An engineering method for constraint based fractureassessment of welded structural components with surface

cracks

é. Ranestada,*, Z.L. Zhangb, C. Thaulowa

aKvaerner Oil and Gas Pb. 222, N-1324, Lysaker, NorwaybSINTEF Materials Technology, N-7034, Trondheim, Norway

Received 16 December 1997; received in revised form 26 April 1999; accepted 12 May 1999

Abstract

In this study it has been shown that accurate descriptions of crack-tip stress-®elds in surface crackedwelded plates can be obtained without large 3D FEA models. When a fracture mechanics FE analysis isrequired in a large construction, existing shell models can be used in combination with a plane strainsubmodel. The 2D plane strain model is driven by displacements from the global shell model. Thistechnique has been used to simulate crack-tip stress-®elds in a surface cracked plate. The crack-tip stress®elds are characterised with the J-integral and the constraint parameter, Q. The crack in the global shellmodel was simulated with line-spring elements. The global behaviour as well as the crack-tip stress-®eldsof the plane strain submodel have been compared to a 3D solid model. Initially, the crack-tip stress-®elds in the plane strain model and the 3D model with surface crack were compared, using the same in-plane mesh and element type. It was found that when ®rst order elements were used, the constraint washigher in 3D than in plane strain. For second order elements, however, the trend was the opposite. Byusing a correction factor for the load, the load vs. J behaviour and the crack-tip stress-®elds of a surfacecracked plate can be predicted from a shell analysis with line-spring elements and a plane strain model.Accurate predictions of J and Q were obtained using the shell + submodel technique for homogeneousmaterial and for a weldment with fusion line crack. The shell + submodelling technique was used toassess brittle fracture in two steel weldments with a surface crack using the RKR failure criterion byRitchie et al. [16]. For the investigated case, the toughness requirements could be relaxed signi®cantlybased on the two parameter analysis compared to conventional fracture mechanics analyses. # 1999Elsevier Science Ltd. All rights reserved.

Engineering Fracture Mechanics 63 (1999) 653±674

0013-7944/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.PII: S0013-7944(99)00054-5

www.elsevier.com/locate/engfracmech

* Corresponding author. Tel.: +47-67595004; fax: +47-73-59-41-29.E-mail address: oyvind.ranestad@kvaerner.com (é. Ranestad)

Keywords: Weldment; FEA; Cleavage; Line spring; Constraint; Steel

1. Introduction

The development of two parameter fracture mechanics has improved the ability to predictbrittle fracture in various geometries. The constraint parameters make it possible to quantifythe di�erence in toughness between low and high constraint geometries.

The two most popular constraint indexing parameters are the Q-parameter [1] and the elasticT-stress [2]. It has been argued by O'Dowd [3] that because the T-stress is an elastic parameter,it is not capable to describe the constraint under large-scale yielding conditions.

Finite element analyses using shell elements are widely used in design of large structures, andthese models can in many cases be used for fracture mechanics evaluations. The line springmodel introduced by Rice and Levy [4], and extended to elastic-plastic calculations by Parks [5]and Parks and White [6], is a simple and computationally inexpensive way to analyse cracks inshell models. By representing the crack with line spring elements, the change in sti�ness due tothe crack is included, and elastic-plastic line spring elements also provide good estimates of theJ-integral.

Wang and Parks [7] have presented a methodology for using the line spring model tocalculate J and T in shell models. The T-stress has, however, been questioned as a constraintparameter under large-scale yielding conditions [3], and to our knowledge it is not possible touse the T-stress to assess cracks on the fusion line of weldments. Due to the low fracturetoughness in the fusion line area, this crack location is often the most critical in weldedconstructions [8].

The J±Q methodology gives a direct measure of the crack-tip stress-®eld, and can thereforedescribe the evolution of constraint throughout the loading to large-scale yielding. The J±Qmethodology has been extended by Zhang et al. [9] to a J±Q±M methodology by including thematerial mismatch constraint (M ) for fusion line cracks in weldments. This methodology givesa framework to assess cracks located on the fusion line in welded constructions.

Through a failure criterion, the M parameter can be used to quantify the mismatch e�ect onthe fracture toughness, and possibly M can be used to transfer test results from weld thermalsimulated specimens to real weldments [10]. Previous studies by Zhang et al. [9] and Ranestadet al. [11] have shown that the interaction between the geometry constraint (Q ) and themismatch constraint (M ) is weak.

