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Prediction of Ductile Fractures in Metal-Forming Processes: an Approach Based on the Damage Mechanics
Prediction of Ductile Fractures in Metal-Forming Processes: an Approach Based on the Damage Mechanics N. Alberti (l), A. Barcellona, L. Cannizzaro (21,F. Micari (21, University of Palermo, Italy Received on January 15,1994
The Authors propose a new approach for the prediction of ductile fractures in bulk metal forming processes: the approach is based on a numerical analysis able to take into account damage occurrence and evolution in constitutive equations. The model supplies the distribution of the void volume fraction in the workpiece during the deformation path: consequently, the comparison to a critical value, determined by means of a simple tension test, allows to predict the growth of defects. The proposed approach has been applied to the drawing process: the numerical results have been compared with a set of experimental tests showing a good predictive capability of the model. Key words: Failure, Forging, Damage. 1. INTRODUCTION During bulk forming processes failure phenomena are often encountered, mainly consisting in internal or surface cracking. The fracture mechanism which limits the ductility of a structural alloy is generally explained taking into account the nucleation and the growth of a large number of small voids. The voids nucleate mainly at second phase particles by decohesion of the particle-matrix interface and grow due to the presence of a tensile hydrostatic stress; as a consequence the ligaments between them thin down until fracture occurs by coalescence [7,18]. In the recent years several approaches have been proposed in order to predict the growth of ductile fractures; among them two main groups can be individuated: the former is based on the numerical analysis of the forming process carried out employing the yield condition and the associated flow rule of the sound, "undamaged", material; the stress and strain fields calculated at each step of the deformation path are introduced in a proper ductile fiacture criterion, able to suggest if and where fracture occurs. Such an approach has been applied to several forming processes, and a large number of ductile fracture criteria has been proposed, among which those proposed by Oyane [1'7], Osakada [16], Ayada [5] and Cockrofi and Latham have shown to be the most suitable. On the other hand other researchers have proposed to take into account damage occurrence and its evolution in the constitutive equations used for the analysis of metal forming processes. Thus this approach requires a proper definition of damage and the development of an accurate damage model able to describe the influence of damage evolution on the plastic flow properties of the material [8,9]; mainly the presence of voids gives rise to plastic volume variation and to the dependence of yielding on the hydrostatic stress. Moreover the model should be able to describe the interaction between voids and the rapid decay followed by the complete loss of load carrying capability occurring in the material when the voids coalesce. The first set of constitutive equations including ductile damage was suggested by Gurson [ll], who proposed a yield criterion depending on the first invariant of the stress tensor, on the second invariant of the deviatoric stress tensor and on a scalar parameter f , (void volume fraction). The void volume fraction is defined as the ratio between the volume of the cavities in an aggregate and the volume of the aggregate. The Gurson model has been subsequently modified and developed by Tvergaard [I81 and by Tvergaard and Needleman [ 141, which have introduced Annals of the ClRP Vol. 43/1/1994
some additional parameters in order to model the decay of the load carrying capability. In the paper the authors apply a ductile damage model to the prediction of central bursting insurgence in the drawing process; as it is well known, central bursts are internal defects which cause very serious problems to the quality control of the products, since it is impossible to detect them by means of a simple surface inspection of the workpiece [l]. In the past several experimental and theoretical studies [2,3,4,5,13,15] have been performed in order to determine the influence of the main operating parameters (reduction in area, die cone angle, friction at the die-workpiece interface and material properties) on the occurrence of central bursts. However these models and empirical rules concern materials with assigned properties and processes for which the lubricating conditions are known and constant. Viceversa it is well known that specimens belonging to the same production batch can be characterised by different structural inhomogeneities, second phase particles distributions, inclusion and void contents, all factors which affect the workability conditions. In this paper the prediction of central bursts occurrence in drawing is carried out by means of a ductile damage model: such a model in fact allows to describe in a very accurate way the plastic flow of the material, including damage effects. The numerical analysis has been carried out employing the yield criteria proposed by Gurson, Tvergaard and Needleman and finally by Shima and Oyane. A simple tension test has been firstly simulated, and the comparison with the experimental evidence has allowed to calculate the critical value of the void volume fraction for the considered material, corresponding to the point at which the coalescence of voids occurs and the material undergoes a rapid decay of its stress-carrying capability. Subsequently the numerical analysis has been applied to several drawing processes, characterised by different reductions in area and die cone angles: for each of them the maximum achieved value of the void volume fraction has been calculated, and, by the comparison with the critical value, it has been evaluated if the coalescence of voids, i.e. the insurgence of the ductile fracture, occurs or not. The numerical results have shown a good agreement with the experimental ones, thus confirming the validity of the proposed approach.
