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Constraint effects on crack tip stress fields for cracks located at the fusion line of weldments
Computational Materials Science 15 (1999) 275±284
Constraint eects on crack tip stress ®elds for cracks located at the fusion line of weldments a,* ,
C. Thaulow a
Z.L. Zhang b, M. Hauge c, W. Burget d, D. Memhard
d
Norwegian University of Science and Technology, N-7465 Trondheim, Norway b SINTEF, Norway c STATOIL, Norway d Fraunhofer Institut f ur Werkstomechanik, Germany Received 20 January 1999; accepted 1 March 1999
Abstract The case of a deeply notched (a/W 0.3) surface crack positioned at the fusion line of a weldment is considered. The tensile properties of the base material and the heat aected zone (HAZ) are kept constant, while the weld metal properties are changed. First the weld metal yield strength overmatches both the base material and the HAZ, and in the second case there is yield strength evenmatch between the HAZ and weld metal and overmatch with respect to the base material. The eect of these strength mismatch conditions has been examined for two fusion line geometries: straight and slant fusion line. The eect of the crack tip constraint has been characterized with the J±Q±M approach where both geometrical and material mismatch eects can be considered with respect to the critical conditions for cleavage fracture initiation. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Brittle fracture; Mismatch; Constraint; Weldments
1. Introduction High strength steels are now frequently applied for oshore structures, with a speci®ed yield strength of about 450 MPa and a required weld metal global yield strength overmatch of about 10%. But as the base metal strength level is further increased, it has been shown that it will be more and more dicult to obtain the required combination of high tensile properties and toughness for the weld metal. It is therefore a need to de®ne the optimum combination of strength mis-match and
* Corresponding author. Tel.: +47 7359 3821; fax: +47 7359 2931; e-mail: [email protected]
weld metal fracture toughness as a function of the base metal strength level. The lowest toughness values for oshore steels are, however normally experienced in the heat aected zone (HAZ), close to the fusion line, and it has been observed that weld metal strength overmatch can be detrimental with respect to the toughness of the HAZ [1,2]. Fracture mechanics test procedures are designed to obtain as large geometrical constraint as possible and hence give conservative estimates of the toughness. The question is now how strength mismatch (and geometry) in¯uence on the constraint, and then how to transfer the test results from the fracture mechanics test condition to a structure.
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The authors have developed a new frame-work for including the strength mismatch in the constraint parameter, called J±Q±M, and this approach is applied in the present paper. 2. Theory The basic assumption for the single parameter fracture mechanics is that K or J uniquely scales the amplitude of the near crack tip ®elds. However, the relation between the scaling parameter and the near tip ®elds loses the one to one correspondance for large scale yielding in ®nite bodies. Based upon observations from detailed FE calculations, a two parameter approach has been proposed [3,4]. rij FiniteBody rij Reference J Drij Q;
1
where J sets the size scale of the local deformation while the hydrostatic stress parameter Q quanti®es the level of stress triaxiality over a signi®cant distance in which the failure microprocess occurs. In other words, Q only alters the magnitude, but not the shape of the near tip stress ®elds. In view of the self-similarity in the stress ®elds observed above, recent FE calculations of the mismatch eect have been examined [5,6]. The observed similarity in the contours of the maximum principle stresses around the crack gave rise to a formulation of a constraint parameter M to scale the near tip stress ®eld caused by material mismatch [7,8]. Calculations with a bi-material modi®ed boundary layer model (MBL) has shown that the crack tip ®eld can be separated into two parts, one is a reference ®eld of a reference material (rI0 ij ) controlled by J and another one is a dierence ®eld controlled by a mismatch property M. The reference material will be the zone where the stress ®elds are investigated, e.g. where fracture is expected to take place. M is de®ned as the amplitude of the dierence ®eld rij rI0 ij J Mr0 dij fij ;
2
where d are constants, dhh drr 1 and drh ÿ2, r0 is the reference yield stress for the reference
material and f are the dierence ®eld functions which depend upon angular positions and on the properties of the reference material only. There is a dierence between the Q and M ®elds. The Q ®elds are symmetric and do not change the shape of the stress contours but the magnitude. Although M ®elds are self-similar, they are not symmetric around the interface and will change both the shape and the magnitude of the stress contours.
