Theoretical Determination of Forming Limit Diagram for Steel , Brass and Aluminum alloy sheets

Sheet metal forming is defined as the ability of metal to deform plastically (deformation by Stretching or drawing) or changing the shape of the sheet into a new desirable shape with out necking or crack . To control the operation of sheet metal forming with out failure. A diagram is used in which the range accepted , failure and critical deformation range are shown . This diagram is known as the Forming limit diagram. It is considered as one of the important tool to determine the formability of sheet metals. Every sheet metal has its own forming limit diagram which determines its formability, strain limit and the forming regions. These diagrams can be assessed using theoretical and experimental approaches, In this paper, the FLD is determined using different yield criteria Hill1948, Hosford1979 and modified Hosford 1985. It is shown that the determination of forming limit curve using the modified Hosford 1985 criterion with the (M-K) analysis , gave the best results compared with the other used criteria .Using this criterion gave the closest forming limit curve to that obtained experimentally, but with different criterion index for different alloy . The value of the index (a=6) gave the best results for brass, while (a=8) gave the best results for aluminum alloy and mild steel. لأاو صاربلاو بلصلا حئافصل لیكشتلا دح ينحنمل يرظنلا نییعتلا موینمل د . سیردإ دیبع سنا يلع للاج دیلو ةیكیناكیملا ةسدنھلا مسق لصوملا ةعماج ةصلاخلا ) بحسلا وأ طملاب لیكشتلا ( رسك وأ ، ھیف نیبی ططخم مادختسا متی لشف ثودح نود حئافصلا لیكشت ةیلمع ىلع رطیسن يكلو ، ملا حئافصلا لیكشت ةیلباق دیدحت يف ةمھملا لئاسولاو تاودلأا نم لیكشتلا دح ططخم . لكل . عوضخلل ) Hill1948 ( ، ) Hosford1979 ( ، ) Hosford1985 ( و ایلمع . ) Hosford 1985 ( ) M-K ( ملا نداعمل ا نم لكل عوضخل ا ةمدختس ، ) a=6 ( سلأا مادختسا دنع امنیب ، صاربلل ) a=8 ( بلصلاو موینمللأل ةبسنلاب لضفلأا ناك . Notation 1,2 ,3 Principle stresses ε1 ε2 ε3 Principle strains m Strain rate sensitivity n Strain hardening exponent

