Optimization and Mapping of the Deep Drawing Force Considering Friction Combination

Abstract

Deep drawing is characterized by extremely complex deformation that is influenced by process characteristics such as die and punch shapes, blank shape, blank holding force, material properties, and lubrication. The optimization of the deep drawing process is a challenging issue due to the complicated functions that define and relate the process parameters. However, the optimization is essential to enhance the productivity and the product cost in the deep drawing process. In this paper, a MATLAB toolbox (Pattern Search) was employed to minimize the maximum deep drawing force (F_d-min) at different values of the operating and the geometrical parameters. As a result, a minimum deep drawing force chart (carpet plot) was generated to show the best combination of friction coefficients at the blank contact interfaces. The extracted friction coefficients guided the selection of proper lubricants while minimizing the deep drawing force. A finite element analysis (FEA) was applied through 3D model to simulate the deep drawing process. The material modeling was implemented utilizing the ABAQUS/EXPLICIT program with plastic anisotropy. The optimization results showed that the deep drawing force and the wrinkling decrease when compared with experimental and numerical results from the literature.

1. Introduction

In terms of production capacity, energy consumption, and process improvement potential, the deep drawing process is one of the most significant technologies in current manufacturing [1,2]. During the deep drawing process, a variety of factors might contribute to product failure [3,4]. This failure typically manifests as wrinkling, tearing, earing, and rupture. These studies contained descriptions of material properties, die design, and operating factors such as blank holder load, friction coefficients, deep drawing ratio, and maximum drawing force; careful control of these parameters helps lower the likelihood of failure for drawing parts [5]. Singh et al. [6] discovered the effectiveness of deep drawing variables, including die shoulder radii, punch nose radii, friction coefficients, and deep drawing ratios for cylinder-shaped parts. Using a genetic algorithm, this research created an evolutionary neural system, i.e., error back transmission. The evolutionary algorithm uses the results of the system to find factors that are preferable to maintaining a consistent thickness in the final cup. A complex blank stamping procedure may be examined under particular assumptions, allowing the use of the multi-objective optimization approach to be used [7]. After multi-objective optimization methods are created for a sheet metal forming problem, special considerations are given to conflicting objectives [8]. The maximal punch load (Fd) to perform the process is a main factor in the deep drawing process. This load is utilized in the choice of the instrument press and equipment, and it controls the creation of wrinkling on the lip of the drawn product. There exists no special mathematical calculation to compute the necessary deep drawing force in the drawing process [9]. Generally, the drawing load for the first draw of circular components can be achieved using two methods, one from theoretical analysis taking into account the plasticity principle, and the other by utilizing well-established practical equations. The generalized function or expression takes the following form:

Fd = f (σ₁, FBH, μ, h, t, n, K, r),

(1)

where σ₁ is the principal stress value, FBH is the blank holder load, μ is the friction coefficient, h is the drawing cup height, t is the initial thickness of the sheet metal, n is the exponent of the strain hardening, K is the coefficient of the strain hardening, and r is the coefficient of normal anisotropy. Suitable records of σ1 and t can be achieved by differentiating Fd with regard to effective stress, such that

dF/dσ = 0.

(2)

