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Branched crack growth behavior of mixed-mode fatigue for an austenitic 304L steel
International Journal of Fatigue 22 (2000) 457–465 www.elsevier.com/locate/ijfatigue
Branched crack growth behavior of mixed-mode fatigue for an austenitic 304L steel Hui-Ji Shi
a,*
, Li-Sha Niu a, Gerard Mesmacque b, Zhong-Guang Wang
c
a
c
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, People’s Republic of China b Laboratoire de Me´canique de Lille, Universite´ de Lille 1, Villeneuve d’Ascq 59650, France The State Key Laboratory for Fatigue and Fracture of Materials, Institute of Metal Research, Academia Sinica, Shenyang 110015, People’s Republic of China Received 18 July 1999; received in revised form 5 December 1999; accepted 21 February 2000
Abstract Experiments on mixed-mode fatigue crack initiation and propagation in an austenitic stainless steel 304L were carried out using a circular ring specimen containing a V-notch on the internal radius. The branched crack behavior was experimentally investigated with respect to the direction of the inclined loading and the stress distribution near the notch root or crack tip, especially, for the second and third branched cracks. The behavior of the crack initiating from the notch root under the mixed-mode condition is identified well by the maximum tangential stress criterion. The energy release rate criterion can be used to simulate the path of the branched cracks initiating from the parent crack. Optical microscope observation was performed to examine the microstructure of the material for the tested specimens and all crack growth was typically transgranular mode. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Mixed-mode fatigue; Branched crack; Crack growth behavior; Numerical simulation; Virtual crack extension
1. Introduction Recently, many studies focus on mixed-mode fatigue-fracture behavior of materials, because, in reality, the macroscopic cracks or microscopic flaws in engineering structures are always subjected to multiaxial stress states that are more complicated than a mode I opening deformation state. In order to reveal crack initiation and propagation mechanisms in combinedmode fatigue, it is necessary to analyze the nature of near-tip stress and deformation fields by both theoretical and experimental methods. In the aspect of the crack initiation and propagation under mixed-mode cyclic loading, attention has been paid to the determination of the critical load, the crack growth rate and direction, and the relationship between the angle of inclined loading and the angle of branched crack. For
* Corresponding author. Tel.: +86-10-6277-2731; fax: +86-106278-1824. E-mail address: [email protected] (H.-J. Shi).
an in-plane crack that initiates from a notch or a precrack under mixed-mode loading, the criteria most often used are the maximum tangential stress criterion [1], the minimum strain energy density theory [2] and the maximum energy release rate criterion [3]. These theories are generally available for the brittle fracture of materials when the microscopic mechanism responsible for crack extension is transgranular cleavage. In the respect of metals, fatigue crack growth behavior was investigated under mode I and/or mode II loading [4–7]. The explanation and prediction of crack growth rates and paths were the main concerns [8,9]. Chambers et al. [10] performed mixed-mode fatigue crack growth tests in a high chromium steel and indicated that the direction of fatigue crack growth was dependent on the KI/KII ratio and could be predicted by the maximum tangential stress theory. Qian et al. [11] studied mixed-mode fatigue crack propagation in stainless steel cruciform specimens in which a center initial crack oriented at 45° was loaded biaxially. It was found that the branched cracks propagated in mode I, namely ⌬KII was almost always zero along the crack paths. Paul [12] studied crack growth paths obtained
0142-1123/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 0 ) 0 0 0 2 7 - X
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Nomenclature a B F G knee K u U j ⌸ q q1 q2 sqq
length of crack thickness of a body global force matrix energy release rate turning point from the first crack to the second crack global stiffness matrix global displacement matrix total strain energy of a body angle of loading potential energy of a body angle of the crack growth inclined to its original direction angle of the first crack growth inclined to the plane of the notch’s axis angle of the second crack growth inclined to the first crack direction tangential stress
by subsequently changing the orientation of a straight crack at the center of a thin circular disk in an aluminum alloy with respect to uniaxially cyclic loading. In predicting the number of load cycles to failure, the fatigue crack growth criterion based on the strain energy density theory was in close agreement with the experiments. But in predicting the successive kink initiation angles, significant deviation from the experiments was observed. Tong et al. [13] investigated mixed-mode fatigue behavior in a weldable structural steel using an asymmetrical four point bend arrangement. Various aspects of the subject, including fatigue thresholds, branch angles, path and branched crack growth rates, were studied. However, the mechanism of the mixed-mode fatigue crack initiation and propagation is so complicated that many characteristics of crack branching are far less well understood. Mixed-mode fatigue tests by suitable specimens and simulation by theoretical and numerical methods are needed on more materials. In this work, experiments on mixed-mode fatigue crack initiation and propagation in an austenitic stainless steel were carried out using a circular ring specimen containing a V-notch on the internal radius. The arrest and branch behavior of cracks was studied experimentally with respect to the direction of the inclined loading and the stress distribution of the crack tip. Especially, the second and third branched cracks have been investigated by changing the angle of cyclic loading. The behavior of the crack initiating from notch root under mixed-mode condition was identified by the maximum tangential stress criterion. The direction of the branched cracks initiating from the parent crack was simulated by using the energy release rate criterion.
2. Experimental procedure 2.1. Tested material and special specimen The material investigated in this work is a type of austenitic stainless steel 304L. The percentage of mass in the composition of material is as follows: 0.03 C, 0.80 Si, 1.42 Mn, 0.03 P, 0.01 S, 18.7 Cr, 10.2 Ni, and remainder Fe. The physical properties are listed in Table 1. The average grain size was about 20 µm. The circular ring specimen with an internal V-notch is shown in Fig. 1. The external and internal diameters of the specimen are 30 mm and 14 mm, respectively. The notch depth is 2.5 mm, the notch width is 1.0 mm and the radius of the notch root is 0.3 mm. The specimens were taken from a same rod of 35 mm in diameter and carefully manufactured by a milling machine. The notch surfaces of the rings were carefully polished by a grinding machine. It is known that when a solid containing a crack or a notch is subjected to purely uniaxial far-field loading, mixed-mode conditions may also prevail ahead of the crack tip or the notch root, if it is inclined at some arbitrary angle to the loading axis. So this special type of specimen, which is easily tested by uniaxial loading, is developed to simulate mixed-mode stress and strain distribution in the vicinity of a crack or a notch. This kind of geometry has already been used to study different fatigue-fracture problems. Ahmad et al. [14] used it to investigate the crack propagation in the case of the constant stress intensity factor. Cheverton et al. [15] performed the tests of the crack propagation and arrest under thermal shock loading conditions. More recently, Nunomura et al. [16] and Shi et al. [17] also used this type of
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459
Table 1 Physical behavior of the material at 20°C Thermal conductivity (W/mm.K)
Specific heat (J/g.K)
Thermal expansion Modulus of Poisson’s ratio coefficient (1/K) elasticity (MPa)
Yield stress s0.2 Tensile strength Elongation (%) (MPa) sb (MPa)
14.8×10⫺3
0.48
16.4×10⫺6
240
1.92×105
0.29
2.
