EVALUATION OF LIMITING DRAWING RATIO ( LDR ) IN DEEP DRAWING PROCESS

Part production by deep drawing technology brings important economic advantages. Cupping test is used to determine material suitability for deep drawing. The main principle of the test is redrawing of cylinder metal test piece to the cup. The result of the test is calculation of limiting drawing ratio (LDR) which states the ratio between the largest blank diameter and final cup diameter. Many cupping tests for various materials were performed in order to determine maximal value of LDR that would still allow deep drawing without failure of material integrity.Paper presents the theoretical determination of the limiting drawing ratio (LDR) in dependence on d0/t0 ratio and friction coefficient between blank and punch.Limiting drawing ratio (LDR) converges to Euler’s number (e) for high test piece depth and final cup diameter ratio values. Theoretical analysis of the cupping on drawing die with tractrix curve shows that maximal value of LDR may achieve value LDR>e. This is possible at lower values of test piece depth to final cup diameter ratio. Coefficient of friction has determining influence on LDR value.


Introduction
Deep drawing technology is used to produce metal parts from various materials such as steel, copper, aluminum and their alloys [1][2][3].The technology is used to make car body parts in automotive industry, cartridges in armament industry, mining blasting cups and various boxes and complex shaped parts [4].Materials given for deep drawing must be tested for capability to be deep drawn and to determine limit condition for deep drawing [5,6].Testing is performed by the cupping test.Geometrical scheme of cupping process is shown in Fig. 1.Cupping is performed without blank holder by two toolsdrawing die and punch.Cylinder metal piece (roundel) with diameter D0 and thickness t0 is drawn by punch with diameter d0.Outer diameter of final cup dk is determined by the formula The thickness of material is not changing during the cupping process.That describes the ratio between the largest blank diameterD0 and drawing punch diameter d0.
Every forming technology has some material and technological limitation.If these limits are overrun, it leads to material failure [7,8].That means crack creation or even material destruction (breakaway of cup bottom) during cupping process.Therefore we are trying to find limits that would guarantee technological process without failure.It results from the formula (2) that the material has higher ability for deep drawing with increasing values of LDR.Many cupping tests were performed in laboratories to determine maximal value of limiting drawing ratio (LDR) designed as max 0 whereD0,max is maximal diameter of blank(roundel)at which the drawing cup is not yet damaged.Tests are performed in the step where the roundel diameter D0 is gradually increased until the cup breaks down.Maximal value of limiting drawing ratio max 0 β warns about technological limits of cupping; and overrun of the LDR may lead to technological failure [1].Generic annular element with marked stress effecting particular areas was chosen to define the state of stress (see Fig. 2).Cupping is performed without the blank holder.Designation of particular stress is as follows: radial stress r, tangential stress t and normal stress n.According to Shawki [9], the balance formula for forces influencing annular element (Fig. 2) is where the geometrical position of annular element is defined by its radius r and the angle r.The deformation resistance is expressed by stress p and friction between drawing material and die by friction coefficient f.The differential equation ( 4) is based on the condition that the wall thickness of a cup is not changing in the cupping process.The wall thickness may be considered as constant because of differential of the wall thickness that is equal to zero (dt=0).This assumption allows to define t=t0.The differential equation ( 4) is constructed with using of plasticity equation ) Construction of the equation ( 4) in more detail with the commentary was published by Pernis [10].Cupping material is expressed as p (yield point of the material) in the differential equation.
We exclude this material constant (p) from differential equation ( 4) and substitute it with relative radial stress r  (6) due to general use of the equation [11]   p r r     (6.) Differential equation ( 4) after the substitution ( 6) is as follows ) The relative radial stress is given by the equation ( 7) as a function , where the friction coefficient is a constant.Geometrical shape of drawing die profile curve is generally defined as a function g(,r) [12].It is necessary to define this function to find out the solution of the differential equation (7).Various curves or their parts were used in history of the cupping: line, sinusoid, circle, ellipse or tractrix curve.Real tests and measurements show that evolvent catenary curve known as tractrix is the optimal drawing curve.Therefore the tractrix curve is used to solve the differential equation (7).

Experimental Material and Methods
This paper is aimed at materials that are processed by deep drawing technology.These materials include the group of steel materials DC01andDC06of the fieldEN10130/91; and non-ferrous metals such as copper, brass CuZn30 and selected aluminum alloys.General differential equation ( 7) describing cupping without blank holder has no analytical solution.To get a specific solution of differential equation ( 7), the geometrical shape of drawing die with tractrix curve was applied.Analytical solution in the first step of differential equation was used and numerical solution was used for the second step.The evaluation of LDR was determined for relative radial stress that must meet the condition 1  r  .