2. THE DUCTILE DAMAGE MODELS As previously reported, the numerical analysis of the drawing process has been carried out employing three different damage models, namely those based on the yield criteria proposed by Gurson [ 111, Tvergaard and Needleman
207
[14], and finally by Shima and Oyane [17]; as it is known the last one has been largely employed in the recent years for the analysis of powder materials forming. Starting from a rigid-perfectly plastic upper bound solution for spherically symmetric deformations around a single spherical void, Gurson proposed a yield function for a solid with a randomly distributed volume fraction of voids f in the form:
where
is the macroscopic equivalent stress, with C Y ~macroscopic stress deviator, and 5 is the yield stress of the matrix (voidfiee) material. Eq.1 clearly shows the effect of the mean stress on the plastic flow when the void volume fraction is non-zero, while for f-0 the Gurson criterion reduces to the von Mises one. The components of the macroscopic plastic strain rate vector can be calculated applying the normality rule to the yield criterion above written: in fact as shown by Bishop and Hill [6] the validity of the normality rule for the matrix material implies the validity of a macroscopic normality rule. Following these considerations the components of the macroscopic , plastic . strain rate vector can be expressed as:
(3) where the parameter ican be calculated if monoaxial stress conditions are assumed. Subsequently the Gurson yield criterion has been modified by Tvergaard [IS] and by Tvergaard and Needleman [141, which introduced other parameters obtaining the following expression: cTz
@=3+2f'q,cosh (4) E* In the above expression the parameter q, allows to consider the interactions between neighbouring voids: Tvergaard in fact obtained the above criterion analysing the macroscopic behaviour of a doubly periodic array of voids using a model which takes into account the nonuniform stress field around each void. The value of q1 was assumed equal to 1.5 by the same Tvergaard. On the other hand the parameter f( f ) was introduced in substitution off in order to describe in a more accurate way the rapid decrease of stresscarrying capability of the material associated to the coalescence of voids: in fact f(0 is defined as: If for fS fc
where f, =l/q,, fc is a critical value of the void volume fraction at which the material stress-carrying capability starts to decay very quickly (easily discernible in a tensile test curve) and finally fF is the void volume fraction value corresponding to the complete loss of stress-carrying capability. Again the constitutive equations associated to the Tvergaard and Needleman yield criterion can be determined by means of the normality rule. Both the Gurson and the Tvergaard and Needleman damage models have been implemented in a finite element code based on an Updated Lagrangian formulation; an
208
incremental procedure has been employed in order to follow the transient process from the workpiece input phase to the achievement of steady state conditions. In fact to evaluate the possibility of a ductile fracture insurgence, it has been necessary to follow the whole deformation path for each point of the material. At the end of each step of the deformation process the values of the void volume fractions calculated at the integration points within each element should be updated: in fact both the existing voids can grow, depending on the stress conditions, and new voids can nucleate. The increasing rate of the void volume fraction in fact can be considered partly due to the growth of the existing voids and partly . . to the nucleation of new voids [7]: (6) f = fm+ fnucieation AS far as the first aspect is concerned the rate of change of the void volume fraction is related to the volumetric strain rate 4, by the relation: fPWh= ( l - f ) + ? (7) On the other hand, several models have been proposed in order to evaluate the nucleation of new voids; in these models the void nucleation rate is assumed as depending on the matrix equivalent plastic strain rate and on the rate of increase of the hydrostatic stress, following the expression: fnucleation = AE, + B(& + 6 , ) (8) where A and B are material dependent constants. In particular Chu and Needleman suggested to calculate A and B assuming that the void nucleation follows a normal distribution and have distinguished two different cases: the case of plastic strain controlled nucleation (B=O) and the case of stress controlled nucleation (A=O). Moreover in the proposed expressions for A and B [ 141, Chu and Needleman have introduced the parameter f,.,, which is the volume fraction of void nucleating particles. In the developed code a simplifjling hypothesis has been made: it has been assumed that the voids nucleate once a material element enters the plastic zone: subsequently, depending on the strain conditions, they can grow or close, following the equation (7). This hypothesis, which allows a significant reduction of the necessary computations, can be interpretated as a particular plastic strain controlled nucleation case, i.e. it has been assumed that the voids nucleate at strains of the order of the tensile yield strain, as suggested by some researchers [lo]. In particular, when an element enters the plastic zone an initial porosity equal to 0.04, i.e. &=0.04, is assumed, in agreement with the values reported in [2] and with the value of the volume fraction of void nucleating particles firstly suggested by Chu and Needleman and generally accepted [14,IS]. Following the above described approach it has been possible to calculate the void volume fraction evolution during several drawing processes characterised by different operating parameters (reduction in area and semicone die angle). Anyway, in order to predict the occurrence of a ductile fracture a critical value of the void volume fraction should be determined. This value has been calculated applying the damage model to the tension test and comparing the numerical with the experimental results. In particular the value o f f calculated for an elongation ratio and a necking corresponding to the one reached in the specimen at the beginning of the rapid decay of the material stress-canying capability, has been assumed as the critical void volume fraction, fc. On the basis of this critical value, for each of the drawing simulations it has been verified if the maximum
reached value of f overcomes f,; in this case a ductile fracture should occur. The obtained results, compared with the experimental verifications, are reported in the next paragraph. It should be outlined that the simplifjing hypothesis about void nucleation previously described has been employed both for the tension test and for the drawing simulations: consequently it does not affect the validity of the ductile fracture prediction. In other words the value of the initial porosity surely influences the critical void volume fraction above defined and the maximum value o f f reached in the drawing simulations, but does not affect the comparison between these values. Finally a further damage model has been employed, based on the Shima-Oyane yield criterion:
;.(+I
(ox- oy)*+(cry -oz)2+(o,- o x )2
1--
DEFORM Post -PI 0 1 Densir)
H
F
ObJCCl
504
d I
A = ,94500
B = .9.'000 c = .YS500 D = 9GOOO
I
+T; +T;)
Rdii
Fig. 1 . Relative density distribution i n the tension test
I3
+
carrying capability of the material occurs, this ratio has been derived from the experimental tests.
