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The influence of material mismatch on the evaluation of time-dependent fracture mechanics parameters
Engineering Fracture Mechanics 64 (1999) 765±780
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The in¯uence of material mismatch on the evaluation of time-dependent fracture mechanics parameters Shan.-Tung Tu a,*, Kee.-Bong Yoon b a
Department of Mechanical Engineering, Nanjing University of Chemical Technology, Nanjing, 210009, People's Republic of China b Department of Mechanical Engineering, Chung Ang University, Seoul, 156-756, South Korea Received 25 January 1999; received in revised form 29 August 1999
Abstract The in¯uence of material mismatch on the stress ®eld of uniaxial specimens and the time-dependent fracture mechanics parameters is studied in the present work. The applicability of the conventional C equation based on the load line displacement is re-examined by using the ®nite element method. It is found that under the extensive creep condition the C value of hard weld/soft parent metal specimen can be signi®cantly higher than that of a single weld metal specimen, and the material mismatch has little in¯uence on C(t ) in small scale creep in the studied cases. It is proposed that the C formula based on the load line displacement recommended by ASTM E1457 needs to be modi®ed to interpret the CCG behaviour of welded specimens. Candidate modi®cation factors have been proposed. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Material mismatch; Time-dependent fracture; Weld; Creep crack growth; Testing standard
1. Introduction In engineering practice, premature failures of weldments due to creep crack growth are a common occurrence. This indicates the detrimental eect of welding on structural integrity and has thus been a problem entangling engineers and researchers since it was introduced in industry. Many fatigue strength data of weldments have been well documented for normal * Corresponding author. Tel.: +86-25-3316755; fax: +86-25-3211316. E-mail address: [email protected] (S.T. Tu). 0013-7944/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 9 9 ) 0 0 0 8 3 - 1
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temperatures since the 1950s [1]. In contrast, at elevated temperatures there are only limited studies that can aid in design and life assessment of the weldments. Over the past 30 years, considerable progress has been made on the creep tests [2±4] and theoretical calibration of damage development in the weldments [5±7]. However, creep crack growth (CCG) studies seemingly have mostly concentrated on homogeneous materials where the crack growth rate is correlated with C(t ) in the extensive creep regimes according to ASTM E1457 [8], and with Ct in the transition regime from small scale creep to extensive creep [9]. Many available creep crack growth tests on weldments were correlated with C(t ) or Ct which is generally suitable for homogeneous materials [10,11]. Some recent studies have found that the time-dependent fracture parameters can be signi®cantly in¯uenced by material mismatch indicating that the direct use of the parameters may not be possible [12,13]. Hence a rigorous mechanics examination of the parameters is needed for predictions of CCG behaviour in weldments where the base metal and weld metal have dissimilar creep deformation rates. In order to provide some mechanics basis for the interpretation of creep crack growth rates, the ®nite element method is used to study the in¯uence of material mismatch on the stress ®eld of a uniaxial specimen and thus on the time-dependent fracture mechanics in the present work. The applicability of the conventional C equation, based on the load line displacement, is examined.
2. The in¯uence of material mismatch on the stress distribution in uniaxial specimen There have been many examples showing the stress redistributions of a welded pipe subjected to internal pressure [2,14]. In fact, as long as there is a material mismatch, the stress redistribution is inevitable as creep and damage develop in the materials. Even in a uniaxial cross-weld specimen of a uniform loading section area, the stress redistribution can be signi®cant. In order to have a basic idea of creep stress distribution in the specimens, computations by using the ®nite element method [15] are performed on two types of uniaxial cross-weld specimen. In all the computations the materials are assumed to follow the following constitutive equation: e_
s_ Bsn E
1
where s and e are the stress and strain respectively, and the dots designate their respective derivatives with time; E is the elastic modulus and B and n are regression constants. The ®rst term on the right-hand side of Eq. (1) represents the elastic strain rate which is only pertinent when stress redistribution is occurring. The second term represents the secondary creep portions of the creep deformation behaviour. A corresponding material mismatch factor is de®ned as: M Bp =Bw where Bp and Bw are material constants in Eq. (1) with subscripts p and w denoting parent
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metal and weld metal, respectively. If M < 1, the weld is referred to as creep-soft weld while if M > 1, it is a creep-hard weld. The ®rst case studied is a plate specimen of a complete weld pro®le. The specimen has a width of 12 mm and a length of 40 mm and is loaded in tension at a stress of 100 MPa. The two-material model is used, with B = 1.772 10ÿ14, n = 3.87 for weld and B = 1.772 10ÿ13, n = 3.87 for parent metal. The weld metal has a creep strain rate 10 times lower than the parent metal and is a creep-hard weld. Plane strain condition is assumed. Half of the specimen is modelled with eight-node bi-quadratic elements (CPE8R). Computation is carried out until a stationary stress state is reached. In the case of normal temperature, no creep occurs. It is expected that the stress must remain uniform and constant with time. In the high temperature creep case, however, stress redistribution is inevitable. Fig. 1 illustrates the von Mises stress contours in the specimen. Fig. 2 shows the stress along the specimen. It is seen that the stress distribution is not uniform in the welded specimen. The stress in weld metal is relatively higher while the stress in parent metal is lower than the applied stress. At the weld root the stress increases to about 1.7 times the applied stress. This is due to deformation compatibility that requires higher stress for the material of smaller creep rate. It can thus be understood that the creep-hard material will generally experience a higher stress while the creep-soft material will experience a lower stress. The other case analyzed is the creep of a cross-weld bar specimen. The specimen has a diameter of 8 mm, a length of 40 mm and a HAZ width of 2.5 mm and is loaded with a tension stress of 110 MPa. The constitutive parameters for dierent zones are obtained based on material data of 1Cr0.5Mo weldments from creep tests at 5508C, as listed in Table 1 (stress
Fig. 1. Von Mises stress contours in plate specimen.
