Advanced Issues in springback

International Journal of Plasticity 45 (2013) 3–20

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International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas

Advanced Issues in springback Robert H. Wagoner a,⇑, Hojun Lim b, Myoung-Gyu Lee c a

Department of Materials Science and Engineering, Ohio State University, Columbus, OH 43210, USA Computational Materials Science and Engineering Department, Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185, USA Graduate Institute of Ferrous Technology, Pohang University of Science and Technology (POSTEC), San 31, Hyoja-dong, Nam-gu, Pohang, Gyeongbuk 790-784, South Korea

b c

a r t i c l e

i n f o

Article history: Received 8 March 2012 Received in final revised form 24 July 2012 Available online 24 August 2012 Keywords: Springback B. Constitutive behavior B. Metallic material C. Finite elements C. Numerical algorithms

a b s t r a c t For purposes of this review, springback is the elastically driven change of shape of a metal sheet during unloading and following forming. Scientific advances related to this topic have accelerated dramatically over roughly the last decade, since the publication of two reviews in the 2004–2006 timeframe (Wagoner, 2004; Wagoner et al., 2006). The current review focuses on the period following those publications, and on work in the first author’s laboratory. Much of this recent work can be categorized into five main topics. (1) (2) (3) (4) (5)

Plastic constitutive equations Variable Young’s modulus Through-thickness integration of stress Magnesium Advanced high strength steels (AHSS)

The first two subjects are related to accurate material representation, the third to numerical procedures, and the last two to particular classes of sheet materials. The principal contributions in these areas were summarized and put into context. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction ‘‘Springback’’ in the present context refers to the elastically-driven change of shape that occurs following a sheet forming operation when the forming loads are removed from the work piece. It is usually undesirable, causing problems such as increased tolerances and variability in the subsequent forming operations, in assembly, and in the final part. These effects typically degrade the appearance and quality of the products being manufactured. Springback involves small strains, similar in magnitude to other elastic deformation of metals. As such, it was formerly considered a simple phenomenon relative to the large-strain deformation required for forming. Nonetheless, appreciation for the subtleties of springback in two areas has grown dramatically. In particular, high precision is needed for the largestrain plastic response that directly affects the stresses in the body before removal of external forces. The unloading, while nominally linear elastic for most cases, can show remarkable departures from an ideal linear law. Interest in springback as a research area and application area is substantial and is growing rapidly. A previous review in April 2005 (Wagoner et al., 2006) showed that the word ‘‘springback’’ appeared in virtually no standard dictionaries at the time although the term had been in use since at least the 1940’s. A search of the ISE Web of Science database (Thomson Scientific) identified 334 published technical papers published since 1980. A Google search found 26,800 references to the word ‘‘springback.’’ ⇑ Corresponding author. Tel.: +1 614 292 2079; fax: +1 614 292 6530. E-mail address: [email protected] (R.H. Wagoner). 0749-6419/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijplas.2012.08.006

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R.H. Wagoner et al. / International Journal of Plasticity 45 (2013) 3–20

Today, many dictionaries include the term ‘‘springback.’’ (Example: Free Merriam-Webster – http://www.merriam-webster.com/dictionary/springback). A search conducted on February 29, 2012, shows the following changes over the intervening 7 years using topics in the Web of Knowledge (Thomson Reuters) and in a World Wide Web search with Google: 2005 334 26,800 N/A

Web of Knowledge Publications Google/‘‘springback’’ Google/‘‘spring-back’’

2012 1428 499,000 1759,000

Increase 428% 1862% –

It appears that there has been more information and interest in springback in the past 7 years than in all years previously. The current review starts from where two review papers published in 2004 and 2006 (Wagoner, 2004; Wagoner et al., 2006) left off. It focuses on new advances since then, more particularly on work performed in the first author’s laboratory. (Neither of these limitations is strictly followed, but this simple statement captures the intent.) Five recent topics in springback are discussed in sections of this document: (1) (2) (3) (4) (5)

Plastic constitutive equations. Variable Young’s modulus. Through-thickness integration of stress. Magnesium. Advanced high strength steels (AHSS).

The first two subjects are related to accurate material representation, the third to numerical procedures, and the last two to particular classes of sheet materials. 2. Overview Before looking at new advances, it will be useful to summarize briefly the field of springback in a general way. For a more comprehensive treatment, the previous reviews are recommended (Wagoner, 2004; Wagoner et al., 2006). Many reports of springback data must be critically evaluated before accepting the results. While many kinds of experiments have been performed, most of them do not control the sheet tension directly, thus making reproducibility and accuracy problematic. Examples of the kinds of geometries used include cylindrical tooling (Yu and Johnson, 1983; Yuen, 1990; Sanchez et al., 1996), L-bending (Livatyali and Altan, 2001; Mkaddem and Saidane, 2007; Gau and Kinzel, 2001), U-bending (Chen and Koç, 2007; Liu et al., 2002; Hino et al., 1999; Sudo et al., 1974; Chakhari et al., 1984), V-bending (Hino et al., 1999; Chakhari et al., 1984; Zhang et al., 1997; Tekaslan et al., 2006; Tekiner, 2004) and stretch-bend tests (Hino et al., 1999; Ueda et al., 1981; Kuwabara et al., 1996). In order to reproduce springback under mechanical conditions similar to industrial practice while at the same time providing the ability to control sheet tension, tool radius (R/t) and contact friction, the draw- bend springback (DBS) test was developed (Carden et al., 2002) from earlier designs used for friction testing (Wenzloff et al., 1992; Haruff et al., 1993; Vallance and Matlock, 1992). As shown in Fig. 1, a strip of sheet metal is formed around a circular tool (roller) and the front actuator applies a constant displacement rate while the back actuator provides for a constant restraining force. Rollers can be set to free, driven or fixed conditions to alter the friction condition between the tool and the specimen. Draw-bend Typical specimen dimension Length = 660 mm Width = 51 mm Friction control Roller: free or fixed

Available tool radii (mm) Fixed (15 sets) 1.0, 1.5, 2.0, 2.5, 3.2, 4.0, 4.8, 5.5, 6.4, 7.0, 7.9, 9.5, 11.1, 14.3, 19.0 Rotating/fixed (8 sets) 3.2, 4.8, 6.4, 7.9, 9.5, 11.1, 19.0 Draw restraint Constant back force

Right stroke Controls velocity and displacement

Fig. 1. Schematic of the draw-bend springback (DBS) test.


