An analytical approach for the prediction of forming limit curves subjected to combined strain paths

International Journal of Mechanical Sciences 53 (2011) 365–373

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An analytical approach for the prediction of forming limit curves subjected to combined strain paths Rohith Uppaluri, N. Venkata Reddy n, P.M. Dixit Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India

a r t i c l e i n f o

abstract

Article history: Received 8 March 2010 Received in revised form 3 February 2011 Accepted 10 February 2011 Available online 21 February 2011

In the present work, an analytical approach for the prediction of forming limit curves is proposed to incorporate the effect of combined strain paths. The effect of combined strain paths (considering the directional dependency of pre-straining and further straining) is addressed by integrating the incremental equivalent strain expression in two stages i.e., pre-strain and subsequent loading. In each stage, the strain path is assumed to be linear and different combinations of pre-strain (uniaxial, planestrain and equi-biaxial) are considered. Material anisotropy is taken into account in each stage. The predictions of the present model are compared with the experimental results on forming limit curves under combined strain paths for Al6111-T4 [24] and are found to be in good agreement with each other. & 2011 Published by Elsevier Ltd.

Keywords: Forming limit curve (FLC) Localized necking Combined strain paths

1. Introduction Sheet metal forming is an extensively used press working process in automotive industry, as it reduces secondary operations and enables to produce components at a very high rate. The sheet metal may have inherent voids/imperfections present, because of pre processing. New voids/imperfections initiate and the existing voids/imperfections grow under the applied stress resulting into instability (localized necking) followed by fracture. The occurrence of localized necking is often a limiting factor in stamping processes. Prediction of the initiation of the localized neck allows a prior modification to the process, which can result in a defect-free final product with financial savings. The knowledge of formability of a sheet metal prior to the actual forming operation is of interest for automotive industry to minimize the costly experimental trials before finalizing the die design of auto body panels. Forming limit curve (FLC), in general, represents the formability of sheet metals. The forming limit corresponds to the state when localized thinning of the sheet starts. Early attempts to generate the FLC experimentally were made by Keeler and Backofen [1]. Measured strain data by conducting standard tests [2] are used for constructing the forming limit curve and is represented in a plot of major strain vs. minor strain. It covers the possible strain domain that occurs in industrial sheet metal forming processes. Experimental generation of an FLC consumes both time and resources; it would be very useful if a developed

n

Corresponding author. Tel.: +91 5122597362; fax: + 91 5122597408. E-mail address: [email protected] (N. Venkata Reddy).

0020-7403/$ - see front matter & 2011 Published by Elsevier Ltd. doi:10.1016/j.ijmecsci.2011.02.006

theoretical model can replace those experimental trials. There have been many attempts (analytical and numerical) for the generation of an FLC from the past 50 years of developments in plasticity. The literature available on forming limit curves can be broadly classified into five groups. They are: (a) models that represent the strain instability as a bifurcated state in an initial homogeneous material; (b) forming limit stress curves; (c) models that represent the strain instability as a result of an imperfection already present in the material [3]; (d) porous plasticity, damage and ductile fracture based models; (e) models based on non-planar state of stress; and (f) models considering the phenomenon of cavitation. Hill [4] developed the theory for localized necking in sheet metals assuming that localization band develops along the zero extension direction in a sheet metal. However, in case of a sheet subjected to positive biaxial stretching, zero extension direction does not exist. Hence, the above criterion is only applicable to the negative region of an FLC i.e., from uniaxial to plane strain state of stress. Swift [5] developed the diffuse necking theory for biaxially stretched sheets, introducing the concept of the maximum in-plane force condition in the necking and localization prediction. According to him, neck initiates when the total differentials of force become zero or negative at the same instant of deformation. However, this criterion is often too conservative and it underestimates the experimentally observed forming limit strains. In order to improve Swift0 s [5] model, Hora et al. [6] considered the experimentally observed fact that the onset of necking depends significantly on the strain rate ratio and proposed a model called modified maximum force criterion (MMFC).