Fig. 1. mm).
Geometry of the notched ring specimen (all dimensions in
specimen to identify fatigue crack propagation behavior under inclined loading. Tests were conducted on an electro-mechanical tension-compression machine. Cyclic compressive loading was performed on the external radius with an inclined angle towards the plane of the V-notch’s axis. It resulted in a mixed mode I-II stress field (tensile opening and inplane shearing) at the notch root. The cyclic compressive loading, denoted as P in Fig. 1, was controlled by constant amplitude during the test. A load ratio, R, was equal to 20 and a triangle waveform of the frequency of 20 Hz was applied. During the tests, a video camera with a magnification of 200 times connected to an image acquisition system were applied to observe the crack initiation and propagation sites and to measure the length of crack growth. 2.2. Steps of fatigue crack growth experiments Ten specimens were used to investigate the mixedmode crack initiation and propagation behavior under an inclined loading, and then the crack arrest and branch behavior in the case of changing loading directions. We define j as the angle of loading, that is an angle of the applied loading inclined to the plane of the notch’s axis (see Fig. 1). It is considered here that a counterclockwise angle to the plane of the notch’s axis is positive and, on the contrary, a clockwise angle is negative. The following testing steps were implemented: 1. Two specimens (numbered A01 and A02) were subjected to the cyclic compressive loading with the angle of loading j=0. The other eight specimens were subjected to the cyclic compressive loading with the
3.
4.
5.
565
56
angle of loading of 5°, 7.5°, 15° and 20°, respectively (refer to Table 2). All specimens were tested until a crack (the first crack) arose from the notch root and propagated about 1.0苲1.5 mm in length. The tests were performed on the specimens A01 and A02 that had been subjected to the symmetrical loading, j=0, and had a 1.0苲1.5 mm straight crack propagating from the notch root. These two specimens were turned to a position that j=15°. At this position the specimens were subjected to the cyclic loading until a branched crack (the second crack) initiated at the parent crack tip and propagated about 1.0苲1.5 mm (refer to Table 3). The tests were performed on the specimens, A04, A05 and A06, that had been subjected to the cyclic inclined loading of j=7.5° and had a 1.0苲1.5 mm inclined crack growing from the notch root. This time the cyclic compressive loading of j=⫺7.5° (the opposite position of the former position of loading) was applied on the specimens until another inclined crack (the second crack) initiated at the parent crack tip and propagated about 1.5 mm length (refer to Table 3). The specimens A03 and A07–A010 were not continued to test for getting branched cracks. The results were used to obtain more data about the first crack and also the relationship between the angle of loading and the first crack growth direction. Finally, the specimens, A01, A02, A04, A05 and A06, which had both the first and second cracks, were returned to the primary position as in the testing condition of the first step. They were continuously subjected to the same loading as at the second or third step in order to observe new crack growth (refer to Table 4).
3. The first crack growth from notch root 3.1. Results of experiments Ten ring specimens with internal V-notches were subjected to the cyclic compressive loading with the different angle of loading, 0°, 5°, 7.5°, 15° and 20°, respectively. It was found that the amplitude of cyclic loading needed to crack initiation increased with the increment
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Table 2 Experimental data of crack growth from the notch root Specimen’s number
A01
A02
A03
A04
A05
A06
A07
A08
A09
A10
Amplitude of loading (kN) Angle of loading (degree) Angle of crack growth (degree) Length of the branched crack (mm) Number of cycles to crack initiation (cycle) Average crack growth rate (mm/cycle)
1.5 0° 0° 1.02 4.4×105
1.5 0° 0° 1.34 6.2×105
1.6 5° 18° 1.26 8.9×105
1.7 7.5° 27° 1.28 7.9×105
1.7 7.5° 27° 1.48 5.5×105
1.7 7.5° 25° 1.12 8.4×105
2.1 15° 46° 1.40 4.7×105
2.1 15° 42° 1.52 4.1×105
2.5 20° 56° 1.32 3.6×105
2.5 20° 54° 1.42 3.2×105
8.7×10⫺7 7.8×10⫺7 4.6×10⫺7 6.1×10⫺7 6.4×10⫺7 7.1×10⫺7 9.8×10⫺7 10.3×10⫺7 9.2×10⫺7 12.5×10⫺7
Table 3 Experimental data of the second crack growth
Table 4 Experimental data of the third crack growth
Specimen’s number
A01
A02
A04
A05
A06
Specimen’s number
A01
A02
A04
A05
A06
Amplitude of loading (kN) Angle of loading to the plane of the notch’s axis (degree) Angle of the second crack growth from away the first crack direction (degree) Length of the second inclined crack (mm) Cyclic numbers of the second crack initiation (cycle) Average crack growth rate of the second crack (mm/cycle)
1.7
1.7
1.7
1.7
1.7
1.7
1.7
1.7
1.7
1.7
15°
15°
⫺7.5°
⫺7.5°
⫺7.5°
0°
0°
7.5°
7.5°
7.5°
57°
52°
⫺55°
⫺53°
⫺56°
Amplitude of loading (kN) Angle of loading to the plane of the notch’s axis (degree) Cyclic numbers of the third crack initiation (cycle) Average crack growth rate of the third crack (mm/cycle) Positions of the cracks arising from the second crack Position of the longest crack arising from the second crack Length of the longest crack (mm)
9.6×105
5.7×105
9.1×105
6.6×105
10.8×105
1.30
1.06
1.28
1.04
2.20
1.5×105
1.9×105
1.7×105
2.0×105
2.4×105
3.7×10⫺7 4.6×10⫺7 4.4×10⫺7 4.4×10⫺7 4.9×10⫺7
of the angle of loading, and the angle of inclined crack also increased with the angle of loading. Table 2 gives the experimental data: the amplitude of loading, the angle of inclined crack with respect to the angle of loading, the length of the first crack, the number of cycles to crack initiation, as well as the average crack growth rate for the first 1 mm of growth from the notch root. 3.2. Numerical simulation with the maximum tangential stress criterion From Fig. 1, one can assume that the stress states of the notched ring can be described as a two-dimensional domain occupied by a homogeneous body. Since the
1.6×10⫺7 1.2×10⫺7 0.7×10⫺7 1.5×10⫺7 0.8×10⫺7
middle middle middle part, tip part and part and knee knee
tip and knee
middle part
knee
knee
middle part
tip
middle part
1.38
1.96
0.72
1.44
1.32
applied loading does not change across the specimen thickness and parallels to the ring’s surface, the deformation field is assumed to be planar and the plane strain condition is applied. In consideration of the non-singular stress field at the blunt notch root, it is convenient to use the maximum tangential stress criterion for identifying a crack growth from the notch root. It is based on a hypothesis that crack growth will occur at the notch boundary in a radial direction along which the tensile stress is a maximum. To realize the identification, a numerical evaluation has been carried out by a finite element method. The finite element mesh is shown in Fig. 2 and the mesh around the notch root, consisting of isoparametric eight-
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461
Fig. 2. Finite element mesh of the specimen: (a) mesh of total configuration and (b) dense mesh at the notch root.
