A SENSITIVITY ANALYSIS OF MATERIAL PARAMETERS FOR THE GURSON CONSTITUTIVE MODEL

Transcription

1 Fatigue Frat. Engng Muter. Struct. Vol. 19, No. 5, pp , 1996 Printed in Great Britain. All rights reserved /96 $ Copyright Fatigue t Fracture of Engineering Materials & Structures Ltd A SENSITIVITY ANALYSIS OF MATERIAL PARAMETERS FOR THE GURSON CONSTITUTIVE MODEL Z. L. ZHANG SINTEF Materials Technology, N-7034 Trondheim, Norway Received injnal form 14 November 1995 Abstract-Following a convincing demonstration of the prediction power of the Gurson model for ductile fracture, it is now required to select realistic material parameters for practical applications of the model. In this paper, using studies on a smooth tensile specimen, a notched tensile specimen, a centre-cracked tensile panel and an analytical cell model, a sensitivity analysis of the material parameters is performed that includes the initial void volume fraction of the primary inclusions and the void volume fraction of the secondary inclusions when fitting the critical void volume fraction. Voids that nucleated from primary and secondary inclusions have been considered separately. It has been found that in either case the selection of material parameters for the finite element analyses is not unique, and the most significant parameter for the predictions is the nucleation burst strain. Some general conclusions concerning the selection of material parameters for the Gurson model have also been made. Keywords-Gurson model; Ductile damage parameters; Non-uniqueness problem INTRODUCTION Ductile tearing is normally controlled by microvoid nucleation, growth, and coalescence mechanisms. Micro-mechanical model based approaches have shown promise when tackling the ductile fracture problem. Within these approaches, a model originally introduced by Gurson [ 1,2] and later modified by Tvergaard [3,4] and Tvergaard and Needleman [S] is attractive in that it is endowed with a yield function, a flow law and rules for void nucleation and growth. In recent years, much pioneer work has demonstrated that the Gurson model can be applied for the prediction of plastic localization and ductile fracture [ Besides the distinct material characteristic length dependent problem, there are two important aspects in applying the Gurson model. One is how to implement accurately and efficiently the model into a computer, the other one is how to interpret the material parameters for the Gurson model. Until now, how to select the material parameters for the Gurson model in a practical application has been an open question, and in practice the determination of the material parameters is often made arbitrarily. Efforts have been made recently for the practical application of the Gurson model by developing a special numerical scheme for implementing the Gurson model in ABAQUS via a user material subroutine, and developing a new failure criterion for the Gurson model [ The new failure criterion, which is based on Thomason s void coalescence mechanism [14-161, is novel in that it could be used for the establishment of damage parameters and also for reducing the non-uniqueness of damage parameters [ 9,171. With the convincing demonstration of its predictive power and the advance of the numerical algorithms for implementing the model, the time has perhaps come to study the selection of realistic material parameters for the Gurson model, and to relate the parameters for the Gurson model to the toughness of homogenous as well as inhomogenous materials such as weldments. In this study, the problems of selecting material parameters for the Gurson model are outlined. Two kinds of void nucleation have been considered. In the first case, 561