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testing has been used widely for springback experiments and corresponding simulations (Carden et al., 2002; Li et al., (1998, 1999a, 1999b, 2002); Geng et al., 2000; Geng and Wagoner, 2002; Wagoner et al., 2000). Springback predictions have been conducted using analytical methods and finite element analysis (FEA). In general, the analytical approach assumes simplified process and material properties. Analytical solutions for pure bending (Gardiner, 1957; Boklen, 1953; Queener and De Angelis, 1968; Marciniak et al., 1992; Hosford et al., 1993; Chan and Wang, 1999; Yu et al., 1996) and bending with tension (Zhang et al., 1997; Kuwabara et al., 1996; Hosford et al., 1993; Chan and Wang, 1999; Yu et al., 1996; Crandall et al., 1961; Woo and Marshall, 1959; Mickalich et al., 1988; Wenner, 1983; Duncan and Bird, 1978; Kuwabara et al., 1995; Takahashi et al., 1996; Kuwabara et al., 1999) for various different hardening laws have been presented. More recent development of analytical solutions includes springback in creep age-forming processes (Jeunechamps et al., 2006), prediction of sidewall curl (Moon et al., 2008; Zhang et al., 2007), springback of magnesium alloys (Lee et al., 2008, 2009), analytical models based on Hill’s yielding criterion and plane strain condition for U-bending and V-bending (Zhang et al., 2007a, b), and prediction of the draw-bend springback by a semi-analytical model (Lee et al., 2007). Finite element analysis (FEA) is a well-established tool for analyzing and predicting sheet forming strains for various materials and test conditions. FE simulation of springback, however, is much more sensitive to numerical tolerances and to material model than forming simulations (Li et al., 1999b; Wagoner et al., 1997; Mattiasson et al., 1995; Lee and Yang, 1998). Numerical procedures that must be considered more critical for springback simulation include the spatial integration scheme (Li et al., 1999a, 1999b; Lee and Yang, 1998; He et al., 1996; Focellese et al., 1998; Narasimhan and Lovell, 1999; Wagoner et al., 1999), element type (Li et al., 1998, 1999a; Wang and Wagoner, 2005) and time integration scheme such as implicit/implicit (Li et al., 1999a; Wagoner et al., 1997, 1999; Hu et al., 1999; Lee et al., 2009b; Guo et al., 2002), explicit/implicit (Mattiasson et al., 1995; Lee and Yang, 1998; He et al., 1996; Narasimhan and Lovell, 1999; Papeleux and Ponthot, 2002; Lee, 2005; Noels et al., 2004; Valente et al., 1999; Park et al., 1999), explicit/explicit (Montmayeur et al., 1999; Xu et al., 2004; Li et al., 1999), one-step approaches (Abdelsalam et al., 1999). Various material representations affect springback simulations significantly: the unloading scheme (Yuen, 1990; Li et al., 1999b; Wagoner et al., 1999; Tang, 1987), strain hardening rule (Mickalich et al., 1988; Wenner, 1983; Zhang and Lee, 1995; Han et al., 1999), evolution of elastic properties (Chakhari et al., 1984; Morestin and Boivin, 1996; Yu, 2009; Eggertsen and Mattiasson, 2009), plastic anisotropy (Chakhari et al., 1984; Geng et al., 2000; Geng and Wagoner, 2002; Wagoner et al., 2000; Ragai et al., 2005; Verma and Haldar, 2007; Gomes et al., 2005), Bauschinger effect (Gau and Kinzel, 2001; Kuwabara et al., 1996; Geng et al., 2000; Kuwabara et al., 1999; Focellese et al., 1998; Zhang and Lee, 1995; Baba and Tozawa, 1964; Tozawa, 1990; Pourboghrat and Chu, 1995; Yoshida and Uemori, 2002; Firat, 2007; Firat and Kaftanoglu, 2008; Bouvier et al., 2005; Dongjuan et al., 2006) and anticlastic curvature (Carden et al., 2002; Li et al., 2002) are prominent examples. In practice, springback is controlled in only two basic ways: (1) by increasing sheet tension (Carden et al., 2002; Mickalich et al., 1988; Kuwabara et al., 1995; Zhang and Lee, 1995; Cho et al., 2003; Moon et al., 2003; Song et al., 2007; Padmanabhan et al., 2008; Sunseri et al., 1996; Liu, 1988) to reduce springback, and/or (2) by compensating the shape of tooling to achieve a final target shape after springback (Gan and Wagoner, 2004; Lingbeek et al., 2005; Wagoner et al., 2007; Karafillis and Boyce, 1996; Cheng et al., 2007). The first method has dominated the solution to the problem of springback for the past century, and is still the mainstay. Upon increased sheet tension, the stress gradient through the thickness of the sheet is reduced and hence the bending moment and the total springback is decreased. In addition to reducing springback itself, it greatly reduces the variability inherent with changes in material behavior. The only downside – it is a significant one – is that increasing sheet tension promotes sheet splitting, particularly with newer high strength materials that typically have lower formability. Die compensation avoids this problem, but requires very accurate springback prediction and measurement, and it may not reduce the scatter of springback caused by typical process and material variations. Perhaps the major driving force for the rapid increase of interest in springback is the rapidly increasing specification of AHSS by automakers, particularly to grades such as dual-phase (DP) steel, transformation-induced plasticity (TRIP) steel, and twinning-induced plasticity (TWIP) steel. These steels represent unique challenges because in general they have higher strength/ductility combinations than traditional autobody steels and they make use of either very coarse microstructures (DP steels) or strain-induced transformations and complex hardening behavior (TRIP and TWIP steels). These differences manifest themselves in very large hardening transients following a stress reversal, large changes of elastic ‘‘modulus’’ following plastic deformation, and high temperatures attained by the plastic work in areas of large strain. It is not coincidental that much of the current work on springback focuses on these aspects.