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This model was successfully able to predict forming limits in the entire region varying from uniaxial to equi-biaxial conditions. To improve the accuracy of the above model, Hora and Tong [7] proposed the enhanced Modified Maximum Force Criterion (eMMFC) taking sheet thickness and the curvature into consideration. Storen and Rice [8] proposed an alternative concept for localized necking, caused by the vertex developed on the subsequent yield surface. This method can predict localized necking over the entire range of forming limit curve. However, it underestimates the limit for localized necking at the negative region of the FLC. Zhu et al. [9] considered moment equilibrium in addition to the force equilibrium adopted by Storen and Rice [8] for the prediction of the FLC over the entire region (LHS and RHS). Chow et al. [10] used the modified vertex theory of Storen and Rice [8] to predict the FLC for strain rate dependent metals. The modified vertex theory uses the strain rate dependent power-hardening rule. Very recently, Min et al. [11] constructed the LHS of the FLC for boron steel sheets at elevated temperatures assuming that the localized necking band is always perpendicular to the major strain direction (based on the experimental evidence). They used Storen and Rice [8] vertex theory to develop the model. Gillis and Jones [12] assumed that neck appears in three distinct phases: (1) homogeneous deformation up to the maximum load (diffuse necking); (2) localization of deformation under constant load; and (3) local necking with a sudden drop in the load. Different conditions are enforced in each of the different phases to predict the localized necking. Gillis and Jones [12] model were restricted to the Hill [13] yield criterion, which is not suitable for aluminum alloys (Ro1). Khafri and Mahmudi [14] used the Logan and Hosford [15] criterion, which is widely used for materials with Ro1, in conjunction with the power-law. Their predictions were in good agreement when the power law proposed by Tian and Zhang [16] is used. Bressan and Williams [17] proposed that shear instability is initiated in a through thickness direction at which the material element experiences no change of length. The instability occurs when the local shear stress exceeds a critical value ðtcr Þ. Their analysis is in good agreement with the results of [8] for materials with strain hardening index ðn 40:4Þ. Hagbart et al. [18] proposed an analytical criterion for the prediction of the FLC known as BWH criterion by combing Hills localized necking theory for an LHS of the FLC and Bressan and Williams [17] shear instability criterion for RHS of the FLC. Bai and Wierzbicki [19] employed the phenomenological form of ductile fracture for predicting the neck formation in sheets subjected to non-proportional loading history. They accounted the loading history in three steps. In the first step, the model uses the FLC of the as received material as the input. In the second step, the FLC is transformed to the space of equivalent strain at necking vs. Lode0 s angle parameter [20]. Finally, the loading history is taken into account by considering the onset of necking as a nonlinear accumulative process of forming severity. However, the calibration of some parameters present in their model (on which the accuracy of the results is dependent) needs the experimental data of two stage proportional loading process. Major limitation of the FLC models presented above is their dependence on the strain path. Both experimental and numerical results have indicated that FLCs are very sensitive to strain path changes [21–24]. Therefore, finding a single path-independent parameter to characterize forming limits is of considerable interest. Knowing the drawback of the conventional FLC0 s, Kleemola and Pelkkikangas [25] and Arrieux et al. [26] represented the formability based on the state of stress rather than the state of strain. They constructed a forming limit stress curve (FLSC) by plotting the calculated principal stresses at necking. Stoughton [27] showed that the forming limit for both proportional and non-proportional loadings could be explained from a

single criterion, which is based on the state of stress by mapping the strain values from different strain paths into stress space assuming the plane stress condition. Later, Stoughton [28] studied the influence of material model on the stress based forming limit criterion. Stoughton and Zhu [29] reviewed the theoretical strain based models of Swift [5], Hill [4] and Storen and Rice [8] and their relevance to the stress based FLC using plane stress conditions. Stoughton and Yoon [30] extended the work of Arrieux [31] and proposed the concept of anisotropic forming limit curve. They proposed that the representation of a forming limit can no longer be done using a curve, but needs a surface in the strain or stress space, and therefore it is no longer appropriate to represent these limits using the convenience of two-dimensional curves. Stoughton [32] developed a generalized failure criterion, which uses the stress distribution through the thickness of the sheet to identify the mode of failure, including localized necking prior to fracture, surface cracking, and the through-thickness fracture, with or without a preceding neck. Simha et al. [33] proposed an extended stress based limit curve (XSFLC).The stress based limit curve is transformed into equivalent stress and mean stress space to obtain an XSFLC. However, both FLC and stress based forming limit curve (SFLC) are measured and derived, for plane stress loading conditions. None of the above-mentioned attempts for the usage of an SFLC was verified with experimental evidence. Marin et al. [34] measured the forming limit stresses of a tube by subjecting it to combined axial load and internal pressure to obtain linear and nonlinear complex stress paths. Recently Yoshida and Kuwabara [35] have conducted experiments similar to that of Marin et al. [34] using a tension-internal pressure testing machine and concluded that the path dependence of forming the limit stress is strongly affected by the strain hardening behavior of the material for given loading paths. Yoshida et al. [36] observed that the SFLC is strain path dependent by considering a combined loading, in which the strain path is abruptly changed without unloading, using numerical simulations. Marciniak and Kuczynski [3] considered sheet metals subjected to in-plane biaxial loading and proposed a model taking into account that sheet metals are non-homogeneous from both geometric and structural point of view. The RHS of the FLC for a material was determined by introducing a pre-existing material imperfection that lies perpendicular to the major stress axis to explain the development of localized necking during biaxial loading. Marciniak and Kuczynski [3] model was extended to the LHS region of the FLC0 s by Hutchinson and Neale [37] by considering an imperfection at an angle to the major stress axis. The angle selected is the one that minimizes the limit strain. Friedman and Pan [38] investigated the influence of different yield functions on the FLC using the M–K model. They introduced a parameter (i.e., the angle of an imperfection) to characterize the influence of the shape of the yield locus on the FLC. Their results indicate the significance of the yield function used in the FLC analysis. Cao et al. [39] incorporated the anisotropic yield criterion of Karafillis and Boyce [40] in the M–K model [3] to predict localized necking in sheet metal alloys for linear and non-linear strain paths. Their results are validated with the experimental observations of Graf and Hosford [23,24]. Yao and Cao [41] developed the FLC0 s, using M–K model [3] considering the effects of pre-strains and kinematic hardening. The exponent of the yield function used in their work is assumed to decrease with increasing pre-strain. Yoshida and Suziki [42] analyzed the forming limits of a sheet metal subjected to linear and combined loading paths, using M–K model [3] and found that the stress based FLC is independent of strain path only if the work hardening behavior is not affected by the strain path change. Aretz [43] extended Hills [4] localized necking model to predict forming limit curve