3. Numerical analysis 3.1. FE model In the present analysis the model applied in Ref. [9] was modi®ed to 50 mm thickness, Fig. 1. The ®gure shows the FEM mesh in a global scale and the details of the core mesh and the crack tip arrangement. There are 28 element rings in the core mesh in order to have a sucient number of elements within the range 1 6 r= J =r0 6 5 for a signi®cant part of the loading. The initial notch radius is 0.005 mm and non-singular elements have been used. The model consists of about 1300 8-node plane strain elements. The HAZ thickness is 2 mm. Later the model was modi®ed to also include a straight fusion line under the crack tip for a length of about 2.3 mm, Fig. 2, and the HAZ thickness was slightly changed. Based on previous calculations, where the eect of the HAZ thickness has been examined, it is not expected that this moderate change in thickness will in¯uence the crack tip stress ®eld signi®cantly [10]. In the following sections the two cases considered, are called slant and straight fusion line. 3.2. Material properties The stress±strain curves from one structural steel weldment was considered, Table 1. In order to obtain weld metal over- and evenmatch, the steel was welded with two dierent consumables, while the heat input was kept constant. Hence, only the tensile properties of the
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277
Fig. 1. Slant fusion line model. Three point bend specimen with a/W 0.3, W 50 mm and l 200 mm (a), mesh in a global scale (b), core mesh (c) and the crack tip arrangement in the core mesh (d).
weld metal were changed. The tensile properties for the HAZ were set equal to the coarse grained zone, and were obtained from weld simulated tensile specimens. The curves are ®tted to Ramberg±Osgood parameters, but incremental plasticity was applied in the present calculations.
In both cases a global weld metal overmatch with respect to the base material is obtained. But the most important characteristic with respect to brittle fracture is the local mismatch at the crack tip. And because of dierent strain hardening for the weld metal and the HAZ there will be
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Fig. 2. Straight fusion line model. The de®nition of the HAZ is slightly changed compared with Fig. 1, to include a 2.3 mm long straight fusion line under the crack tip.
deviations from the original yield strength mismatch conditions as the specimens are loaded. 4. Results 4.1. Global load and displacement The J integral calculations from the 28 contours revealed path independence. Fig. 3 displays the relation between the calculated J and the global load and displacement for the four cases considered. For a given global load, evenmatch yields slightly higher J integral than overmatch. But for a given deformation, there is no dierence between the over- and evenmatch condition. 4.2. J±Q±M calculations In order to express the constraint, the J±Q±M approach outlined in Section 2, will now be applied.
Fig. 3. Global load vs J (a), and global displacement vs J (b).
Table 1 Materials data (n is the Ramberg±Osgood strain hardening exponent) Strength mis-match
Base material
HAZ
Weld metal
Global and local overmatch
Re 450 MPa n 15 Re 450 MPa n 15
Re 530 MPa n 9.5 Re 530 MPa n 9.5
Re 643 MPa n 14 Re 530 MPa n 14
Global overmatch and local evenmatch
C. Thaulow et al. / Computational Materials Science 15 (1999) 275±284
The ®rst step is to calculate the small scale yielding HAZ reference ®eld. The coarse grained HAZ is selected as reference material since the lowest fracture toughness for the steels tested are expected to be located at the coarse grained HAZ. A MBL model was applied with the same mesh as in the core model, Fig. 1. The MBL model was loaded to about J 200 N/mm, and the resulting boundary displacements calculated from the elastic stress ®eld of a plane strain mode I crack: r 1ÿm r h cos 3 ÿ 4m ÿ cos h; u r; h KI E 2p 2 3 r 1ÿm r h sin 3 ÿ 4m ÿ cos h; v r; h KI E 2p 2 4 p 2 where KI EJ = 1 ÿ m under plane strain condition, and r and h are polar coordinates centred at the crack tip where h 0 corresponds to the fusion line. In the J±Q±M approach the crack tip stress ®eld is normalized by the yield strength and the J integral. The reference ®eld solution was ®tted by a second degree polynomial giving the following equation x 97:673 ÿ 52:973y 7:283y 2 ; where y r1 /r0 and x r/(J/r0 ). r is the distance from the crack tip and r0 the yield strength of the coarse grained HAZ. The valid range of x is set to 1 6 x 6 5. This range has been selected since it ®ts the microscopically signi®cant zone with respect to brittle fracture [12]. The blunted crack tip zone is avoided by specifying x P 1, corresponding to r P 2d, where d is the crack tip opening displacement. After the homogenous HAZ case has now been determined, the next step is to compare this reference solution with the mis-match calculations. The principle of the procedure is outlined in Fig. 4. First Fig. 4(a) is constructed, where the reference solution is compared with a number of load increments from the mis-matched case. In this paper only the maximum principal stresses at the nodes closest to the fusion line in the HAZ were