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Fig. 2. Von Mises stress variation along plate specimen.
in MPa, time in h). Under the applied stress, the HAZ has a creep strain rate of about 10 times higher than that of parent metal and about 6.4 times that of the weld metal. Thus the HAZ material is creep-soft in the specimen. Due to its axisymmetric feature, a quarter of the specimen is modelled with eight-node biquadratic axisymmetric elements (CPX8R). The stress distributions after creep for 2000 h are given in Figs. 3 and 4. It is found that there is a signi®cant variation of the stresses across the HAZ. The stress in HAZ decreases while the stress in the weld and parent metals increases. Maximum Mises stress Table 1 Material constants for the PM, HAZ and WM for 1Cr0.5Mo steel Location
B
n
E
m
PM HAZ WM
1.940 10ÿ15 1.540 10ÿ13 1.772 10ÿ14
4.354 3.925 3.870
1.47 105 1.47 105 1.47 105
0.3 0.3 0.3
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Fig. 3. Von Mises stress contours in bar specimen.
occurs in the weld adjacent to the HAZ which is about 1.4 times the applied stress. In contrast, in the HAZ the stress is decreased to about 70% of the applied stress. The stress redistribution feature due to the mismatch of the material properties suggests that the time dependent fracture parameters in a solid of material mismatch can be very much dierent from those in a homogenous solid depending on the location of the crack tip. No matter whether it is a three-material model or a two-material model, it has the same feature that the stress in the creep-soft material is decreased while the stress in the creep-hard material is enhanced during the creep process. In order to have a reasonable simplicity in computation the following discussion is therefore limited to two-material model.
3. The in¯uence of material mismatch on time-dependent fracture parameters It has been shown that the elastic strain rates become negligible compared with creep rates as the crack tip is approached and the near tip stress and strain rate ®elds are thus of HRR type [16,17]. The near tip stress and strain ®elds are expressed as:
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Fig. 4. Mises strain variation across HAZ in bar specimen.
sij e_ij
C t In Br
C t In Br
1= n1
1= n1
s~ ij y, n
2
e~ ij y, n
3
where In is a factor accounting for stress state and s~ ij y, n, e~ ij y, n are non-dimensional functions [18]. C(t ) represents the intensity of the stress singularity and is de®ned as a contour integral. Analogue to the plastic analysis, a creep zone can also be de®ned by assuming the stress ®eld outside the zone is essentially the same as elastic stress ®eld and its size can be estimated as: n1=nÿ1 2=nÿ1 K2 In 4 rc y, t bc n, y 2p BC t where rc is the creep zone size, K is the elastic stress intensity factor and bc(n, y ) a nondimensional angular function. In the long term, as the creep zone grows and extensive creep condition is achieved, the C(t ) approaches a constant value C. It had been shown that over
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the entire transition regime C(t ) can be represented as follows within an error of 5% tT C C t 1 t
771
5
with tT
K 2 1 ÿ n2 n 1EC
6
As it is demonstrated in Section 2, the creep stress ®eld in a uniaxial cross weld specimen can be quite non-uniform. It is easy to understand that the change in the primary stress ®eld due to material mismatch will certainly in¯uence the evaluation of fracture mechanics parameters. The local crack tip HRR stress ®eld and the primary stress ®eld may also have interacting eect. In order to evaluate the eect of material mismatch, a number of ®nite element computations are carried out for compact tension (CT) and single edge notched tension (SENT) specimens with and without material mismatch. The ®nite element models are shown in Fig. 5. The CT specimen has been widely used and recommended in ASTM E1457. The SENT specimen is generally used for life assessment of welded pipes in-service since it simulates better the constraint and prior damage ahead of a circumferential crack in the weld. Eight-node bi-
Fig. 5. Finite element model: (a) CT specimen, a/w = 0.5, h = 1.5 mm, 918 elements, 2941 nodes; (b) SENT specimen, a/w = 0.5, 704 elements, 2233 nodes.