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for negative minor strains and its predictions are closer to that of the M–K model [3]. Thus, it is concluded that the extended model is an efficient alternative to the M–K model [3] in the negative minor strain regime. Brunet and Morestin [44] coupled Hora0 s [6] FLC model with Gurson0 s [45] porous plasticity model and obtained good predictions for the forming limits. Fahrettin and Lee [46] investigated the forming limits of sheet metal, using the phenomenological ductile fracture criteria by performing the finite element simulation of an out-of-plane formability test. Their predictions for the LHS of the FLC are in good agreement with experimental results. However, they were not successful for the RHS of the FLC. Jain et al. [47] and Vallellano et al. [48,49] also reported similar observations. Rohith [50] employed the continuum damage mechanics model (CDM) of Lemaitre [51] to study the instability strains in metals by simulating a out of plane formability test. His study concluded that critical damage value (Dc) should not be used for the prediction of localized necking in an RHS of the FLC. Haddag et al. [52] combined the Khelifa0 s [53] CDM model and the strain localization condition of Rice [54] to obtain the FLC qualitatively. Xue [55] proposed the maximum power localization criterion for proportional loadings and combined it along with the damage plasticity theory [56], to predict the FLC for the entire region. He concluded that the governing factor for localization is not the damage itself, but the resulting effect of the rate of weakening from the plasticity induced damage. By extending CDM model to anisotropic materials, Chow and his co-workers [57–60] developed a unified anisotropic damage approach for predicting the forming limit curves. Their theory predicts the FLC for the damage coupled, kinematic-isotropic hardening material model under non-proportional loading. In some metal forming processes, such as hydro, stretch flange and incremental formings, the onset of necking occurs under loading conditions that are not plane stress. Analytical/numerical attempts have been made in the literature to observe the effect of non-planar state of stress. Gotoh [61] extended the analysis of Swift [5] to incorporate the effect of third normal compressive stress on the forming limits of sheet metal and observed approximately 20% increase in the minima of the FLC (i.e., the plane strain ordinate of the FLC denoted by an FLC0). Smith et al. [62] studied the influence of transverse normal stress on strain space forming limit by assuming that the stress space-forming limit is relatively insensitive to the transverse normal stress. Their model predicts an increase in the formability in strain space that varies nonlinearly with an increasing magnitude of the compressive transverse normal stress ratio. Smith and Matin [63] modified their previous model [62] by assuming the stress ratio to be constant even under the influence of transverse normal stress, instead of assuming the strain ratio to be constant. The behavior of both the models is the same, but the second model [63] is observed to be simple and independent of the material model. Banabic and Soare [64] developed a simple method for including normal stress component to the plane of the sheet in the M–K model. Their model allows for studying the influence of the fluid pressure upon the forming limit strains of orthotropic thin sheets. In practice, it is observed that the formability of the sheet is improved if the sheet is drawn/stamped under the fluid pressure. Allwood and Shouler [65] extended the M–K model [3] by considering all the six components of the symmetric stress tensor and assuming proportional loading conditions. They observed that the generalized forming limits of a sheet metal increase significantly with both the normal compressive stress and through-the-thickness shear. The increase in formability is validated with experiments by a specially designed ‘‘linear paddle testing’’ apparatus. Wu et al. [66] studied the effect of hydrostatic pressure on FLC0 s using the M–K model [3] and found that

