In-plane and out-of-plane unified constraint-dependent creep crack growth rate of 316H steel

In-plane and out-of-plane uni?ed constraint-dependent creep crack growth rate of 316H

steel

H.S.Ma,G.Z.Wang ?,S.Liu,S.T.Tu,F.Z.Xuan

Key Laboratory of Pressure Systems and Safety,Ministry of Education,East China University of Science and Technology,Shanghai 200237,China

a r t i c l e i n f o Article history:

Received 24June 2015

Received in revised form 19October 2015Accepted 13January 2016

Available online 28January 2016Keywords:

In-plane creep constraint

Out-of-plane creep constraint Creep crack growth rate Specimen geometry Equivalent creep strain

a b s t r a c t

The uni?ed characterization parameter A c of in-plane and out-of-plane creep constraints along three-dimensional crack front in different specimen geometries has been analyzed.Based on the parameter A c and the experimental creep crack growth rate data of different specimen geometries in the literature,the in-plane and out-of-plane uni?ed constraint-dependent creep crack growth rate equation of 316H steel has been obtained.The predicted creep crack growth rate by using the equation for each specimen agrees well with the experimental data.Based on the two-parameter C ?àA c concept,the equation may be used in creep crack growth life assessments incorporating in-plane and out-of-plane constraint effects.

ó2016Elsevier Ltd.All rights reserved.

1.Introduction

Under creep conditions,a lot of experimental and theoretical evidences have shown that the crack-tip constraint can affect creep crack growth (CCG)rate [1–11].The constraint includes in-plane and out-of-plane constraints.The in-plane con-straint is directly affected by crack depth,length of the un-cracked ligament and loading con?guration of specimens,while the out-of-plane constraint is affected by the specimen thickness.To accurately predict creep life and achieve structural integrity assessments for high temperature components,it is important to quantify the creep crack-tip constraint levels.The creep crack-tip stress and strain rate ?elds are often described by the C ?àQ two-parameter under plane strain or plane stress conditions and the Q was used to quantify the constraint [12–14].The effect of in-plane constraint on CCG has been examined by using Q parameter [13].Combined the C ?àQ two-parameter concept with the NSW model,Nikbin [3]inves-tigated the effect of constraint on the CCG rate.Based on the C ?àQ two-parameter concept and ?nite element analysis,Bet-tinson et al.[14]examined the effect of specimen type and load level on the Q from short to long term creep conditions for elastic-creep materials.Recently,Zhao et al.[8]quanti?ed the constraint effect induced by specimen geometry on creep crack growth behavior in P92steel by using the parameter Q .The two-parameter C (t )àT Z and the three-parameter C (t )àT Z àQ descriptions for crack-tip ?elds were proposed by Xiang et al.[15]and Xiang and Guo [16]for small and exten-sive creeping,respectively.The T Z is an out-of-plane constraint factor,and the in-plane constraint is characterized by the parameter Q .In the de?nition of the Q parameter above,the HRR stress ?eld is usually taken as the reference ?eld.

In the previous work of authors [17],it has been suggested that the HRR stress ?eld may not be suitable to be used as a reference ?eld for de?ning the constraint parameter under creep condition due to larger crack-tip blunting and creep damage.Based on the reference ?eld of the standard C(T)specimen in plane strain with deep crack (a/W =0.5)and high https://www.360docs.net/doc/ce16335152.html,/10.1016/j.engfracmech.2016.01.0170013-7944/ó2016Elsevier Ltd.All rights reserved.

?Corresponding author.Tel./fax:+862164252681.

E-mail address:gzwang@https://www.360docs.net/doc/ce16335152.html, (G.Z.Wang).

constraint,a constraint parameter R was proposed to characterize the creep crack-tip constraint,and the constraint effects induced by the crack depths [17,18],specimen thicknesses [19]and loading con?gurations [20]have been investigated in detail.For the convenience of application,a load-independent creep constraint parameter R ?has been de?ned by modifying the parameter R [21].Based on the parameter R ?,the characterization and correlation of two-dimensional and three-dimensional creep constraint between axially cracked pipelines and test specimens were studied [22,23],and the constraint-dependent CCG rate equations were established for Cr–Mo–V steel [21].

