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Effect of residual stresses on ductile crack growth resistance
Engineering Fracture Mechanics 77 (2010) 1325–1337
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Effect of residual stresses on ductile crack growth resistance X.B. Ren a, Z.L. Zhang a,*, B. Nyhus b a b
Department of Structural Engineering, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway SINTEF Materials and Chemistry, Department of Applied Mechanics and Corrosion, N-7465 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 29 October 2009 Received in revised form 19 February 2010 Accepted 5 March 2010 Available online 18 March 2010 Keywords: Residual stresses Ductile crack growth resistance Complete Gurson model Eigenstrain method
a b s t r a c t It is well known that residual stresses influence the ductile fracture behaviour. In this paper, a numerical study was performed to assess the effect of residual stresses on ductile crack growth resistance of a typical pipeline steel. A modified boundary layer model was employed for the analysis under plane strain, Mode I loading condition. The residual stress fields were introduced into the finite element model by the eigenstrain method. A sharp crack was embedded in the center of the weld region. The complete Gurson model has been applied to simulate the ductile fracture by microvoid nucleation, growth and coalescence. Results show that tensile residual stresses can significantly reduce the crack growth resistance when the crack growth is small compared with the length scale of the tensile residual stress field. With the crack growth, the effect of residual stresses on the crack growth resistance tends to diminish. The effect of residual stress on ductile crack growth resistance seems independent of the size of geometrically similar welds. When normalized by the weld zone size, the ductile crack growth resistance collapses into one curve, which can be used to assess the structural integrity and evaluate the effect of residual stresses. It has also been found that the effect of residual stresses on crack growth resistance depends on the initial void volume fraction f0, hardening exponent n and T-stress. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction It is widely accepted that the presence of residual stresses can influence the failure characteristics of components and structures. A significant influence due to the residual stress on the failure and load bearing capacity of components under predominantly elastic conditions was demonstrated by Ainsworth et al. [1], who also showed that the effect on load-bearing characteristics reduced with the increasing plasticity within the component. A quantitative understanding of how residual stresses affect the failure behaviour is thus required to provide accurate integrity assessment and reduce the conservatism inherent in many assessment codes. Ductile crack growth plays an important role in the analysis of the fracture behaviour of structures [2]. Crack extension reduces the load-bearing ligament and will influence the capacity of the structures. Ductile crack growth may also change the near-tip stress/strain fields and promote the transition to unstable cleavage fracture. The mechanism of ductile fracture in metallic materials may be considered as a three-stage process: nucleation, growth and coalescence of microvoids. The ductile fracture process is influenced by the local conditions of stress triaxiality and plastic strain within the vicinity of a stress concentrator such as a notch or a crack-tip [3]. Liu et al. [4] and Ren et al. [5] have demonstrated that residual stresses can induce an additional crack-tip constraint, and a parameter R was defined based on the difference between the full stress
* Corresponding author. Tel.: +47 73 59 25 30. E-mail address: [email protected] (Z.L. Zhang). 0013-7944/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2010.03.007
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field and the reference field to quantify the residual stress-induced crack-tip constraint. It is thus interesting to further investigate how residual stresses influence the local failure mechanisms and global ductile crack resistance. Panontin and Hill [6] predicted the brittle and ductile initiation by micromechanical models and showed that the effect of residual stress on the ductile fracture initiation toughness is negligible. Sherry et al. [3] demonstrated that a high strength low toughness aluminum alloy AL2024-T351 showed a marked reduction in initiation and tearing toughness for specimens containing a mechanically induced residual stress field. It should be noted that the fracture behaviour of AL2024-T351 remains in ductile fracture regime. Experimental work undertaken by Sharples et al. [7] on wide plate specimens has also demonstrated that residual stress can significantly influence the ductile tearing behaviour of engineering materials. However, the experiments performed by Mirzaee-Sisan et al. [8] on the AISI Type 361H stainless steel indicates a negligible impact on ductile tearing toughness at load ratio Lr close to 1, i.e. close to the plastic collapse of the specimen. The observation is in contrast to their previous experiments performed on the A533B unstable cleavage fracture in ferritic steels at 150 °C [9]. A reduction of approximately 46% was observed in apparent fracture toughness for specimens containing a residual stress field compared with specimens in the as-received condition. Mahmoudi et al. [10] employed a local out-of-plane compression method to introduce residual stresses into C(T) specimens for aluminum alloys A12650 and A12024. Their experimental results showed that tensile residual stresses reduced the initiation cleavage fracture toughness of A12650 by about one half, and the ductile tearing resistance of A12024 also decreased when tensile residual stresses present. To date, the fundamental understanding of the effect of the residual stresses on ductile fracture resistance remains an open challenge. To this end, we utilized a micromechanical damage model in a Modified Boundary Layer (MBL) model to investigate the problem. The MBL model has been widely used to investigate the crack problems [11], and the transferability to real components can be well handled by simply changing the boundary conditions. In recent years, damage mechanics models have become widely used in numerical simulation of ductile crack growth. The Gurson model [12] is often used to assess ductile tearing behaviour under primary loading, where the effect of the void growth is taken into account in the constitutive model. The Gurson model has later been modified by Tvergaard and Needleman [13–15], thus, it is most often referred to as the Gurson–Tvergaard–Needleman (GTN) model. The GTN model is in fact a void growth model. The effect of void coalescence can be considered with the introduction of a so-called critical void volume fraction, which is not a physical mechanism-based coalescence criterion. Thomason [16] proposed a plastic limit load model for coalescence, which was easy to implement and very accurate [17,18] compared with the finite elements results by Koplik and Needleman [19]. This work has been further improved by considering the plastic strain hardening [20]. By combining the GTN model for void growth and Thomason’s plastic limit load model for coalescence a so-called ‘‘complete Gurson model” has been proposed by Zhang et al. [21], with which ductile fracture is exclusively linked to the void nucleation parameters and the mesh size. Two scenarios have been considered to study the effect of the residual stresses on the crack growth resistance in this paper. Firstly, the effect of residual stresses on ductile crack resistance in a large round weld region was investigated. In such a way, the final crack growth length still locates in weld region. Secondary, a much smaller rectangular weld region was constructed, where the crack can propagate through the whole weld region. The effect of the weld zone size was also taken into account and studied. In addition, the effects of initial void volume fraction and material hardening have also been studied. 2. Numerical procedure 2.1. Problem description This study concerns an ideal problem. A large cylinder with a weld in the center was studied. The cylinder was simulated by a 2D plane strain MBL model with a remote boundary governed by the elastic K-field and T-stress. A sharp crack was embedded in the weld region. The analysis procedure, as illustrated in Fig. 1, consists of the following steps: (1) enforce a welding procedure in un-cracked body, which introduces a residual stress field; (2) introduce a sharp crack; and (3) apply the external load. The contact between the upper and lower free surfaces of the crack is considered when the residual stress was introduced.
Fig. 1. Illustration of the problem. (a) The round cylinder; (b) welding at the center and a sharp crack is introduced; and (c) apply the external load.
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Fig. 2. Modified boundary layer model, (a) global view and (b) crack-tip mesh.
2.2. Finite element model The present analyses were carried out for the conditions of small-scale-yielding. The MBL model used for this study consists of a weld region located in the center of the model, an outer base metal region, and a sharp crack in the center of weld. The load was applied to the remote edges of the model through a displacement field (u, v) controlled by the elastic asymptotic stress field of a plane strain Mode I crack:
rffiffiffiffiffiffiffi r 1 1 m2 cos h ð3 4m cos hÞ þ T r cos h 2p 2 E rffiffiffiffiffiffiffi 1þm r 1 mð1 þ mÞ sin v ðr; hÞ ¼ K I h ð3 4m cos hÞ T r sin h 2p 2 E E uðr; hÞ ¼ K I
1þm E
ð1Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where K I ¼ EJ=ð1 m2 Þ under plane strain condition, E is Young’s modulus, m is Poisson’s ratio, and r and h are polar coordinates centered at the crack tip with h = 0 corresponding to the crack tip. The finite element computations were performed using ABAQUS [22]. Due to symmetry, only the upper half of the geometry is modeled. The mesh is shown in Fig. 2. The radius of the MBL model is taken as 1000 mm to ensure the small-scaleyielding condition is fulfilled. Close to the crack tip, there is a rectangular region (9.4 mm ahead of the initial crack tip and 1.6 above the symmetry line) with uniform mesh sizes of 0.1 0.05 mm for the first layer and 0.1 0.1 mm for the rest of the layers are created. In following context, lc = 0.1 mm represents the size of the uniform elements. Full integration four-node 2D plane strain elements are used. The finite element model has 2616 elements. Nonlinear geometry effects (NLGEOM in ABAQUS) are accounted for in the analyses. An initial opening of 0.02 mm is applied for the upper half model. When the residual stresses are introduced into the model, the crack faces can be partly closed. Thus, a rigid analytical plane is defined in the model to simulate the contact of the crack surfaces. 2.3. Materials As described in Section 1, the complete Gurson model is utilized to simulate the crack growth, which is implemented into ABAQUS using a set of generalized-midpoint algorithms developed by Zhang via the material user subroutine UMAT1 [23– 25]. Both the weld metal and base metal are modeled as a power-law hardening material: 1
A free copy of the ABAQUS UMAT Fortran code can be obtained from the corresponding author.