279
considered. The lines have to be more or less parallel in order to satisfy the self-similarity requirement in the J±Q±M approach. The amplitude of the dierence ®eld can now be readily calculated by subtracting the results obtained for dierent mis-match load increments from the reference solution, Fig. 4(b). Again these dierence ®elds should be parallel. A simple cleavage fracture micromechanism is applied as a fracture criterion. This mechanism implies that fracture will occur once the maximum principal stress has reached a critical level over a critical distance [11]. This model was introduced for the J±Q ®elds [12], and the same procedure is now applied for the mis-matched case. The procedure to determine Jref includes the following steps 1. Select the critical dimensionless distance for x r= Japplied =r0 ) for the actual load increment and applied J value. In our case x 2 was selected. Find the corresponding stress level y r1 =r0 . 2. Find the corresponding dimensionless distance xref for the reference case at the same stress level by applying the reference stress ®eld equation. 3. Find the corresponding Jref Japplied x=xref . This relation implies that the critical distance r is the same in both cases as assumed in the micromechanical model. The relation can be derived from the following equation r1 r; Japplied r1 r; Jref : r0 r0
5
Jref is now interpreted as the J integral to which the reference model must be loaded to achieve the same stress level over the same length as in the case considered. The result of the calculation is shown in Fig. 4(c). The straight line (Jref Japplied ) is included in the ®gure. By comparing the calculated line with the straight one, the loss of constraint due to mis-match and geometry can be quanti®ed. The degree of self-similarity is evaluated by calculating M as a function of the normalized distance, Fig. 4(b). In the case of ``perfect'' selfsimilarity, the M-values should be constant and the lines parallel. The deviation from a parallel pattern precludes the J±Q±M approach as Q and
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Fig. 4. Principle steps in the J±Q±M calculations. Construction of normalized stress distribution along the HAZ side of the fusion line (a), calculation of the dierence ®eld M (b) and derivation of the toughness correction model (c). The applied load is expressed as CMOD, where CMOD 0.30 mm is the upper line and CMOD 0.80 mm is the lower line in (a) and (b).
M can no longer be regarded as measures of a uniform shift in the stress ®eld. An acceptable deviation has been proposed [13] by introducing the mean gradient between the stress distribution in the specimen and the reference solution M0
M 5 ÿ M 1 ; 4
6
where M(5) and M(1) refers to the M values at dimensionless distance 5 and 1. The limits on the maximum dierence has been proposed to be <0.1 [12].
4.3. Constraint eects on the crack tip ®elds The normalized calculations for the four cases considered in this paper, are presented in Fig. 5. In most cases the M value is fairly constant for the loading conditions considered (expressed as CMOD values in the ®gure). There is a tendency of increasing M values with increasing normalized distance, most pronounced for the cases with slant fusion line. However, all data in Fig. 5 satis®es the requirement of jM 0 j < 0:1:
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Fig. 5. Normalized crack tip stress- and dierence ®elds as a function of normalized distance, for the four cases considered. The reference solution is included in the ®gures.
In the case of the limited straight fusion line, with absolute length 2.3 mm, an increasing part of the normalized distance will fall outside the straight fusion line as the load is increased. This restricts the validity range with respect to applied load for the full length to about J 300 N/mm. The calculations of Jref vs Japplied are presented in Fig. 6. These results will be discussed more in detail in the next section.
5. Discussion In Fig. 6(a) and (b), the straight- and slant fusion line cases are plotted separately. In both cases overmatch gives the highest constraint, but the eect is much more pronounced for straight fusion line. In case of the straight fusion line, Fig. 6(a), the constraint at a reference level of 50 N/mm, requires an applied loading close to 50 N/mm for the overmatch and about
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Fig. 6. Toughness correction for straight fusion line with over- and evenmatch (a), slant fusion line with over- and evenmatch (b), overmatch with straight- and slant fusion line (c) and evenmatch with straight- and slant fusion line (d). The one to one solution, Jref Japplied , is also included in the ®gure.
