Materials and Design 34 (2012) 185–191

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Materials and Design

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Technical Report

Calibration of forming limit diagrams using a modified Marciniak–Kuczynskimodel and an empirical law

Amir Ghazanfari, Ahmad Assempour ⇑Center of Excellence in Design, Robotics and Automation, Department of Mechanical Engineering, Sharif University of Technology,Azadi Avenue, P.O. Box 11365-9567, Tehran, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 May 2011Accepted 23 July 2011Available online 9 August 2011

0261-3069/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.matdes.2011.07.057

⇑ Corresponding author. Tel.: +98 21 6616 5529; faE-mail address: assem@sharif.edu (A. Assempour).

The major problem in determining the forming limit diagram (FLD) with the Marciniak–Kuczynski (M–K)model is the necessity of an experimental point in order to find the initial inhomogeneity coefficient andcalibrate the diagram. The purpose of the present work is to eliminate this requirement. To do this, theusual assumption of geometrical inhomogeneity has been replaced with material inhomogeneity and ithas been shown that the sensitivity of the diagram to variations of the inhomogeneity factor is reducedgreatly with the new assumption. Using this advantage and collecting enough experimental data fordifferent materials, an empirical law in terms of sheet thickness has been proposed which estimatesthe initial material inhomogeneity. Thus it is possible to determine the FLD in the absence of experimen-tal data. The results show good agreement with experiments and this method has some advantages overother calibration methods.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In all processes involving sheet metal forming, the major con-cerns of the designer are to avoid wrinkling and necking of thesheet. Until 1960s, the only reliable test of formability waswhether or not the formed product was free of fractures andthrough-thickness necks [1]. To preclude the necking, one mustknow the limiting strains or stresses that result in necking; conse-quently several theoretical and experimental researches have beencarried out in order to predict diagrams indicating these limitstrains or stresses, namely forming limit diagrams (FLDs) andforming limit stress diagrams (FLSDs).

The concept of FLD was introduced by Keeler and Backhofen in1964 [2] for biaxial stretching (e1 > 0; e2 > 0); and 4 years later,Goodwin [3] extended the curve for tension–compression domain(e1 > 0; e2 < 0). They sketched circular mesh on the surface of a sheetand drew it up to the fracture point for various stress ratios. After thedeep drawing process, circles changed shape to ellipses and by mea-suring the major and minor axes of the ellipses nearest to fracturepoint, limiting strains were obtained and the FLD was plotted.

Many investigations have been performed to find easier ways ofconstruction of FLDs; perhaps the most renowned one is that pro-posed by Hecker [4] in which a hemispherical punch with differentsample widths and lubricants is used to determine the curve withfewer tests. However, the experimental ways are still very time-consuming and expensive; even in some cases the acquired points

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x: +98 21 6600 0021.

are such few and scattered that it is hard to pass a reliable linethrough them. Accordingly there has been a great effort to deter-mine the FLDs theoretically.

In numerical field, the first and the most well known model isthe one proposed by Marciniak and Kuczynski [5]. This modelwas further developed by Marciniak et al. [6] and was extendedto tension–compression domain (e1 > 0; e2 < 0) by Hutchinsonand Neale [7–9]. In this model the failure is attributed to an initialgeometrical inhomogeneity in the form of a groove on the surfaceof the sheet. In the groove zone, thickness is slightly smaller thanthe safe zone and as a result of the applied force, groove growsand changes its direction with respect to initial coordinates. Manyresearchers have worked on the M–K model and made beneficialmodifications to it some of which are mentioned here.

Tardos and Mellor [10] developed a model based on M–K modelin which the limit strains were the sum of both diffused and local-ized necking. Some investigators (e.g. [11]) explored the effect ofdifferent constitutive features on FLD. Arrieux et al. [12] demon-strated that using a stress based diagram (FLSD) solves the prob-lem of strain path dependency in conventional FLDs. In severalworks, the influence of yield functions of the diagram has beenshown. Recently the normal stress and through thickness shearhave also been taken into account [13–16].