(ox+oy+o,)*=~*
being A=2+R2
6 = 2- R2- 1 where R is the relative density. This criterion, largely employed for the analysis of forming processes on porous materials, is implemented in the finite element code DEFORM [12] which has been used for these simulations. An initial relative density equal to 0.96 has been assumed: at the end of each step of the drawing process R is updated taking into account that the relative density variation rate R and the volumetric strain rate i, are linked by:
R R
EV = --
By integrating equation (1 1) the updated relative density can be calculated as: R = R, exp(-j ivdt) z R,( 1 - A+) (10) being R,, the relative density at the previous step and A&, the volumetric strain increment during the analysed step. A critical value of the relative density has been obtained by simulating the tension test and comparing the numerical with the experimental results; this value has been used for the comparison with the R values obtained in the simulation of the drawing processes. Central bursting occurrence has been predicted for some process parameters, with a good agreement with the experimental results.
3. DISCUSSION OF THE RESULTS All the tests have been carried out on alluminum alloy (UNI-3571) specimens: firstly five tension tests have been performed, both to obtain the constitutive equation of the material to be used in the numerical simulation, and to determine the elongation ratio and the necking coefficient at fracture; these values in fact, as above said, are necessary in order to evaluate, by the comparison with the numerical simulation of the tensile test, the critical value of the void volume fraction. In particular the constitutive equation for the analysed material was: Q~ = 574 [N] with an elongation ratio at fracture of 15.9% and a necking coefficient equal to 14.9%. The numerical simulation of the tension test has been carried out employing all the three damage models previously described and discretizing only a quarter of the specimen. The analysis has been stopped at the elongation ratio for which the rapid decay of the stress-
EY'
Fig. 1 shows the corresponding relative density distribution: in this case in fact the Shima and Oyane yield criterion was employed. The minimum value of the relative density is equal to 0.943, value which has been consequently assumed as the critical relative density. At the same way, by simulating the tension test employing the ductile damage models of Gurson and of Tvcrgaard .and Needleman, it has been obtained a value of the critical void volume ratio equal to 0.053 and 0.056 respectively, in good agreement between themselves and with the critical relative density previously determined. In order to verify the predictive capability of the proposed approach, the results of a set of 300 drawing tests, reported in a previous paper [I], were employed. In the tests the combined action of the semicone die angle and of the reduction in area was investigated. Tab. 1 reports, for some couples of values a,RA, the frequency P of central bursting occurrences. Table I
a
5
5
5 7.5 7.5 7.5 10 10 10 15 15 15
RA .20 .25 .30 .20 .25 .30 .20 .25 .30 .20 .25 .30 P .O .O .O .32 .20 .08 .96 .76 .68 1.O .96 .92 On the basis of the above results, in order to test the goodness of the employed damage models, it has been decided to perform the numerical simulation of three drawing processes in which a percentage of defects equal to 1.0, 0.76 and 0.0 has been experimentally obtained. Consequently the processes characterised by the following a and RA have been simulated: a=15, RA=0.20; a=10,RA=0.25; a=5,RA=0.30. Table 2 reports the results obtained with the three damage models. For the Gurson and the Tvergaard and Needleman models the maximum achieved value of the void volume fraction is reported, while for the model based on the Shima and Oyane yield condition the minimum relative density is shown. The comparison of these results with the critical values, previously determined and listed in the table, shows that the occurrence of central bursting in the first analysed case is predicted all over the tested damage models, as well as the non occurrence in the third case. In the case with a=10 and RA=0.25, the obtained values are very close to the critical ones: these results are in very good agreement with the experimental ones, thus allowing to validate the predictive capability of the proposed approach.