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superimposed pressure delays the initiation of the neck for any strain path. Later Wu et al. [67,68] studied the effect of hydrostatic pressure on the fracture of axisymmetric and planar specimens under tension and concluded that superimposed hydrostatic pressure has no noticeable effect on necking, but delays the fracture. It is well known that most superplastic materials cavitate during deformation, and the presence of cavitation leads to the premature failure. Cavitation behavior of superplastic materials is shown to relate to the size and morphology of grains, distribution of inter metallic particles, strain rate, temperature range of deformation, stress state and strain levels. Chow and Chan [69] studied the cavitation behavior of a commercially available coarse-grained Al5052 alloy under hot uniaxial and equibiaxial tensions. They observed that the spread of cavity size and the total number of cavities were found to increase with increasing strain. Chan and Chow [70] studied the cavitation behavior of a commercial superplastic Al5083 alloy under different stress ratios and they observed similar cavitation behavior at low strain level. However, the number of cavities under equi-biaxial tension is significantly larger than that under uniaxial tension. Chow and Chan [71] used the available experimental cavitation data [69] and proposed an analytical model based on the M–K model and the damage mechanics approach. An experimental forming limit curve is constructed for coarse-grained Al5052 alloy and the shape of curve is found to be significantly different from that obtained in conventional sheet metal forming at room temperature. Their predictions are found to be in reasonable agreement with the experimental findings. Critical literature review presented above reveals that most of the researchers employed the famous M–K model [3] along with some numerical technique to predict the forming limits curves under combined strain paths. It is well known that the M–K model [3] is highly sensitive to the size of the imperfection chosen and involves a good amount of computational cost. As mentioned earlier, very few analytical attempts like Bai and Wierzbicki [19] have been made to predict the forming limit curves under combined strain paths, but they need more experimental data such as the FLC of the as received material. The present work focuses on developing a simple and cost efficient analytical methodology for the prediction of forming limit curves of a material subjected to combined strain paths. In the present work, the modified maximum force criterion (MMFC) developed by Hora et al. [6] is used as an initial instability criterion. The reason for employing the above model is twofold. This criterion needs only material plasticity data and does not require any other uncertain parameters like the imperfection size (which is required in the well-known M–K model). In addition, a single expression predicts the FLC over the entire strain domain. Further, the present methodology, unlike other analytical approaches, requires only the minima (i.e., the plane strain ordinate FLC) of the as received FLC. In the present work, a combined strain path is first decomposed into two parts: (i) the strain path of pre-straining and (ii) the strain path of further testing (along with the influence of directional dependence of pre-straining). Next, the modified maximum force criterion (MMFC) of Hora et al. [6] is used to generate the forming limit curve (FLC). Finally, the strains are mapped into the stress domain and the validity of the path independent behavior of the stress based FLC is studied. Experimental results of Graf and Hosford [24] are utilized for the validation of the present model. Present approach successfully predicts the forming limit strains under all combinations of combined strain paths presented by Graf and Hosford [24] and the results are in good agreement qualitatively as well as quantitatively.

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2. Formulation In the present work, the modified maximum force criterion (MMFC) proposed by Hora et al. [6] for proportional loading is used as a basis to develop the forming limit curve for combined strain paths. Here, the material is assumed to yield according to Logan and Hosford [15] in-plane non-quadratic yield criterion, which has a better agreement with the yield data for the materials with anisotropy (R) values less than unity, compared to the criterion of Hill0 s [13]. Logan and Hosford [15] criterion is given by R0 R90 R0 R90 ðs1 Þa þ ðs2 Þa þ ðs1 s2 Þa 1 þ R0 1 þ R0 1 þ R0

ðsy Þa ¼

ð1Þ

where R0 and R90 are the material anisotropy coefficients and a ¼8 for aluminum. Here ð1Þ, ð2Þ, ð3Þ directions coincide with transverse, rolling and through-thickness directions, respectively. The expression for the equivalent stress, in terms of stress ratio ða ¼ s2 =s1 Þ and the anisotropy values, is given by

seq ¼ Ha s1 , where Ha ¼



R0 R90 a R0 R90 þ a þ ð1aÞa 1 þ R0 1 þ R0 1 þ R0

1=a

expression for the major strain  0 H b e1 ¼ n þ a 0 Ha b

The above equation implies that, in the plane strain condition (i.e., b ¼0), the ordinate of the FLC (i.e., e1 ) becomes equal to the strain-hardening exponent (n). Therefore, in Eq. (9), the strainhardening exponent (n) is replaced by the experimentally measured plane strain ordinate of the as received FLC ðe^ Þ. This modifies Eq. (9) as    0  H Ha @b e^ ab  ¼1 ð11Þ 0 eeq Ha b @e1 As stated earlier, in the present work, the equivalent strain of Eq. (5) is divided into two parts: (i) the first part corresponding to the pre-strain (PS) and (ii) the second part corresponding to the further testing

