracture Mechanics, vol. 77, pp. 1439 – 1453. Zhang M., Zhang J., McDowell D.L, 2007 "Microstructure – based crystal plasticity modeling of cyclic deformation of Ti-6Al-4V," Int. Journal of Plasticity, vol. 23, pp. 1328-1348.
ScienceDirect Procedia Materials Science 3 (2014) 1860 – 1865
20th European Conference on Fracture (ECF20)
Extension of a Probabilistic Mesomechanics Based Model for Fatigue Notch Factor to Titanium Alloy Components Gbadebo Owolabi*, Oluwamayowa Okeyoyin, Oluwakayode Bamiduro, Horace Whitworth Department of Mechanical Mechanical Engineering, Howard University, Washington, DC, USA, 20059
Abstract This paper extends a recently developed probabilistic mesomechaniccs based model for fatigue notch factor to titanium alloy components. The notch size effects and notch root and inelastic behaviour are combined with probability distributions of microscale stress-strain gradient and small crack initiation to inform minimum life design methods. The fatigue notch factors predicted using the new model are in good agreements with the experimental results for the notched titanium alloy specimens. © 2014 Elsevier Ltd. Open access under CC BY-NC-ND license. © 2014 The Authors. - Published by Elsevier Ltd. Selection and peer-review under responsibility of the University of Science and Technology (NTNU), (NTNU), Department Selection and peer review under responsibility ofNorwegian the Norwegian University of Science and Technology Department of Structural Engineering. of Structural Engineering
Keywords: weakest link, fatigue notch factor, titanium alloy, fatigue damage process zone
1. Introduction Titanium is widely known for its good resistance to corrosion and high strength to weight ratio. When alloyed with other metals and heat treated, it can achieve a wide range of attractive properties both at low and high temperatures. Thus, it is widely used in engineering applications from airframe components and fans to compressor blades of jet engines. Ingestion of debris into the engine of aircraft during takeoff and landing causes nicks and dents to form on the leading and trailing edge of turbine blades as shown in Fig. 1. These dents and nicks can be treated as
* Corresponding author. Tel.: 202-806-6594; fax: 202-483-1396. E-mail address: [email protected]
2211-8128 © 2014 Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Structural Engineering doi:10.1016/j.mspro.2014.06.300
1861
Gbadebo Owolabi et al. / Procedia Materials Science 3 (2014) 1860 – 1865
small notches and thus serve as stress raiser and favorable zones for crack initiation therefore reducing the fatigue strength of the material. This phenomenon is referred to as foreign object damage (FOD) and can be modeled as small notches with notch root radius and notch depth [see Haritos et. al. and Yamashita et. al.].
Fig. 1. Image of FOD damage; (a) Fan blade schematics (b) FOD damage example on edges of airfoil [from Yamashita et. al.].
The fatigue notch factor, kf, is used in the estimation of fatigue life and strength of notched structural components. Different expressions based on different assumptions have been developed for kf in the past. The Neuber, Kuhn et al., and Peterson expressions are all based on average stress assumptions. One of the drawbacks of these models is that they do not incorporate explicit sensitivity to the combined effects of microstructure and strength of the notch root stress field gradient. Recent approaches have been developed to incorporate the stress gradient field at notches [see Adib-Ramezeani et. al; Ren and Nicholas], but are deterministic. It is therefore very difficult to link the kf obtained using these methods to realistic microstructure of the material such as grain size and orientation. Owolabi et al. recently developed a probabilistic model for fatigue notch factor where elements of crystal plasticity were combined with new probabilistic methods for notch sensitivity based on computed slip at the notch root for homogenous oxygen free high conductivity copper. The purpose of this paper is to extend this model to heterogeneous multiphase titanium alloy components. 2. Material and Crystal Plasticity Modeling of Notched Components The HCP structure of titanium, unlike for FCC structured materials, has several planes which are favorable for occurrence of slip or twinning. In most HCP materials, basal (0001) and prismatic ^1 010 ` have been identified as the primary slip planes with a closed packed direction 1 12 0 for the slip vector. The crystal plasticity model used in this work follows the existing work of Mayeur and McDowell and Zhang et.al. The relationship between the slip system shearing rate and the resolved shear stress of the D slip system is described by the power law flow rule given as: D
J
W
J0
D
D
F
N
D
D
D
M
s g n W
D
F
D
(1)
.
where, J o is the reference shearing rate, M is the inverse strain-rate sensitivity exponent which controls the rate sensitivity of flow, W D is the resolved shear stress, F D is the back stress, N D is the length scale-dependent threshold stress and D D is the drag stress. In Zhang et al., the drag stress is taken as a non-evolving constant, i.e. D D 0 , while the back stress evolves according to an Armstrong-Frederick direct hardening/dynamic recovery type of equation, i.e., F
D
hJ
D
hD F
D
J
D
(2)
The threshold stress is expressed as N
D
N
y
d
D
N
D s
(3)
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Gbadebo Owolabi et al. / Procedia Materials Science 3 (2014) 1860 – 1865
3. Three Dimensional Finite Element Implementation Procedure for Notched Titanium Alloy Component The crystal plasticity model in Section 2 is coded into ABAQUS 2006 UMAT. The notched geometry modeled in this work is a v-notched cylindrical component shown in Fig. 2. Finite element simulations were performed on three different geometries, meshed using 3D stress four-node linear tetrahedron element type (C3D4) to estimate the stress distribution and possible plastic straining that occur in the notched specimens. The dimensions of the specimens used and the different test cases are as given in Table 1.
