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Cleavage fracture prediction and KIc assessment of a nuclear pressure vessel carbon steel using local approach criteria
Nuclear Engineering and Design 144 (1993) 1-7 North-Holland
1
Cleavage fracture prediction and Kic assessment of a nuclear pressure vessel carbon steel using local approach criteria P.P. Milella a, C. Maricchiolo a, A. Pini ~, N. B o n o r a b a n d M. M a r c h e t t i b a ENEA-DISP, F'~a V. Brancati 48, 00144 Rome, Italy b Aerospace Dept., University of Rome "La Sapienza", Rome, Italy
Received 8 January 1992, revised version 8 March 1993
This paper presents a theoretical and experimental work carried out by the Aerospace Department of the University of Rome and ENEA-DISP on local approach. The main goal was to apply the local approach criteria to cleavage fracture prediction in a structural material. The material considered was a low alloy steel used in the nuclear industry, type 22NiMoCr37, of German production provided by KWU in the framework of an international round robin on local approach to fracture in steels, promoted by the European Structural Integrity Society (ESIS). Twelve tension tests on round notched bars were performed at low temperature (-90"C) in order to get the experimental data needed to calculate the theoretical Weibull stress and exponent and infer the probability of failure by a finite element analysis. Results have been applied to a three point bend specimen of the same material to predict cleavage fracture.
1. Introduction It is well known that the theoretical strength of a material to cleavage is about one tenth of the Young's modulus E. At variance, practice indicates that it rarely exceeds few thousandth of that value. Griffith [1] was the first to point out that responsible for that discrepancy were defects like cracks, always present in the real materials even though very small. Indeed, the effective reduction of the strength of a real material can be ascribed to the non-homogeneous plastic deformation that takes place somewhere in the material under stress when dislocations activated in unfavourable oriented grains pile-up against a barrier, like an inclusion or grain-boundary, building up a very high local stress which originates the tiny crack responsible for the early failure of the material. In the brittle fracture of a body containing a crack, that happens when the material is still behaving elastically on a macroscopic scale and the nominal stress is sometimes well below the yield strength. Outstanding works by Low [2], Knott [3], Petch [4] and others [5] have definitely demonstrated the local nature of the plastic deformation preceeding the generation of the crack that may start the brittle fracture. These observations suggest to investigate the fracture resistance on a microscopic base developing approaches that can be de-
fined as local since they relate the overall resistance of a structure to a local behaviour of the material that may trigger the unstable fracture.
2. Local approach A more recent local approach to the fracture process has been developed by Mudry and Pineau [6,7]. Its basic hypothesis is that cracks originate in the material where a plastic deformation occurs while an elastic region may be considered crack free. If a flow is generated instability occurs as the stress % normal to the crack plane reaches the critical value:
ro
o'. = ~-a
(2.1)
where a is the half crack length and Kc is the fracture toughness of the material. Equation (2.1) represents the Griffith principle. What makes local approach differ from classical Griffith principle that applies to an existing crack is the fact that now the crack is not known "a priori". Hypotheses must be made on flaws distribution in terms of size and orientation within the material. We assume that in the zone of the material that reaches the yield strain a population of micro-de-
0 0 2 9 - 5 4 9 3 / 9 3 / $ 0 6 . 0 0 © 1993 - Elsevier Science P u b l i s h e r s B.V. All rights reserved
2
P.P. Milellaet al. / Cleavagefractureprediction If the function g(O) takes the most general form of a Gauss' distribution, then:
Tr/2 fo~/Zg(o) dO = 1
(2.6)
and the failure probability distribution function is then related only to the crack length a. Furthermore assuming for the density probability function P(a) the form:
P(a) = a-",
(2.7)
eq. (2.5) yields: Fig. 1. Boundary of stability domain (shaded area).