eeq ¼

ð2Þ Assuming negligible elastic strains and using the principle of incremental plastic work, the incremental equivalent plastic strain, in terms of strain rate ratio ðb ¼ de2 =de1 Þ, is given as deeq ¼

s1 de1 þ s2 de2 1 þ ab ¼ Hab de1 , where Hab ¼ seq Ha

ð3Þ

Integration of the above equation results in

eeq ¼ Hab e1 ðproportional loadingÞ eeq ¼

Z

1 de1 þ Ha

Z

a Ha

ð4Þ

de2 ðnonproportional loadingÞ

ð5Þ

The relation between stress ratio (a) and strain rate ratio (b) can be obtained using an associated flow rule as

bðaÞ ¼

de2 R90 aa1 R0 R90 ð1aÞa1 ¼ de1 R0 þ R0 R90 ð1aÞa1

ð6Þ

Strain hardening relation for materials used in the present work is assumed to obey power law and is given as

seq ¼ Kðeeq Þn

ð7Þ

Hora et al. [6] modified the diffuse necking condition (ds1 =de1 ¼ s1 ) of Swift [5] by considering an experimental observation that the onset of necking depends significantly on the strain rate ratio (b) and named it as modified maximum force criterion (MMFC) given as @s1 @seq @eeq @s1 @b þ ¼ s1 @seq @eeq @e1 @b @e1

ð8Þ

The first term in the above equation corresponds to the point of uniform deformation, whereas the second term corresponds to non-uniform deformation. The above equation should be interpreted so that, in the first term eeq is treated as variable, while a and b are constants. In the second term a and b are variables, while eeq and seq are constants. Using Eqs. (2) and (7), the above equation becomes n

Hab

eeq



Ha0 @b ¼ 1, 0 Ha b @e1

where

Ha0 ¼

@Ha , @a

b0 ¼

@b @a

Z ePS   Z e1   1 1 1 de1 þ de 1 Ha a ¼ aPS Ha aUA r a r aBA 0 ePS 1 Z ePS   Z e2   2 a a þ de2 þ de2 Ha a ¼ aPS Ha aUA r a r aBA 0 ePS 2

For the proportional loading conditionðb ¼ e2 =e1 , @b=@e1 ¼ b=e1 , Hab =eeq ¼ 1=e1 Þ, the above equation leads to the following

ð12Þ

During the first stage of loading (i.e., during the pre-strain (PS) stage), the stress ratio ðaPS Þ is assumed to be constant. In the second stage, the proportional loading condition is assumed and the stress ratio (a) is varied from uniaxial ðaUA ¼ 0Þ to equi-biaxial ðaBA ¼ 1Þ states of stress. Then, the above equation leads to       1 1 eeq ¼ ePS ðe1 ePS 1 þ 1 Þ Ha a ¼ aPS Ha aUA r a r aBA       a a ePS þ ð13Þ þ ðe2 ePS 2 Þ Ha a ¼ aPS 2 Ha aUA r a r aBA Eq. (13) reduces to the proportional loading condition (Eq. (4)) PS of Hora et al. [6] in the absence of pre-strain. Strain rate ratio ðb Þ during the pre-strain stage is defined as

bPS ¼

incremental strain ? to the direction of prestraining Incremental strain along the prestrain direction

ð14Þ

Two cases, based on the directions of pre-straining and further testing, are presented below. First, both the pre-straining and further testing are considered in the transverse direction (TD, taken as direction 1). When the material is pre-strained in the transverse direction, the induced pre-strain ePS 2 in the rolling direction (RD, taken as direction 2) is obtained from the definition of strain rate ratio (Eq. (14)) during the pre-strain (PS) stage PS PS ePS 2 ¼ ðb Þe1

ð15Þ

PS

where ðb Þ is assumed to be constant. Similarly, the total strain e2 in the rolling direction is obtained by integrating the strain rate ratio ðbÞ Z e2 Z e1 e2 ¼ de2 ¼ bde1 ð16Þ 0

0

Division of the integral in the two parts leads to

e2 ¼

Z ePS 1 0 PS

ð9Þ

ð10Þ

ðbÞb ¼ bPS de1 þ

Z e1 ePS 1

ðbÞbUA r b r bBA de1

PS ¼ ðb ÞePS 1 þ ððbÞbUA r b r bBA Þðe1 e1 Þ

ð17Þ

Substitution of the expressions (15) and (17) for ePS 2 and e2 in Eq. (13) and the use of expression (3) for Hab leads to the following expression for the equivalent plastic strain in terms of