The creep constraint parameters described above only can quantify the in-plane or out-of-plane constraint separately,and they may not characterize both of them and overall level of constraints.In actual high-temperature components,there exist both in-plane and out-of-plane constraints.In order to describe their interaction and the overall level of constraints,a uni?ed creep constraint parameter which can characterize both in-plane and out-of-plane creep constraint together is

Nomenclature a crack length

_a creep crack growth rate

_a 0creep crack growth rate of the standard specimen A constant in Norton creep model

A c

uni?ed characterization parameter of in-plane and out-of-plane creep constraint A CEEQ area surrounded by equivalent creep strain isoline

A p uni?ed characterization parameter of in-plane and out-of-plane constraint

A ref area surrounded by equivalent plastic strain isoline at fracture measured in a standard test,or the area sur-rounded by equivalent creep strain isoline in a standard specimen

B specimen thickness B N

net specimen thickness

C 1,C 2constants in constraint-dependent CCG rate equations C (t )C (t )integral

C ?C ?integral analogous to the J integral E Young’s modulus

h stress triaxiality factor

L half length of the test specimens.

n stress exponent in Norton creep model P applied load

Q constraint parameter

R creep constraint parameter

R ?load-independent creep constraint parameter Tz out-of-plane constraint parameter t creep time

t red creep redistribution time v Poisson’s ratio W specimen width

z distance along the crack front

_e 0creep strain rate at normalizing stress _e

c creep strain rate

e c equivalent creep strain r 0

normalizing stress

Abbreviations 2D two-dimensional 3D three-dimensional CCG creep crack growth C(T)compact tension CS(T)C-shaped cracked tension CEEQ equivalent creep strain in ABAQUS code DE(T)double-edge notched tension FEM ?nite element method HRR Hutchinson–Rice–Rosengren M(T)middle cracked tension SE(B)single edge-notched bend SE(T)single-edge notched tension TF stress triaxial factor

H.S.Ma et al./Engineering Fracture Mechanics 155(2016)88–10189

90H.S.Ma et al./Engineering Fracture Mechanics155(2016)88–101

required.In the case of steady-state creep,the power law creep is analogous to power law plasticity,and the equivalent plas-tic strain is analogous to equivalent creep strain.By analogy to the uni?ed constraint parameter A p based on the equivalent plastic strain[24–27],a uni?ed creep constraint parameter A c based on crack-tip equivalent creep strain was de?ned in the previous work of authors[28]:

A c?A CEEQ=A ref at t=t red?1e1Twhere A CEEQ is the area surrounded by the equivalent creep strain(e c)isoline ahead of a crack tip and A ref is the reference area surrounded by the e c in a standard reference specimen,t is creep time,and the t red is stress redistribution time.The A c rep-resents a ratio of the A CEEQ of a specimen to the reference area A ref at the same e c isoline,the same creep time t/t red=1and C?level.An increase in the parameter A c implies loss of constraint.The analysis results in C(T)specimens in the previous work [28]show that a monotonic correlation line between the CCG rate and the parameter A c can be formed regardless of the in-plane and out-of-plane constraints.This indicated that the parameter A c may be equally sensitive to both in-plane and out-of-plane constraints,and it is a uni?ed characterization parameter of in-plane and out-of-plane creep constraint.It also has been proved that the parameter A c is independent on the choice of e c isolines and load level C?,and for t/t red>0.7the A c essentially does not change with the creep time t/t red.These features of the parameter A c may bring convenience for its use.However,the parameter A c along three-dimensional(3-D)crack fronts in a wide range of specimen geometries with dif-ferent in-plane and out-of-plane constraints has not been fully analyzed,and its applicability for establishing a wide range of in-plane and out-of-plane uni?ed constraint-dependent CCG rate equations for different materials has not been widely investigated.

