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First spinning cylinder test analysis using a local approach to fracture
ELSEVIER
Nuclear Engineering and Design 152 (1994) 11-18
Nuclear F.ngi.neng addDmgn
First spinning cylinder test analysis using a local approach to fracture C. Eripret, G. Rousselier Electrit~ de France (EDF), Service R~acteurs Nucl~aires et Echangeurs (RNE), Les Renardi~res, BP 1, 77250 Moret sur Loing, France
Abstract
In recent years, several experimental programmes on large-scale specimens have been organized to evaluate the capabilities of the fracture mechanics concepts employed in structural integrity assessment of pressurized water reactor pressure vessels. During the first spinning cylinder test, a geometry effect was revealed experimentally showing the difficulties of transferring toughness data from small-scale to large-scale specimens. An original analysis of this test, by means of a local approach to fracture, is presented in this paper. Both compact tension specimen and spinning cylinder fracture behaviour were computed using a continuum damage mechanics model developed at EDF. We confirmed by numerical analysis that the cylinder's resistance to ductile tearing was considerably larger than in small-scale fracture mechanics specimen tests, about 50%. The final crack growth predicted by the model was close to the experimental value. Discrepancies in J R curves seemed to be due to an effect of stress triaxiality and plastic zone evolution. The geometry effect inducing differences in resistance to ductile tearing of the material involved in the specimens can be investigated and explained using a local approach to fracture methodology.
1. Introduction In recent years, several experimental programmes on large-scale specimens have been organized to evaluate capabilities of the fracture mechanics concepts employed in structural integrity assessment of pressurized water reactor (PWR) pressure vessels (Bryan, 1987; Kussmaul, 1985; Lacey, 1989; Okamura, 1990). Most of them aimed to investigate the upper shelf toughness fracture behavior of low alloyed steels, and to assess the validity of the J-integral and J-resistance curve concepts regarding ductile crack propagation. Thus, it is now universally accepted that toughness may depend on thickness, loading patterns, Elsevier Science S.A. SSDI 0029-5493(94)00798-4
degree of triaxiality, and geometry of the structure. The J R curve concept was shown not to be an intrinsic characteristic of the material properties, but may vary in some circumstances. It is therefore very important to identify the different factors that can affect this toughness, and to quantify their influence. The following underlying questions should be posed. Is the J R curve a representative measurement of tearing toughness when crack propagation occurs? Is it possible to transfer the J R curve obtained from laboratory specimen tests to other geometries or configurations? This paper does not fully answer the questions above, but highlights the importance of the prob-
12
E. Eripret, G. Rousselier / Nuclear Engineering and Design 152 (1994) 11-18
1300 mm
lem and proposes an explanation of the geometry effect revealed in the first spinning cylinder experiment. An interesting analysis performed using a local approach to fracture methodology showed the influence of the near crack tip stress and strain fields on the fracture behaviour of steels, and explained the geometry effect observed.
i I i J ~==K ~--:
--
2. The first spinning cylinder experiment
The first spinning cylinder test, performed by Northern Research Laboratories, involved the rotation of a 200 mm thick cylinder containing a full length axial flaw to an angular speed of 2600 rev min-1 at a temperature of 290°C (Lacey, 1989). This test aimed at generating ductile crack growth by increasing progressively the rotational speed, creating a membrane hoop stress loading across the thickness. The first objective of this test was to provide experimental data that would permit the construction of a J-resistance curve. This has been achieved by using the measurement of an a.c. drop potential to determine the crack growth, and by performing a finite element analysis to associate to each rotational speed a corresponding value of J-integral. In addition, a J R curve was derived from experimental results corresponding to small-scale compact tension specimen tests and gave considerably lower J-values than the cylinder's J-resistance curve (Fig. 1). A geometry effect (scale, load, or size effect?), was revealed experimentally and showed the problem of transferability of toughness data from small-scale to large-scale specimens.
3. Analysis by local approach methodology An original analysis of this test, by means of a local approach to fracture is presented in this paper. The model used in this paper refers to the generalized standard material constitutive relations (Rousselier, 1981), and enables modelling of material tearing and crack propagation without
__ -- 2.~/f
/ Slot depth : 107 mm
Fig. l. Cylinder geometry for first spinning cylinder experiment.
using any numerical technique such as node release. The main advantage of this approach is assessment of crack initiation and growth using criteria derived from the near crack tip stress and strain fields (local values), which control the material damage. The evolution of the damage is governed by competition between material hardening and softening. These effects are included in the constitutive relations by modifying the expression of the plastic potential as follows: F = Fhardening + Fdamage ]
= ~Yeq - - R ( p ) -]- D B ( f l ) e x p ( am
p where
\ptr~/
3 O'eq =
~1/2
5 U#O"/j )
1 O"m = ~
(Uii)
The constitutive relations are derived from F, and from the yield criterion F = 0, using the normality rule. In this expression, D and tr~ are constants, p is the hardening variable, and fl the damage variable. Material hardening is assumed to be isotropic, as well as damage. The second term R(p) represents the true stress-true strain curve of the material, and the function B(fl) is equal to
B(fl) =
aLfo exp fl 1 - f o +)Coexp fl
where )co is also a scalar that defines the initial volume fraction of cavities. Material softening caused by growth of cavities is taken into account
E. Eripret, G. Rousselier / Nuclear Engineering and Design 152 (1994) 11-18
13
//
900 800
True stress, true strain curve
// /
700 600 .= 500
400 300 200 H !