When the fracture toughness from the HAZ/Fusion line testing of a weldment is available,the mismatch constraint e�ect is included in the fracture toughness value. Assuming that theinteraction between Q and M can be neglected, no further corrections for mismatch constraintare needed for assessing this weldment in other geometries. In order to transfer fracturetoughness between di�erent geometries, the geometries are compared to a reference. In thisstudy, we have applied a MBL model as a reference. In order to obtain good correspondence(self-similarity) between the reference ®eld and the stress-®eld in the specimen, the same

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material properties are used in the modi®ed boundary layer (MBL) reference model as in theweldment.When structural components are assessed, 3D analyses of the structural component with

cracks are often required. 3D models are, however, very expensive both in terms of modellingtime and computation time. In order to calculate Q for a wide load range, a very re®ned meshis needed.For analyses of fusion line cracks, the mesh must also be suitable for de®ning the necessary

material zones close to the fusion line. This means that the mesh must be carefully designed inorder to get reliable results, and the design of the mesh can be very time consuming. Thecomplexity of the 3D mesh also means that changes of i.e., crack size and local geometry canbe complicated.The shell model gives good predictions of global parameters like load and displacement, and

the line spring elements allow accurate estimations of the J-integral. The shell model does not,however, provide the necessary resolution of the crack-tip stress ®eld required fordetermination of Q and M. By introducing a plane strain submodel, the details of the crack-tipstress-®eld can be studied. Fig. 1 shows schematically the submodelling technique forprediction of constraint in 3D models.In this study, a plane strain submodel will be used as a submodel driven by the global shell

model. The plane strain model is relatively inexpensive computationally, and easier to buildand modify. The 2D mesh can also be very highly re®ned without excessive computationalcosts.The procedure with a 2D plane strain submodel applies to cracks in shell models where a

symmetry plane exists perpendicular to the crack plane and the shell plane. For many cases,the cracked geometry includes a symmetry plane, and the load can be approximated assymmetrical with respect to the crack. In very complicated situations, the 2D plane strainmodel may not apply, because of lacking symmetry or complicated distribution of the remoteload. Then a 3D solid submodel is required, with the increased computational cost andmodelling work involved in this analysis. Another assumption for using the plane strainsubmodel is that the J-integral has the peak value at the symmetry plane. The shell modelanalysis with line spring elements gives su�cient information to determine whether a planestrain submodel can be used or if a 3D submodel is required to determine the crack-tip stress-®eld.Previous studies have compared plane strain models to 3D models of fracture mechanics

specimens. Anderson and Dodds [12] showed that for SENB specimens with deep cracks andligament length less than the specimen thickness, the constraint level is maintained to a higher

Fig. 1. Schematic illustration of the submodelling technique used to predict the 3D solid model behaviour. The 3Dmodel is replaced by two 2D models.

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load level in 3D models than in plane strain models. They explained this from the 3D natureof plastic deformation, causing out of plane constraint that is not captured in plane strainmodels. Similar results have been presented by Nevalainen and Dodds [13]. In their results, theinitial high constraint Q is higher in 3D models than in plane strain for deep cracked bendspecimens with B=W � 1 and 2. In a compact tension specimen, however, the plane strainmodel produced higher constraint. Recently Dodds et al. [14] reviewed the 3D constraint e�ectson cleavage, and they concluded that the plane strain and 3D predictions di�er for deepcracked SENB and CT specimens. In this study we compare the constraint in a 3D surfacecracked plate and a corresponding plane strain model.The global behaviour and the constraint in a surface cracked panel modelled with 3D solid

elements, plane strain elements and a shell model combined with a plane strain submodel, havebeen compared. The analyses were carried out for both homogeneous material and a weldmentwith a crack at the fusion line.The purpose of the study is to establish a procedure for assessment of brittle fracture in

weldments where the global quantities (load, displacements and J-integral) are determined fromthe global shell model and the local quantities of the stress-®eld are determined from thesubmodel analyses.It is not recommended to use the line spring elements when the material properties on each

side of the crack are di�erent [15]. Therefore, the line spring model will be used to ®nd theload vs. J behaviour for homogeneous material, and the change of J when the weld isintroduced will be estimated from the submodels.

Fig. 2. Geometry and boundary conditions of the surface cracked plate.

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2. F.E. models and numerical procedures

2.1. Geometry and boundary conditions

The geometry analysed in this study was a 30 mm thick surface cracked plate withdimensions 600 � 600 mm, Fig. 2. The plate contains a surface crack with dimensions a = 9mm and 2c = 180 mm (a=c � 10 and a=W � 0:3). The models were loaded in displacementcontrol. The surface crack front was straight except for the ends which are modelled as a circlesector with radius 9 mm, see Fig. 3.Because of the symmetry, one quarter of the plate was modelled for the homogeneous

material case, and half the plate was modelled in the weldment case.The F.E. calculations were carried out in ABAQUS. The shell model was made with 8-node

thick shell elements with reduced integration scheme (S8R elements). The crack was simulatedwith symmetrical 3 node line spring elements (LS3S) in the shell model. The shell model isshown in Fig. 4.A 3D solid model was generated and meshed with both 8-node brick elements (C3D8) and

