Annette D. Karstensen (1), Anthony Horn (2) and Martin Goldthorpe (1)
(1)TWI, Cambridge, UK
Paper presented at 2nd International Symposium on High Strength Steel (PRESS), Stiklestad, Verdal, Norway, 23-24 April, 2002
This paper is concerned with using the Beremin Model to predict cleavage failure within the grain coarsened heat affected zone (GCHAZ) of a multi-pass weld. The GCHAZ and the intercritically reheated grain coarsened heat affected zone (ICGCHAZ) are considered to be the most important microstructures of welds with regard to cleavage fracture. The material parameters of the Beremin Model are measured using small-scale single edge notch bend (SENB) specimens. These specimens are thermally cycled to simulate the thermal history of the welding process in order to reproduce the tensile and toughness properties of the GCHAZ. The material parameters are subsequently used to predict cleavage fracture in SENB specimens of real multi-pass welds. The predicted CTOD values are slightly lower than those obtained in the experiments. At lower failure probabilities the prediction is reasonable, but at higher probabilities the under-estimate is more significant.
Micromechanical or Local Approach models such as the Beremin Model  can be used to predict the conditions required for cleavage fracture in structural components. These models use a combination of finite element analysis and the results of small-scale experimental tests. One advantage of the Beremin Model, compared with predictions made using conventional fracture mechanics, is that the predictions are probabilistic rather than deterministic. A further advantage is that the model does not suffer restrictions encountered by conventional fracture mechanics with regard to the transfer of results between the small-scale specimen and the large-scale structure. For example, differences in plastic constraint between different cracked structures, leading to differences in apparent values of conventional fracture toughness, are fully taken into account using the Beremin Model. This means that highly pessimistic assessments using conventional fracture mechanics can be avoided for defects situated in regions of low constraint.
2. The Beremin model of cleavage failure
In the Beremin Model 
it is assumed that, for a particular load applied to a structure, the cumulative probability of failure by cleavage is described by a Weibull distribution as follows:
where σ w is the Weibull stress defined below, and σ u and m are known as the Weibull parameters that are assumed to be characteristic of the material and independent of temperature. The stress σ u is the scale parameter of the Weibull distribution, and m is the shape parameter describing the scatter of the distribution.
The Weibull stress σ w is defined by
The above integration is taken over the volume of the plastic zone V p (usually associated with a stress raising feature such as a notch or a crack). V o is a material volume of such a size that there is a finite probability that it contains an existing micro-crack that can trigger cleavage fracture. V o is usually set equal to an arbitrary size of (100µm) 3 since its precise value does not affect the predicted results. σ 1 is the maximum principal stress acting on the volume element dV p .
In a finite element analysis the calculation of Weibull stress in Eq. is actually carried out as follows:
where the summation is taken over the n finite element integration stations within the instantaneous plastic zone, Δ V j is the volume associated with integration station j and σ 1 j is the maximum principal stress there.
3. Method of determining the Weibull parameters
In order to make predictions of cleavage failure in engineering structures, it is necessary to determine the Weibull parameters of the constituent material, or materials, most likely to fail by cleavage. This is usually accomplished by carrying out cleavage tests using a relatively large number of notched or pre-cracked specimens of the material to obtain precise values of the Weibull parameters. Tests should be done at a sufficiently low temperature in order to obtain pure cleavage fracture, avoiding any pre-cleavage ductile tearing. Accompanying finite element simulations of these tests are also needed, as described in more detail below.
As noted earlier, this work is concerned with the GCHAZ of a multi-pass weld. Cleavage tests are therefore carried out on several SENB specimens that have previously undergone a thermal simulation of the welding process. A finite element analysis of the test geometry is subsequently undertaken in order to calculate the Weibull stress at the point of failure of each specimen. Weibull stresses are calculated for each test using a range of trial values of m. in Eq.. The appropriate Weibull parameters of the material are then determined using the maximum likelihood method described more fully in Ref 
In the present work the CTOD is used as the linking parameter. The value is calculated for the test results and the finite element solution using the equations given in BS 7448:Part 1: 1991.  In this way, Weibull stresses at the point of failure of each test are based partly on the measured load and partly on the crack mouth opening displacement.