2 562 z. L. ZHANG the voids were nucleated at the beginning of plastic deformation from the primary inclusions. In the second case, the voids were assumed to nucleate from the secondary inclusions. It has been found that the selection of the initial void volume fraction or the volume fraction of void nucleating particles is not unique in parameter fitting when using finite element models. THE GURSON THEORY In order to maintain consistency, the Gurson model is introduced briefly here. The Gurson model was based on the observation that the nucleation and growth of voids in a ductile metal may be described macroscopically by extending the classical plasticity theory to cover the effects of plastic dilantancy and pressure sensitivities of plastic flow. By considering the limit yield surface of a spherical cell containing a spherical void where the matrix material was perfectly plastic with a Mises yield condition, Gurson [ 1,2] proposed the following yielding functions: where the constants q1 and q2 were introduced by Tvergaard [3,4] to bring predictions of the model into closer agreement with full numerical analyses of a periodic array of voids, o;, and 4 are the mean normal and effective part of the average macroscopic Cauchy stress a, 0 is the yield stress of the matrix material, and f is the void volume fraction. There are two parts which contribute to the increase of the void volume fraction, one is the growth of the existing voids and the other is the nucleation of new voids during the loading, The void growth is described by, df=dfbrowth + dfnuoleation. (2) dfgrowh = ( 1 -f) dp : I (3) which is an outcome of plastic incompressibility of the matrix material. In Eq. (3), d.9 and I are the plastic strain increment tensor and the second order unit tensor, respectively. The large particles or inclusions that nucleate voids at relatively small strains could be taken approximately as the initial voids, while the small particles or inclusions that nucleate by a plastic strain or stress controlled mechanism could be taken as the intermediately nucleated voids. A strain-controlled normal distribution model for the intermediately nucleated voids, introduced by Chu and Needleman [ 181, is widely used in the literature, dfnucleation = A (4) A = exp [ -5 sn $71 1 (s,)1] 9-&n where fn is the volume fraction of void nucleating particles, E, is the mean void nucleation burst strain and Sn is the corresponding standard deviation. Unfortunately very limited knowledge concerning the selection of the void nucleation parameters is available and almost no work has yet been directed to the micro-mechanics of the nucleation process itselfin plastically deforming solids [ 191. What makes the nucleation parameters important is that the critical parameterf, (see Eq. (6)) is very dependent on the initial parameters one chooses. When deriving the Gurson model, it is assumed that the voids do not interact even at the later stage of deformation, hence the Gurson model is not able to predict void coalescence realistically. (5)

3 Sensitivity analysis of material parameters for the Gurson constitutive model 563 An extra criterion should be incorporated into the model to manage the void coalescence issue. Practically, the treatment of the void coalescence aspect relates to the determination of the critical void volume fraction. A function has been introduced by Tvergaard and Needleman [S] to model the post-coalescence behaviour (rapid decay of the load carrying capacity) in the Gurson model: for f<f, Here,f, is the so-called critical void volume fraction at which voids coalesce,f, is the void volume fraction at final failure of the material, and f,* = l/ql. It should be mentioned that the absolute value of f, does not play a significant role in the numerical modelling once f, is determined. CLASSIFICATION OF THE PARAMETERS IN THE GURSON MODEL Besides the matrix yielding stress 5, the parameters involved in the Gurson model can be classified into three categories. The first one can be called constitutive parameters (ql and q2) which decide the constitutive equation form and they can be taken as constant for different materials. Numerical investigations by Koplik and Needleman [20] suggest that q2 = 1, and q1 = are good choices. The second category of parameters can be defined as the initial material parameters, for example, the initial void volume fraction (fo) and the void nucleation parameters (in, E, et d.). The third one is the critical parameter, i.e. the critical void volume fraction at void coalescence (f,). The second and third categories of parameters are material specific. In a common way, the critical parameter is usually taken as a material constant (independent of stress triaxiality) and can be fitted from simple tensile specimens based on the initial parameters assumed. However, whether the critical material parameter is independent of stress triaxiality is questionable. Recent studies [13,20,21] indicate that f, does not depend significantly on the triaxiality if the initial void volume fraction is small and there are no secondary voids. Furthermore, the critical parameter is a function of the initial parameters chosen for the material. The dependency on the initial parameters is even more pronounced when the void nucleation of secondary particles is taken into account. PARAMETER FI G BY FINITE ELEMENT ANALYSES In a practical application, the seven parameters ql, q2,fo,f,, %, S, andf, in the Gurson model are usually selected beforehand and the parameter f, is fitted from numerical simulations. As defined previously, the Tvergaard parameters 41 and q2 are constitutive parameters which are usually taken as being independent of the material applied. Since the introduction of the normal distribution model (Eq. (5)) by Chu and Needleman [18], the values of %=0.3, S, =0.1 have been used for a wide range of materials, even though the real void nucleation distribution could be very different. The value offf has no significant effect on the parameter fitting as long as it is kept the same in the fitting and can be usually taken as being between 0.1 and 0.2 [9,17]. The case dependent parameters are the initial void volume fraction fo and the void nucleation parameter f,. In principle, fo and f, could be determined from metallurgical examinations [6,7]. However, this is difficult in reality and recommendations have yet to be seen. The difficulty here is that the metallurgically determined parameter cannot directly be used in the Gurson model because of the idealizations in the model. This problem has partly been discussed in Ref. [13]. It has been