The major problem in determining the FLD in all numericalmodels is requirement of some experimental results other thanusual material properties. In case of M–K model, requirement isleast and only one point of experimental FLD (usually FLD0

1) is

1 FLD0 is the major strain in plane strain state that results in necking.

186 A. Ghazanfari, A. Assempour / Materials and Design 34 (2012) 185–191

needed to find the initial inhomogeneity coefficient, f0, and calibratethe diagram. Unfortunately, as demonstrated in the following sec-tions, sensitivity of the diagram to variations of f0 is very high andif it is presumed inaccurately, the resulting FLD will differ greatlyfrom experimental data.

Some investigations have been done to reduce the requirementof experimental data in determining the FLD. These investigationsfall into three main categories:

1. Using empirical laws: these methods are very practical anduseful in industrial applications because of their simplicityand requirement of few parameters. However, these lawshave limitations and are suitable only for a certain group ofmaterials. Perhaps the most renowned relation is the one pro-posed by Keeler and Brazier [17] and known as NADDRG2

relation in the literature. This empirical law is suitable forlow carbon steels. The value of FLD0 in terms of engineeringstrain is obtained from the following equation:

2 Nor

FLD0 ¼ ð23:3þ 14:13 t ðmmÞÞðn=0:21Þ ð1Þ

2. Using grain size and surface roughness: another method forobtaining the FLD is to relate the grain size and surface rough-ness of the sheet to geometrical inhomogeneity. An example ofsuch relations is equation (2) [18]:

f0 ¼ta

0 � 2ðRþ Kd0:50

�ebÞta

0ð2Þ

where R is the initial surface roughness, d0 is the grain sizeand K is a material constant. The resultant FLDs show goodagreement with experimental results but requirement ofmany experimental constants which are difficult and timeconsuming to determine, makes this method less popular.

3. Using strain gradient theory [19]: in this method the straingradient theory of plasticity is incorporated into the M–Kmethod to determine the diagram. It has been proved thatvariations of initial inhomogeneity factor have a negligibleeffect on the FLD. On the other hand, mathematical complex-ity and some required experimental parameters affect thepopularity of this method. It should be noted that the mainpurpose of this method is not to calibrate the FLD.

In the present work, it has been tried to eliminate the require-ment of experimental data by the following procedure: first thegeometrical inhomogeneity has been replaced with material inho-mogeneity. It has been shown that this assumption reduces thesensitivity of the diagram to variations of inhomogeneity coeffi-cient greatly. Next using this important property, an empiricallaw is proposed to predict the initial material inhomogeneity. Thisempirical law is only a function of sheet thickness. The new meth-od has some advantages over other methods which will be dis-cussed in the following sections.

2. Numerical work

Ganjiani and Assempour [20] predicted the FLD based on theM–K model by using Hosford [21] and BBC2000 [22] yield func-tions and introducing an energy equation in the groove zone asEq. (4) to find the groove parameters. In the present paper, the gen-eral procedure and equations are the same, but instead of applyinggeometrical inhomogeneity (f0 ¼ tb=ta) in zone ‘‘b’’ (groove or weakzone) relative to zone ‘‘a’’ (safe zone), some types of material inho-mogeneity have been examined, namely K ¼ kb

=ka, R ¼ rb=ra and

th American Deep Drawing Research Group.

N ¼ nb=na; and the FLDs for some materials have been predicted.The effect of applying different values of ‘‘k’’ or ‘‘n’’ for zone ‘‘a’’and ‘‘b’’ shows itself in the hardening law, where applying a smal-ler value for ‘‘k’’ or a larger value for ‘‘n’’ in zone ‘‘b’’, results in asofter material. Similarly, the effect of applying different valuesof ‘‘r’’ for zone ‘‘a’’ and ‘‘b’’ shows itself in the yield function whereapplying a smaller value for ‘‘r’’ in zone ‘‘b’’, results in a materialwith decreased drawability. In what follows the numerical workhas been reviewed briefly. The reader may refer to Ganjiani andAssempour [20] for more details.

