constant related to the multi-axial stress state within the material.Integration of Eq.(4)under uniaxial conditions leads to:
x ?à12Ln 1à1àe àq 2eTt f
e6T
where
t f ?1
B e1t/Tr v
e7T
A ,n ,
B ,v ,q 2and /are material constants which can be obtained by curve ?tting to the uniaxial creep curves.The value of the multi-axial parameter a is obtained by ?tting the failure times of notched bar specimens,made of the PM,the WM and across the weld,to those of FE notched bar models of the PM,WM and across the weld,respectively.Procedures used to obtain these constants are given in [11]and summarized in [12].The values of the material constants are given in Table 2[12].
As the value of x approaches unity within an element,the creep strength of the material reduces very rapidly.To model the effects of damage,the modulus of elasticity,E ,is reduced as the x increases according to the relationship E i +1=(1àx i )E i ,where E i +1is the modulus of elasticity at time increment i +1and E i is the modulus of elasticity at time increment i .4.Finite element models
Geometries and dimensions of the CT models used throughout this study are shown in Fig.2.Single material CT models,shown in Fig.2(a),are the PM model and the WM model.Experimentally,it is possible to cut out PM and WM CT specimens,of a weldment,and to test them under creep conditions.However,due to the small size of the HAZ region (normally 2–4mm width),it is very dif?cult to cut out (and hence to test)HAZ CT specimens,unless simulated HAZ material is used.Therefore,only the PM and WM CT’s were modeled using the single-material model.In all cases,the initial crack length is 15.5mm and the specimen width is 32mm and,hence,the ratio a /w is about 0.48.This ratio conforms with the speci?cations of the CT specimens which were mentioned in ASTM-1457-00[13].In all of the CT models,the initial crack was located at the mid-way between the two loading point.
Two bi-material CT models were used in the current study.The ?rst model consists of PM and HAZ;namely PM–HAZ model.The second model consists of PM and WM;namely PM–WM model.In the bi-material models,the initial crack was located on the materials interface,as shown in Fig.2(b).
The three-material CT model,shown in Fig.2(c),consists of PM,HAZ,and WM.Dimensions and geometry of this model are the same as those used in the experimental program [11].The width of the HAZ in the model is 2.4mm.Three different con?gurations of the model were used,as summarized in Table 3.In the ?rst con?guration,the initial crack was located at the PM–HAZ boundary,namely 3mat-PM/HAZ model.The crack in this con?guration represents the Type IV crack in a weld.In the second con?guration,the initial crack was located at the WM–HAZ boundary,namely 3mat-WM/HAZ model.
The
Table 1
Chemical compositions of P91PM and WM,wt%[10].Element C Mn Si Cr Mo N Ni PM 0.110.360.0228.740.980.0480.12WM
0.087
1.07
0.28
8.6
1.02
0.04
–
74M.Saber et al./Engineering Fracture Mechanics 154(2016)72–82
crack in this con?guration represents the Type III crack in a weld.In the third con?guration,the initial crack was located in the middle of the sandwiched HAZ,namely 3mat-middle-HAZ.The three models were analyzed under the same loading conditions.
A ?nite element commercial package (ABAQUS)[14]was used to carry out the FE analyses.2D plane stress elements were used to model all of the CT models,see Fig.3.In order to minimize the processing time,?ne elements were used in the vicin-ity of the crack tip while coarse elements were used elsewhere.
Mesh study was carried out on single material CT models.It was found that the single material CT models experienced less mesh sensitivity when the element size was 2mm or more,see Fig.4.Therefore,the element size of 2mm was used for the area of interest ahead of the crack tip in all of the analyses.Moreover,a uni?ed CT model was created,partitioned and meshed such that it can be used for all of the analyses without the need to change any of its features.
