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Most of the existing local approaches for cleavage fracture derive from the assumptions that global failure is a weakest-link event and that only the tail of the size distribution of micro-crack initiating features is significant. This appears to be

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A novel particle failure criterion for cleavage fracture modellingallowing measured brittle particle distributions
P.M. James
a,
⇑
, M. Ford
a
, A.P. Jivkov
b
a
AMEC Clean Energy, Walton House, Birchwood Park, Risley, Warrington, Cheshire WA3 6GA, UK
b
School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Manchester M13 9PL, UK
a r t i c l e i n f o
Article history:
Received 13 June 2013Received in revised form18 December 2013Accepted 12 March 2014Available online 21 March 2014
Keywords:
Local approach modellingToughness shiftIrradiation embrittlementRPV steelLocal failure criterionTransition temperature regime
a b s t r a c t
Most of the existing local approaches for cleavage fracture derive from the assumptionsthat global failure is a weakest-link event and that only the tail of the size distribution of micro-crack initiating features is signiﬁcant. This appears to be sufﬁcient in predictinglower shelf toughness under high constraint conditions, but may fail when attemptingto predict toughness in the transition region or for low constraint conditions when usingthe same parameters. While coupled ductile damage models with Beremin-like failureprobability could be useful in the transition region, uncoupled models with ‘‘a posteriori’’probability calculations are advantageous to the engineering community. Cleavagetoughness predictions in the transition regime, which can be extended to low constraintconditions, are here made with improved criterion for particle failure and experimentallybased size distribution of initiators for speciﬁc RPV steel. The model is shown to predictexperimentally measured locations of cleavage initiators. Further, the model predicts thefracture toughness in a large part of the transition region and accurately predicts themeasured shift with irradiation, although further works is required to improve the mod-el’s predictions at higher levels of fracture toughness and for low constraint conditions,which are expected to be achieved by a better accounting for plasticity. All results areobtained without changes in model parameters. This suggests that the model can beused as a tool for analysing toughness changes due to constraint and temperature drivenplasticity changes.Crown Copyright
2014 Published by Elsevier Ltd. All rights reserved.
1. Introduction
Safety assessment and life extension decisions in nuclear plant require reliable methodologies for predicting changes incleavage fracture toughness behaviour of ferritic reactor pressure vessel (RPV) steels due to irradiation and defect geometryeffects. Local approaches (LA) to cleavage fracture are promising as these could account for the micro-mechanisms involvedinthecleavagefailurephenomenon,suchasthenucleationofmicro-cracksatsecond-phasebrittleparticles,thepropagationof such micro-cracks within grains and the propagation of a critical micro-crack leading to component failure [1]. The
pioneering LA to cleavage, proposed by the Beremin group [2], is based on two main assumptions: that the global failureprobability is a weakest-link event and that the individual failure probabilities are dictated by local mechanical ﬁelds andspeciﬁc microstructure data such as the size distribution and number density of cleavage-initiating particles. Assuming that
http://dx.doi.org/10.1016/j.engfracmech.2014.03.0050013-7944/Crown Copyright
2014 Published by Elsevier Ltd. All rights reserved.
⇑
Corresponding author. Tel.: +44 (0) 1925 253901; fax: +44 (0) 1925 252285.
