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M. G. Lee

C. KimE. J. Pavlina

F. Barlat1e-mail: f.barlat@postech.ac.kr

Graduate Institute of Ferrous Technology,

Pohang University of Science and Technology,

San 31 Hyoja-dong, Nam-gu, Pohang,

Gyeongbuk 790-784, Republic of Korea

Advances in SheetForming—Materials Modeling,Numerical Simulation, and Press

Technologies Forming modern advanced high strength steels poses challenges that were not of real im- portance in the previous decades. These challenges are the result of the steels’ complexmicrostructures and hardening behaviors, and the problems directly related to the highstrength of the material, especially springback. New methodologies and processes arerequired to overcome these challenges and to produce formed panels via optimized form-ing processes. This paper reviews the key developments in the fields of numerical simula-tion of sheet forming processes, the material models required to obtain accurate results,and the advanced stamping presses and approaches for shaping modern steel sheet mate-rials into desired shapes. Present research trends are summarized, which point to further developmental possibilities. Within the next decade, it is predicted that numerical simula-tions will become an integral part of the developmental and optimization process for stamping tools and forming processes. In addition to predicting the strains in the formed  panel and its shape after trimming and springback, the simulation technology will also

determine the optimum displacement path of the forming tool elements to realize mini-mum springback. Toward those goals, digital servo presses are expected to become an in-tegral element of the overall forming technology.  [DOI: 10.1115/1.4005117]

1 Introduction

The previous 50 years has been a period of remarkable advanceswith respect to all facets of sheet steels and how they are formed,including the varieties of steel materials, their production methods,forming equipment and processes, and the technologies by which theforming processes are optimized. All of these developments werefueled by the world-wide socio-economic needs to reduce the energyrequired to produce steels and to reduce the environmental pollutionand energy consumption of operating the automobile, which is one

of the most important steel products. These developments did notoccur individually; rather, they have been inseparably tied to oneanother. For example, the demand for more fuel-efficient automo-biles necessitated lower-mass body structures which could only befabricated using thinner gauge and higher strength steels. Thesehigher strength steels could only be economically produced follow-ing innovations in continuous-casting technology and continuous-annealing lines. Furthermore, the high strength of the new steels andassociated forming difficulties spurred development of new genera-tions of stamping presses and forming tools. Finally, the unprece-dented demands associated with forming the new higher strengthsteels required process optimization via numerical simulation whichwas only facilitated by an explosive growth in computing power over the previous three decades.

In this review, the authors will describe the key developmentsin the fields of numerical simulation of sheet forming processes,in particular the material models required to obtain accurateresults, and the advanced stamping presses and approaches for shaping modern steel sheet materials into desired shapes. This pa-per is not intended to present an exhaustive and comprehensivereview of the topics covered. Rather, the authors wish to cover thesubjects only to such sufficient details as to substantiate the pres-ent research trends and to point to major developmental possibil-ities for the future. The core of this paper consists of three parts:

advances in materials modeling; advances in numerical simulationof sheet forming; and advances in stamping presses and processes.This structure was selected only for the sake of convenience,because a truly separate treatment of any of these subjects is bothmeaningless and impossible due to the obvious inseparable natureof these developments in their recent history.

2 Historical Background

Only three decades ago, the automotive industry of the United

States was beginning to learn how to form sheet steels at the350 MPa tensile strength level. At that time, forming panels withoutnecking failure or splitting was the main goal of production engi-neers. Springback was seldom a concern during this period. Engi-neers usually depended on experience and simple strain estimatesto design stamping tools and the forming processes. Gridded sam-ple blanks were also used to determine the strains in prototype pan-els for some applications. During this same time period, dual-phase(DP) steels at the 600 MPa tensile strength level were being intro-duced, but despite significant research and development activitiesand high expectations, these steels rarely found high-volume appli-cations. Application was limited because of inconsistent mechani-cal properties of the available DP steels of that era and sub-optimum design practices. Old design methods attempted to mini-mize strain in the formed panel, however, the high initial strain

hardening of DP steels and inconsistent mechanical propertiesresulted in variable springback, which was unacceptable in partsformed following the old practices. It took another two decades anddevelopments in other areas before the stage was set for substantialapplication of DP steels in automotive applications.

Sheet forming simulations demonstrated in the 1960s and1970s served to show the feasibility of these methods and technol-ogy, but wide-scale deployment in the stamping industry was lim-ited. It was not until the early 1980s when a finite-element codewas developed to simulate forming realistic part geometries. Atthat time, Norman Wang, at General Motors, developedGMFORM which was later improved and renamed PANELFORM. GMFORM was introduced with great expectations, but itdid not find significant use for real applications because even a

1Corresponding author.Contributed by the Manufacturing Engineering Division of ASME for publication

in the JOURNAL OF   MANUFACTURING   SCIENCE AND   ENGINEERING. Manuscript receivedApril 15, 2011; final manuscript received August 19, 2011; published online Novem-ber 28, 2011. Assoc. Editor: Gracious Ngaile.

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simple geometry required specially trained experts and long com-putation times even using supercomputers. It took another 20 yearsof advances in computing power and the proliferation of economicworkstations for numerical simulation of the forming processes tobecome commonplace.

GMFORM was not introduced outside of General Motors, butits core principles were the foundation of the commercial codes,which themselves had been the results of many efforts, mostlyfrom academia. The decades leading into the 1980s also mark theintroduction of other multipurpose commercial software codes.

However, in their early days, most of these codes suffered fromslow speeds because they did not have special versions tailored tospecific applications. At that time, the users were discussing theuse of supercomputers and massive parallel computers, but for obvious reasons the world waited for the coming of fast, inexpen-sive computers.

At the material forefront, Thyssen Steel of Germany led the firstexercise of fabricating an automotive body-in-white (BIW) from350 MPa grade steels in the 1980s. More recently, Porsche Engi-neering, along with thirty-five participating organizations, coordi-nated the ultra-light steel autobody BIW program. By the year 2000, the use of steels at the 350 MPa strength level became com-monplace and the automotive industry was actively investigatingsteels with strengths exceeding 350 MPa. Some of these newsteels have complex microstructures, often with austenite as a sig-nificant constituent, that were not found in steels of earlier peri-

ods. Transformation-induced plasticity (TRIP) steels andtwinning-induced plasticity (TWIP) steels are two notable exam-ples of this group. The term advanced high-strength steels(AHSS) was coined relatively recently to describe those steelswith nontraditional microstructures and forming behaviors. Morerecently, the term ultra-high-strength became openly used to des-ignate steels with strength levels exceeding those of previoushigh-strength sheet steels. The impetus behind these developmentsin sheet steel metallurgy is the ever increasing world-wide needfor lighter, more fuel-efficient automobiles.