It is assumed that in zone ‘‘a’’, strain path is linear. The stressand strain calculations are performed for this zone and then forzone ‘‘b’’. The required material properties of safe region andgroove region depend on hardening law and yield function. Apply-ing Swift’s hardening law and Hill’s 1948 yield function [23], therequired material properties are thickness ratio (tb/ta), strengthconstant ratio (kb/ka), strain hardening exponent (nb, na), stain ratesensitivity exponent (mb, ma), anisotropy factor (rb, ra) and initialstrain (e0).

To start the analysis, a small value for d�ea (the equivalent strainincrement in the safe zone) is set. The equivalent strain in eachstep will be obtained by:

�eanew ¼ �ea

old þ d�ea ð3Þ

Now using hardening law, the equivalent stress can be found:

�ra ¼ kaðe0 þ �eaÞna _�ema ð4Þ

Having the equivalent stress, the stress ratio a ¼ r2=r1 and theyield function, ra

1 and ra2 are calculated and the stress tensor in

principal coordinates is:

½ra�xyz ¼ra

1 00 ra

2

� �ð5Þ

To obtain the stress tensor in the groove coordinates which isshown in Fig. 1, rotation matrix is used:

T ¼cos h sin h

� sin h cos h

� �; ra½ �ntz ¼ T � ra½ �xyz � TT ¼

rann ra

nt

rant ra

tt

� �ð6Þ

Using flow rule and incompressibility condition, the compo-nents of strain in zone ‘‘a’’ are determined. Thus zone ‘‘a’’ is com-pletely solved and now all stresses and strains in zone ‘‘b’’should be calculated. In this region, unknown parameters arerb

nn;rbnt ;rb

tt; ebnn; eb

nt; ebtt and eb

3 but using flow rule and incompress-ibility condition, they reduce to rb

nn;rbnt;rb

tt; d�eb. The four equationsrequired to find these unknowns are the energy equation, straincompatibility equation and two force equilibrium equations:

debnnrb

nn þ debttrb

tt þ debntrb

nt

d�eb �rb� 1 ¼ 0

debtt

deatt� 1 ¼ 0

frb

nn

rann� 1 ¼ 0

frb

nt

rant� 1 ¼ 0

ð7Þ

where f changes according to:

f ¼ f0 expðeb3 � ea

3Þ ð8Þ

This set of equations is solved using Newton–Raphson methodas follows:

A general problem gives N functional relations in terms of vari-ables xi, i = 1, 2, . . ., N to be zeroed:

Fiðx1; x2; :::; xNÞ ¼ 0; i ¼ 1;2; :::;N ð9Þ

α

ad ε

aε aσ 1aσ 2

aσ [ ]antzσ [ ]a

ntzd ε aiε

, , ,b b b bnn nt tt dσ σ σ ε

10b ad dε ε >

1aε 2

aε

1aε 2

aε

Fig. 2. Flow chart of analysis.

Table 1Mechanical properties of selected material.

Material ka (MPa) na ra

STKM-11A [24] 1500 0.15 2.14

Table 2Chemical composition of selected material.

Fig. 1. Geometrical model of the modified M–K theory.

A. Ghazanfari, A. Assempour / Materials and Design 34 (2012) 185–191 187

where x denotes the vector of values of xi and F denotes the vectorof functions Fi. In the neighborhood of x, each of functions Fi can beexpanded in Taylor series:

Fiðxþ dxÞ ¼ FiðxÞ þXN

j¼1

@Fi

@xjdxj þ Oðdx2Þ ð10Þ

In matrix notation, this equation reduces to:

Fðxþ dxÞ ¼ FðxÞ þ J � dxþ Oðdx2Þ ð11Þ

The matrix of partial derivatives appearing in the previousequation is the Jacobin matrix J:

Jij ¼@Fi

@xjð12Þ

By neglecting terms of order dx2 and higher, and settingF(x + dx) = 0, we obtain a set of equations:

J � dx ¼ �F or dx ¼� J�1 � F ð13Þ

The variable dx is then added to the solution vector,

xnew ¼ xold þ dx ð14Þ

and the process is iterated up to the convergence point.After d�eb is obtained in each step, it is compared to d�ea and if

d�eb=d�ea > 10 the necking has begun and ea1, ea

2 are saved; otherwisea greater value for d�ea is assumed and the process repeats. Thisprocedure is done for different values of stress ratios and grooveangles until the whole diagram is determined. The flow chart ofthe algorithm is shown in Fig. 2.

