A new failure criterion for the Gurson-Tvergaard dilational constitutive model
Z.L. Z H A N G * and E. NIEMI Department of Mechanical Engineering, Lappeenranta University of Technology, P.O. Box 20, 53851 Lappeenranta, Finland
Received 24 November 1993; accepted in revised form 11 November 1994
Abstract. In the Gurson-Tvergaard model a failure criterion has to be used to signify the void coalescence. In the literature, a constant critical void volume fraction criterion has been widely used. However, it is questionable whether the critical void volume fraction is a material constant and, furthermore, it is also difficult in practice to determine the 'constant'. By modifying Thomason's plastic limit-load model, a new failure criterion which is fully compatible with the Gurson-Tvergaard model, is presented in this study. In the present criterion, the void coalescence failure mechanism by internal necking has been considered and the material failure is a natural result of the development of dual constitutive, stable and unstable, responses. In practical application of the presentcriterion, no critical void volume fraction needs to be pre-determined either numerically or experimentally. Furthermore, according to the new criterion, the void volume fraction corresponding to void coalescence is not a material constant, rather a function of stress triaxiality. The predictions using the present criterion have been compared with the finite element results by Koplik and Needleman, and very good agreement is observed. The potential advantage of this criterion and other related issues are discussed.
1 . I n t r o d u c t i o n
It has long been recognized that the mechanism of ductile fracture is a damage accumulation process and there are three distinct idealized stages in the development of ductile fracture known as void nucleation, growth and coalescence, the last stage being the most critical and least understood. Due to the limitations of conventional global fracture criteria, there has recently been considerable interest and research on 'local approaches' to fracture [1-5]. These local approaches either assume conventional material behaviour supplemented by methods of local failure process, or use continuum damage mechanics. The later trend has become increasingly popular.
In general, there are two problems involved in the application of cont inuum damage mechanics to ductile fracture. One is a constitutive relation which must reflect the softening behaviour of material, the other is a failure criterion. A micro-mechanical model-based consti- tutive equation introduced by Gurson [6-7] and modified by Tvergaard [8-9] and Tvergaard and Needleman [ 10] has been applied more than any other as a dilational constitutive equation in theoretical analyses and, increasingly, in practical applications. Gurson's theory is endowed with a yield condition, a flow law, a measure of void volume fraction, a rule for nucleating voids, and a law for evolution of the voids [11]. Although the Gurson-Tvergaard dilational model demonstrates the softening effect of the material the model, however, itself does not constitute a fracture criterion. As we will see later, i f we expect the Gurson-Tvergaard model to naturally lose load-carrying capacity, a very large and unrealistic void volume fraction has
* Current address: SINTEF Materials Technology, N-7034 Trondheim, Norway
322 Z.L. Zhang and E. Niemi
to be reached. Therefore, a criterion of void coalescence which determines a critical void volume fractionfc has to be used to simulate the material failure.
A constant critical void volume fraction, 0.15, was suggested by Tvergaard and Needleman [10] and this has been widely used in the theoretical analysis of ductile fracture and the investigation of fracture initiation of real materials [12-14]. However, recent numerical study of void coalescence using cell models by Koplik and Needleman [15] has shown that the critical void volume fraction depends on the initial void volume fraction of the material and is generally smaller than the value of 0.15. Tvergaard [ 16] has hence pointed out that the most realistic predictions are obtained by using a critical void volume fraction that depends on the initial void volume fraction.
