Quantifying the constraint effect induced by specimen geometry on creep crack growth behavior in P9

Quantifying the constraint effect induced by specimen geometry
on creep crack growth behavior in P92steel
Lei Zhao a,b,Lianyong Xu a,b,n,Yongdian Han a,b,Hongyang Jing a,b
a School of Materials Science and Engineering,Tianjin University,Tianjin300072,China
b Tianjin Key Laboratory of Advanced Joining Technology,Tianjin300072,China
a r t i c l e i n f o
Article history:
Received15November2014
Received in revised form
7February2015
Accepted17February2015
Available online25February2015
Keywords:
Specimen geometry
Creep crack growth
P92steel
In-plane constraint
a b s t r a c t
In this paper,the effect of the specimen geometry on the creep crack growth behavior in P92steel was
quantified and six different types of cracked specimens(including C-ring in tension CS(T),compact
tension C(T),single notch tension SEN(T),single notch bend SEN(B),middle tension M(T),and double
edge notch bend tension DEN(T))were employed.Results revealed that the creep crack growth rate
against C n relations varied with changing specimen geometry.For a given C n value,C(T)and CS
(T)showed the highest crack growth rates,which were three times as the lowest crack growth rates in M
(T).This revealed that distinctions in specimen geometry influenced the in-plane constraint level ahead
of crack tip.Moreover,constraint parameter Q was introduced to quantify the in-plane constraint.During
the creep crack propagation,the distributions of Q varied as specimen geometry changed.The specimen
order in terms of Q values from high to low was CS(T),C(T),DEN(T),SEN(B),SEN(T)and M(T),which
basically represented the constraint level in these specimens.
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1.Introduction
Many components servicing at elevated temperatures are continu-
ally exposed to high temperatures.Creep crack growth is a principal
failure mechanism of components within the high temperature regime
due to the detectedflaws generated during manufacturing and serving,
which usually results in the early failure prior to the designed lives
[1–4].The issue of assessing creep cracks in high temperature
components is becoming increasingly important for assuring the
reliability of in-service components.Nowadays,extensive experimental
works had been performed to study the creep crack growth behavior
[5–13],but found that the creep crack initiation and growth character-
istics could be affected by the in-plane constraint ahead of crack tip
imposed by crack size,specimen geometry,loading model and loading
magnitude.Zhao et al.[5]investigated the crack growth behavior using
the compact tension(C(T))specimen with the different crack depths
and found that the creep crack growth rates at the same C n values
increased as the crack depth increased.Yokobori et al.[6]studied the
effects of component geometry on the embrittling behavior of creep
crack growth using C(T)and circular notched specimens and found that
the creep crack growth rates were dependent on the stress triaxiality
ahead of crack tip.Bettinson et al.[7]performed creep growth tests
using ex-service316H stainless steel and stated that constraint effects
were observed in the data,with the C(T)specimens having the fastest
crack growth rate and the small middle tension(M(T))specimens
having the slowest.Davies et al.[8]reported that for a given value of C n,
the long-term C(T)data exhibited higher creep crack growth rates and
shorter creep crack initiation times,compared to short-term tests on C
(T)geometry but the creep crack growth behavior of the long-term
double edge notch bend tension(DEN(T))test was similar to that of the
shorter term DEN(T)test.A test result of a low free nitrogen C–Mn steel
for the specimens with different loading geometries such as single
notch tension(SEN(T)),single notch bend(SEN(B)),M(T)and C
(T)specimens showed the similar tendency,but the highest creep
crack rates were occurred in C(T)specimens[9].Matvienko et al.[10]
investigated the variations of the crack-front stressfield under creep
loading with M(T),SEN(B)and C(T)specimens.
