Engineering Fracture Mechanics 135 (2015) 81–93
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Numerical simulation of fish-eye fatigue crack growth in very high cycle fatigue H.Q. Nguyen a,b, L. Gallimard a,⇑, C. Bathias a a b
Laboratoire Energétique, Mécanique, Electromagnétisme, Université Paris Ouest Nanterre-La Défense, 50 rue du Sèvre-92410, Ville d’Avray, France Institute of Construction Engineering, University of Transport and Communications, HaNoi, Vietnam
a r t i c l e
i n f o
Article history: Received 28 January 2014 Received in revised form 29 September 2014 Accepted 10 January 2015 Available online 30 January 2015 Keywords: Fish-eye crack Very high cycle fatigue Virtual crack closure technique Finite element analysis Fatigue crack growth
a b s t r a c t In this paper, we numerically study the fish-eye fatigue crack growth after crack nucleation for very high cycle fatigue. The crack growth rate is modeled by the Paris–Hertzberg law. An iterative procedure based on three dimensional finite element analyses is developed to conduct crack growth simulations. The virtual crack closure technique is used to calculate the stress intensity factor for each step of the crack growth process. These stress intensity factors are then used to estimate the fatigue crack growth by integrating the fatigue crack law between the initial and final crack lengths. Our objectives are twofold: first is to study the variation of the fish-eye shape when the crack tip approaches the edge of structure, second is to study the evolution of the fatigue crack growth life when analytical solutions are not available or not sufficiently reliable. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction In fatigue tests, depending on the strain or stress level, three regimes are observed: the low cycle fatigue regime (LCF) (N < 105 cycles), the high cycle fatigue regime (HCF) (105 < N < 107 cycles, megacycle regime) and the very high cycle fatigue regime (VHCF) (N > 107 cycles, gigacyle regime). According to the fatigue regime, different types of crack initiation can occur. In LCF, several cracks nucleate from the surface, while in HCF, there is only one initiation site on the surface, but for the VHCF, the initiation is often located at internal zone in alloys [1]. When the crack initiation site is in the interior, the crack growth leads to the formation of a structure so-called fish-eye which could be described as a circular pattern that is observed on the fracture surface. A specific characteristic of specimens which failed at high numbers of cycles from subsurface inclusions is a rough area around the inclusion within the fish-eye. The mechanisms forming this area are still unclear, but there are indications that this area is responsible for the failure at high numbers of cycles. In literature different names like ODA (optical dark area) [2], FGA (fine granular area) [3] or GBF (granular bright facet) [4] exist for this area, depending on the postulated mechanism or the observation method. The term of FGA is used in following text of this paper. Now it is generally accepted that the formation of FGA consumes most of the fatigue life [6,7]. The total life N f can be separated in two parts [7], a part due to the interior crack nucleation denoted N FGA and a part due to the fish-eye crack growth N cg .
Nf ¼ N FGA þ Ncg
⇑ Corresponding author. E-mail address:
[email protected] (L. Gallimard). http://dx.doi.org/10.1016/j.engfracmech.2015.01.010 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved.
ð1Þ
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Nomenclature a ao af a1 ; b1 b C e E n Nf NFGA Ncg m p q L R1 Sr Dai Damax DaIA DK DK eff DK i DK IA DK max j DK max Dr
distance from the point of simulated crack front to the center of initial circular crack (lm) radius of initial circular crack (lm) radius of final circular crack (lm) semi-major and demi-minor lengths of initial elliptical crack (lm) Burgers vector (m) constant for Paris fatigue crack growth law crack eccentricity Young’s modulus (GPa) number of steps in the crack growth process number of cycles to failure number of cycles to crack nucleation number of cycles to propagate the crack exponent in Paris fatigue crack growth law number of the corner nodes used to constitute the crack front number of integration steps half-height of cylinder (mm) radius of cylinder (mm) relative standard deviation local crack advance at an arbitrary point i (lm) maximum local crack advance along the crack front (lm) local crack advance at point A (lm)pffiffiffiffiffi stress intensity factor range (MPa) m pffiffiffiffiffi effective stress intensity factor range (MPa) m pffiffiffiffiffi stress intensity factor range at an arbitrary point i of the crack front pffiffiffiffiffi (MPa) m stress intensity factor range at point A of the crack front (MPa) m pffiffiffiffiffi maximum stress intensity factor range along the crack front (MPa) m pffiffiffiffiffi maximum stress intensity factor range along the crack front at step j of crack growth process (MPa) m stress range (MPa)
After the FGA is formed, and when a critical stress intensity threshold is reached, the crack enters the Paris regime, i.e. the formation period of the fish-eye outside the FGA. Various estimations of this treshold can be found in the literature [8]. In this paper we use an estimation of this threshold proposed by Paris et al. [9].