In this work,extensive three-dimensional?nite element analyses have been conducted for six types of test specimen geometries with different in-plane and out-of-plane constraints in the literature[29],and the uni?ed characterization parameter A c of in-plane and out-of-plane creep constraints was calculated and analyzed for these specimens.Based on the parameter A c and the experimental creep crack growth rate data in the different specimen geometries in the literature [29],the applicability of the parameter A c for establishing a wide range of in-plane and out-of-plane uni?ed constraint-dependent CCG rate equation for316H steel was investigated.

2.Finite element model and numerical procedures

2.1.Material

The material used in this work was the316H stainless steel which is the same as that used in the literature[29].An elastic–plastic-power law creep material model was used,and the creep strain rate_e is given by:

r n?_e0er=r0Tne2T_e

c?A

The creep and elastic–plastic material parameters of316H stainless steel at550°C in the literature[29]were used.The values of n and A are11.3and3.0?10à34MPaàn hà1,respectively.The Young’s Modulus E of the steel at550°C is140GPa, and the Poisson’s ratio v was taken to be0.3.The normalizing stress r0was taken as170MPa,and the corresponding _e0=4.80Eà9hà1.The true stress–strain curve of the316H stainless steel at550°C is shown in Fig.1[29].

2.2.Finite element model for test specimens

The six different types of cracked specimens with a wide range of geometries and sizes(including a wide range of in-plane and out-of-plane constraints)used in the literature[29]for measuring creep crack growth rate of the316H steel have been modeled using3-D FE analyses.The test specimens include compact tension(C(T))specimens of width W=50mm and

W=25mm(denoted as1T and1/2T specimens,respectively),C-shaped cracked tension(CS(T))specimen with X/W=0.5

H.S.Ma et al./Engineering Fracture Mechanics155(2016)88–10191 and R i/R0=0.5,single-edge cracked bend(SE(B))specimen with2L=4W,single-edge cracked tension(SE(T))and middle cracked tension(M(T))specimens with2L=4W.Schematic illustrations of the specimens are shown in Fig.2.Detailed geometries of specimens and the information on creep crack growth tests in the literature[29]are listed in Table1.The rel-ative initial crack depth a/W is ranged from0.17to0.53.Some1T C(T)specimens had40%side-grooves and M(T)specimens had no side-groove.All other specimens had20%side-grooves[29].The B N in Table1is the thickness of specimens with side-

Fig.2.Schematic illustrations of specimens:(a)C(T),(b)CS(T),(c)SE(B),(d)SE(T),and(e)M(T).

the element size.Hence the minimum mesh size of 40l m around the crack tips was used in this study.The local mesh dis-tribution around the crack tip is shown in Fig.3(f).

The FE analyses were performed using large-displacement with the NLGEOM option ON,and the blunting of crack tip was taken into consideration.The contour integral C (t )and C ?were evaluated by using the in-built ABAQUS routines.Ten con-tours were set around the crack tips.The C ?is usually path-independent,and C (t )is not path-independent,but with increas-ing time the C (t )gradually becomes path-independent.The C (t )was de?ned as the value from the sixth contour near the crack tip for the different specimens with the same size of meshes and contours (The C (t )de?ned by other contours has sim-ilar value as the sixth contour).The areas surrounded by the equivalent creep strain e c isolines,C ?and constraint parameter A c ahead of crack tips were calculated.2.3.Calculation of creep constraint parameter A c