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rf ¼ r0 1 þ
ep
n ð2Þ
e0
where rf is the flow stress, ep is the equivalent plastic strain, the Poisson’s ratio m is 0.3, Young’s modulus is 200,000 MPa, yield stress r0 is 400 MPa, e0 = r0/E is the yield strain and n is the plastic strain hardening exponent. The yield function of the Gurson model has the following form:
; f ; rm Þ ¼ /ðq; r
3q2 rm 1 ðq1 f Þ2 ¼ 0 þ 2q fcosh 1 2r r2f q2
ð3Þ
where f is the void volume fraction, rm the mean macroscopic stress, q the von Mises stress, rf the flow stress, and q1 and q2 are parameters introduced by Tvergaard [13,14]. Fixed values of q1 = 1.5 and q2 = 1.0 have been used in this study. In this study, a cluster void nucleation model has been used [21]. The critical void volume fraction (fc) is automatically determined by Thomason’s plastic limit load criterion [16,20]. The post-coalescence deformation behaviour of the Gurson model is numerically simulated by an artificial acceleration of void growth, as suggested by Tvergaard and Needleman [15]:
(
f ¼
f
for f 6 fc f f
fc þ fuF fcc ðf fc Þ for f > fc
ð4Þ
where fu ¼ 1=q1 and fF is the void volume fraction at the end of void coalescence. Here, fF = 0.15 + 2f0, where f0 is the initial void volume fraction. When the coalescence starts and f > fc, f* replaces f in Eq. (3). 2.4. Eigenstrain method Different thermal expansion coefficients of the base metal (ab) and weld metal (aw) were used to introduce the residual stresses into the model by the so-called eigenstrain method. It should be noted that the thermal expansion coefficients here are not physical thermal coefficients, but are used to introduce residual stresses into the FE model. The eigenstrain method was also called ‘‘inherent strain” method when first introduced by Ueda et al. [26]. The concept of the eigenstrain method is that the source of residual stress is an incompatible strain field caused by plastic deformation, thermal strains and phasetransformation etc. [27]. Thus, if the distribution of the eigenstrain is known, the distribution of residual stresses can be obtained through linear elastic calculation by using the finite element method. Our approach is to set the eigenstrain values equal to the thermal expansion coefficients for different regions. The model is then loaded by applying a unit temperature decrease, thereby introducing the residual stress field of interest. Similar approaches were also applied in the literature [28,29], and their results showed that the distributions of residual stresses obtained by such approaches agreed well with the experimental results. 2.5. The crack driving force In fracture mechanics, the J-integral [30] or Crack Tip Opening Displacement (d, CTOD) are used to characterize the cracktip driving force of a cracked body made of an elastic-plastic material. The J-integral is a path-independent integral based on the assumption that the strain energy density is a single-valued function of the strain (or stress) [31]. However, non-proportional loading may occur in the region where the J-integral is evaluated and lead to path-dependence, for example, in the case of ductile crack growth [2], or in the presence of a residual stress field [32]. In this study, the J-integral is evaluated on contours far away from the zone of highly non-proportional loading, and displays practically path independence in both cases with and without residual stresses. Residual stresses as an additional stress field induce an initial J-integral. However, the residual stress-induced J-integral is negligible compared to the final J-integral caused by the combination of residual stress and external load. Also, under the small-scale-yielding conditions, the difference between calculated J-integral and applied J-integral is less than 2.78% for the case without residual stress, and 3.25% for the case with residual stresses [5]. Thus, J -integral should be applicable in the present study, and the J in the following context represents far-field J-integral. 3. Results and discussion 3.1. Residual stress fields As described in Section 2.4, we used the eigenstrain method to introduce residual stresses into the finite element model. A round weld region was introduced in the center of the model, as illustrated in Fig. 3. According to the eigenstrain method, the thermal expansion coefficients of both the weld metal (aw) and base metal (ab) were assumed to be isotropic and equal to the eigenstrain values respectively. In this section, we assume ab=0. Four residual stress fields were generated by setting aw= 0.001, 0.001, 0.002, and 0.003 and designated as RsField0, RsField1, RsField2, and RsField3 respectively. Fig. 4 shows the distribution of the residual stresses both before and after the crack was inserted. Note that the stresses are normalized by the yield stress, and the distance from the crack tip x is normalized by the size of the uniform element lc.
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Fig. 3. Illustration of the weld region (gray area). The radius of the weld is 20 mm.
3
RsField0 RsField0
RsField2 RsField2
4 3
0
RsField0 RsField0
crack
RsField1 RsField1
crack
RsField2 crack RsField2
crack
RsField3 crack RsField3
1
2
RsField3 crack RsField3
1 0
−1 −2 0
b
RsField1 crack RsField1
2
σ11/σ0
crack
σ22/σ0
a
−1
100
200
300
400
x/lc
500
−2 0
100
200
300
400
500
x/lc
Fig. 4. Comparison of the residual stress distribution in MBL model with a large round weld before (bold line) and after (thin line) the crack was inserted, (a) components parallel to the crack plane and (b) normal to the crack plane. Four different residual stress cases were considered, where RsField0 is compressive and the remaining 3 are tensile. ab = 0; aw = 0.001, 0.001, 0.002, and 0.003; lc = 0.1 mm.
It can be seen that the negative eigenstrain value introduces the compressive residual stress at the weld region while the positive ones generate tensile residual stresses. Both tensile and compressive residual stresses parallel to the crack front converge to zero far from the crack tip. The opening residual stresses are self-balanced ahead of the crack tip. There is a sharp turning point in the distribution of the opening residual stress, which is the region where eigenstrain discontinuity has been introduced into the FE model, namely the weld metal-base metal boundary. The tensile residual stresses also show similarity, and the level of the tensile residual stress increases with the increasing aw. Due to the singularity, r11 is about 800 MPa and r22 is about 1200 MPa at the crack tip for RsField2. Fig. 4 also shows that residual stress components are smaller than the yield stress before the crack was inserted. 3.2. Effect of residual stresses on ductile crack resistance Ductile crack growth resistance is important for structural integrity assessment, and it is interesting to investigate the effect of residual stresses on it. In this section, effects of residual stress fields on the crack growth resistance were studied. The initial void volume fraction f0 is fixed to be 0.1% and the strain hardening exponent n = 0.1. The crack growth resistance described by the J-integral is showed in Fig. 5. For the cases chosen, the residual stresses seem to have significant effects on the ductile crack growth resistance. Fig. 5a shows that the compressive residual stress enhances the crack growth resistance while the tensile residual stresses have the
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a
b
25
1.6
RsField0 RsField1 RsField2 RsField3
1.4
20
without residaul stress
15
J/Jwithout
J/(σ0lc)
1.2
10 no/res RsField0 RsField1 RsField2 RsField3
5 0 0
5
10
15
Δa/lc
20
25
30
1 0.8 0.6 0.4 0.2 0
5
10
15
20
25
30
Δa/lc
Fig. 5. Effect of residual stresses on crack growth resistance, (a) absolute crack resistance curves and (b) normalized crack resistance curves. f0 = 0.1%; n = 0.1; E/r0 = 500.
Fig. 6. Illustration of the weld region (gray area). The length of the weld region along the ‘‘11” direction is 9.6 mm, and the length along the ‘‘22” direction is 6.4 mm.
opposite effect. With the increase of tensile residual stress, the crack growth resistance decreases. The absolute difference between the case with and without residual stress increases with the increase of crack growth. Fig. 5b presents the normalized crack growth resistance. The resistances with residual stresses were normalized by the resistance at the same crack growth without residual stresses. The distance from the original crack tip was normalized by lc. It can be seen that the tensile residual stresses significantly decrease the initiation toughness while the compressive residual stress increases it. With advancing crack growth, the effect of the residual stresses decreases and approaches to a constant value. Note that the current crack grows only to 3 mm, which is very small compared to the size of the residual stress dominant length scale shown in Fig. 4. As shown in Fig. 4, beyond the singularity affected zone (x/lc > 30, i.e. 3 mm), the residual stress fields approach a hydrostatic stress state. Thus, the residual stresses cannot be easily released by the crack growth, and the effect of residual stresses retains. In order to better understand the effect of residual stress on crack growth resistance, a smaller rectangular weld region was constructed, as illustrated in Fig. 6. In this section, ab=0, and aw was assumed to be orthogonal and characterized by a11 and a22 in following context. The ratio a11/a22 was fixed to be 2. Four residual stress fields with a22=-0.001, 0.001, 0.002, and 0.003 were generated and represented by RsField0, RsField1, RsField2, and RsField3 respectively, as shown in Fig. 7. It should be noted that the eigenstrain values selected here are taken from experimental measurement results in literature [33,34]. The residual stress fields generated by the selected values have similar distribution as the residual stress fields showed in Ref. [35]. To obtain accurate distributions of the residual stress fields by the eigenstrain method, one should carry out the experiments to measure the distribution of the eigenstrain. However, the main objective of this study is to investigate the effect of the residual stresses, and prediction of the real distribution of the residual stress field is outside of the scope. The absolute and normalized crack growth resistances are shown in Fig. 8. It can be seen that the compressive residual stress increases the crack growth resistance while tensile residual stresses decrease the crack growth resistance, as shown in
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a
3
RsField0 RsField1 RsField2 RsField3
2
0
3
RsField0 RsField1 RsField2 RsField3
2
σ22/σ0
σ11/σ0
1
b
1 0 −1
−1 −2
−2 0
50
100
150
200
0
50
100
150
200
x/lc
x/lc
Fig. 7. Residual stress distribution in MBL model with a rectangular weld after the crack was inserted, (a) components parallel to the crack plane and (b) normal to the crack plane. Four different residual stress cases were considered, where RsField0 is compressive and the remaining 3 are tensile. a11/a22 = 2; a22 = 0.001, 0.001, 0.002, and 0.003.
a
b
25
1.2 1.1
20
J/Jwithout
J/(σ0lc)
1 15 10 no/res RsField0 RsField1 RsField2 RsField3
5 0
0
5
10
15
20
25
30
35
Δa/lc
40
0.9 0.8 0.7 RsField0 RsField1 RsField2 RsField3
0.6 0.5
0
10
20
30
40
Δa/lc
Fig. 8. Effect of residual stresses on crack growth resistance, (a) absolute crack resistance curves and (b) normalized crack resistance curves. f0 = 0.1%; n = 0.1; E/r0 = 500.
