Analysis of incremental sheet metal forming using the upper-bound approach

Abstract

The incremental sheet metal forming is a flexible process producing various parts with no need for dedicated dies. In this article, the upper-bound approach is used to study the deformation zone of single point incremental forming of truncated cones. The velocity field and the dissipated power of the process are achieved using an assumed deformation zone and streamlines defined by Bezier curves. The tangential force acting on the tool is attained by optimizing the presented upper-bound solution. Then, influences of the effective parameters including vertical pitch, initial thickness, tool diameter, and wall angle on the tangential force and the predicted equivalent strain are investigated. A strain hardening law is utilized to consider the work hardening behavior of sheet metals. In order to validate the presented upper-bound solution, predicted tangential forces are compared with experimental data reported in literature for forming AA3003 truncated cones. The comparison shows an appropriate agreement between experimental and predicted tangential forces.

Introduction

The utilization of traditional sheet metal forming processes, such as deep drawing for manufacturing parts, consumes time and money to prepare dedicated tools. In some cases, manufacturing a part with the least possible time and cost is necessary. In these situations, processes with simple tools and high performances are the point of interest. The incremental sheet metal forming (ISMF) process can be introduced as an appropriate choice to achieve these aims. In this process, in the simplest form, a hemispherical head tool is used instead of a milling cutter on a computer numerically controlled (CNC) milling machine. The periphery of a blank is fully fixed, then the hemispherical head tool incrementally forms the blank along a predetermined path generated using computer-aided manufacturing (CAM) software according to the desired shape (Figure 1). In this way, it is possible to form various shapes without dedicated dies for purposes of rapid prototyping. The ISMF process has some alternatives, among them the process described above called single point incremental forming (SPIF) depicted in Figure 1. An overview of incremental sheet forming patents and their technological developments can be found in Emmens et al. ¹

Figure 1.

Typical presentation of single point incremental forming a truncated cone.

Besides the mentioned advantage, the formability of sheet metals can be improved using ISMF in comparison with traditional forming processes. The improvement of formability in ISMF has been proved by several experimental works. Shim and Park ² studied the formability of aluminum 1050 sheet in ISMF. They indicated that forming limit curves (FLC) in ISMF has a distinct difference with traditional forming processes. The FLC is a straight line with a negative slope in the positive region of the minor strain. It is illustrated for different shapes that FLC in ISMF is higher than in traditional forming processes. For this reason, several researches on mechanisms of the formability improvement in ISMF are conducted. Ham and Jeswiet ³ studied the effect of the critical process parameters, including material type, material thickness, tool diameter, vertical pitch, and formed shape on forming limits using a design of experiment. Silva et al. ⁴ presented an analytical model based on membrane analysis for SPIF considering different deformation modes. They also concluded that neck formation is suppressed in SPIF, then fracture forming limit diagrams should be considered instead of traditional forming limit diagrams (FLDs). Through thickness shear in ISMF was experimentally investigated by Jackson and Allwood. ⁵ They studied the mechanism of deformation using copper sheets. Results showed the existence of shears in planes parallel and perpendicular to the tool direction, and mentioned that these shears are the main difference between the deformation mechanisms of ISMF and press working of sheet metal. Allwood and Shouler ⁶ and Eyckens et al. ⁷ modified the Marciniak–Kuczynski (M–K) model to consider the through thickness shear, and concluded that the through thickness shear can lead to the formability improvement. There are other mechanisms mentioned in literature for explanation of higher forming limits in ISMF. Several mechanisms that may enable stable deformation above the FLC in incremental sheet forming including shear, contact stress, bending under tension, cyclic straining, restriction of neck growth, and hydrostatic stress, have been discussed by Emmens and van den Boogaard. ⁸

One of the limitations of ISMF is the obtainable geometrical accuracy. Micari et al. ⁹ studied various sources resulting to errors and effective parameters on the dimensional accuracy. They concluded that the optimized tool path is an effective approach to reduce dimensional errors. Bambach et al. ¹⁰ formed a pyramidal part using multi-stage SPIF. It was shown that multi-stage forming can yield an increased accuracy compared with single-stage forming. They showed that a combination of multi-stage forming and stress-relief annealing before trimming the part can improve the geometric accuracy of the final part.

Since in ISMF the blank is incrementally formed to its final shape, at any moment there is a small deformation zone in the vicinity of the forming tool leading to lower forming forces than one in traditional forming processes, such as deep drawing. Also, unlike traditional forming processes, the size of the desired part does not effect the forming forces. However, prediction of forming forces is essential to design tools and to determine the necessary power of the machine. Iseki ¹¹ obtained the forming forces for the incremental forming of the pyramid using a simply approximate deformation analysis. The plane strain deformation was assumed and this assumption was verified using a finite element method (FEM). Experimental results on aluminum sheet were in good agreement with those obtained by the approximate analysis and FEM. Filice et al. ¹² utilized the gradient of tangential forming force to control the online ISMF for manufacturing AA1050-O truncated cones. Using statistical analysis they obtained a critical value for the gradient of the tangential force. Then they indicated that it is possible to control the online gradient of tangential force to be lower than the critical value by changing process parameters for preventing the fracture. Aerens et al. ¹³ studied the incremental forming of truncated cones of AA3003, AA5754, DC01, AISI304, and 65Cr2 using experiments and statistical analyses. They obtained regression formulae to predict the triple forming force components including axial, radial, and tangential components. These formulae are functions of wall angle, initial thickness, tool diameter, and vertical pitch. Finally, an approximate formula was deduced for predicting the axial component for forming any material based on the tensile strength only.