20-node brick elements (C3D20R). The crack was modelled with an initial opening D0 = 10mm, and the smallest element edge length was 0.7 mm. The FEA model is shown in Figs. 5 and6.The plane strain full size model and the plane strain submodel were made by cutting one

element layer from the 3D model, and imposing plane strain conditions. Hence, the planestrain full size model represents the symmetry plane of the plate, whereas the submodel isshorter in the plate length direction. Initial investigations showed that ABAQUS plane strainelements (CPE8R) give identical results as the layer model with 20-node brick elements. The®rst order 8-node brick elements, however, gave di�erent result to the second order elements.Therefore, all homogeneous models were analysed with both ®rst order and second orderelements. The results are presented in Section 3.1.Due to limited computational resources, the 3D model of the weldment was considered too

large for analysis with 20-node brick elements. Therefore the ®rst order elements was used forthe weldment.The sizes of the models are shown in Table 1 The required CPU time was about 20 times

higher for the 3D solid model than for the shell + submodel analyses.Kinematic constraints (MPC) were applied on the submodel edges in order to constrain the

Fig. 3. Cross section of the panel showing the dimensions of the surface crack.

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edge nodes to stay in the same plane during the deformation. Displacements from the shellmodel were applied to the centre node (u, v ) and the upper corner node (u ) of the submodeledge. Fig. 7 illustrates the boundary conditions of the submodel.

2.2. Material properties

The material properties are based on testing of steel weldments for ships, Table 2. For theveri®cation of the shell + submodel analysis, the inhomogeneity of the weld was increased byan arti®cial increase of the yield strength in the weld metal and the HAZ.ABAQUS small strain formulation was applied to all the solid element models. The material

behaviour was described with J2 deformation plasticity theory for the solid and shell elements.ABAQUS only supports incremental plasticity for line spring elements. Therefore, incremental

Fig. 4. The shell model with line spring elements to simulate the crack.

Fig. 5. 3D solid model of the surface cracked plate (1/4).

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Fig. 6. Details of the mesh in the crack-tip area.

Table 1Overview of the model sizes for each analysis

Model No. of elements No. of variables

Homogeneous3D brick 8-nodes 9542 33339

3D brick 20-nodes 9542 1284273D plane strain 789 50823D plane strain submodel 773 4974

Shell 108 1920Weld3D brick 19084 64056

3D plane strain 1578 98343D plane strain submodel 1546 9618

Shell 466 8724

Fig. 7. The submodel is loaded with displacements taken from the shell model. The displacements (u, v ) are appliedon two nodes and the other (`follower') nodes on the edge are forced to stay in the plane de®ned by the twoconstrained nodes.

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plasticity was used for the line spring elements, and the true stress (s) vs. plastic strain (ep)curves were approximated from the power-law relation

ss0��1� ep

e0

�1=n

�1�

where s0 denotes the yield stress, e0 is the yield strain s0=E, where E represents Young'smodulus, and n is the plastic hardening exponent.

3. Results

3.1. In¯uence of element type on the results

Initial calculations with the models indicated that the results obtained were dependent on thetype of element used in the model. Therefore, the plane strain full size model and the 3Dmodel was calculated with both ®rst order (Brick 8-nodes with full integration) and second

Table 2Material properties for the F.E.M analyses

Property Base metal Heat a�ected zone Weld metal

Young's modulus E 210,000 (MPa) 210,000 (MPa) 210,000 (MPa)Poisson ratio n 0.3 0.3 0.3Yield strength s0 400 (MPa) 450 (MPa) 500 (MPa)

Strain hardening exponent n 12 12 12

Fig. 8. Comparison of: (a) load versus displacement and (b) gross stress vs. J-integral, in 2D and 3D with ®rst andsecond order elements. J was determined in the centre in the 3D model.

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order elements (Brick 20-nodes with reduced integration scheme). The calculations were carriedout with homogeneous base metal, see Table 2 for the material properties. A comparison ofglobal (J vs. load) and local (J vs. Q ) parameters between a 3D model, the 2D plane strain fullsize model and a shell model of a surface cracked plate is presented in this section.