Materials properties were implemented into the analyses by using CREEP subroutine.The CREEP subroutine works in junction with the FE analyses to calculate creep strain and creep damage using the Liu and Murakami material model.Load
Table 2
P91material constants for damage constitutive equations at 650°C (r in MPa and time in h)[12].Material A
n B
/v
q 2a
PM 1.092?10à208.462 3.537?10à177.346 6.789 3.20.3125WM 1.370?10à207.65 1.600?10à2011.4637.950 5.00.81HAZ
2.300?10à20
8.462
1.522?10à14
7.346
5.502
2.8
%
0.5
Table 3
Locations of the initial cracks in three-material models.Model
Location of the initial cracks 3mat-PM/HAZ On the PM–HAZ interface 3mat-middle-HAZ At the middle of HAZ
3mat-PM/WM
On the PM–WM interface
M.Saber et al./Engineering Fracture Mechanics 154(2016)72–82
75
model;?ne elements were used in the vicinity of the crack tip while
and boundary conditions were applied to the CT model via rigid pins which were modeled and placed at pin holes.Details of load application to CT models using rigid pins are given in[11].
5.Results and discussion
A set of FE analyses was carried out on different con?gurations of CT models using material properties for the P91weld,at 650°C.The Liu and Murakami model,given in Eq.(2),was used to calculate creep damage,x,at each integration point of each element,where x=0indicates that the material is not damaged and x=1indicates that the material is fully damaged (failure).
Throughout this study,the maximum value of x was set to0.99.This is to avoid numerical errors that could arise from using the maximum value of x=1.As the value of x approaches the critical value,0.99,at an integration point within an element,its modulus of elasticity,E,approaches zero,and hence,it cannot support any more load and is,therefore,consid-ered to be removed from the model,i.e.the element failed.When this failed element is located on the crack path,the crack length is considered to increase by the amount of the size of that?led element.
5.1.Results of single-material models
single-material CT models is shown in Fig.5.It can be seen
the PM CT model.For the WM CT model,the damage
damage accelerates the damage rate and,therefore,reduces
where x=0.99,represent the crack extension.The resulted
6.It can be seen that the CCG curves for the two models
followed by tertiary CCG.The primary CCG has not been modeled
Fig.4.Mesh sensitivity of single material model.
simplicity.It can also be seen that the failure time of the WM CT model is about 5%of that for the PM CT model.This can be attributed the brittleness of the WM compared to the PM.This brittleness can be noticed from the uniaxial behavior of the two materials which is shown in Fig.7[10].Fig.8shows that the average creep ductility of the WM is about 3.8%which is about one tenth that of the PM (%36%).This remarkable difference in failure ductility justi?es the remarkable difference in CCG,failure time between the PM and WM.
Fig.5.Damage prediction in (a)PM CT model and (b)WM CT model.
15
17
192123
0100200300400500600700
C r a c k L e n g t h (m m )
Time (h)
WM
PM
Fig.6.FE creep crack growths (CCG)for the PM and the WM CT models.
Fig.7.Creep strain curves for P91parent material and weld material.
M.Saber et al./Engineering Fracture Mechanics 154(2016)72–8277
78M.Saber et al./Engineering Fracture Mechanics154(2016)72–82
Fig.10.FE damage prediction in(a)PM–HAZ CT model and(b)PM–WM CT model.
Fig.9compares the CCG rates from the PM model to those from the WM model.It can be seen that the creep crack growth rates for the WM are more than an order of magnitude higher than those for the PM model.It is worth mentioning that it is impractical to attain the damage shown in the WM case as the specimen yield when the crack length reaches a value where the stress in the vicinity of the crack is more than the yield stress of the CT material.
