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A ductile fracture analysis using a local damage model
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International Journal of Pressure Vessels and Piping 85 (2008) 219–227 www.elsevier.com/locate/ijpvp
A ductile fracture analysis using a local damage model N. Benseddiqa,, A. Imadb a
Laboratoire de Me´canique et de Rhe´ologie de Tours, Ecole Nationale d’Inge´nie´urs du Val de Loire (ENIVL), Rue de la Chocolaterie, 41000 Blois Cedex, France b Laboratoire de Me´canique de Lille (UMR CNRS 8107), USTL, Ecole Polytechnique Universitaire de Lille Cite´ Scientifique, Avenue P. Langevin, 59655 Villeneuve d’Ascq Cedex, France Received 26 July 2006; received in revised form 27 September 2007; accepted 27 September 2007
Abstract In this study, the Gurson–Tvergaard–Needleman (GTN) model is used to investigate ductile tearing. The sensitivity of the model parameters has been examined from literature data. Three types of parameters have been reported: the ‘‘constitutive parameters’’ q1, q2 and q3, the ‘‘initial material and nucleation parameters’’ and the ‘‘critical and final failure parameters’’. Each parameter in this model has been analysed in terms of various results in the literature. Both experimental and numerical results have been obtained for notched round and CT specimens to characterize ductile failure in a NiCr steel (12NC6) with a small initial void volume fraction f0 (f0 ¼ 0.001%). Ductile crack growth, defined by the J–Da curve, has been correctly simulated using the numerical calculations by adjusting the different parameters of the GTN model in the calibration procedure. r 2007 Elsevier Ltd. All rights reserved. Keywords: Ductile tearing; Local damage; Numerical simulation
1. Introduction It is well known that the ductile failure of materials is controlled by three stages, i.e. micro-void nucleation, growth and coalescence mechanisms. In order to describe this damage and fracture process, several models using local approaches have been proposed in the literature. Globally, local approach modelling may be described by two main families. Firstly, uncoupled models are based upon void growth, such as McClintock [1], Thomason [2,4], Rice and Tracey [3]. For these models, the material failure stage is characterized by a critical void growth ratio (R/R0)c, corresponding to crack initiation. This parameter is considered as a material intrinsic property and it is obtained by a combination of numerical results and experimental data. Crack growth simulation is achieved by a node release technique when the critical void growth ratio is reached [5–8].
Secondly, coupled models are based upon continuum damage mechanics such as Rousselier [9] and Gurson [10]. The use of these models is based upon a constitutive equation introducing the material behaviour and a failure criterion. In this type of model, the crack growth simulation is automatically performed using the complete deterioration of the element at the crack tip [6,11–14]. Among these models, the micro-mechanical model proposed by Gurson is the most widely used for ductile porous materials. The Gurson model employs equations to describe the constitutive response of the metal [10]. These expressions are based upon a continuum elastic–plastic model that accounts for micro-void nucleation and growth. Tvergaard and Needleman [15] have subsequently modified this model introducing the following yield condition (Gurson–Tvergaard–Needleman (GTN) model):
Fðsy ; seq ; f Þ ¼ Corresponding author.
E-mail address: [email protected] (N. Benseddiq). 0308-0161/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2007.09.003
s2eq 3 sm q þ 2f q cosh 1 2 2 sy s2y ð1 þ q3 ðf Þ2 Þ ¼ 0,
ð1Þ
ARTICLE IN PRESS N. Benseddiq, A. Imad / International Journal of Pressure Vessels and Piping 85 (2008) 219–227
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f* f u q1, q 2, SN d sy seq sm E
Nomenclature J a Da f f0 fc fF fN
J-integral crack length crack extension void volume fraction initial void volume fraction critical void volume fraction void volume fraction at final failure volume fraction of void-forming particles
where q1 q2 and q3 are the constitutive parameters with q3 ¼ (q1)2; sm is the mean normal stress defined by sm ¼ skk/3, skk is the trace of the stress tensor; seq is the conventional pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi von Mises equivalent stress defined by seq ¼ ð3=2Þsij sij (sij is the stress deviator); sy is the yield stress; f*(f) is a function of the void volume fraction f and represents the modified damage parameter. This function is defined by 8 f for f pf c ; > < f þ dðf f Þ for f c pf pf F ; (2) f ðf Þ ¼ c c > :f f Xf : F u The expression d ¼ ðf u f c Þ=ðf F f c Þ is considered as an accelerating factor, which is introduced in order to describe the final stage of ductile failure, where fc is the critical void volume fraction corresponding to void coalescence which at first occurs. This is considered as a material intrinsic parameter, f u is the ultimate value of the damage parameter which can be written as a function of q1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and q3 by f u ¼ ðq1 þ ðq1 Þ2 q3 Þ=q3 . If q3 ¼ (q1)2 the ultimate void volume fraction f u equals (1/q1). fF is the final void volume fraction. According to Eq. (2), f ¼ fF corresponds to the failure stage, f ¼ f u . In general, the evolution equation for f consists of two terms describing the nucleation and growth of voids: df ¼ df nucleation þ df growth with f ðt0 Þ ¼ f 0 ,
(3)
where f0 is the initial void volume fraction. Nucleation is considered to depend exclusively on the effective strain of the matrix material: df nucleation ¼ Adpeq þ Bðdseq þ cdsm Þ.
(4)
The parameter ‘‘A’’ is chosen so that void nucleation follows a normal distribution around a critical plastic strain ep, which is estimated using the following expression: " # fN 1 p N 2 pffiffiffiffiffiffi exp A¼ , (5) 2 SN SN 2p where fN is the volume fraction of void nucleating particles, eN is the mean void nucleation strain and SN is the corresponding standard deviation.
modified damage parameter ultimate value of damage parameter q3 constitutive parameters standard deviation accelerating factor yield stress of material equivalent stress hydrostatic stress elastic modulus
For strain-controlled void nucleation B ¼ 0. But in the case of macroscopic ductility, the hydrostatic stress dependence of void nucleation leads to a strong nonnormality in the plastic flow rule, which promotes early flow localization. It is worth noting that a stress-controlled nucleation criterion was found to be important in capturing the effect of stress state on damage initiation. In this case A ¼ 0 and " # fN 1 seq þ csm sN 2 pffiffiffiffiffiffi exp , B¼ 2 SN S N 2p c is a constant parameter and can take values between 0.3 and 0.4. The micro-void volume fraction rate due to growth is given by df growth ¼ ð1 f Þ dpkk ,
(6)
where ekkp is the plastic hydrostatic strain. 2. Scope of study In the current paper, the GTN model is used to analyse ductile tearing. The sensitivity of the model parameters is analysed from literature data. Three types of parameters have been reported: ‘‘the constitutive parameters’’ q1, q2 and q3, ‘‘the initial material and nucleation parameters’’ and ‘‘the critical and final failure parameters’’. Both experimental and numerical results have been obtained using notched round and CT specimens to characterize rupture behaviour in a NiCr steel (12NC6) with a small initial void volume fraction f0 (f0 ¼ 0.001%). The parameters in this model are determined by a calibration procedure using three types of mechanical tests: classical tensile (true stress versus true strain), tensile with an axisymmetric notched (AN) specimen AE (load versus diameter reduction) and ductile tearing tests (J-parameter versus crack growth Da). The results obtained by the GTN model are discussed in the context of the extensive literature on model parameters. 3. Analysis of GTN model parameters Generally, the GTN model is used to predict the damage of ductile materials [4–43]. To apply this model nine
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Table 1 GTN model parameter values from the literature Ref. q1
f0
fc
fF
[6] [12]
0.002 0.00016 0.0001 0.0015 0.001 0.114 0.114 0.114 0.114 0.114 0.00033 0.002 0.077 0.0023 0.001 – 0.0 0.08 0 0 0 0.00057 0.0025 0.005 0 0 0 0.00033 0.002
0.004 0.0005 0.0003 0.035 0.02 0.13 0.2 0.3 0.19 0.175 0.026 0.033 0.12 0.004 0.003 0.15 0.04 0.15 0.02 0.2 0.15 0.03 0.021 – 0.06 0.04 0.03 0.026 0.028
– 23 – 2.8 – 4.3 0.15 5.49 1.141 67 0.272 4 0.35 4.73 0.44 4.35 0.235 14.3 0.235 10.97 0.15 5.17 0.15 – 0.2 6.8 3 – – 0.25 5.17 0.195 – 0.28 4 0.34 2 0.32 4 0.28 4 1.3 3.4 0.2 – 0.212 4 0.197 4 0.189 4 0.15 6.24 – –
[13] [14]
[17]
[29] [31] [32] [33] [34] [35] [36] [37]
[38] [39] [40] [41]
[42] [43]
1.5 1.47 1.47 1.5 1.5 1.43 1.1 1.1 1.2 1.2 1.25 1.15 1.5 1.5 – 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.25 1.5 1.5 1.5 1.25 1.47
d
fN
eN
SN
Le (mm) Material
sy (MPa) Rm (MPa) Specimen
– – – 0.00085 0.04 – – – – – 0.006 0.004 – – 0.01 0.04 0.008 0 0.04 0.4 0.08 0.004 0.02 0.001 0.002 0.002 0.002 0.006 –
– – – 0.3 0.3 – – – – – 0.3 0.3 – – – 0.3 0.3 0 0.5 0.05 0.025 0.3 0.3 0.04 0.3 0.3 0.3 0.3 –
– – – 0.1 0.1 – – – – – 0.1 0.1 – – 0.01 0.1 0.1 0 0.2 0.02 0.025 0.1 0.1 0.01 0.1 0.1 0.1 0.1 –
0.2 0.5 0.8 – 0.1 0.1 – – – – – 0.033 – 0.2 0.5 – – –
– 471 470 779
– 593 608 840
SENT25 CT22.5 AE2 AE2 AE2 SENB (B ¼ 25)
366 – – – – 370 – 230 190 260 – 366
– – – – – – – 433 500 – – –
AE4
–
0.2 0.2 0.1 0.1 0.1 0.033 –
parameters are necessary. They can be classified into two principal families:
the ‘‘constitutive parameters’’: q1, q2 and q3 the ‘‘material parameters’’, which are classified in two parts. Firstly, ‘‘the initial material and nucleation parameters’’, which are determined as the initial void volume fraction f0 and the void nucleation parameters fn, en and Sn. Secondly, ‘‘the critical and final failure parameters’’, the critical void volume fraction fc and fF.