E-mail address:
peter.james2@amec.com (P.M. James).Engineering Fracture Mechanics 121–122 (2014) 98–115
Contents lists available at ScienceDirect
Engineering Fracture Mechanics
journal homepage: www.elsevier.com/locate/engfracmech
Nomenclature
Latin symbolsa
crack length
B
crack front length
B
0
reference crack front length
E
Young’s modulus
f
(
r
) probability density of initiator size
J J
-integral
k
strain scaling coefﬁcient
K
stress intensity factor
K
0
cleavage fracture toughness
m
shape parameter
n
power law exponent
N
number of possible weakest links
N
i
number of particles capable of forming micro-cracks
p
c,i
probability of micro-crack nucleation
p
f,i
probability of failure at integration point
P
f
(
V
) global probability of failure
p
s
survival probability
r
radius
r
0
average particle radius
r
c,i
critical micro-crack radius at integration point
r
f
modiﬁed radius
T
temperature
T
0
ductile to brittle transition temperature
u
(
x
) crack opening displacement of penny shaped crack
V
volume
V
i
integration point volume
V
0
reference volume
W
width parallel to crack direction
x
distance from the centre of a penny shaped crack
Greek symbols
b
particle shape parameter
c
particle shape parameter
c
p
plastic work associated with crack propagation
c
s
surface energy of the matrix
e
matrix
strain in matrix
e
p,I
plastic strain at integration point
e
p
a
principal plastic strain
q
particle density
R
I
maximum principal stress
R
a
principal stress in particle
r
I
principal stress
r
iI
principal stress at integration point
r
matrix
stress in matrix
r
part
stress in particle
r
th
threshold stress
r
u
scaling stress
r
w
Weibull stress
r
ys
yield stress
r
ys
,0
reference yield stress
r
a
stress acting normally to the micro-crack plane
W
strain energy lost
w
0
scaling strain energy density
w
c
strain energy density
P.M. James et al./Engineering Fracture Mechanics 121–122 (2014) 98–115
99
thetailofthesizedistributioncanbeapproximatedbyapower-law,theweakest-linkstatisticsleadstoaglobalfailureprob-ability expressed as a Weibull-type function of a generalised stress, the Weibull stress
r
w
[2,3]:
P
f
ð
V
Þ ¼
1
exp
r
w
r
u
m
ð
1
Þ
The shape parameter,
m
, and scaling stress,
r
u
, of the Weibull function can be calibrated using fracture toughness dataobtainedwithaparticularcrackgeometryatagiventemperatureandcorrespondingﬁniteelement(FE)analysis[4]. Forcal-ibrations performed with geometries with high constraint at the crack tip at or below the ductile-to-brittle transition tem-perature,
T
0
, whichdeﬁnesthetemperatureatwhichthe63.2%cleavagefracturetoughness,
K
0
=108MPa
p
m,theshapeandscale parameters obtained can be used with sufﬁciently good accuracy to predict cleavage fracture toughness of other high-constraint geometries at temperatures below
T
0
[2–4]. However, attempts to use the same parameters to calculate cleavagefracture toughness under lower constraint conditions or at higher temperatures leads to predictions that do not matchexperimental values. For cleavage fracture toughness data obtained at a given temperature with low and high constraintgeometries, independent calibrations would typically show that the low-constraint
m
is smaller than the high-constraint
m
when
r
u
is constant, or the low constraint
r
u
is higher than the high constraint
r
u
, where
m
is constant, see e.g. [5]. Thissuggests that the increase in plastic deformation introduced by the lower constraint requires a reduction of
m
to make reli-able predictions. The methodology proposed in [5] for cross-calibration of
m
between high and low constraint geometrieshas shown promise [5,6], but remains limited to temperatures well below
T
0
, where the lower constraint conditions donot introduce a substantial increase in plastic deformations relative to the high constraint case. This is problematic whenused to predict cleavage fracture toughness for short cracks in irradiated material, where the
T
0
temperature is shifted tohigher values. To predict the irradiation effects on cleavage one needs a reliable model for the ductile-to-brittle transition(DBT) regime. Here, however, the Weibull parameters need to be varied to match experimental data which allows two op-tions. One possibility is to keep the shape parameter constant, which leads to temperature dependence of the scale param-eter,
r
u
[7,8]. Another possibility is to keep the scale parameter constant, which leads to temperature dependence of theshape parameter,
m
[9,10]. For the low-constraint situation,
m
needs to be reduced with increasing temperature, i.e. withenhanced plasticity.ConsideringthephysicalbasisfortheBereminLA[2,3], thecurrentstateofaffairsisnot satisfactory, becausetheWeibullparametersmustdependexclusivelyonthematerialmicrostructure.Ifthemodelforindividualfailureprobabilityaccountedadequatelyforthelocalﬁeldsandmicrostructureeffectsonparticlefailure,andglobalfailurewasaweakest-linkevent,thenthe changes in plasticity due to changes in constraint or temperature should be already accounted for, leaving the Weibullparametersconstants.Inparticular,