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quadratic reduced integration elements (CPE8R) are used in the modelling. The meshes are re®ned until a converged C solution is obtained. Let us ®rst look into the CT specimen. The specimen has a width of 30 mm and a thickness of 13 mm. The material constants are B = 1.772 10ÿ14, n = 3.87 for weld metal. The B value is assigned to parent metal according to material mismatch factor while n is kept unchanged. The crack in the model is located at h = 1.5 mm (the distance between the crack line and the interface). The applied load is 3920 N. For comparison purpose, single weld metal specimens are ®rstly calculated (M = 1). According to Eq. (6), the transition time is 219.6 h. Fig. 6 shows the development of creep zone in the case M = 10. As compared to the single weld material case, in the ®rst 10 h the shape of the creep zone is symmetrical and no evident in¯uence of material mismatch is observed. When creep time reaches 12 h, part of the parent metal also enters the creep region as the result of higher creep strain rate. Two separated creep zones are thus formed. An unbalance development of the creep zone in the crack tip is found. The zone close to the interface is slightly larger than the other due to higher creep stress level created in the weld metal. The zones are then connected to each other. Fig. 7 shows the dierence of creep zones between weld/parent metal specimen and the single weld metal specimen. The transition time has been reached for weld/parent metal specimen while the single weld metal specimen is still in small scale creep.
Fig. 6. Development of creep zone in specimen M = 10: (a) t = 10 h; (b) t = 12 h; (c) t = 41.9 h; (d) t = 62 h.
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Fig. 7. Comparison of creep zones between weld/parent metal specimen and the single weld metal specimen: (a) weld/parent metal specimen, t = 102 h; (b) single weld metal specimen, t = 102 h.
Fig. 8 shows the evolution of C(t ) values with time. Values from dierent contour integrals are illustrated in the Figure at the same time. Both creep-soft and creep-hard welds are calculated. It is obvious that in the small scale creep region the C(t ) is a path-dependent variable and curves for specimens of dierent mismatch factors overlap each other indicating
Fig. 8. Evolution of C(t ) values with time in CT specimen.
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that the material mismatch has little in¯uence on C(t ) in the small scale creep. After the transition time is ®nished, all integrals fall in a same curve showing the path independent feature of C(t ). When the material mismatch takes eect, i.e. the stress redistribution due to material mismatch plays its role, the C(t ) curves deviates from those of the single material specimen. It is seen the transition completes earlier for creep hard weld and later for creep soft weld. Under the extensive creep condition, C value of hard weld/soft parent metal specimen is 3.24 times the value of single weld metal specimen, while the value of soft weld/hard parent metal specimen is about 2/3 of the value of single weld metal specimen. For the SENT specimen, similar computations are carried out. The specimen has a width of 12 mm, a thickness of 5 mm and a half length of 20 mm. The same material properties as CT specimen are used. The applied load is a tension stress of 60 MPa on the top of the specimen. For the single weld metal specimen, the transition time given by Eq. (6) is 52.87 h. Fig. 9 gives a comparison of creep zones between the weld/parent metal specimen and the single weld metal specimen at the same time. No obvious dierence is found when the creep is in a small scale. It is seen, however, that in the longer time the creep zone in the weld/parent metal specimen is enlarged due to stress enhancement in the weld. Creep zones are developed both in the weld and parent metals and are connected in the later time. Fig. 10 shows the evolution of C(t ) values with time in the SENT specimen. The same trend as the CT specimen is observed. However, the in¯uence of material mismatch is not as signi®cant as that in the CT specimen. This is mainly due to the specimen geometry and the distance between the crack tip and the interface of the two materials. When a welded specimen of a very soft parent metal is encountered (M = 100), the C is about 3 times that of the single weld metal specimen. However, for a welded specimen of a very hard parent metal (M = 0.01), the decrease of the C value is not dramatic and is not more than 20%.
Fig. 9. Comparison of creep zones between the weld/parent metal specimen and the single weld metal specimen: (a) t = 20.6 h; (b) t = 77 h.
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Fig. 10. Evolution of C(t ) values with time in SENT specimen.
Tentative conclusions can therefore be drawn from the computations: the material mismatch can signi®cantly in¯uence the time-dependent fracture parameters in the extensive creep condition; a higher parent metal creep strain rate will increase the C value while a lower parent metal creep strain rate decrease the C value. 4. The applicability of measured C(t ) for two-material specimen In the CCG experiments, C or Ct are generally calculated in terms of measured load line displacement rather than time-consuming ®nite element computations. Under steady state creep condition, C can be expressed as: C Z a=w, n
PV_ c BW
7
where Z is a function of a/w and n, Vc the load line displacement, P the applied load, B the thickness and W the width of the specimen. It is apparent that this formula is established
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without considering the material mismatch. As Vc is also directly measured from the welded specimen, it may re¯ect partially the in¯uence of material mismatch. But the problem is to what extent that the formula can describe the in¯uence of the material mismatch and how to modify it for welded specimens. A material mismatch coecient can thus be introduced to modify Eq. (7): C W g M, a=w, hZ a=w, n
PV_ wC BW
8
where g(M, a/W, h ) describes the material mismatch eect which can be the function of M, a/ w, and h. C W is C-integral for welded specimen, VwC is the load line displacement. A division of Eq. (8) by Eq. (7) results in: g M, a=w, h f n, a=w, P=BW
C W V_ wC
9
where f n, a=w, P=BM V_ c =C , is determined from a single material specimen. To quantify the g value, a number of ®nite element computations are carried out for dierent n values and also a/w values. The previous CT and SENT ®nite element models are used.