In the optimization of the operation of deep drawing, Gharib et al. [10] developed an optimization system to be used in determining the optimum blank holder force (BHF) scheme that minimizes punch force without running into any of the two operation restrictions: wrinkling and tearing. The design factors of the deep drawing machine included the geometrical factors for the punch and the die, in addition to the blank holder load. Hsiao-Chu et al. [11] described the forecast of the drawing load and forming rise in perspective of optimizing the operating parameters included in the deep drawing operation. A model was employed to discover that friction has substantial effects on the punch force, and the low friction between the die and blank holder can significantly decrease the failure probability and increase the quality of the deep drawing process. The lubrication system and the type of lubricant used can significantly affect the process output and the amount of energy consumed [12]. Plevy [13] studied the minimization of drawing power used in the deep drawing operation by reducing the effect of friction and proposed the utilization of a solid lubricating polymer film barrier rather than oil lubrication. This lubricating barrier significantly decreases the drawing energy of precoated sheet metal at small tolerances. The polymer film contributed successfully in saving surface reliability of the sheet metal [14]. An optimized lubricant application method through a Taguchi and regression model during deep drawing was proposed [15]. The optimum combination of lubricant and sheet surface microstructure was investigated experimentally. The experimental results were analyzed statistically to select the optimum combination of the two factors. Minimum work done while avoiding tearing, wrinkling, and spring back was achieved [16,17,18]. Zhang et al. [19] mentioned that variability in the deep drawing operation may result in effects which the deterministic models are not able to expect. A common method was applied to determine the doubts and combine them through the response surface method (RSM) in order to direct probabilistic established optimization [20]. The probabilistic design model was used to discover the optimum arrangement of blank holder load and friction coefficient through the existence of a variety of material characteristics. The outcome demonstrated that, by utilizing the probabilistic model, the wrinkling and failure decreased by 42% compared with the conventional design [19]. The uncertainness and unreliability of the deterministic optimization models utilized in deep drawing encouraged further researchers to implement quality enhancement applications, e.g., six sigma [21]. Atrian and Fereshteh [22] examined the influence of diverse factors on the deep drawing operation of steel/brass blanks. Finite element and experimental investigations of the deep drawing process of steel/brass blanks were conducted to display the positions of tearing for the cup. In this research, a straight-line relationship was obtained between the maximum punch load and the initial blank diameter. Manoochehri and Kolahan [23] developed a combined finite element (FE), artificial neural network (ANN), and simulated annealing (SA) algorithm as a method to model and optimize the deep drawing operation for stainless steel 304 (SUS304). In this method, the parameters (die radius, punch radius, blank holder load, and frictional coefficients) were considered as input factors. Kakandikar and Nandedkar [24] suggested a new procedure combining two methods for optimizing and decreasing sheet metal thickness for the cover of automotive sealing. Mathematical equations were generated to join input operation factors and thinning. The optimum process was expressed for thinning using a genetic algorithm. Volk et al. [25] generated a new optimizing method for optimizing the deep drawing operation utilizing the finite element technique incorporated with the response surface method (RSM). This optimizing method was a procedure for obtaining the optimal values of blank holder loads and then the desirable final product. Dienemann et al. [26] presented an innovative method for optimizing shell structures taking into consideration their middle surface areas, such as undercuts. The authors presented a production restriction to the 3D topology optimization depending on the stiffness technique to be able to get an optimum structure with no undercuts, along with consistent thickness. Then, these shell structures could be produced by deep drawing within one stage. Lastly, the thickness of the shell enhanced the efficiency of the shell structure using this technique. Kottayam et al. [27] suggested a procedure to simultaneously identify the ideal blank shape to minimize the earing and the ideal value of the blank holder load in the deep drawing operation for the circular cup. The total production cost was decreased by utilizing this procedure.

In the present work, the optimization MATLAB toolbox (Pattern Search) was utilized to minimize the maximum deep drawing force. The deep drawing force and the value of the dimensionless ratio of die shoulder radius to sheet thickness (r_d/t) were optimized to minimize the forming work. The other operating parameters and geometrical parameters were constrained according to logical and practical limits documented in the literature [28]. A carpet plot chart for a given value of the best interfacial frictional combinations at different values of the sheet metal thickness was extracted, showing that the proper selection of equivalent conditions led to the minimum deep drawing force (F_dmin). Furthermore, in this research, a finite element analysis (FEA) 3D model was created to simulate the deep drawing operation by utilizing the ABAQUS/EXPLICIT FEA program with proper boundary conditions. As a final point, the optimization results using the finite element analysis (FEA) model were compared with the results of the recommended values from the literature by using the experimental work for validation of the optimization results.