On the other hand, for the same material with unique initial void volume fraction, whether the critical void volume fraction is a material constant, or whether critical void volume fraction is independent of stress triaxiality, is questionable. Experimental studies by Shi et al. [17] and Thomson and Hancock [ 18] have indicated a variation of the real critical void volume fraction of material as a strong function of stress triaxiality. Numerical study by Koplik and Needleman [15] has also indicated that, when the initial void volume fraction is small, the real critical void volume fraction corresponding to the shift to a uniaxial straining deformation mode of their model depends slightly on the stress triaxiality. However, when the initial void volume fraction is large, there is a large spread in the observed critical void volume fraction. The general trend shown is that the critical void volume fraction decreases with the increase in stress triaxiality. Another interesting result found in [ 15] is that the shift to a uniaxial straining deformation mode of the model is very sensitive to the uniformity of void distribution or void-matrix dimensions. In other words, if the distribution of voids in the cell model is not equal in radial and axial directions in their model, the void volume fraction can hardly be taken as a failure criterion. As will be discussed later, it is important to note that there is a difference between the critical void volume fraction in the Gurson-Tvergaard model fc and the one observed either experimentally [17] or numerically [15], which is denoted here asfc*. Nevertheless, despite these findings, in the absence of appropriate failure criterion and in line with mathematical convenience, the criterion using constant critical void volume fraction has almost always been used in the analysis using the Gurson-Tvergaard constitutive equation.
Taking the critical void volume fractionf~ as a 'constant', there are usually two methods to determine this 'constant'. One is using the cell model by Koplik and Needleman [ 15]. Because fc is not equal to f~* which varies with the change of stress triaxiality, especially when the initial void volume fraction is large,fc has to be guessed or obtained by 'trial and error' method to get the best fit. No specific procedure was given in [15] for the choice off~. The limitation of this method is that it can only be used to determine the critical volume fraction for a specific initial void volume fraction. This is because the intermediate void nucleation process which is very important in ductile fracture, is very difficult to incorporate into the cell model. Furthermore, in practical application the critical void volume fraction obtained by this method still has to be re-examined in order to obtain realistic simulation of the global behaviour. Quite recently, Sun et al. [19-20] have suggested thatfc can be numerically obtained from smooth axisymmetric tensile tests and then applied to a more general stress status case. The advantage of Sun et al.'s method is that the intermediate void nucleation can be taken into account. This method has been verified by simulating tensile bars with different notch radii for a steel with small initial void volume fraction. It is worth noting that in Sun et al.'s method [20], the critical void volume fraction is not unique. It depends on the choice of void nucleation model
Gurson-Tvergaard dilational constitutive model 323
l z
S !--- I
I ? y , I 4
Fig. 1. Current void geometry in a plastic matrix unit cell, symmetric tension in z-plane (~= = ~u) and ellipsoidal void (R= = Ry) are assumed.
and parameter(s). However, the problem is that there is no sound theory or method at present available in the literature for the choosing of void nucleation model and parameter(s).
In this study, a new criterion for the Gurson-Tvergaard constitutive model is presented. The criterion is based on the ductile fracture model by Thomason and his proposals to the Gurson- Tvergaard model [21]. Thomason has developed a 2D [22] and a 3D [23] micromechanical model, called plastic limit-load or the internal necking model for material failure by void coalescence. Recently, in a separate study [24], the authors have compared three different local failure criteria and shown that, with modifications, Thomason's plastic limit load model can be combined with Gurson's model to give very reasonable predictions of ductility. What is unique in the plastic limit-load criterion is that fracture is not only related to void volume fraction, but also to void-matrix geometry and stress triaxiality. It will be seen in this study that the results of the void distribution effect studied in [15] can be well predicted by the present criterion. Contrary to the original plastic limit-load criterion where the void geometry was calculated from the Rice-Tracy void growth equations [25], in the present criterion, we use the void volume fraction from the Gurson-Tvergaard model to calculate the void geometry changes. It should be noted that no 'law of mixtures' [21] relation between the macroscopic and microscopic values of the yield stress is needed in the present criterion to take into account the dilational effect of voids. This criterion is fully compatible with the Gurson-Tvergaard model and can be applied wherever the Gurson-Tvergaard model can be applied. It will be discussed later that one advantage of this criterion is that the critical void volume fraction need not be pre-determined. On the other hand, in practical application, the void nucleation parameters which are difficult to monitor in experiments and are usually 'arbitrarily' chosen, can be numerically calibrated from, for example, simple axisymmetric tensile tests.
