## Abstract

The paper reports three-dimensional creep continuum damage mechanics (CDM) analyses of creep failure in a medium bore Cr–Mo–V low alloy ferritic steel welded branched-pressure vessel that has been tested under a constant pressure of 4 MPa, at a uniform temperature of 590 °C. The use of the CDM computer software Damage XXX to analyse the initiation and growth of creep damage and subsequent failure in the branch weld is reported for a five-material model that includes: parent, Type IV, refined heat affected zone (R-HAZ), coarse grained heat affected zone (CG-HAZ) and weld materials. The results of the analyses are presented for two cases: the first without the CG-HAZ; and, the second with the CG-HAZ included. For both cases, lifetimes are conservatively, yet accurately predicted. It is shown that it is necessary to use a Type IV thickness of 0.7 mm to accurately predict the failure location and mode. The results of metallographic examinations of a tested vessel and the predicted damage fields are in close accord. Failure is predicted to take place, by steam leakage, from the interior of the vessel, through the Type IV zone adjacent to the main pipe, connecting through the R-HAZ to the CG-HAZ, where leakage takes place at the weld toe in the crotch plane.

## 1. Introduction

Design codes (BS 806: 1993; BS 1113: 1989; BS 5500: 1991) for high-temperature, pressurized welded pipes are based on uni-axial stress rupture properties of the parent pipe material, and do not account for the effects of the multi-axial stress states that arise from the different mechanical properties of the parent material, weld material and the associated phases. In these codes, the pipe wall thickness, , is determined from the thin pressure vessel formulae for the mean diameter hoop stresswhere *D*_{m} is the mean diameter and *D*_{in} and *P*_{int} are internal diameter and pressure respectively, coupled with weld strength reduction factors. Owing to these limitations, creep failure in weldments can occur earlier than the expected rupture time. In addition, the weld strength reduction factors, typically 1.3 on stress, have been found to be non-conservative (Hayhurst & Goodall 2003).

In an attempt to overcome this shortcoming, Hall & Hayhurst (1991) developed a two-dimensional continuum damage mechanics (CDM)-based finite-element (FE) solver, Damage XX, (Hayhurst *et al*. 1984*a*,*b*), which incorporates the physics of the creep deformation and rupture of the individual phases of the weldment. They studied thick section steam pipes fabricated from 0.5Cr, 0.5Mo, 0.25V parent pipe butt-welded using 2.25Cr, 1Mo weld material (Coleman *et al*. 1985). This approach has been shown to successfully predict the deformation, damage and failure history of the tests of Coleman *et al*. (1985). The research highlighted the important role of the difference in the creep characteristics of the weld metal, the heat affected zone (HAZ) material and the parent material. It was shown that in butt-welded pipes the mismatch between material phases results in a marked redistribution of stress from the weld metal into the HAZ and the parent material. And also, that the location of the maximum axial and hoop stresses shifts from the inner surface to the outer surface of the pipe in all micro-structural regions of the weld during primary–secondary creep. This leads to the initiation and evolution of creep damage in the outer third of the weld metal and close to the HAZ, which is in the region known as *fusion boundary*. It was also observed that during the secondary creep stage, the maximum effective stress occurs at the inner surface of the pipe and the maximum principal stress occurs at the outer surface. Hence, the multi-axial stress rupture criterion, *α* (Hayhurst 1972), has an important role in further stress redistribution, in the determination of the damage distribution and on the predicted weldment lifetime. Depending upon the value of the multi-axial stress rupture criterion, the weldment can show either strengthening or weakening relative to the behaviour of the material phase in which failure occurs. Hayhurst *et al*. (1984*b*) have also observed this phenomenon in the estimation of the creep rupture times for notched bars. To investigate the effect of the material properties on the rupture time of butt-welded pressurized pipes, Wang & Hayhurst (1994) introduced a set of normalized material parameters to define the creep behaviour of the weld metal and the HAZ material relative to the parent metal. This permitted the determination of the constants in the CDM-based constitutive equations (Wang & Hayhurst 1994). They used these equations in axi-symmetric FE studies of pressurized butt-welded pipes to identify an optimal set of weld and HAZ material properties, which results in an improvement in the lifetime prediction of 30% over that obtained for the initial material property dataset. Hence, CDM-based FE methods have been found to be a valuable tool for the analysis of the creep rupture behaviour of pipe weldments and for the selection of optimal material properties.