3.1.1. Comparison of the global parametersFig. 8 shows the global behaviour of the plane strain layer model and the surface cracked

plate with ®rst order and second order elements. Shell + linespring results are also included inthe ®gure. The load versus displacement curves, Fig. 8a, are identical for all the 3Dcalculations, including the shell model. The plane strain model, however, shows a signi®cantdi�erence in sti�ness between ®rst and second order elements.The gross stress, sGross, de®ned as sGross � P=A, where P is the sum of the reaction forces

and A is the gross section area, was taken as the load parameter. The comparison of grossstress vs. J, Fig. 8b, shows that the J becomes higher for a given load when second orderelements are used. This e�ect is more pronounced in the plane strain model than in the 3Dmodel. The trend for the di�erence between plane strain and 3D is the same for ®rst andsecond order elements. This suggests that similar results may be obtained using di�erentelement types, but a direct comparison between a 3D model with ®rst order elements and aplane strain model with second order elements may give misleading results.It is noticed that at high load levels, when Jpl is much larger than Jel, the di�erence between

2D and 3D is almost constant. Also, the shape of all the sGross vs. J curves are very similar.This suggests that the di�erence in load between plane strain and 3D for a given element typecan be described with a constant. This is addressed further in Section 3.2.The J calculated in the shell model with line spring elements lies between the results from

®rst and second order solid elements. In this case, the line spring model gives a goodapproximation for the load vs. J behaviour for both ®rst order and second order elements.

3.1.2. Crack-tip stress-®eldsIn this section, we compare the crack tip stress-®elds in plane strain and 3D, using ®rst and

second order elements. The crack-tip stress-level has been characterised with the constraintparameter Q, de®ned as:

Q � s1�X � 2� ÿ sSSY1�X � 2�

s0�2�

where s1�X � 2� denotes the maximum principal stress measured in front of the crack tip atnormalised distance, X � rs0=J � 2 for the case to be assessed. In the 3D model, Q was outputfrom the symmetry plane in the centre of the plate. sSSY

1 was determined from the stress-®eld ina MBL model with a pure KI ®eld (zero T-stress). In order to make sure that Q isrepresentative of a uniform shift of the crack-tip stress ®eld, the radial dependence of Q waschecked with the parameter Q ' de®ned as

Q 0 �����Q�1� ÿQ�5�

4

���� �3�

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The maximum allowable Q ' was set to 0.1, but the deviation was below 0.05 for most of thecalculations.Fig. 9 shows the constraint parameter Q as a function of: (a) J and (b) remote load. The

results in Fig. 9a show that the plane strain model predicts lower constraint (Q ) than 3D atthe same J with ®rst order elements, whereas the constraint (Q ) is higher in plane strain whensecond order elements are used. The results also show that for the same applied J, secondorder elements produce higher constraint than ®rst order elements.Fig. 9b shows the constraint Q as a function of the gross stress. The results show that the

constraint (Q ) in 3D is higher than in plane strain for both the investigated element types.From Fig. 8b it can however be seen that the J is higher in plane strain for the same remoteload. Also, it can be seen in Fig. 9b that the ®rst order elements predict higher constraint thanthe second order elements.From the results it can be seen that use of ®rst and second order elements may give di�erent

conclusions regarding the constraint in the 3D and the plane strain model. The element type isespecially important in plane strain. First order elements predict lower constraint than both®rst and second order elements in the 3D model. The predicted constraint in plane strain withsecond order elements is however higher than both ®rst and second order elements in 3D.The results also show that it is di�cult to compare the two models directly, because the load

vs. J relations are di�erent. If the constraint is compared at the same J, the remote stress willbe higher in the 3D model than in plane strain. Also, the load is higher in ®rst order elementsthan in second order elements at the same J. Likewise, the J will be di�erent if the constraint iscompared at the same remote stress.

3.2. Correction of the load between 2D and 3D models

The results in Fig. 8b show that the development of J vs. gross stress in the plane strain

Fig. 9. Comparison of: (a) Q vs. J and (b) Q vs. gross stress, for plane strain and 3D with ®rst and second orderelements. Homogeneous material.

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model and the 3D model are di�erent. However, since the shape of the curves are very similar,the di�erence between plane strain and 3D can be characterised by a scaling parameter. Wede®ne this load scaling parameter l as the di�erence between the gross stress level at J = 500N/mm in the shell model and the plane strain model:

l � sShellGross�J � 500�

sPlanestrainGross

�J � 500� �4�

By multiplying the gross stress in the plane strain model with l, the J vs. gross stress curve forplane strain coincides with the shell model curve. The scaling implies that the 3D gross stress isused for the plane strain model, and l re¯ects the di�erence in sti�ness between 3D and planestrain. This di�erence is assumed to be geometry dependent. Assuming that l is the same in aweldment as in homogeneous material, l can be used to predict the J vs. gross stress curve fora weldment. In order to predict this curve for a weldment, we only need to carry out a shellmodel analysis with homogeneous material properties, and two plane strain analyses. First weanalyse the plane strain model with the same material properties as used in the shell model.Then we carry out a plane strain analysis with the actual weldment properties, and predict theJ vs. gross stress curve for 3D as:

J 3Dweld

ÿs3D

Gross

� � J 2Dweld

ÿs2D

Gross � l� �5�

The accuracy of this prediction depends on how much l is in¯uenced by the materialproperties of the weldment and how accurate the shell + line spring model determines the J-integral. According to the ABAQUS user manual [15] the line spring predictions of J arereliable when the plastic part of J dominates (Jpl � Jel). By determining l at J = 500 N/mm,this requirement should be ful®lled.By correcting the homogeneous plane strain calculations with l, we get a prediction of the J

Fig. 10. J-integral (J ) and constraint (Q ) as functions of the gross stress. The plane strain load has been correctedwith the scaling parameter l.

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in the homogeneous 3D specimen throughout the loading, Fig. 10. It can be seen that thecorrected plane strain prediction of J ®ts the line spring prediction at high J values, and at lowJ values, the prediction is closer to the 3D solid results than the line spring model. Therefore,this approach may be used instead of the line spring model for prediction of J at low loadlevels, where the line spring model is not reliable.Table 3 shows the l factors for the ®rst and second order plane strain elements.Fig. 10 shows that by correcting the gross stress in the plane strain results with l, the

di�erences between the constraint predictions in plane strain and 3D are reduced. The loadcorrection also reduces the di�erence in load vs. J signi®cantly. The deviation after thecorrection re¯ects the di�erence of calculated J between the shell model and the 3D solidmodel. This deviation also rearranges the curves in Fig. 10b and the Q vs. J curves in Fig. 9a.If the J vs. gross stress curves were identical, the same trend would be expected in the twocomparisons.The predicted constraint with plane strain elements presented in Figs. 9 and 10 should

however provide a su�ciently accurate description of the 3D constraint, especially whensecond order elements are used.From the results it can be concluded that for the same J, the second order elements predict

higher constraint than ®rst order elements in the investigated model. Also, the correspondencebetween plane strain and 3D is closer when second order elements are used. It was found thatfor second order elements, the plane strain model predicts higher constraint than 3D.The computation time is 2±3 times higher for second order elements. The model will become

too large (more than 250,000 variables) and time consuming for our computational resources ifsecond order elements are used. Therefore, we had to use ®rst order elements in the nextsection for the comparison between the shell + submodel analysis and the 3D analysis. Thiscomparison is ®rst carried out with homogeneous material, and then for a weldment with acrack on the fusion line.

3.3. The shell + submodelling technique

In this section, a plane strain submodel driven by the global shell model is introduced. Whenthe boundary conditions are taken from a shell model, the plane strain model can be small,and details of the global construction can be omitted. Therefore, the shell + plane strainsubmodel has a more general applicability than the shell model combined with a global planestrain model.This paper considers a simple wide plate specimen. For this simple case, the advantage of

Table 3Correction factor l for prediction of J vs. load in 3D

sGross�J � 500� l

Shell 1.064Plane strain ®rst order elements 1.043 1.020

Plane strain second order elements 0.974 1.092

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using the submodel instead of the full size plane strain model is not so obvious. The purpose inthis study is however only to demonstrate the principle of the shell + submodel technique.Initial investigations with di�erent submodel length revealed that the actual length of the

submodel is not important. However, the length should be large enough to avoid interactionsbetween the edge and the crack tip plastic zone. As a rule of thumb, the distance between thecrack plane and the edge of the submodel is not recommended to be less than two times theplate thickness.

3.3.1. Homogeneous materialFirst the shell model was analysed with homogenous material, (base metal in Table 2).

Displacements and rotations for the submodel were derived from one node in the shell model70 mm from the crack plane. We were prepared to use more than one load step in order todescribe the displacement history in the shell model, and we found that two load steps give agood description of the loading. The boundary conditions were applied to the model as shownin Fig. 7.The gross stress vs. J plots are shown in Fig. 11a for the shell model, the submodel and the

3D solid model. The correction factor l was calculated from the gross stress values at J = 500N/mm in Fig. 11a, and in this case we found that l � 1:025.The constraint Q was calculated in the submodel, and the results are shown in Fig. 11b. The

gross stress for the submodel has been corrected with l. The prediction of the constraint isreasonably good, but the submodel prediction is slightly non-conservative at high load levels.The deviation is approximately 6% at sGross=s0 � 1, which is regarded as su�cient forengineering purposes. Based on the results from the previous section, higher constraint can beexpected if the plane strain model is made with second order elements.

3.3.2. Weldment with fusion line crackA 3D model of a weldment with fusion line crack was prepared in order to investigate if the

submodelling technique can be used in this case. The model has the same geometry as the

Fig. 11. J and Q as a function of the gross stress for the solid 3D model and the shell + submodel approximation.

