Finite element simulation of welding and residual ... - SAGE Journals

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Finite element simulation of welding and residual stresses in a P91 steel pipe incorporating solid-state phase transformation and post-weld heat treatment A H Yaghi*, T H Hyde, A A Becker, and W Sun Department of Mechanical Engineering, University of Nottingham, Nottingham, UK The manuscript was received on 12 November 2007 and was accepted after revision for publication on 19 February 2008. DOI: 10.1243/03093247JSA372

Abstract: The finite element (FE) method has been applied to simulate residual axial and hoop stresses generated in the weld region and heat-affected zone of an axisymmetric 50-bead circumferentially butt-welded P91 steel pipe, with an outer diameter of 145 mm and wall thickness of 50 mm. The FE simulation consists of a thermal analysis and a sequentially coupled structural analysis. Solid-state phase transformation (SSPT), which is characteristic of P91 steel during welding thermal cycles, has been modelled in the FE analysis by allowing for volumetric changes in steel and associated changes in yield stress due to austenitic and martensitic transformations. Phase transformation plasticity has also been taken into account. The effects of post-weld heat treatment (PWHT) have been investigated, including those of heat treatment holding time. Residual axial and hoop stresses have been depicted through the pipe wall thickness as well as along the outer surface of the pipe. The results indicate the importance of including SSPT in the simulation of residual stresses during the welding of P91 steel as well as the significance of PWHT on stress relaxation. Keywords: P91 steel pipe, finite element weld simulation, residual stress, phase transformation, post-weld heat treatment

1

INTRODUCTION

The efficient operation and safety of power generation plants have been the concern of the science community for decades. In particular, the integrity considerations for steel pipelines in power plants delivering steam at a high temperature and pressure have played a critical role in the overall safety assessment of power plant components. Since the 1970s and 1980s, following significant advancement in the metallurgical properties of the steel grade chosen in the construction of power plant pipelines, such as enhanced creep strength, modified 9 per cent Cr (P91) steel has been used for that purpose, with the added complication of having to weld many pipe joints together. The process of welding involves a large number of thermal cycles responsible for *Corresponding author: Department of Mechanical Engineering, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. email: [email protected] JSA372 F IMechE 2008

alterations in the steel microstructure as well as in its mechanical and physical properties. Welded steel joints are sometimes considered the weakest part in a pipeline owing to the possible reduced creep strength of the weld metal and the surrounding heataffected zone (HAZ). Power plant steel pipes are usually joined together by fusion welding, which is a process involving the melting of what solidifies into weld beads by applying thermal fluxes, causing the weld material to reach the material’s melting point or even higher temperatures. In some other engineering components, a joint can be formed with friction welding [1], which is a more recent application, involving friction under pressure of the two metal surfaces to be joined, thereby allowing the produced friction weld to form at substantially lower temperatures than the material’s melting point. This paper is only concerned with the fusion welding process. P91 steels have desirable creep-resistant properties. The efficiency of power plants has been J. Strain Analysis Vol. 43

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A H Yaghi, T H Hyde, A A Becker, and W Sun

improved significantly by being able to raise the temperature of steam to 593 uC in P91 pipes in service [2]. Creep damage analyses and failure predictions have been carried out for representative welded components of P91 steel [3], including repair welds [4], to reach a better understanding of how creep can lead to the failure of pipes. The finite element (FE) simulation of creep has been one of the most effective techniques in the integrity assessment of welded components and pipes. Although creep has been extensively investigated for welded pipe joints, the usual assumption has been to have no residual stresses at the inception of the analyses. The process of welding is responsible for substantial residual stresses, which can be as high as the material yield stress in places, if no post-weld heat treatment (PWHT) is applied. Even after PWHT, some residual stresses will persist, albeit with reduced magnitude. The FE simulation of the process of welding in order to model the thermal cycles and to obtain the final residual stress distribution is complex. In addition to the standard considerations of FE thermomechanical functions, P91 steel characteristically exhibits a solid-state phase transformation (SSPT), which ought to be taken into account in the FE simulation, since the SSPT induces important mechanical effects such as volumetric changes in the material. In order to simulate the process of creep realistically for the prediction of ultimate failure of steel components, residual stresses due to the process of welding should be investigated first, with all relevant considerations included, such as SSPT and PWHT, and only then can the effects of residual stresses on creep be established and included in further creep studies. This paper describes the numerical techniques for obtaining residual stresses in a multi-pass buttwelded P91 steel pipe using the method described in reference [5], starting with an FE thermal analysis and ending with an FE structural analysis, sequentially coupled and modified by a user subroutine to manipulate the temperature field in the material. A specific set of material properties have been generated from literature for P91 steel. The temperature dependence of the material properties used in the FE simulation has been taken into account and so has SSPT. The volumetric change in the form of the variable coefficient of linear thermal expansion as well as the variation in yield stress of the parent and weld materials due to austenitic and martensitic transformations have been incorporated in the FE code through ABAQUS user subroutines. J. Strain Analysis Vol. 43

Transformation plasticity has also been accounted for in the FE analysis. In a previous study by the present authors [6], residual axial and hoop stress fields were determined for a similar FE model. In this paper, further residual stress fields have been determined. In addition, the effects of PWHT have been investigated, including those of the PWHT holding time as well as the constants in the Norton creep law which describes the stress redistribution due to creep during the holding procedure. The results are then presented in the form of axial and hoop stresses throughout the pipe wall thickness and also against distance from the weld centre along the outer surface of the pipe. Although, in reality, welding is a three-dimensional procedure, it is often considered sufficient to represent a pipe weld using an axisymmetric FE model [5–11], which is considerably faster and easier to simulate, and which has been adopted throughout this work.