To establish in-plane and out-of-plane uni?ed constraint-dependent creep crack growth rate for 316H steel at 550°C and at steady-state creep by using the constraint parameter A c ,the three-dimensional FEM calculations were conducted to obtain values of the parameter A c along crack fronts for the test specimens with a wide range of geometries and sizes (a wide range of in-plane and out-of-plane constraints)in Table 1.The parameter A c was calculated by using Eq.(1),and the area A CEEQ sur-rounded by the equivalent creep strain e c (The e c is denoted as CEEQ in ABAQUS code)isoline at creep time t/t red =1and a given C ?level is calculated by FEM for each specimen in Table 1.In order to calculate the A CEEQ accurately for characterizing crack-tip constraints,it must ensure that the e c isolines ahead of a crack-tip at t/t red =1and a given C ?level do not connect those in loading region near the surface of specimens [28].Fig.4gives typical equivalent creep strain e c isolines in the middle plane of the 1T CT1specimen at an applied load of 20kN and MT1specimen at an applied load of 27.5kN and creep time t/t red =1in Table 1.It can be seen that for e c P 0.015in the 1T CT1specimen and MT1specimen,the e c isolines at crack-tip region cannot connect to loading region near specimen surface.The area A CEEQ can characterize creep crack-tip constraint,and a larger A CEEQ indicates constraint loss [28].

In this work,the A CEEQ surrounded by e c =0.02isoline in Eq.(1)was calculated at the same creep time t/t red =1and the tested load levels for the various test specimens with different in-plane and out-of-plane constraints in Table 1.The speci-men with W =2B =50mm and a /W =0.5in plane strain was chosen as the standard specimen,and the reference area A ref surrounded by the e c =0.02in the standard specimen was calculated at t/t red =1and the C ?level which is the same as the C ?level for each specimen at the tested load in Table 1.It should be noted that the parameter A c is independent on the choice of e c isolines and load level C ?[28].

3.Distributions of constraint parameter A c along 3-D crack fronts

The constraint is generally regarded as the resistance of a structure against plastic or creep deformation.The locations with lower constraint along crack fronts may have larger creep deformation.Fig.5gives the distributions of equivalent creep strain (CEEQ)along crack fronts for two typical specimens of 1T CT2specimen at an applied load of 26.5kN and MT1spec-imen at an applied load of 27.5kN in Table 1.The sizes of three-dimensional creep deformation zone may re?ect the overall constraint level composed of in-plane and out-of-plane constraints in the specimens.The largest creep strain zone along crack fronts occurs at the locations near the free surfaces,which indicates the lower constraint there.This result is similar to that of the constraint parameter A p based on crack-tip equivalent plastic strain within elastic–plastic fracture mechanics in the literature [31].

Table 1

Detailed geometries of specimens and information on creep crack growth test [29].

Test name

P (kN)W (mm)B (mm)W/B B N /B a/W (mm)Test duration (h)1T C(T)

CT1205024.9 2.010.810.531221CT226.55025 2.000.600.45323CT3275025 2.000.600.453781/2T C(T)CT18.6726.213 2.020.800.46856CT28.5926.212.9 2.030.800.46933CS(T)CST 16.52525 1.000.800.233830M(T)

MT127.58.988.95 1.00 1.000.34772MT235.58.998.94 1.01 1.000.17678MT3148.99 4.5 2.00 1.000.33308MT4178.98 4.46 2.01 1.000.17655SE(B)SB192512.5 2.000.800.33641SB27.62512.5 2.000.800.336522SE(T)

ST1302512.5 2.000.800.38379ST2352512.5 2.000.800.31601ST3

32

25

12.5

2.00

0.80

0.35

525

92H.S.Ma et al./Engineering Fracture Mechanics 155(2016)88–101

Fig.6shows the distributions of the constraint parameter A c along 3-D crack fronts (specimen thickness)for all test spec-imens in Table 1.It shows that the A c in the center region (from z /B =0to about 0.3)of all specimens has uniform distribution and lower value,and that in the region near free surface (from z /B =0.33to 0.45)has protuberant distribution and higher value.The protuberant distribution is related to the larger creep deformation zone in Fig.5.These results imply that the cen-ter region of all specimens has high and uniform constraint,and the lower constraint occurs at the region near free surface.Fig.6also indicates that the specimen order in terms of crack-tip constraint from high to low is 1T C(T),1/2T C(T),CS(T),SE (B),SE(T)and M(T).For the same type of specimen,the constraint increases with increasing specimen thickness B (out-of-plane constraint)and crack depth a /W (in-plane constraint).The specimens with the same geometry and size have the same constraint.The results in Fig.6further show that the parameter A c is sensitive to both in-plane and out-of-plane creep con-straints,and is a uni?ed creep constraint

parameter.