Unlike the above mentioned work for predicting the forming forces acting on the tool, in this article the tangential force required for SPIF of truncated cones is predicted using an assumed deformation zone and the upper-bound approach. Then the predicted deformation zone is presented. Influences of the most effective parameters, including vertical pitch, initial thickness, tool diameter, and wall angle on the tangential force and the equivalent strain are investigated. In this research, the upper-bound approach is introduced as an appropriate tool to predict the forming forces and the deformation zones in ISMF. In the presented upper-bound solution, the material behavior is assumed to be isotropic, while different material behaviors can influence the predicted tangential force. Generally, for simplicity, the Von Mises yield criterion is used in the upper-bound approach while using the FEM, one can consider other material behaviors easily (e.g. anisotropy, kinematic strain hardening, etc.). In this way, Flores et al. ¹⁴ and Henrard et al. ¹⁵ considered effects of different constitutive laws and yield conditions on predicted forces in SPIF of a cone, including tangential forces using finite element simulations. Henrard et al. ¹⁵ concluded that a proper accuracy can be achieved using a material model that combines the isotropic yield criterion of Von Mises with the mixed isotropic–kinematic hardening model of Voce–Ziegler.

Upper-bound analysis of ISMF

Figure 1 shows a scheme of SPIF for manufacturing a truncated cone, depicting the effective parameters. Here, the tool path is assumed to be circular depending on the amount of ΔZ at the end of each path. The forming forces acting on the tool are in axial, radial, and tangential directions. In this research, the deformation zone is first introduced. Then, the velocity field of the deformation zone, the strain rates, and the dissipated powers in the process are calculated.

The deformation zone of ISMF

Figure 2 shows the deformation zone assumed in this research in the different views. The forming tool has a hemispherical head and the tool center is in r_i off the centerline of the cone (Figure 2(a)). Figure 2(a) shows ISMF in two consecutive passes. If the feed rate of the tool center is equal to F, the sheet metal approaches the tool in the angular velocity of $\text{F}/{\text{r}}_{\text{i}}$ . Three regions are depicted in Figure 2(b). Regions I and III are assumed rigid, so that region I was formed in the previous pass and the region III will be formed in the following pass. Thus, region II is formed when the sheet metal approaches the tool.

Figure 2.

Deformation zone of incremental sheet forming: (a) r–z view, (b) r–Θ view, and (c) deformation of a streamline at rΘ–z view.

In the upper-bound analysis of shear spinning of cones proposed by Kim et al., ¹⁶ it was assumed that the sheet metal is first bent and then starts to make contact with the tool. Similarly, here the deformation zone is assumed to be like Figure 2(c). According to Figure 2(b), when an element in zone II on a streamline with the radius r approaches the tool, after passing angle ${\theta }_{\text{i}}$ (entry surface s_i) its height decreases until it reaches the tool at angle ${\theta }_{\text{c}}$ (contact surface s_c). At angle ${\theta }_{\text{c}}$ the element starts to contact with the tool and then its height decreases based on the tool head until it reaches angle zero (exit surface s_o). The streamline is assumed to be at the center of the sheet thickness. Thus, the variation of the streamline through the thickness vanishes. In this way, s_i, s_o, and s_c are expressing the start of the deforming zone, the end of deforming zone, and the start of contact with the tool, respectively. Accordingly, the radius of a streamline is unchanged in the deformation zone and there is motion only in the vertical and the circumferential directions. The angle ${\theta }_{\text{r}}$ can be calculated geometrically based on the height of the sheet before entering the deforming zone. Figure 2(c) shows the above mentioned contents for a streamline with radius r.

According to Figures 1 and 2, the deformation zone is surrounded by r_i, r_o, s_i, and s_o, which are the interior radius, the exterior radius, the entry surface, and the exit surface of the deformation zone, respectively. The shaded area and the blank area of the deformation zone depicted by Figure 2(b) are the contact and the non-contact regions of the sheet with the tool, respectively. The entry surface s_i is assumed to be a circle with the radius (r_o–r_i) at the center of the tool head in the r–Θ plane. Since there are no thickness changes in the z direction, for simplicity “curves” can be used instead of “surfaces”.