4 564 z. L. ZHANG suggested [9] that for steel, fo could be taken from the inclusion fraction of MnS. After these initial parameters have been selected, then a fitting procedure is applied for the determination of the critical void volume fraction f,. By comparing the numerical simulation and experimental results from a simple tensile specimen, for example a smooth tensile specimen, the void volume fraction which best fits the load drop point is signalled out and taken as the critical void volume fractionf, [ 6,7]. These selected initial parameters and fitted critical parameters are usually verified by different notched tensile specimens and then used in other applications. In the following, all the calculations were carried out using the user material subroutine UMAT for ABAQUS developed by the author [9,10], and 8-node parabolic elements with reduced integration points have been used. The values of q1 = 1.5, q2 = 1.0 and ff = 0.2 are used in all the calculations. Fittingf, from a smooth tensile specimen A European round-robin on micro-mechanical models has recently been completed [ 221. One of the purposes of the round-robin was to characterize the material and identify the critical damage parameters for ductile tearing at room temperature. In the round-robin, an initial void volume fraction of was assumed for the ductile tearing. No void nucleation of secondary particles was considered for the sake of comparison. By comparing the numerical simulation with the experimental results, the critical void volume fractionf, was fitted. It has been shown by the author that f,= is a good fit to the experimental results. In the following we found that several other sets of parameters can predict almost identical results for the same experimental curve. The stress-strain curve used in the round robin was used in the present calculations. Both cases with and without void nucleation of the secondary inclusions have been tested. Figure l(a) shows the finite element mesh used for the smooth axisymmetric tensile specimen. The diameter of the specimen is 6 mm. A small imperfection of times the diameter was applied at the middle of the specimen. First the initial void volume fraction was decreased to half of the one used in the round robin, and then we fit a critical void volume fractionf,=0.005, which is nearly one third of that corresponding to fo= Then, the initial void volume fraction was further decreased to fo = , which is very close to the volume fraction of the MnS inclusions in a typical structural steel [9]. The critical void volume fraction fitted wasf,= Next, we assume that no initial voids are present and that all the voids nucleate from secondary 20 Fig. 1. Finite element meshes for (a) the smooth tensile specimen, (b) the notched tensile specimen and (c) the centre cracked tensile panel. Fig. 2. Parameter fitting for the smooth specimen.