Hayhurst & Miller (1998) have used CDM-based constitutive equations to model the behaviour of the weld, refined heat affected zone (R-HAZ), Type IV, and parent material in a welded pressurized branched connection, operated at 590 °C. The medium bore 0.5Cr, 0.5Mo, 0.25V parent vessel was welded using 2.25Cr, 1Mo weld material to the parent material branch, figure 1. The Type IV material was located between the parent material and the R-HAZ zone. They used the constitutive equations in Damage XX to model the creep deformation and multi-axial rupture behaviour of weldments in the axi-symmetric equivalents of the crotch section (plane containing the axes of the main pipe and branch) and flank section (plane normal to the crotch and coincident with the branch axis) of pressurized branched pipe-work. The crotch section equivalent was predicted to fail in the Type IV region adjacent to the branch in 28 906 h, while the flank section equivalent was predicted to fail in the Type IV zone close to the sphere (main pipe) in 14 759 h. This behaviour is caused by the low effective stress and high first stress invariant in these regions, which themselves are a result of stress redistribution caused by the mismatch in the creep properties of the weld, R-HAZ and Type IV and parent materials. The results of these CDM analyses have not, until recently, been compared with the results of vessel tests (cf. figure 1).

The results of a test carried out on a medium bore welded branch (0.5Cr, 0.5Mo, 0.25V parent vessel welded using 2.25Cr, 1Mo weld material) under a constant pressure of 4 MPa at 590 °C have been reported by Patel (2002, 2003). The experimental set-up is shown in figure 1. The vessel failed after 20 038 h at the crotch section within the coarse grained heat affected zone (CG-HAZ), and Type IV regions on the main pipe side of the weld. This mode of failure is not predicted by the axi-symmetric analyses of Hayhurst & Miller (1998); although the flank section has the lower lifetime, failure is incorrectly predicted to occur in the branch Type IV region.

The accuracy of the predictions made using the two-dimensional equivalent axi-symmetric nozzle–sphere intersections suffers from two deficiencies. Firstly, the assumed approximations of the two-dimensional boundary conditions to the three-dimensional problem involved initial stress systems with inherent errors, but these were unlikely to significantly influence the predicted damage evolution modes; secondly, the two sections were assumed to be unconnected and statically admissible without having to satisfy compatibility of nozzle end displacements. Hayhurst & Miller (1998) concluded that this deficiency could best be rectified by carrying out a full three-dimensional analysis. A crude assessment of what might be achieved can be obtained by comparison of the average of the predicted lifetimes for crotch and flank, i.e. 21 833 h, with the experimental lifetime of 20 038 h; namely, a lifetime estimate of the correct order.

In an attempt to rectify the deficiencies of the two-dimensional model of Hayhurst & Miller (1998), Mustata *et al*. (2003) have modelled the same branch and weld geometry using the three-dimensional CDM solver Damage XXX. The weld geometry defined by Hayhurst & Miller (1998) was used in conjunction with two sets of constitutive parameters for the different phases of the weldment (0.5Cr, 0.5Mo, 0.25V parent welded using 2.25Cr, 1Mo weld material). The first set is that used by Hayhurst & Miller (1998) and the second set has been derived by Hayhurst *et al*. (2003). In both cases, predictions of time to first steam leak were 18 685 h and 18 688 h for the constitutive equations of Hayhurst & Miller (1998) and Hayhurst *et al.* (2003), respectively. In comparison with the test vessel lifetime of 20 038 h this represents an error of 6.8%. However, the distribution of damage within the weldment, its initiation and evolution, did not closely correspond with the results of the metallographic examination of the tested vessel.

There are three possible reasons for this: the first relates to the geometry/thickness of the different weldment phases; the second relates to the presence of an additional weldment phase, known as the CG-HAZ, which is located in the HAZ adjacent to the weld material, it is a narrow zone that has lifetimes similar to the refined HAZ, but has reduced minimum creep rates and ductility; and the third relates to the accuracy of the FE model.

With regard to the weldment phase geometry/thickness, the previous paper used the dimensions reported by Hayhurst & Miller (1998); however, the tested vessel has now been sectioned, and the geometry was measured from the micrographs. There are significant geometrical differences, including the presence of CG-HAZ; and in addition, Hayhurst *et al*. (2004) have shown that weldment behaviour is strongly dependent on Type IV zone thickness. Hence, it is essential that the three-dimensional analysis be carried out with the correct thickness of both the CG-HAZ and Type IV zones.

The CG-HAZ region has been clearly identified from metallographic sections of a welded vessel, from which dimensions have been measured and used to construct a five-material weld model. The thickness of the Type IV zone that has been used is that determined by Hayhurst *et al*. (2004) from studies carried out on butt-welded pipes and cross-welded testpieces. The geometry of the R-HAZ, CG-HAZ and weld materials have been determined from macrographs as outlined in §2.

Concerning the accuracy of the FE model used in the previous paper, only one six-sided brick occupied the thickness of the thinnest Type IV zone; elongated bricks were used around the Type IV band from the crotch to flank planes. Previous experience by Hayhurst *et al*. (2002) indicates that a better distribution of elements is appropriate, and hence a more refined mesh has been used in this study.

In the following sections the branch geometry and boundary conditions are introduced for the five-material weld model; this is followed by a section on constitutive equations and their calibration; and finally, a section is presented on the three-dimensional FE analysis before the CDM branched vessel analysis results are presented.