The advent of sheet steels with high and ultra-high strength lev-els exposed technological issues that were ignored in earlier deca-des. Earlier design practices were focused on producing partswithout splits or other surface defects. However, the new steelsexhibited generally low total elongations and high strengths com-

bined with low deep-drawability (i.e., low r-values), which madeit necessary to design panels and forming processes in such waysthat forming strains were more evenly distributed over the panelsand that the resultant level of springback was tolerable. Becausespringback increases with material strength, it is obvious why theinterest in springback prediction and control has risen in such anexplosive manner in the past decade with the increasing applica-tion of AHSS.

Panels and components made from AHSS are typically formedat or near room temperature, but hot stamping is another area thathas also garnered interest in recent years. Hot stamping involvesforming a steel blank at high temperatures in the austenite phasefield followed by rapid quenching in the die to produce a marten-sitic microstructure. Hot stamping has the benefit of reduced pressloads because the metal’s flow strength is reduced at elevated tem-

perature and springback is greatly reduced. However, thisapproach has some significant drawbacks. First the elevated form-ing temperature requires stamping tools to be fabricated from hotwork diet steels or other expensive high-alloy tool steels. Fabrica-tion, maintenance, and refurbishment of these steels are consider-ably more time-consuming and expensive than those of diematerials for room temperature forming processes. The high form-ing temperatures also require additional means to protect sensitiveparts within the press environment. The maximum panel size of ahot stamped part is also limited. Finally, surface quality of theformed panel is diminished because of high temperature oxidationand decarburization effects that occur during the stamping pro-cess. Thus, this paper will be dedicated only to matters which per-tain to sheet stamping at room temperature. For more detail about

hot stamping, readers are referred to the excellent review by Kar-basian and Tekkaya [1].

In summary of this section, the present authors wish to call thelast decade and the near future as the new age of sheet metal form-ing characterized by numerical simulation which emphasizesspringback prediction and forming processes realized on a newgeneration of presses.

3 Advances in Materials Modeling

Section  2  provides a historical perspective to help emphasizethe need for advanced forming technology combined with accu-rate numerical simulations. For the latter, finite element (FE)methods are now very common. As will be described in Sec.  4,many parameters related to numerical stability, such as elementtype, solution method and contact algorithm, and parametersrelated to the physical problem itself, such as the tool and part ge-ometry and constitutive (material) model, need to be carefullyselected. Moreover, a practical compromise between computationtime, user-friendliness, and accuracy is essential for the successfulapplication of FE methods to industrial sheet forming processes.This section (Sec. 3) provides a brief review of the trends withinconstitutive modelling. In order to develop an accurate materialdescription for a given application, it is necessary to understandthe mechanisms of plastic deformation and to include the relevantparameters as input. Therefore, before discussing approaches to

material modelling for sheet metal forming simulations, certainaspects of the plasticity of metals and alloys at macroscopic andmicroscopic scales are first briefly reviewed.

3.1 Continuum Aspects in Plasticity.   The stress–strainbehavior of metals and alloys within a low strain range is almostalways reversible and linear with the ratio of the stress to strain inthis range given by the elastic, or Young’s, modulus of the mate-rial. However, this elastic range is bounded by the yield limit,which is the stress above which permanent, or inelastic, deforma-tion occurs. The yield stresses in tension and compression are of-ten identical but this behavior is not true for undeformed materialsthat exhibit the strength-differential (SD) effect. In the plasticrange, the flow stress, simply defined as the stress required to con-tinue plastic deformation, usually increases with the total amountof dissipated plastic work or a corresponding measure of accumu-lated plastic strain. The flow stress becomes the new yield stress if the material is unloaded and reloaded.

In general, plastic deformation occurs without any volumechange and is not affected by hydrostatic pressure. This assump-tion is valid when phase transformation and damage (microvoids)in the material are not significant. When a material is deformed upto a given strain, unloaded, and then loaded in the reverse direc-tion, typically, tension followed by compression, the Bauschinger effect can be observed. The Bauschinger effect describes the phe-nomenon of which the yield stress after strain reversal is lower than the flow stress at unloading while the initial strain hardeningrate is higher. This phenomenon is a consequence of the build upof a back-stress, which can be considered a self-equilibrated stressfield in the matrix that remains when the material is freed from

external loads. The Bauschinger effect is different from the SDeffect previously described.

At low homologous temperatures, flow stress, and plasticity ingeneral, exhibit little time dependence. However, higher tempera-tures reduce a material’s flow stress and often introduce signifi-cant strain rate effects. Furthermore, at higher temperatures amaterial can deform by creep even if the stress within the elasticlimit.

3.2 Microstructural Aspects in Plasticity.   Commercialmetals and alloys used in forming operations are polycrystalline.They are composed of numerous grains, each with a given latticeorientation with respect to some arbitrary macroscopic axes. At

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low homologous temperatures, metals and alloys deform by dislo-cation glide, or slip, or by twinning on given crystallographicplanes and directions, both of which produce microscopic shear deformations. Because of the geometrical nature of slip and twin-ning deformations, strain incompatibilities arise between neigh-boring grains and produce microresidual stresses, which arepartially responsible for the Bauschinger effect. Both body-centered cubic (BCC) and face-centered cubic (FCC) materialstend to deform by slip because of the large number of availableslip systems. However, the number of potential slip systems in

hexagonal close-packed (HCP) materials is limited, and thesematerials tend to accommodate an imposed deformation by atwinning mechanism as opposed to dislocation glide or slip. Dur-ing deformation, the dislocation density increases and dislocationsaccumulate at microstructural obstacles, including twin bounda-ries, which increase the slip resistance for further deformation.This process is termed strain hardening and produces the charac-teristic shape of the stress–strain curve. More slip systems areavailable to accommodate deformation at elevated temperatures[2]. Atomic diffusion is also another mechanism that affects plas-tic deformation at high temperature and contributes to creep.Superplastic forming is performed at elevated temperature and de-formation mainly occurs by the sliding of grain boundaries weak-ened by thermal activation. In this case, the grain size andmorphology are important parameters.

Most commercial materials contain second-phases or interme-

tallic particles. These phases are present in materials by designin order to control either the microstructure (e.g., grain refine-ment) or mechanical properties (e.g., increased strength). How-ever, in some cases, large amounts of second-phases areundesired. Regardless, the presence of these inhomogeneitiesalters the material behavior because of their differences in elasticproperties with the matrix as in the case of composite materials,or because of their interactions with dislocations in the case of precipitation strengthened alloys. In both cases, these effects pro-duce incompatibility stresses that contribute to the Bauschinger effect. If the second phase is metastable, then a phase transfor-mation can be induced by the imposed stress or strain. Thiseffect is exploited in TRIP steels where metastable austeniteretained at room temperature is transformed into martensiteunder an applied stress or strain. TRIP steels are noted for their 

enhanced strain hardening behavior and ductility, which resultfrom the additional stress field created by the transformation [3].If a phase transformation occurs, then volume conservation canno longer be assumed.

