Two-parameter (J -M) description of crack tip stress-fields for an idealized weldment in small scale yielding

Transcription

1 International Journal of Fracture 88: , Kluwer Academic Publishers. Printed in the Netherlands. Two-parameter (J -M) description of crack tip stress-fields for an idealized weldment in small scale yielding Ø. RANESTAD 1, Z.L. ZHANG 2 and C. THAULOW 1 1 NTNU, Department of Machine Design and Materials Technology, N-7034 Trondheim, Norway oyvind.ranestad@keo.kvaerner.com 2 SINTEF Materials Technology, N-7034 Trondheim, Norway Received 24 June 1997; accepted in revised form 12 January 1998 Abstract. The crack tip stress-field in a trimaterial finite element model has been examined. The model represents an idealised steel weldment with a crack located at the fusion line. The model was loaded with a K I displacement field to simulate small scale yielding conditions. The effect of changing the weld metal plastic properties and the HAZ layer thickness on the crack tip stress-field was studied, keeping the material properties of the HAZ and base metal constant. The results show that the calculated J -integral remains path independent in the trimaterial model. It is confirmed that the crack tip stress-fields can be normalised by the J -integral. The mismatch constraint can be characterised by a difference field, which is independent of the normalised distance from the crack tip. The results show that changes of HAZ thickness only have a small effect on the stress-fields close to the crack tip. The hardenability of the weld metal influences on the slope of the crack tip stress distribution, but for small changes in hardenability, this effect can be neglected. The results indicate that the difference fields show some radial dependence when a homogeneous reference field is used, but the radial dependence was removed by introducing an inhomogeneous reference field. The effect of changes in the weld metal yield strength has been described with a two parameter (J -M) formulation using the inhomogeneous reference field. 1. Introduction The weldments are usually the weakest part of steel structures, because of the poor toughness of the heat affected zone where cracks are often located. They can grow in fatigue from stress concentrators such as weld defects or surface imperfections into the heat affected zone. In this situation a structural component may fail in brittle fracture at loads below the designed static strength. In fracture mechanics testing of weldments the lowest fracture toughness values are often observed when a crack hits close to the fusion line (C. Thaulow et al. 1994). The fusion line is thus often regarded as the most critical location with respect to brittle fracture. However, the current standards for fracture mechanics testing were developed for homogeneous materials, and failure predictions based on these standards are not accurate for cracks in weldments. The stress-fields surrounding cracks at the fusion line of weldments are influenced by geometric and material constraint, which invalidate single parameter fracture mechanics approaches. The geometry constraint is caused by specimen dimensions, crack size and the mode of loading. This constraint can be taken into account by the parameter Q, introduced by O Dowd and Shih (1991). O Dowd and Shih observed that crack tip stress-fields shift uniformly ahead of the crack tip with the level of the geometry constraint. They suggested that the shift can be quantified approximately by Q. The Q-parameter is defined as the difference in hydrostatic stresses between an actual geometry and the small-scale yielding solution at