1
( K e ) -2(n-1)
e,(0.1) =
;T
l ll fects is generated which does not increase with further increments of plastic strain. For any of those cracks, instability does not occur as long as: [0" 1 c o 8 2 ( 0 ) + 0 " 2 sin2(0)]~a- < K c ,
(2.2)
1-n
(2.8)
KoI
with m = 2(n + 1). Any distribution function can be given the form 1 - exp[-~p(x)] so that Pf(0.1) can be expressed also as: pf = 1 - exp[-~0(0.1)],
where 0.~ > 0.2 are the principal stresses, a is the crack length and K c is the fracture toughness. If P(a) is the probability density function of defects, that is to say the probability that a defect has a length between a and a + da, and g(O) that of the defect's orientation, then the distribution function pf of the probability of failure is given by:
(2.9)
where ~0(0.~) is a proper function that can be determined as follows: if 0.u is the cleavage stress a t w h i c h the material fails, no matter how small the size of a real crack, possibly present in the material, can be, then the probability that it fails under a stress lower or equal to 0.u must be l:a Pf(0.u) = 1,
Pf= ffs(al, crzP(a)g(O) da dO"
(2.3)
pf represents the probability that failure occurs under a combination of principal stresses equal or lower than a given set of stresses 0.1, 0.2, in the failure domain S(0.1, 0.2) defined by eq. (2.2). In the a, 0 plane the non failure locus is shown in Fig. 1 by the shaded area. If one assumes that 0.z = 0 then the boundary of the stability domain moves to the right as depicted in Fig. 1 and the analysis become less conservative, yet the most critical condition is always represented by:
(2.4) The hypothesis cr2 = 0 allows to simplify the integration domain eq. (2.3) that can be rewritten as:
(2.10)
and, recalling eq. (2.8):
l_la/" 1= 1-nkKcl
(2.11)
or:
1 - n = (0.u/Kc) m.
(2.12)
Substituting eq. (2.12) in eq. (2.8) and recalling eq. (2.9), yields: 1 -- e x p [ -
(2.13)
~0(0.1) ] -~- (0.1/0.u) m .
For m :~ 1 and 0.u >0.1, as in our case, i.e. when (0.1/0.,)" is small compared to unit, it can be assumed, as first approximation: ~o(0.1) = (0.1/0.~) m
(2.14)
and eq. (2.9) becomes: Pf(0.1) = 1 - exp[
- (0.1/0.u)m]
,
(2.15)
P.P. Milellaet aL / Cleavagefractureprediction
3
of volumes V0, then, for the theory of the weakest link [8] we can finally write:
Equation (2.15) expresses the probability that a material fails under a stress lower or equal to a given value tr z. Recalling our fundamental hypothesis that instability occurs when a crack generated anywhere in the plastic enclave, of volume V, reaches a critical value, we can split the plastic volume V into s elementary volumes V0, each of which the probability of failure given in eq. (2.15) can be applied to and say that the overall failure occurs when any of those elementary volumes V0 fails. Assuming a statistical independence
p,(o.l,V)=l-exp[-I°'llms]
= 1 - exp -
--
~
.
The statistical independence of the volumes can be
R=2mm 111111 IIIIII IIIIII
) I
R
R=
=
(2.16)
!
L
4 mm
lOmm
Fig. 2. Geometries of the three specimens considered in this study with mesh used in the FE analysis.
4
P.P. Milella et al. / Cleavagefractureprediction
asserted on the basis of the weak stress gradient experienced through the volume V since it is a plastic enclave in which for an elastic-perfect plastic material the stress would be levelled off to the yield stress and in a real material is confined within the strain hardening range.
3. Determination of the probability of failure Equation (2.16) can be solved using a combination of experimental measurements and finite element calculations in which it can be written as:
Pf(~l, V ) = l - e x p [ i=1 .....
(3.1) in which Ne is the number of element used in the mesh and NG is the number of Gauss' points that yielded. Introducing the Weibull stress ¢rw as:
o-~=
E i=1 ..... Ne jfl
o." 1U
(3.2) V0
..... N G
eq. (3.1) finally becomes:
[( t (7"W
p f ( o " I, V) = 1 - exp -
~
.
(3.3)
The finite elements calculations provide the relationship between ~r, and the applied stress for different geometries using the equation (3.2) in which m and V0 are given a first approximation value and A V o are the elementary volumes where a plastic deformation occurs. Parallel to calculation an experimental program was carried out on specimens of equal geometries. For each geometry the failure stress was measured and the Weibull stress inferred entering the theoretical ~rw, ~r relationship provided by eq. (3.2) for that geometry. By doing so the % value can be assumed as the average of the Weibull stress o, at failure for the different geometries. This is just a first approximation since cru, as already said, is the cleavage stress, but it can be assumed from a practical point of view instead of the cleavage stress proposed by Knott [9]. Beremin [10] has shown that for m = 22 ~w = 0.9757 cru.