3. Plastic constitutive equations 3.1. 2006 status Accurate springback prediction requires knowing the stress state throughout the body before unloading, which is controlled by the plastic response of the material during forming. Features of the material model that can often be ignored in

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satisfactory forming simulations must be taken into account for springback simulations. The dominant example of such a feature studied in recent years is the Bauschinger effect,1 which is seldom considered in applied sheet metal forming simulations, but which can change springback magnitudes by a factor of 2 Geng et al. (2000). That result utilized a form of kinematic hardening (Armstrong et al., 1966; Dafalias and Popov, 1976; Krieg, 1975; Chaboche et al., 1979; Ristinmaa, 1995; Jiang and Kurath, 1996) based on Armstrong–Frederick-type hardening rules (Chaboche et al., 1979, 1987; Chaboche, 2008, 1991; Chaboche and Rousselier, 1983; Chaboche and Nouailhas, 1989; Ohno and Wang, 1991, 1993; Jiang and Sehitoglu, 1996; Khan and Huang, 1995), with a two-surface plasticity formulation. It allowed modeling of so-called ‘‘permanent softening’’ following a path change by incorporating a bounding surface that translates and expands according to a mixed hardening rule (Hodge, 1957; Crisfield, 1991). This hardening law has been implemented for plane-stress thin-shell elements in conjunction with three anisotropic yield criteria: Hill’48 (Hill, 1948, 1950; Mellor et al., 1978), Barlat’s three-parameter yield function (Barlat Yld89) Barlat and Lian, 1989 and Barlat’s Yld96 (Barlat et al., 1997). When proper material anisotropy and non-proportional path hardening were incorporated, springback simulations agreed with measurements within the experimental scatter (Geng et al., 2000). 3.2. Advances since 2006 Nonlinear hardening models incorporating kinematic hardening have been widely adopted to improve the accuracy of the sheet metal forming simulations. Three main nonlinear hardening models used to predict springback accurately are: (1) Armstrong-Frederick type hardening models, (2) multi-surface-type hardening models and (3) a novel hardening model without simple kinematic hardening. 3.2.1. Armstrong–Frederick type hardening model Among many models describing the flow stress characteristics with the change of strain paths such as Bauschinger effect, transient behavior, and permanent softening, Armstrong–Frederick type non-nonlinear kinematic hardening models dominated before 2005 and remain prevalent, with new variations still being introduced. Choi et al. (2006) proposed an anisotropic hardening model describing an anisotropic evolution by rotation of the yield function combined with the common non-linear kinematic hardening concept. A comprehensive review on Armstrong–Frederick type hardening models is available in the recent review article by Chaboche (Lee et al., 2005) where the modeling capabilities of various nonlinear kinematic hardening models are compared in the context of predicting ratcheting effect. The role of path changes on applied springback predictions was quantified. Chaboche (2008) assessed various hardening models by the springback of a U-shaped rail. Springback prediction using the kinematic hardening underestimated the springback, while isotropic hardening overestimated it. Oliveira et al. (2007) studied the influence of hardening model on springback and found that not only must strain-path changes be accommodated, but also the magnitude of strain attained by each strain-path segment. 3.2.2. Multi-surface-type hardening model Recently, conventional two-surface models, such as the ones proposed by Krieg (1975) and Dafalias and Popov (1976), were integrated and extended in a unified mathematical context to incorporate anisotropy, the Bauschinger effect, transient behavior, and permanent softening (Lee et al., 2007). As shown in Fig. 2, the inner loading and outer bounding surfaces were decomposed into isotropic and kinematic hardening parts, thus providing a flexible implementation to accommodate complex material behavior. A simple but effective stress-update algorithm was introduced with a supplemental numerical treatment to resolve so-called overshooting problem. The draw-bend springback of aluminum alloy 5754-O sheet was predicted (Lee et al., 2007) within the experimental scatter using the non-quadratic anisotropic yield function Yld2000-2d (Barlat et al., 2003) and tension/compression experiments. Comparisons with standard Chaboche-type (or Armstrong–Frederick type) hardening models showed that it is essential to consider permanent softening as shown in Fig. 3. A standard method to characterize the change of the loading direction is not clear for general stress states and paths. In terms of the single surface model, particularly Armstrong-Frederick type nonlinear kinematic hardening model, various modifications have been made mainly to include the permanent softening observed at large plastic strain upon reversal loading (Geng and Wagoner, 2002; Choi et al., 2006). Note that the Chabochetype hardening model used in Fig. 3 is the standard combined isotropic-nonlinear kinematic hardening model that cannot reproduce the permanent softening. More rigorous models modified from this 2-parameter model have been able to reproduce the permanent softening successfully. Among other hardening models based on two-surface schemes, the Yoshida–Uemori (Y–U) model Yoshida et al. (2002) is gaining popularity in part because it has relatively few parameters to be determined and it has been implemented into the commercial FE software Pam-Stamp (ESI/PSI, 1995). It assumes kinematic hardening of the loading surface and combined isotropic-kinematic hardening for the bounding surface, with both surfaces based on Hill’s yield function (Hill, 1948). The model reproduces the Bauschinger transient similar to a general two-surface model, but it allows for work hardening stag1 We refer to the Bauschinger effect in its most general meaning as the evolution of hardening under non-proportional paths, particularly after a path reversal such as is encountered in a tension/compression test.


