- Email: [email protected]
Studies on the ductility predictions by different local failure criteria
Engineering Fracture Mechanics Vol. 48, No. 4, pp. 529-540, 1994
OOW7944(94)E0023-A
Copyright 0 1994 Elsevier Science Ltd. Printed in Great Britain. All rights reserved 0013-7944/94 $7.00+ 0.00
STUDIES ON THE DUCTILITY PREDICTIONS DIFFERENT LOCAL FAILURE CRITERIA
BY
Z. L. ZHANG and E. NIEMI Department of Mechanical Engineering, Lappeenranta University of Technology, P.O. Box 20, 53851 Lappeenranta, Finland Abstract-It is well known that the ductile fracture of metals has frequently been observed to result from the growth and coalescence of microscopic voids. Correspondingly, local approach methodologies have been developed in the modelhng of ductile fracture. Three local ductile fracture criteria, a critical void volume fraction criterion based on a Gurson-type constitutive relation, a critical void growth criterion based on Rice-Tracey void growth equations, and Thomason’s plastic limit-load criterion, have been studied and examined against the predictions of ductility with respect to stress triaxiality. A calibration for the predictions of the plastic limit-load criterion has been made, which yields a modification to the criterion to give more realistic predictions at low stress triaxiality than the original criterion. Various comparisons of the criteria have been made. It has been found that there are considerable differences in the predictions by the three criteria. Finally, a method has been introduced which can correlate the predictions well using dual dilational constitutive models.
1. INTRODUCTION DURING the past 20 years, many research works have been devoted to the study of ductile fracture, and now it is well known that the ductile fracture of metals occurs by the nucleation, growth and coalescence of voids and that plastic deformation is not the only determinant of ductile fracture. Notched axisymmetric tensile tests have demonstrated that plastic deformation at fracture is highly dependent on the state of stresses [l]. It has been shown that ductile fracture surfaces are formed by the sudden catastrophic coalescence of voids which grow by plastic deformation under the influence of a prevailing triaxial stress system [2]. Naturally, therefore, the macroscopic ductile fracture strain should be composed of two parts, the void nucleation strain and void growth strain. Although void nucleation is obviously important in ductile fracture, it is difficult to determine in normal practice. In this study, in common with most other workers, we assume that the strain necessary to nucleate a void is negligible compared to the strain needed to cause coalescence [3]. Once a void has been nucleated in a plastically deforming matrix, the resulting stress-free surface of the void causes a localized stress and strain concentration in the adjacent plastic field. With continuing plastic loading of the matrix, the void will undergo a volumetric growth and shape change. Rice and Tracey [4] have successfully established a growth model of an isolated spherical void in an infinite non-hardening von Mises matrix. A fracture criterion named “critical void growth ratio” based on Rice and Tracey’s work has been formulated and widely used in the local approach to fracture [3,5]. In contrast to developing a single void growth model, a theory of dilational plasticity has been developed by Gurson [6,7] and modified by Tvergaard [8,9] to incorporate the inelastic straining resulting from the nucleation and growth of voids. Without physical proof, the “critical void volume fraction” criterion simulating the void coalescence based on the Gurson model has been increasingly popular both in theoretical analyses and practical applications [lO-151. On the basis of detailed analysis of the mechanism of ductile fracture by the nucleation, growth and coalescence of voids, Thomason [2] has presented a plastic limit-load or internal necking criterion for ductile fracture which leads to a strong dilational yield surface. A critical load condition at incipient void coalescence has been established based on dual dilational constitutive responses. Although the three criteria are relatively well known, no examination of the consistency of the predictions, especially with respect to stress triaxiality, has been found in the literature. In the application of the first two criteria, usually the critical void volume fraction and critical void growth ratio are obtained from simple axisymmetric tensile tests and then applied to cases under various 529
530
Z. L. ZHANG and E. NIEMI
stress conditions without proof. The advantages of Thomason’s plastic limit-load criterion are that, once the initial conditions are given, ductile fracture is a natural result of the dual dilational responses and is solely determined. The purpose of this study is not to examine physically whether critical void volume fraction and void growth ratio are material constants and independent of stress states as done in ref. [16], but to show the difference and inconsistency among the criteria. A simple geometric model which simulates the axisymmetric loading is used for various studies (see Fig. 1). Firstly, a calibration of Thomason’s plastic limit-load criterion is made which leads to a modification of the model, to bring realistic predictions at low stress triaxiality. Then various comparisons of the criteria are made. It has been found that there are considerable differences in the predictions of the three criteria for large stress triaxiality, under the condition that the three criteria are calibrated for the same lower stress triaxiality case. Here and subsequently, in order to simplify the presentation, Rc, Fc and Sc are used to identify the critical void growth ratio, critical void volume fraction and plastic limit-load criteria, respectively, and plastic strain in the figures means the equivalent plastic strain. 2. THREE LOCAL FAILURE CRITERIA 2.1. Critical void growth ratio criterion (Rc) For the axisymmetric case (6, = uY; Fig. 1) and constant stress triaxiality (proportional loading), the Rice-Tracey void growth theory can be expressed as: !!!?%A % R --f=A+B Rll R R i=f=” -$B, where A = exp(DcP) B
l+E =.(A+
D = 5 f$
for linear hardening
D = 0.283 exp 1 + Em 5/3 1+ E x 2
for non-hardening
for linear hardening or low 6, with non-hardening for high values of a,,, with non-hardening.
Fig. I. A spherical void in a plastic matrix unit ceil.
(1)
(3)
531
Ductilitypredictionsby local failun criteria
R,,, R,, R,, and R,_ are the initial void radius, current radii in x and y directions and mean radius, respectively, tP is the equivalent plastic strain, Q,,,is the first invariant of the stress tensor and 8 the flow stress of the matrix surrounding the void, which is equal to the von Mises equivalent stress q. It must be noted that the above equations were derived for a spherical void located inside an infinite perfectly plastic matrix material. Although the void growth model does not itself constitute a fracture criterion, and a modelling of the coalescence must be added, a general remark has been made [5]: as a first approximation, deformation at fracture in tension is independent of the size of and distance between voids and only depends on void growth ratio. With this observation, a fracture criterion based on the assumption of a critical ratio of void growth (R,,/R,,),, specific to the material for a given direction of the load, has been formulated and widely used in the local approach of fracture. In general, the critical void growth fracture criterion modified to take account of hardening and the non-constant stress triaxiality case can be written as [3,51 II 0.283 exp($!!)VO”Mimd@ = ln(%)c
= R,,
where E” and t, are the strains at void nucleation and coalescence, respectively. The value of (R_,,/R,,), is usually determined through simple uniaxial or axisymmetric tests and then applied to complex stress cases, for example, notched or cracked specimens. 2.2. Critical void volume fraction criterion (Fc) Based on an approximation of a solid with a volume fraction,f, of voids, by a spherical body with a concentric spherical void, and carrying out an approximate rigid-plastic upper bound analysis of this thick-walled spherical cell, with matrix material idealised as rigid-perfectly plastic and obeying the von Mises yield criterion, Gurson [6,7j obtained the following dilational yield condition (q, = 1 and q2 = 1) for a porous plastic solid, which is a good fit to his upper bound yield load locus:
d(u,f, 6) =
$ +2q,f
cash
- 1 - (q&Z = 0,
where constants q, and q2 are introduced by Tvergaard [8,9] to bring predictions of the model into closer agreement with full numerical analyses of a periodic array of voids. q is the effective part of the average macroscopic Cauchy stress u. It is easy to see that the material loses its load carrying capacity if f reaches the limit l/q,, because all the stress components have to vanish in order to satisfy eq. (5). However, even if q1 = 1.5, the void volume fraction, f = l/q,, is still too large to be realistic in practice to simulate the final material failure. Therefore, a modification usingp to replace f in eq. (5) in order to account for the void coalescence effect on final failure has been proposed [lo]: p=
f ( f,+ W-f,)
forf Gf, forf >A.