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homogeneous model. However, the crack plane is not a symmetry plane in this model, andtherefore the number of elements is doubled.The material properties are shown in Table 2. The weldment was modelled with 1 mm HAZ

thickness and 18 mm weld width. Since the material properties are ®xed in this case, we canuse the same properties in the SSY reference as in the plate. This means that the mismatchconstraint e�ect is included in the reference. The reference stress-®eld was established from aMBL analysis carried out using the same material properties and HAZ thickness as in theinvestigated weldment.The yield strength of the HAZ was used to normalise the stress-®elds, and the yield strength

of the base metal was used to normalise the gross stress. The submodel displacements weretaken from the shell model with homogeneous base material properties. We assume that thechanges in global behaviour due to the weld only have a minor e�ect on the crack-tip stress-®eld.We used Eq. (5) to estimate the gross stress vs. J behaviour in the weldment for the shell +

submodel analysis, with l � 1:025, as determined in the previous section for homogeneousmaterial.Fig. 12 shows J and Q as functions of the gross stress, and it can be seen that the shell +

submodel results are in agreement with the 3D results.The results show that the shell + submodel technique can provide good estimations of the J

and Q, for the weldment with fusion line crack.After these parameters have been determined, they can be used together with a local failure

criterion to predict brittle fracture. In the next section, we present such an analysis, withfracture toughness data from CTOD testing of two weldments.

Fig. 12. Comparison of J and Q vs. gross stress in the 3D solid model and the shell + submodel simpli®cation, forthe weldment with fusion line crack. Weldment with 25% overmatch and FL crack.

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4. Assessment of brittle fracture with the shell + submodel technique

In this section, we present a case study that demonstrates how the shell + submodeltechnique can be used to assess brittle fracture in steel weldments.

4.1. Failure criterion for the assessment procedure

In this case, we apply the RKR criterion [16] to predict the brittle fracture. An e�ective J,Jref can be de®ned and used to predict fracture in geometries with di�erent constraint. Jref isdetermined as:

Jref � 2Japp

Xref

�Xref

�X � 2� �Q� �6�

where Xref�Sref� and Sref�Xref� are 4th order polynomial curve-®ts of the SSY reference stress®eld. Sref � �s1=s0�ref , Xref � rs0=J for the reference curve, and Japp is the applied J ascalculated by ABAQUS.When the RKR criterion is used, we only need the J±Q trajectory and the reference stress-

®eld in order to carry out an assessment of brittle fracture. In [11] it was shown that moreadvanced failure criteria give similar predictions as the RKR criterion for weldments withfusion line cracks.

4.2. Geometry and material properties

The example case is the same plate as we have analysed in the previous section, but thematerial properties have been obtained from mechanical testing of two ship steel weldments.Two sample weldments were produced with electro-gas welding, using high heat input (17 MJ/m). The plate thickness was 30 mm.The weldments are regarded as typical ship weldments, with high heat input in order to

satisfy the high productivity demands in the shipbuilding industry [17]. The results from themechanical testing are reported in a SINTEF report [18], and only the results necessary for theassessment are included here. The mechanical properties of base metal, weld deposit and theHAZ were ®tted to the Ramberg±Osgood power-law function used in ABAQUS fordeformation plasticity analysis:

Table 4

Mechanical properties used for the F.E.M. analyses of the two weldments

Base metal HAZ Weld metal

Weldment s0 (MPa) n s0 (MPa) n s0 (MPa) n

W1 320 12 390 12 410 12

W2 400 12 390 12 380 12

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ee0� s

s0� a

�ss0

�n

�7�

where s and e denotes the ¯ow stress and strain. s0 and e0�e0 � s0=E � are the yield stress andstrain, n is the hardening exponent and a is a dimensionless parameter which is usually setequal to 1 for steels. The properties of the two weldments, W1 and W2 are shown in Table 4.The strain hardening exponent was only determined for the weld metal, and the same

hardening was assumed for the HAZ and the base metal. The base metal yield strength wastaken from the steel certi®cate, and the yield strength of the HAZ was estimated from hardnessmeasurements. It can be seen in Table 4 that W1 has about 30% overmatch, whereas W2 has aslight undermatch.Six CTOD bend specimens with notch on the fusion line were prepared for each weldment.

The crack to thickness ratio a=W was 0.5 for all specimens. The J integral and the CTOD wasrecorded during the testing. The J integral was determined according to ASTM E 1152-87 [19]from the load versus load-line displacement record. The CTOD was determined from theCMOD recorded with clip-gauges, in accordance with BS 7448:1991 [20]. The minimumfracture toughness values from the tests are shown in Table 5.