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Fig. 2. A schematics of the two-surface model; the two surfaces usually have the same shape to avoid penetration and the correspondence between points ‘a’ and ‘A’ on loading and bounding surfaces, respectively, is defined by common yield surface normal directions. The hardening rate is determined by the gap distance between current stress and corresponding stress on bounding surface, that is initially very stiff but gradually decreases (Lee et al., 2007).

Fig. 3. U-draw bending springback predictions with various hardening models (Lee et al., 2007).

nation at large plastic strain at the expense of additional complexity. It is unclear whether accounting for work hardening stagnation is critical for the accurate prediction of springback or not. Recently, Ghaei et al. (2010) proposed an implicit stress integration algorithm for Y–U model for ABAQUS (ABAQUS, 2006) and applied it to a channel draw springback of DP 600 steel sheet. Predicted springback angles using Y–U and Chaboche-type hardening models were in good agreement with experiments, although the accuracy was enhanced by taking into account reduced elastic modulus. 3.2.3. Hardening model without simple kinematic hardening approach Kinematic hardening models, whether involving one or two yield surfaces, still invoke the translation and expansion of a fixed surface shape, thus preserving the direction of normals at given locations. A new approach presented by Barlat et al. (2011) describes the smooth change of yield surface shape by homogenous yield function-based anisotropic hardening (HAH). The homogeneous yield function consists of a stable component associated with a general anisotropic yield function and a fluctuating component which distorts the overall shape of the yield surface. The yield surface shape is flattened opposite from the active stress state during the proportional loading, but this fluctuating component does not affect the shape of the yield surface near the active stress state. The HAH approach leads identical plastic flow stress response to the isotropic hardening for the monotonous loading, but the Bauschinger effect and transient hardening behavior can be efficiently reproduced by the appropriate control of fluctuating component during continuous and reverse loading. In this regard, the HAH model is similar to the combined isotropic-kinematic hardening, but does not involve the translation of the yield surface. The new feature in the HAH model is an introduction of a microstructure deviator which memorizes the previous deformation history and controls the continuous evolution of yield surface distortion during multiple strain path changes. For

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Fig. 4. Homogeneous anisotropic hardening model in p-plane. The yield locus evolution during uni-axial tension (UT) followed by equi-biaxial tension (BT). The stress is normalized by the uni-axial tension (Barlat et al., 2011).

example, the yield surface locus evolution during uni-axial tension followed by equi-biaxial tension is shown in Fig. 4 where microstructure deviator continuously saturates from initial uni-axial loading direction to equi-biaxial direction. The FE simulations for the NUMISHEET’93 U-channel benchmark showed that the new approach captures the Bauschinger effect, transient hardening behavior and permanent softening in cyclic stress–strain curves, which resulted in good agreement with experimentally measured springback (Lee et al., 2012, in press). Other approaches have also introduced the distortion of the yield function (Francois, 2001; Feigenbaum and Dafalias, 2007, 2008). In real forming process, the deformation is more complex such that neither stress state nor the strain path does not follow that of the simple uni-axial deformation. Instead, the deformation usually involves a significant strain change which leads to the development of more complicate constitutive models for the finite element simulation of sheet metal forming. For this purpose, physical descriptions of the microstructure changes during deformation (Teodosiu and Hu, 1998; Haddadi et al., 2006) to account for the Bauschinger effect were proposed. For example, Teodosiu and Hu. (1998) and Haddadi et al. (2006) used a tensorial form of dislocation structures developed during reverse loading and combined this microstructure change effect with the conventional nonlinear kinematic hardening. The microstructure change by the description of the dislocation structure during strain path change was also considered in the crystal plasticity approach to explicitly investigate the slip activities among slip systems in polyscrystals (Peeters et al., 2000; Li et al., 2003; Holmedal et al., 2008; Kim et al., 2012). Based on this dislocation density based hardening model, the effect of strain-path change on the springback prediction or on the two-stage deep drawing was investigated by using finite element analysis (Oliveira et al., 2007; Haddag et al., 2007; Thuillier et al., 2010). The advanced constitutive models could predict the complex material behavior during stain path change under uni-axial condition. However, there was no significant influence of the stagnated hardening or stress overshooting during reverse or cross-loading on springback compared to that predicted by the phenomenological isotropic-nonlinear kinematic hardening models which could also predict the Bauschinger effect. Although the physically-based hardening models or dislocation-based models have been applied to the forming and springback simulations, these models usually require many parameters. For example, the model by Teodosiu and Hu (1998) has 13 parameters to describe the evolutions of state variables including fourth- and second-order tensors. Recently, the HAH model was successfully extended to describe more complex material behavior during strain path change such as cross-loading (Barlat et al., in press). With 8 hardening parameters, the extended model could predict measured stress-strain responses of an EDDQ steel sheet sample during various two-step tension tests. Especially, the flow stress overshooting and work hardening stagnation with very low or negative strain hardening rate for cross-loading conditions could be captured reasonably. 4. Variable Young’s modulus 4.1. 2006 status Morestin and Boivin (1996) identified the significant role of an ‘‘elastic modulus effect’’ in springback prediction while Ghosh et al. (Luo and Ghosh, 2003; Cleveland et al., 2002) made detailed measurements for an aluminum alloy and