2. Simulation of Deep Drawing Process

2.1. The Simulation Model

The deep drawing operation is displayed in several stages: the round blank was placed between the blank holder and the die; the punch traveled downward; the blank diameter reduced with the blank was drawn into the die hole. Figure 1 shows the main dimensions of the blank, die, punch, and the blank holder. Furthermore, the main dimensions with the recommended values of friction coefficients are displayed in

Table 1. Geometrical and the operating parameters.

Item	Parameter	Value
1. Blank:	Blank radius (D/2)	45 mm
	Blank thickness (t)	0.8 mm
2. Punch:	Punch radius (d1/2)	26 mm
	Punch nose radius (rp)	5 mm
	Punch stroke (h1)	26.5 mm
3. Die:	Die radius (dD/2)	27.286 mm
	Die shoulder radius (rd)	4 mm
	Radial clearance (wc)	1.286 mm
4. Operating parameters	Blank holder force (FBH)	0.375 ton
	Friction coefficient between punch and blank (μp)	0.25
	Friction coefficient between holder and blank (μh)	0.125
	Friction coefficient between die and blank (μd)	0.125

, according to an FEA simulation from an earlier study [28].

The 3D FE model is shown in Figure 2. The discrete rigid form was utilized to model the punch, the blank holder, and the die, whose movements were represented by the movement of a single node, identified as the rigid body reference node. The punch, the blank holder, and the die were meshed with R3D4 elements. On the other hand, the round blank sheet metal (45 mm radius × 0.8 mm thickness) was modeled as a deformable planar shell base and meshed with shell-type elements of reduced integration S4R elements [29].

2.2. Material Properties

The sheet metal blank was created from mild steel, and modeling was implemented as an elastic–plastic material with isotropic elasticity [30]. The Hill anisotropic yield theory was employed for the plasticity to depict the anisotropic properties of the sheet metal blank in the interior of the FEA simulation program (ABAQUS/EXPLICIT) [31]. The true stress–true strain curve of the material is shown in Figure 3, and

Table 2. The material properties of sheet metal blank [1].

Property	Value
Young’s modulus (E)	206 GPa
Poisson’s ratio (ν)	0.3
Density (ρ)	7800 kg/m3
Yield strength (σyo)	167 MPa
Anisotropy characteristic values	r0 = 1.79, r45 = 1.51, r90 = 2.27

presents the material properties [32].

2.3. Forming Limit Stress Diagram (FLSD)

The failure at different values of the geometrical and operating parameters was investigated by using the damage initiation criteria in ABAQUS for the forming limit stress diagram (FLSD) to determine the mechanical properties of the sheet metal. The material was assumed to be anisotropic with three perpendicular similarity planes. The yield theory is presented as the following quadratic function [33,34,35]:

$$2f\left({\sigma }_{ij}\right)\equiv F{\left({\sigma }_y-{\sigma }_z\right)}^2+G{\left({\sigma }_z-{\sigma }_x\right)}^2+H{\left({\sigma }_x-{\sigma }_y\right)}^2+2L{\tau }_{yz}^2+2M{\tau }_{zx}^2+2N{\tau }_{xy}^2=1,$$

(3)

where f is the yield function, F, G, H, L, M, and N are factors particular to the anisotropic condition of the material, and x, y, and z are the principal anisotropic axes. For the plane stress condition (σz = τxz = τyz = 0; σx ≠ 0; σy ≠ 0; τxy ≠ 0), the yield criterion function becomes

$$2f\left({\sigma }_{ij}\right)\equiv \left(G+H\right){\sigma }_x^2-2H{\sigma }_x{\sigma }_y+\left(H+F\right){\sigma }_y^2+2N{\tau }_{xy}^2=1.$$

(4)

