Developments of the Marciniak-Kuczynski model for sheet metal formability: A review

Journal Pre-proof Developments of the Marciniak-Kuczynski Model for Sheet Metal Formability: a Review Dorel Banabic, Abdolvahed Kami, Dan-Sorin Comsa, Philip Eyckens

PII:

S0924-0136(19)30419-4

DOI:

https://doi.org/10.1016/j.jmatprotec.2019.116446

Reference:

PROTEC 116446

To appear in:

Journal of Materials Processing Tech.

Received Date:

10 May 2019

Revised Date:

14 August 2019

Accepted Date:

30 September 2019

Please cite this article as: Banabic D, Kami A, Comsa D-Sorin, Eyckens P, Developments of the Marciniak-Kuczynski Model for Sheet Metal Formability: a Review, Journal of Materials Processing Tech. (2019), doi: https://doi.org/10.1016/j.jmatprotec.2019.116446

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Developments of the Marciniak-Kuczynski Model for Sheet Metal Formability: a Review Dorel Banabica*, Abdolvahed Kamib, Dan-Sorin Comsaa, Philip Eyckensc a

CERTETA Research Centre, Technical University of Cluj-Napoca, Str. C. Daicoviciu nr. 15, 400020 Cluj-Napoca, Romania

Corelab CodesignS, Flanders Make, Gaston Geenslaan 8, 3001 Leuven, Belgium

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Mechanical Engineering Department, Semnan University, Semnan, Iran

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* Corresponding author: Dorel Banabic, Email: [email protected]

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Abstract

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This paper presents a historical perspective on the development of the model that was initially proposed by Prof. Marciniak in 1965 for calculating the Forming Limit Curves (FLC's) of metallic sheets. Based on a pre-existing geometric and/or structural non-homogeneity of the metallic sheet, it is at present the most widely used framework for the theoretical prediction of FLC’s. This is a consequence of the fact that this modelling approach is well funded from the mechanical and intuitive points of view. By consulting lesser known papers (in English and Polish), the authors revisit the origins of this model that has become commonly known as the Marciniak-Kuczyński (M-K) model in the literature. To put these developments in historical context, the first experimental test for assessing the formability of metallic sheets, the evolution of the Forming Limit Diagram (FLD) concept and the first models for evaluating limit strains (Swift and Hill's models) are summarized as an introduction. The model, initially published in 1965 by Marciniak as sole author, was further developed by Marciniak, Kuczynski and co-workers as explained next in this review. Since then, the following lines of development of the M-K model have been identified and are systematically addressed in this review: implementation of new constitutive equations, polycrystalline and ductile damage models, extending the models to take into account new material or process parameters, taking into account of non-planar stress states and non-linear strain-paths.

Keywords:

Forming limit diagram, Forming limit curve, Marciniak model, Marciniak-Kuczynski model, Formability

Contents 1. Introduction

3. Marciniak-Kuckzynski Model: A historical perspective

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4. Developments of the M-K model

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2. Forming limit models

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4.1 Implementation of new constitutive equations 4.2 Implementation of polycrystalline models

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4.3 Implementation of ductile damage models

4.4 Model extensions towards additional material and process parameters

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4.5 Model extensions for non-linear strain paths 4.6 Model extensions for non-planar stress conditions

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4.7 Advanced numerical methods for the solution of limit strain models

5. Conclusions References

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1. Introduction

The accelerated development of sheet metal forming technology in the nineteenth century demanded a more thorough knowledge of the phenomena associated to the mechanical behavior and plasticity of metallic materials. Consequently, a series of laboratory tests for evaluating the formability of sheet metals were elaborated during the first decades of the twentieth century. In general, the idea was to simulate different types of forming processes (deep-drawing, punch stretching, bending, etc.) with the aim of allowing a global estimation of the limiting strain level.

One could mention the following tests: Erichsen (Erichsen, 1914), Olsen (Olsen, 1920), Siebel (Siebel and Pomp, 1929), Jovignot (Jovignot, 1930), Sachs (Sachs, 1934), Fukui (Fukui, 1938), etc. Gensamer (1946) was the first researcher who emphasized the need of a local strain analysis. In 1946, he proposed a method by means of which the formability index was related to the mechanics of the forming process at local level, but not to the particular type of forming process (Gensamer, 1946). In his paper, Gensamer analyzed the ductility (defined by him as the maximum strain at

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which the fracture occurred) associated to loads corresponding to different ratios of the principal stresses. He plotted the limit strains associated to each stress ratio on a rectangular diagram using

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the longitudinal and transverse principal strains as abscissa and ordinate, respectively (Fig. 1 (Gensamer, 1946)). By connecting the fracture points, Gensamer obtained a curve separating the

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acceptable strains (located below the curve) from the unacceptable ones (located on the curve and above it). One of the most important conclusions drew by Gensamer was that “ductility (…) is strongly influenced by local forming conditions”. The diagram in Gensamer's paper is nothing else

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but a mirrored representation of the Forming Limit Diagram (FLD) later given by Keeler (Keeler

concept instead of Keeler.

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and Backofen, 1963). One may thus consider that Gensamer was the first who proposed the FLD

The scientific community showed little interest in Gensamer's paper at the time it was published.

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Fifteen years later, the Forming Limit Diagram concept was revisited and valued by Keeler in his PhD thesis supervised by Prof. Backofen (Keeler, 1961). A paper giving a synthetic presentation of the results obtained by Keeler was published in a journal having a worldwide audience (Keeler and Backofen, 1963). For a long time, this paper has been considered as giving the first graphical representation of the FLD (Fig. 2 (Keeler, 1965)).

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A significant contribution to the development of the Forming Limit Diagram concept is due to Goodwin who extended the curve defined by Keeler into the negative strain domain (Fig. 3 (Goodwin, 1968). The research in this field was later focused on the methods used for measuring the limit strains and on assessing the influence of different material and process parameters on the shape and position of the Forming Limit Curves. During those pioneering times, the Forming Limit Curve (FLC) concept has been defined as being the curve represented in the plane of principal strains and obtained by connecting the points associated to some limit strains producing the

occurrence of a defect (necking or fracture). The first research on the theoretical determination of limit strains started at about the same time. Using the maximum force criterion proposed by Considère (1885) for evaluating the limit strain in uniaxial traction, Swift (1952) elaborated a theoretical model which was able to predict the limit strains associated to biaxial loads. Hill (1952) developed another model for the theoretical determination of limit strains under planar loads by adopting the hypothesis that strain localization occurred in a band with stationary extension. In this paper, we review the different developments made in the Marciniak-Kuczynski’s (M-K)

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model and its application for prediction of formability in different forming condition. The theoretical models of FLC calculation are described in short (in section 2). Then a historical

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perspective on the progress of the M-K model is presented in section 3. The M-K model has been enriched with new constitutive equations and polycrystalline models, also combined with ductile

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damage models. Furthermore, the M-K model has been extended to take into account new material and process parameters, non-planar stress conditions and non-linear strain-paths. All these M-K

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developments are detailed in section 4.