A combination of numerical results and experimental data is necessary in order to determine some of the parameters. It is important to note that most of the parameters are not easy to define. Globally, there is no unique method to determine these parameters. For this reason, a global analysis of the large data available in the literature has been performed, in order to examine the validity of the choices of these parameters, and suggest a global method to determine them (see Table 1). 3.1. Constitutive parameters: q1, q2, q3 In order to describe material ductile fracture, Tvergaard [15] has suggested fixing the q1 and q2 values at q1 ¼ 1.5
C–Mn steel (at 300 1C) A508 C13 A A508 C13 B E690 20MnMoNi5 5
GGG40
22NiMoCr37 GGG40 CMn steel (300 1C) AlMgSi alloy – – Composite Al-Al3Ti
E460 steel A533B Steel
CMn steel Cu
54.7
80
– CT25 AE4 AE2, 4 and 10 – – – .
612
707
AE0.25 and 4 CT
– 440 620 320 360 312
– 600 750 580 – 325
– SENB – AE1 2 4 8
and q2 ¼ 1. These values have been used in several investigations (see Table 1). In a recent study, Perrin et al. [16] have determined a correlation between the parameter q and porosity f: q ¼ q(f). The authors show that when the porosity tends to zero, the q value tends to 1.47. This value is similar to that suggested by Tvergaard (q ¼ 1.5) [15]. In the modelling of nodular cast iron, Steglich et al. [17] have proposed a yield condition determined from a cell model that results in using a quadratic equation to estimate the parameter q1. The calculations of this parameter for various stress triaxiality ratios T give an average value of q1 equal to 1.5. The authors have noted that the parameter q1 does not depend on the stress triaxiality ratio T but little physical significance is given to this value. Gao et al. [18] have performed several calculations considering a wide range of material flow properties (hardening N, sy/E) and stress triaxiality ratios (2–3.3) with sy/E ¼ 0.001–0.004. For a specific set of flow properties, the authors have obtained various q1 and q2 parameters. It is interesting to note that the parameter q ¼ q1 q2 is approximately always constant and equal to about 1.5 for all cases studied (Table 2). Faleskog [19] indicates that the two parameters depend on the material hardening exponent. The q parameters depend strongly on strain hardening and
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Table 2 Optimal values for micro-mechanics parameters (q1, q2) [19] Hardening (N) sy/E ¼ 0.001 q1 5 6.7 10
q2
sy/E ¼ 0.002
q1 q2 q1
1.96 0.78 1.531 1.78 0.83 1.483 1.58 0.9 1.425
q2
sy/E ¼ 0.004
q1 q2 q1
1.87 0.8 1.496 1.68 0.86 1.438 1.46 0.93 1.359
q2
q1 q2
1.71 0.84 1.43 1.49 0.9 1.342 1.29 0.98 1.265
material strength; for a material of moderate hardening and strength, e.g., N ¼ 0.1 and sy/E ¼ 0.002, the calibration gives q1 ¼ 1.46 and q2 ¼ 0.93. High strain hardening reduces the effect of stress triaxiality on void growth rate, resulting in q2 values much below unity [19] (Table 2). It is important to note that the value of q1 may be considered as essential to describe the failure stage. Because of Eq. (2), fracture of material occurs (i.e. f ¼ fF) when f* reaches its ultimate value f u ¼ 1=q1 (when q1 ¼ 1.5, f u ¼ 0:666). The physical validity of this value is not proven. Table 1 shows that the value of the constitutive parameter q1 varies from 1.1 to 1.5 for many materials. However, this parameter is often fixed to 1.5 and q2 ¼ 1. 3.2. Material parameters: f0, fN, fc, fF, eN and SN 3.2.1. Initial material and nucleation parameters Generally, the initial void volume fraction f0 and volume fraction of void nucleating particles fN are evaluated by microscopical examination of the undamaged material. As a first approximation, f0 can be obtained from the MnS inclusions by Franklin’s formula using the chemical composition [20]. Table 1 also shows that the values of the nucleation parameters fN, eN and SN can be arbitrarily fixed. The values eN ¼ 0.3 and SN ¼ 0.1 have been used in several studies, and they are determined by fitting. 3.2.2. Critical and final failure parameters In most investigations, only the critical void volume fraction fc is considered as a parameter, obtained by fitting the numerical calculations with experimental results, and the other parameters are fixed arbitrary. According to Zhang et al. [21], fc is not a constant but decreases when the stress triaxiality ratio T increases. However, other authors note that fc can be taken as a constant only for small f0 values. In a similar study, Steglich et al. [17] confirm that the fc value depends on stress triaxiality T: fc decreases with the increase of T. In a recent experimental study [22], these observations were confirmed and the author proposed a linear correlation between fc and T for a high f0 value (f0 ¼ 13%): fc ¼ 0.2550.146 T. According to Koplik and Needleman [23], the fc value, which signifies the onset of coalescence, seems to vary slowly with stress triaxiality ratio but depends strongly on the initial void volume fraction f0 and is generally smaller
Fig. 1. Dependence of fc on f0.
than the value of 0.15. Fig. 1 shows a dependence of fc on f0. Only for small f0, as a first approximation, can the value of fc be taken as a constant. Tvergaard and Needleman [15] suggested that the value of fc can be taken as 0.15. These observations show that fc is not a material parameter but is a parameter determined by fitting. Table 1, which gives numerous values from the literature, shows the variability of this parameter for a considered material. On the other hand, the final failure void volume fraction fF is considered a parameter that may be experimentally determined [6]. The mentioned authors show that fF affects the post-initiation load diameter–reduction curve and that the smaller the fF the faster the load decreases. In FE modelling, Chambert [24] has analysed curves illustrating the evolution of equivalent stress with equivalent plastic strain using three fF values: 0.075, 0.15 and 0.225. The author noted that this parameter controls the final slope of these curves. Originally, Tvergaard and Needleman [15] proposed a value equal to 0.25. From Eq. (2), the final failure void volume fraction fF may be determined if the values of q1, d and fc are previously known using fF ¼ fc+(1/q1fc)/d. This shows that the final failure parameter may be given to a first approximation by f F ¼ 1=dq1 ¼ ð1=dÞ f u when the value of fc is low. Recently, Zhang et al. [28] have proposed an empirical expression for fF and f0, which is written as a linear equation: fF ¼ 0.15+2 f0. This signifies that fF can be fixed to a first approximation at 0.15 for low f0 values. As shown in Table 1, this parameter can take values between 0.15 and 0.44. 4. Experimental technique and results 4.1. Material The nickel and chromium steel used in this investigation is the 12NC6 steel. Its chemical composition was (wt%): 0.12C, 0.007S, 0.32Si, 0.6Mn, 1.6M, 0.85Cr and 0.076A1.