R. Uppaluri et al. / International Journal of Mechanical Sciences 53 (2011) 365–373



the major strains ðe1 , ePS 1 Þ 0 1 0

1

B

C

B

C

@

A

@

A

C B C PS eeq ¼ B BðHab Þ a ¼ aPS CePS 1 þ BðHab Þ aUA r a r aBA Cðe1 e1 Þ

b ¼ bPS

þ

bUA r b r bBA

ð18Þ

Here, the strain rate ratio (b), during the second stage of UA BA loading, varies from the uniaxial ðb Þ to equi-biaxial ðb Þ states UA BA of stress. The strain rate ratios ðb , b Þ are calculated from Eq. (6) using the respective stress ratios ðaUA , aBA Þ. A shift is observed at the FLC0 (i.e., the plane strain ordinate) of the forming limit curve of the combined strain paths due to the presence of the first term in Eq. (18). Rearranging Eq. (17), one can obtain PS

b¼

PS 1

e2 ðb Þe e1 ePS 1

ð19Þ

Partial differentiation of the above expression with respect to major strain e1 and using Eq. (17) results in

ePS 2

0

ePS 1

PS PS e2 ¼ ePS 2 þ ððbÞbUA r b r bBA Þðe1 b e2 Þ

ð27Þ

Substitution of expressions (22) and (27) for ePS 1 and e2 along  ab in Eq. (25) and the use of expression (3) for Hab with Ha0 b ¼ 1 þ 0 H a

leads to the following expression for the equivalent plastic strain in terms of the strains ðe1 , ePS 2 Þ 0 1 0 1 B

C

B

A

@

C

C B C PS PS eeq ¼ B BðHa0 b Þ a ¼ aPS CePS 2 þ BðHab Þ aUA r a r aBA Cðe1 b e2 Þ @

ð20Þ

b ¼ bPS

bUA r b r bBA

A

ð28Þ Rearranging Eq. (27), one can obtain

b¼

2

ð30Þ

2

Substitution of Eqs. (28) and (30) in Eq. (11) leads to 0 1 B

C

C e^ B BðHab Þ aUA r a r aBA C

ð21Þ

@

ð22Þ

Similarly, the relation between stresses in transverse (TD) and rolling (RD) directions during pre-strain is given by

s1 ¼ ðaPS Þs2

ð29Þ

e2 ePS @b b 2 ¼ ¼ PS @e1 ðe1 b ePS Þ2 e1 bPS ePS

The above equation for e1 is quadratic in nature with two roots. One root gives the major strain, required to construct the FLC. Once the major strain is determined, the minor strain e2 can be obtained from Eq. (17). Thus, Eq. (21) can be used to generate the FLC for a material pre-strained in the transverse direction and further tested in the same direction. Next, the case of pre-straining along rolling direction (RD) and further testing in transverse direction (TD) is discussed. The induced pre-strain in the transverse direction due to pre-straining along the rolling direction is obtained from the definition of the strain rate ratio (Eq. (14)) during the pre-strain (PS) stage PS PS ePS 1 ¼ ðb Þe2

e2 ePS 2 e1 bPS ePS 2

Differentiation of the above expression with respect to major strain e1 and using Eq. (29) results in

1

B C B C  B C B C þ BðH Þ BðHab Þ a ¼ aPS CePS C e ePS @ A 1 @ ab aUA r a r aBA A 1 1 b ¼ bPS bUA r b r bBA 0 1 ! B H 0  C ðbÞ UA b r b r bBA B C a þB ¼1 C @ Ha b0 aUA r a r aBA A e1 ePS 1 UA BA b rbrb

0

1

0

bUA r b r bBA

A 1

B C B C B 0 C B C PS PS þ BðHab Þ UA BðHab Þ a ¼ aPS CePS BA Cðe1 b e2 Þ 2 a rara A @ A @ b ¼ bPS bUA r b r bBA 0 1 ! B H0  C ðbÞ UA b r b r bBA B C a þB ¼1 C UA BA @ Ha b 0 a r a r a A e1 bPS ePS 2 UA BA b rbrb

ð31Þ

Eq. (31) is a quadratic in e1 with two roots. In the present work, the root that predicts the shape of the FLC is chosen. Once the major strain e1 is determined, the minor strain e2 can be obtained from Eq. (27). Thus, Eq. (31) can be used to generate the FLC for a material pre-strained in the rolling direction (RD) and further tested in the transverse direction (TD).

