- Email: [email protected]
Forming limit diagrams: a micromechanical approach
International Journal of Mechanical Sciences 42 (2000) 2041}2054
Forming limit diagrams: a micromechanical approach H.-P. GaK nser *, E.A. Werner, F.D. Fischer CEMEF, Ecole des Mines de Paris, rue Claude Daunesse, B.P. 207, F-06904 Sophia-Antipolis, France Christian Doppler Laboratory for Modern Multiphase Steels, Technical University of Munich, Boltzmannstra}e 15, D-85748 Garching, Germany Institute of Mechanics, Christian Doppler Laboratory for Micromechanics of Materials, Montanuniversita( t Leoben, Franz-Josef-Stra}e 18, A-8700 Leoben, Austria Received 22 December 1997; accepted 26 June 1999
Abstract A method for obtaining the forming limit diagram (FLD) from a micromechanical approach is proposed. Periodic representative volume elements (RVEs) characteristic for the microstructure of a two-phase material containing particles are subjected to biaxial stretching with di!erent strain paths. In the course of straining, a groove forms and grows, "nally leading to a stress drop in the overall response. This deformation instability is obtained as a natural result of the strain "eld #uctuations caused by the inclusions in the two-phase material. For this approach, only the geometry of the microstructure and the material data of the constituents have to be known; no additional assumptions about the deformation process or about a macroscopic inhomogeneity are necessary. The in#uence of the RVE representation on the possible deformation patterns is investigated. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction There exists a great variety of methods for calculating the forming limit diagram (FLD) for stretch forming of sheet materials. An extensive review of experimental work as well as of theoretical approaches is given by Wagoner et al. [1]. Much of the theoretical work is based on the concept of imperfection analysis as originally proposed by Marciniak and Kuczynski [2]. The Marciniak}Kuczynski (MK) approach assumes an initial notch-like imperfection extending across the sheet. During the stretch-forming operation, the plastic strain accumulates in the
* Corresponding author now at: Numerical Simulation Department, Corporate Research, Hilti Corp., FL-9494 Schaan, LmH echtenstein. Tel.: #423 236 2316; Fax: #423 236 3844. E-mail address: [email protected] (H.-P. GaK nser). 0020-7403/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 9 9 ) 0 0 0 5 7 - 0
2042
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
Nomenclature a e e G e
h hM l ,l p p
l G l G l n t z E ,E R < =
principal strain ratio e /e equivalent macroscopic strain macroscopic strain in direction `ia equivalent strain of the matrix angle of inclination of the deformation bands in the RVE angle of inclination of the deformation band according to Hill's analysis Poisson's ratio (inclusions, matrix) equivalent macroscopic stress equivalent stress of the matrix initial edge length of the RVE in direction `ia current edge length of the RVE in direction `ia spacing of the deformation bands in the RVE strain hardening exponent of the matrix thickness and number of elements in thickness direction of a 3D RVE number of deformation bands in the RVE Young's modulus (inclusions, matrix) yield stress of the matrix volume of the RVE external work applied to the RVE
grooved region rather than in the rest of the specimen, thus leading to localized thinning in the notch. Finally, loss of stability occurs due to the reduction of the minimum cross-sectional area. However, the MK analysis proves to be very sensitive to the choice of the shape and size of the initial imperfection. One obvious remedy is to subdivide the imperfection analysis into three stages * homogeneous deformation, localization under constant load, and localized necking with a precipitous drop in load * and to "t those stages to the tensile test [3]. Another possibility is to conduct an MK-type analysis based on local #uctuations of microstructural quantities, as, e.g., internal damage from void growth and coalescence [4] or inhomogeneities of the crystallographic texture [5]. Both approaches use a suitable mesoscopic model to describe the microstructure and, therefore, cannot account exactly for the local stress and strain #uctuations on the microscale. On the other hand, numerous e!orts have been devoted to determine the e!ective properties of composites directly from their microstructure via the periodic micro"eld approach using either periodic arrangements of inclusions [6,7] or quasi-random arrangements [8]. It is well-known that, even in the plane strain state, band-like strain concentrations may develop near precipitates [9], and that the geometrical arrangement of the particles in#uences the evolution of the plastic #ow patterns [10]. It may be expected that these e!ects will be still more pronounced in a plane stress state, as prevailing near the surface of thin sheets. In the sequel, the in#uence of such quasi-randomly arranged precipitates on the formation and growth of a groove, and thus on the limits for stable plastic deformation in sheet metal forming, will
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
2043
be investigated using the periodic micro"eld approach. A feature inherent to this approach is that, in contrast to the usual MK-type analyses, the computations are carried out directly on the microscale by modeling the matrix-inclusion topology of the material explicitly. Thus, no prede"ned imperfection is necessary; rather, the notch develops in the course of deformation due to the interaction of the strain "elds caused by the individual inclusions.