200 N/mm for the evenmatch. If the applied load is ®xed at 200 N/mm, Jref is increased from about 40 to 80 N/mm as we go from even to overmatch, indicating a factor of two with respect to fracture toughness. The detrimental eect of weld metal overmatch with respect to the HAZ has also been reported previously [1,14]. In the case of slant fusion line, Fig. 6(b), the volume of weld metal included in the crack tip process zone will be reduced, and hence the mismatch eect will also be reduced. The overmatch
will not be so detrimental, and the evenmatch not so bene®cial. In Fig. 6(c) and (d), the over- and evenmatch cases are plotted separately. For overmatch there is a tendency that the two curves are coming closer with increasing load, while for evenmatch the curves are more or less identical. This behaviour can be explained on the basis of (1) the dierent hardenability and (2) the limited length of the straight fusion line. In the case of weld metal overmatch the HAZ has a steeper hardening gradient than the weld
C. Thaulow et al. / Computational Materials Science 15 (1999) 275±284
metal, and at a true strain of about 0.4 local undermatch will occur. From the discussion above, this eect should be most pronounced in the case of straight fusion line, giving a more rapid reduction in constraint than for the slant fusion line. The declined slope of straight fusion line/overmatch can also be due to increased in¯uence from the slant fusion line as the load is increased. It is dicult to judge which eect is most pronounced, and calculations with identical hardenablity will be necessary to separate the two eects. In the case of evenmatch, Fig. 6(d), the mismatch eect is minor, and the eect of local geometry negligible. 6. Conclusions The J±Q±M approach has been applied to quantify the constraint of crack tip stress ®elds for cracks located at the fusion line of weldments. 1. For a given global load, local overmatch gives slightly lower J than the local even(under)match. For a given global deformation there is no dierence between the over- and evenmatch. 2. Stress ®eld self-similarity was observed within some limits, and the J±Q±M approach was applied. 3. The highest constraint was observed for weld metal overmatch and straight fusion line and the lowest for weld metal even(under)match and both slant- and straight fusion line. In order to obtain the same reference J (a typical value of 50 N/mm has been selected), the applied J was four times larger in the case of even(under)match compared with the overmatch. 4. When the angle of the fusion line is changed so that less weld metal is included in the process zone, the eect of mis-match is reduced. Acknowledgements The work has been supported by the research project ACCRIS (Acceptance Criteria and Level of Safety for High Strength Steel Weldments), and
283
the support from ECSC, the Norwegian Research Council and the sponsoring industry are highly acknowledged.
References [1] M. Toyoda, F. Minami, C. Ruggieri, C. Thaulow, M. Hauge, Fracture property of HAZ-notched weld joint with mechanical mis-matching, Part I. Analysis of strength mismatching of welds on fracture initiation resistance of HAZnotched joint, in: K.-H. Schwalbe, M. Kocak (Eds.), Symposium Mis-Matching of Welds, ESIS, 17, 1994, pp. 399±415. [2] C. Thaulow, M. Hauge, A. Larsen, F. Minami, M. Toyoda, Eect of extraction procedure on the fracture behaviour of HAZ-notched welded joints, in: Second Workshop on Constraint Eects on the Structural Performance of Welded Joints, Osaka, 1994a, pp. 87±100. [3] N.P. O'Dowd, C.F. Shih, Family of crack-tip ®elds characterized by a triaxiality parameter: Part I. Structure of ®elds, J. Mech. Phys. Solids 39 (1991) 989±1015. [4] N.P. O'Dowd, C.F. Shih, Family of crack-tip ®elds characterized by a triaxiality parameter: Part II. Fracture applications, J. Mech. Phys. Solids 40 (1992) 939±963. [5] C. Thaulow, M. Toyoda, Strength mismatch eect on fracture behaviour of HAZ, in: K.H. Schwalbe, M. Kocak, (Eds.), Keynote Lecture. Mis-Matching of Interfaces and Welds, GKSS Research Center Publications, Geestacht, FRG, 1997, pp. 75±98. [6] F. Minami, M. Toyoda, C. Thaulow, M. Hauge, Eect of strength mis-match on fracture mechanical behaviour of HAZ-notched weld joint, in: Proceedings of the Second Workshop on Constraint Eects on the Structural Performance of Welded Joints, Osaka, September 1994. [7] Z.L. Zhang, M. Hauge, C. Thaulow, Two parameter characterisation of the near tip stress ®elds for a bi-material elastic±plastic interface crack, Int. J. Fracture 79 (1996) 65±83. [8] C. Thaulow, Z.L. Zhang, é. Ranestad, M. Hauge, J±Q±M approach for failure assessment of fusion line cracks: two material and three material models, ASTM STP 1360, Fatigue and Fracture Mechanics, vol. 30, St. Louis, 1998. [9] Z.L. Zhang, C. Thaulow, M. Hauge, Eects of crack size and weld metal mismatch on the HAZ cleavage toughness of wide plates, Engrg. Fracture Mech. 57/6 (1997) 653±664. [10] é. Ranestad, Z.L. Zhang, C. Thaulow, Quanti®cation of mismatch constraint for an interface crack in an elastic plastic tri-material system. in: Presented at the Second International Symposium on Mis-Matching of Welds, Geesthacht, Germany, 24±26 April 1996. [11] O. Ritchie, J.F. Knott, J.R. Rice, J. Mech. Phys. Solids 21 (1973) 395±410. [12] R.H. Dodds, C.F. Shih, T.L. Anderson, Continuum and micromechanics treatment of constraint in fracture, Int. J. Fracture 64 (1993) 101±133.