node quadrilateral elements, is much denser than the other elements in order to obtain the necessary accuracy. The elements around the notch root are specially arranged: the two opposite sides of an element perpendicular to the tangent line of the arched boundary. Therefore, the tangential stress can be easy to obtain at the arched boundary of the notch. The direction of the first crack growth, q1, is determined by the point at the notch root where the tangential stress sqq is maximum: ∂sqq ∂2sqq | ⱕ0 |q=q1⫽0 and ∂q ∂q2 q=q1
(1)
The maximum value of stress sqq at the arched boundary is shown in Fig. 3a with different angles of loading. It is found that the maximum tangential stress has the highest value at j=0° and it decreases with the increment of the angle of loading. This tendency explains the experimental results that crack initiation from notch root is easier with a smaller angle of loading under the same amplitude of applied loading. When the angle of loading is equal to zero degrees, the maximum tangential stress corresponds to the angle of branched crack q1=0°. When the angle of loading is different from zero degrees, the larger the angle of loading is, the larger the angle of branched crack will be. The relationship between the angle of loading and the angle of branched crack is given in Fig. 3b. It is noted that the results of these calculations and experiments are comparable and the differences between them are less than 15%.
4. Inclined crack growth from parent crack 4.1. Experimental data of the second crack growth Tests were performed on two specimens, numbered as A01 and A02, which had been subjected to the symmetrical loading of amplitude of 1.5 kN and had possessed a 1.02 and 1.34 mm straight crack, respectively. The cyclic compressive loading with an amplitude of 1.7
Fig. 3. The calculated results of the crack initiating from the notch root: (a) the maximum tangential stress versus the inclined angle of loading under the same amplitude load and (b) the relationship between the direction of branched crack and the inclined angle of loading.
kN applied on the external ring of the specimens with an inclined angle of loading of 15° to the plane of the notch’s axis (turning the specimen counter clockwise). In this experiment, the change of loading direction gave rise to an arrest of crack growth along the original direction. It was necessary to use 1.5苲1.9×105 cycles for initiating a branched crack from the tip of the first crack. The second crack propagated along a direction that was approximately inclined at 52苲57° towards the direction of the first crack and the average crack growth rate was about 3.7苲4.6×10⫺7 mm/cycle. Testing was continued on three specimens, numbered as A04, A05 and A06, that had been subjected to the 7.5° inclined loading with the amplitude of 1.7 kN and had possessed a 1.28, 1.48 and 1.12 mm inclined crack, respectively. To obtain the second inclined crack, a loading of 1.7 kN amplitude was applied on the specimens with an inclined angle of loading ⫺7.5° to the plane of
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the notch’s axis. It was found that 1.7苲2.4×105 cycles were needed for initiating a branched crack from the tip of the first crack. The second crack propagated along a direction that was approximately inclined at 53苲56° towards the direction of the first crack and the average crack growth rate was about 4.4苲4.9×10⫺7 mm/cycle. Table 3 gives the experimental data of the second crack growth behavior. The length and the inclined angle of the second crack, the number of cycles to the second crack initiation and the average crack growth rate are listed. 4.2. Experimental data of the third crack growth In the fourth step, the specimens (numbered as A01, A02, A04, A05 and A06) that had both first and second cracks were continuously subjected to the same loading as at the second or third step, namely the amplitude of loading was 1.7 kN. However, the direction of applied loading came back to the original direction, that is the notch’s axis (j=0°) for the specimens A01 and A02 and the angle of 7.5° to the notch’s axis for the specimens A04, A05 and A06. The change of loading direction again created an arrest of crack growth. It was more difficult than the second or third step to initiate a branched crack, and a longer period of 5.7×105苲10.8×105 cycles was needed. The average crack growth rate obviously slowed down during this step. The position of initiation was not always situated at the tip of the second crack. The experimental data are listed in Table 4. In the case of specimen A01, cracks initiated in succession from the tip, middle and knee of the second crack. The crack that arose from the tip of the second crack propagated the shortest period and then stopped. The crack arising from the knee of the second crack continued propagating for the longest period. In the case of specimen A02, no crack arose from the tip of the second crack. However, there were three cracks arising from the middle part of the second crack. The longest crack also appeared from the knee of the second crack and did not stop propagating. In the case of specimen A05, cracks initiated simultaneously from both tip and knee of the second crack. The crack appearing from the tip propagated a very short period and then stopped. The crack from the knee of the second crack continued propagating. In the case of specimens A04 and A06, there was only one branched crack that initiated from the middle part of the second crack, respectively. In most cases, the angles of branched crack were approximately 65苲70° towards the second crack direction. It appeared that the branched crack growth rate was higher for the crack growing from the tip and knee of the former crack than that growing from the middle part of the former one. Fig. 4 shows schematically the crack paths of the experiments. An optical microscope observation was performed to
Fig. 4. Schematic drawing of the crack paths for the specimens having prossessed branched cracks (A01, A02 and A04-A06).
examine the fatigue-fracture behavior. Figs. 5 and 6 show the microstructure of the material and the branched crack paths of specimens A01 and A06. The bigger figure shows the total branched cracks and the two smaller figures give the microstructure in detail of parts A and B indicated in the bigger figure. It is noted that all crack growth is typically transgranular mode. Many short cracks may be found on the second cracks, it appears that cracks may initiate at any position of the former crack at the last step.