2. Computational approach The material investigated consists of an isotropic elastic * J -plastic matrix with quasi randomly distributed elastic inclusions of cubical shape (the overall particle distribution is only quasi-random because the material is represented by one repeating RVE, even though the particle distribution within the RVE is random). The elastic properties of the matrix are characterized by a Young's modulus of E "210 000 N/mm and a Poisson's number of l "0.3, while the plastic
behavior is described by an initial yield stress of R "150 N/mm and isotropic hardening according to a power law p "532eL , where p denotes the matrix equivalent stress, e the matrix
equivalent plastic strain, and the strain hardening exponent has the value n"0.23. The respective data for the inclusions are E "600 000 N/mm, l "0.15. An in"nitely extending plane sheet is modeled by the periodic micro"eld approach, using representative volume elements (RVEs) with periodic boundary conditions. This type of boundary conditions enforces that the corresponding nodes of two associated boundaries (left/right and lower/upper boundaries) may di!er only by some identical vector; i.e., the boundary lines must remain parallel but need not remain straight. Fig. 1 displays both a three- and a two-dimensional "nite element (FE) discretization of a typical RVE. Both models consist of 25;25 second-order, reduced integration, Serendipity-type isoparametric elements in the `1a}`2a plane. Furthermore, in the three-dimensional model the thickness (`3a) direction is subdivided into "ve elements; in addition, 3D models with 10 and 15 elements will be investigated. Periodic boundary conditions are applied in the directions `1a and `2a in order to simulate an in"nitely extending sheet. No constraint is applied in the thickness direction, which corresponds to the free lower and upper boundary planes of a thin sheet. In the two-dimensional model, this condition is accounted for by using plane stress elements. To be truly representative of a material with quasi-randomly distributed inclusions, the RVE must contain a su$cient number of precipitates. For this purpose, one inclusion is discretized by one single element. Although the stress "eld near the phase boundaries cannot be captured precisely, this rather coarse approximation allows for the required amount of inclusions and for particle}particle interactions while keeping the computational cost reasonably low. On the other hand, some e!ects will not be captured: a trustworthy estimation of the in#uence of the inclusion shape, e.g., would require a much more accurate description of the stress and strain "elds around the inclusions than the one achieved by a single-element discretization of an inclusion. The main e!ect of the inclusions in this model is that they trigger deformation inhomogeneities in the matrix; the development of these inhomogeneities will be studied. Several RVEs with inclusion volume fractions between 2 and 8% are generated and loaded with prescribed biaxial stretches of di!erent ratios. The principal in-plane strains are denoted by e and e . Given the current edge lengths l and the edge length increments dl of the RVE, the G G
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
2044
Fig. 1. (a) 3D and (b) 2D FE mesh of an RVE describing a sheet material with randomly distributed inclusions (2 vol% inclusions).
macroscopic principal strain increments in the directions `1a and `2a are obtained as dl dl de " , de " . (1) l l Neglecting the elastic strains, the macroscopic strain increment in the direction `3a is calculated from the conservation of volume as de "!(de #de ), and the equivalent macroscopic strain increment is
(2)
de"( (de #de #de ). (3) The principal strains in the direction `ia, e , and the equivalent strain e are obtained by G integration:
e" G
JG
l de "ln G , G l JG G
(4)
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
2045
where l denotes the initial RVE edge length in direction `ia, and G
e" de.
(5)
For convenience of notation in the following discussion, the principal strain ratio is de"ned as e a" . (6) e The macroscopic equivalent stress p follows from the applied external work increment d= and from the volume <"l l l of the RVE as d= . (7) p" < de The deformation instability is assumed to occur when the macroscopic stress}strain curve reaches its maximum, dp "0, de
(8)
and the respective principal strains e and e give the corresponding point in the FLD. So, by prescribing di!erent strain ratios a in the range from !0.5 (uniaxial stress) to 1 (equibiaxial strain), the entire FLD is constructed point by point. This approach permits any path in principal strain space to be prescribed; though, the simulations in the present work were performed using radial strain paths only.
3. Comparison of 2D and 3D models To compare the deformation behavior of the 2D and the 3D models, both models are subjected to uniaxial tension in the direction `1a. The deformation patterns are depicted in Fig. 2; Fig. 3 shows the resulting stress}strain curves. In addition, Fig. 3 displays the matrix behavior and the stress}strain curves for a number of randomly generated 2D and 3D RVEs (with thicknesses t of 5, 10, and 15 units and elements, respectively). In all cases, the inclusion volume fraction was 0.02. It can be seen in Fig. 2 that the inhomogeneities of the strain "eld caused by the inclusions link up to form a series of inclined parallel grooves. The parallel pattern results from the periodic boundary conditions; the interdependence between the RVE geometry, the angle and the spacing of the grooves will be investigated below. While in the earlier stages of the deformation the composite shows slightly higher hardening than the matrix * due to the reinforcement by the particles *, in the later stages the geometric softening caused by the groove outweighs the reinforcement by the inclusions, "nally leading to a drop in the stress}strain curve, Fig. 3. This behavior is essentially the same for the 2D and the 3D models. However, the 2D RVEs show the instability at lower strains because of the somewhat poorer representation of the underlying geometry: while a 3D model can easily capture the exact geometry of the cell, in a 2D model the cubic inclusions become "bers with square cross-section
2046
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
Fig. 2. Equivalent plastic strain distribution of the (a) 3D and the (b) 2D RVE (2% inclusions) from Fig. 1 subjected to uniaxial tension in the `1a direction. Failure occurs by inclined grooves. The undeformed shape is drawn as wireframe for comparison.
extending in the thickness direction; thus, the necking between the inclusions is facilitated signi"cantly. These di!erences between the 2D and the 3D models have been investigated in detail in Refs. [7,8,11]. A pronounced scatter between the stress}strain curves from di!erent RVEs of the same inclusion volume fraction is noted for the 2D as well as for the 3D models. The limit strains for instability (horizontal tangent of the curves in Fig. 3) are spread over a range of about 0.1 for the 2D RVEs and as much as 0.3 for the 3D RVEs with 5 elements. This shows that there exist inclusion arrangements which favor the development of deformation concentrations, and such which delay them.
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
2047
Fig. 3. Macroscopic stress}strain curves obtained from 3D and 2D RVEs (2% inclusions). The stress}strain curve of the pure matrix is also shown. The 3D models exhibit thicknesses t of 5, 10 and 15 units, each of them containing cubic inclusions of unit edge length. Note the scatter between the di!erent random RVEs (scatter of the strain at instability of 0.1 for the 2D and of 0.3 for the 3D RVEs with t"5).
Clearly, for the 3D RVE, a discretization of only 5 elements in thickness direction is a very coarse description. To investigate the in#uence of the ratio between the size of the inclusion and the sheet thickness, Fig. 3 displays also the results from two other RVEs of the same inclusion volume fraction, but of double and three-fold thickness t (10 and 15 elements/units in thickness direction, respectively). It is seen that an increase of the sheet thickness results in a shift of the point of instability towards higher strains; however, the 10 element RVE reaches instability at roughly the same strain as two of the 5 element RVEs, which indicates that even in rather thick sheets necking may occur by the mechanism discussed above. The strains at instability will be lower for thinner foils, though. In the sequel, we use the 2D plane stress model as an inexpensive approximation of a thin foil in order to demonstrate that, in principle, the FLD can be constructed from a micromechanical model using a unit cell (RVE) representation of the microstructure, and to discuss the in#uence of the periodic boundary conditions on the failure modes.