2

WELDING PROCESS SIMULATION

The process of welding inherently subjects the welded material to thermal cycles which cause the metal to experience high temperatures and even to melt and resolidify at the weld region, inducing nonlinear mechanical behaviour in the weld region and HAZ, resulting in microstructural alteration, and giving rise to undesirable residual stresses. With the most recent advances in technology, the numerical simulation of such stresses is becoming more readily achievable, making it one of the most effective methods to determine the stress and strain fields for the integrity assessment of welded industrial components, and in particular in the case of determining the residual stress distributions through thick cross-sections. The FE simulation of the process of fusion welding can be performed by adopting either of two different mechanical approaches. Either the complex fluid and thermodynamics local to the weld pool and HAZ are numerically analysed or, alternatively, the solid mechanics approach is applied. The thermodynamics method relies on the conservation of mass, momentum, and heat together with the latent heat and surface tension boundary conditions to represent the physical phenomena of the molten weld pool and thermal behaviour of the HAZ. The solid mechanics method, which is used in this paper, models the global thermomechanical behaviour of the weld structure, paying special attention to the heat source, the accuracy of which relies on the JSA372 F IMechE 2008

FE simulation of welding and residual stresses

Fig. 1

Axisymmetric FE mesh of a multi-pass butt-welded P91 steel pipe

theoretical and empirical parameters describing the weld pool size and shape. 2.1

Model geometry and material properties

The process of fusion welding has been numerically simulated for a P91 steel pipe with an outer diameter of 145 mm, a wall thickness of 50 mm, and a total length of 350 mm. The FE modelled joint consists of two P91 steel pipe sections circumferentially welded with 50 beads of similar P91 weld metal, using gas– tungsten arc welding (GTAW) (or tungsten–inert gas (TIG) welding) for weld beads 1 to 7 and shielded metal arc welding (SMAW) (or manual metal arc (MMA) welding) for weld beads 8 to 50. The axisymmetric FE mesh used in the simulation is depicted in Fig. 1(a), which represents a crosssection of the pipe wall, showing the full length of the welded pipe, the axis of symmetry, and the specified boundary conditions of constraint. An enlarged view of the FE mesh for the weld and the surrounding regions is depicted in Fig. 2(b). The sequence of laying the weld passes is shown in Figs 2 JSA372 F IMechE 2008

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(a), (b), and (c), and the welding specifications are provided in Table 1. During the process of welding, the two pipe sections were held together lightly until the first few weld beads were applied; then the gripping jaws were released to minimize the effect on the stress field. Similarly, during the process of PWHT, the pipe was supported by avoiding bending moments and any significant stresses. In the FE simulation, the root bead (bead 1) protrudes by 1 mm (see Fig. 2(a)), and the last layer of beads (weld crown) protrudes by 5 mm (see Fig. 2(c)). In the FE mesh, each bead is considered to be a pass, so that the number of passes in the FE model is equal to the number of beads in the simulated weld. Each pass (or bead) in the mesh consists of a discrete number of elements, ranging from a minimum of 96 elements for pass 1 and a maximum of 324 elements for each of passes 6 and 7 (see Fig. 2(a)). The weld region has a total of 7704 elements. The whole FE model contains 13 736 elements and 39 873 nodes. The FE mesh represents a welded pipe with the weld crown machined off at the end of the welding process before PWHT is applied. This is modelled J. Strain Analysis Vol. 43

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field. The resulting stress redistribution is independent of the machining operation time. The FE simulation incorporated two sets of mechanical and thermal material property data corresponding to the weld metal and the parent metal, using general P91 parent metal data and modified, where possible, to allow for actual cast and the weld metal manufacturing specifications, as described in a previous publication [5]. The chemical compositions for both materials are specified in Table 2. The material property data used in the numerical simulation are shown in Fig. 3. The coefficient of linear thermal expansion, a, shown in the figure, does not take into account the effects of SSPT, which will be described in detail in section 2.3. In addition to the properties given in Fig. 3, a latent heat capacity of 260 kJ/kg has been assumed together with a solidus temperature of 1420 uC and a liquidus temperature of 1500 uC. A kinematic hardening law has been used in the structural analysis. The hardening modulus, which is equal to the gradient of the plastic portion of the stress–strain curve for the material, as defined in the ABAQUS user manual [12], and which has been implemented in the FE analysis, is shown in Fig. 4. Two polynomial functions have been fitted to the hardening modulus data for the weld and parent metal to facilitate the incorporation of the hardening moduli in the appropriate user subroutine. The density of the material has been assumed to be constant at a value of 7770 kg/m3 [5]. The FE simulation has been performed using the ABAQUS FE commercial software [12].

2.2

Fig. 2

Weld sequence of the numerically simulated P91 steel pipe

in the FE structural analysis by removing the elements which make up the weld crown in one step, inducing a redistribution of the residual stress

Table 1

Thermal and structural FE modelling

An FE simulation of the fusion welding process, including the determination of the resulting residual stresses, consists in principle of a thermal analysis, which represents the thermal process during welding, followed by a sequentially coupled structural analysis, based on the temperature history obtained during the thermal analysis, which has all been described in detail in reference [5]. In the thermal analysis, the welding process is simulated

Welding process specifications

Process

Bead(s)

Current (A)

Voltage (V)

GTAW (or TIG welding)

Root bead (bead 1 (see Fig 2(a)) Beads 2 to 7

95–105 120–150

12–14 12–14

5–10 5–10

SMAW (or MMA welding)

Beads 8 to 50

140–160

26–28

21–23

J. Strain Analysis Vol. 43

Speed (cm/min)

JSA372 F IMechE 2008


Table 2

279

Chemical composition of the modelled P91 steel pipe

Element

C

Ni

Mn

Cr

Mo

V

Si

Cu

Fe

Amount (wt %)

0.109

0.165

0.443

8.35

0.946

0.22

0.307

0.152

Balance

Fig. 3

Fig. 4

Mechanical and thermal material properties for the parent metal (PM) and weld materials metal (WM) against temperature, used in the FE analysis of the P91 steel pipe

Hardening moduli for the parent and weld materials against temperature, used in the FE analysis of the P91 steel pipe

by applying a distributed heat flux (DFLUX) for each pass, which is a triangular function of heat per unit volume against time, as can be seen in Fig. 5. The FE element type used in the thermal part of the simulation is a quadratic axisymmetric diffusive heat transfer continuum solid quadrilateral which is mostly eight noded with only a limited use of a sixnode element. The temperature field produced by the thermal analysis can verify that the molten zones throughout the analysis are realistic and the temperatures reached in the HAZ are reasonable and so is the penetration distance of the molten weld metal into the surrounding material. It is assumed that the HAZ, which is typically 4 mm deep, starts with the JSA372 F IMechE 2008

Fig. 5

DFLUX against time for five weld pass depositions, the time being depicted from the beginning of each weld pass application

region of the parent metal which melts due to the penetration of the weld metal and ends where the parent metal reaches an approximate maximum temperature of 800 uC [5]. There are usually four distinct microstructural regions in the HAZ, described as coarse grained, fine grained, intercritical, and overtempered. The HAZ microstructure is dependent on the peak temperatures reached during welding thermal cycles as well as cooling rates and exposure time at peak temperatures [13]. In particular, the grain evolution and size will depend on these factors. Coarse grains will form adjacent to the fusion boundary and finer grains will be found on J. Strain Analysis Vol. 43