The anisotropic constants F, G, H, and N can be formulated in terms of the r-values as follows:

$$\begin{gathered} F=\frac{r_0}{r_{90}\left(1+r_{90}\right)}, \\ G=\frac{1}{\left(1+r_0\right)}, \\ H=\frac{r_0}{\left(1+r_0\right)}, \\ N=\frac{\left(r_0+r_{90}\right)\left(1+2r_{45}\right)}{2r_{90}\left(1+r_0\right)}. \end{gathered}$$

(5)

The r-values r₀, r₄₅, and r₉₀ were 1.79, 1.51, and 2.27, respectively as presented in Table 2. When the principal directions of the stress tensor correspond with the anisotropic axes (σx = σ1, σy = σ2, τxy = 0), the yield theory can be presented as the principal stress.

$${\sigma }_1^2-\frac{2r_0}{\left(1+r_0\right)}{\sigma }_1{\sigma }_2+\frac{r_0\left(1+r_{90}\right)}{r_{90}\left(1+r_0\right)}{\sigma }_2^2={\sigma }_0^2,$$

(6)

where σ₁ is a major stress, σ₂ is a minor stress, and σ₀ is the yield stress in the rolling direction (σ_o) of the sheet metal. Figure 4 shows the forming limit stress diagram for the sheet metal with a thickness equal to 0.8 mm.

3. Optimization of the Maximum Deep Drawing Force (F_dmax)

The optimization MATLAB toolbox (Pattern Search) was utilized to minimize the deep drawing load at the optimal values of the deep drawing process parameters for cylindrical cups. Figure 5 shows the basic components and configuration of the process.

The drawing force can be obtained using the equation below [2].

$$F_d=\left\{\left({\sigma }_{f,m,flange}.t.\ln \frac{d_o^{\prime }}{d_m}\right)+\left(\frac{2{\mu }_{h . F_{BH}}}{\pi . d_o^{\prime }}\right)+\left(\frac{{\sigma }_{f,m,die-ring} . t^2}{2.r_d+t}\right)\right\} \pi . d_m.\sin \alpha .e^{\varnothing {\mu }_d},$$

(7)

where ${\sigma }_{f,m,flange}$ is the mean flow stress in the flange of the cup (MPa), $d_o^{\prime }$ is the instantaneous blank diameter (mm), $d_m$ is the mean diameter of the cup (=$d_1+t$) (mm), d₁ is the punch diameter (mm), t is the sheet metal thickness (mm), μ_h is the friction coefficient between holder and blank, $F_{BH}$ is the blank holder force (N), ${\sigma }_{f,m,die-ring}$ is the mean flow stress in the die ring of the cup (MPa), r_d is the shoulder radius (mm), and μ_d is the friction coefficient between die and blank. At the maximum deep drawing force (F_d-max),

$$d_{o }^{\prime }=d_{F,max}\cong 0.77 D,$$

(8)

where $d_{F,max}$ is the outside diameter of the flange when the drawing force is a maximum (mm), and D is the diameter of the developed blank (mm); at the maximum deep drawing force (F_d-max)$, {\sigma }_{f,m,flange}$= ${\sigma }_{f,m,I}$ is the mean flow stress in the flange zone (MPa), ${\sigma }_{f,m,die-ring}$= ${\sigma }_{f,m,II}$ is the mean flow stress in the die ring zone (MPa), and α and $\varnothing$ are the angles, see Figure 5, assuming that α = $\varnothing$ = $\pi$/2 at the maximum deep drawing force (F_d-max). Then,

$$F_{d-max}=\left\{\left({\sigma }_{f,m,I}.\ln \frac{d_{F,max}}{d_m} \right)+\left(\frac{2{\mu }_{h . F_{BH}}}{\pi .t . d_{F,max}}\right)+\left( \frac{{\sigma }_{f,m,II} . t}{2.r_d+t} \right)\right\} \pi .t . d_m.e^{0.5 \pi {\mu }_d}.$$

(9)

4. Determining the Mean Flow Stresses (${\sigma }_{f,m,I}$ and ${\sigma }_{f,m,II}$)

In order to calculate the maximum flow stresses, three critical controlling points 1, 2, and 3 were selected, as shown in Figure 6.

