Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 A comparative study of the extended forming limit diagrams considering strain path, through-thickness normal and shear stress Mozhdeh Erfanian, Ramin Hashemi* School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran *Corresponding author: e-mail: rhashemi@iust.ac.ir (R. Hashemi) Tel.: +98-21-77240068, Fax: +98-21-77240540 Abstract The substitute presentations of the conventional forming limit diagram (FLD) are stress-based forming limit diagram (FLSD), extended stress forming limit diagram (XSFLD), and polar effective plastic strain forming limit diagram (PEPS-FLD). These diagrams have already been proposed as alternative criteria to the conventional FLD for predicting forming limits in processes that sheet does not experience proportional or in-plane loading conditions. The present study provides a complete comparison of different forms of forming limit diagrams. For this purpose, FLD and extended FLDs are predicted based on the modified Marciniak and Kuczynski (M-K) model with Yld2011 anisotropic yield function while various factors that cause a change in the FLD of a specific alloy including initial imperfection coefficient (f0), non-linear strain path, through-thickness normal and shear stresses are considered. The results indicate that all forms of extended diagrams are independent of strain path and the diagrams in stress space (i.e., FLSD and XSFLD) are less sensitive to the strain path than PEPS-FLD. In this regard, the main weakness of XSFLD, especially on the right-hand side, is that the safety margin cannot be visualized easily. By increasing the initial imperfection coefficient, the level of all forms of diagram increases. However, the effect of through thickness stresses on different diagrams is not the same. Increase in normal and shear stresses results in a downward shift in FLSD and an upward shift in other types of forming limit diagrams. Keywords Extended forming limit diagram; Prestraining; Through-thickness normal stress; Through-thickness shear stress; M-K model. Introduction In the process of sheet metal forming, knowing the maximum allowable strains before necking is a critical requirement. For this purpose, forming limit diagram has been introduced. For many years, this diagram has been obtained both experimentally and theoretically with assumptions of plane stress state and linear strain path. But soon it became clear that due to these simplifying assumptions, the forming limit diagram was not accurate enough and no longer could predict forming limits in manufacturing processes that impose a non-linear and 3D state of stress on the sheet metal. The experimental evidence and theoretical results proved that the conventional FLD was highly sensitive to the strain path. Graf and Hosford investigated the effect of bi-linear strain paths on the aluminium alloys AA2008-T4 and AA6111 [1,2] through careful experiments. Kohara [3] 1 Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 considered the effect of non-proportional strain paths and annealing on the FLD of AA1050. According to Yao and Cao [4] due to the Bauschinger effect, isotropic hardening was not suitable for modeling the behavior of prestrained sheet therefor they considered back stress and calculated right side of FLD using the M-K model. Butuc et al. [5] studied the application of the M-K model for predicting FLD for a sequence of three linear strain path stages (double strain path changes). They concluded that similar to the one strain path change, for this particular loading case, formability increased if the first prestraining was uniaxial tension and plane-strain tension, respectively. On the contrary, the first prestraining in biaxial stretching decreased the forming limits. Shakeri et al. [6] introduced a theoretical model utilizing the many slices approach to simulate neck growth and investigated the effect of uniaxial prestraining on the forming limits of St12 low carbon steel. Li et al. [7] proposed a model based on Hill’s 1948 local necking criterion and affirmed the ability of the model in predicting forming limits for loading condition with nonlinear strain path. Furthermore, they proposed formability evaluation index and verified the application of the index in the two-stage forming processes with drawing and reverse-drawing. They concluded that this index predicts forming limit of the forming process with non-linear strain path more accurately than the traditional FLD. In addition to the effect of non-linear strain path, through-thickness normal and shear stresses also change forming limits. Hydroforming and incremental sheet forming processes are practical examples in which a sheet can withstand higher strains than under operations that are mainly tensile. Bridgeman and Fuchs [8,9] were among