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moving away into the parent material. On moving further into the HAZ, the intercritical region is reached, where austenitization of the metal during the SSPT is only partial with varying degrees depending on the peak temperature reached. Beyond the intercritical region, peak temperatures remain below the critical value at which austenitization begins; nevertheless, some limited carbide growth occurs because of overtempering, which is not easily detectable using optical means. The structural analysis uses the temperature distributions obtained from the thermal analysis as input thermal loading. The FE element type used in the structural part of the simulation is an axisymmetric stress–displacement continuum solid element which is mostly eight-noded biquadratic quadrilateral with reduced integration with only a limited use of a six-node quadratic triangular element. Large displacement effects have been incorporated in the simulation by including the non-linear geometry option, NLGEOM [12]. The ‘element birth’ technique is used, simulating weld elements during their application to bring them into existence in the structural analysis without incurring strain incompatibilities, details of which have been described in previous publications [5, 7, 11]. The ‘element birth’ technique generates the mesh representing the weld part from the start of the FE analysis, keeping the weld elements at an assumed softening temperature, TSOFT, via a user subroutine [12], until the moment of weld application in the structural analysis. An appropriate temperature of 1200 uC has been set for TSOFT, at which the metal is sufficiently soft, preventing any significant stresses from existing inside the material [5, 7, 11]. Hence, the weld can be applied without suffering from any significant stress step change, since Young’s modulus and the yield stress of the material are so low at TSOFT. Although material properties have been plotted in Fig. 3 up to a temperature of 2000 uC, in the structural analysis the maximum temperature has been truncated at 1200 uC in the user subroutine to avoid unnecessary and excessive computing time.

2.3

volumetric change. Martensitic transformation, also known as shear or displacive transformation, is produced by shear deformation on a multi-grain scale, and it is diffusionless since the chemical composition of martensite is identical with that of parent austenite [14]. P91 steel is assumed to be pearlite–ferrite before any thermal cycles take place [15]. During welding, when the steel reaches the temperature A1, it begins to transform from pearlite–ferrite to austenite (Fig. 6). When the temperature reaches A3, the austenite transformation is assumed to be complete and the material fully austenitic. Pearlite–ferrite steel has a b.c.c. structure, whereas austenite has an f.c.c. structure. During austenite transformation, the steel undergoes a reduction in volume, as can be seen schematically in Fig. 6. When the material cools just after welding and a certain temperature is reached, martensitic transformation is expected to take place. The rate of cooling can influence such transformation. When the cooling rate is sufficiently rapid, which applies to the welding of most pipes, at the end of the martensitic transformation, the material is fully martensitic [8, 16]. At the start of martensitic transformation, the temperature Ms is calculated [17] using the values given in Table 2 by substituting for the chemical composition in the equation Ms ~454{210½Cz

4:2 {27½Ni{7:8½Mn ½C

{9:5ð½Crz½Moz½V zWz1:5½SiÞ{21½Cu ð1Þ where the square brackets indicate the amount of each element. To allow for a discrepancy between

Solid-State phase transformation

Thermal cycles during the process of fusion welding are responsible for the SSPT in P91 steel, inducing changes in both the volume and the yield stress in the metal. Martensitic transformation, which occurs during relatively rapid cooling immediately after each intense welding heat cycle, is the phase transformation associated with the most significant J. Strain Analysis Vol. 43

Fig. 6 Schematic diagram of the volume change due to phase transformation JSA372 F IMechE 2008


calculated and empirical values [17], an adjustment by 11 uC gives a value of 375 uC for Ms. When the temperature during cooling reaches Ms (5 375 uC), austenite begins to transform into martensite, which has a b.c.t. structure, causing an increase in volume of the material, as can be seen schematically in Fig. 6. When the temperature Mf (5 200 uC) is reached, the martensitic transformation is considered to be full and complete. Whenever the material experiences further thermal cycles, austenitic and martensitic transformations continue to take place according to the schematic presentation in Fig. 6. Martensitic transformation can be partial or complete depending on the temperature reached upon cooling. If Mf is not reached during cooling, only partial martensitic transformation takes place. During the process of welding of the P91 pipe, an interpass temperature between 250 uC and 300 uC was maintained, which has been modelled in the FE simulation by using a sink temperature of 250 uC in the thermal analysis and by choosing a cooling time between applying the successive weld beads, ensuring that the interpass temperature in the FE simulation remains between 250 uC and 300 uC. This results in a partial martensitic transformation during cooling. The martensitic transformation becomes complete in the final stage of the welding process, when all the beads have been applied and the welded pipe is allowed to cool to ambient temperature. The multi-pass nature of applying the weld beads is responsible for multiple cycles of heating and cooling in the weld region and also in the surrounding HAZ. The temperature history of each point in the FE model is followed by the user subroutine. The fraction of martensite in the steel is determined as described in the following paragraphs. Martensite is assumed to form upon cooling only. During heating, the fraction of martensite is assumed to remain unaltered, until the metal undergoes austenitic transformation between the temperatures A1 and A3, shown in Fig. 6. This will set the martensitic fraction back to zero, and then, in turn, the steel goes through martensitic transformation on cooling. Martensitic transformations during thermal cycles are assumed to take place during cooling only. Similarly, austenitic transformations are assumed to occur during heating only. Accordingly the fractions of each type of steel phase are calculated in the numerical simulation. These conditions are assigned in the following transformation-governing equations and are satisfied in the user subroutine so that each phase transformation occurs in proportion to the temperature reached and also in the one applicable direction of either cooling or heating only. JSA372 F IMechE 2008

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In order to trace the martensite fraction in the material during martensitic transformation, it is assumed that the transformation obeys the Koistinen–Marburger relationship, given for carbon steel in the equation [8, 14–16] fm ~1{exp½{0:011ðMs {T Þ ðT ¡Ms Þ

ð2Þ

where fm is the fraction of martensite at the current temperature T during cooling. So that this relationship can be used in the FE analysis to trace the martensite fraction numerically, the differential of the above equation in the form of increments is expressed as Dfm ~f{0:011 exp½0:011ðT {Ms Þg DT

ðT ¡Ms Þð3Þ

where DT is the temperature increment during cooling. However, it is assumed that, when martensitic transformation begins at Ms (5 375 uC) and ends at Mf (5 200 uC), the corresponding martensite fraction goes from 0 to 1. For this condition to be satisfied, it is necessary to include a correction factor for Dfm of 1.17 [6]. Hence, the above equation becomes Dfm ~1:17f{0:011 exp½0:011ðT {Ms Þg DT ðMf ¡T ¡Ms Þ