(3.4)
4. Results of finite element calculation and experiments Three different geometries have been considered simulating a round traction specimen of 18 mm diameter carrying circumferential surface notches of different tip radius, namely 2 ram, 4 mm and 10 mm whose depth was always equal to 4 mm, as proposed by Beremin [11]. The relative meshes are depicted in Fig. 2. The true stress-strain relationship for the material under investigation, a low alloy steel 22 NiMoCr37 of German production used in the nuclear industry, is shown in Fig. 3. Its chemical composition and mechanical properties are shown in Table 1 and 2. The results of the analysis performed with MARC code were used as input data for an ad hoc program to calculate the corresponding Weibull stress given by eq. (3.2) through an iterating process in which m is given an initial value, equal to 15 in our case, to which it corresponds to a slope of the failure distribution function pf(crl, V). If the experimental data does not match the computed distribution curve, m is changed and the iteration continues to find the agreement between experimental points and theoretical prediction. In our case, it was found a good agreement, as also suggested by Mudry, for m = 22 and the iteration stopped. The experimental program was carried out with 12 tensile specimens having the same notch tip radii selected for the FE calculation. Specimen were broken at - 9 0 ° C since the final goal was to predict the brittle fracture of three point bend specimen. Using the iterating procedure just mentioned it was possible to construct the applied load versus Weibull stress curve, Fig. 4, to derive the Weibull stress at failure and then cru, as the average Weibull stress, that turned out to be equal to 2478 MPa. Finally equation (3.3) was used to calculate the
1000
" 900"
800700"
600"
~_
- i/ 500
2
steel
22NiMoCr37
!
400
.... 0.0
, .... , .... , .... ~ .... , .... , .... , .... , .... 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 TRUE PLASTIC STRAIN
F i g . 3. T r u e s t r e s s - t r u e
strain curve of the 22NiMoCr37
steel.
P.P. Milella et aL / Cleavagefracture prediction Table 1 Material chemical composition Mn
P
W
Cr
S
Mo
C
Si
Ni
0.22
0 . 2 3 0 . 8 8 0.006 0.004 0 . 3 9 0 . 5 1 0.84
1.o
5
/
__J
2 0.8
CTu =
2478
MPo
b_
R=2mm m R = 4rnm A
o
0.5 d
Table 2 Material properties R~. 2
Rm
NDT
USE-CV energy
[MPa]
A5 [%]
Z
[MPa]
[%]
[°C]
[J]
980
1015
24
40
- 40
189
< 0.3 ~3 o t~ G_
R = 10 mm X
0.0
n i
1500
n ,
,
i
i
,
,
i
n i
,
I
u
I
n 1
2000
u I
I
I
u *
,
2500
I
F n
n t
I
I
3000
I
,
r
u
n
u
n
,.3500
WEIBULL STRESS (MPo) Fig. 5. Calculated probability of failure versus Weibuli stress. failure probability distribution versus the Weibull stress shown in Fig. 5 together with the data point. The agreement is quite good.
90 Applied Lood versus Welbull stress m=22
R-2mm
5. Brittle fracture prediction
El
70 0 __J
R-4mm
~ 6o ....I 0..
R
•~ 50 40
-
10
mm
0
, J U J l , J J , l l U n , l u u u n l J l U ' l , r u n l u ' ~ l ' ' ' , l , = ' '
1000 1250 1500 1750 2000 2250 2500 2750 3000 3250
WEIBULL STRESS (MPo) Fig. 4. Applied load versus Weibull stress for the three geometries considered.
The purpose of this investigation was to apply the local approach methodology to predict the brittle fracture, i.e. the Kxc toughness, of the material considered in this program using a three point bend specimen containing a crack. This specimen, an ASTM E399 bend one [12], has been analysed with F.E. using the MARC code with the 724 elements mesh shown in Fig. 6 having 1554 degrees of freedom. At each load step the program calculated the K] value. The correspond-
nnmnmunn nmnlunnun
nmmb':;J~,~ ~."dnE ;.:- ;!!!!i
~'-qL~-_:':'/iF~ nnn,~'-.~-I uummnmmml Inuununl innnnnnl Y
l, Fig. 6. Mesh of the Kic specimen used in the FE analysis.
