A continuum damage mechanics model for ductile fracture

International Journal of Pressure Vessels and Piping 77 (2000) 335±344

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A continuum damage mechanics model for ductile fracture S. Dhar a, P.M. Dixit a, R. Sethuraman b,* a b

Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600036, India Received 17 June 1999; accepted 1 February 2000

Abstract Continuum damage mechanics theory together with large deformation elastic±plastic ®nite element analysis has been used to predict crack growth initiation in ductile materials. The damage growth law is based on experimental observations reported in the literature. A local crack growth initiation criterion is proposed. The criterion makes use of the critical damage as the continuum parameter and the average austenite grain size as the characteristic length. To test the validity of this criterion, experiments have been conducted on standard specimens and have also been simulated numerically. The proposed criterion has been used to predict the values of the critical load at crack growth initiation and the fracture toughness. The numerically predicted values compare favourably with the experimental values. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: Ductile fracture; Continuum damage mechanics model; Crack growth initiation; Elasto plastic analysis

1. Introduction It has been observed from metallurgical test results that ductile fracture occurs due to void nucleation, growth and ®nally coalescence into a micro-crack. Earlier models of McClintock [1] and Rice and Tracey [2] predict the critical strain at fracture from the study of an isolated void growth in a remote uniform plastic ®eld. The major limitation of this approach is that it ignores effects of void nucleation (as it assumes a pre-existing ®nite size void) and coalescence on the fracture process. The porous plasticity model of Berg [3] and Gurson [4], in which the plastic potential is represented in terms of void volume fraction, incorporates both void nucleation and growth. In this model, ductile fracture is regarded as the result of instability in the dilatational plastic ®eld localised in a small band. Yamamoto [5] derived the instability conditions and represented the fracture criterion as a graph of critical localization strain versus the critical void volume fraction with strain hardening exponent as a parameter. Void coalescence by internal necking of intervoid matrix is considered as a secondary effect in this model. It is this failure to appreciate the dominant effect of internal necking that severely limits the validity of the Berg±Gurson model [6].

* Corresponding author. Tel.: 191-44-445-8538; fax: 191-44-235-0509. E-mail address: [email protected] (R. Sethuraman).

Thomason [6] combined the results of Goods and Brown [7] on void nucleation, those of Rice and Tracy [2] on void growth and his own on void coalescence to arrive at a fracture criterion in the form of a graph of the fracture strain versus the mean stress. The expressions for void dimensions, which he used in his model, were derived on the basis of an assumption that the principal directions of strain rates remain ®xed throughout the strain path. This is true only for the case of in®nitesimal strain and rotation. As a result, this approach cannot be used when the strains and/or rotations are large. In continuum damage mechanics (CDM) model, the effect of void growth on material behaviour is incorporated by introducing a (internal) damage variable in the constitutive relation. The effect of void nucleation can also be included by modifying the damage growth law appropriately. But as far as void coalescence is concerned, it has to be incorporated as an additional condition (in terms of continuum parameters) based on a suitable micro model. Recently, Dhar et al. [8] combined Lemaitre's [9] CDM model and Thomason's [6] void coalescence condition in a large deformation ®nite element analysis of different case studies to show that the critical value of damage variable is a geometry independent material parameter and can be used for predicting micro-crack initiation. When a micro-crack initiation criterion is to be used for predicting the onset of growth of an existing crack (i.e.

0308-0161/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S 0308-016 1(00)00019-3

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Nomenclature a a1 a2 a A b B c EP Cijkl [C] [C] EP D Dc eij E {f} F {F} F1 FD H J JI JIc KIc K [K] lc n p

effective crack length, Fig. 1 coef®cient of damage growth equation coef®cient of damage growth equation ¯ow vector area under the load de¯ection curve ligament length, Fig. 1 thickness of fracture test specimen, Fig. 1 coef®cient in damage growth equation fourth-order elastic±plastic tensor elasticity matrix elastic±plastic matrix damage variable critical damage Green±Lagrange strain tensor Young's modulus internal force factor plastic potential external force factor plastic potential associated with yielding plastic potential associated with damage slope of the hardening curve Rice's J integral J integral in mode-I fracture critical value of the J integral plane strain fracture toughness hardening parameter stiffness matrix critical length parameter hardening exponent hardening variable