ð4Þ

The full volumetric change strain for P91 steel has been reported as 3.7561023 [6]. Hence, the volumetric change strain increment is given by DeVOL ~3:75|10{3 Dfm

ðMf ¡T ¡Ms Þ

ð5Þ

Austenitic transformation is less significant for residual stresses than martensitic transformation is. The former occurs during heating at a relatively high temperature (820–920 uC), therefore its influence on stresses is relatively small owing to the weaker mechanical properties at such high temperatures. In order to incorporate the effects of volumetric reduction due to austenitic transformation into the FE code, a simplified linear relationship for the volumetric change strain has been adopted [15]. The volumetric change strain due to a full austenitic transformation is assumed to be 22.28861023 [15, 16]. The relationship is expressed in the equation for the volumetric change strain increment given by DeVOL ~

{2:288|10{3 DT A3 {A1

ðA1 ¡T ¡A3 Þ

ð6Þ

where DT is the temperature increment during heating and A1 and A3 are 820 uC and 920 uC respectively for P91 steel. J. Strain Analysis Vol. 43

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Thermal strain during thermal cycles is accounted for by using the linear thermal expansion coefficient a, which is additional to the elastic and plastic components of strain. During phase transformation, the volumetric change can be taken into account by equating it to an additional strain. Phase transformation is a shear deformation which takes place at the lattice planes as the steel goes through microscopic restructuring, resulting in transformationinduced plasticity in addition to the volumetric change. This can be represented analytically by adding a separate strain component to the total strain function. Hence, the total strain rate for a steel undergoing phase transformation can be written as e_ TOTAL ~_eE z_eP z_eT z_eVOL z_eTRP

ð7Þ

where e˙E, e˙P, e˙T, e˙VOL, and e˙TRP are the strain rate components due to elastic loading, plastic loading, thermal loading, volumetric change, and transformation plasticity respectively. For the purpose of numerical simulation, the above equation can be written in the form of increments to give an expression for the total strain increment as DeTOTAL ~DeE zDeP zDeT zDeVOL zDeTRP

ð8Þ

where DeE, DeP, DeT, DeVOL, and DeTRP are strain increments due to elastic loading, plastic loading, thermal loading, volumetric change, and transformation plasticity respectively. The first three terms in the above equation have already been incorporated in the simulation in the thermal and mechanical analyses sections of this paper. The volumetric strain increment DeVOL is calculated and then added to the thermal strain component DeT, in order to obtain the overall effect of volumetric change on the linear thermal transformation coefficient, a. Hence, for martensitic transformation, strain due to both thermal loading and volumetric change is given by DeT zDeVOL ~a DT z3:75|10{3 Dfm ðMf ¡T ¡Ms Þ

ð9Þ

Substituting for Dfm and then Ms gives DeT zDeVOL ~a DT z3:75|10{3 |1:17f{0:011 exp½0:011ðT {Ms Þg DT ~ a{4:825|10{5 exp½0:011ðT {375Þ DT ðMf ¡T ¡Ms Þ

ð10Þ

For austenitic transformation, strain due to both J. Strain Analysis Vol. 43

thermal loading and volumetric change is given by DeT zDeVOL ~a DT z

{2:288|10{3 DT A3 {A1 ð11Þ

ðA1 ¡T ¡A3 Þ Substituting for A1 and A3 gives DeT zDeVOL ~ a{2:288|10{5 DT ðA1 ¡T ¡A3 Þ

ð12Þ

The thermal and volumetric change strain incremental components from equations (10) and (12) can be combined and expressed as a function of the linear thermal expansion coefficient. The resultant function is the original thermal coefficient with the effect of volumetric change superimposed on the original curve over the corresponding phase transformation temperature ranges. This is relevant to the FE simulation since it makes it possible to model the volumetric change effect as a numerical modification to the linear thermal expansion coefficient a. This is achieved through a user subroutine which interacts with the FE structural analysis to determine the function of a which is to be adopted in the FE strain and stress calculations. The yield stress of the material is influenced by the microstructural phase of steel. In this work, it is assumed that the yield stress of the pipe material is that of the parent material prior to any welding thermal cycles [8]. During the austenitic phase, the yield stress of the material, whether parent or weld, is given the value of the parent metal. The material goes through heating, followed by cooling, and, when the martensitic transformation takes place, the yield stress of the material is assumed to become that of the weld steel. Parts of the weld and parent materials experience multiple thermal cycles due to the multi-pass nature of the weld. Once the material transforms into martensite, the yield stress of the material acquires the weld value and then it is kept at that value unless an austenitic transformation takes place. In the FE simulation, through a user subroutine, the temperature history of each point in the FE model is traced in order to establish the steel phases at that point throughout the analysis, according to Fig. 6, which gives a schematic representation of the steel phases against temperature. Polynomial equations have been fitted and implemented in the FE code to represent the parent and weld yield stresses and hardening properties in the user subroutine which prescribes yield stress values for the material in the FE structural analysis. JSA372 F IMechE 2008


Unlike austenitic transformation, the formation of martensite, which involves shear and is displacive in nature, induces transformation plasticity. It is usual to describe transformation plasticity by equating the induced strain rate under constant and small applied stress in the equation 0 _3 e_ TRP ij ~kQ ðfÞf Sij 2

ð13Þ

where k is the Greenwood–Johnson parameter, Q9(f) is a function expressing the kinetics of transformation plasticity, f is the volume proportion of the forming phase (martensite), f˙ is the rate of f, and Sij represents the deviatoric stresses [18–21]. During martensitic transformation, stresses in the material are expected to vary, making transformation plasticity a dynamic phenomenon and causing its governing laws also to vary. A simplified and approximate approach is to assume that transformation plasticity produces a mechanical effect on the stresses and strains equivalent to that induced when the yield stress of the steel is reduced by a certain extent over the phase transformation temperature range. It has been reported that a reduction of 30 MPa in the value of yield stress can represent the mechanical effect of transformation plasticity [9, 22], which has been implemented in the appropriate subroutine, thereby modifying the yield stress value of the material throughout the martensitic transformation temperature range.