2 316 Ø. Ranestad et al. normalised distance X = 2, where X is defined as the absolute distance (r), normalised by the J -integral (J ) and the yield strength (σ 0 ), X = r/j/σ 0. Zhang, Hauge and Thaulow (1996) studied a bi-material system in a modified boundary layer (MBL) model (T = 0) with an interface crack. In their model the materials differed in plastic properties, i.e. the elastic constants of the two materials were identical. For this system they concluded that the stress-fields at the crack tip can be separated into a reference field and a difference field that describes the mismatch constraint. The stress-field in the region r/j/σ 10 1, 5 θ 45, 45 can be described by the equation σij M = σij Ref (n 1,J,r,θ)+ σ 10 M(m,n 2 )f ij (θ + 12β,n 1 ). (1) In (1) σij M are the components of the stress-field, σij Ref are the components of the homogeneous reference material or the material in focus. M is the amplitude of the difference field caused by the mismatch constraint, f ij are angular functions, n 1 and n 2 are hardening exponents of the reference material and mismatched material respectively. J denotes the J -integral, r and θ define the polar coordinate system with origin at the crack tip. The constant σ 10 is the yield strength of the reference material. The mismatch ratio m is defined as m = σ 20 /σ 10,whereσ 20 is the yield strength of the mismatched material. β is defined as β = 1form<1andβ = 0 for m > 1. This description implies that the mismatch constraint can be described by the M parameter in the same way as the geometry constraint can be described by the Q parameter. Recent results (Zhang et al. 1997) indicate that geometry constraint and mismatch constraint are independent of each other, and a J -Q-M formulation of the stress-fields in a geometry with interface crack and mismatch in the plastic material properties has been presented. This formulation has been applied to assess fracture in wide plate specimens (Zhang et al. 1997a; 1997b) using the RKR (Ritchie et al. 1973) failure criterion. However, the bimaterial model might not give a sufficient description of a weldment. A more accurate description would be to include a thin layer of heat affected zone (HAZ) between the base material and the weld material. The HAZ is known to contain brittle microstructures that can cause low fracture toughness of the weldment. In order to study the effect of the HAZ and make the description of the weldment more realistic a study on a trimaterial model has been carried out. The thin HAZ layer in the trimaterial model represents the most brittle part of the heat affected zone. The first results from the tri-material model (Ranestad et al. 1997) showed that the stress-fields in the HAZ close to the fusion line can be described by a formulation similar to (1). The first results also revealed that the reference field should be modified if the weld metal has different hardening exponent n than the reference material (HAZ). The present work aims to describe the stress-field in the forward sector r/j/σ 10 1, 5 θ 45, 45. The effect of changes in yield strength of the weld metal σ0 WM, the HAZ thickness and hardening of the weld metal, n WM on the near tip stress-field, will be addressed. In all cases the properties of two of the materials, namely HAZ and base metal, are kept constant, and the effect of changes in weld metal properties has been quantified within the J -M framework.

3 2. Materials and procedures Two-parameter (J -M) description of crack tip stress-fields NUMERICAL PROCEDURES The finite element calculations were carried out using ABAQUS. Deformation plasticity theory was applied, using the Ramberg Osgood power hardening law (2) ε i = σ ( ) 1/ni i σi + α. (2) ε i0 σ i0 σ i0 σ i denotes the flow stress for material i, σ i0 and ε i0 are the yield stress and strain of material i, α is a dimensionless constant and n i is the hardening exponent of material i. Small strain theory was used for improved convergence of the solution. The results from the crack tip elements were discarded. The finite element model contains node 2D quadrilateral plane strain elements with reduced integration scheme, Figure 1. The model was loaded by a K I displacement field, with displacements (u, ν) defined by (3). 1 + ν r u(r, θ) = K I E ν(r,θ) = K I 1 + ν E 2π cos ( ( 1 1 ν 2 2 θ) (3 4ν cos θ)+ T E r 2π sin ( ( 1 ν(1 + ν) 2 θ) (3 4ν cos θ) T E ) r cos θ, ) r sin θ. In (3) ν denotes the Poisson ratio, E is Young s modulus. The stress intensity K I can be written K I = EJ/(1 ν 2 ) for plane strain. r and θ are the polar co-ordinates with origin at the crack tip, and T denotes the T -stress which is zero for all calculations in this study. The effect of T -stress on the trimaterial model has also been studied and will be reported elsewhere. The finite element model contains 3 materials. The mesh is very focused in the vertical direction close to the crack tip in order to allow changes of the HAZ thickness. Figure 1(b) shows the material zones in the model, and the crack tip mesh is shown in Figure 2. The crack tip has an initial opening of 0.02 mm. The size of the smallest element in the model was mm. Figure 2(b) shows the crack tip mesh deformed at the maximum applied load, J = 200 N/mm. Since the MBL mesh used in this study is not a standard radial focused mesh, the stress-fields have been compared to corresponding fields of a standard mesh model. The results were identical for the investigated region MATERIAL PROPERTIES The three material zones in the model had identical elastic properties, with Young s modulus E = MPa, and Poisson ratio ν = 0.3. The plastic properties of the base metal and the HAZ were kept constant in all calculations, and the plastic properties of the weld metal were changed. The yield strength of the base metal was σ0 BM = 500 MPa and the yield strength of the HAZ σ0 HAZ = 600 MPa. Both materials had the same hardening exponents n BM = n HAZ = The investigated weld metal yield strengths (σ0 WM ) were 700, 650, 600 and 550 MPa, Table 1. The local mismatch (Thaulow et al. 1994) is also shown in the table. The hardening exponents applied to the weld metal were n WM = 0.033, 0.07 and 0.2. Unless otherwise mentioned, all calculations have the same hardening exponents n i = 0.07 in this (3)