crack growth initiation), an additional (local) length parameter is needed to characterize the distance from the crack tip at which the micro crack initiation criterion is to be applied. The literature contains quite a few local criteria for crack growth initiation. Ritchie et al. [10] proposed critical stress criterion for cleavage fracture. While working with mild steel at low temperature, they found that predictions from this criterion agreed well with the experimental results when it was applied at a distance of 2±3 times the ferrite grain diameter of the material. For ductile fracture, Rice and Johnson (RJ) [11] expressed the critical crack tip opening displacement in terms of an inter-particle distance. For structural steels, the predictions from RJ model match well with experimental results if the inter-particle distance is taken as the average spacing of MnS particles, which are considered as the major void nucleating particles. Ritchie et al. [12] used a strain based criterion to predict the fracture toughness of A533B and A302B alloy steels. Their ®ndings show a

R r Sij t Du v W 2Y Greeks d D {Du} {DF} De ij e_ pij epeq e_ peq h Dh ij l n s ij s 0ij s_ ij sm s eq s ijp s0 sy V ij

hardening force virtual work of external forces in ®nite element formulation 2nd Piola±Kirchoff stress tensor time incremental displacement vector volume width of fracture test specimen, Fig. 1 energy release rate due to damage variation increment in a quantity incremental displacement vector incremental force vector linear part of incremental strain tensor plastic strain rate tensor equivalent plastic strain equivalent plastic strain rate Ernest factor non-linear part of the incremental strain tensor scalar multiplier in ¯ow rules Poisson's ratio Cauchy stress tensor deviatoric part of Cauchy stress tensor Jaumann stress rate mean stress equivalent stress effective equivalent stress initial yield stress current yield stress spin tensor

critical length varying from one to six times the average Mn particle distance. In this paper, a local criterion for crack growth initiation is proposed which makes use of the critical damage as the critical continuum parameter and the average austenite grain size as the (local) length parameter. To test the validity of this criterion, experiments have been conducted on standard specimens and have also been simulated numerically. For numerical simulation of mode-I ductile fracture in AISI1095 spheroidized steel, Lemaitre's [9] CDM model and large deformation elastic±plastic ®nite element method are used. The damage growth law used is the same as proposed by earlier author's [8]. The critical load for crack growth initiation and the fracture toughness are predicted by using the proposed criterion for crack growth initiation. Comparison of numerical and experimental results shows that the proposed criterion is reasonably good in predicting the crack growth initiation in mode-I ductile fracture.

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337

to damage. It is given by [9] ! 2 s eq sm 2Y ˆ f 2E s eq where

sm f s eq

"

! ˆ

2 sm …1 1 n† 1 3…1 2 2n† s eq 3

…2†

!2 # …3†

Here E is Young's modulus, n is Poisson's ratio, s m is the mean part of the Cauchy stress tensor, s ij, and s eq is the equivalent stress related to the deviatoric part s 0ij by the relation: q s eq ˆ 32 s 0ij s 0ij …4† p is the effective equivalent stress related to s eq Further, s eq by s eq p …5† s eq ˆ …1 2 D†

Following Lemaitre [9], the plastic potential for a damaged material can be decomposed as F ˆ F1 …s ij ; R; D† 1 FD …2Y; p; D†

…6†

where F1 is the plastic potential associated with yielding and FD is due to damage. R is the hardening stress corresponding to the hardening variable p. For the case of strain hardening, p is identi®ed as the equivalent plastic strain epeq de®ned by Zt …7† p ˆ epeq ˆ e_ peq dt 0

e_ peq ˆ

q 2 _p _p 3 e ij e ij

…8†

Here e_ peq is called as the equivalent plastic strain rate and e_ pij is the plastic part of the strain rate tensor. For a material yielding according to the von Mises criterion, the form of F1 is F1 ˆ Fig. 1. Geometry of (a) TPB specimens; (b) CT specimens.

