Approximate creep rupture lifetimes for butt welded ferritic steel pressurised pipes

Eur. J. Mech. A/Solids 19 (2000) 223–258

2000 Éditions scientifiques et médicales Elsevier SAS. All rights reservedS0997-7538(00)00160-1/FLA

Approximate creep rupture lifetimes for butt welded ferritic steel pressurised pipes

Ian J. Perrina,* , David R. Hayhursta,1, Robert A. Ainsworthb

a Department of Mechanical Engineering, UMIST, P.O. Box 88, Manchester M60 1QD, UKb British Energy, Barnett Way, Barnwood, Gloucestershire GL4 3RS, UK

(Received 15 November 1996; revised and accepted 20 October 1999)

Abstract – The creep rupture of butt welded ferritic steel pipes composed of a range of weld and heat affected zone materials has been previouslyanalysed in detail (Leckie and Hayhurst, 1994). These analyses required substantial computational resources which cannot be justified during thepreliminary phases of the design process. To reduce cost and improve speed an approximate method for the analysis of kinematically determinatestructures, known as the modal method (Leckie and Hayhurst, 1974), has been developed to compute creep rupture lifetimes. This paper reports anextension of the method for the analysis of multi-material structures, such as weldments, and its implementation as a post processor to a stationary statefinite element creep analysis. Histories of stress and damage have been determined using the modal method for weldments with a range of heat affectedzone and weld material combinations. The lifetimes determined in this way are compared with those determined using complete continuum damagemechanics analyses, and are shown to be conservative; in addition the regions of intense damage have been shown to be accurately predicted. The modalmethod is also shown to be superior to more approximate reference stress methods. 2000 Éditions scientifiques et médicales Elsevier SAS

creep rupture / welds / CDM / lifetimes

Notation

m, n,G,M,χ,φ, θ material constants in primary-secondary-tertiary creep law

ε0, ψ0, σ0, n, ν, ζ, β material constants in secondary-tertiary creep law

α multi-axial stress damage parameter

A, V area and volume, respectively

u displacement

ω damage state variable

ψ continuity state variable(= 1−ω)εij , σij creep strain and stress tensors

sij stress deviation tensor(= σij − δij σkk/3)p hydrostatic stress(= σkk/3)σI maximum principal stress

σeff effective stress(= 3sij sij /2)1/2

εeff effective strain(= 2εij εij /3)1/2

D creep energy dissipation rate(= σij εij )t time

* Present address: ALSTOM Energy Technology Centre, Cambridge Road, Whetstone, Leicester, LE8 6LH, UK1 Correspondence and reprints.

224 I.J. Perrin et al.

τ normalised time

f (t) kinematical determinacy time function

σ0 normalising stress

σy yield stress

σD representative deformation stress

σR representative rupture stress

σL limit load reference stress

κ normalised primary creep strain

2 normalised secondary creep strain

γ normalised rupture strain ratio

ξ normalised rupture time ratio

C stationary state convergence parameter

tp unwelded parent pipe lifetime

tw welded pipe lifetime

Tw normalised pipe weldment lifetime(= tw/tp)2,1 stress functions for creep deformation and damage evolution

Superscripts

b refers to a property of the base or parent material

s refers to the stationary state

· differentiation with respect to time

Subscripts

l lower bound value

m mean value

u upper bound value

1. Introduction

Within design codes, excessive creep deformation and creep rupture are avoided in the design life by theimposition of limits on elastically calculated stresses (Marriott, 1992). This use of elastically calculated stressescan, however, be overly conservative and simplified reference stress methods have been developed to providemore accurate estimates (Goodall et al., 1979). These methods use limit load analysis to provide an approximatedescription of creep behaviour in terms of overall structural response, such as creep rupture life and deflectionsunder applied loads. Where more detailed information is required, the finite element creep continuum damagemethod has been developed by a number of authors. For example, Hall and Hayhurst (1991) have used themethod not only to predict the deformation and lifetime of a butt welded ferritic steel pipe, for which thegeometry and the loading is shown infigure 1, but have also predicted the location and mode of failureas propagation of a macrocrack in the weld metal close to the fusion boundary. This is in agreement withexperimental data on a steam pipe weldment (Coleman et al., 1979, 1985).

The success of the calculations of Hall and Hayhurst (1991) was attributed to the accurate constitutivecharacterisation of the material zones and in particular to the accurate representation of the multi-axial stress

Approximate creep rupture lifetimes for butt welded ferritic steel pressurised pipes 225

Figure 1. Diagram showing the axisymmetric finite element model, the boundary conditions and the cylindrical polar co-ordinate system(z, r, θ) usedto represent the thick steam pipe weld. All dimensions are in millimetres.

damage growth criteria. It was recognised that future application of such techniques relied upon the availabilityof material data for the characterisation of each constituent material.

With this in mind, Wang and Hayhurst (1995) developed a means of transforming the normalised materialparameters commonly measured in engineering practice into the constants required in the constitutive lawsused to model the creep strain and damage evolution. These parameters allowed the weld and HAZ materialsto be characterised using the parent material as a reference. A material data sheet (Wang and Hayhurst, 1993)was also developed to convey this information in a form that could be readily comprehended by practisingengineers. Wang and Hayhurst (1994) then considered the likely variations in the properties of the weld andHAZ achievable by current welding technology and proceeded to find the effects of the parameters on weldmentlifetimes. A number of analyses carried out to deduce an optimum set of parameters for the weld and HAZ tomaximise the creep rupture lifetime. The result was that a 30% increase in lifetime was obtained over that forthe material properties used in the original weldment study by Hall and Hayhurst (1991).

The finite element continuum damage method has thus been shown to be a valuable tool for the analysisof the behaviour of pipe weldments. The results obtained agree well with test data and through the use ofnormalised material parameters Wang and Hayhurst (1995) have provided a means of obtaining the materialdata required for the constitutive equations.


The drawback with this approach is that the calculations are expensive and time consuming to perform atthe early or conceptual stage of design where there is a need for analysis methods which quickly give goodpredictions of the behaviour of structures, and allow design variants to be ranked, without committing toomuch expense.

In this paper approximate techniques, in particular a so-called modal method based on the assumptionof kinematic determinacy, will be developed that draw upon the methods of Hall and Hayhurst (1991)and Wang and Hayhurst (1994). Their work will also be used extensively to validate the approximateapproaches considered. The constitutive relations which are appropriate for use with the modal method will bedescribed, and the phases of stress redistribution and damage evolution will be embodied within the approach.Approximate methods will be presented, and evaluated, that describe the behaviour of a steam pipe. Finally, theresults will be presented of the modified modal approach that incorporates the essential features of the previouswork of Hall, Wang and Hayhurst.

2. Strategy for simplified methods

Stress redistribution that results from creep may be divided into two phases. The first phase, primary-secondary, is a result of stress redistribution as creep strains are accumulated in the presence of low damagelevels. During this phase stresses redistribute from regions of high stress to other more lowly stressed parts ofthe structure while maintaining global equilibrium. Calculations to determine the stress distribution at stationarystate, when the rate of change of stress with respect to time is zero, have been undertaken for a variety ofstructures (Penny and Marriott, 1971). In addition to this, estimates of the time for the redistribution to takeplace have been made (Calladine, 1969). All of this work was based on analytical or semi-analytical calculationsthat have now been largely superseded by the finite element method.

The second phase of stress redistribution is that due to secondary-tertiary creep. During this period the strainrates increase as a result of material softening due to microstructural change and damage. The softening of thematerial reduces its load bearing capacity and consequently stress is off-loaded onto stronger, less damaged,material. The analysis of this process is complex, generally requiring a complete finite element continuumdamage mechanics analysis. However, numerous attempts have been made to obtain approximate solutions forparticular structures where simplifying assumptions can be made (Hayhurst, 1973).

In reality some interaction may occur between the two phases. Primary-secondary stress redistribution canoccupy sufficient time for some material softening to occur; however, the degree of softening is often small.This is particularly so for plane stress structures such as thin plates containing holes (Penny and Marriott, 1971),where redistribution times are small. Because of this it is possible to separate the two phases and study themindependently. In structures that are under conditions nearer to plane strain than to plane stress, the effectivestress which controls creep deformation is often suppressed relative to the multi-axial stresses which producecreep damage—and times for redistribution are much longer; thus, interaction effects may be more significant.

Assuming that stress redistribution due to the primary-secondary phase takes place with a negligible amountof damage, then the stationary state stress distribution may be determined and the time for redistribution totake place may be found. The stationary state stress field may then be used as a starting point for a secondary-tertiary analysis. This splits the problem into the two phases outlined and—providing the interaction betweenphases is insignificant, then the calculation method should be reasonably accurate. However, the work of Halland Hayhurst (1991) shows that accuracy can only be expected if the correct mechanism-based multi-axialconstitutive equations, and the corresponding material constants, are used. Therefore, the relevant constitutiveequations are developed in the next section.


Table I. Material constants for the 0.5Cr 0.5Mo 0.25V parent material at 565◦C.

Material Stress m n G χ M θ α σ

range

Parent σ 6 σ −0.2031 4.8971 2.8531×10−14 3.0110 1.4522×10−10 0.2365 0.5955 120

σ > σ −0.2031 10.3442 1.3485×10−25 6.9613 8.8846×10−19 0.2365 0.5955 MPa

3. Constitutive equations for weldment analysis

3.1. General relations

To describe the creep behaviour of materials using continuum damage mechanics, the following equationscan be used (Hayhurst et al., 1984) for uni-axial creep strain rateε and damage rateω at the stressσ and timet

ε=G(

σ

1−ω)ntm, (1)

ω= M

1+ φσχ

(1− ω)φ tm, (2)

wherem, n, G, M, χ andφ are material constants over particular stress ranges.