4 318 Ø. Ranestad et al. Figure 1. (a) The MBL model used for the calculations, and (b) the material zones in the crack tip area. The crack tip mesh is shown in Figure 2. Figure 2. Crack tip mesh (a) undeformed and (b) deformed to J = 200 N/mm. The material case shown in Figure (b) is (confer Table 1). study. The investigated yield strength combinations represents typical mismatch conditions for steel weldments where global overmatch (σ0 WM >σ0 BM ) has been specified. In addition a calculation with homogeneous HAZ material properties (6-6-6) was used as a reference. The effect of the HAZ layer thickness was investigated on material combinations and (Table 1), representing local overmatch and undermatch respectively. Three finite thicknesses were examined, 0.9 mm, 0.6 mm and 0.3 mm. In addition two bimaterial models were included, WM-BM, representing no HAZ, and WM-HAZ representing infinite HAZ thickness RESULTS OUTPUT Figure 3 shows how the results have been output from the model. The results were output from the integration points along 4 radial lines with constant angle θ = 45,θ = 0 (WM/FL), θ = 0 1 (HAZ/FL) and θ = 45. Furthermore, angular results between 45 and 45 were output from integration points at normalised distance about X = r/j/σ 0 = 2. Integration point layer parallel to the crack plane with an offset of 0.02 mm from the fusion line.

5 Two-parameter (J -M) description of crack tip stress-fields 319 Table 1. Yield strength combinations in the study. Material σ0 WM σ0 HAZ σ0 BM Local mismatch combination [MPa] [MPa] [MPa] m L = σ0 WM /σ0 HAZ Figure 3. Results output in front of the crack tip. The arrows show the location of integration point layers used to extract radial stress distributions. The circle sector illustrates where angular plots were extracted from the model. 3. Results and discussion In order to use the J -integral to characterise the crack tip stress-field, we have to make sure that J is path independent in the model. The path dependence is addressed in Section 3.1. We are going to compare stabilised stress-distributions, which can be normalised by the J -integral to remove the load dependence. In order to normalise the stress-distributions with the J -integral, the stress distribution must exhibit a J/r singularity. As noted by McMeeking (1977), the crack tip must open in a self-similar way in order to obtain stabilised stress-fields. This means that some loading is required before stabilisation, and that the mismatch can influence the stabilisation. The stabilisation of the stress-fields is investigated in Section 3.2. Radial and angular maximum principal stress distributions for different mismatch conditions are presented in Section 3.3. The weld metal hardening influence on the stress-field is shown in Section 3.4, and Section 3.5 addresses the effect of changes in the HAZ layer thickness on the stress distributions. The mismatch constraint effect is systematically studied in Section 3.6, and formulations for the stress-distributions are derived in Section 3.7.

6 320 Ø. Ranestad et al. Figure 4. Path independence of the J -integral (a) and development of calculated J during the loading for three material combinations (b) PATH INDEPENDENCE OF THE J -INTEGRAL Outside the first crack tip contours the J -integral is path independent, Figure 4(a). The results are taken from material combination 7-6-5, and similar curves were obtained from other material combinations. In this study, only one node at the crack tip was used as seed for J calculations. By including all nodes of the blunted crack tip, the J -integral becomes stable from the first contours. This indicates that the cause for the path dependence lies in the finite strain zone close to the crack tip. Path-independence of J was also noted by Shih and Asaro (1988), Asaro et al. (1993) and Aoki et al. (1992), in their bi-material studies. In all these cases the crack is parallel to the material borders. However, Zhang et al. (1997a; 1997b) have also noted path independence of the J -integral for cases where the material borders are not parallel to the crack. In the trimaterial model used in this study, the paths for calculation of the J -integral will include two materials in the first contours, and three materials when the base material is included in the path for determination of the J -integral. The results in Figure 4(a) show that the material border between the HAZ and base metal does not affect the calculated J -integral. The development of the applied and calculated J is shown in Figure 4(b) for material combinations 7-6-5, and The results show that the calculated J is slightly lower than the applied J for all cases. The maximum deviation between applied and calculated J was 6 percent for material combination These results demonstrate that J is path independent and can be taken as the loading parameter for this study NORMALISATION OF THE STRESS-FIELDS WITH THE J -INTEGRAL In Figure 5 the stress distributions at increasing load levels have been normalised by the J - integral. The stress distributions shown in Figure 5 are from θ = 0 in the HAZ close to the fusion line. The material combination is which represents the worst case among the investigated combinations. The results show that the normalisation of the stress-field is slightly affected by the J at normalised distance X = 2, and independent of the load level at X = 4.