For simplicity and convenience, here and subsequently, we denote Sc the original plastic limit-load model by Thomason, Fc the failure criterion for Gurson-Tvergaard model using constant critical void volume fraction and NSc the new failure criterion.
2. A new failure criterion by modified plastic limit-load model
2.1. THOMASON'S ORIGINAL 3D PLASTIC LIMIT-LOAD MODEL
After a detailed analysis of the ductile fracture mechanics by void coalescence, Thomason [21 ] has observed that the critical condition for ductile fracture by void coalescence is fully
324 Z.L. Zhang and E. Niemi
equivalent to the attainment of a state where the sufficient condition for plastic stability of a body containing voids is no longer satisfied. Based on this observation, a critical condition called the plastic limit-load condition for incipient void coalescence by plastic limit-load failure of the intervoid matrix has been developed for a three-dimensional unit cell, shown in Fig. 1 [21].
In the plastic limit-load model, the current void-matrix dimensions with the development of plastic flow should be calculated. By using the Rice-Tracy [25] void growth equations and assuming that the material containing voids consists of a non-hardening rigid-plastic von Mises solid and the initial voids are of spherical shape and are centred at the nodal points of a regular cubic network, the variations in the geometry of the intervoid matrix with continuing plastic flow are calculated [21 ]. Then the upper-bound theorem was applied to obtain a close overestimate of the plastic constraint factors, Crn/# which corresponds to the development of an incipient model of plastic limit-load failure (internal necking) in the intervoid matrix of the regular three-dimensional porous solid. Here, ~rn is the mean stress on the plane of maximum principal stress (in the present case, the z-plane) and # is the uniaxial yield stress of the matrix. By approximating the ellipsoidal void by an equivalent square-prismatic void and using assumed velocity fields, Thomason [21] obtained the following closed-form empirical expression for the plastic constraint factor for the plastic limit-load failure cr , /#:
an 0.1 1.2
which gives close agreement with his upper-bound results. In (1) Rz, Rz and 2R are the current radii of the ellipsoidal void in the x- and z-axes and current length of the cell in the x-axis, respectively, see Fig. 1. Rx and Rz are calculated from the Rice-Tracy void growth theory [25]. Thomason [21] stated that a necessary condition for continuing homogenous ductile flow in a body containing voids is that a state of plastic limit-load failure (unstable response) cannot develop in the intervoid matrix. It is further noted in [21] that, for a three-dimensional plastic field in a void containing body, the current homogeneous macroscopic flow-field displays a weak but stable dilational-plastic response, while the virtual mode of incipient void coalescence represents a strong but unstable dilational-plastic response. Therefore the resulting critical condition for material failure by incipient void coalescence can be written in the form:
0"~ tr°ng -- ~r] veak ---- 0 , (2)
where cr~ veak is the current maximum principal stress in the homogenous flow field on the weak dilational-plastic yield surface, which for a real problem can be calculated by any analytical or numerical method, for example, finite element method, and o'sl tr°ng is the incipient void coalescence stress on the strong dilational-plastic yield surface which is directly related to the plastic constraint factor (1) by [21]
O'Sl trOng = crnAn, (3)
where An is the net area fraction of the intervoid matrix in the maximum principal stress direction. It must be noted that a~ veak depends on the current stress state, internal state variables
strong. and material constitutive model used, whereas ~r 1 is solely determined by the current void- matrix dimensions and the yield stress of the matrix material.
Gurson-Tvergaard dilational constitutive model 325
R ' q l lb .
i Fig. 2. Current dimensions in a plastic axisymmetric cell model [15].
Table 1. Void spacing data at the shift to a uniaxial straining state
"The original value of 0.28 in p.844 of [15] is a typographical error, as is evident from Fig. 7 on p.845. The current value was confirmed by Professor A. Needleman in private communica- tion. b Value was not given in [15]. Same value as M2 was used here. Actually no significant effect can be expected.