Fig. 8a. Fig. 8b shows that the reduction of the crack growth resistance converges with crack growth to the case without residual stress. As known, the quantity rm/re defines a convenient measure of triaxiality linked to the growth rate of micro-scale voids consistent with the subsequently introduced damage measures [36]. Therefore, we investigated the distribution of triaxiality ahead of the crack tip. Fig. 9 presents the triaxiality values on the ligament ahead of the original crack tip for different crack growth. The distance from the original crack tip are scaled by J/r0, or equivalently (KI/r0)2, which defines approximately the value of crack-tip opening displacement and provides a physical meaningful length scale for normalization. It can be seen that tensile residual stresses enhance the triaxiality values while the compressive residual stress reduces the triaxiality value at the crack initiation, see Fig. 9a, which corresponds to our previous findings [4,5]. However, the effect of residual stress on triaxiality tend to be negligible when the crack advanced to 3.5 mm, as shown in Fig. 9b. Higher stress triaxility corresponds to a smaller plastic zone. Thus, the energy dissipated by the plastic deformation is smaller, which in turn result in a lower crack growth resistance. 3.3. Effect of weld zone size on crack growth resistance The length scale of the residual stress field may play an important role on the effect of residual stress on ductile crack growth resistance, as shown in Section 3.2. To better demonstrate this, four geometrically similar rectangular weld regions were constructed, as illustrated in Fig. 10. The size of the weld is designated as c. The orthogonal thermal expansion coefficient was used with a value of a11=0.004 and a22=0.002 for all welds. The residual stress fields designated as Size1, Size2, Size3, and Size4 respectively are showed in Fig. 11. The residual stresses are tensile in the weld metal and show similar features as the previous residual stresses shown in Fig. 7. With the increase of the weld size, the residual stresses parallel to the crack plane increase; the size of tensile dominated region of the opening stress also increases.
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b
3
3
2.75
2.75
2.5
2.5
σm/σe
σm/σe
a
2.25
2
2 no/res RsField0 RsField1 RsField2 RsField3
1.75 1.5
2.25
1
2
3
4
no/res RsField0 RsField1 RsField2 RsField3
1.75
5
1.5
1
2
x/(J/σ0)
3
4
5
x/(J/σ0)
Fig. 9. Triaxiality values ahead the crack tip for different crack propagation, (a) Da = 0.1 mm; (b) Da = 3.5 mm. f0 = 0.1%; n = 0.1; E/r0 = 500.
Fig. 10. Illustration of weld size. The size increases by a scale of 2.
Fig. 12a presents the absolute crack growth resistance. In Fig. 12b, crack growth length was normalized by łc. It can be seen that the residual stresses generated in the larger welds influence more significantly the crack growth resistance. With increasing crack growth, the effect of the residual stresses on the crack growth resistance converges to the case without residual stresses. However, an interesting pattern emerges when normalizing the crack growth length by the weld zone size c, as shown in Fig. 12c. Surprisingly, the normalized crack growth resistances collapse into one curve, which indicates that the effect of the residual stresses on the crack growth resistance is nearly independent of the weld zone size. For the cases specified in this section, when the crack grows approximately to 3 times of c, the effect of residual stresses can be neglected. For geometrically similar welds, the uniform curve can be used to roughly predict the length scale of residual stress-affected region beyond which the effect of the residual stress can be neglected. In our previous study [5], it has been shown that residual stress fields generated in round weld regions with three sizes respectively can be normalized by the weld zone size and collapsed into a uniform field. Thus, the ‘‘master” curve as shown in Fig. 12c can be expected. 3.4. Effect of residual stresses on crack growth resistance for different hardening exponents It has been shown that the effect of strain hardening on the ductile crack resistance is not fully understood. Xia and Shih [37] demonstrated that the ductile resistance increases with increasing hardening capacity. However, Eikrem et al. [38] and Østby et al. [2,39] reported that decreasing the hardening exponent will significantly raise the resistance curve. In this study, the effect of strain hardening on the crack growth resistance was investigated for cases both with and without residual stress. It should be noted that in this particular study, the initial volume fraction f0 was fixed to be 0.05%, and the residual stress field was introduced by the rectangular weld with orthogonal thermal expansion coefficient a11 = 0.004 and a22 = 0.002, i.e. RsField2 in Fig. 7. The crack growth resistance was plotted as a function of hardening exponent n in Fig. 13. Fig. 13a shows that with decreasing hardening exponent the crack growth resistance increases for both with and without residual stress cases, which corresponds to the results demonstrated in Ref. [2,38,39]. For a given hardening exponent, residual stresses reduce both the initial toughness and crack growth resistance. Also, the effect of residual stress on the ductile resistance becomes stronger for stronger hardening material, as shown in Fig. 13b. With increasing crack growth, the reduction of the crack growth resistance decreases and tends to converge to the case without residual stress. Residual stresses reduce the equivalent plastic strain significantly at crack initiation, which indicates smaller plastic deformation. Hence,
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a
2
Size1 Size2 Size3 Size4
b
2 1.5 1
σ22/σ0
σ11/σ0
1.5
1
Size1 Size2 Size3 Size4
0.5
0.5
0 −0.5
0 0
50
100
150
200
0
50
100
150
200
x/lc
x/lc
Fig. 11. Residual stress distribution in MBL model with geometrically similar welds after the crack was inserted, (a) components parallel to the crack plane and (b) normal to the crack plane. a11 = 0.004, a22 = 0.002.
a
b
18 16
1 0.95
J/Jwithout
J/(σ0lc)
14 12 10
0.9 0.85
8
no/res Size1 Size2 Size3 Size4
6 4
1.05
0
5
10
15
20
Size1 Size2 Size3 Size4
0.8 0.75
25
0
5
10
c
15
20
25
Δ a/lc
Δ a/lc
1.05 1
J/Jwithout
0.95 0.9 0.85 Size1 Size2 Size3 Size4
0.8 0.75
0
1
2
3
4
Δa/c Fig. 12. Effect of weld size on crack growth resistance, (a) absolute crack growth resistance, (b) crack growth normalized by lc, and (c) normalized by the weld zone size c. f0 = 0.1%; n = 0.1; E/r0 = 500.
lower crack growth resistance curves can be expected. With the crack growth, the effect of residual stresses on the equivalent plastic strain becomes negligible for different hardening exponents. 3.5. Effect of residual stresses on ductile crack resistance for different initial void volume fraction The initial void volume fraction f0 represents the degree of damage in the material. The larger the initial void volume fraction is, the larger damage the material has. In this section, the effect of residual stress fields on crack growth resistance was
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investigated for three initial void volume fractions, f0 = 0.05%, 0.1% and 0.2%. The residual stress was introduced into a rectangular weld with a11 = 0.004 and a22 = 0.002, i.e. RsField2 in Fig. 7. The crack growth resistance curves are shown in Fig. 14. As shown in Fig. 14a, for both with and without residual stress cases, the crack growth resistance increases with the decrease of initial void volume fraction, which can be expected because the ductility becomes better when initial void volume fraction decreases. Also, it can be observed that the residual stress reduces the crack growth resistance for fixed f0. Normalized crack resistance curves shown in Fig. 14b indicate that with increasing crack growth, the effects of residual stresses decrease and become less dependent on f0. The residual stress enhances the opening stress beyond the larger strain effect region compared with the case without residual stress at crack initiation, which induces an increase of crack-tip constraint [4,5], and a lower fracture toughness can then be expected. However, it has been found that the effect of the residual stress on opening stress becomes negligible when crack growth becomes larger. 3.6. Effect of residual stresses on ductile crack resistance for different T/r0 Crack tip constraint effects on fracture toughness have received considerable attention recently. In our earlier work [5], we have found that the residual stress-induced crack-tip constraint is lower for a higher geometric constraint. Xia and Shih [37], Tvergaard and Hutchinson [40] studied the effect of T-stress on the crack growth resistance and showed that a negative T-stress results in a rapidly rising resistance curve while the positive T-stress lowers the fracture resistance. It is thus interesting to investigate how residual stress combined with varying T-stress affect the crack growth resistance. In this paper, T/ r0 = 0.5, 0, and 0.5 were studied. The initial void volume fraction is fixed to be 0.1%, and the residual stress with a11 = 0.004 and a22 = 0.002, i.e. RsField2 in Fig. 7, was introduced. The absolute and normalized resistance curves are shown in Fig. 15. It can be seen that the tensile residual stress reduces the crack growth resistance for all T/r0 cases, as shown in Fig. 15a. Fig. 15b shows that the normalized crack growth resistance is lower for smaller T-stress. However, the differences between
a
30 25
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Fig. 13. Effect of residual stresses on crack growth resistance for different hardening, (a) absolute crack growth resistance curves and (b) normalized crack resistance curves. a11 = 0.004; a22 = 0.002; f0 = 0.05%; E/r0 = 500.
45
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0
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10
15
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Fig. 14. Effect of residual stresses on crack growth resistance for different initial void volume fraction, (a) absolute crack growth resistance curves and (b) normalized crack resistance curves. a11 = 0.004; a22 = 0.002; n = 0.1; E/r0 = 500.
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a
b
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6
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0
5
10
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15
20
0.65
T/σ =0.5 0
0
5
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15
20
Δ a/lc
Fig. 15. Effect of the residual stress on normalized crack growth resistances for different T/r0, (a) absolute crack growth resistance curves and (b) normalized crack resistance curves. a11 = 0.004; a22 = 0.002; f0 = 0.1%; n = 0.1; E/r0 = 500.
