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Stress intensity factor expression for center-cracked butt joint considering the effect of joint shape
Materials and Design 35 (2012) 72–79
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Stress intensity factor expression for center-cracked butt joint considering the effect of joint shape T. Wang ⇑, J.G. Yang, X.S. Liu, Z.B. Dong, H.Y. Fang State Key Laboratory of Advanced Welding and Joining, Harbin Institute of Technology, Harbin 150001, China
a r t i c l e
i n f o
Article history: Received 4 June 2011 Accepted 8 September 2011 Available online 16 September 2011 Keywords: H. Failure analysis F. Defects E. Fatigue
a b s t r a c t In this paper, the stress intensity factor (SIF) expression for center-cracked butt joint under tensile load was derived and the SIF differences with or without considering weld reinforcement were discussed. For the sake of discussion, the curve of weld reinforcement is regarded as circle, the reinforcement circle and the other two weld toe smooth transition circles are assumed to be tangential. The influences of such joint shape parameters as weld width, weld toe smooth transition radius and base metal thickness on SIF were also studied. The relationship between SIF for center-cracked butt joint and its shape parameters was finally obtained based on the integration of analytical and finite element method. It showed that it is wrong to perform failure analysis on center-cracked butt joint without considering weld reinforcement, the derivation method of SIF developed here can be use to design the shape of the butt joint with defects under static loads or fatigue condition. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Various welding processes, such as arc welding, electron beam welding, friction stir welding (FSW), have been developed and widely used for many years in numerous industrial applications. During the welding process, weld defects may be formed in the joints. For instance, incomplete penetration has been found during arc welding tests on stainless steels [1]. When electron beam welding is performed in a vacuum, pores and hot cracks can be formed in the joint due to inappropriate welding conditions and the metallurgical state of materials [2,3]. Various weld defects can be also formed in the FSW joint if inappropriate primary FSW process parameters are applied [4]. The appropriate welding process parameters are generally considered as the key factors to avoid weld defects, but the suitable weld material also plays a significant role to avoid weld defects while for some structure materials or function materials. Although the weldability of them are not good and they are prone to weld defects, they need to joint to each other. As one of such materials, high strength steel is commonly used in engineering applications. However, there are several problems in welding joints of high strength steel. One of the major problems is the cold crack [5]. The problem of weld cold crack can be prevented if lower strength filler metal with higher plasticity and ductility is selected for welding such kind of high strength steel [6]. ⇑ Corresponding author. Tel.: +86 451 86418422; fax: +86 451 86416186. E-mail addresses: [email protected] (T. Wang), [email protected] (J.G. Yang). 0261-3069/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2011.09.020
It has been found that weld defects can significantly reduce such structural properties as tensile strength and toughness, and thus deteriorates the load-carrying capacity of joints [7,8]. Typical brittle fracture is often observed in the defect-existent welded joints [9,10]. To avoid weld defects and improve the load-carrying capacity of joints, suitable weld materials and appropriate welding parameters conditions should be selected [11,12]. Fracture mechanics can be used to deal with the problem of defect-existent welded structures. In engineering application of fracture mechanics, the stress intensity factor (SIF) is an important mechanical parameter which can be used to evaluate safety and calculate the fatigue remaining life of structures with defects. It is therefore very important to obtain the expression of SIF. There are three main methods to obtain SIF: analytical, numerical and experiment standardization method. Analytical method is used mainly in infinite plate and regular shape plate [13]. Numerical method can obtain SIF in almost all the cases, especially the finite element method, it has been widely applied in engineering fields [14–16]. Experiment standardization method is reliable but it is inefficient and relative expensive [17–19]. Crack is the most dangerous defect in welded joints, and the crack tip is the stress concentration point which will be the fracture source of the structure when the structure is under load. Center crack always occurs in welded joints [20], and butt joint is a common joint form in welded structures [21,22]. In this paper, the expression of SIF for centercracked butt joint was obtained through combining analytical and finite element method. It is well known that weld reinforcement is usually ignored in the calculation of SIF. However this shape parameter of the welded
T. Wang et al. / Materials and Design 35 (2012) 72–79
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joint may have some significant effects on SIF [23,24]. To clarify the influence of weld reinforcement on SIF, SIF with or without considering weld reinforcement were calculated and studied in the present investigation. The influences of other shape parameters such as weld width, weld toe smooth transition radius on SIF were also discussed based on the given expression of SIF. Fig. 2. Shape parameters of modified butt joint.
2. Basic concept and theory There are three fracture modes, i.e. mode I, II and III. The SIF of each fracture mode has a universal expression, which can be described as [25],
pffiffiffi K ¼ Yr a
ð1Þ
The expression includes the influence of work stress r, crack dimension a and shape factor Y, which depends on structure shape parameters, loading mode and crack location, respectively. The structure failure would occur when K reaches a critical value. The critical value is designated as KIC, a material property named fracture toughness. Therefore, the smaller the K is, the safer the structure is. The mode I is the most critical fracture mode, its SIF can be described as,
pffiffiffi KI ¼ Yr a
ð2Þ
where r is work stress, a is crack dimension, Y is shape factor. So far, joint with no reinforcement is always used to calculate the SIF for normal butt joint [26,27]. As we know, normal butt joint has at least two indispensable shape parameters, i.e. weld reinforcement h and weld width 2w, as shown in Fig. 1. Obviously, it is unscientific to ignore the influence of weld reinforcement on SIF. As a matter of fact, weld reinforcement will reduce the fatigue strength of normal joint. Some researches showed that shape optimization methods such as smooth transition of weld toe can improve the fatigue property of joint [28,29]. Therefore, the fatigue property of normal butt joint can be improved through the addition of shape parameter, weld toe smooth transition radius r. The modified butt joint is shown in Fig. 2. The modified center-cracked butt joint under tensile load, as illustrated in Fig. 3, is symmetrical and has four shape parameters: weld reinforcement h, weld width 2w, weld toe smooth transition radius r and base metal thickness 2t. The center crack length is 2a. The shaded area represents the weld deposited metal. r can be obtained by machining or any other ways. For the sake of discussion, the curve of weld reinforcement is regarded as circle, the reinforcement circle and the other two weld toe smooth transition circles are assumed to be tangential. 3. SIF expression for center-cracked butt joint and its application 3.1. SIF expression for center-cracked butt joint The SIF K is a factor which varies as a function of crack location and dimension, loading mode and magnitude, and the structure
Fig. 3. Shape parameters of center-cracked modified butt joint under tensile load.
shape parameters as well. Thus, the expression of K will vary from one condition to another. The calculation expressions of K for various conditions have been calculated and are available in some handbooks [30,31]. If a long-distance tensile stress r is exerted on a center crack with length of 2a and a plate width of b, its SIF calculation expression can be expressed as [25],
pffiffiffiffiffiffi 1 K I ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a2 r pa 1 b
Therefore, for a center-cracked butt joint (Fig. 3) with complicated shape and is subjected to a tensile stress r far away from the crack normal, its analytic expression of SIF can be described as Eq. (4) where weld reinforcement h is not considered [23,24]. The SIF value calculated by Eq. (4) must be larger than the actual value. But if weld reinforcement h is considered completely, the calculation expression of SIF for center-cracked butt joint showed in Fig. 3 can be described as Eq. (5). The SIF value calculated by Eq. (5) must be smaller than the actual value.