## 2. Branch geometry and boundary conditions

### (a) Test conditions, material and main vessel geometry

The medium bore 0.5Cr, 0.5Mo, 0.25V parent vessel was welded using 2.25Cr, 1Mo material to the parent material branch. The main body of the vessel (pipe) was 465 mm in outer-diameter with a wall thickness of 20 mm; the branch was 111 mm in inner-diameter with a wall thickness of 8 mm. The mean radius to thickness ratio of the main pipe *R*/*t*_{p} is 11.125, and the mean radius to thickness ratio of the branch *r*/*t*_{b} is 7.438. Both the main pipe and branch were pressurized to a constant level of 4.0 MPa, at a constant uniform temperature of 590 °C. The internal pressure generates a main pipe mean diameter hoop stress *PR*/*t* of 44.5 MPa. The parent and weld materials have the same Young's modulus of 160 GPa at 590 °C.

### (b) Residual stresses due to welding

It is well established (Xu & Burdekin 1998) that welding processes induce high levels of residual stress in weldments and surrounding parent material; and, that the resulting plasticity can influence subsequent fracture behaviour. For the class of low alloy ferritic steels studied here, it is established practice to stress relieve welds using an appropriate heat treatment. The test vessel under investigation has been processed in this way; and, in the CDM analyses, the welds have been assumed to be stress free.

### (c) Weld geometry

The weld geometry was specified at two geometrical locations, namely the crotch and flank sections. The crotch plane contains the axes of both main pipe and branch; figure 1 defines two such diametrically opposite locations as 0 and 180°. The flank plane is normal to the crotch plane, and contains the axis of the branch. The same terminology has been used by Hayhurst & Miller (1998).

The geometry of the weld on the flank section is shown in figure 2*a*. The details of how the weld, CG-HAZ, R-HAZ and Type IV materials interface with the main pipe are shown in figure 2*b*. The geometry of the weld on the crotch section is shown in figure 2*c*. It may be seen that there are two main differences between the geometry specified in figure 2, and that used by Mustata *et al*. (2003). They are the thickness of the Type IV zone, and the presence of the CG-HAZ. These are now discussed in turn.

#### (i) Thickness of Type IV zone

In a study of the effect of Type IV zone thickness on lifetime of thin butt-welded pipes and uni-axial crosswelded testpieces in the same ferritic steel as studied in this paper, Hayhurst *et al*. (2004) have shown testpiece lifetimes, predicted by CDM methods, to be very sensitive to Type IV zone thickness. Following a computational study of a range of testpiece geometries, and comparisons with experimental results, they concluded that the Type IV thickness is 0.7 mm.

#### (ii) Coarse grained heat affected zone

The testing procedures for the vessel shown in figure 1 have been reported by Patel (2002, 2003), and the results of post-testing metallographic investigation have been presented by S. M. Chilcott (2001, private communication). Independent metallographic examinations of both crotch and flank weldments sections have been carried out by British Energy and the resultant macrographs have been used to: (i) identify the presence of the CG-HAZ and (ii) determine the mean CG-HAZ thickness at different locations. On both the crotch and flank sections, two CG-HAZ regions were identified; one associated with the branch and the other with the main pipe, yielding a total of four macrographs. For the crotch and flank main pipe macrographs, the areas of the CG-HAZ zones have been determined, and divided by the respective CG-HAZ through-thickness lengths to yield an average thickness for the crotch and flank sections. The same procedure was followed for the branch CG-HAZ. The thicknesses of the CG-HAZ regions have been averaged to yield one value for the main pipe, 0.50 mm, as shown in figure 2*b*,*c*; and one value for the branch, 0.57 mm, as given in figure 2*a*,*c*. The average CG-HAZ thickness values were assumed to be the same for all branch diametral sections between the crotch and flank planes.

While the complexities of FE mesh generation necessitate the use of constant averaged values of CG-HAZ thickness, variations in thickness from crotch to flank were observed. For example, for the main pipe, the ratio of the CG-HAZ thickness at the weld toe and inner bore was 0.6 and 0.5, for the crotch and flank planes, respectively; and, for the branch CG-HAZ the corresponding ratio was 2.2 on the flank section, and no CG-HAZ was evident on the inner bore of the crotch section. The significance of this will be discussed in a later section.