2. Forming limit models

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Since the 1950's, several models have been developed with the aim of predicting the forming limits of metallic sheets. The first notable contributions were the models of localized and diffuse necking proposed by Hill (1952) and Swift (1952), respectively. Swift's model was later extended by Hora

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in the form of the Modified Maximum Force Criterion – MMFC (Hora, 1994). All these theoretical approaches assume the perfect homogeneity of the metallic sheet; the limit state being associated to the maximization of a traction load. A very different approach was developed by Marciniak (1965a): the thickness and/or micro-structural non-homogeneity of the metallic sheet causes the strain localization in a necking band. Over the years, other theoretical approaches have been

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proposed for analyzing the strain localization and evaluating the limit strains, including the bifurcation model developed by Stören and Rice (1975), the Through-Thickness Shear Instability Criterion elaborated by Bressan and Williams (1983), as well as the perturbation method used by Dudzinski and Molinari (1988). The reader is directed to the monograph published by Banabic (2010b) for a detailed presentation of the formability models.

3. Marciniak-Kuckzynski Model: A historical perspective

On the basis of an idea previously formulated by Swift (1952) and Tomlenov (1958), Marciniak (1965b) elaborated a new model for the determination of limit strains. Marciniak's model assumes that the strain localization is initiated by a preexisting thickness non-homogeneity of the metallic sheet. The model has been further developed in 1966 and published in Polish in the paper (Marciniak, 1966).The schematic representation of the non-homogeneity is shown in Fig. 4 (Marciniak, 1965b). The model consists of the following ingredients: - Yield criterion and hardening law for both regions of the metallic sheet: A – non-defective region

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and B – region characterized by a diminished thickness

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- Equilibrium of the loads acting at the interface between regions A and B

- Strain continuity along the groove at the interface between regions A and B

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- Linearity of the strain path in region A.

By adopting a plasticity model based on Hill's 1948 yield criterion (Hill, 1948) and Swift's

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hardening law, Marciniak numerically solved his model using the finite difference method. Fig. 5 (Marciniak, 1965b) shows the evolution of the ratio thickness in region B vs thickness in region A

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for two cases: non-hardening material (dashed line) and hardening material (continuous line). The diagram in Fig. 5 allows detecting the strain localization in region B and thus finding the associated limit strain in region A. The paper (Marciniak, 1965b), though written in English, was published

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in the Polish review journal Archiwum Mechaniki Stosowanej. Due to this fact, the paper had little international visibility and did not attract the interest of the scientific community. In 1967, following Prof. W Johnson's suggestion (Marciniak, 1990), Prof. Marciniak published together with Kuczyński an extended version of his previous paper in International Journal of Mechanical Sciences (Marciniak and Kuczyński, 1967). As a consequence, the model immediately

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entered the international circuit being the most cited paper published in this journal, having more than 2000 citations to date, and it became commonly known as the Marciniak-Kuczyński or M-K model. Besides the mathematical formulation of the formability model, the paper published in 1967 contains a sensitivity analysis for different material parameters (anisotropy coefficient and hardening exponent) and thickness non-homogeneity ratio. The geometric model used in this paper is shown in Fig. 6 (Marciniak and Kuczyński, 1967).

Later on, Marciniak (Marciniak, 1967) extended his model by admitting an inclination of the thickness non-homogeneity band at an angle α with respect to the direction of the maximum principal stress (Fig. 7 (Marciniak, 1967)). In this way, he was able to obtain more realistic predictions of the limit strains in the traction-compression domain (left branch of the Forming Limit Curve). After being extended, Marciniak's model covered the entire range of strain paths, from uniaxial traction (load 1), going through plane strain (load 2), biaxial traction (loads 3 and 4), up to balanced biaxial traction (load 5), as shown in Fig. 8. The paper (Marciniak, 1967) was

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published in Polish by the journal Mechanika Teoretyczna i Stosowana (Journal of Theoretical and Applied Mechanics). Due to this fact, it had little audience in the scientific community outside Poland. The extended model was also presented in 1968 at the IDDRG Conference in Turin and

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published in the same year as a paper in the journal La Metallurgia Italiana (Marciniak, 1968). Despite this publication was in English, the extended model that includes the traction-compression

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domain (the left-hand side of the FLC) had little impact on the scientific community due to the limited circulation of the previously mentioned journal. Even nowadays, this extension is usually

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considered to be a contribution of Hutchinson and Neale (1978); who published their model 11 years after the Polish paper published by Marciniak (1967) and 10 years after the English version

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(Marciniak, 1968).

The results obtained by Marciniak in the field of FLC modeling during the 1960's were presented in an excellent book entitled “Sheet Metal Forming Limits” (Marciniak, 1971). Unfortunately, this

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valuable book has only a Polish version until now.

The M-K model was further enhanced by Marciniak and his co-workers by including a viscoplastic constitutive model in its formulation (Marciniak et al., 1973). The influence of different mechanical parameters of the metallic sheet (anisotropy coefficient, hardening exponent, and

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strain-rate sensitivity exponent) an the Forming Limit Curve was thoroughly analyzed in the paper (Marciniak et al., 1973). In the same paper, Marciniak proposed a new laboratory test for the experimental determination of FLC's. The test uses a carrier sheet that drastically reduces the friction interaction between punch and specimen (Fig. 9 (Marciniak et al., 1973)). This methodology allowed Marciniak to determine FLC's for several materials (steel, aluminum and copper), the experimental FLC's being then used for the validation of his theoretical model of strain localization.