100 I
1 0.04
I~f
I 0.08
I
t 0.12
I
I 0.16
I
I 0.20
True strain Fig. 2. S t r e s s - s t r a i n
T Initial crack tip /
I
1 rnm
I
(a)
curve accounting for damage.
2500
through the third term Fdamage,which competes with the hardening part Fhardening. As loading is increased, the plastification effects make the cavities grow, and damage the material. When the damage becomes important, softening of the material takes place and the stress-strain relation decreases (Fig. 2). The material resistance becomes lower and lower, until failure occurs. From the results of calculation, we can determine the instants at which the crack growth initiates, as well as the position of the crack tip during propagation when the opening stress reaches a maximum just before collapsing (Fig. 3). The first maximum observed defines crack initiation, the second maximum occurs when the first element fails, the third maximum defines the failure of the second element, and so on (Rousselier, 1989). Then, combining numerical results and J-contour integral calculation provides a numerical J R curve characteristic of the structure behaviour with regard to ductile tearing. Using Rousselier's model, one needs to identify the three parameters that control the fracture behaviour: the initial volume fraction of cavities fo, the metallic matrix stiffness a~, and the characteristic length l~ of the finite elements. Usually, fo is estimated from the chemical composition of the material (Mudry, 1982). In fact, manganese sulphide inclusions play an essential
~Oyy (MPa) A
._
oJ (rounds/mn) (b)
Fig. 3. (a) Stable crack growth (damaged zone in grey). (b) Hoop stress at the crack tip (the symbols correspond to the location in (a)).
role in cavity growth and we can relate directly the initial volume fraction of cavities fv with the percentage of Mn and S through Franklin's formula: f0 =fv = 0-054( S°/o-- 0.001/Mn°/o) The second parameter a] may be estimated from the value of flow stress but this gives a poor evaluation. In practice, mechanical testing is necessary to calibrate al (Mudry, 1982). The basic specimens used are axisymmetric notched tension specimens, .for which the calibration procedure is
E. Eripret, G. Rousselier / Nuclear Engineering and Design 152 (1994) 11-18
14
: Diametral contraction
AO
J (kJ/m2)
P
6OO
A, A ' : Crack initiation /
5OO 4OO
,oad
a
3OO
200
100
AO= 0 o - 0
model simulatio I"1 Experimental
~0= 0o - • I
(a) Effect of 01 (or fo )
(b) Effect of ~c
Fig. 4. Numerical load-displacement curves of a n o t c h e d tension specimen (schematic diagram).
depicted in Fig. 4. However, we could not follow the same procedure in this study, becasue we had neither coupons of material for machining notched specimens, nor experimental results (provided from axisymmetrical notched specimen tension tests) to compare with numerical computations. Thus, calibration was done with the help of results from compact tension (CT) specimens, and the model parameters were determined so that the calculated J R curve (determined from CT speci-
0
1
i
~oints i 2
3 da (ram)
Fig. 5. Comparison of experimental and numerical J R curves for CT specimens.
men test simulation) fitted the experimental curve. The two curves, which are in good agreement, are plotted in Fig. 5. The mechanical properties were taken from Lacey and Leckenby (1989) as well as the chemical composition of the material (Table 1). The parameters were found to be f0 = 6 x 10 -4, al = 350 MPa, and lc = 0.55 mm.
Table 1 Mechanical properties of spinning cylinder test material
Identification Nominally A508 Class 3 composition in a non-standard quenched and tempered condition
Chemical analysis C 0.22
Thermal
Si 0.20
Mn 1.32
S 0.012
P 0.012
Cr 0.08
Mo 0.57
Ni 0.78
treatment
Austenitize Quench Temper
6 H at 1065 °C in water from 1065 °C 7 h at 590 _ 10 °C
Test temperatures 290 °C
Engineering and'true stress-strain tensile data E-modulus (MPa) Rpo.2 (MPa) R M (MPa) v, Poisson ratio
193 000 (measured using an electrostatic resonance technique) 540 710 0.275 (determined from biaxial strain gauge measurements of material strips loaded in tension)
E. Eripret, G. Rousselier / Nuclear Engineering and Design 152 (1994) 11-18
Then, a two-dimensional finite element computation was performed to analyse the fracture behavior of the spinning cylinder. Obviously, the same parameters of Rousseher's model were used to carry out this computation. The same size of element was kept to mesh the crack tip area, according to Rousselier et al. (1989). This condition ensures that the crack propagation speed in the CT specimen and in the cylinder will be close. As our finite element code did not account for body forces, we replaced the rotation load by internal pressure which provided an equivalent hoop stress profile for an uncracked structure (Dexter, 1990) (at most 3.6% error at inner surface). However, these loadings are not equivalent regarding the radial stresses; the contraction due to pressure is very different from the contraction caused by spinning. This difference may influence the stress and strain fields in the vicinity of the crack, and also the evolution of damage. The influence of the radial stress on ductile tearing will have to be clarified in further studies. The equivalence between loadings provided from pressure or rotational speed is given by (Pissarenko, 1975) trr,~ =
tr=~ ~
+
-
R,)
Ri
where a~.. and a=.~ are the average values of hoop stress through the thickness generated respectively by internal pressure and by rotational speed:
;
J (kJ/m 2) 600
5OO
-/"
A
A
A
4OO
300
( 200
Numerical J-R curve derived frorr Rousselier's mode
100
A C-0
,
simulation Experimental Pts
I
I
1
2
3 da (mm)
Fig. 6. Comparison of numerical and experimental J R curves for spinning cylinder experiment.
about 50%. The final crack growth (about 2.5 mm) obtained at 2600 rev min -t (corresponding to an internal pressure of 85 MPa) was close to the experimental value (2.75 mm). Scatter in J R curves seemed to be due to an effect of stress triaxiality and plastic zone evolution, which are very different in the two situations. It should be Table 2 Typical stress-strain data at 290°C for first spinning cylinder test material
P =f(co)
2 (Ro +
=P 3
15
.