DK eff pffiffiffi ¼ 1 E b
ð2Þ
where b is the Burgers vector and E is the elastic modulus, DK eff is the amplitude of the effective stress intensity factor. The crack propagation rate at the threshold is estimated by da=dN ¼ b and the Paris–Hertzberg crack growth law is used to estimate the life of the small cracks in the fish-eye range:
DK eff 3 da ¼ b pffiffiffi dN E b
ð3Þ
The crack growths from the size of the FGA a0 to a final crack size af (see Fig. 1). The fatigue crack growth life can be obtained by integrating Eq. (3)
Ncg ¼
Z
af
ao
pffiffiffi !3 1 E b da b DK eff
ð4Þ
In [6], Marines-Garcia et al. have used Eq. (4) and an analytical approximation of the stress intensity factor (SIF) for a circular crack of radius a in an infinite medium (Eq. (5)) to compute the number of crack growth cycles consumed in the fish eye where the crack closure effect could be neglected.
DK eff ¼ DK ¼
2
p
pffiffiffiffiffiffi Dr p a
ð5Þ
However, Guiniea et al. [10] showed numerically that when the planar circular crack placed eccentrically in the specimen approaches the surface, the SIF differs from the analytical solution. The SIF is not distributed regularly along the crack front. It obtains the maximum value at the point of the crack front which is the closest to the specimen surface. There are two problems to solve here. Firstly, because of the irregular distribution of SIF along the crack front when the crack is near the specimen surface, the crack shape may deviate from the circular form. However, in literature, the crack shape is assumed to be
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83
Fig. 1. Fish-eye crack model.
circular during the crack growth [9,11,12]. It is of interest to numerically investigate the crack shape taking into account the irregular distribution of SIF. Secondly, to obtain a precise integration of Eq. (4), the computation of SIF must be based on numerical methods. On the other hand, Nakasone and Hara [13] used FEM in order to simulate the crack growth of internal da crack initiation using the classical Paris law dN ¼ CðDKÞm . However, they did not show the effect of crack position in their work. The purpose of this work is firstly to use three dimensional finite element analyses to simulate the fish-eye crack growth and then to use the stress intensity factor calculated along the crack front in each step to compute the crack growth life (Eq. (4)). A further advantage of the numerical method is its applicability to any structure. In this regard, some initial elliptical cracks having various aspect ratios are also considered in this paper. This paper is structured in two parts: the first part describes the numerical procedure to simulate the fish-eye crack growth and life. In the second part, the numerical results are shown. In this part, firstly, the symmetric crack configuration is taken as a reference to check the performance of the numerical procedure. Secondly, the numerical estimation of fatigue crack growth life is compared with the analytical expression for two configurations of fish-eye, obtained from experiments [6]. Finally, we simulate the propagation of some initial elliptical cracks of various aspect ratios. The corresponding number of cycles to failure is also calculated. 2. Numerical modeling Evidence of internal fatigue crack initiation from an inclusion is found in [5,6] (Fig. 2). As shown in Fig. 1, all crack parameters ao ; af ; e are measured after failure.
Fig. 2. Fish-eye crack.
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Δσ
Fig. 3. Model description.