## 3. Constitutive equations

### (a) Physical background

The set of constitutive equation developed by Perrin & Hayhurst (1996, 1999) make use of a number of state variables to describe the softening mechanisms (Ashby & Dyson 1984) that ferritic steels are known to suffer, namely, creep cavitation and the coarsening of the carbide precipitates. A state variable is also used to describe the interaction between strain hardening and recovery that is associated with primary creep. The constitutive equations in multi-axial form, that relate the creep strain rate, , to the stress, *σ*_{ij}, are now presented(3.1)where *N*=1 for *σ*_{1}>0 and *N*=0 for *σ*_{1}< 0. The material parameters *A*, *B*, *C*, *h*, *H* * and *K*_{c} define the uni-axial creep behaviour and the stress index *ν* defines the multi-axial stress rupture criterion. The parameter *σ*_{1} is the maximum principal stress, is the effective stress, where is the stress deviator and is the effective creep strain. *H*, *Φ* and *ω* are the three state variables. The first state variable, *H*, represents the strain hardening that occurs during primary creep; initially *H* is zero and, as strain is accumulated, increases to a value *H* * when secondary creep would be expected to occur. The second state variable, *Φ*, describes the evolution of the spacing of the carbide precipitates, which is known to lead to a progressive loss in the creep resistance of particle hardened alloys such as ferritic steels. This state variable is defined, for mathematical convenience, to lie in the range of zero to unity. The third state variable, *ω*, represents intergranular cavitation damage and varies from zero, for the material in virgin state, to 1/3, when all of the grain boundaries normal to the applied stress have completely cavitated (Dyson & Gibbons 1988), at which time the material is considered to have failed. The parameter *ν* is the multi-axial stress sensitivity parameter determined from independent multi-axial stress state tests.

As outlined by Leckie & Hayhurst (1975), equation set (3.1) makes use of the damage state variable *ω* to define identical states in situations where stresses are either increasing or decreasing. In this way, evolution of strain and damage is calculated for variable stress histories.

### (b) Selected constitutive datasets

In this paper, the welded medium bore branch pressure vessel will be analysed in three dimensions using the software Damage XXX (Hayhurst *et al*. 2002). The set of constitutive parameters reported by Hayhurst *et al*. (2005) will be used for 590 °C, the details of which are now discussed. Since all phases of the weld are from the same generic group, the constitutive equation (3.1) has been calibrated by Hayhurst *et al*. (2004, 2005), for each material component of the weld. The constitutive parameters are summarized in table 1 for all five materials at the temperatures of 590 °C. Hayhurst *et al*. (2004) have shown that the multi-axial stress rupture parameter *ν* for the Type IV material, given in the fourth line of equation set (3.1), is dependent upon the level of the average effective stress (*σ*_{e}) in the minimum section of the plane strain crossweld testpiece in the absence of the weld. The variation of *ν* with the normalized effective stress (*σ*_{e}/*σ*_{o}), where *σ*_{o} is the uni-axial stress required to give 1000 h lifetime in the Type IV material at the operating temperature, is given by(3.2)where values of the empirical factors are *a*=2.2133 and *b*=1.9861 for 590 °C. The value of the effective stress *σ*_{e}=25.76 MPa used in the vessel analysis is that calculated for the branch membrane using the mean branch diameter; and with the value of *σ*_{o}=136.58 MPa this yields the value of *ν*=3.2192. The constitutive parameters given in table 1 have been used with equations (3.1) and (3.2) to carry out analyses on the welded branch with the newly created parallel architecture software Damage XXX, which is now discussed.

## 4. Three-dimensional CDM finite-element analyses of welded branch

### (a) Damage XXX–CDM software

The first three-dimensional creep CDM analysis was carried out by Wong (1999) and Hayhurst *et al*. (2002). The drawbacks to the use of the software for problems of industrial significance have been, firstly, the inability to solve problems with large numbers of FEs and degrees of freedom, and secondly, the speed of the numerical equation solver, which occupies a major part of the computer resource required.

The solution route taken has been to use the Frontal numerical solver written by Scott (2002) for use on parallel architecture computers; and, to employ a heuristic time-step algorithm due to Vakili-Tahami (2002). Both of these developments have resulted in a speed-up of the running of the code by a factor of 35. Also, the availability of a computer with increased RAM and hard drive storage capability has meant that the number of degrees of freedom and FEs that can be analysed has been considerably increased from 5000 to 200 000 degrees of freedom.

The constitutive equations established by Hayhurst *et al*. (2004), for ferritic steel and associated weldment phases for a range of temperatures, have been used by Hayhurst *et al*. (2003) to model three-dimensional plane strain cross-welded testpieces studied experimentally by Fairman (1995). The predictions made using the new three-dimensional parallel computer architecture software Damage XXX have been compared with the two-dimensional plane-strain predictions of Hayhurst *et al*. (2004) and with experimental results of Fairman (1995); and in this way, the Damage XXX software has been quality assured.

### (b) Finite-element idealization of branch

A single quadrant of symmetry of the branch is shown in figure 3; the planes of symmetry are the flank and crotch planes. The volume has been modelled with six-sided bricks using the Femsys1 software. Each brick is then filled with 24 constant strain tetrahedra; the total number of nodes is 54 687, the number of degrees of freedom is 164 061 and the number of elements is 240 048.

The internal pressure conditions of 4 MPa were applied using nodal forces normal to the internal surfaces of the pipe and branch; and the pipe and branch endloads were applied using nodal forces parallel to the pipe and branch axes. Zero normal displacements were applied to nodes on the two planes of symmetry.