The strain state at the rim is given as follows:

$${\epsilon }_{\theta }=\ln \left(\frac{d_o^{\prime }}{D}\right) \mathrm{Circumferential}\mathrm{strain},$$

(10)

$${\epsilon }_t=\ln \left(\frac{t_{}^{\prime }}{t}\right) \mathrm{Thickness}\mathrm{strain},$$

(11)

$${\epsilon }_r=\ln \left(\frac{D.t}{d_o^{\prime }. t_{}^{\prime }}\right) \mathrm{Radial}\mathrm{strain},$$

(12)

where $d_o^{\prime }$ is the instantaneous blank diameter (mm) (see Figure 6), $D$ is the diameter of the developed blank (mm), $t_{}^{\prime }$ is the instantaneous sheet metal thickness (mm), and t is the sheet metal thickness (mm). Assuming that

$${\epsilon }_t={\epsilon }_r=-\frac{1}{2}{\epsilon }_{\theta },$$

(13)

where the volume continuity condition requires that

$${\epsilon }_{\theta }+{\epsilon }_t+{\epsilon }_r=0,$$

(14)

then the equivalent uniaxial strain (${\epsilon }_{\mathrm{e}}$) is as follows [3]:

$${\epsilon }_e=\sqrt{\frac{2}{3} \left({\epsilon }_{\theta }^2+{\epsilon }_t^2+{\epsilon }_r^2\right)}={\epsilon }_{\theta }.$$

(15)

Then, at point 1 in the cup flange,

$${\epsilon }_{\theta max,1}=\ln \left(\frac{d_{F, max}}{D}\right),$$

(16)

where $d_{F, max}$ is the outside diameter of the flange when the drawing force is a maximum (mm), and D is the diameter of the developed blank (mm). The drawing ratio at maximum drawing force occurs according to the following relationship [4]:

$$d_{F, max}\cong 0.77 D.$$

(17)

The material between points 1 and 2 formed an annulus with outside diameter (D) and unknown inside diameter (d_i) in the blank. Assuming constant sheet thickness, the unknown diameter (d_i) can be calculated from the condition that the volume remains constant.

$$\frac{\pi \left(D^2-d_i^2\right)}{4}=\frac{\pi \left[d_{F,max }^2-{\left(d_D+2r_d\right)}^2\right] }{4},$$

(18)

or

$$d_i=\sqrt{D^2+{\left(d_D+2r_d\right)}^2-d_{F,max }^2 }.$$

(19)

Then, the strain ${\epsilon }_{\theta max,2}$ at point 2 when the load is a maximum is given as follows [5]:

$${\epsilon }_{\theta max,2}=\left|{\epsilon }_{\theta ,2}\right| n di - \ln \left(d_D+2r_d\right),$$

(20)

$${\epsilon }_{\theta max,2} =\ln \frac{\sqrt{D^2+{\left(d_D+2r_d\right)}^2-d_{F,max }^2 }}{d_D+2r_d},$$

(21)

where d_D is the die diameter (mm),

d_D = d₁ + (2w_c),

(22)

where d₁ is the punch diameter (mm), w_c is the radial clearance (mm), and r_d is the die shoulder radius (mm). For highly strain-hardening materials, the ratio r_d/t < 10; hence, the increase in ${\sigma }_f$ should be taken into account. In bending, the incremental strain $\left({\epsilon }_s\right)$ of the outside fibers can be determined from the radius of curvature (r_M) of the neutral fiber and the sheet thickness (t) [6].

$${\epsilon }_s =\frac{t}{2r_M}.$$

(23)

Applying this relationship to the die radius yields

$${\epsilon }_s =\frac{t}{2r_d+t}.$$

(24)

Since the strain distribution is linear across the sheet thickness, the mean bending strain $\left(\overline{\epsilon }\right)$ in the cross-section is given by

$$\overline{\epsilon }=\frac{{\epsilon }_s}{2}.$$

(25)