By substituting tr,~/# into (2), the critical void coalescence condition can be finally writ- ten:
0.1 1.2 ] e~ veak (R___~2 + ( ~ 0 . 5 Z , ~ - \ n - R x ]
(4)
It should be noted that according to the plastic limit-load theory, the material failure is the result of a 'competition' process between the weak but stable and the strong but unstable responses and the satisfaction of condition (4). In the beginning of plastic flow the left-hand side of (4) (strong dilational response) exceeds the right-hand side (weak dilational response) and failure is prevented. During the plastic flow, ductile fracture failure by void coalescence only occurs when the equality of (4) is just satisfied after sufficient strain is reached.
326 Z.L. Zhang and E. Niemi
1,5
R z/(R-R x)
0,5
I° \ I . . . . . M2 I
I I I
1 2 3
R ~/R z
Fig. 3. Predictions by the plastic limit-load model versus FE results from [15], for the case with stress triaxiality T = 2 .
1,5
R ~/(R-R x) 1
0,5
!
|
i l - - . , To,> I
• M,-FE i \ , . . . . . M2 i
~ ~ Mises l
0 I I I 0 1 2 3 4
R x / R z
Fig. 4. Predictions by the plastic limit-load model versus FE results from [15], for the case with stress triaxiality T = I .
2.2. COMPARISON BETWEEN THE PREDICTIONS BY SC AND THE FE RESULTS BY KOPLIK AND
NEEDLEMAN [15]
Axisymmetric cell models shown in Fig. 2 have been used in [15] to study the void growth and coalescence in porous plastic solids. Three models with two initial void volume fractions f0 are considered: Ml(f0 = 0.0013, Bo/Ro -- 1.0), M2(f0 = 0.0013, Bo/Ro -- 8.0) and M3(f0 = 0.0104, Bo/Ro = 1.0), where/30 and R0 are the initial axial and radial dimensions of the model (see Fig. 2). Koplik and Needleman found that, until the stress drops, the stress-strain response is the same for both calculations with f0 = 0.0013 (M1 and M2). However, the void volume fraction fc, corresponding to the shift to a uniaxial straining state which is associated with the accelerated void growth accompanying coalescence, for M2 is about one tenth of that for M1. Hence, they concluded that the initial stress-strain response is primarily a function of the void volume fraction while the onset of localization primarily depends on void spacing.
Gurson-Tvergaard dilational constitutive model 327
In the following, we will try to compare the predictions by the plastic limit-load model and the finite element results [15]. Table 1 shows the void spacing data at which the shift to a uniaxial straining state occurs, for two stress triaxiality cases. The values offc were suggested in the original paper. The comparison of the predictions and FE results is shown in Fig. 3 for the case with stress triaxiality T = 2 and in Fig. 4 with stress triaxiality T = 1. In Figs. 3 and 4, curves are the void spacing traces at failure predicted by the plastic limit-load model (4), according to a specific material model and a stress state; FE means the FE analysis results from [15]. It should be noted that if the von Mises model is used in the evaluation of (4), the prediction curves in Figs. 3 and 4 only depend on the stress triaxiality. However, the prediction curve is also dependent on the void volume fraction, if the Gurson-Tvergaard model is used as formulated in Section 3. Figure 3 shows that the von Mises model gives lower prediction compared with the FE results. However, by using the Gurson-Tvergaard model (ql = 1.25, qa = 1.0) and taking the critical void volume fraction fc into account, it is seen from Fig. 3 that the predictions have been improved very much for M1 with a slight underestimate for M2 and a slight overestimate for M3. Figure 4 shows that the von Mises model gives rather good predictions for M1 and M2, while the Gurson-Tvergaard model (ql = 1.25, qa = 1.0) improves the prediction for M2 and gives a slightly worse prediction for M1 than the von Mises model. The good agreement between the predictions by plastic limit-load model (4) and FE results shown in Figs. 3 and 4 indicates that the void spacing effect in which the criterion using constant void volume fraction does not work at all, can be well predicted by the original plastic limit-load model with a reasonable accuracy.