In metal forming, ductile fracture generally results from themechanisms of void nucleation, growth, and coalescence [4]. Theassociated microporosity leads to volume changes and hydrostaticpressure will affect the deformation behavior of the material,although the metal itself is still plastically incompressible. At lowhomologous temperature, hard second-phases are the main sites of damage. The stress concentration at these phases initiates voidnucleation, and void growth continues by plasticity. Void coales-cence results from plastic flow microlocalization of the ligamentsbetween voids. At higher temperature, when creep becomes domi-nant, cavities nucleate at grain boundaries by various mechanisms

including grain sliding and vacancy concentration [5]. Generally,materials which have undergone creep or superplastic forming ex-hibit higher porosity levels than those deformed at lower temperature.

3.3 Constitutive Modelling Formulation.   Plasticity can bestudied at various scales [6] but for industrial sheet forming appli-cations, macroscopic models are most appropriate. Because of thescale difference between the microstructure and an engineeredcomponent, the amount of microscopic material information nec-essary to store in a forming simulation would be monumental. Itsimply would not be possible to track all of the relevant micro-structural features in detail during the simulation. Therefore, it ismore appropriate to integrate all of the relevant microstructural

features and effects into a few macroscopic variables.Microscopic models still have utility and are more suitable as amethodology for material design, as a tool for fundamental under-standing of plasticity, and as a guide for inferring suitable formu-lations at the macroscopic scale.

Most of the constitutive equations that describe plasticity havebeen developed in differential forms and are assumed to containsufficient information about the material’s deformation history.Constitutive equations have been developed in scalar forms for uniaxial loading and in tensorial forms for multiaxial loading,

which are more applicable to most real forming applications. Theconstitutive laws generally consist of a state equation and someset of evolution equations, for instance

_e ¼ ðr;H; xiÞ

_ xi ¼   _ xiðr;H; xk Þ(1)

The state equation provides the relationship between the strainrate,   _e, stress, r, temperature, H, and state variables, xi, which rep-resent the microstructural state of the material. The amount of plastic work or an associated measure of the effective strain is oneof these state variables. The evolution equations represented by   _ xi

in Eq. (1)  describe the development of the microstructure throughthe changes of the state variables. When a material model is for-mulated, it is necessary to implement it in a FE code, which

requires specific incremental formulations. This issue will beaddressed briefly in Sec. 4.

3.4 Yield Surface and Plastic Potentials.   For time-independent plasticity in multiaxial stress space, plastic deformationis well described by a yield surface, which defines the initiation of plastic deformation, and a flow rule and a hardening law, which cor-respond to the state and evolution equations, respectively [7]. Anextensive review of yield functions, /, for many types of materials isgiven by Yu [8]. For metals and alloys, the associated (or normality)flow rule, i.e.,   _e ¼   _k@ /=@ r, where   _k   is the plastic multiplier, is agood approximation and is discussed by Bishop and Hill [9]. Strainhardening can be isotropic or anisotropic. The former corresponds toan expansion of the yield surface without distortion due to anincrease of the dislocation density. Isotropic hardening is completely

defined by a single stress–strain curve. Any other form of hardening,such as kinematic hardening, which corresponds to the translation of the yield surface, is anisotropic.

It is necessary to describe yielding as a function of the stresstensor because stress states are multidimensional. For cubic met-als, there are usually enough potentially active slip systems toaccommodate any shape change. Moreover, compressive and ten-sile yield strengths are virtually identical. In the isotropic case,yielding of such materials is usually represented adequately by aneven function of the principal values of the stress deviator,  sk ,such as

/ ¼ js1   s2ja þ js2   s3ja þ js3   s1ja ¼ 2ra (2)

This isotropic yield function,  /, was proposed by Hershey [10]

for a FCC polycrystal with a random distribution of grain orienta-tions. In fact, Eq.   (2)   is an example in which a microstructuralmodel guided the development of a macroscopic description sinceHershey based his proposed function on self-consistent crystalplasticity results. The yield condition is fulfilled when   r ¼  hðeÞ,where hðeÞ is a function of the dissipated plastic work or the corre-sponding accumulated plastic strain,   e, and possibly some other state variables. The exponent, a, is connected to the crystal struc-ture of the material, and takes a value of 6 for BCC materials anda value of 8 for FCC materials [11]. Besides Eq.  (2), other iso-tropic forms that provide a good description of FCC and BCC pol-ycrystals can be found in Refs. [12,13].

In sheet forming, plastic anisotropy is an important aspectbecause it influences the strain distribution in a part and,

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consequently, the critical failure spots. Proper anisotropic plastic-ity formulations can be obtained when they are developed withina suitable framework [14]. For instance a method suitable for con-vex formulations was developed based on linear transformationsof the stress tensor [15]. This approach is detailed below for incompressible materials for which a linear transformation is per-formed on the stress deviator, s, leading to the transformed tensor,~s, given by

~s ¼  Cs   (3)

where C  is a fourth order tensor that contains the anisotropy coef-ficients and accounts for the macroscopic symmetries of the mate-rial. It reduces to the 4th order identity tensor in the case of isotropic materials. In this theory, an anisotropic yield condition isexpressed with an isotropic yield function, /, but with the princi-pal values of ~s, i.e.,  ~s1,  ~s2, and  ~s3, as arguments.

In the absence of anisotropic hardening, plastic anisotropy iscontained in the shape of the yield surface, which is influenced bythe tensor  C  in Eq. (3). Two anisotropic generalizations of Eq. (2)were proposed recently, each using two linear transformations of the type given by Eq. (3)  with fourth order tensors  C0 and C00. Thefirst, Yld2000-2d [16], is restricted to plane stress loading, whilethe second, Yld2004-18p [15], applies to any general stress state.These yield functions were successfully implemented into FEcodes [17,18]. Simulation procedures as well as application resultscan be found in Yoon et al. [17 – 19]. Other recent approaches tomodel plastic anisotropy can be found in Refs. [20,21].

For most HCP metals (e.g., Ti, Mg, Zr, etc.), at low tempera-tures or high strain rates, twinning plays an important role in plas-tic deformation. Unlike slip, twinning is sensitive to the sign of the applied stress, which is conducive to a strength differentialeffect [22]. Furthermore, the strong crystallographic texture dis-played by HCP materials leads to pronounced anisotropy. Todescribe the yield asymmetry of HCP materials, Cazacu et al.[23 – 25] proposed two yield functions, one based on the generaltheory of tensor representation [14] and the other based on a linear transformation of the stress deviator. Both approaches were ableto capture the SD effect in HCP materials very well.

