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A Methodology for Ductile Fracture AnalysisBased on Damage Mechanics: Anlllustration of a Local Approach
of Fracture
Gilles Rousselier, Jean-Claude Devaux, Gérard Mottet, Georges Devesa
To cite this version:
Gilles Rousselier, Jean-Claude Devaux, Gérard Mottet, Georges Devesa. A Methodology for Ductile Fracture AnalysisBased on Damage Mechanics: Anlllustration of a Local Approach of Fracture. Non- linear fracture mechanics: Volume I, Elastic-Plastic fracture, STP995V2-EB, ASTM International, 1989. �hal-02081538�
Aulhorized Reprint 1989lrom Special Technical Publication 995'19æ
Copyright American Sociely ior Testing and Materials, 1916 Race Streel, Philadelphia, PA 19103
Gilles Rousselier,t Jean-Claude Devaux,2 Gérard Mottet,z and Georqes Devesar
A Methodology for Ductile Fracture Analysis Based on Damage Mechanics: An
lllustration of a Local Approach of Fracture
REFTRENCEr Rousselier, G.. Devaux, J.-C.. Mottet. G.. and Devesa. G.. "A MelhodoloSy for Dùcaile Froature Àn|llsis B|sed on Dtmsge Mechanics: Àn lllusllition of o Local Ap.
prolrch ofFlaclùre," Nonlinear Fructwe Mechanics: volume ll Elastic'Plaslic F,aclure, ASTM STP 995, J. D. Landes. A. Saxena. and J. G. Merkle. Eds-, Amcrican Society for Testing and Materials. Philadelphiâ. 1989, pp. 332-354.
ABSTRACT: Major solutions needed in frlcture analysis are (d) simple and accurate matedal characterization and (ô) easy transferring of malerial data lo cracked structures. ln !h€ pro- posed methodology. constitutive relationships. including cavity growth and coalescence, are used. Material characterization is based on the simple notched tension lest. Several structural steels have been characterized. especially 4508 steel. and the direct transferring of material data hâs been demonstrated wilh tests on circumf€rentially cracked tension specimens.
In addition. the extrapolation of material data io different inclusion conl€nls and temper- atures was attempted, with favorabl€ rcsults for the first factor through a specific parameter of the model. Th€ temperature dcpendence of 4508 steel ductiliiy is related to an inverle strain rale effect on the flow curv€: the modeling of this effect gives encoùraging results, but it must be refined to producc an effeclive prediction.
KEY WORDSI ductile fracture. damâge mechanics. local approach to fracture, crack iniriation, slable crack growth, 4508 steel. inclusion contenls, notched tension test. fracture mechanics.
nonlinear fracture mcchanics
Some of the major problems engineers have to cope with in fracture analysis are the following:
(a) material characterization, that is. the generation of adequate data from specimen testing, and
(ô) the transferring of fracture mechanics data (o the struclural analysis of components.
As a mattcr of fact, the generation of matedal data can be money and time consuming:
for example, the determination ofJ-Ad resistance curves is still a toilsome task, though some progress was gained with partial unloading cornpliance methods. Frequently, existing ma- terial data do not correspond to the specific application (in reference to temperature, strain rate, irradiation. aging, and so forth), and hazardous extrapolations are necessary.
L Senior engineer and engineer, respectively, Electricité de France, Service Réacteurs Nucléaires et Echangeu6. Les Renardières. 77250 Moret-sur-Loing, France
'?Senior engineer and engineer, respectively, Framatome, Centre de Calcul de la Division des Fab- rications. BP 13.71380 Saint-Marcel. France.
With permission of ASTM (March 11, 2019)
in Landes, J., Saxena, A., Merkle, J., Eds., Nonlinear Fracture Mechanics:
Volume II, Elastic-Plastic Fracture, STP995V2-EB, ASTM International, West Conshohocken, PA, 1988.
https://doi.org/10.1520./
STP995V2-EB.
ROUSSELIER ET AL. ON DUCTILE FRACÎURE ANALYSIS 333
On the other hand, the transferring of ftacture mechanics data to industrial components may be questionable, especially if complex situations are involved, including heterogeneous materials, residual stresses, thermomechanical loadings, historical effects, or othel factors.
A local approach of ftacture is particularly suited to solving these two difficulties lt can be defined very generally as a combination of the following:
(a) the computation of local stress and strain in the most loaded parls of a component or a structure and
(b) physical fracture models coresponding to various mechanisms: cleavage, duclile frac- ture, creep, and so forlh.
Actually, the generation of material data is simple in a local approach 1o fracture The application of a local approach 1o ftacture in complex structural situations is direct, as is shôwn in the accompanying paper by Devaux et al. Ul' and the progress of numerical analysis software and hardware makes the computational cost a less and less significant consideration.
The present paper deals only with ductile fracture. The development of a new methodology for duciile fracturc analysis was underlaken a few years ago in a cooperative program between El€ctricité de France and Framatome; this methodology is based on a local approach to fracture and damage mechanics. The paper focuses on material characterization, de ving from the very simple notched lension test (a) tbe ductile ftacture model and parameters calibration and (ô) the prcdiclion of the inclusion content and temperature effects on ductile fracture properties. In addition, the abilily of the model to predict crack initiation and growth in a structure is briefly presented in the last section of this paper'
the Ductile Frâcture Model-Parameten Calibration
Ductile ftacture results from the formation, growth, and coalescence of cavities. The case of intergranular cavities, which corresponds more specifically to creep damage, is not con- sidered here.