Later on, Marciniak et al. (1980) analyzed the effect of incremental (pulsating) deformation on the limit strain for the case of a geometrically non-homogeneous bar subjected to uniaxial traction. The geometric model of the bar used in the analysis is shown in Fig. 10. The influence of the strain increments on the limit strain is presented in Fig. 11. One may notice a significant dependence of the limit strain on the strain increment size. More precisely, there is a value of the strain increment for which the limit strain is maximized. An increase of the strain increment beyond this value leads to an asymptotic evolution of the limit strain toward the level corresponding to a continuous load

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of the bar. Marciniak's approach was extended to the case of a biaxial loading by Banabic and Valasutean (1992) who emphasized the effect of pulsating loads on the FLC's.

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The last contribution of Prof. Marciniak in the field of FLC modeling was presented in the meeting of IDDRG Working Group 2 organized at Helsinki in 1983 (Marciniak and Ike, 1983). In that

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paper, Marciniak and Ike used the non-quadratic Hill 1979 yield criterion (Hill, 1979) to analyze the influence of its exponent on the FLC.

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A synthetic presentation of Prof. Marciniak's contribution to the FLC modeling was given by himself in the frame of the symposium organized by Koistinen and Wang at Waren, Mi in 1977.

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That paper was published in the proceedings volume entitled Mechanics of Sheet Metal Forming (Wang and Koistinen, 1978). The Marciniak (1984) paper also presents Prof. Marciniak's achievements in the field of sheet metal formability and recommends the use of some indices for

(1971)).

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assessing the formability (these indices had already been defined in the book written by Marciniak

4. Developments of the M-K model In this section, different modifications made in the M-K model are presented. These modifications

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include the implementation of new constitutive equations, polycrystalline models, and ductile damage models. Furthermore, the extensions of the M-K model by taking into account new material and process parameters, non-planar stress conditions and non-linear strain-paths are discussed.

4.1.

Implementation of new constitutive equations

The predictive performances of the M-K model are significantly influenced by the constitutive equations that control the shape of the yield surface and its evolution in the stress space. In classical

plasticity, the equivalent stress defines the shape of the yield surfaces, while the hardening law defines its evolution. Many researchers focused their efforts on testing different expressions of the equivalent stress in order to obtain more accurate predictions from the M-K model. Banabic and his co-workers performed such tests using the Hill 1993 equivalent stress (Banabic, 1999), as well as the BBC2003 equivalent stress (Banabic and Siegert, 2004). Fig. 12 shows a comparison between the predictions obtained using the BBC2003 equivalent stress and experimental data for an AA5182-O aluminum

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alloy (Banabic and Siegert, 2004). Similar results were reported by Mattiasson and Sigvant (2008) after testing several expressions of the equivalent stress. Butuc and her co-workers also

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investigated the predictive capabilities of the M-K model when the Barlat 1997 (Butuc et al., 2003) and BBC expressions of the equivalent stress were used (Butuc et al., 2002). The Karafillis-Boyce

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equivalent stress was used by Cao and her co-workers to analyze the effect of strain path changes on computed FLC's (Cao et al., 2000). Kuroda and Tvergaard (2000b) compared experimental data with FLC's predicted by four expressions of the equivalent stress included in the M-K model. Aretz

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(2004) used the Yld2000 equivalent stress to investigate the influence of the biaxial anisotropy coefficient on computed FLC's. Vegter et al. (1999, 2008) implemented Vegter's expression of the

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equivalent stress in an M-K model. Ahmadi et al. (2009) analyzed the FLC's predicted by the MK model combined with the BBC2003 equivalent stress calibrated with different sets of material data. Soare and Banabic (2009) studied the predictive performances of the M-K model in which

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Soare's polynomial expressions of the equivalent stress were incorporated. The influence of the yield surface evolution on the FLC's predicted by the M-K model was investigated by Butuc et al. (2011) and Haddag et al. (2008) using Teodosiu's hardening model in combination with different expressions of the equivalent stress.

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All the authors mentioned in this section of the paper agree by concluding that the accuracy of the FLC's predicted by the M-K model depends on the quality of the constitutive model. In general, better results can be obtained by using expressions of the equivalent stress and hardening laws able to give more accurate descriptions of the yield surface shape and its evolution.

4.2.

Implementation of polycrystalline models

The accurate FLC prediction relies on the type of the implemented yield criterion and its shape. On the other hand, the yield surface shape and of course the other mechanical properties of a material are controlled by its microstructural characteristics. These facts motivated a lot of researches to incorporate microstructure-based constitutive models in the FLC calculation procedures based on the M-K model. The works done by Da Costa Viana et al. (1978), and Bate (1984) could be regarded as the pioneering studies conducted in the forgoing subject area, which then followed by Barlat (1987), Zhou and Neale (1995), Hiwatashi et al. (1998) and many others.

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Wu et al. (2005) examined the path-dependency of strain and stress based forming limit diagrams (FLD and FLSD, respectively) based on crystal plasticity theory in conjunction with the M-K

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model. They showed that in comparison with the FLD, the FLSD is much less sensitive to the strain path change. Almost the same results were reported by Wang et al. (2011) in computation

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of the FLD and FLSD of AZ31B magnesium sheet. Cyr et al. (2017) calculated the FLC’s of the AA5754 and AA3003 aluminum alloys at elevated temperatures with a thermo-elastoviscoplastic crystal plasticity constitutive framework developed by Cyr et al. (2015). Hiwatashi et al. (1998)

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and Van Houtte (2005) combined the Van Houtte model (1994) with the anisotropic hardening constitutive model of Teodosiu and Hu (1998) and predicted the limit strains corresponding to

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change in strain paths. Chiba et al. (2013) determined the FLC of AA1100-H24 sheet using phenomenological and crystal plasticity theories. It was observed that the crystal plasticity theory predicted higher forming limit strains in the drawing region and lower limit strains in the bulging

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region than those obtained from the phenomenological theory (see Fig. 13). Zhang et al. (2018a) implemented the crystal plasticity framework developed by Wu et al. (1997) in the M-K model and used this combined model to predict the FLC of a TRIP assisted AHSS. They studied the effect of the mechanism of transforming from low strength austenite to high strength martensite and they found that incorporating the transformation mechanism enhances formability compared to a

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material without the mechanism.