(6)
Here, f, is the critical void volume fraction at which void coalescence is first observed and K is a constant determined from the void volume fraction at final failure of the material which is much smaller than f = l/q,. Due to the lack of physical proof and in line with mathematical convenience, this modification has actually induced another fracture criterion-the critical void volume fraction criterion, which is perhaps one of the most widely applied criteria [lO-151. Because of the difficulty in obtaining f, physically, Sun et al. [14] have suggested that f, can be determined from smooth axisymmetric tensile tests by numerical simulation. In this study, we are only interested in the plastic strain corresponding to f,, therefore eq. (6) is not used.
Z. L. ZHANG and E. NIEMI
532
For the axisymmetric proportional loading (0; = uv = ccr,, c is a constant), we obtain, after some algebraic manipulations, the following analytical equations: ip = awl, i; =
G,wi,
ciz= E,(l
- G,w)L,
f= @it,
(7)
where
4 Gs a4 ah !!f E, GzGs - B --af ma5 G -1 a0 : W--c) CO=
7%
“-3&F, G s=-
(1 + 2c) 86, 3 ao,+
2fl - c>* 62 0,
&G* Q,= (1 -f)$ B=(l E,
-r$
m
=E
1 - 2cv
ef: is the plastic strain in the z direction, E is Young’s modulus, v is Poisson’s ratio; ab/ao,, @/af and &$/A? can be easily calculated from eq. (5). In the following calculations, eq. (7) with g1 = 1.5 and gz = 1.0 is numerically integrated by an Euler forward algorithm using very small time steps. 2.3. Pfaftic limit-load criterion (SC) After analysing the mechanics of ductile fracture by void coalescence, Thomason [2] observed that the sudden transformation for a macroscopically homogeneous state of plastic flow, to the highly localized internal necking of the intervoid matrix across a single “weakest” sheet of voids, coincides with the attainment of the plastic limit-load condition for localized plastic failure of the intervoid matrix. Based on this observation, a critical condition for incipient void coalescence by internal necking failure of the intervoid matrix can therefore be given by: 0; = a;,
(8)
where o; is the applied maximum principal stress on the current yield surface, called the weak constitutive surface; a; is the virtual maximum principal stress to initiate the localized internal necking of the intervoid matrix material, which represents the strong dilational-plastic response. For a real problem, cr; can be calculated by any analytical or numerical method, for example, finite element method. The critical condition, eq. (8), for the incipient void coalescence in a unit cell with an ellipsoidal void can be written: a; = 6, A, = o;,
(9)
where A, is the net area fraction of intervoid matrix in the maximum principal stress direction which in the present case is the z direction (Fig. 1), and a,, is the mean stress required to initiate the internal necking in the intervoid matrix of a porous solid. By assuming that the initial lengths of the unit
Ductility predictions by local failure criteria
533
cell in three directions are unit (Fig. I), and denoting the current lengths of the cell by 2X, 2 Y and 22 and current radii of the ellipsoidal void by Rx, Ry and R,, we can write:
where 2X = 2Y = e&C= e‘; has been used. For von Mises material, EP,+ 6; = -6: and A n = 1 - RR: e’f. A “law-of-mixtures” has been used to relate the microscopic yield stress 6 and macroscopic yield stress ts, 121: 6, = (1 -f&F,
(11)
wheref, is the initial void volume fraction. Using the above two equations we obtain the following non-dimensionalized equation:
4 -= CT,
02
(1
-&)-‘A,
=
2, ?#I
where a,/6 is the plastic constraint factor for incipient plastic limit-load failure of the intervoid matrix. By appro-~rna~ng the ellipsoidal void by the equivalent square-prismatic void and assuming two velocity fields, namely parallel and triangular fields in the intervoid matrix of the three-dimensional unit cell, Thomason [2] obtained the following closed-form empirical expression for the constraint factor a,/6 which gives close agreement with his upper-bound results:
where RXand R, are calculated from Rice-Tracey theory, eqs (l)--(3). Substituting A, and c~/C into eq. (12), the critical void coalescence condition can be finally written:
It must be noted that in the beginning of plastic flow the left-hand side of eq. (14) {strong dilational response) exceeds the right-hand side (weak dilational response) and ductile fracture is thus prevented. Ductile fracture by void coalescence only occurs when the equality (14) is just satisfied after sufIicient void growth strain, If the matrix material is characterized by the von Mises model and for the fixed stress triaxiality case, eq. (14) clearly shows that the plastic strain solely depends on the initial void volume fraction. This demonstrates the si~ifi~nt difference between the plastic limit-load criterion and the other two criteria, where for a given initial void volume fraction the critical values still have to be determined by experiment. 3. CALIBRATION OF THE PLASTIC LIMIT-LOAD CRITERION Two matrix materials are used in this study: one is rigid-perfectly plastic, with initial yield stress, rS = 404.7 MPa; the other one is linear hardening with the same initial yield stress and a constant plastic tangent modulus of 400.6 MPa. In the presentation, stress triaxiality is defined by: h=O”. 4
(15)
h can be determined by uniquely choosing c in the axisymmetric condition, u, = oy = err,. Please note that in the definition (15) the von Mises equivalent stress, not the actual flow stress, is used. There is no difference between the two stresses in von Mises material, because q = CT;however, as we will see in the next section, a considerable difference exists in Gurson material, especially when the stress triaxiality is high. The stress triaxiality used in the comparison ranges from 0.7 to 3.0, which is believed to represent most practical cases. In the original Thomason criterion, the real void radius changes calculated from the Rice--Tracey theory, eqs (l)-(3), were used. The results for two initial void volume fraction cases
534
Z. L. ZHANG and E. NIEMI 2
u
-@-Caoo2FO-aOQC _-o-Cam
1.FO =O.Ol
Plastic , sbrin
Qs
0
0s
1
10 Stress
2
3
zb
a6
trlaxiallty
Fig. 2. Predictions by Sc criterion, where in case 1, R, and R, were calculated by Rice-Tracey equations; in case 2, the mean void radius was used.