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The steel was austenitized at 880 1C for 1 h, and then quenched in an oven. The final steel micro-structure is illustrated in Fig. 2. This shows the ferritic–pearlitic bands. The initial void volume fraction f0 is small (f0 ¼ 0.001%). This micro-structural parameter is necessary to use the GTN model. 4.1.1. Mechanical properties Three tensile tests were carried out at ambient temperature on cylindrical specimens with a diameter d ¼ 11 mm. The mechanical properties of this material are as follows: 0.2% proof stress of 340 MPa, ultimate stress of 489 MPa
Fig. 2. Micro-structure of the steel used in this study.
223
and Young’s modulus of 194 GPa. The uniaxial plastic flow behaviour was assumed to follow a Ramberg–Osgood stress–strain law: 8 ¼ e þ p > < s ¼ Ee if spsy ; (7) > : s ¼ sy þ kn if s4sy ; p with the hardening exponent N ¼ 0.45 and ductility coefficient k ¼ 544 MPa. To experimentally determine the crack initiation point, other tensile tests have been carried out using three sets of standard axisymmetrically notched bar specimens: AE2, AE4 and AE10. This specimen is often used in order to vary the stress triaxiality ratio T with the notch radius according to Brigdman’s formula [26]: sm 1 r T¼ ¼ þ ln 1 þ , (8) 2r seq 3 where r is notch radius and r is the minimum radius of the notched round specimen. As shown in Fig. 3, the specimens used in this study have a notch radius r ¼ 2, 4 and 10 mm, and minimum radius r ¼ 3 mm. This leads to three values of stress traixiality ratio T ¼ 0.89, 0.65 and 0.47. During this kind of test, it is difficult to correctly measure the diameter reduction. In our study, the variation of diameter has been measured using an image technique, which has the advantage of not requiring contact. An example of an image is shown in Fig. 4. This technique can follow the evolution of the diameter reduction exactly at the notch bottom and at the minimum diameter. In
Fig. 3. Notched round specimens: (a) NR2, (b) NR4 and (c) NR10.
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Fig. 5. Load versus diameter reduction for three notched axisymmetric specimens.
Fig. 6. Failure strain versus stress triaxiality ratio.
when T increases, i.e., when the minimum diameter decreases. This observation is in agreement with several literature results. 4.2. Fracture toughness tests Fig. 4. An example of typical images to measure the diameter reduction. Global profile corresponding to (a) first image, (b) at initial diameter and (c) at crack initiation.
addition, the image analysis technique easily detects the point where crack initiation is localized. Fig. 5 shows the evolution of applied load as a function of diameter reduction DF for the three configurations examined. These curves give the critical diameter reduction DFc at crack initiation. The ductility er is defined as the average longitudinal strain at fracture: F0 r ¼ 2 ln . F0 DFc Fig. 6 shows the variation of failure strain with stress triaxiality ratio T. The failure strain is sensitive to the stress triaxiality ratio T. Indeed, the coalescence strain decreases
Six fracture mechanics tests were performed at room temperature using side-grooved CT specimens (width W ¼ 50 mm, thickness BN ¼ 20 mm) pre-cracked to a/W ¼ 0.5. The parameter J was determined according to the ASTM standard procedure [25] using the area under the load–displacement curve and crack extension was deduced from the partial unloading compliance method. Fig. 7 shows the evolution of J with crack extension. The J0.2 value, corresponding to a crack extension Da ¼ 0.2 mm, is about 210 kJ/m2. The high value of this parameter can be explained by the small value of initial void volume fraction of the steel used in this study. The stable growth stage, within the valid domain of crack extension defined by two parallel offset lines at 0.15 and 1.5 mm crack growth, is described by a power function J ¼ J1.0Dan, with J1.0 ¼ 529 kJ/m2 and n ¼ 0.89.
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Fig. 7. J-integral versus crack extension Da (experimental results).
5. Numerical investigation The GTN model has been applied in the FE simulation using the SYSTUS software. The material is assumed to be homogeneous, elastoplastic with isotropic hardening. Five parameters of this model are necessary: q1 ¼ q (i.e., q2 ¼ 1), A, f0, d and fc. Two configurations have been analysed: the tensile AN specimen and the CT specimen. Fig. 8 shows the meshing corresponding to each specimen using quadratic elements with eight nodes. In the CT specimen, the mesh size around the crack tip is refined and chosen equal to Le ¼ 0.2 mm, which is a size usually used for ductile tearing simulation. It is well known that the Le value affects the computation results using the local approach [9,26]. In the literature, proposals have been made to estimate the mesh size as a function of the number of inclusions per unit volume Nv [9,26]. For the ductile tearing computation, the whole specimen must be considered even where the specimen has geometrical symmetry [24,27]. At first, the following parameter set has been chosen in order to simulate the P–DF curve of the AN specimen:
f0: initial void volume fraction is evaluated by observations f0=0.001%, q: interaction between void parameters, q=q1=1.5 and q2=l ((with q3(q1)2), d: amplifying factor of f in the coalescence phase is arbitrarily fixed to 3 with d ¼ ðf u f c Þ=ðf F f c Þ where fu*=1/q1=0.666, A: continuous void nucleation parameter is arbitrarily fixed to 0.001 and fc: critical void volume fraction or critical porosity (adjustable parameter) with fF determined as a function of 1/q1, d and fc from Eq. (2).
From this set of parameters, the critical void volume fraction fc has been computed by fitting the experimental load–diameter reduction curves for different stress triaxi-
Fig. 8. (a) Tensile axisymmetric notched, (b) typical meshing for CT specimen and (c) detail around crack tip (Le ¼ 0.2 mm).
Fig. 9. Critical void volume fraction versus stress triaxiality ratio.
ality ratio values. As expected, fc decreases with increasing stress triaxiality ratio T (Fig. 9). The value of fc depends on the set of parameters initially fixed and the average value in this case is fc ¼ 0.004. Using this set of parameters (f0 ¼ 0.001%, q ¼ q1 ¼ 1.5, A ¼ 0.001, d ¼ 3 and fc ¼ 0.004), crack growth simulation has been carried out
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parameters characterizing this model have been studied using literature data and have been classified as follows:
Fig. 10. J–Da curves. Comparison between numerical results and experimental data.
The constitutive parameters (q1, q2, q3) which are always fixed and not dependent on material. In the majority of cases q1 ¼ 1.5. The material parameters (f0, fN, fc, fF, eN and SN), where f0 can be determined experimentally and it is possible to determine the relationship between f0, fc and fF.