the normalized crack growth resistances are insignificant. It should be noted that a relatively small initial void fraction was used in this study, i.e. f0 = 0.1%. With increasing crack growth, the effect of residual stresses on crack growth resistance tends to be independent of the T-stress. The positive residual stresses reduce the plastic strain compared to the case without residual stresses. Lower equivalent plastic strain indicates a smaller plastic zone and thus the energy dissipated by the plastic deformation decreases. In turn, the ductile crack toughness decreases. When the crack growth becomes larger, the effect of residual stresses on the equivalent plastic strain becomes negligible. 4. Conclusions This study describes a systematic computational investigation of the effect of residual stresses on ductile crack growth resistance. 2D plane strain finite element analysis have been carried out in a modified boundary layer model with the remote boundary condition governed by K-field and T-stress. The large strain effect has been taken into account in the analysis. The complete Gurson model was utilized to predict the crack growth resistance for both with and without residual stress situations. Residual stresses were generated by the eigenstrain method and introduced into the FE model. Both the tensile and compressive residual stress fields were introduced. However, tensile residual stress was known to be detrimental for the fracture. Thus, the effect of the tensile residual stress on crack growth resistance was mainly presented. Residual stresses as an additional field do not alter the material constitutive relation, however, the residual stress may influence the neartip stress state and enhance the level of damage along the ligament ahead of an advancing crack tip. The following conclusions can be made: 1. Tensile residual stresses reduce the crack growth resistance while compressive residual stress can enhance the crack growth resistance. With the increase of crack growth, the effect of residual stresses decreases and converges to the case without residual stress. The initial damage of the ligament elements caused by the residual stress is negligible for the cases studied. 2. Under certain circumstances, the effect of the residual stresses on ductile crack growth resistance can be normalized by the size of the geometrically similar weld, and the normalized crack growth resistance curves collapse into a single curve. One can use this curve to evaluate the effect of the residual stress on the structural integrity and simplify the assessment procedure. It can also be used to predict the length scale of the residual stress-affected range beyond which the effect of the residual stresses can be neglected. For the cases specified in this study, when the crack grows to a length of 3c the effect of the residual stresses can be neglected. 3. The reduction of the crack growth resistance caused by residual stresses is smaller for weaker hardening materials, and the influence of the residual stresses decreases with crack propagation. 4. Residual stress reduces the crack growth resistance more significantly for the materials with larger initial void volume. The influence of the residual stresses decreases with the crack growth and becomes independent of f0. 5. The reduction of the crack growth resistance induced by the residual stress increases with increasing T/r0 at the early stage of crack growth and then tend to be negligible when the crack growth is larger. The results are in line with our earlier findings that the effect of residual stress on the crack-tip constraint becomes weaker for higher T/r0. The present study indicates that residual stress as an additional stress field can alter the stress state near the crack tip and further influence the ductile crack growth resistance. Triaxility or crack-tip constraint was shown to play an important role in the ductile fracture behaviour. Therefore, the residual stress-induced crack-tip constraint is a very important factor to consider in structural integrity assessment. In our previous studies [4,5], we have defined a parameter R to quantify the residual
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stress-induced crack-tip constraint. However, the effect of residual stress on the ductile crack resistance has not been linked with R by a quantitative way in this study. The reason for this is that the residual stress field in reality is very complicated, hence one should find an efficient way to standardize the distribution of the residual stresses for typical weld joints. To reduce the conservatism of current integrity assessments, a proper description of the residual stress field is very important. Once a proper description of the residual stress is obtained, the ‘‘master‘‘ curve, though obtained from the MBL model, can be applied to real engineering structures and components that contain similar residual stress fields under smallscale-yielding conditions. The transferability of the results from MBL studies to real structures should be verified in further studies.
Acknowledgment The funding from the Research Council of Norway through the ‘‘STORFORSK” Project No. 167397/V30 is greatly acknowledged. The first author would like to acknowledge Dr. Erling Østby and Dr. Sigmund Kyrre Ås of SINTEF Materials and Chemistry for their valuable comments and discussions.
References [1] Ainsworth RA, Sharples JK, Smith SD. Effects of residual stresses on fracture behaviour–experimental results and assessment methods. J Strain Anal Engng Des 2000;35(4):307–16. [2] Østby E, Thaulow C, Zhang ZL. Numerical simulations of specimen size and mismatch effects in ductile crack growth–Part I. Tearing resistance and crack growth paths. Engng Fract Mech 2007;74:1770–92. [3] Sherry AH, Wilkes MA, Sharples JK, Budden PJ. The assessment of residual stress effects on ductile tearing using continuum damage mechanics. J Pressure Ves Technol 2008;130. 041212–1–8. [4] Liu J, Zhang ZL, Nyhus B. Residual stress induced crack tip constraint. Engng Fract Mech 2008;75(14):4151–66. [5] Ren XB, Zhang ZL, Nyhus B. Effect of residual stress on crack-tip constraint in a boundary layer model. Int J Solids Struct 2009;46(13):2629–41. [6] Panontin TL, Hill MR. The effect of residual stresses on brittle and ductile fracture initiation predicted by micromechanical models. Int J Fract 1996;82(4):317–33. [7] Sharples JK, France CC, Ainsworth RA. Experimental validation of R6 treatment of residual stresses. ASME PVP 1999;392:225–38. [8] Mirzaee-Sisan A, Truman CE, Smith DJ, Smith MC. Interaction of residual stress with mechanical loading in an austenitic stainless steel. Fatigue Fract Engng Mater Struct 2008;31(3-4):223–33. [9] Mirzaee-Sisan A, Truman CE, Smith DJ, Smith MC. Interaction of residual stress with mechanical loading in a ferritic steel. Engng Fract Mech 2007;74(17):2864–80. [10] Mahmoudi AH, Truman CE, Smith DJ. Using local out-of-plane compression (LOPC) to study the effects of residual stress on apparent fracture toughness. Engng Fract Mech 2008;75(6):1516–34. [11] Larsson SG, Carlsson AJ. Influence of non-singular stress terms and specimen geometry on small-scale yielding at crack tips in elastic–plastic materials. J Mech Phys Solids 1973;21:263–77. [12] Gurson AL. Continuum theory of ductile rupture by void growth. Part I–yield criteria and flow rules for porous ductile media. J Engng Mater Technol 1977;99(2):2–15. [13] Tvergaard V. Influence of voids on shear band instabilities under plane strain conditions. Int J Fract 1981;17(4):389–407. [14] Tvergaard V. On localization in ductile materials containing spherical voids. Int J Fract 1982;18(4):237–52. [15] Tvergaard V, Needleman A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 1984;32(1):157–69. [16] Thomason P. Ductile fracture of metals. Pergamon; 1990. [17] Zhang ZL, Niemi E. A new failure criterion for the Gurson–Tvergaard dilational constitutive model. Int J Fract 1994;70(4):321–34. [18] Zhang ZL, Niemi E. Analyzing ductile fracture using dual dilational constitutive equations. Fatigue Fract Engng Mater Struct 1994;17(6):695–707. [19] Koplik J, Needleman A. Void growth and coalescence in porous plastic solids. Int J Solids Struct 1988;24(8):835–53. [20] Pardoen T, Hutchinson JW. An extended model for void growth and coalescence. J Mech Phys Solids 2000;48(12):2467–512. [21] Zhang ZL, Thaulow C, Ødegård J. A complete Gurson model approach for ductile fracture. Engng Fract Mech 2000;67(2):155–68. [22] ABAQUS Version 6.7 User’s Manual. Hibbit, Karlsson and Sorensen, Inc.; 2007. [23] Zhang ZL, Niemi E. A class of generalized mid-point algorithms for the Gurson–Tvergaard material model. Int J Numer Methods Engng 1995;38(12):2033–53. [24] Zhang ZL. On the accuracies of numerical integration algorithms for Gurson-based pressure-dependent elastoplastic constitutive models. Comput Methods Appl Mech Engng 1995;121(1-4):15–28. [25] Zhang ZL. Explicit consistent tangent moduli with a return mapping algorithm for pressure-dependent elastoplasticity models. Comput Methods Appl Mech Engng 1995;121(1-4):29–44. [26] Ueda Y, Fukuda K, Nakacho K, Endo S. A new measuring method of residual stresses with the aid of finite element method and reliability of estimated values. Trans Jpn Weld Res Inst 1975;4(2):123–31. [27] Hill MR, Nelson DV. The inherent strain method for residual stress determination and its application to a long welded joint. ASME, Pressure Ves Pip Div 1995;318:343–52. [28] Hill MR, Nelson DV. Determining residual stress through the thickness of a welded plate. ASME, Pressure Ves Pip Div 1996;327:29–36. [29] Mochizuki M, Hayashi M, Hattori T. Residual stress analysis by simplified inherent strain at welded pipe junctures in a pressure vessel. J Pressure Ves Technol 1999;121(4):353–7. [30] Rice JR. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 1968;35:379–86. [31] Lei Y. J-integral evaluation for cases involving non-proportional stressing. Engng Fract Mech 2005;72(4):577–96. [32] Lei Y, O’Dowd NP, Webster GA. Fracture mechanics analysis of a crack in a residual stress field. Int J Fracture 2000;106:195–216. [33] Ueda Y, Fukuda K. New measuring method of three-dimensional residual stresses in long welded joints using inherent strains as parameters–Lz method. J Engng Mater Technol 1989;111(1):1–8. [34] Ueda Y, Nakacho K, Yuan M. Application of FEM to theoretical analysis, measurement and prediction of welding residual stresses. Trans JWRI (Jpn Weld Res Inst) (Jpn) 1991;20(1):97–107. [35] Bouchard PJ, Withers PJ. Identification of residual stress length scales in welds for fracture assessment. In: Residual stress and its effects on fatigue and fracture: proceedings of a special symposium held within the 16th European conference of fracture-ECF16, Alexandroupolis, Greece; 3–7 July, 2006. p. 163. [36] Sobotka J, Dodds R, Sofronis P, Effects of hydrogen on steady, ductile crack growth: computational studies. Int J Solids Struct. doi:10.1016/ j.ijsolstr.2009.08.002 .
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[37] Xia L, Shih CF. Ductile crack growth–I. A numerical study using computational cells with microstructurally-based length scales. J Mech Phys Solids 1995;43(2):233–59. [38] Eikrem PA, Zhang ZL, Østby E, Nyhus B. Numerical study on the effect of prestrain history on ductile fracture resistance by using the complete Gurson model. Engng Fract Mech 2008;75:4568–82. [39] Østby E, Thaulow C, Zhang ZL. Numerical simulations of specimen size and mismatch effects in ductile crack growth. Part II: near-tip stress fields. Engng Fract Mech 2007;74:1793–809. [40] Tvergaard V, Hutchinson JW. Effect of T-stress on mode I crack growth resistance in a ductile solid. Int J Solids Struct 1994;31(6):823–33.