7 Two-parameter (J -M) description of crack tip stress-fields 321 Figure 5. Normalisation of the σ 1 stress distributions with J for the material combination. Figure 6. Convergence of the normalised stress distributions. The open symbols show normalised distance X = r/j/σ HAZ 0 = 2 and the filled symbols X = 4. In order to investigate the evolution of the normalised stress distribution when J increases, the deviations between the normalised stress distribution values at J = 200 N/mm and lower J values have been plotted for distances X = 2 and X = 4, Figure 6. The figure shows how the normalised stress distributions approach the values obtained at J = 200 N/mm for the four investigated angles. At distance X = 4, the stress distribution is stable and independent of the applied J, whereas the convergence is slower at X = 2. Since the HAZ layer has a finite thickness of 0.6 mm, the influence of the HAZ layer on the normalised stress distribution in the near tip area will change during the loading. The stressfield expands in size as the load (J ) increases, and consequently the HAZ thickness become relatively smaller (t HAZ σ 0 /J decrease) with increasing loading. This can explain the slower convergence at X = 2. At low J values, the relative size of the HAZ is large. The trimaterial model then approaches the WM/HAZ bimaterial model, and this causes an elevation of the stress-level in front of the crack. In the same way, the stress level is slightly decreased towards the WM/BM bimaterial as J become high. The effect of the HAZ thickness is further addressed in Section 3.5. Whereas Figures 5 and 6 demonstrates that the effect of the HAZ thickness on the normalised stress-distributions decrease when J increases, all further results are taken from applied J = 200 N/mm.

8 322 Ø. Ranestad et al. Figure 7. Maximum principal stress along 4 lines with constant θ in front of the crack tip. The dotted line for the HAZ 45 location shows the material border between HAZ (left-hand side) and BM (right-hand side) for applied J = 200 N/mm MAXIMUM PRINCIPAL STRESS DISTRIBUTIONS The stabilised and normalised stress distributions have been output for the four angles (θ) shown in Figure 3. The stress distributions in the radial directions, Figure 7, move up and down in a parallel pattern when the mismatch changes. From Figure 7 is that the parallel pattern is more pronounced in the fusion line area than in the 45 locations. The homogeneous HAZ stress distributions ( ) has slightly different slope than the stress distributions in the mismatch cases close to the crack tip, Figure 7. The deviation close to the crack tip originates from the HAZ layer, that causes an elevation of the stresses in a confined region in the vicinity of the crack tip WELD METAL HARDENING EFFECT The effect of changing the hardening exponent of the WM (n WM ) has been investigated, keeping the hardening of the HAZ and base metal constant. The maximum principal stress distributions are shown in Figures 8 and 9 for local overmatch (7-6-5) and undermatch (55-6-5) respectively. The general trend is that increased hardening results in steeper stress distributions. This means that the effect of plastic hardening is stronger in the near tip area than further away from the crack tip. However, if the difference in hardening is small, the effect of different hardening on the slope of the stress-distributions might be neglected. This can be seen by comparing stress distributions for n WM = 0.03 and n WM = In steel weldments the hardening