pffiffiffiffiffiffi pffiffiffiffiffiffi 1 t ffi r pa K I ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a2 r pa ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 a t 1 2t
ð4Þ
pffiffiffiffiffiffi pffiffiffiffiffiffi tþh t t K I ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r pa ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r pa t þ h 2 2 ðt þ hÞ a2 ðt þ hÞ a2
ð5Þ
where h is weld reinforcement, t is half of the base metal thickness, a is half of the center crack length. It is well known that joint shape parameters can affect SIF. If joint shape parameters, i.e. weld reinforcement h, weld width 2w, weld toe smooth transition radius r and base metal thickness 2t are all considered, the calculation expression of SIF for centercracked butt joint showed in Fig. 3 should be the modifying Eq. (5) with a modified coefficient function. Thus, the calculation expression of SIF for center-cracked butt joint showed in Fig. 3 should be
pffiffiffiffiffiffi t K I ¼ f ðh; w; r; tÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r pa ðt þ hÞ2 a2
Fig. 1. Shape parameters of normal butt joint.
ð3Þ
ð6Þ
where f(h,w,r,t) is the modified coefficient function, the influence of butt joint shape parameters on SIF. Thus, the SIF value calculated by Eq. (6) is the closest to the actual SIF value of center-cracked butt joint under tensile load.
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Fig. 4. Finite element model of 1/4 center-cracked butt joint.
The modified coefficient function f(h,w,r,t) is the key factor in order to obtain accurate calculation expression of SIF for centercracked butt joint. Apparently, it can be obtained using inverse analysis through finite element method and regression analysis. The process consists of four steps. Step (1): Computing J-integral of series of different joint shape by finite element method. Step (2): Converting the J-integral into corresponding SIF KI. Step (3): Calculating the modified coefficient f of each joint shape through Eq. (6). Step (4): Using regression analysis to obtain all the modified coefficient f of different joint shape. A detailed description of each step follows. Step (1): The basic geometry of center-cracked butt joint in this paper is shown in Fig. 3. The joint is subjected to a remote tensile load. Plane strain conditions are assumed because the joint length is much larger than the base metal thickness. The joint geometry, center crack and loading are symmetrical, so a quarter of finite element model shown in Fig. 4 is established. The analysis involves a two-dimensional finite element method by using the finite element software MSC.MARC. The J-integral value can be extracted from the result document. The joint shown in Fig. 4 has a thickness (2t) of 16 mm, a weld width (2w) of 32 mm, a center crack length (2a) of 4 mm, a reinforcement (h) of 2 mm and a weld toe smooth transition radius (r) of 32 mm. To get an accurate result, meshes near the crack tip are refined. The ratio of minimum mesh size near the tip to half of the crack length is only 0.005. As a matter of fact, a lot of center-cracked butt joints with different shapes should be analyzed to obtain the corresponding J-integral by FEM. Step (2): Under plane strain condition, the relationship of SIF and J-integral is [32]
KI ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi EJ 1 t2
(a) w/t=2
(b) w/t=4
ð7Þ
where E is the Young’s modulus, J is J-integral obtained in step (1), t is Poisson’s ratio. Step (3): Based on Eq. (6), the modified coefficient f can be obtained as follows,
f ¼
KI
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt þ hÞ2 a2 pffiffiffiffiffiffi tr pa
ð8Þ
Here, KI is the SIF value calculated in step (2), t is half of the base metal thickness, h is weld reinforcement, a is half of the center crack length, r is work stress. Fig. 5 shows the influence of butt joint shape parameters on modified coefficient f. r ranges from 0 mm to 32 mm, h in relation to t ranges from 0 to 1. Three different w/t, 2, 4, 6 are discussed. The modified coefficient f is obtained by Eq. (8). The analysis indicated that f increases with the increase of h and r, f decreases significantly with the increase of w when other shape parameters are fixed. Step (4): The modified coefficient function f(h,w,r,t) can be obtained by regression analysis of all the modified coefficient f which are calculated in step (3),
(c) w/t=6 Fig. 5. Influence of shape parameters on modified coefficient f: (a) w/t = 2, (b) w/ t = 4 and (c) w/t = 6.
2
3 w 2hwr 2 6 7 h 6 h þ t ðw2 þ h Þðh þ tÞ7 exp 6 f ðh; w; r; tÞ ¼ 1 þ A 7 4 5 hþt Bðht Þ2 þ C ht þ D
ð9Þ
where A, B, C and D are 1.77633, 0.49536, 0.66789 and 0.75115, respectively.
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T. Wang et al. / Materials and Design 35 (2012) 72–79
Substituting (9) into (6), the calculation expression of SIF for center-cracked butt joint can be expressed as follows:
8 39 2 w 2hwr > > > > > < = 6 h þ t ðw2 þ h2 Þðh þ tÞ7> h 7 6 exp 6 KI ¼ 1 þ A 7 2 > 5> 4 hþt Bðht Þ þ C ht þ D > > > > : ; pffiffiffiffiffiffi t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r pa 2 2 ðt þ hÞ a
ð10Þ
where A, B, C and D are 1.77633, 0.49536, 0.66789 and 0.75115, respectively. 3.2. Shape factor for center-cracked butt joint From Eq. (2), it can be seen that KI is only dependent on shape factor Y when r and a are known. In the mean time, Y is dependent on loading mode, crack location and joint shape parameters. For center-cracked butt joint under tensile load, the loading mode is tensile load and the crack location is in the center. So, Y is only dependent on joint shape parameters, which means KI is only dependent on the joint shape parameters when r and a are given. Obviously, the smaller the shape factor is, the safer the joint is. There must be certain joint shape parameters which can lead to the shape factor small. Combining Eqs. (2) and (6), the shape factor of center-cracked butt joint can be obtained as follows,
pffiffiffiffi t p Y ¼ f ðh; w; r; tÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt þ hÞ2 a2
(a) w/t=2
ð11Þ
where f(h,w,r,t) is the modified coefficient function in Eq. (9). Fig. 6 shows the influence of modified butt joint shape parameters on shape factor Y. Here, t is 10 mm, a is 2 mm, r ranges from 0 to 20 mm, h in relation to t ranges from 0 to 1. Three different w/t, 2, 4, 6 are discussed. Shape factor Y is obtained by Eq. (11). The results show that Y decreases with the increase of h and w. Y increases with the increase of r, especially when h is large or w is small. The results without considering weld reinforcement (h/t is 0) are larger than the ones with considering weld reinforcement. This means that calculating KI without considering weld reinforcement is conservative and unreasonable in actual engineering application. From Fig. 6c, it can also be seen that Y will be stable at a relatively small value when w and h are both large enough. So, center-cracked butt joint with large enough weld reinforcement and width, and also an appropriate weld toe smooth transition radius will have a high load-carrying capacity.