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Effect of residual stresses on ductile crack growth resistance X.B. Ren a, Z.L. Zhang a,*, B. Nyhus b a b
Department of Structural Engineering, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway SINTEF Materials and Chemistry, Department of Applied Mechanics and Corrosion, N-7465 Trondheim, Norway
a r t i c l e
i n f o
Article history: Received 29 October 2009 Received in revised form 19 February 2010 Accepted 5 March 2010 Available online 18 March 2010 Keywords: Residual stresses Ductile crack growth resistance Complete Gurson model Eigenstrain method
a b s t r a c t It is well known that residual stresses influence the ductile fracture behaviour. In this paper, a numerical study was performed to assess the effect of residual stresses on ductile crack growth resistance of a typical pipeline steel. A modified boundary layer model was employed for the analysis under plane strain, Mode I loading condition. The residual stress fields were introduced into the finite element model by the eigenstrain method. A sharp crack was embedded in the center of the weld region. The complete Gurson model has been applied to simulate the ductile fracture by microvoid nucleation, growth and coalescence. Results show that tensile residual stresses can significantly reduce the crack growth resistance when the crack growth is small compared with the length scale of the tensile residual stress field. With the crack growth, the effect of residual stresses on the crack growth resistance tends to diminish. The effect of residual stress on ductile crack growth resistance seems independent of the size of geometrically similar welds. When normalized by the weld zone size, the ductile crack growth resistance collapses into one curve, which can be used to assess the structural integrity and evaluate the effect of residual stresses. It has also been found that the effect of residual stresses on crack growth resistance depends on the initial void volume fraction f0, hardening exponent n and T-stress. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction It is widely accepted that the presence of residual stresses can influence the failure characteristics of components and structures. A significant influence due to the residual stress on the failure and load bearing capacity of components under predominantly elastic conditions was demonstrated by Ainsworth et al. [1], who also showed that the effect on load-bearing characteristics reduced with the increasing plasticity within the component. A quantitative understanding of how residual stresses affect the failure behaviour is thus required to provide accurate integrity assessment and reduce the conservatism inherent in many assessment codes. Ductile crack growth plays an important role in the analysis of the fracture behaviour of structures [2]. Crack extension reduces the load-bearing ligament and will influence the capacity of the structures. Ductile crack growth may also change the near-tip stress/strain fields and promote the transition to unstable cleavage fracture. The mechanism of ductile fracture in metallic materials may be considered as a three-stage process: nucleation, growth and coalescence of microvoids. The ductile fracture process is influenced by the local conditions of stress triaxiality and plastic strain within the vicinity of a stress concentrator such as a notch or a crack-tip [3]. Liu et al. [4] and Ren et al. [5] have demonstrated that residual stresses can induce an additional crack-tip constraint, and a parameter R was defined based on the difference between the full stress
* Corresponding author. Tel.: +47 73 59 25 30. E-mail address: [email protected] (Z.L. Zhang). 0013-7944/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2010.03.007
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field and the reference field to quantify the residual stress-induced crack-tip constraint. It is thus interesting to further investigate how residual stresses influence the local failure mechanisms and global ductile crack resistance. Panontin and Hill [6] predicted the brittle and ductile initiation by micromechanical models and showed that the effect of residual stress on the ductile fracture initiation toughness is negligible. Sherry et al. [3] demonstrated that a high strength low toughness aluminum alloy AL2024-T351 showed a marked reduction in initiation and tearing toughness for specimens containing a mechanically induced residual stress field. It should be noted that the fracture behaviour of AL2024-T351 remains in ductile fracture regime. Experimental work undertaken by Sharples et al. [7] on wide plate specimens has also demonstrated that residual stress can significantly influence the ductile tearing behaviour of engineering materials. However, the experiments performed by Mirzaee-Sisan et al. [8] on the AISI Type 361H stainless steel indicates a negligible impact on ductile tearing toughness at load ratio Lr close to 1, i.e. close to the plastic collapse of the specimen. The observation is in contrast to their previous experiments performed on the A533B unstable cleavage fracture in ferritic steels at 150 °C [9]. A reduction of approximately 46% was observed in apparent fracture toughness for specimens containing a residual stress field compared with specimens in the as-received condition. Mahmoudi et al. [10] employed a local out-of-plane compression method to introduce residual stresses into C(T) specimens for aluminum alloys A12650 and A12024. Their experimental results showed that tensile residual stresses reduced the initiation cleavage fracture toughness of A12650 by about one half, and the ductile tearing resistance of A12024 also decreased when tensile residual stresses present. To date, the fundamental understanding of the effect of the residual stresses on ductile fracture resistance remains an open challenge. To this end, we utilized a micromechanical damage model in a Modified Boundary Layer (MBL) model to investigate the problem. The MBL model has been widely used to investigate the crack problems [11], and the transferability to real components can be well handled by simply changing the boundary conditions. In recent years, damage mechanics models have become widely used in numerical simulation of ductile crack growth. The Gurson model [12] is often used to assess ductile tearing behaviour under primary loading, where the effect of the void growth is taken into account in the constitutive model. The Gurson model has later been modified by Tvergaard and Needleman [13–15], thus, it is most often referred to as the Gurson–Tvergaard–Needleman (GTN) model. The GTN model is in fact a void growth model. The effect of void coalescence can be considered with the introduction of a so-called critical void volume fraction, which is not a physical mechanism-based coalescence criterion. Thomason [16] proposed a plastic limit load model for coalescence, which was easy to implement and very accurate [17,18] compared with the finite elements results by Koplik and Needleman [19]. This work has been further improved by considering the plastic strain hardening [20]. By combining the GTN model for void growth and Thomason’s plastic limit load model for coalescence a so-called ‘‘complete Gurson model” has been proposed by Zhang et al. [21], with which ductile fracture is exclusively linked to the void nucleation parameters and the mesh size. Two scenarios have been considered to study the effect of the residual stresses on the crack growth resistance in this paper. Firstly, the effect of residual stresses on ductile crack resistance in a large round weld region was investigated. In such a way, the final crack growth length still locates in weld region. Secondary, a much smaller rectangular weld region was constructed, where the crack can propagate through the whole weld region. The effect of the weld zone size was also taken into account and studied. In addition, the effects of initial void volume fraction and material hardening have also been studied. 2. Numerical procedure 2.1. Problem description This study concerns an ideal problem. A large cylinder with a weld in the center was studied. The cylinder was simulated by a 2D plane strain MBL model with a remote boundary governed by the elastic K-field and T-stress. A sharp crack was embedded in the weld region. The analysis procedure, as illustrated in Fig. 1, consists of the following steps: (1) enforce a welding procedure in un-cracked body, which introduces a residual stress field; (2) introduce a sharp crack; and (3) apply the external load. The contact between the upper and lower free surfaces of the crack is considered when the residual stress was introduced.
Fig. 1. Illustration of the problem. (a) The round cylinder; (b) welding at the center and a sharp crack is introduced; and (c) apply the external load.
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Fig. 2. Modified boundary layer model, (a) global view and (b) crack-tip mesh.
2.2. Finite element model The present analyses were carried out for the conditions of small-scale-yielding. The MBL model used for this study consists of a weld region located in the center of the model, an outer base metal region, and a sharp crack in the center of weld. The load was applied to the remote edges of the model through a displacement field (u, v) controlled by the elastic asymptotic stress field of a plane strain Mode I crack:
rffiffiffiffiffiffiffi r 1 1 m2 cos h ð3 4m cos hÞ þ T r cos h 2p 2 E rffiffiffiffiffiffiffi 1þm r 1 mð1 þ mÞ sin v ðr; hÞ ¼ K I h ð3 4m cos hÞ T r sin h 2p 2 E E uðr; hÞ ¼ K I
1þm E
ð1Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where K I ¼ EJ=ð1 m2 Þ under plane strain condition, E is Young’s modulus, m is Poisson’s ratio, and r and h are polar coordinates centered at the crack tip with h = 0 corresponding to the crack tip. The finite element computations were performed using ABAQUS [22]. Due to symmetry, only the upper half of the geometry is modeled. The mesh is shown in Fig. 2. The radius of the MBL model is taken as 1000 mm to ensure the small-scaleyielding condition is fulfilled. Close to the crack tip, there is a rectangular region (9.4 mm ahead of the initial crack tip and 1.6 above the symmetry line) with uniform mesh sizes of 0.1 0.05 mm for the first layer and 0.1 0.1 mm for the rest of the layers are created. In following context, lc = 0.1 mm represents the size of the uniform elements. Full integration four-node 2D plane strain elements are used. The finite element model has 2616 elements. Nonlinear geometry effects (NLGEOM in ABAQUS) are accounted for in the analyses. An initial opening of 0.02 mm is applied for the upper half model. When the residual stresses are introduced into the model, the crack faces can be partly closed. Thus, a rigid analytical plane is defined in the model to simulate the contact of the crack surfaces. 2.3. Materials As described in Section 1, the complete Gurson model is utilized to simulate the crack growth, which is implemented into ABAQUS using a set of generalized-midpoint algorithms developed by Zhang via the material user subroutine UMAT1 [23– 25]. Both the weld metal and base metal are modeled as a power-law hardening material: 1
A free copy of the ABAQUS UMAT Fortran code can be obtained from the corresponding author.