2. Damage growth model The damage variable (D) at a point is de®ned as the area void fraction in a plane. That is Dˆ

DAv DA

…1†

where DA is an in®nitesimal area around the point in some plane and DAv is the area of the void traces (contained in DA) in the plane. The conjugate variable corresponding to D is 2Y, the strain energy release rate (at constant stress) due

s eq 2 R p 2 s 0 ˆ s eq 2 sy 12D

…9†

where s 0 is the initial yield stress (in tension) and s y is the current yield stress given by   R 1 s0 sy ˆ …10† 12D When D ˆ 0; F1 reduces to the original form due to von Mises and FD becomes zero. The constitutive equation and damage growth law are obtained as the plastic ¯ow rules:

e_ pij ˆ l p_ ˆ l

2F1 l 3s 0ij ˆ 2s ij 1 2 D 2s eq

2F1 l ˆ 2…2R† 12D

…11†

…12†

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S. Dhar et al. / International Journal of Pressure Vessels and Piping 77 (2000) 335±344

Table 1 Chemical composition (% wt) of the test material C

Mn

S

P

0.92

0.80

0.035

0.013

2FD D_ ˆ l 2…2Y†

…13†

The scalar l can be obtained by combining Eqs. (7) and (12)

l ˆ …1 2 D†e_ peq

…14†

The constitutive Eq. (11) is really not convenient for the updated Lagrangian formulation for which an incremental stress±strain relationship is needed. It is discussed in the next section. Unlike F1, FD is not well established in the literature. As a result, we cannot use Eq. (13) as the damage growth law. Based on the experimental results of LeRoy et al. [13] for AISI-1090 spheroidized steel on the measurements of area void fraction versus strain, the following damage growth law has been proposed by the authors [8]: D_ ˆ ce_ peq 1 …a1 1 a2 D†…2Y†e_ peq

…15†

The material constants c, a1 and a2 are evaluated from the experimental results. The ®rst term, which is independent of 2Y, represents the damage growth due to void nucleation while the next two terms represent the evolution of damage due to void growth. To express 2Y in terms of the equivalent plastic strain, the relationship describing the strain hardening characteristic of the material is required. Here, it is assumed that it is given by a power law p s eq ˆ s y ˆ K…epeq †n

Fig. 2. (a) Austenite grain structure of material AISI-1095 (C% 0.92). (b) Spheroidized structure of material AISI-1095 (C% 0.92).

formulation is given by [14] Z t1Dt d…t et1Dt † dvt ˆ r t1Dt t Sij ij

…18†

vt

Here the right superscript indicates the current con®guration (i.e. the con®guration at time t 1 Dt† and the left subscript represents the reference con®guration (i.e. the con®guration at time t). Sij denotes the second Piola±Kirchoff stress tensor, eij denotes the Green±Lagrange strain tensor (which is conjugate to Sij), d is the variation, r is the virtual work done by the surface tractions and v denotes the domain. Decomposing the stress and strain tensors, linearizing the resulting expressions and using the incremental stress±strain relation, Eq. (18) simpli®es to Z Z EP t s ijt d…t Dhij † dvt t Cijklt Dekl d…t Deij † dv 1

…16†

where K and n are the hardening parameters. Using the p the expressions (2) and (16), respectively, for 2Y and s eq form of the damage growth law (Eq. (15)) becomes ! K 2 p 2n s m p _ e_ p …17† …e † f D ˆ ce_ eq 1 …a1 1 a2 D† 2E eq s eq eq

3. Updated Lagrangian formulation

vt

ˆ r t1Dt 2

The virtual work expression in the updated Lagrangian

Z vt

vt

s ijt d…t Deij † dvt

…19†

Table 2 Mechanical properties of the test material Young's modulus, E (GPa)

Poisson's ratio, n

Yield stress, s 0 (MPa)

Ultimate stress, s ut (MPa)

Ultimate strain e ut (%)

Fracture strain e f (%)

Hardness, Rc

K (MPa)

n

210

0.3

497

690

15

59

34

1211

0.22

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Fig. 4. Stretch zone of (a) TPB1 specimen; (b) CT1 specimen.