(b) w/t=4
3.3. Engineering application The critical crack length, critical stress and the fatigue remaining life of joint are the most concerned problems in engineering application [33,34]. For butt joint, as shown in Fig. 3, its critical crack length ac and critical stress rc can be obtained through its calculation expression of SIF showed in Eq. (10) respectively.
ac ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðpt 2 r2 f ðh; w; r; tÞ2 Þ2 þ 4K 4IC ðt þ hÞ2 pt2 r2 f ðh; w; r; tÞ2
(c) w/t=6
2K 2IC ð12Þ
rc
K IC ¼ f ðh; w; r; tÞt
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt þ hÞ2 a2 pa
ð13Þ
where KIC is fracture toughness of weld metal, f(h,w,r,t) is the modified coefficient function calculated by Eq. (9), h, w, r, t are butt joint shape parameters which are shown in Fig. 3. r in Eq. (12) is the
Fig. 6. Influence of shape parameters on shape factor Y of center-cracked butt joint: (a) w/t = 2, (b) w/t = 4 and (c) w/t = 6.
maximum work stress. a in Eq. (13) is half of center crack length which can be obtained through nondestructive examination. When butt joint showed in Fig. 3 endures under fatigue load, its remaining life Nf can be obtained by combination of Paris law and its calculation expression of SIF showed in Eq. (10),
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Nf ¼
T. Wang et al. / Materials and Design 35 (2012) 72–79
Z
ac
a0
0
da
1m
ð14Þ
pffiffiffiffiffiffiC t B A@f ðh; w; r; tÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dr paA ðt þ hÞ2 a2
where a0 is half of the initial center crack length which can be obtained through nondestructive examination, ac is the critical crack length which can be calculated by Eq. (12), Dr is cyclic stress, A and m are material properties. 4. Verification of SIF calculation expression for center-cracked butt joint The accuracy of SIF calculation expression for center-cracked butt joint is evaluated by both FEM analysis and static tensile experiment. Fatigue experiment of center-cracked butt joint with different shape is also tested. 4.1. FEM verification SIF values calculated by Eq. (10) are compared with that from FEM analysis result calculated by Eq. (7). Here, r is 100 MPa, a is 2 mm, t is 8 mm. Fig. 7 shows the comparison of K calculated by
(a) r=0
Eq. (10) and FEM analysis, where ‘formula’ represent K calculated by Eq. (10) and ‘simulation’ represent K obtained by FEM analysis. Fig. 7a shows the normal butt joint with no weld toe smooth transition radius(r = 0) and Fig. 7b shows the modified butt joint with weld toe smooth transition radius (r = 8 mm). The results demonstrate that SIF value calculated by Eq. (10) and the FEM analysis are in good agreement. So, it is proved that the calculation expression of SIF for center-cracked butt joint, showed in Eq. (10), can provide clear guidance for shape design of center-cracked butt joint. 4.2. Experimental verification Modified coefficient function f(h,w,r,t) is the determinant factor of Eqs. (6), (11)–(13), (and) (14). The calculation expression of KI, Y, rc, ac and Nf can be accurate only if f(h,w,r,t) is precise. The accuracy of f(h,w,r,t) can be verified by comparing expression with experiment result. The modified coefficient f of expression result can be obtained through substituting the shape parameters of each joint into Eq. (9). From Eq. (13), modified coefficient f can be obtained according to the following equation,
K IC f ðh; w; r; tÞ ¼ t rc
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt þ hÞ2 a2 pa
ð15Þ
So, the modified coefficient f of experiment result can be obtained through substituting the fracture toughness KIC of weld metal, half of the base metal thickness t, critical stress of tensile test rc, weld reinforcement h and initial crack length a into Eq. (15). Three types of center-cracked butt joints were tested (see Fig. 8) by utilizing the material testing machine (CSS-44300). Mechanical properties of joint material are shown in Table 1. Fracture toughness KIC of weld metal is 34.8 MPa (m)1/2. The base metal thickness is 10 mm, which means t is 5 mm. X groove was first made on the plate, and then manual arc welding was adopted to produce the joints. All center cracks were obtained through wire–electrode cutting before fatigue pre-cracking to a same crack length, a0 is 2 mm. The comparison of expression results with experiment results are shown in Table 2. The comparison illustrates that all of the three types of butt joints show little differences between expression results with experiment results. It means that the calculation expression of SIF and its applications given in this paper could be
Fig. 8. Three types of center-cracked butt joints.
Table 1 Mechanical properties of joint material. Joint material
(b) r=8 mm Fig. 7. Comparison of SIF values calculated by expression calculation and FEM analysis: (a) r = 0 and (b) r = 8 mm.
Yield strength
rs (MPa)
Tensile strength
Elongation et (%)
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T. Wang et al. / Materials and Design 35 (2012) 72–79 Table 2 Comparison of expression results with experiment results. Specimen
h (mm)
w (mm)
r (mm)
rc (MPa)
f(h,w,r,t) by experiment
f(h,w,r,t) by FEM
Differences (%)
Normal butt joint Modified butt joint 1 Modified butt joint 2
1.5 1.5 2.5
6.5 15 15
0 12 12
456 498 550
1.1904 1.0900 1.1535
1.1361 1.0482 1.1268
4.78 3.99 2.37
Table 3 Fatigue test results of three types of joints. Specimen
h w r Average Alternating Fatigue (mm) (mm) (mm) load load (MPa) remaining (MPa) life (cycle)
Normal butt joint Modified butt joint 1 Modified butt joint 2
1.5
6.5
0
100
100
100,122
1.5
15
12
100
100
157,627
2.5
15
12
100
100
967,569
used to guide the shape design of center-cracked butt joint under tensile load. From the result, it can also be seen that appropriate shape design can improve the static load-carrying capacity of center-cracked butt joint.
(a) Normal butt joint
4.3. Fatigue experiment Fatigue experiments of these three types of joints (see Fig. 8) were tested by using high frequency fatigue testing machine (PLG-100C). The fatigue test results of three types of joints are shown in Table 3. It can be seen that fatigue remaining life of modified butt joint is much larger than that of the normal butt joint. Meanwhile, fatigue remaining life of modified butt joint with larger reinforcement (modified butt joint 2) is larger than that of modified butt joint 1. The corresponding SEM results of three types of center-cracked butt joints are shown in Fig. 9. All the SEM results include three different zones, i.e. wire cutting zone, pre-crack zone and fatigue propagation zone. For these three different joints, wire cutting zone and pre-crack zone are the same. Comparing the fatigue propagation zone of the three joints, it can be seen that the roughness of fracture surface follows such a sequence that normal butt joint > modified butt joint 1 > modified butt joint 2. The more serious the fracture surface roughness is, the larger the fatigue propagation rate is. Therefore, the fatigue life of modified joint 2 is the largest among the three joints. So, appropriate joint shape design can also improve the fatigue load-carrying capacity of centercracked butt joint.