4.3. F.E.M. results

A plane strain SENB model was prepared with the same core mesh as shown in Fig. 6.ABAQUS 2 order plane strain elements with reduced integration scheme were used for boththe SENB model and the wide plate submodel. In Section 3.1 it was found that thecorrespondence between plane strain and 3D is improved for this mesh when second orderelements are used. It has been reported by Dodds et al. [12,13] that plane strain models predictlower constraint than 3D solid models in deep notched bend specimens. For the assessment inthis case the applied J in the bend specimens at fracture is about 100 N/mm. We assume thatthe plane strain model gives reliable constraint predictions for these rather low J values. Also,if the plane strain model predicts too low a constraint, the failure predictions in the surfacecracked plate will become conservative, because the Jref is underestimated.The J±Q description of the stress-®eld can be precluded in SENB specimens with deep

notches because of the in¯uence from the global bending ®eld. Fig. 13 shows the developmentof: (a) the crack-tip stress-®elds in the SENB a=W � 0:5 specimen and (b) the surface crackedplate submodel. The stress-®eld in the surface cracked plate is parallel to the reference, whilethe stress-®eld of the SENB specimen deviates from the reference.The radial dependence of the bending ®eld is quanti®ed with Q ', de®ned in Eq. (3). In the

Table 5Fracture toughness values for the two weldments

Weldment CTODC (mm) JC (N/mm)

W1 0.16 92.4

W2 0.20 116.3

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bend specimens the limit for Q ' was set to 0.3, and this enabled us to derive results for J up toabout 130 N/mm. The limit Q ' = 0.3 is very loose, and this a�ects the accuracy of the results.As Fig. 13a indicates, the actual stress-®eld decreases compared to the level indicated by the Qvalue. The Q value becomes a conservative estimate for the constraint at distances abovers0=J � 2 from the crack-tip. However, when the bend specimen is used to obtain fracturetoughness results, the conservative estimate of Q means that the predicted reference toughnessbecomes too high, and this gives non-conservative predictions for structural components. Inorder to get a lower bound estimate of the reference toughness, we also determined theconstraint Q in the outer part of the fracture process zone, Q�rs0=J � 4�.The strong radial dependence of the bending ®eld on the crack tip stress ®eld limits the

Fig. 13. (a) The development of the crack-tip stress-®eld in the SENB specimen and (b) in the submodel, duringloading of weldment W1. The SSY reference ®eld is also shown in the ®gure.

Fig. 14. Predictions of the reference toughness, Jref for weldments: (a) W1 and (b) W2, in the SENB specimens andthe surface cracked plate.

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allowable deformation of J±Q characterisation of the stress ®elds. Results by Nevalainen andDodds [13] and Gullerud and Dodds [21] indicate that the deformation limit is higher for J±Qcharacterisation for Compact Tension (CT) specimens. Thus, CT specimens may beadvantageous for obtaining the fracture toughness for J±Q assessments.

4.4. Fracture assessment

The predicted Jref vs. Japplied curves are shown in Fig. 14 for: (a) W1 and (b) W2. In bothweldments there is a signi®cant di�erence in the prediction for the SENB specimen based on Qand Q(4). The Jref vs. Japplied curves for the surface cracked plate are also shown in Fig. 14. Themuch steeper curves for the SENB specimens indicate that the constraint is much higher thanin the surface cracked plate.The reference toughness of the weldment is found from the fracture toughness results in

Table 5, by entering the fracture toughness JC from Table 5 as Japplied, and determine thecorresponding Jref from the SENB curves. The uncertainty of the predictions preclude anaccurate determination of Jref . We therefore pick conservative Jref values from the curves, anduse Jref � 60 N/mm for W1 and Jref � 65 N/mm for W2. With these values it can be seen thatthe surface cracked plate is not expected to fracture before J = 500 N/mm in either of theweldments. At such high loads ductile crack growth is expected, and the validity of thepredictions are uncertain. However, we may conclude that with the measured toughness, brittlefracture is not expected in the surface cracked plates.

4.5. Comparison with conventional fracture assessment methods

The Jref vs. Japplied relations for the SENB specimens in Fig. 14(a) and (b) are approximatelylinear until fracture occurs. Based on this assumption, Jref can be translated to a critical CTODvalue. Many engineers are more familiar with the CTOD than the J-integral. The J-integral isrelated to CTOD (d) through the equation:

J � mds0 �8�where m is a constant, and s0 denotes the yield stress of the HAZ. mref can be determined fromEq. (8), the fracture toughness values in Table 5 and the predicted Jref values at fracture. Inaddition, m was predicted from J without constraint corrections from the critical CTOD and Jvalues in Table 5. The calculated m and mref values are shown in Table 6.Two comparisons of the constraint corrected predictions with conventional predictions will

Table 6m factors for transformation between J and CTOD

Weldment m mref

W1 1.48 0.96

W2 1.49 0.83

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be presented. According to J or CTOD based one-parameter fracture mechanics, the fracturetakes place in any geometry when the fracture parameter (J or CTOD) reaches a critical value.In addition, a level 2 F.A.D. analysis according to PD 6493:1991 [22] has been carried out.