based on the original Thomason criterion (case 1) are shown in Fig. 2. It is assumed in the calculation that radii R, and R,, will be fixed once they decrease, because otherwise it will predict very large, sometimes even infinite void coalescence strain. For the case with stress triaxiality h = 0.7, the plastic limit-load criterion predicts equivalent plastic strain of 1.722 and 1.272 corresponding to two cases withf, = 0.005 and 0.01, respectively. By the Gurson model, it can be easily shown that the two plastic strains correspond to J, = 0.322 and 0.224, which are far from realistic compared with the cases in refs [lO-151. It therefore means that the original Thomason criterion gave too large predictions at low stress triaxiality. A modification is tested which uses the mean void radius R,, to replace R,and R,in eq. (14). The results after this modification are shown in Fig. 2 as case 2 for comparison with the original predictions. Surprisingly, this modification greatly decreases the prediction at lower stress triaxiality, while for the high stress triaxiality case, the predictions are almost the same. For stress triaxiality h = 0.7, the new predictions give plastic strains of 0.749 and 0.561, which correspond to f, = 0.045 and 0.038 by the Gurson model, respectively. These two f,values are very similar to the values determined by Sun et al. [13, 141. The convergence behaviour of the plastic limit-load criterion (14) is shown in Fig. 3, for the case h = 0.7, where the horizontal line represents the right-hand side of eq. (14). In the beginning
Nondlmentional stress
a2
41
9b
ob
Plastic strain Fig. 3. Convergence of plastic limit-load criterion for two initial void volume fractions for the case h = 0.7. Horizontal line represents right-hand side of eq. (14).
Ductility predictions by local failure criteria
535
Table 1 fo 0.005 0.01
Plastic strain
L
R,
0.748 0.561
0.045 0.038
0.6051 0.4537
of plastic loading, the left-hand side of eq. (14) is almost four times larger than the right-hand side for the case with& = 0.005. Void coalescence only occurs when the two lines cross, or the equality (14) is satisfied. It is obvious that according to the plastic limit-load criterion, small initialf, results in large void growth strain, and vice versa. Linear hardening matrix material has also been used for calculation; however, although the absolute values of the stress are different from those of non-hardening material, no significant difference has been found corresponding to Figs 2 and 3. Therefore, subsequently, only non-hardening material is used. 4. COMPARISON OF THE PREDICTIONS In the following, all the predictions are calibrated at stress triaxiality h = 0.7, based on modified SC. This means that all the plastic strain predictions are the same at stress triaxiality h = 0.7 and the critical void growth ratio R, and critical void volume fractionf, are then obtained as shown in Table 1. 4.1. Comparison of the three criteria Two initial void volume fractions have been compared, with results shown in Figs 4 and 5. In Figs 4 and 5, predictions are given by the original criteria, except that the modification R, = R, = R_, was used in SC. It is found that there are large differences in the predictions by the three criteria. It can be seen that Rc and Fc criteria give almost the same predictions at low stress triaxiality (less than 1.25). In general, Fc gives the highest prediction, Sc gives the lowest, with Rc in the middle. Although the Rc prediction is close to Fc at low stress triaxiality, it is near the Sc prediction at high stress triaxiality. Comparing Figs 4 and 5, it can be seen that a small initial void volume fraction reduces the difference between Fc and SC predictions. 4.2. Comparison between Rc and Fc criteria It was found above that when the stress triaxiality exceeds 1.25, a very big difference appears in the prediction between Rc and Fc. The difference is caused by the fact that, in the Fc criterion, the effect of void volume fraction on the flow behaviour has been taken into account, and the larger 0.0
Plraic Q* sbrin
Fig. 4. Comparison of the predictions by different criteria for the case of initial void volume fraction 0.01. R,=R,=R,, was assumed in the Sc criterion.
2. L. ZHANG and E. NIEMI
536
Plastic Q* swain
.
0 a6
1
13
Stress
3
tLxia~lft
*
Fig. 5. Comparison of the predictions by different criteria for the case of initial void votume fraction 0.005. was used in the SC criterion. R,=R,=R_
the stress triaxiality, the more significant the effect. With the above understanding, the Rc criterion has been recalculated using the Gurson model by the following equation:
Please note the difference between the present eq. (16) and the original eq. (4). The results are shown in Fig. 6 for two values off0 and good consistency at low as well as high stress triaxiality is obtained. This finding suggests that as long as Gut-son material is used in the caiculations, the Rc criterion is virtually equivalent to the Fc criterion. Figure 6 also shows that the initial void volume fraction has less effect on the prediction at high stress triaxiality than at low stress triaxiality. 4.3. Predictions by SC using d@erent methods to calculate void growth Figures 4 and 5 have shown that the SCcriterion using Rice-Tracey equations to calculate the void growth even under the modification RX= R, = R,, gave much smaller predictions than both the Rc and Fc criteria. In order to see whether the predictions can be narrowed, another modification is tried here: (a) use the Gurson constitutive model to represent the weak dilational response of the material [2]; (b) assume the void grows spherically and use the void
0s
1
1s Strim
3
2s
3
sb
trlaxiality
Fig. 6. Comparison between predictions by Rc and Fc, where Gurson material was used in Rc calculation.
Ductiiity p~ictions
537
by local failure criteria
Plastic qrl swain
0
a#
1
2
lb stress
2
es
2#
triaxwty
Fig. 7. Predictions by SC criterion using different methods to calculate the void radius change.
volume fraction from the Gurson model to calculate the void radius changes. This proposal has the following reasons. l The void itself in the Gut-son model is approximate as spherical [6,7j. l The proposal is backed by Rice and Tracey’s work at least at higher stress triaxiality [4], where they found that the spherically symmetric volume changing part of void growth far overwhelms the shape changing part when the mean stress is large. The effect of this modification on SC prediction is shown in Fig. 7 for two initial void volume fraction cases. Obviously, after this modi~cation the SC criterion si~ificantly increases its predictions when the stress triaxiality is high, while the prediction remains more or less the same at low stress triaxiality. It is noteworthy that, after the modification, the convergence behaviour is different (see Fig. 8, where the weak response is not a horizontal line any more). Instead, it declines with the plastic strain because of the effect of void volume fraction considered in the Gurson model. Using the modification just proposed, we can compare the three criteria again. After a new calibration with the prediction by modified SC at stress triaxiality h = 0.7, the results are as shown in Figs 9 and 10, where in Rc the original von Mises material is used; see eq. (4). It is interesting
7
NondlmenUO~SI 8tr088
-t 02
OA
w
a0
Plastic strain Fig. 8. Convergence behaviour after the modification for the case with A -0.7.