5. Crack growth simulation with the energy release rate method A numeral simulation of the first crack that initiates from the notch root has been carried out by finite element analysis with the maximum tangential stress criterion, described in Section 3.2. Here, we studied by numerical method the behavior of the second crack that initiated from the first crack tip. It is noted that a more complicated mixed-mode stress field arises in this case than that in the case of the first crack growth. Therefore, we introduced the energy release rate method described by virtual crack extension and used it to the inclined crack extension problem. 5.1. The energy release rate calculated by virtual crack extension method In order to simulate a crack extending away from its original direction by changing the direction of external loading, the energy release rate method can be used [18]. It is assumed that an original crack in length a turns sharply at some angle q to its original direction and extends an increment da, then the energy release rate G is defined as U1−U2 da→0 da
G⫽ lim
(2)
where U1 is the total strain energy of the body with the
H.-J. Shi et al. / International Journal of Fatigue 22 (2000) 457–465
Fig. 5.
The microstructure of the material and the branched crack path of specimen A01.
Fig. 6.
The microstructure of the material and the branched crack path of specimen A06.
original crack a and U2 is the total strain energy of the body with the original crack a plus a small extension da. In a linear elastic system, we use the stiffness derivative procedure to appropriate the finite element analysis. The potential energy ⌸ of a body with a crack a is given by 1 ⌸⫽ uT Ku⫺uT F 2
(3)
where u, K and F are the global displacement, stiffness and force matrices, respectively. The energy release rate can be expressed as
G⫽⫺
冉
463
冊
∂F 1 ∂uT 1 ∂K ∂uT 1 ∂⌸ ⫽⫺ Ku⫹ uT u⫺ F⫺uT B ∂a B ∂a 2 ∂a ∂a ∂a
(4)
where B is the thickness of the body. If supposing that the external force is nearly constant and F=Ku, then G⫽⫺
1 ∂⌸ 1 ∂K 1 dK ⫽⫺ uT u⬇⫺ uT u B ∂a 2B ∂a 2B da
(5)
In actual calculation, da is taken as very small but finite and displacement u is calculated for a crack configuration with the original crack a plus a small extension da.
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dK is the difference between the stiffness matrices before and after the crack extension. With the energy release rate method, the crack could extend towards any direction, called virtual crack extension. It is well known that the actual crack extension is one of the virtual crack extensions and its direction is that along which the energy release rate achieves a maximum value. Therefore, the direction of an actual crack extension, indicated by the angle of branched crack q2, can be obtained by following formula ∂G ∂2 G |q=q1⫽0 and 2 |q=q1ⱕ0 ∂q ∂q
(6)
5.2. Results of numerical simulation In order to obtain the actual crack extension direction, the energy release rate of the virtual crack extension should be calculated in all divided directions with sufficiently small intervals. We used eight-node collapsed quadrilateral elements in the crack extension region to realize more dense mesh. Fig. 7 describes the meshing technique: among the figures, Fig. 7(a) gives the mesh only with an original crack (the first crack in our case) before the virtual crack extension; Fig. 7(b) shows the mesh that has a virtual crack extension in a direction of ⫺30°; Fig. 7(c) is an enlarged image of the virtual crack extension area in Fig. 7(b); Figs. 7(d), (e) and (f) indicate the deformed mesh. A calculated result of Eq. (6) is shown in Fig. 8 by indicating the crack growing direction (in angle), along which the energy release rate is a maximum, versus the inclined angle of loading for the specimen having the first crack. The superior curve gives the state that the original crack (the first crack) has an angle q1=0° to the notch’s axis, and the lower curve gives the state that the
Fig. 7. Mesh technique for calculating the energy release rate of virtual crack extension.
Fig. 8. The crack growth direction versus the inclined angle of loading for the specimen having an angle of crack q1=0° and q1=25°, respectively.
original crack has an angle q1=25° to the notch’s axis. It is found that the calculated results of the angle of branched crack are larger than that of our experiments and the difference between them is about 12苲15%. It has been studied that for a cracked ring under compression, the stress intensity factor is almost constant for a wide range of crack length [14,19]. So it is reasonable in our case that the crack growth rate for each step is quasi-constant. The differences of the crack growth rates between the steps may be the influence of the former branched cracks and the roughness-induced closure effect [20, p. 254–63]. Such investigation in detail is beyond the scope of the present study.
6. Conclusions In this paper, mixed mode (combing mode I and II) crack initiation and propagation under far-field uniaxial loading for an austenitic 304L steel have been investigated by using a special type of notched ring specimen. The arrest and branch of cracks are analyzed with respect to the direction of the inclined loading. Some conclusions can be stated as follows: 1. The orientation of crack initiation and propagation is the function of the inclined angle of loading. When the far-field cyclic loading changes an angle during the test, the fatigue crack growth appear to arrest, and a great number of cycles are necessary to initiate a new crack from the former crack. The direction of the new crack propagation is determined by mixed mode conditions. 2. The behavior of the crack initiating from the notch root under mixed-mode conditions for the tested material can be identified well by the maximum tangential stress criterion. The energy release rate cri-
H.-J. Shi et al. / International Journal of Fatigue 22 (2000) 457–465
terion has been used to simulate the path of the branched crack initiating from the first crack tip. There is about a 12苲15% difference between the crack growth directions obtained by experiment and calculation. 3. The third branched cracks can initiate at any position of the second crack when the direction of external loading changes and more than one macrocrack growing from the second crack may be found. 4. Optical microscope observation was performed to examine the microstructure of the material and the branched crack paths of specimens. It is noted that all crack growth is typically transgranular mode. Therefore, the simulation of fatigue-fracture behavior by elasticity theory is available.