2.4

Post-weld heat treatment

Residual stresses in welded pipes can impose a risk of causing premature cracking during service in power plants or even prior-service cracking during transportation if the pipes are not handled gently. Although residual stresses are expected to decay during service, under creep conditions, they may still contribute towards the fracture or damage of the welded pipes, which may reduce the life expectancy of the component. If the weldment undergoes PWHT, the risk of premature cracking caused by residual stresses is mitigated. PWHT cannot completely eliminate residual stresses, since the maximum attained temperature during heat treatment should not reach A1, the temperature at which austenitic transformation begins. For P91 steel, PWHT is typically conducted at 760 uC, which is kept fixed during the holding time (typically 2–3 h). In the reported FE simulation, PWHT is modelled by taking into consideration the holding time and by JSA372 F IMechE 2008

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assuming that the material experiences creep at the elevated temperature. Although the heating and cooling procedures are important in real applications, in the FE simulation it is sufficient to allow for the creep of the material during the holding time. In addition, during the heating and cooling of the material, yield stress changes in value, inducing a corresponding plastic change in residual stresses and a redistribution of the stress field. Hence, PWHT is numerically simulated by allowing for the temperature dependence of the material’s mechanical properties during the heating and cooling stages and by including creep during the holding time. Creep during the PWHT holding time, in the FE simulation, is assumed to obey the Norton law [23], given in the multi-axial form by 3 n{1 e_ CR ij ~ Aseq Sij 2

ð14Þ

where e_ CR ij represents the creep strain rate components, seq is the equivalent stress, and A and n are non-zero material constants, which are not available for the P91 parent material at 760 uC and therefore have to be estimated from those obtained from creep tests performed at 650 uC with uniaxial stress ranging from 120 MPa to 140 MPa [4]. It is assumed that the n value at 760 uC is the same as that at 650 uC (n 5 10.836, which is considered to be the baseline value), and therefore the difference between the properties at 650 uC and 760 uC can be quantified by the difference between the values of A. The ratio of A for 760 uC to A for 650 uC can be estimated by using a temperature-dependent Norton law of the uniaxial form 0

e_ ~A0 e{Q0 =RT sn

0

ð15Þ

where e˙ is the creep strain rate, s is the stress, A9 and n9 are constants, T9 is temperature in kelvins, R is the gas constant, and Q0 is the activation energy. This is achieved using R 5 2 and Q0 5 60 000 cal/mol, which is typical for ferritic steels at low stress levels, although the Q0 values for P91 steels at high stress levels could be higher [24]. The uniaxial form of Norton law is given by e_ ~Asn

ð16Þ

Assuming that n remains unchanged between the temperatures of 650 uC (< 923 K) and 760 uC (< 1033 K), it can be seen from equations (15) and (16) that J. Strain Analysis Vol. 43

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A760 expð{Q0 =1033RÞ ~ expð{Q0 =923RÞ A650

ð17Þ

where A760 and A650 are the values of A at 760 uC and 650 uC respectively. By substituting the published empirical value for A of 1.782610228 [3, 25] in equation (17), a value for A of 56.8610228 is obtained (for stress in megapascals and time in hours), which is considered to be the baseline value. The effect of the holding time on residual stresses has been investigated by running a series of PWHT analyses subsequent to the modelling of the welding process including SSPT. The holding time has been varied from a minimum of zero to a maximum of 100 h. When the holding time is zero, the elastic– plastic temperature-dependent redistribution of the stress field is considered during the heating and cooling stages but creep is ignored. The effects of varying the values of A and n in the Norton law as well as the holding time have also been considered.

3

FE RESIDUAL STRESSES

The FE thermal analysis has provided a temperature history which is consistent with the expected P91 material behaviour. The structural part of the simulation starts with an analysis which does not allow for SSPT. This is followed by analyses which include the effects of SSPT on the predicted stress field, by allowing for volumetric changes and yield stress changes due to SSPT, as well as transformation plasticity. The crown part of the weld has been effectively removed at the end of the structural

Fig. 7

analyses to simulate the case when the weld crown is machined off after welding. The effects of PWHT on the residual stress field have also been modelled. Maximum principal residual stress contours, obtained from the initial structural analysis, ignoring the effects of SSPT, are plotted in Fig. 7, at the weld region and HAZ. In a previous study on the same FE model [6], residual axial and hoop stress contours were plotted for the same pipe section, revealing that both residual stresses were tensile at the outside surface and compressive at the inside surface in the weld region and HAZ. In comparison, by depicting maximum principal stresses, Fig. 7 effectively shows the higher tensile stress field out of the axial and hoop stress fields.

3.1

Effects of SSPT

Following the initial FE structural analysis, all the subsequent structural analyses have incorporated the modelling of SSPT. Figure 8 shows residual maximum principal stress contours including the effects of SSPT. The plot of the maximum principal stress depicts the higher tensile stress field out of the axial and hoop stresses, providing information on the tensile range of the two most important stresses. The residual axial and hoop stress fields for the same FE model can be found in a previous study [6]. Figure 8 shows that the most significant influence of including SSPT in the analysis is that the residual maximum principal stress at the outer surface of the weld goes from tensile to compressive in the vicinity of the last weld pass, which reflects the effect on both the axial stress field and the hoop stress field.

Residual maximum principal stress contours (MPa) at the weld region and HAZ, without allowing for SSPT (PM, parent metal; WM, weld metal)




Fig. 8

Residual maximum principal stress contours (MPa) at the weld region and HAZ, including the effects of SSPT (PM, parent metal; WM, weld metal)

Figures 9 and 10 present the residual axial stresses and residual hoop stresses respectively, using fivepoint averaging, against distance from the weld centre, both with and without allowing for SSPT, plotted along a straight line on the outer surface in the axial direction, starting from the side farther away from bead 50. The midpoint of the straight line coincides with the weld centre-line. The figures show a third curve representing PWHT, which is the subject of the next section. The weld boundaries, which are those between the weld and the HAZ, are identified with two vertical dashed lines in the

Fig. 9

285

figures. The stresses are depicted after the weld crown has been removed. The residual axial stress curve without SSPT in Fig. 9 shows that, at the outside surface of the pipe, the weld region has areas with a tensile stress magnitude significantly higher than that occurring in the parent metal also on the outside surface of the pipe. The stress is tensile with the exception of a moderate dip in the compressive range only at the interface between the weld metal and the parent material. The residual hoop stress curve without SSPT in Fig. 10 shows that, at the outside surface of the pipe, except at the interface

Residual axial stress variations against distance from the weld centre, along a straight line on the outside surface in the axial direction, after removing the weld crown



286


Fig. 10

Residual hoop stress variations against distance from the weld centre, along a straight line on the outside surface in the axial direction, after removing the weld crown

between the weld metal and parent material, the weld region has a hoop stress which is significantly higher than that found in the parent metal on the outside surface of the pipe in the pipe section shown in the figure, and also the surface stresses are all tensile. By comparing the curves in Figs 9 and 10, it can be seen that SSPT causes both the tensile residual axial stresses and the residual hoop stresses on the outer surface of the pipe to reduce in magnitude significantly but moderately in the parent metal and approximately half the weld region. The other half of the weld region, in the vicinity of the last weld pass, has compressive residual axial and hoop stresses, which are mostly small in magnitude, with the exception of two narrow regions with higher compressive stresses. The importance of including the effects of SSPT in the FE simulation for accurate representation of material behaviour is evident in the comparison between the residual stress curves in Figs 9 and 10.