Fig. 11. g value vs mismatch factor, M for CT specimen.
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In order to obtain the f value, the ®nite element computation is ®rst carried out for a single weld specimen. Based upon the calculated C and Vc value, the f value is determined. Then computations are carried out for a number of welded specimens of the same geometry with material mismatches. Fig. 11 is the variation of g versus material mismatch factor, M for a CT specimen. It is seen that 0 < g R 1 for Mr 1 and g > 1 for M < 1 which implies that Eq. (7) may overestimate C for creep-hard weld but slightly underestimate it for creep-soft weld. The n value has a signi®cant in¯uence on the g value. For a same M value, the smaller the n value, the larger the deviation of g. The distance between the crack tip and the interface line is also a major factor. It seems that reducing the distance may reduce the deviation. Fig. 12 is the variation of g for the SENT specimen when n = 3.87. More signi®cant deviation of g is found as compared with CT specimen. For M = 10 and a/w = 5/12, the value of g is 0.48 which implies that Eq. (7) may overestimate the C value of the welded specimen by one time. Eq. (7) may underestimate the C value for M <1. A factor of 1.58 is required to modify Eq. (7) in case that M = 0.1 and a/w = 5/12. Fig. 13 is the variation of g for the SENT specimen when n = 11.475. It is obvious that a longer crack requires less modi®cation due to less interaction between the crack tip and the material interface. As compared with the lower n value case in Fig. 12, the deviation of g is less signi®cant.
Fig. 12. g value vs mismatch factor, M for SENT specimen, n = 3.87.
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Fig. 13. g value vs mismatch factor, M for SENT specimen, n = 11.475.
5. Conclusions In order to provide some mechanics basis for the interpretation of creep crack growth rates in solids of inhomogeneous materials, the ®nite element method is used to study the in¯uence of material mismatch on the stress ®eld of uniaxial specimens and the time-dependent fracture mechanics parameters in the present work. The applicability of conventional C equation based on the load line displacement is re-examined. Some conclusions have been drawn from the study. 1. Even in a uniaxial tension specimen, the stress redistribution can be very signi®cant as long as material mismatch exists in the high temperature components. 2. Under the extensive creep condition, the C value of hard weld/soft parent metal specimen can be more than 3 times the value of single weld metal specimen, while the value of soft weld/hard parent metal specimen can be less than 15% of the value of single weld metal specimen. The material mismatch has little in¯uence on C(t ) in small scale creep in the cases studied. 3. The C formula based on the load line displacement recommended by ASTM E1457 needs to be modi®ed to interpret the CCG behaviour of welded specimens. A modi®cation factor, g is introduced. It is found that 0 < g R 1 for Mr 1 and g > 1 for M < 1 which implies
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that the recommended formula may overestimate C for creep-hard weld but underestimate it for creep-soft weld. It is suggested that further computations shall be carried out in order to quantify the relationship between the g function and h value.
Acknowledgements The supports provided by the China Natural Science Foundation (59875039) and Korean Science and Engineering Foundation (1997 Brain Pool Program) are gratefully acknowledged. The authors also wish to thank colleagues who have contributed to the preparation of this paper.
References [1] Maddox SJ. Fatigue strength of welded structures. Abington Publishing, 1985. [2] Coleman MC, Parker JD, Walters DJ. The behavior of ferritic weldment in thick section 1/2Cr1/2Mo1/4 V pipe at elevated temperature. Int J Pres Ves & Piping 1985;18:277±310. [3] DeWitte M, Cossement G. Creep behavior of 2.25Cr1Mo base material and weldments. In: Proceedings of VGB-Conferemce ``Residual Service Life 1992'', Mannheim, 6±7 July 1992, 1992. p. 38.1±38.28. [4] Prager M. Issues in life assessment of chrome-moly welds. In: Serviceability of petroleum, process and power equipment, vol. PVP-Vol.239. New York: ASME, 1992. p. 253±65. [5] Tu ST, Wu R, Sandstrom R. Design against creep failure for weldments in 0.5Cr0.5Mo 0.25 V pipe. Int J Pres Ves & Piping 1994;58:345±54. [6] Hall FR, Hayhurst DE. Continuum damage mechanics modeling of high temperature deformation and failure in a pipe weldment. Proc R Soc Lond A 1991;433:383±403. [7] Tu ST, Sandstrom R. Numerical simulation of creep exhaustion of weldments and some design considerations. In: Wilshire B, Evans RE, editors. Proceedings of the Fifth International Conference on Creep and Fracture of Engineering Materials and Structures. Institute of Materials, 1993. p. 695±704. [8] ASTM Standard E1457. Standard test method for measurement of creep crack growth rates in metals, 1992. [9] Saxena A, Liaw PK. Remaining life assessment of boiler pressure parts Ð crack growth studies. Report No. EPRI-CS 4688, Electric Power Research Institute, Palo Alto, CA, 1986. [10] Saxena A, Han J, Banerji K. Creep crack growth behaviour in power plant boiler and steam pipe steels. EPRI Report on Project 2253-10, Electric Power Research Institute, Palo Alto, CA, April, 1987. [11] McCarthy PR, Shibli IA, Raynor D. A survey of European creep crack growth data generation from weldments and recommendations for a draft code of practice. In: Proceedings of International HIDA Conference on Creep and Fatigue Crack Growth in High Temperature Plant, 15±17 April, Paris, France, 1998. p. S2±12. [12] Segle P, Andersson P, Samuelson L. A parameteric study of creep growth in heterogenous CT specimens by use of ®nite element simulations. In: Proceedings of International HIDA Conference on Creep and Fatigue Crack Growth in High Temperature Plant, 15±17 April, Paris, France, 1998. p. S2±17. [13] Tu ST, Yoon KB. Welding eect on the structural integrity at high temperature. In: Proceedings of 1998 Annual Meeting of Korean Welding Society. KWS, 1998. p. 11±24 (Invited Lecture). [14] Tu ST, Sandstrom R. The evaluation of weldment creep strength reduction factors by experiment and numerical simulations. Int Pres Ves & Piping 1994;57:335±44. [15] ABAQUS, Version 5.7. Hibbit, Karlsson and Sorensen, Inc, 1997. [16] Riedel R, Rice JR. Tensile cracks in creeping solids. In: Proceedings of Fracture Mechanics, 12th Conference, ASTM STP 700, 1980. p. 112±30.