We performed the numerical simulation of the behaviour of the cracked cylinder under internal pressure and translated results of both crack propagation and J-integral into values depending on the rotational speed. The J R curve obtained numerically is presented in Fig. 6, and compared with the J R curve derived from CT specimen testing. We confirmed by numerical analysis that the cylinder's resistance to ductile tearing was considerably larger than in small-scale fracture mechanics specimen tests,
True strain (%)
True stress (N mmj-2)
0.003906 0.05662 0.1054 0.1542 0.2007 0.228 0.2572 0.3118 0.3916 0.4985 0.6054 0.714 0.8225 0.9329 1.043 1.153 1.207 1.261
7.9 111.1 207 303.5 393 433.8 461.8 496.2 525.3 548.4 564.5 576.4 587.5 595.6 604.4 611.8 614.4 617.9
16
E. Eripret, G. Rousselier / Nuclear Engineering and Design 152 (1994) 11-18
4
am/%q am/%q ~ r - I ~ (Cylinder)
~_/tr_
4
, 0
I 1
,
I 2
, 3 Di=tance (rnrn)
Fig. 7. Evolution of stress triaxiality along the ligament for CT specimens and spinning cylinder experiment.
noted that we denote stress triaxiality by the ratio Um/Ueq.
The CT specimen tests involved a quasi-pure bending loading of the structure, which is totally different to membrane loading as generated by internal pressure or spinning. This loading effect may be responsible for changing the J-resistance curve level, at the initiation point as well as during crack propagation. The stress triaxiality around the crack tip is larger in the CT specimen than in the hollow structure (Fig. 7). Then, the material damage will increase earlier in the crack tip area of a CT specimen: the growth of cavities which is directly dependent on the stress triaxiality level (Bethmont, 1989), will be quicker. The resistance of steel to ductile tearing will be lower in that case.
4. Discussion of the dependence of finite element analyses on mesh size Through this numerical analysis carried out with a local approach to fracture, it has been shown that an essential parameter of this kind of model is the mesh size. This finite element size plays an important role in application of local fracture mechanics concepts, and the effect of finite element analyses has to be highlighted and
explained. In fact, this parameter is the most controversial one because intuitively it is thought that increasing the mesh refinement will provide more accurate results. Thus, the use of a fixed mesh scale which may be large compared with the microstructure or with the stress and strain gradients, may upset any physical reasoning. However, it must be noticed that introducing a distance criterion for failure at the crack tip is absolutely necessary when developing a model based on microstructured controlled fracture process. As far as the local approach modelling is based on microscopic observations of damage mechanisms, and tries to relate the macroscopic fracture behaviour of an homogeneous material to microscopic metallurgical heterogeneities, this way of modelling obeys the rule and has to introduce a scale factor that averages the microscopic mechanisms and microstructural effects. Previous works have already exhibited this conclusion. Rice and Johnson (1970) and later Ritchie et al. (1973) mentioned that for cleavage fracture, where failure occurs on microstructural initiation sites, the critical fracture stress has to be achieved at a distance which is characteristic of the material microstructure. More recently, Neville (1988) introduced a new definition of that critical distance, but demonstrating the same conclusion: introducing microstructural effects of failure in a continuum mechanics analysis requires a characteristic distance that relates the mechanical behaviour at microscopic scale to the macroscopic scale. The physical reason for this is that, for a sharp crack as well as for a blunt crack, the stresses or strains in the highly stretched zone will always exceed a critical value. Then, any failure criterion expressed in terms of critical stress for cleavage or critical strain for ductile tearing will be achieved in a process volume near the crack tip. Therefore, if no critical distance had to be introduced, the minimum toughness for any microstructured material would be zero. As far as a scale factor must be introduced, the solution that has been retained for the local approach to fracture models is to introduce it directly through the mesh size. This is the simplest solution, but may not be the most satisfactory from a physical point of view. Recent works on strain or
E. Eripret, G. Rousselier / Nuclear Engineering and Design 152 (1994) 11-18
damage localization have shown that it is possible to make the results o f local models independent of the mesh size by using a redistribution function in the finite element analysis (Pijaudier-Cabot, 1988). Then, the critical distance appears in the "delocalization" procedure by determining the width of the Gaussian shaped redistribution function. This way of modelling may be more satisfactory, but is still time consuming in numerical analysis, makes the finite element code more difficult to operate, and lastly exhibits the same difficulties in relating this characteristic distance to any microstructural scale. Thus, although it is not physically justified, introducing this scale through the mesh size seems to be the simplest solution and the most convenient for today's industrial applications. This parameter must be fitted numerically in order to account for coalescence o f growing cavities (interactions between elementary cells containing an insolated cavity). The mesh size has therefore a limited influence on crack initiation, and a large influence on propagation. Another point deserves to be highlighted; it concerns the role played by the microstructure on the resistance of material or industrial structure to fracture. In order to predict structural integrity in connection with microstructural fracture processes, four different scales of observations must be considered. The first related to microstructure and material material microscopic heterogeneities, the second concerns the scale of continuum mechanics, the third represents the scale of the process zone (damaged zone or yielded area), and the last is the size of the structure. The microstructural distance can be related to mean spacing between inclusions or carbides, or any other particles that play a role in the microstructure fracture process. When comparing this distance with the process zone size, two cases must be considered. If the plastic zone, or crack tip opening displacement, is much larger than the mean spacing between inclusions, the effect of microstructure on material failure is very limited. It can be considered that the material, observed at the scale of C T O D or mesh size, is rather homogeneous. In that case, the macroscopic tensile properties of the material, even if including damage, are determined at a scale which already averages micro-
IIIllllll
I I
17
1
Fig. 8. Mesh of CT specimen.