The vertical component of s_c (i.e. the contact position of the sheet with the tool) is unknown and is given as equation (1)

$${\text{z}}_{\text{c}}=\text{m}({\text{z}}_{\text{i}}-{\text{z}}_{\text{o}})+{\text{z}}_{\text{o}}0\le \text{m}\le 1$$

(1)

Where m is an unknown parameter obtainable using the optimization of the upper-bound solution. Accordingly z_c is limited to values between the entry height z_i and the exit height z_o of the deformation zone.

According to Figure 2(c), the height variation of the streamline from the entry angle ${\theta }_{\text{i}}$ to the contact angle ${\theta }_{\text{c}}$ (i.e. non-contact region) is unknown and can be predicted by optimizing the upper-bound solution. The flow of the sheet from the contact angle ${\theta }_{\text{c}}$ to the angle zero (i.e. contact region) is based on the geometry of the tool head. Accordingly, the vertical component of a streamline at radius r in the deformation zone is given as

$${\text{z}}_{\text{d}}=\text{f}(\text{r},\theta )=\text{n}(\text{r},\theta )\text{g}(\text{r},\theta )$$

(2)

$$\text{n}(\text{r},\theta )=\{\begin{array}{cc}\hfill 1\hfill & \hfill 0\le \theta \le {\theta }_{\text{c}}\hfill \\ \multicolumn{1}{c}{0\le n(r,\theta )\le 1}& \hfill {\theta }_{\text{c}}\le \theta \le {\theta }_{\text{i}}\hfill \end{array}$$

(3)

$$\text{g}(\text{r},\theta )=\{\begin{array}{cc}\hfill {\text{z}}_{\text{T}}\hfill & \hfill \theta \le {\theta }_{\text{r}}\hfill \\ \multicolumn{1}{c}{{\text{z}}_{\text{i}}}& \hfill \theta \ge {\theta }_{\text{r}}\hfill \end{array}$$

(4)

where the vertical component of the tool head z_T is obtained geometrically. The value of function $\text{n}(\text{r},\theta )$ is between 0 and 1. For simplicity $\text{n}(\text{r},\theta )$ is defined in a way that there is no velocity discontinuity on s_i and s_c. Thus, $\text{n}(\text{r},\theta )$ should satisfy the boundary conditions as

$$\{\begin{array}{cc}\hfill \frac{\partial \text{n}}{\partial \theta }=0\hfill & \hfill \theta ={\theta }_{\text{i}},{\theta }_{\text{c}}\hfill \\ \multicolumn{1}{c}{n=1}& \hfill \theta ={\theta }_{\text{i}},{\theta }_{\text{c}}\hfill \end{array}$$

(5)

According to the research of Abrinia and Mirnia ¹⁷ for the streamlines in the equal channel angular extrusion (ECAE) process, Bezier curves are utilized here, since they can define easily using their control points and their characteristic polygons. Accordingly, $\text{n}(\text{r},\theta )$ is defined as a second order Bezier curve that satisfies equation (5) as

$${\theta }_{\text{c}}\le \theta \le {\theta }_{\text{r}}:\{\begin{array}{c}\hfill \text{n}(\text{r},\text{t})={(1-\text{t})}^2+2\text{t}(1-\text{t})+{\text{t}}^{2}N\prime \hfill \\ \multicolumn{1}{c}{\theta (\text{r},\text{t})={(1-\text{t})}^2{\theta }_{\text{c}}+2\text{t}(1-\text{t})(\frac{{\theta }_{\text{c}}+{\theta }_{\text{r}}}{2})+{\text{t}}^2{\theta }_{\text{r}}}\end{array}$$

$${\theta }_{\text{r}}\le \theta \le {\theta }_{\text{i}}:\{\begin{array}{c}\hfill \text{n}(\text{r},\text{t})={(1-\text{t})}^{2}N\prime +2\text{t}(1-\text{t})+{\text{t}}^2\hfill \\ \multicolumn{1}{c}{\theta (\text{r},\text{t})={(1-\text{t})}^2{\theta }_{\text{r}}+2\text{t}(1-\text{t})(\frac{{\theta }_{\text{r}}+{\theta }_{\text{i}}}{2})+{\text{t}}^2{\theta }_{\text{i}}}\end{array}$$

(6)

where $0\le \text{t}\le 1$ , and ${\text{N}}^{\prime }$ is given as

$$\text{N}\prime =\frac{(\text{r}-{\text{r}}_{\text{i}})(\text{N}-1)}{{\text{r}}_{\text{i}}-{\text{r}}_{\text{o}}}+\text{N}$$

(7)

where

$$0\le \text{N}\le 1$$

(8)

In equation (7), N is the other unknown parameter obtainable using the optimization of the upper-bound solution. Details for computing $\text{n}(\text{r},\theta )$ are given in the Appendix.

Accordingly, the shape of a streamline in the deformation zone (see Figure 2(c)) is a function of m and N obtained by optimizing the upper-bound solution. In the present analysis the effect of friction is neglected owing to a small contact zone. Now, using the definition of the deformation zone and the streamlines, the velocity field can be computed, as described in the following section.