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high-strength steel. Various mechanisms for the nonlinear unloading behavior following plastic deformation have been proposed over the years: residual stress (Hill, 1950), anelasticity (Lubahn et al., 1961; Zener, 1948), damage evolution (Yeh and Cheng, 2003; Halilovic et al., 2009), twinning or kink bands in HCP alloys (Caceres et al., 2003; Zhou et al., 2008; Zhou and Barsoum, 2010a, 2010b), and piling up and relaxation of dislocation arrays (Morestin and Boivin, 1996; Luo and Ghosh, 2003; Cleveland et al., 2002; Yang et al., 2004). AHSS show large deviations from linear unloading according to the usual Young’s modulus following plastic deformation (Morestin and Boivin, 1996; Luo and Ghosh, 2003; Cleveland et al., 2002; Yeh and Cheng, 2003; Caceres et al., 2003; Yang et al., 2004; Augereau et al., 1999). Springback prediction can be improved significantly by adjusting the value of an apparent ‘‘elastic modulus’’ (Morestin and Boivin, 1996; Yu, 2009; Halilovic et al., 2009; Fei and Hodgson, 2006; Zang et al., 2007; Li et al., 2002; Pourboghrat et al., 1998; Eggertsen and Mattiasson, 2010; Vrh et al., 2008; Ghaei et al., 2008; Kubli et al., 2008; Sun and Wagoner, 2011; Kim et al., 2009; Andersson et al., 2002; Yamamura et al., 2002; Bjorkhaug et al., 2004; Yao et al., 2002; Nguyen et al., 2004; Wagoner and Li, 2007). Up to 2006 and beyond, nearly all proposed practical approaches adopt a chord modulus (Morestin and Boivin, 1996; Yu, 2009; Luo and Ghosh, 2003; Fei and Hodgson, 2006; Zang et al., 2007; Li et al., 2002; Ghaei et al., 2008; Kubli et al., 2008) that varies with plastic strain. The chord modulus uses a linear slope of stress–strain data from joining two points obtained from just before unloading and zero applied stress. It is readily incorporated in finite element (FE) simulations by adopting a different value of Young’s modulus (which may or may not be allowed to vary with the plastic strain before unloading). Unfortunately, such treatments, while convenient, do not capture the nonlinearity of the unloading response. This means that partial unloading (to a final value of residual stress, for example) will occur with significant errors of strain (corresponding to errors of final shape after springback). 4.2. Advances since 2006 Li et al. Sun and Wagoner (2011) examined the nature of the so-called ‘‘modulus effect’’ and then proposed and developed a corresponding novel continuum description. Simple tensile experiments of loading and unloading of DP 780 and DP 980 steels showed that the effect was not markedly strain-rate dependent, as would be expected on the basis of classical anelasticity. The observed unloading chord modulus was reduced in some cases by 30% relative to Young’s modulus. Simple considerations rule out damage evolution as the origin of such a large effect; such alloys show very little damage before fracture under sheet forming conditions (Kim et al., 2009). A new class of strain was identified. Named ‘‘Quasi-Plastic–Elastic (QPE) strain,’’ it is recoverable (elastic-like) but energy dissipative (plastic-like). A 3-D continuum constitutive law was devised to account for the three types of deformation behavior separated by two transition surfaces, one between linear elastic and QPE modes, and one between QPE and plastic modes. The QPE theory reproduced nonlinear loading and unloading curves following stress/strain path changes in a natural and highly accurate way. The magnitude of the QPE strain was found to be proportional to flow stress or elastic strain. The stress–strain behavior upon unloading depended only on the magnitude of the stress change after plastic deformation. Most surprising, the QPE relationship was found to be identical for two alloys when unloaded from the same flow stress, Fig. 5(a). Fig. 5(b) compares experimental unloading-loading data in tension with various constitutive models. The QPE model captures accurately all known features of the modulus effect whereas elastic or chord models do not. The initial unloading following plastic deformation occurs elastically according to the handbook value of Young’s modulus, until some critical stress

(a) Fig. 5. (a) Relationship between dissipated energy and stress at unloading of DP 780 and DP 980. (b) Comparison of QPE model with existing constitutive approaches for unloading and reloading following tensile deformation for DP 980 (Roberts, 1960).

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Table 1 Average draw-bend springback angle simulation errors of DP980 for normalized sheet tensions of 0.3, 0.6, 0.8, and 0.9.