Fig. 11, which reflects the high tensile stress distribution parts of both the axial stress field and the hoop stress field. By comparing Fig. 11 with Fig. 8, it can be seen that PWHT reduces the highest tensile residual stresses to approximately one third their original values. Another observed influence of PWHT is the more even distribution of the high tensile stress profile in the weld region and HAZ on the outside surface as well as in the weld region through half the wall thickness from the outside surface. The effect of PWHT on the residual stress field on the outside surface along the same line defined in the previous section is shown in Figs 9 and 10 for the axial stresses and the hoop stresses respectively. It can be seen that PWHT is responsible for a total reduction in the highest tensile stresses of approximately half of the original value for axial stress and two thirds for hoop stress.

3.3 3.2

Effects of PWHT

In PWHT modelling, after including the effects of SSPT, the temperature of the FE model is increased to 760 uC, followed by holding the temperature for 3 h, during which time creep stress redistribution obeys the Norton law, and finally the temperature is decreased to 20 uC. The final residual maximum principal stress distribution after PWHT is shown in J. Strain Analysis Vol. 43

Effects of the PWHT holding time and creep constants

The FE simulation of PWHT assumes that the stress field experiences redistribution during the heating and cooling stages due to the change in the stress yield value. Any redistribution due to creep during those stages is considered negligible. The redistribution during the holding stage is assumed to be due to creep, the simulation of which requires the material constants in the Norton law. In order to obtain JSA372 F IMechE 2008


Fig. 11

Residual maximum principal stress contours (MPa) at the weld region and HAZ, including the effects of SSPT, after PWHT (PM, parent metal; WM, weld metal)

values for A and n at the holding temperature of 760 uC, extrapolation is made from a temperature of 650 uC, at which A and n have been experimentally determined. Extrapolation over such a temperature range makes the values of the two constants approximate. Therefore, it is deemed apt to consider the effect on the stress field of varying the holding time as well as the values of A and n in the FE simulation for a parametric analysis purpose.

Fig. 12

287

In a series of PWHT simulations, the holding time is varied from 0 to 100 h, the value of A is increased by a factor of 10 and 100 with holding times of 3 h and 1 h respectively, and the value of n is increased from 10.836 to 11.418 and then to 12, keeping A at the baseline value with a holding time of 3 h. Figures 12 and 13 show the effect of holding time from 0 to 100 h on the residual axial stress field and the residual hoop stress field respectively, keeping

Residual axial stress variations against distance from the weld centre, along a straight line on the outside surface in the axial direction, after removing the weld crown, and after PWHT with the holding time ranging from 0 to 100 h (A 5 56.8610228 for stress in megapascals and time in hours; n 5 10.836)



288


Fig. 13

Residual hoop stress variations against distance from the weld centre, along a straight line on the outside surface in the axial direction, after removing the weld crown, and after PWHT with the holding time ranging from 0 to 100 h (A 5 56.8610228 for stress in megapascals and time in hours; n 5 10.836)

the Norton law material constants A and n at their baseline values and using five-point averaging. The reduction in the highest residual axial stresses, in Fig. 12, is approximately one third due to plasticity and a further one third of the new value of stress due to creep for 3 h. In Fig. 13, the reduction in the highest residual hoop stress is approximately half due to plasticity and a further one third of the new value of stress due to creep for 3 h. It can also be seen in both figures that generally the first 30 min of holding time induce a more substantial reduction in residual stresses than that induced by what remains of the first 3 h, which in turn is more substantial than the reduction induced by the rest of the 100 h holding time. This effect of holding time is even more pronounced in Fig. 14, which shows the peak tensile residual axial and hoop stresses on the outside surface plotted against holding time, for which the curves behave exponentially. When the plot in the figure is changed into a log–log graph, as can be seen in Fig. 15, the curves virtually become straight lines, allowing simple and approximate linear interpolation and extrapolation. The gradient of the log–log plot in Fig. 15 can be related to the stress index n of the material, and the intersection with the vertical axis of the graph can be related to the material constants A, J. Strain Analysis Vol. 43

n, and E, when the uniaxial form of the Norton law (equation (16)) is applied. The value of A in the FE simulation is increased from 56.8610228 to 56.8610227 for a holding time of 3 h and then to 56.8610226 for a holding time of 1 h. The two sets of residual stress results are identical with those obtained by keeping the value of A at 56.8610228 and increasing the holding time to 30 h and 100 h respectively. This observation of proportionality is in line with the creep behaviour governed by the Norton law. The effect of changing the value of n from 10.836 to the intermediate value of 11.418 and then to 12 has also been considered. When n is changed, keeping A at the baseline value, as can be seen in Figs 16 and 17, the reductions in the residual axial stress profile and the residual hoop stress profile respectively becomes much more substantial. Not only is the change in the residual stresses more pronounced in magnitude, but also it is more significant to the shape of the residual stress profile, making residual stresses extremely sensitive to the n value in the Norton law during PWHT. 4

DISCUSSION

The significance of residual stresses in welded structures can be attributed to a number of JSA372 F IMechE 2008


289

Fig. 14

Peak tensile residual axial and hoop stresses against PWHT holding time (A 5 56.8610228 for stress in megapascals and time in hours; n 5 10.836)

Fig. 15

Logarithm of the peak tensile residual axial and hoop stresses against logarithm of the PWHT holding time (A 5 56.8610228 for stress in megapascals and time in hours; n 5 10.836)

associated assessment risks. Induced weaknesses due to the presence of residual stresses can range from possible fabrication or transportation cracking to a faster high-temperature microstructural degradation at peak tensile stress locations in the weld region and HAZ, including the possibility of enhancing early crack growth [26]. Once operating creep conditions are prevalent in a welded power plant JSA372 F IMechE 2008

pipeline, adverse effects of the residual stress distribution usually become less significant, since high-temperature conditions are likely to reduce stress magnitudes substantially in a relatively short period of time because of creep deformation. Depending on the type of risk assessment to be conducted, a simulated residual stress distribution can be incorporated in the life assessment of a J. Strain Analysis Vol. 43