In the classical flow theory of plasticity, the yield functionserves as a potential for the plastic strain rates (or strain incre-

ments) as implied by the normality flow rule. Hill [26] proved theexistence of the dual conjugate, w, to a stress potential, /, for rateindependent perfect plasticity. This potential can be expressed interms of the dual strain rate variables and its gradient leads to thestress deviator, i.e.,   sij  ¼  l@ w=@ _eij , where   l   is analogous to theplastic multiplier. Arminjon et al. [27,28] and Van Houtte [29,30]introduced strain rate potentials for a polycrystal. The strain ratepotential proposed by Barlat et al. [31,32] has a structure similar to the stress potential Yld91 [33] (Eq.   (2)) but with one linear transformation in strain rate space. Recently, Kim et al. [34]developed the following strain rate potential

w ¼ j ~ E002 þ   ~ E00

3 jb þ j ~ E003 þ   ~ E00

1 jb þ j ~ E001 þ   ~ E00

2 jb þ j ~ E01jb þ j ~ E0

2jb

þ j ~ E03jb ¼ ð22b þ 2Þ_eb

(4)

where   _e is the corresponding effective strain rate and b  is an expo-nent recommended to be 3=2 for BCC materials and 4=3 for FCCmaterials. The values   ~ E0 and   ~ E00

i   are the principal values of twotensors,   ~_e0 and   ~_e00, resulting from two linear transformations of theplastic strain rate tensor,  _e.

3.5 Strain Hardening.   Empirical plastic flow descriptionshave been postulated to represent the stress–strain behavior of metals and the well-known power laws by Hollomon and Swift,among others, using a strain hardening exponent,  n, are suitablefor a number of steels. For aluminium alloys, it is often reportedthat the Voce law given as

rðeÞ ¼  rs  ðrs   r yÞ expðe=esÞ   (5)

where rs, r y, and es  are constants that depend on material and testconditions, provides a better approximation of the stress–straindata compared to power laws. The Voce law is obtained directlyby integration of the dislocation-based state and evolution equa-tions developed by Kocks and Mecking [35,36], thus providing aphysical validation for its use. However, a drawback of this hard-ening law is that extrapolated flow stresses beyond the data rangeused for fitting are usually underestimated (e.g., [37]). In HCP

metals, twinning leads to a hardening rate that is strongly influ-enced by the mode of deformation (i.e., tension or compression)[38]. In general, the accuracy of a constitutive model’s ability topredict the stress–strain curve depends on the ability of the modelto capture the physical phenomena involved in plastic deforma-tion. For instance, TWIP steels exhibit high strength (UTS of 1400 MPa), large elongation (about 0.5) and almost linear strainhardening behavior. This combination of properties is the result of very fine twinned regions produced during straining, which leadto a dynamic Hall–Petch effect. It is likely that the formulation of an accurate strain hardening law for this class of materials willrely on a deep understanding of the underlying mechanisms of plastic flow for these materials.

Nonisotropic hardening effects can be described classically by kine-matic hardening and are usually related to the microstresses resultingfrom strain incompatibilities between grains, dislocation–dislocationinteractions, or by the interactions between the matrix and second-phases. This type of hardening captures the Bauschinger effect veryefficiently and can be described in its simplest form by the followingrelationship

/ðr   aÞ ¼  hðeÞ   (6)

where  r is the applied stress tensor and a is the back stress tensor,which controls translation of the yield surface. Evolution laws for this tensor can take many forms, including that given in Sec.  4.Teodosiu and Hu [39] proposed an evolution law with several ten-sorial state variables to account for the dislocation microstructurein a material. Yoshida et al. [40] published kinematic hardeningrules specifically for sheet forming applications. Kinematic hard-ening can be successfully applied in forming simulations where

the loading direction is changed abruptly (e.g., Ref. [41]) such asfor springback prediction in sheet forming.

As an alternative approach, the Homogeneous yield function-based Anisotropic Hardening (HAH) model (Barlat et al. [42])was formulated using a microstructure deviator,  hs, as a state vari-able that encapsulates and tracks the prior material deformationhistory [2]. In general, this tensor can be viewed as the continuumrepresentation of a given set of active slip systems, irrespective of the slip direction, for a representative volume element of a poly-crystal. The microstructure deviator is governed by its own set of evolution equations but the deviator is always normalized. In theHAH approach, a first degree homogeneous function of the stress,defined as a combination of a stable component, /, and a fluctuat-ing component, /h, was proposed as

UðsÞ ¼ ½/q þ /qh

¼ ½/q þ f q

1 jhs: s  jhs

: sjjq þ f q

2 jhs: s þ jhs

: sjjq1q ¼   r

(7)

Any isotropic or anisotropic yield function, /, homogenous of anarbitrary degree, may be used as the stable component in thistheory. However, it must be first reduced to a yield function of degree one and written in the form of an effective stress such as/ðsÞ ¼   r. In the above equation,  q  is a constant exponent, and f 1and f 2 are two state variables whose evolution laws depend on thesign of the double dot product s  :  hs [42]. As an illustration, Fig. 1represents the experimental stress–strain data of an uniaxial ten-sion-compression-tension test sequence of an interstitial-free steel

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sheet and the predicted curves assuming HAH or isotropic harden-ing. Figure 2  schematically shows the distortion of the yield sur-face during the first tension leg [43]. The HAH approach is

believed to simplify the material description for numerical simula-tions of springback.

3.6 Strain Rate and Temperature Effects.   Viscoplasticitydescribes the time-dependent material behavior that occurs whenthe homologous temperature is less than one half. In the case of viscous plastic flow, plastic deformation still occurs by the motionof dislocations and the models used for classical plasticity are stillvalid, but it is necessary to include terms containing the strain rateinto the constitutive models. A widely used approach to modeltime effects on inelastic behavior is called viscoplastic regulariza-tion (for instance, see Perzyna [44]). The basic assumption is thatthe viscous properties become noticeable only after the initiationof plastic deformation. Thus, the strain rate can be additively

decomposed into elastic and viscoplastic parts. In this case, theexpressions of the constitutive equations (e.g., yield function andinelastic potential) are the same as those used to describe elasto-plastic time-independent behavior. The inelastic strain is propor-

tional to the overstress (see also Ref. [45]) which is the differencebetween the applied stress and the quasi-static stress. Note thatPlunkett et al. [46] developed an anisotropic elasto-viscoplasticmodel based on the overstress concept [44]. This model is able totake into account the simultaneous influence of strain-rate, tem-perature, and anisotropy on the inelastic response of a texturedmetal for monotonic loading paths.