Ferritic steels are known to behave differently at low stress and at high stress. This behaviour may beapproximately modelled using two sets of constants for each material with a break stress(σ ) defining thetransition point between levels. As may be explained by the work of Cocks and Ashby (1982), the break stresscorresponds to a change in the physical mechanisms of deformation and rupture. The stress levels to whichthe pipe weldments considered here are subjected, in general, lie below the break stress and are thereforeassociated with the regime of intergranular cavitation damage with a relatively low strain to failure(< 10%).Material constants for the parent material are given intable I for stress in units of MPa, strain in percent andtime in hours. The constantφ may be obtained from the values ofn andθ by θ = 1− n/(φ + 1).

For multi-axial conditions the creep strain rates are governed by an effective stress function and the damagerates by a combination of maximum principal and effective stress (Hayhurst, 1972). A parameter,α, isintroduced as a material constant to determine the bias of the multi-axial rupture behaviour between themaximum principal(σI) and effective stress(σeff). The normalised stress function for damage evolution,1,is given by

1(σij/σ0)= ασI/σ0+ (1− α)σeff/σ0, (3)

whereσ0 is introduced as a normalising stress.

3.2. Secondary-tertiary relations

The general relations described by equations (1)–(3) cover both primary and tertiary behaviour. For situationsin which the creep behaviour can be separated into the two phases described earlier, it is possible to expressthe secondary-tertiary phase of behaviour by constitutive equations of the simpler form (Leckie and Hayhurst,1974)

εij

ε0=2n

(σij

ψσ0

)∂2/∂

(σij

ψσ0

), (4)

ψ/ψ0=−1ν(σij /σ0)/ψζ , (5)


whereε0, ψ0, σ0, ν, ζ andn are material constants. The transformationψ = 1−ω is introduced for convenience,and in the same way thatω is some dimensionless measure of deterioration, thenψ can be regarded as adimensionless measure of material continuity (Rabotnov, 1967). The dimensionless function2 is convex andhomogeneous of degree one in(σij /σ0) and has the value unity whenσij is the uniaxial stressσ . The function1,which relates damage evolution to stress state, is that defined by equation (3).

Equations (4) and (5) are of similar form to those presented for general creep deformation but excludethe term describing the primary behaviour (tm). A mathematical transformation has been derived to enablethe constants of equations (4) and (5) to be determined from the material constants of the general relations(1) and (2). Details are given in Appendix A. The transformation ensures that rupture times and strains arematched and minimum creep rates are equal for the secondary-tertiary and the full constitutive equations. Thebreak stress (σ ) remains unaffected by the transformation and two sets of material constants are obtained as forthe primary-secondary-tertiary constitutive equations.

3.3. Weldment relations

Wang and Hayhurst (1995) developed a set of material parameters that allow the weld and HAZ materialsto be characterised using the base material, namely the parent pipe material. These normalised parameters aredefined as

κ =[εpri

εR

](σ1,σ2)

γ =[εR

εbR

](σ1,σ2)

γ = [εR]σ=σ[εb

R]σ=σb

2 =[εsec/εR

εbsecε

bR

](σ1,σ2)

ξ =[tR

tbR

](σ1,σ2)

ξ = [tR]σ=σ[tbR]σ=σb,

whereεpri and εsec are the primary and secondary components of creep strain, respectively,εR is the creeprupture strain, andtR is the time to rupture. The definition of these parameters is shown graphically infigure 2.The superscript ‘b’ indicates the quantities for the base or parent pipe material; while the symbol− denotesthe average values calculated over the stress range(σ1, σ2) of normalised primary creep strainκ , normalisedsecondary creep strain ratio2, rupture creep strain ratioγ , and the ratio of times to ruptureξ . The symbolˆ is used to denote values ofγ and ξ at the break stress. An additional parameterλ = σ /σ b may be usedto represent the difference between the break stress of the base and non-base materials. These normalisedmaterial parameters can be calculated from commonly measured data for weld and HAZ materials. The materialconstants for weld and HAZ materials, listed intable II, are used as the mean data of a parameter matrix (Wangand Hayhurst, 1994) which is given in Appendix B. These datasets, which form a parameter matrix, allow theeffects of the normalised material parameters to be examined in a systematic way. Individual parameters varyin value from the lower bound, through the mean, to the upper bound. The range of values for each of thenormalised material property ratios has been selected to cover most practical situations with ferritic weldments(Wang and Hayhurst, 1995). These ranges may be expressed in terms of the mean, or central, value given intable II. The ranges of the parametersγ andγ have been set equal to[12 γm,2γm] and[12 γm,2γm], respectively;

while the ranges of the parametersξ andξ have been set equal to[12 ξm,10ξm] and [12 ξm,10ξm], respectively,where the subscript ‘m’ denotes mean values. The parameterκ varies in the range[0.001,0.05] and2 isallowed to vary in the range[0.5,2.0]. A more detailed account is given by Wang and Hayhurst (1995) of theconstruction of the parameter matrix used here.

By grouping the parameters a material dataset can be formed. The parameter groupingκ2 combines thenormalised primary creep strainκ with the normalised secondary creep strain parameter2; it relates physicallyto the contribution to the creep strain which is not due to the effects of damage or material softening. The


Figure 2. Schematic representation of a uni-axial creep curve giving definitions of primary,εpri, secondary,εsec, tertiary,εter, and failure,εR, creepstrains.

Table II. Central data for the 2.25Cr 1Mo weld and HAZ materials.

Material κ 2 γ ξ γh ξh γ ξ λ

Weld 0.008304 1.1224 2.3887 0.1678 1.2051 0.1447 1.4472 0.3613 0.8333

HAZ 0.009891 1.1799 0.5642 1.2243 0.07144 8.7891 0.6268 0.5018 1.5000

parameter groupingγ ξ combines the normalised rupture ductilityγ with the normalised rupture lifetimeξ ;it relates physically to the rate of damage accumulation and to the extent to which damage enhances strainaccumulation. For the weld material, dataset 1 is comprised of the central valuesκm2m (1) and γmξm (1);and for the HAZ material, dataset 18 is comprised of the central valuesκm2m (18) andγmξm (18). For eachmaterial,κ2 forms one matrix, datasets 1 to 9 for the weld metal; and a second matrix, datasets 18 to 26 for theHAZ material. The combinationγ ξ forms another matrix, datasets 10 to 17 plus dataset 1 for the weld materialand datasets 27 to 34 plus set 18 for the HAZ material. When forming a matrix, the parameters which are notcontained within it remain equal to their mean values.

Wang and Hayhurst (1994) presented a mapping technique to obtain the material constants for the fullcontinuum damage constitutive equations (1) and (2) from the normalised material parameters. The equations todetermine the constants for the secondary-tertiary constitutive equations form an additional mapping as shownin figure 3. A direct mapping can be formed between the material parameters expressed in terms of the basematerial properties and the secondary-tertiary material constants. This mapping does not require the parameterκ , given in the first group of parameters infigure 3, which is often the most difficult parameter to specify. Theother material parameters(2, γ , ξ )may be found from the minimum creep rate, rupture strain and rupture timeas discussed by Wang and Hayhurst (1995).

The remaining sections of the paper are concerned with the use of methods for creep rupture lifetimeestimation, particularly for weldments. Approximate methods of lifetime prediction are first considered.


Figure 3. Schematic representation of non-linear mapping from material property ratios expressed in terms of the base material properties, to thematerial constants used in the full constitutive equations, and to the material constants used in the secondary-tertiary constitutive equations.

4. Approximate techniques for creep rupture analysis

The most common methods for analysing the secondary-tertiary behaviour of engineering components andstructures, and more particularly for obtaining rupture lifetime estimates, are the reference stress and the modalanalysis methods. These are now reviewed in the context of weldment analysis.

4.1. Representative deformation and rupture stresses

A representative deformation stress may be defined as that uni-axial stress,σD, which causes a creep energydissipation rate equal to the average dissipation rate in the structure. Hence

σD/σ0={

1

V

∫v

2n+1(σ sij

σ0

)dV}1/(n+1)

(6)

whereV is the volume of deforming material andσ sij refers to the stationary state stress distribution. The

stressσD is based on the function2(σij/σ0) = σeff/σ0. In practice the value ofσD is almost independent ofthe stress index,n, particularly whenn is high. This representative deformation stress has been widely usedto estimate structural behaviour, in conjunction with uni-axial creep strain data (Leckie et al., 1976). Strictly,this stress should not be used to estimate creep rupture lifetimes but it is used here for comparative purposesbecause of its synergy with the limit load reference stress discussed below.

Ponter and Leckie (1970) have shown that an upper bound onσD is given by a limit load reference stress,σL,determined from the ratio of the applied load,P , to the limit load,PL, giving

σL/σ0= (P/PL)σy/σ0 (7)

whereσy is the yield stress used to definePL, and the limit load is based on a von-Mises yield surface describedby σeff. This method of determining a reference stress is favoured in practice because limit load solutions areavailable for many structures and the requirement for material properties is reduced from the definition inequation (6).


The determination of a representative rupture stress,σR, has been considered by Leckie and Wojewodzki(1975) who proposed the expression

σR/σ0={

1

V

∫V

1ν

(σ sij

σ0

)dV}1/ν

(8)

It may be noted that if functions1 and2 are of the same form and ifν = n+ 1, then equations (6) and (8) arethe same. Somewhat earlier, Goodall and Cockroft (1973) had proposed that the effects of the rupture surfaceembodied in equation (8) could be incorporated in the limit load reference stress of equation (7) through theuse of a yield function that takes the form of the isochronous rupture surface.

The definitions of reference stress presented here depend on whether rupture and deformation are governedby the same multi-axial stress function. For materials whose deformation and rupture surfaces are of similarform the methods are likely to give similar results. Their application to weldment lifetime estimation will nowbe considered.