5.2.Results of bi-material models
Two bi-material CT models,i.e.the PM–HAZ model and the PM–WM model,were analyzed under the same loading conditions.For the P91weldment,it was found that the uniaxial tensile creep strength of the WM is higher than that of the PM and the creep strength of the PM is higher than that of the HAZ[12].Consequently,each of the studied bi-material models consists of a stronger material and a weaker material.Fig.10shows the damage predicted in the
M.Saber et al./Engineering Fracture Mechanics154(2016)72–8279
PM–HAZ and in the PM–WM models,respectively.Higher damage can be seen in the vicinity of the crack tip in the weaker material,i.e.the HAZ in the PM–HAZ model and the PM in the PM–WM model.Therefore,it can be said that for the bi-material models,the creep strength of the weaker material largely affects the creep crack growth and,hence,the failure life of the model(and the weldment).Fig.11compares the FE CCG for the bi-material model to that of the single-material models.It can be seen that the CCG of the PM–WM models is the same as that of the PM model.Furthermore,although the CCG for the PM–HAZ model has the same trend as that for the PM model,the failure time for the PM–HAZ model is about one half of that for the PM model.
Fig.12compares the CCG rates for the single-material and bi-material models.It can be seen that,except that for the WM model,the CCG rates for the all of the other models are similar.
5.3.Results of three-material models
For the three-material CT models,three FE analyses were carried out under the same loading conditions but with different crack position(see Table3).The FE analyses were left running until the time increment,within the analyses,dropped to impractical values then the analyses was stopped by the user.
Fig.13compares the FE predicted damage,and hence the CCG,in the three-material CT models.In all of the three cases,higher damage can be seen in the HAZ while low damage appears in the PM but almost no damage appears in the WM.The fully damaged elements,i.e.where x=0.99,are considered as CCG.The CCGs for the three-material models are shown in Fig.14.Results of the single-material models,i.e.PM and WM,are also included.It can be seen that,for the bi-material model,the CCG of a crack that is located on the interface of two materials is the same as the CCG for the single-material model made of the weaker material.For the three-material models,as seen in Fig.14,the CCGs for both the3mat-PM/HAZ and the3mat-WM/HAZ models are determined by the HAZ properties.These CCGs are similar to each others and different from those of the PM or the WM.This can be attributed to the relatively small width of the HAZ, 2.4mm,and the constancy of material properties across the HAZ.It can also be seen that the failure time for the 3mat-PM/HAZ and for the3mat-WM/HAZ are less than that for the3mat-middle-HAZ model.This is because the location of initial crack on material interface accelerates its growth,due to high stress triaxiality found at the materials interface. The creep crack growth rates for the three-material models are similar to that for the PM model,see Fig.15.In Fig.15,it can be also seen that the CCG rate for the3mat-middle-HAZ model is slightly less than that for3mat-PM/HAZ and 3mat-WM/HAZ models.
6.Effects of the width of the HAZ
In order to study the effects of the HAZ width on CCG,a set of FE analyses was carried out on 3mat-middle-HAZ models with different values of HAZ width.Effects of the HAZ width on the specimens’life time are also addressed.The HAZ widths were taken as 0.4,0.8,1.2,1.6,2.0,2.4,4.8mm.The CCG’s for those analyses are shown in Fig.16along with that for a single-material model which made entirely of HAZ and that for the two bi-material models.Firstly,it can be seen that the CCG rates are high,and hence life time is low,for models of the smallest and the largest HAZ widths,0.4and 4.8mm,respectively.Moreover,the life time for both models is about 406h.The time for crack initiation for the HAZ =0.4model is about 82%of the model life time and hence the time from the crack initiation to complete failure is 18%.This justi?es the high CCG rate
15.5
16.517.518.519.520.521.522.523.5
100200300400500600700
C r a c k L e n g t h (m m )
Time (h)
PM
WM Middle HAZ PM-HAZ WM-HAZ
Fig.14.FE creep crack growth of three-material CT models compared to that for single-material models.
1.0010.00
Δt
PM WM
Middle HAZ 3mat -PM-HAZ 13.FE damage prediction in three-material CT models when the crack tip was located (a)on the PM–HAZ interface,(b)at the middle of the HAZ and the WM–HAZ interface.
80M.Saber et al./Engineering Fracture Mechanics 154(2016)72–82
Fig.16.Creep crack growth for different HAZ width.