Fig. 3. Experimental load±de¯ection curve of (a) specimen TPB3; (b) specimen CT2. EP Here, t Cijkl is the elastic±plastic tensor appearing in the incremental stress±strain relation, s ijt is the Cauchy stress tensor at time t and the linear and non-linear parts of the Green±Lagrange strain tensor are given by t Deij

ˆ

1 2 …t Dui; j

t Dhij

ˆ

1 2 …t Duk;it Duk; j †

1 t Duj;i †

…20† …21†

where t Dui; j stands for the derivative of t Dui (the incremental displacement vector at time t) with respect to t xj (the EP position vector at time t). The matrix form of t Cijkl is given by ( ) ‰CŠa a T ‰CŠ EP …1 2 D† …22† ‰CŠ ˆ ‰CŠ 2 H 1 a T ‰CŠa where [C] is the elasticity matrix, a is the ¯ow vector (i.e.

the derivative of the plastic potential with respect of the stress expressed in a vector form), and H is the slope of the hardening curve. Dividing the domain into a number of elements and substituting a suitable approximation, t Du in Eq. (19) for displacement ®eld, we get the following algebraic equation governing t{Du}, the vector of nodal values of t Du : t ‰KŠt {Du}

ˆ t {DF}

…23†

Here, t[K] is the stiffness matrix corresponding to the lefthand side terms of Eq. (19). The vector t{DF} is the incremental external force vector for the ®rst iteration but in subsequent iterations, it denotes the unbalanced force vector, that is the difference of the external force vector {F}t1Dt (related to Rt1Dt † and the internal force vector {f }t1Dt which corresponds to the second term of the righthand side. Eq. (23) is solved iteratively till the unbalanced force is zero. After that t Deij is determined from Eq. (20) and the Jaumann stress rate is calculated from the following incremental equation: EP s ijt Dt ˆ t Cijklt Dekl

…24†

The Cauchy stress at time t 1 Dt is obtained from the

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S. Dhar et al. / International Journal of Pressure Vessels and Piping 77 (2000) 335±344 Table 4 Experimental results of CT specimens Specimen a/W

Pcr (kN) Ucr (mm)

JIc (kN/m) Average JIc (kN/m)

CT1 CT2

6.98 7.70

64.52 66.85

0.58 0.60

D ˆ Dc Jaumann stress rate by the following expression:

s ijt1Dt ˆ s ijt 1 s ijt Dt 1 s ipt V pjt Dt 1 s jpt V pit Dt

5. Experiments

2 t Dui; j †

…26†

A local fracture criterion can be stated by two parameters, namely a critical continuum parameter and a length parameter. The critical length parameter denotes the distance ahead of the crack tip where the continuum parameter has to reach its critical value for micro-crack initiation. Thus, the critical length is a characteristic dimension of the material when the crack growth initiates. It is obvious that this characteristic dimension depends on the microstructure of the material and hence it is a material property. Local criteria based on the critical stress or critical strain is widely used to predict crack growth initiation in cleavage or ductile fracture [10,12]. But, it has been found from experimental works of Mackenzie et al. [15] and Hancock and Brown [16] that triaxiality also plays an important role in ductile fracture. In general, micro-crack initiation due to void growth and coalescence in ductile material depends not just on the critical strain or critical triaxiality but on a critical combination of these two parameters [15]. Since the damage incorporates both the parameters, in the present study, the critical damage (Dc) is used as the critical continuum parameter. The length parameter (lc) is taken as the average austenite grain size of the material. The choice of austenite grain size is not unjusti®ed because the larger particles restrict the austenite grain growth. Thus the proposed crack growth initiation criterion can be stated as

Pcr (kN) Ucr (mm) JIc (kN/m) Average JIc (kN/m)