The critical void volume fraction fc decreases with increasing stress triaxiality. In numerical calculations, related to experimental data on a nickel–chrome steel, an average value of this parameter has been used. The crack growth resistance curve has been simulated by using the following parameters: the constitutive parameters are constant, q1 ¼ 1.5 and q2 ¼ 1, the amplifying factor in the coalescence phase d is 3, the void nucleation factor is arbitrarily fixed at 0.001. The first calculations lead to a difference between the numerical results and experimental data. This may be due to using a value for the coalescence phase, d, which is too high. Indeed, a value of this chosen parameter equal to 2 leads to the correct results. In this model case, it is difficult to determine a unique solution. References
Fig. 11. Load versus diameter reduction for d ¼ 2 and 3.
using the GTN model, but underestimates the resistance curve. This may be explained by the high value of the amplifying factor d (Fig. 10). Fig. 10 also shows that the crack growth curve is correctly simulated when d is reduced to 2. This indicates that the amplifying factor plays an important role in crack growth simulation. On the other hand, d plays a weak part on the end of the curve (load versus diameter reduction) for AN numerical result (see Fig. 11). In our case we have chosen arbitrarily this parameter (d ¼ 3), but we suggest taking smaller values (1–2) to obtain good results. 6. Conclusions This study has examined ductile fracture analysis using the GTN model based on local damage. The nine
[1] McClintock FA. A criterion for ductile fracture by growth of holes. J Appl Mech 1968;35:363–434. [2] Thomason PF. Ductile fracture and the stability of incompressible plasticity in the presence of microvoids. Acta Metall 1968;29:360–5. [3] Rice JR, Tracey DM. On ductile enlargement of voids in triaxial stress fields. J Mech Phys Solids 1969;17:201–17. [4] Thomason PF. Ductile fracture of metals. Oxford: Pergamon Press; 1990. [5] Wilsius J, Inad A, Nait Abdelaziz M, Mesmacque G, Eripret C. Void growth and damage models for predicting ductile fracture in welds. Fatigue Fract Eng Mater Struct 2000;23:105–12. [6] Bauvineau L, Burlet H, Eripret C, Pineau A. Modelling ductile stable crack growth in a C–Mn steel with local approaches. In: First European mechanics of materials conference on local approach to fracture, Fontainebleau, September 1996. [7] Imad A, Wilsius J, Na-t Abdelaziz M, Mesmacque G. Experiments and numerical approaches to ductile tearing in an 2024-T351 aluminium alloy. Int J Mech Sci 2003;45:1849–61. [8] Dutta BK, Saini S, Arora N. Application of a modified damage potential to predict ductile crack initiation in welded pipes. Int J Press Vessels Piping 2005;82(11):833–9. [9] Rousselier G. Ductile fracture models and their potential in local approach to fracture. Nucl Eng Design 1987;105:97–111. [10] Gurson AL. Continuum theory of ductile rupture by void nucleation and growth: Part I—yield criteria and flow rules for porous ductile media. ASME J Eng Mater Technol 1977;99:2–15. [11] Wilsius J. Etude experimentale et numerique de la dechirure ductile basee sur des approches locales en mecanique de la rupture. PhD thesis, Universite des Sciences et Technologies de Lille, France, 1999. [12] Devaux J, Joly P, Leblond JB. Simulation by the local approach of ductile crack growth in a pressure vessel steel using an improved Gurson–Tvergaard model. 21st MPA seminar, October 1995. [13] Betegon C, Rodriguez C, Belzunce FJ. Analysis and modelisation of short crack growth by ductile fracture micromechanisms. Fatigue Fract Eng Mater Struct 1997;20(5):633–44.
ARTICLE IN PRESS N. Benseddiq, A. Imad / International Journal of Pressure Vessels and Piping 85 (2008) 219–227 [14] Hao S, Brocks W, Heerens J, Hellmann D. Simulation of 3D ductile crack growth by the Gurson–Tvergaard–Needleman model. ECF11 Mechanisms and mechanics of damage and failure, September 1996. p. 805–10. [15] Tvergaard V, Needleman A. Analysis of the cup–cone fracture in a round tensile bar. Acta Metall 1984;32:157–69. [16] Perrin G, Leblond JB. Analytical study of a hollow sphere made of plastic porous material and subjected to hydrostatic tension— application to some problems in ductile fracture metals. Int J Plasticity 1990;6:677–98. [17] Steglich D, Brocks W. Micromechanical modelling of damage and fracture of ductile materials. Fatigue Fract Eng Mater Struct 1998; 21(10):1175–88. [18] Gao X, Faleskog J, Shih CF, Dodds RH. Ductile tearing in partthrough cracks: experiments and cell-model predictions. Eng Fract Mech 1998;59:761–77. [19] Faleskog J, Gao X, Shih CF. Cell model for nonlinear fracture mechanics—I. Micromechanics calibration. Int J Fract 1998;89:365–73. [20] Franklin AG. Comparison between a quantitative microscope and chemical methods for assessment of non-metallic inclusions. J Iron Steel Inst 1969:181–6. [21] Zhang ZL, Niemi E. A new failure criterion for the Gurson–Tvergaard dilatation constitutive model. Int J Fract 1995;70:321–34. [22] Guillemer-neel C. Comportement mecanique et endommagement de la fonte a graphite spheroidal. PhD thesis, UTC France, 1999. [23] Koplik J, Needleman A. Void growth and coalescence in porous plastic solids. Int J Solids Struct 1988;24(8):835–53. [24] Chambert J. Etude parametrique d’un modele d’endommagement de type Gurson. Prise en compte d’un critere de coalescence. In: Proceeding du 14eme Congres de Mecanique, Toulouse, 1999. [25] Chung WN, Williams JG. Elastic–plastic fracture test methods: the user’s experience (second volume). In: Joyce JA, editor. ASTM STP, 1114, vol. 2. ASTM; 1991. p. 320–39. [26] Devaux JC, Mudry F, Pineau A, Rousselier G. Experimental and numerical validation of a ductile fracture local criterion based on a simulation of cavity growth. In: Saxena A, Merkle JG, editors. Non linear fracture mechanics, vol. 2II. Philadelphia: American Society for Testing and Materials; 1989. p. 332–54. [27] Zhang ZL, Niemi E. Analyzing ductile fracture using dual dilational constitutive equations. Fatigue Fract Eng Mater Struct 1994;17(6): 695–707. [28] Zhang ZL, Thaulow C, Phidegard J. A complete Gurson model approach for ductile fracture. Eng Fract Mech 2000;67:155–68. [29] Zhang ZL. A sensitivity analysis of material parameters for the Gurson constitutive model. Fatigue Fract Eng Mater Struct 1996; 19(5):561–70.
227
[30] Brocks W, Hao S, Steglich D. Micromechanical modelling of the damage and toughness behaviour of nodular cast iron materials. In: First European mechanics of materials conference on local approach to fracture, Fontainebleau, September 1996. [31] Brocks W, Sun DZ, Honig A. Verification of micromechanical models for ductile fracture by cell model calculations. Comput Mater Sci 1996;7:235–41. [32] Dong MJ, Berdin C, Beranger AS, Prioul C. Damage effect in the fracture toughness of nodular cast iron. In: First European mechanics of materials conference on local approach to fracture, Fontainebleau, September 1996. [33] Decamp K, Bauvineau L, Besson J, Pineau A. Size and geometry effects on ductile rupture of notched bars in a C–Mn steel. Experiments and modelling. Int J Fract 1997;88:1–18. [34] Sovik OP. Experimental and numerical investigation of void nucleation in an AlMgSi alloy. In: First European mechanics of materials conference on local approach to fracture, Fontainebleau, September 1996. [35] Ghosal AK, Narasimhan R. Numerical simulations of hole growth and ductile fracture initiation under mixed-mode loading. Int J Fract 1996;77:281–304. [36] Kuna M, Sun DZ. Three-dimensional cell model analyses of void growth in ductile materials. Int J Fract 1996;81:235–58. [37] He R, Steglich D, Heerens J, Wang G-X, Brocks W, Dahms M. Influence of particle size and volume fraction on damage and fracture in Al-A13Ti composites and micromechanical modelling using the GTN model. Fatigue Fract Eng Mater Struct 1998;21(10): 1189–201. [38] Sun DZ, Siegle D, Voss B, Schmitt W. Application of local damage models to the numerical analysis of ductile rupture. Fatigue Fract Eng Mater Struct 1989;12(3):201–12. [39] Siegmund T, Bernauer G, Brocks W. Two models of ductile fracture in contest: porous metal plasticity and cohesive elements. ECF12 fracture from defects, September 1998. p. 933–8. [40] Xia L, Cheng L. Transition from ductile tearing to cleavage fracture: a cell-model approach. Int J Fract 1997;87(3):289–306. [41] Schmitt W, Sun DZ, Blauel JG. Damage mechanics analysis (Gurson model) and experimental verification of the behaviour of a crack in a weld-cladded component. Nucl Eng Design 1997;74:237–46. [42] Skallerud B, Zhang ZL. A 3D numerical study of ductile tearing and fatigue crack growth under nominal cyclic plasticity. Int J Solids Struct 1997;34(24):3141–61. [43] Pardoen T, Doghri I, Delannay F. Experimental and numerical comparison of void growth models and void coalescence criteria for the prediction of ductile fracture in copper bars. Acta Mater 1998;46(2):541–52.