In a loca| approach to ductile fractule, a damage variable is introduced. This variable is computed at évery point on the structure. Its evolution is a function of local stresses and strains. For exampl;, if isotropic cavity growth only is considered' the damage variable can be the equivaleni cavity radius R; its evolution can be given by th€ well-known equation by Rice and Tracey
fi: o."r,r"-o (ft)
where o,, = o,,/3 is the mean hydrostatic stress, and o.,r and è€q are the equivalent stress and plastic strain rate, respectively. Then, a very simple criterion for cavity coalescenc€ can be fÀrmulated: it is based on the assumption of a c tical cavity growth (À/R0).' where À0 is the initial cavity radius. This criterion has been widely used in local approach to ftacture
11,21.
' In the proposed methodology a somewhat different formulation is used The model refers to the b;sic assumptions of "continuum damage mechanics," that is, the damage variable is included in the constitutive relalionships of the material; there is no conceptual difference between the hardening va able related to plastic delormation and the damage variable
The constitutive relationships are derived from a plastic potential F, and yield criterion ( 1 )
334 NoNLINEAR FRAoTURE MEoHANICS: VOLUME II F - 0, with the normality rule [J,41. The function F is
r ( a , p , Ê ) : a* - R(p)
where D and or are constants, p is the hardening variable (the cumulated plastic strain if È : é3q), and p is the damage va able. p is defined in the APpendix bY Ê = û/u, where u is the average volume of the cavities. Both p and p are scalar, as isotropic hardening and damage are assumed. The function R(p) defines the hardening curv€ of the material (stress- strain cuwe). The function 8(B) is
rtBt:;-l!-lL =
",/ (3)
t - t o i J o e x p p
where j6 is a constant, indicating the initial volume ftaction of cavities, and J is the actual volume fraction of cavities.
The derivation of Eqs 2 and 3 is detailed in the Appendix; it is not significant to the following results. wltat bas to be remembered is that damage results in softening of the material and lracture proceeds from the compelition between hardening and damage: when damage overcomes the hardening of the material at the tip of a crack or in the center of a tcnsion specimen, there is strain localization, which rapidly results in tremendous strains and damage. The stresses decrease abruptly and vanish, and the zone of strain and damage localization can be assimilated to a crack.
Therefore, it is not necessary to introduce a critical value of the damage variable. As a matter of fact, ftacture can occur for different values of the damage variable, depending on the previous history of the material. The only material parameters are the constants of Eqs 2 al:'d 3: D, o,, and fi, When steep gradients of stress and srrain exist, as at the tip of a crack, it is acknowledged that a c tical distance l. has to be added to the local ftacture criterion: L. reflects the stalistical aspects of fracture, the interaction between the inclusions and the crack tip in the case of ductile fracture.
From cavity growth measurcments and theoretical considerations, the autho$ concluded that the constant D does not depend on the mat€rial and can be taken to be equal to 2, at least for the initial volume fraction of cavities /0 equal toorsmallerthan 10 I [4]. Thethree remaining material parameten to be calibrated are these:
(a) /0, related lo the volume fraction of inclusions f";
(à) ûr, related to the resistance of the metal matrix to the growth and coalescence of cavities; and
(c) (. related to interinclusion spacing.
Cslibration Procedure
How can the paramet€$ fr, o,, and f. be calibrated? Good eslimates of j' and f. are obtained ftom metallographic examinalions of the inclusions [4]
+ r1p;o *p (:.) (2)
. . (d"d")'''
. 5
' (N")'"
ROUSSELIER ET AL. ON DUCTILE FRACTURE ANALYSIS 335
where 4, d, , and d, are the average dimensions of the inclusions, z is thc direction of the load, and N" the number of inclusions per unit,of volume. The value J, can be calculated either from the metallographic examinations or from the chemical analysis (Franklin's for- mula [5]).
The stress level of the hardening curve of the material gives a poor estimate of o' [4]. ln practice, mechanical testing is necessary to calibrate o'. The basic specimens are axisym- metric notched tension specimens. The use of these specimens might seem surprising as far as fracture mechanics is concerned, but the following must be noted:
l. In a local approach to fraclure, uncracked geometries as well as cracked geometries can be analyzed, and material parameters can be transferred from one geometry to the other.
2. Once a crack has been initiated in the center ofa notched tension specimen, it responds like a cracked specimen.
The calibration procedure is depicted in Fig. L In notched tension specimens, the initiation of a crack results in a marked change of the slope of the load-displacement cu e, Point A in Fig. 1, as shown by lests inlerrupted before and after this point [6]. In the numerical simulation of the test, the location of this point depends on the parameters o, and Jn only, and not on f., as there is initially no steep gradients of stress and strain. If J0 is known, the comparison of experimental and numerical curves makes it possible to calibrate the param- e l e r û r .
The crack propagation rate, which depends on 0., is related to the slope ofthe postinitiation curve, AB in Fig. 1. In a numerical calculation t'. is the length of the finite elements in fiont of the crack tip. So, the estimated f. = 5(N ) '/3 can be checked with the comparison of numerical and experimenlal postinitiation curves.