The specimen geometry is shown in Fig. 3. To model the propagation of this crack we consider a cylinder f which has a circular cross section of radius R1 and a height 2L. An initial eccentrically positioned penny-shape crack of radius ao perpendicular to the cylinder axis lies at the center of cylinder. f is submitted to a tensile uniform stress range Dr. Given the symmetry of the problem, only one quarter of the cylinder is modeled. 2.1. Fatigue crack growth simulation There are two main methodologies which can be used to deal with the fatigue crack growth simulation. The first one proposed by Newman and Raju [14,15] considers only a few crack front keypoints (one or two nodes) and assumes that a particular crack shape (circular, elliptical, . . .) is maintained during all the propagation phase. The crack can only change its aspect ratio. This approach is not appropriate in the case of fish-eye crack growth, because when the crack approaches the surface of the specimen, there is a complex stress distribution along the crack front. In this case, the use of this approach may lead to an important error in the simulation of fatigue crack growth. The crack may significantly deviate from the assumed shape. A more sophisticated methodology was developed by Lin and Smith [16] who considered several points along the crack front. This approach avoids the assumption of a fixed shape and enables the crack front of growing cracks to be traced directly and therefore, provides more precise simulation. For this reason, this second approach is used to simulate the fish-eye crack growth. In order to simulate the fatigue crack growth, the following iterative procedure is widely used in literature: – Estimation of the SIF at a set of points along the crack front. – Calculation of local crack growth advances by integrating a type of Paris fatigue crack growth law at this set of points and construction of the new crack front. – Modification of the mesh to accommodate crack growth. The whole procedure is repeated up to final fracture. In this work, the virtual crack closure integral technique (VCCT) is used to estimate the SIF. This technique was originally proposed in 1977 by Rybicki and Kannien [17] for two dimensional crack problems, and was extended to three dimensional cases by Shivakumar et al. [18]. With the VCCT, most of the difficulties encountered while using other methods can be avoided, e.g. the required singularity elements at the crack front or creating elements normal to the curved crack front. Therefore, with the VCCT, the remeshing is quite simple. The 3D mesh is created by initially defining a two-dimensional cracked mesh on its base plane and expanding it into a 3D mesh. Fig. 4 shows the structure meshed. When the SIFs at a set of points are known, the local crack growth is calculated using the fatigue crack growth rate Eq. (3). This law can be used at any points along the crack front as follows:
dai DK i 3 b ¼ b pffiffiffi ¼ pffiffiffi 3 ðDK i Þ3 dN E b ðE bÞ
ð6Þ
where dai and DK i are, respectively, the local crack growth and the SIF at an arbitrary point i. The local crack growth is calculated numerically by the following equations:
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85
(a)
Crack front
(b)
Fig. 4. Finite element model for an eccentrically positioned crack: (a) two dimensional crack mesh, and (b) three dimensional crack mesh.
Dai ¼
DK i DK max
3
Damax ;
DN ¼
Damax b pffiffi 3 ðE bÞ
ðDK max Þ3
ð7Þ
where DK max and Damax are, the maximum SIF range along the crack front and the maximum local crack growth at the point where DK max occurs, respectively. In numerical applications, the value of Damax is usually a small constant throughout the whole crack growth process in order to achieve a good numerical accuracy. According to these local crack growth rates, a set of new points can be obtained. A new crack front is established by joining each point with a straight line to form a polygonal shape. Repeating the calculation enables the growth of a fatigue crack to be followed step-by-step. The procedure is stopped when the distance between the center of the crack and the point of the simulated crack front which is the closest to the specimen surface is equal to af . In this work, the local increment at each step is taken as a constant Damax ¼ ðaf a0 Þ=n, n is the number of steps.