Drawing die with tractrix curve
The tractrix curve is defined as evolvent upon general catenary curve and it is described by the function ( 8  The location of point B is defined by radius r and angle .Their correlation is defined by rectangular triangle ABE (see Fig. 3) where parameter a is given by hypotenuse AB.The radius of tractrix curve r for point B is given by addition of cup external radius rk and triangle cathetus EB Radius r (9) and its differential dr (11) are then both inserted into differential equation (7) where the result differential equation ( 12) is ) There are two geometrical constants a and rk in the equation (12).The substitution is used to reduce these constants to one non-dimensional constant k.
a It is necessary to express k constant through t0/d0 ratio and also limiting drawing ratio before the substitution. ) Final differential equation suitable for evaluating is made by inserting the k constant ( 14) into the differential equation (12). ) The final equation ( 15) belongs to a group of differential equations that could be described by general expression ) Where P() a Q()are known functions defined by equation (15) as The method of constant variation is used for analytical evaluation of the differential equation ( 16) with regard to its type.The result of the evaluation is the expression Where C is integral constant (determined from initial conditions) and function Integral constant C is determined according to Fig. 3 The evaluation of relative radial stress ) 0 ( r  is described by sequence in (26).The relative radial stress at point D in Fig. 3   The formula for evaluation of limiting drawing ratio is expressed from the equation ( 14) what could be used to evaluate theoretical value of the LDR.
where the expression (k+1)/k is substituted by the equation ( 29).Limit value of LDR is then determined according to the equation (30) for condition  

Discussion
The definition of LDR shows that material has better ability to be drawn with increasing coefficient of friction.Many publications and examinations deal with that topic [17][18][19][20] to determine LDR limit.Various geometrical shapes of drawing die were examined such as are line, circle, parabola, sinusoid, ellipse and tractrix curve.In a view of minimal drawing force and maximal value of LDR, the best results were achieved with the use of the tractrix curve.This fact is documented in the publication of Shawki [9,21].His results are shown in Fig. 7. Limit value of LDR is expressed as a function of aggregated element . Maximal value of LDR reached limit value LDR = 2.85.The author of publication [7] did not define complete measurement conditions.The publication does not define for which value of LDR the curves were created.Marciniak et al. [22] indicated 2.72 e LDR    as the limit value.The value was determined differently than it is described in this paper.Morawiecki et al. [23] published Table 1 where limit value achieves value of LDR=2.72 what corresponds with the equation (31).Theoretical limit value is LDR =2.25÷2.50.This value is reduced by the friction influence (the friction is not defined).Drastík and Elfmark [24] indicated the limit value of LDR as much as LDR=2.9.Theoretical analysis proves the possibility to reach such value as it is documented on graph in Fig. 6.The real measurements of LDR limit value prove the advisability of differential equation ( 15) application for relative radial stress calculation in the process of cupping without blank holder.Marumo [25] et al. and Sekhara-Reddy et al. [26] confirmed the dependency of LDR value on friction coefficient and also on depth of the material.Inoue and Takasui [27] documented the dependency of LDR value on the sheets anisotropy at rolled aluminum alloys sheets.Halkaci et al. [28] used finite element method (FEM) to calculate the LDR value.

Fig. 1
Fig. 1 Scheme of the cup deepdrawing process

Fig. 3 .
The curve starts at point C, then continues to point B and ends at point D. Zero point of coordinate system lies at point S. Parameter a describes the length of tangent to tractrix curve.The length of tangent is considered as the length form apposing point B on tractrix curve to point A. The value of angle is point D. The annular element (see Fig.2) is located at point B.

(
25.)    Numerical integration is then used to evaluate definite integral in the equation (25) for 0  [15].The functions F(α) and Q(α) are in convergence to infinity for 0

4 .
is independent of constant k and friction DOI 10.12776/ams.v21i4.642p-ISSN 1335-1532 e-ISSN 1338-1156 coefficient f.The absolute value of radial stress at point D is equal to deformation resistance equation (25) is visualized in Fig.The distribution of relative radial stress is valid for friction coefficient f=0.2 and initial dimensions of roundel D0=125 mm and a t0=1 mm.The parameter for particular curves is limiting drawing ratio.All solutions of differential equation (12) for any constants run through point ] has a character of tensile strength; therefore, it cannot exceed the value of deformation resistance in deep drawing process.The curves where relative radial stress is 1 σ r  make conditions for destruction of the cup in deep drawing process.The values of relative radial stress have to be destruction of drawn material.It results from Fig.4that the value of limiting drawing ratio is 2entire high range along the drawing die.

Fig. 4 Fig. 5
Fig. 4 Distribution of relative radial stress along the high of cupping die

Fig. 7
Fig.7Dependence of the LDR on d0/t01,5 ratio and drawing die profile[9] ) where t is a parameter.Drawing die with tractrix curve is shown in

Table 1
[23]t values of LDR[23]It is necessary to describe measurement conditions for evaluating of LDR limit value of specific real material.The basic conditions are ratio of roundel (blank) diameter to cup diameter, friction coefficient and geometrical shape of cupping die.The differential equation (15) stated above properly describes state of the stresses along the high of cupping die in cupping process.Needed cupping force could be calculated based on stated relative radial stress.The paper documents the importance of friction coefficient for cupping test.If friction influence conditions are not defined, the verification of LDRlimit value is problematic.This publication was created within the project "Alexander Dubček University of Trenčín wants to offer high-quality and modern education", ITMS code 26110230099, based on the Operational Programme Education.Modern education for knowledge society / The project is co-funded by the European Social Fund.