Let us point out that no crack growth measurements are necessary, which makes the calibration procedure a very simple one compared with 'I-Ad resistance curve testing. On the other hand, a numerical simulation of the tests is required, though. fortunately, existing
À O : O I A M € T R A L C O N T F A C T I O N A , A ' I C n A C K lN l l l A T l O N
A 8 . A ' 8 ' : C R A C K P n o P A G A T I O N
l r ) E F F Ë C T O F o t ( O R to ) FlG. l-Numericol load-displacemenl
( b ) EFF€CT O F Q c curves of a notched tension specimen (schemalic).
LOAO
A O = o o - o a @ = @ o - @
ooo NONLINEAR FRACTURE MECHANICS: VOLUME ll
calculations can often be used. Anyway, two-dimensional elastoplastic analyses are no longer a problem with recent computers. The us€ of axisymmetric specimens makes these two- dimensional analyses perfecrly surted.
The constitutive relationships with ductile fracture damage were implemented in two finite- element programs:
(d) the TITUS program, a general system lor mechanical analysis developed by Framatome; and
(ô) the ALIBABA program, a two-dimensional program developed by Electricité de F ance.
Numerous cross-checks of the two programs demonstrated that both give practically iden- tical results. So the results obtained with the two programs will not be differentiated in the following sections. With both, the large changes in geometry are taken into account by modifying the coordinates at each load step.
Duclile Fraclûre Charrclerization of Strùctural Steels
The foregoing calibration procedure was applied to several structural steels:
1. A low-alloy rotor steel-The specimens were taken in the tangential direction of a 500-mm-thick turbine disk.
2. An austenitic weld-The specimens were taken at the bottom ofthe 70-mm-thick weld in the transverse direction.
3. A first heat, Heat A subsequently, of A508 Class 3 steel-The specimens were taken from a nozzle dropout of a 200-mm-thick pressurized water reactor (PWR) vessel shell. in the long (that is tangential) direction.
The chemical composition of these steels is given in Table 1. The metallographic exam- inations resulted in f,, : 10 ', l'. = 0.+ mm for the rotor steel [4] and the austenitic weld [7], and /6 = 1.6 x 10 ', I = 0.55 mm for Heat A of the A508 steel [8
The diameter of the minimum cross section ofthe axisymmetric notched tension specimens was $u = 10 mm, and the outer diameter was l8 mm. These specimens only differed in the notch radius, which was 2,4,5. and l0 mm for the AE2, AE4, AE5. and AEl0 specimens, respectively.
For the rotor steel, two AE5 specimens were tested at 40'C. The right location of the initiation point was obtained with 01 = 490 MPa. The expe mental and numerical load displacement curves are compared in Fig. 2. The agreement is excellent, and thus the calibration of the parameterc is satisfactory. ln Fig. 3. the highly damaged zone (B > 4.5.
that is, the void growth ratio u/u0 > 90), which is assimilated into the cracked zone. is localized in the minimum cross section of the specimen. Its radius is in good agreement wilh the experimental cracks derived from tests interrupted at the same diametral contraction Aô : 1.5 mm. The numerical crack resistance curve (loading parameter versus crack area) is shown in Fig. 4; it is very close to a straight line. This feature, which is also observed with other steels. makes it possible to determine the crack resistance curve of a material with a single specimen: the load-displacement curve gives the initiation point. and the interrupted test gives the final crack area.
For the austenitic weld, AE2, AE4. and AEl0 specimens were tested at 20"C [7]. The minimum cross sections of the specimens were located in the middle of the weld line. The strong anisotropy of the weld metal made the diametral contraction measurements hazard-
337
é
o
z
|:
ROUSSELIER ET AL. ON DUCTILE FFACTURE ANALYSIS
: 9 : : : :
b
; È : i : e
, E E e x E
^ n
Q ! t = = R- - v - - R F h s s S
c i ô i d d d o
s R E c E F
: 6 O O O O
6 r ) ê
ë : F R F Ê
. . . i - d c i d o
R s R S R R
d é c t c i d c ;
Ë s { È î q
C ) - Ê È - :
Ê q Ë F E Ê
d d c j d d o
. â = A ç t f t r €
R 9 ç € $ G
. J q . _ ' - - - o o o o o o
< o ( J â 6 _ > 6
E . e q b 9 e ç Ë : ù t d < <
.:p
i
I ô ûI
lI]
J
NONLINÊAR FRACTURE MECHANICS: VOLUME ll
60
l o
J $
æ
20 l 0 0
\
\
\
R O T O R S T E E T NOTCH AAO||, TEST T€MP€fi
. EXPÉNIMT
l S 5 . n m { A E s l AlURE 4OOC I N T o = 2, o, = 49O Mf'.
t ^ = t g ' 4 , ç =q4 rr.
0 0.2 0.4 0.6 0-8 | r.2 1.1 t.6
oIAMETRAL coNTFAcTIoN, Ao = oo . o, ôl'l FlG. 2-Load-ditplûcement curve of a rctor steel notched tension tpecirnen.
p > 2 . 8 9 > 3 s p > 4.3
0 r ? 3 4 5
FlG.3-Damaged zonet in o rotor tteel notched te sion specimen.