9 Two-parameter (J -M) description of crack tip stress-fields 323 Figure 8. Radial distributions of σ 1 along 4 lines ahead of the crack tip for different hardening (n) of WM. Material combination exponents for different material zones can in many cases be assumed to the same (Zhang and Hauge, 1995), but if the strain hardening of the zones is very different, the differences will affect the validity of the J -M formulation HAZ THICKNESS EFFECT Three thicknesses of the HAZ layer have been investigated. Two additional bimaterial configurations representing no HAZ (WM-BM bimaterial) and infinite HAZ (WM-HAZ bimaterial) were also analysed. These investigations were carried out for both local overmatch (7-6-5) and local undermatch (55-6-5) material combinations. The stress-fields have been normalised with the HAZ yield strength in all cases, also when t HAZ = 0. All results were obtained at J = 200 N/mm. The thickness of the HAZ becomes relatively smaller during deformation in the MBL model. The tσ 0 /J ratios for J = 200 N/mm are shown in Table 2. Evidently the HAZ layer up to 0.9 mm only affects the stresses in a small area of the HAZ, within X<3 for the investigated cases, Figures Figures 10 and 11 include two bimaterial calculation for each location. If the weldment is modelled as a bimaterial, one has to choose between the WM/HAZ bimaterial and the WM/BM bimaterial. In this case the HAZ is always stronger than the BM, and therefore the stress level is higher for the WM/HAZ bimaterial. It can be seen that for both material combinations in Figures 10 and 11, the WM/HAZ represent the upper bound, and WM/BM bimaterial represent the lower bound for the stress levels. A bimaterial model with WM/BM properties is more close to the investigated trimaterial model and HAZ thicknesses, but the

10 324 Ø. Ranestad et al. Figure 9. Radial distributions of σ 1 along 4 lines ahead of the crack tip for different hardening (n) of WM. Material combination Table 2. rσ HAZ 0 /J at J = 200 N/mm for the investigated HAZ thicknesses. HAZ thickness tσ HAZ 0 /J 0.0 mm mm mm mm 2.7 Infinite Infinite predictions are on the non-conservative side. The WM/HAZ bimaterial, however is clearly conservative compared to the trimaterial model. The area of elevated stresses increases with the thickness of the HAZ layer. Thus, statistical fracture mechanics approaches would suggest increasing failure probability with the HAZ layer thickness, even though the peak stresses are little affected by the changes of the HAZ thickness. The calculations indicate that a certain thickness is necessary to affect the stress field. The smallest HAZ size (t HAZ = 0.3 mm) only has a small effect on the stresses in the WM-45

11 Two-parameter (J -M) description of crack tip stress-fields 325 Figure 10. HAZ thickness effect on σ 1 for material configuration. position. The stress distributions in the fusion line area and in the HAZ-45 location are almost identical to the corresponding distributions in the BM-WM bimaterial. The effect of the HAZ on the maximum principal stress is very similar for the two investigated material cases, indicating that the effect of the HAZ is the same whether HAZ is the weaker or stronger material adjacent to the fusion line. The thickness effect on the stress distributions seems to be more pronounced for the WM 45 distribution than for the other investigated distributions. These results suggest that an increase of the HAZ thickness might be more harmful to the WM side than to the HAZ itself, or even the fusion line area QUANTIFICATION OF MISMATCH CONSTRAINT The mismatch constraint was quantified with the amplitude of the difference fields, M, as defined in (1). The difference field is defined as the difference between the actual stress distribution and a reference stress distribution. The difference field must be approximately independent of the normalised distance (X) in order to be characterised by a constant value (M). This means that the actual stress distribution must be approximately parallel to the reference stress-distribution, and the choice of reference field is therefore important.

12 326 Ø. Ranestad et al. Figure 11. HAZ thickness effect on σ 1 for material configuration. Figure 12. Difference fields calculated using reference (open symbols) and (filled symbols). The results are from a radial line in the HAZ-45 position Choice of reference field Zhang et al. (1996) showed that a solution for the stress-field of a homogeneous material can be applied as the reference for bi-materials. The maximum principal stress distributions from the trimaterial model, Figure 7, indicate that the mismatched stress distributions are not parallel to the homogeneous stress distribution in the trimaterial model. Therefore, the difference fields will not be independent of the normalised distance (X) from the crack tip.