International Journal of Pressure Vessels and Piping 85 (2008) 219–227 www.elsevier.com/locate/ijpvp
A ductile fracture analysis using a local damage model N. Benseddiqa,, A. Imadb a
Laboratoire de Me´canique et de Rhe´ologie de Tours, Ecole Nationale d’Inge´nie´urs du Val de Loire (ENIVL), Rue de la Chocolaterie, 41000 Blois Cedex, France b Laboratoire de Me´canique de Lille (UMR CNRS 8107), USTL, Ecole Polytechnique Universitaire de Lille Cite´ Scientifique, Avenue P. Langevin, 59655 Villeneuve d’Ascq Cedex, France Received 26 July 2006; received in revised form 27 September 2007; accepted 27 September 2007
Abstract In this study, the Gurson–Tvergaard–Needleman (GTN) model is used to investigate ductile tearing. The sensitivity of the model parameters has been examined from literature data. Three types of parameters have been reported: the ‘‘constitutive parameters’’ q1, q2 and q3, the ‘‘initial material and nucleation parameters’’ and the ‘‘critical and final failure parameters’’. Each parameter in this model has been analysed in terms of various results in the literature. Both experimental and numerical results have been obtained for notched round and CT specimens to characterize ductile failure in a NiCr steel (12NC6) with a small initial void volume fraction f0 (f0 ¼ 0.001%). Ductile crack growth, defined by the J–Da curve, has been correctly simulated using the numerical calculations by adjusting the different parameters of the GTN model in the calibration procedure. r 2007 Elsevier Ltd. All rights reserved. Keywords: Ductile tearing; Local damage; Numerical simulation
1. Introduction It is well known that the ductile failure of materials is controlled by three stages, i.e. micro-void nucleation, growth and coalescence mechanisms. In order to describe this damage and fracture process, several models using local approaches have been proposed in the literature. Globally, local approach modelling may be described by two main families. Firstly, uncoupled models are based upon void growth, such as McClintock [1], Thomason [2,4], Rice and Tracey [3]. For these models, the material failure stage is characterized by a critical void growth ratio (R/R0)c, corresponding to crack initiation. This parameter is considered as a material intrinsic property and it is obtained by a combination of numerical results and experimental data. Crack growth simulation is achieved by a node release technique when the critical void growth ratio is reached [5–8].
Secondly, coupled models are based upon continuum damage mechanics such as Rousselier [9] and Gurson [10]. The use of these models is based upon a constitutive equation introducing the material behaviour and a failure criterion. In this type of model, the crack growth simulation is automatically performed using the complete deterioration of the element at the crack tip [6,11–14]. Among these models, the micro-mechanical model proposed by Gurson is the most widely used for ductile porous materials. The Gurson model employs equations to describe the constitutive response of the metal [10]. These expressions are based upon a continuum elastic–plastic model that accounts for micro-void nucleation and growth. Tvergaard and Needleman [15] have subsequently modified this model introducing the following yield condition (Gurson–Tvergaard–Needleman (GTN) model):
Fðsy ; seq ; f Þ ¼ Corresponding author.
E-mail address: [email protected] (N. Benseddiq). 0308-0161/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2007.09.003
s2eq 3 sm q þ 2f q cosh 1 2 2 sy s2y ð1 þ q3 ðf Þ2 Þ ¼ 0,
ð1Þ
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220
f* f u q1, q 2, SN d sy seq sm E
Nomenclature J a Da f f0 fc fF fN
J-integral crack length crack extension void volume fraction initial void volume fraction critical void volume fraction void volume fraction at final failure volume fraction of void-forming particles
where q1 q2 and q3 are the constitutive parameters with q3 ¼ (q1)2; sm is the mean normal stress defined by sm ¼ skk/3, skk is the trace of the stress tensor; seq is the conventional pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi von Mises equivalent stress defined by seq ¼ ð3=2Þsij sij (sij is the stress deviator); sy is the yield stress; f*(f) is a function of the void volume fraction f and represents the modified damage parameter. This function is defined by 8 f for f pf c ; > < f þ dðf f Þ for f c pf pf F ; (2) f ðf Þ ¼ c c > :f f Xf : F u The expression d ¼ ðf u f c Þ=ðf F f c Þ is considered as an accelerating factor, which is introduced in order to describe the final stage of ductile failure, where fc is the critical void volume fraction corresponding to void coalescence which at first occurs. This is considered as a material intrinsic parameter, f u is the ultimate value of the damage parameter which can be written as a function of q1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and q3 by f u ¼ ðq1 þ ðq1 Þ2 q3 Þ=q3 . If q3 ¼ (q1)2 the ultimate void volume fraction f u equals (1/q1). fF is the final void volume fraction. According to Eq. (2), f ¼ fF corresponds to the failure stage, f ¼ f u . In general, the evolution equation for f consists of two terms describing the nucleation and growth of voids: df ¼ df nucleation þ df growth with f ðt0 Þ ¼ f 0 ,
(3)
where f0 is the initial void volume fraction. Nucleation is considered to depend exclusively on the effective strain of the matrix material: df nucleation ¼ Adpeq þ Bðdseq þ cdsm Þ.
(4)
The parameter ‘‘A’’ is chosen so that void nucleation follows a normal distribution around a critical plastic strain ep, which is estimated using the following expression: " # fN 1 p N 2 pffiffiffiffiffiffi exp A¼ , (5) 2 SN SN 2p where fN is the volume fraction of void nucleating particles, eN is the mean void nucleation strain and SN is the corresponding standard deviation.
modified damage parameter ultimate value of damage parameter q3 constitutive parameters standard deviation accelerating factor yield stress of material equivalent stress hydrostatic stress elastic modulus
For strain-controlled void nucleation B ¼ 0. But in the case of macroscopic ductility, the hydrostatic stress dependence of void nucleation leads to a strong nonnormality in the plastic flow rule, which promotes early flow localization. It is worth noting that a stress-controlled nucleation criterion was found to be important in capturing the effect of stress state on damage initiation. In this case A ¼ 0 and " # fN 1 seq þ csm sN 2 pffiffiffiffiffiffi exp , B¼ 2 SN S N 2p c is a constant parameter and can take values between 0.3 and 0.4. The micro-void volume fraction rate due to growth is given by df growth ¼ ð1 f Þ dpkk ,
(6)
where ekkp is the plastic hydrostatic strain. 2. Scope of study In the current paper, the GTN model is used to analyse ductile tearing. The sensitivity of the model parameters is analysed from literature data. Three types of parameters have been reported: ‘‘the constitutive parameters’’ q1, q2 and q3, ‘‘the initial material and nucleation parameters’’ and ‘‘the critical and final failure parameters’’. Both experimental and numerical results have been obtained using notched round and CT specimens to characterize rupture behaviour in a NiCr steel (12NC6) with a small initial void volume fraction f0 (f0 ¼ 0.001%). The parameters in this model are determined by a calibration procedure using three types of mechanical tests: classical tensile (true stress versus true strain), tensile with an axisymmetric notched (AN) specimen AE (load versus diameter reduction) and ductile tearing tests (J-parameter versus crack growth Da). The results obtained by the GTN model are discussed in the context of the extensive literature on model parameters. 3. Analysis of GTN model parameters Generally, the GTN model is used to predict the damage of ductile materials [4–43]. To apply this model nine
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Table 1 GTN model parameter values from the literature Ref. q1
f0
fc
fF
[6] [12]
0.002 0.00016 0.0001 0.0015 0.001 0.114 0.114 0.114 0.114 0.114 0.00033 0.002 0.077 0.0023 0.001 – 0.0 0.08 0 0 0 0.00057 0.0025 0.005 0 0 0 0.00033 0.002
0.004 0.0005 0.0003 0.035 0.02 0.13 0.2 0.3 0.19 0.175 0.026 0.033 0.12 0.004 0.003 0.15 0.04 0.15 0.02 0.2 0.15 0.03 0.021 – 0.06 0.04 0.03 0.026 0.028
– 23 – 2.8 – 4.3 0.15 5.49 1.141 67 0.272 4 0.35 4.73 0.44 4.35 0.235 14.3 0.235 10.97 0.15 5.17 0.15 – 0.2 6.8 3 – – 0.25 5.17 0.195 – 0.28 4 0.34 2 0.32 4 0.28 4 1.3 3.4 0.2 – 0.212 4 0.197 4 0.189 4 0.15 6.24 – –
[13] [14]
[17]
[29] [31] [32] [33] [34] [35] [36] [37]
[38] [39] [40] [41]
[42] [43]
1.5 1.47 1.47 1.5 1.5 1.43 1.1 1.1 1.2 1.2 1.25 1.15 1.5 1.5 – 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.25 1.5 1.5 1.5 1.25 1.47
d
fN
eN
SN
Le (mm) Material
sy (MPa) Rm (MPa) Specimen
– – – 0.00085 0.04 – – – – – 0.006 0.004 – – 0.01 0.04 0.008 0 0.04 0.4 0.08 0.004 0.02 0.001 0.002 0.002 0.002 0.006 –
– – – 0.3 0.3 – – – – – 0.3 0.3 – – – 0.3 0.3 0 0.5 0.05 0.025 0.3 0.3 0.04 0.3 0.3 0.3 0.3 –
– – – 0.1 0.1 – – – – – 0.1 0.1 – – 0.01 0.1 0.1 0 0.2 0.02 0.025 0.1 0.1 0.01 0.1 0.1 0.1 0.1 –
0.2 0.5 0.8 – 0.1 0.1 – – – – – 0.033 – 0.2 0.5 – – –
– 471 470 779
– 593 608 840
SENT25 CT22.5 AE2 AE2 AE2 SENB (B ¼ 25)
366 – – – – 370 – 230 190 260 – 366
– – – – – – – 433 500 – – –
AE4
–
0.2 0.2 0.1 0.1 0.1 0.033 –
parameters are necessary. They can be classified into two principal families:
the ‘‘constitutive parameters’’: q1, q2 and q3 the ‘‘material parameters’’, which are classified in two parts. Firstly, ‘‘the initial material and nucleation parameters’’, which are determined as the initial void volume fraction f0 and the void nucleation parameters fn, en and Sn. Secondly, ‘‘the critical and final failure parameters’’, the critical void volume fraction fc and fF.