m
shouldbelinkedtotheshapeofsizedistributionofcleavageinitiationparticles, while
r
u
should depend on the elastic properties and surface energy of the material, in addition to the scale of the particle sizedistribution. Since these parameters do not change noticeably with constraint or temperature, the need to vary
m
and/or
r
u
to accurately predict the fracture toughness as plastic deformation increases suggests that the link between physicsandmechanicsdoesnotadequatelycapturetheeffectsofhighlevelsofplasticity.Onepossibilityisthattheindividualfailureprobabilitymodel does not account adequatelyfor the mechanical and particle size effects. A secondpossibility is that, withthe increase of plastic deformation, the population of micro-cracks that need to be accounted for in the weakest-link statis-tics becomes larger than the tail of the distribution, approximated by power-law in the Weibull-stress models. A third pos-sibility is that the weakest-link assumption becomes increasingly invalid with increasing plasticity, and micro-crackinteraction effects need to be accounted for.To improve the individual failure probability, local plastic strains has been incorporated in the calculation of
r
w
in mod-iﬁcationsoftheBereminModel[11,12]aswellasinincrementalformulations[13].Recently,inadditiontoplasticstrainsthe
local stress triaxiality has also been incorporated [14] into
r
w
. In a previous work [15], these models have been comparedwiththeoriginalBereminModelanddemonstratedthattheyprovideimprovementsinthepredictedfailureprobabilitypro-ﬁles ahead of cracks with different levels of constraint. This comparison has been done against a large set of experimentaldataforthelocationsof cleavageinitiationsitesreportedin[14]. However,thepredictionof thecleavagefracturetoughnesstemperature dependence with the Weibull based models remained unsatisfactory when performed with single set of Weibull parameters calibrated at
T
0
. The effect of particle size on local failure probability was ﬁrst proposed in [16]. Themodel now known as WST, however, remained as a microstructure-informed local model and to the authors’ knowledgehas not previously been applied for global failure predictions. A different direction of work considers ductile damage prior
Abbreviations
CTOD crack tip opening displacementDBT ductile to brittle transitionFE ﬁnite elementLA local approachesPCCV pre-cracked Charpy V notch specimenRPV reactor pressure vessel
100
P.M. James et al./Engineering Fracture Mechanics 121–122 (2014) 98–115
tocleavageintheDBTregime, employingdamagemodelsforthematerialbehaviourcoupledwithsubsequentcalculationof failureprobabilityfollowingBeremin[17,18]. This approachhasthe potential tobetter capturethefracturebehaviourintheDBT, but the models are presently too simpliﬁed with no interaction of void growth and micro-cracking at the micro-struc-tural level [19]. Until advanced coupled models are developed and validated, the wider engineering community would ben-eﬁtfromasimpleruncoupledcleavagefracturemodelfortheDBTregionwhichisabletopredictfracturetoughnessthroughtheDBTregionandtakeintoaccountchangesinmaterialduetoirradiationandchangesincracktipconstraint.Onedirectionof work on uncoupled modelling, known as the Prometey model, has been in improving the local failure criterion by takinginto account the probability of particle rupture and micro-crack propagation [20]. The effect of particle size, however, stillneeds to be accounted for in this model.In [15] a model incorporating mechanical and particle size effects was initially considered and it demonstrated that itprovidesimprovedpredictionsforfailureprobabilityproﬁles. Inthisworkapplicationoffurtherdevelopmentstothismodelare applied to predict cleavage fracture toughness in the DBT regime. The focus of the model development was on the ﬁrsttwo possibilities mentioned above: to improve individual probability of failure and to account directly for the real particlesizedistributionratherthanapproximatingthetailwithapowerlawrelation.However,theweakest-linkassumptionforthecalculation of global probability is maintained. The proposedmodel, as the discussed local approaches, is applicable to mac-roscopically homogeneous materials.It is noted that the modiﬁed Beremin, Bordet and WST model can incorporate the effects of constraint and irradiationthrough the variation of explicit terms. The goal of our development is to have a model where no terms require variation.Our investigation calibrates the different approaches at a single temperature in the un-irradiated condition with high con-straint and evaluates how the approaches perform when used predictively, i.e. can the change in fracture toughness due toirradiation and changes in constraint be predicted using only the changes in tensile properties and crack geometry.