Graphing software was used to segment the dataset into four zones with minimal interconnecting regions; the datasets for each of these four zones were assigned to individual computer processors, and hence used to achieve optimal parallel processing speed with the four processors of SUN E450 computer workstation. Without the use of computer parallelism, it would not have been possible to solve the massive combined boundary-initial-value problem posed by the welded branch problem.

### (c) Definition of material phases in the weldment

The four weldment material phases: Type IV, R-HAZ, GG-HAZ and weld, are shown on the flank and crotch cross-sections in figure 4*a*,*b*. There are two Type IV two R-HAZ and two CG-HAZ zones; one associated with the branch (cf. upper illustration of figure 5) and the other with the main pipe. Each zone has been treated as a twisted band; for example, consider the branch Type IV band, it may be represented as a six-sided body, the upper-face is in contact with the parent material, the lower face with the R-HAZ region; one face is common with the flank section and another face is common with the crotch section. The two remaining faces are common with the inner and outer cylindrical surfaces of the branch. For the purpose of creating the FE mesh, the pre-processor Femsys automatically calculates the intersection between cylinders whose axes are at right angles; and this facility has been used. Each edge of the zone or body, except those in the flank and crotch planes, has been approximated by the intersection of cylinders at right angles. Hence, for each edge of the band in the branch Type IV zone a separate cylinder is selected whose axis is in the crotch plane and parallel to that of the main pipe, and which is separated from the axis of the main pipe by a distance *α*, the radius of the cylinder is *ρ*; both *α* and *ρ* have been selected, first by calculation, and then by human interaction, to give the best approximation. For the Type IV zone, four such intersections were defined and optimized to give a close approximation to a zone or band of constant thickness. All other edges of the CG-HAZ band have been treated in a similar way. The edges of the Type IV and CG-HAZ zones in the main pipe have been determined in the same way. However, at the toe of the weld the interfaces between the main pipe Type IV, R-HAZ and CG-HAZ zones all have radii whose centre is at the point of intersection between the weld zone and the main pipe CG-HAZ (cf. figure 2*b*); this condition is maintained for all planar sections, passing through the axis of the branch, between the flank and crotch planes. Each band, zone or body has been segmented into 30° sectors, one coincident with crotch section *θ*=0°, a second central sector *θ*=30–60°, and a third sector *θ*=60–90°, which is coincident with the flank section. Each of the three sectors, for each material zone, is used to define FE brick size and their gradation with angular position in the band, zone or body.

For clarity the branch and pipe Type IV, R-HAZ and CG-HAZ bands have been isolated and are shown in figure 5. Only the bricks are shown without the constant strain tetrahedral elements. Each 30° section can be seen to comprise eight bricks. On the right of the figure is the *θ*=0° crotch plane and on the left is the *θ*=90° flank plane. This material zone layout will be used to present field variable results in subsequent sections.

### (d) Numerical procedures

A three-dimensional CDM FE solver, Damage XXX (Hayhurst *et al*. 2002) was used for the numerical computations. The numerical procedure used to solve the boundary-value problem for creep damage deformation was that used by Hayhurst *et al*. (1984*a*). It is based on the FE method and employs constant strain tetrahedral FEs. The procedure takes the elastic solution as its starting point and integrates with respect to the normalized time, the normalized creep strains, *λ*_{ij}(=*ϵ*_{ij}/*ϵ*_{o}, where *ϵ*_{o}=*σ*_{o}/*E*) and creep damage variable *ω*, the particle coarsening parameter *Φ*, and the hardening/recovery parameter *H*. The integration is carried out over a series of discrete normalized time-steps using a fourth-order Runge–Kutta technique; this procedure involves the repeated solution of the boundary-value problem to determine the field quantities required for the numerical solution. Creep damage, as represented by the two damage state variables, develops monotonically with time throughout the structure, and failure of an element is deemed to have occurred when the damage state variable, *ω*, attains the value of 0.33. For the case when *ω*>0.33, and provided that *σ*_{e}>20 MPa, *σ*_{1}>20 MPa and *σ*_{1}/*σ*_{e}>0.3 the element is retained within the system until *ω*=0.7 to permit complete redistribution of the significant stresses that exceed 20 MPa. Material elements which satisfy these conditions are assumed to have failed, and are then unable to transmit or sustain load, hence they are removed from the model. This is justified by virtue of the kinematical determinacy of the structure, as opposed to the statical determinacy of the uni-axial testpieces that have been used to calibrate the equations. The boundary-value problem is then redefined to allow either a crack, or damage zone, to develop and spread. Once the boundary-value problem is redefined, the time integration is continued by taking the field variables just before the local failure occurred as the new starting point. The procedure is then repeated until complete failure of the damage vessel occurs.