Z. L. ZHANG and E. NIEMI
538
PlrrUc Q* strain
Fig. 9. Predictions by different criteria with fo = 0.01. Sc was modified according to the proposal.
to note that the modified SC has made its prediction very close to the original Rc prediction and also narrowed the difference with the Fc prediction. This closeness is even better for a small initial void volume fraction. It is interesting to make another comparison for an even smaller initial void volume fraction, 0.0005, which is close to the values used by refs [14, 11. The results are shown in Fig. 11. Surprisingly, the difference nearly disappears and the three criteria give almost identical predictions. 5. DISCUSSIONS
AND SUMMARY
5.1. Rc and Fc criteria In the comparison of the three criteria, results have demonstrated that there are very large differences in the predictions. Rc and Fc are relatively more widely used than SC in the literature and Rc is usually taken as a continuum damage mechanics approach [3]. However, it has been shown that the original criteria compel each other, i.e. iff, is constant, prediction shows R, is not. The difference is mainly attributed to the effect of void volume fraction. In the original Rc criterion, an infinite matrix surrounding the voids is assumed and the remote stress changes by the void growth and void interaction are neglected, while in Fc, although only approximately, the effect of void growth is reflected into the constitutive equation. It has been demonstrated that if Rc is
Plrsdc strain
Fig. 10. Predictions by different criteria with fo = 0.005. Sc was modified according to the proposal.
Ductility predictions by local failure criteria
539
Plastic swain
Fig. 11. Predictions by different criteria with fo = 0.0005. SC was modified according to the proposal.
calculated using the Gurson model, the predictions by Rc and Fc are very similar. This implies that if the void effect is taken into consideration using the Gurson-type material model, the critical void fraction criterion and critical void growth criterion are identical to a first approximation. 5.2. SC and the other criteria Two modifications have been tried to the SC criterion. The original SC gives very large predictions, sometimes even infinite, at low stress triaxiality, which corresponds to unrealistic critical volume fractions. This leads to the first modification, which uses the mean radius (RX= R, = R,,,) to calculate the strong dilational yield surface, the left-hand side of eq. (14). This modification significantly decreases the predictions at low stress triaxiality, while it keeps the predictions at high stress triaxiality almost without change, which well reproduces the fact that symmetrical volume growth dominates the shape changing at large stress triaxiality [4]. Compared with predictions by Rc and Fc, it is found that SC gives very much smaller predictions than Fc and Rc at high triaxiality. Noticing this inconsistency, another modification has been tested. In the second modification, it is proposed to use the void volume fraction from the Gurson model to calculate the intervoid matrix geometry. This modification yields an interesting result in that it has brought the SC prediction very close to the one by the original Rc and significantly narrowed the difference between SC and Fc. This difference is found to decrease with a smaller initial void volume fraction. Surprisingly, in an extra comparison for a very small initial void fraction, the three criteria give virtually the same predictions. 5.3. General remarks As mentioned in the Introduction, the purpose of this work is not to justify which criterion is physically right or which one is the best. Actually, until now experimental studies have not given concrete confirmation of the ductile fracture criteria [16]. In the Rc and Fc criteria considered here no real “mechanics” for void coalescence has been incorporated, and only the “effect” of coalescence was reflected. This explains why the critical values have to be determined from simple tests. The SC criterion itself is a condition which represents the “mechanism” of void coalescence and is a natural result of the plastic deformation. No parameter needs to be obtained from tests, except the initial conditions. Once the initial conditions are given, the fracture strain is solely determined. Therefore, the SC criterion is physically more sound than Fc and Rc. It must be noted that all the predictions in the present paper are for the void growth strain only, with void nucleation neglected. In general, the ductility equals the void nucleation strain plus the void growth strain. Finally, further work which is currently underway is needed to verify the ductile fracture criteria. Acknowledgement-Z.
L. Zhang acknowledges the financial support from the Ministry of Education, Finland.
Z. L. ZHANG and E. NIEMI
540
REFERENCES [l] A. C. Mackenzie, J. W. Hancock and D. K. Brown, On the influence of state of stress on ductile failure initiation in high strength steels. Engne Fracture Me&. 9, 167-188 (1977). [Z] P. F. Thomason, Ductile Fracfure of Mefals. Pergamon Press, Oxford (1990). [3] M. Zheng, Z. J. Luo and X. Zheng, A new damage model for ductile materials. Engng Fracture Mech. 41, 103-110 (1992). [4] J. R. Rice and D. M. Tracey, On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solidr 17, 201-207 (1969). [S] G. Rousselier, Ductile fracture models and their potential in local approach of fracture. Nucl. Engng Des. 105.97-I 11 (1987). [6] A. L. Gurson, Plastic Row and fracture behaviour of ductile materials incorporating void nucleation, growth, and interaction. Ph.D. dissertation, Brown University (1975). [7] A. L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: part I-yield criteria and flow rules for porous ductile media. J. Engng Mafer. Technof. 99, 2-15 (1977). [8] V. Tvergaard, Influence of voids on shear band instabilities under plane strain conditions. Inr. J. Fracfure 17,389~407 (1981). [9] V. Tvergaard, On localization in ductile materials containing spherical voids. Int. J. Fracture 18, 237-252 (1982). [lo] V. Tvergaard and A. Needleman, Analysis of the cup-cone fracture in a round tensile bar. Actu Metall. 32, 157-169 (1984). [l l] X.-P. Xu and A. Needleman, Simulations of ductile failure with two size scales of voids. Eur. J. Mech., A/Solids 10, 459-484 (1991). [12] V. Tvergaard, A numerical analysis of 3D localization failure by a void-sheet mechanism. Engng Fracture Mech. 41, 787-803 (1992). [13] D. Z. Sun, D. Siegele, B. Voss and W. Schmitt, Application of local damage models to the numerical analysis of ductile rupture. Fatigue Fracture Engng Mater. Structures 12, 200-212 (1989). [14] D.-Z. Sun, B. Voss and W. Schmitt, Numerical prediction of ductile fracture resistance behaviour based on micromechanical models, in Defect Assessment in Components-Fundamentals and Applications, ESIS/EGF9 (Edited by J. G. Blauel and K.-H. Schwalbe), pp. 447-458. Mechanical Engineering Publications, London (1991). [15] R. Narasimhan, A. J. Rosakis and B. Moran, A three-dimensional numerical investigation of fracture initiation by ductile mechanisms in a 4340 steel. Inf. J. Fracture 56, 1-24 (1992). [16] Y. W. Shi, J. T. Bamby and A. S. Nadkami, Void growth at ductile crack initiation of a structural steel. Engng Fracture Mech. 39, 37-44 (1991). [17] R. Batisse, M. Bethmont, G. Devesa and G. Rousseher, Ductile fracture of A 508 CI 3 steel in relation with inclusions content: the benefit of the local approach of fracture and continuum damage mechanics. Nucl. Engng Des. 105,113-120 (1987).