The workpiece undergoes twofold bending in the zone of the die shoulder radius such that the total mean bending strain after unbending is

$${\overline{\epsilon }}_{total}=2\overline{\epsilon }={\epsilon }_s.$$

(26)

The corresponding natural strain $({\epsilon }_{total})$ is given by

$${\epsilon }_{total}=\ln \left(1+{\overline{\epsilon }}_{total}\right)=\ln \left(1+{\epsilon }_s\right).$$

(27)

Using this result, the strain at point 3 for the maximum load can now be determined as follows, if ${\epsilon }_{\theta -max,2}$ is known:

$${\epsilon }_{\theta max,3}={\epsilon }_{total}+{\epsilon }_{\theta max,2}.$$

(28)

Then, from the flow stress equation, we can calculate the maximum flow stresses at points 1, 2, and 3.

$${\sigma }_{f,1max}=K . {\left({\epsilon }_{\theta max,1}\right)}^n,$$

(29)

$${\sigma }_{f,2max}=K . {\left({\epsilon }_{\theta max,2}\right)}^n,$$

(30)

$${\sigma }_{f,3max}=K . {\left({\epsilon }_{\theta max,3}\right)}^n,$$

(31)

where K is the strength coefficient for the sheet metal material (MPa), and n is the exponent of the strain hardening for the sheet metal material; then, we can calculate the mean flow stresses ${\sigma }_{f,m,I}$ and ${\sigma }_{f,m,II}$ for (r_d/t < 10).

$${\sigma }_{f,m,I}=0.5\left({\sigma }_{f,1max}+{\sigma }_{f,2max}\right),$$

(32)

$${\sigma }_{f,m,II}=0.5\left({\sigma }_{f,2max}+{\sigma }_{f,3max}\right).$$

(33)

5. Objective Function

The objective function is simply to minimize the maximum deep drawing force from Equation (9).

$$F_{dmax}=\left\{\left({\sigma }_{fm,I}.\ln \frac{d_{F,max}}{d_m}\right)+\left(\frac{2{\mu }_{h . F_{BH}}}{\pi .t . d_{F,max}}\right)+\left( \frac{{\sigma }_{fm,II} . t}{2.r_d+t} \right)\right\} \pi .t . d_m.e^{0.5 \pi {\mu }_d}.$$

(34)

All values in the above equation are calculated at the position to get the maximum drawing force, where

$${\sigma }_{f,m,I}=0.5\left({\sigma }_{f,1max}+{\sigma }_{f,2max}\right),$$

(35)

$${\sigma }_{f,m,I}=0.5\left[K . {\left({\epsilon }_{\theta -max,1}\right)}^n+K . {\left({\epsilon }_{\theta max,2}\right)}^n\right],$$

(36)

$${\sigma }_{f,m,I}=0.5\left\{K . {\left(\ln \left((d_{F, max}D)\right) \right)}^n+K . {\left(\ln \frac{\sqrt{D^2+{\left(d_D+2r_d\right)}^2-d_{F,max }^2 }}{d_D+2r_d}\right)}^n\right\},$$

(37)

$${\sigma }_{f,m,II}=0.5\left({\sigma }_{f,2max}+{\sigma }_{f,3max}\right),$$

(38)

$${\sigma }_{f,m,II}=0.5\left[K . {\left({\epsilon }_{\theta max,2}\right)}^n+K . {\left({\epsilon }_{\theta max,3}\right)}^n\right],$$

(39)

$${\sigma }_{f,m,II}=0.5\left[K . {\left({\epsilon }_{\theta max,2}\right)}^n+K . {\left({\epsilon }_{total}+{\epsilon }_{\theta max,2}\right)}^n\right],$$

(40)

$${\sigma }_{f,m,II}=0.5\left\{K . {\left({\epsilon }_{\theta max,2}\right)}^n+K . {\left(\ln \left(1+{\epsilon }_s\right)+{\epsilon }_{\theta max,2}\right)}^n\right\},$$