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2.2. Fatigue crack growth life estimation In order to numerically integrate Eq. (3) to obtain the fatigue crack growth life, the trapezoidale rule is used:
NFE cg ¼
Z
af
ao
0 pffiffiffi!3 j¼q1 X Damax B 1 E b da ¼ @ b DK 2 j¼0
1 1 b pffiffi 3 ðE bÞ
j ðDK max Þ
3
þ
1 b pffiffi 3 ðE bÞ
ðDK jþ1 max Þ
C
3A
ð8Þ
where q is the number of divisions, DK j ; DK jþ1 are, the maximum SIF range in the step j and the step j + 1 of the crack growth process, respectively. The interval Damax is the maximum increment at each step. It is seen that the required integration accuracy of Eq. (8) is achieved by increasing the number of divisions q. During each step of fatigue crack growth, the maximum SIF along the crack front is stored in order to calculate the fatigue crack growth life. Thus, in this work, the number of steps of fatigue crack growth is coincident with the number of divisions. 3. Numerical results 3.1. Circular internal crack embedded in an infinite solid subjected to uniform tension In order to check the performance of the numerical procedure, a symmetric crack under uniform tension is used as a reference, Fig. 5. In this particular case, due to symmetry considerations, only one eighth of cylinder is modeled and shown in Fig. 6. The radius of cylinder is equal to R1 ¼ 1:5 mm and its height is defined by 2L ¼ 6 mm. Young’s modulus and Poisson’s ratio are taken as 204:8 GPa and 0.3 respectively. The stress range is equal to Dr ¼ 310 MPa. The initial and final circular crack radii are taken to be 10 lm and 100 lm, respectively. The length of elements across the crack front is always equal to a twentieth of the crack radius a during the crack growth process. As the maximum crack radius a < 0:1 R and a < 0:05 L, such cracks are considered to be in an infinite medium. In this case, the SIF for a circular crack of radius a in an infinite medium is used, as shown in Eq. (5). For all analyses presented, the 20-node hexahedron element is used. This paper does not consider the short crack to long crack transition and uses only the short crack growth rate law. Marines-Garcia et al. [6] showed that this assumption has a negligible influence on the prediction of the number cycles corresponding to the fish-eye crack growth. Therefore, the number of cycles is analytically calculated between the initial circular crack of radius ao to the final circular crack of radius af by the formula:
Fig. 5. Circular internal crack embedded in an infinite solid subjected to uniform tension.
H.Q. Nguyen et al. / Engineering Fracture Mechanics 135 (2015) 81–93
87
(a)
(b)
crack front a/20
a/20
Fig. 6. The finite element used to calculate SIF for this case: (a) finite element model, and (b) detailed mesh around the crack front.
"
Nanal cg
sffiffiffiffiffi#
pE2 ao ¼ 1 af 2ðDrÞ2
ð9Þ
Substituting the crack parameters in this formula, N fisheye;anal is equal to 4:68 105 cycles. In the numerical procedure, the effect of the number of divisions is firstly investigated. The effect of n on the number of cycles is shown in Table 1. A relative FE anal error between the numerical solution and the analytical formula can be defined by e ¼j N anal j. The convercg N cg j = j N cg gence of this error as a function of the number of subdivision n is shown on Fig. 7. It can be seen that N cg converges to analytical solution. Thirty subdivisions are sufficient to obtain an error less than 1%. The SIF computed from the VCCT is normalized by dividing its numerical value by the value obtained from the analytical solution Eq. (5), and its evolution as a function of the crack growth (for n ¼ 30) is presented in Fig. 8. It can be seen that the numerical error is well under 1% for all the cases. These results show the performance of the meshing procedure. Therefore, mesh structure, size and quality are kept similar for the eccentrically positioned crack.
3.2. Circular crack placed eccentrically in the cylinder In order to show the effect of the eccentricity on the number of cycles calculated when a crack grows from a size ao to a size af , several calculations are made with various values of eccentricity e using the same data as in Section 3.1. The result is shown in Fig. 9. It can be seen that, in this simple test case, when the eccentricity increases, the number of cycles calculated,
Table 1 Convergence of number of cycles calculated. Number of divisions
N cg
5
5:89 105
10
5:02 105
20
4:78 105
30
4:72 105
40
4:71 105
80
4:69 105
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Relative error in %
Relative error in %
10 5
1 0.5
0.1 5
10
20
40
80
Number of divisions Fig. 7. Evolution of the error
e as a function of the number of subdivision.