Fig. 9. Mesh of the cylinder. structural effects. However, if the continuum mechanics scale and the mean spacing are within the same order o f magnitude, the characteristic distance will obviously play a greater role on crack initiation as well as crack growth. Moreover, if the distance between initiation sites is greater than C T O D or continuum mechanics scale, the effects o f microstructure on fracture will be enhanced and any attempt to model the fracture process will have to include statistics on the geometrical distribution of inclusions in order to be able to account for structural resistance as well as for scatter associated with crack initiation and toughness measurements. Once again, the simplest solution is to use the mesh size as an averaging tool for stress and strain gradients, and also for the effects of microstructure on failure.
5. Conclusions This modelling, instead of applying criteria based on global loading parameters, describes the damage evolution from local values of stress and strain fields. For this reason, this method is able to account for local effects o f the crack area loading factors, such as stress triaxiality.
18
E. Eripret, G. Rousselier / Nuclear Engineering and Design 152 (1994) 11-18
The geometry effect including differences in resistance to ductile tearing of the specimens can be investigated and explained by using a local approach to fracture methodology. Rousselier's model proved to be an efficient tool for understanding ductile tearing behaviour of steel, and gives answers where classified fracture mechanics concepts fail. References M. Bethmont, G. Devesa and G. Rousselier, A methodology for ductile fracture analysis based on damage mechanics: an application of local approach to fracture to the NKS-3 thermal shock experiment, 15th MPA Seminar, Stuttgart, 5-6 October 1989. R.H. Bryan et al., Pressurized thermal shock experiment of 6 in-thick pressure vessels--PTSE-2: investigation of low tearing resistance and warm-prestressing, NUREG CR4888, ORNL-6377 (Oak Ridge National Laboratory, Oak Ridge, TN, 1987). R.J. Dexter, Analysis of the first spinning cylinder experiment, CSNI/FAG Specialists Meet., FALSIRE Project, Boston, MA, 8-10 May 1990. K. Kussmanl and A. Sauter, Application of ductile fracture mechanics to large-scale experiment simulation and analyses for pressurized thermal shock behavior of RPV's, Int. Conf. on Fatigue, Corrosion Cracking, Freture Mechanics and Failure Analysis, American Society for Metals, Salt Lake City, 2-6 December 1985.
D.J. Lacey and R.E. Leckenby, Determination of upper shelf fracture resistance in the spinning cylinder test facility, Trans. 10th Int. Conf. on Structural Mechanics in Reactor Technology, Division F, pp. 1-6, Anaheim, CA, August 1989, Division F, pp. 1-6. F. Mudry, Etude de la rupture ductile et de la rupture par clivage d'aciers faiblement alli6s, Th6s¢ d'Etat, Universit6 de Technologic de Compi~gne, 23 March 1982. D.J. Neville, On distance criterion for failure at the tips of Cracks, minimum fracture toughness, and non-dimensional toughness parameters, J. Mech. Phys. Solids 36(4) (1988) 443-457. H. Okamura et al., Japanese pressurized thermal shock experiment program, Specialists Meet., CSNI/FAG Project for Fracture Analyses of Large-Scale International Reference Experiments (Project Falsire), Boston, MA, 8-10 May 1990. G. Pijaudier-Cabot and Z.P. Bazant, Nonlocal damage theory, ASCE J. Eng. Mech. 113 (1988) 1512-1533. G. Pissarenko, A. Yakovlev and V. Matveev, Material Strength Memorandum, MIR, Moscow, 1945. J.R. Rice and M.A. Johnson, in Kanninen et al. (eds.) Inelastic Behaviour of Solids, McGraw Hill, New York, 1970, p. 641. R.O. Ritchei, J.F. Knott and J.R. Rice, J. Mech. Phys. Solids 21 (1973) 395. G. Rousselier, Finite deformation constitutive relations including ductile fracture damage. In Three Dimensional Constitutive Relations and Ductile Fracture, North-Holland, Amsterdam, 1981. G. Rousselier et al., A methodology for ductile fracture analysis based on damage mechanics: an illustration of a local approach of fracture, Nonlinear Fracture Mechanics, Vol. II, Elastic-Plastic Fracture, ASTM STP 995, ASTM, Philadelphis, PA, 1989, pp. 332-354.