4.2. The use of reference stress methods in weldment analysis

The evaluation of a reference stress for a weldment is complicated by the geometry and the multi-materialconstruction. Although analytical solutions for limit loads are possible for some simple geometries (Joch etal., 1993) more generally numerical techniques, such as the finite element method, have to be used eitherto compute the limit load, or the stationary state stress distribution. In either case a reference stress may bedetermined. Because of the associated complexity simpler approaches have been developed that attempt tofind the weldment lifetime from a knowledge of the reference stress for a plain, unwelded, pipe for whichanalytical solutions exist. The first such method considered here, designated method 1, involves the calculationof reference stresses for a plain pipe made entirely of the parent, the weld and the HAZ materials in turn. Thesereference stresses are each compared with the appropriate stress rupture data for the corresponding material andthe smallest resulting lifetime is used to define the overall weldment lifetime. The shortcoming of this methodis that it does not account for stress redistribution within the weldment due to the different creep properties ofthe constituent materials. Coleman et al. (1985) have shown that, compared to the parent material, the weldand the HAZ materials sustain lower and higher proportions of the stress, respectively, owing to the complexstresses generated therein. To consider this effect another approach may be adopted which involves using theparent pipe reference stress and factoring it to give reference stresses for the weld and HAZ (Ainsworth andBudden, 1994). The weldment lifetime is given by the smallest lifetime obtained when the reference stressesare compared with stress rupture data. The factors used depend on the material properties, and on the geometryof the weldment, and so their specification can be difficult.

Another approach, referred to as method 2, is to evaluate a reference stress for each material zone of theweldment using equation (6) or (8), whereV is taken as the volume of the material zone for which the referencestress is being evaluated. The weldment lifetime is then given by the smallest lifetime determined from the threereference stresses. This approach allows for stress redistribution but the use of a local volume in equations (6)and (8) means that the bounding properties of these expressions no longer hold.

The methods 1 and 2 have been used to determine weldment lifetimes for the geometry offigure 1underinternal pressure for the 34 datasets of Wang and Hayhurst (1994) described in Section 3.3. The results of thetwo methods for determining reference stress, and corresponding lifetime estimates, are discussed in the nextsection.


4.3. Prediction of weldment lifetimes by reference stress methods

The methods 1 and 2 are assessed here by comparison of their respective lifetime estimates with those givenby Wang and Hayhurst (1994) using full continuum damage mechanics analysis. The reference stresses wereevaluated using equations (6) and (7) for method 1 and equations (6) and (8) for method 2. The normalisingstress was set equal to the applied internal pressure (σ0= 45.5 MPa).

4.3.1. Method 1: behaviour independent of weldment materials

Reference stresses have been evaluated for the plain pipe assuming that it was made from each of the threeconstituent materials of the weldment. The energy dissipation reference stress, equation (6), gives a valueof σD/σ0 = 2.052 for the parent material, between 2.034 and 2.057 for the weld metal and between 2.033and 2.055 for the HAZ material. These changes inσD/σ0 reflect the small effect of the stress index,n, whichvaried from 2.08 to 9.88 for the weld metal, and from 2.04 to 7.91 for the HAZ material. The limit loadreference stress determined using equation (7) yielded a value ofσL/σ0 = 2.063; which was independent ofmaterial properties. The value ofσL/σ0 corresponds to a very largen and reflects that the limit load providesan upper bound on the energy dissipation reference stress.

The lifetimes determined from the limit load reference stress, equation (7), were between 50 and 80%conservative when compared to the results of Wang and Hayhurst (1994). The smaller magnitudes of thereference stresses determined from the energy dissipation calculation, equation (6), gave lifetimes that wereapproximately 1.5% longer than those obtained from the limit load reference stress. The lifetimes were onlyinfluenced by the normalised rupture strain ratio (γ ), which caused the changes in stress index ‘n’, and thenormalised rupture time ratio (ξ ), which caused the changes in the material constantsψ0, ζ andν in the lifetimedetermination equation (A4) given in Appendix A.

The lifetimes are extremely conservative and insensitive to the material properties. The weld material, whichhas the lowest creep strength of the three material zones, limited the overall weldment lifetime for all thedatasets specified in Appendix B. This shows that the stress redistribution which takes place between theweldment materials is significant and must be included if a realistic lifetime prediction is to be obtained.

4.3.2. Method 2: effects of creep behaviour of weldment materials

Stationary state stress distributions were determined by performing finite element creep calculations usingthe constitutive equation (4) with the appropriate stress index,n, and normalising creep rate,ε0, for eachmaterial. The continuity,ψ , was set equal to unity for the calculations. The results, using reference stressequations (6) and (8), are now described.

4.3.2.1. Lifetimes determined from (σD) using equation (6). All of the 38 datasets given in Appendix Bwere considered. The reference stress determined for the parent material wasσD/σ0 = 2.041 to within 2%.This value provides a convenient reference to assess the significance of the stress redistribution. For datasets 1to 17—which vary the weld material properties while the HAZ material retains mean properties—the weldmaterial reference stress varied between 0.837 and 1.650; and the HAZ material reference stress variedbetween 2.587 and 2.891, despite retaining mean properties. For datasets 18 to 34—which vary the HAZmaterial properties while the weld material retains mean properties—the HAZ material reference stress variedbetween 2.024 and 3.285; and the weld material reference stress remained close to 1.33. This shows thesignificance of the stress redistribution that takes place in the primary-secondary creep phase with the weldmaterial taking a lower stress than the parent material and the HAZ material a higher stress.


The lifetimes determined were, on average, conservative by 20% when compared to the results of Wang andHayhurst (1994). Unconservative lifetimes were predicted for five of the datasets, with a maximum lifetimeoverestimate of 11%. The most conservative lifetime prediction was 50% shorter than the lifetime reportedby Wang and Hayhurst (1994). The lifetimes were unaffected by the normalised primary creep strain ratio (κ)because the primary creep characteristics of the materials have not been included in the constitutive equation (4)as explained in section 3.2. Conservative lifetimes are determined when the weld metal rupture strength (ξ ) isat least 0.15 times that of the parent material and, for the HAZ material, when the normalised secondary creepstrain parameter (2) is below 1.6. If the rupture ductility (γ ) of the HAZ material is greater than 0.75 of that ofthe parent material then the rupture strength (ξ ) should be less than 5 times that of the parent material to obtaina conservative lifetime.

4.3.2.2. Lifetimes determined from (σR) using equation (8). The parent material reference stress remainedclose to 1.982 for all datasets. For datasets 1 to 17—which vary the weld material properties while theHAZ material retains mean properties—the weld material reference stress varied between 1.047 and 1.658;and the HAZ material reference stress varied from 2.434 to 2.643. This again shows that the HAZ materialreference stress varies despite the material properties remaining constant. For datasets 18 to 34—which varythe HAZ material properties while the weld material retains mean properties—the HAZ reference stress variedbetween 1.902 and 3.015; the weld material reference stress remained at about 1.33. This again shows thesignificance of stress redistribution with the weld and the HAZ materials having different reference stresses tothe parent material. The rupture reference stress, equation (8), is also seen to lead to a slightly lower referencestress than the energy dissipation reference stress, equation (6), for the same set of material properties.

On average the lifetimes were conservative by 7% when compared to the results of Wang and Hayhurst(1994); unconservative lifetimes were obtained for twelve of the 38 datasets. The maximum conservative andunconservative lifetime estimates were 36 and 18%, respectively. The rupture reference stress determines aconservative lifetime when the weld metal rupture ductility (γ ) is at least twice that of the parent material andthe rupture strength (ξ ) is at least a quarter of that of the parent material. For the HAZ material the normalisedsecondary creep strain parameter (2) should be less than 1.2, and if the rupture strength (ξ ) of the HAZmaterial is greater than that of the parent material then the rupture ductility (γ ) should be less than 0.7 of theparent material value to predict conservative lifetimes. In addition, the method is unconservative when the HAZmaterial rupture strength (ξ ) is greater than five times that of the parent material.

4.4. Assessment of results and recommendations

Method 1 may be used to obtain a reliably conservative estimate of the weldment lifetime. It is, however,extremely conservative and does not take advantage of the potential life of the weldment. The method appliesessentially the same value of reference stress to each of the constituent materials, regardless of their relativecreep ductilities and rupture strengths, which means that the weld metal limits the lifetime for every dataset.The lifetime estimates from method 2, which allow for the effects of the stress redistribution that takes placein reaching a stationary state, are more realistic although numerous constraints must be placed on the range ofmaterial properties to ensure the lifetime estimates are conservative, and, more particularly, the method is notable to systematically distinguish between material properties that are beneficial in terms of lifetime and thosethat are not. Indeed, for homogeneous structures the method is known to provide an overestimate of rupture life(Leckie and Wojewodzki, 1975). This means that the method is of limited value during the embodiment phaseof the design process when a variety of material combinations are to be considered in an effort to optimise theweldment lifetime and obtain a design for more detailed evaluation.


The variations in the reference stress, given in Sections 4.3.2.1 and 2, show the significance of the stressredistribution required to achieve a stationary state. Further to this, the inability of either the deformation or therupture reference stresses to predict the lifetimes accurately suggests that the stress redistribution that occurs inthe secondary-tertiary phase of creep is also significant.

The reference stress methods have been shown to be of limited value in multi-material creep rupture problemswhere the material properties and all stages of stress redistribution are important. This suggests that a morephysically based analysis method is required. One such method which is investigated here is the modal method(Martin and Leckie, 1972; Leckie and Hayhurst, 1974), which overcomes the above difficulties at the expense ofsome simplicity. Here, the term modal refers to the mode of deformation of the structure, i.e. it is assumed to bekinematically determinate, and should not be confused with modal methods for impulsively loaded structures.The method takes the stationary state creep solution as a starting point, assumes kinematical determinacy, andallows damage evolution. It therefore takes account of all the stress redistribution and multi-axial stress effects.It has been successfully applied to the analysis of the secondary-tertiary behaviour of numerous structures,with notable examples being: torsion bars and thick cylinders (Leckie and Hayhurst, 1974) and notched bars(Hayhurst et al., 1978). The method will now be described in more detail.