290


Fig. 16

Residual axial stress variations against distance from the weld centre, along a straight line on the outside surface in the axial direction, after removing the weld crown, and after PWHT with a holding time of 3 h and n values of 10.836, 11.418, and 12 (A 5 56.8610228 for stress in megapascals and time in hours)

Fig. 17

Residual hoop stress variations against distance from the weld centre, along a straight line on the outside surface in the axial direction, after removing the weld crown, and after PWHT with a holding time of 3 h and n values of 10.836, 11.418, and 12 (A 5 56.8610228 for stress in megapascals and time in hours)

serving component, such as a repaired pipe weldment for example, as part of a comprehensive integrity assessment approach, which may include PWHT or solid-mechanics creep damage simulation. J. Strain Analysis Vol. 43

In this study, the process of welding has been numerically simulated for a P91 steel pipe having an axisymmetric single-U butt weld with 50 beads. The FE simulation consists of a thermal analysis which JSA372 F IMechE 2008


gives a temperature history consistent with thermal expectations, such as molten zones, weld penetration into the parent metal and temperatures at the HAZ, followed by sequentially coupled structural analyses, initially without accounting for SSPT, subsequently including its effects, and finally allowing for PWHT with a range of holding times. The numerically determined residual maximum principal stresses are depicted through the pipe wall thickness and the residual axial and hoop stresses are plotted along a line on the outside surface at the weld region and the HAZ. The effects of SSPT are incorporated in the FE structural analysis to model the characteristic mechanical and metallurgical behaviour of P91 steel more accurately during the welding thermal cycles. The effects of SSPT on residual stresses are clearly demonstrated in the large differences between the values and shapes of the contours and curves of residual stresses when comparing results with and without SSPT. It can be seen in Figs 9 and 10 that the influence of SSPT on the residual stress profiles is a moderate but significant reduction in magnitude of both the residual axial stresses and the residual hoop stresses on the outer surface of the pipe in the parent material at the HAZ and beyond and also in approximately half of the weld region. The other half of the weld, in the vicinity of the last weld pass, experiences a large drop in both residual stresses, making them moderately compressive in general. It is usual for P91 welded pipes to be heat treated before being put into service in order to reduce the magnitude of the residual stress field as well as to temper the steel and to improve its ductility. The reduction in high tensile residual stresses caused by PWHT is particularly important since high tensile stresses are usually responsible for early microcracks and damage. In a previous publication [27], FE modelling of a welded steel component experiencing PWHT has allowed for the redistribution of stresses during thermal cycles due to the associated changes in the material yield stress, but it has not included the effects of creep. In another publication [23], the redistribution of stresses due to creep during PWHT holding time, obeying the Norton law, has been accounted for in the FE simulation in addition to including plasticity effects, brought about thermally, owing to a temperature-dependent yield stress. In this study, PWHT is simulated by allowing for stress redistribution due to both plasticity and creep, depicting the effect of plasticity alone and then the effect of both together. Furthermore, the creep constants in the Norton law for P91 steel are varied JSA372 F IMechE 2008

291

and so is the holding time, determining how each parameter influences the change occurring in the magnitude and shape of the most relevant residual stresses. The residual stress fields obtained from both the FE structural analyses and the FE modelling of PWHT are shown in Figs 7 to 17. It can be seen in Figs 12 and 13 that PWHT reduces the highest tensile stresses by approximately a third in the axial case and a half in the hoop case during the timeindependent temperature-dependent elastic–plastic redistribution. It reduces stresses by a further one third of the new stress profiles owing to the timeand temperature-dependent creep redistribution during the PWHT holding time. It can also be seen, by a comparison with Figs 16 and 17, that the effect of changing the value of n in the Norton law is much more pronounced than those caused by changing the value of A or the holding time. The effect of changing the value of A in the Norton law can be predicted by altering the holding time instead by the same factor, as stated in the previous section.

5

CONCLUDING REMARKS

1. Residual stresses have been numerically determined in an axisymmetric single-U multi-pass butt weld of a P91 steel pipe, taking the temperature dependences of the material properties into consideration as well as allowing for the SSPT. The PWHT has also been numerically modelled, showing how changes in PWHT holding time and material creep constants can affect the residual stress profiles. 2. The residual maximum principal stress, reflecting the superimposed tensile axial and hoop stresses, is tensile and substantial on the outside surface of the welded pipe and in its vicinity. When the SSPT is taken into account, the residual maximum principal stress moderately reduces in magnitude on the pipe outer surface at the HAZ and approximately half of the weld region; in the other half of the weld, in the vicinity of the last weld bead, the residual stresses become moderately compressive. 3. The PWHT has been shown to reduce the highest tensile residual axial and hoop stresses to approximately half the original value or even less, the reduction being caused, firstly, by plasticity due to the temperature dependence of the material yield stress and, secondly, by stress redistribution due to the creep of the material at the holding temperature of 760 uC. J. Strain Analysis Vol. 43

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4. The PWHT holding time affects the residual stresses exponentially, with more than half the reduction in the stress profile, during a holding time of 3 h, occurring in the first 30 min. During a holding time of 100 h, more than half the reduction in stresses occurs in the first 3 h. Increasing the constant A in the Norton law in the numerical analysis has the same effect as increasing the holding time on reducing residual stresses, whereas changing the value of the constant n has a much more pronounced effect.

11

12 13

14

REFERENCES 1 Yaghi, A. H. and Becker, A. A. State of the art review – weld simulation using finite element methods, 2005 (NAFEMS, Glasgow). 2 Klueh, R. L. and Nelson, A. T. Ferritic/martensitic steels for next-generation reactors. J. Nucl. Mater., 2007, 371, 37–52. 3 Hyde, T. H., Becker, A. A., Sun, W., Yaghi, A., Thomas, A., and Seliger, P. Finite element creep failure analyses of P91 large tensile cross-weld specimens tested at 625 uC. In Proceedings of the Fifth International Conference on Mechanics and Materials in Design, Porto, Portugal, 24–26 July 2006, pp. 169–170 (INEGI, Lec¸a do Balio, Portugal). 4 Hyde, T. H., Sun, W., Becker, A. A., and Williams, J. A. Creep behaviour and failure assessment of new and fully repaired P91 pipe welds at 923 K. Proc. Instn Mech. Engrs, Part L: J Materials: Design and Applications, 2004, 218(3), 211–222. 5 Yaghi, A. H., Hyde, T. H., Becker, A. A., Williams, J. A., and Sun, W. Residual stress simulation in welded sections of P91 pipes. J. Mater. Processing Technol., 2005, 167, 480–487. 6 Yaghi, A. H., Hyde, T. H., Becker, A. A., and Sun, W. Numerical simulation of P91 pipe welding including the effects of solid state phase transformation on residual stresses. Proc IMechE, Part L: J. Materials: Design and Applications, 2007, 221(4), 213–224. 7 Brickstad, B. and Josefson, B. L. A parametric study of residual stresses in multi-pass butt-welded stainless steel pipes. Int. J. Pressure Vessels Piping, 1998, 75, 11–25. 8 Deng, D. and Murakawa, H. Prediction of welding residual stress in multi-pass butt-welded modified 9Cr–1Mo steel pipe considering phase transformation effects. Comput. Mater. Sci., 2006, 37, 209–219. 9 Karlsson, R. I. and Josefson, B. L. Three-dimensional finite element analysis of temperatures and stresses in a single-pass butt-welded pipe. J. Pressure Vessel Technol., 1990, 112, 76–84. 10 Deng, D. and Murakawa, H. Numerical simulation of temperature field and residual stress in multipass welds in stainless steel pipe and comparison J. Strain Analysis Vol. 43