the first who carried out a number of experiments and reported the forming limits increasing as a result of hydraulic pressure. Later on, many researchers considered this fact. Smith et al. [10] assumed that normal stress has no effect on the FLSD and by employing the Stoughton’s strain-to-stress space mapping procedure [11], proposed a new formability model based on Hill’s 1948 yield criterion, which could consider the effect of through-thickness normal stress. Results showed that through-thickness normal stress could increase the formability. Afterward, Matin and Smith [12] modified the previous Smith’s model and stated that the modified model correlates well with experiments when considering the evolution of work hardening of metal and ratio of through-thickness normal stress during the deformation. Gotoh et al. [13] generalized Swift analysis for considering normal stress and plotted FLD as a function of the ratio of through-thickness normal stress to major in-plane stresses. Assempour et al. [14] developed the M-K model using new equations, which included throughthickness normal stress in the M-K model. They showed formability improvement of AA6011 and STKM-11A as a result of through-thickness normal stress. Hashemi et al. [15] used the model proposed by Assempour et al. [14] in conjunction with finite element simulations and predicted the necking pressure and axial feeding in hydroforming of the T-shaped tube. Banabic and Soare [16] also studied the effect of normal stress, applying the M-K model and Soar2008-Poly6 yield function for considering anisotropy behavior of sheet metal. Allwood and Shouler [17] generalized the M-K model for considering all six components of the symmetric stress tensor. They proposed a new general forming limit diagram (GFLD) and demonstrated that both through-thickness normal compressive stress and shear stress increase formability. In this regard, the theoretical results were supported by the experimental data from the specially designed test. Eyckens et al. [18] numerically studied the effect of through-thickness shear stress on forming limits using the M-K model. It was found out that through-thickness shear stress distinctly increased formability depending on the planes in which shear stress imposed. Fatemi and Dariani [19] considered through-thickness stresses acting in planes perpendicular to minor and major strain directions simultaneously. They concluded that exertion of through-thickness stresses led to increasing strain 2 Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 hardening, changing the strain state in the groove region and increasing forming limits, consequently. In recent years, several new forms of forming limit diagrams have been developed, in which forming limits are presented by variables other than ε1 and ε2 and the goal is predicting forming limits in the manufacturing processes with high non-linearity and 3D loading conditions. Yoshida et al. [20] experimentally ascertained path-independency of FLSD through measuring the forming limit stresses in biaxial tests for an aluminum alloy tube, using a servo-controlled internal pressure-axial load type testing device. In [21], the researchers investigated the effect of prestraining on FLSD considering two types of combined loading, with and without unloading between the first and second stages of loading. The results demonstrated that without unloading, the calculated FLSD strongly depended on the strain paths. Stoughton et al. [22] reviewed various theoretical models of sheet metal forming instability, including bifurcation analyses of Swift, Hill, and Storen and Rice as well as the microscopic damage and the M-K defect model. They demonstrated that in stress space, the models were less complicated and led to a criterion that only was a function of the current state of stress and thus, path-independent. Butuc et al. [23] performed a detailed theoretical study on FLSD, using wide range of material models, from Von Mises, Hill and advanced Barlat Yld96 yield function with the conjunction of Swift and Voce law stress-strain relations. They also verified the influence of work hardening coefficient, strain rate sensitivity, and strain path changes. Simha et al. proposed [24] the idea of the XSFLD for the first time, assuming that the stress states at the onset of necking under plane-stress loading are equivalent to those under three-dimensional loading. They used the XSFLD to predict the onset of necking and final failure location in tubular hydroforming of both pre-bent and straight tubes. Hashemi and Abrinia [25] reexamined the path independency of the XSFLD and analyzed the effect of throughthickness normal stress on XSFLD. The results were inconsistent with the assumption taken by Simha et al. [24] that the XSFLD is not affected by normal