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[13] C.F. Shih, N.P. O'Dowd, A fracture mechanics approach based on a toughness locus, in: International Conference Shallow Crack Fracture Mechanics, Toughness Tests and Applications, Paper 31, Cambridge, UK, 23±24 September 1992. [14] C. Thaulow, A.J. Paauw, M. Hauge, M. Toyoda, F. Minami, Fracture property of HAZ-notched weld joint
with mechanical mis-matching, Part II. Eect of local mechanical mis-matching on fracture initiation in steel weldment. in: K.-H. Schwalbe, M. Kocak (Eds.), Symposium Mis-Matching of Welds, ESIS, 17, 1994b, pp. 417± 432.
Constraint eects on crack tip stress ®elds for cracks located at the fusion line of weldments a,* ,
C. Thaulow a
Z.L. Zhang b, M. Hauge c, W. Burget d, D. Memhard
d
Norwegian University of Science and Technology, N-7465 Trondheim, Norway b SINTEF, Norway c STATOIL, Norway d Fraunhofer Institut f ur Werkstomechanik, Germany Received 20 January 1999; accepted 1 March 1999
Abstract The case of a deeply notched (a/W 0.3) surface crack positioned at the fusion line of a weldment is considered. The tensile properties of the base material and the heat aected zone (HAZ) are kept constant, while the weld metal properties are changed. First the weld metal yield strength overmatches both the base material and the HAZ, and in the second case there is yield strength evenmatch between the HAZ and weld metal and overmatch with respect to the base material. The eect of these strength mismatch conditions has been examined for two fusion line geometries: straight and slant fusion line. The eect of the crack tip constraint has been characterized with the J±Q±M approach where both geometrical and material mismatch eects can be considered with respect to the critical conditions for cleavage fracture initiation. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Brittle fracture; Mismatch; Constraint; Weldments
1. Introduction High strength steels are now frequently applied for oshore structures, with a speci®ed yield strength of about 450 MPa and a required weld metal global yield strength overmatch of about 10%. But as the base metal strength level is further increased, it has been shown that it will be more and more dicult to obtain the required combination of high tensile properties and toughness for the weld metal. It is therefore a need to de®ne the optimum combination of strength mis-match and
* Corresponding author. Tel.: +47 7359 3821; fax: +47 7359 2931; e-mail: [email protected]
weld metal fracture toughness as a function of the base metal strength level. The lowest toughness values for oshore steels are, however normally experienced in the heat aected zone (HAZ), close to the fusion line, and it has been observed that weld metal strength overmatch can be detrimental with respect to the toughness of the HAZ [1,2]. Fracture mechanics test procedures are designed to obtain as large geometrical constraint as possible and hence give conservative estimates of the toughness. The question is now how strength mismatch (and geometry) in¯uence on the constraint, and then how to transfer the test results from the fracture mechanics test condition to a structure.