0.595 12.0 0.526 15.1 0.531 15.0

0.65 0.60 0.59

The experiments were performed on AISI-1095 spheroidized steel whose chemical composition is given in Table 1. The specimens tested for experimentation are three-point bend (TPB) specimens and compact tension (CT) specimens. The dimensions of the specimens were chosen according to the ASTM standard. Fig. 1a and b shows the geometry and dimensions of the specimens. The bulk material was austenitized by heating it to 10008C for 2 h and then furnace cooled. Fig. 2a shows the austenitized grain structure of the material. The average grain size was measured by interception method. The grain size was found within a range of 30±60 mm with 33 mm as the average value. After machining, the specimens were annealed by heating them to 7508C for 2 h to relieve the residual stresses due to machining. The ®nal heat treatment was done to spheroidize the material by heating at 7108C for 20 h. Fig. 2b shows the spheroidized structure of the material. A tension test was performed to ®nd the mechanical properties of the material. Table 2 shows the results of the test. After pre-cracking, the specimens were steadily loaded by controlling the displacement of the ram. In all the cases, a ram speed of 0.1 mm/min was maintained. The load and the load point displacement were recorded in X±Y plotter throughout the loading history. The crack growth initiation was indicated by a distinct pop-in in all the cases. The measured values of the load and the load point displacement and the calculated value of JI at this point are termed as their critical values and are denoted respectively by Pcr, Ucr and JIc. For calculating JI value at a certain load level, the following formulae were used [17]. For the TPB specimen: JI ˆ

Table 3 Experimental results of TPB specimens

TPB1 TPB2 TPB3

67.50 70.99 65.77

…27†

…25†

4. Crack growth initiation criterion

Specimen a/W

at l ˆ lc

the crack growth initiates.

where the spin tensor V ijt is de®ned by 1 2 …t Duj;i

65.68

follows. Whenever

Fig. 5. Modelling of crack tip.

V ijt Dt ˆ

0.45 0.425

2A Bb

…28†

where A is the area under the load versus load point de¯ection curve. For the CT specimen:

68.08

JI ˆ

hA Bb

…29†

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341

Fig. 6. Load versus load point de¯ection curve for TPB1 specimen.

where Ernest factor h is given by [17]

h ˆ 2:0 1 0:522

b W

mentioned here …30†

Here, B is the thickness of the specimen, b is the ligament length and a is the effective crack length (see Fig. 1a and b). Fig. 3a and b shows the experimental load±de¯ection curves for a TPB and a CT specimen, respectively. The fractured surfaces of broken specimens were scanned by 15 kV JEOL JSM 840A scanning electron microscope. Fig. 4a and b reveals the stretch zones. 6. Numerical simulation of fracture tests The updated Lagrangian ®nite element formulation of an elastic±plastic material, developed in Section 3, is used to numerically simulate the fracture mechanics tests conducted in this study. The arc length method [18] together with the Newton±Raphson scheme is used for iteration purpose. Since the chemical composition and the mechanical properties of the test material are close to that of AISI-1090 spheroidized steel [8], the values of the coef®cients a1, a2, and c of the damage growth law (Eq. (17)) are taken to be the same as that of AISI-1090 spheroidized steel. These values are given in Ref. [8]. For the sake of completeness, those are

a1 ˆ 9:8 £ 10204 MPa21

a2 ˆ 1:86 MPa21

c ˆ 1:848 £ 10202 Also, the critical value of the damage variable is taken the same as that of AISI-1090 spheroidized steel, i.e. Dc ˆ 0:05: The average austenite grain size of the test material, as mentioned in the previous section, is 33 mm. For crack growth initiation, the criterion proposed in Section 4 is used. According to this criterion, whenever the value of the damage variable reaches Dc at a distance lc (austenite grain size) ahead of the crack tip, the crack growth initiates (Eq. (27)). The size of the crack tip element is chosen so as to facilitate the application of this criterion. Eight-noded isoparametric elements are used everywhere except at the crack tip where the collapsed crack tip elements are used. A 2 £ 2 Gauss integration scheme is used throughout. The specimen are deformed by giving a prescribed incremental displacement at a ®xed point which is the load point in actual experiment. Load is calculated from nodal reactions. At the end of each increment, the increment in damage is calculated by using the damage growth law (Eq. (17)) which is added to its previous value to get the total damage. During elastic unloading the damage

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Fig. 7. Load versus load point de¯ection curve for CT1 specimen.

calculation is bypassed. At the end of each increment, the damage at the Gauss point nearest to the crack tip is compared with its critical value. As soon as `D' reaches `Dc' the programme is terminated. All values at this stage are termed as the critical values of the respective quantities. The details of numerical procedure are given in Ref. [8]. The `JI' value is calculated after computing the area under the load±de¯ection curve using the relations (28) and (29). The stretch zone width is calculated from the stretching of the initial crack, i.e. SZW ˆ …ai 2 a0 †