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with experimental measurements. Comput. Mater. Sci., 2006, 37, 269–277. Yaghi, A. H., Hyde, T. H., Becker, A. A., Williams, J. A., and Sun, W. Residual stress simulation in thin and thick-walled stainless steel pipe welds including pipe diameter effects. Int. J. Pressure Vessels Piping, 2006, 83, 864–874. ABAQUS user manual, version 6.3, 2002 (Hibbitt, Karlsson & Sorensen, Pawtucket, Rhode Island). Hyde, T. H., Sun, W., and Williams, J. A. Creep analysis of pressurized circumferential pipe weldments – a review. J. Strain Analysis, 2003, 38(1), 1–29. Bhadeshia, H. K. D. H. Martensite in steels. 2000, available from http://www.msm.cam.ac.uk/phasetrans/2000/C9/lectures45.pdf. Deng, D., Lou, Y., Serizawa, H., Shibahara, M., and Murakawa, H. Numerical simulation of residual stress and deformation considering phase transformation effect. Trans. JWRI, 2003, 32(2), 325–333. Cho, S. H. and Kim, J. W. Analysis of residual stress in carbon steel weldment incorporating phase transformations. Sci. Technol. Weld. Joining, 2002, 7(4), 212–216. Be´res, L., Balogh, A., and Irmer, W. Welding of martensitic creep-resistant steels. Weld. J., 2001, 80(8), 191-s–195-s. Taleb, L. and Petit, S. New investigations on transformation induced plasticity and its interaction with classical plasticity. Int. J. Plasticity, 2006, 22, 110–130. Wolff, M., Bohm, M., Lowisch, G., and Schmidt, A. Modelling and testing of transformation-induced plasticity and stress-dependent phase transformations in steel via simple experiments. Comput. Mater. Sci., 2005, 32, 604–610. Taleb, L. and Sidoroff, F. A micromechanical modeling of the Greenwood–Johnson mechanism in transformation induced plasticity. Int. J. Plasticity, 2003, 19, 1821–1842. Fischer, F. D., Reisner, G., Werner, E., Tanaka, K., Cailletaud, G., and Antretter, T. A new view on transformation induced plasticity (TRIP). Int. J. Plasticity, 2000, 16, 723–748. Karlsson, C. T. Finite element analysis of temperatures and stresses in a single-pass butt-welded pipe – influence of mesh density and material modeling. Engng Compututions, 1989, 6, 133–141. Berglund, D., Alberg, H., and Runnemalm, H. Simulation of welding and stress relief heat treatment of an aero engine component. Finite Elements Analysis Des., 2003, 39, 865–881. Eggeler, G., Ramteke, A., Coleman, M., Chew, B., Peter, G., Burblies, A., Hald, J., Jefferey, C., Rantala, J., deWitte, M., and Mohrmann, R. Analysis of creep in a welded P91 pressure vessel. Int. J. Pressure Vessel Piping, 1994, 60, 237–257. Hyde, T. H., Becker, A. A., Sun, W., Yaghi, A., Williams, J. A., and Concari, S. Determination of creep properties for P91 weldment materials at JSA372 F IMechE 2008


625 uC. In Proceedings of the Fifth International Conference on Mechanics and Materials in Design, Porto, Portugal, 24–26 July 2006, pp. 165–166 (INEGI, Lec¸a do Balio, Portugal). 26 Nikbin, K. Evaluating creep cracking in welded fracture mechanics specimens. Engng Fracture Mechanics, 2007, 74(6), 853–867. 27 Cho, J. R., Lee, B. Y., Moon, Y. H., and Van Tyne, C. J. Investigation of residual stress and post weld heat treatment of multi-pass welds by finite element method and experiments. J. Mater. Processing Technol., 2004, 155–156, 1690–1695.

R SMAW SSPT Sij TSOFT T T9 WM a Dfm DeE, DeP, DeT, DeTOTAL

APPENDIX Notation A, n A9, n9

A650, A760 A1 A3 C DFLUX E fm FE GTAW HAZ Mf

Ms

PM PWHT Q0


material constants in the Norton creep law material constants in the temperature-dependent Norton creep law values of A at 650 uC and 760 uC respectively temperature at the start of austenitic transformation (uC) temperature at the end of austenitic transformation (uC) specific heat capacity (kJ/kg K) distributed heat flux (W/m3) elastic modulus (MPa) martensite fraction finite element gas–tungsten arc welding heat-affected zone temperature at the end of martensitic transformation (uC) temperature at the start of martensitic transformation (uC) parent metal post-weld heat treatment activation energy (cal/mole)

DeTRP, DeVOL

e˙E, e˙P, e˙T, e˙TOTAL e˙TRP, e˙VOL

e_ , e_ CR ij e_ TRP ij f f˙ Q9(f)

k l n s, seq sy

293

gas constant shielded metal arc welding solid-state phase transformation deviatoric stresses (MPa) softening temperature (uC) temperature (uC) temperature (K) weld metal coefficient of linear thermal expansion (K21) martensite fraction increment elastic, plastic, thermal, and total strain increments respectively strain increments due to transformation plasticity and volumetric change respectively elastic, plastic, thermal, and total strain rates respectively strain rates due to transformation plasticity and volumetric change respectively creep strain rate and creep strain rate components respectively strain rate components due to transformation-induced plasticity volume proportion of the forming phase of steel rate of f function expressing the kinetics of transformation plasticity Greenwood–Johnson parameter thermal conductivity (W/m K) Poisson’s ratio stress and equivalent stress respectively (MPa) yield stress (MPa)