At higher temperatures, such as those which are common in hotpress forming, the flow stress can be empirically represented byfunctions of the accumulated strain, strain rate, and temperature,

and include parameters such as the melting point. Furthermore, inthis temperature range, recrystallization and phase transformationmay strongly affect the material properties and necessarily requirethe use of additional constitutive parameters. Since temperatureinfluences the kinetics of microscopic deformation mechanisms, ithas an effect on plasticity, similar to that of the strain rate.Increasing the temperature under which an experiment is con-ducted is equivalent to decreasing the strain rate without a changein temperature. Therefore, in processes involving hot deformation,these two variables, strain rate and temperature, can be combinedinto a single quantity called the Zener–Hollomon parameter (seeHosford et al. [6]), thus reducing the number of variables in theconstitutive relationships.

3.7 Damage Approaches.   Modelling approaches that

include damage are useful for the prediction of failure. An exam-ple of such a constitutive model was proposed by Gurson [47] andlater extended by Tvergaard [48] (see also Ref. [49]) and is givenas

/ ¼  re

r

2

þ2q1 f  cosh  3q2rm

2r

 1 þ ðq1 f Þ2 ¼ 0   (8)

where the porosity, f , (i.e., void volume fraction) and the effectivestress,   r, are state variables. The term  re  is the von Mises equiva-lent stress of the damaged material and q1 and  q2 are material con-stants, both equal to 1 in the original Gurson model [47]. Thistype of formulation can include other physical aspects that affectvoid growth such as void morphology and alignment, the presenceof hard inclusions inside voids, and the nucleation and coales-

cence of cavities. If these other aspects are included, then addi-tional state variables and their evolution laws are necessary[49 – 51]. Pardoen and Hutchinson [52] studied the simultaneousinfluence of a number of relevant variables, including coales-cence, on porosity growth. A critical review of the nonlinear mechanics of materials containing voids, and the estimation of their overall properties, was recently published by Huang andWang [53]. Another approach to account for porosity or other forms of material degradation is called continuum damagemechanics and it modifies the applied stress tensor,   r, by somedamage tensor, D, (e.g., Refs. [54 – 56]) following

~r ¼ ½I  Dj1r   (9)

The modified stress tensor,   ~r, can then be used in the classical

mechanics and constitutive formulations used to describe plastic-ity of damaged materials.

4 Advances in Numerical Simulation of Sheet

Forming

4.1 Introduction.  The design of sheet metal forming proc-esses has generally been experience-based and relied on multiplefield trials conducted by experts. However, as the lead time fromdesign to production is shortened and as material strengthincreases, traditional trial-and-error approaches are too time-consuming and expensive, and they cannot be effectively appliedto complex geometries or materials. Thus, with these

Fig. 1 Representative experimental stress–strain data of aninterstitial-free tested in a tension-compression-tension sequence.Simulation results using the HAH model and a simple isotropichardening model are also shown. Reproduced from Ref. [42].

Fig. 2 Schematic representation of the distortion of the yieldsurface for the HAH model at different strain levels during ten-sile deformation

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considerations, the simulation of sheet metal forming has becomeessential for designing tools and processes.

With the rapid development of computational power and solutiontechniques, the finite element method (FEM), along with other nu-merical approaches, has been widely utilized to predict and under-stand sheet metal forming processes. Early work by Woo [57] ledinto pioneering investigations by Wifi [58], Gotoh and Ishise [59],and Wang and Budiansky [60] who incorporated elasto-plastic mate-rial laws into their FE analyses. Their early work was later extendedto three-dimensional applications (see Tang et al. [61] and Toh and

Kobayashi [62]). Commercial FEM software programs such as LS-DYNA3D and LS-NIKE3D represented a common finite elementapproach using explicit algorithms in the forming stage and implicitalgorithms for springback, respectively [63,64]. Large strain prob-lems in LS-DYNA3D use the thin shell elements developed byHughes and Liu [65] and Belytschko and Tsay [66]. Tekkaya [67]provides a more thorough review on the early development of FEanalyses of sheet metal forming.

Finite element simulations of sheet metal forming processes areperformed to predict the deformed shape of the workpieces, thedistribution of strain and stress within the part, the change of sheetthickness, and even defects or failure modes such as fracture,necking, or fatigue. FE simulations are also utilized to optimizeforming processes while minimizing experimental cost and time.Therefore, the demands for accurate, efficient, and robust finiteelement procedures are increasing. In this section, various FE

methodologies applied to sheet metal forming are brieflyreviewed. Some useful examples will also be demonstrated tofacilitate understanding of the discussion. Finally, several impor-tant aspects of FE analysis required to achieve common simula-tion objectives are presented.

4.2 Solution (or Program) Type.  The goals of sheet metalforming simulations are to reduce the time and cost associated with aprocess or product design, to increase the accuracy of the final part,and finally to increase the overall quality of the product. These goalscan be achieved via different numerical approaches that have beenproposed for FE analysis. These approaches include the static or dynamic implicit, dynamic or static explicit, and inverse one-stepmethods. Among these methods, dynamic explicit and implicit

schemes are most frequently used for sheet metal forming simula-tions and they are briefly summarized in this section.

Dynamic explicit FE codes have been favored for sheet metalforming simulations because explicit codes do not suffer fromdivergence problems which are frequently encountered in implicitcodes. In addition, explicit codes require less memory and can beeasily parallelized. The explicit time integration method is condi-tionally stable, which means that the computational time isdirectly related to the stability limit governed by the Courant crite-rion [68,69]. When the explicit method is applied to a quasi-staticproblem, computational time becomes problematic unless time or mass scaling are employed. Currently, computation speed is faster and the memory requirement is less than that for static implicitcodes. One disadvantage of the explicit code is that the explicitcharacter of the numerical scheme is fulfilled only if the mass ma-trix is lumped. Furthermore, the speed advantage of explicit codecan be realized only if the element computations are minimized asmuch as possible. Minimization can be maintained by using asingle-quadrature rule for defining finite elements, but may reducethe solution accuracy.

The dynamic explicit code is based on the solution of the virtualwork equation with an inertia term given by

ð V 

T ij dui; j dV  ¼

ð  A

t iduidA 

ð V 

q€uiduidV    (10)

where T ij  is the Cauchy stress, u i,j   is the displacement gradient, t iis the traction vector, d  is the variational operator,  q  is the density,and  €ui  is the acceleration of a material particle.

The discretized form of Eq. (10) leads to

fFg fIg ¼ ½Mf€ug   (11)

where fFg and  fIg are vectors representing the external force andthe internal force, respectively, and  ½M   is the lumped mass ma-trix, which includes the mass value only in the diagonal [70].

Implicit finite element codes solve an equilibrium equationwith an iterative procedure which is unconditionally stable, there-fore allowing a larger time increment to be used. For implicit

code the equilibrium equations isð V 

T ij dui; j dV  ¼

ð  A

t iduidA   (12)

The discretized form of Eq. (12) leads to

½KðuÞfug ¼ fFg   (13)

where   ½KðuÞ   is the nonlinear stiffness matrix in displacementspace and should be solved by standard nonlinear solutionprocedures.