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[17] Ohji K, Kubo S. Fracture mechanics evaluation of crack growth behaviour under creep and creep fatigue conditions. In: Ohtani R, Ohnami M, Inoue T, editors. High temperature creep-fatigue. London: Elsevier, 1988. p. 91±113. [18] Shih CF. Tables of Hutchinson±Rice±Rosengren singular ®eld quantities. Material Research Laboratories, Brown University, MRL E-147, 1983.
www.elsevier.com/locate/engfracmech
The in¯uence of material mismatch on the evaluation of time-dependent fracture mechanics parameters Shan.-Tung Tu a,*, Kee.-Bong Yoon b a
Department of Mechanical Engineering, Nanjing University of Chemical Technology, Nanjing, 210009, People's Republic of China b Department of Mechanical Engineering, Chung Ang University, Seoul, 156-756, South Korea Received 25 January 1999; received in revised form 29 August 1999
Abstract The in¯uence of material mismatch on the stress ®eld of uniaxial specimens and the time-dependent fracture mechanics parameters is studied in the present work. The applicability of the conventional C equation based on the load line displacement is re-examined by using the ®nite element method. It is found that under the extensive creep condition the C value of hard weld/soft parent metal specimen can be signi®cantly higher than that of a single weld metal specimen, and the material mismatch has little in¯uence on C(t ) in small scale creep in the studied cases. It is proposed that the C formula based on the load line displacement recommended by ASTM E1457 needs to be modi®ed to interpret the CCG behaviour of welded specimens. Candidate modi®cation factors have been proposed. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Material mismatch; Time-dependent fracture; Weld; Creep crack growth; Testing standard
1. Introduction In engineering practice, premature failures of weldments due to creep crack growth are a common occurrence. This indicates the detrimental eect of welding on structural integrity and has thus been a problem entangling engineers and researchers since it was introduced in industry. Many fatigue strength data of weldments have been well documented for normal * Corresponding author. Tel.: +86-25-3316755; fax: +86-25-3211316. E-mail address: [email protected] (S.T. Tu). 0013-7944/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 9 9 ) 0 0 0 8 3 - 1
766
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temperatures since the 1950s [1]. In contrast, at elevated temperatures there are only limited studies that can aid in design and life assessment of the weldments. Over the past 30 years, considerable progress has been made on the creep tests [2±4] and theoretical calibration of damage development in the weldments [5±7]. However, creep crack growth (CCG) studies seemingly have mostly concentrated on homogeneous materials where the crack growth rate is correlated with C(t ) in the extensive creep regimes according to ASTM E1457 [8], and with Ct in the transition regime from small scale creep to extensive creep [9]. Many available creep crack growth tests on weldments were correlated with C(t ) or Ct which is generally suitable for homogeneous materials [10,11]. Some recent studies have found that the time-dependent fracture parameters can be signi®cantly in¯uenced by material mismatch indicating that the direct use of the parameters may not be possible [12,13]. Hence a rigorous mechanics examination of the parameters is needed for predictions of CCG behaviour in weldments where the base metal and weld metal have dissimilar creep deformation rates. In order to provide some mechanics basis for the interpretation of creep crack growth rates, the ®nite element method is used to study the in¯uence of material mismatch on the stress ®eld of a uniaxial specimen and thus on the time-dependent fracture mechanics in the present work. The applicability of the conventional C equation, based on the load line displacement, is examined.