13 Two-parameter (J -M) description of crack tip stress-fields 327 Table 3. Quantification of the deviation from the parallel pattern of the crack tip difference fields with the homogeneous HAZ (6-6-6) reference field. M 1 M 1 HAZ-45 M 1 HAZ-X M 1 WM-X M 1 WM M 11 M 11 HAZ-45 M 11 HAZ-X M 11 WM-X M 11 WM M 12 M 12 HAZ-45 M 12 HAZ-X M 12 WM-X M 12 WM M 22 M 22 HAZ-45 M 22 HAZ-X M 22 WM-X M 22 WM The deviation is most pronounced in the 45 directions. The distributions obtained close to the fusion line only deviate from the parallel pattern in a small area near the crack tip. The weld metal hardening and the thickness of the HAZ will also affect the stress distributions to some extent. The use of a homogeneous reference field will therefore give lower accuracy in the trimaterial model than in the bimaterial model of Zhang et al. (1996). An alternative is to use a more complicated reference field, like local evenmatch, In addition to better accuracy this introduces a possibility to include the actual hardening exponents and HAZ thickness in the reference field. The mismatch constraint can still be quantified, it is only compared to local evenmatch instead of homogeneous material Difference fields Figure 12 shows the maximum principal stress difference fields obtained from homogeneous (6-6-6) reference field (O) and the local evenmatch (6-6-5) reference field ( ). The results shown are from HAZ 45. The figure demonstrates that the value of M 1 (M obtained from the σ 1 stress component) is less dependent on the radial distance X = r/j/σ 0 when the reference is used.

14 328 Ø. Ranestad et al. In order to study the difference-fields and quantify the deviation from the parallel pattern of the stress-fields in the mismatch cases, a measure for the deviation has been defined M ij = Mmax ij (X) Mij min (X), X 1, 5. (4) The parameter M ij in (4) is similar to a parameter Q defined by Shih and O Dowd (1992) to investigate the radial dependence of the Q difference fields in fracture mechanics test specimens. M ij measures the difference fields, and the value is zero if the difference fields are independent of the radial distance. The acceptable error is chosen to 10 percent of the yield strength of the HAZ (M ij 6 0, 1). This is similar to the limit Q < 0.04 or 16 percent of the yield strength which was used by Shih and O Dowd (1992). Dodds et al. (1993) emphasise that Q much larger than 0.1 is unacceptable for predictions of brittle fracture. The calculated values are tabulated below, for the case with homogeneous HAZ (6-6-6) (Table 3) and the case with local evenmatch (6-6-5) as the reference (Table 4). The notations M ij HAZ-X and M ij WM-X are the deviation from the parallel pattern (M ij ) in HAZ and WM along fusion line, and M ij HAZ-45 and M ij WM-45 are the M ij along 45 lines in HAZ and WM respectively. The accepted values according to the criterion are printed with bold characters. By comparing Tables 3 and 4 we find that the reference field gives less radial variations in the difference fields than the homogeneous reference (6-6-6). All results in the two tables are within the limit mentioned by Dodds et al. (1993), for Q difference fields. However, the magnitude of Q is much larger than M, and the relative change of M is therefore larger for the same deviation. Since the M values in Tables 3 and 4 are larger when the homogeneous reference is applied, the inhomogeneous reference should be applied in cases where a high accuracy is required. Table 3 shows that the difference fields are less accurate in the HAZ side than in the WM side when the homogeneous reference is applied. There is also a tendency that the strongest overmatch (7-6-5) yields less accurate results. Table 4 shows that the results are much better when the reference is used. Especially in the HAZ the results are much more accurate when the reference is applied, and that is the most important area when the HAZ is brittle. The only unacceptable values according to the criterion were found in the WM-45 location for the material combination. For determination of fracture toughness from testing it is disadvantageous to replace the homogeneous reference field with a more complicated one. Then the advantage of testing homogeneous material and use the J -M approach to take the mismatch effect into account by calculating a corrected toughness is lost STRUCTURE OF THE CRACK TIP STRESS-FIELDS In order to have a more complete description of the stress-fields, the in-plane stress components σ 11,σ 22 and σ 12 were investigated in addition to the maximum principal stress. The component σ 11 is the normal stress acting parallel to the crack, and σ 22 is the opening stress or the normal stress component acting perpendicular to the direction of the crack. The angular stress distributions of these components and their difference fields are shown in Figure 13. The difference fields were obtained using the combination as the reference stress-field. We observe that the stress-fields can be organised in the following form σ ij (r, θ) = σij SSY (r, θ) + σ0 REF M ij f ij (θ). (5)