It should be noted that the numerical procedure proposed byGanjiani and Assempour [20] and described briefly here, does notaffect the results and one can easily implement these assumptionsfor inhomogeneity in other numerical methods and acquire thesame results.

Material C (%) Si (%) Mn (%) P (%) S (%)

STKM-11A [24] 0.12 0.35 0.60 0.04 0.04

3. Effects of using various inhomogeneity types

To evaluate the effects of different inhomogeneity types, the de-scribed algorithms were implemented to determine the FLDs formaterial of Table 1 and the results were compared with the exper-imental data in the literature [24]. The material is STKM-11Awhich is a carbon steel used extensively in tubes. The chemicalcomposition of this material is presented in Table 2. Differentassumptions for the initial inhomogeneity were made to obtainthe diagram and effect of variations of each of them has been stud-ied and compared with other parameters.

In this section (as conventional M–K model), the method offinding the appropriate value for the initial inhomogeneity factor(i.e. the calibration method) is by try-and-error process. In this pro-cess, different initial inhomogeneity values are set to acquire themost compatible results with the experimental data. Thus at leastone experimental point in the FLD space is required for calibrationof the theoretical model.

In Section 4, a relation is proposed to find the initial inhomoge-neity factor without requirement of experimental data.

3.1. Thickness inhomogeneity (f0)

The absolute value of sheet thickness, which has a great influ-ence on the level of FLD (see e.g. [17,25,26]), is neglected in theM–K method and only the thickness ratio, f0, has efficacy. Thusfor the same material with different thicknesses, the initial thick-ness inhomogeneity factor differs and should be obtained in sepa-rate tests. In addition, the diagram is very sensitive to f0, whichcould be seen in Fig. 3. The calibrated f0 and FLD0 for STKM-11Aare 0.97 and 0.0855 respectively [24]. At this point, the derivativeof FLD0 with respect to f0 (see Fig. 4), which is a measure of the

Fig. 3. Sensitivity of diagram to variations of f0 for STKM-11A (f0 ¼ tb=ta).

Fig. 4. Effect of variations of different inhomogeneities on FLD0 for STKM-11A (thetrue value is 0.0855 [24]).

Fig. 5. Sensitivity of diagram to variations of R for STKM-11A (R ¼ rb=ra).

Fig. 6. Sensitivity of diagram to variations of N for STKM-11A (N ¼ nb=na).

Fig. 7. Comparison between experimental data for STKM-11A [24] and predictedcurves for different inhomogeneities.

Fig. 8. Demonstration of Eq. (15).

188 A. Ghazanfari, A. Assempour / Materials and Design 34 (2012) 185–191

sensitivity of diagram to thickness inhomogeneity, is 1.7 which is anoticeable value.

3.2. Strength inhomogeneity (K)

Similar to thickness, the absolute value of strength constant isalso neglected in the M–K method and only the ratio of strengthconstant in zone ‘‘b’’ relative to zone ‘‘a’’, K ¼ kb

=ka, is important.But unlike thickness, this is in good agreement with experimentaldata and empirical laws (see e.g. [17]), in the other hand the valueof strength constant does not have a significant effect on FLD.

Introducing strength inhomogeneity is not an innovation; evenin the first work of Marciniak and Kuczynski [5], the inhomogene-ity is defined as kbtb=kata. Strength inhomogeneity is used in someother investigations as well, especially in those pertaining to finiteelement simulations (see e.g. [27]). However, it has exactly thesame effect of thickness inhomogeneity (i.e. assuming K = 0.97 isidentical to f0 = 0.97). Thus this technique would not help us toeliminate the requirement of experimental data.