The implicit method requires iterative solution of systems of nonlinear equations at each time increment. Problems involvingmany degrees of freedom require a large amount of memory for 

the stiffness matrix, and correspondingly long computation time.One of the most critical disadvantages of implicit codes is the dif-ficulty caused by contact nonlinearities in obtaining convergedsolutions for complex forming problems. The divergence problembecomes more pronounced when a large number of contact nodesare involved, which is critical for sheet metal forming applica-tions. The computational time also increases nearly quadraticallywith increasing number of elements. One approach to overcomethese limitations is to use the advantages of both explicit andimplicit numerical schemes. For example, in this approachdynamic explicit code can be used to solve the forming, or plasticdeformation, problem and implicit code can then be used to solvethe problem of post-forming springback. However, despite thecomputational difficulties and disadvantages inherent to implicitsolution schemes, the demands for increased accuracy may lead tofurther development of the static implicit approach in the longterm. This future expectation should be accompanied by the de-velopment of new constitutive models and enhanced computa-tional efficiency.

4.3 Element Choice and Related Numerical Parameters.   Auser of the finite element method is required to select the mostappropriate type of element for the problem at hand. The generalelement categories available are beam, shell, and solid. Sheetmetal forming processes frequently involve bending and stretch-ing over a tool radius which is usually greater than ten times thesheet thickness. Therefore, membrane-type elements are effectiveto model the planar shape of the sheet and the most common ele-ment used for sheet metal forming process is the shell element.Most commercial FE programs provide various types of shell ele-

ments with detailed formulations and descriptions available in theuser’s manual of each program. Elasticity-based simulations areusually more sensitive to numerical effects than plasticity-basedsimulations and thus, springback is a good example to illustratethe effect of element type in FE simulation predictions [71,72]. Ina recent study, Li et al. [71] performed a variety of simulations toinvestigate the effect of element type on the accuracy of spring-back predictions. Springback accuracy was evaluated for simula-tions using 2D and 3D solid elements and shell elements. Figure 3shows that accuracy was remarkably improved when the simula-tion employed either 3D shell elements or higher-order 3D 20-node solid elements. However, the simulations using 2D solid ele-ments or 3D solid elements with only 8 nodes did not show satis-factory accuracy for the springback predictions.

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Once an appropriate finite element type is selected, other criti-cal numerical parameters need to be carefully considered. Thesenumerical parameters include the number of integration pointsthrough the thickness and the number of elements at the corner of the tool radius. A generally accepted guideline for common simu-lations of forming and springback is the following. The number of integration points through the thickness should be 5–9 for a typi-cal forming simulation, while more points may be requireddepending on the ratio of tool radius to thickness ratio and for springback simulations. Li et al. [71] suggested as many as 15–25

integration points for accurate simulations of springback. How-ever, the proper number of integration points for sheet metal form-ing simulations is still debated among many researchers [73 – 76].In general, more contact nodes are needed for springback simula-tions than for forming simulations, i.e., one node per 5 deg of turnangle for springback simulations, while one node per 10 deg of turn angle for forming simulations is suggested by Li et al. [71].

4.4 Contact Models.   The complex tool geometry requiredfor realistic forming simulations leads to difficulty in achievingconverged FE solutions in sheet metal forming simulations. Thetools are generally assumed to be a rigid body and only the toolsurfaces are described by one of four possible schemes. Theseschemes are the analytical function, the parametric patch, the

triangular-jewel, and the mesh-patch scheme. The analytical func-tion scheme can only describe tool surfaces in the shape of acylinder, sphere, or other simple mathematical functions. Theparametric-patch scheme describes the tool surface using varioustechniques, such as NURBS, Bezier, or B-spline functions, tosmooth the discretized surfaces [77 – 79]. The major disadvantageof this scheme is that the contact search requires significant effortwhich can often lead to increased computation time and instabilityduring the simulations [80]. The triangular-jewel schemedescribes tool surfaces with triangular facets defined by evenly

distributed points [81,82]. The advantage of this scheme is thatthere is no need for a global contact search. However, this schemehas difficulty in describing tools with vertical regions, which areoften encountered in typical tooling used for sheet metal forming.The mesh-patch scheme describes tool surfaces with mesh patchesand arbitrarily complex tool surfaces can be easily described byFE preprocessors. Therefore, this scheme has been adopted inmost commercial FE software packages and can be successfullyapplied to 3D tool geometries with efficient contact searchalgorithms.

Recently, Zhuang et al. [83] developed a contact searching algo-rithm incorporating global and local search procedures within their N-CFS framework for 3D contact tool surfaces. In this approach theequilibrium and contact search utilize the mesh normal direction(Fig. 4), providing accuracy and stability at the same time over theconventional tool-based normal formulations. A related modification

takes sheet thickness into account for enforcing contact with theforming tools during forming. Figure 5  shows that incorporating thesheet thickness into the shell contact algorithm significantly improvesthe solution accuracy while maintaining almost identical efficiency.This study demonstrated that the contact treatment between the sheetand tools can be one of the most important issues necessary toachieve accurate predictions in sheet metal forming. Studies investi-gating techniques to improve the tool modeling accuracy are rare tothe best knowledge of the authors, and so the development of accu-rate and efficient tool descriptions along with a robust contact algo-rithm should be future topics in the advancement of sheet metalforming simulations.

4.5 Material Models.   The steel sheets used in the automo-

tive industry usually exhibit initial anisotropy that results from therolling process. As the microstructure of these steels becomesmore complex in order to enhance the strength and formability,

Fig. 3 Effect of element choice on the accuracy of drawbendspringback prediction for aluminum alloy 6022-T4; (a ) predic-tions using 2D elements and linear 3D solid elements and (b )predictions using higher order 3D solid elements and 3D shellelements [71]

Fig. 4 Mesh normal contact scheme proposed by Zhuang et al.[83] where sheet thickness (h ) is considered where   X 0  corre-sponds to the sheet mid-plane position and  X 1  corresponds tothe offset surface plane closest to a particular tool surface

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the mechanical properties have also become highly complicated.For example, the large Bauschinger effect and complex transientbehavior observed during reverse loading in AHSS are major hur-dles to overcome when conventional material models based on

isotropic yield and hardening laws are applied. Therefore, thisstandard behavior of advanced steels requires significant investi-gations on the development of advanced material models.