ð23Þ

Using the above expression (Eq. (23)), the equivalent stress expression for pre-strain (Eq. (2)) can be obtained as

seq ¼ Ha0 s2 , where Ha0 ¼

ð25Þ

Combination of Eqs. (22) and (26) results in

Substitution of Eqs. (18) and (20) in Eq. (11) results in 0 1 B C C e^ B BðHab Þ aUA r a r aBA C @ A bUA r b r bBA 1 0

     1 a PS e þ ðe2 ePS 2 Þ Ha0 a ¼ aPS 2 Ha aUA r a r aBA

To obtain the expression for the total strain e2 in the rolling direction, Eq. (16) is used by considering the appropriate change in the limits of an integration as follows: Z e2 Z e1 de2 ¼ bde1 ð26Þ

PS

e2 ðb ÞePS @b b 1 ¼ ¼ 2 @e1 e1 ePS ðe1 ePS 1 1 Þ

369



R0 R90 R0 R90 ðaPS Þa þ þ ð1aPS Þa 1 þ R0 1 þ R0 1þ R0

1=a

ð24Þ Integration of Eq. (3) after substituting Eq. (24) for prestraining and Eq. (2) for further loading results in       a 1 PS eeq ¼ e þ ðe1 ePS 1 1 Þ Ha aUA r a r aBA Ha0 a ¼ aPS

3. Results and discussion To test the validity of the model developed to generate forming limit curves for combined strain paths, its predictions are compared with the experimental [24] results available in the literature. The comparison shows that the FLC predictions of the present model are in good agreement with the experimental results reported by Graf and Hosford [24]. Given material can be pre-strained to different levels in rolling or transverse direction under an uniaxial or plane strain state of deformation and the further straining can be done either in the

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same direction or in the other. Fig. 1 shows the FLC0 s of the material pre-strained in an uniaxial tension along the transverse direction (TD) and further tested along the same direction. It can be clearly seen from Fig. 1 that the pre-straining in uniaxial condition enhances the forming limits of the sheet metal. This observation is in conformity with the experimental results of Graf and Hosford [24]. Note that the pre-straining increases the forming limits in uniaxial tension even though no strain path change occurs. The maximum percentage deviation varies from 4.9% at an uniaxial state and increases to 20.9% at plane strain state and further decreases to 1.62% at an equibiaxial stress state for pre-strain value of 0.18. Similar percentage differences are obtained at other pre-strain values. Fig. 2 shows the FLC0 s of a material pre-strained in an uniaxial tension along the rolling direction and further tested along the transverse direction of the sheet. One can observe from Fig. 2 that there is a significant reduction in the formability when the prestraining and further testing are performed in different directions. Note that the decrease in formability increases with an increase in the pre-strain. Fig. 3 shows the FLC0 s of the material pre-strained in plane strain tension along the transverse direction and further tested

Fig. 1. Comparison between the predicted and experimental FLC’s for Al6111-T4 prestrained in uniaxial tension and further tested along the transverse direction (TD).

Fig. 2. Comparison between the predicted and experimental FLC’s for Al6111-T4 pre-strained in uniaxial tension along the rolling direction (RD) and further tested along the transverse direction (TD).

along the same direction. Results presented in Fig. 3 indicate that the formability is enhanced due to the plane-strain pre-straining. The predicted FLC shows no change in the value of the FLC minima (i.e., the plane strain ordinate FLC0). However, the experimental results of Graf and Hosford [24] indicate an increase in the FLC minima. This can be attributed to the experimental PS difficulty of making the value of b zero. Fig. 4 shows the FLCs of the material pre-strained in plane strain tension along the rolling direction and further tested in the transverse direction. Similar to that of an uniaxial pre-strain loading (Fig. 2), significant reduction in the formability is observed when the prestraining and further testing are in different directions. Fig. 5 shows the FLCs of the material pre-strained in equi-biaxial tension and further tested. From figure, one can observe that for the case of equi-biaxial tension, pre-straining increases the forming limits of the material and this observation is in conformity with the experimental results of Graf and Hosford [24]. If the strain rate ratio (b) of the initial pre-straining and further loading are the same, no rise in the limiting strains is expected, as the strain path is not changed. Most theoretical models [19,36,39] and some experimental results [72] have predicted no rise in the forming limits for the uniaxial and equi-biaxial tensions, when the paths of

Fig. 3. Comparison between the predicted and experimental FLC’s for Al6111-T4 pre-strained in plane strain tension and further tested along the transverse direction (TD).

Fig. 4. Comparison between the predicted and experimental FLC’s for Al6111-T4 pre-strained in plane-strain tension along the rolling direction (RD) and tested along the transverse direction.

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Fig. 5. Comparison between the predicted and experimental FLC’s for Al6111-T4 pre-strained in equi-biaxial tension and tested along the transverse direction (TD).