2.3. MODIFICATIONS TO THOMASON'S PLASTIC LIMIT-LOAD MODEL
In Thomason's original plastic limit-load model, von Mises material was used together with a 'law of mixtures' to take into account the void effect on the relation between macroscopic and microscopic yield stresses, and the void geometry changes were calculated from the Rice-Tracy void growth theory [25], in which the effect of void growth on the remote stress changes were neglected. It has been shown by the authors[24] that Thomason's original model gives very large, sometimes even infinite, predictions of the equivalent plastic strain which corresponds to unrealistic critical void volume fractions in the Gurson-Tvergaard material, at lower stress triaxiality. A modification which uses the mean void radius (R~ = Rz = Rmean) in Rice-Tracy equations [25] to calculate the strong response, o'1 str°ng w a s tried by the authors. This modification significantly decreased the predictions at low stress triaxiality, while it keeps the predictions at high stress triaxiality almost unchanged, which reproduced the observation that symmetrical volume growth dominates the shape changing at large stress triaxiality [25, 26]. In order to combine the plastic limit-load model with the Gurson-Tvergaard model and treat the plastic limit-load model as a failure criterion, further modifications have been tested by the authors [24]:
• assume that the void grows spherically (Rz = R~:) • as originally suggested by Thomason [21], use the Gurson-Tvergaard model to char-
acterize the material as the weak but stable response. Furthermore, calculate the void and matrix geometry changes using the current strain and void volume fraction from the Gurson-Tvergaard model.
If the initial and current volumes of a unit void containing cell are 1 and V, according to the first modification suggested, the current void radius Rz, and current half intervoid distance
328 Z.L. Zhang and E. Niemi
Plas t i c 6
constraint factor
g(Rx/R) 4
10
0 0,2 0,4 0,6 0,8
R JR
Fig. 5. Plastic constraint factor g(R=/R) as a function of R=/R.
R, in the direction (in the present case, the x-axis in Fig. 1) perpendicular to the maximum principal stress are calculated
/ G = v , (5)
(6) R = ROeCX~
where ex is the current strain in the z-axis in Fig. 1, which is perpendicular to the maximum principal stress, and f is the current void volume fraction. It must be noted that, after the modification, the plastic constraint factor (1) is solely determined by Rx/R as
¢Y g (7)
which is shown in Fig. 5. Because A~ in (4) can also be determined by R=/R, the left-hand-side of (4) is only a function of R=/R. Then the plastic limit-load condition (4) after modification can be re-written as
9 A~ _ ~ (8)
Equation (8) represents the new criterion NSc we presented in this study.
2.4. PROCEDURES FOR USING NSC IN THE GURSON-TVERGAARD MODEL
With the above formulations, NSc can now be incorporated into the Gurson-Tvergaard mod- el:
• Calculate the maximum principal stress and the principal strains in the directions per- pendicular to the maximum principal stress from the output of the Gurson-Tvergaard model.
• Calculate the current void and matrix dimensions and plastic limit-load factor according to (5)-(7) from the output of the Gurson-Tvergaard model.
Gurson-Tvergaard dilational constitutive model 329
• Evaluate the plastic limit-load condition (8). • Once the condition (8) is satisfied, the void coalescence starts to occur and the void volume
fraction at this point is the critical void volume fraction fc in the Gurson-Tvergaard model.
It should be mentioned that the modifications suggested in Section 2.3 make the plastic limit-load model fully compatible with the Gurson-Tvergaard model. From (5), (6) and (8), it can be seen that the left hand side of (8) is a function of f, ex and V.
3. Gurson-Tvergaard material model
In contrast to developing a single void growth model as in Rice-Tracy [25], Gurson [6, 7] has proposed a yield function for a porous solid with a randomly distributed volume f rac t ionfof voids. This function was obtained based on an approximate analysis of spherical voids. In the present contextGurson's yield function (ql = ql = 1) is written
q2 ( 3q2am'~ ¢(tr, f , 0") = ~-~ + 2q l fcosh - 1 - (q l f ) 2 = O,
\ 2~ ) (9)
where constants ql 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, am and q are the mean normal and effective part of the average macroscopic Cauchy stress tr. The void volume fraction increase in the model is written
d f = dfgrowth + d.fnucleation.