(b) Modified butt joint 1
5. Discussion When using the fracture mechanics approach, an appropriate SIF expression must be proposed. A number of formulas for center-cracked butt joint under tensile load can be found in the SIF handbooks [25,30,31]. However, in these formulas, the joint reinforcement was not taken into account. It is known that SIF has been demonstrated to be strongly related to the joint shape parameters [23,24]. Meanwhile, it is not possible to solve the SIF analytically for joint with complex shape. For these reasons mentioned above, the SIF expression for center-cracked butt joint must be further amended by a modified coefficient function. Tedious numerical computations are needed in order to generate an accurate modified coefficient function. By adopting the ordinary SIF expression and combining with suitable modified coefficient function, the SIF expression for center-cracked butt joint under tensile load proposed in this research is extendable to normal center-cracked butt
(c) Modified butt joint 2 Fig. 9. SEM results of three types of center-cracked butt joints: (a) normal butt joint, (b) modified butt joint 1 and (c) modified butt joint 2.
joint with certain reinforcement, thus the expression is obtained and it is adopted to guide engineering applications and joint shape designs. Eq. (4) shows the original SIF expression for center-cracked butt joint under tensile load. The joint reinforcement is not considered here. When it comes to the flush center-cracked butt joint, its SIF can be calculated by the Eq. (4). Nevertheless, when joint has certain reinforcement, it is not appropriate to calculate SIF by using the Eq. (4). In the proposed SIF expression (Eq. (10)), the SIF correlates with all the joint geometries (reinforcement h, half of the weld width w, weld toe smooth transition radius r and half of
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the base metal thickness t), crack dimension (half of the center crack length a) and work stress r. In Section 4, it can be seen that the proposed SIF expression with considering the effect of joint shape in this paper is more accurate than the ordinary one. In addition, it works well for both engineering applications and joint shape designs. The calculation expression of SIF for center-cracked butt joint described in Eq. (10) is aimed at central through crack. When it comes to central part through crack, its crack length can be converted to an equivalent length on central through crack according to different conditions. Because the residual stress here is transversal stress which is usually much smaller than the material yield strength, welding residual stress is not considered to calculate the SIF in this paper. It can be treated as an extra work stress when calculating the SIF of welded joint. Isotropic material is used in the FEM analysis in this paper. Actually, the welded joint should be treated as anisotropic material. In the case of anisotropic material, the calculation expression of SIF is ought to be modified. When welded joint works under low stress which belongs to linear elastic fracture mechanics (LEFM) situation, its SIF values calculated by anisotropic material and isotropic material are almost the same [25]. But there are some limitations of LEFM because it assumes that no plastic deformation exists in the structure. As a matter of fact, the material near the crack tip would deform plastically. However, a correction term for the presence of the crack tip plastic zone can be incorporated when the size of the plastic zone is small with respect to the crack length. In regard to the plane stress situation, the approximate plasticzone radius will be [25]
ry ¼
1 2p
K
2 ð16Þ
ry
While for the plane strain situation, the approximate plasticzone radius will be
ry ¼
1 6p
K
2
ry
ð17Þ
The effective crack length can be expressed as
2ðaÞeff ¼ 2ða þ r y Þ
ð18Þ
Substituting (a + ry) for a in Eq. (10) gives an adequate adjustment for low work stress situations. With this adjustment, the calculation expression of SIF showed in Eq. (10) is more reasonable and more widely used for center-cracked butt joint under tensile load. 6. Conclusions (1) The stress intensity factor expression for center-cracked butt joint under tensile load is revised by adding a modified coefficient function to common expression. The analytical method, finite element method and regression analysis are applied to obtain the modified coefficient function. (2) The accurate stress intensity factor expression for centercracked butt joint under tensile load is given to guide its shape design under static load and fatigue load conditions. (3) It is unreasonable to calculate the stress intensity factor for center-cracked butt joint under tensile load without considering weld reinforcement. The stress intensity factor decreases with the increase of weld reinforcement or weld width, especially the weld reinforcement. The stress intensity factor will be stable when weld reinforcement and weld width are large enough. The influence of weld toe smooth
transition radius can be ignored when calculating the stress intensity factor. (4) The applications of stress intensity factor expression for center-cracked butt joint under tensile load are discussed such as acquiring the shape factor, critical crack length, critical stress and fatigue remaining life of the joint. (5) Modified center-cracked butt joint with certain weld reinforcement, weld width and also an appropriate weld toe smooth transition radius has greatly improved static and fatigue load-carrying capacity than that of normal butt joint. It is much meaningful to improve the load-carrying capacity of center-cracked butt joint by choosing appropriate joint shape parameters.
Acknowledgment The authors are grateful to be supported by Sino-Russian Intergovernmental Cooperation Projects (2007DFR70070). References [1] Mirapeix J, Garcia-Allende PB, Cobo A, Conde OM, Lopez-Higuera JM. Real-time arc-welding defect detection and classification with principal component analysis and artificial neural networks. NDT&E Int 2007;40:315–23. [2] Fujii H, Umakoshi H, Aoki Y, Nogi K. Bubble formation in aluminium alloy during electron beam welding. J Mater Process Technol 2004;155(156): 1252–5. [3] Durocher A, Ayrault D, Chagnot Ch, Lipa M, Saikaly W. CuCrZr alloy hot cracking during electron beam welding. J Nucl Mater 2007;367(370):1208–12. [4] Kim YG, Fujii H, Tsumura Y, Komazaki T, Nakata K. Three defect types in friction stir welding of aluminum die casting alloy. Mat Sci Eng A-Struct 2006;415:250–4. [5] Lee HW, Choe WH, Park JU, Kang CY, Park WJ. Weld metal hydrogen assisted cracking in 50 mm TMCP steel plate with SAW process. Sci Technol Weld Joi 2006;11:243–9. [6] Hao S, Schwalbe K-H, Cornec A. The effect of yield strength mis-match on the fracture analysis of welded joints: slip-line field solutions for pure bending. Int J Solids Struct 2000;37:5385–411. [7] Li B, Shen YF, Hu WY. The study on defects in aluminum 2219-T6 thick butt friction stir welds with the application of multiple non-destructive testing methods. Mater Des 2011;32:2073–84. [8] Dubourg L, Merati A, Jahazi M. Process optimisation and mechanical properties of friction stir lap welds of 7075-T6 stringers on 2024-T3 skin. Mater Des 2010;31:3324–30. [9] Hu LH, Huang J, Li ZG, Wu YX. Effects of preheating temperature on cold cracks, microstructures and properties of high power laser hybrid welded 10Ni3CrMoV steel. Mater Des 2011;32:1931–9. [10] Ayatollahi MR, Torabi