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Acknowledgements We are grateful for the financial support provided by the Special Funds for the Major State Basic Research Projects G19990650.
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Branched crack growth behavior of mixed-mode fatigue for an austenitic 304L steel Hui-Ji Shi
a,*
, Li-Sha Niu a, Gerard Mesmacque b, Zhong-Guang Wang
c
a
c
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, People’s Republic of China b Laboratoire de Me´canique de Lille, Universite´ de Lille 1, Villeneuve d’Ascq 59650, France The State Key Laboratory for Fatigue and Fracture of Materials, Institute of Metal Research, Academia Sinica, Shenyang 110015, People’s Republic of China Received 18 July 1999; received in revised form 5 December 1999; accepted 21 February 2000
Abstract Experiments on mixed-mode fatigue crack initiation and propagation in an austenitic stainless steel 304L were carried out using a circular ring specimen containing a V-notch on the internal radius. The branched crack behavior was experimentally investigated with respect to the direction of the inclined loading and the stress distribution near the notch root or crack tip, especially, for the second and third branched cracks. The behavior of the crack initiating from the notch root under the mixed-mode condition is identified well by the maximum tangential stress criterion. The energy release rate criterion can be used to simulate the path of the branched cracks initiating from the parent crack. Optical microscope observation was performed to examine the microstructure of the material for the tested specimens and all crack growth was typically transgranular mode. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Mixed-mode fatigue; Branched crack; Crack growth behavior; Numerical simulation; Virtual crack extension
1. Introduction Recently, many studies focus on mixed-mode fatigue-fracture behavior of materials, because, in reality, the macroscopic cracks or microscopic flaws in engineering structures are always subjected to multiaxial stress states that are more complicated than a mode I opening deformation state. In order to reveal crack initiation and propagation mechanisms in combinedmode fatigue, it is necessary to analyze the nature of near-tip stress and deformation fields by both theoretical and experimental methods. In the aspect of the crack initiation and propagation under mixed-mode cyclic loading, attention has been paid to the determination of the critical load, the crack growth rate and direction, and the relationship between the angle of inclined loading and the angle of branched crack. For
* Corresponding author. Tel.: +86-10-6277-2731; fax: +86-106278-1824. E-mail address: [email protected] (H.-J. Shi).
an in-plane crack that initiates from a notch or a precrack under mixed-mode loading, the criteria most often used are the maximum tangential stress criterion [1], the minimum strain energy density theory [2] and the maximum energy release rate criterion [3]. These theories are generally available for the brittle fracture of materials when the microscopic mechanism responsible for crack extension is transgranular cleavage. In the respect of metals, fatigue crack growth behavior was investigated under mode I and/or mode II loading [4–7]. The explanation and prediction of crack growth rates and paths were the main concerns [8,9]. Chambers et al. [10] performed mixed-mode fatigue crack growth tests in a high chromium steel and indicated that the direction of fatigue crack growth was dependent on the KI/KII ratio and could be predicted by the maximum tangential stress theory. Qian et al. [11] studied mixed-mode fatigue crack propagation in stainless steel cruciform specimens in which a center initial crack oriented at 45° was loaded biaxially. It was found that the branched cracks propagated in mode I, namely ⌬KII was almost always zero along the crack paths. Paul [12] studied crack growth paths obtained
0142-1123/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 0 ) 0 0 0 2 7 - X
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Nomenclature a B F G knee K u U j ⌸ q q1 q2 sqq
length of crack thickness of a body global force matrix energy release rate turning point from the first crack to the second crack global stiffness matrix global displacement matrix total strain energy of a body angle of loading potential energy of a body angle of the crack growth inclined to its original direction angle of the first crack growth inclined to the plane of the notch’s axis angle of the second crack growth inclined to the first crack direction tangential stress
by subsequently changing the orientation of a straight crack at the center of a thin circular disk in an aluminum alloy with respect to uniaxially cyclic loading. In predicting the number of load cycles to failure, the fatigue crack growth criterion based on the strain energy density theory was in close agreement with the experiments. But in predicting the successive kink initiation angles, significant deviation from the experiments was observed. Tong et al. [13] investigated mixed-mode fatigue behavior in a weldable structural steel using an asymmetrical four point bend arrangement. Various aspects of the subject, including fatigue thresholds, branch angles, path and branched crack growth rates, were studied. However, the mechanism of the mixed-mode fatigue crack initiation and propagation is so complicated that many characteristics of crack branching are far less well understood. Mixed-mode fatigue tests by suitable specimens and simulation by theoretical and numerical methods are needed on more materials. In this work, experiments on mixed-mode fatigue crack initiation and propagation in an austenitic stainless steel were carried out using a circular ring specimen containing a V-notch on the internal radius. The arrest and branch behavior of cracks was studied experimentally with respect to the direction of the inclined loading and the stress distribution of the crack tip. Especially, the second and third branched cracks have been investigated by changing the angle of cyclic loading. The behavior of the crack initiating from notch root under mixed-mode condition was identified by the maximum tangential stress criterion. The direction of the branched cracks initiating from the parent crack was simulated by using the energy release rate criterion.
2. Experimental procedure 2.1. Tested material and special specimen The material investigated in this work is a type of austenitic stainless steel 304L. The percentage of mass in the composition of material is as follows: 0.03 C, 0.80 Si, 1.42 Mn, 0.03 P, 0.01 S, 18.7 Cr, 10.2 Ni, and remainder Fe. The physical properties are listed in Table 1. The average grain size was about 20 µm. The circular ring specimen with an internal V-notch is shown in Fig. 1. The external and internal diameters of the specimen are 30 mm and 14 mm, respectively. The notch depth is 2.5 mm, the notch width is 1.0 mm and the radius of the notch root is 0.3 mm. The specimens were taken from a same rod of 35 mm in diameter and carefully manufactured by a milling machine. The notch surfaces of the rings were carefully polished by a grinding machine. It is known that when a solid containing a crack or a notch is subjected to purely uniaxial far-field loading, mixed-mode conditions may also prevail ahead of the crack tip or the notch root, if it is inclined at some arbitrary angle to the loading axis. So this special type of specimen, which is easily tested by uniaxial loading, is developed to simulate mixed-mode stress and strain distribution in the vicinity of a crack or a notch. This kind of geometry has already been used to study different fatigue-fracture problems. Ahmad et al. [14] used it to investigate the crack propagation in the case of the constant stress intensity factor. Cheverton et al. [15] performed the tests of the crack propagation and arrest under thermal shock loading conditions. More recently, Nunomura et al. [16] and Shi et al. [17] also used this type of
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459
Table 1 Physical behavior of the material at 20°C Thermal conductivity (W/mm.K)
Specific heat (J/g.K)
Thermal expansion Modulus of Poisson’s ratio coefficient (1/K) elasticity (MPa)
Yield stress s0.2 Tensile strength Elongation (%) (MPa) sb (MPa)
14.8×10⫺3
0.48
16.4×10⫺6
240
1.92×105
0.29
2.