3.3. Anisotropy inhomogeneity (R)

As the normal anisotropy factor of a material decreases, thedrawability of that material also decreases; thus it is reasonableto define inhomogeneity as the ratio of anisotropy of zone ‘‘b’’ to‘‘a’’ (i.e. R ¼ rb=ra). As demonstrated in Fig. 5, the sensitivity of dia-gram to variations of R is much less than that of f0 which provesthat this model could be a good substitution for conventional M–K model. The calibrated R is 0.85 for STKM-11A. At this point, thederivative of FLD0 with respect to R (see Fig. 4) is 0.22 (compareto 1.7 for f0).

3.4. Strain hardening inhomogeneity (N)

Another idea is to introduce the inhomogeneity as the ratio ofstrain hardening exponent of zone ‘‘b’’ to ‘‘a’’ (i.e. N ¼ nb=na). As

Table 3Collected data for typical materials in metal forming processes.

Material t (mm) n r m k (MPa) e0 FLD0a FLD0

b FLD0c Error of FLD0

b (%) Error of FLD0c (%)

Al-1.25Mn AR85 [44] 1.4 0.032 1* 0.011 0.10 0.064 0.1103 36.42 10.30Al 1100 [32] 1* 0.04 0.75 0.003 0.05 0.069 0.054 37.73 8.80Al-1.25Mn SR85 [44] 1.4 0.062 1* 0.014 0.18 0.120 0.163 33.48 9.44DX 53D [35] 1 0.154 1.515 0.012 0.265 0.243 0.247 8.30 6.79ZStE 340 [35] 1.25 0.165 1.076 0.012 0.253 0.279 0.271 10.28 7.11Al 5182 [38] 1 0.17 0.685 0* 371.2 0.003 0.17 0.265 0.160 55.69 5.88AA3003-H16 [37] 0.7 0.173 0.603 0* 0.01 0.155 0.242 0.152 55.94 1.81Al 5754 [38] 1 0.177 0.775 0* 309.1 0.002 0.17 0.274 0.173 61.30 1.76Micro Alloy Steel [34] 1.2 0.188 0.932 0.012* 824.5 0 0.27 0.307 0.299 13.7 10.74DP steel [28] 1.5 0.19 0.83 0.012* 0.328 0.338 0.322 3.05 1.83ZStE 180 BH [28] 0.77 0.19 1.4 0.012* 0.318 0.270 0.271 15.09 14.78Cu–Ni [45] 0.7 0.19 0.8 0.004 0.24 0.263 0.210 9.41 12.50Al 2024 [40] 0.813 0.191 0.507 0* 0.20 0.275 0.182 37.45 9.00ZStE 220P [35] 0.8 0.197 1.68 0.012 0.256 0.281 0.277 9.81 8.20AA 8014 S330 [46] 1.3 0.2 1* 0.01 0.32 0.334 0.309 4.44 3.43Carbon Manganese [34] 1.4 0.205 1.30 0.012* 777 0 0.30 0.350 0.330 16.67 10.00High strength sttel [31] 1 0.21 0.88 0.012* 0.31 0.318 0.312 2.58 0.65AA 3105-U [36] 1 0.21 0.284 0.002 0.20 0.318 0.206 58.97 3.00A 5182-O [39] 1 0.21 0.93 0* 570 0.22 0.318 0.210 44.52 4.55Rimming steel [51] 1 0.21 1.37 0.01 0.28 0.318 0.2946 13.55 5.21DX 54D [35] 0.8 0.215 2.385 0.012 0.341 0.303 0.300 11.14 12.02AK Steel [47] 0.97 0.217 1.632 0.017 0.36 0.324 0.3468 10.05 3.67EDD [25] 2 0.22 1.37 0.012 0.463 0.432 0.410 6.70 11.45Al 6016-T4 [38] 1 0.22 0.7 0* 388.3 0.01 0.20 0.331 0.205 65.42 2.50LPG Steel B [48] 3.15 0.22 0.85 0.0086 0.53 0.537 0.504 1.268 4.91Al 1050 [14] 1 0.222 1 0 0.203 0.333 0.215 64.23 5.91IF-HS [28] 0.84 0.23 1.86 0.012* 0.327 0.326 0.321 0.31 1.83VDIF [29] 0.85 0.23 1.53 0.012* 0 0.38 0.327 0.343 13.95 9.74Mild steel fep04 [30] 0.7 0.237 1.95 0.012* 521 0.005 0.30 0.318 0.312 6.00 4.00Mild steel [31] 1 0.24 2.33 0.012* 0.40 0.356 0.350 11.00 12.50AA 8011-U [36] 1 0.24 0.43 0.003 0.20 0.356 0.220 78.06 10.00AA 8014 F480 [46] 1.3 0.24 1* 0.009 0.36 0.389 0.3468 8.19 3.67AZ31 magnesium [33] 0.8 0.24 0.95 0* 0.25 0.333 0.231 33.29 7.60Al 1100 [32] 1* 0.25 0.62 0.003 0.24 0.369 0.242 53.55 0.83EDD [25] 0.8 0.25 1.19 0.012 0.36 0.345 0.341 4.17 5.28EDD [25] 1.6 0.25 1.58 0.014 0.425 0.436 0.427 2.59 0.47LPG Steel A [48] 3.15 0.25 0.897 0.0097 0.57 0.592 0.551 3.83 3.33AZ31 magnesium [33] 0.8 0.253 1.1 0* 0.21 0.348 0.235 65.94 11.9A 1100-O [39] 1 0.26 0.78 0* 171 0.268 0.381 0.254 42.08 5.22IF [41] 1* 0.26 2.13 0.012 536 0.01 0.39 0.381 0.373 2.31 4.35IF [28] 1.01 0.27 1.88 0.012* 0.377 0.394 0.388 4.51 2.92DDQ [51] 0.8 0.28 1.8 0.01 0.35 0.379 0.3624 8.40 3.54Brass [45] 0.7 0.28 0.77 0.002 0.28 0.366 0.2875 30.86 2.68AA 3105-H [36] 1 0.29 0.36 0 0.30 0.417 0.290 38.89 3.33St 12 [42] 1* 0.3 1.21 0.01 238 0.01 0.41 0.428 0.406 4.39 0.98A 5182-O [43] 1 0.323 0.887 0 585.2 0.285 0.455 0.3111 59.63 9.16Tough pith Cu [45] 0.7 0.34 0.88 0.005 0.37 0.430 0.3785 16.24 2.30Oxygen-free Cu [45] 0.7 0.44 0.98 0.002 0.39 0.528 0.454 35.37 16.4170-30 brass tube [49] 0.508 0.55 1.51 0* 0.505 0.587 0.5191 16.20 2.79TWIP940 steel [50] 1.47 0.62 1.33 �0.007 0.064 0.38 0.833 0.398 119.32 4.74