stress. Nasiri et al. [26] theoretically analyzed FLSD and XSFLD based on the M-K model and Yld2011 yield criterion considering through-thickness normal and shear stress. Zeng et al. [27] proposed the effective plastic strain to assess formability. They chose an approach based on the M-K model and calculated effective strain using Hill quadratic yield function and introduced a new form of forming limit presentation (eFLD), which was constructed based on effective strains and material flow direction at the end of forming. They investigated the influence of strain path on their proposed forming limit diagram and concluded that regardless of the strain history, the onset of necking only depends on the final strain ratio. Zhang et al. [28] studied the effect of through-thickness normal stress on different forms of forming limit diagrams, including FLD, FLSD, XSFLD, and eFLD under both linear and nonlinear strain paths with the M-K model and Yld2003 yield function. Stoughton and Yoon [29] also chose the effective plastic strain to represent formability. They introduced new diagram named polar effective plastic strain (PEPS) and successfully mapped all Graf and Hosford experimental diagrams [1] into effective plastic strain space and verified the utility of PEPS diagram as an insensitive criterion to strain path. Basak et al. [30] proposed a mathematical framework and converted the experimental limiting strain data of the extra deep drawing (EDD) steel sheets prestrained with seven different types of loading to PEPS-FLD applying the Yld96 anisotropic yield function. This diagram was next incorporated as a criterion in finite element simulation to evaluate forming behaviors of particular two-stage stretch forming test. Nguyen et al. [31] applied the PEPS diagram of free-expansion hydroforming of as-received tube in finite element simulation to predict necking in free-expansion hydroforming of the tubes with tensile prestraining in the axial direction. They demonstrated that although PEPS diagram slightly 3 Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 underestimated the failure, it captured the forming severity accurately. Regarding the above literature review, some studies have been devoted to investigating the effect of different factors on the FLSD and XSFLD, but the PEPS diagram has been less explored. Besides, various forms of forming limit diagram have rarely been compared with each other. The effect of shear stress on the forming limits is the other subject that needs further investigation. With these goals, in the present work, the Yld2011 advanced anisotropic yield criterion is employed in the modified M-K model to predict forming limit diagrams and extended forming limit diagrams for sheets that undergo various non-proportional loadings and through-thickness normal and shear stresses. The reason to choose this specific yield criterion is its high accuracy in describing the behavior of aluminum sheet [26,32]. Theoretical forming limit diagrams for the plane-stress state are compared with available experimental data for AA2008-T4 sheets [1] and the predicted FLD with considering through-thickness normal stress is validated with published data for hydroforming of AA6063 tube [33]. In each loading condition, FLDs are next converted into PEPS-FLD, FLSD and XSFLD. The term “sensitivity” has been used to compare dependency of different forms of the diagram to the investigated factor (imperfection coefficient, prestrain, normal stress or shear stress) and to show how much these factors cause the obtained diagram to be different from the as-received diagram. Results demonstrated that FLSD and XSFLD are less sensitive to strain path than PEPS-FLD. Initial imperfection coefficient, through-thickness normal and shear stresses cause all forms of forming limit diagram to vary. Thus, these parameters should be carefully considered to obtain precise theoretical forming limit diagrams. Yld2011 yield criterion The advanced yield functions can more accurately predict the actual forming limits of a material better than classic criteria [26]. Such a higher accuracy is owning to the set of terms and parameters that makes these novel criteria much more precise in the prediction of the directional anisotropy and stress values. Artez and Barlat [32] introduced the anisotropic Yld2011-18p yield criterion in the form of linear transformations of derivative stresses, which required 18 measured parameters to calibrate the function. This yield function is expressed as: 1/ m 1 3 3 m     [   S i  S j ] .1   i 1 j 1    (4 / 3) m  4(2 / 3) m  4(1 / 3) m , m  1 where m is the yield function exponent and S' and S'' are the linear transformation of the stress deviator, which is defined as: S   C : S , 0   S11  0 0 C12 C13  S11   0 S   C  0  S22  0 0 0 C23  22   21  S33   