5 Sensitivity analysis of material parameters for the Gurson constitutive model 565 particles during the loading process. The nucleation was strain-controlled by Eq. (5), with fixed values E, =0.3 and S, = 0.1. Three values offn, , and 0.003, were tried. The corresponding fitted values of fc are , and , respectively. Table 1 documents all the sets of parameters tried and the fittedf,. It is interesting to note that if the initial void volume fraction or the volume fraction of nucleating particles is doubled, the fittedf, nearly tripled. Table 1 also shows that for the same fitted critical void volume fraction, the required value off, is larger than the value off,. This is because, for the parameter sets FN1-FN3, the effect of voids was delayed while in the sets F01-FO3 the void effect was present from the start of plastic deformation. Figure 2 presents the numerical results. From Fig. 2, we can observe that for all the six parameter sets tried, the numerical results are almost identical, which implies that from a continuum mechanics point of view, the six parameter sets are all valid for the smooth tensile specimen. This means that the selection of material parameters for the smooth tensile specimen is not unique. Figure 2 indicates that there is no difference if the voids are nucleated from primary inclusions or from secondary inclusions. It is also not important on how large are the values of fo or f, as long as they are small in the smooth tensile specimen. It can be expected that more sets of parameters can be found in this smooth tensile specimen case to give similiar predictions. Application of the fitted critical pararneterf, The six material parameter sets (Table 1) have been applied in a notched axisymmetric tensile specimen where the stress triaxiality is higher than in the smooth tensile specimen. The specimen AX2 and the corresponding finite element mesh used in [9,17] are used in this study. Fig. l(b) shows the finite element mesh. The minimum diameter of the specimen is 4.0mm with a notch radius of mm. Figure 3(a) shows the numerical results of the six sets of parameters. First, Fig. 3(a) indicates that there is a great difference between the predictions by parameter sets F01-FO3 in which the voids were nucleated at the beginning of plastic deformation from primary inclusions, Table 1. Material parameter sets in the finite element models Parameter sets FO I F02 F03 FN I FN2 FN3 Critical parameter Initial parameters (E,=O 3. Sn=O.l) f,, 1; f< 0.OoOs I (a) I s 6 4 f-'. FN3 0' I 0 ' I ,s Diameter reduction [mm] CMOD [mm] Fig. 3. Numerical results for (a) the notched specimen and (b) the centre-cracked tensile panel.

6 566 z. L. ZHANG Crack growth [mm] Fig. 4. CMOD vs. crack growth for the centre-cracked panel. and the predictions by sets FN1-FN3 in which the voids were nucleated during the loading from secondary inclusions. Secondly, for the parameter sets F01-FO3 or for sets FN1-FN3 (where the difference is the value off,, or f,) the results are almost identical, as we similarly observed in the smooth tensile specimen. The numerical results in Fig. 3(a) imply that even in the notched tensile specimen the selection offo orfn is not unique for the finite element analyses. Furthermore, a centre cracked plane-strain panel under tension has been analyzed by using the above sets of parameters. The crack length to plate width ratio is 0.5. Figure l(c) shows the finite element mesh used in the calculations. The mesh has a minimum mesh size of mm at the crack tip. The width of the panel is 10 mm and the half length of the panel is 60 mm. Only one quarter of the panel was modelled because of symmetry. Regular elements with no singularity were used at the crack tip. Figure 3(b) presents the numerical results of tensile force vs. crack mouth opening displacement. The results of CMOD versus numerical crack growth are shown in Fig. 4. The numerical crack was defined as the part in front of the crack where the load carrying capacity has disappeared. Very similar results to those observed in the notched tensile specimen can be found for this real crack case. The difference between the ways of treating void nucleation is distinct. Once the void nucleation distribution pattern is determined, the numerical prediction to a great extent is not sensitive to the value of the initial void volume fraction or the volume fraction of nucleating particles. In the case where the voids were solely nucleated from primary inclusions, the CMOD values rise rapidly in the initial stage of growth, then more slowly as crack growth continues. Because of the later involvement of void effects in the case where voids were nucleated from the secondary inclusions, a fully developed plastic zone is formed in front of the crack tip resulting in a large CMOD value as the crack advances. PARAMETER F I"G BY A CELL MODEL In parameter fitting using the finite element analyses, it has been demonstrated that numerical predictions in the three specimens are not sensitive to the choice of the value of fo or off,. It is interesting to check whether it is the same when applied to a unit cell with constant stress triaxiality. In the following, the Gurson model is applied to an analytical cell model with constant stress triaxiality [Ill]. In the cell model, the Gurson equations can be integrated in a semi-analytical way. The integration equations presented in Refs. [9,11] have been used here. All six sets of parameters have been applied to an arbitrary equivalent plastic strain of 1.0 at a stress triaxiality