Average error

Elastic (Eo), Isotropic hardening

Elastic (Eo), Chaboche

Chord [E(e)], Chaboche

QPE, Chaboche

18.0o

8.5o

5.3o

2.6o

is reached. At that point, a new constitutive behavior representing QPE is entered, with corresponding nonlinear unloading and energy absorption. Reloading follows a similar process, again transitioning from elastic behavior to QPE behavior and finally to plastic/elastic/QPE behavior at yielding. Table 1 show that springback predictions based on the QPE constitutive behavior are much more accurate than existing representations. A standard Chaboche-type plastic model, even with three component back stress evolution rule, cannot be fitted to reproduce the QPE effect adequately. However, the plastic work hardening behavior on reloading might be able to be modeled more accurately by introducing additional back-stress tensor. Draw-bend springback simulations using the QPE model agreed with experiments to within 3 degrees whereas other standard model formulations showed 5–18 degrees in average error. 5. Through-thickness integration 5.1. 2006 status Industrial FE simulation of complex sheet is carried out using shell elements almost exclusively because they make the simulations much faster by reducing the number of degrees of freedom relative to solid elements. This was true in 2006 and remains so today. For a given number of degrees of freedom, shell elements offer the possibility to treat stress variations through the thickness very accurately by using large number of integration points (IP). (The equivalent treatment with solid elements adds more elements and degrees of freedom, thus making the computation time unrealistically long.) In spite of this advantage, most simulations of industrial sheet forming operations rely on small numbers of integration points through the thickness, typically 3 or 5. Some commercial programs for sheet forming analysis in the past did not make provision for more than 7 or 9 integration points. Reports were made between 1999 and 2002 by Li et al. (2002) and Wagoner et al. (1999) that 25-to-51 IP were required to assure 1% numerical accuracy for springback simulations, depending on the bend ratio, R/t, and the magnitude of sheet tension. They recommended use of 25 IP for general springback analysis. This work set off a flurry of activity on the subject, including conflicting and contrary reports in the literature, particularly among users and developers of commercial software. Examples include reports that there is no difference in simulated springback for 3 to 10 IP (Andersson et al., 2002) or 7 to 15 IP (Yamamura et al., 2002), and only marginal differences between 5 and 20 IP (Bjorkhaug et al., 2004). Some authors recommended 7 or 9 IP (Xu et al., 2004; Yao et al., 2002; Nguyen et al., 2004) and one (Xu et al., 2004) even stated that 7 IP was optimal and larger number decreased the accuracy! (This would seem to be a mathematical impossibility, as was later verified. See below.) 5.2. Advances since 2006 In order to address this controversy, Wagoner and Li (2007) conducted analytical and numerical integration of bending moments for bending-under-tension of a beam. This removed numerical uncertainties associated with FE modeling. They evaluated the role of number of integration points (NIP) on the accuracy of the numerical integration alone under various conditions of varying R/t, sheet tension and strain hardening behavior. The simulations showed that the springback error limit must be considered, not the error for a single, particular simulation. The actual integration error is oscillatory in terms of small changes of R/t and sheet tension, which has the appearance of scatter if not computed systematically. This means that it is possible to get a very accurate answer within the limiting envelope, but such a result is purely fortuitous. (This may explain the much better agreement between simulations and experiments when the experimental results are known in advance!) As shown in Fig. 6(a), the numerical integration error limit depends on integration scheme and increases with increased sheet tension. No single integration scheme always performed best, but Gauss integration is preferred for the higher tension force range, where the interaction errors are largest. Fig. 6(b) and Table 2 show the effect of NIP and R/t on the accuracy of springback prediction. It was shown that the maximum error increases with decreasing bending radius and NIP. More IP are required for low-strength steel than for high strength aluminum, because the lower bending moment corresponds to larger fractional errors, and because the stress-strain curve has more curvature. These results confirmed the original recommendations of Li and Wagoner generally, although the predictions can be refined for particular cases. Therefore, reports of the adequacy of small NIP (3–9) for springback analysis should be critically examined before being adopted. Larger NIP can always reproduce a continuous stress distribution, and therefore the post-forming bending moment, more accurately, but at the expense of increased computational time. Burchitz and Meinders (2008) developed an adaptive through-thickness integration method which showed negligible difference for NIP larger than 20–25. Xia et al. (2006) showed that the details of the results can be modified by the boundary conditions adopted.

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(a)

(b)

Fig. 6. Effect of integration method and NIP. (a) Variation of unsigned limiting error with normalized tension force for three integration methods. (b) Variation of maximum fractional springback errors with R/t for low strength steel (IF) (Wagoner and Li, 2007).

Table 2 Number of integration points (NIP) required to limit the springback values for low strength steel using Gauss integration. Max error

1%

5%

10%

50%

R/t = 5 R/t = 20 R/t = 100

68 38 22

26 18 10

16 13 6

4 4 3

6. Magnesium 6.1. 2006 Status Magnesium alloys were being increasingly considered for sheet forming applications because of their low density (lower than aluminum) and high strength (similar to aluminum). The principal problem was the low ductility in tension at room temperature (Roberts, 1960; Bettles and Gibson, 2005). Therefore, the main areas of research focused on the deformation mechanisms, improving ductility, and possible warm forming applications. The limited ductility of magnesium alloys made the solution of other application problems moot. For example, springback would be expected to be large because the Young’s modulus of magnesium is less than 2/3 that of aluminum, which itself exhibits large and problematic springback. A search of the Web of Knowledge (Thomson Reuters) through 2005 shows only two papers combining ‘‘springback’’ and ‘‘magnesium’’ as topics (a third paper found in the search is mostly about aluminum-magnesium alloys). Chen and Huang (2003) showed that, as expected, forming at higher temperatures (and thus at lower stresses) reduced springback. In the other paper, Boger et al. introduced a large-strain tension-compression test that is particularly suited for revealing the widely differing strain hardening in tension and compression for magnesium alloys (Boger et al., 2005). 6.2. Advances since 2006 The number of papers combining ‘‘springback’’ and ‘‘magnesium’’ since 2006 is 38, as compared to 2 for all years previously. Clearly, while still small, interest in the subject is increasing. Most of these more-recent papers focus on the special asymmetric plastic constitutive response of magnesium alloys (Lee et al., 2009a, b, 2007; Kim et al., 2011, 2009; Hama and Takuda, 2011, 2012; Supasuthakul et al., 2011; Hama et al., 2011; Tadano, 2010), on the effects of elevated temperature or warm forming (Xiao et al., 2011; Greze et al., 2010; Hama et al., 2010; Gao et al., 2010; Ozturk et al., 2009; Palumbo et al., 2009; Kim et al., 2008; Bruni et al., 2006), or on special forming processes to mitigate problems (Kuo and Lin, 2012; Gisario et al., 2011; Han and Lee, 2011; Bunget et al., 2010; McNeal et al., 2009; Hino et al., 2009, 2008; Palumbo et al., 2008). Lou et al. (2007) measured the continuous, room temperature, large-strain, reverse-path hardening of Mg AZ31B and through metallography and acoustic emission associated the unusual features to twinning, untwining, and their exhaustion at room temperature. That work made use of the Boger tension–compression device (Boger et al., 2005) and the results were incorporated in complex plasticity formulations for FE implementation based on combined hardening models (Li et al., 2008, 2010), on two-surface models (Lee et al., 2007, 2008), and on visco-plastic polycrystal models (Choi et al., 2009). From approaches such as these, the room-temperature springback was computed using FE modeling (Lee et al., 2009b) and analytical models (Lee et al., 2008, 2009a; Kim et al., 2009). The two-surface approach (Lee et al., 2008) is based on a modified Drucker–Prager yield function to incorporate the strength differential effect. Simulations based on this approach predict the springback of magnesium at room temperature