Fig. 15 shows the predicted critical CTOD as a function of the gross stress ratio. The criticalCTOD is interpreted as the required fracture toughness to avoid fracture in the surface crackedplate.The results show that the toughness requirements are lower in the constraint based analysis

than in the conventional analyses. For a given toughness, i.e., d � 0:10 mm, the critical loadbecomes signi®cantly higher for the constraint based analysis. In weldment W1, the constraintbased analysis predicts higher toughness requirements at low loads. This shows that theconstraint is rather high at low load levels in this weldment. The applied J is, however low atthis load level. Therefore, the non-conservativeness of conventional analyses at this stage hasminor practical implications. The F.A.D. analysis has a combination of a fracture mechanicsfailure criterion and a plastic collapse failure criterion. At high load levels, the plastic collapselimits the critical load. Plastic collapse must therefore be checked separately for the two otherfailure analyses.It is noted that the F.A.D. analysis is more critical than the single parameter fracture

mechanics prediction in weldment W1, whereas the F.A.D. analysis is more conservative inW2. The reason is that the mechanical properties of the weld metal is used as input in theF.A.D. analysis. When the applied load is related to the yield strength of the base metal, theanalysis becomes more conservative for overmatched weldments.In practise, the weldments W1 and W2 could both have been applied in a construction as a

grade 32 steel. It is therefore interesting to compare the CTOD requirement as a function ofthe applied gross stress, Fig. 16. The CTOD calculated from single parameter fracture

Fig. 15. Predicted CTOD requirement as a function of the gross stress ratio for the investigated case. Comparisonof the constraint based analysis with F.A.D. level 2 and one-parameter fracture mechanics; (a): weldment W1; (b):weldment W2.

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mechanics is shown in Fig. 16a. Fig. 16b shows the constraint based predictions and theF.A.D. level 2 predictions for the two weldments.The F.E.M. calculations show that the required fracture toughness is higher for W1 at the

same remote load, whereas the F.A.D. predicts a minor di�erence in the required toughness.The F.A.D. predicts the same required toughness because the stress intensity factor KI is afunction of the applied load and the geometry, and thus it is not a�ected by the change inmaterial properties. The J and the CTOD are however in¯uenced by the change of the materialproperties, and this is re¯ected in Fig. 16a. The di�erence between the two cases evolves froma di�erent Jpl, and therefore, the curves are identical at low load levels where the elasticconditions dominate.The constraint based analysis re¯ects the stress-®eld, and the results in Fig. 16b show that

the required fracture toughness is higher in W1 than in W2. The di�erence can be attributed tothe mismatch constraint that elevates the stress-®eld locally close to the fusion line.

5. Conclusions

The crack-tip stress-®elds in a plane strain model and a 3D model with surface crack havebeen compared, using identical in-plane mesh and element type. It was found that when ®rstorder elements are used, the constraint was higher in 3D than in plane strain. For second orderelements, however, the trend was reversed.By using a correction factor for the load, the load vs. J behaviour was predicted for 3D

weldments from a shell analysis line-spring model and two plane strain analyses. The analysesshow good correlation in both global behaviour and crack-tip constraint between the shell +submodel analysis and 3D analysis. Accurate predictions of J and Q were obtained with the

Fig. 16. Comparison of the toughness requirements in W1 and W2 as a function of the applied load; (a): CTODfrom single parameter fracture mechanics (applied J ); (b): F.A.D. prediction and constraint based analysis.

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shell + submodel technique for homogeneous material and for a weldment with fusion linecrack.The shell + submodelling technique was used to assess brittle fracture in two steel

weldments with a surface crack using the RKR failure criterion. Analyses of a SENB specimenwith deep crack revealed strong radial dependence of Q. It has been concluded that thisgeometry is not well suited for J±Q characterisation.It was shown that the constraint based failure analysis allows an increase of the utilisation of

the material in terms of brittle fracture. The fracture toughness requirements could besigni®cantly relaxed compared to conventional analyses.

Acknowledgements

This work is supported by Kvñrner ASA through a PhD grant for éyvind Ranestad, withinthe research project `Ship for the Future'.Kvñrner Masa Yards in Turkku, Finland have supplied test material for this work. Their

contribution is gratefully acknowledged.This work has also received support from the research council of Norway, (Programme for

Supercomputing) through a grant of computation time.

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