a The experimental value.b The value predicted by NADDRG empirical law (using Eq. (1) and converting engineering strain to true strain).c The value predicted by the present model (using Eq. (9) to find the initial inhomogeneity factor).

* The exact value is not mentioned in the specified reference.

A. Ghazanfari, A. Assempour / Materials and Design 34 (2012) 185–191 189

the strain hardening exponent of a material increases, the level ofstress–strain curve decreases and the material softens.

As Fig. 6 shows, the sensitivity of diagram to variations of N ismuch less than that of f0 which proves that this model could bea good substitution for conventional M–K model. The calibratedN is 1.14 for STKM-11A. At this point, the derivative of FLD0 withrespect to N (see Fig. 4) is 0.29 (compare with 1.7 for f0).

3.5. Comparison and verification

In Fig. 4 the effect of variations of different types of inhomoge-neities on the value of FLD0 for STKM-11A is shown. It should benoted that the true value of FLD0 acquired from [24] is 0.0855. Asmentioned in previous sections, the derivative of FLD0 with respectto inhomogeneities at the calibration point, which is a measure of

the sensitivity of diagram to inhomogeneity, for f0 is 7.7 and 5.9times more than that of R and N respectively. This shows the effec-tiveness of the proposed models. Fig. 7 illustrates the comparisonbetween experimental data for STKM-11A, obtained from [24],and predicted curves for different inhomogeneities. Patently, theoverall form of all curves are very similar and therefore it is notpossible to categorically state that which one is the nearest tothe experimental results.