C31  C32  0   S33  0 0 0        0  0  S32  C44 0 0 0 S32  S31   0 0   S31  0 0 0 C55         S21     0 C66 0 0 0 0  S21 4 .2 Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 0   S11  0 0  S11   0 C12 C13 S  C  0  S22  0 C23 0 0  22   21  S33  C31  C32  0   S33  0 0 0 S  C : S ,        0  0  S32  0 0 C44 0 S32  S31   0 0   S31  0 0 0 C55         0    S21  0 0 0 0 C66  S21 where C' and C'' are the linear transformation matrices [32]. The method to calculate these 18 so-called anisotropy parameters Cij is the minimization of the following error function proposed by Banabic et al. [34]. e   ((  /   exp )  1) 2  (( b /  b exp )  1) 2   ((r / r exp )  1)2  ((rb / rbexp )  1)2 .3 where and are respectively the experimental yield stresses and anisotropy values obtained from the uniaxial tensile test in different directions (φ) to the rolling direction. Also, and are the experimental yield stress and anisotropy value obtained from the equi-biaxial tensile test. The genetic algorithm (GA) was applied for finding the optimum of the error function in this article. Also, it is assumed that out-of-plane behavior is isotropic and as a result, C′44, C′′44, C′55, and C′′55 are equal to 1 [32]. Modified M-K theory The traditional M-K model was proposed by Marciniak and Kuczynski. This model is based on the existence of imperfection that is modeled by the reduction of thickness in a part of the sheet. The safe region, imperfection region and initial ratio of the groove thickness to the safe zone thickness are denoted by region ‘a’, region ‘b’ and ‘f0’, respectively [35]. The traditional M-K model can only predict forming limits for proportional strain paths and inplane state of stress; however, it can be modified for considering strain path change, and throughthickness normal and shear stresses, as explained in the following. According to previous studies [18,26], between the σ32 and the σ31 shear stresses, σ32 has more impact on formability improvement of the sheet metal. Thus, in the present study, σ23 is chosen to investigate the effect of through-thickness shear stress on the extended forming limit diagrams. In fact, with the existence of σ32, the strain mode in the groove and matrix will not be the same and changing the groove strain mode toward the plane strain delays the onset of necking. Nevertheless, with the existence of σ31, this mechanism does not happen because of an unchanged groove strain mode in this loading condition [18]. In the presented model both shear and normal stresses are taken into account but the effect of each loading condition is investigated separately. The base algorithm for prediction of forming limit diagram and solving method is similar to that of Ganjiani and Assempour [36]. At first, all initial strains are assumed to be zero and a small value is assigned to the effective plastic strain increment in the safe region. Effective stress is determined by applying hardening law. With definite values of different stress ratios (α22 = σ22/σ11, α32 = σ32/σ11, and α33 = σ33/σ11) and effective stress, using yield function, σ11 and as a result all components of the stress tensor in the safe region can be obtained and written as follows: 0   11 0  a .4 [ ]123   0  22  32   0  32  33  5 Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 With assigning σ21= 0, there will be no rotation in the axes of stress tensor so strain increments can be integrated during the deformation and strains values in each stage can be obtained according to the previous stage [17]. ε11, ε22, and ε32 are calculated by the flow rule and ε33 is calculated using the volume constancy as: .5  33  (11   22 ) These values are also used to obtain strains and stresses in the groove region, but they should be first transferred from the global coordinate system to the groove coordinate system (n, t, 3) using the following relationship:  a   T T  a  T nt 3 123 .6 a T a    T   T 123 nt 3 where rotation matrix T in Eq. 6 for the general form of stress tensor is defined as:  cos  T   sin   0  sin  cos  0 0 0  1  .7 Utilizing the equilibrium of forces over the boundary between the safe and groove regions, leads to a set of equations that describes  nnb ,  ntb and  nb3 [26]: a a b b  nnb   nn a / f  nn t   nn t  b  a a a b b .8  nt t   nt t    nt   nt / f  a a  b a b b  n3t   n3t  n3   n3 / f where f should be updated in every single step based on the following equation [37]: b .9 f  f 0 exp( 33   33a ) It is assumed that the surface traction on the surface in the safe and groove regions are uniform [26]. Therefore: .10  ta3   tb3 ,  33a   33b In the groove zone d  nn , d  nt , d  n3 and d  33 are calculated by applying the flow rule. Also, d  tt is computed using the strain compatibility condition between the two regions, which leads to: d  ttb  d  tta . 