3.Typical 3D FE meshes for specimens:(a)C(T),(b)M(T),(c)CS(T),(d)SE(B),(e)SE(T)and (f)local meshes around the crack

4.In-plane and out-of-plane uni?ed constraint-dependent creep crack growth rate of 316H steel

Under creep conditions,a lot of experimental and theoretical evidences have shown that the geometry constraint of spec-imens can affect creep crack growth (CCG)rate [1–11].For a given C ?value (creep fracture mechanics parameter),the model predications showed that the CCG rates in plane strain are signi?cantly greater than those in plane stress [3].Some exper-imental results have shown that there is an effect of specimen thickness on the CCG rate [2,6],and the specimens with larger thickness exhibit higher CCG rate.It also has been found that at the same C ?value the creep crack growth rates measured in the middle tension (M(T))specimens are lower than those obtained from deep crack compact tension (C(T))specimens for the austenitic stainless steels [1]and ferritic steels [4].Yokobori et al.proposed a parameter Q ?for correlating creep crack growth rate [10,11,32–36],and their work showed that the creep crack growth rate for a thick specimen is higher than that of a thin specimen [10,34].This geometry constraint effect is related to multi-axial stress state around the creep crack-tip,and was de?ned as ‘structural brittleness’[11,34].The effect of the ‘structural brittleness’on CCG rate of materials could be analyzed by a stress triaxial factor TF [11,34,35].The creep ductility and constraint effects can be estimated by using the parameter Q ?[33,34].Because the geometry constraint of specimens can affect CCG rate,the use of the C(T)specimen with high constraint for testing and analysis is expected to provide a safe upper bound to the CCG rate of the material [37].Thus,a wider range of test geometries is needed in order to apply the results from experimental laboratory test specimens to the range of conditions experienced by engineering components with different constraint levels under service conditions [37].The geometry related g factor for determining experimental C ?parameter for a range of specimen geometries (C(T),SE(T),SE(B),DE(T),M(T)and CS(T))has been examined and presented [36,37].This will allow a user to determine values of C ?for a wide range of cracked geometries.However,to accurately predict the creep life and achieve structural integrity assessments for high temperature components with any constraint levels (it includes both in-plane and out-of-plane con-straints),it may be important to quantify the creep crack-tip constraint levels in specimens or cracked components by using appropriate constraint parameter and obtain constraint-dependent creep crack growth rate of materials,and to develop methodology of creep life assessments based on two-parameter creep fracture mechanics.

In the recent work by Kim et al.[29],the effects of specimen geometry and loading mode on CCG rate of 316H steel have been investigated.In present paper,the uni?ed characterization parameter A c of in-plane and out-of-plane creep constraints along three-dimensional crack front in different specimen geometries used in the work of Kim et al.[29]was analyzed.Based on the parameter A c and the experimental CCG rate data of the different specimen geometries,the in-plane and out-of-plane uni?ed constraint-dependent creep crack growth rate equation of 316H steel was investigated.The relations between

the

isolines in the middle plane for 1T CT1specimen at an applied load of 20kN 1.

equivalent creep strain (CEEQ)along crack fronts for 1T CT2specimen at an applied load of 26.5kN in Table 1at creep time t/t red =1.