2. The in¯uence of material mismatch on the stress distribution in uniaxial specimen There have been many examples showing the stress redistributions of a welded pipe subjected to internal pressure [2,14]. In fact, as long as there is a material mismatch, the stress redistribution is inevitable as creep and damage develop in the materials. Even in a uniaxial cross-weld specimen of a uniform loading section area, the stress redistribution can be signi®cant. In order to have a basic idea of creep stress distribution in the specimens, computations by using the ®nite element method [15] are performed on two types of uniaxial cross-weld specimen. In all the computations the materials are assumed to follow the following constitutive equation: e_
s_ Bsn E
1
where s and e are the stress and strain respectively, and the dots designate their respective derivatives with time; E is the elastic modulus and B and n are regression constants. The ®rst term on the right-hand side of Eq. (1) represents the elastic strain rate which is only pertinent when stress redistribution is occurring. The second term represents the secondary creep portions of the creep deformation behaviour. A corresponding material mismatch factor is de®ned as: M Bp =Bw where Bp and Bw are material constants in Eq. (1) with subscripts p and w denoting parent
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metal and weld metal, respectively. If M < 1, the weld is referred to as creep-soft weld while if M > 1, it is a creep-hard weld. The ®rst case studied is a plate specimen of a complete weld pro®le. The specimen has a width of 12 mm and a length of 40 mm and is loaded in tension at a stress of 100 MPa. The two-material model is used, with B = 1.772 10ÿ14, n = 3.87 for weld and B = 1.772 10ÿ13, n = 3.87 for parent metal. The weld metal has a creep strain rate 10 times lower than the parent metal and is a creep-hard weld. Plane strain condition is assumed. Half of the specimen is modelled with eight-node bi-quadratic elements (CPE8R). Computation is carried out until a stationary stress state is reached. In the case of normal temperature, no creep occurs. It is expected that the stress must remain uniform and constant with time. In the high temperature creep case, however, stress redistribution is inevitable. Fig. 1 illustrates the von Mises stress contours in the specimen. Fig. 2 shows the stress along the specimen. It is seen that the stress distribution is not uniform in the welded specimen. The stress in weld metal is relatively higher while the stress in parent metal is lower than the applied stress. At the weld root the stress increases to about 1.7 times the applied stress. This is due to deformation compatibility that requires higher stress for the material of smaller creep rate. It can thus be understood that the creep-hard material will generally experience a higher stress while the creep-soft material will experience a lower stress. The other case analyzed is the creep of a cross-weld bar specimen. The specimen has a diameter of 8 mm, a length of 40 mm and a HAZ width of 2.5 mm and is loaded with a tension stress of 110 MPa. The constitutive parameters for dierent zones are obtained based on material data of 1Cr0.5Mo weldments from creep tests at 5508C, as listed in Table 1 (stress
Fig. 1. Von Mises stress contours in plate specimen.
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Fig. 2. Von Mises stress variation along plate specimen.
in MPa, time in h). Under the applied stress, the HAZ has a creep strain rate of about 10 times higher than that of parent metal and about 6.4 times that of the weld metal. Thus the HAZ material is creep-soft in the specimen. Due to its axisymmetric feature, a quarter of the specimen is modelled with eight-node biquadratic axisymmetric elements (CPX8R). The stress distributions after creep for 2000 h are given in Figs. 3 and 4. It is found that there is a signi®cant variation of the stresses across the HAZ. The stress in HAZ decreases while the stress in the weld and parent metals increases. Maximum Mises stress Table 1 Material constants for the PM, HAZ and WM for 1Cr0.5Mo steel Location
B
n
E
m
PM HAZ WM
1.940 10ÿ15 1.540 10ÿ13 1.772 10ÿ14
4.354 3.925 3.870
1.47 105 1.47 105 1.47 105
0.3 0.3 0.3
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Fig. 3. Von Mises stress contours in bar specimen.
occurs in the weld adjacent to the HAZ which is about 1.4 times the applied stress. In contrast, in the HAZ the stress is decreased to about 70% of the applied stress. The stress redistribution feature due to the mismatch of the material properties suggests that the time dependent fracture parameters in a solid of material mismatch can be very much dierent from those in a homogenous solid depending on the location of the crack tip. No matter whether it is a three-material model or a two-material model, it has the same feature that the stress in the creep-soft material is decreased while the stress in the creep-hard material is enhanced during the creep process. In order to have a reasonable simplicity in computation the following discussion is therefore limited to two-material model.
3. The in¯uence of material mismatch on time-dependent fracture parameters It has been shown that the elastic strain rates become negligible compared with creep rates as the crack tip is approached and the near tip stress and strain rate ®elds are thus of HRR type [16,17]. The near tip stress and strain ®elds are expressed as:
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Fig. 4. Mises strain variation across HAZ in bar specimen.