7. Results and discussion 7.1. Experimental results Experimental load±de¯ection curves are shown in Fig. 3a and b. It is observed that the curves show a distinct pop-in which is taken as the crack growth initiation point. This is followed by ductile crack growth. At crack growth

…31†

where ai is the stretched crack length and a0 is the initial crack length. The value of the stretch zone width at the critical condition is regarded as critical stretch zone width or (SZW)c. Out of all collapsed nodes, the ®rst node is vertically restrained and others are allowed to move during deformation (see Fig. 5). CTOD is regarded as the vertical opening of the last collapsed node (node 9). Its value at the critical condition is regarded as the critical crack tip opening displacement or (CTOD)c. Of all test specimens, TPB1 and CT1 were completely simulated using the procedure cited above. The plane strain condition was assumed in both the cases.

Fig. 8. Geometry of full CT specimen.

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Table 4 shows the experimental results of CT specimens. The average value of JIc is found as 65.68 kN/m. Fractographs (Fig. 4a and b) reveal the critical stretch zone. The critical stretch zone width for TPB1 specimen is found to be 35 mm and that of CT1 specimen to be 50 mm. 7.2. Comparison between experimental and numerical results

Fig. 9. Growth of plastic zone.

initiation, there is profuse void growth and coalescence in the highly localised plastic zone at the crack tip and this causes the initial instability. This tendency reduces and ®nally arrests due to a fall in triaxiality as the free surface is approached. Table 3 shows the experimental results of TPB specimens. It is found that, for a/W ranging from 0.526 to 0.595, JIc value varies from 65.77 to 70.99 kN/m. The average experimental value of JIc is 68.08 kN/m. The table shows a decrease in critical load with the increase of a/W ratio. This is due to decrease in the ligament length b.

Fig. 6 shows a comparison of the computer simulated load versus load point de¯ection curve with that of the experimental curve for the TPB1 specimen. The computer simulated curve depicts similar trend as that of the experimental curve. The difference in critical load prediction is 11.0%. Fig. 7 is a comparison of load versus load point de¯ection curve with that of the experimental one for CT1 specimen. The difference in critical load values is 5.0%. Both the computer simulated curves have higher critical loads than those of experimental curves because the simulated curves are for the plane strain condition. The computer simulated JIc value of TPB1 specimen is 72.44 kN/m. On the other hand, the average experimental value of JIc for TPB specimens is 68.08 kN/m (Table 3). The simulated value is 6.3% higher than the experimental value. The computer simulated value of JIc for CT1 specimen is 73.78 kN/m which is 11.0% higher than that of the average experimental value (65.68 kN/m) for CT specimens (Table 4). Both the simulated cases yield higher values of JIc than

Fig. 10. J1 versus CTOD curve.

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those from experiments. Since the computer simulations are for plane strain condition, the simulated values are expected to be less than those of experiments. The discrepancy may be due to the reason that the critical value of damage variable (Dc) of the material is determined by using Thomason's limit load analysis [6] which uses the upper bound theorem. Thus, the value of Dc equal to 0.05 is an over-estimation of the actual value. The values of critical stretch zone width, calculated for TPB1 and CT1 specimens from Eq. (31), are 31.26 and 32.09 mm, respectively. These computer simulated values are almost equal to the average austenite grain size (33 mm) of the material. The experimental values of critical stretch zone width (35 mm for TPB1 specimen and 50 mm for CT1 specimen) are within the range of measured grain size (30± 60 mm). The results show that the critical stretch zone width is of the order of grain size lc. Comparison of the computer simulated results with the experimental data show that the overall agreement is satisfactory. Since the computer simulation uses the proposed crack growth initiation criterion, Eq. (27), for calculating the critical values of the quantities, it shows that the proposed criterion is reasonably good in explaining mode-I fracture for the class of materials considered here. Some additional numerical results are computed from the plane strain analysis of a CT specimen (Fig. 8). Fig. 9 shows the growth of plastic zone at different deformation levels. The critical plastic zone size is found to be 0.176 mm. A graph of JI versus CTOD is show in Fig. 10. The relation is linear up to crack growth initiation point. The value of critical crack tip opening displacement (CTOD)c is 56 mm which is 1.8 times the calculated value of critical stretch zone width. The slope of the curve is found to be 2.4s 0. The theoretical value of the slope is 1.6s 0 (in plane strain conditions) for an ideally plastic material in small scale yielding. In large scale yielding, this value can go up to 2.6s 0 depending on the hardening exponent (n) [19]. Thus, the value of the slope is in conformity with the existing results. 8. Conclusions In light of the above discussion the following conclusions can be drawn: 1. Ductile fracture process is in¯uenced by both the plastic strain and triaxiality. 2. The critical damage parameter (Dc) is a good measure of the ductile failure initiation due to void growth.