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rf ¼ r0 1 þ
ep
n ð2Þ
e0
where rf is the flow stress, ep is the equivalent plastic strain, the Poisson’s ratio m is 0.3, Young’s modulus is 200,000 MPa, yield stress r0 is 400 MPa, e0 = r0/E is the yield strain and n is the plastic strain hardening exponent. The yield function of the Gurson model has the following form:
; f ; rm Þ ¼ /ðq; r
3q2 rm 1 ðq1 f Þ2 ¼ 0 þ 2q fcosh 1 2r r2f q2
ð3Þ
where f is the void volume fraction, rm the mean macroscopic stress, q the von Mises stress, rf the flow stress, and q1 and q2 are parameters introduced by Tvergaard [13,14]. Fixed values of q1 = 1.5 and q2 = 1.0 have been used in this study. In this study, a cluster void nucleation model has been used [21]. The critical void volume fraction (fc) is automatically determined by Thomason’s plastic limit load criterion [16,20]. The post-coalescence deformation behaviour of the Gurson model is numerically simulated by an artificial acceleration of void growth, as suggested by Tvergaard and Needleman [15]:
(
f ¼
f
for f 6 fc f f
fc þ fuF fcc ðf fc Þ for f > fc
ð4Þ
where fu ¼ 1=q1 and fF is the void volume fraction at the end of void coalescence. Here, fF = 0.15 + 2f0, where f0 is the initial void volume fraction. When the coalescence starts and f > fc, f* replaces f in Eq. (3). 2.4. Eigenstrain method Different thermal expansion coefficients of the base metal (ab) and weld metal (aw) were used to introduce the residual stresses into the model by the so-called eigenstrain method. It should be noted that the thermal expansion coefficients here are not physical thermal coefficients, but are used to introduce residual stresses into the FE model. The eigenstrain method was also called ‘‘inherent strain” method when first introduced by Ueda et al. [26]. The concept of the eigenstrain method is that the source of residual stress is an incompatible strain field caused by plastic deformation, thermal strains and phasetransformation etc. [27]. Thus, if the distribution of the eigenstrain is known, the distribution of residual stresses can be obtained through linear elastic calculation by using the finite element method. Our approach is to set the eigenstrain values equal to the thermal expansion coefficients for different regions. The model is then loaded by applying a unit temperature decrease, thereby introducing the residual stress field of interest. Similar approaches were also applied in the literature [28,29], and their results showed that the distributions of residual stresses obtained by such approaches agreed well with the experimental results. 2.5. The crack driving force In fracture mechanics, the J-integral [30] or Crack Tip Opening Displacement (d, CTOD) are used to characterize the cracktip driving force of a cracked body made of an elastic-plastic material. The J-integral is a path-independent integral based on the assumption that the strain energy density is a single-valued function of the strain (or stress) [31]. However, non-proportional loading may occur in the region where the J-integral is evaluated and lead to path-dependence, for example, in the case of ductile crack growth [2], or in the presence of a residual stress field [32]. In this study, the J-integral is evaluated on contours far away from the zone of highly non-proportional loading, and displays practically path independence in both cases with and without residual stresses. Residual stresses as an additional stress field induce an initial J-integral. However, the residual stress-induced J-integral is negligible compared to the final J-integral caused by the combination of residual stress and external load. Also, under the small-scale-yielding conditions, the difference between calculated J-integral and applied J-integral is less than 2.78% for the case without residual stress, and 3.25% for the case with residual stresses [5]. Thus, J -integral should be applicable in the present study, and the J in the following context represents far-field J-integral. 3. Results and discussion 3.1. Residual stress fields As described in Section 2.4, we used the eigenstrain method to introduce residual stresses into the finite element model. A round weld region was introduced in the center of the model, as illustrated in Fig. 3. According to the eigenstrain method, the thermal expansion coefficients of both the weld metal (aw) and base metal (ab) were assumed to be isotropic and equal to the eigenstrain values respectively. In this section, we assume ab=0. Four residual stress fields were generated by setting aw= 0.001, 0.001, 0.002, and 0.003 and designated as RsField0, RsField1, RsField2, and RsField3 respectively. Fig. 4 shows the distribution of the residual stresses both before and after the crack was inserted. Note that the stresses are normalized by the yield stress, and the distance from the crack tip x is normalized by the size of the uniform element lc.
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Fig. 3. Illustration of the weld region (gray area). The radius of the weld is 20 mm.
3
RsField0 RsField0
RsField2 RsField2
4 3
0
RsField0 RsField0
crack
RsField1 RsField1
crack
RsField2 crack RsField2
crack
RsField3 crack RsField3
1
2
RsField3 crack RsField3
1 0
−1 −2 0
b
RsField1 crack RsField1
2
σ11/σ0
crack
σ22/σ0
a
−1
100
200
300
400
x/lc
500
−2 0
100
200
300
400
500
x/lc
Fig. 4. Comparison of the residual stress distribution in MBL model with a large round weld before (bold line) and after (thin line) the crack was inserted, (a) components parallel to the crack plane and (b) normal to the crack plane. Four different residual stress cases were considered, where RsField0 is compressive and the remaining 3 are tensile. ab = 0; aw = 0.001, 0.001, 0.002, and 0.003; lc = 0.1 mm.
It can be seen that the negative eigenstrain value introduces the compressive residual stress at the weld region while the positive ones generate tensile residual stresses. Both tensile and compressive residual stresses parallel to the crack front converge to zero far from the crack tip. The opening residual stresses are self-balanced ahead of the crack tip. There is a sharp turning point in the distribution of the opening residual stress, which is the region where eigenstrain discontinuity has been introduced into the FE model, namely the weld metal-base metal boundary. The tensile residual stresses also show similarity, and the level of the tensile residual stress increases with the increasing aw. Due to the singularity, r11 is about 800 MPa and r22 is about 1200 MPa at the crack tip for RsField2. Fig. 4 also shows that residual stress components are smaller than the yield stress before the crack was inserted. 3.2. Effect of residual stresses on ductile crack resistance Ductile crack growth resistance is important for structural integrity assessment, and it is interesting to investigate the effect of residual stresses on it. In this section, effects of residual stress fields on the crack growth resistance were studied. The initial void volume fraction f0 is fixed to be 0.1% and the strain hardening exponent n = 0.1. The crack growth resistance described by the J-integral is showed in Fig. 5. For the cases chosen, the residual stresses seem to have significant effects on the ductile crack growth resistance. Fig. 5a shows that the compressive residual stress enhances the crack growth resistance while the tensile residual stresses have the
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a
b
25
1.6
RsField0 RsField1 RsField2 RsField3
1.4
20
without residaul stress
15
J/Jwithout
J/(σ0lc)
1.2
10 no/res RsField0 RsField1 RsField2 RsField3
5 0 0
5
10
15
Δa/lc
20
25
30
1 0.8 0.6 0.4 0.2 0
5
10
15
20
25
30
Δa/lc
Fig. 5. Effect of residual stresses on crack growth resistance, (a) absolute crack resistance curves and (b) normalized crack resistance curves. f0 = 0.1%; n = 0.1; E/r0 = 500.
Fig. 6. Illustration of the weld region (gray area). The length of the weld region along the ‘‘11” direction is 9.6 mm, and the length along the ‘‘22” direction is 6.4 mm.
opposite effect. With the increase of tensile residual stress, the crack growth resistance decreases. The absolute difference between the case with and without residual stress increases with the increase of crack growth. Fig. 5b presents the normalized crack growth resistance. The resistances with residual stresses were normalized by the resistance at the same crack growth without residual stresses. The distance from the original crack tip was normalized by lc. It can be seen that the tensile residual stresses significantly decrease the initiation toughness while the compressive residual stress increases it. With advancing crack growth, the effect of the residual stresses decreases and approaches to a constant value. Note that the current crack grows only to 3 mm, which is very small compared to the size of the residual stress dominant length scale shown in Fig. 4. As shown in Fig. 4, beyond the singularity affected zone (x/lc > 30, i.e. 3 mm), the residual stress fields approach a hydrostatic stress state. Thus, the residual stresses cannot be easily released by the crack growth, and the effect of residual stresses retains. In order to better understand the effect of residual stress on crack growth resistance, a smaller rectangular weld region was constructed, as illustrated in Fig. 6. In this section, ab=0, and aw was assumed to be orthogonal and characterized by a11 and a22 in following context. The ratio a11/a22 was fixed to be 2. Four residual stress fields with a22=-0.001, 0.001, 0.002, and 0.003 were generated and represented by RsField0, RsField1, RsField2, and RsField3 respectively, as shown in Fig. 7. It should be noted that the eigenstrain values selected here are taken from experimental measurement results in literature [33,34]. The residual stress fields generated by the selected values have similar distribution as the residual stress fields showed in Ref. [35]. To obtain accurate distributions of the residual stress fields by the eigenstrain method, one should carry out the experiments to measure the distribution of the eigenstrain. However, the main objective of this study is to investigate the effect of the residual stresses, and prediction of the real distribution of the residual stress field is outside of the scope. The absolute and normalized crack growth resistances are shown in Fig. 8. It can be seen that the compressive residual stress increases the crack growth resistance while tensile residual stresses decrease the crack growth resistance, as shown in
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a
3
RsField0 RsField1 RsField2 RsField3
2
0
3
RsField0 RsField1 RsField2 RsField3
2
σ22/σ0
σ11/σ0
1
b
1 0 −1
−1 −2
−2 0
50
100
150
200
0
50
100
150
200
x/lc
x/lc
Fig. 7. Residual stress distribution in MBL model with a rectangular weld after the crack was inserted, (a) components parallel to the crack plane and (b) normal to the crack plane. Four different residual stress cases were considered, where RsField0 is compressive and the remaining 3 are tensile. a11/a22 = 2; a22 = 0.001, 0.001, 0.002, and 0.003.
a
b
25
1.2 1.1
20
J/Jwithout
J/(σ0lc)
1 15 10 no/res RsField0 RsField1 RsField2 RsField3
5 0
0
5
10
15
20
25
30
35
Δa/lc
40
0.9 0.8 0.7 RsField0 RsField1 RsField2 RsField3
0.6 0.5
0
10
20
30
40
Δa/lc
Fig. 8. Effect of residual stresses on crack growth resistance, (a) absolute crack resistance curves and (b) normalized crack resistance curves. f0 = 0.1%; n = 0.1; E/r0 = 500.
Fig. 8a. Fig. 8b shows that the reduction of the crack growth resistance converges with crack growth to the case without residual stress. As known, the quantity rm/re defines a convenient measure of triaxiality linked to the growth rate of micro-scale voids consistent with the subsequently introduced damage measures [36]. Therefore, we investigated the distribution of triaxiality ahead of the crack tip. Fig. 9 presents the triaxiality values on the ligament ahead of the original crack tip for different crack growth. The distance from the original crack tip are scaled by J/r0, or equivalently (KI/r0)2, which defines approximately the value of crack-tip opening displacement and provides a physical meaningful length scale for normalization. It can be seen that tensile residual stresses enhance the triaxiality values while the compressive residual stress reduces the triaxiality value at the crack initiation, see Fig. 9a, which corresponds to our previous findings [4,5]. However, the effect of residual stress on triaxiality tend to be negligible when the crack advanced to 3.5 mm, as shown in Fig. 9b. Higher stress triaxility corresponds to a smaller plastic zone. Thus, the energy dissipated by the plastic deformation is smaller, which in turn result in a lower crack growth resistance. 3.3. Effect of weld zone size on crack growth resistance The length scale of the residual stress field may play an important role on the effect of residual stress on ductile crack growth resistance, as shown in Section 3.2. To better demonstrate this, four geometrically similar rectangular weld regions were constructed, as illustrated in Fig. 10. The size of the weld is designated as c. The orthogonal thermal expansion coefficient was used with a value of a11=0.004 and a22=0.002 for all welds. The residual stress fields designated as Size1, Size2, Size3, and Size4 respectively are showed in Fig. 11. The residual stresses are tensile in the weld metal and show similar features as the previous residual stresses shown in Fig. 7. With the increase of the weld size, the residual stresses parallel to the crack plane increase; the size of tensile dominated region of the opening stress also increases.