4. Periodic grooves and the RVE approach Taking a closer look on Fig. 2, one remarks that the 2D and the 3D simulations di!er with respect to the number of grooves and to their inclination angle. Evidently, the periodic boundary conditions of an RVE impose some constraints on the possible con"gurations of the band-like deformation concentrations. Fig. 4 depicts schematically such an RVE containing parallel deformation bands, as observed in the simulation results (Fig. 2). Periodicity requires that an RVE of current length l must contain
2048
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
Fig. 4. Deformed RVE with z parallel grooves at an angle of h with respect to the principal tensile direction.
an integer number z of deformation bands of length l so that l "l /z, and the inclination angle of the grooves is given by
(9)
l l tan h" "z . (10) l l The logarithmic strains e give the link between the initial and the current edge lengths l and l of G G G the RVE, l "l exp e . (11) G G G Inserting the latter relation and the de"nition of the principal strain ratio a"e /e into Eq. (10), we obtain l tan h"z exp(a!1)e . (12) l The resulting admissible angles h are shown in Fig. 5 for some numbers z of deformation bands and principal strain ratios a of !0.5 and 0, the values of a which delimit the left-hand side of the FLD, for an RVE with l /l "1 as used in the present study. For the right-hand side of the FLD, a*0, the grooves are expected to be perpendicular to the major tensile axis, h"903. In this case, the RVE admits * contrary to Eq. (12), which predicts zPR for hP903 * also a single perpendicular groove. The additional constraints introduced by the use of periodic RVEs may be illustrated best by the following simple example. Let us assume that the deformation bands occur exactly as predicted by Hill's analysis [12] of localized necking in thin sheets, applicable to the left-hand side of the FLD. It must be emphasized that this assumption is made only for the purpose of illustration. Of course, the present approach is entirely di!erent from Hill's concept of localized necking and is thus expected to give somewhat di!erent results, apart from the fact that the deformation bands in the FE
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
2049
Fig. 5. Admissible angles h in an RVE with l /l "1, from Eq. (12), for di!erent numbers z of deformation bands and di!erent principal strain ratios a.
simulations manifest themselves gradually so that the onset of necking cannot be determined unambiguously. However, a comparison between Hill's analysis and the present results (see the following section and Fig. 7) will show that they di!er not too much one from the other. According to Hill's analysis [12], the left-hand side of the FLD is given by (cf. also [13]) e #e "e (1#a)"n, (13) where n denotes the strain hardening exponent of the material, and the corresponding angle hM of the shear band at the onset of localized necking is obtained from 1 tan hM " . (!a
(14)
Assuming that the onset of necking follows Eq. (13), the angle admitted by the RVE, from Eq. (12), has to satisfy the relation l a!1 tan h"z exp n. (15) l a#1 The resulting angles h and hM are compared in Fig. 6 for the present case of a matrix strain hardening exponent (because it is in the matrix where the neck develops) of n"0.23 and an RVE geometry of l /l "1 as employed in the FE simulations. Obviously, the RVE can accommodate Hill's results only at a limited number of points for special values of a and z; this illustrates quite well the additional constraints which result from the periodic boundary conditions. It may, however, be expected that the FE simulations will favor the angles that are close to `physical realitya; in other words, that the number of the bands will be z"3 for uniaxial tension (a"!0.5) and augment successively as the principal strain ratio a approaches zero.
2050
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
Fig. 6. Angles obtained in an RVE with l /l "1, accommodating Hill's analysis (strain hardening exponent n"0.23), Eq. (15), for di!erent numbers z of deformation bands and di!erent principal strain ratios a. Hill's result, i.e. hM from Eq. (14), is shown for comparison.
5. Forming limit diagram To obtain the complete FLD, randomly generated RVEs are subjected to straining with di!erent ratios a, and the points (e , e ) where instability occurs are assembled into the diagram. Fig. 7a shows such FLDs for three di!erent RVEs with inclusion volume fractions of 2, 5, and 8%; for comparison, Fig. 7b shows the corresponding predictions from Hill's and Swift's analyses (see Ref. [13]). One recognizes in Fig. 7a the typical V shape with the highest strains for uniaxial tension (a"!0.5) and equibiaxial stretching (a"1), and the smallest strains for uniaxial stretching (a"0). Also, as is to be expected, higher precipitate volume fractions yield lower strains in the FLD. The predictions of the present model for the left-hand side of the FLD are not too di!erent from the one obtained by Hill's model (which justi"es a posteriori the simple estimate of the band angles by means of Fig. 6), while on the right-hand side far higher limit strains than in Swift's analysis are reached. For the left-hand side of the FLD with strain ratios a between !0.5 and about !0.15, one obtains inclined grooves as already shown in Fig. 2. The number of grooves z varies, as predicted by the simpli"ed analysis of the previous section, between 3 and 4; the higher number of grooves prevailing E for less negative strain ratios, so that Eq. (15) may accommodate the higher angles to be expected from Eq. (14), cf. Fig. 6, and E in cases where failure occurs later than predicted by Hill's analysis (like in the 3D simulation of uniaxial tension in Fig. 2a), in order to give still roughly the same angle for the deformation bands in such highly stretched RVEs like in other cases where the instability is reached at lower strains, cf. Eq. (12).
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
2051
Fig. 7. (a) Forming limit diagrams obtained from 2D RVEs with 2, 5, and 8% inclusions. The straight lines depict the strain paths of uniaxial tension (e "!e /2), uniaxial stretching (e "0), and equibiaxial stretching (e "e ). (b) FLD constructed from Hill's analysis of localized necking (left-hand side) and Swift's analysis of di!use necking (right-hand side) for n"0.23, corresponding to the strain hardening of the matrix in the RVE computations.