The driving force for creep crack growth was dominated by the
local elastic–plastic stress in the creep damage zone around a crack tip
[14].Specimen geometry,crack size and microstructure played an
important role in the stress state and hence affected the cracking
behavior in the creep range.For the aim of quantify the constraint on
the crack front,the two parameter approach was introduced in to
provide a more accurate characterization of the crack-tip stressfields,
such as C n–T[10],C n–T z[15],C n–Q[16–18],C n–R[19]and C n–A2[20].In
these approaches,thefirst parameter C n integral set the size scale over
which high stresses and strains developed,and the secondary para-
meters T,T z,Q,R and A2were introduced to quantify the crack-tip
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International Journal of Mechanical Sciences
/10.1016/j.ijmecsci.2015.02.009
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reserved.
n Corresponding author.Tel./fax:þ862227402439.
E-mail address:xulianyong@(L.Xu).
International Journal of Mechanical Sciences94-95(2015)63–74
constraint.Among the above mentioned constraint parameters,the parameter Q could be easily obtained and could be incorporated into the creep crack prediction model to give an accurate creep crack growth bined the C n –Q two-parameter concept with a NSW model,Budden [16],Nikbin [17]and Yatomi [21]investigated the effect of constraint on the creep crack growth rates.Based on the C n –Q two-parameter concept and finite element analysis,Bettinson et al.[22]examined the effect of specimen type and load level on the Q from short to long term creep conditions for elastic-creep materials and Zhao et al.[18]predicted the creep crack growth behavior with different crack sizes by considering the constraint effect.
Analogous to J –Q in power law plasticity [23–26],under creep conditions,the stress fields can be given by C n –Q ,which are de fined as [16]:
σij ¼σ0C n
_ε0σ0I n r 1=ðn þ1Þ
~σij θ;n ÀÁþQ σ0δij ð1Þ
Q ¼
σθθðr ;0ÞÀσHRR
θθðr ;0Þσ0
ð2Þ
where σij is the stress tensor;σ0is the yield stress;_ε
0is the strain rate at the σ0;r and θare polar coordinates centered at the crack tip,which correspond to the radial distance from the crack tip and the angle from the crack plane,respectively;I n is an integration
constant;~σ
ij θ;n ÀÁ,~εij θ;n ÀÁare dimensionless functions of n and θ.σθθðr ;0Þis the actual crack-opening stress in specimens.σHRR
θθðr ;0Þ
is the analytical crack-opening stress determined from the HRR field,which is determined by [27]:
σij ¼σ0C n
_ε0σ0I n r 1=ðn þ1Þ
~σij θ;n ÀÁð3Þ
Because creep crack resistance could be affected by in-plane constraint conditions,a proper method should be developed to quantify in-plane constraint.Up to the present,several works on these issues have been reported,but no systematic investigations based on detailed elastic –plastic-creep finite element (FE)analyses to study the in-plane constraint induced by specimen geometries and the variations of constraint during the whole creep crack process have been yet reported.
The material studied in this work is a high strength ferritic steel,P92steel,which has been widely employed in high temperature components of ultra super critical fossil power plant,i.e.main steam pipes,headers and so on,due to its high creep strength and corrosion resistance at high temperature [28].The variations of the creep crack growth behavior in P92steel with different specimen geometries are studied,in which six different types of cracked specimens are employed,i.e.C(T),CS(T),SEN(T),SEN(B),M(T),and DEN(T)specimen.On the basis of the creep crack data from FE and the two parameter approach,the creep crack-tip constraint induced by the specimen geometry during the creep crack growth process has been investigated in detail.