AR. Brittle fracture in rounded-tip V-shaped notches. Mater Des 2010;31:60–7. [11] Edwards KL. Selecting materials for optimum use in engineering components. Mater Des 2005;26:469–73. [12] Edwards KL, Deng YM. Supporting design decision-making when applying materials in combination. Mater Des 2007;28:1288–97. [13] Wang JG, Ju DY, Sun MJ, Li SL. A new analytical method for stress intensity factors based on in situ measurement of crack deformation under biaxial tension. Mater Des 2011;32:664–70. [14] Bouiadjra BB, Achour T, Berrahou M, Ouinas D, Feaugas X. Numerical estimation of the mass gain between double symmetric and single bonded composite repairs in aircraft structures. Mater Des 2010;31:3073–7. [15] Ouinas D, Hebbar A, Bouiadjra BB, Benderdouche N, Serier B. Interaction effect crack-interfacial crack using finite element method. Mater Des 2010;31:375–81. [16] Shahani AR, Amini Fasakhodi MR. Finite element analysis of dynamic crack propagation using remeshing technique. Mater Des 2009;30:1032–41. [17] Konsta-Gdoutos M. Limitations in mixed-mode stress intensity factor evaluation by method of caustics. Eng Fract Mech 1996;55:371–82. [18] Newman Jr JC. Stress-intensity factors and crack-opening displacements for round compact specimens. Int J Fract 1981;17:567–78. [19] Sarangi H, Murthy KSRK, Chakraborty D. Radial locations of strain gages for accurate measurement of mode I stress intensity factor. Mater Des 2010;31:2840–50. [20] Liu HJ, Fujii H, Maeda M, Nogi K. Tensile properties and fracture locations of friction-stir-welded joints of 2017-T351 aluminum alloy. J Mater Process Techol 2003;142:692–6. [21] Sharifitabar M, Halvaee A, Khorshahian S. Microstructure and mechanical properties of resistance upset butt welded 304 austenitic stainless steel joints. Mater Des 2011;32:3854–64. [22] Xu WF, Liu JH, Zhu HQ. Analysis of residual stresses in thick aluminum friction stir welded butt joints. Mater Des 2011;32:2000–5.
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Materials and Design journal homepage: www.elsevier.com/locate/matdes
Stress intensity factor expression for center-cracked butt joint considering the effect of joint shape T. Wang ⇑, J.G. Yang, X.S. Liu, Z.B. Dong, H.Y. Fang State Key Laboratory of Advanced Welding and Joining, Harbin Institute of Technology, Harbin 150001, China
a r t i c l e
i n f o
Article history: Received 4 June 2011 Accepted 8 September 2011 Available online 16 September 2011 Keywords: H. Failure analysis F. Defects E. Fatigue
a b s t r a c t In this paper, the stress intensity factor (SIF) expression for center-cracked butt joint under tensile load was derived and the SIF differences with or without considering weld reinforcement were discussed. For the sake of discussion, the curve of weld reinforcement is regarded as circle, the reinforcement circle and the other two weld toe smooth transition circles are assumed to be tangential. The influences of such joint shape parameters as weld width, weld toe smooth transition radius and base metal thickness on SIF were also studied. The relationship between SIF for center-cracked butt joint and its shape parameters was finally obtained based on the integration of analytical and finite element method. It showed that it is wrong to perform failure analysis on center-cracked butt joint without considering weld reinforcement, the derivation method of SIF developed here can be use to design the shape of the butt joint with defects under static loads or fatigue condition. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Various welding processes, such as arc welding, electron beam welding, friction stir welding (FSW), have been developed and widely used for many years in numerous industrial applications. During the welding process, weld defects may be formed in the joints. For instance, incomplete penetration has been found during arc welding tests on stainless steels [1]. When electron beam welding is performed in a vacuum, pores and hot cracks can be formed in the joint due to inappropriate welding conditions and the metallurgical state of materials [2,3]. Various weld defects can be also formed in the FSW joint if inappropriate primary FSW process parameters are applied [4]. The appropriate welding process parameters are generally considered as the key factors to avoid weld defects, but the suitable weld material also plays a significant role to avoid weld defects while for some structure materials or function materials. Although the weldability of them are not good and they are prone to weld defects, they need to joint to each other. As one of such materials, high strength steel is commonly used in engineering applications. However, there are several problems in welding joints of high strength steel. One of the major problems is the cold crack [5]. The problem of weld cold crack can be prevented if lower strength filler metal with higher plasticity and ductility is selected for welding such kind of high strength steel [6]. ⇑ Corresponding author. Tel.: +86 451 86418422; fax: +86 451 86416186. E-mail addresses: [email protected] (T. Wang), [email protected] (J.G. Yang). 0261-3069/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2011.09.020
It has been found that weld defects can significantly reduce such structural properties as tensile strength and toughness, and thus deteriorates the load-carrying capacity of joints [7,8]. Typical brittle fracture is often observed in the defect-existent welded joints [9,10]. To avoid weld defects and improve the load-carrying capacity of joints, suitable weld materials and appropriate welding parameters conditions should be selected [11,12]. Fracture mechanics can be used to deal with the problem of defect-existent welded structures. In engineering application of fracture mechanics, the stress intensity factor (SIF) is an important mechanical parameter which can be used to evaluate safety and calculate the fatigue remaining life of structures with defects. It is therefore very important to obtain the expression of SIF. There are three main methods to obtain SIF: analytical, numerical and experiment standardization method. Analytical method is used mainly in infinite plate and regular shape plate [13]. Numerical method can obtain SIF in almost all the cases, especially the finite element method, it has been widely applied in engineering fields [14–16]. Experiment standardization method is reliable but it is inefficient and relative expensive [17–19]. Crack is the most dangerous defect in welded joints, and the crack tip is the stress concentration point which will be the fracture source of the structure when the structure is under load. Center crack always occurs in welded joints [20], and butt joint is a common joint form in welded structures [21,22]. In this paper, the expression of SIF for centercracked butt joint was obtained through combining analytical and finite element method. It is well known that weld reinforcement is usually ignored in the calculation of SIF. However this shape parameter of the welded
T. Wang et al. / Materials and Design 35 (2012) 72–79
73
joint may have some significant effects on SIF [23,24]. To clarify the influence of weld reinforcement on SIF, SIF with or without considering weld reinforcement were calculated and studied in the present investigation. The influences of other shape parameters such as weld width, weld toe smooth transition radius on SIF were also discussed based on the given expression of SIF. Fig. 2. Shape parameters of modified butt joint.