Fig. 1. mm).
Geometry of the notched ring specimen (all dimensions in
specimen to identify fatigue crack propagation behavior under inclined loading. Tests were conducted on an electro-mechanical tension-compression machine. Cyclic compressive loading was performed on the external radius with an inclined angle towards the plane of the V-notch’s axis. It resulted in a mixed mode I-II stress field (tensile opening and inplane shearing) at the notch root. The cyclic compressive loading, denoted as P in Fig. 1, was controlled by constant amplitude during the test. A load ratio, R, was equal to 20 and a triangle waveform of the frequency of 20 Hz was applied. During the tests, a video camera with a magnification of 200 times connected to an image acquisition system were applied to observe the crack initiation and propagation sites and to measure the length of crack growth. 2.2. Steps of fatigue crack growth experiments Ten specimens were used to investigate the mixedmode crack initiation and propagation behavior under an inclined loading, and then the crack arrest and branch behavior in the case of changing loading directions. We define j as the angle of loading, that is an angle of the applied loading inclined to the plane of the notch’s axis (see Fig. 1). It is considered here that a counterclockwise angle to the plane of the notch’s axis is positive and, on the contrary, a clockwise angle is negative. The following testing steps were implemented: 1. Two specimens (numbered A01 and A02) were subjected to the cyclic compressive loading with the angle of loading j=0. The other eight specimens were subjected to the cyclic compressive loading with the
3.
4.
5.
565
56
angle of loading of 5°, 7.5°, 15° and 20°, respectively (refer to Table 2). All specimens were tested until a crack (the first crack) arose from the notch root and propagated about 1.0苲1.5 mm in length. The tests were performed on the specimens A01 and A02 that had been subjected to the symmetrical loading, j=0, and had a 1.0苲1.5 mm straight crack propagating from the notch root. These two specimens were turned to a position that j=15°. At this position the specimens were subjected to the cyclic loading until a branched crack (the second crack) initiated at the parent crack tip and propagated about 1.0苲1.5 mm (refer to Table 3). The tests were performed on the specimens, A04, A05 and A06, that had been subjected to the cyclic inclined loading of j=7.5° and had a 1.0苲1.5 mm inclined crack growing from the notch root. This time the cyclic compressive loading of j=⫺7.5° (the opposite position of the former position of loading) was applied on the specimens until another inclined crack (the second crack) initiated at the parent crack tip and propagated about 1.5 mm length (refer to Table 3). The specimens A03 and A07–A010 were not continued to test for getting branched cracks. The results were used to obtain more data about the first crack and also the relationship between the angle of loading and the first crack growth direction. Finally, the specimens, A01, A02, A04, A05 and A06, which had both the first and second cracks, were returned to the primary position as in the testing condition of the first step. They were continuously subjected to the same loading as at the second or third step in order to observe new crack growth (refer to Table 4).
3. The first crack growth from notch root 3.1. Results of experiments Ten ring specimens with internal V-notches were subjected to the cyclic compressive loading with the different angle of loading, 0°, 5°, 7.5°, 15° and 20°, respectively. It was found that the amplitude of cyclic loading needed to crack initiation increased with the increment
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Table 2 Experimental data of crack growth from the notch root Specimen’s number
A01
A02
A03
A04
A05
A06
A07
A08
A09
A10
Amplitude of loading (kN) Angle of loading (degree) Angle of crack growth (degree) Length of the branched crack (mm) Number of cycles to crack initiation (cycle) Average crack growth rate (mm/cycle)
1.5 0° 0° 1.02 4.4×105
1.5 0° 0° 1.34 6.2×105
1.6 5° 18° 1.26 8.9×105
1.7 7.5° 27° 1.28 7.9×105
1.7 7.5° 27° 1.48 5.5×105
1.7 7.5° 25° 1.12 8.4×105
2.1 15° 46° 1.40 4.7×105
2.1 15° 42° 1.52 4.1×105
2.5 20° 56° 1.32 3.6×105
2.5 20° 54° 1.42 3.2×105
8.7×10⫺7 7.8×10⫺7 4.6×10⫺7 6.1×10⫺7 6.4×10⫺7 7.1×10⫺7 9.8×10⫺7 10.3×10⫺7 9.2×10⫺7 12.5×10⫺7
Table 3 Experimental data of the second crack growth
Table 4 Experimental data of the third crack growth
Specimen’s number
A01
A02
A04
A05
A06
Specimen’s number
A01
A02
A04
A05
A06
Amplitude of loading (kN) Angle of loading to the plane of the notch’s axis (degree) Angle of the second crack growth from away the first crack direction (degree) Length of the second inclined crack (mm) Cyclic numbers of the second crack initiation (cycle) Average crack growth rate of the second crack (mm/cycle)
1.7
1.7
1.7
1.7
1.7
1.7
1.7
1.7
1.7
1.7
15°
15°
⫺7.5°
⫺7.5°
⫺7.5°
0°
0°
7.5°
7.5°
7.5°
57°
52°
⫺55°
⫺53°
⫺56°
Amplitude of loading (kN) Angle of loading to the plane of the notch’s axis (degree) Cyclic numbers of the third crack initiation (cycle) Average crack growth rate of the third crack (mm/cycle) Positions of the cracks arising from the second crack Position of the longest crack arising from the second crack Length of the longest crack (mm)
9.6×105
5.7×105
9.1×105
6.6×105
10.8×105
1.30
1.06
1.28
1.04
2.20
1.5×105
1.9×105
1.7×105
2.0×105
2.4×105
3.7×10⫺7 4.6×10⫺7 4.4×10⫺7 4.4×10⫺7 4.9×10⫺7
of the angle of loading, and the angle of inclined crack also increased with the angle of loading. Table 2 gives the experimental data: the amplitude of loading, the angle of inclined crack with respect to the angle of loading, the length of the first crack, the number of cycles to crack initiation, as well as the average crack growth rate for the first 1 mm of growth from the notch root. 3.2. Numerical simulation with the maximum tangential stress criterion From Fig. 1, one can assume that the stress states of the notched ring can be described as a two-dimensional domain occupied by a homogeneous body. Since the
1.6×10⫺7 1.2×10⫺7 0.7×10⫺7 1.5×10⫺7 0.8×10⫺7
middle middle middle part, tip part and part and knee knee
tip and knee
middle part
knee
knee
middle part
tip
middle part
1.38
1.96
0.72
1.44
1.32
applied loading does not change across the specimen thickness and parallels to the ring’s surface, the deformation field is assumed to be planar and the plane strain condition is applied. In consideration of the non-singular stress field at the blunt notch root, it is convenient to use the maximum tangential stress criterion for identifying a crack growth from the notch root. It is based on a hypothesis that crack growth will occur at the notch boundary in a radial direction along which the tensile stress is a maximum. To realize the identification, a numerical evaluation has been carried out by a finite element method. The finite element mesh is shown in Fig. 2 and the mesh around the notch root, consisting of isoparametric eight-
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461
Fig. 2. Finite element mesh of the specimen: (a) mesh of total configuration and (b) dense mesh at the notch root.