11 By equating effective stresses obtained from hardening law and yield function in groove zone and d  tt of region ‘a’ and region ‘b’, two nonlinear equations for calculating two left unknown parameters (i.e.,  ttb andRaphson method -the numerical Newton. In this study, derived are) d  is used to solve the nonlinear set of equations. Also, the gradient descent technique is applied to guarantee the convergence of the solution with the Newton-Raphson method [36]. b a In the M-K model necking occurs when d  10d [37]. The strains in the safe zone related to this condition are considered as the limiting strains. This numerical process is repeated for different groove angles in each stress ratio to obtain the minimum limit strain. Considering different stress ratios, all the limit strains are obtained. Non-proportional loading is modeled as subsequent of two linear strain paths described by the following relation:   1 If 1   * .12  If 1   *    2 b 6 Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 Where α1 and α2 are respectively stress ratio related to prestrain and second loading stage and ε* is the value of prestrain [5]. Also the values of the effective strain and the strains in the safe and groove region at the beginning of the second stage loading should be modified according to the corresponding values at the end of the prestraining stage. Extended forming limit diagram The extended diagrams are diagrams that are defined by variables other than major and minor strains. By transferring ε1 and ε2 into effective strain  , all needed variables for plotting extended diagrams are obtained. Effective strain for loading included the prestrain can be calculated by the algebraic sum of effective strains through the prestrain stage  pre and final forming stage   . If limiting strains at the end of the prestrain stage are (ε1pre, ε2pre) and finial strains are (ε1, ε2), the effective strain can be expressed by Eq. (13) by incorporating the plane stress condition in the definition of plastic work:    ( 1 /  ).(( 1   1 pre )   ( 2   2 pre )) .13      pre where the numerical value of the parameter α' should be calculated for the second stage deformation. In Eq. (13), the ratio of the major stress to the effective stress can be obtained as a function of the stress ratio by dividing the yield function by σ1 [30]: .14 1 /    /  1   (1,  ) For a linear strain path under 3D stress condition, effective strain can also be computed using the plastic work principle as long as the normal stress is taken into account. With the effective strain calculated, the effective stress can be derived using an appropriate hardening law. Finally σ1 and σ2 can be defined by Eq. (15):  1   . .15  2   1 .  In this way, all variables needed for plotting the extended diagrams are obtained. The extended stress-based necking prediction diagram (XSFLD) uses information of the effective stress and mean stress defined as Eq. (16) [24]:   f ( ij )  g ( ) .16  m  ( 1   2   3 ) / 3 In PEPS-FLD, each point creates a path by a line projecting back to the origin. The magnitude of path is determined by effective plastic strain and the direction is defined as the arctangent of the ratio of the principal strain rates of the latest loading condition. In other words:   tan1 ( 2   2 pre ) / (1  1 pre ) .17   In a Cartesian system, the variables of the diagram are presented as [29]: ( x, y )  ( sin  ,  cos  ) .18 Accordingly, all strain points can be mapped to extended coordinate and then joined with a smooth curve. Forming limits evaluation This analysis is performed with the purpose of investigating the influence of non-proportional and out-of-plane loading on FLD and extended FLD. In this section, to validate the model the FLDs in 7 Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 plane-stress condition are compared with the available experimental data for aluminium alloy AA2008-T4 reported by Graf and Hosford [1]. For the case with through-thickness normal stress due to lack of experimental data for aluminium alloy AA2008-T4, validation is performed by experimental data of hydroforming process of the AA6063 tube, mentioned in Hashemi et al. [33]. The directional anisotropy, yield stress and the material properties for aluminium alloys AA2008T4 and AA6036 are listed in Table 1. According to these directional data, the corresponding calculated anisotropy coefficients of Yld2011 yield function are obtained and presented in Table 2. It is assumed that anisotropy of the sheet does not change with prestraining. Hence, calculated coefficients of anisotropy can be used for predicting the forming limit of both as-received and prestrained sheet. Table 1: The material properties for the aluminium alloys AA2008-T4 and AA6063 ε0 σ0 σ45 σ90 r0 r45 n m K AA2008-T4[1] 535 AA6063[38] 177 0.00005 r90 0.27 -0.003 160 150 146 0.85 