(Received 17 June 1993)
OOW7944(94)E0023-A
Copyright 0 1994 Elsevier Science Ltd. Printed in Great Britain. All rights reserved 0013-7944/94 $7.00+ 0.00
STUDIES ON THE DUCTILITY PREDICTIONS DIFFERENT LOCAL FAILURE CRITERIA
BY
Z. L. ZHANG and E. NIEMI Department of Mechanical Engineering, Lappeenranta University of Technology, P.O. Box 20, 53851 Lappeenranta, Finland Abstract-It is well known that the ductile fracture of metals has frequently been observed to result from the growth and coalescence of microscopic voids. Correspondingly, local approach methodologies have been developed in the modelhng of ductile fracture. Three local ductile fracture criteria, a critical void volume fraction criterion based on a Gurson-type constitutive relation, a critical void growth criterion based on Rice-Tracey void growth equations, and Thomason’s plastic limit-load criterion, have been studied and examined against the predictions of ductility with respect to stress triaxiality. A calibration for the predictions of the plastic limit-load criterion has been made, which yields a modification to the criterion to give more realistic predictions at low stress triaxiality than the original criterion. Various comparisons of the criteria have been made. It has been found that there are considerable differences in the predictions by the three criteria. Finally, a method has been introduced which can correlate the predictions well using dual dilational constitutive models.
1. INTRODUCTION DURING the past 20 years, many research works have been devoted to the study of ductile fracture, and now it is well known that the ductile fracture of metals occurs by the nucleation, growth and coalescence of voids and that plastic deformation is not the only determinant of ductile fracture. Notched axisymmetric tensile tests have demonstrated that plastic deformation at fracture is highly dependent on the state of stresses [l]. It has been shown that ductile fracture surfaces are formed by the sudden catastrophic coalescence of voids which grow by plastic deformation under the influence of a prevailing triaxial stress system [2]. Naturally, therefore, the macroscopic ductile fracture strain should be composed of two parts, the void nucleation strain and void growth strain. Although void nucleation is obviously important in ductile fracture, it is difficult to determine in normal practice. In this study, in common with most other workers, we assume that the strain necessary to nucleate a void is negligible compared to the strain needed to cause coalescence [3]. Once a void has been nucleated in a plastically deforming matrix, the resulting stress-free surface of the void causes a localized stress and strain concentration in the adjacent plastic field. With continuing plastic loading of the matrix, the void will undergo a volumetric growth and shape change. Rice and Tracey [4] have successfully established a growth model of an isolated spherical void in an infinite non-hardening von Mises matrix. A fracture criterion named “critical void growth ratio” based on Rice and Tracey’s work has been formulated and widely used in the local approach to fracture [3,5]. In contrast to developing a single void growth model, a theory of dilational plasticity has been developed by Gurson [6,7] and modified by Tvergaard [8,9] to incorporate the inelastic straining resulting from the nucleation and growth of voids. Without physical proof, the “critical void volume fraction” criterion simulating the void coalescence based on the Gurson model has been increasingly popular both in theoretical analyses and practical applications [lO-151. On the basis of detailed analysis of the mechanism of ductile fracture by the nucleation, growth and coalescence of voids, Thomason [2] has presented a plastic limit-load or internal necking criterion for ductile fracture which leads to a strong dilational yield surface. A critical load condition at incipient void coalescence has been established based on dual dilational constitutive responses. Although the three criteria are relatively well known, no examination of the consistency of the predictions, especially with respect to stress triaxiality, has been found in the literature. In the application of the first two criteria, usually the critical void volume fraction and critical void growth ratio are obtained from simple axisymmetric tensile tests and then applied to cases under various 529
530
Z. L. ZHANG and E. NIEMI
stress conditions without proof. The advantages of Thomason’s plastic limit-load criterion are that, once the initial conditions are given, ductile fracture is a natural result of the dual dilational responses and is solely determined. The purpose of this study is not to examine physically whether critical void volume fraction and void growth ratio are material constants and independent of stress states as done in ref. [16], but to show the difference and inconsistency among the criteria. A simple geometric model which simulates the axisymmetric loading is used for various studies (see Fig. 1). Firstly, a calibration of Thomason’s plastic limit-load criterion is made which leads to a modification of the model, to bring realistic predictions at low stress triaxiality. Then various comparisons of the criteria are made. It has been found that there are considerable differences in the predictions of the three criteria for large stress triaxiality, under the condition that the three criteria are calibrated for the same lower stress triaxiality case. Here and subsequently, in order to simplify the presentation, Rc, Fc and Sc are used to identify the critical void growth ratio, critical void volume fraction and plastic limit-load criteria, respectively, and plastic strain in the figures means the equivalent plastic strain. 2. THREE LOCAL FAILURE CRITERIA 2.1. Critical void growth ratio criterion (Rc) For the axisymmetric case (6, = uY; Fig. 1) and constant stress triaxiality (proportional loading), the Rice-Tracey void growth theory can be expressed as: !!!?%A % R --f=A+B Rll R R i=f=” -$B, where A = exp(DcP) B
l+E =.(A+
D = 5 f$
for linear hardening
D = 0.283 exp 1 + Em 5/3 1+ E x 2
for non-hardening
for linear hardening or low 6, with non-hardening for high values of a,,, with non-hardening.
Fig. I. A spherical void in a plastic matrix unit ceil.
(1)
(3)
531
Ductilitypredictionsby local failun criteria
R,,, R,, R,, and R,_ are the initial void radius, current radii in x and y directions and mean radius, respectively, tP is the equivalent plastic strain, Q,,,is the first invariant of the stress tensor and 8 the flow stress of the matrix surrounding the void, which is equal to the von Mises equivalent stress q. It must be noted that the above equations were derived for a spherical void located inside an infinite perfectly plastic matrix material. Although the void growth model does not itself constitute a fracture criterion, and a modelling of the coalescence must be added, a general remark has been made [5]: as a first approximation, deformation at fracture in tension is independent of the size of and distance between voids and only depends on void growth ratio. With this observation, a fracture criterion based on the assumption of a critical ratio of void growth (R,,/R,,),, specific to the material for a given direction of the load, has been formulated and widely used in the local approach of fracture. In general, the critical void growth fracture criterion modified to take account of hardening and the non-constant stress triaxiality case can be written as [3,51 II 0.283 exp($!!)VO”Mimd@ = ln(%)c
= R,,
where E” and t, are the strains at void nucleation and coalescence, respectively. The value of (R_,,/R,,), is usually determined through simple uniaxial or axisymmetric tests and then applied to complex stress cases, for example, notched or cracked specimens. 2.2. Critical void volume fraction criterion (Fc) Based on an approximation of a solid with a volume fraction,f, of voids, by a spherical body with a concentric spherical void, and carrying out an approximate rigid-plastic upper bound analysis of this thick-walled spherical cell, with matrix material idealised as rigid-perfectly plastic and obeying the von Mises yield criterion, Gurson [6,7j obtained the following dilational yield condition (q, = 1 and q2 = 1) for a porous plastic solid, which is a good fit to his upper bound yield load locus:
d(u,f, 6) =
$ +2q,f
cash
- 1 - (q&Z = 0,
where constants q, and q2 are introduced by Tvergaard [8,9] to bring predictions of the model into closer agreement with full numerical analyses of a periodic array of voids. q is the effective part of the average macroscopic Cauchy stress u. It is easy to see that the material loses its load carrying capacity if f reaches the limit l/q,, because all the stress components have to vanish in order to satisfy eq. (5). However, even if q1 = 1.5, the void volume fraction, f = l/q,, is still too large to be realistic in practice to simulate the final material failure. Therefore, a modification usingp to replace f in eq. (5) in order to account for the void coalescence effect on final failure has been proposed [lo]: p=
f ( f,+ W-f,)
forf Gf, forf >A.