2. Theory and model
2.1. Statistical basis
LA methods share a common philosophy based on two distinct components. Firstly, the local mechanical ﬁelds, deﬁnedassuming a macroscopically homogenous material, provide a local or ‘individual’ probability of failure when linked to thesize distribution of the micro-crack initiators. The individual probability of failure at location
i
is
p
f
;
i
¼
Z
1
r
c
;
i
p
c
;
i
f
ð
r
Þ
dr
ð
2
Þ
where
f
(
r
) is the probability density of initiators’ sizes,
p
c
,
i
is the probability of micro-crack nucleation (and propagation,assumingweakestlink),and
r
c
,
i
isthecriticalmicro-cracksizeatlocation
i
.Notethat
p
c
f
(
r
)istheprobabilitydensityofnucle-ated micro-crack sizes. Existing LAs canbe recast intoEq. (2) albeit withdifferent deﬁnitions of
p
c
and
r
c
. For examplein [2–4]
p
c
=0or1forzeroandnon-zeroplasticstrains.In[11,12]
p
c
scaleswiththeequivalentplasticstrain,whilein[14]
p
c
scaleswith the equivalent plastic strain and exponent of the stress triaxiality. In [13]
p
c
is a more complex function of stress andplastic strainincrements. Inall cases
r
c
is deﬁnedviaGrifﬁthor plasticity-modiﬁedGrifﬁthcriterion. Commonfeatureof [2–4,11–14] is that
p
c
is independent of particle size.Secondly, it is assumed that the individual failure events are independent. This allows the weakest-link argument to beinvoked for calculating the global failure probability, so that
P
f
ð
V
Þ ¼
1
Y
N i
¼
1
ð
1
p
f
;
i
Þ ð
3
Þ
where
N
is the number of possible weakest-links, i.e. potential active micro-cracks, in volume
V
. In practice, LAs are appliedto FE solutions of crackedcomponents, where the mechanical ﬁelds are constant within anintegrationpoint volume,
V
i
. Thefailure probability of such a volume is thus
P
f
ð
V
i
Þ ¼
1
ð
1
p
f
;
i
Þ
N
i
¼
1
ð
1
p
f
;
i
Þ
q
i
V
i
ð
4
Þ
where
N
i
=
q
i
V
i
is thenumberof particlescapableof failingandformingmicro-cracksin
V
i
, and
q
i
is the densityof thesepar-ticles. Strictly speaking
q
i
=
q
p
c
,
i
, where
q
is the density of initiating particles in the material. The component probability of failure is then calculated by repeated application of Eq. (4) to get
P
f
ð
V
Þ ¼
1
Y
IP i
¼
1
ð
1
p
f
;
i
Þ
q
i
V
i
ð
5
Þ
wheretheproductistakenoverallintegrationpoints.When
p
c
isindependentofparticlesize, Eq.(2)canhaveaclosed-formsolutionintermsofcriticalmicro-cracksizes,
r
c
.Withapowerlawapproximationforparticlesizesandassumingsmallindi-vidual failure probabilities, Eq. (5) yields the Weibull-stress function in Eq. (1). Here Eq. (5) is used directly to calculate the
global failure probability, because the experimental particle size distribution, given in Section 2.2, and a new expression for
P.M. James et al./Engineering Fracture Mechanics 121–122 (2014) 98–115
101
individualfailureprobability, derivedinSection2.3, donot allowfor closedformsolutionof Eq. (2) andhenceacorrespond-
ing deﬁnition of a generalised stress.