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b
3
3
2.75
2.75
2.5
2.5
σm/σe
σm/σe
a
2.25
2
2 no/res RsField0 RsField1 RsField2 RsField3
1.75 1.5
2.25
1
2
3
4
no/res RsField0 RsField1 RsField2 RsField3
1.75
5
1.5
1
2
x/(J/σ0)
3
4
5
x/(J/σ0)
Fig. 9. Triaxiality values ahead the crack tip for different crack propagation, (a) Da = 0.1 mm; (b) Da = 3.5 mm. f0 = 0.1%; n = 0.1; E/r0 = 500.
Fig. 10. Illustration of weld size. The size increases by a scale of 2.
Fig. 12a presents the absolute crack growth resistance. In Fig. 12b, crack growth length was normalized by łc. It can be seen that the residual stresses generated in the larger welds influence more significantly the crack growth resistance. With increasing crack growth, the effect of the residual stresses on the crack growth resistance converges to the case without residual stresses. However, an interesting pattern emerges when normalizing the crack growth length by the weld zone size c, as shown in Fig. 12c. Surprisingly, the normalized crack growth resistances collapse into one curve, which indicates that the effect of the residual stresses on the crack growth resistance is nearly independent of the weld zone size. For the cases specified in this section, when the crack grows approximately to 3 times of c, the effect of residual stresses can be neglected. For geometrically similar welds, the uniform curve can be used to roughly predict the length scale of residual stress-affected region beyond which the effect of the residual stress can be neglected. In our previous study [5], it has been shown that residual stress fields generated in round weld regions with three sizes respectively can be normalized by the weld zone size and collapsed into a uniform field. Thus, the ‘‘master” curve as shown in Fig. 12c can be expected. 3.4. Effect of residual stresses on crack growth resistance for different hardening exponents It has been shown that the effect of strain hardening on the ductile crack resistance is not fully understood. Xia and Shih [37] demonstrated that the ductile resistance increases with increasing hardening capacity. However, Eikrem et al. [38] and Østby et al. [2,39] reported that decreasing the hardening exponent will significantly raise the resistance curve. In this study, the effect of strain hardening on the crack growth resistance was investigated for cases both with and without residual stress. It should be noted that in this particular study, the initial volume fraction f0 was fixed to be 0.05%, and the residual stress field was introduced by the rectangular weld with orthogonal thermal expansion coefficient a11 = 0.004 and a22 = 0.002, i.e. RsField2 in Fig. 7. The crack growth resistance was plotted as a function of hardening exponent n in Fig. 13. Fig. 13a shows that with decreasing hardening exponent the crack growth resistance increases for both with and without residual stress cases, which corresponds to the results demonstrated in Ref. [2,38,39]. For a given hardening exponent, residual stresses reduce both the initial toughness and crack growth resistance. Also, the effect of residual stress on the ductile resistance becomes stronger for stronger hardening material, as shown in Fig. 13b. With increasing crack growth, the reduction of the crack growth resistance decreases and tends to converge to the case without residual stress. Residual stresses reduce the equivalent plastic strain significantly at crack initiation, which indicates smaller plastic deformation. Hence,
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a
2
Size1 Size2 Size3 Size4
b
2 1.5 1
σ22/σ0
σ11/σ0
1.5
1
Size1 Size2 Size3 Size4
0.5
0.5
0 −0.5
0 0
50
100
150
200
0
50
100
150
200
x/lc
x/lc
Fig. 11. Residual stress distribution in MBL model with geometrically similar welds after the crack was inserted, (a) components parallel to the crack plane and (b) normal to the crack plane. a11 = 0.004, a22 = 0.002.
a
b
18 16
1 0.95
J/Jwithout
J/(σ0lc)
14 12 10
0.9 0.85
8
no/res Size1 Size2 Size3 Size4
6 4
1.05
0
5
10
15
20
Size1 Size2 Size3 Size4
0.8 0.75
25
0
5
10
c
15
20
25
Δ a/lc
Δ a/lc
1.05 1
J/Jwithout
0.95 0.9 0.85 Size1 Size2 Size3 Size4
0.8 0.75
0
1
2
3
4
Δa/c Fig. 12. Effect of weld size on crack growth resistance, (a) absolute crack growth resistance, (b) crack growth normalized by lc, and (c) normalized by the weld zone size c. f0 = 0.1%; n = 0.1; E/r0 = 500.
lower crack growth resistance curves can be expected. With the crack growth, the effect of residual stresses on the equivalent plastic strain becomes negligible for different hardening exponents. 3.5. Effect of residual stresses on ductile crack resistance for different initial void volume fraction The initial void volume fraction f0 represents the degree of damage in the material. The larger the initial void volume fraction is, the larger damage the material has. In this section, the effect of residual stress fields on crack growth resistance was
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investigated for three initial void volume fractions, f0 = 0.05%, 0.1% and 0.2%. The residual stress was introduced into a rectangular weld with a11 = 0.004 and a22 = 0.002, i.e. RsField2 in Fig. 7. The crack growth resistance curves are shown in Fig. 14. As shown in Fig. 14a, for both with and without residual stress cases, the crack growth resistance increases with the decrease of initial void volume fraction, which can be expected because the ductility becomes better when initial void volume fraction decreases. Also, it can be observed that the residual stress reduces the crack growth resistance for fixed f0. Normalized crack resistance curves shown in Fig. 14b indicate that with increasing crack growth, the effects of residual stresses decrease and become less dependent on f0. The residual stress enhances the opening stress beyond the larger strain effect region compared with the case without residual stress at crack initiation, which induces an increase of crack-tip constraint [4,5], and a lower fracture toughness can then be expected. However, it has been found that the effect of the residual stress on opening stress becomes negligible when crack growth becomes larger. 3.6. Effect of residual stresses on ductile crack resistance for different T/r0 Crack tip constraint effects on fracture toughness have received considerable attention recently. In our earlier work [5], we have found that the residual stress-induced crack-tip constraint is lower for a higher geometric constraint. Xia and Shih [37], Tvergaard and Hutchinson [40] studied the effect of T-stress on the crack growth resistance and showed that a negative T-stress results in a rapidly rising resistance curve while the positive T-stress lowers the fracture resistance. It is thus interesting to investigate how residual stress combined with varying T-stress affect the crack growth resistance. In this paper, T/ r0 = 0.5, 0, and 0.5 were studied. The initial void volume fraction is fixed to be 0.1%, and the residual stress with a11 = 0.004 and a22 = 0.002, i.e. RsField2 in Fig. 7, was introduced. The absolute and normalized resistance curves are shown in Fig. 15. It can be seen that the tensile residual stress reduces the crack growth resistance for all T/r0 cases, as shown in Fig. 15a. Fig. 15b shows that the normalized crack growth resistance is lower for smaller T-stress. However, the differences between
a
30 25
1.05 1 0.95
J/Jwithout
20
J/(σ0lc)
b
n=0.05 no/res n=0.05 with/res n=0.1 no/res n=0.1 with/res n=0.15 no/res n=0.15 with/res
15
0.9
10
0.85
5
0.8
0
0
5
10
15
0.75
20
n=0.05 n=0.1 n=0.15
0
5
Δa/lc
10
Δa/lc
15
20
Fig. 13. Effect of residual stresses on crack growth resistance for different hardening, (a) absolute crack growth resistance curves and (b) normalized crack resistance curves. a11 = 0.004; a22 = 0.002; f0 = 0.05%; E/r0 = 500.
45
0
1
f =0.01% with/res 0
f0=0.05% no/res
35
0.95
f0=0.05% with/res f0=0.1% no/res
30
J/(σ0lc)
b
f =0.01% no/res
40
f0=0.1% with/res
25
J/Jwithout
a
20
0.9
15 0.85
10
f =0.01% 0
f =0.05%
5 0
0
0
5
10
Δ a/lc
15
20
0.8
f0=0.1%
0
5
10
15
20
Δ a/lc
Fig. 14. Effect of residual stresses on crack growth resistance for different initial void volume fraction, (a) absolute crack growth resistance curves and (b) normalized crack resistance curves. a11 = 0.004; a22 = 0.002; n = 0.1; E/r0 = 500.
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a
b
16
0.95
14
10 8
T/σ 0=−0.5 no/res T/σ0=−0.5 with/res
6
0.85
0.75
T/σ0=0 no/res T/σ =0 with/res
T/σ =−0.5
T/σ =0.5 no/res
T/σ =0
0
0
4 2
J/Jwithout
J/(σ0lc)
12
0
0
T/σ0=0.5 with/res
0
5
10
Δ a/lc
15
20
0.65
T/σ =0.5 0
0
5
10
15
20
Δ a/lc
Fig. 15. Effect of the residual stress on normalized crack growth resistances for different T/r0, (a) absolute crack growth resistance curves and (b) normalized crack resistance curves. a11 = 0.004; a22 = 0.002; f0 = 0.1%; n = 0.1; E/r0 = 500.