Forming processes are among the most important metal working operations.The industrial process of sheet-metal forming is strongly dependent on numerous interactive variables such as material behavior, lubrication, forming equipment, etc.
Forming limit Diagram is a representation of the critical combination of the two principal surface strains major and minor above which localized necking instability is observed.Forming limit curve (FLC) provides excellent guidelines for adjusting material, tooling and lubrication conditions.Also it is strongly dependent on material parameters, The idea of forming limit diagrams was first introduced by Keeler [1], when he observed that the maximum local elongation was not enough to determine the possible straining rate of a sheet .He established that the plotting of the principal strains at fracture ε1, and ε2 on two axes of a same diagram gave a curve : the forming limit curve.This curve, first restricted on the area ε2>0, was made complete for ε2< 0 by Goodwin's works [2].This curve is interesting because it divides the plane into two zones .The success area under the forming limit curve and the fail area above it, for a deep drawing operation .The criteria to reject the drawn parts is now the onset of localized necking.Hill [3] was the first who proposed a general criterion for localized necking in thin sheets under plane stress states.His analysis predicts localized plastic deformation in the negative minor strain region.Marciniak and Kuckzinsky (M-K) [4] have proposed the first realistic mathematical model for theoretical determination of FLDs that suppose an infinite sheet metal containing a region of local imperfection where heterogeneous plastic flow develops and localizes.Hutchinson and Neale [5] extended M-K theory using a J2 deformation theory.Therefore, the left and right hand sides of the forming limit diagram can be calculated by M-K analysis.
Sheet metals exhibit a highly anisotropic material behavior by cold rolling.It is therefore of major importance to extend the plastic instability analysis to anisotropic materials.Constitutive relations for the plastic yielding and deformation of anisotropic metals at a macroscopic level were proposed long ago by Hill 1948[6] This theory was the simplest conceivable one for anisotropic materials, however, inevitable limitations of its range of validity have eventually became apparent.The original M-K analysis [4] was based on Hill's 1948 yield criterion [6].However , it can be seen from the comparison with experiments and predicted results of Painter and Pearce 1974 [7] that this analysis overestimates the limit strains towards the equibiaxial strain region, and underestimates the limit strains towards the plane strain region, particularly for materials with R values less than unity such as aluminum or brass .In addition, the calculated limit strains for the right hand side of the FLDs are very sensitive to the material anisotropy, a phenomenon that has not been observed in experiment .Sowerby and Duncan 1971 [8] argued that the difference between these two stress states depends on the yield criterion and the shape of the corresponding yield locus.The effect of R on the FLDs depends on how the R-value affects the yield locus shape.Using (Hill's 1948) yield criterion, the stress ratios for positive strain ratios depend strongly on the value of R.
Hill's 1979 [9] yield criterion, taken with the assumption of the principle of equivalence of plastic work, was proposed to account for the so-called "anomalous behavior" of aluminum.This yield criterion has undergone application.One line of attack is represented by the work of Parmar and Mellor 1978 [10].
Hosford 1979 [11] developed an extension of Hill's 1948 yield criterion , which is also found to be a special case of Hill's 1979 yield criterion.This criterion has been used by Graf and Hosford 1990[12] for sheet metals with normal anisotropy.Later, Padwal and Chaturvedi 1992 [13] also used Hosford's 1985 [14] planar anisotropy yield criterion to analyze the insatiable behavior of strain localization.They found that the effect of planar anisotropy is negligible while the predictions are strongly dependent on exponent "a" an exponent in yield criterion .Predictions with a=5,6 or 8 match the experimental results much better than the predictions that were obtained from Hill's yield criterion.
Friedman and Pan 2000 [15] studied the effect of different yield criteria of (Hill1948),(Hill1979)and (Hosford 1979) on the right hand side of the forming limit diagram .
Dariani and Azodi 2003 [16] showed the agreement between theoretical and experimental results by changing the index of Hill1979 yield criterion for right and left hand of FLD.
Banabic2004 [17] determined the FLD of Aluminum alloy (Al-2008) using new yield criterion (BBC2000) [18] , showed the best agreement between theoretical and experiment result of right hand side of FLD.
In this paper the FLDs of different sheet metals: Steel, Brass and Aluminum alloy are obtained theoretically using, the Marciniak-Kuczynski Theory, the following yield criteria :Hill1948 , Hosford 1979 and modified Hosford 1985 [14] , These FLDs were compared with the experimentally obtained FLDs of the same sheets metals to obtain the best agreement between the calculated FLDs and the experimental FLDs.
The geometry of neck formation and the element of sheet undergoing plastic deformation are shown in Fig. 1.Following the MK analysis , based on a simplified model with assumed pre-existing thickness imperfection in the form of a groove perpendicular to the principal strain directions , The sheet is composed of the nominal area and weak groove area, which are denoted by `a' and `b', respectively.The initial imperfection factor of the groove, ƒ0, is defined as the thickness ratio ƒ0=(tbo/tao); where `t' denotes the thickness and subscript `0' denotes the initial state.A biaxial stress state is imposed on the nominal area and causes the development of strain increments in both the nominal (a) and the weak area (b).
The yield criterion proposed recently by modified Hosford was used in the calculation [14] in the plane stress state , this criterion is obtained as follows :  8)and( 9),( 11 from eq.( 10),( 15) and ( 16)