7 Sensitivity analysis of material parameters for the Gurson constitutive model 561 Table 2. Parameter sets in the cell model Parameter set No. F FN FNZ FN , 5, (b) v _.._. ~03 FN1 ---FN3 0 0, ,75 1 Equivalent plastic strain E f l - s FO FO3 FNI ---M ,2 03 Eauhraient Dlastlc strain Fig. 5. Predictions by the six sets of parameters for the cell model at stress triaxiality 0.58 (a), and 2( b). of 0.58, which approximately represents the smooth axisymmetric tension case. We can then calculate the critical void volume fractions, according to the initial void volume fraction (fo) or the nucleation parameter f,. The parameter sets determined for the cell model case are listed in Table 2. Comparing Table 1 with Table 2, it is interesting to note that in the cell model, when the initial void volume fraction or the volume fraction of nucleating particles doubles, the critical void volume fraction also doubles, rather than nearly triples as found in Table 1. The six sets of parameters have also been applied to a higher stress triaxiality case which represents a typical crack tip condition. Figures 5(a) and 5(b) show the predictions of the axial stress-equivalent plastic strain relation of the six sets of parameters. As can be expected, the results in Fig. 5(a) show that for the low stress triaxiality case, the six parameter sets give nearly identical predictions. However, for the high stress triaxiality case, Fig. 5( b), a distinct difference can be found in the predictions by the parameter sets. For either the case where voids nucleated from the primary inclusions or the case where voids nucleated from secondary inclusions, large values offo or f, will decrease the maximum stress significantly. DISCUSSION Primary and secondary inclusions In this study, both the analytical cell-model study and the finite element analyses have demonstrated that at low stress triaxiality, it may not be necessary to distinguish the primary and secondary inclusions in practical applications. From a continuum mechanics point of view, identical results can be obtained. However, when the stress triaxiality is high, where the voids grow significantly faster than at low stress triaxiality, the way of treating the inclusions is important. Assuming voids from secondary inclusions will delay the material failure process and increase the crack resistance capacity. In a previous study [ 171, it has been found that a small change of the

8 568 z. L. ZHANG nucleation burst strain will result in a significant difference in the stress-strain relation in a high stress triaxiality case. From these results, it can be seen that correct modelling of void nucleation of secondary inclusions, particularly, the burst strain is very important in the application of the Gurson model. Cell model and finite element model It has been shown in the cell model study that, when the stress triaxiality is low, the effect of voids is not distinct. However, the effect of voids can be clearly seen in the cell model with a constant high stress triaxiality. This implies that the material parameters in high stress triaxiality cases can be uniquely determined. In a real cracked specimen, where the stress triaxiality is only high at the local crack tip, the selection of material parameters is not unique according to the usual way of parameter fitting. If a nucleation distribution pattern is fixed, the values of fo or f, are not important, as long as they are kept small. The reason for this kind of behaviour is that the stress triaxiality in the specimen is not homogenous. Before void coalescence, the contribution of the local large void effects in a limited number of elements to the global behaviour is not significant. Non-uniqueness of material parameters Recently Li et al. [23] have pointed out that the possibility of non-uniqueness of damage parameters might be a fundamental feature of all damage theories involving tunable internal damage parameters. In the above, the same non-uniqueness of material parameters for the Gurson model has been demonstrated. The results indicate that, without a failure mechanism, the nonuniqueness problem can not be solved by the ordinary parameter fitting technique. The good thing is that it is not important to have the exact value of fo or f,, once the nucleation pattern is settled. Because of the non-uniqueness aspect, there is no direct link between the micromechanical parameters and the material failure behaviour. It is also not possible to judge or classify the material behaviour according to the fitted damage parameters. The above problem can be, to a large extent, solved by using the new failure criterion which is based on the physical void coalescence mechanism suggested by Thomason [ 14-16]. The nonuniqueness problem is due to the fact that there is no physical void coalescence mechanism in the Gurson model, which means that failure is just taken into consideration by the fitted critical void volume fraction. It has been realized [9,13,17] that if a physical mechanism is incorporated into the Gurson model, the material failure is solely determined by the assigned initial material Dlameter reduction [mm] Fig. 6. Effect of the initial parameter fo on failure by the new failure criterion. No value off, was fitted.