## 5. Investigation overview

As discussed in §1 there are three drivers to this investigation. The first concerns the thickness of the Type IV zone. The study of Mustata *et al*. (2003) carried out on a similar branch, assumed a Type IV zone thickness of 1 mm; this investigation will use the true value of 0.7 mm identified by Hayhurst *et al*. (2004). The second driver is to assess how the presence of the CG-HAZ zones and their different properties influences the predicted lifetimes, damage initiation and damage evolution crack/growth. To achieve this end, analyses will be carried out, both with and without the CG-HAZ regions. The third driver is to help rectify the deficiency of the relatively coarse FE meshes used in the three-dimensional analyses carried out by Mustata *et al*. (2003); this has been achieved by almost doubling the number of bricks through material zones in the pipe thickness direction; and, doubling the number of bricks in the material bands from the crotch plane to the flank plane; however, the minimum number of bricks through the Type IV thickness remains one. In the next section the results of the computational predictions are presented, and the above three points addressed.

## 6. Presentation of results

The results of two CDM calculations are now presented for the branch using the constitutive equation set (3.1) with the parameters given in table 1: firstly, for a four-material model without CG-HAZ, and, secondly, for the five-material model with CG-HAZ. In both cases, the elastic solutions are the same, since all material phases have the same value of Young's modulus and Poisson's ratio. The elastic fields have been quality assured by comparison with the elastic solutions of Patel (2002, 2003) obtained using the FE package Abaqus for the same geometry, and extremely close agreement has been obtained.

CDM solutions are now presented for both cases. The three material phases for which extensive damage occurs are the Type IV, R-HAZ and CG-HAZ; small amounts of damage were observed in the weld and parent materials, although these were not significant and will not be reported.

### (a) Four-material model: without CG-HAZ

Failure was predicted to take place in the main pipe Type IV region, and this is shown in figure 6. The figure shows an outline of the region with those elements which have failed, and removed from the calculation, shown in grey. The first element fails as shown in table 2 at 9111 h. Figure 6*a* is for the life fraction *t*/*t*_{r}=0.81 and shows damage to be concentrated, subsurface, in the radius of the main pipe Type IV zone between the diametrical planes 0<*θ*<15°, and located close to the crotch plane. Figure 6*b* shows the distribution of the Type IV failed elements at the life fraction *t*/*t*_{r}=0.94. Two features are evident: firstly, growth along the band from crotch to flank, an extension of significant growth from *θ*=15 to 26°, and secondly, damage growth through the wall thickness to over one half the wall thickness in the Type IV zone. At this stage, there is still a significant ligament of unfailed elements to avoid the formation of a through-thickness steam leak path. In figure 6*c* the domain of failed elements is shown for the life fraction *t*/*t*_{r}=0.99. The extent of damage in the Type IV band has intensified close to the crotch plane and a steam leak path has almost formed over the region 0<*θ*<15°; and, growth of damage has taken place around the Type IV band to beyond *θ*=37°. The branch diametral plane, defined by *θ*=37°, coincides with the location of predicted failure, in the branch Type IV zone, made using the four-material model by Mustata *et al*. (2003).

Shown in figure 7 are damage contour plots of diametral sections of the branch close to failure at the life fraction *t*/*t*_{r}=0.99; both the branch and main pipe Type IV regions are presented. Three sections are shown: figure 7*a* is the crotch section; figure 7*b* is a section on the plane defined by *θ*=37°, selected for comparison with damage on the identical plane examined in the analysis carried out by Mustata *et al*. (2003) and figure 7*c* is the flank section. Figure 7*a* shows crotch plane R-HAZ damage, which took place at the weld toe, shortly after loading. This damage has not developed. By contrast, the growth of damage through the Type IV region is most evident. Damage on the diametrical plane defined by *θ*=37° is shown in figure 7*b*, the lower intensities of damage at the weld toe, and in the Type IV region can be seen; however, damage in the branch Type IV can be observed as predicted by Mustata *et al*. (2003). Figure 7*c* shows the absence of damage, except in the R-HAZ region at the weld toe; this damage formed early in life and was subsequently benign—its formation serving to provide a stress-relief mechanism.

Figure 8 shows the predictions made for the four-material model presented in the same format as used in figure 6 for the main pipe Type IV region. Figure 8*a* shows the results for the life fraction *t*/*t*_{r}=0.81; damage has initiated subsurface on diametral planes defined by 15°<*θ*<37°; this does not coincide with the location of predicted failure, *θ*=37°, made by Mustata *et al*. (2003). Figure 8*b*,*c* shows that damage in this region has propagated both through the branch thickness and around the band, in the directions of the crotch and flank with increasing life fractions. In comparison with the predictions of Mustata *et al*. (2003), damage growth/intensification takes place some 11° closer to the crotch plane at *θ*=26.