Fig. 8. Comparison of SIF for numerical and analytical solutions during the crack growth (n ¼ 30).
decreases. This is explained by the fact that when the crack closes to the specimen surface, SIF at the point which is the nearest to the specimen surface of the same crack size increases. The aim of this paper is to give an estimation of the number of cycles corresponding to fatigue crack growth in the gigacycle regime considering the effect of crack position on the SIF. Here, the numerical results are compared with the analytical results for two specimens, obtained from steel of several strengths that failed by internal cracking in very high cycle fatigue regime [6]. The mesh used in the case of a circular crack placed eccentrically in the cylinder is presented in Fig. 4. The results are shown in Table 2. It is seen that the fish-eye crack growth life N cg is very small compared to the total fatigue life N f in very high cycle fatigue. This result is consistent with previous work [6,21,22] which show that the very high number of cycles is primarily due to nucleation and that crack growth is not a significant portion of life. It is seen that the number of cycles calculated numerically is less than the number of cycles calculated analytically. It is due to an increase of SIF when the crack approaches the specimen surface. Figs. 10 and 11 show the maximum SIF variations, pffiffiffiffiffiffi normalized by DK ¼ ð2=pÞDr pa. The distance from the point of simulated crack front to the center of initial circular crack is denoted by a. It can be seen that when the crack is far from the specimen surface, the SIF is close to the SIF for a circular crack in an infinite medium. The SIF begin to increase when the crack front approaches the edge of the specimen. When the crack approaches the specimen surface, the irregular distribution of SIF along the crack front leads to a deviation from the circular form. In order to quantify the shape deviation of crack fronts simulated by numerical analysis from circular form, we use the definition of relative standard deviation Sr proposed by Lin and Smith [16]:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2ffi u u1 Xp OP OQ t Sr ¼ i¼1 p OQ
ð10Þ
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e Fig. 9. Influence of eccentricity on the number of cycles calculated (ao ¼ 10 lm;af ¼ 100 lm).
Table 2 Fish-eye crack growth numerical estimation vs analytical estimation. Test
Dr [6] (MPa)
E [6] (GPa)
a0 [6] (lm)
af [6] (lm)
e [6]
1
310
204.3
8.3
86
0.93
2
320
208.3
58
296
0.78
N FE cg
N anal cg 4:72 10 3:70 10
5
5
N f [6] 5
3:86 109
5
8:22 107
4:64 10 3:40 10
Fig. 10. Maximum stress intensity factor variation during crack growth for test 1: eccentricity e ¼ 0:93.
where p is the number of the corner nodes used to constitute the cracks front in FE models. Sr represents the relative deviation between the simulated crack profiles and a circle whose center is coincident with the center of an initially circular crack and passes through the point of the simulated crack which is the nearest to the specimen surface (see Fig. 12). The results in Figs. 13 and 14 present the variation of relative standard deviation Sr during crack growth for two specimens. It is seen that Sr increases as the crack progresses. However, the maximum of Sr is only 6%. This means that the crack profiles are very close to being circular as observed from experiments. 3.3. Initial elliptical crack placed eccentrically in the cylinder The hypothesis that the shape of initial crack is circular is only a simplification. In fact, the initial crack has not necessary circular shape. However, a crack which initiates at inclusion quickly tends to have a penny shape regardless of the original shape of the inclusion [19].
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Fig. 11. Maximum stress intensity factor variation during crack growth for test 2: eccentricity e ¼ 0:78.
Fig. 12. Comparison of a numerically simulated crack front with the circular shape.
Fig. 13. Variation of the relative standard deviation during crack growth for test 1: eccentricity e ¼ 0:93.