2. Basic concept and theory There are three fracture modes, i.e. mode I, II and III. The SIF of each fracture mode has a universal expression, which can be described as [25],
pffiffiffi K ¼ Yr a
ð1Þ
The expression includes the influence of work stress r, crack dimension a and shape factor Y, which depends on structure shape parameters, loading mode and crack location, respectively. The structure failure would occur when K reaches a critical value. The critical value is designated as KIC, a material property named fracture toughness. Therefore, the smaller the K is, the safer the structure is. The mode I is the most critical fracture mode, its SIF can be described as,
pffiffiffi KI ¼ Yr a
ð2Þ
where r is work stress, a is crack dimension, Y is shape factor. So far, joint with no reinforcement is always used to calculate the SIF for normal butt joint [26,27]. As we know, normal butt joint has at least two indispensable shape parameters, i.e. weld reinforcement h and weld width 2w, as shown in Fig. 1. Obviously, it is unscientific to ignore the influence of weld reinforcement on SIF. As a matter of fact, weld reinforcement will reduce the fatigue strength of normal joint. Some researches showed that shape optimization methods such as smooth transition of weld toe can improve the fatigue property of joint [28,29]. Therefore, the fatigue property of normal butt joint can be improved through the addition of shape parameter, weld toe smooth transition radius r. The modified butt joint is shown in Fig. 2. The modified center-cracked butt joint under tensile load, as illustrated in Fig. 3, is symmetrical and has four shape parameters: weld reinforcement h, weld width 2w, weld toe smooth transition radius r and base metal thickness 2t. The center crack length is 2a. The shaded area represents the weld deposited metal. r can be obtained by machining or any other ways. For the sake of discussion, the curve of weld reinforcement is regarded as circle, the reinforcement circle and the other two weld toe smooth transition circles are assumed to be tangential. 3. SIF expression for center-cracked butt joint and its application 3.1. SIF expression for center-cracked butt joint The SIF K is a factor which varies as a function of crack location and dimension, loading mode and magnitude, and the structure
Fig. 3. Shape parameters of center-cracked modified butt joint under tensile load.
shape parameters as well. Thus, the expression of K will vary from one condition to another. The calculation expressions of K for various conditions have been calculated and are available in some handbooks [30,31]. If a long-distance tensile stress r is exerted on a center crack with length of 2a and a plate width of b, its SIF calculation expression can be expressed as [25],
pffiffiffiffiffiffi 1 K I ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a2 r pa 1 b
Therefore, for a center-cracked butt joint (Fig. 3) with complicated shape and is subjected to a tensile stress r far away from the crack normal, its analytic expression of SIF can be described as Eq. (4) where weld reinforcement h is not considered [23,24]. The SIF value calculated by Eq. (4) must be larger than the actual value. But if weld reinforcement h is considered completely, the calculation expression of SIF for center-cracked butt joint showed in Fig. 3 can be described as Eq. (5). The SIF value calculated by Eq. (5) must be smaller than the actual value.
pffiffiffiffiffiffi pffiffiffiffiffiffi 1 t ffi r pa K I ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a2 r pa ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 a t 1 2t
ð4Þ
pffiffiffiffiffiffi pffiffiffiffiffiffi tþh t t K I ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r pa ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r pa t þ h 2 2 ðt þ hÞ a2 ðt þ hÞ a2
ð5Þ
where h is weld reinforcement, t is half of the base metal thickness, a is half of the center crack length. It is well known that joint shape parameters can affect SIF. If joint shape parameters, i.e. weld reinforcement h, weld width 2w, weld toe smooth transition radius r and base metal thickness 2t are all considered, the calculation expression of SIF for centercracked butt joint showed in Fig. 3 should be the modifying Eq. (5) with a modified coefficient function. Thus, the calculation expression of SIF for center-cracked butt joint showed in Fig. 3 should be
pffiffiffiffiffiffi t K I ¼ f ðh; w; r; tÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r pa ðt þ hÞ2 a2
Fig. 1. Shape parameters of normal butt joint.
ð3Þ
ð6Þ
where f(h,w,r,t) is the modified coefficient function, the influence of butt joint shape parameters on SIF. Thus, the SIF value calculated by Eq. (6) is the closest to the actual SIF value of center-cracked butt joint under tensile load.
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Fig. 4. Finite element model of 1/4 center-cracked butt joint.
The modified coefficient function f(h,w,r,t) is the key factor in order to obtain accurate calculation expression of SIF for centercracked butt joint. Apparently, it can be obtained using inverse analysis through finite element method and regression analysis. The process consists of four steps. Step (1): Computing J-integral of series of different joint shape by finite element method. Step (2): Converting the J-integral into corresponding SIF KI. Step (3): Calculating the modified coefficient f of each joint shape through Eq. (6). Step (4): Using regression analysis to obtain all the modified coefficient f of different joint shape. A detailed description of each step follows. Step (1): The basic geometry of center-cracked butt joint in this paper is shown in Fig. 3. The joint is subjected to a remote tensile load. Plane strain conditions are assumed because the joint length is much larger than the base metal thickness. The joint geometry, center crack and loading are symmetrical, so a quarter of finite element model shown in Fig. 4 is established. The analysis involves a two-dimensional finite element method by using the finite element software MSC.MARC. The J-integral value can be extracted from the result document. The joint shown in Fig. 4 has a thickness (2t) of 16 mm, a weld width (2w) of 32 mm, a center crack length (2a) of 4 mm, a reinforcement (h) of 2 mm and a weld toe smooth transition radius (r) of 32 mm. To get an accurate result, meshes near the crack tip are refined. The ratio of minimum mesh size near the tip to half of the crack length is only 0.005. As a matter of fact, a lot of center-cracked butt joints with different shapes should be analyzed to obtain the corresponding J-integral by FEM. Step (2): Under plane strain condition, the relationship of SIF and J-integral is [32]
KI ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi EJ 1 t2
(a) w/t=2
(b) w/t=4
ð7Þ
where E is the Young’s modulus, J is J-integral obtained in step (1), t is Poisson’s ratio. Step (3): Based on Eq. (6), the modified coefficient f can be obtained as follows,
f ¼
KI
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt þ hÞ2 a2 pffiffiffiffiffiffi tr pa
ð8Þ
Here, KI is the SIF value calculated in step (2), t is half of the base metal thickness, h is weld reinforcement, a is half of the center crack length, r is work stress. Fig. 5 shows the influence of butt joint shape parameters on modified coefficient f. r ranges from 0 mm to 32 mm, h in relation to t ranges from 0 to 1. Three different w/t, 2, 4, 6 are discussed. The modified coefficient f is obtained by Eq. (8). The analysis indicated that f increases with the increase of h and r, f decreases significantly with the increase of w when other shape parameters are fixed. Step (4): The modified coefficient function f(h,w,r,t) can be obtained by regression analysis of all the modified coefficient f which are calculated in step (3),
(c) w/t=6 Fig. 5. Influence of shape parameters on modified coefficient f: (a) w/t = 2, (b) w/ t = 4 and (c) w/t = 6.
2
3 w 2hwr 2 6 7 h 6 h þ t ðw2 þ h Þðh þ tÞ7 exp 6 f ðh; w; r; tÞ ¼ 1 þ A 7 4 5 hþt Bðht Þ2 þ C ht þ D
ð9Þ
where A, B, C and D are 1.77633, 0.49536, 0.66789 and 0.75115, respectively.