0.485 0.78 0.2 - 71 41 36 0.59 0.83 0.79 Table 2: Yld2011 parameters for the aluminium alloys AA2008-T4 and AA6063 C′12 C′13 C′21 C′23 C′31 C′32 C′44 C′55 C′66 AA2008-T4 -0.675 -2.195 -1.264 -2.953 0.499 -0.875 1 1 1.235 AA6063 0.631 C′′12 2.048 C′′13 0.421 C′′21 0.297 C′′23 0.166 C′′31 0.245 C′′32 1 C′′44 1 C′′55 0.732 C′′66 AA2008-T4 2.222 0.147 -0.908 3.447 3.211 -2.035 1 1 1.293 AA6063 0.566 0.488 0.056 0.527 0.861 -1.232 1 1 0.994 Fig. 1 shows the calculated FLD for as-received AA2008-T4 using Swift hardening law and initial imperfection ratio f0=0.993. It is observed that the theoretical FLD predicted based on yld2011 is in better agreement with the experimental data than that predicted based on Hill’s 48 yield function. Yld2011 yield criterion presents a good accuracy in the right side of FLD; however, Hill’s48 function significantly overestimates strain limits. In negative strain ratios both yield functions predict almost the same values and, as mentioned by Zhang et al. [28], the left hand-side of FLC is not sensitive to yield function. Fig. 1: The FLD of As-received AA2008-T4 8 Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 Predicted FLDs using modified M-K model and Yld2011 yield function with different uniaxial and equi-biaxial tension prestrain is shown in Fig. 2a and with preloading in-plane strain are shown in Fig. 2b. Comparing the theoretical FLDs with experimental results in Graf and Hosford [1] revealed that, the displacements of the FLD as a result of various pre-strains are well-predicted using the present model. Fig. 2: The FLDs of the prestrained AA2008-T4 Fig. 3 illustrates both computed and experimental forming limit diagrams considering normal stress. Swift hardening law and f0 = 0.96 is adopted for theoretical prediction. The experimental data used to verify the model in this case, are obtained from free hydroforming of the aluminium alloy AA6063 with the internal pressure of 15 Mpa [33]. Since the hydraulic pressure was applied to one side of the tube, σ3 is regarded as -7.5 MPa in M-K model. Due to the lack of experimental data for the right-hand side of FLD, validation is performed only on the left side. As illustrated in Fig. 3, the model introduces satisfactory accurate results compared to the experimental data. Fig. 3: The FLD under normal stress of -7.5MPa for AA6063 Comparison of different types of forming limit diagrams In the previous section, it was shown that the Yld2011 yield function introduces results with reasonable accuracy. Also, theoretical FLDs considering the effect of prestrain and normal stress were compared with the experimental data and the validity of the proposed modified M-K model 9 Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 was confirmed. In the next four sub-sections, the effects of initial imperfection coefficient, prestrain, through thickness normal and shear stresses are studied theoretically on different forms of forming limit diagram of AA2008-T4 using Yld2011 yield function. The term “sensitivity” has been used to compare dependency of different forms of diagram to the investigated factor (imperfection coefficient, prestrain, normal stress or shear stress) and to show how much these factors cause the obtained diagram to be different from the as-received diagram. Effect of initial imperfection coefficient One of the factors that affect the shape and relative level of a theoretical FLD obtained from M-K model is material imperfection. The effect of initial imperfection coefficient on FLD and extended FLD is studied by allocating three values 0.99, 0.993, and 0.999 to f0. As shown in Fig. 4, for this alloy the effect of f0 on the right side of FLD is more pronounced than on the left side. FLSD and XSFLD shift upward and right with an increase in f0. The sensitivity of different diagrams to f0 is evaluated by measuring the percentage increase in the lowest point with respect to the curve obtained by f0 = 0.99. Results in Table 3 show that the diagrams in stress space (i.e., FLSD and XSFLD) are less sensitive to f0. Fig. 4: The effects of initial imperfection values on different forming limit diagrams for AA2008-T4 sheet Table 3: the percentage increase in the lowest point concerning the f0 = 0.99 curve FLD PEPS-FLD FLSD XSFLD 0.993 4.6 4.6 1.2 1.2 0.999 9.8 9.8 2.4 2.4 10 Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 Effect of prestrain To re-inspect the path independency of extended forming limit diagrams and determine a presentation with the lowest sensitivity to strain path, the theoretical FLDs described in the strain space with various prestrains (Fig. 2) were mapped to the extended diagrams. As can be seen in Fig. 5, a dynamic feature of FLD due to changes in loading path no longer exists in the extended diagrams and all of three configurations, all curves