Fig. 2. Gage section of the cylindrical specimen with a circumferential V-notch. Table 1. Notch Root Geometries and Load Test Cases.
Test Case 1 2 3 4 5 6 7
Kt 2.78 2.78 2.78 2.78 2.78 2.78 2.78
Notch radius, ૉ (mm) 0.330 0.330 0.330 0.203 0.203 0.127 0.127
Notch depth, h (mm) 0.729 0.729 0.729 0.254 0.254 0.127 0.127
R-ratio -1 0.1 0.5 0.1 0.5 0.1 0.5
Average alternating HCF 6 strength at 10 cycles (MPa) 173.6 158.9 104.6 167.2 105.2 144.7 111.0
4. Probabilistic Framework Following the framework presented in Owolabi et.al., the probability of survival of a smooth component having a fatigue damage process zone of volume V, divided into small volume elements, dV is given as: Ps
§ 1 exp ¨ ¨ V0 ©
³ Vd
§§V V ¨¨ ¨ ©© V0
b
th
·· ¸ ¸¸ d V ¹¹
· ¸ ¸ ¹
(4)
where, b, Vth and σ0 represent 3-parameter Weibull shape, location and scale parameters respectively. V is the stress distribution. The cumulative probability of HCF failure of the component, can be obtained from Equation (5) as:
Pf
§ 1 1 exp ¨ ¨ V0 ©
³ Vd
§§V V ¨¨ ¨ ©© V0
b
th
·· ¸ ¸¸ d V ¹¹
· ¸ ¸ ¹
(5)
Gbadebo Owolabi et al. / Procedia Materials Science 3 (2014) 1860 – 1865
1863
To facilitate development of the expression for the fatigue notch factor from Equation (5), the concept of stress homogeneity factor is introduced here. Thus, Equation (5) could be re-written as, b
Pf
§ k V d § V m ax · · 1 exp ¨ ³ ¨ ¸ ¸ ¨ V0 © V 0 ¹ ¸ © Vd ¹
(6)
where
k
1 § §V V ¨ ¨ ³ Vd ¨ V © V 0 © d
th
· ¸ ¹
b
· ¸ dv ¸ ¹
(7)
is regarded as the stress homogeneity factor. Conventionally, the fatigue notch factor is the ratio of unnotched to notched fatigue strength at the same probability of failure (usually 50%). Using Equation (6), the probability of failure of unotched specimen and a notched specimen will be the same when ° k s V s § V m a x , s · °½ exp ® ¨ ¸¾ °¯ V 0 © V 0 ¹ °¿
° k n V n § V m a x , n · °½ exp ® ¨ ¸¾ °¯ V 0 © V 0 ¹ °¿
(8)
where the subscripts n and s represent the respective value of the variable for notched and smooth (unmatched) specimens. The ratio of the smooth to notch fatigue driving force parameters (i.e., the stress amplitude) is used to define a new fatigue notch factor given as: kf
V
m ax, s
V
m ax, n
§ kn · ¨ ¸ © ks ¹
1
b
§ Vn · ¨ ¸ © Vs ¹
1
(9)
b
For smooth specimen that is loaded at a very low stress or strain amplitude in the HCF regime, the number of critically stressed grains (or elements) is very small. Thus for the life limiting case in which only one grain or element is critically stressed above the threshold, Vs = Ve (i.e. volume of element or grain) and k s = 1; thus Equation (9) becomes
k
f
ª 1 º « » ³ ¬ V d ¼ Vd
§§V V th ¨¨ ¨¨ V n m a x , ©©
· ¸ dv ¸ ¹
· ¸ ¸ ¹
1
b
§ Vn · ¨ ¸ © Ve ¹
1
b
(10)
The yield stress Vth for titanium alloy it is 990 MPa, b is determined to be 7.7, while the scale parameter V 0 is determined to 3205.03 MPa. From the geometry of the specimens, the reference volume V0 is calculated to be 16.96 mm3. 5. Results and Discussion The stress distribution obtained from the finite element analysis was used in the developed model to determine the probability of failure for each test case. Also, the average fatigue notch factor is computed for each test case using the developed probabilistic framework. Figure 3 shows probability of failure plotted against the notch root radii for ten different grain orientations. Figure 3 shows that the probability of failure increases with increasing notch root radius. It is interesting to note that for titanium alloy specimen with 0.127 mm notch root radius, some of the probabilities of failure for some of the grain orientations are greater than for the remaining higher notch root radii of 0.203 mm and 0.33 mm. This is an indication that grain orientation around the notch root region also plays
1864
Gbadebo Owolabi et al. / Procedia Materials Science 3 (2014) 1860 – 1865
an equally important role in determining the occurrence of fatigue failure in the notched titanium alloy specimen. The fatigue notch factor determined using the developed method is compared to experimentally obtained values from Naik et al., close from solution, and other existing conventional method such as Neuber as shown in Figure 4 and Table 2. It can be noted from Table 2 that the percentage variance for Weibull fatigue notch is lower compared to Neuber’s predictions. Thus, the developed probabilistic model is more accurate in predicting fatigue notch factor of titanium alloy compared to the Neuber method.