Velocity field of the deformation zone

Since there is no flow in the radial direction for a streamline at radius r, it flows only in the vertical and the circumferential directions, thus

$${\stackrel{·}{u}}_r=0$$

(9)

If the feed rate of the tool head center is equal to F, the sheet metal rotates at angular velocity $\text{F}/{\text{r}}_{\text{i}}$ round the centerline of the cone, thus

$${\stackrel{·}{u}}_{\theta }=\text{r}\frac{\text{F}}{{\text{r}}_{\text{i}}}$$

(10)

For computing the vertical component of the velocity using equation (2), the following equation is obtained

$$\text{df}=\frac{\partial \text{f}}{\partial \text{r}}\text{dr}+\frac{\partial \text{f}}{\partial \theta }\text{d}\theta \stackrel{÷\text{dt}}{\Rightarrow }{\stackrel{·}{u}}_{\text{z}}=\frac{\partial \text{f}}{\partial \text{r}}{\stackrel{·}{u}}_{\text{r}}+\frac{\partial \text{f}}{\partial \theta }\stackrel{·}{\theta }$$

(11)

Therefore, by substituting equation (9) into equation (11) and as regards $\stackrel{·}{\theta }=\text{F}/{\text{r}}_{\text{i}}$ the vertical component of the velocity is obtained

$${\stackrel{·}{u}}_z=\frac{\text{F}}{{\text{r}}_{\text{i}}}\frac{\partial \text{f}}{\partial \theta }$$

(12)

Strain rates of the deformation zone

The strain rates in the cylindrical polar coordinates are defined as

$${\stackrel{·}{\epsilon }}_{\mathrm{rr}}=\frac{\partial {\stackrel{·}{\text{u}}}_{\text{r}}}{\partial \text{r}}{\stackrel{·}{\epsilon }}_{\theta \theta }=\frac{{\stackrel{·}{u}}_r}{\text{r}}+\frac{1}{\text{r}}\frac{\partial {\stackrel{·}{\text{u}}}_{\theta }}{\partial \theta }{\stackrel{·}{\epsilon }}_{\mathrm{zz}}=\frac{\partial {\stackrel{·}{\text{u}}}_z}{\partial \text{z}}$$

$${\stackrel{·}{\epsilon }}_{r\theta }=\frac{1}{2}(\frac{1}{\text{r}}\frac{\partial {\stackrel{·}{\text{u}}}_{\text{r}}}{\partial \theta }+\frac{\partial {\stackrel{·}{\text{u}}}_{\theta }}{\partial \text{r}}-\frac{{\stackrel{·}{u}}_{\theta }}{\text{r}}){\stackrel{·}{\epsilon }}_{\theta z}=\frac{1}{2}(\frac{\partial {\stackrel{·}{\text{u}}}_{\theta }}{\partial \text{z}}+\frac{1}{\text{r}}\frac{\partial {\stackrel{·}{\text{u}}}_z}{\partial \theta }){\stackrel{·}{\epsilon }}_{\mathrm{rz}}=\frac{1}{2}(\frac{\partial {\stackrel{·}{\text{u}}}_{\text{r}}}{\partial \text{z}}+\frac{\partial {\stackrel{·}{\text{u}}}_z}{\partial \text{r}})$$

(13)

By substituting the velocity field into equation (13), the strain rates are obtained as

$${\stackrel{·}{\epsilon }}_{\theta z}=\frac{\text{F}}{2\text{r}{\text{r}}_{\text{i}}}\frac{{\partial }^2\text{f}}{\partial {\theta }^2}$$

(14)

$${\stackrel{·}{\epsilon }}_{\text{rz}}=\frac{\text{F}}{2{\text{r}}_{\text{i}}}\frac{{\partial }^2\text{f}}{\partial \theta \partial \text{r}}$$

(15)

$${\stackrel{·}{\epsilon }}_{\text{ij}}=0\left(\text{for}\text{other}\text{strain}\text{rates}\right)$$

(16)

Power dissipated in ISMF

The dissipated power during ISMF computed using the strain rates can be expressed as

$$\stackrel{·}{\text{w}}={\stackrel{·}{\text{w}}}_{\text{d}}+{\stackrel{·}{\text{w}}}_{\text{sr}}$$

(17)

where ${\stackrel{·}{\text{w}}}_{\text{d}}$ and ${\stackrel{·}{\text{w}}}_{\text{sr}}$ are the dissipated powers in the deformation zone and on the velocity discontinuity surface, respectively. ${\stackrel{·}{\text{w}}}_{\text{d}}$ is computed as

$${\stackrel{.}{\text{w}}}_{\text{d}}=\frac{2{\text{y}}_0}{\sqrt{3}}\int \int \int {\left(\frac{1}{2}{\stackrel{.}{\epsilon }}_{\text{ij}}{\stackrel{.}{\epsilon }}_{\text{ij}}\right)}^{\frac{1}{2}}\text{dv}$$

(18)

$${\stackrel{·}{\text{w}}}_{\text{d}}=\frac{2{\text{y}}_0{\text{t}}_0}{\sqrt{3}}\iint {\left({{{\stackrel{·}{\epsilon }}_{\theta \text{z}}}^2+{\stackrel{·}{\epsilon }}_{\text{rz}}}^2\right)}^{\frac{1}{2}}\text{ds}$$

(19)

For computing the above equation, the deformation zone is divided to six sub-regions, as shown in Figure 3 owing to the discontinuity of equation (2).