5. The modal analysis method

Leckie and Hayhurst (Leckie and Hayhurst, 1974) presented the modal method for single material structuresand showed that a lower bound estimate of the rupture lifetime could be obtained. This formulation will beextended to describe the behaviour of multi-material structures, such as weldments, and a method of solutionwill be proposed.

5.1. Formulation

5.1.1. Kinematical determinacy and stress deviators

Consider the secondary-tertiary behaviour of the structure. If it is assumed that at stationary state the structureis undamaged and the distributions of stress and strain rate areσ s

ij and εsij , respectively, then as material

deterioration takes place the stress distribution changes but if the structure is kinematically determinate thenthe displacement and strain rates must have the form

u= usf (t), εij = εsij f (t), (9)

wheref (t) is a function of time only and is unity when time is zero. If the structure is composed of more thanone material then these expressions (9) must hold for each of the constituent materials and all of the materialsmust operate on the same time scale,f (t). So for a two material structure

u1= us1f (t), u2= us

2f (t) (10a,b)

and

εij1 = εsij1f (t), εij2 = εs

ij2f (t), (11a,b)

where the subscripts 1 and 2 denote two different materials.


The constitutive equations for the secondary-tertiary creep phase are given by the following equations forthe two materials

εij1

ε01

=2n11

(σij

ψσ0

)∂21/∂

(σij

ψσ0

),

εij2

ε02

=2n22

(σij

ψσ0

)∂22/∂

(σij

ψσ0

). (12a,b)

These equations may be rewritten to express the stresses in terms of the strain rates. Or more particularly, formaterials which deform at constant volume only the stress deviator,sij , can be expressed in terms of the strainrates; and, for the first material, equation (12a) becomes

sij

ψσ0=�1/n1

1

(εij

ε01

)∂�1/∂

(εij

ε01

). (13)

At stationary statesij = ssij andψ = 1 and, following Leckie and Hayhurst (1974), the following relationship

may be obtained

sij /σ0=ψ(ssij /σ0

)f 1/n1(t) (14a)

and by analogy, for material 2

sij /σ0=ψ(ssij /σ0

)f 1/n2(t). (14b)

This allows the deviatoric stress field at any time during the secondary-tertiary phase to be found from thestationary state deviatoric stress field, and the continuity at the given instant andf (t). The determinationof f (t) will now be presented.

5.1.2. Internal and external energy dissipation rate balance and the determination of the kinematicaldeterminacy time function,f (t)

An expression forf (t) may be found by equating internal and external work rates for the structure. Theexternal work rate due to the applied loading is given by

Wext=∫A

Piui dA. (15)

Using equation (10) and noting that at stationary state the external work must be balanced by the dissipation ofviscous energy within the structure then

Wext= f (t)∫V

DsdV, (16)

whereDs is the stationary state rate of dissipation of viscous energy per unit volume. In the multi-material casethe integral will be formed for each material zone. This external work rate is balanced by the general internalwork rate

Wint =∫V

D dV. (17)

The dissipation rate per unit volumeD = σ ε, can be found from the constitutive relations (12a, b)

D1

σ0ε01

=ψ2(n1+1)1

(σij

ψσ0

),

D2

σ0ε02

=ψ2(n2+1)2

(σij

ψσ0

). (18a,b)


At stationary state,ψ = 1, σij = σ sij , hence

Ds1

σ0ε01

=2(n1+1)1

(σ sij

σ0

),

Ds2

σ0ε02

=2(n2+1)2

(σ sij

σ0

). (19a,b)

Equations (19a, b) can be used to evaluateWext (equation (16)).

To obtain an expression forWint in terms ofDs andf (t) the following approach may be adopted. Consideringmaterial 1 the stress deviator at stationary state is given by

ssij

σ0=�1/n1

1

(εsij

ε01

)∂�1/∂

(εsij

ε0

). (20)

Using equation (20) in conjunction with equation (14a) yields

D1

σ0ε01

=ψ�(n1+1)/n11

(εsij

ε01

)f (n1+1)/n1(t) (21)

but,

Ds1

σ0ε01

=�(n1+1)/n11

(εsij

ε01

)(22)

and, hence combining equations (21) and (22) gives, for material 1

D1

σ0ε01

=ψ Ds1

σ0ε01

f (n1+1)/n1(t) (23a)

and by analogy, for material 2

D2

σ0ε02

=ψ Ds2

σ0ε02

f (n2+1)/n2(t). (23b)

Recalling thatDs is given by equations (19a, b) for materials 1 and 2, respectively, and using the work ratebalance between equations (16) and (17) and noting that the dissipation rate of the second material may benormalised with respect to the first in the following way

Ds2

σ0ε01

= Ds2

σ0ε02

ε02

ε01

then

f 1/n1(t)

∫V1

ψDs

1

σ0ε01

dV1+ f 1/n2(t)

∫V2

ψDs

2

σ0ε01

dV2−[∫

V1

Ds1

σ0ε01

dV1+∫V2

Ds2

σ0ε01

dV2

]= 0 (24a)

or more generally for a multi-material structure

∑i

[f 1/ni (t)

∫Vi

ψDsi

σ0ε01

dVi

]−∑

i

∫Vi

Dsi

σ0ε01

dVi = 0. (24b)

Equation (24) may be solved to findf (t).


5.1.3. Time scales for damage/continuity evolution in several materials

The evolution of the continuity,ψ , within the structure is also required. Returning to the secondary-tertiaryrelationships presented earlier the rate of change of continuity is given by equation (5) which is

dψ/dt =−ψ01ν(σij /σ0

)/ψζ . (25)

By defining a normalising time,τ , so that

dτ = ψ0 dt (26)

then equation (25) may be rewritten for material 1 as

dψ1/dτ1=−1ν11

(σij /σ0

)/ψ

ζ11 (27a)

and for material 2 as,

dψ2/dτ2=−1ν22

(σij /σ0

)/ψ

ζ22 . (27b)

The time scales may be normalised as follows so that the second material is on the same time scale as the first(Hall and Hayhurst, 1991)

dψ2

dτ1= dψ2

dτ2

dτ2

dτ1(28)

and by using equation (26) this becomes

dτ2/dτ1= ψ02/ψ01. (29)

The coupling with real time is achieved through equation (26) by

t = τ1/ψ01. (30)

In the continuity evolution equations (27a, b) it is necessary to evaluate the stress function1 given by equation(3). Since the modal method allows the deviatoric component of stress to be found at any time, it is convenientto express equation (3) in terms of deviatoric and hydrostatic components of stress giving

1(σij /σ0

)= α(sI/σ0)+ α(p/σ0

)+ (1− α)(3sij sij /2)1/2. (31)

In general the determination of the hydrostatic stress component is straightforward for plane stress situations,but for plane strain and axi-symmetric situations the solution to the problem requires special treatment whichwill be considered in a later section. However, if the function,1, which describes the isochronous rupturesurface is described by effective stress alone then no such problem arises and the solution can be obtained interms of the stress deviators alone.

5.2. Method of solution

The basis of the method used is first to determine the stationary state stress distribution using the finiteelement method, and second to use the modal method as a post processor to this solution. This approachhas been adopted since finite element solvers are readily accessible, and form the major component of thesoftware required. The finite element mesh provides a convenient definition of the volumes of material at which


continuity and stress are to be evaluated; volume integrals may be determined for each element, and summed asrequired for each material zone. Having evaluated energy dissipation rates from the stresses the rates of changein continuity are calculated using equation (27). Values of continuity after an interval of time,1τ , may bedetermined from

ψ(τ+1τ) =ψ(τ) + (dψ/dτ)(τ)1τ. (32)

The time increment1τ can be evaluated from

1τ =−ψi/Q(dψ/dτ)i, (33)

wherei refers to the position where dψ/dτ is a maximum andQ is a constant chosen to determine valuesof 1τ sufficiently small to validate the approximations inherent in equation (32).

Equation (24) may be solved, by the Newton–Raphson method, to givef (t) and the stress deviatorsrecalculated from equation (14). The time integration procedure is repeated until the continuity,ψ , fallsbelow 0.01 on an element. The weldment lifetimetw is then determined from the normalised timeτ . Themodal calculation may, however, be continued until the continuity of a number of elements has fallen belowthe criteria for failure (ψ 6 0.01) such that the equilibrium of the structure is lost.

In the presentation of the modal method the effects of a break stress (Section 3.1) have not been explicitlyincluded. However, such effects may be incorporated quite simply by defining the volume integrals ofequation (24b) for both different material regions and for the parts of those regions which are above, or below,the break stress. The equation for the evolution of continuity (25) is used with the material constants appropriateto the regions of high stress or low stress and the solution progresses in the manner described above. For brevity,the details of this have not been presented, and in any case for the pipe weldments considered here the stressesonly exceed the break stress on initial loading; by the time the stationary state has been reached the stresses arebelow the break stress and they remain so until the development of a distinct macrocrack.

In the following section the application of the modal method to the analysis of the weldment will be justifiedby assessment of the assumptions that are implicit within the method.

6. Justification of proposed strategy

The modal method is based on several approximations: kinematic determinacy; no damage accumulated inreaching a stationary state; and the time to achieve a stationary state is not a significant fraction of componentlife. These approximations will now be assessed in the context of the pipe weldment and in addition to this, theproblem of determining the hydrostatic stress for use in the damage or continuity evolution equations will beexamined.