node quadrilateral elements, is much denser than the other elements in order to obtain the necessary accuracy. The elements around the notch root are specially arranged: the two opposite sides of an element perpendicular to the tangent line of the arched boundary. Therefore, the tangential stress can be easy to obtain at the arched boundary of the notch. The direction of the first crack growth, q1, is determined by the point at the notch root where the tangential stress sqq is maximum: ∂sqq ∂2sqq | ⱕ0 |q=q1⫽0 and ∂q ∂q2 q=q1
(1)
The maximum value of stress sqq at the arched boundary is shown in Fig. 3a with different angles of loading. It is found that the maximum tangential stress has the highest value at j=0° and it decreases with the increment of the angle of loading. This tendency explains the experimental results that crack initiation from notch root is easier with a smaller angle of loading under the same amplitude of applied loading. When the angle of loading is equal to zero degrees, the maximum tangential stress corresponds to the angle of branched crack q1=0°. When the angle of loading is different from zero degrees, the larger the angle of loading is, the larger the angle of branched crack will be. The relationship between the angle of loading and the angle of branched crack is given in Fig. 3b. It is noted that the results of these calculations and experiments are comparable and the differences between them are less than 15%.
4. Inclined crack growth from parent crack 4.1. Experimental data of the second crack growth Tests were performed on two specimens, numbered as A01 and A02, which had been subjected to the symmetrical loading of amplitude of 1.5 kN and had possessed a 1.02 and 1.34 mm straight crack, respectively. The cyclic compressive loading with an amplitude of 1.7
Fig. 3. The calculated results of the crack initiating from the notch root: (a) the maximum tangential stress versus the inclined angle of loading under the same amplitude load and (b) the relationship between the direction of branched crack and the inclined angle of loading.
kN applied on the external ring of the specimens with an inclined angle of loading of 15° to the plane of the notch’s axis (turning the specimen counter clockwise). In this experiment, the change of loading direction gave rise to an arrest of crack growth along the original direction. It was necessary to use 1.5苲1.9×105 cycles for initiating a branched crack from the tip of the first crack. The second crack propagated along a direction that was approximately inclined at 52苲57° towards the direction of the first crack and the average crack growth rate was about 3.7苲4.6×10⫺7 mm/cycle. Testing was continued on three specimens, numbered as A04, A05 and A06, that had been subjected to the 7.5° inclined loading with the amplitude of 1.7 kN and had possessed a 1.28, 1.48 and 1.12 mm inclined crack, respectively. To obtain the second inclined crack, a loading of 1.7 kN amplitude was applied on the specimens with an inclined angle of loading ⫺7.5° to the plane of
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the notch’s axis. It was found that 1.7苲2.4×105 cycles were needed for initiating a branched crack from the tip of the first crack. The second crack propagated along a direction that was approximately inclined at 53苲56° towards the direction of the first crack and the average crack growth rate was about 4.4苲4.9×10⫺7 mm/cycle. Table 3 gives the experimental data of the second crack growth behavior. The length and the inclined angle of the second crack, the number of cycles to the second crack initiation and the average crack growth rate are listed. 4.2. Experimental data of the third crack growth In the fourth step, the specimens (numbered as A01, A02, A04, A05 and A06) that had both first and second cracks were continuously subjected to the same loading as at the second or third step, namely the amplitude of loading was 1.7 kN. However, the direction of applied loading came back to the original direction, that is the notch’s axis (j=0°) for the specimens A01 and A02 and the angle of 7.5° to the notch’s axis for the specimens A04, A05 and A06. The change of loading direction again created an arrest of crack growth. It was more difficult than the second or third step to initiate a branched crack, and a longer period of 5.7×105苲10.8×105 cycles was needed. The average crack growth rate obviously slowed down during this step. The position of initiation was not always situated at the tip of the second crack. The experimental data are listed in Table 4. In the case of specimen A01, cracks initiated in succession from the tip, middle and knee of the second crack. The crack that arose from the tip of the second crack propagated the shortest period and then stopped. The crack arising from the knee of the second crack continued propagating for the longest period. In the case of specimen A02, no crack arose from the tip of the second crack. However, there were three cracks arising from the middle part of the second crack. The longest crack also appeared from the knee of the second crack and did not stop propagating. In the case of specimen A05, cracks initiated simultaneously from both tip and knee of the second crack. The crack appearing from the tip propagated a very short period and then stopped. The crack from the knee of the second crack continued propagating. In the case of specimens A04 and A06, there was only one branched crack that initiated from the middle part of the second crack, respectively. In most cases, the angles of branched crack were approximately 65苲70° towards the second crack direction. It appeared that the branched crack growth rate was higher for the crack growing from the tip and knee of the former crack than that growing from the middle part of the former one. Fig. 4 shows schematically the crack paths of the experiments. An optical microscope observation was performed to
Fig. 4. Schematic drawing of the crack paths for the specimens having prossessed branched cracks (A01, A02 and A04-A06).
examine the fatigue-fracture behavior. Figs. 5 and 6 show the microstructure of the material and the branched crack paths of specimens A01 and A06. The bigger figure shows the total branched cracks and the two smaller figures give the microstructure in detail of parts A and B indicated in the bigger figure. It is noted that all crack growth is typically transgranular mode. Many short cracks may be found on the second cracks, it appears that cracks may initiate at any position of the former crack at the last step.