For the entire right-hand side of the FLD, a*0, a single groove perpendicular to the major principal strain axis (Fig. 8a and b) is obtained. For strain ratios close to unity, failure may occur in a more di!use mode (Fig. 8b), but still exhibiting the perpendicular groove. Also for the part of the left-hand side of the FLD near the center (!0.1)a)0), the FE analyses favor a single perpendicular groove instead of a high number of near-perpendicular grooves as could be expected from Fig. 6. In fact, none of the simulations led to a number of grooves greater than z"4.
2052
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
Fig. 8. Equivalent plastic strain distribution of a 2D RVE (2% inclusions) subjected to (a) uniaxial (a"0) and (b) equibiaxial (a"1) stretching. Failure occurs by perpendicular grooves.
Regarding the in#uence of the inclusion volume fraction, the present results have to be interpreted with caution. As mentioned above, the inclusions serve mainly as triggers for the deformation inhomogeneities which coalesce subsequently to grooves. Thus, the point where instability occurs depends strongly on the clustering of the inclusions, the latter being in#uenced by inclusion rearrangements in randomly generated RVEs as well as by changes in their volume
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
2053
fraction. These dependencies have been pointed out in the experimental work by Dubensky and Koss [14]. Consequently, the inclusion rearrangements for di!erent RVEs of the same volume fraction will produce a scatter in the FLD; this has already been demonstrated in Ref. [16] for FLD predictions using an MK-like model with random sheet thickness distributions. In fact, in the present case it turns out that inclusion rearrangements may produce a scatter of the strain at instability of the order of 0.1 (cf. the 2D results in Fig. 3; the scatter reported in Ref. [16] is of the same order of magnitude). If we plotted such scatter bands in Fig. 7a, the FLDs for the three di!erent inclusion volume fractions would overlap one another to a great extent. This is partially due to the aforementioned interdependence, but also due to the rather coarse FE discretization; if one aimed at a detailed FE simulation of the mutual in#uences of voids or inclusions, this task would need a much more re"ned * and computationally much more expensive * numerical treatment (see, e.g., the case study by Becker and Smelser [15]). 6. Concluding remarks A method for obtaining the FLD from a micromechanical approach was proposed. The deformation instability is obtained as a natural result of the strain "eld #uctuations caused by the precipitates in a two-phase material. For this approach, only the geometrical arrangement of the second phase particles in the microstructure and the material data of the constituents have to be known; no additional assumptions about deformation history or about a macroscopic inhomogeneity are necessary. The analysis was conducted for a two-phase material with quasi-randomly distributed, ideally bonded precipitates. In this way it was shown that instabilities in sheet metal forming are not necessarily due to the growth and coalescence of the voids, but can also be a consequence of the two-phase nature of most materials. This may be important for the deformation behavior of thin foils made from materials with strong matrix-inclusion bonds which remain intact up to rather high strains. The in#uence of the sheet thickness with respect to the inclusion diameter was discussed as well as the additional constraints resulting from the approximation of the material structure by RVEs with periodic boundary conditions. Due to the limitations of the rather simple model, the conclusions have to be of a qualitative nature, only. Once computational resources allow for 3D calculations of real microstructures, quantitative results can easily be obtained using the procedure outlined above. Acknowledgements The FE simulations were performed with ABAQUS V5.6 (Hibbitt, Karlsson & Sorensen, Inc.) under academic license. References [1] Wagoner RH, Chan KS, Keeler SP, editors. Forming limit diagrams: concepts, methods, and applications. Warrendale: The Minerals, Metals & Materials Society, 1989.
2054
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
[2] Marciniak Z, Kuczynski K. Limit strains in the processes of stretch-forming sheet metal. International Journal of Mechanical Sciences 1967;9:609. [3] Jones SE, Gillis PP. An analysis of biaxial stretching of a #at sheet. Metallurgical Transactions A 1984;15A:133. [4] Barlat F. Forming limit diagrams * Predictions based on some microstructural aspects of materials. In: Wagoner RH, Chan KS, Keeler SP, editors, Forming limit diagrams: concepts, methods, and applications, Warrendale: The Minerals, Metals & Materials Society, 1989, p. 275. [5] Lee WB, Yang W. Methodology and applications of mesoplasticity in manufacturing sciences. International Journal of Mechanical Sciences 1993;35:1079. [6] Tvergaard V. Analysis of tensile properties for a whisker-reinforced metal}matrix composite. Acta Metallurgica et Materialia 1990;38:185. [7] Weissenbek E, Rammerstorfer FG. In#uence of the "ber arrangement on the mechanical and thermo-mechanical behavior of short "ber reinforced MMCs. Acta Metallurgica et Materialia 1993;41:2833. [8] Iung T, Petitgand H, Grange M, Lemaire E. Mechanical behavior of multiphase materials. Numerical simulations and experimental comparisons. In: Pineau A, Zaoui A, editors, IUTAM Symposium on Micromechanics of Plasticity and Damage of Multiphase Materials. Dordrecht, Kluwer: 1996, p. 99. [9] Needleman A, Tvergaard V. Comparison of crystal plasticity and isotropic hardening predictions for metal}matrix composites. Transactions ASME, Journal of Applied Mechanics 1993;60:70. [10] Shen Y-L, Finot M, Needleman A, Suresh S. E!ective plastic response of two-phase composites. Acta Metallurgica et Materialia 1995;43:1701. [11] GaK nser H-P, Fischer FD, Werner EA. Large strain behavior of two-phase materials with random inclusions. Computational Materials Science 1998;11:221. [12] Hill R. On discontinuous plastic states, with special reference to localized necking in thin sheets. Journal of the Mechanics and Physics of Solids 1952;1:19. [13] Hosford WF, Caddell RM. Metal forming. Mechanics and metallurgy. Englewood Cli!s, NJ: Prentice-Hall, 1983. [14] Dubensky EM, Koss DA. Void/pore distributions and ductile fracture. Metallurgical Transactions A 1987;18:1887. [15] Becker R, Smelser RE. Simulation of strain localization and fracture between holes in an aluminum sheet. Journal of the Mechanics and Physics of Solids 1994;42:773. [16] Narasimhan K, Zhou D, Wagoner RH. Application of the Monte Carlo and "nite element methods to predict the scatter band in forming limit strains. Scripta Metallurgica et Materialia 1992;26:41.