P.P. Milella et al. / Cleavage fracture prediction
6 1.00
f
by ASME [13]. They are shown in Fig. 8. It can be seen that toughness data at 70°C below the Reference Transition Temperature (RTNDx), as it is in our case, range between 80 MPa~fm- and 30 M P a v ~ . It must be considered, however, that the experimental data refer to different heats of materials which increases the spread of the results.
13
i,i a,0.75 b,._ 0
0.50
m 0.25 < 0 t3~ Q_ 0.00
6. Conclusions uu),ll;lll
0
,i, l,~11.,[u~
10
20
r~F,l
ii*)
IIIUlII[ITIIIII
30 40 KI (MPo ~r~)
r IllUl)'II'
50
'
50
Fig. 7. Probability of failure versus applied K x.
ing Weibull stress % was calculated for the different load steps using the ad hoc program described in the previous section. Once the Kr-xrW relationship is obtained, the probability of failure of the specimen versus the applied K z can be derived entering equation (3.3) as shown in Fig. 7. Fig. 7 says that the probability that a material fails under an applied K I lower or equal to 25-30 MPavrm is practically negligible. These values should be considered, then, as lower bound toughness values. Conversely, the probability to get a fracture at or below 60 MPavCm- reaches the maximum value. We have, then, derived a scatter band for the material toughness at -90°C. Experimental data on the material used are not yet available, but very significative are the data obtained on analogous material and published
3OO
u
a.
2oOloo i
er bound
Lit.
0
-200
i
l
)
-100
0
100
200
Temperature ( T-RTNDT ) C Fig. 8. Fracture toughness data obtained at various temperatures (relative to RTNDT) for carbon steels (ref. ASME).
Even though the experimental program to measure the KI¢ toughness using three bend specimens of the carbon steel used in this study has not been carded out yet, experimental data already obtained and published by the American Society of Mechanical Engineers (ASME) on a large variety of carbon steels show quite an excellent agreement with the theoretical predictions based on local approach. This seems to be encouraging the use of the local approach criterion to predict a brittle fracture event. Also the m value suggested by Mudry, m = 22, is likely to be the most appropriate one for the material under investigation. The K k testing on the specific material used in this study will provide further information on the applicability of the local approach method.
References
[1] A.A. Griffith, The phenomena of rupture and flow in solids, Phi. Trans. Roy. Soc. of London A 221 (1921). [2] G.R. Low, IUTAM Madrid Colloquium, Deformation and Flow of Solids, pp. 60 (Springer Verlag, Berlin, 1956). [3] J.F. Knott and A.H. Cottrell, Iron Steel Institute, pp. 201-249, 1963. [4] N.J. Petch, The fracture of metals, Prog. Met. Phy. 5 (1954) 1. [5] P.J. Worthington, E. Smith, Acta Metall. (1966) 14-35. [6] F. Mudry, Local approach to cleavage fracture, Nucl. Engng. Des. 105 (1987) 65-76. [7] A. Pineau, Review of fracture micro-mechanisms and local approach to predicting crack resistance in low strength steels, Proc. of Fifth Int. Conf. of Fract. (ICF5) Cannes, France, 1981. [8] W. Weibull, A statistical distribution function of wide applicability, J. of App. Mech. (Sept. 1951) 293-297. [9] J.F. Knott, On stress intensification in specimen of charpy geometry prior general yield, J. Mech. Phys. and Solids 15 (1969) 97. [10] F.M. Beremin, A local criterion for cleavage fracture of a nuclear vessel steel, Met. Trans. 14A (Nov 1983) 22772287.
P.P. Milella et aL / Cleavagefracture prediction [11] F.M. Beremin, Calculs elasto-plastic par la m6thod des elementes finis d'Eprouvettes axisymmetriques entaile circulairement, J. de M6ch. App. 4 (1980) 397-325. [12] Annual Book of ASTM Standards, Vol 3.01. [13] W. Oldfield and W.L. Server, Fracture toughness predic-
tion for pressure vessel steels: the development of statistically based method, in Reference Fracture Toughness Procedures Applied to Pressure Vessel Materials, MCP24 ed. T.R. Mager, ASME, N.Y. (1984) pp. 9.