Figure 3.

Divided deformation zone.

Regarding the assumed deformation zone, surface s_r is the only discontinuity surface as shown in Figure 4. Therefore, ${\stackrel{·}{\text{w}}}_{\text{sr}}$ is computed as

Figure 4.

Velocity discontinuity on shear surface s_r in thickness direction ( $v=\stackrel{·}{u}$ ).

$${\stackrel{·}{\text{w}}}_{\text{sr}}=\underset{{\text{s}}_{\text{r}}}{\iint }\text{k}\left|\Delta \text{v}\right|\text{ds}$$

(20)

Then by rearranging

$${\stackrel{·}{\text{w}}}_{\text{sr}}=\text{k}{\text{t}}_0\int \left|\Delta \text{v}\right|\text{d}{\text{l}}_{\text{sr}}$$

(21)

The velocity discontinuity on the surface s_r in the vertical direction is equal to

$$\Delta \text{v}=\Delta {\stackrel{·}{u}}_{\text{z}}=\frac{\text{F}}{{\text{r}}_{\text{i}}}{\frac{\partial \text{f}}{\partial \theta }}_{\theta ={\theta }_{\text{r}}^{+}}-\frac{\text{F}}{{\text{r}}_{\text{i}}}{\frac{\partial \text{f}}{\partial \theta }}_{\theta ={\theta }_{\text{r}}^{-}}\Rightarrow$$

$$\Delta \text{v}=\frac{\text{F}}{{\text{r}}_{\text{i}}}(\frac{2(1-{\text{N}}^{\prime })}{{\theta }_{\text{i}}-{\theta }_{\text{r}}}{\text{z}}_{\text{i}}-\frac{2({\text{N}}^{\prime }-1)}{{\theta }_{\text{r}}-{\theta }_{\text{C}}}{\text{Z}}_{\text{T}}-\text{N}\prime {\frac{\partial {\text{z}}_{\text{T}}}{\partial \theta }}_{\theta ={\theta }_{\text{r}}})$$

(22)

The length element is given as

$$\text{d}{\text{l}}_{\text{sr}}=\sqrt{1+{(\text{r}\frac{\text{d}{\theta }_{\text{r}}}{\text{dr}})}^2}\text{dr}$$

(23)

In the above equations the shear strength k is equal to

$$\text{k}=\frac{{\text{y}}_0}{\sqrt{3}}$$

(24)

Equivalent strain

The equivalent strain induced in an element on a streamline is computed using the strain rates. Then an average equivalent strain is introduced. Finally, using this average equivalent strain, the average yield strength is obtained for work hardening materials.

The equivalent strain is given as

$$\overline{\epsilon }={\overline{\epsilon }}_{\text{s}}+{\overline{\epsilon }}_{\text{d}}$$

(25)

where ${\overline{\epsilon }}_{\text{d}}$ is the equivalent strain induced in the deformation zone and ${\overline{\epsilon }}_{\text{s}}$ is the equivalent strain induced on the discontinuity surface s_r.

${\overline{\epsilon }}_{\text{d}}$ can be computed as

$${\overline{\epsilon }}_{\text{d}}=\int \overline{\stackrel{·}{\epsilon }}\text{dT}$$

(26)

where dT is the time element. The length element of the streamline in (r,Θ) coordinates

$$\text{ds}=\text{rd}\theta ={\stackrel{·}{u}}_{\theta }\text{dT}$$

(27)

therefore, dT is given by

$$\text{dT}=\frac{{\text{r}}_{\text{i}}\text{d}\theta }{\text{F}}$$

(28)

Also, the equivalent strain rate is given by

$$\overline{\stackrel{·}{\stackrel{·}{\epsilon }}}=\frac{2}{\sqrt{3}}{\left(\frac{1}{2}{\stackrel{·}{\epsilon }}_{\text{ij}}{\stackrel{·}{\epsilon }}_{\text{ij}}\right)}^{\frac{1}{2}}=\frac{2}{\sqrt{3}}{({{\stackrel{·}{\epsilon }}_{\theta \text{z}}}^2+{{\stackrel{·}{\epsilon }}_{\text{rz}}}^2)}^{\frac{1}{2}}$$

(29)