Forming limit diagrams: a micromechanical approach H.-P. GaK nser *, E.A. Werner, F.D. Fischer CEMEF, Ecole des Mines de Paris, rue Claude Daunesse, B.P. 207, F-06904 Sophia-Antipolis, France Christian Doppler Laboratory for Modern Multiphase Steels, Technical University of Munich, Boltzmannstra}e 15, D-85748 Garching, Germany Institute of Mechanics, Christian Doppler Laboratory for Micromechanics of Materials, Montanuniversita( t Leoben, Franz-Josef-Stra}e 18, A-8700 Leoben, Austria Received 22 December 1997; accepted 26 June 1999
Abstract A method for obtaining the forming limit diagram (FLD) from a micromechanical approach is proposed. Periodic representative volume elements (RVEs) characteristic for the microstructure of a two-phase material containing particles are subjected to biaxial stretching with di!erent strain paths. In the course of straining, a groove forms and grows, "nally leading to a stress drop in the overall response. This deformation instability is obtained as a natural result of the strain "eld #uctuations caused by the inclusions in the two-phase material. For this approach, only the geometry of the microstructure and the material data of the constituents have to be known; no additional assumptions about the deformation process or about a macroscopic inhomogeneity are necessary. The in#uence of the RVE representation on the possible deformation patterns is investigated. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction There exists a great variety of methods for calculating the forming limit diagram (FLD) for stretch forming of sheet materials. An extensive review of experimental work as well as of theoretical approaches is given by Wagoner et al. [1]. Much of the theoretical work is based on the concept of imperfection analysis as originally proposed by Marciniak and Kuczynski [2]. The Marciniak}Kuczynski (MK) approach assumes an initial notch-like imperfection extending across the sheet. During the stretch-forming operation, the plastic strain accumulates in the
* Corresponding author now at: Numerical Simulation Department, Corporate Research, Hilti Corp., FL-9494 Schaan, LmH echtenstein. Tel.: #423 236 2316; Fax: #423 236 3844. E-mail address: [email protected] (H.-P. GaK nser). 0020-7403/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 9 9 ) 0 0 0 5 7 - 0
2042
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
Nomenclature a e e G e
h hM l ,l p p
l G l G l n t z E ,E R < =
principal strain ratio e /e equivalent macroscopic strain macroscopic strain in direction `ia equivalent strain of the matrix angle of inclination of the deformation bands in the RVE angle of inclination of the deformation band according to Hill's analysis Poisson's ratio (inclusions, matrix) equivalent macroscopic stress equivalent stress of the matrix initial edge length of the RVE in direction `ia current edge length of the RVE in direction `ia spacing of the deformation bands in the RVE strain hardening exponent of the matrix thickness and number of elements in thickness direction of a 3D RVE number of deformation bands in the RVE Young's modulus (inclusions, matrix) yield stress of the matrix volume of the RVE external work applied to the RVE
grooved region rather than in the rest of the specimen, thus leading to localized thinning in the notch. Finally, loss of stability occurs due to the reduction of the minimum cross-sectional area. However, the MK analysis proves to be very sensitive to the choice of the shape and size of the initial imperfection. One obvious remedy is to subdivide the imperfection analysis into three stages * homogeneous deformation, localization under constant load, and localized necking with a precipitous drop in load * and to "t those stages to the tensile test [3]. Another possibility is to conduct an MK-type analysis based on local #uctuations of microstructural quantities, as, e.g., internal damage from void growth and coalescence [4] or inhomogeneities of the crystallographic texture [5]. Both approaches use a suitable mesoscopic model to describe the microstructure and, therefore, cannot account exactly for the local stress and strain #uctuations on the microscale. On the other hand, numerous e!orts have been devoted to determine the e!ective properties of composites directly from their microstructure via the periodic micro"eld approach using either periodic arrangements of inclusions [6,7] or quasi-random arrangements [8]. It is well-known that, even in the plane strain state, band-like strain concentrations may develop near precipitates [9], and that the geometrical arrangement of the particles in#uences the evolution of the plastic #ow patterns [10]. It may be expected that these e!ects will be still more pronounced in a plane stress state, as prevailing near the surface of thin sheets. In the sequel, the in#uence of such quasi-randomly arranged precipitates on the formation and growth of a groove, and thus on the limits for stable plastic deformation in sheet metal forming, will
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
2043
be investigated using the periodic micro"eld approach. A feature inherent to this approach is that, in contrast to the usual MK-type analyses, the computations are carried out directly on the microscale by modeling the matrix-inclusion topology of the material explicitly. Thus, no prede"ned imperfection is necessary; rather, the notch develops in the course of deformation due to the interaction of the strain "elds caused by the individual inclusions.
2. Computational approach The material investigated consists of an isotropic elastic * J -plastic matrix with quasi randomly distributed elastic inclusions of cubical shape (the overall particle distribution is only quasi-random because the material is represented by one repeating RVE, even though the particle distribution within the RVE is random). The elastic properties of the matrix are characterized by a Young's modulus of E "210 000 N/mm and a Poisson's number of l "0.3, while the plastic
behavior is described by an initial yield stress of R "150 N/mm and isotropic hardening according to a power law p "532eL , where p denotes the matrix equivalent stress, e the matrix
equivalent plastic strain, and the strain hardening exponent has the value n"0.23. The respective data for the inclusions are E "600 000 N/mm, l "0.15. An in"nitely extending plane sheet is modeled by the periodic micro"eld approach, using representative volume elements (RVEs) with periodic boundary conditions. This type of boundary conditions enforces that the corresponding nodes of two associated boundaries (left/right and lower/upper boundaries) may di!er only by some identical vector; i.e., the boundary lines must remain parallel but need not remain straight. Fig. 1 displays both a three- and a two-dimensional "nite element (FE) discretization of a typical RVE. Both models consist of 25;25 second-order, reduced integration, Serendipity-type isoparametric elements in the `1a}`2a plane. Furthermore, in the three-dimensional model the thickness (`3a) direction is subdivided into "ve elements; in addition, 3D models with 10 and 15 elements will be investigated. Periodic boundary conditions are applied in the directions `1a and `2a in order to simulate an in"nitely extending sheet. No constraint is applied in the thickness direction, which corresponds to the free lower and upper boundary planes of a thin sheet. In the two-dimensional model, this condition is accounted for by using plane stress elements. To be truly representative of a material with quasi-randomly distributed inclusions, the RVE must contain a su$cient number of precipitates. For this purpose, one inclusion is discretized by one single element. Although the stress "eld near the phase boundaries cannot be captured precisely, this rather coarse approximation allows for the required amount of inclusions and for particle}particle interactions while keeping the computational cost reasonably low. On the other hand, some e!ects will not be captured: a trustworthy estimation of the in#uence of the inclusion shape, e.g., would require a much more accurate description of the stress and strain "elds around the inclusions than the one achieved by a single-element discretization of an inclusion. The main e!ect of the inclusions in this model is that they trigger deformation inhomogeneities in the matrix; the development of these inhomogeneities will be studied. Several RVEs with inclusion volume fractions between 2 and 8% are generated and loaded with prescribed biaxial stretches of di!erent ratios. The principal in-plane strains are denoted by e and e . Given the current edge lengths l and the edge length increments dl of the RVE, the G G
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
2044
Fig. 1. (a) 3D and (b) 2D FE mesh of an RVE describing a sheet material with randomly distributed inclusions (2 vol% inclusions).