In justifying these approximations the results of Hall and Hayhurst (1991) and Wang and Hayhurst (1994)will be used extensively. For simplicity the material dataset numbering system introduced by Wang andHayhurst will be used to refer to the material combination of a particular weldment. The system is explainedin Appendix B, where the material combination of each dataset is given. Primarily dataset 1, correspondingto the properties used by Hall and Hayhurst (1991), and dataset 37, corresponding to the optimal propertiesdetermined by Wang and Hayhurst (1994), will be used.


6.1. Assessment of kinematic determinacy

The kinematic determinacy of the weldment was assessed by studying the time variation of the spatialdistributions of the total effective strain,εeff, where the bar denotes total strain. The total effective strain wasexamined at three radial positions within each of the three material zones. The total effective strain at anyposition(r, z) can be expressed in terms of the initial elastic value,εe

eff, at that position by the relation

εeff(r, z)=Kεeeff(r, z), (34)

whereK is a constant which is a function of time alone. If the weld response is truly kinematically determinate,thenK is also independent of position. Values ofK were obtained, for a particular time, by averaging valuesdetermined at each of the nine points throughout the weldment. The use of equation (34) with an averagevalue ofK at the life fraction of 72% resulted in a maximum error of 18.4% for dataset 1 and 11.4% fordataset 37. These errors occurred in the weld metal, for the point with the greatest normalised radial position((r − a)/(b − a) = 0.8) located centrally between the HAZ and the weld metal centre line. They may beattributed to the differences in the ductilities of the parent and the weld metals, and to the geometry of theweldment, which result in the deformation of the weld metal being less constrained in this region. The errorsinK at the other positions considered were less than 10%. At life fractions lower than 72% similar trends wereobserved, but with lower errors. For life fractions in excess of 72% the kinematic determinacy breaks down asa distinct macro-crack forms.

Equation (34) with a value ofK independent of position is accurate to within 20% for 75% of the weldmentlife for the two datasets considered. This corresponds to errors of approximately 4% in effective stress.Therefore, the assumption of kinematical determinacy would appear reasonable for the butt welded pipe.Hayhurst et al. (1978) performed a similar evaluation in their study of notched bar behaviour. Errors of up to45% for effective strain were obtained but it was noted that this corresponded to an error in stress of about 8%,which, for the purposes of preliminary design, is acceptable. In addition Leckie and Hayhurst (1974) suggestedthat the modal method could be applied to kinematically indeterminate structures. They compared predictionswith the results of experiments on plane stress plates with holes and slots and found that theoretical predictionswere always conservative. The differences, interpreted by comparing reference stresses, were never greaterthan 10%.

6.2. Stress redistribution in the primary-secondary phase

Stationary state occurs when the rate of change of stress with respect to time is zero. To obtain a stationarystate stress distribution for use with the modal method a finite element creep calculation is required because ofthe geometric and multi-material complexities of the weldment. Hayhurst and Brown (1984) used the followingparameter ‘C ’ to determine the achievement of stationary state in their finite element calculations

C = maxi{||DBi u−Dv||∞}maxi{||DBiu||∞, ||Dv||∞} , (35)

where||a||∞ =maxj {|aj |} is the maximum norm of any vectora. The matrix,D, relates stresses to strains; thematrix, Bi , relates displacement rates,u, and strain rates; and,v, is the vector of creep strain rates;i refers tothe ith finite element. The stationary state convergence parameterC varies between 2 and 0: the closer it is tozero the nearer the solution is to the stationary state. Practically it is found thatC does not fall to zero, becauseof approximations associated with the finite element mesh, but converges to a near constant value of about 0.05for the weldments studied here.


Figure 4. Variation ofC parameter with normalised time for material dataset 1.

The use of theC parameter to monitor convergence to a stationary state is illustrated infigure 4where thevariation ofC with normalised time is presented for dataset 1. The curve shows that initiallyC falls rapidlyand then gradually reaches a lower plateau. The stationary state was taken at the start of this plateau which,for the majority of datasets, occurred whenC fell below 0.05. The exception was dataset 15 where the plateaustarted at aC value of 0.075. It may be seen fromfigure 4thatC continues to reduce, but at a much decreasedrate, on the lower plateau. If stationary states are taken at later times, corresponding to smaller values ofC,then the stress fields were found to be locally perturbed owing to the high numbers of iterations and associatedcumulative error propagation.

In the initial finite element calculations a primary-secondary constitutive equation(ε = Gσntm) was used.However, it was found that because of the large mismatch in primary creep strains between adjacent materialswithin particular datasets, the resulting stress fields were unrealistic. The stresses observed in the HAZ werehigher than those observed at any time in the full continuum damage solution. This is because the primarymismatch causes particularly long redistribution times, in excess of 16600 h for dataset 37, and at these timesdamage would, in reality, be sufficient to cause some redistribution of stress in regions such as the HAZ. Usingsuch stress fields in the modal calculation would result in erroneous damage evolution and lifetime estimates.

To overcome this problem the primary behaviour of each material was neglected since Wang and Hayhurst(1994) had shown that primary strain had a small overall effect on weldment lifetime. Stationary state stressdistributions were found with constitutive relations of the form of equation (4), withψ = 1. Times to achievea stationary state are about 5000 h, which, for most of the datasets considered, represents a small fraction ofweldment lifetime. Thus, the calculations suggest that the approximation of damage accumulated in reaching astationary state being small is reasonable and that the stationary state distribution is, therefore, suitable for usein the modal approach.

6.3. Determination of hydrostatic stress

The modal method only allows the stress deviators to be determined, and when the function1 in equation (3)contains onlyσI (α = 1) then the hydrostatic stress must be evaluated separately. For the study of the creeprupture of notched bars, Hayhurst et al. (1978) overcame the problem by solving the equilibrium equations forthe region of interest at a given instant in time. To perform similar calculations in the case of the pipe weldment


Figure 5. Variation of normalised hydrostatic stress for different times along a line through the centre of the HAZ for material dataset 1.

for highly refined finite element meshes would detract from the essence of the current work, which is to providesolutions with the minimum of computational effort. As a result simplifications were sought.

A full finite element continuum damage mechanics analysis was used to determine the variation of thehydrostatic stress spatially and with time. The results of this for a line through the centre of the HAZ materialfor dataset 1 are shown infigure 5. This is the location where the largest variations of the hydrostatic stressfrom its steady state value are observed. It is evident from the plots that the hydrostatic stress in the majorityof the pipe does not change significantly from the stationary state value over the life of the weldment up to thetime when first failure occurs. This suggests that the stationary state distribution of hydrostatic stress could beused in the modal computations hence avoiding the extra effort in calculating it. The errors involved in makingthis assumption will now be assessed.

The largest variation in hydrostatic stress is observed on the internal boundary(r = a). However, in thisregion the stationary state value of hydrostatic stress is an over-estimate of the actual value; the effect being toaccumulate more damage than would otherwise be the case. This is not a major drawback because failure rarelyoccurs on the internal boundary and since damage evolution is over-estimated, the result would be conservative.Quite large variations were also observed on the outer boundary. These were mainly due to softening effects inthe material that cause the stress to be redistributed. This again represents a conservative assumption becausethe use of a steady state field overestimates the hydrostatic stress and consequently damage evolution.

Ignoring variations of hydrostatic stress at the inner and outer boundaries the most significant deviationsfrom stationary state are found a small distance in from the outer boundary (5–10 mm). These variations aredue to the material being slightly less damaged than that on the outer boundary and thus being able to sustainmore load. Consideration of the variations of hydrostatic stress at this point enables an estimate to be madeof the likely error in the calculation of the stress function,1. The difference in the actual hydrostatic stressand the value at steady state was found for datasets 1 and 37 at a time near to first failure, this was of theorder of 10%. Typical values of the principal stress deviator and effective stress were used in conjunction withhydrostatic stress values to estimate the error in1, which was underestimated by 3.5 and 1.7% for datasets 1and 37, respectively. These small errors justify the approach of assuming the hydrostatic stress to remain at itsstationary state value up to the point of first failure.


7. Results and discussion

The stationary state stress field was obtained from finite element computations with the solver Damage XXwith zero damage growth. The finite element mesh was identical to that used by Wang and Hayhurst (1994)consisting of linear displacement triangular elements. The constitutive equation was of secondary creep form,equation (4) withψ = 1. Convergence to a stationary state, monitored using the ‘C ’ parameter, was deemed tohave occurred whenC 6 0.05; the stress field was then used as the starting point for the modal method. Themodal computation was performed until the continuity on an element was less than 0.01, at which time failureof the element was said to have occurred and the assumptions of the method were deemed to have been violated.This time was used to define the modal method weldment lifetime. The computation of a stationary state stressfield and that of the lifetime from the modal method was performed in under 5 min using a workstation, whereasa full finite element continuum damage computation carried out on a Cray XMP/416 with Damage XX can takein excess of 30 min (Wang and Hayhurst, 1994). The results of the computations are presented and discussedin this section.

To assess the lifetime predictions of the modal method each of the 38 datasets of Wang and Hayhurst (1994)were analysed. The material combination of each dataset is given in Appendix B. The modal method lifetimeshave been compared with the times to first and final failure (rupture) determined by the creep continuumdamage program Damage XX. The modal method lifetimes are conservative when compared to the DamageXX rupture lifetime, on average by 14%. The most conservative estimate, by 42%, was obtained for dataset15. This dataset corresponds to a weld material with an upper bound rupture strain ratio and a lower boundrupture time ratio (γuξl ) whilst all other properties remain at their mean values. The large error for this datasetcan be attributed to the assumption of kinematic determinacy not being satisfied. Evaluation of this using themethod outlined in Section 6.1 showed that after 30% of weldment lifetime the errors in predicted kinematicallydeterminate strains were greater than 50%. Dataset 32 gave the least conservative result, with an error of 0.6%.This dataset corresponds to a HAZ material with an upper bound rupture strain ratio and a lower bound rupturetime ratio (γuξl ) whilst all other properties retain their mean values.