Over the last decade, a significant attention has been given to idealized textures, in particular for aluminum alloys (FCC structure). Such theoretical textures consist of randomly-generated orientations with certain spread around a single orientation (Wu et al., 2004) or orientation fiber (Yoshida and Kuroda, 2012). The chosen ideal orientation, the texture spread around the central orientation, which is linked to the sharpness of yield locus in the biaxial range (Wu et al., 2004)

and also the change in the yield locus shape due to texture evolution, i.e. distortional hardening effect (Signorelli et al., 2009) are important factors in study of formability (Banabic et al., 2016). In aluminum alloys for example, the cube fiber textures gives high forming limits due to the beneficial distortional hardening effect, in spite of an unfavorable initial yield locus shape with low r-value (Yoshida and Kuroda, 2012, Banabic et al., 2016). In FCC materials, rolling texture components generally have a negative effect on biaxial formability (Yoshida et al., 2007), whereas the cube texture (recrystallization texture component)

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can be beneficial (Wu et al., 2004, Yoshida et al., 2007). However, Inal et al. (2005) showed that the FLC of BCC materials is less sensitive to texture evolution in comparison to FCC materials.

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The FLC may also be affected by the variation of texture through the thickness of polycrystalline metals (Bhattacharyya et al., 2019). In the case of Mg sheet alloy (HCP crystal structure), besides

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dislocation-based plasticity, the mechanical twinning needs to be considered (Neil and Agnew, 2009, Tadano, 2016). Additional hardening due to twinning can promote the resistance to localized

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necking (Banabic et al., 2016).

Lebensohn and Tomé (1993) developed a viscoplastic self-consistent (VPSC) anisotropic model

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for modeling the plastic deformation of polycrystals. This model has been widely used in conjunction with the M-K model (M-K-VPSC model) for computation of FLC of different types of sheet metals and to study the effect of microstructural parameters on the FLC. For instance,

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Signorelli et al. (2009) calculated the FLCs of AA6116-T4 and AA5182-O aluminum alloys using the M-K-VPSC model. They also investigated the effects of slip hardening, strain-rate sensitivity, anisotropy and initial texture on the FLC. Serenelli et al. (2011) analyzed the limit strains of FCC and BCC materials using M-K-VPSC model. They compared their results with the FLC’s calculated by Inal et al. (2005), who used M-K-FC model (a generalized Taylor-type polycrystal

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model) in their work. As shown in Fig. 14, the FLC of AA5182-O alloy calculated using the MK-VPSC model has good agreement with the experimental FLC, while the M-K-FC model fails to predict the correct limit strains on biaxial stretching region of the FLC (Serenelli et al., 2011). Almost similar results have been obtained for the FLC of a DQ-type steel-sheet (Serenelli et al., 2011). Steglich and Jeong (2016) employed the M-K-VPSC model to study the formability of AZ31 and ZE10 magnesium alloys at 200 °C. They analyzed the effects of the sample orientation with respect to rolling direction, pre-straining and the initial crystallographic texture on the limit

strains. Using the M-K-VPSC model, Steglich and Jeong (2016) were able to demonstrate the effects of sample orientation and the initial crystallographic texture on the limit strains. An important aspect in determination of FLCs using crystal plasticity approaches, is the computational efficiency of those approaches (Gupta et al., 2018). Gupta et al. (2018) developed a computationally efficient method for FLC prediction based on rate-dependent crystal plasticity models using discrete Furrier transform (DFT) spectral database. They claim that a significant improvement in the accuracy of the crystal plasticity computations were obtained using the DFT

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database. Nagra et al. (2018) also proposed a computational efficient model for computation of FLC’s of FCC polycrystals, which uses the rate tangent-fast Fourier transform-based elasto-

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viscoplastic crystal plasticity constitutive framework (RTCP-FFT) developed by Nagra et al. (2017). Using this model, they could successfully predict the FLC’s of AA3003-O and AA5754-

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O alloy sheets.

For complex microstructures such as multi-phase materials, a microstructural representative

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volume element (RVE) may be defined through FE modelling to produce the constitutive behavior in formability modelling framework (Banabic et al., 2016). Such RVE material modelling

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phase steel sheets, respectively.

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approach has been used by Tadano et al. (2013) and Srivastava et al. (2016) for FCC and dual-

Implementation of ductile damage models

Assuming initial thickness imperfection in the M-K model may in some cases be unrealistic if it is required to select large values for this thickness imperfection to reduce the discrepancy between the predictions of the M-K model and experimental results. To redeem this drawback some

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researchers adopted M-K based models by introducing initial voids in the sheet and taking into account the growth of voids (e.g. see Kim and Kim, 1983). Necking and consequent fracture of sheet due to growth (and nucleation) of voids could be modeled with damage constitutive equations like the model developed by Gurson (1977). The performance of the M-K model could be improved by characterizing the material deformation with damage models (just in the imperfection zone or in both the imperfection and non-defective zones). This approach provides a modified M-K model, which could be called coupled M-K-damage model. This M-K-damage

model has been discussed briefly in preceding reviews (Banabic et al., 2016, Zhang et al., 2018b). In the coupled M-K-damage model, both the thickness heterogeneity and heterogeneity due to higher volume of voids could be assumed. The schematic of the coupled M-K-damage model is illustrated in Fig. 15. In this figure, zone B (the imperfection zone) has higher value of initial void volume fraction with respect to zone A (the non-defective zone), i.e.

B

A

f0  f0

, while a difference

in initial thickness between both zones is not assumed (Gologanu et al., 2013). Needleman and Triantafyllidis (1978) employed Gurson model (Gurson, 1977) in conjunction with

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the M-K model to study the effect of void growth in local necking generation in biaxially stretched sheets. The elastoplastic deformation of material in both imperfection and non-defective zones was

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defined by the Gurson model and it was assumed that the imperfection zone has a greater volume of initial voids. The results of this study attribute the shape of FLC to the weakening effect of void

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growth. In addition to void growth, nucleation of voids is an important mechanism of damage evolution in ductile materials and it affects the predicted limit strains. Chu and Needleman (1980)

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investigated the effect of void nucleation on the FLC of sheet metals. They found that the shape of in-plane FLC’s is affected by the void nucleation parameters. The lower limit strains were

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obtained when nucleation is considered in the evolution of void volume fraction. In another approach, the initial thickness defect of M-K model is characterized by randomly distributed voids (Jalinier and Schmitt, 1982, Barlat et al., 1984). By introducing a damage function and void growth

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equation, the limit strains may be calculated with good accuracy. Melander (1983) proposed a model based on M-K and Gurson models for prediction of FLC’s of copper-based alloys. He assumed that the onset of necking occurs in a region with higher concentration of particles and voids than average. The effect of the strain rate and the normal