Our motivation is to study this fact considering some initial elliptical cracks with various aspect ratios. The influence of initial crack shape on the number of cycles is also calculated. Fig. 15 shows the initial elliptical crack dimension. In order to compare with the case of the initial circular crack, the data obtained in test 1 is used. The numerical analysis is made for various aspect ratios a1 =b1 . The value of a1 is kept constant and equal to 8.3 lm. Furthermore, we suppose that the radius of final circular crack is af ¼ 86 lm. In this way, the formula of the local crack growth, Eq. (7), and the formula of the
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Fig. 14. Variation of the relative standard deviation during crack growth for test 2: eccentricity e ¼ 0:78.
Fig. 15. Initial elliptical crack dimension.
0.7
0.6
Residual deviation, S r
b1 = 2a1
0.5
b =1.5a
0.4
b1 = a1
1
1
b1 = 0.75a1
0.3
b1 = 0.5a1
0.2
0.1
0 0
10
20
30
40
50
60
70
80
90
Crack length (μm) Fig. 16. Variations of the relative standard deviation during the crack growth for some initial elliptical crack with an eccentricity e ¼ 0:93.
number of cycles are modified. DK max is changed by DK IA which is the SIF at point A (as shown on Fig. 15). Damax is change by DaIA ¼ ðaf ao Þ=n which is the local crack growth at point A in each step of crack growth process.
Dai ¼
DK i DK IA
3
DaIA ;
DN ¼
DaIA b pffiffi 3 ðE bÞ
ðDK IA Þ3
ð11Þ
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H.Q. Nguyen et al. / Engineering Fracture Mechanics 135 (2015) 81–93 Table 3 Number of cycles for some initial elliptical cracks. N cg
b1 =a1
K IA pffiffiffiffiffiffi Dr pa1 1:2111 pffiffiffiffiffiffi Dr pa1 1:3182 pffiffiffiffiffiffi Dr pa1 1:5707 pffiffiffiffiffiffi Dr pa1 1:8387 pffiffiffiffiffiffi Dr pa1 2:4222
5
2
3:52 10
1.5
3:97 105
1
4:64 105
0.75
5:19 105
0.5
6:30 105
and
Ncg ¼
Z
af
ao
0 pffiffiffi!3 j¼q1 X 1 E b DaIA B da ¼ @ b DK 2 j¼0
b pffiffi 3 ðE bÞ
1 1 ðDK IAj Þ
3
þ
1 b pffiffi 3 ðE bÞ
ðDK jþ1 IA Þ
C
3A
ð12Þ
Fig. 16 shows the relative standard deviation Sr for various aspect ratios. The results show that, in all cases, the cracks become rapidly circular. When the circular form is obtained, the cracks propagate in the same manner as the initial circular crack. Table 3 shows the number of cycles to failure for various initial elliptical cracks and the SIF at point A for each configuration calculated in [20]. It can be seen that as the aspect ratio b1 =a1 increases, the number of cycles decreases. This is explained by the increase of SIF at point A when the aspect ratio increases.
4. Conclusion and prospects 1. The simulated crack during the fish-eye crack growth is demonstrated numerically to be very close to the circular form as observed by experiment. 2. The number of cycles corresponding to the fish-eye crack growth calculated numerically is smaller than the number of cycles calculated analytically. This calculation does not change the conclusion in which, with VHCF ‘‘fish-eye’’ failures, the crack growth is not a significant portion of total life. However, this calculation gives a good estimation of fish-eye crack life. 3. Considering some initial elliptical cracks with various aspect ratios, we demonstrate that the cracks quickly tend to a circular shape. The influence of the aspect ratio of the initial elliptical crack on the number of cycles is calculated. It is seen that, the higher aspect ratio is, the less number of cycles is calculated. 4. The next step in this work would be study the influence of various factors on internal crack propagation: crack shape, crack closure effect, specimen geometry, and loading. The study of the thermomechanical effects proposed in [12] would be extended to an elliptic initial crack. Moreover, bounding techniques should be developed to compute certified bounds on N cg by introducing an error estimator on the computation of the stress intensity factor by the finite element method [23].
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