The void growth law is described by
dfgrowth = (1 - f ) d e p : I,
(10)
(11)
which is an outcome of the plastic incompressibility of the matrix material. In (11), de p and I are the plastic strain increment tensor and the second-order unit tensor, respectively. The nucleation of new voids is usually taken to be either strain-controlled or stress-controlled. In the literature, a strain-controlled nucleation model is widely used [8, 9, 14, 19-20]
(12) dfnucleation = A dE p,
where ~P is the equivalent plastic strain in the matrix material, with parameter A chosen so that void nucleation follows a normal distribution as suggested by Chu and Needleman [27]. Although void nucleation is very important, our attention here is confined to void growth only. The effect of void nucleation on the critical void volume fraction will be discussed in the final part of the paper.
Furthermore, the equivalence of the overall rate of plastic work and that in the matrix material leads to
o" : de p = (1 - f ) # d ~ p, (13)
or equivalently
tr : d~ p d~ p _ (1 - f )~" (14)
330 Z.L. Zhang and E. Niemi
1
I ° F E M I X ~ - - FC
0,8 • 1 ~ " Fc(T=I)I ~ l ' Fc{T=3}
EffectiVestrain 0,6 ~, [. NSC [
o,4 ~ , , fo = 0 . 0 0 1 3
0,2
0 I I I I I
0,s 1 1,s a 2,s a a,s
Stress triaxiality
Fig. 6. Effective strain at failure as a function of stress triaxiality for the case withfo = 0.0013.
The flow rule is usually assumed to obey macroscopic normality, so that,
04, (15) d fj =
It is easy to see that the material loses load carrying capacity i f f reaches the limit 1/ql, because all the stress components have to vanish in order to satisfy (9). However, even if ql = 1.5 as suggested by Tvergaard [8, 9], the void volume fraction, f = 1/1.5, is still too large to be realistic in practice to simulate the final material failure. Therefore, it can be seen that although the Gurson-Tvergaard model displays the softening effect of the material, the model itself does not constitute a fracture criterion. Once failure (void coalescence) has been determined to appear according to a specific criterion, numerically it is preferred in the Gurson-Tvergaard model that the material separation is simulated gradually, rather than suddenly. Hence, a modification using the following f* in (16) to replace f i n (9) in order to take into account the void coalescence effect on final failure has been proposed by Tvergaard and Needleman [10]
f . = ( f for f ~< fc f~ + K ( f - f~) for f > f~"
(16)
Here, fc is the so-called critical void volume fraction at which voids coalesce, which in the present study was determined by the new failure criterion NSc, and K is a constant determined from the void volume fract ionfr at final failure of the material
I(=fg-fc fF - fc ' (17)
where f~* = 1 / ql. In this study, the calculation was stopped once (8) was satisfied. Therefore, (16) is not used.
The implementation of the new criterion (8) is simple. The only extra effort involved is the calculation of the maximum principal stress and principal strains and the evaluation of (8).
Gurson-Tvergaard dilational constitutive model 331
0 ,6
• FEM • Fc
• l" " " F c f r = l ) l
~ ' l - - " Fc t t=a> l 0.+ - ~ , I msc I
~ t
Effective "~,~, strain fo = 0 . 0 1 0 4
0,2 • • •
0 I • ~ 0,5 1 1,5 2 2,5 3 3,5
Stress triaxiality
Fig. 7. Effective strain at failure as a function of stress triaxiality for the case withfo = 0.0104.
0,06
0,04
fc
0,02
I ---0-- f0=0.0013 ---Q-- f0=0.0104
0 I I I I I
0,5 1 1,5 2 2,5 3
Stress triaxiality
Fig. 8. fc as a function of stress triaxiality.