macroscopic principal strain increments in the directions `1a and `2a are obtained as dl dl de " , de " . (1) l l Neglecting the elastic strains, the macroscopic strain increment in the direction `3a is calculated from the conservation of volume as de "!(de #de ), and the equivalent macroscopic strain increment is
(2)
de"( (de #de #de ). (3) The principal strains in the direction `ia, e , and the equivalent strain e are obtained by G integration:
e" G
JG
l de "ln G , G l JG G
(4)
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
2045
where l denotes the initial RVE edge length in direction `ia, and G
e" de.
(5)
For convenience of notation in the following discussion, the principal strain ratio is de"ned as e a" . (6) e The macroscopic equivalent stress p follows from the applied external work increment d= and from the volume <"l l l of the RVE as d= . (7) p" < de The deformation instability is assumed to occur when the macroscopic stress}strain curve reaches its maximum, dp "0, de
(8)
and the respective principal strains e and e give the corresponding point in the FLD. So, by prescribing di!erent strain ratios a in the range from !0.5 (uniaxial stress) to 1 (equibiaxial strain), the entire FLD is constructed point by point. This approach permits any path in principal strain space to be prescribed; though, the simulations in the present work were performed using radial strain paths only.
3. Comparison of 2D and 3D models To compare the deformation behavior of the 2D and the 3D models, both models are subjected to uniaxial tension in the direction `1a. The deformation patterns are depicted in Fig. 2; Fig. 3 shows the resulting stress}strain curves. In addition, Fig. 3 displays the matrix behavior and the stress}strain curves for a number of randomly generated 2D and 3D RVEs (with thicknesses t of 5, 10, and 15 units and elements, respectively). In all cases, the inclusion volume fraction was 0.02. It can be seen in Fig. 2 that the inhomogeneities of the strain "eld caused by the inclusions link up to form a series of inclined parallel grooves. The parallel pattern results from the periodic boundary conditions; the interdependence between the RVE geometry, the angle and the spacing of the grooves will be investigated below. While in the earlier stages of the deformation the composite shows slightly higher hardening than the matrix * due to the reinforcement by the particles *, in the later stages the geometric softening caused by the groove outweighs the reinforcement by the inclusions, "nally leading to a drop in the stress}strain curve, Fig. 3. This behavior is essentially the same for the 2D and the 3D models. However, the 2D RVEs show the instability at lower strains because of the somewhat poorer representation of the underlying geometry: while a 3D model can easily capture the exact geometry of the cell, in a 2D model the cubic inclusions become "bers with square cross-section
2046
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
Fig. 2. Equivalent plastic strain distribution of the (a) 3D and the (b) 2D RVE (2% inclusions) from Fig. 1 subjected to uniaxial tension in the `1a direction. Failure occurs by inclined grooves. The undeformed shape is drawn as wireframe for comparison.
extending in the thickness direction; thus, the necking between the inclusions is facilitated signi"cantly. These di!erences between the 2D and the 3D models have been investigated in detail in Refs. [7,8,11]. A pronounced scatter between the stress}strain curves from di!erent RVEs of the same inclusion volume fraction is noted for the 2D as well as for the 3D models. The limit strains for instability (horizontal tangent of the curves in Fig. 3) are spread over a range of about 0.1 for the 2D RVEs and as much as 0.3 for the 3D RVEs with 5 elements. This shows that there exist inclusion arrangements which favor the development of deformation concentrations, and such which delay them.
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
2047
Fig. 3. Macroscopic stress}strain curves obtained from 3D and 2D RVEs (2% inclusions). The stress}strain curve of the pure matrix is also shown. The 3D models exhibit thicknesses t of 5, 10 and 15 units, each of them containing cubic inclusions of unit edge length. Note the scatter between the di!erent random RVEs (scatter of the strain at instability of 0.1 for the 2D and of 0.3 for the 3D RVEs with t"5).
Clearly, for the 3D RVE, a discretization of only 5 elements in thickness direction is a very coarse description. To investigate the in#uence of the ratio between the size of the inclusion and the sheet thickness, Fig. 3 displays also the results from two other RVEs of the same inclusion volume fraction, but of double and three-fold thickness t (10 and 15 elements/units in thickness direction, respectively). It is seen that an increase of the sheet thickness results in a shift of the point of instability towards higher strains; however, the 10 element RVE reaches instability at roughly the same strain as two of the 5 element RVEs, which indicates that even in rather thick sheets necking may occur by the mechanism discussed above. The strains at instability will be lower for thinner foils, though. In the sequel, we use the 2D plane stress model as an inexpensive approximation of a thin foil in order to demonstrate that, in principle, the FLD can be constructed from a micromechanical model using a unit cell (RVE) representation of the microstructure, and to discuss the in#uence of the periodic boundary conditions on the failure modes.