almost coincide into one single curve and create a narrow band. For more accurate investigation throughout each curve, in addition to the plane-strain state, three other strain ratios (β = -0.1, 0, 0.2, and 0.36) were arbitrarily selected and the percentage increase in these four points concerning the as-received diagram in the same configuration was measured. The average of these 4 data for each prestrained curve is reported in Table 4. Results indicate that the sensitivity of FLSD and XSFLD to strain path are almost at the same level and the PEPS-FLD diagram shows a higher sensitivity to the strain path. Fig. 5: The mapping of different calculated prestrained paths of AA2008-T4 in extended FLD Table 4: The deviation of extended FLDs with respect to the as-received diagram Uniaxial Uniaxial Plane-strain Plane-strain Equi-biaxial 0.05 0.12 0.08 0.13 0.07 FLSD 0.012 0.012 0.020 0.001 0.010 mean 0.011 XSFLD 0.013 0.011 0.020 0.001 0.013 0.012 PEPS-FLD 0.038 0.044 0.081 0.006 0.023 0.038 11 Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 According to Fig. 5, it can be deduced that in the format of the extended diagram there is no difference between the as-received and the prestrained curve. So, the as-received curve can be used as a general criterion for predicting formability of sheet in complex forming processes. To ensure this conclusion, it is necessary to check whether the points after mapping are in the same position relative to the as-received curve as they were relative to prestrained FLD before being mapped. The importance of this investigation is that in the FLD the location of a point relative to the diagram determines its severity. Therefore, it is necessary to check the data to locate correctly in the extended diagram after mapping. For this purpose, as illustrated in Fig. 6, two pairs of points, one on the left and another on the right side of each theoretical curve (Fig. 2), with a distance of 0.02 from that are randomly selected. These points were next converted into PEPS-FLD, FLSD, and XSFLD locus as illustrated in Fig. 7a, 7c, and 7c respectively. Fig. 6: How points are chosen from prestrained FLDs to check safety margin 12 Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 Fig. 7: The location of different mapped prestrained points to the extended FLD for the as-received AA2008-T4 It is observed that the points are placed in the correct position relative to the as-received curve; i.e., the points above the prestrained FLD are still located above the as-received curve and the points below the prestrained FLD are located below the as-received curve after being mapped into the extended diagrams. In comparison, on the right side of XSFLD, the points extend along the asreceived curve such that the points located along the vertical direction (Fig. 6) are located along the oblique direction after mapping. Consequently, the interspace between points and curve becomes inappreciable and makes it difficult to interpret the diagram to obtain solutions for formability problems. On the contrary, points in PEPS-FLD and FLSD have an acceptable distance from the reference curve; therefore, it is easy to see or quantify the margin of safety visually. However, in the case of non-linear strain path, the PEPS-FLD is slightly more sensitive to the strain path than the stress-based diagrams. But some advantages such as arranging based on strain, similar shape to the FLD, having a radial direction that corresponds to different forming modes parallel to the corresponding directions in the FLD, and having a safety margin that is easily visible make this diagram more attractive. Effect of through-thickness normal stress In this section, the effect of through-thickness normal stress on different forming limit diagrams is analyzed considering two arbitrary normal stresses of -30 and -40 MPa. It should be noted that assigning fewer values was not appropriate because in order to compare the effect of normal stress with effect of shear stress, same values are given to them and since the effect of shear stress on increasing the forming limit is much less than that of normal stress (as it will be shown later), if fewer values were chosen, this increase would not be easily visible in the FLDs. The increase in compressive through-thickness normal stress improves the formability [8-16], as shown in Fig. 8, but the variations of forming limit diagrams in different platforms depending on the variables of the coordinate plane. The level of FLD increases, meanwhile, moves right slightly with an increase in normal stress. Similar to the influence of normal stress on traditional FLDs, normal stress increases the level of PEPS diagram. The sensitivity of FLD and PEPS diagram to the normal stress can be evaluated by measuring the percentage increase of the lowest