Graphical comparison of lifetimes predicted by the modal and Damage XX techniques are made infigures 6–13, for datasets 1–34. The figures show the effects of the normalised material properties (Wangand Hayhurst, 1994) upon the normalised weldment lifetime. The normalised weldment lifetime was definedasTw = tw/tp, wheretw is the weldment lifetime, as computed either by Damage XX or by the modal method,and tp is the rupture time of the pipe assuming that it is made entirely of parent material, as computed byDamage XX(tp = 69853 h, as quoted by (Wang and Hayhurst, 1994). In the following sections the effectsof the normalised material parameters on the predicted lifetimes are discussed, as are the stress and damagedistributions from the modal method.

7.1.1. Effect of the variation of the weld material properties

The conservative nature of the predictions made by the modal method are shown infigures 6–13wherethe lines indicating lifetimes from the modal calculation, labelled with a subscript M, are always below thecorresponding lines from the Damage XX computation, labelled with a subscript D.Figures 6and 7 showthe effects on lifetime of the normalised primary and secondary creep strain ratios (κ2) for the weld metal.The independence of the modal method results on primary creep strain ratio (κ) is shown infigure 6, wherelines AM, BM and CM are co-incident, and also infigure 7 where the lines for the modal technique areinvariant to the primary strain ratio.Figure 6 shows that the modal method becomes more conservative asthe secondary creep strain ratio (2) is increased. With the possible exception of the line AD, figure 7shows thatthe Damage XX lifetimes are relatively insensitive to the primary creep strain ratio, justifying the neglection


Figure 6.Variation of normalised weldment lifetime with average normalised secondary strain ratio,2, for different values of normalised primary strainratio, κ, for the weld material. Mean parameter combinationsκm2m andγmξm are assumed for the HAZ material.

of this parameter.Figure 7 also shows that the modal method predicts shorter lifetimes for the upper boundsecondary creep strain ratio (2= 1.75 given by CM) than for the mean secondary creep strain ratio (2= 1.122given by BM), whereas Damage XX predicts similar lifetimes for the respective secondary creep strain ratiosdenoted by lines BD and CD. The variation of weldment lifetime with the weld metal rupture time ratio (ξ ) andthe average rupture strain ratio (γ ) is described well by the modal method infigures 8and9 where the modalmethod predictions of lifetime accurately follow those computed by Damage XX. Both figures show that, ingeneral, the lifetimes predicted by the modal method become more conservative asγ andξ are increased.

7.1.2. Effect of the variation of the HAZ material properties

The changes of lifetime with the HAZ material properties are shown infigures 10–13. The independence ofthe modal method on the primary creep strain ratio (κ) is shown infigure 10where lines AM, BM and CM are co-incident. This figure also shows the modal method predicts increasing lifetime with increasing secondary creepstrain ratio (2), whereas Damage XX predicts that the weldment lifetimes reduce when the secondary creepstrain ratio exceeds its mean value.Figure 11again illustrates the independence of lifetimes from the modalmethod on the primary creep strain ratio (κ) and also that the Damage XX lifetimes are relatively insensitive tothe primary creep strain ratio. When the average rupture time ratio (ξ ) and the average rupture strain ratio (γ )for the HAZ are varied,figures 12and 13, it may be seen that the agreement between the lifetimes fromDamage XX and the modal method is not as good as when the same material parameters are varied for the


Figure 7. Variation of normalised weldment lifetime with average normalised primary strain ratio,κ , for different values of normalised secondary strainratio,2, for the weld material. Mean parameter combinationsκm2m andγmξm are assumed for the HAZ material.

weld (figures 8and9). The modal method therefore can be seen to give more conservative lifetimes as theHAZ material average rupture time ratio (ξ ) is increased (figure 12). This figure also shows that when the HAZaverage rupture time ratio is at its lower bound (ξ = 0.500) and the average rupture strain ratio is at is upperbound (γ = 1.0, lines CD and CM) the least conservative result is obtained. This can also be seen onfigure 13where lines AD and AM converge.

7.1.3. Extreme material property combinations

In their studies Wang and Hayhurst (1994) showed the additive effect of material properties. They combineddatasets that give long lifetimes to further improve the weldment lifetime, hence the existence of datasets 35and 37. The converse is also true and datasets giving poor lifetimes may be combined to determine even shorterlifetimes, hence the creation of datasets 36 and 38. Details of these datasets are given intable BIII of AppendixB. This additive effect is described well by the modal method with the long and the short lifetimes beingcorrectly predicted for datasets 35–38 as shown intable III. The lifetimes predicted by the modal method canbe seen to be conservative when compared to the Damage XX rupture lifetimes. The comparison of first failuretimes shows that the modal method overestimates for datasets 35 and 36 and underestimates for datasets 37 and38. The modal method predicts the same lifetime for datasets 35 and 37; this is because these datasets differonly in the primary creep strain ratio (κ) which is neglected in the modal solution. Wang and Hayhurst (1994)showed that the long lifetime for dataset 37 results from the region of highly damaged material being located


Figure 8. Variation of normalised weldment lifetime with average rupture time ratio,ξ , for different values of average rupture strain ratio,γ , for theweld material. Mean parameter combinationsκm2m andγmξm are assumed for the HAZ material.

Table III. Lifetimes from modal method and Damage XX compared for datasets 35 to 38.

Data Normalised Normalised Normalised % Difference % Difference

set modal Damage XX Damage XX between between

lifetime time to first rupture modal modal

Tw failure lifetime lifetime and lifetime and

Tw Damage XX Damage XX

time to first rupture life

failure

35 0.712 0.668 0.843 −6.6 15.5

36 0.277 0.273 0.296 −1.4 6.35

37 0.712 0.729 0.878 2.4 18.9

38 0.235 0.239 0.253 1.4 7.2

in the parent material, as opposed to in the weld metal. Examination of the modal method damage distributionfor dataset 37 similarly showed intense damage in the parent material.


Figure 9. Variation of normalised weldment lifetime with average rupture strain ratio,γ , for different values of average rupture time ratio,ξ , for theweld material. Mean parameter combinationsκm2m andγmξm are assumed for the HAZ material.

The lifetime predictions of the modal method, from datasets 1 to 34, may be used to select an optimal set ofmaterial properties for the weldment in the same way that Wang and Hayhurst (1994) obtained dataset 37. Ifthis selection is made the mean secondary creep strain ratio (2m), the mean rupture strain ratio (γm) and theupper bound rupture strength ratio (ξu) are chosen for the weld metal; and the upper bound secondary creepstrain ratio (2u), the upper bound rupture strain ratio (γu) and the lower bound rupture time ratio (ξl ) for theHAZ material. The primary creep strain (κ) has not been specified because it is not considered by the modalmethod. The selection of properties for the weld metal is similar to that of dataset 37 except the mean rupturestrain ratio (γm) is chosen instead of the upper bound rupture strain ratio (γu) present in dataset 37. The effect ofthis is likely to be small because the change in lifetime is negligible between the mean rupture strain ratio (γm)and the upper bound rupture strain ratio (γu) for the upper bound rupture strength (ξu) as shown infigure 9. Forthe HAZ material the selection again differs from that of dataset 37. The upper bound secondary creep strainratio (2u) is selected instead of the mean secondary creep strain ratio (2m) used in dataset 37, but the effect ofthis parameter on lifetime is not significant as shown infigure 10. The modal method indicates that the selectionof the lower bound rupture strength (ξl ) should be made rather than the upper bound rupture strength (ξu) as


Figure 10. Variation of normalised weldment lifetime with average normalised secondary strain ratio,2, for different values of normalised primarystrain ratio,κ , for the HAZ material. Mean parameter combinationsκm2m andγmξm are assumed for the weld material.

Table IV. Optimal material properties selected from the modalmethod analyses and the corresponding weldment lifetimes.

Weld HAZ Normalised Normalised

properties properties modal Damage XX

lifetime, Tw rupture

lifetime, Tw

(κm)2mγmξu (κm)2uγuξl 0.746 0.850

specified in dataset 37. It is expected that this selection would give a poor result since the predictions from themodal method are against the general trend for the combination of the lower bound rupture strength (ξl ) andthe upper bound rupture ductility (γu), as shown infigure 12.

The modal method and Damage XX were used to determine the lifetime and the damage distributions for theoptimal properties selected from the modal method analyses. For the Damage XX computation each materialwas assigned a mean primary creep strain ratio (κm). The material properties and computed lifetimes are givenin table IV.


Figure 11. Variation of normalised weldment lifetime with average normalised primary strain ratio,κ , for different values of normalised secondarystrain ratio,2, for the HAZ material. Mean parameter combinationsκm2m andγmξm are assumed for the weld material.

The lifetime determined by the modal method, which is greater than that predicted for datasets 1–34, showsthe additive effect of the material properties on the weldment lifetime. The lifetime computed by Damage XXis 3% below the lifetime obtained for dataset 37 which shows that the material properties selected from themodal analyses do give a near optimal lifetime. The damage distributions from the modal and the Damage XXcomputations agree well; the region of intense damage is predicted to occur near the outer boundary in theparent material.

If the lifetimes predicted by the modal method, for datasets 1–34, are used to determine a set of materialproperties to give the shortest creep rupture lifetime then the selected parameters are identical to those obtainedfor dataset 38 by Wang and Hayhurst (1994).