Nuclear Engineering and Design 152 (1994) 11-18
Nuclear F.ngi.neng addDmgn
First spinning cylinder test analysis using a local approach to fracture C. Eripret, G. Rousselier Electrit~ de France (EDF), Service R~acteurs Nucl~aires et Echangeurs (RNE), Les Renardi~res, BP 1, 77250 Moret sur Loing, France
Abstract
In recent years, several experimental programmes on large-scale specimens have been organized to evaluate the capabilities of the fracture mechanics concepts employed in structural integrity assessment of pressurized water reactor pressure vessels. During the first spinning cylinder test, a geometry effect was revealed experimentally showing the difficulties of transferring toughness data from small-scale to large-scale specimens. An original analysis of this test, by means of a local approach to fracture, is presented in this paper. Both compact tension specimen and spinning cylinder fracture behaviour were computed using a continuum damage mechanics model developed at EDF. We confirmed by numerical analysis that the cylinder's resistance to ductile tearing was considerably larger than in small-scale fracture mechanics specimen tests, about 50%. The final crack growth predicted by the model was close to the experimental value. Discrepancies in J R curves seemed to be due to an effect of stress triaxiality and plastic zone evolution. The geometry effect inducing differences in resistance to ductile tearing of the material involved in the specimens can be investigated and explained using a local approach to fracture methodology.
1. Introduction In recent years, several experimental programmes on large-scale specimens have been organized to evaluate capabilities of the fracture mechanics concepts employed in structural integrity assessment of pressurized water reactor (PWR) pressure vessels (Bryan, 1987; Kussmaul, 1985; Lacey, 1989; Okamura, 1990). Most of them aimed to investigate the upper shelf toughness fracture behavior of low alloyed steels, and to assess the validity of the J-integral and J-resistance curve concepts regarding ductile crack propagation. Thus, it is now universally accepted that toughness may depend on thickness, loading patterns, Elsevier Science S.A. SSDI 0029-5493(94)00798-4
degree of triaxiality, and geometry of the structure. The J R curve concept was shown not to be an intrinsic characteristic of the material properties, but may vary in some circumstances. It is therefore very important to identify the different factors that can affect this toughness, and to quantify their influence. The following underlying questions should be posed. Is the J R curve a representative measurement of tearing toughness when crack propagation occurs? Is it possible to transfer the J R curve obtained from laboratory specimen tests to other geometries or configurations? This paper does not fully answer the questions above, but highlights the importance of the prob-
12
E. Eripret, G. Rousselier / Nuclear Engineering and Design 152 (1994) 11-18
1300 mm
lem and proposes an explanation of the geometry effect revealed in the first spinning cylinder experiment. An interesting analysis performed using a local approach to fracture methodology showed the influence of the near crack tip stress and strain fields on the fracture behaviour of steels, and explained the geometry effect observed.
i I i J ~==K ~--:
--
2. The first spinning cylinder experiment
The first spinning cylinder test, performed by Northern Research Laboratories, involved the rotation of a 200 mm thick cylinder containing a full length axial flaw to an angular speed of 2600 rev min-1 at a temperature of 290°C (Lacey, 1989). This test aimed at generating ductile crack growth by increasing progressively the rotational speed, creating a membrane hoop stress loading across the thickness. The first objective of this test was to provide experimental data that would permit the construction of a J-resistance curve. This has been achieved by using the measurement of an a.c. drop potential to determine the crack growth, and by performing a finite element analysis to associate to each rotational speed a corresponding value of J-integral. In addition, a J R curve was derived from experimental results corresponding to small-scale compact tension specimen tests and gave considerably lower J-values than the cylinder's J-resistance curve (Fig. 1). A geometry effect (scale, load, or size effect?), was revealed experimentally and showed the problem of transferability of toughness data from small-scale to large-scale specimens.
3. Analysis by local approach methodology An original analysis of this test, by means of a local approach to fracture is presented in this paper. The model used in this paper refers to the generalized standard material constitutive relations (Rousselier, 1981), and enables modelling of material tearing and crack propagation without
__ -- 2.~/f
/ Slot depth : 107 mm
Fig. l. Cylinder geometry for first spinning cylinder experiment.
using any numerical technique such as node release. The main advantage of this approach is assessment of crack initiation and growth using criteria derived from the near crack tip stress and strain fields (local values), which control the material damage. The evolution of the damage is governed by competition between material hardening and softening. These effects are included in the constitutive relations by modifying the expression of the plastic potential as follows: F = Fhardening + Fdamage ]
= ~Yeq - - R ( p ) -]- D B ( f l ) e x p ( am
p where
\ptr~/
3 O'eq =
~1/2
5 U#O"/j )
1 O"m = ~
(Uii)
The constitutive relations are derived from F, and from the yield criterion F = 0, using the normality rule. In this expression, D and tr~ are constants, p is the hardening variable, and fl the damage variable. Material hardening is assumed to be isotropic, as well as damage. The second term R(p) represents the true stress-true strain curve of the material, and the function B(fl) is equal to
B(fl) =
aLfo exp fl 1 - f o +)Coexp fl
where )co is also a scalar that defines the initial volume fraction of cavities. Material softening caused by growth of cavities is taken into account
E. Eripret, G. Rousselier / Nuclear Engineering and Design 152 (1994) 11-18
13
//
900 800
True stress, true strain curve
// /
700 600 .= 500
400 300 200 H !