4. Parent plate material
The parent material is a quenched and tempered grade 450EMZ bainitic steel in the form of 50mm thick plate. This is a structural steel used in offshore applications. It is produced to meet the requirements of BS 7191:1991 and has a specified minimum yield strength of 450N/mm 2. The main chemical components are given in Table 1.
Table 1 Chemical composition of grade 450EMZ bainitic steel
5. Small-scale thermal simulation test specimens
5.1. Thermal simulation and specimen preparation
Fifty-two single edge notch bend (SENB) 11x11 mm specimens are machined from the parent plate. The GCHAZ microstructure is simulated using a Gleeble 1500 thermal simulator. The thermal cycle is programmed to give a heating rate of 470°C/s, a peak temperature of 1350°C and a heat input of 3.5 kJ/mm for a hold time of 0.3s. After thermal simulation the specimens are ground to a cross section of 10x10 mm, notched in the rolling direction and pre-cracked by fatigue.
5.2. Cleavage fracture toughness tests
The cleavage test results are shown in terms of CTOD versus temperature in Fig.1. Three of the specimens show non-linear δ u type behaviour. One specimens exhibit pop-in. The remaining specimens show predominantly linear behaviour prior to failure by cleavage, giving valid values of δ c, the critical CTOD at the onset of brittle crack extension with ductile tearing Δa less than 0.2 mm according to BS 7448:Part 1.
Fig.1. The fracture toughness results for GCHAZ thermal simulation specimens in the form of CTOD versus temperature
5.3. Finite element modelling The SENB specimen is modelled using version 5.8-8 of the finite element program ABAQUS.  By taking advantage of two planes of symmetry, only one quarter of the specimen is modelled using the three-dimensional finite element mesh shown in Fig.2. Eight-noded, first order brick elements of type C3D8H are used throughout the mesh. Six rows of elements are used to represent the half thickness of the specimen. The elements are thinner near the free surface to represent the loss of constraint due to plane stress conditions.
The crack is modelled as a very narrow notch with a semi-circular tip to accommodate the effect of blunting. The mesh refinement increases near the crack tip to improve the accuracy of results and therefore the accuracy of the Weibull stress. The first element in front of the crack has a size of 37µm.
Fig.2. The finite element mesh of the GCHAZ thermal simulation 10x10mm SENB specimens: complete mesh (top) and mesh in crack tip region (below)
Appropriate boundary conditions are applied to restrain the specimen and properly represent the two planes of symmetry. The loading is applied by means of prescribed nodal displacements.
A large strain, elastic-plastic incremental analysis is carried out of the loading of the specimen at each of the three test temperatures of -80°C, -100°C and -120°C. Figure 3 hows the true stress versus true plastic strain behaviour used at these three temperatures. These curves are interpolated from tensile tests carried out for a range of test temperatures.
Fig.3. True stress versus true plastic strain behaviour of the thermally simulated GCHAZ at the three test temperatures: full range used in the analyses (top), behaviour at small plastic strain (bottom)
Approximately 120 increments of prescribed displacement loading are used to reach the maximum level of deformation measured at failure during the associated tests. By modelling a knife edge 2mm above the specimen surface ( Fig.2), the crack mouth opening displacement (CMOD) is determined for each increment. The CTOD is then calculated from the numerical clip gauge records according to BS 7448:Part 1, as for the actual tests. The highest prescribed displacement applied in the analyses gives a value of CTOD just beyond the maximum measured in the cleavage tests. At each load increment results are written to file for later post-processing.
5.4. Determination of Weibull parameters
A purpose-written computer program is used to post-process the three finite element analyses. Results for the Weibull stress at each load increment are calculated using Eq. for a range of trial values of the Weibull parameter m
. In Eq. the maximum principal stress, σ 1 j
, is taken to be the highest value achieved during the previous history of loading of each finite element integration station contained in the zone of plastic deformation around the crack front. Only two of the three parameters of the Beremin Model (the shape parameter, m
, scaling parameter, σ u
and material volume V o
) are independent, thus V o
is arbitrarily taken to be (100µm) 3
A set of Weibull stresses at the point of failure of each cleavage test is determined by linking the value of CTOD, measured at the point of cleavage failure during the test, with the corresponding CTOD calculated in the analysis at the same temperature. This involves the interpolation of results between load increments. The resulting three sets Weibull stresses at failure (corresponding to the three test temperatures) are combined into one set and ranked according to the value of the Weibull stress. This statistical sample of Weibull stress is then used in conjunction with the maximum likelihood method to determine, by iteration, the values of m and σ u that best describe the experimental sample of Weibull stress. The process is described fully in Ref  . It usually involves two or three iterations to arrive at the final value of m and σ u . The resulting values of m and σ u are given in Table 2.