2.Creep crack growth behavior calculation 2.1.Specimen geometry
To investigate the constraint effect induced by the specimen geo-metry,six cases of creep crack growth testing specimens are considered in numerical analyses,which are depicted in Fig.1.They respectively are C(T)specimen (see Fig.1(a)),CS(T)specimen (see Fig.1(b)),SEN(T)specimen (see Fig.1(c)),SEN(B)specimen (see Fig.1(d)),M(T)specimen (see Fig.1(e)),and DEN(T)specimen (see Fig.1(f)).For all cases,the specimen thickness is denoted by B .For M(T)and DEN(T)specimens,the half crack length and half width are denoted by a and W ,respectively.For other specimens with an single edge
crack such as C(T),CS(T),SEN(T)and SEN(B),the crack length and width are denoted by a and W ,respectively.In the present study,for all the specimens,B is 25mm;W is 50mm and a /W is 0.5.The detailed conditions for six cracked specimens are shown in Table 1.2.2.Material model
Creep crack growth simulations are performed using elastic –plastic –creep FE damage analyses.The total strain εt is calculated by:
εt ¼εe þεp þεc
ð4Þ
where εe ,εp and εc stand for elastic,plastic and creep strain components,respectively.The deformation theory of plasticity and the von Mises yield criterion are adopted.The elastic –plastic beha-vior follows the stress –strain relation:
εe þεp
¼σE þασσ0 m
ð5Þ
where E is the Young's modulus,m is the hardening exponent and α
is the strain hardening coef ficient.A power law creep model determined by the average creep strain rate is employed to describe entire creep curves for producing more representative deformation rates with longer creep times [1,29]:_εc ave ¼
εf
r
¼A σn
ð6Þ
where _εc ave is the average creep strain rate;εf is the uniaxial failure
strain;t r is the rupture time in a uniaxial creep test;A and n are material constants which can be calculated by plotting the uniaxial minimum creep strain rates against applied stresses (log –log scale).εf is the uniaxial creep ductility and depends on the material and the temperature.In this paper,the constant εf of 20%is employed,which is the measured value of uniaxial creep tests for the present ASME P92steel at 6501C.
On the basis of the creep ductility exhaustion concept,an uncoupled damage model is introduced to calculate the creep damage accumulation and then simulate crack growth during the transient and steady state creep crack growth regimes.Creep
damage rate _ω
is de fined by the ratio of average creep strain rate,_εc
ave and multi-axial creep ductility,εn f as follows:_ω
¼_εc ave εn f
ð7Þ
In this model,multi-axial creep ductility depends on the uniaxial creep ductility of material and the stress triaxiality ahead of crack tip which is the ratio of the principle stress σm and the equivalent stress σe .In previous study,it was found that the creep crack growth in P92steel is mainly caused by the nucleation,growth and coalescence of creep voids by investigation the microstructures [5].Although there are several models to quantify the multi-axial stress effect on creep ductility,the model proposed by Cocks and Ashby on the basis of the void growth and coalescence [27]is employed and is de fined as follow:
εn f
εf
¼sinh 2n À0:5 !=sinh 2n À0:5
σm σe !ð8Þ
The damage accumulated is calculated using a simple time
integral rule:
ω¼Z t 0
_ωdt ¼Z t 0
_εc
ave εn
f
dt ð9Þwhen the accumulated creep damage calculated from Eq.(9)
becomes critical (attains 0.999),local failure is considered to occur and progressive creep crack is simulated.Note that within an FE analysis the value of εn
f changes for a fixed material point since it
depends on the stress triaxiality through Eq.(8),which changes
L.Zhao et al./International Journal of Mechanical Sciences 94-95(2015)63–74
64
with creep time as the stress redistributes local to the crack tip during creep crack process.
The tensile properties and creep properties for P92steel were measured at high temperature of 6501C using standard specimens oriented in the longitudinal direction derived from P92pipes,which are shown in Table 2.2.3.Finite element model
In this paper,creep crack propagation was simulated using a node-release approach.Based on the methods modeling crack extension,existing methods to predict creep crack growth can be clari fied into two groups.Firstly,when the accumulated damage at a gauss point became unity,the failure of the element was simulated by reducing the load-carrying capacity of the point to a small level [29,30].However,the phenomenon in actual creep crack growth that a sharp crack formed and grew within a creeping material could not be
accurately represented.Secondly,when the accumulated damage of the node at the crack tip reached to the critical value,the node at the crack tip was released and then the crack propagated [2,18].It was assumed in the FE analysis that the crack grew in the plane of the initial crack front,i.e.along the symmetry plane.Prior to simulation,the nodes ahead of the initial crack tip along the plane were all fixed.As the creep damage extent in the nodes at the crack tip reached a critical value,the nodes ahead of the crack tip were released and thus the current crack tip propagated further and a new sharp crack tip was occurred through the mesh along the symmetry plane in the FE models.This approach could simulate the creep crack growth beha-vior similar as the actual propagation.However,this method had limitations on the three dimensional creep crack simulation.