4. Periodic grooves and the RVE approach Taking a closer look on Fig. 2, one remarks that the 2D and the 3D simulations di!er with respect to the number of grooves and to their inclination angle. Evidently, the periodic boundary conditions of an RVE impose some constraints on the possible con"gurations of the band-like deformation concentrations. Fig. 4 depicts schematically such an RVE containing parallel deformation bands, as observed in the simulation results (Fig. 2). Periodicity requires that an RVE of current length l must contain
2048
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
Fig. 4. Deformed RVE with z parallel grooves at an angle of h with respect to the principal tensile direction.
an integer number z of deformation bands of length l so that l "l /z, and the inclination angle of the grooves is given by
(9)
l l tan h" "z . (10) l l The logarithmic strains e give the link between the initial and the current edge lengths l and l of G G G the RVE, l "l exp e . (11) G G G Inserting the latter relation and the de"nition of the principal strain ratio a"e /e into Eq. (10), we obtain l tan h"z exp(a!1)e . (12) l The resulting admissible angles h are shown in Fig. 5 for some numbers z of deformation bands and principal strain ratios a of !0.5 and 0, the values of a which delimit the left-hand side of the FLD, for an RVE with l /l "1 as used in the present study. For the right-hand side of the FLD, a*0, the grooves are expected to be perpendicular to the major tensile axis, h"903. In this case, the RVE admits * contrary to Eq. (12), which predicts zPR for hP903 * also a single perpendicular groove. The additional constraints introduced by the use of periodic RVEs may be illustrated best by the following simple example. Let us assume that the deformation bands occur exactly as predicted by Hill's analysis [12] of localized necking in thin sheets, applicable to the left-hand side of the FLD. It must be emphasized that this assumption is made only for the purpose of illustration. Of course, the present approach is entirely di!erent from Hill's concept of localized necking and is thus expected to give somewhat di!erent results, apart from the fact that the deformation bands in the FE
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
2049
Fig. 5. Admissible angles h in an RVE with l /l "1, from Eq. (12), for di!erent numbers z of deformation bands and di!erent principal strain ratios a.
simulations manifest themselves gradually so that the onset of necking cannot be determined unambiguously. However, a comparison between Hill's analysis and the present results (see the following section and Fig. 7) will show that they di!er not too much one from the other. According to Hill's analysis [12], the left-hand side of the FLD is given by (cf. also [13]) e #e "e (1#a)"n, (13) where n denotes the strain hardening exponent of the material, and the corresponding angle hM of the shear band at the onset of localized necking is obtained from 1 tan hM " . (!a
(14)
Assuming that the onset of necking follows Eq. (13), the angle admitted by the RVE, from Eq. (12), has to satisfy the relation l a!1 tan h"z exp n. (15) l a#1 The resulting angles h and hM are compared in Fig. 6 for the present case of a matrix strain hardening exponent (because it is in the matrix where the neck develops) of n"0.23 and an RVE geometry of l /l "1 as employed in the FE simulations. Obviously, the RVE can accommodate Hill's results only at a limited number of points for special values of a and z; this illustrates quite well the additional constraints which result from the periodic boundary conditions. It may, however, be expected that the FE simulations will favor the angles that are close to `physical realitya; in other words, that the number of the bands will be z"3 for uniaxial tension (a"!0.5) and augment successively as the principal strain ratio a approaches zero.
2050
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
Fig. 6. Angles obtained in an RVE with l /l "1, accommodating Hill's analysis (strain hardening exponent n"0.23), Eq. (15), for di!erent numbers z of deformation bands and di!erent principal strain ratios a. Hill's result, i.e. hM from Eq. (14), is shown for comparison.
5. Forming limit diagram To obtain the complete FLD, randomly generated RVEs are subjected to straining with di!erent ratios a, and the points (e , e ) where instability occurs are assembled into the diagram. Fig. 7a shows such FLDs for three di!erent RVEs with inclusion volume fractions of 2, 5, and 8%; for comparison, Fig. 7b shows the corresponding predictions from Hill's and Swift's analyses (see Ref. [13]). One recognizes in Fig. 7a the typical V shape with the highest strains for uniaxial tension (a"!0.5) and equibiaxial stretching (a"1), and the smallest strains for uniaxial stretching (a"0). Also, as is to be expected, higher precipitate volume fractions yield lower strains in the FLD. The predictions of the present model for the left-hand side of the FLD are not too di!erent from the one obtained by Hill's model (which justi"es a posteriori the simple estimate of the band angles by means of Fig. 6), while on the right-hand side far higher limit strains than in Swift's analysis are reached. For the left-hand side of the FLD with strain ratios a between !0.5 and about !0.15, one obtains inclined grooves as already shown in Fig. 2. The number of grooves z varies, as predicted by the simpli"ed analysis of the previous section, between 3 and 4; the higher number of grooves prevailing E for less negative strain ratios, so that Eq. (15) may accommodate the higher angles to be expected from Eq. (14), cf. Fig. 6, and E in cases where failure occurs later than predicted by Hill's analysis (like in the 3D simulation of uniaxial tension in Fig. 2a), in order to give still roughly the same angle for the deformation bands in such highly stretched RVEs like in other cases where the instability is reached at lower strains, cf. Eq. (12).
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
2051
Fig. 7. (a) Forming limit diagrams obtained from 2D RVEs with 2, 5, and 8% inclusions. The straight lines depict the strain paths of uniaxial tension (e "!e /2), uniaxial stretching (e "0), and equibiaxial stretching (e "e ). (b) FLD constructed from Hill's analysis of localized necking (left-hand side) and Swift's analysis of di!use necking (right-hand side) for n"0.23, corresponding to the strain hardening of the matrix in the RVE computations.
For the entire right-hand side of the FLD, a*0, a single groove perpendicular to the major principal strain axis (Fig. 8a and b) is obtained. For strain ratios close to unity, failure may occur in a more di!use mode (Fig. 8b), but still exhibiting the perpendicular groove. Also for the part of the left-hand side of the FLD near the center (!0.1)a)0), the FE analyses favor a single perpendicular groove instead of a high number of near-perpendicular grooves as could be expected from Fig. 6. In fact, none of the simulations led to a number of grooves greater than z"4.