209
Table 2
5. REFERENCES
Gurson
Tvergaard Shima adNeedleman Oyane (fc=O.053) (fc=O.056) (Rc=.943)
a
RA
15 10 5
0.20 0.25
0.066 0.05 1
0.071 0.055
0.920 0.940
0.30
0.045
0.045
0.957
Fig. 2 reports the relative density and void volume fraction distribution obtained respectively with the Shima and Oyane and the Gurson damage models with reference to the case a=l5 ,RA=0.20. In conclusion it is possible to affirm that the proposed approach, is able to supply an effective prediction of ductile fracture insurgence; moreover the characterisation of the material is quite simple, since only a set of tension tests is required in order to obtain the critical value of the void volume fraction' (or of the relative density) to be employed for the comparison with the values obtained in the simulation of the analysed forming process. Finally, due to the fact that the damage effects are introduced directly in the constitutive equations, the model supplies a very accurate description of the process mechanics.
I n 38.0,
DEFORM Post -PI o
t
D-iY object# I A-.92000
-
B - . m C
.93m
D- .9uM) E .95Mo F-.%OOO
-21.
-36.
I=MO 2 = ,345
3 = ,050 4 = .055 5=.060 6 = 065
Fig.2 Relative density distribution in drawing according to: top: Shima and Oyane; bottom: Gurson yield criteria. 4. ACKNOWLEDGEMENTS This work has been made using MURST 40% funds.
210
[l] Alberti, N., Barcellona, A,, Masnata A., Micari F., 1993, Central Bursting Defects in Drawing and Extrusion: Numerical and Ultrasonic Evaluation, Annals of CIRP, ~01.42/1:269-272. [2] Aravas, N., 1986, The Analysis of Void Growth that Leads to Central Bursts during Extrusion, J. of Mech. Phys. Solids, vol. 34, No. 1: 55-79. [3] Avitzur, B., 1968, Analysis of Central Bursting Defects in Extrusion and Wire Drawing, Trans. ASME, ser. B,~01.90:79-91. [4] Avitzur, B., Choi, C. C., 1986, Analysis of Central Bursting Defects in Plane Strain Drawing and Extrusion, Trans. ASME, ser. B, ~01.108:317-321. [5] Ayada, M., Higashino, T., Mon, K., 1987, Central Bursting in Extrusion of Inhomogeneous Materials, Advanced Technology of Plasticity, vol. 1: p.553-558. [6] Berg, C. A., 1970, Inelastic Behavior of Solids, McGraw-Hill, New York. [7] Gelin, J. C., Predelanu, M., 1992, Recent Advances in Damage Mechanics: Modelling and Computational Aspects, Proc. ofNumiform '92: 89-98. [8] Gelin, J. C., Predeleanu, M., 1989, Finite Strain ElastoPlasticity including Damage - Applications to Metal Forming Problems, Proc. ofNumiform 89: 151-157. [9] Gelin, J., C., Oudin, J., Ravalard, Y., 1985, An Improved Finite Element Method for the Analysis of Damage and Ductile Fracture in Cold Forming Processes, Annals of the CIRP, vol. 34/1: 209-213. [lo] Green, G.,Knott, J., F., 1976, Trans. ASME, Series H, J. Eng. Mat. Tech. [ l l ] Gurson, A. L., 1977, Continuum Theory of Ductile Rupture by Void-Nucleation and Growth: Yield Criteria and Flow Rules for Porous Ductile Media, J. of Eng. Mat. Tech., ~01.99:2-15. [12] Kobayashi, S., Oh, S. I., Altan, T., 1989, Metal Forming and the Finite Element Method, Oxford University Press. [13] Moritoki, H., 1990, Central Bursting in Drawing and Extrusion under Plane Strain, Advanced Technology of Plasticity, vol. 1: p.441-446. [14] Needleman, A., Tvegaard, V., 1984, An Analysis of Ductile Rupture in Notched Bars, J. of Mech. Phys. Solids, vol. 32, No. 6: 461-490. [ 15J Orbegozo, J. I., 1968, Fracture in wire drawing, Annals ofCIRP, vo1.16/1: 319. [16] Osakada, K., Mori, K., Kudo, H., 1978, Prediction of Ductile Fracture in Cold Forging, Annals of CIRP: 135-139. [17] Oyane, M., Sato, T., Okimoto, S., Shima, S., 1980, Criteria for Ductile Fracture and their Applications, Jnl. of Mech. Work. Tech., vo1.4: 65-8 1. j18] Tvergaard, V., 1982, Influence of Void Nucleation on Ductile Shear Fracture at a Free Surface, J. Mech Phys. Solids, vol. 30, no. 6: 339-425.