0927-0256/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 9 9 ) 0 0 0 1 6 - 6
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C. Thaulow et al. / Computational Materials Science 15 (1999) 275±284
The authors have developed a new frame-work for including the strength mismatch in the constraint parameter, called J±Q±M, and this approach is applied in the present paper. 2. Theory The basic assumption for the single parameter fracture mechanics is that K or J uniquely scales the amplitude of the near crack tip ®elds. However, the relation between the scaling parameter and the near tip ®elds loses the one to one correspondance for large scale yielding in ®nite bodies. Based upon observations from detailed FE calculations, a two parameter approach has been proposed [3,4]. rij FiniteBody rij Reference J Drij Q;
1
where J sets the size scale of the local deformation while the hydrostatic stress parameter Q quanti®es the level of stress triaxiality over a signi®cant distance in which the failure microprocess occurs. In other words, Q only alters the magnitude, but not the shape of the near tip stress ®elds. In view of the self-similarity in the stress ®elds observed above, recent FE calculations of the mismatch eect have been examined [5,6]. The observed similarity in the contours of the maximum principle stresses around the crack gave rise to a formulation of a constraint parameter M to scale the near tip stress ®eld caused by material mismatch [7,8]. Calculations with a bi-material modi®ed boundary layer model (MBL) has shown that the crack tip ®eld can be separated into two parts, one is a reference ®eld of a reference material (rI0 ij ) controlled by J and another one is a dierence ®eld controlled by a mismatch property M. The reference material will be the zone where the stress ®elds are investigated, e.g. where fracture is expected to take place. M is de®ned as the amplitude of the dierence ®eld rij rI0 ij J Mr0 dij fij ;
2
where d are constants, dhh drr 1 and drh ÿ2, r0 is the reference yield stress for the reference
material and f are the dierence ®eld functions which depend upon angular positions and on the properties of the reference material only. There is a dierence between the Q and M ®elds. The Q ®elds are symmetric and do not change the shape of the stress contours but the magnitude. Although M ®elds are self-similar, they are not symmetric around the interface and will change both the shape and the magnitude of the stress contours.
3. Numerical analysis 3.1. FE model In the present analysis the model applied in Ref. [9] was modi®ed to 50 mm thickness, Fig. 1. The ®gure shows the FEM mesh in a global scale and the details of the core mesh and the crack tip arrangement. There are 28 element rings in the core mesh in order to have a sucient number of elements within the range 1 6 r= J =r0 6 5 for a signi®cant part of the loading. The initial notch radius is 0.005 mm and non-singular elements have been used. The model consists of about 1300 8-node plane strain elements. The HAZ thickness is 2 mm. Later the model was modi®ed to also include a straight fusion line under the crack tip for a length of about 2.3 mm, Fig. 2, and the HAZ thickness was slightly changed. Based on previous calculations, where the eect of the HAZ thickness has been examined, it is not expected that this moderate change in thickness will in¯uence the crack tip stress ®eld signi®cantly [10]. In the following sections the two cases considered, are called slant and straight fusion line. 3.2. Material properties The stress±strain curves from one structural steel weldment was considered, Table 1. In order to obtain weld metal over- and evenmatch, the steel was welded with two dierent consumables, while the heat input was kept constant. Hence, only the tensile properties of the
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277
Fig. 1. Slant fusion line model. Three point bend specimen with a/W 0.3, W 50 mm and l 200 mm (a), mesh in a global scale (b), core mesh (c) and the crack tip arrangement in the core mesh (d).
weld metal were changed. The tensile properties for the HAZ were set equal to the coarse grained zone, and were obtained from weld simulated tensile specimens. The curves are ®tted to Ramberg±Osgood parameters, but incremental plasticity was applied in the present calculations.
In both cases a global weld metal overmatch with respect to the base material is obtained. But the most important characteristic with respect to brittle fracture is the local mismatch at the crack tip. And because of dierent strain hardening for the weld metal and the HAZ there will be
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Fig. 2. Straight fusion line model. The de®nition of the HAZ is slightly changed compared with Fig. 1, to include a 2.3 mm long straight fusion line under the crack tip.
deviations from the original yield strength mismatch conditions as the specimens are loaded. 4. Results 4.1. Global load and displacement The J integral calculations from the 28 contours revealed path independence. Fig. 3 displays the relation between the calculated J and the global load and displacement for the four cases considered. For a given global load, evenmatch yields slightly higher J integral than overmatch. But for a given deformation, there is no dierence between the over- and evenmatch condition. 4.2. J±Q±M calculations In order to express the constraint, the J±Q±M approach outlined in Section 2, will now be applied.
Fig. 3. Global load vs J (a), and global displacement vs J (b).
