In cup drawing many different processes take place simultaneously. The punch pushes the sheet into the die (Fig. 11.13). The sheet under the punch pulls the remaining sheet in flange under the blank holder plate. As the punch advances, the sheet in flange is pulled over the die profile radius into cup wall.
In doing so the sheet in the flange suffers circumferential compression, bends under tension into die profile radius, slides over die profile radius and then it unbends into cup wall. Also the sheet on the punch profile radius bends under tension. All these processes add to tension in cup wall. The maximum stress thus occurs at the punch profile radius. The stress σw in the cup wall is due to the cumulative effect of following processes taking place during deep drawing.
(i) Drawing of flange under friction between the die and blank holding plate.
(ii) Bending of sheet at the die profile radius.
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(iii) Slipping of sheet over the die profile radius (die hole corner radius).
(iv) Unbending of sheet into the cup wall.
(v) Bending over punch profile radius (punch corner radius).
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Let us consider the stresses acting on a small element of flange contained between two radial lines inclined at dθ to each other and two circumferential curves at radius r and r + dr as shown in Fig. 11.14(a, b). Taking the equilibrium of forces in the radial direction we get the following equation. The frictional force due to pr is not shown in (Fig. 11.14).
During drawing of the flange the rim (outer edge) thickens more than the interior of flange, with the result that blank holding plate touches mainly on a small band on the outer periphery and consequently the blank holding force may be assumed to be acting only on the outer rim. This produces radial tensile stress in the sheet. This is considered separately.
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Therefore, for time being we remove the last term involving pr from the Eqn. (11.7). The equation is simplified by neglecting the terms having product of two or more differentials. And by substituting sin (dθ/2) ≈ dθ/2, the above equation can be simplified to following-
where C is the constant of integration. Its value may be determined by the condition at the edge of flange. At the start of the process the blank holding force is evenly distributed over the contact area. However, with drawing in of flange the thickness of outer rim becomes bigger compared to the remaining sheet.
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Thus the contact area of sheet and blank holding plate gets limited to a band at the outer rim. It is difficult to estimate the width of the band, therefore, it is assumed that the entire blank holding force Fb = pa × area is acting at the outer edge.
Stress Due to Bending at Die Profile Radius:
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Here we assume that the material remains perfectly plastic (does not work harden) during bending. As illustrated (Fig. 11.15) an element of width dx and unit length at position I goes to position 2 after bending. The strain at a distance y from the neutral section which is at radius rd + t/2 is proportional to y, the maximum strain occurs at y = t/2 and it is zero at y = 0.
where θ is the angle of bend over which the slip takes place. If ‘α’ is the angle that cup wall makes with the punch axis then θ = (π/2 – α).
The stress in sheet after sliding but before unbending into cup wall is equal to T2 which is given below-
where α is the angle between the punch axis and the cup wall, σw is the stress in cup wall during drawing, t is the thickness of cup wall and Rc. is the mean radius of the cup at the section where σw is determined. The failure generally occurs at the punch profile radius where there is maximum stress.
A sheet blank of 80 mm diameter and 1.2 mm thick is drawn into cup of 44 mm internal diameter. Die profile radius = 6 mm. Determine the blank holding force and stress in the sheet due to flange drawing only. The ultimate strength of material = 350 N/mm2 and yield strength as 280 N/mm2. The co-efficient of friction = 0.15.
Let us first determine the blank holding pressure. For this we use Eqn. (11.6a) which is repeated below-