1
Cleavage fracture prediction and Kic assessment of a nuclear pressure vessel carbon steel using local approach criteria P.P. Milella a, C. Maricchiolo a, A. Pini ~, N. B o n o r a b a n d M. M a r c h e t t i b a ENEA-DISP, F'~a V. Brancati 48, 00144 Rome, Italy b Aerospace Dept., University of Rome "La Sapienza", Rome, Italy
Received 8 January 1992, revised version 8 March 1993
This paper presents a theoretical and experimental work carried out by the Aerospace Department of the University of Rome and ENEA-DISP on local approach. The main goal was to apply the local approach criteria to cleavage fracture prediction in a structural material. The material considered was a low alloy steel used in the nuclear industry, type 22NiMoCr37, of German production provided by KWU in the framework of an international round robin on local approach to fracture in steels, promoted by the European Structural Integrity Society (ESIS). Twelve tension tests on round notched bars were performed at low temperature (-90"C) in order to get the experimental data needed to calculate the theoretical Weibull stress and exponent and infer the probability of failure by a finite element analysis. Results have been applied to a three point bend specimen of the same material to predict cleavage fracture.
1. Introduction It is well known that the theoretical strength of a material to cleavage is about one tenth of the Young's modulus E. At variance, practice indicates that it rarely exceeds few thousandth of that value. Griffith [1] was the first to point out that responsible for that discrepancy were defects like cracks, always present in the real materials even though very small. Indeed, the effective reduction of the strength of a real material can be ascribed to the non-homogeneous plastic deformation that takes place somewhere in the material under stress when dislocations activated in unfavourable oriented grains pile-up against a barrier, like an inclusion or grain-boundary, building up a very high local stress which originates the tiny crack responsible for the early failure of the material. In the brittle fracture of a body containing a crack, that happens when the material is still behaving elastically on a macroscopic scale and the nominal stress is sometimes well below the yield strength. Outstanding works by Low [2], Knott [3], Petch [4] and others [5] have definitely demonstrated the local nature of the plastic deformation preceeding the generation of the crack that may start the brittle fracture. These observations suggest to investigate the fracture resistance on a microscopic base developing approaches that can be de-
fined as local since they relate the overall resistance of a structure to a local behaviour of the material that may trigger the unstable fracture.
2. Local approach A more recent local approach to the fracture process has been developed by Mudry and Pineau [6,7]. Its basic hypothesis is that cracks originate in the material where a plastic deformation occurs while an elastic region may be considered crack free. If a flow is generated instability occurs as the stress % normal to the crack plane reaches the critical value:
ro
o'. = ~-a
(2.1)
where a is the half crack length and Kc is the fracture toughness of the material. Equation (2.1) represents the Griffith principle. What makes local approach differ from classical Griffith principle that applies to an existing crack is the fact that now the crack is not known "a priori". Hypotheses must be made on flaws distribution in terms of size and orientation within the material. We assume that in the zone of the material that reaches the yield strain a population of micro-de-
0 0 2 9 - 5 4 9 3 / 9 3 / $ 0 6 . 0 0 © 1993 - Elsevier Science P u b l i s h e r s B.V. All rights reserved
2
P.P. Milellaet al. / Cleavagefractureprediction If the function g(O) takes the most general form of a Gauss' distribution, then:
Tr/2 fo~/Zg(o) dO = 1
(2.6)
and the failure probability distribution function is then related only to the crack length a. Furthermore assuming for the density probability function P(a) the form:
P(a) = a-",
(2.7)
eq. (2.5) yields: Fig. 1. Boundary of stability domain (shaded area).