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much better than conventional approaches, including the presence of a knee in the variation of springback angle with back force, Fig. 7. The drawback of conventional hardening model is clear from the comparison between measured and simulated springback results shown in Fig. 7. Various Armstrong–Frederick type isotropic-kinematic hardening models have been reasonably well applied for the springback prediction of cubic crystals such as aluminum and steels. The evolution of back stress in the AF-type nonlinear hardening model is fundamentally exponential for a monotonic uni-axial loading, which is well suited for the standard plastic flow stress behavior shown in cubic crystals. However, HCP materials such as magnesium alloys and titanium alloys having strong basal texture show significant strength differential, asymmetric plastic flow stress between tension and compression. Moreover, the shape of flow stress curves during twinning (or untwining) dominant deformation such as tension followed by compression is unusually inflected, which makes the conventional hardening law difficult to be directly applicable. Measurements of the springback of magnesium under controlled draw-bend conditions have been carried out at room temperature (Lee et al., 2009b, 2007). Such springback measurements, Fig. 8, show the expected large magnitudes corresponding to the low Young’s modulus (maximum angles of 100–120 degrees vs. 60 degrees for aluminum (Carden et al., 2002), but also some unusual features. In contrast to aluminum alloys (Carden et al., 2002), AZ31B magnesium alloys show an increase or flat aspect of springback for increasing sheet tension at lower values but a more rapid decrease after a critical point, such as was presented in Fig. 8.

(a)

(b)

Fig. 7. Comparison of springback angles predicted by four different models with various normalized back force for (a) R/t = 19.05, and (b) R/t = 11.1. (Chaboche IK model denotes the conventional nonlinear kinematic hardening model, Iso-Ten is the isotropic hardening model based on the tensile stressstrain curve, Iso-Comp. is the isotropic hardening model based on compressive stress-strain curve) (Lee et al., 2009).

(a)

(b)

Fig. 8. Effect of restrain back force on total springback angle (Dh), springback from the section of the strip in contact with the tool radius (Dh1 ), and the springback from sidewall curl (Dh2 ): (a) R/t=19.1, and (b) R/t = 22.2 (Lee et al., 2007). Note that arrows represent that fracture occurred in the specimens during applying the corresponding back forces.


13

Measurements of springback have also been performed at elevated temperature (Hama et al., 2010). For temperatures above 200 °C, which coincide with large reductions of flow stress and ductility increases to practical magnitudes, springback is drastically reduced or eliminated. The most recent work in the area of springback of magnesium alloys combines several of the aspects noted above. In particular, Piao et al. (2011) developed an elevated-temperature, tension–compression testing capability suitable to for measuring the strain hardening under non-proportional loading as a function of temperature (i.e. as twinning disappears and slip becomes easier). A new procedure (Piao et al., in press) makes use of this procedure in a single test to determine the presence or absence of twinning and a semi-quantitative estimate of the area of fraction of twins. 7. Advanced high-strength steels (ahss) 7.1. 2006 Status Springback was known to be one of the major problems limiting their widespread adoption of advanced high-strength steels (AHSS Wagoner, 2006. Fig. 9 compares the springback of traditional high-strength steel and AHSS with nominally the same yield stress. Evidently AHSS’s have more dramatic behavior that needed to be understood. Advances in plastic constitutive equations for AHSS have been addressed implicitly in earlier sections of this paper and will not be addressed again here. Pertinent to springback, AHSS, and in particular dual-phase (DP) steels, have several aspects that make them distinct from traditional mild and high strength/low alloy steels:

Strength-to-modulus ratios similar to aluminum alloys. High strain hardening concurrent with high strength. Very large plastic work products and thus high temperatures developed during deformation. Very large non-isotropic hardening effects after path reversals. Very large ‘‘modulus’’ changes when unloading after large plastic strains.

By 2006, steels of the era2 had been shown not to exhibit time-dependent springback, even at periods up to 7 years after forming (Wang et al., 2004). This intriguing phenomenon, first reported in 1997 (Wagoner et al., 1997) for aluminum alloys, refers to the change of a formed part’s shape for up to a year after forming. Results and analysis (Wang et al., 2004) showed that time-dependent springback is caused by room-temperature creep driven by residual stresses in the equilibrated body after forming. Conditions that produce high internal stresses relative to the yield stress favor time-dependent springback. 7.2. Advances since 2006 With the exception of time-dependent springback, which will be addressed below, only a few recent key AHSS references will be cited. Through-thickness variations of material properties (‘‘banding’’) in AHSS were associated with significant springback differences (Gan et al., 2006). Constitutive equations for 1-D (Sung et al., 2010) and 3-D plasticity (Sun and Wagoner, 2011; Sun et al., 2009; Kim et al., 2011; Sun and Wagoner (submitted for publication) were introduced and shown to allow accurate simulation of springback (Kim et al., 2011; Chung et al., 2008) and formability (Kim et al., 2011). Modulus changes were found to fit into a new constitutive framework (Kim et al., 2011) and patterns of multi-axial path hardening were identified and found amenable to interpretation by nonlinear kinematic hardening theories (Sun et al., 2009, submitted for publication). The remainder of this section will finish with a closer look at time-dependent springback of AHSS. Contrary to the status in 2006, Lim et al. (2012) reported the first observations of time-dependent springback of steels, in particular AHSS grades of DP 600, DP 800, DP 980, and TRIP 780. Similar to aluminum alloys, but opposite of traditional steels, all AHSS showed a linear increase of the springback angles with log time for the first few days to weeks (Fig. 10(b)). The magnitude of time-dependent springback decreased with increasing back force and tool radius, consistent with the behavior of aluminum and with a roomtemperature creep mechanism. The final time-dependent shape change of AHSS was approximately 1/3 of that observed for aluminum alloys under similar test conditions and was up to 18% and 6% of the total springback for Al 6022-T4 and DP 600, respectively. The mechanism of time-dependent springback of AHSS was found to be room-temperature creep, similar to aluminum alloys. The principal alternative, anelasticity, was found to exhibit kinetics more than an order of magnitude faster than time-dependent springback. Details of room temperature creep predictions for DP 600 was carried out, Fig. 11. Although good qualitative agreement was found, simulations over-estimated the time-dependent springback angle. However, predicted time-dependent springback using time hardening and strain hardening creep laws showed better prediction than the model adopting steady-state creep law, by a factor of 2–3 (Lim et al., 2012). While the choice of creep law is important (and standard ones from tensile tests do not give perfect predictions), other constitutive effects were minor, in all cases less that 2% on initial springback angle, and less than 7% on final time-dependent 2