3,5
4. Verification of the new criterion
In this section, we will compare the ductility as a function of stress triaxiality predicted by the Gurson-Tvergaard model using the new criterion and the finite element analysis results in [15]. Consistent with the results in [15], here ductility means the effective strain at failure. The constitutive equations of the Gurson-Tvergaard model were solved for various proportional stressing histories using a Euler forward integration scheme with very small increment steps. The material properties are the same as [15]: E/tro = 500, v = 0.3. Both hardening and non-hardening materials were considered in [15], however, we only compare the results of non-hardening material here. Two pairs of Tvergaard's q parameters in (9): qt = 1.5, q2 = 1.0 and ql = 1.25, q2 = 1.0 were tested by Koplik and Needleman [15] and they found that the last pair of parameters gave better results. In this study ql = 1.25, q2 = 1.0 is used in (9).
It should be mentioned that in predictions using NSc, the net area fraction of the intervoid matrix and the volume of the void containing cell model are calculated according to the cell
332 Z.L. Zhang and E. Niemi
model in Fig. 2 as An = 1 - (Rx/R) 2 and V = 27rR2B. The comparison for the case with small initial void volume fraction, f0 = 0.0013 is shown in Fig. 6. It is surprising to find that a very good fit to the finite element results [15] has been given by the predictions using the new criterion. In Fig. 6, the predictions of Fc using constant valuesfc = 0.03 proposed in [15] for the case f0 = 0.0013, and two f* in [15] which correspond to the cases with stress triaxiality T -- 1 and T = 3 are also presented. The three predictions are indicated in the legend as Fc, Fc(T -- 1) and Fc(T -- 3), respectively, f* of the case with T = 2 is very close to fc, so its prediction is not presented. Because the values offc and f~* are near each other in this small initial void volume fraction case, no large difference in the predictions has been observed. From Fig. 6, it can also be concluded that, for a small initial void volume fraction case, predictions using both NSc and Fc can well correlate the effective strain at failure as a function of stress triaxiality.
Figure 7 shows the same comparison as in Fig. 6 but for a large initial void volume fraction f0 = 0.0104. In this case, theft* corresponding to T -- 2 is also quite close tof~ = 0.055 proposed in [15], so its prediction is not shown. It can be seen from Fig. 7 that there is a reasonably good agreement between the predictions of NSc and the finite element results [15]. Furthermore, Fig. 7 shows that the predictions of NSc provide a lower bound to the finite element results, while the predictions of Fc using fc -- 0.055 provide an upper bound to the finite element results. A large scatter between the predictions of Fc and the finite element results [15] can be seen in Fig. 7. Figure 7 also indicates the difficulty of choosing f~ from theft*. A general trend can be observed from Fig. 7 that the predictions of Fc usually overestimate the ductility at high stress triaxiality for a case with large initial void volume fraction. Compared with Fig. 6, it can be found that this trend diminishes when the initial void volume fraction becomes small.
The critical void volume fractions obtained in the new criterion versus stress triaxiality are shown in Fig. 8 for the two initial void volume fractions. Obviously fc is not exactly a constant but decreases as the stress triaxiality increases. From Figs. 6 and 8, it can be seen that, only for the small f0 case, as a first approximation, fc can be taken as a constant.
5. Discussion
5.1. CONDITIONS FOR THE USE OF NSC
In this study, a failure criterion, NSc, was introduced and verified with the finite element results of non-hardening matrix material [15]. Rather good agreement has been observed. However, it should be mentioned that various assumptions and approximations have been made in obtaining the plastic constraint factor (1). Thomason [23] has pointed out that the plastic constraint factor (1) is only expected to be accurate when the void volume fraction is less than 0.2. Due to the assumptions made in Fig. 1, it can also be expected that NSc is accurate only when the loading conditions are similar to the assumptions shown in Fig. 1. In fact, the order of 0.2 does not lead to any serious limitation in the application of NSc, because the critical void volume fraction is usually less than or equal to 0.15.