1
( K e ) -2(n-1)
e,(0.1) =
;T
l ll fects is generated which does not increase with further increments of plastic strain. For any of those cracks, instability does not occur as long as: [0" 1 c o 8 2 ( 0 ) + 0 " 2 sin2(0)]~a- < K c ,
(2.2)
1-n
(2.8)
KoI
with m = 2(n + 1). Any distribution function can be given the form 1 - exp[-~p(x)] so that Pf(0.1) can be expressed also as: pf = 1 - exp[-~0(0.1)],
where 0.~ > 0.2 are the principal stresses, a is the crack length and K c is the fracture toughness. If P(a) is the probability density function of defects, that is to say the probability that a defect has a length between a and a + da, and g(O) that of the defect's orientation, then the distribution function pf of the probability of failure is given by:
(2.9)
where ~0(0.~) is a proper function that can be determined as follows: if 0.u is the cleavage stress a t w h i c h the material fails, no matter how small the size of a real crack, possibly present in the material, can be, then the probability that it fails under a stress lower or equal to 0.u must be l:a Pf(0.u) = 1,
Pf= ffs(al, crzP(a)g(O) da dO"
(2.3)
pf represents the probability that failure occurs under a combination of principal stresses equal or lower than a given set of stresses 0.1, 0.2, in the failure domain S(0.1, 0.2) defined by eq. (2.2). In the a, 0 plane the non failure locus is shown in Fig. 1 by the shaded area. If one assumes that 0.z = 0 then the boundary of the stability domain moves to the right as depicted in Fig. 1 and the analysis become less conservative, yet the most critical condition is always represented by:
(2.4) The hypothesis cr2 = 0 allows to simplify the integration domain eq. (2.3) that can be rewritten as:
(2.10)
and, recalling eq. (2.8):
l_la/" 1= 1-nkKcl
(2.11)
or:
1 - n = (0.u/Kc) m.
(2.12)
Substituting eq. (2.12) in eq. (2.8) and recalling eq. (2.9), yields: 1 -- e x p [ -
(2.13)
~0(0.1) ] -~- (0.1/0.u) m .
For m :~ 1 and 0.u >0.1, as in our case, i.e. when (0.1/0.,)" is small compared to unit, it can be assumed, as first approximation: ~o(0.1) = (0.1/0.~) m
(2.14)
and eq. (2.9) becomes: Pf(0.1) = 1 - exp[
- (0.1/0.u)m]
,
(2.15)
P.P. Milellaet aL / Cleavagefractureprediction
3
of volumes V0, then, for the theory of the weakest link [8] we can finally write:
Equation (2.15) expresses the probability that a material fails under a stress lower or equal to a given value tr z. Recalling our fundamental hypothesis that instability occurs when a crack generated anywhere in the plastic enclave, of volume V, reaches a critical value, we can split the plastic volume V into s elementary volumes V0, each of which the probability of failure given in eq. (2.15) can be applied to and say that the overall failure occurs when any of those elementary volumes V0 fails. Assuming a statistical independence
p,(o.l,V)=l-exp[-I°'llms]
= 1 - exp -
--
~
.
The statistical independence of the volumes can be
R=2mm 111111 IIIIII IIIIII
) I
R
R=
=
(2.16)
!
L
4 mm
lOmm
Fig. 2. Geometries of the three specimens considered in this study with mesh used in the FE analysis.
4
P.P. Milella et al. / Cleavagefractureprediction
asserted on the basis of the weak stress gradient experienced through the volume V since it is a plastic enclave in which for an elastic-perfect plastic material the stress would be levelled off to the yield stress and in a real material is confined within the strain hardening range.
3. Determination of the probability of failure Equation (2.16) can be solved using a combination of experimental measurements and finite element calculations in which it can be written as:
Pf(~l, V ) = l - e x p [ i=1 .....
(3.1) in which Ne is the number of element used in the mesh and NG is the number of Gauss' points that yielded. Introducing the Weibull stress ¢rw as:
o-~=
E i=1 ..... Ne jfl
o." 1U
(3.2) V0
..... N G
eq. (3.1) finally becomes:
[( t (7"W
p f ( o " I, V) = 1 - exp -
~
.
(3.3)
The finite elements calculations provide the relationship between ~r, and the applied stress for different geometries using the equation (3.2) in which m and V0 are given a first approximation value and A V o are the elementary volumes where a plastic deformation occurs. Parallel to calculation an experimental program was carried out on specimens of equal geometries. For each geometry the failure stress was measured and the Weibull stress inferred entering the theoretical ~rw, ~r relationship provided by eq. (3.2) for that geometry. By doing so the % value can be assumed as the average of the Weibull stress o, at failure for the different geometries. This is just a first approximation since cru, as already said, is the cleavage stress, but it can be assumed from a practical point of view instead of the cleavage stress proposed by Knott [9]. Beremin [10] has shown that for m = 22 ~w = 0.9757 cru.