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T. Wang et al. / Materials and Design 35 (2012) 72–79
Substituting (9) into (6), the calculation expression of SIF for center-cracked butt joint can be expressed as follows:
8 39 2 w 2hwr > > > > > < = 6 h þ t ðw2 þ h2 Þðh þ tÞ7> h 7 6 exp 6 KI ¼ 1 þ A 7 2 > 5> 4 hþt Bðht Þ þ C ht þ D > > > > : ; pffiffiffiffiffiffi t qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r pa 2 2 ðt þ hÞ a
ð10Þ
where A, B, C and D are 1.77633, 0.49536, 0.66789 and 0.75115, respectively. 3.2. Shape factor for center-cracked butt joint From Eq. (2), it can be seen that KI is only dependent on shape factor Y when r and a are known. In the mean time, Y is dependent on loading mode, crack location and joint shape parameters. For center-cracked butt joint under tensile load, the loading mode is tensile load and the crack location is in the center. So, Y is only dependent on joint shape parameters, which means KI is only dependent on the joint shape parameters when r and a are given. Obviously, the smaller the shape factor is, the safer the joint is. There must be certain joint shape parameters which can lead to the shape factor small. Combining Eqs. (2) and (6), the shape factor of center-cracked butt joint can be obtained as follows,
pffiffiffiffi t p Y ¼ f ðh; w; r; tÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt þ hÞ2 a2
(a) w/t=2
ð11Þ
where f(h,w,r,t) is the modified coefficient function in Eq. (9). Fig. 6 shows the influence of modified butt joint shape parameters on shape factor Y. Here, t is 10 mm, a is 2 mm, r ranges from 0 to 20 mm, h in relation to t ranges from 0 to 1. Three different w/t, 2, 4, 6 are discussed. Shape factor Y is obtained by Eq. (11). The results show that Y decreases with the increase of h and w. Y increases with the increase of r, especially when h is large or w is small. The results without considering weld reinforcement (h/t is 0) are larger than the ones with considering weld reinforcement. This means that calculating KI without considering weld reinforcement is conservative and unreasonable in actual engineering application. From Fig. 6c, it can also be seen that Y will be stable at a relatively small value when w and h are both large enough. So, center-cracked butt joint with large enough weld reinforcement and width, and also an appropriate weld toe smooth transition radius will have a high load-carrying capacity.
(b) w/t=4
3.3. Engineering application The critical crack length, critical stress and the fatigue remaining life of joint are the most concerned problems in engineering application [33,34]. For butt joint, as shown in Fig. 3, its critical crack length ac and critical stress rc can be obtained through its calculation expression of SIF showed in Eq. (10) respectively.
ac ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðpt 2 r2 f ðh; w; r; tÞ2 Þ2 þ 4K 4IC ðt þ hÞ2 pt2 r2 f ðh; w; r; tÞ2
(c) w/t=6
2K 2IC ð12Þ
rc
K IC ¼ f ðh; w; r; tÞt
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt þ hÞ2 a2 pa
ð13Þ
where KIC is fracture toughness of weld metal, f(h,w,r,t) is the modified coefficient function calculated by Eq. (9), h, w, r, t are butt joint shape parameters which are shown in Fig. 3. r in Eq. (12) is the
Fig. 6. Influence of shape parameters on shape factor Y of center-cracked butt joint: (a) w/t = 2, (b) w/t = 4 and (c) w/t = 6.
maximum work stress. a in Eq. (13) is half of center crack length which can be obtained through nondestructive examination. When butt joint showed in Fig. 3 endures under fatigue load, its remaining life Nf can be obtained by combination of Paris law and its calculation expression of SIF showed in Eq. (10),
76
Nf ¼
T. Wang et al. / Materials and Design 35 (2012) 72–79
Z
ac
a0
0
da
1m
ð14Þ
pffiffiffiffiffiffiC t B A@f ðh; w; r; tÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dr paA ðt þ hÞ2 a2
where a0 is half of the initial center crack length which can be obtained through nondestructive examination, ac is the critical crack length which can be calculated by Eq. (12), Dr is cyclic stress, A and m are material properties. 4. Verification of SIF calculation expression for center-cracked butt joint The accuracy of SIF calculation expression for center-cracked butt joint is evaluated by both FEM analysis and static tensile experiment. Fatigue experiment of center-cracked butt joint with different shape is also tested. 4.1. FEM verification SIF values calculated by Eq. (10) are compared with that from FEM analysis result calculated by Eq. (7). Here, r is 100 MPa, a is 2 mm, t is 8 mm. Fig. 7 shows the comparison of K calculated by
(a) r=0
Eq. (10) and FEM analysis, where ‘formula’ represent K calculated by Eq. (10) and ‘simulation’ represent K obtained by FEM analysis. Fig. 7a shows the normal butt joint with no weld toe smooth transition radius(r = 0) and Fig. 7b shows the modified butt joint with weld toe smooth transition radius (r = 8 mm). The results demonstrate that SIF value calculated by Eq. (10) and the FEM analysis are in good agreement. So, it is proved that the calculation expression of SIF for center-cracked butt joint, showed in Eq. (10), can provide clear guidance for shape design of center-cracked butt joint. 4.2. Experimental verification Modified coefficient function f(h,w,r,t) is the determinant factor of Eqs. (6), (11)–(13), (and) (14). The calculation expression of KI, Y, rc, ac and Nf can be accurate only if f(h,w,r,t) is precise. The accuracy of f(h,w,r,t) can be verified by comparing expression with experiment result. The modified coefficient f of expression result can be obtained through substituting the shape parameters of each joint into Eq. (9). From Eq. (13), modified coefficient f can be obtained according to the following equation,
K IC f ðh; w; r; tÞ ¼ t rc
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt þ hÞ2 a2 pa
ð15Þ
So, the modified coefficient f of experiment result can be obtained through substituting the fracture toughness KIC of weld metal, half of the base metal thickness t, critical stress of tensile test rc, weld reinforcement h and initial crack length a into Eq. (15). Three types of center-cracked butt joints were tested (see Fig. 8) by utilizing the material testing machine (CSS-44300). Mechanical properties of joint material are shown in Table 1. Fracture toughness KIC of weld metal is 34.8 MPa (m)1/2. The base metal thickness is 10 mm, which means t is 5 mm. X groove was first made on the plate, and then manual arc welding was adopted to produce the joints. All center cracks were obtained through wire–electrode cutting before fatigue pre-cracking to a same crack length, a0 is 2 mm. The comparison of expression results with experiment results are shown in Table 2. The comparison illustrates that all of the three types of butt joints show little differences between expression results with experiment results. It means that the calculation expression of SIF and its applications given in this paper could be
Fig. 8. Three types of center-cracked butt joints.
Table 1 Mechanical properties of joint material. Joint material
(b) r=8 mm Fig. 7. Comparison of SIF values calculated by expression calculation and FEM analysis: (a) r = 0 and (b) r = 8 mm.