On the surface s_r the shear strain ${\text{γ}}_{\theta \text{z}}$ is induced in sheet metal, which is obtained by dividing the velocity discontinuity by the velocity component perpendicular to s_r (see Figure 4)

$${\text{γ}}_{\theta \text{z}}=\frac{\left|\Delta \text{v}\right|}{{\stackrel{·}{u}}_{\theta }}$$

(30)

The equivalent strain is given by

$$\overline{\text{ε}}=\frac{2}{\sqrt{3}}{\left(\frac{1}{2}{\text{ε}}_{\text{ij}}{\text{ε}}_{\text{ij}}\right)}^{\frac{1}{2}}$$

(31)

By substituting equation (30) into equation (31)

$${\overline{\epsilon }}_s=\frac{2}{\sqrt{3}}{\text{ε}}_{\theta \text{z}}=\frac{1}{\sqrt{3}}{\text{γ}}_{\theta \text{z}}$$

(32)

Equation (25) is expressing the equivalent strain for a streamline at radius r. Hence, the average equivalent strain is given as

$${\overline{\text{ε}}}_{\text{avg}}=\frac{{\int }_{{\text{r}}_{\text{i}}}^{{\text{r}}_{\text{o}}}\overline{\text{ε}}\text{dr}}{{\text{r}}_{\text{o}}-{\text{r}}_{\text{i}}}$$

(33)

Equation (33) is expressing the average equivalent strain induced in sheet metal in a pass (i.e. a complete circle path).

By assuming the swift type work hardening law ${\text{σ}}_{\text{eq}}=\text{K}({\text{ε}}_0+\text{ε}{)}^{\text{n}}$ for a sheet metal, the average yield strength is obtained as

$${\text{y}}_0=\frac{{\int }_0^{{\overline{\text{ε}}}_{\text{avg}}}{\text{σ}}_{\text{eq}}\text{d}\epsilon }{{\int }_0^{{\overline{\text{ε}}}_{\text{avg}}}\text{d}\epsilon }$$

(34)

where $\text{ε}$ is the plastic strain. Also, K, n, and ${\text{ε}}_0$ are the material parameters.

Optimization of the upper-bound solution

Equation (17) expresses the dissipated power in ISMF. This expression is the function of two optimizing parameters m and N. By varying these parameters in their defined range (i.e. $0\le \text{m},\text{N}\le 1$ ), the minimum upper-bound solution closer to the experiment is obtained. The following equation can be written for dissipated power

$$\stackrel{·}{\text{w}}={\text{F}}_{\text{t}}{\stackrel{·}{\text{u}}}_{\theta }$$

(35)

Then it is assumed that the tangential force is acting at radius $\overline{\text{r}}=({\text{r}}_{\text{i}}+{\text{r}}_{\text{o}})/2$ on the tool head. Accordingly, the tangential force is expressed as

$${\text{F}}_{\text{t}}=\frac{{\text{r}}_{\text{i}}\stackrel{·}{\text{w}}}{\overline{\text{r}}\text{F}}$$

(36)

Results and discussion

The tangential force necessary to form a truncated cone is computed using the optimization of the upper-bound solution proposed in the previous section. Then the shape of the deformation zone is obtained by two optimizing parameters m and N. Regarding to the upper-bound solution, the effective parameters on the tangential force are vertical pitch $\Delta \text{z}(=\Delta \text{cos}\alpha )$ , initial thickness ${\text{t}}_0$ , tool diameter ${\text{r}}_{\text{t}}$ , and wall angle $\text{β}(=(\pi /2)-\alpha )$ .

It is assumed that a cone of AA3003 sheet is formed using SPIF. The above mentioned parameters are assumed as Table 1. Also, the swift work hardening law obtained by Aerens et al., ¹³ and the tangential force predicted by optimizing the upper-bound solution, are shown in Table 1. Now, the shape of the deformation zone is obtained using two optimizing parameters m and N as shown in Figure 5.

Table 1.

Assumed parameters, optimizing parameters, and predicted tangential force for incremental forming of a cone of AA3003.

β°	dt (mm)	Δz (mm)	t (mm)	K (MPa)	${\epsilon }_0$	n	m	N	Ft (N)
50	10	0.5	1.2	180.39	$0.00068$	0.218	0.1	0.7	113.88

Figure 5.

Deformation zone predicted using upper-bound for incremental forming of AA3003. Process parameters are β = 50°, d_t = 10 mm, Δz = 0.5 mm, and t₀ = 1.2 mm.

The six sub-regions shown in Figure 5 are the same as shown in Figure 3. Figure 6 shows the deformation zone depicted in Figure 5, in three views. It is notable to comprise Figure 6 with Figure 2. Each line in Figures 5 and 6 is a streamline. Hereafter the value of the four effective parameters are the same as the one mentioned previously, only if variations are mentioned.

Figure 6.

Deformation zone relevant to Figure 5 in three different views. Each curve is a streamline.