(6)
Here, f, is the critical void volume fraction at which void coalescence is first observed and K is a constant determined from the void volume fraction at final failure of the material which is much smaller than f = l/q,. Due to the lack of physical proof and in line with mathematical convenience, this modification has actually induced another fracture criterion-the critical void volume fraction criterion, which is perhaps one of the most widely applied criteria [lO-151. Because of the difficulty in obtaining f, physically, Sun et al. [14] have suggested that f, can be determined from smooth axisymmetric tensile tests by numerical simulation. In this study, we are only interested in the plastic strain corresponding to f,, therefore eq. (6) is not used.
Z. L. ZHANG and E. NIEMI
532
For the axisymmetric proportional loading (0; = uv = ccr,, c is a constant), we obtain, after some algebraic manipulations, the following analytical equations: ip = awl, i; =
G,wi,
ciz= E,(l
- G,w)L,
f= @it,
(7)
where
4 Gs a4 ah !!f E, GzGs - B --af ma5 G -1 a0 : W--c) CO=
7%
“-3&F, G s=-
(1 + 2c) 86, 3 ao,+
2fl - c>* 62 0,
&G* Q,= (1 -f)$ B=(l E,
-r$
m
=E
1 - 2cv
ef: is the plastic strain in the z direction, E is Young’s modulus, v is Poisson’s ratio; ab/ao,, @/af and &$/A? can be easily calculated from eq. (5). In the following calculations, eq. (7) with g1 = 1.5 and gz = 1.0 is numerically integrated by an Euler forward algorithm using very small time steps. 2.3. Pfaftic limit-load criterion (SC) After analysing the mechanics of ductile fracture by void coalescence, Thomason [2] observed that the sudden transformation for a macroscopically homogeneous state of plastic flow, to the highly localized internal necking of the intervoid matrix across a single “weakest” sheet of voids, coincides with the attainment of the plastic limit-load condition for localized plastic failure of the intervoid matrix. Based on this observation, a critical condition for incipient void coalescence by internal necking failure of the intervoid matrix can therefore be given by: 0; = a;,
(8)
where o; is the applied maximum principal stress on the current yield surface, called the weak constitutive surface; a; is the virtual maximum principal stress to initiate the localized internal necking of the intervoid matrix material, which represents the strong dilational-plastic response. For a real problem, cr; can be calculated by any analytical or numerical method, for example, finite element method. The critical condition, eq. (8), for the incipient void coalescence in a unit cell with an ellipsoidal void can be written: a; = 6, A, = o;,
(9)
where A, is the net area fraction of intervoid matrix in the maximum principal stress direction which in the present case is the z direction (Fig. 1), and a,, is the mean stress required to initiate the internal necking in the intervoid matrix of a porous solid. By assuming that the initial lengths of the unit
Ductility predictions by local failure criteria
533
cell in three directions are unit (Fig. I), and denoting the current lengths of the cell by 2X, 2 Y and 22 and current radii of the ellipsoidal void by Rx, Ry and R,, we can write:
where 2X = 2Y = e&C= e‘; has been used. For von Mises material, EP,+ 6; = -6: and A n = 1 - RR: e’f. A “law-of-mixtures” has been used to relate the microscopic yield stress 6 and macroscopic yield stress ts, 121: 6, = (1 -f&F,
(11)
wheref, is the initial void volume fraction. Using the above two equations we obtain the following non-dimensionalized equation:
4 -= CT,
02
(1
-&)-‘A,
=
2, ?#I
where a,/6 is the plastic constraint factor for incipient plastic limit-load failure of the intervoid matrix. By appro-~rna~ng the ellipsoidal void by the equivalent square-prismatic void and assuming two velocity fields, namely parallel and triangular fields in the intervoid matrix of the three-dimensional unit cell, Thomason [2] obtained the following closed-form empirical expression for the constraint factor a,/6 which gives close agreement with his upper-bound results:
where RXand R, are calculated from Rice-Tracey theory, eqs (l)--(3). Substituting A, and c~/C into eq. (12), the critical void coalescence condition can be finally written:
It must be noted that in the beginning of plastic flow the left-hand side of eq. (14) {strong dilational response) exceeds the right-hand side (weak dilational response) and ductile fracture is thus prevented. Ductile fracture by void coalescence only occurs when the equality (14) is just satisfied after sufIicient void growth strain, If the matrix material is characterized by the von Mises model and for the fixed stress triaxiality case, eq. (14) clearly shows that the plastic strain solely depends on the initial void volume fraction. This demonstrates the si~ifi~nt difference between the plastic limit-load criterion and the other two criteria, where for a given initial void volume fraction the critical values still have to be determined by experiment. 3. CALIBRATION OF THE PLASTIC LIMIT-LOAD CRITERION Two matrix materials are used in this study: one is rigid-perfectly plastic, with initial yield stress, rS = 404.7 MPa; the other one is linear hardening with the same initial yield stress and a constant plastic tangent modulus of 400.6 MPa. In the presentation, stress triaxiality is defined by: h=O”. 4
(15)
h can be determined by uniquely choosing c in the axisymmetric condition, u, = oy = err,. Please note that in the definition (15) the von Mises equivalent stress, not the actual flow stress, is used. There is no difference between the two stresses in von Mises material, because q = CT;however, as we will see in the next section, a considerable difference exists in Gurson material, especially when the stress triaxiality is high. The stress triaxiality used in the comparison ranges from 0.7 to 3.0, which is believed to represent most practical cases. In the original Thomason criterion, the real void radius changes calculated from the Rice--Tracey theory, eqs (l)-(3), were used. The results for two initial void volume fraction cases
534
Z. L. ZHANG and E. NIEMI 2
u
-@-Caoo2FO-aOQC _-o-Cam
1.FO =O.Ol
Plastic , sbrin
Qs
0
0s
1
10 Stress
2
3
zb
a6
trlaxiallty
Fig. 2. Predictions by Sc criterion, where in case 1, R, and R, were calculated by Rice-Tracey equations; in case 2, the mean void radius was used.