sij e_ij
C t In Br
C t In Br
1= n1
1= n1
s~ ij y, n
2
e~ ij y, n
3
where In is a factor accounting for stress state and s~ ij y, n, e~ ij y, n are non-dimensional functions [18]. C(t ) represents the intensity of the stress singularity and is de®ned as a contour integral. Analogue to the plastic analysis, a creep zone can also be de®ned by assuming the stress ®eld outside the zone is essentially the same as elastic stress ®eld and its size can be estimated as: n1=nÿ1 2=nÿ1 K2 In 4 rc y, t bc n, y 2p BC t where rc is the creep zone size, K is the elastic stress intensity factor and bc(n, y ) a nondimensional angular function. In the long term, as the creep zone grows and extensive creep condition is achieved, the C(t ) approaches a constant value C. It had been shown that over
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the entire transition regime C(t ) can be represented as follows within an error of 5% tT C C t 1 t
771
5
with tT
K 2 1 ÿ n2 n 1EC
6
As it is demonstrated in Section 2, the creep stress ®eld in a uniaxial cross weld specimen can be quite non-uniform. It is easy to understand that the change in the primary stress ®eld due to material mismatch will certainly in¯uence the evaluation of fracture mechanics parameters. The local crack tip HRR stress ®eld and the primary stress ®eld may also have interacting eect. In order to evaluate the eect of material mismatch, a number of ®nite element computations are carried out for compact tension (CT) and single edge notched tension (SENT) specimens with and without material mismatch. The ®nite element models are shown in Fig. 5. The CT specimen has been widely used and recommended in ASTM E1457. The SENT specimen is generally used for life assessment of welded pipes in-service since it simulates better the constraint and prior damage ahead of a circumferential crack in the weld. Eight-node bi-
Fig. 5. Finite element model: (a) CT specimen, a/w = 0.5, h = 1.5 mm, 918 elements, 2941 nodes; (b) SENT specimen, a/w = 0.5, 704 elements, 2233 nodes.
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quadratic reduced integration elements (CPE8R) are used in the modelling. The meshes are re®ned until a converged C solution is obtained. Let us ®rst look into the CT specimen. The specimen has a width of 30 mm and a thickness of 13 mm. The material constants are B = 1.772 10ÿ14, n = 3.87 for weld metal. The B value is assigned to parent metal according to material mismatch factor while n is kept unchanged. The crack in the model is located at h = 1.5 mm (the distance between the crack line and the interface). The applied load is 3920 N. For comparison purpose, single weld metal specimens are ®rstly calculated (M = 1). According to Eq. (6), the transition time is 219.6 h. Fig. 6 shows the development of creep zone in the case M = 10. As compared to the single weld material case, in the ®rst 10 h the shape of the creep zone is symmetrical and no evident in¯uence of material mismatch is observed. When creep time reaches 12 h, part of the parent metal also enters the creep region as the result of higher creep strain rate. Two separated creep zones are thus formed. An unbalance development of the creep zone in the crack tip is found. The zone close to the interface is slightly larger than the other due to higher creep stress level created in the weld metal. The zones are then connected to each other. Fig. 7 shows the dierence of creep zones between weld/parent metal specimen and the single weld metal specimen. The transition time has been reached for weld/parent metal specimen while the single weld metal specimen is still in small scale creep.
Fig. 6. Development of creep zone in specimen M = 10: (a) t = 10 h; (b) t = 12 h; (c) t = 41.9 h; (d) t = 62 h.
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Fig. 7. Comparison of creep zones between weld/parent metal specimen and the single weld metal specimen: (a) weld/parent metal specimen, t = 102 h; (b) single weld metal specimen, t = 102 h.
Fig. 8 shows the evolution of C(t ) values with time. Values from dierent contour integrals are illustrated in the Figure at the same time. Both creep-soft and creep-hard welds are calculated. It is obvious that in the small scale creep region the C(t ) is a path-dependent variable and curves for specimens of dierent mismatch factors overlap each other indicating
Fig. 8. Evolution of C(t ) values with time in CT specimen.
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that the material mismatch has little in¯uence on C(t ) in the small scale creep. After the transition time is ®nished, all integrals fall in a same curve showing the path independent feature of C(t ). When the material mismatch takes eect, i.e. the stress redistribution due to material mismatch plays its role, the C(t ) curves deviates from those of the single material specimen. It is seen the transition completes earlier for creep hard weld and later for creep soft weld. Under the extensive creep condition, C value of hard weld/soft parent metal specimen is 3.24 times the value of single weld metal specimen, while the value of soft weld/hard parent metal specimen is about 2/3 of the value of single weld metal specimen. For the SENT specimen, similar computations are carried out. The specimen has a width of 12 mm, a thickness of 5 mm and a half length of 20 mm. The same material properties as CT specimen are used. The applied load is a tension stress of 60 MPa on the top of the specimen. For the single weld metal specimen, the transition time given by Eq. (6) is 52.87 h. Fig. 9 gives a comparison of creep zones between the weld/parent metal specimen and the single weld metal specimen at the same time. No obvious dierence is found when the creep is in a small scale. It is seen, however, that in the longer time the creep zone in the weld/parent metal specimen is enlarged due to stress enhancement in the weld. Creep zones are developed both in the weld and parent metals and are connected in the later time. Fig. 10 shows the evolution of C(t ) values with time in the SENT specimen. The same trend as the CT specimen is observed. However, the in¯uence of material mismatch is not as signi®cant as that in the CT specimen. This is mainly due to the specimen geometry and the distance between the crack tip and the interface of the two materials. When a welded specimen of a very soft parent metal is encountered (M = 100), the C is about 3 times that of the single weld metal specimen. However, for a welded specimen of a very hard parent metal (M = 0.01), the decrease of the C value is not dramatic and is not more than 20%.
Fig. 9. Comparison of creep zones between the weld/parent metal specimen and the single weld metal specimen: (a) t = 20.6 h; (b) t = 77 h.