Fig.17.Specimens life time against the width HAZ in3mat-middle HAZ specimens.
for this model during the tertiary region.For the HAZ=4.8mm model,the time for crack initiation is about64%of the life time and hence,the time from crack initiation to complete failure is36%of the life time.Here the alarming of complete fail-ure lasts for36%of the model life time which is two times that for the HAZ=0.4model.The short life time and steeper CCG of HAZ=0.4mm can be attributed to the highest stress triaxiality acting on the tinny sandwiched HAZ,between PM and WM. While for HAZ=4.8mm the effect of the plastic strain was signi?cant and when combined with the creep strain it results in higher CCG rate with shorter life time.The effect of the accumulated plastic strain increased by increasing the width of the HAZ.This is clearly evident when the model is entirely made of HAZ;the model achieves the lowest life time of all the middle HAZ tested models,see Fig.17.Fig.16also shows that for the models of HAZ=0.8–2.4mm,the average life time is451h and the time for the crack initiation is about69%of the life time.
When compared to middle-HAZ models,the bi-material models occupy the two extremes of Fig.16.The minimum life and the highest CCG rates are achieved when the crack was located on the PM/HAZ boundary.While the maximum life is obtained when the crack is located on the PM/WM boundary.To sum up,if the crack is found to exist on the PM/HAZ (i.e.type IV)it affects the failure time greatly when compared to other situations.Moreover,the thin HAZ can resist CCG for long time compared to other middle HAZ models;crack initiation is long.However,once the cracking starts it lasts for shorter time compared to the other middle HAZ models.
7.Conclusions
The effects of the location of cracks on creep crack growth,in a P91weldment,were investigated.Single-material,bi-material and three-material CT models were used.The Liu and Murakami damage model was used to predict the damage, and hence the CCG,in the models.For the bi-material and three-material models,the initial cracks were located on the boundary of the two materials of different creep strengths.The results obtained showed that,the CCG and the CCG rates in the WM CT models are much higher than those in the PM CT models.This may be attributed to the creep brittleness of the WM when compared to that of the PM.For the bi-material models,since the initial crack was located on the material interface,the CCG and CCG rates are similar to those for the weaker material of the two materials.At the material interface, high triaxiality exists and,therefore,enhances the CCG and CCG rates.Crack tip is subjected to further triaxiality due plane
82M.Saber et al./Engineering Fracture Mechanics154(2016)72–82
strain conditions arose at the crack tip.This cumulative triaxiality at the crack tip makes the material behaving at a very brittle material.This is obvious in three-material models where the CCG and the CCG rates for the3mat-PM/HAZ and for the3mat-WM/HAZ are higher than those for the3mat-middle-HAZ models.
Damage distribution depends on material properties.The results shown in Figs.5,10and13for single materials,bi-material and three-material models,respectively,indicate that the damage is uniformly distributed in both the PM and HAZ materials while it is localized in the WM.This localization of damage leads to the increase in the CCG and CCG rates and decrease in the failure life of the WM CT model.
The thickness of the HAZ greatly affects the CCG,CCG rates and the life time of weldment.It can be concluded that the very small width of the HAZ causes the model to fail early.This is due to the high triaxiality found in that thin HAZ region. However,on the other hand,the substantial large width of the HAZ shortens the life time of weldment.This can be attributed to the combined effect of high plastic strain and creep strain in the vicinity of the crack tip.It is recommended to control the width of the HAZ such that the life time of the weldment is maximum.
In order to evaluate the simulated behavior of the multiple material weld CT specimens for validation,it is important to use small scale testing methods to determine the full or some of creep properties across the heat-affected zones in welds, using for example,the nano-indentation tests(e.g.[15])or miniature specimen testing(e.g.[16]).The authors have been developing miniature creep specimen creep test techniques,and intend to use such methods(e.g.[17])in the future to deter-mine the full creep strain curves using the specimens removed from weld metal and heat-affected zone(s)in power plant main steam pipes.
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