NOTCX RAOIUS 5 m|tr {AE5) OIAUÊlRAL CONTRAClIOfl A O - O o - o - r . 5 r r r û
E X P € R I M E N T A L C R A C I < S ( I N T E R R U P T E O T E S I S )
ROUSSELIER ET AL. ON DUCTILE FRACTURE ANALYSIS ?eo
E X P E R I M € N T S I R O T O R S T E E L ) l . ô
- t . 5
o caacK rNlTlallolt tr INTEFRUPÎEO ÎESTS d
d
ll 0 2 F
2
:
l l
C c  C < A F E A n r 2
FIG.4-Crock rcsistance curve of a rotor steel notched tension specimen'
ous. As a malter of fact, only the diameter at ftacture of the specimens, coinciding with crack initiation for this material, could be measured on the broken test pieces. A reasonable agreement between these measurements and rhe numedcal iniliation points of the three specimens is oblained wilh ûr = 565 MPa (see Fig. 5) The v€ry steep posliniliation curve c;lculated for the AE2 specimer is coherent with the exp€rimental coincidence of initiation and ftacture (aking into account that the stiffness of the testing machine is finite, which promot€s inslability). Neve h€less, stable crack growth measurements would be useful to check the value f" = 0.4 mm.
For A508 steel, Heat A, two AE2 specimens were tested at 100"C. The numerical curve with the calibraled value o, = 445 MPa is compared in Fig. 6 with the experimental cury€s' The calibration of the parameters is satisfactory. In Fig. 7, the localization of damage and the corresponding collapse of longitudinal stress in the vicinity of the minimum closs section of the specimen is shown. (Tbe diametral contraction AÔ is about 2.6 times the end dis- placement r,; crack initiation is at !. - 0 61 mm.)
In the conclusion of this section the following can be asse ed:
1. Th€ material characterization can be performed on various steels with simple mea- surements and tests (some additional specimen testing would be necessary 10 confirm the charactedzation of the austenitic weld).
2. The numerical simulation of the notcbed lension tesl is in good agreement with the experimental results, for both crack initiation and propagation.
2 sp€cirnens AEs
NONLINEAR FRACTURE MECHANICS: VOLUME ll
€ X P E R I M E N T A L I N I Ï I A T I O I { A E 2
0raMEl8aL CONTRACTloN, AO =Oo - O, flm FlG. s-Nume cal load-alisplacement curves and erpe mental initiation points of lhe ous' tenitic h/eld.
o 0 . 5 1 1 . 5 2
0IAMETFAL CONTFACTION, AO =Oo - O, mln FIG. 6-Load-displacement curve of A508 steel notched tension sPecimen.
z Â= { o
auslEiûÎlc wELo
N O T C H R A O I I 2 . a . n d l0 n m TESÎ lEMPÉRAlURE 2OOC o = 2 , o t = 5 6 5 M P . r o
- t o { , 1 " = 6 . 4 ' ' ' .
a50€ Cl.r 3, HEAT a N O I C H R A O T U S 2 m m ( A E 2 ) TEST TEMP€RA'U8€ IOOOC
. O E X P É F I M E I T T S - N U M E n t c { q L C U R V E o = 2, ot = a/t5 MP.
to = I 51 1g-4, q. =g.55 rt
BOUSSELIER ET AL, ON DUCTILE FRACTUBE ANALYSIS 341
0.5 0.6 0.7
€No OISPLACEMÉNÎ, u., mm
E N 0 0 I S P L A C Ê M E N T , ! r , n h
FlG. 1-Longiudinal stress ond domage at Bouss integration points in the vicinity of the minimum cross section.
Prediction of lhe lnclùsion Cont€nl Elfect on Ductile Fractùre
In this section two additional heats of A508 Class 3 steel were tested, Heats B and C.
For Heat B, the specimens were taken from a nozzle dropout of a PWR vessel shell, in the transverse (that is, axial) direction. For Heat C, the specimens were taken in the longitudinal direction from a 200-mm-thick cylindrical lorging, used for large-scale pressurized thermal shock testing at the Staatliche Materialprùfungsanstalt (MPA) in Stuttga [9]. The chemical compositions of both steels are given in Table 1. From the metallographic examination of Heat B, a good estimate of the initial volume fraction of cavities to be used in the calculations is f,, - 19 t
[8]. No metallographic analysis of Heat C has been performed yet, but in this very clean steel the inclusions are expected 10 be more or less spherical, so that J,, - f, can
. r 5(x) t
E roæ
=
= I soo
0.4
0.4 0 6
u2
NONLINEAR FRACTURE MECHANICS: VOLUME tlbe assumed. According to Franklin's formula [5], the volume fraction of inclusions is J, - 7 x 1 0 5 .
Considering that the key difference betweeh the three heats of A508 steel is their inclusion content, the point ol most interest is whether or not it is possible, with the sole parameter f0, to estimate the ductility of the two "unknown" heats. B and C. That is why the first finite-element analyses were performed with the mechanical properties of Heat A (same stress-strain curve. or, li. and other factors) but with different J,, [10 r, 1.6 x l0 r (as in Figs. 6 and 7), and 7 x l0 rl.
The results are given in Fig. 8, together with the mean experimental fracture initiation points deduced from the experiments on AE2 notched specimens. We can conclude that:
l. The parameter J,, represents with a good accuracy the eftect of inclusion content in the rang€ of interest (10 I to l0 r).
2. As first estimates, i, : tO . and 7 x l0 i can be retained for Heats B and C.
Note that from Fig. ti an approximation formula can be proposed between f,, and the numerical ductility at fracture initiation e, = 2 ln (ô"/ôr)
€ f = 0 . 1 0 4 J , , r ' r r
7 r l O - 5
0laMÊlRAL CONTÂACTTON. ÂO =Oo - 0. nn FlG. \-Numetical load-displacement curves and etpeimental iûitiation points of A508
E X P E R I M E N T A L I N I l I A T I O N
n € a r s
I r e r r a
! . 6 x 1 O - 4
A5æ CIti3 3
N O T C T i F A o I U S 2 m m ( A E 2 ) r E S T T E M P Ê R A l U R E I O O O C N U M E F I C A L C U f i V E S : H E A T A , O = 2 , I O ol =445 MP.. lc =0.55 mm
S.d3 sters-strâin curve for.ll h€ats.