100 I
1 0.04
I~f
I 0.08
I
t 0.12
I
I 0.16
I
I 0.20
True strain Fig. 2. S t r e s s - s t r a i n
T Initial crack tip /
I
1 rnm
I
(a)
curve accounting for damage.
2500
through the third term Fdamage,which competes with the hardening part Fhardening. As loading is increased, the plastification effects make the cavities grow, and damage the material. When the damage becomes important, softening of the material takes place and the stress-strain relation decreases (Fig. 2). The material resistance becomes lower and lower, until failure occurs. From the results of calculation, we can determine the instants at which the crack growth initiates, as well as the position of the crack tip during propagation when the opening stress reaches a maximum just before collapsing (Fig. 3). The first maximum observed defines crack initiation, the second maximum occurs when the first element fails, the third maximum defines the failure of the second element, and so on (Rousselier, 1989). Then, combining numerical results and J-contour integral calculation provides a numerical J R curve characteristic of the structure behaviour with regard to ductile tearing. Using Rousselier's model, one needs to identify the three parameters that control the fracture behaviour: the initial volume fraction of cavities fo, the metallic matrix stiffness a~, and the characteristic length l~ of the finite elements. Usually, fo is estimated from the chemical composition of the material (Mudry, 1982). In fact, manganese sulphide inclusions play an essential
~Oyy (MPa) A
._
oJ (rounds/mn) (b)
Fig. 3. (a) Stable crack growth (damaged zone in grey). (b) Hoop stress at the crack tip (the symbols correspond to the location in (a)).
role in cavity growth and we can relate directly the initial volume fraction of cavities fv with the percentage of Mn and S through Franklin's formula: f0 =fv = 0-054( S°/o-- 0.001/Mn°/o) The second parameter a] may be estimated from the value of flow stress but this gives a poor evaluation. In practice, mechanical testing is necessary to calibrate al (Mudry, 1982). The basic specimens used are axisymmetric notched tension specimens, .for which the calibration procedure is
E. Eripret, G. Rousselier / Nuclear Engineering and Design 152 (1994) 11-18
14
: Diametral contraction
AO
J (kJ/m2)
P
6OO
A, A ' : Crack initiation /
5OO 4OO
,oad
a
3OO
200
100
AO= 0 o - 0
model simulatio I"1 Experimental
~0= 0o - • I
(a) Effect of 01 (or fo )
(b) Effect of ~c
Fig. 4. Numerical load-displacement curves of a n o t c h e d tension specimen (schematic diagram).
depicted in Fig. 4. However, we could not follow the same procedure in this study, becasue we had neither coupons of material for machining notched specimens, nor experimental results (provided from axisymmetrical notched specimen tension tests) to compare with numerical computations. Thus, calibration was done with the help of results from compact tension (CT) specimens, and the model parameters were determined so that the calculated J R curve (determined from CT speci-
0
1
i
~oints i 2
3 da (ram)
Fig. 5. Comparison of experimental and numerical J R curves for CT specimens.
men test simulation) fitted the experimental curve. The two curves, which are in good agreement, are plotted in Fig. 5. The mechanical properties were taken from Lacey and Leckenby (1989) as well as the chemical composition of the material (Table 1). The parameters were found to be f0 = 6 x 10 -4, al = 350 MPa, and lc = 0.55 mm.
Table 1 Mechanical properties of spinning cylinder test material
Identification Nominally A508 Class 3 composition in a non-standard quenched and tempered condition
Chemical analysis C 0.22
Thermal
Si 0.20
Mn 1.32
S 0.012
P 0.012
Cr 0.08
Mo 0.57
Ni 0.78
treatment
Austenitize Quench Temper
6 H at 1065 °C in water from 1065 °C 7 h at 590 _ 10 °C
Test temperatures 290 °C
Engineering and'true stress-strain tensile data E-modulus (MPa) Rpo.2 (MPa) R M (MPa) v, Poisson ratio
193 000 (measured using an electrostatic resonance technique) 540 710 0.275 (determined from biaxial strain gauge measurements of material strips loaded in tension)
E. Eripret, G. Rousselier / Nuclear Engineering and Design 152 (1994) 11-18
Then, a two-dimensional finite element computation was performed to analyse the fracture behavior of the spinning cylinder. Obviously, the same parameters of Rousseher's model were used to carry out this computation. The same size of element was kept to mesh the crack tip area, according to Rousselier et al. (1989). This condition ensures that the crack propagation speed in the CT specimen and in the cylinder will be close. As our finite element code did not account for body forces, we replaced the rotation load by internal pressure which provided an equivalent hoop stress profile for an uncracked structure (Dexter, 1990) (at most 3.6% error at inner surface). However, these loadings are not equivalent regarding the radial stresses; the contraction due to pressure is very different from the contraction caused by spinning. This difference may influence the stress and strain fields in the vicinity of the crack, and also the evolution of damage. The influence of the radial stress on ductile tearing will have to be clarified in further studies. The equivalence between loadings provided from pressure or rotational speed is given by (Pissarenko, 1975) trr,~ =
tr=~ ~
+
-
R,)
Ri
where a~.. and a=.~ are the average values of hoop stress through the thickness generated respectively by internal pressure and by rotational speed:
;
J (kJ/m 2) 600
5OO
-/"
A
A
A
4OO
300
( 200
Numerical J-R curve derived frorr Rousselier's mode
100
A C-0
,
simulation Experimental Pts
I
I
1
2
3 da (mm)
Fig. 6. Comparison of numerical and experimental J R curves for spinning cylinder experiment.
about 50%. The final crack growth (about 2.5 mm) obtained at 2600 rev min -t (corresponding to an internal pressure of 85 MPa) was close to the experimental value (2.75 mm). Scatter in J R curves seemed to be due to an effect of stress triaxiality and plastic zone evolution, which are very different in the two situations. It should be Table 2 Typical stress-strain data at 290°C for first spinning cylinder test material
P =f(co)
2 (Ro +
=P 3
15
.