... exp 3 3
Substitute equations( 10),( 12), (15) in equilibrium equation ( 25), an equation can be found and solved numerically.Imposing a loading path (ρa), a finite increment of strain is also imposed in region (a), and by numerical computation is performed by using computer program (Fortran power Station) to determine the limit strain of a strain path in the FLD , and the limit strain is determined when [(dεb/dεa)> 10] in the range of strain ratios from (-0.5 to 1.0).
In the experimental study, mild Steel, brass and aluminum alloy sheets their chemical composition are shown in were used .The FLDs of the sheets are determined using stretch forming tests with a hemispherical punch of (50mm) diameter and Die [19] with blank holder as shown in .Using two sets of specimens with constant length (100mm) and having various widths with radius in one set for negative minor strain

Al%
.By changing the sheet width , major and minor strains were measured following varied deformation paths.Circular grids of (2mm) diameter were initially printed on the surface of the specimens for the purpose of strain measurements.For each specimen the strain were directly measured from deformed grids.
In the experimental work lubrication type(C140 Oil) was used .The FLD was obtained by drawing a line between the necked region and the un-necked (acceptable) region .
The mechanical properties of sheets metals were obtained from tensile test, by using specimens at different angles ( ، ) to the rolling direction .After testing , the engineering stress-strain curve and true stress-strain curve were drawn.The value of strain hardening exponent ( ) was determined from the slope of line in the (log coordinate of true stress strain curve) by selection two points one before ultimate stress and the other after yield point .The intersection of this line with unit strain gives the stress value that define the magnitude of strength coefficient ( ) ( ) [20] .For anisotropy plastic property (R) the same specimens of tensile test were used with different angles and using the equation and to determine the plastic anisotropic ratio [20] .The strain rate sensitivity (m) was determined by using the same tensile test specimens , the cross-head speed is suddenly changed during the uniform deformation region of a tensile test and a small jump in the load may be observed , the exponent (m) is then calculated ( ) from load and cross-head speed before and after the speed change , which is denoted by suffixes 1 and 2 in ( [20].show the experimental Forming limit curves of Aluminum alloy , Mild Steel and α-Brass as received , it is clear from the FLC for Brass is the highest while that for Aluminum alloy is the lowest , this ensure that the strain hardening exponent has the dominate effect on the formability of sheet metal . show the comparison between the experimental and theoretical Forming limit Diagrams using (Hill 1948) yield criterion, It can be seen that this analysis overestimates the limit strain towards the equibiaxial strain path and underestimate the limit strains towards the plane strain and uniaxial regions , show that the determination of forming limit curves using modified Hosford 1985 criterion with the (M-K) analysis , gave the best results compared with the other used criteria .Using this criterion gave the closest forming limit curve to those obtained experimentally, but with different criterion index for different alloy ie, the value of the index (a=6) gave the best result for brass, while (a=8) gave the best results for aluminum alloy and mild steel.
This work provides an experimental and theoretical analysis for the determination of the FLD using Hill1948, Hosford 1979 and modified Hosford 1985 yield criteria with Marciniak and Kuczynski (M-K) theory to find the best yield criterion (index of yield criterion) for the different materials, used the following conclusions can be obtained justified : Efficiency of stretch forming by using hemispherical punch (50mm) and using eight type of specimens to cover the whole of forming limit curve .
Strain hardening exponent (n) has higher effect compared with other parameters this makes Brass has highest FLD while Aluminum alloy has the lowest.

Table (
criterion , gave the best results compared with the other used criteria , but with different index to different sheet metal.The index (a=6)gave best result for Brass while the index (a=8)gave best result for Aluminum and mild Steel.1.Keeler S.P. and Backofen W.A., (1963) Plastic Instability and Fracture in Sheets Stretched Over Rigid Punches, Transactions of The ASM ,Vol.56, 25-48.2.Goodwin, G.M. (1968), Application of Strain Analysis to Sheet Metal Forming Problems in the Press Shop, Society of Automotive Engineers, technical paper no.680093 .3.Hill, R., (1952), On Discontinuous Plastic States, With Special Reference to Localized Necking in Thin sheets, Journal of the Mechanics and Physics of Solids, Vol. 1, pp. 19-30.