15 Two-parameter (J -M) description of crack tip stress-fields 329 Table 4. Quantification of the deviation from the parallel pattern of the crack tip difference fields using local evenmatch (6-6-5) reference field. M 1 M 1 HAZ-45 M 1 HAZ-X M 1 WM-X M 1 WM M 11 M 11 HAZ-45 M 11 HAZ-X M 11 WM-X M 11 WM M 12 M 12 HAZ-45 M 12 HAZ-X M 12 WM-X M 12 WM M 22 M 22 HAZ-45 M 22 HAZ-X M 22 WM-X M 22 WM In (5) M ij are understood as the coefficients of f ij, and the conventions for summations of matrices are not applied to the product M ij f ij. M ij denotes the amplitude of the difference field caused by the mismatch constraint for stress component σ ij. By dividing the difference fields by M ij,definedas( σ ij /σ 0 ) at X = 2 in HAZ close to FL, the angular functions f ij appear. This definition of M ij is in correspondence with the definition of Q as proposed by O Dowd and Shih (1991). This definition of M differs from the definition of M proposed by Zhang et al. (1996) in (1). Zhang et al. used the maximum values of the difference fields along a circle sector at X = 2todefineM. Figure 14 shows the definitions of M in this study and in Zhang et al. (1996). In Zhang et al. (1996) it was found that the M ij coefficients were almost identical and can be represented by one value (M) for all stress components. In the present study the fixed position was chosen to make sure that the constraint is quantified in the most critical material which is assumed to be the HAZ. In this study, the differences between the M ij coefficients were too large to justify the use of the same value for all M ij. For predictions of cleavage fracture it is in most cases sufficient to evaluate one of the stress components, in most cases the opening stress σ 22 or the maximum principal stress σ 1. The angular functions f ij are shown in Figure 15 for the components f 11,f 12,f 22 and f 1. The figure shows that the angular functions are approximately constant for different mismatch configurations. Figure 16 shows the mismatch effect on the stress-fields of σ 1,σ 11,σ 22 and σ 12 measured by the mismatch constraint factors M ij. The figure shows that the mismatch effect is almost identical on the components σ 1 and σ 12, slightly stronger on σ 11 and weaker on the opening

16 330 Ø. Ranestad et al. Figure 13. Angular distributions of σ 22,σ 1,σ 11,σ 12 and their difference fields. Normalised distance X = r/j/σ 0 = 2. Figure 14. Definition of M in the trimaterial model (a) and in the bimaterial model of Zhang et al. (1996), (b). stress σ 22. The mismatch constraint M expresses the shift of the stress field compared to the reference, and M = 0.1 indicates that the stress shift at the fusion line is 10 percent compared to the reference. From Figure 16 it can be seen that the shift in constraint (M) in the investigated case approximately corresponds to the shift in mismatch (m L ). An overmatch of 10 percent (m L = 1.1) results in a mismatch constraint M of 10 percent (M = 0.1). In a similar study by Ganti et al. (1997), it was found that for a bimaterial the mismatch

17 Two-parameter (J -M) description of crack tip stress-fields 331 Figure 15. Normalised angular functions f ij derived at r/j/σ 0 = 2. Figure 16. M ij from X = r/j/σ 0 = 2 in HAZ close to fusion line as a function of the local mismatch m L. constraint increases almost linearly with the mismatch until an overmatch of 42.1 percent for non-hardening materials. 4. Conclusions The J -integral was found to be path independent in this tri-material system, and the radial distributions of the crack tip stress field were scaled by the J -integral. Based on these observations, the J -integral was taken as the loading parameter for the study.