A combination of numerical results and experimental data is necessary in order to determine some of the parameters. It is important to note that most of the parameters are not easy to define. Globally, there is no unique method to determine these parameters. For this reason, a global analysis of the large data available in the literature has been performed, in order to examine the validity of the choices of these parameters, and suggest a global method to determine them (see Table 1). 3.1. Constitutive parameters: q1, q2, q3 In order to describe material ductile fracture, Tvergaard [15] has suggested fixing the q1 and q2 values at q1 ¼ 1.5
C–Mn steel (at 300 1C) A508 C13 A A508 C13 B E690 20MnMoNi5 5
GGG40
22NiMoCr37 GGG40 CMn steel (300 1C) AlMgSi alloy – – Composite Al-Al3Ti
E460 steel A533B Steel
CMn steel Cu
54.7
80
– CT25 AE4 AE2, 4 and 10 – – – .
612
707
AE0.25 and 4 CT
– 440 620 320 360 312
– 600 750 580 – 325
– SENB – AE1 2 4 8
and q2 ¼ 1. These values have been used in several investigations (see Table 1). In a recent study, Perrin et al. [16] have determined a correlation between the parameter q and porosity f: q ¼ q(f). The authors show that when the porosity tends to zero, the q value tends to 1.47. This value is similar to that suggested by Tvergaard (q ¼ 1.5) [15]. In the modelling of nodular cast iron, Steglich et al. [17] have proposed a yield condition determined from a cell model that results in using a quadratic equation to estimate the parameter q1. The calculations of this parameter for various stress triaxiality ratios T give an average value of q1 equal to 1.5. The authors have noted that the parameter q1 does not depend on the stress triaxiality ratio T but little physical significance is given to this value. Gao et al. [18] have performed several calculations considering a wide range of material flow properties (hardening N, sy/E) and stress triaxiality ratios (2–3.3) with sy/E ¼ 0.001–0.004. For a specific set of flow properties, the authors have obtained various q1 and q2 parameters. It is interesting to note that the parameter q ¼ q1 q2 is approximately always constant and equal to about 1.5 for all cases studied (Table 2). Faleskog [19] indicates that the two parameters depend on the material hardening exponent. The q parameters depend strongly on strain hardening and
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Table 2 Optimal values for micro-mechanics parameters (q1, q2) [19] Hardening (N) sy/E ¼ 0.001 q1 5 6.7 10
q2
sy/E ¼ 0.002
q1 q2 q1
1.96 0.78 1.531 1.78 0.83 1.483 1.58 0.9 1.425
q2
sy/E ¼ 0.004
q1 q2 q1
1.87 0.8 1.496 1.68 0.86 1.438 1.46 0.93 1.359
q2
q1 q2
1.71 0.84 1.43 1.49 0.9 1.342 1.29 0.98 1.265
material strength; for a material of moderate hardening and strength, e.g., N ¼ 0.1 and sy/E ¼ 0.002, the calibration gives q1 ¼ 1.46 and q2 ¼ 0.93. High strain hardening reduces the effect of stress triaxiality on void growth rate, resulting in q2 values much below unity [19] (Table 2). It is important to note that the value of q1 may be considered as essential to describe the failure stage. Because of Eq. (2), fracture of material occurs (i.e. f ¼ fF) when f* reaches its ultimate value f u ¼ 1=q1 (when q1 ¼ 1.5, f u ¼ 0:666). The physical validity of this value is not proven. Table 1 shows that the value of the constitutive parameter q1 varies from 1.1 to 1.5 for many materials. However, this parameter is often fixed to 1.5 and q2 ¼ 1. 3.2. Material parameters: f0, fN, fc, fF, eN and SN 3.2.1. Initial material and nucleation parameters Generally, the initial void volume fraction f0 and volume fraction of void nucleating particles fN are evaluated by microscopical examination of the undamaged material. As a first approximation, f0 can be obtained from the MnS inclusions by Franklin’s formula using the chemical composition [20]. Table 1 also shows that the values of the nucleation parameters fN, eN and SN can be arbitrarily fixed. The values eN ¼ 0.3 and SN ¼ 0.1 have been used in several studies, and they are determined by fitting. 3.2.2. Critical and final failure parameters In most investigations, only the critical void volume fraction fc is considered as a parameter, obtained by fitting the numerical calculations with experimental results, and the other parameters are fixed arbitrary. According to Zhang et al. [21], fc is not a constant but decreases when the stress triaxiality ratio T increases. However, other authors note that fc can be taken as a constant only for small f0 values. In a similar study, Steglich et al. [17] confirm that the fc value depends on stress triaxiality T: fc decreases with the increase of T. In a recent experimental study [22], these observations were confirmed and the author proposed a linear correlation between fc and T for a high f0 value (f0 ¼ 13%): fc ¼ 0.2550.146 T. According to Koplik and Needleman [23], the fc value, which signifies the onset of coalescence, seems to vary slowly with stress triaxiality ratio but depends strongly on the initial void volume fraction f0 and is generally smaller
Fig. 1. Dependence of fc on f0.
than the value of 0.15. Fig. 1 shows a dependence of fc on f0. Only for small f0, as a first approximation, can the value of fc be taken as a constant. Tvergaard and Needleman [15] suggested that the value of fc can be taken as 0.15. These observations show that fc is not a material parameter but is a parameter determined by fitting. Table 1, which gives numerous values from the literature, shows the variability of this parameter for a considered material. On the other hand, the final failure void volume fraction fF is considered a parameter that may be experimentally determined [6]. The mentioned authors show that fF affects the post-initiation load diameter–reduction curve and that the smaller the fF the faster the load decreases. In FE modelling, Chambert [24] has analysed curves illustrating the evolution of equivalent stress with equivalent plastic strain using three fF values: 0.075, 0.15 and 0.225. The author noted that this parameter controls the final slope of these curves. Originally, Tvergaard and Needleman [15] proposed a value equal to 0.25. From Eq. (2), the final failure void volume fraction fF may be determined if the values of q1, d and fc are previously known using fF ¼ fc+(1/q1fc)/d. This shows that the final failure parameter may be given to a first approximation by f F ¼ 1=dq1 ¼ ð1=dÞ f u when the value of fc is low. Recently, Zhang et al. [28] have proposed an empirical expression for fF and f0, which is written as a linear equation: fF ¼ 0.15+2 f0. This signifies that fF can be fixed to a first approximation at 0.15 for low f0 values. As shown in Table 1, this parameter can take values between 0.15 and 0.44. 4. Experimental technique and results 4.1. Material The nickel and chromium steel used in this investigation is the 12NC6 steel. Its chemical composition was (wt%): 0.12C, 0.007S, 0.32Si, 0.6Mn, 1.6M, 0.85Cr and 0.076A1.