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Fig. 10. Evolution of C(t ) values with time in SENT specimen.
Tentative conclusions can therefore be drawn from the computations: the material mismatch can signi®cantly in¯uence the time-dependent fracture parameters in the extensive creep condition; a higher parent metal creep strain rate will increase the C value while a lower parent metal creep strain rate decrease the C value. 4. The applicability of measured C(t ) for two-material specimen In the CCG experiments, C or Ct are generally calculated in terms of measured load line displacement rather than time-consuming ®nite element computations. Under steady state creep condition, C can be expressed as: C Z a=w, n
PV_ c BW
7
where Z is a function of a/w and n, Vc the load line displacement, P the applied load, B the thickness and W the width of the specimen. It is apparent that this formula is established
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without considering the material mismatch. As Vc is also directly measured from the welded specimen, it may re¯ect partially the in¯uence of material mismatch. But the problem is to what extent that the formula can describe the in¯uence of the material mismatch and how to modify it for welded specimens. A material mismatch coecient can thus be introduced to modify Eq. (7): C W g M, a=w, hZ a=w, n
PV_ wC BW
8
where g(M, a/W, h ) describes the material mismatch eect which can be the function of M, a/ w, and h. C W is C-integral for welded specimen, VwC is the load line displacement. A division of Eq. (8) by Eq. (7) results in: g M, a=w, h f n, a=w, P=BW
C W V_ wC
9
where f n, a=w, P=BM V_ c =C , is determined from a single material specimen. To quantify the g value, a number of ®nite element computations are carried out for dierent n values and also a/w values. The previous CT and SENT ®nite element models are used.
Fig. 11. g value vs mismatch factor, M for CT specimen.
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In order to obtain the f value, the ®nite element computation is ®rst carried out for a single weld specimen. Based upon the calculated C and Vc value, the f value is determined. Then computations are carried out for a number of welded specimens of the same geometry with material mismatches. Fig. 11 is the variation of g versus material mismatch factor, M for a CT specimen. It is seen that 0 < g R 1 for Mr 1 and g > 1 for M < 1 which implies that Eq. (7) may overestimate C for creep-hard weld but slightly underestimate it for creep-soft weld. The n value has a signi®cant in¯uence on the g value. For a same M value, the smaller the n value, the larger the deviation of g. The distance between the crack tip and the interface line is also a major factor. It seems that reducing the distance may reduce the deviation. Fig. 12 is the variation of g for the SENT specimen when n = 3.87. More signi®cant deviation of g is found as compared with CT specimen. For M = 10 and a/w = 5/12, the value of g is 0.48 which implies that Eq. (7) may overestimate the C value of the welded specimen by one time. Eq. (7) may underestimate the C value for M <1. A factor of 1.58 is required to modify Eq. (7) in case that M = 0.1 and a/w = 5/12. Fig. 13 is the variation of g for the SENT specimen when n = 11.475. It is obvious that a longer crack requires less modi®cation due to less interaction between the crack tip and the material interface. As compared with the lower n value case in Fig. 12, the deviation of g is less signi®cant.
Fig. 12. g value vs mismatch factor, M for SENT specimen, n = 3.87.
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Fig. 13. g value vs mismatch factor, M for SENT specimen, n = 11.475.
5. Conclusions In order to provide some mechanics basis for the interpretation of creep crack growth rates in solids of inhomogeneous materials, the ®nite element method is used to study the in¯uence of material mismatch on the stress ®eld of uniaxial specimens and the time-dependent fracture mechanics parameters in the present work. The applicability of conventional C equation based on the load line displacement is re-examined. Some conclusions have been drawn from the study. 1. Even in a uniaxial tension specimen, the stress redistribution can be very signi®cant as long as material mismatch exists in the high temperature components. 2. Under the extensive creep condition, the C value of hard weld/soft parent metal specimen can be more than 3 times the value of single weld metal specimen, while the value of soft weld/hard parent metal specimen can be less than 15% of the value of single weld metal specimen. The material mismatch has little in¯uence on C(t ) in small scale creep in the cases studied. 3. The C formula based on the load line displacement recommended by ASTM E1457 needs to be modi®ed to interpret the CCG behaviour of welded specimens. A modi®cation factor, g is introduced. It is found that 0 < g R 1 for Mr 1 and g > 1 for M < 1 which implies
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that the recommended formula may overestimate C for creep-hard weld but underestimate it for creep-soft weld. It is suggested that further computations shall be carried out in order to quantify the relationship between the g function and h value.
Acknowledgements The supports provided by the China Natural Science Foundation (59875039) and Korean Science and Engineering Foundation (1997 Brain Pool Program) are gratefully acknowledged. The authors also wish to thank colleagues who have contributed to the preparation of this paper.
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[17] Ohji K, Kubo S. Fracture mechanics evaluation of crack growth behaviour under creep and creep fatigue conditions. In: Ohtani R, Ohnami M, Inoue T, editors. High temperature creep-fatigue. London: Elsevier, 1988. p. 91±113. [18] Shih CF. Tables of Hutchinson±Rice±Rosengren singular ®eld quantities. Material Research Laboratories, Brown University, MRL E-147, 1983.