(3.4)
4. Results of finite element calculation and experiments Three different geometries have been considered simulating a round traction specimen of 18 mm diameter carrying circumferential surface notches of different tip radius, namely 2 ram, 4 mm and 10 mm whose depth was always equal to 4 mm, as proposed by Beremin [11]. The relative meshes are depicted in Fig. 2. The true stress-strain relationship for the material under investigation, a low alloy steel 22 NiMoCr37 of German production used in the nuclear industry, is shown in Fig. 3. Its chemical composition and mechanical properties are shown in Table 1 and 2. The results of the analysis performed with MARC code were used as input data for an ad hoc program to calculate the corresponding Weibull stress given by eq. (3.2) through an iterating process in which m is given an initial value, equal to 15 in our case, to which it corresponds to a slope of the failure distribution function pf(crl, V). If the experimental data does not match the computed distribution curve, m is changed and the iteration continues to find the agreement between experimental points and theoretical prediction. In our case, it was found a good agreement, as also suggested by Mudry, for m = 22 and the iteration stopped. The experimental program was carried out with 12 tensile specimens having the same notch tip radii selected for the FE calculation. Specimen were broken at - 9 0 ° C since the final goal was to predict the brittle fracture of three point bend specimen. Using the iterating procedure just mentioned it was possible to construct the applied load versus Weibull stress curve, Fig. 4, to derive the Weibull stress at failure and then cru, as the average Weibull stress, that turned out to be equal to 2478 MPa. Finally equation (3.3) was used to calculate the
1000
" 900"
800700"
600"
~_
- i/ 500
2
steel
22NiMoCr37
!
400
.... 0.0
, .... , .... , .... ~ .... , .... , .... , .... , .... 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 TRUE PLASTIC STRAIN
F i g . 3. T r u e s t r e s s - t r u e
strain curve of the 22NiMoCr37
steel.
P.P. Milella et aL / Cleavagefracture prediction Table 1 Material chemical composition Mn
P
W
Cr
S
Mo
C
Si
Ni
0.22
0 . 2 3 0 . 8 8 0.006 0.004 0 . 3 9 0 . 5 1 0.84
1.o
5
/
__J
2 0.8
CTu =
2478
MPo
b_
R=2mm m R = 4rnm A
o
0.5 d
Table 2 Material properties R~. 2
Rm
NDT
USE-CV energy
[MPa]
A5 [%]
Z
[MPa]
[%]
[°C]
[J]
980
1015
24
40
- 40
189
< 0.3 ~3 o t~ G_
R = 10 mm X
0.0
n i
1500
n ,
,
i
i
,
,
i
n i
,
I
u
I
n 1
2000
u I
I
I
u *
,
2500
I
F n
n t
I
I
3000
I
,
r
u
n
u
n
,.3500
WEIBULL STRESS (MPo) Fig. 5. Calculated probability of failure versus Weibuli stress. failure probability distribution versus the Weibull stress shown in Fig. 5 together with the data point. The agreement is quite good.
90 Applied Lood versus Welbull stress m=22
R-2mm
5. Brittle fracture prediction
El
70 0 __J
R-4mm
~ 6o ....I 0..
R
•~ 50 40
-
10
mm
0
, J U J l , J J , l l U n , l u u u n l J l U ' l , r u n l u ' ~ l ' ' ' , l , = ' '
1000 1250 1500 1750 2000 2250 2500 2750 3000 3250
WEIBULL STRESS (MPo) Fig. 4. Applied load versus Weibull stress for the three geometries considered.
The purpose of this investigation was to apply the local approach methodology to predict the brittle fracture, i.e. the Kxc toughness, of the material considered in this program using a three point bend specimen containing a crack. This specimen, an ASTM E399 bend one [12], has been analysed with F.E. using the MARC code with the 724 elements mesh shown in Fig. 6 having 1554 degrees of freedom. At each load step the program calculated the K] value. The correspond-
nnmnmunn nmnlunnun
nmmb':;J~,~ ~."dnE ;.:- ;!!!!i
~'-qL~-_:':'/iF~ nnn,~'-.~-I uummnmmml Inuununl innnnnnl Y
l, Fig. 6. Mesh of the Kic specimen used in the FE analysis.