ROUSSELIER ET AL. ON DUCTILE FRACTURE ANALYSIS 343
a
I
I \ 9^
ASO8 Cl.s 3, EEAT C NOTCH aADluS 2 ôm (AE2) IEST TEMPERATUsE l OOOC
. O E X P Ê N I M E I I T S _ N U M E R I C A L C U F V E o = 2 . o 1 = 4 0 0 M P r f o = 7 x l 0 - 5 , q = 0 . 5 6 | î h
O l a M € T R A L C O N T R A C T I O N , A O = O o - O. mm FlG. 9-Load-displacement curve of A508 steel notched tension specimen.
where the three points corresponding to lo : 'l x 10 5, 1.6 x 10-', and 10 I are in per.
alignment in a log-log diagram.
Additional calculations were performed with the lrue slress/strain curves of Heats B C. The stress-strain curyes of Heats A and B are very close, so il was not necessarl calibrate the param€ter o1 (o1 = 445 MPa) again. The stress-strain curve of Heat C significantly lower than lhose of Heats A and B, so the parameter ûr had to be reduced relation for Heal C. The resulting numerical curve, with or = 400 MPa, is plotted F i g . 9 .
In conclusion, a good prediction of the effect of inclusion content on the ductile ftactr properties of a steel is obtained in the proposçd methodology, through the parameter fo.
Prediction ol lhe Temperaturc Eftect on Ductile Fracture
Carbon steels can pr€sent a drop iû duclility and fracture toughness in the temperature range 100 to 400"C, known as the "blue brittleness." Earlier work [10] has shown that A508 Class 3 steel presents this phenomenon; it has been demonstrated that the temperature of minimum fracture loughness depends on the strain rate and that there is a clear correlation between this phenomenon and dynamic strain aging, resulting from dynamic interacrions between dislocations and intenticial atoms such as nitrogen 111,121.'fhe macroscopic man- ifestations of dynamic strain aging are the serrated flow observed on smooth tension spec- imens (Portevin-Le Chatelier effect) and an inverse effecÎ of strain rate on the flow curve (hardening curve) of tbe material. This inverse effect promotes flow localization between cavities, that is, cavity coalescence, which explains the lower ductility and fracture toughness.
Amar and Pineau have performed tests in the temperature range 100 to 450'C on notched zr - 1 0
344 NONLINEAR FRACTURE MECHANICS: VOLUME ll
tension specimens, taken in the longitudinal direction from a nozzle dropout of a PWR vessel shell [12]. The chemical composition qf the steel (A508, Heat D) is given in Table 1. Mean deformation in the minimum cross section and diametral contraction at ftacture initiation are plotted in Fig. 10. In terms of a local approach to fracture, the authon used the critical cavity growth c terion (R/R0).and, ofcourse, the same temperatur€ dependence of (À/Ào). as in Fig, 10.
An equivalent approach can be performed with the proposed methodology. The param- eters t0 and [., related to the inclusions, should not depend on the temperature. The dynamic strain aging effect, resulting in an easier flow localization in the metal matrix between cavilies, is naturally expressed through the parameter o,, which is related to the resistance of the metal matrix to the growth and coalescence of cavilies. The calibration procedure presented in the first section can be used, and 01 will present the same temperalure dependence as in Fig. 10.
)wever, a more ambitious objective is to really predict the effecl of temperature (and rain rate) on the ductile fracture properties of the material. For this, the inverse effect rain rate on the flow curve À(p) is introduced (see the Appendix)
n ( p , h ) : R ( p ) + h ( p )
rm the smooth and notched tension tests performed at 300'C and at various strain rates eats B and D, including those in Ref .12, the following relationship can be used as a rpproximation
R(p.p) : R, (p) - Kp h (fi)
50 MPa, and p0 = 10 r s ' is the reference strain rate of the characte zation step-by-step elastoplastic numerical calculation, the equivalent strain rate p
(4) e K :
. I n t h e
1- o.r:
I a ^ -
:
E 3 0.25 z
:
0
a5G Cl.n 3. HEAT O A M A R .ô d P T N E A U l r 2 l
" /
-8--.-o-
"./ ,, - \ -
.-"--"i:-"-Z-'/,"/>
- E-.--o -À'--a-l--
"|f, '2"
--\o-.-o_E--o/
O A É 2  A Ê 4 o  E l 0
E
eo e-
; r 5 6
0.5 E
s
l()0
T E M P € R A T U R Ê . ù t C FlG. l\-Temperature dependence of fracture inùiation in notched tension specimens.
ROUSSELIER ET AL. ON DUCTILE FRACTUBE ANALYSIS 345
l r M € , r FlG. ll-Inverse strain rate effect on damage in a notched tension specimen.
calculated at the preceding step is used in Eq 4 to determine the bardening curye of the current step.