The Authors propose a new approach for the prediction of ductile fractures in bulk metal forming processes: the approach is based on a numerical analysis able to take into account damage occurrence and evolution in constitutive equations. The model supplies the distribution of the void volume fraction in the workpiece during the deformation path: consequently, the comparison to a critical value, determined by means of a simple tension test, allows to predict the growth of defects. The proposed approach has been applied to the drawing process: the numerical results have been compared with a set of experimental tests showing a good predictive capability of the model. Key words: Failure, Forging, Damage. 1. INTRODUCTION During bulk forming processes failure phenomena are often encountered, mainly consisting in internal or surface cracking. The fracture mechanism which limits the ductility of a structural alloy is generally explained taking into account the nucleation and the growth of a large number of small voids. The voids nucleate mainly at second phase particles by decohesion of the particle-matrix interface and grow due to the presence of a tensile hydrostatic stress; as a consequence the ligaments between them thin down until fracture occurs by coalescence [7,18]. In the recent years several approaches have been proposed in order to predict the growth of ductile fractures; among them two main groups can be individuated: the former is based on the numerical analysis of the forming process carried out employing the yield condition and the associated flow rule of the sound, "undamaged", material; the stress and strain fields calculated at each step of the deformation path are introduced in a proper ductile fiacture criterion, able to suggest if and where fracture occurs. Such an approach has been applied to several forming processes, and a large number of ductile fracture criteria has been proposed, among which those proposed by Oyane [1'7], Osakada [16], Ayada [5] and Cockrofi and Latham have shown to be the most suitable. On the other hand other researchers have proposed to take into account damage occurrence and its evolution in the constitutive equations used for the analysis of metal forming processes. Thus this approach requires a proper definition of damage and the development of an accurate damage model able to describe the influence of damage evolution on the plastic flow properties of the material [8,9]; mainly the presence of voids gives rise to plastic volume variation and to the dependence of yielding on the hydrostatic stress. Moreover the model should be able to describe the interaction between voids and the rapid decay followed by the complete loss of load carrying capability occurring in the material when the voids coalesce. The first set of constitutive equations including ductile damage was suggested by Gurson [ll], who proposed a yield criterion depending on the first invariant of the stress tensor, on the second invariant of the deviatoric stress tensor and on a scalar parameter f , (void volume fraction). The void volume fraction is defined as the ratio between the volume of the cavities in an aggregate and the volume of the aggregate. The Gurson model has been subsequently modified and developed by Tvergaard [I81 and by Tvergaard and Needleman [ 141, which have introduced Annals of the ClRP Vol. 43/1/1994
some additional parameters in order to model the decay of the load carrying capability. In the paper the authors apply a ductile damage model to the prediction of central bursting insurgence in the drawing process; as it is well known, central bursts are internal defects which cause very serious problems to the quality control of the products, since it is impossible to detect them by means of a simple surface inspection of the workpiece [l]. In the past several experimental and theoretical studies [2,3,4,5,13,15] have been performed in order to determine the influence of the main operating parameters (reduction in area, die cone angle, friction at the die-workpiece interface and material properties) on the occurrence of central bursts. However these models and empirical rules concern materials with assigned properties and processes for which the lubricating conditions are known and constant. Viceversa it is well known that specimens belonging to the same production batch can be characterised by different structural inhomogeneities, second phase particles distributions, inclusion and void contents, all factors which affect the workability conditions. In this paper the prediction of central bursts occurrence in drawing is carried out by means of a ductile damage model: such a model in fact allows to describe in a very accurate way the plastic flow of the material, including damage effects. The numerical analysis has been carried out employing the yield criteria proposed by Gurson, Tvergaard and Needleman and finally by Shima and Oyane. A simple tension test has been firstly simulated, and the comparison with the experimental evidence has allowed to calculate the critical value of the void volume fraction for the considered material, corresponding to the point at which the coalescence of voids occurs and the material undergoes a rapid decay of its stress-carrying capability. Subsequently the numerical analysis has been applied to several drawing processes, characterised by different reductions in area and die cone angles: for each of them the maximum achieved value of the void volume fraction has been calculated, and, by the comparison with the critical value, it has been evaluated if the coalescence of voids, i.e. the insurgence of the ductile fracture, occurs or not. The numerical results have shown a good agreement with the experimental ones, thus confirming the validity of the proposed approach.