0.0

0.1

0.20.30.4

0.5

1.0

1.21.41.61.8

2.02.2A c

z/B (z/B n )

1T CT1 1T CT2 1T CT3 1/2T CT1 1/2T CT2 CST SB1 SB2 ST1 ST2 ST3 MT1 MT2 MT3 MT4

Fig.6.The distributions of the constraint parameter A c along thickness for all test specimens in Table

experimental creep crack growth rate da/dt and C?for all the specimens of the316H steel in Table1are shown in Fig.7[29]. The CCG rate is usually expressed in the form[38],

_a?D

0C?qe3T

Table2

The CCG rate correlation equations in the tested C?range for different specimens of316H steel.

Specimen CCG rate equation Tested C?range(MPa m hà1) 1T C(T)CT1_a?15:38C?0:84 4.78Eà66C?69.58Eà5

CT2,CT3_a?7:84C?0:9 1.27Eà46C?64.77Eà3?TC(T)CT1,CT2_a?7:17C?0:849.53Eà66C?61.26Eà3 CS(T)CST_a?9:47C?0:779.61Eà76C?62.90Eà5 M(T)MT1_a?7:27C?0:85 2.30Eà56C?61.00Eà3

MT2_a?18:8C?0:95 3.84Eà56C?61.57Eà3

MT3_a?5:82C?0:85 6.97Eà56C?67.39Eà4

MT4_a?9:53C?0:87 2.56Eà56C?64.91Eà4 SE(B)SB1_a?85:82C?1:12 1.11Eà56C?63.01Eà3

SB2_a?192:2C?0:98 5.76Eà76C?61.04Eà5 SE(T)ST1_a?5:22C?0:858.46Eà56C?68.79Eà4

ST2_a?25:09C?0:85 1.38Eà56C?61.65Eà2

ST3_a?14:64C?0:84 2.98Eà56C?62.63Eà4

Table3

The applied load P,C?and average constraint parameter A c along crack front for each specimen.

Specimens Applied load P(KN)C?(MPa m hà1)A c

1T C(T)CT1208.68?10à5 1.16 CT226.5 1.84?10à4 1.31

CT327 2.29?10à4 1.32

1/2T C(T)CT18.67 4.14?10à4 1.44 CT28.59 3.58?10à4 1.45

CS(T)CST16.5 3.62?10à5 1.49 M(T)MT127.5 6.42?10à5 1.81 MT235.5 4.61?10à5 1.86

MT314 3.14?10à5 1.83

MT417 2.41?10à5 1.91

SE(B)SB19 3.67?10à5 1.49 SB27.67.93?10à6 1.48

SE(T)ST130 2.48?10à5 1.69 ST235 1.09?10à5 1.56

ST332 1.48?10à5 1.51 96H.S.Ma et al./Engineering Fracture Mechanics155(2016)88–101

where _a

is CCG rate and in mm h à1,and C ?is crack-tip fracture parameter and in MPa m h à1.The D 0and q are material con-stants which are usually measured experimentally.From the experimental results in Fig.7[29],the da/dt -C ?correlation equations in tested C ?range were ?tted for all specimens,as shown in Table 2.When the constraint effect is considered,the CCG rate can be expressed as a function of C ?and constraint parameter.In this study,the constraint parameter A c is used

and the constraint dependent CCG rate _a

can be described in the following form [13],_a

?_a 0f eA c Te4T

where _a

0is the CCG rate from the standard specimen with high constraint,and f eA c Tis a function of A c .

H.S.Ma et al./Engineering Fracture Mechanics 155(2016)88–10197

Fig.6shows that the constraint A c is not uniform along3-D crack front.In the3-D constraint analysis,in addition to the constraint parameter being evaluated at mid-thickness of a specimen,an average value of constraint parameter along the crack front also is usually determined to characterize the average constraint level[12,39].The average constraint may re?ect overall constraint level and incorporate both in-plane and out-of-plane constraint effects.Thus,the average value of the con-straint parameter A c along the crack front(specimen thickness)for each specimen in Table1was calculated and tabulated in Table3.The applied load and C?level for each specimen are also listed in Table3.