In the first attempt to model the strain aging effect, the reference calculation is that of an AE2 specimen of A508, Heat A (see Fig.6). The diametral contraction al fracture initiation is L6r = 1-62 mm, without strain aging; K = 0 in Eq 4. Two calculations wilh the inve$e slrain rate effect were pedormed, with K : 50 and 250 MPa. The displacement rate i, imposed at the end of the mesh is such that the mean strain rate in the minimum cross section is Ê = -2ôlô = 10 3 s '; at the larger strains it coresponds to ll, : 1.4 x l0 I mrn/s.
As rl(p) = 0 forp : l0 r s r, the load-displacement curves resulting from the three calculations are almost indistinguishable. But, as was expected, the local values of the damage va able, plolted in Fig. 11, are progressively more important in the case of inverse strain rate effect, and a forward fracture initiation is obtained: Aôr = 1.58 mm (time = 343 s) for K = 50 MPa, and Aèr = 1.47 mm (time = 309 s) for K = 250 MPa. Note that the same parameler, ûr : 445 MPa, is used in the three calculations. As was also expected, the inve$e strain rate effect gives smaller values of the damage va able at fracture initiation, that is, smaller critical cavily growlhs.
But even with the arlificially enlarged inverse strain rate effect (K = 250 MPa), the decrease in ductility A$o is smaller than that measured between 100 and 3m'C (Fig. 10).
So, before a real prediction of the temperature effect on ductile fracture can be used, mole experimenlal work on matedal characterization at 300"C and a more refined numerical alsodthm for the inverse strain rate effect are needed.
AsG CI.'t 3, HEAT A N O T C H R A O I U S 2 m m ( A E 2 t 0 =2. oi ={45 Mtt r o = r , 6
' ' 1 6 { , q = 6 5 5 , " .
Applicslion to Crâcked Slruct[res
The preceding sections were devoted to material characterization and to the prediction of material properties as they are affected by various facto$. The transferring of the char- acterization data 10 structural situations has still to be illustrated-that is to say, to what extent has the model, calibrated with notched lension tests, the ability to simulate the behavior ol a cracked structure? In the present application, the structures are simple cir- cumferentially fatigue precracked tension specimens (see Figs. 12 and 13); this geometry was chos€n because it can be analyzed witb a two-dimensional calculation. Other applications to different geometries and thermomechanical loadings are in the making and cannot be reported in this short paper.
Two different specimen sizes were tested at 100"C; the outer diameters were ô = 30 and 50 mm. All specimens were made of A508 Class 3 steçI, Heat A. The radii of the uncracked sections after fatigue precracking were rû = 8.3 t 0.2 mm (ô = 30 mm) and /0 : 13.8 + 0.2 mm (g : 50 mm); the initial crack lenglh was oo : 612 - ru. The erongarron was measured on gage lenglhs L0 : 78.4 mm (ô : 30 mm) and Ln : 130.6 mrn (ô = 50 mm).
The interrupted t€st method was used: the specimens were loaded up to various elongations, unloaded, heat-tinted, and broken in liquid nitrogen. This allowed a direct measurement of the crack growth Aa. Note that Ac includes the "stretch zone', due to the initial bluntins of the crack tip. More details can be found in Ref 13. Each specimen gives only one poini (plotled in Fig. 12 or Fig. 13) of the exfre mental crack resistance curve-the loading parameter versus crack growth. The loading parameter is the elongation and not the
C R A C K G s O W T H
A5(B Cl|r! 3, 8EAl A .rEsT
TEMPEFATUn€ I OOoc . I X T E R E U P I E D l E S l S - 1 { U M € F T C A L C U R v E o = 2 , o , = 4 4 5 M P r t o = t . 6 x l0_., Qc = 0 . 5 5 mh
346 NONLINEAR FRACTURE MECHANTCS: VOLUME tl
I ' ! I I I A T I O N
\ ST8€TC|I ZOr.tE(at : c100/2 (_)
ICRACX TIP OPENING OISPLACEM€NT'
STAgLE CaACK GaOwTH.
^.
=. _ âo, nh FlG. l2-Cruck resittance curve ol a circumferentially Necracked tension specimen (fi 30
ROUSSELIER ET AL. ON DUCTILE FRACTURE ANALYSIS 347
ASO8 C1|'.3, H€AT A I E S T T E M P E R A T U s E I O O P C
. I N T E R N U P l E D T E S T S _ N U M E R I C A L C U N V E o = 2 , o t = 4 4 5 M f l t o = 1 . 6 ; t O - ' , Q c = 0 . 5 5 ôm
' C Ê A C K G N O W T H - nttrtlrtotrl
STFETCH ZOI{E {.I
=c100/2 (-)
ICNACK 1I? OPCNII{G OISPLACEMENT)
The initiation points coincide almost exactly with the experimental ones The stable crack growth is predicted with very good accuracy up to Aa = 1 mm.
For Àa ) I mm. the calculation overestimates the experimental crack growth.
At least two factors can explain this latter discrepancy:
1. ln real three-dimensional ductile crack growth, the crack surface, initially flat, becomes more and more distorled. The different planes along the crack front have to be reconnected by shear bands for the crack to continue its growth. This shear fracture, and the corresponding additional energy, are not taken into account in the model, based on cavity growth.
2. There is a statistical effect. in which the volume of material involved in crack initiation and early crack growth is more important than that involved in subsequent crack growth.
Thus. the probability of having large inclusions in that volum€ of material is lower for large crack growth, and a statistically lower value of f,, should be used. Note that both statistical
E
i z
S T A B L E C F A C K G Â O W T H . ô . : a - . o , m ù FlG. l3-Crack resktance curve of a circumferettially precracked tension specimen (6 50 mm).