based on the original Thomason criterion (case 1) are shown in Fig. 2. It is assumed in the calculation that radii R, and R,, will be fixed once they decrease, because otherwise it will predict very large, sometimes even infinite void coalescence strain. For the case with stress triaxiality h = 0.7, the plastic limit-load criterion predicts equivalent plastic strain of 1.722 and 1.272 corresponding to two cases withf, = 0.005 and 0.01, respectively. By the Gurson model, it can be easily shown that the two plastic strains correspond to J, = 0.322 and 0.224, which are far from realistic compared with the cases in refs [lO-151. It therefore means that the original Thomason criterion gave too large predictions at low stress triaxiality. A modification is tested which uses the mean void radius R,, to replace R,and R,in eq. (14). The results after this modification are shown in Fig. 2 as case 2 for comparison with the original predictions. Surprisingly, this modification greatly decreases the prediction at lower stress triaxiality, while for the high stress triaxiality case, the predictions are almost the same. For stress triaxiality h = 0.7, the new predictions give plastic strains of 0.749 and 0.561, which correspond to f, = 0.045 and 0.038 by the Gurson model, respectively. These two f,values are very similar to the values determined by Sun et al. [13, 141. The convergence behaviour of the plastic limit-load criterion (14) is shown in Fig. 3, for the case h = 0.7, where the horizontal line represents the right-hand side of eq. (14). In the beginning
Nondlmentional stress
a2
41
9b
ob
Plastic strain Fig. 3. Convergence of plastic limit-load criterion for two initial void volume fractions for the case h = 0.7. Horizontal line represents right-hand side of eq. (14).
Ductility predictions by local failure criteria
535
Table 1 fo 0.005 0.01
Plastic strain
L
R,
0.748 0.561
0.045 0.038
0.6051 0.4537
of plastic loading, the left-hand side of eq. (14) is almost four times larger than the right-hand side for the case with& = 0.005. Void coalescence only occurs when the two lines cross, or the equality (14) is satisfied. It is obvious that according to the plastic limit-load criterion, small initialf, results in large void growth strain, and vice versa. Linear hardening matrix material has also been used for calculation; however, although the absolute values of the stress are different from those of non-hardening material, no significant difference has been found corresponding to Figs 2 and 3. Therefore, subsequently, only non-hardening material is used. 4. COMPARISON OF THE PREDICTIONS In the following, all the predictions are calibrated at stress triaxiality h = 0.7, based on modified SC. This means that all the plastic strain predictions are the same at stress triaxiality h = 0.7 and the critical void growth ratio R, and critical void volume fractionf, are then obtained as shown in Table 1. 4.1. Comparison of the three criteria Two initial void volume fractions have been compared, with results shown in Figs 4 and 5. In Figs 4 and 5, predictions are given by the original criteria, except that the modification R, = R, = R_, was used in SC. It is found that there are large differences in the predictions by the three criteria. It can be seen that Rc and Fc criteria give almost the same predictions at low stress triaxiality (less than 1.25). In general, Fc gives the highest prediction, Sc gives the lowest, with Rc in the middle. Although the Rc prediction is close to Fc at low stress triaxiality, it is near the Sc prediction at high stress triaxiality. Comparing Figs 4 and 5, it can be seen that a small initial void volume fraction reduces the difference between Fc and SC predictions. 4.2. Comparison between Rc and Fc criteria It was found above that when the stress triaxiality exceeds 1.25, a very big difference appears in the prediction between Rc and Fc. The difference is caused by the fact that, in the Fc criterion, the effect of void volume fraction on the flow behaviour has been taken into account, and the larger 0.0
Plraic Q* sbrin
Fig. 4. Comparison of the predictions by different criteria for the case of initial void volume fraction 0.01. R,=R,=R,, was assumed in the Sc criterion.
2. L. ZHANG and E. NIEMI
536
Plastic Q* swain
.
0 a6
1
13
Stress
3
tLxia~lft
*
Fig. 5. Comparison of the predictions by different criteria for the case of initial void votume fraction 0.005. was used in the SC criterion. R,=R,=R_
the stress triaxiality, the more significant the effect. With the above understanding, the Rc criterion has been recalculated using the Gurson model by the following equation:
Please note the difference between the present eq. (16) and the original eq. (4). The results are shown in Fig. 6 for two values off0 and good consistency at low as well as high stress triaxiality is obtained. This finding suggests that as long as Gut-son material is used in the caiculations, the Rc criterion is virtually equivalent to the Fc criterion. Figure 6 also shows that the initial void volume fraction has less effect on the prediction at high stress triaxiality than at low stress triaxiality. 4.3. Predictions by SC using d@erent methods to calculate void growth Figures 4 and 5 have shown that the SCcriterion using Rice-Tracey equations to calculate the void growth even under the modification RX= R, = R,, gave much smaller predictions than both the Rc and Fc criteria. In order to see whether the predictions can be narrowed, another modification is tried here: (a) use the Gurson constitutive model to represent the weak dilational response of the material [2]; (b) assume the void grows spherically and use the void
0s
1
1s Strim
3
2s
3
sb
trlaxiality
Fig. 6. Comparison between predictions by Rc and Fc, where Gurson material was used in Rc calculation.
Ductiiity p~ictions
537
by local failure criteria
Plastic qrl swain
0
a#
1
2
lb stress
2
es
2#
triaxwty
Fig. 7. Predictions by SC criterion using different methods to calculate the void radius change.
volume fraction from the Gurson model to calculate the void radius changes. This proposal has the following reasons. l The void itself in the Gut-son model is approximate as spherical [6,7j. l The proposal is backed by Rice and Tracey’s work at least at higher stress triaxiality [4], where they found that the spherically symmetric volume changing part of void growth far overwhelms the shape changing part when the mean stress is large. The effect of this modification on SC prediction is shown in Fig. 7 for two initial void volume fraction cases. Obviously, after this modi~cation the SC criterion si~ificantly increases its predictions when the stress triaxiality is high, while the prediction remains more or less the same at low stress triaxiality. It is noteworthy that, after the modification, the convergence behaviour is different (see Fig. 8, where the weak response is not a horizontal line any more). Instead, it declines with the plastic strain because of the effect of void volume fraction considered in the Gurson model. Using the modification just proposed, we can compare the three criteria again. After a new calibration with the prediction by modified SC at stress triaxiality h = 0.7, the results are as shown in Figs 9 and 10, where in Rc the original von Mises material is used; see eq. (4). It is interesting
7
NondlmenUO~SI 8tr088
-t 02
OA
w
a0
Plastic strain Fig. 8. Convergence behaviour after the modification for the case with A -0.7.