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anisotropy of the material were incorporated in the model. The effect of texture and void evolution on limit strains of biaxially stretched aluminum alloy sheets have been investigated by Hu et al. (1998). Ragab and co-workers developed M-K based models for prediction of FLC of planar isotropic (Ragab and Saleh, 2000) and kinematically hardened voided sheets (Ragab et al., 2002). Huang et al. (2000) and Chien et al. (2004) used an enhanced Gurson model (developed by Liao et al. (1997)) in combination with M-K model to study effects of anisotropy, inhomogeneity of material and geometry and surface curvature on FLC calculation. Fig. 16 shows the FLC’s calculated by Huang et al. (2000) for different hardening

theories. According to this figure, the isotropic hardening (b = 1) and the kinematic hardening theories (b = 0) provide upper-bound and lower-bound FLC’s, respectively. Whereas, the combined isotropic and kinematic hardening (b = 0:25) the calculated FLC shows a very good agreement with the experimental FLC of mild steel (Huang et al., 2000). Son and Kim (2003) incorporated the void shape effects in the M-K model using Gologanu–Leblond–Devaux damage model (Gologanu et al., 1993, Gologanu et al., 1994) and employed this model to calculate FLC of anisotropic voided sheets.

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Simha et al. (2007) developed a finite element (FE) model of M-K combined with GTN model (Tvergaard and Needleman, 1984) for construction of extended stress-based formability curve

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(XSFLC), a limit curve in terms of mean and equivalent stresses. In this FE model, the zone A is homogeneous with no void and it was modeled by the J2 flow theory and the isotropic hardening.

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While the zone B was modeled using the GTN model. Results obtained from this approach in straight tube hydroforming were satisfactory. A similar M-K-GTN model was utilized by Hu et al.

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(2017) for construction of the FLC of 22MnB5 high-strength steel at 600C, 700C and 800C temperatures. In the GTN model, Hosford anisotropic yield criterion (Hosford, 1979) with modified Norton-Hoff hardening law (Merklein and Lechler, 2009) were used. The applicability

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of the M-K-Gurson model in the prediction of limit strains of sheet metals under superimposed double-sided pressure was studied by Liu et al. (2012). A certain volume fraction of initial voids was assumed in the imperfection zone. Reasonable values of limit strains were obtained when the

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volume fraction of initial voids was set to be 0.007. Zadpoor et al. (2009) determined limit strains by phenomenological ductile fracture modeling, the M-K model, a modified Gurson damage model and a combined M-K-Gurson model. In the M-KGurson model, an initial imperfection was assumed and a population of initial voids was

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considered in both imperfection and non-defective zones. The value of strain in the non-defective zone when the void volume fraction of the imperfection zone reaches the failure void volume fraction was recorded as the limit strain. It was found that the M-K-Gurson and the modified Gurson models are more computationally expensive in comparison with the M-K and the phenomenological ductile fracture models. Furthermore, it was concluded that the phenomenological ductile fracture models can predict the forming limits with better accuracy and lower computational costs in comparison with the M-K-Gurson model. In another study, the ability

of M-K-Gurson model in prediction of FLC’s of AK and IF steels was examined by Hosseini et al. (2017). It may be noted that in contrast to above-mentioned works, damage modelling outside of the M-K model framework has also been applied to determine the FLC of sheet metals. The manuscripts related to this topic is not covered in this manuscript, as they fall out of the scope of the present

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review.

Model extensions towards additional material and process parameters

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Apart from the various material modelling approaches discussed so far, additional modifications have been made over the years to improve the accuracy of the M-K model for specific sheet metal

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forming processes such as hydrostatic deep-drawing, high-speed sheet forming, incremental sheet forming and superplastic forming (Banabic, 2010a). Furthermore, these developments enable to

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study the effect of different material and process parameters on the formability of sheet metals using the M-K model. In this section, we will try to give a brief review of the enhancements made

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in the M-K model through introduction of the following parameters: strain rate, temperature, equibiaxial r-value, rb, surface defects and grain size. Other modifications on the M-K model for taking into account the effects of yield criteria, void growth and nucleation, through thickness

sections.

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normal stress and shear stress and complex loading path on FLC prediction are discussed in other

The forming temperature is one important factor that affects the formability of sheet metals. In most cases, by increasing the forming temperature the formability improves. Especially in the case of magnesium alloys, it is usually required to form the material at elevated temperatures and

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knowledge of the FLC at forming temperature is then essential to design successful forming operations. The M-K model has demonstrated to have good capability for calculating FLC’s at high temperatures. A few examples of the FLC’s calculations by the M-K model at elevated temperatures are as follows: 5083-O alloy sheet at 20 C – 300 C (Naka et al., 2001), 22MnB5 steel at 200 C, 250 C and 300 C (Chan and Lu, 2014) and Mg-Al-Ca-Gd alloy sheets at 210 C ((Yan et al., 2018), see Fig. 17). In the last case, the M-K model was adopted by using a modified

Grosman equation (Gronostajski, 2000), which takes into account both temperature and strain rate. The FLC’s computed by this modified M-K model is shown in Fig. 17 (Yan et al., 2018). The formability of sheet metals varies with the change of the forming speed and the rate of straining. Predictive models of localized necking for strain-rate-dependent sheet metals have been developed by several researchers (see (Li et al., 2014) and the references cited in that paper). Aretz conducted studies on prediction of FLC’s of AA6016-T4, AA5182-O and AA3104-H19 materials and showed that the equibiaxial r-value rb has a significant impact on the FLC (Aretz,

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2006). The M-K model has been used to calculate the FLC of superplastic materials (Chan and Tong, 1998) and also advanced sheet materials such as multi-layered sheets (Wang et al., 2019)

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and tailor-welded blanks (TWBs) (Safdarian, 2015).

In addition to the abovementioned parameters, the effect of the following parameters on FLC

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prediction has been investigated using the M-K model: the grain size (Peng et al., 2017), the surface defects (Hiroi and Nishimura, 1997), the topology (Wu et al., 2007) the roughness and the

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sheet thickness (Assempour et al., 2009) and the sheet bending (He et al., 2018). An overview of

Model extensions for non-linear strain paths

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4.5.

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parameters included in the M-K model and their values are presented in Table 1.