Steels tested included DQSK (drawing quality, silicon-killed steel), AKDQ (aluminum-killed, drawing quality steel) and HSLA (high strength low alloy steel).

14


Fig. 9. Comparison of shapes after U-channel forming and springback of an AHSS (DP 600) and traditional high-strength steel (HSLA 450) having equal yield stresses (IISI, 2006).

(a)

(b)

Fig. 10. Measured springback (a) initial springback (t = 30 s) for tested AHSS and (b) time-dependent springback angles for DP 600 (Lim et al., 2012).

(a)

(b)

Fig. 11. (a) Measured and fitted creep laws and (b) measured and simulated time-dependent springback using three creep laws for DP 600 (Lim et al., 2012).


15

springback angle. Examples that were tested include detailed strain hardening law, plastic anisotropy, strain rate sensitivity and FE mesh size. Deformation-induced heating is another complication for draw-bend springback of AHSS (Sung et al., 2010; Kim et al., 2011). A thermo-mechanical FE model of the DBS test for DP 980 showed that incorporating the thermal effect changed the springback angle by 8%.

8. Concluding remark Although the capability of the springback prediction has been significantly improved in terms of the elastic–plastic constitutive equations and relevant FEM modeling as described in the previous sections, there are still remaining sources of uncertainty that should be considered in the future investigations. The topics that might be further improved in the springback analysis are: (1) (2) (3) (4)

Constitutive model considering the effect of different strain paths Friction Temperature New press technology

The first subject becomes more important as the emergence of the advanced high-strength steels since the conventional forming technology might not be applicable to successfully form parts within expected shape accuracy. In this case, the multi-stage forming technology might be necessary to improve both formability and shape accuracy, which may be optimized by the finite element modeling that takes the effect of different strain paths into account. Several plastic constitutive models have been proposed to capture the stress-strain responses under load reversal, cross-loading condition with previous deformation histories (Oliveira et al., 2007; Haddag et al., 2007; Thuillier et al., 2010; Barlat et al., in press). For example, the springback of U-channel type with or without pre-tension was proposed as a benchmark problem of Numisheet2011 (2011), which resulted in significant scatters among participants depending on their applied constitutive models. The work on the stain path effect may be especially needed in areas related to the strain-induced transformation of austenite to martensite and the consequent creation of anisotropic mechanical behavior, and the resultant effect of such anisotropic behavior on springback (Lee et al., 2007). The second and third topics, friction and temperature, are usually coupled to each other. Besides an understating of constitutive behavior of advanced high strength steels, it is also necessary to understand the interaction between material and tool surface, or frictional behavior, and temperature dependency on the springback. Especially for the AHSS, it might be imperative to investigate the change of friction conditions during forming because the sheet materials experience variable deformation rate, temperature and pressure between blank and tool. The influence of the temperature in the cold forming was proved to be significant in terms of the temperature dependent plastic flow curve and friction coefficient (Barlat et al., 2003; Yoshida et al., 2002). For example, in the real continuous forming conditions the temperature rise might be as high as 60–80°C with the AHSS, which influences the flow stress curves, friction, necking and springback. Therefore, more robust measurements of stress–strain behavior and friction by taking the temperature effect into account will be necessary for better simulation of springback in the sheet metal forming. To introduce more flexible control of the forming speed, frictional behavior and materials’ temperature during forming of advanced high strength steels, a new generation of presses, so called servo presses, emerged with the capability to control the stroke of all of the moving components of the stamping tool. The advantages of servo press were well documented by the previous articles (Osakada, 2010; Tamai et al., 2010). The primary feature of the servo press is able to control slide action independently by a set of powerful servo motors, which is not possible with traditional presses. Therefore with this servo press technology the displacement path can be liberally defined with respect to not only the position and speed but also the direction. Another advantage of servo press is that slide velocity can be controlled throughout the forming process. Since the number of achievable slide motions is indefinite, the computer simulation technology to optimize them is more crucial for the sheet metal forming with servo press. The constitutive equations should be compatible with path dependent mechanical behavior, including the effect of strain rate and the friction change as a function of forming speed and temperature.

Acknowledgements This work was supported by the National Science Foundation (Grant CMMI 0727641), the Department of Energy (Contact DE-FC26-02OR22910), the Auto/Steel Partnership, and the National Research Foundation of Korea (Grant NRF-2010-220D00037). Thanks are due the many authors whose work was cited here. Special acknowledgement to collaborators who contributed to this work extensively over many years: Kwansoo Chung (Seoul National University), David K. Matlock (Colorado School of Mines), Michael L. Wenner (G. M. Research, retired), James G. Schroth (G.M. Research), James R. Fekete (formerly General Motors, now at NIST Boulder), Sean R. Agnew (University of Virginia), Thomas B. Stoughton (G. M. Research), Fredric Barlat (Pohang University of Science and Technology), and Myoung-Gyu Lee (formerly of Ohio State University, now at Pohang University of Science and Technology).

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