Comparison of figures 6 and 8 suggests that damage in the main pipe Type IV occurs in the region 0°<*θ*<15°, while in the branch Type IV it forms in the region 15°<*θ*<37°. In terms of load paths along the axis of the branch, these two regions do not overlap and each provides a local mechanism for unloading of the axial load in the branch. Hence, it is unlikely that dominant growth will take place in the branch Type IV in the region 0°<*θ*<15° and in the main pipe Type IV region 15°<*θ*<37°. This is observed until the latter stages of crack growth.

In comparison with the results of Mustata *et al*. (2003), the time to first element failure of 8906 h compares well with the 9111 h prediction (cf. table 2), and the rupture lifetime of Mustata *et al*. (2003), 18 688 h, compares well with the prediction of 18 546 h (cf. table 2), giving a difference of 0.76%. The damage growth phase predicted by Mustata *et al*. (2003) is 52% (=((18 688−8906)/18 688)100), whereas that predicted by the present study is 51% (=((18 546−9111)/18 546)100). These results are close considering that the analysis of Mustata *et al*. (2003) used 12 480 FEs and 87 048 degrees of freedom; and the present study used 240 048 FEs and 164 061 degrees of freedom. However, the predicted failure mechanisms are different. The fact that first failure times are so close, a phenomenon which would be expected to be mesh sensitive, indicates that the dramatic difference in the predicted failure location is due, not to FE mesh size, but to the correct modelling of the Type IV thickness, i.e. 0.7 mm as opposed to the 1.0 mm used by Mustata *et al*. (2003).

### (b) Five-material model: with CG-HAZ

Failure was predicted to take place in the main pipe Type IV region, as for the four-material model. The equivalent damage fields to those given in figures 6 and 8 have been obtained and found to be almost identical; hence, they are not reproduced here. It is very clear that despite the presence of the weaker CG-HAZ region, relative to the parent material: ; and ; (Hayhurst *et al*. 2005) the predicted failure mode is unchanged.

Figure 9 shows damage contours on diametral sections of the branch close to failure at the life fraction *t*/*t*_{r}=0.99. The layout of the presentation is the same as that in figure 7, with the exception that figure 9*b* is for the diametral plane *θ*=22° and figure 7*b* is for *θ*=37°. Figure 9*a* shows failure to take place in the main pipe Type IV region, as indicated in figure 6, with damage localized at the weld toe in the CG-HAZ region. The weld toe CG-HAZ damage formed very early in life and spread from crotch to flank very quickly, the reason for this is that it provides a stress-relief mechanism; but, having relieved the stress the CG-HAZ damage is benign and does not propagate until near end of life. Growth of damage then predominates in the Type IV region; however, evidence of linkage between CG-HAZ and Type IV damage through the main pipe R-HAZ is apparent (cf. figures 9*a* and 10*b*).

Intense damage may be observed in figure 9*a*,*b* in the branch CG-HAZ, and some evidence of ‘fusion boundary’ damage in the adjacent weld region, although this occurs near the end of life.

Figure 9*b* shows damage on the diametral plane defined by *θ*=22° and its location is shown in figure 6*c*; this plane has been selected since it resides between the regions of dominant Type IV damage in the branch and main pipe as indicated in figure 6. Subsurface damage can be seen in the branch Type IV region, figure 9*b*; CG-HAZ damage at both weld toes is evident, a predominance of main pipe Type IV damage can be seen and evidence of linkage between CG-HAZ and Type IV damage through the main pipe R-HAZ is apparent. Figure 9*c* for the flank section shows no evidence of damage other than the initial CG-HAZ damage referred to earlier.

The growth of damage in the branch Type IV is the same as for the four-material case, and is shown in figure 8. Again it shows a predominant growth around the band 15°<*θ*<37°; a less predominant growth through the band; and, a general intensification of damage.

Figure 10 compares the three-dimensional distributions of failed elements for the four-material model, figure 10*a*, and the five-material model, figure 10*b*. The principal difference is that figure 10*b* includes the orange CG-HAZ elements that have failed. They tend to take over the role of the blue R-HAZ failed elements of figure 10*a*. For both four- and five-material simulations, the weld toe R-HAZ/CG-HAZ damage initiates and rapidly spreads from crotch to flank sections. In both simulations the presence of fusion boundary damage can be seen adjacent to the branch R-HAZ/CG-HAZ as green weld material elements.

The times to first element failure for the four material (without CG-HAZ) and the five material (with CG-HAZ) are 9111 and 8826 h, respectively, a difference of 3.1%; and the lifetimes for four- and five-material models are 18 546 and 18 242 h, a difference of 1.6%. In addition, the predicted failure modes are little different, except that the five-material model clearly shows an increase in the size of the damage zones at both weld toes, predominantly in the CG-HAZ regions, cf. figures 7*a* and 9*a* and figure 10*a*,*b*.

The effect of the inclusion of the fifth material CG-HAZ zone is to define a further zone, with different material properties, in which damage is predicted to occur; however, it does not significantly affect the time-scales or nature of the observed behaviour. This is mainly due to the fact that damage occurs early in the vessel life, but does not begin to propagate through the vessel thickness until close to structural failure.