To decide which of the two proposed models (R or N) is morepragmatic and beneficial, it should be noted that in spite of the factthat the FLD is slightly less sensitive to variations of R, the value ofr has some uncertainties in determination and is merely an averagefor different directions. Also the n-value of materials could be moreeasily found in the literature with more precision. Additionally,some researchers neglect the effect of r-value on FLD and use

190 A. Ghazanfari, A. Assempour / Materials and Design 34 (2012) 185–191

von Mises yield criterion, so they are not able to use anisotropyinhomogeneity. Therefore, it is suggested to use the strain harden-ing inhomogeneity for predicting the forming limit diagram.

4. Empirical law for calibration of FLD

To benefit from the advantage of assuming material inhomoge-neity rather than geometrical inhomogeneity, it has been tried tofind an empirical law to predict the value of N for different mate-rials. It could have been tried to find an empirical law for f0, buta small error in determination of f0 would result in a large errorin predicted FLD, whereas using the empirical law for N will resultin FLDs which are in good agreement with experimental diagrams.

To derive the empirical law, a large number of experimentaldata for different materials used in forming processes were col-lected from the literature and the appropriate values of N weredetermined for them with trial and error method. Surprisinglythe value of N is only a function of sheet thickness and the samelaw applies for different types of materials. To obtain the mostappropriate relation, various curves were fitted to the diagram ofN versus t, and the best results were acquired by the followingequation:

N ¼ 1þ 0:0292 expð�2:02� t ðmmÞ þ 0:7466Þ ð15Þ

The plot of this equation is shown in Fig. 8 to demonstrate thevariations of N versus t. The success and accuracy of this law topredict the initial inhomogeneity and its advantages over othermethods will be discussed in the following sections.

5. Verification of the empirical law

To examine the accuracy of the proposed formula for N (Eq.(15)), several experimental FLDs for typical materials used in metalforming processes were collected from different references. Thevalues of FLD0 were predicted for them with the modified M–Kmodel using the value of N suggested by Eq. (15). The predictedvalues by the well known NADDRG law [17] are also shown tocompare the accuracy of the two methods. It should be noted thatthe predicted values by the NADDRG relation are obtained usingEq. (1) and converting engineering strain to true strain. The resultsare charted in Table 3. In this table, the term denoted by FLDa

0 cor-responds to the experimental value of FLD0 acquired from experi-ments; FLDb

0 corresponds to the value predicted by Eq. (1) andconverting engineering strain to true strain; and FLDc

0 correspondsto the value predicted by the present model (using Eq. (15) to findthe initial inhomogeneity factor).

Evidently the error of the proposed method is considerably lessthan that of NADDRG in most cases. It should be noted that thematerials of Table 3 are different from those used to obtain Eq.(15). Thus results charted in Table 3 show the actual capability ofthe proposed model to predict the value of FLD0 for variousmaterials.

6. Conclusions

The sensitivity of FLD to variations of initial geometrical inho-mogeneity, f0, has been shown to be very high and some materialinhomogeneities have been proposed to be replaced with conven-tional assumption to reduce this sensitivity. Among the proposedmodels, the N model which assumes a hardening exponent inho-mogeneity has been chosen and proved to be a practical and ben-eficial substitution for conventional M–K model. Finally, using theN model, an empirical law has been developed which predicts thevalue of N as a function of sheet thickness. It has been shown thatusing this law, the FLD of the material can be determined in the ab-

sence of experimental data. This method has the following advan-tages over other calibration methods and empirical laws:

1. The only additional required parameter is the thickness ofthe sheet which is very easy to measure.

2. Calculating the value of N and implementing this value in themodified M–K model are very easy.

3. The same law can be used for different materials rather thanonly a small group of them.

4. The effect of some important parameters such as e0, r and mvalue on FLD can be illustrated.

5. The predicted FLDs are in good agreement with the experi-mental FLDs.

Furthermore, since it has been proved that the initial inhomoge-neity is only a function of sheet thickness, it can be concluded that:‘‘if the effect of thickness is somehow included in the analyticalmodel, there is no need to estimate the initial inhomogeneityand the same value of it can be used for all cases’’.

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