Table 1 Materials data (n is the Ramberg±Osgood strain hardening exponent) Strength mis-match
Base material
HAZ
Weld metal
Global and local overmatch
Re 450 MPa n 15 Re 450 MPa n 15
Re 530 MPa n 9.5 Re 530 MPa n 9.5
Re 643 MPa n 14 Re 530 MPa n 14
Global overmatch and local evenmatch
C. Thaulow et al. / Computational Materials Science 15 (1999) 275±284
The ®rst step is to calculate the small scale yielding HAZ reference ®eld. The coarse grained HAZ is selected as reference material since the lowest fracture toughness for the steels tested are expected to be located at the coarse grained HAZ. A MBL model was applied with the same mesh as in the core model, Fig. 1. The MBL model was loaded to about J 200 N/mm, and the resulting boundary displacements calculated from the elastic stress ®eld of a plane strain mode I crack: r 1ÿm r h cos 3 ÿ 4m ÿ cos h; u r; h KI E 2p 2 3 r 1ÿm r h sin 3 ÿ 4m ÿ cos h; v r; h KI E 2p 2 4 p 2 where KI EJ = 1 ÿ m under plane strain condition, and r and h are polar coordinates centred at the crack tip where h 0 corresponds to the fusion line. In the J±Q±M approach the crack tip stress ®eld is normalized by the yield strength and the J integral. The reference ®eld solution was ®tted by a second degree polynomial giving the following equation x 97:673 ÿ 52:973y 7:283y 2 ; where y r1 /r0 and x r/(J/r0 ). r is the distance from the crack tip and r0 the yield strength of the coarse grained HAZ. The valid range of x is set to 1 6 x 6 5. This range has been selected since it ®ts the microscopically signi®cant zone with respect to brittle fracture [12]. The blunted crack tip zone is avoided by specifying x P 1, corresponding to r P 2d, where d is the crack tip opening displacement. After the homogenous HAZ case has now been determined, the next step is to compare this reference solution with the mis-match calculations. The principle of the procedure is outlined in Fig. 4. First Fig. 4(a) is constructed, where the reference solution is compared with a number of load increments from the mis-matched case. In this paper only the maximum principal stresses at the nodes closest to the fusion line in the HAZ were
279
considered. The lines have to be more or less parallel in order to satisfy the self-similarity requirement in the J±Q±M approach. The amplitude of the dierence ®eld can now be readily calculated by subtracting the results obtained for dierent mis-match load increments from the reference solution, Fig. 4(b). Again these dierence ®elds should be parallel. A simple cleavage fracture micromechanism is applied as a fracture criterion. This mechanism implies that fracture will occur once the maximum principal stress has reached a critical level over a critical distance [11]. This model was introduced for the J±Q ®elds [12], and the same procedure is now applied for the mis-matched case. The procedure to determine Jref includes the following steps 1. Select the critical dimensionless distance for x r= Japplied =r0 ) for the actual load increment and applied J value. In our case x 2 was selected. Find the corresponding stress level y r1 =r0 . 2. Find the corresponding dimensionless distance xref for the reference case at the same stress level by applying the reference stress ®eld equation. 3. Find the corresponding Jref Japplied x=xref . This relation implies that the critical distance r is the same in both cases as assumed in the micromechanical model. The relation can be derived from the following equation r1 r; Japplied r1 r; Jref : r0 r0
5
Jref is now interpreted as the J integral to which the reference model must be loaded to achieve the same stress level over the same length as in the case considered. The result of the calculation is shown in Fig. 4(c). The straight line (Jref Japplied ) is included in the ®gure. By comparing the calculated line with the straight one, the loss of constraint due to mis-match and geometry can be quanti®ed. The degree of self-similarity is evaluated by calculating M as a function of the normalized distance, Fig. 4(b). In the case of ``perfect'' selfsimilarity, the M-values should be constant and the lines parallel. The deviation from a parallel pattern precludes the J±Q±M approach as Q and
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Fig. 4. Principle steps in the J±Q±M calculations. Construction of normalized stress distribution along the HAZ side of the fusion line (a), calculation of the dierence ®eld M (b) and derivation of the toughness correction model (c). The applied load is expressed as CMOD, where CMOD 0.30 mm is the upper line and CMOD 0.80 mm is the lower line in (a) and (b).
M can no longer be regarded as measures of a uniform shift in the stress ®eld. An acceptable deviation has been proposed [13] by introducing the mean gradient between the stress distribution in the specimen and the reference solution M0
M 5 ÿ M 1 ; 4
6
where M(5) and M(1) refers to the M values at dimensionless distance 5 and 1. The limits on the maximum dierence has been proposed to be <0.1 [12].