We performed the numerical simulation of the behaviour of the cracked cylinder under internal pressure and translated results of both crack propagation and J-integral into values depending on the rotational speed. The J R curve obtained numerically is presented in Fig. 6, and compared with the J R curve derived from CT specimen testing. We confirmed by numerical analysis that the cylinder's resistance to ductile tearing was considerably larger than in small-scale fracture mechanics specimen tests,
True strain (%)
True stress (N mmj-2)
0.003906 0.05662 0.1054 0.1542 0.2007 0.228 0.2572 0.3118 0.3916 0.4985 0.6054 0.714 0.8225 0.9329 1.043 1.153 1.207 1.261
7.9 111.1 207 303.5 393 433.8 461.8 496.2 525.3 548.4 564.5 576.4 587.5 595.6 604.4 611.8 614.4 617.9
16
E. Eripret, G. Rousselier / Nuclear Engineering and Design 152 (1994) 11-18
4
am/%q am/%q ~ r - I ~ (Cylinder)
~_/tr_
4
, 0
I 1
,
I 2
, 3 Di=tance (rnrn)
Fig. 7. Evolution of stress triaxiality along the ligament for CT specimens and spinning cylinder experiment.
noted that we denote stress triaxiality by the ratio Um/Ueq.
The CT specimen tests involved a quasi-pure bending loading of the structure, which is totally different to membrane loading as generated by internal pressure or spinning. This loading effect may be responsible for changing the J-resistance curve level, at the initiation point as well as during crack propagation. The stress triaxiality around the crack tip is larger in the CT specimen than in the hollow structure (Fig. 7). Then, the material damage will increase earlier in the crack tip area of a CT specimen: the growth of cavities which is directly dependent on the stress triaxiality level (Bethmont, 1989), will be quicker. The resistance of steel to ductile tearing will be lower in that case.
4. Discussion of the dependence of finite element analyses on mesh size Through this numerical analysis carried out with a local approach to fracture, it has been shown that an essential parameter of this kind of model is the mesh size. This finite element size plays an important role in application of local fracture mechanics concepts, and the effect of finite element analyses has to be highlighted and
explained. In fact, this parameter is the most controversial one because intuitively it is thought that increasing the mesh refinement will provide more accurate results. Thus, the use of a fixed mesh scale which may be large compared with the microstructure or with the stress and strain gradients, may upset any physical reasoning. However, it must be noticed that introducing a distance criterion for failure at the crack tip is absolutely necessary when developing a model based on microstructured controlled fracture process. As far as the local approach modelling is based on microscopic observations of damage mechanisms, and tries to relate the macroscopic fracture behaviour of an homogeneous material to microscopic metallurgical heterogeneities, this way of modelling obeys the rule and has to introduce a scale factor that averages the microscopic mechanisms and microstructural effects. Previous works have already exhibited this conclusion. Rice and Johnson (1970) and later Ritchie et al. (1973) mentioned that for cleavage fracture, where failure occurs on microstructural initiation sites, the critical fracture stress has to be achieved at a distance which is characteristic of the material microstructure. More recently, Neville (1988) introduced a new definition of that critical distance, but demonstrating the same conclusion: introducing microstructural effects of failure in a continuum mechanics analysis requires a characteristic distance that relates the mechanical behaviour at microscopic scale to the macroscopic scale. The physical reason for this is that, for a sharp crack as well as for a blunt crack, the stresses or strains in the highly stretched zone will always exceed a critical value. Then, any failure criterion expressed in terms of critical stress for cleavage or critical strain for ductile tearing will be achieved in a process volume near the crack tip. Therefore, if no critical distance had to be introduced, the minimum toughness for any microstructured material would be zero. As far as a scale factor must be introduced, the solution that has been retained for the local approach to fracture models is to introduce it directly through the mesh size. This is the simplest solution, but may not be the most satisfactory from a physical point of view. Recent works on strain or
E. Eripret, G. Rousselier / Nuclear Engineering and Design 152 (1994) 11-18
damage localization have shown that it is possible to make the results o f local models independent of the mesh size by using a redistribution function in the finite element analysis (Pijaudier-Cabot, 1988). Then, the critical distance appears in the "delocalization" procedure by determining the width of the Gaussian shaped redistribution function. This way of modelling may be more satisfactory, but is still time consuming in numerical analysis, makes the finite element code more difficult to operate, and lastly exhibits the same difficulties in relating this characteristic distance to any microstructural scale. Thus, although it is not physically justified, introducing this scale through the mesh size seems to be the simplest solution and the most convenient for today's industrial applications. This parameter must be fitted numerically in order to account for coalescence o f growing cavities (interactions between elementary cells containing an insolated cavity). The mesh size has therefore a limited influence on crack initiation, and a large influence on propagation. Another point deserves to be highlighted; it concerns the role played by the microstructure on the resistance of material or industrial structure to fracture. In order to predict structural integrity in connection with microstructural fracture processes, four different scales of observations must be considered. The first related to microstructure and material material microscopic heterogeneities, the second concerns the scale of continuum mechanics, the third represents the scale of the process zone (damaged zone or yielded area), and the last is the size of the structure. The microstructural distance can be related to mean spacing between inclusions or carbides, or any other particles that play a role in the microstructure fracture process. When comparing this distance with the process zone size, two cases must be considered. If the plastic zone, or crack tip opening displacement, is much larger than the mean spacing between inclusions, the effect of microstructure on material failure is very limited. It can be considered that the material, observed at the scale of C T O D or mesh size, is rather homogeneous. In that case, the macroscopic tensile properties of the material, even if including damage, are determined at a scale which already averages micro-
IIIllllll
I I
17
1
Fig. 8. Mesh of CT specimen.