7.1.4. Comparison of modal lifetime with Damage XX first failure time

The time to first failure is defined as the time taken for the first element of the finite element mesh to fail;this occurs for the modal method whenψ = 0.01, and for Damage XX whenω = 0.999. Comparison of themodal lifetimes with the Damage XX times to first failure reveals that the modal method lifetimes are notalways conservative. On average the time to first failure was overestimated by 5.2%, with the extremes being amaximum overestimate of 17.5% and a maximum underestimate of 15.3%. The best correlation of first failuretimes was found for datasets 10–17 where the weld metal average rupture time and strain ratios,ξ and γ ,


Figure 12. Variation of normalised weldment lifetime with average rupture time ratio,ξ , for different values of average rupture strain ratio,γ , for theHAZ material. Mean parameter combinationsκm2m andγmξm are assumed for the weld material.

respectively, were varied. For these datasets the largest error in first failure time was 7% with several of thedatasets accurate to within 3%.

Hence, the modal method may be used to provide an estimate of the first failure time, but this time may notbe conservative. The predictions of damage growth are discussed in the following section.

7.1.5. First failure location and damage growth

The modal method predicted the incorrect initial failure position for four of the 38 datasets considered. Fordatasets where Damage XX predicted failure a few elements removed from the internal diameter, the modalmethod predicted failure of an element on the internal diameter. This can be explained by the assumption ofconstant hydrostatic stress; in the Damage XX solution the hydrostatic stress falls on the internal diameter(figure 5) and consequently less damage is accumulated there.

The assumptions inherent in the modal method are no longer valid once an element fails and a crack startsto form. However, it is possible to continue the modal calculation past first failure by removing an elementfrom the solution when its continuity falls below 0.01. This revealed that elements continued to fail in similarlocations to those predicted by Damage XX. Good agreement between the two methods was found whenfive to ten elements were permitted to fail progressively, even if the location of failures moved from oneposition, or material zone, to another. The lifetime determined by the modal method, after several failures


Figure 13. Variation of normalised weldment lifetime with average rupture strain ratio,γ , for different values of average rupture time ratio,ξ , for theHAZ material. Mean parameter combinationsκm2m andγmξm are assumed for the weld material.

had occurred, exceeded the rupture lifetime calculated by Damage XX for many datasets. This was because themodal method did not model the increase in the hydrostatic stress that occurs as elements progressively failedto form a macro-crack, and consequently cracks did not propagate as rapidly as in reality. Comparison of thedamage distributions after several elements had progressively failed revealed an improved correlation betweenthe Damage XX and the modal solutions for the datasets where the modal method had predicted the position ofinitial failure in a different location to that of Damage XX. Hence it would appear that the modal method mayalso be used to provide an indication of the zones of intense damage that would cause final rupture. However,the lifetimes associated with these damage distributions are likely to be unconservative.

7.1.6. Prediction of stress fields

Comparison of the stress distributions from Damage XX and the modal method at first failure show thatthe modal method accurately predicts the stress redistribution that takes place in the secondary-tertiary phase.The modal method does, however, tend to underestimate the effective stress in the HAZ, and in the weldmetal close to the fusion boundary; the errors being about 5%. Comparison of the maximum principal stressesproduced errors of a similar order of magnitude for many of the datasets. The largest principal stress errors,underestimates, were found in locations where an element had failed; and also in locations near the internalboundary of the HAZ, where the stress was generally overestimated. These discrepancies can be attributed


Figure 14.Field distribution of the damage variable over a diametral section of the weldment for material dataset 1: (a) and (b), where one and five ofthe finite elements have failed, respectively, are from a complete continuum damage mechanics analysis; (c) and (d) are from the modal calculation and

are correspondingly for one and five failures, respectively.

to the assumption of constant hydrostatic stress and to the fact that the stresses determined from the modalmethod do not necessarily satisfy equilibrium. The results for datasets with closely matched material properties,notably dataset 37, revealed that the principal stresses predicted by the modal method on the outer boundary ofthe parent material were over-estimated by about 5%. These, generally small, errors in the stress fields showthat the modal method describes the essential features of the stress redistribution that occurs during the tertiarycreep phase. This, in turn, confirms the kinematic determinacy of the weldment since the stresses will only becorrect if the assumed strain rate field is the exact field.

At this juncture it is worth observing that an assessment of the quality of the modal predictions could beobtained by checking that the computed stresses are in equilibrium with the applied loads. Such an assessment,which has not been performed in the present work, could enhance the confidence in the predictions of the modalmethod. The prediction of damage distributions will now be discussed.

7.1.7. Prediction of damage fields

Damage distributions after one and five element failures have taken place are compared onfigure 14 fordataset 1 which corresponds to mean values for all the material parameters for both the weld and the HAZmaterials. In this figure the upper two plots(a, b) were predicted by Damage XX and the lower two(c, d)were predicted by the modal method. The left hand plots(a, c) show the damage distributions after one finiteelement failure and the right hand plots(b, d) after five finite element failures. Common colour contouringhas not been used so reference should be made to the key on each plot; damage varies from zero initially to


unity at final failure. The damage values,ω = 1−ψ , have been averaged to the nodes from element centroids,this smoothing has the effect of slightly lowering the values of damage in failed regions. First and subsequentfailures are predicted to occur in the weld metal, close to the fusion boundary, for the Damage XX and themodal solutions (figure 14a, bandc, d, respectively). The magnitude of the damage levels predicted by themodal method are lower than those predicted by Damage XX, and additionally the damage is more localised(figure 14a, c). However, both methods predict the same failure mechanism, a crack growing along the fusionboundary (figure 14b, d). The localised nature of damage predicted by the modal method, especially ahead ofthe crack tip (figure 14d), can be explained by the inability of the assumption of constant hydrostatic stress topredict the stress concentrating effect of the crack. Other datasets showed similar agreement between the twomethods.

Failures were predicted to occur, by the modal method, in both the weld metal and the parent material,adjacent to the HAZ, and close to the outer boundary of the pipe for dataset 37, consisting of the optimalcombination of properties to give the longest life. This compared well with the failure pattern predicted byDamage XX; however, the damage predicted by the modal method was in general much more localised, andwas not accurately modelled in the HAZ region located on the internal boundary of the pipe. This can beexplained by the independence of the modal method predictions on primary creep strain. The large primarycreep strain mismatch between the materials of dataset 37 causes the stress in the HAZ material to be higherthan that predicted by the modal method, particularly near the internal boundary of the pipe. Further evidencethat the primary creep strain causes this discrepancy is provided by examination of the results for dataset 35;datasets 35 and 37 only differ in their primary creep strain characteristics. The modal method gave the samepredictions for these two datasets, which when compared with the Damage XX predictions, for dataset 35,gives improved agreement with the damage in the HAZ being less significant.

7.1.8. The influence of the volume of parent material

The effect of the volume of parent material included in the modal calculation has been assessed because inthe limit, as the volume of parent material tends to infinity, and if damage is considered to be confined to theweld metal, then equation (24) predicts thatf (t) tends to unity. This effect has been examined by variation ofthe ratio of the length of parent material considered to the length of weld metal, on the outer diameter of thepipe, between a factor of 4 and 20. For this variation in the volume of the parent material the modal calculationpredicts a variation in lifetime of less than 2% for datasets 1 and 37 due to Hall and Hayhurst (1991) and Wangand Hayhurst (1994), respectively. This variation in lifetime may be associated with the approximations of thefinite element mesh size which becomes relatively large for long lengths of pipe. The results show that thevolume of parent material included in the calculation is not critical providing that the length of parent pipe issufficient for local perturbations in the stress field, associated with the inhomogeneity of the weldment, to havedied away so that the stresses in the parent material are similar to those for a homogeneous pipe.

The modal method has, therefore, been shown to describe the essential features of weldment behaviour fora large range of material properties. Sensibly conservative lifetime predictions have been obtained using thetime for the first element to fail in the modal solution to define the overall rupture lifetime of the weldment.In addition, the modal calculation may be continued, allowing elements to fail progressively, to give a goodindication of the regions of intense damage that would be likely to cause the final rupture of the weldment. Theuse of the method for weldment design studies is discussed in the next section.


8. The use of the modal method in weldment design

The modal method has been shown to give good predictions of the weldment lifetimes and the associatedstress and damage distributions for the range of material properties considered. The following generalrecommendations may be made to assist in weldment material selection.

a) The material properties of the weld metal are particularly important when trying to achieve long lifetimes.Good rupture ductilities (γ ) and good rupture strength (ξ ) are the most important requirements. The modalmethod indicates that the highest possible rupture strength (ξ ) for the weld metal is desirable and that increasingthe rupture strain ratio (γ ) beyond about twice that of the parent material gives little change in the lifetime.

b) The properties of the HAZ material do not strongly influence the lifetime provided that the rupture strength(ξ ) is approximately equal to that of the parent material and the ductility (γ ) is greater than about one half ofthat of the parent material. Increasing the rupture strength (ξ ) beyond that of the parent material is shown togive a slight reduction in lifetime; this is contrary to the results of Wang and Hayhurst (1994) where a slightincrease in lifetime was predicted. The modal method predicts an increase in lifetime when the rupture strength(ξ ) of the HAZ material changes from being equal to that of the parent material to being about one half ofthat for the parent material when the rupture ductility (γ ) is equal to that of the parent material. This trend isopposite to that reported by Wang and Hayhurst (1994), which suggests that the modal method should be usedwith caution for these material properties.

c) For both the weld and the HAZ materials the secondary creep strain ratio (2) does not have a significanteffect on the weldment lifetime, particularly if it is above unity.

d) The primary creep strain ratio (κ) does not affect the lifetimes computed by the modal method. This is nota major drawback because the primary creep strain ratio (κ) has a relatively small effect on overall lifetime, asshown by Wang and Hayhurst (1994), particularly if the secondary creep strain ratio (2) is above unity. Thestress and damage distributions predicted by the modal method may be inaccurate if the primary creep straincharacteristics of the materials are dissimilar.

e) Long weldment lifetimes are obtained when the highly damaged region is moved into the parent material.This can be achieved through the use of the modal method to select suitable material parameters.