P.P. Milella et al. / Cleavage fracture prediction
6 1.00
f
by ASME [13]. They are shown in Fig. 8. It can be seen that toughness data at 70°C below the Reference Transition Temperature (RTNDx), as it is in our case, range between 80 MPa~fm- and 30 M P a v ~ . It must be considered, however, that the experimental data refer to different heats of materials which increases the spread of the results.
13
i,i a,0.75 b,._ 0
0.50
m 0.25 < 0 t3~ Q_ 0.00
6. Conclusions uu),ll;lll
0
,i, l,~11.,[u~
10
20
r~F,l
ii*)
IIIUlII[ITIIIII
30 40 KI (MPo ~r~)
r IllUl)'II'
50
'
50
Fig. 7. Probability of failure versus applied K x.
ing Weibull stress % was calculated for the different load steps using the ad hoc program described in the previous section. Once the Kr-xrW relationship is obtained, the probability of failure of the specimen versus the applied K z can be derived entering equation (3.3) as shown in Fig. 7. Fig. 7 says that the probability that a material fails under an applied K I lower or equal to 25-30 MPavrm is practically negligible. These values should be considered, then, as lower bound toughness values. Conversely, the probability to get a fracture at or below 60 MPavCm- reaches the maximum value. We have, then, derived a scatter band for the material toughness at -90°C. Experimental data on the material used are not yet available, but very significative are the data obtained on analogous material and published
3OO
u
a.
2oOloo i
er bound
Lit.
0
-200
i
l
)
-100
0
100
200
Temperature ( T-RTNDT ) C Fig. 8. Fracture toughness data obtained at various temperatures (relative to RTNDT) for carbon steels (ref. ASME).
Even though the experimental program to measure the KI¢ toughness using three bend specimens of the carbon steel used in this study has not been carded out yet, experimental data already obtained and published by the American Society of Mechanical Engineers (ASME) on a large variety of carbon steels show quite an excellent agreement with the theoretical predictions based on local approach. This seems to be encouraging the use of the local approach criterion to predict a brittle fracture event. Also the m value suggested by Mudry, m = 22, is likely to be the most appropriate one for the material under investigation. The K k testing on the specific material used in this study will provide further information on the applicability of the local approach method.
References
[1] A.A. Griffith, The phenomena of rupture and flow in solids, Phi. Trans. Roy. Soc. of London A 221 (1921). [2] G.R. Low, IUTAM Madrid Colloquium, Deformation and Flow of Solids, pp. 60 (Springer Verlag, Berlin, 1956). [3] J.F. Knott and A.H. Cottrell, Iron Steel Institute, pp. 201-249, 1963. [4] N.J. Petch, The fracture of metals, Prog. Met. Phy. 5 (1954) 1. [5] P.J. Worthington, E. Smith, Acta Metall. (1966) 14-35. [6] F. Mudry, Local approach to cleavage fracture, Nucl. Engng. Des. 105 (1987) 65-76. [7] A. Pineau, Review of fracture micro-mechanisms and local approach to predicting crack resistance in low strength steels, Proc. of Fifth Int. Conf. of Fract. (ICF5) Cannes, France, 1981. [8] W. Weibull, A statistical distribution function of wide applicability, J. of App. Mech. (Sept. 1951) 293-297. [9] J.F. Knott, On stress intensification in specimen of charpy geometry prior general yield, J. Mech. Phys. and Solids 15 (1969) 97. [10] F.M. Beremin, A local criterion for cleavage fracture of a nuclear vessel steel, Met. Trans. 14A (Nov 1983) 22772287.
P.P. Milella et aL / Cleavagefracture prediction [11] F.M. Beremin, Calculs elasto-plastic par la m6thod des elementes finis d'Eprouvettes axisymmetriques entaile circulairement, J. de M6ch. App. 4 (1980) 397-325. [12] Annual Book of ASTM Standards, Vol 3.01. [13] W. Oldfield and W.L. Server, Fracture toughness predic-
tion for pressure vessel steels: the development of statistically based method, in Reference Fracture Toughness Procedures Applied to Pressure Vessel Materials, MCP24 ed. T.R. Mager, ASME, N.Y. (1984) pp. 9.