Parameters study on the tangential force

Figure 7 shows the variations of the tangential force with the vertical pitch predicted by the upper-bound solution and compared with the experimental values reported by Aerens et al. ¹³ For all vertical pitches, the theoretical values are more than the experimental values. The more the vertical pitch, the larger the deformation zone. Hence, the tangential force reaches higher values. By increasing $\Delta \text{z}$ , the difference between the theoretical and experimental values increases owing to more severe deformations predicted by the upper-bound than experiments with a higher $\Delta \text{z}$ . In reality the plastic deformations are less than the ones predicted by the upper-bound that may be addressed to the assumption of considering a rigid material.

Figure 7.

Variations of the tangential force with the vertical pitch obtained by the presented upper-bound solution and experimental work of Aerens et al. ¹³ Process parameters are β = 50°, d_t = 10 mm, and t₀ = 1.2 mm.

Figure 8 shows the tangential forces predicted using upper-bound and those experimentally obtained by Aerens et al., ¹³ for various wall angles. According to Figure 8, the predicted forces show a minimum value for $\text{β}=35\circ$ . By increasing the wall angle from $\text{β}=35\circ$ , the deformation zone becomes larger and, subsequently, the deforming tangential force increases. An increase in the tangential force as increasing the wall angle from 60° was reported by Filice et al. ¹² Also, for wall angles lower than 35°, the tangential force increases owing to the increase in the contact zone between the sheet and the tool, while the deformation zone becomes smaller. The deformation zones for forming wall angles 20°, 40°, and, 60° are depicted in Figure 9. The lower regions decreasing owing to the increase in wall angle are contact zones. As depicted in Figure 9, by increasing the wall angle, the deformation zone becomes smoother and larger.

Figure 8.

Variations of tangential force with wall angle obtained by presented upper-bound solution and experimental work of Aerens et al. ¹³ Process parameters are d_t = 10 mm, Δz = 0.5 mm, and t₀ = 1.2 mm.

Figure 9.

Deformation zones predicted using upper-bound for wall angles 20°, 40°, and 60°.

According to Figures 7 and 8, the upper-bound solution based on the assumed deformation zone is in proper agreement with the experiment reported by Aerens et al. ¹³ Hence, influences of the effective parameters on the tangential force and the equivalent strain can be evaluated using the upper-bound solution.

Variations of the tangential force with the forming tool diameter for various vertical pitches are shown in Figure 10. These curves show a minimum. For a fixed vertical pitch, by increasing the forming tool diameter, the deformation zone becomes larger and results in higher forces, while by decreasing the forming tool diameter from one showing the minimum, the contact zone between the tool and the sheet becomes larger and results in higher forces owing to severe shear strains. Figure 11 shows that a small tool diameter compared with a vertical pitch results in a severe shear deformation. By increasing the tool diameter, the deformations become moderate and lower forces are achieved. But the larger tool diameter results in a wider deformation zone and subsequently the tangential force reaches higher values.

Figure 10.

Variations of tangential force with the forming tool diameter for various vertical pitches.

Figure 11.

Comparison of the deformation zone predicted using the upper-bound solution for two tool diameters.

Using Figure 10 it can be concluded that the tangential force always remains at a minimum for a constant parameter named R and defined by $\text{R}={\text{d}}_{\text{t}}/\Delta \text{z}$ . Figure 12 shows that for R = 20 the tangential force is minimum. The higher tangential forces are reached linearly by increasing the vertical pitch at a constant R owing to wider deformation zones. Hence, knowing the optimum R helps to obtain a minimum tangential force.

Figure 12.

Variations of tangential force with vertical pitch for various $\text{R}={\text{d}}_{\text{t}}/\Delta \text{z}$ .

As shown in Figure 13, the tangential force increases linearly with an increase in initial blank thickness since the material volume under deformation becomes larger.

Figure 13.

Variations of tangential force with initial blank thickness.

Parameters study on the equivalent strain

Using equation (25) the equivalent strain distribution can be obtained. Influences of the effective parameters on the average equivalent strain induced in one pass ${\overline{\epsilon }}_{\text{avg}}$ can be investigated using equation (33). Inasmuch as the wall of a part is formed in successive passes, the equivalent strain of a point on the wall ${\overline{\epsilon }}_{\text{total}}$ is the sum of the equivalent strains induced in previous passes. Therefore, the number of passes necessary to form a point on the wall and the average equivalent strain induced in one pass ${\overline{\epsilon }}_{\text{avg}}$ are effective on ${\overline{\epsilon }}_{\text{total}}$ . Influences of the effective parameters on ${\overline{\epsilon }}_{\text{total}}$ are investigated. The values of ${\overline{\epsilon }}_{\text{total}}$ are calculated for point A (r = r_i). Figure 14 shows the formation of point A in successive passes. The number of passes to form point A can be calculated using r_o-r_i divided by Δr.

Figure 14.

A number of sequences necessary to form point A placed at r_i from the centerline.