During sheet metal forming processes, the blank usually evolves along non-linear strain paths. Nakazima et al. (1968) demonstrated experimentally that the shape and position of FLC's are changed under such conditions. Therefore, the FLC calculation for non-linear strain paths is essential for industrial applications. The elaboration of M-K models for non-linear strain paths

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became an active research field in the early 1980’s (Barata da Rocha and Jalinier, 1984). With the aim of refining these models, Butuc et al. (2002) implemented several hardening models in a general computer code for the FLC computation along non-linear strain paths. Rajarajan et al. (2005) also validated the predictive capabilities of the CRACH model for non-linear strain paths. Cao et al. (2000), as well as Yao and Cao (2002) analyzed the influence of the changing strain paths on the forming limit of metallic sheets. Hiwatashi et al. (1998) used Teodosiu’s hardening model to study the influence on the strain-path changes on FLC's. Kuroda and Tvergaard (2000a)

analyzed the effect of the strain-path changes on the forming limits using four anisotropic yield criteria.

4.6.

Model extensions for non-planar stress conditions

This section firstly reviews M-K developments with non-zero thickness stress resulting from sheet contact pressure. Conditions with through-thickness shear stresses that may arise under sliding

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contact conditions is treated next. There is a beneficial formability effect when applying compressive sheet normal pressure during

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forming. This has been long time known and exploited in industrial practice (Keeler, 1970).

Examples for which the thickness compressive stress can become relevant, include hydroforming

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(Hashemi et al., 2016) and forming operations with relatively sharp tooling features compared to sheet thickness that induce sheet curvature (Ma et al., 2016). Examples of the latter include

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Bending-Under-Tension (He et al., 2013) and Incremental Sheet Forming. Banabic and Soare (2008) were first to extend the Marciniak-Kuczynski model for a non-zero sheet normal pressure,

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whereby this contact pressure is assumed to be imposed, constant through deformation, and applied to both matrix and groove constituents, resulting in the FLC’s shown in Fig. 18. In this work, it was also shown how a plane stress yield function may be adopted based on invariability of plastic

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yield to the hydrostatic pressure component. Other early work was conducted by Allwood and Shouler (2009) and Wu et al. (2009). Allwood and Shouler (2009) were first to consider a general stress state within M-K model’s constituting matrix and groove, including non-zero normal stresses. In the extended M-K analysis of Wu et al. (2009), the authors point out the principal difference in boundary conditions between an imposed and fixed sheet normal pressure and a

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imposed and fixed hydrostatic pressure leading to distinct formability predictions, whereas the trends in formability improvement are seen to be similar to a large extent. In the course of the next years, several studies have scrutinized the aspect of material modelling in M-K analysis for non-zero sheet normal stresses, e.g. adoption of Hill 1948 anisotropic flow criterion in (Assempour et al., 2010). A continuum damage material model is considered by Liu et al. (2012), and Liu and Meng (2012) in a finite element set-up of a M-K-like grooved sheet material. Nurcheshmeh and Green (2012) found that among common mechanical parameters, the

strain hardening capacity has the largest influence on the pressure dependence of the FLC. The beneficial formability effect by applying normal pressure may be limited material with strain hardening stagnation (Zhang et al., 2014b). Wang et al. (2015) adopt a multi-scale material approach (Elasto-Viscoplastic self-consistent model) for the hydroforming of a magnesium alloy at room temperature, whereas in (Lang et al., 2015) a temperature-dependent material modelling is considered for application to hydroforming at elevated temperatures. Regarding non-proportion loading, Nurcheshmeh and Green (2014) present M-K formability

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predictions with pre-strain and conclude that the known strain-path dependence of the FLC is little affected by sheet normal pressure. Zhang et al. (2014a) and Hashemi and Abrinia (2014) study the

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sheet normal stress in the context of various strain- and stress-based formability representations; furthermore, in (Zhang et al., 2014a) a correcting Forming Limit Stress Criterion (FLSC) relative

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insensitive to normal stress is proposed.

In punch forming with relatively small tooling, sheet curvature is significant and a strong gradient

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of sheet normal pressure across the thickness builds up. He et al. (2013) and later Bettaieb et al. (2018) propose extensions to the M-K model with gradients of stress and strain along the sheet

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thickness direction. Whereas, He et al. (2013) use Finite Element simulations of the BendingUnder-Tension test to feed into the M-K model the strain history, including thickness gradient. Bettaieb et al. (2018) considers pure bending as a pre-deformation before subsequent stretching.

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Predicted forming limit curves are higher compared to the absence of curvature if the total strain of bending and stretching is considered in the FLD (Bettaieb et al., 2018). Contact pressure is markedly higher for thicker sheet stretched over a fixed tool; extension of the M-K model with a tool contact pressure that depends on sheet thickness and tool radius is presented in (Dong et al., 2019), and applied to the Nakazima forming limit test set-up.

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Comparison of the extended M-K model to other forming limit modelling approaches is made in (Bettaieb and Abed-Meraim, 2017) and (Hu et al., 2019), both with a focus on sheet normal pressure. Bettaieb and Abed-Meraim (2017) demonstrate that for vanishing initial groove imperfection, the forming limit predictions by M-K analysis tend towards those found by the bifurcation theory. Hu et al. (2019) make an elaborate comparison of M-K against the perturbation approach for formability analysis.

Through-thickness shear stresses may be induced in sheet metal through sliding friction, especially if curved tools are involved and normal contact pressure is high, as can for instance be found in deep drawing over relatively sharp die corners and across draw beads. Research on the formability under these conditions have been originally initiated by the unusually high formability observed in incremental sheet forming processes in (Allwood and Shouler, 2009) and (Eyckens et al., 2009). Allwood and Shouler (2009) make the first generalization of the M-K model to a six component stress tensor and propose a generalized forming limit diagram that shows the positive influence of

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both of the normal and through thickness stresses. In (Eyckens et al., 2009), Eyckens and coauthors focus on the shear stresses and propose extension to the classic M-K equations of

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geometric compatibility and force equilibrium between matrix and groove constituents. This study is extended further in (Eyckens et al., 2011) for anisotropic materials and introduction of the

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concept of direction of imposed through-thickness shear.

More recently, general M-K models for normal and through-thickness shear stresses have been

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presented in (Fatemi and Dariani, 2015, Fatemi and Dariani, 2016, Nasiri et al., 2018). Combined normal and through-thickness shear stresses are most beneficial to formability under plane strain

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forming conditions (Fatemi and Dariani, 2015). Furthermore, formability enhancement is more sensitive to applied normal stress on the sheet surface than the shear stress component (Nasiri et

4.7.