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The steel was austenitized at 880 1C for 1 h, and then quenched in an oven. The final steel micro-structure is illustrated in Fig. 2. This shows the ferritic–pearlitic bands. The initial void volume fraction f0 is small (f0 ¼ 0.001%). This micro-structural parameter is necessary to use the GTN model. 4.1.1. Mechanical properties Three tensile tests were carried out at ambient temperature on cylindrical specimens with a diameter d ¼ 11 mm. The mechanical properties of this material are as follows: 0.2% proof stress of 340 MPa, ultimate stress of 489 MPa
Fig. 2. Micro-structure of the steel used in this study.
223
and Young’s modulus of 194 GPa. The uniaxial plastic flow behaviour was assumed to follow a Ramberg–Osgood stress–strain law: 8 ¼ e þ p > < s ¼ Ee if spsy ; (7) > : s ¼ sy þ kn if s4sy ; p with the hardening exponent N ¼ 0.45 and ductility coefficient k ¼ 544 MPa. To experimentally determine the crack initiation point, other tensile tests have been carried out using three sets of standard axisymmetrically notched bar specimens: AE2, AE4 and AE10. This specimen is often used in order to vary the stress triaxiality ratio T with the notch radius according to Brigdman’s formula [26]: sm 1 r T¼ ¼ þ ln 1 þ , (8) 2r seq 3 where r is notch radius and r is the minimum radius of the notched round specimen. As shown in Fig. 3, the specimens used in this study have a notch radius r ¼ 2, 4 and 10 mm, and minimum radius r ¼ 3 mm. This leads to three values of stress traixiality ratio T ¼ 0.89, 0.65 and 0.47. During this kind of test, it is difficult to correctly measure the diameter reduction. In our study, the variation of diameter has been measured using an image technique, which has the advantage of not requiring contact. An example of an image is shown in Fig. 4. This technique can follow the evolution of the diameter reduction exactly at the notch bottom and at the minimum diameter. In
Fig. 3. Notched round specimens: (a) NR2, (b) NR4 and (c) NR10.
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Fig. 5. Load versus diameter reduction for three notched axisymmetric specimens.
Fig. 6. Failure strain versus stress triaxiality ratio.
when T increases, i.e., when the minimum diameter decreases. This observation is in agreement with several literature results. 4.2. Fracture toughness tests Fig. 4. An example of typical images to measure the diameter reduction. Global profile corresponding to (a) first image, (b) at initial diameter and (c) at crack initiation.
addition, the image analysis technique easily detects the point where crack initiation is localized. Fig. 5 shows the evolution of applied load as a function of diameter reduction DF for the three configurations examined. These curves give the critical diameter reduction DFc at crack initiation. The ductility er is defined as the average longitudinal strain at fracture: F0 r ¼ 2 ln . F0 DFc Fig. 6 shows the variation of failure strain with stress triaxiality ratio T. The failure strain is sensitive to the stress triaxiality ratio T. Indeed, the coalescence strain decreases
Six fracture mechanics tests were performed at room temperature using side-grooved CT specimens (width W ¼ 50 mm, thickness BN ¼ 20 mm) pre-cracked to a/W ¼ 0.5. The parameter J was determined according to the ASTM standard procedure [25] using the area under the load–displacement curve and crack extension was deduced from the partial unloading compliance method. Fig. 7 shows the evolution of J with crack extension. The J0.2 value, corresponding to a crack extension Da ¼ 0.2 mm, is about 210 kJ/m2. The high value of this parameter can be explained by the small value of initial void volume fraction of the steel used in this study. The stable growth stage, within the valid domain of crack extension defined by two parallel offset lines at 0.15 and 1.5 mm crack growth, is described by a power function J ¼ J1.0Dan, with J1.0 ¼ 529 kJ/m2 and n ¼ 0.89.
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Fig. 7. J-integral versus crack extension Da (experimental results).
5. Numerical investigation The GTN model has been applied in the FE simulation using the SYSTUS software. The material is assumed to be homogeneous, elastoplastic with isotropic hardening. Five parameters of this model are necessary: q1 ¼ q (i.e., q2 ¼ 1), A, f0, d and fc. Two configurations have been analysed: the tensile AN specimen and the CT specimen. Fig. 8 shows the meshing corresponding to each specimen using quadratic elements with eight nodes. In the CT specimen, the mesh size around the crack tip is refined and chosen equal to Le ¼ 0.2 mm, which is a size usually used for ductile tearing simulation. It is well known that the Le value affects the computation results using the local approach [9,26]. In the literature, proposals have been made to estimate the mesh size as a function of the number of inclusions per unit volume Nv [9,26]. For the ductile tearing computation, the whole specimen must be considered even where the specimen has geometrical symmetry [24,27]. At first, the following parameter set has been chosen in order to simulate the P–DF curve of the AN specimen:
f0: initial void volume fraction is evaluated by observations f0=0.001%, q: interaction between void parameters, q=q1=1.5 and q2=l ((with q3(q1)2), d: amplifying factor of f in the coalescence phase is arbitrarily fixed to 3 with d ¼ ðf u f c Þ=ðf F f c Þ where fu*=1/q1=0.666, A: continuous void nucleation parameter is arbitrarily fixed to 0.001 and fc: critical void volume fraction or critical porosity (adjustable parameter) with fF determined as a function of 1/q1, d and fc from Eq. (2).
From this set of parameters, the critical void volume fraction fc has been computed by fitting the experimental load–diameter reduction curves for different stress triaxi-
Fig. 8. (a) Tensile axisymmetric notched, (b) typical meshing for CT specimen and (c) detail around crack tip (Le ¼ 0.2 mm).
Fig. 9. Critical void volume fraction versus stress triaxiality ratio.
ality ratio values. As expected, fc decreases with increasing stress triaxiality ratio T (Fig. 9). The value of fc depends on the set of parameters initially fixed and the average value in this case is fc ¼ 0.004. Using this set of parameters (f0 ¼ 0.001%, q ¼ q1 ¼ 1.5, A ¼ 0.001, d ¼ 3 and fc ¼ 0.004), crack growth simulation has been carried out
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parameters characterizing this model have been studied using literature data and have been classified as follows:
Fig. 10. J–Da curves. Comparison between numerical results and experimental data.
The constitutive parameters (q1, q2, q3) which are always fixed and not dependent on material. In the majority of cases q1 ¼ 1.5. The material parameters (f0, fN, fc, fF, eN and SN), where f0 can be determined experimentally and it is possible to determine the relationship between f0, fc and fF.