Variations of ${\overline{\epsilon }}_{\text{avg}}$ and ${\overline{\epsilon }}_{\text{total}}$ at point A with the wall angle are shown in Figure 15. By increasing the wall angle, ${\overline{\epsilon }}_{\text{avg}}$ decreases since the contact zone becomes smaller and the smoother deformation zone is formed (see Figure 9). According to Figure 16, passes necessary for point A to form increase by increasing the wall angle and higher ${\overline{\epsilon }}_{\text{total}}$ is obtained. It is noteworthy that values of ${\overline{\epsilon }}_{\text{total}}$ predicted using the upper-bound is larger than the one reported in literature owing to accumulated errors for predicting ${\overline{\epsilon }}_{\text{avg}}$ , but here the trends of ${\overline{\epsilon }}_{\text{total}}$ and ${\overline{\epsilon }}_{\text{avg}}$ are investigated.

Figure 15.

Variations of ${\overline{\epsilon }}_{\text{avg}}$ and ${\overline{\epsilon }}_{\text{total}}$ at point A with the wall angle.

Figure 16.

The increase of ${\overline{\epsilon }}_{\text{total}}$ by increasing the number of passes for various wall angles.

An increase in Δz, increases ${\overline{\epsilon }}_{\text{avg}}$ as well and reduces the forming passes of point A. Accordingly, a minimum can be obtained for ${\overline{\epsilon }}_{\text{total}}$ , depicted in Figure 17.

Figure 17.

Variations of ${\overline{\epsilon }}_{\text{avg}}$ and ${\overline{\epsilon }}_{\text{total}}$ at point A with vertical pitch.

By increasing the forming tool diameter, ${\overline{\epsilon }}_{\text{avg}}$ decreases as shown in Figure 18 and passes necessary to form point A increase. In Figure 19 a minimum is shown for ${\overline{\epsilon }}_{\text{total}}$ at each Δz. By a tool diameter, at smaller Δz, the forming passes are more and subsequently ${\overline{\epsilon }}_{\text{total}}$ is large (see Figure 19 for diameters more than 10 mm). For diameters smaller than 10 mm, larger ${\overline{\epsilon }}_{\text{avg}}$ overcomes fewer forming passes. Hence, ${\overline{\epsilon }}_{\text{total}}$ increases at larger Δz.

Figure 18.

Variations of ${\overline{\epsilon }}_{\text{avg}}$ with tool diameter for different vertical pitches.

Figure 19.

Variations of ${\overline{\epsilon }}_{\text{total}}$ with tool diameter for different vertical pitches.

It can be concluded from Figure 20 that a simultaneous increase in the vertical pitch and the tool diameter (i.e. at constant R) has no effect on ${\overline{\epsilon }}_{\text{total}}$ . Also, Figure 20 shows a minimum for ${\overline{\epsilon }}_{\text{total}}$ at a given R. This can be explained using the mutual interactions of forming passes and equivalent strains.

Figure 20.

Variations of ${\overline{\epsilon }}_{\text{total}}$ with vertical pitch for various $\text{R}={\text{d}}_{\text{t}}/\Delta \text{z}$ .

Conclusions

In the present article, for the first time, the SPIF of a truncated cone was studied using the upper-bound approach. In the presented upper-bound solution a deformation zone was introduced using assumed streamlines defined by Bezier curves. By optimizing the upper-bound solution, the deformation zone, the tangential force, and the equivalent strain was predicted for forming a truncated cone from a known material based on the predetermined values of the vertical pitch, the tool diameter, the sheet thickness, and the wall angle. The following was concluded.

• Based on the variations of tangential force with vertical pitch, a good agreement was found between the present upper-bound solution and the experimental results for vertical pitches lower than 0.6 mm (the mean difference is 9%). Also, according to the variations of the tangential force with the wall angle, a reasonably good agreement was found for wall angles lower than 60°, at which the mean difference is 10%.

• By increasing the vertical pitch, the tangential force increases as the larger deformation zone is formed. Also, it was shown that the more the thickness, the higher the tangential force.

• By increasing the wall angle from 20°, the tangential force decreases and then increases to form a larger deformation zone.

• The tangential force becomes smaller as the tool diameter increases from 4 mm and by further increase of it, the tangential force increases. Also, it was concluded that a ratio of the tool diameter to the vertical pitch (i.e. R) can be obtained to minimize the tangential force.

• For determining the total equivalent strain of a point on the wall, the number of forming passes necessary to form the point, and the equivalent strain induced in each pass, are to be considered simultaneously.

There are other tool paths where a sheet can be formed to a non-circular shape using SPIF. Such a tool path is seen in SPIF of a truncated pyramid in which a sheet is formed using straight paths in the z-plane except at corners. The model presented here can be applied to this kind of tool path easily by defining the deformation zone and streamlines in a Cartesian coordinate instead of the polar coordinate (equation (2)). Also, the sheet metal approaches the tool at a feed rate instead of an angular velocity.

In future work, using a modification of the presented upper-bound solution and the geometrical characteristics of the deformation zone, the vertical and the radial components of the forming force will be predicted and will be verified by experiments.