As discussed in Sec.  3.4, the proper description of the initialyield surface and its evolution with deformation are essential for constitutive modeling of advanced steels to be applicable in asheet metal forming process. In particular, when the materialsundergo nonmonotonic deformation, isotropic hardening, whichassumes proportional expansion of the yield surface during plasticdeformation, may not be very effective. A purely kinematic hard-ening assumption was proposed to introduce a translation of theinitial yield surface and could successfully predict the Bau-schinger effect, but not the transient hardening behavior [84,85].Recently, models combining isotropic hardening and nonlinear kinematic hardening have been commonly used to describe the

expansion and translation of the yield surface during plasticdeformation [86 – 92].A numerical formulation for FE simulations using the com-

bined isotropic-kinematic hardening rules based on the Chabochemodel [93 – 95] were established within the deformation theoryframework. The evolution rules for the kinematic hardening lawcan be generalized for any anisotropic yield functions as

_a ¼  d a1

d e_e

ðr   aÞ

riso

  d a2

d e_e

a   (14)

where the two scalar coefficients   a1   and   a2   have exponentialforms which are empirically determined and functions of equiva-lent plastic strain,  e 

Р d e. The hardening parameters can be con-

veniently obtained from experiments which involve forward-

reverse loading with varying amount of prestrain [92,95].In a series of papers, a constitutive law proposed by Lee et al.[93 – 95] was applied to springback simulations of complexdouble-S rails and other benchmark problems. Figure 6  shows theamount of springback predicted by pure isotropic and kinematichardening models and a combined isotropic-kinematic hardeningmodel. The combined model results in a more accurate predictionof springback and sidewall curl than either of the pure models,and this result clearly demonstrates the importance of advancedconstitutive models in predicting macroscopic deformation behav-ior of materials, especially when they experience nonmonotonicdeformation.

As mentioned in Sec.   3, stress and strain rate potentials havebeen proposed to describe plastic anisotropy in metals and alloys.

Recently, comprehensive studies were reported by Kim et al. [96]for elasto-plastic constitutive models using both plastic strain ratepotentials and yield functions for anisotropic cubic metals. Theyused circular cup drawing tests to validate estimations of the

proper representation of anisotropy of an aluminum alloy sheet,AA2090-T3. Figure 7 shows the dimensions and the finite elementmodel of the circular cup drawing test. Figure  8(a)   shows thedeformed configuration and the equivalent stress contour of thedrawn cup calculated using the elasto-plastic strain rate potentialmodel. The cup height is not uniform and shows a significant ear-ring profile because of the anisotropy. Figure 8(b)  compares pre-dictions of the anisotropy with the experimental cup drawing data.Predictions were made using the plane stress (2D) and 3DYld2004-18p models and the Srp2004-18p anisotropic models.All of the material models successfully predicted the cup ear height and showed good agreement with experimental measure-ment. This particular example clearly shows that the finite elementmethod can be an efficient numerical tool to analyze the anisot-ropy in sheet metal forming when it is implemented with proper 

descriptions of material properties.

Fig. 5 Comparison of measured and predicted (a ) punch force and (b ) draw-in distances withvarious sheet thickness contact treatments for a cup drawing forming operation [83]

Fig. 6 Comparisons of FE predictions of springback in a dou-ble S-rail simulated using a pure isotropic (Iso), a pure kinetic(Kine), and a combined isotropic-kinematic (Iso-kine) hardeningmodel for aluminum alloy 6111-T4 [95]

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4.6 Process Chain Simulation.   Process chain simulationdescribes the integration of distinct and separate simulations intoa global approach to model the final manufactured componentperformance from the initial material processing parameters andbehavior [97 – 101]. In a process chain simulation, the outputsfrom one process step simulation are used as input into the subse-quent processing step or performance simulation. For example,local thickness strain, local plastic prestrain, and some predamagefactor from a forming simulation can be used as input into a crash-worthiness simulation to predict crack formation [99]. The finiteelement software packages used for each simulation step of theprocess chain simulation are not necessarily the same. Thus, theoutput data of a previous step must be mapped to new materialand failure models and a new mesh for input into the subsequentstep of the process chain simulation. It is critical that static and

kinematic compatibility is maintained during the mapping processand that the mapping process does not introduce any additionalerror into the simulation. Process chain simulations can becoupled with optimization tools to automatically simulate andoptimize the entire process chain or product [97,101].

5 Advances in Stamping Processes and PressesAs seen in Sec.  3, numerical forming simulations can be tai-

lored to any specific material if the constitutive model isadvanced enough and if the input data are carefully selected. Asa result of the simulations, key information about the suitableprocess parameters can be provided very effectively for a suc-cessful operation. Beyond this result, numerical simulationsusing advanced constitutive models have the potential to accu-rately predict the deformation paths that critical areas of theblank must follow in order to avoid defects and failure. Thesedeformation paths can be implemented in the press shop only if the forming press technology is advanced enough to practicallyreproduce the predicted deformation paths. This potential leadsto the following discussion about advances in process and stamp-ing press technology.

Traditional, relatively simple presses that rely on potentialenergy stored in massive flywheels or pneumatics, released in asemicontrolled mode, are not adequate to create the optimum strainpath required to form AHSS with complex microstructures and de-formation behavior, notably the TRIP family and most likely theTWIP family, as well. First, the older generation of presses issuited to forming steels which are insensitive to forming speed, or more specifically those steels with nonnegative strain-rate sensitiv-ity exponents. However, many modern AHSS are not insensitive tostrain rate and old rules and assumptions regarding best formingpractices may no longer be valid. Second, the mechanical proper-ties and deformation behavior of AHSS requires the formingstrains to be distributed over a greater portion of the final formedshape. Therefore, it is necessary to design the forming tools andprocesses in such a way as to create the optimal strain path for 

each combination of material and component geometry, whichconsequently necessitates greater control of the press environment.Any change in the forming process will alter the deformation

and strain history of a formed part. Consequently, two geometri-cally identical parts formed via two different deformation pathswill exhibit different amounts of springback. This phenomenon isexploited in multistage forming processes, such as progressive dieforming and restriking, which can result in substantially lessspringback compared to that from a single-stage stamping process.Springback is reduced in the multistage process because the mate-rial experiences strain reversal at many regions of the panel, for example, by bending and unbending while simultaneously spread-ing the forming strain to the greatest volume of the formedsurface.

Fig. 7 Schematic of the circular cup drawing test showing (a ) tooling dimensions (inmillimeters) and (b ) the corresponding FE model [96]

Fig. 8 (a ) Deformed configuration and equivalent stress at theouter surface of the drawn cup predicted with by the Srp2004-18p anisotropic model and (b ) comparison of the experimentalcup ear profile of a drawn cup with predictions by three aniso-tropic models [96]

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The history of forming speed, or the strain rate during forming,is an increasingly important parameter in the forming of AHSS.High deformation rates can generate significant heat in AHSSbecause of the amount of heat is generally proportional to thesteel’s strength. If the heat generated is not efficiently dissipatedwhen forming advanced and ultra-high-strength steels, additionalattention must be given to the forming speed in order that theamount of heat generated by the deforming sheet does not signifi-cantly raise the tool surface temperature and cause the process tostray from its initially intended outcome. These factors indicate

the need to optimize the tools and processes that are best suited toforming AHSS. Consequently, a new generation of presses isneeded with the capability to control the stroke or displacement of all of the moving components of the stamping tool, thereby con-trolling the strain history of the formed part. The blank holder detail is a possible exception, which may be controlled better withhydraulic means. Digital servo presses are a new generation of presses that have emerged in the last decade and they have suchcontrol capability.