The critical void volume fraction fc decreases with increasing stress triaxiality. In numerical calculations, related to experimental data on a nickel–chrome steel, an average value of this parameter has been used. The crack growth resistance curve has been simulated by using the following parameters: the constitutive parameters are constant, q1 ¼ 1.5 and q2 ¼ 1, the amplifying factor in the coalescence phase d is 3, the void nucleation factor is arbitrarily fixed at 0.001. The first calculations lead to a difference between the numerical results and experimental data. This may be due to using a value for the coalescence phase, d, which is too high. Indeed, a value of this chosen parameter equal to 2 leads to the correct results. In this model case, it is difficult to determine a unique solution. References
Fig. 11. Load versus diameter reduction for d ¼ 2 and 3.
using the GTN model, but underestimates the resistance curve. This may be explained by the high value of the amplifying factor d (Fig. 10). Fig. 10 also shows that the crack growth curve is correctly simulated when d is reduced to 2. This indicates that the amplifying factor plays an important role in crack growth simulation. On the other hand, d plays a weak part on the end of the curve (load versus diameter reduction) for AN numerical result (see Fig. 11). In our case we have chosen arbitrarily this parameter (d ¼ 3), but we suggest taking smaller values (1–2) to obtain good results. 6. Conclusions This study has examined ductile fracture analysis using the GTN model based on local damage. The nine
[1] McClintock FA. A criterion for ductile fracture by growth of holes. J Appl Mech 1968;35:363–434. [2] Thomason PF. Ductile fracture and the stability of incompressible plasticity in the presence of microvoids. Acta Metall 1968;29:360–5. [3] Rice JR, Tracey DM. On ductile enlargement of voids in triaxial stress fields. J Mech Phys Solids 1969;17:201–17. [4] Thomason PF. Ductile fracture of metals. Oxford: Pergamon Press; 1990. [5] Wilsius J, Inad A, Nait Abdelaziz M, Mesmacque G, Eripret C. Void growth and damage models for predicting ductile fracture in welds. Fatigue Fract Eng Mater Struct 2000;23:105–12. [6] Bauvineau L, Burlet H, Eripret C, Pineau A. Modelling ductile stable crack growth in a C–Mn steel with local approaches. In: First European mechanics of materials conference on local approach to fracture, Fontainebleau, September 1996. [7] Imad A, Wilsius J, Na-t Abdelaziz M, Mesmacque G. Experiments and numerical approaches to ductile tearing in an 2024-T351 aluminium alloy. Int J Mech Sci 2003;45:1849–61. [8] Dutta BK, Saini S, Arora N. Application of a modified damage potential to predict ductile crack initiation in welded pipes. Int J Press Vessels Piping 2005;82(11):833–9. [9] Rousselier G. Ductile fracture models and their potential in local approach to fracture. Nucl Eng Design 1987;105:97–111. [10] Gurson AL. Continuum theory of ductile rupture by void nucleation and growth: Part I—yield criteria and flow rules for porous ductile media. ASME J Eng Mater Technol 1977;99:2–15. [11] Wilsius J. Etude experimentale et numerique de la dechirure ductile basee sur des approches locales en mecanique de la rupture. PhD thesis, Universite des Sciences et Technologies de Lille, France, 1999. [12] Devaux J, Joly P, Leblond JB. Simulation by the local approach of ductile crack growth in a pressure vessel steel using an improved Gurson–Tvergaard model. 21st MPA seminar, October 1995. [13] Betegon C, Rodriguez C, Belzunce FJ. Analysis and modelisation of short crack growth by ductile fracture micromechanisms. Fatigue Fract Eng Mater Struct 1997;20(5):633–44.
ARTICLE IN PRESS N. Benseddiq, A. Imad / International Journal of Pressure Vessels and Piping 85 (2008) 219–227 [14] Hao S, Brocks W, Heerens J, Hellmann D. Simulation of 3D ductile crack growth by the Gurson–Tvergaard–Needleman model. ECF11 Mechanisms and mechanics of damage and failure, September 1996. p. 805–10. [15] Tvergaard V, Needleman A. Analysis of the cup–cone fracture in a round tensile bar. Acta Metall 1984;32:157–69. [16] Perrin G, Leblond JB. Analytical study of a hollow sphere made of plastic porous material and subjected to hydrostatic tension— application to some problems in ductile fracture metals. Int J Plasticity 1990;6:677–98. [17] Steglich D, Brocks W. Micromechanical modelling of damage and fracture of ductile materials. Fatigue Fract Eng Mater Struct 1998; 21(10):1175–88. [18] Gao X, Faleskog J, Shih CF, Dodds RH. Ductile tearing in partthrough cracks: experiments and cell-model predictions. Eng Fract Mech 1998;59:761–77. [19] Faleskog J, Gao X, Shih CF. Cell model for nonlinear fracture mechanics—I. Micromechanics calibration. Int J Fract 1998;89:365–73. [20] Franklin AG. Comparison between a quantitative microscope and chemical methods for assessment of non-metallic inclusions. J Iron Steel Inst 1969:181–6. [21] Zhang ZL, Niemi E. A new failure criterion for the Gurson–Tvergaard dilatation constitutive model. Int J Fract 1995;70:321–34. [22] Guillemer-neel C. Comportement mecanique et endommagement de la fonte a graphite spheroidal. PhD thesis, UTC France, 1999. [23] Koplik J, Needleman A. Void growth and coalescence in porous plastic solids. Int J Solids Struct 1988;24(8):835–53. [24] Chambert J. Etude parametrique d’un modele d’endommagement de type Gurson. Prise en compte d’un critere de coalescence. In: Proceeding du 14eme Congres de Mecanique, Toulouse, 1999. [25] Chung WN, Williams JG. Elastic–plastic fracture test methods: the user’s experience (second volume). In: Joyce JA, editor. ASTM STP, 1114, vol. 2. ASTM; 1991. p. 320–39. [26] Devaux JC, Mudry F, Pineau A, Rousselier G. Experimental and numerical validation of a ductile fracture local criterion based on a simulation of cavity growth. In: Saxena A, Merkle JG, editors. Non linear fracture mechanics, vol. 2II. Philadelphia: American Society for Testing and Materials; 1989. p. 332–54. [27] Zhang ZL, Niemi E. Analyzing ductile fracture using dual dilational constitutive equations. Fatigue Fract Eng Mater Struct 1994;17(6): 695–707. [28] Zhang ZL, Thaulow C, Phidegard J. A complete Gurson model approach for ductile fracture. Eng Fract Mech 2000;67:155–68. [29] Zhang ZL. A sensitivity analysis of material parameters for the Gurson constitutive model. Fatigue Fract Eng Mater Struct 1996; 19(5):561–70.
227
[30] Brocks W, Hao S, Steglich D. Micromechanical modelling of the damage and toughness behaviour of nodular cast iron materials. In: First European mechanics of materials conference on local approach to fracture, Fontainebleau, September 1996. [31] Brocks W, Sun DZ, Honig A. Verification of micromechanical models for ductile fracture by cell model calculations. Comput Mater Sci 1996;7:235–41. [32] Dong MJ, Berdin C, Beranger AS, Prioul C. Damage effect in the fracture toughness of nodular cast iron. In: First European mechanics of materials conference on local approach to fracture, Fontainebleau, September 1996. [33] Decamp K, Bauvineau L, Besson J, Pineau A. Size and geometry effects on ductile rupture of notched bars in a C–Mn steel. Experiments and modelling. Int J Fract 1997;88:1–18. [34] Sovik OP. Experimental and numerical investigation of void nucleation in an AlMgSi alloy. In: First European mechanics of materials conference on local approach to fracture, Fontainebleau, September 1996. [35] Ghosal AK, Narasimhan R. Numerical simulations of hole growth and ductile fracture initiation under mixed-mode loading. Int J Fract 1996;77:281–304. [36] Kuna M, Sun DZ. Three-dimensional cell model analyses of void growth in ductile materials. Int J Fract 1996;81:235–58. [37] He R, Steglich D, Heerens J, Wang G-X, Brocks W, Dahms M. Influence of particle size and volume fraction on damage and fracture in Al-A13Ti composites and micromechanical modelling using the GTN model. Fatigue Fract Eng Mater Struct 1998;21(10): 1189–201. [38] Sun DZ, Siegle D, Voss B, Schmitt W. Application of local damage models to the numerical analysis of ductile rupture. Fatigue Fract Eng Mater Struct 1989;12(3):201–12. [39] Siegmund T, Bernauer G, Brocks W. Two models of ductile fracture in contest: porous metal plasticity and cohesive elements. ECF12 fracture from defects, September 1998. p. 933–8. [40] Xia L, Cheng L. Transition from ductile tearing to cleavage fracture: a cell-model approach. Int J Fract 1997;87(3):289–306. [41] Schmitt W, Sun DZ, Blauel JG. Damage mechanics analysis (Gurson model) and experimental verification of the behaviour of a crack in a weld-cladded component. Nucl Eng Design 1997;74:237–46. [42] Skallerud B, Zhang ZL. A 3D numerical study of ductile tearing and fatigue crack growth under nominal cyclic plasticity. Int J Solids Struct 1997;34(24):3141–61. [43] Pardoen T, Doghri I, Delannay F. Experimental and numerical comparison of void growth models and void coalescence criteria for the prediction of ductile fracture in copper bars. Acta Mater 1998;46(2):541–52.