Z. L. ZHANG and E. NIEMI
538
PlrrUc Q* strain
Fig. 9. Predictions by different criteria with fo = 0.01. Sc was modified according to the proposal.
to note that the modified SC has made its prediction very close to the original Rc prediction and also narrowed the difference with the Fc prediction. This closeness is even better for a small initial void volume fraction. It is interesting to make another comparison for an even smaller initial void volume fraction, 0.0005, which is close to the values used by refs [14, 11. The results are shown in Fig. 11. Surprisingly, the difference nearly disappears and the three criteria give almost identical predictions. 5. DISCUSSIONS
AND SUMMARY
5.1. Rc and Fc criteria In the comparison of the three criteria, results have demonstrated that there are very large differences in the predictions. Rc and Fc are relatively more widely used than SC in the literature and Rc is usually taken as a continuum damage mechanics approach [3]. However, it has been shown that the original criteria compel each other, i.e. iff, is constant, prediction shows R, is not. The difference is mainly attributed to the effect of void volume fraction. In the original Rc criterion, an infinite matrix surrounding the voids is assumed and the remote stress changes by the void growth and void interaction are neglected, while in Fc, although only approximately, the effect of void growth is reflected into the constitutive equation. It has been demonstrated that if Rc is
Plrsdc strain
Fig. 10. Predictions by different criteria with fo = 0.005. Sc was modified according to the proposal.
Ductility predictions by local failure criteria
539
Plastic swain
Fig. 11. Predictions by different criteria with fo = 0.0005. SC was modified according to the proposal.
calculated using the Gurson model, the predictions by Rc and Fc are very similar. This implies that if the void effect is taken into consideration using the Gurson-type material model, the critical void fraction criterion and critical void growth criterion are identical to a first approximation. 5.2. SC and the other criteria Two modifications have been tried to the SC criterion. The original SC gives very large predictions, sometimes even infinite, at low stress triaxiality, which corresponds to unrealistic critical volume fractions. This leads to the first modification, which uses the mean radius (RX= R, = R,,,) to calculate the strong dilational yield surface, the left-hand side of eq. (14). This modification significantly decreases the predictions at low stress triaxiality, while it keeps the predictions at high stress triaxiality almost without change, which well reproduces the fact that symmetrical volume growth dominates the shape changing at large stress triaxiality [4]. Compared with predictions by Rc and Fc, it is found that SC gives very much smaller predictions than Fc and Rc at high triaxiality. Noticing this inconsistency, another modification has been tested. In the second modification, it is proposed to use the void volume fraction from the Gurson model to calculate the intervoid matrix geometry. This modification yields an interesting result in that it has brought the SC prediction very close to the one by the original Rc and significantly narrowed the difference between SC and Fc. This difference is found to decrease with a smaller initial void volume fraction. Surprisingly, in an extra comparison for a very small initial void fraction, the three criteria give virtually the same predictions. 5.3. General remarks As mentioned in the Introduction, the purpose of this work is not to justify which criterion is physically right or which one is the best. Actually, until now experimental studies have not given concrete confirmation of the ductile fracture criteria [16]. In the Rc and Fc criteria considered here no real “mechanics” for void coalescence has been incorporated, and only the “effect” of coalescence was reflected. This explains why the critical values have to be determined from simple tests. The SC criterion itself is a condition which represents the “mechanism” of void coalescence and is a natural result of the plastic deformation. No parameter needs to be obtained from tests, except the initial conditions. Once the initial conditions are given, the fracture strain is solely determined. Therefore, the SC criterion is physically more sound than Fc and Rc. It must be noted that all the predictions in the present paper are for the void growth strain only, with void nucleation neglected. In general, the ductility equals the void nucleation strain plus the void growth strain. Finally, further work which is currently underway is needed to verify the ductile fracture criteria. Acknowledgement-Z.
L. Zhang acknowledges the financial support from the Ministry of Education, Finland.
Z. L. ZHANG and E. NIEMI
540
REFERENCES [l] A. C. Mackenzie, J. W. Hancock and D. K. Brown, On the influence of state of stress on ductile failure initiation in high strength steels. Engne Fracture Me&. 9, 167-188 (1977). [Z] P. F. Thomason, Ductile Fracfure of Mefals. Pergamon Press, Oxford (1990). [3] M. Zheng, Z. J. Luo and X. Zheng, A new damage model for ductile materials. Engng Fracture Mech. 41, 103-110 (1992). [4] J. R. Rice and D. M. Tracey, On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solidr 17, 201-207 (1969). [S] G. Rousselier, Ductile fracture models and their potential in local approach of fracture. Nucl. Engng Des. 105.97-I 11 (1987). [6] A. L. Gurson, Plastic Row and fracture behaviour of ductile materials incorporating void nucleation, growth, and interaction. Ph.D. dissertation, Brown University (1975). [7] A. L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: part I-yield criteria and flow rules for porous ductile media. J. Engng Mafer. Technof. 99, 2-15 (1977). [8] V. Tvergaard, Influence of voids on shear band instabilities under plane strain conditions. Inr. J. Fracfure 17,389~407 (1981). [9] V. Tvergaard, On localization in ductile materials containing spherical voids. Int. J. Fracture 18, 237-252 (1982). [lo] V. Tvergaard and A. Needleman, Analysis of the cup-cone fracture in a round tensile bar. Actu Metall. 32, 157-169 (1984). [l l] X.-P. Xu and A. Needleman, Simulations of ductile failure with two size scales of voids. Eur. J. Mech., A/Solids 10, 459-484 (1991). [12] V. Tvergaard, A numerical analysis of 3D localization failure by a void-sheet mechanism. Engng Fracture Mech. 41, 787-803 (1992). [13] D. Z. Sun, D. Siegele, B. Voss and W. Schmitt, Application of local damage models to the numerical analysis of ductile rupture. Fatigue Fracture Engng Mater. Structures 12, 200-212 (1989). [14] D.-Z. Sun, B. Voss and W. Schmitt, Numerical prediction of ductile fracture resistance behaviour based on micromechanical models, in Defect Assessment in Components-Fundamentals and Applications, ESIS/EGF9 (Edited by J. G. Blauel and K.-H. Schwalbe), pp. 447-458. Mechanical Engineering Publications, London (1991). [15] R. Narasimhan, A. J. Rosakis and B. Moran, A three-dimensional numerical investigation of fracture initiation by ductile mechanisms in a 4340 steel. Inf. J. Fracture 56, 1-24 (1992). [16] Y. W. Shi, J. T. Bamby and A. S. Nadkami, Void growth at ductile crack initiation of a structural steel. Engng Fracture Mech. 39, 37-44 (1991). [17] R. Batisse, M. Bethmont, G. Devesa and G. Rousseher, Ductile fracture of A 508 CI 3 steel in relation with inclusions content: the benefit of the local approach of fracture and continuum damage mechanics. Nucl. Engng Des. 105,113-120 (1987).
(Received 17 June 1993)