5. Crack growth simulation with the energy release rate method A numeral simulation of the first crack that initiates from the notch root has been carried out by finite element analysis with the maximum tangential stress criterion, described in Section 3.2. Here, we studied by numerical method the behavior of the second crack that initiated from the first crack tip. It is noted that a more complicated mixed-mode stress field arises in this case than that in the case of the first crack growth. Therefore, we introduced the energy release rate method described by virtual crack extension and used it to the inclined crack extension problem. 5.1. The energy release rate calculated by virtual crack extension method In order to simulate a crack extending away from its original direction by changing the direction of external loading, the energy release rate method can be used [18]. It is assumed that an original crack in length a turns sharply at some angle q to its original direction and extends an increment da, then the energy release rate G is defined as U1−U2 da→0 da
G⫽ lim
(2)
where U1 is the total strain energy of the body with the
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Fig. 5.
The microstructure of the material and the branched crack path of specimen A01.
Fig. 6.
The microstructure of the material and the branched crack path of specimen A06.
original crack a and U2 is the total strain energy of the body with the original crack a plus a small extension da. In a linear elastic system, we use the stiffness derivative procedure to appropriate the finite element analysis. The potential energy ⌸ of a body with a crack a is given by 1 ⌸⫽ uT Ku⫺uT F 2
(3)
where u, K and F are the global displacement, stiffness and force matrices, respectively. The energy release rate can be expressed as
G⫽⫺
冉
463
冊
∂F 1 ∂uT 1 ∂K ∂uT 1 ∂⌸ ⫽⫺ Ku⫹ uT u⫺ F⫺uT B ∂a B ∂a 2 ∂a ∂a ∂a
(4)
where B is the thickness of the body. If supposing that the external force is nearly constant and F=Ku, then G⫽⫺
1 ∂⌸ 1 ∂K 1 dK ⫽⫺ uT u⬇⫺ uT u B ∂a 2B ∂a 2B da
(5)
In actual calculation, da is taken as very small but finite and displacement u is calculated for a crack configuration with the original crack a plus a small extension da.
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dK is the difference between the stiffness matrices before and after the crack extension. With the energy release rate method, the crack could extend towards any direction, called virtual crack extension. It is well known that the actual crack extension is one of the virtual crack extensions and its direction is that along which the energy release rate achieves a maximum value. Therefore, the direction of an actual crack extension, indicated by the angle of branched crack q2, can be obtained by following formula ∂G ∂2 G |q=q1⫽0 and 2 |q=q1ⱕ0 ∂q ∂q
(6)
5.2. Results of numerical simulation In order to obtain the actual crack extension direction, the energy release rate of the virtual crack extension should be calculated in all divided directions with sufficiently small intervals. We used eight-node collapsed quadrilateral elements in the crack extension region to realize more dense mesh. Fig. 7 describes the meshing technique: among the figures, Fig. 7(a) gives the mesh only with an original crack (the first crack in our case) before the virtual crack extension; Fig. 7(b) shows the mesh that has a virtual crack extension in a direction of ⫺30°; Fig. 7(c) is an enlarged image of the virtual crack extension area in Fig. 7(b); Figs. 7(d), (e) and (f) indicate the deformed mesh. A calculated result of Eq. (6) is shown in Fig. 8 by indicating the crack growing direction (in angle), along which the energy release rate is a maximum, versus the inclined angle of loading for the specimen having the first crack. The superior curve gives the state that the original crack (the first crack) has an angle q1=0° to the notch’s axis, and the lower curve gives the state that the
Fig. 7. Mesh technique for calculating the energy release rate of virtual crack extension.
Fig. 8. The crack growth direction versus the inclined angle of loading for the specimen having an angle of crack q1=0° and q1=25°, respectively.
original crack has an angle q1=25° to the notch’s axis. It is found that the calculated results of the angle of branched crack are larger than that of our experiments and the difference between them is about 12苲15%. It has been studied that for a cracked ring under compression, the stress intensity factor is almost constant for a wide range of crack length [14,19]. So it is reasonable in our case that the crack growth rate for each step is quasi-constant. The differences of the crack growth rates between the steps may be the influence of the former branched cracks and the roughness-induced closure effect [20, p. 254–63]. Such investigation in detail is beyond the scope of the present study.
6. Conclusions In this paper, mixed mode (combing mode I and II) crack initiation and propagation under far-field uniaxial loading for an austenitic 304L steel have been investigated by using a special type of notched ring specimen. The arrest and branch of cracks are analyzed with respect to the direction of the inclined loading. Some conclusions can be stated as follows: 1. The orientation of crack initiation and propagation is the function of the inclined angle of loading. When the far-field cyclic loading changes an angle during the test, the fatigue crack growth appear to arrest, and a great number of cycles are necessary to initiate a new crack from the former crack. The direction of the new crack propagation is determined by mixed mode conditions. 2. The behavior of the crack initiating from the notch root under mixed-mode conditions for the tested material can be identified well by the maximum tangential stress criterion. The energy release rate cri-
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terion has been used to simulate the path of the branched crack initiating from the first crack tip. There is about a 12苲15% difference between the crack growth directions obtained by experiment and calculation. 3. The third branched cracks can initiate at any position of the second crack when the direction of external loading changes and more than one macrocrack growing from the second crack may be found. 4. Optical microscope observation was performed to examine the microstructure of the material and the branched crack paths of specimens. It is noted that all crack growth is typically transgranular mode. Therefore, the simulation of fatigue-fracture behavior by elasticity theory is available.
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Acknowledgements We are grateful for the financial support provided by the Special Funds for the Major State Basic Research Projects G19990650.
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[17] [18] [19] [20]
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