The features and advantages of digital servo presses were welldescribed by Osakada [102] and Tamai et al. [103] in the proceed-ings of Metal Forming 2010. The primary and unique characteris-tic of servo presses compared with traditional presses is that, withthe exception of the hydraulic pressure used to apply the blankholding force, slide action is independently driven by a set of powerful servo motors under displacement control. The forming

process can then be defined as a sequence of displacement-timesegments, and the displacement path can be liberally defined withrespect to not only the position and speed but also the direction.Aside from slide displacement, other features of the press such asthose required for restriking processes can be monitored and con-trolled by the main control computer.

Highly accurate displacement control of the slide offers severaladvantages. First, slide inclination under eccentric loading condi-tion can be substantially reduced because each servo motor is in-dependently monitored and controlled. Hasegawa et al. [104]showed that slide inclination in a servo press could be reduced toa mere 20  lm=m by using a feed-forward control scheme. Withtraditional presses, this level of slide inclination could only beapproached by making the press frames and slides much moremassive and rigid. Reduced slide inclination reduces damage to

the forming tools and facilitates forming large, thin plates such asfuel cell separators. Another advantage of servo presses is thatslide velocity can be controlled throughout the forming process.Figure 9(a) shows an example of slide displacement control possi-ble with a servo press. In this example, the slide and toolingapproach the workpiece at a high velocity (Stage I) which helpsmaintain production speed and efficiency. The slide velocity isthen reduced just prior to contact between the tooling and work-piece (Stage II) and this new speed can be maintained duringforming or some other displacement profile can be used duringforming. A reduction in contact speed reduces noise and canextend die life because impact loading is minimized [102]. Further reductions in noise can be realized by controlling the tooling exitspeed in blanking operations [105]. Once forming is complete thetooling can be retracted (Stage III) and an appropriate exit speed

can be selected to maintain some process objective.Slide motion during Stage II in Fig.  9(a) deserves special atten-tion. There is no reason that the slide must progress at a constantrate toward the bottom dead point (BDP) after initial contact withthe workpiece. The flexible program control capability allows theslide to follow a stepwise displacement path toward BDP.Figure   9(b)   shows an example of slide displacement during theforming stage of a stamping process after contact is made. In thisexample, the downward movement of the slide is accompanied byperiodic segments during which the slide retreats slightly. Slidedisplacement profiles of this type have been shown to reduceforming loads in drawing and upsetting processes and to reducewall thinning (i.e., strain localization) in stamped parts [103,106].Wrinkling was also reported to be eliminated at lower flange

clamping forces compared to conventional presses when stepwiseslide displacement was utilized in a servo press [107]. Theseresearchers propose that these improvements are primarily relatedto lubrication effects at the tooling-workpiece interface. It is sug-gested that relubrication of the workpiece occurs when the toolingbriefly retracts and that this relubrication results in a decrease in

the coefficient of friction between the tooling and workpiece dur-ing subsequent deformation steps. Preliminary FE simulationresults by Tamai et al. [103] lend some support to this mechanismalthough strong experimental evidence has not been established. Itis also highly probable that the cyclic loading-unloading of thematerial during stepwise slide motion plays some role in theobserved performance of the servo press. Therefore, these charac-teristics of modern servo presses capable of generating enhanceddeformation profiles and complex strain paths are ideal for form-ing AHSS and other complex high-strength steels.

It has been said that stamping technology is presently under-going the same kind of revolution which happened in the world of machining three to four decades ago when computer-numerically-controlled machines were first introduced. Computer control of amodern servo press can enable extra actions, such as a program-

mable cushion, integrated into the forming process. For instance,during forming the blank holder force will be able to be monitoredand changed in parallel with slide displacement in such ways thatthe forming process is optimized and springback is reduced. It isenvisioned that the process designer will use this capability to cre-atively add more operations or functionality to the same press tomeet the needs of forming new advanced materials, while poten-tially resulting in savings in cost and floor space for the press shop.

6 Concluding Remarks

Within the next decade, the present authors predict that it willbecome commonplace to conduct numerical simulations to designstamping tools for advanced high-strength steels and to optimize

Fig. 9 Schematic illustration of (a ) slide displacement profileof a metal forming operation conducted by a digital servo pressand (b ) example of a stepwise slide displacement profile possi-ble in a servo press

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the forming process. In addition to predicting the strains in theformed panel, its shape after springback, and the final shape of thepart after trimming, the simulation technology will also track thefastest moving elements in the model to determine the optimumdisplacement path of the forming tool elements.

At the forefront of knowledge of advanced high-strength steels,more work is needed to understand the effects that different strainpaths may cause. This work is especially needed in areas relatedto the well known phenomenon of strain-induced transformationof retained austenite to martensite and the consequent creation of 

anisotropic mechanical behavior, and the resultant effect of suchanisotropic behavior on springback. In addition to a greater under-standing of the steel materials, it is also necessary to gain greater understanding of the interaction between materials and tool surfa-ces. It is imperative to conduct research on the change of frictionconditions of panels experiencing repeated high-stress contact,and this research alone may signify a minor revolution in the sci-ence of friction. The authors predict that, along with greater use of advanced high-strength and ultra-high-strength steels, there willbe development of new tool steels which are far stronger than tra-ditional cast or water-hardening tool steels. These advances willalso necessitate further studies on the surface interaction betweensheet steels and the tool materials.

Finally, in their 1999 review, Hosford and Duncan [108] appro-priately summarized the previous era’s classical approach of usingcontinuum mechanics to understand strain development in sheet

stamping. The present authors wish to consider this paper to markthe advent of a new era in sheet forming. In this era, classical con-tinuum mechanics alone can no longer satisfy industrial needs andnumerical simulations coupled with advanced material deforma-tion models are used to predict springback in formed panels andto optimize complex forming processes.

Acknowledgments

This research was supported by the World Class University(WCU) program through the National Research Foundation of Koreafunded by the Ministry of Education, Science and Technology (R32-10147). M.G. Lee appreciates the support of grants from the Indus-trial Source Technology Development Program (#10040078) of theRepublic of Korea’s Ministry of Knowledge Economy (MKE) andfrom the Basic Science Research Program (#2011-0009801).

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