2052
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
Fig. 8. Equivalent plastic strain distribution of a 2D RVE (2% inclusions) subjected to (a) uniaxial (a"0) and (b) equibiaxial (a"1) stretching. Failure occurs by perpendicular grooves.
Regarding the in#uence of the inclusion volume fraction, the present results have to be interpreted with caution. As mentioned above, the inclusions serve mainly as triggers for the deformation inhomogeneities which coalesce subsequently to grooves. Thus, the point where instability occurs depends strongly on the clustering of the inclusions, the latter being in#uenced by inclusion rearrangements in randomly generated RVEs as well as by changes in their volume
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
2053
fraction. These dependencies have been pointed out in the experimental work by Dubensky and Koss [14]. Consequently, the inclusion rearrangements for di!erent RVEs of the same volume fraction will produce a scatter in the FLD; this has already been demonstrated in Ref. [16] for FLD predictions using an MK-like model with random sheet thickness distributions. In fact, in the present case it turns out that inclusion rearrangements may produce a scatter of the strain at instability of the order of 0.1 (cf. the 2D results in Fig. 3; the scatter reported in Ref. [16] is of the same order of magnitude). If we plotted such scatter bands in Fig. 7a, the FLDs for the three di!erent inclusion volume fractions would overlap one another to a great extent. This is partially due to the aforementioned interdependence, but also due to the rather coarse FE discretization; if one aimed at a detailed FE simulation of the mutual in#uences of voids or inclusions, this task would need a much more re"ned * and computationally much more expensive * numerical treatment (see, e.g., the case study by Becker and Smelser [15]). 6. Concluding remarks A method for obtaining the FLD from a micromechanical approach was proposed. The deformation instability is obtained as a natural result of the strain "eld #uctuations caused by the precipitates in a two-phase material. For this approach, only the geometrical arrangement of the second phase particles in the microstructure and the material data of the constituents have to be known; no additional assumptions about deformation history or about a macroscopic inhomogeneity are necessary. The analysis was conducted for a two-phase material with quasi-randomly distributed, ideally bonded precipitates. In this way it was shown that instabilities in sheet metal forming are not necessarily due to the growth and coalescence of the voids, but can also be a consequence of the two-phase nature of most materials. This may be important for the deformation behavior of thin foils made from materials with strong matrix-inclusion bonds which remain intact up to rather high strains. The in#uence of the sheet thickness with respect to the inclusion diameter was discussed as well as the additional constraints resulting from the approximation of the material structure by RVEs with periodic boundary conditions. Due to the limitations of the rather simple model, the conclusions have to be of a qualitative nature, only. Once computational resources allow for 3D calculations of real microstructures, quantitative results can easily be obtained using the procedure outlined above. Acknowledgements The FE simulations were performed with ABAQUS V5.6 (Hibbitt, Karlsson & Sorensen, Inc.) under academic license. References [1] Wagoner RH, Chan KS, Keeler SP, editors. Forming limit diagrams: concepts, methods, and applications. Warrendale: The Minerals, Metals & Materials Society, 1989.
2054
H.-P. Ga( nser et al. / International Journal of Mechanical Sciences 42 (2000) 2041}2054
[2] Marciniak Z, Kuczynski K. Limit strains in the processes of stretch-forming sheet metal. International Journal of Mechanical Sciences 1967;9:609. [3] Jones SE, Gillis PP. An analysis of biaxial stretching of a #at sheet. Metallurgical Transactions A 1984;15A:133. [4] Barlat F. Forming limit diagrams * Predictions based on some microstructural aspects of materials. In: Wagoner RH, Chan KS, Keeler SP, editors, Forming limit diagrams: concepts, methods, and applications, Warrendale: The Minerals, Metals & Materials Society, 1989, p. 275. [5] Lee WB, Yang W. Methodology and applications of mesoplasticity in manufacturing sciences. International Journal of Mechanical Sciences 1993;35:1079. [6] Tvergaard V. Analysis of tensile properties for a whisker-reinforced metal}matrix composite. Acta Metallurgica et Materialia 1990;38:185. [7] Weissenbek E, Rammerstorfer FG. In#uence of the "ber arrangement on the mechanical and thermo-mechanical behavior of short "ber reinforced MMCs. Acta Metallurgica et Materialia 1993;41:2833. [8] Iung T, Petitgand H, Grange M, Lemaire E. Mechanical behavior of multiphase materials. Numerical simulations and experimental comparisons. In: Pineau A, Zaoui A, editors, IUTAM Symposium on Micromechanics of Plasticity and Damage of Multiphase Materials. Dordrecht, Kluwer: 1996, p. 99. [9] Needleman A, Tvergaard V. Comparison of crystal plasticity and isotropic hardening predictions for metal}matrix composites. Transactions ASME, Journal of Applied Mechanics 1993;60:70. [10] Shen Y-L, Finot M, Needleman A, Suresh S. E!ective plastic response of two-phase composites. Acta Metallurgica et Materialia 1995;43:1701. [11] GaK nser H-P, Fischer FD, Werner EA. Large strain behavior of two-phase materials with random inclusions. Computational Materials Science 1998;11:221. [12] Hill R. On discontinuous plastic states, with special reference to localized necking in thin sheets. Journal of the Mechanics and Physics of Solids 1952;1:19. [13] Hosford WF, Caddell RM. Metal forming. Mechanics and metallurgy. Englewood Cli!s, NJ: Prentice-Hall, 1983. [14] Dubensky EM, Koss DA. Void/pore distributions and ductile fracture. Metallurgical Transactions A 1987;18:1887. [15] Becker R, Smelser RE. Simulation of strain localization and fracture between holes in an aluminum sheet. Journal of the Mechanics and Physics of Solids 1994;42:773. [16] Narasimhan K, Zhou D, Wagoner RH. Application of the Monte Carlo and "nite element methods to predict the scatter band in forming limit strains. Scripta Metallurgica et Materialia 1992;26:41.