Due to the employed node-release method and the geometrical and loading symmetries,two dimensional plane strain FE analyses were performed using the general purposed software package,ABAQUS Standard,to calculate the creep crack growth behavior and then to determine the full-field stress and strain fields which were used to derive the constraint level ahead of crack tip for all specimens.Fig.2depicts the typical FE meshes employed for different specimen geometries in the present work.Because of symmetry,one half of C(T),CS(T),SEN(T)and SEN(B)specimens (shown in Fig.2(a)–Fig.2(d))and one quarter of M(T)and DEN (T)specimens (shown in Fig.2(e)and Fig.2(f))were considered.Regular square elements were used in the vicinity of the crack tip (see Fig.2(g))so the crack grew through a region of uniform elements.For different specimen geometries,a same fine mesh with the smallest element size of 0.015mm focused on the crack tip was used,with an increasingly coarse mesh generated elsewhere.The smallest size of mesh was designed in accordance with the observed average grain size in the adopted P92steel.On the other hand,the size of the smallest mesh on the crack front could eliminate mesh dependency effects in the analysis and give a good prediction [18,31].As well known,the convergence dif ficulty was easy to occur during the creep crack growth simulation.The mesh size and the increment should be as small as possible to improve the convergence.In addition,due to the same smallest mesh applied in the crack plane,the number of total mesh for different specimen geometries was similar,which was in the range of 5200–
5400.
Fig.1.Schematic illustrations of geometries and dimensions for cracked specimens employed in the present work:(a)C(T),(b)CS(T),(c)SEN(T),(d)SEN(B),(e)M(T)and (f)DEN (T)specimens.
Table 1
Numerical creep crack growth conditions employed in FE.Specimen B (mm)W (mm)a 0/W P (N)K in (MPa m 1/2)C(T)25500.501041718CS(T)25500.50661818SEN(T)25500.502514118SEN(B)25500.50994818M(T)25500.5014345218DEN(T)
25
50
0.50
138091
18
Table 2
Mechanical properties and creep properties of employed P92steel at 6501C.Material
Elastic modulus E (GPa)Yield strength σ0(MPa)A
(MPa Àn h À1)
n
εf (%)
α
m
P92steel 125
180
2.6353E-16 5.2320 1.78E-411
L.Zhao et al./International Journal of Mechanical Sciences 94-95(2015)63–7465
The load was applied by an analytical rigid circle representing the pin in the experiment.This pin had the geometry and dimensions of the hole.This pin was joined to the center of the hole from one side and to the inner nodes of the hole from the other side.Contact pairs were created between the rigid circle and the inner nodes of the hole.The center of the hole was constrained in X-direction to
prevent
Fig.2.Typical FE meshes for the different specimen geometries used in the numerical creep crack simulation (a)C(T),(b)CS(T),(c)SEN(T),(d)SEN(B),(e)M(T),(f)DEN (T)and (g)fine mesh employed on the crack front.
L.Zhao et al./International Journal of Mechanical Sciences 94-95(2015)63–74
66
rigid body translation and rotation of the model.Furthermore,symmetric boundary conditions were utilized in other planes for all six types of specimens.2.4.Loading
The loading is chosen such that the employed initial stress intensity factor K is equal to 18MP am 1/2regardless of specimen geometry.This loading is stabilized during the simulations,which is provided in Table 1.The stress intensity factor is calculated accord-ing to ASTM E1457standard [32]and is de fined as follow:
K ¼Y a =W ÀÁσffiffiffia p ð10Þwhere σdenotes the applied nominal stress and Y (a /W )is the function of geometry,which are determined by specimen con fig-urations and calculated according to ASTM E1457.2.5.Calculation of creep crack growth rate
In FE analyses,the regular elements in the crack tip plane are employed and the crack only grows in the plane of the initial crack front (see Fig.2).Hence,the crack growth increment is uniform and the crack growth length can be calculated by the number of released regular elements on the crack plane.Record the relation between the number of the released elements and the corresponding creep time,which is used to derive the creep crack growth rate in FE.2.6.Calculation of C n
In general,the creep parameter,C n ,is used to correlate the creep crack growth rate.In experiment,for different specimen geometries,the C n could be derived using the recorded creep load line displacement and creep crack growth data according to ASTM E1457[32].The C n is de fined as follows:C n
¼
F _ΔFLD
c
B W Àa ðÞ
H FLD ηFLD ð11Þ
where _ΔFLD
c
is the creep component of force line displacement;H FLD is derived by the material constant n and ηFLD is determined by specimen geometry,dimension and size of specimen,and crack length,which is determined by the requirements in ASTM E1457for different specimens.