According to the previous studies[28],the f(A c)in Eq.(4)can be taken as the form of the C1eA cTC2,then the Eq.(4)can be written as:

_a?C

1eA cTC2_a

0e5T

The constants C1and C2can be determined by?tting the relation curve between the CCG rate ratio_a=_a0and A c on log–log scale.The constraint parameter A c was used to establish the constraint-dependent CCG rate equations.Because the geometry and crack depth of the1T CT1specimen meet the test standard,the high constraint1T CT1specimen was taken to be the stan-dard reference specimen and its CCG rate was taken to be_a0=15.38C?0.84(Table2).The CCG rate ratio_a=_a0can be calculated by the equations in Table2at different C?values(It can be typically taken to be1?10à5,1?10à4and1?10à3MPa m hà1) for different specimens,and the corresponding A c for the specimens can be calculated by Eq.(1).It should be noted that the constraint parameter A c is independent on the choice of e c isolines and load level C?[28].So the constraint parameter A c can be calculated at different C?.The relations calculated between the CCG rate ratio_a=_a0and A c on log–log scale for different spec-imens at three C?values are shown in Fig.8.The power law relation can be used to?t the curve and the Eq.(5)can be written as:

_a=_a

0?1:895eA cT

à2:667e6Twhere_a0=15.38C?0.84is for the reference specimen.It can be seen that a monotonic correlation line between the da/dt and A c on log–log scale is formed for the three C?levels for all specimens with different geometries and sizes,loading modes and crack sizes.This further suggests that the parameter A c is regardless of the in-plane and out-of-plane constraints and load levels,and is a uni?ed characterization parameter for a wide of specimen geometries with different in-plane and out-of-plane constraints.However,the results in the previous work of authors[28]have showed that compared with the parameter A c,the constraint parameters R?and stress triaxiality h based on crack-tip stress?elds cannot form a monotonic correlation

98H.S.Ma et al./Engineering Fracture Mechanics155(2016)88–101

H.S.Ma et al./Engineering Fracture Mechanics155(2016)88–10199 line with the CCG rates for the specimens with different in-plane and out-of-plane constraints due to a larger scatter in data. The reason for this is that the parameters R?and h based on crack-tip stress?elds can effectively incorporate the in-plane creep constraint effect,but they are inadequate for capturing the out-of-plane creep constraint[28].

In CCG tests and life assessments,the CCG data in different specimen geometries may be measured and used,and high-temperature components may have different degrees of in-plane and out-of-plane constraints.Thus,it is necessary to estab-lish in-plane and out-of-plane uni?ed constraint-dependent creep crack growth rate for a material,and it may be used in accurate creep life assessments for specimens or cracked components with different in-plane and out-of-plane constraint levels.Because the parameter A c based on crack-tip equivalent creep strain can effectively capture in-plane and out-of-plane creep constraints[28],the Eq.(6)represents the relation between the CCG rate and the overall constraint level com-posed of in-plane and out-of-plane constraint for the316H steel.For a specimen or component with a certain geometry and loading mode,as long as the constraint parameter A c is calculated under an arbitrary loading level C?using FE analysis,the in-plane and out-of-plane uni?ed constraint-dependent-dependent CCG rate can be predicted at different C?levels using Eq.

(6).The applicability of the Eq.(6)for components with different in-plane and out-of-plane constraint levels needs to be fur-ther investigated.

https://www.360docs.net/doc/ce16335152.html,parison of predicted CCG rates with experimental data

For analyzing and validating the uni?ed constraint-dependent CCG rate equation in Eq.(6),the predicted CCG rates by Eq.