9 Sensitivity analysis of material parameters for the Gurson constitutive model 569 parameters. In other words, there is a one-to-one relation between the initial parameters and the material failure. If different initial material parameters are assumed, different failure results will be obtained. By comparing the experimental results, the exact material parameters can be obtained, [ 241. Figure 6 shows an example in which different initial void volume fractions, according to the new failure criterion [9,13], will give different results. Two values off,,, i.e and have been tried. Fig. 6 shows that the failure results are very different. By the new failure criterion, the parameters to be fitted are the initial parameters. The critical parameter is a natural result of the failure process. However, it should be mentioned that if there are more than one unknown initial parameters some parameters should be pre-determined. CONCLUSIONS The following conclusions can be drawn, noting that the characteristic length-dependent problem in damage mechanics is not the subject of this study. (1) Great attention should be paid to the selection of material parameters in the application of the Gurson model. When the initial void volume fraction and the volume fraction of nucleating particles are small, these values are not unique once a nucleation distribution pattern is selected or determined. For the case where the nucleation of secondary inclusions is involved, the most significant parameter is the burst strain. (2) The non-uniqueness problem of the values of the material parameters for the Gurson model can be improved by using a physical coalescence mechanism. (3) The calibration of the material parameters should be in a high stress triaxiality case. Different ways of treating voids in the Gurson model could yield almost identical results in a low stress triaxiality case, but at a high stress triaxiality, the simulated behaviours are significantly different. Under a high stress triaxiality case, the effect of the burst of void nucleation becomes distinct and therefore by adjusting the void nucleation burst parameters for the high stress triaxiality case, a more realistic modelling of void nucleation can be obtained. (4) More research effort is required to understand the micro-mechanics of void nucleation. Acknowledgements-The work reported here is one part of the project Application of Damage Mechanics to Aluminum Constructions at SINTEF. Financial support of the Royal Norwegian Research Council (NFR) is greatly appreciated. REFERENCES 1. A. L. Gurson (1975) Plastic flow and fracture behavior of ductile materials incorporating void nucleation, growth, and interaction. Ph.D. Dissertation, Brown University. 2. A. L. Gurson (1977) Continuum theory of ductile rupture by void nucleation and growth: part I-yield criteria and flow rules for porous ductile media. J. Engng Mater. Tech. 99, V. Tvergaard (1981) Influence of voids on shear band instabilities under plane strain conditions. Int. J. Fract. 17, V. Tvergaard (1982) On localization in ductile materials containing spherical voids. Znt. J. of Fract. 18, V. Tvergaard and A. Needleman (1984) Analysis of the cup-cone fracture in a round tensile bar. Acta Metall. 32, D. Z. Sun, D. Siegele, B. Voss and W. Schmitt (1989) Application of local damage models to the numerical analysis of ductile rupture. Fatigue Fract. Engng Mater. Struct. 12, D.-Z. Sun, B. Voss and W. Schmitt (1991) Numerical prediction of ductile fracture resistance behavior based on micromechanical models. In: Defect Assessment in Components-Fundamental and Applications. ESIS/EGF9 (Edited by J. G. Blauel and K.-H. Schwalbe) Mechanical Engineering Publications, London. pp D. Klingbeil, G. Kunecke and J. Schicker (1993) On the application of Gursonk model to various fracture mechanics specimens. Report BAM-1.31, BAM, Berlin.

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