## 7. Comparison of damage predictions with the results of micro-structural examinations of tested vessel

Detailed tested vessel examination reports have been presented by Patel (2002) and by S. M. Chilcott (2001, private communication). The first observation is that lifetime predicted by the five-material model, 18 242 h, is conservative and in good agreement with the experimental lifetime of 20 038 h, with an error of 9%.

With regard to the location of the first damage, and its development through the thickness of the vessel, these have been reported by S. M. Chilcott (2001, private communication) and Patel (2002, 2003). A summary of their findings is provided in the following paragraphs.

During the latter stages of testing, steam leakage was reported from two crotch defects located close to the weld toes in the main pipe. The defects/cracks were located in planes almost parallel to the crotch plane. Following visual inspection at the weld toe regions in the crotch locations, it was reported that the majority of the damage was in the R-HAZ on the main pipe side, with cracks running into the parent pipe material. Limited spread was observed into the weld material. Visual inspection of the inner bore crotch locations revealed cracking in the parent, R-HAZ and weld materials.

More detailed metallurgical examination confirmed the leakage took place due to the two crotch defects located close to the weld toes in the main pipe. The cracks propagated at the two locations in the crotch plane, within the CG-HAZ on the main pipe side of the weld. Significant defects were also observed in the Type IV zones on the main pipe side of the weld and also on the branch side of the weld.

With regard to the predictions, figures 7*a* and 9*a* show damage growth from the weld toe R-HAZ and CG-HAZ regions in the main pipe; however, spread of damage to the parent material is not predicted. Failure on the inner bore is predicted to be within the Type IV region, figures 7*a* and 9*a*, with no evidence of damage in the parent, R-HAZ and weld materials.

As noted in §2*c*(i), the thickness of the CG-HAZ in the main pipe was not a uniform 0.5 mm, as assumed in figure 2*b*,*c*; instead, at the crotch section it varied from 0.47 mm at the weld toe to 0.78 mm at the inner bore. However, it is known that for CDM modelling of the Type IV region (Hayhurst et al. 2004) that the thickness should be approximately 0.7 times the thickness measured from micrographs. If this also applies to the CG-HAZ, it would be expected to generate a higher local constraint with the possibility of earlier formation of damage in the CG-HAZ at the weld toe. This could well result in the predicted damage fields being closer to those observed.

In summary, the metallographic evidence provides confirmation that both the four and five materials analyses correctly predict crotch plane failure in the main pipe Type IV. Also, that the steam leakage path is correctly determined to be from the inner bore crotch plane locations, along the Type IV regions, through the R-HAZ regions close to the weld toe, to produce leakage through the CG-HAZ and Type IV regions at the weld toe.

## 8. Conclusions

Three-dimensional CDM modelling has been carried out for a medium bore branch which has been tested at a constant pressure of 4 MPa, and a uniform temperature of 590 °C.

Two analyses have been made, one without the CG-HAZ, and one with it present. Both analyses essentially predict the same behaviour in almost the same time-scales.

The analyses have been carried out with a Type IV thickness of 0.7 mm rather than the value of 1.0 mm used by Mustata

*et al*. (2003). Consequently, the correct failure mode has been predicted, in the main pipe CG-HAZ, R-HAZ and Type IV.The increased numbers of degrees of freedom, used in the analyses, relative to those used by Mustata

*et al*. (2003), did not produce significantly different times to first failure, and failure times.The predicted steam leak failure path of the Type IV, inner bore, diffusing through main pipe R-HAZ, and leaking out through CG-HAZ at the weld toe, on the crotch plane, would appear to be in accord with the spirit of the metallographic reports.

## Acknowledgments

R.J.H. acknowledges funding provided by EPSRC (RAIS) during secondment from UMIST to British Energy, Barnwood. The provision of creep data and metallographic information, together with support regarding data interpretation, by Dr D. Dean and Mr M. Spindler of British Energy, Barnwood, Gloucester, is gratefully acknowledged. Acknowledgement for permission to publish the creep test results is given to the sponsors of the ERA Technology Project-4080, from which the data originate, in particular, ALSTOM Power (UK) Ltd, British Energy plc, and PowerGen plc. The provision of EPSRC, CSAR, computer resource on the Silicon Graphics Origin 2000 (Fermat) is gratefully acknowledged. The final stages of this paper were completed whilst D.R.H. was on sabbatical leave at The Materials and Mechanical Engineering Departments, University of California at Santa Barbara, USA; he acknowledges the financial support provided, through a Global Research Award, by The Royal Academy of Engineering of the United Kingdom.

## Footnotes

↵Femgen and Femview are tradenames of Femsys, 158 Upper New Walk, Leicester LE1 7QA, UK.

- Received July 15, 2004.
- Accepted April 1, 2005.

- © 2005 The Royal Society