(6)can be compared with the experimental data of each test specimen in Table1.As long as the constraint parameter A c for each specimen and the normalized CCG rate(_a0?15:38C?0:84in Table2)for reference specimen are obtained,the CCG rate_a for each specimen in the Table1can be predicted at different C?levels by using Eq.(6)(_a?1:895eA cTà2:667_a0).The average A c value along crack front for each specimen in Table3was used in Eq.(6),and Figs.9–11show the predicted CCG rates and their comparisons with experimental data.Fig.9shows that the comparison of predicted CCG rates with experimental data for1T C(T)specimen(CT1,CT2and CT3)and1/2T C(T)specimen(CT1and CT2).It shows that the predicted da/dt-C?line for each specimen agrees well with the experimental data.Fig.10shows that the comparison of predicted CCG rates with exper-imental data for CS(T)specimen and SE(B)specimen(SB1and SB2).The predictions of CS(T)specimen and SB1specimen agree well with the experimental data.But the test data of the SB2specimen(Fig.10(b))are located above the predicted line. This may be caused by faster CCG rate during long term test.Table1shows that the test duration of the SB2specimen

100H.S.Ma et al./Engineering Fracture Mechanics155(2016)88–101

(6522h)is the longest at lower load,and the corresponding CCG rate data are mainly located at lower C?region less than 1?10à5MPa m hà1(Fig.10(b)).It has been shown that at lower C?region,the creep fracture mechanism of the316H steel [40]and the Cr–Mo–V steel[6]is creep brittle fracture,and the corresponding CCG rate is higher than that at middle-high C?region.Fig.11shows that the comparison of predicted CCG rates with experimental data for SE(T)specimen(ST1,ST2and ST3)and M(T)specimen(MT1and MT2).It can be seen that the predicted CCG rates for most specimens also agree well with the experimental data.The experimental data of ST1specimen are below the predicted line.This may be caused by exper-imental error.Because the three specimens ST1,ST2and ST3have similar sizes,they should have similar constraint levels and CCG rates.

Figs.9–11indicate that the uni?ed constraint-dependent CCG rate equation in Eq.(6)based on the constraint parameter A c and C?can predict the CCG rates of specimens with different in-plane and out-of-plane constraints.It can be inferred that the Eq.(6)may also be used to predict the CCG rates of high-temperature components with different in-plane and out-of-plane constraints by using the two-parameter C?àA c concept.The use of the two-parameter concept C?àA c in the CCG life prediction and assessment of specimens or cracked components may signi?cantly improve accuracy and decrease the degree in conservatism or non-conservatism.The constraint parameter A c of the cracked components may be calculated by the method similar to specimens.The application methodology of the two-parameter C?àA c concept needs to be further investigated.

6.Conclusion

Extensive three-dimensional?nite element analyses have been conducted for six types of test specimen geometries(1TC (T),1/2C(T),CS(T),M(T),SE(B)and SE(T))with different in-plane and out-of-plane constraints in the literature[29],and the uni?ed characterization parameter A c of in-plane and out-of-plane creep constraints was calculated and analyzed for these specimens.Based on the parameter A c and the experimental creep crack growth rate data of the six types of specimen geometries for316H steel in the literature[29],the applicability of the parameter A c for establishing a wide range of in-plane and out-of-plane uni?ed constraint-dependent CCG rate equation for316H steel was investigated.The main results obtained are summarized as follows:

(1)The three-dimensional?nite element analyses show that the center regions of all specimen geometries have higher

and uniform constraint,and the lower constraint occurs at the regions near free surface.

(2)A monotonic correlation line between the CCG rate_a=_a0and the constraint parameter A c on log–log scale can be

formed for different specimen geometries and sizes.This further indicates that the parameter A c can effectively char-acterize both in-plane and out-of-plane creep constraints for a wide range of specimen geometries with different in-plane and out-of-plane constraint levels.

(3)Based on the parameter A c,the in-plane and out-of-plane uni?ed constraint-dependent creep crack growth rate equa-

tion of316H steel has been obtained.The predicted CCG rate by using the equation for each specimen agrees well with the experimental data.

(4)Based on the two-parameter C?àA c concept,the uni?ed constraint-dependent creep crack growth rate equation may

be used for accurate CCG life prediction and assessment incorporating in-plane and out-of-plane constraint effects. Acknowledgments

This work was?nancially supported by the Projects of the National Natural Science Foundation of China(51375165, 51575184,51325504).

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