2. THE DUCTILE DAMAGE MODELS As previously reported, the numerical analysis of the drawing process has been carried out employing three different damage models, namely those based on the yield criteria proposed by Gurson [ 111, Tvergaard and Needleman
207
[14], and finally by Shima and Oyane [17]; as it is known the last one has been largely employed in the recent years for the analysis of powder materials forming. Starting from a rigid-perfectly plastic upper bound solution for spherically symmetric deformations around a single spherical void, Gurson proposed a yield function for a solid with a randomly distributed volume fraction of voids f in the form:
where
is the macroscopic equivalent stress, with C Y ~macroscopic stress deviator, and 5 is the yield stress of the matrix (voidfiee) material. Eq.1 clearly shows the effect of the mean stress on the plastic flow when the void volume fraction is non-zero, while for f-0 the Gurson criterion reduces to the von Mises one. The components of the macroscopic plastic strain rate vector can be calculated applying the normality rule to the yield criterion above written: in fact as shown by Bishop and Hill [6] the validity of the normality rule for the matrix material implies the validity of a macroscopic normality rule. Following these considerations the components of the macroscopic , plastic . strain rate vector can be expressed as:
(3) where the parameter ican be calculated if monoaxial stress conditions are assumed. Subsequently the Gurson yield criterion has been modified by Tvergaard [IS] and by Tvergaard and Needleman [141, which introduced other parameters obtaining the following expression: cTz
@=3+2f'q,cosh (4) E* In the above expression the parameter q, allows to consider the interactions between neighbouring voids: Tvergaard in fact obtained the above criterion analysing the macroscopic behaviour of a doubly periodic array of voids using a model which takes into account the nonuniform stress field around each void. The value of q1 was assumed equal to 1.5 by the same Tvergaard. On the other hand the parameter f( f ) was introduced in substitution off in order to describe in a more accurate way the rapid decrease of stresscarrying capability of the material associated to the coalescence of voids: in fact f(0 is defined as: If for fS fc
where f, =l/q,, fc is a critical value of the void volume fraction at which the material stress-carrying capability starts to decay very quickly (easily discernible in a tensile test curve) and finally fF is the void volume fraction value corresponding to the complete loss of stress-carrying capability. Again the constitutive equations associated to the Tvergaard and Needleman yield criterion can be determined by means of the normality rule. Both the Gurson and the Tvergaard and Needleman damage models have been implemented in a finite element code based on an Updated Lagrangian formulation; an
208
incremental procedure has been employed in order to follow the transient process from the workpiece input phase to the achievement of steady state conditions. In fact to evaluate the possibility of a ductile fracture insurgence, it has been necessary to follow the whole deformation path for each point of the material. At the end of each step of the deformation process the values of the void volume fractions calculated at the integration points within each element should be updated: in fact both the existing voids can grow, depending on the stress conditions, and new voids can nucleate. The increasing rate of the void volume fraction in fact can be considered partly due to the growth of the existing voids and partly . . to the nucleation of new voids [7]: (6) f = fm+ fnucieation AS far as the first aspect is concerned the rate of change of the void volume fraction is related to the volumetric strain rate 4, by the relation: fPWh= ( l - f ) + ? (7) On the other hand, several models have been proposed in order to evaluate the nucleation of new voids; in these models the void nucleation rate is assumed as depending on the matrix equivalent plastic strain rate and on the rate of increase of the hydrostatic stress, following the expression: fnucleation = AE, + B(& + 6 , ) (8) where A and B are material dependent constants. In particular Chu and Needleman suggested to calculate A and B assuming that the void nucleation follows a normal distribution and have distinguished two different cases: the case of plastic strain controlled nucleation (B=O) and the case of stress controlled nucleation (A=O). Moreover in the proposed expressions for A and B [ 141, Chu and Needleman have introduced the parameter f,.,, which is the volume fraction of void nucleating particles. In the developed code a simplifjling hypothesis has been made: it has been assumed that the voids nucleate once a material element enters the plastic zone: subsequently, depending on the strain conditions, they can grow or close, following the equation (7). This hypothesis, which allows a significant reduction of the necessary computations, can be interpretated as a particular plastic strain controlled nucleation case, i.e. it has been assumed that the voids nucleate at strains of the order of the tensile yield strain, as suggested by some researchers [lo]. In particular, when an element enters the plastic zone an initial porosity equal to 0.04, i.e. &=0.04, is assumed, in agreement with the values reported in [2] and with the value of the volume fraction of void nucleating particles firstly suggested by Chu and Needleman and generally accepted [14,IS]. Following the above described approach it has been possible to calculate the void volume fraction evolution during several drawing processes characterised by different operating parameters (reduction in area and semicone die angle). Anyway, in order to predict the occurrence of a ductile fracture a critical value of the void volume fraction should be determined. This value has been calculated applying the damage model to the tension test and comparing the numerical with the experimental results. In particular the value o f f calculated for an elongation ratio and a necking corresponding to the one reached in the specimen at the beginning of the rapid decay of the material stress-canying capability, has been assumed as the critical void volume fraction, fc. On the basis of this critical value, for each of the drawing simulations it has been verified if the maximum
reached value of f overcomes f,; in this case a ductile fracture should occur. The obtained results, compared with the experimental verifications, are reported in the next paragraph. It should be outlined that the simplifjing hypothesis about void nucleation previously described has been employed both for the tension test and for the drawing simulations: consequently it does not affect the validity of the ductile fracture prediction. In other words the value of the initial porosity surely influences the critical void volume fraction above defined and the maximum value o f f reached in the drawing simulations, but does not affect the comparison between these values. Finally a further damage model has been employed, based on the Shima-Oyane yield criterion:
;.(+I
(ox- oy)*+(cry -oz)2+(o,- o x )2
1--
DEFORM Post -PI 0 1 Densir)
H
F
ObJCCl
504
d I
A = ,94500
B = .9.'000 c = .YS500 D = 9GOOO
I
+T; +T;)
Rdii
Fig. 1 . Relative density distribution i n the tension test
I3
+
carrying capability of the material occurs, this ratio has been derived from the experimental tests.