Table 2 Results for Weibull parameters of the Beremin Model for grain coarsened heat affected zone material resulting from cleavage tests on small-scale thermal simulation SENB specimens.
|m||σ u (N/mm 2)||V o (µm 3)|
6. Multi-pass weld test specimens
6.1. Welding procedure
Full-scale submerged arc butt welds are produced. Special weld procedures, designed to maximise the amount of GCHAZ, are used for the welding. 
The weld preparation used is a half 'K' weld. This gives a vertical edge to the weld into which the pre-crack is machined, thus maximising the length of the target GCHAZ microstructure ahead of the crack tip.
6.2. Testing of SENB specimens
In total of 13 full thickness, 50x50mm, SENB, fracture toughness specimens are machined from the welded plate. After final machining, the thickness of the specimens actually range from 47.4mm to 50mm. Each specimen is surface notched to place the crack in the GCHAZ of the vertical fusion boundary. The specimens are tested according to BS 7448:Part 1. The ratio of fatigue crack depth to section width ( α/W
) varies from about 0.24 to 0.31. One specimen is tested at -70°C, two at -100°C, eight at -130°C, one at -160°C and one at -190°C.
After the tests the broken specimens are sectioned and scanning electron micrographs taken to determine the point of cleavage initiation and the subsequent fracture. These investigations are described fully in Ref  .
6.3. Finite element modelling
The geometry of the weld, including the width of the HAZ, are measured on the macro section . These measurements are used to construct finite element models of the SENB specimen. Slightly different models are set up for each temperature in order to match as closely as possible the average ratio of crack depth to width at the temperature.
By taking advantage of the plane of symmetry through the mid-thickness, only one half thickness of the specimen is modelled. Figure 5 shows the mesh. Eight-noded, first order brick elements of type C3D8H are used throughout the mesh. Six rows of elements are used to represent the half thickness of the specimen. The elements are thinner near the free surface to better represent the loss of constraint.
The crack is modelled as a very narrow notch with a semi-circular tip to accommodate the blunting that occurs there. The mesh refinement is increased near the crack tip to improve the accuracy of results there for Weibull stress. This is done by constructing a box of size 1.25x1.25mm (in plan view) around the crack tip, within which the elements are heavily focused towards the crack tip. The first element in front of the crack has a size of 5.3µm. The region modelled with GCHAZ properties extends the full thickness of the mesh (along the whole half crack front modelled), about 1.9mm ahead of the crack front and 0.95mm on either side of the plane of the crack. As noted below, this region dominates the cleavage fracture behaviour.
Fig.5. Finite element mesh of the 50x50mm SENB specimen of the weld, overall geometry of mesh (top), and close up of crack tip region (below)
Appropriate boundary conditions are applied to restrain the specimen and properly represent the two planes of symmetry. The loading is applied by means of prescribed nodal displacements. The maximum displacement applied ensures that a failure probability of at least 90% is achieved in all analyses (see the next sub-section).
A large strain, elastic-plastic incremental analysis is carried out of the loading of the specimen at each of the five test temperatures of -70°C, -100°C, -130°C, -160°C and -190°C. True stress versus plastic strain curves are obtained at all temperatures and as an example Figure 6 shows the true stress versus true plastic strain behaviour used for the parent, weld and GCHAZ at -130°C.
Fig.6. Tensile properties used for the parent, weld metal and simulated GCHAZ at -130°C: full range used in the analyses (top), behaviour at small plastic strain (bottom)
A knife edge is modelled 2mm above the specimen surface to allow the CMOD to be determined at each load increment. The CTOD is calculated from the numerical clip gauge records according to BS 7448:Part 2;  the same as the actual tests. At each load increment results are written to file for later post-processing.