the normalized crack growth resistances are insignificant. It should be noted that a relatively small initial void fraction was used in this study, i.e. f0 = 0.1%. With increasing crack growth, the effect of residual stresses on crack growth resistance tends to be independent of the T-stress. The positive residual stresses reduce the plastic strain compared to the case without residual stresses. Lower equivalent plastic strain indicates a smaller plastic zone and thus the energy dissipated by the plastic deformation decreases. In turn, the ductile crack toughness decreases. When the crack growth becomes larger, the effect of residual stresses on the equivalent plastic strain becomes negligible. 4. Conclusions This study describes a systematic computational investigation of the effect of residual stresses on ductile crack growth resistance. 2D plane strain finite element analysis have been carried out in a modified boundary layer model with the remote boundary condition governed by K-field and T-stress. The large strain effect has been taken into account in the analysis. The complete Gurson model was utilized to predict the crack growth resistance for both with and without residual stress situations. Residual stresses were generated by the eigenstrain method and introduced into the FE model. Both the tensile and compressive residual stress fields were introduced. However, tensile residual stress was known to be detrimental for the fracture. Thus, the effect of the tensile residual stress on crack growth resistance was mainly presented. Residual stresses as an additional field do not alter the material constitutive relation, however, the residual stress may influence the neartip stress state and enhance the level of damage along the ligament ahead of an advancing crack tip. The following conclusions can be made: 1. Tensile residual stresses reduce the crack growth resistance while compressive residual stress can enhance the crack growth resistance. With the increase of crack growth, the effect of residual stresses decreases and converges to the case without residual stress. The initial damage of the ligament elements caused by the residual stress is negligible for the cases studied. 2. Under certain circumstances, the effect of the residual stresses on ductile crack growth resistance can be normalized by the size of the geometrically similar weld, and the normalized crack growth resistance curves collapse into a single curve. One can use this curve to evaluate the effect of the residual stress on the structural integrity and simplify the assessment procedure. It can also be used to predict the length scale of the residual stress-affected range beyond which the effect of the residual stresses can be neglected. For the cases specified in this study, when the crack grows to a length of 3c the effect of the residual stresses can be neglected. 3. The reduction of the crack growth resistance caused by residual stresses is smaller for weaker hardening materials, and the influence of the residual stresses decreases with crack propagation. 4. Residual stress reduces the crack growth resistance more significantly for the materials with larger initial void volume. The influence of the residual stresses decreases with the crack growth and becomes independent of f0. 5. The reduction of the crack growth resistance induced by the residual stress increases with increasing T/r0 at the early stage of crack growth and then tend to be negligible when the crack growth is larger. The results are in line with our earlier findings that the effect of residual stress on the crack-tip constraint becomes weaker for higher T/r0. The present study indicates that residual stress as an additional stress field can alter the stress state near the crack tip and further influence the ductile crack growth resistance. Triaxility or crack-tip constraint was shown to play an important role in the ductile fracture behaviour. Therefore, the residual stress-induced crack-tip constraint is a very important factor to consider in structural integrity assessment. In our previous studies [4,5], we have defined a parameter R to quantify the residual
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stress-induced crack-tip constraint. However, the effect of residual stress on the ductile crack resistance has not been linked with R by a quantitative way in this study. The reason for this is that the residual stress field in reality is very complicated, hence one should find an efficient way to standardize the distribution of the residual stresses for typical weld joints. To reduce the conservatism of current integrity assessments, a proper description of the residual stress field is very important. Once a proper description of the residual stress is obtained, the ‘‘master‘‘ curve, though obtained from the MBL model, can be applied to real engineering structures and components that contain similar residual stress fields under smallscale-yielding conditions. The transferability of the results from MBL studies to real structures should be verified in further studies.
Acknowledgment The funding from the Research Council of Norway through the ‘‘STORFORSK” Project No. 167397/V30 is greatly acknowledged. The first author would like to acknowledge Dr. Erling Østby and Dr. Sigmund Kyrre Ås of SINTEF Materials and Chemistry for their valuable comments and discussions.
References [1] Ainsworth RA, Sharples JK, Smith SD. Effects of residual stresses on fracture behaviour–experimental results and assessment methods. J Strain Anal Engng Des 2000;35(4):307–16. [2] Østby E, Thaulow C, Zhang ZL. Numerical simulations of specimen size and mismatch effects in ductile crack growth–Part I. Tearing resistance and crack growth paths. Engng Fract Mech 2007;74:1770–92. [3] Sherry AH, Wilkes MA, Sharples JK, Budden PJ. The assessment of residual stress effects on ductile tearing using continuum damage mechanics. J Pressure Ves Technol 2008;130. 041212–1–8. [4] Liu J, Zhang ZL, Nyhus B. Residual stress induced crack tip constraint. Engng Fract Mech 2008;75(14):4151–66. [5] Ren XB, Zhang ZL, Nyhus B. Effect of residual stress on crack-tip constraint in a boundary layer model. Int J Solids Struct 2009;46(13):2629–41. [6] Panontin TL, Hill MR. The effect of residual stresses on brittle and ductile fracture initiation predicted by micromechanical models. Int J Fract 1996;82(4):317–33. [7] Sharples JK, France CC, Ainsworth RA. Experimental validation of R6 treatment of residual stresses. ASME PVP 1999;392:225–38. [8] Mirzaee-Sisan A, Truman CE, Smith DJ, Smith MC. Interaction of residual stress with mechanical loading in an austenitic stainless steel. Fatigue Fract Engng Mater Struct 2008;31(3-4):223–33. [9] Mirzaee-Sisan A, Truman CE, Smith DJ, Smith MC. Interaction of residual stress with mechanical loading in a ferritic steel. Engng Fract Mech 2007;74(17):2864–80. [10] Mahmoudi AH, Truman CE, Smith DJ. Using local out-of-plane compression (LOPC) to study the effects of residual stress on apparent fracture toughness. Engng Fract Mech 2008;75(6):1516–34. [11] Larsson SG, Carlsson AJ. Influence of non-singular stress terms and specimen geometry on small-scale yielding at crack tips in elastic–plastic materials. J Mech Phys Solids 1973;21:263–77. [12] Gurson AL. Continuum theory of ductile rupture by void growth. Part I–yield criteria and flow rules for porous ductile media. J Engng Mater Technol 1977;99(2):2–15. [13] Tvergaard V. Influence of voids on shear band instabilities under plane strain conditions. Int J Fract 1981;17(4):389–407. [14] Tvergaard V. On localization in ductile materials containing spherical voids. Int J Fract 1982;18(4):237–52. [15] Tvergaard V, Needleman A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 1984;32(1):157–69. [16] Thomason P. Ductile fracture of metals. Pergamon; 1990. [17] Zhang ZL, Niemi E. A new failure criterion for the Gurson–Tvergaard dilational constitutive model. Int J Fract 1994;70(4):321–34. [18] Zhang ZL, Niemi E. Analyzing ductile fracture using dual dilational constitutive equations. Fatigue Fract Engng Mater Struct 1994;17(6):695–707. [19] Koplik J, Needleman A. Void growth and coalescence in porous plastic solids. Int J Solids Struct 1988;24(8):835–53. [20] Pardoen T, Hutchinson JW. An extended model for void growth and coalescence. J Mech Phys Solids 2000;48(12):2467–512. [21] Zhang ZL, Thaulow C, Ødegård J. A complete Gurson model approach for ductile fracture. Engng Fract Mech 2000;67(2):155–68. [22] ABAQUS Version 6.7 User’s Manual. Hibbit, Karlsson and Sorensen, Inc.; 2007. [23] Zhang ZL, Niemi E. A class of generalized mid-point algorithms for the Gurson–Tvergaard material model. Int J Numer Methods Engng 1995;38(12):2033–53. [24] Zhang ZL. On the accuracies of numerical integration algorithms for Gurson-based pressure-dependent elastoplastic constitutive models. Comput Methods Appl Mech Engng 1995;121(1-4):15–28. [25] Zhang ZL. Explicit consistent tangent moduli with a return mapping algorithm for pressure-dependent elastoplasticity models. Comput Methods Appl Mech Engng 1995;121(1-4):29–44. [26] Ueda Y, Fukuda K, Nakacho K, Endo S. A new measuring method of residual stresses with the aid of finite element method and reliability of estimated values. Trans Jpn Weld Res Inst 1975;4(2):123–31. [27] Hill MR, Nelson DV. The inherent strain method for residual stress determination and its application to a long welded joint. ASME, Pressure Ves Pip Div 1995;318:343–52. [28] Hill MR, Nelson DV. Determining residual stress through the thickness of a welded plate. ASME, Pressure Ves Pip Div 1996;327:29–36. [29] Mochizuki M, Hayashi M, Hattori T. Residual stress analysis by simplified inherent strain at welded pipe junctures in a pressure vessel. J Pressure Ves Technol 1999;121(4):353–7. [30] Rice JR. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 1968;35:379–86. [31] Lei Y. J-integral evaluation for cases involving non-proportional stressing. Engng Fract Mech 2005;72(4):577–96. [32] Lei Y, O’Dowd NP, Webster GA. Fracture mechanics analysis of a crack in a residual stress field. Int J Fracture 2000;106:195–216. [33] Ueda Y, Fukuda K. New measuring method of three-dimensional residual stresses in long welded joints using inherent strains as parameters–Lz method. J Engng Mater Technol 1989;111(1):1–8. [34] Ueda Y, Nakacho K, Yuan M. Application of FEM to theoretical analysis, measurement and prediction of welding residual stresses. Trans JWRI (Jpn Weld Res Inst) (Jpn) 1991;20(1):97–107. [35] Bouchard PJ, Withers PJ. Identification of residual stress length scales in welds for fracture assessment. In: Residual stress and its effects on fatigue and fracture: proceedings of a special symposium held within the 16th European conference of fracture-ECF16, Alexandroupolis, Greece; 3–7 July, 2006. p. 163. [36] Sobotka J, Dodds R, Sofronis P, Effects of hydrogen on steady, ductile crack growth: computational studies. Int J Solids Struct. doi:10.1016/ j.ijsolstr.2009.08.002 .
X.B. Ren et al. / Engineering Fracture Mechanics 77 (2010) 1325–1337
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[37] Xia L, Shih CF. Ductile crack growth–I. A numerical study using computational cells with microstructurally-based length scales. J Mech Phys Solids 1995;43(2):233–59. [38] Eikrem PA, Zhang ZL, Østby E, Nyhus B. Numerical study on the effect of prestrain history on ductile fracture resistance by using the complete Gurson model. Engng Fract Mech 2008;75:4568–82. [39] Østby E, Thaulow C, Zhang ZL. Numerical simulations of specimen size and mismatch effects in ductile crack growth. Part II: near-tip stress fields. Engng Fract Mech 2007;74:1793–809. [40] Tvergaard V, Hutchinson JW. Effect of T-stress on mode I crack growth resistance in a ductile solid. Int J Solids Struct 1994;31(6):823–33.