"t-integral, usually plotted in a crack resistance diagram. The elongalion data are the direct experimental data to be compared with calculated data.
Just as in the notched specimens, the stable crack growth is obtained numerically by the propagation of a highly damaged zone (see Fig. 14). The numerical crack growth resistance culves are plolled in Figs. i2 and 13 wilh the following results:
1 . 2 . 3 .
348 NONLINEAR FRACTURE MECHANICS: VOLUME tl
A L - L - L o = 1 8 5 m m
R A O I A L C O O n O T N A T Ë . r . m 6 FlG. l4-Damaged zones in a circumferentially prccrucked tension specimen (ô 30 mm\.
and three-dimensional ellects could be taken into account with an a ificially lower value of Jlr. However. as a whole. the numerical simulation of crack initiation and srowth can be considered of good accuracy for industrial applications.
Conclùsions
A new methodology for ductile lracture analysis, based on a local approach to ftacture and damage mechanics, has been applied to the characterization of structural steels: a Ni- Cr-Mo rotor steel, an austenitic weld, and several heats of A508, Class 3 steel. The comolete characterization can be performed with simple measurements and tests on notched rension speclmens.
An effective prediction of the effects of inclusion content and temperature on ductile fracture has been attempted, with favorable resulrs for the inclusion conient effect. throueh a specific parameter of the model. For the temperature effect, a mor€ refined numeriàl modeling of the inv€Ne strain rate effect on the stress-strain curve should be the subiect of future work.
Tbe transferring of material characte zation data to cracked structures is direct, and the numerical simulation of crack initiation and growth in circumferentially cracked tension specimens is in good agreement with the experimental results. Additional appli- cations in the case of thermomechanical loadings are in the making, with favorable prelim- Inary results.
ROUSSELIER ET AL. ON DUCTILE FRACTURE ANALYSIS 349
APPENDIX
Theoreticrl Derivrtion of the Equatons of lhe Ductile Frrc'ture Mod€l C o n t inuam t hermody na m i c s
onlytheinformationneededforthecomprehensionofthemodelisrecalledinthisshort section. Fol more d€tails, refel in particular to Ref 14.
The thermodynamic state of an element of material can be defined by "observable"
variables (elastii deformation €., temperature I, and so forth) and "internal" variables o, which reflàct various dissipalive mechanisms (plastic deformation (hardening), damage, and so forth). Considering specific free en€rgy Ô in the isothermal case, ô : Ô (€', d), it can be shown that dissipated power O > 0 takes lhe form
o = 9 i , - 9 9 op d a
l o l
ô t - l
\ p / a = l i
T h e f i l s t r e l a t i o n s h i p ( E q 8 ) i s t h e c l a s s i c a l n o r m a l i t y t u l e l n t h e s t a n d a r d m o d e l ' t h e behavior of the matèrial ii entirely defired by the choice of the two potentials' è (€" d)
and F (olp, A) or O (o/P, ,4).
(5)
(6) and that
where p is the mass per unit of volume. Entering the "forces".z4 associated with d
Equation 5 is thris written in the form
A = 2Y,Xt
with X = (êp, q) an d Y : (olp, A). The definition of corctitutive relotions consists of linking the X to the Y.
ln the standard mode,f, viscoplastic case, a convex viscoplastic potential O( y ) ' and c tedon of plasticity, F = 0, are entered' so that X = ÀôF/ôf; that is
9 = a o
p a€'
. ô ô (7)
(8)
350 NONLINEAR FRACTURE MECHANICS: VOLUME ll
In the application to ductile fracture, the iotemal variables are divided into:
(a) a hardening variable marked p and (â) a damage variable marked p.
Derivation oJ the Model
Having defined the purpos€ of the model as the plastic deformation and ductile fracture damage of metals, and its scope as the continuum damage mechanics, in the final analysis the model is based on two concepts only:
(a) the "principle of simplicity" (choice of the simplest formulation in a given theoretical framework) and
(à) the standard model (Hypothesis Hl).
The application of the principle of simplicity leads to the hypotheses set out below:
Hypothesis H2-The intemal hardening and damage variables p and p are scalar (hy- pothesis of isotropy), and the "forces" associated with p and P are denoted P aîd B, resp€ctively.
Hypothesis H3-The thermodynamic potential ô (e', p, p) is of tbe form ô = ô . ( e " ) + ô , ( p ) + ô Ê ( g )
which. according to Ëq 7, simply yields P = P(p) aû B = B(Ê). The elastic constitutive relationships are assumed to be linear, hence
6 " : e ' L e " / 2 according to Eq 6. L is the matrix of elastic moduli.
Hypothesis H4-The plastic potential F(ô,P,8), in which ô = o/p, is of the form
where
F : |(ô.q,P) + F1(t-,8)
4 = ô.u + P(p) (von Mises yield criterion)
r ' : B ( 9 ) g ( 4 . ) (10)
Hypothesis H4 calls for some comment.
l. F only depends on the first two invariants of the stress tensor. The sp€cific nature of the model (ductile ftacture) appears through the first invariant û-.
2. Thc additive form (Eq 9), only, yields simple constirutive relationsbips in light of Eq 8 of the slandard model. Equation 10 is also the simplest possible.