Fig. 9. Mesh of the cylinder. structural effects. However, if the continuum mechanics scale and the mean spacing are within the same order o f magnitude, the characteristic distance will obviously play a greater role on crack initiation as well as crack growth. Moreover, if the distance between initiation sites is greater than C T O D or continuum mechanics scale, the effects o f microstructure on fracture will be enhanced and any attempt to model the fracture process will have to include statistics on the geometrical distribution of inclusions in order to be able to account for structural resistance as well as for scatter associated with crack initiation and toughness measurements. Once again, the simplest solution is to use the mesh size as an averaging tool for stress and strain gradients, and also for the effects of microstructure on failure.
5. Conclusions This modelling, instead of applying criteria based on global loading parameters, describes the damage evolution from local values of stress and strain fields. For this reason, this method is able to account for local effects o f the crack area loading factors, such as stress triaxiality.
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E. Eripret, G. Rousselier / Nuclear Engineering and Design 152 (1994) 11-18
The geometry effect including differences in resistance to ductile tearing of the specimens can be investigated and explained by using a local approach to fracture methodology. Rousselier's model proved to be an efficient tool for understanding ductile tearing behaviour of steel, and gives answers where classified fracture mechanics concepts fail. References M. Bethmont, G. Devesa and G. Rousselier, A methodology for ductile fracture analysis based on damage mechanics: an application of local approach to fracture to the NKS-3 thermal shock experiment, 15th MPA Seminar, Stuttgart, 5-6 October 1989. R.H. Bryan et al., Pressurized thermal shock experiment of 6 in-thick pressure vessels--PTSE-2: investigation of low tearing resistance and warm-prestressing, NUREG CR4888, ORNL-6377 (Oak Ridge National Laboratory, Oak Ridge, TN, 1987). R.J. Dexter, Analysis of the first spinning cylinder experiment, CSNI/FAG Specialists Meet., FALSIRE Project, Boston, MA, 8-10 May 1990. K. Kussmanl and A. Sauter, Application of ductile fracture mechanics to large-scale experiment simulation and analyses for pressurized thermal shock behavior of RPV's, Int. Conf. on Fatigue, Corrosion Cracking, Freture Mechanics and Failure Analysis, American Society for Metals, Salt Lake City, 2-6 December 1985.
D.J. Lacey and R.E. Leckenby, Determination of upper shelf fracture resistance in the spinning cylinder test facility, Trans. 10th Int. Conf. on Structural Mechanics in Reactor Technology, Division F, pp. 1-6, Anaheim, CA, August 1989, Division F, pp. 1-6. F. Mudry, Etude de la rupture ductile et de la rupture par clivage d'aciers faiblement alli6s, Th6s¢ d'Etat, Universit6 de Technologic de Compi~gne, 23 March 1982. D.J. Neville, On distance criterion for failure at the tips of Cracks, minimum fracture toughness, and non-dimensional toughness parameters, J. Mech. Phys. Solids 36(4) (1988) 443-457. H. Okamura et al., Japanese pressurized thermal shock experiment program, Specialists Meet., CSNI/FAG Project for Fracture Analyses of Large-Scale International Reference Experiments (Project Falsire), Boston, MA, 8-10 May 1990. G. Pijaudier-Cabot and Z.P. Bazant, Nonlocal damage theory, ASCE J. Eng. Mech. 113 (1988) 1512-1533. G. Pissarenko, A. Yakovlev and V. Matveev, Material Strength Memorandum, MIR, Moscow, 1945. J.R. Rice and M.A. Johnson, in Kanninen et al. (eds.) Inelastic Behaviour of Solids, McGraw Hill, New York, 1970, p. 641. R.O. Ritchei, J.F. Knott and J.R. Rice, J. Mech. Phys. Solids 21 (1973) 395. G. Rousselier, Finite deformation constitutive relations including ductile fracture damage. In Three Dimensional Constitutive Relations and Ductile Fracture, North-Holland, Amsterdam, 1981. G. Rousselier et al., A methodology for ductile fracture analysis based on damage mechanics: an illustration of a local approach of fracture, Nonlinear Fracture Mechanics, Vol. II, Elastic-Plastic Fracture, ASTM STP 995, ASTM, Philadelphis, PA, 1989, pp. 332-354.