2.2. Existing local approach models 2.2.1. Original Beremin Model
The Beremin Model forms the basis of the conventional approach to the prediction of cleavage fracture within the localapproachconstruction.ThebasicpremiseoftheoriginalBereminModel[2]isthattheprobabilityofprobabilityofruptureisdirectlylinkedtothemaximumprincipalstressandthefailureeventfollowstheweakestlinkprinciple,wheretheformationof a single micro-crack ruptures the surrounding matrix and leads to global failure. A measure of the uncertainty in the fail-ure (i.e. that linked to the particle size distribution,
m
, typically a second phase carbide) is adopted in the calculation. If theprincipal stress is seen to go signiﬁcantly beyond the provided value of the reference stress,
r
u
, (which is calibrated) theprobability of rupture is seen to rapidly approach unity from Eqs. (6) and (7),
P
f
¼
1
exp
ð
r
w
=
r
u
Þ
m
ð
6
Þ
r
w
¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃX
i
r
i
m
I
V
i
V
0
m
s
ð
7
Þ
where
r
I
is the principal stress,
V
i
is the volume being considered and
V
0
is the reference volume. Where values are denotedby
i
this is the value associated with a ﬁnite volume, such as that in a ﬁnite element. The only variables available for cali-bration are
r
u
,
m
,
V
0
. Note that the form of Eq. (7) means that
r
u
,
m
and
V
0
are related as
r
u
V
1
m
0
¼
const
. Once calibratedfor a given geometry and irradiation condition the Weibull parameters can be transferred to another to calculate the prob-ability of cleavage fracture.However, some difﬁculties have been observed during the use of the Beremin Model, which are [4]: (1) different shapefactors
m
are obtained for the same material depending on crack geometry (i.e. constraint), temperature or loading rate, (2)the iterative calibration procedure for the shape parameter
m
diverges or produces very high
m
values, which do not relatewell with the recorded scatter in experimental data and (3) the model sometimes gives poor cleavage fracture predictions.
2.2.2. Modiﬁed Beremin Model
The Modiﬁed Beremin Model [2,12] provides a further approach to the prediction of cleavage fracture within a local ap-proach construct, and is based on the srcinal Beremin Model. The modiﬁcation is the replacement of the Weibull stress
r
w
calculation in Eq. (7) with that deﬁned by Eq. (8),
r
w
¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃX
i
r
i
m
I
exp
m
e
p
;
I
k
V
i
V
0
m
s
ð
8
Þ
which includes the principal plastic strains,
e
p
,
I
, and scaling coefﬁcient
k
, which can be used as a calibration parameter. Theaddition of the strain term allows for the Modiﬁed Beremin Model to better take into account the effect of constraint,improving the ability to apply the model to geometries other than the calibration geometry. This can also be adopted withthe constraints that the probability of failure is increasing with a threshold condition when applied to warm pre stressconditions.
2.2.3. Bordet Model
The Bordet Model [13] is a modiﬁcationof thesrcinal BereminModel withthe aimof addressingsome of thedifﬁcultiesthathaveappearedinthelatter’suse.ThemodiﬁcationisthereplacementoftheWeibullstress
r
w
calculationinEq.(6)withthat deﬁned by Eq. (9),
r
w
¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃX
i
r
ys
ð
T
;
_
e
p
Þ
r
ys
;
0
r
i
m
I
ð
e
i p
;
dV
Þ
r
mth
d
e
i p
!
V
i
V
0
m
v uut
ð
9
Þ
where
r
ys
ð
T
;
_
e
p
Þ
istheyieldstressatthetemperatureandstrainrateattheintegrationpoint,
r
ys
,0
isareferenceyieldstressata reference temperature, which has been taken as equal to the
T
0
temperature,
r
th
is a threshold stress,
d
e
i p
is the change instrain.Theterm
r
i
m
I
ð
e
i p
;
dV
Þ
r
mth
onlyapplieswhen
r
i
m
I
ð
e
i p
;
dV
Þ
>
r
mth
(wheretheprincipalstressis greaterthanthethresholdstress), otherwise it is taken as 0. The relation between current yield stress and reference yield stress enables the BordetModel to account for the effects of strain rate and temperature. The addition of the threshold stress parameter allows forthe removal from consideration of integration points where the stress is below this, and the addition of the
d
e
i p
meansthe Bordet Model only considers newly nucleated cracks, ignoring cracks from previous steps which would have blunted,and hence will not trigger brittle fracture.
102
P.M. James et al./Engineering Fracture Mechanics 121–122 (2014) 98–115

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