The FE damage analysis could provide the force line displace-ment and the creep crack growth length as a function of ing these results,C n values of FE are estimated using numerical load line displacement records through Eq.11,exactly same as the way they are determined in actual tests.2.7.Validity of parameter C n
In creep ductile materials,where creep strains dominate and creep crack growth is accompanied by substantial time-dependent creep strains at the crack tip,the crack growth rate is correlated by C n .In ASTM E1457standard,some speci fic criteria that must be satis fied are de fined to ensure that valid correlations between
creep crack growth rate _a
against C n data are obtained [32].First of all,data obtained prior to the transition time,t T ,should be excluded,where the transient time t T is de fined as follow:t T ¼
K 2ð1Àυ2ÞE ðn þ1ÞC n T ð12Þ
The calculation of t T depends on the value of C n (t T ).Because t T cannot be obtained directly,C n (t T )is also cannot be obtained explicitly.In ASTM E1457,it is recommended that the following procedure must be used for its evaluation.For the time t T ,
corresponding to each date point,calculated t 0T using the above equation but replacing C n (t T )by C n (t).Then,t T is the largest value of t 0T in the entire data set.Furthermore,the creep force line
displacement rate _ΔFLD c
and the total force line displacement rate _ΔFLD t should be in the ratio of _ΔFLD c =_ΔFLD t
Z 0:5to assure the validity of the relationship between creep crack growth rate and C n .This ratio represents the degree of creep area spread.
The variations of creep force line displacement rate during the creep crack growth for different specimen geometries are shown in Fig.3.It can be seen that the creep force line displacement rate is all above 0.8,which is much higher than 0.5.In particular,the rates of M(T)and DEN(T)keep constant at 0.97.These phenomena reveal that the calculated results strati fies the speci fication on the creep ductility failure in ASTM E1457.Hence,it can be deduced that the corresponding relationships between C n and creep crack growth rate obtained from the FEM results are effective.
3.Results and discussion
3.1.Validation of simulated creep crack growth
Fig.4gives the comparison of the experimental correlation between creep crack growth rate and C n for C(T)tests with the simulated results from the FE Experimental test conditions are summarized in Table 3,where a C(T)specimen with 10%side groove is employed.In addition,the results under plain stress condition were provided for comparison.Because the effective thickness of
specimen B e (which is de fined as B e ¼ffiffiffiffiffiffiffiffi
BB n p where B is the thickness of specimen and B n is the net thickness of specimen)is used for calculating the applied load of the side groove specimen according to ASTM E1457,the specimen with an effective thickness is adopted in FE for considering the side groove effect.The other conditions such as crack size and loading con figuration employed in FE are identical to the experiment.
It can be noted that creep crack growth curves obtained from FE and experiment show similar tendency.A “tail ”region has occurred at first,in which the creep crack growth rates increase with decreasing C n values and this is mainly occurred in the initial creep crack growth region.Then,a linear relationship between creep crack growth rate and C n is observed when plotted on log –log axes,which corresponds to the steady creep crack growth state.
Moreover,it can noted that the plain strain condition gives a more accurate prediction in comparison with the plain stress con-dition.Although the simulated creep crack growth rates from plain strain condition are slightly higher than experimental data,they agree overall well with experimental results.Although side groove is introduced into specimen,the crack front is not straight
during
Fig.3.Variation of force line displacement during creep crack growth for different specimen geometries.