3. The choice of austenite grain size as the characteristic length parameter in crack growth initiation criterion leads to a good estimation of the fracture parameters. 4. The ®nite deformation elastic±plastic ®nite element analysis together with CDM can estimate the fracture parameters reasonably well. References [1] McClintock FA. A criterion for ductile fracture by the growth of holes. ASME J Appl Mech 1968;35:363±71. [2] Rice JR, Tracey DM. On the ductile enlargement of voids in triaxial stress ®eld. J Mech Phys Solids 1969;17:201±17. [3] Berg CA. Plastic dilation and void interaction. In: Kanninen MF, editor. Inelastic behaviour of solids, New York: McGraw-Hill, 1970. p. 171±210. [4] Gurson AL. Continuum theory of ductile rapture by void nucleation and growth: part IÐyield criteria and ¯ow rules for porous ductile media. ASME J Engng Mater Technol 1977;99:2±15. [5] Yamamoto H. Condition for shear localisation in the ductile fracture of void containing materials. Int J Fract 1978;14:347±65. [6] Thomason PF. Ductile fracture of metals. Oxford, UK: Pergamon Press, 1990. [7] Goods SH, Brown LM. Nucleation of cavities by plastic deformation. Acta Metall 1979;27:1±15. [8] Dhar S, Sethuraman R, Dixit PM. A continuum damage mechanics model for void growth and micro crack initiation. Engng Fract Mech 1996;53:917±28. [9] Lemaitre J. A continuous damage mechanics model for ductile fracture. ASME J Engng Mater Technol 1985;107:83±9. [10] Ritchie RO, Knott JF, Rice JR. On the relationship between critical tensile stress and fracture toughness in mild steel. J Mech Phys Solids 1973;21:395±410. [11] Rice JR, Johnson MA. The role of large crack tip geometry changes in plane strain fracture. In: Kanninen MF, editor. Inelastic behaviour of solids, New York: McGraw-Hill, 1970. p. 641±72. [12] Ritchie RO, Server WL, Wullaert RA. Critical fracture stress and fracture strain models for the prediction of lower and upper self toughness in nuclear pressure vessel steels. Metall Trans 1979;10A:1557±70. [13] LeRoy G, Embury JD, Edward G, Ashby MF. A model for ductile fracture based on nucleation and growth of voids. Acta Metall 1981;29:1509±22. [14] Bathe KJ, Ramm E, Wilson EL. Finite element formulations for large deformation dynamic analysis. Int J Numer Meth Engng 1975;9:353± 86. [15] Mackenzie AC, Hancock JW, Brown DK. On the in¯uence of state of stress on ductile failure initiation in high strength steels. Engng Fract Mech 1977;9:167±88. [16] Hancock JW, Brown DK. On the role of strain and stress in ductile fracture. J Mech Phys Solids 1983;31:1±24. [17] Roberts R, editor. ASTM STP 743, Fracture mechanics. [18] Ramm E. Strategies for tracing nonlinear response near limit points. Europe±US Workshop on Nonlinear Finite Element Analysis in Structural Mechanics, July 28±31, 1980, Bochum, Germany. [19] Paranjape SA, Banerjee S. Inter-relation of crack tip opening displacement and J integral. Engng Fract Mech 1979;11:43±50.