(41)

$$\begin{gathered} {\sigma }_{f,m,II}=0.5\left\{K . {\left(\ln \frac{\sqrt{D^2+{\left(d_D+2r_d\right)}^2-d_{F,max }^2 }}{d_D+2r_d} \right)}^n \\ +K . {\left(\ln \left(1+\frac{t}{2r_d+t}\right) \\ +\ln \frac{\sqrt{D^2+{\left(d_D+2r_d\right)}^2-d_{F,max }^2 }}{d_D+2r_d}\right)}^n\right\}. \end{gathered}$$

(42)

6. Cracking Force

The biggest permissible drawing force is constrained by the force which can be transferred by the sheet metal in the zone of the punch nose radius or at the transmission from cup wall to the bottom cup radius, which is named the cracking force [7]. It should be dependably higher than the maximum value of the deep drawing force. The cracking force can be calculated from the following empirical formula [36]:

F_crack = π t d₁UTS,

(43)

where F_crack is the cracking force (N), t is the sheet metal thickness (mm), d₁ is the punch diameter (mm), and UTS is the ultimate tensile strength of the sheet metal material (MPa).

7. Constraints

The operating parameters and geometrical parameters are constrained according to logical and practical limits documented in the literature to minimize spring back and residual stresses. Die shoulder radius practical limits were based on earlier work [8,37].

4 < r_d/t ≤ 10.

(44)

Grote and Antonsson [8] established a working domain for the deep drawing process, as shown in Figure 7, and the conditions of the drawing process are as follows:

F_BH-wrinkling < F_BH < F_BH-tearing.

(45)

Figure 8 describes the variation in the wrinkling formations with diverse values of the blank holder force/the maximum deep drawing force (F_BH/F_dmax). It is clear that the wrinkling formation is reduced with an increase in the ratio (F_BH/F_d-max). A deep drawing force F_BH of about 30% of the maximum deep drawing force (F_dmax) appears to be a good choice to prevent or minimize wrinkling without causing tearing of the cup wall. Figure 8 shows that the wrinkling force is 2 tons and the tearing force is 2.8 t at μ_d = μ_h = 0.125. In order to avoid wrinkling and tearing, the value of (F_BH) must be between the two values of wrinkling force and tearing force. In addition, by using the finite element simulation model, Figure 9 describes the variation in the wrinkling formations with dissimilar values of the blank holder force/the maximum deep drawing force (F_BH/F_dmax). Moreover, it shows that the wrinkling force is equal to 2 tons and the tearing force is equal to 3.2 t at μ_d = μ_h = 0.05. Note that the recommended values for the friction coefficients used in the finite element analysis and in the optimization were based on logical and practical limits documented in the literature [10].

Table 3. The values of the wrinkling and tearing forces at different friction coefficients.

The Coefficient of Friction	FBH-wrinkling (ton)	FBH-tearing (ton)
μd = μh = 0.125	2	2.8
μd = μh = 0.05	2	3.2

displays the values of the wrinkling force and the tearing force taken from Figure 8 and Figure 9 by utilizing the FEA simulation model.

According to Table 3, the range of F_BH is as follows:

2.4 t < F_BH < 2.8 t.

(46)

8. Results and Discussion

By using the optimization MATLAB toolbox (Pattern Search), the minimum deep drawing force (F_d-min) and the dimensionless ratio of (r_d/t) were determined. Figure 10 shows a layout of the operating parameters and constraints of the deep drawing process considering the friction coefficients μ_d and μ_h as equal within a range of 0.05 to 0.15, and the sheet thickness was varied from t = 0.5 to 1.7 mm. The objective function (Equation (34)) and constraints were as follows:

Objective function:

$$F_{d-max}=\left\{\left({\sigma }_{fm,I}.\ln \frac{d_{F,max}}{d_m}\right)+\left(\frac{2{\mu }_{h.FBH}}{\pi .t . d_{F,max}}\right)+\left(\frac{{\sigma }_{fm,II}.t}{2.r_d+t}\right)\right\}\pi .t . d_m.e^{0.5 \pi {\mu }_d}.$$

Constraints:

x(1) = μ_h (0.1 < μ_h ≤ 0.3) (coefficient of friction between holder/blank),

x(2) = μ_d (0.1 < μ_d ≤ 0.3) (coefficient of friction between die/blank),

x(3) = r_d (4 t < r_d ≤ 10 t) (die shoulder radius),

x(4) = wc (wc > t) (radial clearance),

F_BH-wrinkling < F_BH < F_BH-tearing.

The interface of the Pattern Search optimization toolbox is shown in Figure 11.

As a result of the objective function, the minimum deep drawing force F_d-min can be obtained at optimum r_d/t ratios. The carpet plot shown in Figure 12 was developed to be used as a visual tool for obtaining the minimum deep drawing force considering the best combination of friction coefficients. These operating factors at a given set of design data correspond to a situation of minimum deep drawing force (F_d-min). It is obvious that the minimum deep drawing force (F_d-min) at different sheet metal thickness (t) was not constant. In addition, Figure 12 describes the different coefficient of frictions related to sheet thickness, as well as the dimensionless ratio of die shoulder radius for a wide range of geometrical and operating parameters. During the initial stage of deep drawing, the dimensionless ratio (r_d/t) should not exceed 10 according to the finite element analysis results in earlier work [11]. Therefore, a limiting line was placed to limit the choices within the recommended design domain.

9. Experimental Work

The experimental work was carried out to validate the optimization results of the carpet plot for the deep drawing operation of the cylindrical cup. The experimental work was conducted for eight samples for the recommended values from the literature of the deep drawing process. The experiments were done at room temperature by means of a 250 kN MTS testing machine with 1 mm/s as the velocity of the punch and a mild steel sheet metal of 0.8 mm thickness. Figure 13 and Figure 14 display the deep drawing die set (punch, blank holder, and die) and the setup for the deep drawing process. Table 1 displays the basic geometrical parameters utilized in the cylindrical cup deep drawing experiment.

10. Validation

Figure 15 displays the experimental comparison of the wrinkling using the recommended values from the literature [10] and using the optimum values from minimizing the maximum deep drawing force (optimum case). It shows that wrinkling was reducing by utilizing the values extracted from the carpet plot. Furthermore, by using a parametric modeling of the FEA simulation, Figure 16 shows a comparison of the wrinkling by using the recommended values from the literature and by using the case of minimizing the maximum deep drawing force. It shows that wrinkling was reduced by using the values extracted from the carpet plot. As a final point, from Figure 15 and Figure 16, it is clear that there was a good matching between the experimental work and the FEA simulation model in the case of recommended values from the literature and in the case of optimum values. Figure 17 shows the comparison of the experimental and FE results at optimum conditions and process parameters. It can be noticed that a reasonable agreement was obtained between experimental and FE results. However, a complete matching was achieved at punch travels of 19 mm and 30 mm.

11. Conclusions

The deep drawing force and value of the dimensionless ratio of (r_d/t) were optimized successfully. The other geometrical and operating parameters were considered as constraints according to logical and practical limits. A carpet plot for the given value of best interfacial frictional combinations at different values of the sheet metal thickness was extracted, showing that the proper selection of equivalent conditions led to the minimum deep drawing force (F_d-min).

The optimization results showed that the minimizing of the deep drawing force was achieved satisfactorily. However, by using the friction values quoted in the carpet plot at the same geometrical parameters of the recommended values from the literature, the present work shows that the deep drawing force (F_d) was decreased by applying the values of the coefficients of friction (μ_d and μ_h) extracted from the carpet plot. In addition, wrinkling was reduced by using the values extracted from the carpet plot.