L.Zhao et al./International Journal of Mechanical Sciences 94-95(2015)63–7467
test and thus the obtained crack length is the average length.While only the crack propagation on the surface with highest stress state is simulated in FE and thus the calculated rate is a little higher than the experiment at the same C n value.However,the linear relation-ship under plain strain condition in the steady creep crack region is nearly equal to that of experiment.In contrast,at the same C n values,the creep crack growth rates from plain stress condition are lower that of experiment and are only one third in experiment.However,the slopes for three cases under steady creep crack growth region are nearly same.The phenomena reveal that the results from FE under plain strain condition demonstrate a good agreement with the corresponding experimental results and could be employed to represent the actual creep crack growth in experiment.
The variations of creep crack growth rates vs C n under different initial K values are shown in Fig.5.It can be observed that the similar linear relationship between creep crack rates and C n occur.This demonstrates that the relationship between the CCG rate and C n is independent of the initial K value,which is consistent with the reported experimental results [5].But a high K value would accelerate the stress state ahead of crack tip and improve the force line displacement rate.Hence,the range of C n values at a high K is larger than that at a low K .
Hence,in this paper,the effect of the constraint induced by specimen geometry is carried out under plain strain conditions and a constant initial K value.
3.2.Creep crack growth behavior with different specimen geometries The creep crack growth extension,Δa ,was determined from FE,which was calculated by the number of the released nodes on the crack plane.The relationships between creep time t and non-dimensional creep time t /t f (that is de fined as the ratio of the current creep time t and the life of the creep crack growth t f )virus creep crack growth length are summarized in Fig.6and Fig.7for six specimen geometries.
Besides DEN(T)specimen,the creep crack growth behavior is similar with changing the specimen geometry.The crack propagation is increased linearly with the creep time or the non-dimensional time and then the crack extension is steeply increased at the final stage.More than 50%of crack extension is made in final 20%period.However,for the DEN(T)specimen,the creep crack growth length shows a linear relationship with creep time throughout the creep crack process.In
addition,at the same non-dimensional time,the creep crack growth length in DEN(T)specimen is higher than that in the rest specimens.Creep crack initiation in a ductile material may be considered to occur at the time when micro-cracks that are formed by the nucleation,growth and coalescence of voids first link up,which is an important factor in creep crack growth behavior [33].The creep crack initiation time is recommended to be the time when the crack extension is equal to 0.2mm after full creep loading [32].Fig.8shows the variation of the creep crack initiation time for different specimen geometries at the same initial stress intensity factor.It can be observed that the smallest creep crack initiation time occurs in the C(T)specimen and is about half of the highest creep crack initiation time in the SEN(T)specimen.The specimen order in terms of the creep crack initiation time from low to high is C(T),SEN(B),CS(T),DEN(T),M(T)and SEN(T),where creep crack initiation time of C(T),SEN(B)and CS(T)is similar while that of M (T)and SEN(T)is similar.
The creep crack growth behaviors are evaluated in terms of C n and creep crack growth rate as shown in Fig.9for different specimen geometries.The curves for different geometries show similar tendency,besides that the tail part in SEN(T)is more obviously compared with the other specimens.The creep crack propagation in Fig.9can be divided into three bands.First,the C(T)and CS(T)specimens show similar creep crack propagation behavior and the correlation of creep crack growth rate and C n in steady region is:_a
¼0:06597C n ÀÁ
0:61223ð14Þ
Second,the linear relationships in the steady creep crack state for SEN(B),SEN(T)and DEN(T)specimens are the same,which
can
parison of experimental creep crack growth rate for C(T)specimen with simulated results.
Table 3
Testing conditions of experimental creep crack growth tests.Specimen B (mm)B n (mm)a 0/W P (N)K in (MPa m 1/2)C(T)-25
25.08
20.10
0.50
11789
22
Fig.5.Variations of creep crack growth rates against C n under different initial K
values.
Fig. 6.Relationship between creep crack length and creep time for different specimen geometries.
L.Zhao et al./International Journal of Mechanical Sciences 94-95(2015)63–74
68。