It should be noted that the Gurson-Tvergaard model was originally based on rigid non- hardening matrix material and was extended to hardening matrix material by regarding # in (9) as a measure of the effective (in an appropriate average sense) flow stress of the matrix material in the current state [11]. Like the Gurson-Tvergaard model, the Sc criterion could be readily applied, in principal, to incorporate the effect of work-hardening of the matrix
Gurson-Tvergaard dilational constitutive model 333
material, as argued by Thomason [22]. However, how sensitive the NSc is to work-hardening and especially the strain-rate sensitivity of NSc remains to be seen in our future work.
5.2. f~ ANDf~"
It is essential to point out the difference between the critical void volume fraction in the Gurson-Tvergaard modelfc and the real critical void volume fraction observed either exper- imentally [17] or numerically [15] fc*. Unlike the yield stress and other material constants, fc is an indirect material parameter which depends on the mathematical form of the material constitutive model. Studies [15, 17] show that fc* is not a material constant and depends strongly on stress triaxiality, fc is usually different tof~* because of the inability of the current material model to describe the material behaviour very accurately. If a more accurate and reasonable model appears, f~ should be closer tof~*. So, strictly speaking, f~ is not a material constant. Figures 6 and 8 show that according to the results of NSc, for the small initial void volume fraction case, as a first approximation, f~ can be taken as a constant. In such a case, fc should not be determined experimentally, but numerically.
5.3. THE POTENTIAL ADVANTAGES OF NSc
In the Introduction, we mentioned two weaknesses in Fc:
(1) it is difficult to determinefc even when void nucleation is not present; (2) no sound theory or method is available for the choice of the void nucleation model and
parameter(s).
For an assumed nucleation model, a small variation in the selected void nucleation param- eters which produces very little difference in the low stress triaxiality case, could possibly yield very large different behaviours in a high stress triaxiality case. It is interesting to note that these weaknesses could be partially overcome by the use of NSc, if an assumed nucleation model can be used. Similarly to the Gurson-Tvergaard model, with no extra effort, the void nucleation process can be taken into account in NSc. By NSc, the failure is a natural result of the competition between the stable plastic flow mechanism and unstable void coalescence mechanism, andfc is determined once the unstable response prevails. Therefore, if the initial void volume fraction is fixed, which usually can be deduced from the initial void volume fraction of inclusions, the primary void nucleation parameter(s) can be calibrated based on an assumed nucleation model, from experimental results, for example, notched tensile spec- imens. Here we prefer a notched round tensile specimen, rather than a smooth specimen because the stress triaxiality in a notched specimen is higher than in the smooth specimen and is more suitable for the application of the Gurson-Tvergaard model. Furthermore, the assumed nucleation model itself could be studied and its validity could be assessed by NSc.
In summary, the significant difference between Fc and NSc is that, in Fc, the void coa- lescence parameterfc usually has to be fitted or pre-determined based on the initial material state and an assumed void nucleation model and parameter(s); in contrast, the void nucleation parameter(s), not the void coalescence parameterfc, can be fitted in NSc. It is thus obvious that the proposed criterion NSc is not only accurate in the cases assessed but also versatile.
NSc has recently been implemented into the commercial finite element program ABAQUS [28] via the user material subroutine UMAT with the Gurson-Tvergaard model using the generalized mid-point algorithms developed by the authors [29]. Application of NSc to the
334 Z.L. Zhang and E. Niemi
simulation of a notched axisymmetric tensile specimen [30] and more complicated welded T-joints [31] have obtained many fruitful results.
Acknowledgements
We wish to thank Professor Alan Needleman, Brown University and Dr. Dong-Zhi Sun, FhG/IWM, Germany for their valuable comments and suggestions on the initial manuscript. We also wish to thank Mr. Andreas H6nig, FhG/IWM, for making us aware of a small inconsistency in our initial manuscript. The first author would like to acknowledge the financial support from the Ministry of Education of Finland.
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