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Fig. 6. Comparison between the predicted FLC’s of the modified model and experimental FLC’s of Armco-Iron, pre-strained in uniaxial and equi-biaxial tension along the rolling direction (RD) (R0 ¼ 0:88, R90 ¼ 0:88, e^ ¼ 0:3, a ¼ 2).

pre-strain and further loading are the same. However, the left end of the FLC0 s of Fig. 1 (based on Eq. (21)) shows that the forming limit in uniaxial tension increases even though the directions of pre-strain and further loading are the same. Similarly, the right end of the FLC0 s of Fig. 5 shows that the forming limit in equi-biaxial tension increases even though the paths of pre-strain and further loading are the same. To obtain this trend, the present model is modified by replacing Eqs. (19) and (20) by ðb ¼ e2 =e1 ,@b=@e1 ¼ b=e1 Þ. The above modification results in a minor change in the second term of the FLC model (Eqs. (21) and (31)). When pre-straining and further loading are in the same direction, expression (21) becomes 0 1 B

C

C e^ B BðHab Þ aUA r a r aBA C @

0

1

bUA r b r bBA

A

0

1

B C B C B C B C PS þ BðHab Þ UA BðHab Þ a ¼ aPS CePS BA Cðe1 e1 Þ 1 a r a r a @ A @ A PS UA BA b¼b b rbrb 0 1 ! B H 0  C ðbÞ UA b r b r bBA B C a ¼1 þB C @ Ha b0 aUA r a r aBA A e1 UA BA b rbrb

Fig. 7. FLC’s of Al6111-T4 pre-strained in uniaxial tension and further tested along the transverse direction (TD).

ð32Þ

where the minor strain is given by Eq. (17). When the directions of pre-straining and further testing are different, expression (31) becomes 0 1 B

C

C e^ B BðHab Þ aUA r a r aBA C @

0

1

0

bUA r b r bBA

A 1

B C B C B 0 C B C PS PS þ BðHab Þ UA BðHab Þ a ¼ aPS CePS BA Cðe1 b e2 Þ 2 a rara A @ A @ b ¼ bPS bUA r b r bBA 0 1 ! B H 0  C ðbÞ UA b r b r bBA B C a þB ¼1 C @ Ha b0 aUA r a r aBA A e1 UA BA b rbrb

ð33Þ

where the minor strain is given by Eq. (27). To test the validity of the modified model presented above (Eqs. (34) and (35)), its predictions are compared with the

experimental results of Gronostajski et al. [69]. It can be clearly seen from Fig. 6 that the predictions of the modified model are in good agreement with the experimental results of Armco Iron, prestrained in an uniaxial tension. A value of 2 for the yield function exponent (a) is observed to give a good agreement between the theoretical and experimental results. Note that the modified model predicts no change in the limiting strains at an uniaxial state when compared to the values of the as-received FLC of the material. Fig. 7 shows the FLC0 s of the modified model pre-strained in an uniaxial tension and further tested along the transverse direction. The pre-strain values used to predict the FLCs in Fig. 7 and 8 are the experimental values used by Graf and Hosford [24]. It can be clearly seen from Fig. 8 that there is no increase in the forming limit strain for uniaxial tension compared to that of Fig. 1. Yoshida and Suziki [42] and Bai and Wierzbicki [19] reported similar results. Fig. 8 shows the FLC0 s of the modified model pre-strained in equi-biaxial tension and further tested along the transverse direction. It can be clearly seen from figure that there is no increase in the forming limit strain for equi-biaxial tension compared to that of Fig. 5. The present observations are in good

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Fig. 8. FLC’s of Al6111-T4 pre-strained in equi-biaxial tension and further tested along the transverse direction (TD).

agreement with the numerical results of Cao et al. [39] and the analytical results of Bai and Wierzbicki [19]. The other cases of the above-mentioned modified model such as uniaxial pre strain (Fig. 2) in the rolling direction (RD) and further testing along the transverse direction (TD) and the two types of plane strain pre-straining as shown in Figs. 3 and 4 are not shown in the paper as they are giving a similar trend as expected with original model.

4. Conclusions An analytical approach to predict the forming limit curves under combined strain paths is presented and the results are found to be in good agreement both qualitatively and quantitatively with the experimental observations [24,69] and theoretical predictions [19,39] available in the literature. As the strain path in an actual forming process cannot be determined prior to the deformation, the assumption of bi-linear strain path in our analysis leads to a good agreement given the computational resources and the accuracy of the predictions. The following conclusions are drawn based on the results obtained in the present work:

 Formability of a material pre-strained either in uniaxial or 

plane-strain tension can be increased when the directions of initial pre-strain and further loading coincides with each other. Formability is enhanced over the whole range of the minor strain when the material is pre-strained in the plane strain tension.

Acknowledgment The authors sincerely acknowledge the financial support provided by Tata Steel R&D, Jamshedpur.

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