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al., 2018).

Advanced numerical methods for the solution of limit strain

models

Wagoner and his co-workers used the finite element (FE) method for the numerical determination

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of the limit strains in the frame of the M-K theory (Narasimhan and Wagoner, 1991). The simulation reproduced the standard M-K model with the aim of studying the influence of various mechanical parameters of the metallic sheet (Lankford coefficients, strain hardening exponent, and strain-rate sensitivity) on FLC0. Later on, an internal state variable plasticity/damage model was implemented by Horstemeyer et al. (1994) in explicit and implicit FE codes to predict the FLC. Tai and Lee (1996) calculated the FLC using FEM based on an elastoplastic constitutive model coupled with damage. Narasimhan and Nandedkar (1999) analyzed the influence of changing the

strain paths and work hardening exponent on the FLC using FE modeling. Gänser et al. (2000) proposed a micromechanical approach to predict the FLC. A 3D model was used to study the instability of a two-phase material. A solution algorithm of the M-K model with incremental loading was implemented in an FE code by Evangelista et al. (2002). Van den Boogaard and Huetink (2003) made a comparison between the M-K analysis and an FE analysis with membrane and shell elements of a biaxially loaded plate with a groove. The benefits of the FE model (boundary conditions, non-proportional loading, and friction with tool surfaces) were presented.

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Lademo et al. (2004) made an FE-based calculation of FLC's and the results of the simulation were compared with the predictions of the M-K theory. The authors found a fair agreement between the FE-based and analytical FLC calculations. Lademo et al. (2005) implemented an elastoplastic

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constitutive model coupled with damage in LS-DYNA for plane-stress analysis. The results showed that shell element analysis was able to predict the plastic instability of sheet metals.

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Berstad et al. (2004) analyzed the localized necking phenomenon using non-linear FE models. The FLC of an aluminum alloy sheet was computed by stretching a patch of shell elements with a

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random distribution of the thickness. Brunet et al. (2005) implemented a non-local damage model in a dynamic explicit FE code. The results showed that the strong mesh dependence caused by

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strain softening could be avoided by non-local damage models. Paraianu and Banabic (2005) implemented the BBC 2003 and Cazacu-Barlat yield criteria in the ABAQUS finite element code to predict FLC's. The analysis proved the capability of the FE method to predict the strain

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localization. Teixeira et al. (2006) used Lemaitre's ductile damage model coupled with Hill’s 1948 yield criterion in the frame of Continuum Damage Mechanics (CDM) to calculate FLC's. The results showed that the model was able to capture the damage evolution and predict the location of possible material failure. Hopperstad et al. (2006) analyzed the influence of serrated yielding and Portevin–Le Chatelier effect on the predicted forming limit curve. The results suggested that

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the finite element method was, at least qualitatively, useful, efficient and applicable for calculating forming limit curves.

5. Conclusions The paper gives a historical perspective on the development of the Marciniak-Kuczyński (M-K) model which, for more than 50 years, is the most widely used theoretical model for predicting the

Forming Limit Curves (FLC's) of metallic sheets. A review of the literature referring to the concept of the Forming Limit Diagram (FLD) and the original Marciniak model (less known due to the fact that it was published not only in English but also in Polish) lead to the reinterpretation of the previously mentioned items. The historical evolution of the formability and FLC concepts introduced by Gensamer is synthetically discussed. The FLD published by Gensamer more than 10 years earlier than the one elaborated by Keeler is presented, the paternity of the FLD concept being thus reconsidered. The development of the M-K model is thoroughly analyzed. By using the

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papers published by Marciniak in Polish and international journals, as well in conference proceedings, the authors prove that the so-called Marciniak-Kuczyński model was published in a Polish journal two years before the moment generally accepted by the scientific community. The

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paternity of the theoretical model used for calculating the left branch of FLC's is also revisited by presenting the paper published by Marciniak in the proceedings of the IDDRG Conference

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organized in 1968. Some other significant contributions of Prof. Marciniak are also presented by reintroducing two papers in the international circuit (implementation of a non-quadratic yield

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criterion in the M-K model and extension of the model to the case of incremental straining). A major contribution of this paper consists in the fact that a series of papers and books published by

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Prof. Marciniak only in Polish are mentioned (for example, the monograph dealing with the prediction of limit strains published in 1971). The translation of these fundamental papers and books into English is a duty for the Polish school of plasticity. By reviewing the literature, the

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authors identify the main directions along which the M-K model has evolved since 1965: implementation of new constitutive equations, inclusion of polycrystalline and ductile damage models, extending the model to take into account new material or process parameters, non-planar stress states and non-linear strain paths. In the authors' opinion, the original model proposed by Marciniak in 1965 is one of the most prolific idea in the mechanics of sheet metal forming

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processes and will remain a source of good inspiration for researchers involved in this domain. The present paper aims at proving this statement. The future research in the field of sheet metal formability will be focused on developing and refining the polycrystalline and ductile damage models, as well as improving the capability of M-K models to give accurate predictions under realistic forming conditions including non-linear strain paths.

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List of Figures Fig. 1 a) First FLD published by Gensamer (1946) and b) its mirrored representation Fig. 2 FLD published by Keeler (1965) Fig. 3 FLD published by Goodwin (1968) Fig. 4 Schematic representation of the thickness non-homogeneity postulated by Marciniak as being the cause of strain localization (Marciniak, 1965b)

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Fig. 5 Evolution of the thickness ratio – region B vs region A (Marciniak, 1965b)

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Fig. 6 Model of the geometric non-homogeneity proposed by Marciniak and Kuczyński (1967)

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Fig. 7 Model of the inclined geometric non-homogeneity proposed by Marciniak (1967) Fig. 8 Linear strain paths used for determining a complete FLC: 1-uniaxial traction; 2-plane

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strain; 3 and 4-biaxial traction; 5-balanced biaxial traction

Fig. 9 Model of the inclined geometric non-homogeneity proposed by Hutchinson and Neale

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(1978)

Fig. 10 Principle of the testing device with carrier sheet proposed by Marciniak for the experimental determination of FLC's (Marciniak et al., 1973)

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Fig. 11 Uniaxial geometric model proposed by Marciniak for analyzing the influence of the strain increment size on the limit strains (Marciniak et al., 1980) Fig. 12 Influence of the strain increment size on the limit strains (Marciniak et al., 1980) Fig. 13 Theoretical FLC versus experimental data for an AA5182-O aluminum alloy (Banabic

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and Siegert, 2004)

Fig. 14 Comparison of the FLC’s of AA1100-H24 sheet predicted by phenomenological and crystal plasticity theories with the experimental FLD (Chiba et al., 2013) Fig. 15 Experimental FLD of AA5182-O alloy and the FLC’s calculated using M-K-VPSC and M-K-FC models (Serenelli et al., 2011b)

Fig. 16 Schematic of coupled M-K-damage model. The zone B (imperfection zone) has a higher value of initial void volume fraction with respect to the zone A (non-defective zone), i.e. B

A

A

f 0  f 0 . f 0 could be equal to zero. Also, damage in zone A could be neglected.