Yield strength
rs (MPa)
Tensile strength
Elongation et (%)
77
T. Wang et al. / Materials and Design 35 (2012) 72–79 Table 2 Comparison of expression results with experiment results. Specimen
h (mm)
w (mm)
r (mm)
rc (MPa)
f(h,w,r,t) by experiment
f(h,w,r,t) by FEM
Differences (%)
Normal butt joint Modified butt joint 1 Modified butt joint 2
1.5 1.5 2.5
6.5 15 15
0 12 12
456 498 550
1.1904 1.0900 1.1535
1.1361 1.0482 1.1268
4.78 3.99 2.37
Table 3 Fatigue test results of three types of joints. Specimen
h w r Average Alternating Fatigue (mm) (mm) (mm) load load (MPa) remaining (MPa) life (cycle)
Normal butt joint Modified butt joint 1 Modified butt joint 2
1.5
6.5
0
100
100
100,122
1.5
15
12
100
100
157,627
2.5
15
12
100
100
967,569
used to guide the shape design of center-cracked butt joint under tensile load. From the result, it can also be seen that appropriate shape design can improve the static load-carrying capacity of center-cracked butt joint.
(a) Normal butt joint
4.3. Fatigue experiment Fatigue experiments of these three types of joints (see Fig. 8) were tested by using high frequency fatigue testing machine (PLG-100C). The fatigue test results of three types of joints are shown in Table 3. It can be seen that fatigue remaining life of modified butt joint is much larger than that of the normal butt joint. Meanwhile, fatigue remaining life of modified butt joint with larger reinforcement (modified butt joint 2) is larger than that of modified butt joint 1. The corresponding SEM results of three types of center-cracked butt joints are shown in Fig. 9. All the SEM results include three different zones, i.e. wire cutting zone, pre-crack zone and fatigue propagation zone. For these three different joints, wire cutting zone and pre-crack zone are the same. Comparing the fatigue propagation zone of the three joints, it can be seen that the roughness of fracture surface follows such a sequence that normal butt joint > modified butt joint 1 > modified butt joint 2. The more serious the fracture surface roughness is, the larger the fatigue propagation rate is. Therefore, the fatigue life of modified joint 2 is the largest among the three joints. So, appropriate joint shape design can also improve the fatigue load-carrying capacity of centercracked butt joint.
(b) Modified butt joint 1
5. Discussion When using the fracture mechanics approach, an appropriate SIF expression must be proposed. A number of formulas for center-cracked butt joint under tensile load can be found in the SIF handbooks [25,30,31]. However, in these formulas, the joint reinforcement was not taken into account. It is known that SIF has been demonstrated to be strongly related to the joint shape parameters [23,24]. Meanwhile, it is not possible to solve the SIF analytically for joint with complex shape. For these reasons mentioned above, the SIF expression for center-cracked butt joint must be further amended by a modified coefficient function. Tedious numerical computations are needed in order to generate an accurate modified coefficient function. By adopting the ordinary SIF expression and combining with suitable modified coefficient function, the SIF expression for center-cracked butt joint under tensile load proposed in this research is extendable to normal center-cracked butt
(c) Modified butt joint 2 Fig. 9. SEM results of three types of center-cracked butt joints: (a) normal butt joint, (b) modified butt joint 1 and (c) modified butt joint 2.
joint with certain reinforcement, thus the expression is obtained and it is adopted to guide engineering applications and joint shape designs. Eq. (4) shows the original SIF expression for center-cracked butt joint under tensile load. The joint reinforcement is not considered here. When it comes to the flush center-cracked butt joint, its SIF can be calculated by the Eq. (4). Nevertheless, when joint has certain reinforcement, it is not appropriate to calculate SIF by using the Eq. (4). In the proposed SIF expression (Eq. (10)), the SIF correlates with all the joint geometries (reinforcement h, half of the weld width w, weld toe smooth transition radius r and half of
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T. Wang et al. / Materials and Design 35 (2012) 72–79
the base metal thickness t), crack dimension (half of the center crack length a) and work stress r. In Section 4, it can be seen that the proposed SIF expression with considering the effect of joint shape in this paper is more accurate than the ordinary one. In addition, it works well for both engineering applications and joint shape designs. The calculation expression of SIF for center-cracked butt joint described in Eq. (10) is aimed at central through crack. When it comes to central part through crack, its crack length can be converted to an equivalent length on central through crack according to different conditions. Because the residual stress here is transversal stress which is usually much smaller than the material yield strength, welding residual stress is not considered to calculate the SIF in this paper. It can be treated as an extra work stress when calculating the SIF of welded joint. Isotropic material is used in the FEM analysis in this paper. Actually, the welded joint should be treated as anisotropic material. In the case of anisotropic material, the calculation expression of SIF is ought to be modified. When welded joint works under low stress which belongs to linear elastic fracture mechanics (LEFM) situation, its SIF values calculated by anisotropic material and isotropic material are almost the same [25]. But there are some limitations of LEFM because it assumes that no plastic deformation exists in the structure. As a matter of fact, the material near the crack tip would deform plastically. However, a correction term for the presence of the crack tip plastic zone can be incorporated when the size of the plastic zone is small with respect to the crack length. In regard to the plane stress situation, the approximate plasticzone radius will be [25]
ry ¼
1 2p
K
2 ð16Þ
ry
While for the plane strain situation, the approximate plasticzone radius will be
ry ¼
1 6p
K
2
ry
ð17Þ
The effective crack length can be expressed as
2ðaÞeff ¼ 2ða þ r y Þ
ð18Þ
Substituting (a + ry) for a in Eq. (10) gives an adequate adjustment for low work stress situations. With this adjustment, the calculation expression of SIF showed in Eq. (10) is more reasonable and more widely used for center-cracked butt joint under tensile load. 6. Conclusions (1) The stress intensity factor expression for center-cracked butt joint under tensile load is revised by adding a modified coefficient function to common expression. The analytical method, finite element method and regression analysis are applied to obtain the modified coefficient function. (2) The accurate stress intensity factor expression for centercracked butt joint under tensile load is given to guide its shape design under static load and fatigue load conditions. (3) It is unreasonable to calculate the stress intensity factor for center-cracked butt joint under tensile load without considering weld reinforcement. The stress intensity factor decreases with the increase of weld reinforcement or weld width, especially the weld reinforcement. The stress intensity factor will be stable when weld reinforcement and weld width are large enough. The influence of weld toe smooth
transition radius can be ignored when calculating the stress intensity factor. (4) The applications of stress intensity factor expression for center-cracked butt joint under tensile load are discussed such as acquiring the shape factor, critical crack length, critical stress and fatigue remaining life of the joint. (5) Modified center-cracked butt joint with certain weld reinforcement, weld width and also an appropriate weld toe smooth transition radius has greatly improved static and fatigue load-carrying capacity than that of normal butt joint. It is much meaningful to improve the load-carrying capacity of center-cracked butt joint by choosing appropriate joint shape parameters.
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