3. û = q / p and not o enters the plastic potential, in light of the expression of dissipared power (Eq 5). In classical plasticity, the hypotbesis ol incompressibility makes ir possible to neglect the variations of p, but this hypothesis is no longer acceptable for modeling ductile fracture. ln the present model, in the absence of damage (p = constant), we have p :
(e)
ROUSSELIER ET AL. ON OUCTILE FRACTURE ANALYSIS 351 constant, except for elastic deformations; in the opposite case (p vadable), one can write p = p(p) as a lilst apprcximation. .
The consequences of hypotheses H1 1o H4 are examined below.
Neglecting the variations in volume due to elastic deformations, the mass pleservatron law can be written
ë + 3 P è k = 0 However
p = p ' ( p ) p
and the relations in Eqs 8 yield
3è,- = rB(9)s'('É.) P = re(a-)
Consequently
g ' ( ; - )
- - P ' ( Ê )
8(ô-) 8(P)P(P)
The two members of this equation are functions of distinct variables, ô- and p; they are therefore equal to a constant, the dimension of which is the reciProcal of a stress and which is called 1/o1. The integration of 8'l8 = 1/û' leads to
c(É.) = D *P (%) (11)
in which D is the constant of integration. Here, q' and D are the first two parameters to appear in the model.'in
light of Eq 11, the plastic potential is written as in Eq 2
F = b - R ( p ) + B ( p ) D " * ( # )
in which R(p) = -P(p) is the hardening curve of the mateial The relations in Eqs 8 of the standard model frnallY lead to
(12) è h = p B ' " D * o ( * 9 . )
u = u" *o (*,o)
(13)
352 NONLINEAR FRACTURE MECHANICS: VOLUMÊ ll
i o : ' è P '
and 3
s 4 = o u - o . ô , ; ëlj = èlj - èk6,,
are lhe deviators of o and ÈP, respectively. Computation of é€q yields èlq : p; the intemal hardening vadable thus identilies with the cumulated equivalent plastic deformation.
The function B(p) remains to be specified. Considering cavities of dny shape with an overage yollume u in an incompressible matrix, it is easy to show that the volume fraction of cavities f is such that
J _ :
f 0 - ù - u
Moreover, the mass per unit of volume is given by
To simplify the notation, p is defined as the relalive density obtained by dividing the density of the damaged material by that of the material in its initial state: the initial value of p is therefore p,, = l. Equations 14 and 15 yield
g : - p ' Ê u p ( l - p + p j ' ) As a result of Eq 13 and p = ê&
p
Po
(14)
(1s)
i ) -l p ' ' l
^ . " - _ . , / " , \
- = I - --::--- ltr€r" exD l- |
u L p ( r - p + p J r , ) l ' ' \ p û , / (16) There is a striking analogy with the equation of Rice and Tracey (Eq 1): it should be recalled that the exponential lorm is not a hypothesis in itself but that it flows from Hy- potheses Hl to H4. Through a new and last application of the p ncipl€ of simplicity, it is assumed (Hypothesis H5) thal Eq 16 reduces to
9 : ot,- .*o {&)v
\Por,/
that is, to the Rice and Tlacey equation if D : 3 x 0.283, p = 1,ando,:2v,"13 = 2ool3 in the nonhardening case. (Note lhat D : 2 was used in this paper, in agreement with caviiy growth measurements; the Rice and Tracey coefficient 0.283 is recognized to be too small [4].) Hypothesis H5 implies
p ( p ) =
1 - fo + fo exp Ê (17)
ROUSSELIER ET AL. ON DUCTILE FRACTUBE ANALYSIS 353
and given that -p'lpB : 1/o', Eq 17 yields Eq 3 trl/ a'
-..) = , ojfi,'*l l,- ^
r - . l o + J n e x P F
A third parameter, the initial volume fractiot of cavities J0, is thus added to o' and D. The use of Eq 17 is optional; in a numerical computation, p can be deduced directly ftom the field of displacements.
The model is thus Tully defined with the only guidance being systematic selection of the simples, lormulation.
V iscoplastic Case
Again, according to the p nciple of simplicity, the viscoplastic potential is in the form O(v) : O(F(v)), with F(ô,P,8) I 0 in the viscoplastic case. As
the other components of X = (èP, p, p) are given by X : PôFlôY, exactly as in the plastic case (Eqs 12 and 13), in which also p = Ëtq
The difference is that F : â(p), instead of F = 0 in the plastic case' where ft is the reciprocal function oi dÙldF. F : ,(i) gives
a * - R ( p ) - h ( P ) + a ( P ) D e x P
. ôO dA aF .iO
' A P d F ô P d F
{ % ) = o
One interpreiation of the viscoPlastic model is to handle it numerically like a plastic model, but with a hardeniog curve depending on the strain rate
n ( p , è ) = R ( p ) + h ( i t )
References
[1] Devaux. J. C.. Saillard. P, and Pellissier-Tanon, A., "Elastic-Plastic Assessment of a Cladded ' '
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t2l D'Escitha, Y. and Divaux, j c in Etas;iè'Pt&ttic Fructut. A'TM sTP 668 Amerrcan Sociely for Tesring and Marerials. Philadelphia. 1919- pp 229-248
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77250 Moreiiur'Loing. France' 3-5 June 1986' pp 285-298'
354 Nq{UNE F FR CruRE MECI'I NICS: VOLUI E ll
[9] Kusoaul, K., Sautcr,4., Nguyctr-Huy, T.,8trd Fcybl, D. io Proc..AttgJ, Eighth Intcmatiolst ' '
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