(ox+oy+o,)*=~*
being A=2+R2
6 = 2- R2- 1 where R is the relative density. This criterion, largely employed for the analysis of forming processes on porous materials, is implemented in the finite element code DEFORM [12] which has been used for these simulations. An initial relative density equal to 0.96 has been assumed: at the end of each step of the drawing process R is updated taking into account that the relative density variation rate R and the volumetric strain rate i, are linked by:
R R
EV = --
By integrating equation (1 1) the updated relative density can be calculated as: R = R, exp(-j ivdt) z R,( 1 - A+) (10) being R,, the relative density at the previous step and A&, the volumetric strain increment during the analysed step. A critical value of the relative density has been obtained by simulating the tension test and comparing the numerical with the experimental results; this value has been used for the comparison with the R values obtained in the simulation of the drawing processes. Central bursting occurrence has been predicted for some process parameters, with a good agreement with the experimental results.
3. DISCUSSION OF THE RESULTS All the tests have been carried out on alluminum alloy (UNI-3571) specimens: firstly five tension tests have been performed, both to obtain the constitutive equation of the material to be used in the numerical simulation, and to determine the elongation ratio and the necking coefficient at fracture; these values in fact, as above said, are necessary in order to evaluate, by the comparison with the numerical simulation of the tensile test, the critical value of the void volume fraction. In particular the constitutive equation for the analysed material was: Q~ = 574 [N] with an elongation ratio at fracture of 15.9% and a necking coefficient equal to 14.9%. The numerical simulation of the tension test has been carried out employing all the three damage models previously described and discretizing only a quarter of the specimen. The analysis has been stopped at the elongation ratio for which the rapid decay of the stress-
EY'
Fig. 1 shows the corresponding relative density distribution: in this case in fact the Shima and Oyane yield criterion was employed. The minimum value of the relative density is equal to 0.943, value which has been consequently assumed as the critical relative density. At the same way, by simulating the tension test employing the ductile damage models of Gurson and of Tvcrgaard .and Needleman, it has been obtained a value of the critical void volume ratio equal to 0.053 and 0.056 respectively, in good agreement between themselves and with the critical relative density previously determined. In order to verify the predictive capability of the proposed approach, the results of a set of 300 drawing tests, reported in a previous paper [I], were employed. In the tests the combined action of the semicone die angle and of the reduction in area was investigated. Tab. 1 reports, for some couples of values a,RA, the frequency P of central bursting occurrences. Table I
a
5
5
5 7.5 7.5 7.5 10 10 10 15 15 15
RA .20 .25 .30 .20 .25 .30 .20 .25 .30 .20 .25 .30 P .O .O .O .32 .20 .08 .96 .76 .68 1.O .96 .92 On the basis of the above results, in order to test the goodness of the employed damage models, it has been decided to perform the numerical simulation of three drawing processes in which a percentage of defects equal to 1.0, 0.76 and 0.0 has been experimentally obtained. Consequently the processes characterised by the following a and RA have been simulated: a=15, RA=0.20; a=10,RA=0.25; a=5,RA=0.30. Table 2 reports the results obtained with the three damage models. For the Gurson and the Tvergaard and Needleman models the maximum achieved value of the void volume fraction is reported, while for the model based on the Shima and Oyane yield condition the minimum relative density is shown. The comparison of these results with the critical values, previously determined and listed in the table, shows that the occurrence of central bursting in the first analysed case is predicted all over the tested damage models, as well as the non occurrence in the third case. In the case with a=10 and RA=0.25, the obtained values are very close to the critical ones: these results are in very good agreement with the experimental ones, thus allowing to validate the predictive capability of the proposed approach.