4.3. Constraint eects on the crack tip ®elds The normalized calculations for the four cases considered in this paper, are presented in Fig. 5. In most cases the M value is fairly constant for the loading conditions considered (expressed as CMOD values in the ®gure). There is a tendency of increasing M values with increasing normalized distance, most pronounced for the cases with slant fusion line. However, all data in Fig. 5 satis®es the requirement of jM 0 j < 0:1:
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Fig. 5. Normalized crack tip stress- and dierence ®elds as a function of normalized distance, for the four cases considered. The reference solution is included in the ®gures.
In the case of the limited straight fusion line, with absolute length 2.3 mm, an increasing part of the normalized distance will fall outside the straight fusion line as the load is increased. This restricts the validity range with respect to applied load for the full length to about J 300 N/mm. The calculations of Jref vs Japplied are presented in Fig. 6. These results will be discussed more in detail in the next section.
5. Discussion In Fig. 6(a) and (b), the straight- and slant fusion line cases are plotted separately. In both cases overmatch gives the highest constraint, but the eect is much more pronounced for straight fusion line. In case of the straight fusion line, Fig. 6(a), the constraint at a reference level of 50 N/mm, requires an applied loading close to 50 N/mm for the overmatch and about
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Fig. 6. Toughness correction for straight fusion line with over- and evenmatch (a), slant fusion line with over- and evenmatch (b), overmatch with straight- and slant fusion line (c) and evenmatch with straight- and slant fusion line (d). The one to one solution, Jref Japplied , is also included in the ®gure.
200 N/mm for the evenmatch. If the applied load is ®xed at 200 N/mm, Jref is increased from about 40 to 80 N/mm as we go from even to overmatch, indicating a factor of two with respect to fracture toughness. The detrimental eect of weld metal overmatch with respect to the HAZ has also been reported previously [1,14]. In the case of slant fusion line, Fig. 6(b), the volume of weld metal included in the crack tip process zone will be reduced, and hence the mismatch eect will also be reduced. The overmatch
will not be so detrimental, and the evenmatch not so bene®cial. In Fig. 6(c) and (d), the over- and evenmatch cases are plotted separately. For overmatch there is a tendency that the two curves are coming closer with increasing load, while for evenmatch the curves are more or less identical. This behaviour can be explained on the basis of (1) the dierent hardenability and (2) the limited length of the straight fusion line. In the case of weld metal overmatch the HAZ has a steeper hardening gradient than the weld
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metal, and at a true strain of about 0.4 local undermatch will occur. From the discussion above, this eect should be most pronounced in the case of straight fusion line, giving a more rapid reduction in constraint than for the slant fusion line. The declined slope of straight fusion line/overmatch can also be due to increased in¯uence from the slant fusion line as the load is increased. It is dicult to judge which eect is most pronounced, and calculations with identical hardenablity will be necessary to separate the two eects. In the case of evenmatch, Fig. 6(d), the mismatch eect is minor, and the eect of local geometry negligible. 6. Conclusions The J±Q±M approach has been applied to quantify the constraint of crack tip stress ®elds for cracks located at the fusion line of weldments. 1. For a given global load, local overmatch gives slightly lower J than the local even(under)match. For a given global deformation there is no dierence between the over- and evenmatch. 2. Stress ®eld self-similarity was observed within some limits, and the J±Q±M approach was applied. 3. The highest constraint was observed for weld metal overmatch and straight fusion line and the lowest for weld metal even(under)match and both slant- and straight fusion line. In order to obtain the same reference J (a typical value of 50 N/mm has been selected), the applied J was four times larger in the case of even(under)match compared with the overmatch. 4. When the angle of the fusion line is changed so that less weld metal is included in the process zone, the eect of mis-match is reduced. Acknowledgements The work has been supported by the research project ACCRIS (Acceptance Criteria and Level of Safety for High Strength Steel Weldments), and
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the support from ECSC, the Norwegian Research Council and the sponsoring industry are highly acknowledged.
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with mechanical mis-matching, Part II. Eect of local mechanical mis-matching on fracture initiation in steel weldment. in: K.-H. Schwalbe, M. Kocak (Eds.), Symposium Mis-Matching of Welds, ESIS, 17, 1994b, pp. 417± 432.