18 332 Ø. Ranestad et al. Stress distributions along four lines at different angles in front of the crack tip show that the crack tip stress-field shifts vertically in a parallel pattern when the yield stress of the weld metal changes. The results show that the difference fields are not independent of the radial distance when the homogeneous HAZ is used as the reference calculation. The main reason for this is that the HAZ layer influences the stress distributions. An inhomogeneous reference field representing local evenmatch was introduced in order to obtain difference fields that are independent of the radial distance from the crack tip. Stress distributions for different weld metal yield strengths have been described by a two parameter formulation using this reference. The stress-field can be separated into a reference field and a difference field that describes the mismatch constraint change from the local evenmatch reference. The difference field is independent of the radial distance, and the angular dependence is described by functions f ij that are independent of the local mismatch. The hardening exponent of the weld metal n WM was found to change the slope of the crack tip stress distributions. This change can be neglected if the difference in hardening is small. The thickness of the HAZ layer affects the stress-field close to the crack tip. Based on the investigated HAZ thicknesses, it has been concluded that moderate changes of the HAZ thickness only have a small effect on the stress-field close to the crack tip. Acknowledgements This work has been supported by Kværner AS, through a PhD grant for Øyvind Ranestad, within the research project Ship for the future. The work has also received support from The Research Council of Norway (Programme for Supercomputing) through a grant of computing time. References Aoki, S., Kishimoto, K. and Takeuchi, N. (1992). An elastic-plastic finite element analysis of a blunting interface crack with microvoid damage. International Journal of Fracture 55, Asaro, R.J., O Dowd, N.P. and Shih, C.F. (1993). Elastic-plastic analysis of cracks on bimaterial interfaces: Interfaces with structure. Material Science and Engineering A162, Dodds Jr. R.H., Shih, C.F. and Anderson, T.L. (1993). Continuum and micromechanics treatment of constraint in fracture. International Journal of Fracture. 64, Ganti, S., Parks, D.M. and McClintock, F.A. (1997). Analysis of strength mis-matched interface cracks in SSY. Mis-Matching of Interfaces and Welds (Edited by K.-H. Schwalbe and M. Kocak), GKSS Research Center Publications, Geestacht, FRG, McMeeking, R.M. (1977). Finite deformation analysis of crack tip opening in elastic-plastic materials and implications for fracture. Journal of the Mechanics and Physics of Solids 25, O Dowd, N.P. and Shih, C.F. (1991). Family of crack tip fields characterized by a triaxiality parameter 1. Structure of fields. Journal of the Mechanics and Physics of Solids 39, Ranestad, Ø., Zhang, Z.L. and Thaulow, C. (1997). Quantification of mismatch constraint in an elastic-plastic trimaterial system. Mis-Matching of Interfaces and Welds (Edited by K.-H. Schwalbe and M. Kockak), GKSS Research Center Publications, Geestacht, FRG, pp Ritchie, R.O., Knott, J.F. and Rice, J.R. (1973). On the relationship between critical tensile stress and fracture toughness in mild steel. Journal of the Mechanics and Physics of Solids 21, Shih, C.F. and Asaro, R.J. (1988). Elastic-plastic analysis of cracks on bimaterial interfaces: Part I Small scale yielding. Journal of Applied Mechanics 55, Shih, C.F. and O Dowd, N.P. (1992). A fracture mechanics approach based on a toughness locus. Proceedings of TWI/EWI/IS International Conference on Shallow Crack Fracture Mechanics. TWI, Cambridge, UK.

19 Two-parameter (J -M) description of crack tip stress-fields 333 Thaulow, C., Hauge, M., Paauw, A.J., Toyoda, M. and Minami, F. (1994). Effect of notch tip location in CTOD testing of the heat affected zone of steel weldments. Proceedings from ECF 10: Structural Integrity: Experiments, Models and Applications (Edited by K-H. Schwalbe and C. Berger), Berlin, Thaulow, C., Larsen, A., Ranestad, Ø. and Hauge, M. (1994). Effect of local strength mis-match on CTOD toughness in the HAZ of steel weldments, Proc. IIW Sub. Comm. X-F, Paris, IIW Doc. X-F Zhang, Z.L. and Hauge, M. (1995). Ramberg Osgood parameter fitting for ACCRIS steels and stress-strain curves proposed for FEM calculations, SINTEF report. Zhang, Z.L., Hauge, M. and Thaulow, C. (1996). Two parameter characterization of the near tip stress fields for a bi-material elastic-plastic interface crack. International Journal of Fracture 79, Zhang, Z.L., Hauge, M. and Thaulow, C. (1997). The effect of T stress on the near tip stress fields of an elasticplastic interface crack. Proceedings of the 9th International Conference on Fracture. Sydney (Edited by B.L. Karihalo, Y.-W. Mai, M.I. Ripley and R.O. Ritchie), Zhang, Z.L., Thaulow, C. and Hauge, M. (1997a). Effects of crack size and weld metal mismatch on the HAZ cleavage toughness of wide plates. Engineering Fracture Mechanics 57(6), Zhang, Z.L., Thaulow, C. and Hauge, M. (1997b). On the constraint effects in HAZ cracked wide plate specimens. Mis-Matching of Interfaces and Welds (Edited by K.-H. Schwalbe and M. Kocak), GKSS Research Center Publications, Geestacht, FRG,