6.4. Determining the probability of cleavage failure
A purpose-written computer program is used to post-process the finite element analyses. Results for the predicted failure probability at each load increment are calculated by means of Eq. followed by Eq., using the values of m
, σ u
and V o
given in Table 2
. The volume integral in Eq. is carried out only within the GCHAZ region included in the model ahead of the crack front. This limited region of integration is acceptable, since the principal stresses is this crack tip region provide the dominant contribution to the Weibull stress through Eq.. Including the stresses outside the crack tip neighbourhood makes no significant difference to the result.
7. Results and discussion
compares the predictions with the test results in terms of CTOD versus temperature.
Fig.7. Comparison of test results and predictions of failure probability as a function of temperature for CTOD tests on surface notched 50x50mm SENB specimens of 450 EMZ welds
The solid symbols in Fig.7 show the results of the SENB tests that fail by cleavage. The two curves link the values of CTOD that should give cleavage failure probabilities of 10% and 90% according to the finite element predictions using the Beremin Model. The curve showing the 10% level of predicted failure probability gives a reasonable, though slight under-estimate, of the experimental results for CTOD, since none of the 13 test results lie beneath it. The 90% level of predicted failure probability gives some under-estimate of the test results. Three results out of the 13 lie above the curve, rather than the one or two that might be expected. This under-estimate of CTOD is caused by the over-estimate of failure probability for a given CTOD.
The over-estimate of failure probability, particularly at the 90% level can occur for a number of different reasons:
- Imprecise values of the Weibull parameters of the weld GCHAZ caused, possibly, by scatter in the experimental results due to variation of the tensile properties not taken account in the finite element analyses.
- The use of a uniform 1.9mm wide 'box' of GCHAZ material ahead of the crack. These worst case values of inherent toughness relevant to the GCHAZ might be applied over a volume that is actually larger than occurs in practice, andwill give the greatest over-predictions of failure at the highest CTOD as witnessed in the present study.
- The presence of small amounts of crack tip ductile damage and possible tearing in the tests with the highest CTOD. If present, this damage is not taken into account in the finite element analyses of the weld SENB specimens.
Reason i) cannot be readily eliminated or properly taken into account. However, it is possible that the inherent scatter in tensile properties, within the GCHAZ in particular, is taken somewhat into account in the Weibull parameters. That is, variations in hardness and so yield strength are manifest as variations in inherent toughness and so a reduction in the value of m. Reason ii) can be remedied by determining the Weibull stress by integration of Eq. over a larger volume about the crack tip. However, Weibull parameters for the parent and weld should strictly be used as well as GCHAZ. Finally, item iii) can be taken into account by using a more comprehensive constitutive model of material behaviour that includes ductile damage, but this was outside the scope of the present study.
8. Summary and conclusions
This study reported here uses the Beremin Model to predict cleavage failure within the grain coarsened heat affected zone of a multi-pass weld.
The following conclusions are reached:
- The predicted 10% level of failure probability gives a reasonable, though slight under-estimate of the experimental results for CTOD in the real weld specimens at the various temperatures.
- The 90% level of predicted failure probability gives some under-estimate of the test results. Three results out of the 13 lie above the curve, rather than the one or two that might be expected.
- The predictions have shown benefits in using specimens made from simulated GCHAZ microstructure to obtain fracture toughness value for real multi-pass weld, as evident from the agreement between the actual test results and predictions.
- Beremin F M: 'A Local Criterion for Cleavage Fracture of a Nuclear Pressure Vessel Steel', Met. Trans. A., Vol. 14A, November 1983, pp.2277-2287.
- Wiesner C S: 'The Local Approach to Cleavage Fracture - Concepts and Applications'. Abington Publishing, ISBN 1 85573 261 0, 1996.
- BS 7448:Part 1:1991 'Method for determination of K IC, critical CTOD and critical J values of metallic materials', BSI London.
- ABAQUS/Standard User's Manuals , Version 5.8, Hibbitt, Karlsson and Sorenson Inc., 1080 Main Street, Pawtucket, Rhode Island, 1998.
- Pisarski H G: 'A Review of HAZ Toughness Evaluation', TWI Report No. 566/1996, August 1996.
- Cardinal N and Wiesner C S: 'Fracture Risk Prediction of Welded Joints', ECSC Agreement Nos 7210.MC/805 and 7210.MC/806, Technical Report No. S394-7 961, British Steel plc, Swinden Technology Centre, 1996.
- BS 7448:Part 2:1997 'Method for determination of K IC, critical CTOD and critical J values of welds in metallic materials', BSI London.