These recommendations regarding weldment material selection are in agreement with those of Wang andHayhurst (1994), unless otherwise stated, and show that the modal method provides the capability to rapidlyanalyse the creep rupture of weldments and aid in the selection of the material properties to optimise theweldment lifetime and the damage distributions. Considered next is the use of the method for the analysis ofother weldment geometries and other material property combinations.

The analyses of the weldment considered here shows that the method is valid for the entire range of materialproperties appropriate for ferritic steels, however, the method is most suited to materials that do not showsignificant primary creep and for material property combinations that have similar primary creep characteristics.The suitability of the method for other weldment geometries will be governed by the extent to which thedeformation of the weld metal and the HAZ material are constrained by the deformation of the parent pipewhich will, in turn, dictate whether or not the deformation of the weldment as a whole is kinematicallydeterminate. Deviations from this constrained deformation, and therefore from kinematic determinacy, arelikely to occur when the length of the weld metal is relatively large compared to the thickness of the pipe,and when the angle of the weld boundary is relatively large. The results of the analyses presented in this paperdo not allow specific recommendations to be made and different geometries should be considered on a case bycase basis.


A prudent strategy for the use of the modal method in weldment design would require that each weldmentgeometry, with a representative set of material properties, is assessed for kinematic determinacy in the mannerdescribed in Section 6.1. This would require a full continuum damage mechanics analysis, and provided thatthe requirements of kinematic determinacy are satisfied the results of this analysis may be used to validatethe modal method. Other material property combinations, for the same geometry, can then be analysed withthe modal method to optimise material property selection. This optimal set of parameters, and possibly a fewothers, can then be used in a full continuum damage mechanics calculation to validate the results of the modalmethod, and to obtain detailed predictions.

Therefore, the modal method allows a range of design variants to be evaluated rapidly and accurately. Themethod may be used during the embodiment phase of the design process, where conceptual designs, which mayhave been preliminarily evaluated with reference stress methods, are evaluated and a single design identifiedwhich is then taken forward to the detailed design phase.

9. Conclusions

The modal method has been used as a post processor to a stationary state finite element creep solver toprovide histories of stress and damage, up to the time of first failure, for a butt welded steam pipe. Assumptionsof kinematic determinacy and constant hydrostatic stress for the secondary-tertiary period were made in order touse the modal method, and to compute the multi-axial stress function on which damage evolution depends. Thelifetime estimates determined by the method were conservative for all material combinations considered, onaverage by 14%, when compared with full finite element continuum damage mechanics calculations. Stress anddamage distributions also compared well with those of the more complex calculation made using Damage XX.The assumptions limited the application of the method up to the time at which first failure occurred; however,continuation of the calculation showed the zones of high damage were accurately predicted but the lifetimesbecame unconservative.

The neglect of the primary creep strain characteristics of the materials did not have a large effect onthe predicted weldment lifetimes. However, for datasets where the primary creep strain mismatch betweenmaterials was large the distributions of stress and damage were not as accurately predicted as those wherethe materials have mean primary creep strain ratios. The consequence being that confidence in the stress anddamage predictions of the modal method can only be high when the constituent materials have approximatelythe same primary strain characteristics.

Simpler reference stress analysis techniques have been assessed and shown to be less reliable in predictinga conservative lifetime and do not allow the lifetimes for various material property combinations to be orderedin a way that is possible with the modal method. This can be attributed to the problem of characterising theinteraction that takes place between the constituent materials of the weldment which is very dependent onmaterial properties. Reference stress methods which neglect this distribution of material properties are limitedin providing accurate lifetime estimates for weldments.

The modal method provides a rapid method for the estimation of weldment lifetime accounting for multi-axial stress effects and material properties. The method relies on the assumption of kinematic determinacy;however, the work presented here and the comments of Leckie and Hayhurst (1974) suggest that this doesnot have to be strictly satisfied. Should the isochronous rupture locus, which describes damage evolution, bedependent uponσI and σeff then a suitable assumption (or calculation) of the spatial and time variation ofhydrostatic stress must be provided. Subject to these limitations the method, in the form presented here asa post processor to the finite element creep solution, may be applied to other structural situations to give an


indication of the secondary-tertiary behaviour without committing the resources associated with complex finiteelement creep continuum damage mechanics calculations.

Appendix A. Transformation of the constitutive equations to secondary-tertiary form

The primary-secondary-tertiary damage evolution equation (2) may be rewritten in terms of continuity as

dψ

dt= −M

1+ φσχ

ψφtm. (A1)

This may be integrated between the limitsψ = 1, t = 0 andψ = 0, t = tR to give rupture timetR

tR=[m+ 1

M

]1/(m+1)

σ−χ/(m+1). (A2)

The secondary-tertiary equation for damage evolution (equation (5)) in uniaxial form is

ψ/ψ0=−(σ/σ0)ν/ψζ . (A3)

Integration of this relation between the same limits yields

tR= 1/ψ0(1+ ζ )(σ/σ0)ν. (A4)

The rupture time predicted by equations (A2) and (A4) at a given stress,σ , must be equal, thus

ψ0= σ ν0

(1+ ζ )(m+ 1

M

)−1/(m+1)

(A5)

and by retaining the value of the stress index

ν = χ/(m+ 1). (A6)

From the primary-secondary-tertiary constitutive equations (1, 2) it is possible to obtain an expression for theminimum creep rate (Wang and Hayhurst, 1995)

εmin=GM−m/(m+1)[(θ − 1)(m+ 1)

mθ + θ − 1

]θ−1[m(m+ 1)

mθ + θ − 1

]m/(m+1)

σ n−mm+1χ . (A7)

When thetm term is not present in the constitutive law this minimum creep strain rate must be equal to thestationary state creep strain rate

εss= ε0(σ/σ0)n. (A8)

Thus,

ε0/σn0 =GM−m/(m+1)

[(θ − 1)(m+ 1)

mθ + θ − 1

]θ−1[m(m+ 1)

mθ + θ − 1

]m/(m+1)

(A9)

and

n= n−mχ/(m+ 1). (A10)


Using equation (4) for strain rate under secondary-tertiary conditions in conjunction with equation (A3) yields

ε = ε0(σ/σ0

)n[1− (t/tR)]−n/(ζ+1)

. (A11)

This may be integrated forε = 0 whent = 0 giving

ε = ε0(σ/σ0)n−ν

βψ0(1+ ζ )[1− (1− (t/tR))β], (A12)

where

β = 1− n/(ζ + 1). (A13)

When the time equals the rupture time,t = tR, the strain must equal the failure strain,εR,

εR= ε0(σ/σ0)n−ν/βψ0(1+ ζ ). (A14)

A similar expression may be found for the primary-secondary-tertiary equations giving

εR= G

θMσ n−χ . (A15)

It may be shown thatn− ν = n− χ and hence to achieve equal failure strains at a given stress,σ , then

β = θ(m+ 1)1/(m+1)[(θ − 1)(m+ 1)

mθ + θ − 1

]θ−1[m(m+ 1)

mθ + θ − 1

]m/(m+1)

. (A16)

Thus a full set of material constants may be obtained for the secondary-tertiary constitutive equations from theconstants used in the primary-secondary-tertiary equations. These constants ensure that minimum strain ratesare compatible; and, that the time to rupture and the strain to rupture are equal for both sets of constitutiveequations.

Appendix B. Weldment dataset numbering

A total of 34 basic datasets has been considered by Wang and Hayhurst (1994). In the first 17 the propertiesof the weld metal are varied whilst those of the HAZ are held at their mean. Datasets 18–34 vary the materialproperties of the HAZ whilst the weld retains mean properties. Parameter groupings are based on the normalisedmaterial properties: the first group combines the normalised primary strain,κ , with the normalised secondarystrain,2; the second groups the normalised rupture ductility,γ , with the normalised rupture lifetime,ξ . Each ofthe normalised parameters takes either a lower bound, mean or upper bound value. The parameter combinationsand corresponding dataset numbers are given intables BIandBII for the weld and HAZ materials respectively.Sets 35–38 consider more complex combinations of parameters that lead to very long or very short lifetimes.The details of the material properties used for these datasets, and the corresponding lifetimes obtained by Wangand Hayhurst (1994), are given intable BIII.


Table BI. Parameter combinations for weld material.

κl2l (2) κl2m (3) κl2u (4) γl ξm (11) γl ξu (12)

κm2l (5) κm2m (1) κm2u (6) γmξm (1) γmξu (14)

κu2l (7) κu2m (8) κu2u (9) γuξm (16) γuξu (17)

Table BII. Parameter combinations for HAZ material.

κl2l (19) κl2m (20) κl2u (21) γl ξl (27) γl ξm (28) γl ξu (29)

κm2l (22) κm2m (18) κm2u (23) γmξl (30) γmξm (18) γmξu (31)

κu2l (24) κu2m (25) κu2u (26) γuξl (32) γuξm (33) γuξu (34)

Table BIII. Parameter combinations of datasets 35–38.

Data Parameters for weld Parameters for HAZ Lifetime

set (Wang and Hayhurst, 1994)

(h)

35 κm2mγmξu (Set 14) κm2mγmξu (Set 31) 58886

36 κm2mγl ξl (Set 10) κm2mγl ξl (Set 27) 20676

37 κu2mγmξu (Set 8 & 14) κu2mγmξu (Set 25 & 31) 61330

38 κm2l γl ξl (Set 5 & 10) κm2l γl ξl (Set 22 & 27) 17672

Acknowledgement

The authors gratefully acknowledge the support provided by the United Kingdom SERC and by BritishEnergy. This paper is published with the permission of Nuclear Electric Ltd.

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Approximate creep rupture lifetimes for butt welded ferritic steel pressurised pipes