209
Table 2
5. REFERENCES
Gurson
Tvergaard Shima adNeedleman Oyane (fc=O.053) (fc=O.056) (Rc=.943)
a
RA
15 10 5
0.20 0.25
0.066 0.05 1
0.071 0.055
0.920 0.940
0.30
0.045
0.045
0.957
Fig. 2 reports the relative density and void volume fraction distribution obtained respectively with the Shima and Oyane and the Gurson damage models with reference to the case a=l5 ,RA=0.20. In conclusion it is possible to affirm that the proposed approach, is able to supply an effective prediction of ductile fracture insurgence; moreover the characterisation of the material is quite simple, since only a set of tension tests is required in order to obtain the critical value of the void volume fraction' (or of the relative density) to be employed for the comparison with the values obtained in the simulation of the analysed forming process. Finally, due to the fact that the damage effects are introduced directly in the constitutive equations, the model supplies a very accurate description of the process mechanics.
I n 38.0,
DEFORM Post -PI o
t
D-iY object# I A-.92000
-
B - . m C
.93m
D- .9uM) E .95Mo F-.%OOO
-21.
-36.
I=MO 2 = ,345
3 = ,050 4 = .055 5=.060 6 = 065
Fig.2 Relative density distribution in drawing according to: top: Shima and Oyane; bottom: Gurson yield criteria. 4. ACKNOWLEDGEMENTS This work has been made using MURST 40% funds.
210
[l] Alberti, N., Barcellona, A,, Masnata A., Micari F., 1993, Central Bursting Defects in Drawing and Extrusion: Numerical and Ultrasonic Evaluation, Annals of CIRP, ~01.42/1:269-272. [2] Aravas, N., 1986, The Analysis of Void Growth that Leads to Central Bursts during Extrusion, J. of Mech. Phys. Solids, vol. 34, No. 1: 55-79. [3] Avitzur, B., 1968, Analysis of Central Bursting Defects in Extrusion and Wire Drawing, Trans. ASME, ser. B,~01.90:79-91. [4] Avitzur, B., Choi, C. C., 1986, Analysis of Central Bursting Defects in Plane Strain Drawing and Extrusion, Trans. ASME, ser. B, ~01.108:317-321. [5] Ayada, M., Higashino, T., Mon, K., 1987, Central Bursting in Extrusion of Inhomogeneous Materials, Advanced Technology of Plasticity, vol. 1: p.553-558. [6] Berg, C. A., 1970, Inelastic Behavior of Solids, McGraw-Hill, New York. [7] Gelin, J. C., Predelanu, M., 1992, Recent Advances in Damage Mechanics: Modelling and Computational Aspects, Proc. ofNumiform '92: 89-98. [8] Gelin, J. C., Predeleanu, M., 1989, Finite Strain ElastoPlasticity including Damage - Applications to Metal Forming Problems, Proc. ofNumiform 89: 151-157. [9] Gelin, J., C., Oudin, J., Ravalard, Y., 1985, An Improved Finite Element Method for the Analysis of Damage and Ductile Fracture in Cold Forming Processes, Annals of the CIRP, vol. 34/1: 209-213. [lo] Green, G.,Knott, J., F., 1976, Trans. ASME, Series H, J. Eng. Mat. Tech. [ l l ] Gurson, A. L., 1977, Continuum Theory of Ductile Rupture by Void-Nucleation and Growth: Yield Criteria and Flow Rules for Porous Ductile Media, J. of Eng. Mat. Tech., ~01.99:2-15. [12] Kobayashi, S., Oh, S. I., Altan, T., 1989, Metal Forming and the Finite Element Method, Oxford University Press. [13] Moritoki, H., 1990, Central Bursting in Drawing and Extrusion under Plane Strain, Advanced Technology of Plasticity, vol. 1: p.441-446. [14] Needleman, A., Tvegaard, V., 1984, An Analysis of Ductile Rupture in Notched Bars, J. of Mech. Phys. Solids, vol. 32, No. 6: 461-490. [ 15J Orbegozo, J. I., 1968, Fracture in wire drawing, Annals ofCIRP, vo1.16/1: 319. [16] Osakada, K., Mori, K., Kudo, H., 1978, Prediction of Ductile Fracture in Cold Forging, Annals of CIRP: 135-139. [17] Oyane, M., Sato, T., Okimoto, S., Shima, S., 1980, Criteria for Ductile Fracture and their Applications, Jnl. of Mech. Work. Tech., vo1.4: 65-8 1. j18] Tvergaard, V., 1982, Influence of Void Nucleation on Ductile Shear Fracture at a Free Surface, J. Mech Phys. Solids, vol. 30, no. 6: 339-425.