point. Results in Table 5 show that the sensitivity of PEPS-FLD and traditional FLD are at the same level. As depicted in Fig. 8c and 8d, Floating range of FLSD and XSFLD are more than strainbased FLD. Moreover the trend is somehow misleading. Unlike the strain-based FLD, the level of FLSD decreases with the increase in through-thickness normal stress. This increase can be 13 Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 explained by investigating the effect of normal stress on the yield surface. As shown in Fig. 9, with an increase in normal stress, the level of initial yield surface in the tensile-tensile region decreases. Fig. 8: The effects of through-thickness normal stress on different forming limit diagrams for AA2008-T4 sheet Fig. 9: The effects of through-thickness normal stress on the initial yield surface For loading condition with normal compressive stress the effective stress is the function of σ1, σ2, and σ3. For this alloy, the increase in σ3 and decrease in σ1 and σ2 as a result of normal stress, leads to effective stress to increase and XSFLD to shift up. From Fig. 8, it can be concluded that normal stress changes any forms of forming limit diagram. Hence, it should be included in the theoretical calculation to obtain correct forming limit diagrams. 14 Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 Effect of through-thickness shear stress Fig. 10 illustrates the effect of σ23 on the forming limits of the AA2008-T4 sheet. To compare the effectiveness of through-thickness normal and shear stress, FLDs in Fig. 8a are repeated in this figure. According to the figure, normal stress, in compare with the shear stress, has larger effect in increasing the formability of sheet metal. Fig. 10: The effect of normal and shear through-thickness stress on formability of AA2008-T4 The effect of σ23 on different configurations, including FLD, PEPS-FLD, FLSD and XFLSD is compared in Fig. 11. Similar to the effect of normal stress, by increasing the σ23 shear stress, the stress limit decreases and effective stress increases (Figs. 11c and 11d). Table 5 presents the percentage increase in the lowest point in FLD and PEPS-FLD due to shear stress. The result demonstrates that, the same as the effect of normal stress, FLD and PEPS-FLD are equally sensitive to shear stress. Fig. 11: The effects of through-thickness shear stress on different forming limit diagrams for AA2008-T4 sheet 15 Post-print of “A comparative study of the extended forming limit diagrams considering strain path, throughthickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 Table 5: the percentage increase in the lowest point with respect to in-plane diagram |σ33|= 30 |σ33|= 40 σ32= 30 σ32= 40 FLD 23 31.2 6.5 8.4 PEPS-FLD 23.5 31.6 6.7 9.3 MPa MPa MPa MPa Conclusion In this study, the modified M-K model along with Yld2011 advanced anisotropic yield function is employed to investigate the effect of different factors including initial imperfection ratio, nonlinear strain path, through-thickness normal stress and shear stress on the FLD and extended FLDs of AA2008-T4. Initially, the strain path dependency of FLDs is reexamined and then data points are mapped into extended diagrams. The predicted in-plane FLD for as-received and prestrained conditions and also theoretical FLD considering the through-thickness normal stress effect, are validated with available experimental data. The effect of through-thickness shear stress on FLD is not verified; however, it is consistent with results reported by Nasiri et al. [26]. The main findings of the present investigation are summarized as follows: 1- All of the extended diagrams are independent of the strain path, but the diagrams in stress space (i.e., FLSD and XSFLD) are less sensitive to strain paths than PEPS-FLD. Nevertheless, in the case of non-linear strain path, some advantages such as arranging based on the strain, similar shape to the FLD, having a radial direction that corresponds to different forming modes parallel to the corresponding directions in the FLD, and having a safety margin that is easily visible make PEPS diagram more attractive. 2- Forming limit diagrams in strain space, including FLD and PEPS-FLD, shift upward with an increase of through-thickness normal and shear stresses. The sensitivity of these two diagrams to out-of-plane stresses is at the same level and they are not independent of normal and shear stresses; thus, these parameters should be carefully considered for a correct prediction. 3- Forming limit diagrams in stress space, including FLSD and XSFLD, do not follow the same trend for out-of-plane stresses. With the increase in through-thickness normal and shear stress, both major and minor stresses decrease, but the effective stress increases. 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