Fig. 3. Probability of failure vs. notch radius for notched titanium alloy at load ratio R= 0.1.
Fig. 4. Fatigue notch factor as a function notch root radius and load ratio compared to experimental values.
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Gbadebo Owolabi et al. / Procedia Materials Science 3 (2014) 1860 – 1865 Table 2. Fatigue notch factor % variance from experimental value for Weibull, close form solution and Neuber methods.
Radius
0.127 0.203 0.33
Experiment kf R=0.1 1.98 1.71 1.80
R=0.5 1.65 1.74 1.75
Weibull@R=0.1
Weibull@R=0.5
Kf 2.05 1.86 1.89
kf 1.72 1.83 1.82
% Var -3.54 -8.77 -5.00
% Var -4.24 -5.17 -4.00
Neuber kf 1.79 1.89 2.00
% Var 9.60 -10.53 -11.11
Closed Form Sol. kf % Var 2.00 -1.01 1.80 -5.26 1.88 -4.44
6. Conclusions A recently developed probabilistic model for microstructure-sensitive fatigue notch factor was extended to titanium alloy components. The result shows that the probability of failure and the fatigue notch factor increase with increasing notch root radius. Also it is noted that the grain orientation around the notch root region plays an important role in determining or predicting the fatigue strength of the material; the probability of failure varies at the same notch root radius for different grain orientation. Acknowledgements The authors of this paper express their profound gratitude to the Department of Defense for the financial support provided through contract # W911NF-11-1-041 (Dr. L. Russell, Program Manager, Army Research Office and Dr. D. Stargel, Program Manager, Air Force Office of Scientific Research.) References Adib-Ramezani H. and Jeong J., 2007,"Advanced Volumetric method for fatigue life prediction using stress gradient effects at notch root," Computational Material Science , vol. 39, pp. 649 - 663. Haritos G K, Nicholas T, & Lanning D B, 1999, "Notch Size Effects in HCF Behavior of Ti-6Al-4V," Int. J. Fatigue, vol. 21, pp. 643 – 652. Kuhn P; Hardraht H.F,1952, "An Engineering Method for Estimating the Notch-Size Effect in Fatigue Tests on Steel," in NACA TN2805Langley Aeronautical Laboratory , Washington. Mayeur J. R., McDowell D.L, 2007 "A three-dimensional crystal plasticity model of duplex Ti-6Al-4V.," Int.Journal of Plasticity, vol. 23, pp. 1457-1485. Naik R.A., Lanning D.B., Nicholas T., Kallmeyer A.R.,2005, "A critical plane gradient approach for the prediction of notched HCF life.," Int. J. Fatigue, vol. 27, pp. 481-492. Neuber H.J., 1961 "Theory of Stress Concentration for Shear-Strained Prismatical Bodies with Arbitrary Nonlinear Stress-Strain Law.," J. Appl. Mech., vol. 28, no. 4, pp. 544 - 550, 1961. Owolabi G., Egboiyi B., Shi L., Withworth H., 2011 "Microstructure-Dependent Fatigue Damage Process Zone and Notch Sensitivity Index," International Journal of Fracture , pp. 159-173. Peterson R. E., 1959,"Notch Sensitivity," in Metal Fatigue, G. Sines and J. Waisman, Eds., New York, McGrawHill, pp. 293-306. Ren W., Nicholas T.,2003, "Notch Size Effects on High Cycle Fatigue Limit Stress of Udimet 720," Material Science and Engineering, vol. A357, pp. 141-152. Yamashita Y., Ueda Y., Kuroki H., Shinozaki M., 2010, "Fatigue Life Prediction of Small Notched Ti-6Al-4V Specimens Using Critical Distance," Engineering Fracture Mechanics, vol. 77, pp. 1439 – 1453. Zhang M., Zhang J., McDowell D.L, 2007 "Microstructure – based crystal plasticity modeling of cyclic deformation of Ti-6Al-4V," Int. Journal of Plasticity, vol. 23, pp. 1328-1348.