Fig. 17 The predicted FLC’s for various hardening theories (isotropic hardening (b = 1), kinematic hardening (b = 0) and combined isotropic/kinematic hardening (b= 0.25)) and the experimental FLD of mild steel sheet (Huang et al., 2000)

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Fig. 18 The experimental limit strains of Mg-Al-Ca-Gd alloy sheets and the FLCs predicted by the M–K model (a) at room temperature and (b) at 210 C (Yan et al., 2018)

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Fig. 19 First FLC predictions by MK model with non-zero contact stress p (Banabic and Soare,

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2008)

List of Tables

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Table 1 Parameters considered in study of FLD using M-K model; the abbreviations and units of the parameters are as follows: strain rate (1/s) – SR, forming speed – FS (mm/min), temperature (C) – T, coefficient of biaxial stretching – rb, surface defects – SD, grain size (m) – GS, sheet

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thickness (mm) – Th, superplastic forming – SP, sheet type – ST, Bending – B

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Authors

Yamaguchi and Mellor (1976)

Shakeri et al. (2000) Assempour et al. (2009)

Yield Criteria

-

Materials

Mild steel, annealed aluminum, 70/30 Brass

Hill’48

St12, SUS321

Hill’79 (Hill, 1979b)

St12

Studied parameters and their values/types

GS GS (8 – 60) GS (8 – 50), Th (1, 2.5), SDa (1.4, 6, 20)

von Mises

Cu-FRHC

GS (17.4 – 166.3)

Hill’48

AA1070-O, AA1070-H26

SDb (0 – 0.1)

Chan and Tong (1998)

von Mises

Al6061/20SiCw

SP at T (600)

Chan and Tong (2003)

-

Al-4.4Cu-1.5Mg/21SiCW

SP at T (520)

Naka et al. (2001)

-

AA5083-O

von Mises

AA5083

von Mises

AA5083

Zhang et al. (2008)

Khan and Baig (2011)

Min et al. (2010)

Li et al. (2013)

Logan–Hosford (Logan and Hosford, 1980) BBC2005 (Banabic et al., 2005a)

Logan–Hosford (Logan and Hosford, 1980)

T (550), SR (0.60, 0.67, 0.83, 1.00)

AA5182-O

1997)

ur na

Chan and Lu (2014)

YLD96 (Barlat et al.,

22MnB5

22MnB5

AZ31B

Li et al. (2014)

von Mises

Ti–6Al–4V alloy

Hussaini et al. (2015)

Hill’48

ASS 316

Jo

Chan and Lu (2014)

Logan–Hosford (Logan and Hosford, 1980)

200)

of

(2002)

T (20 – 300), FS (0.2 –

T (240, 300), SR (0.01 –

ro

Chan and Chow

-p

(1997)

re

Hiroi and Nishimura

lP

Peng et al. (2017)

AZ31B

100)

T (20 – 200), SR (10-4 – 1)

T (800) T (600 – 800), SR (10-2 – 1) T (200 – 300), SR (10-3 – 0.01) T (650 – 750), SR (0.0005 – 0.05) T (20 – 400) T (200 – 300), SR (10-3 – 0.01) T (650 – 750), SR

Li et al. (2014)

von Mises

Ti–6Al–4V alloy

Hussaini et al. (2015)

Hill’48

ASS 316

T (20 – 400)

Lang et al. (2015)

Hill’48

5A06

T (150 – 250)

(0.0005 – 0.05)

Bressan et al. (2016)

Hill79 (Hill, 1979b)

AA5083

T (20, 400)

Cao et al. (2016)

von Mises, Hill’48

AZ31

T (200, 250)

Ma et al. (2016b)

Hill’48

TA15

T (810)

Ying et al. (2017)

-

22MnB5

T (700)

Gao et al. (2017b)

Hosford (Hosford,

T (300 – 450), FS (4500

AA2060

1979) Hosford (Hosford,

– 24000) T (300 – 420), FS (4500

AA7075

1979)

– 24000)

Hill’48

B1500HS

Jiang and Xiao (2018)

von Mises

AA2024

Yan et al. (2018)

Hill’48

Mg-Al-Ca-Gd

Wang et al. (2019a)

von Mises

AA7075

Kim et al. (2003b),

GTN with Hosford

Kim et al. (2003c)

(Hosford, 1979)

Yld2003

ur na

(2007)

-p

re

rimmed steel, AA6111-T4

lP

Aretz (2006), Aretz

Tc

ro

Ma et al. (2017)

of

Gao et al. (2017a)

AA6016-T4, AA5182-O, AA3104-H19

T (350 – 450), SR (0.01 – 1) T (20, 210) T (200 – 480), SR (0.01 – 10) Th (0.5 – 1.5)

rb (0.5, 1.2, 1.58)

AKDQ

SR (10-5 – 10)

AA5182, DP600

SR (10-3 – 2500)

Hill’48

DP590

SR (65 – 109)

He et al. (2018)

von Mises

DP780

B

Mohebbi and

Hosford (Hosford,

Akbarzadeh (2012)

1979)

E275, E335, DX54

ST (TWB)

Jie et al. (2009)

Davies (2012)

Jo

Hashemi et al. (2013)

Kim et al. (2015)

Safdarian (2015)

von Mises

Hosford (Graf and Hosford, 1990)

Srp2003-2d (Kim et al.,

AA5182-

2003a), Hill’48

O/polypropylene/AA5182-O

Hill’48

IF steel

ST

ST (TWB)

Darabi et al. (2017)

Wang et al. (2019b)

and Lian, 1989) von Mises

AA3105/St14

ST

AA2B06/SUS321

ST

surface roughness (m), b The ratio of surface defect depth to sheet thickness, c temperature history.

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