Study on Deformation Force of Hard Aluminum Alloy Incremental Forming

Abstract

The deformation force is an important factor affecting the forming accuracy of parts in the incremental forming process of sheet metal. This paper proposes an analytical calculation method of the deformation force based on pure shear deformation. After assuming and simplifying the factors affecting the deformation force, a graphical method is used to approximate the contact area between the forming tool and the sheet metal. A forming test is also designed. In addition, the deformation force is measured in the experiment, and its theoretical analysis value is compared with the actual measurement value of the forming test to validate the analytical method of deformation force calculation. The results show that the radial forming deviations are 28.5% and 22.5%, the axial deformation force deviations are 9.8% and 16.1%, and the forming force deviations are 6.3% and 10.3%, which demonstrates the effectiveness of using the analytical method to calculate the deformation force.

1. Introduction

The traditional stamping and forming process typically requires relatively expensive punch and concave dies. In addition, the die manufacturing cycle is long and expensive, and it is therefore difficult to quickly change the new customer needs. This new market demand can be met using digital incremental forming [1,2,3], which is a new sheet metal flexible rapid prototyping technology [4,5,6]. In contrast to the traditional stamping forming, this technology can produce sheet metal parts with large forming limits and complex shapes without the use of a special die, or by simply using model support, which makes it ideal for rapid prototyping and small batch production [7,8,9,10,11].

The incremental forming process is complicated, and the lack of precision after forming is the main issue impeding the advancement of the process. Studying the forming force is the most direct entry point into the study of the incremental forming process mechanism. It is useful for predicting the forming accuracy in the incremental forming process, and it provides a theoretical foundation and guidance for the trial production of special forming equipment. Consequently, if the deformation force calculation method can be performed through deformation zone mechanics analysis, researchers can predict the forming accuracy in advance, and then modify the forming parameters to improve the precision of the part [12,13,14].

Many researchers calculate the forming force primarily from the mechanism and after analyzing the stress and strain in the plate’s deformation area [15,16,17]. Some studies have used the finite element software to create a finite element model and simulate the forming force. Experiments have also been conducted in this field [18,19,20,21]. Using intelligent algorithms, empirical models were obtained, and forming forces for the incremental forming processes were predicted [22,23]. Silva et al. [24] used the membrane analysis method to calculate the forming force when processing conical and square cones. The basic principle of their study is to ignore the thickness of the sheet and approximate its deformation with a plane strain state. Chang et al. [25,26] proposed the membrane analysis method [24]. Improvements have been made on this basis, and the forming force prediction models of the single-point incremental forming and multi-pass forming have been developed, which greatly improved the prediction accuracy. After observing the shape of the contact area between the plate and the tool head, Aerens et al. [27] developed an empirical model for predicting the forming force. Li et al. [28] and Aerens et al. [27] established the finite element model of the incremental forming process to analyze the forming mechanism of the incremental forming process. Li et al. [28] proposed an efficient forming force prediction analysis model based on the finite element model, which simultaneously considered the influence of bending, shearing, and stretching on the forming force when processing square cones.

Although many studies have been conducted on the forming force of the incremental forming process, these studies have some limitations. In this paper, it is assumed that the stress state of the contact deformation zone during the incremental forming process of sheet metal is the superposition of pure shear stress and hydrostatic pressure, and thus the deformation of the material is a pure shear deformation. The analytical calculation method of the deformation force is deduced after using the graphic method to approximate the contact area between the forming tool and the metal sheet, based on the pure shear deformation. This is performed after making assumptions and simplifications on the factors affecting the deformation force in the forming process. A set of forming tests is then designed, and the deformation force is measured using a Kistler 9443B three-way piezoelectric force measuring instrument. Finally, the theoretical deformation force analysis value is compared with the actual measurement value of the forming test in order to validate the normal deformation force calculations.

2. Analysis of the Sheet Metal Incremental Forming Process

2.1. The Principle of Incremental Forming

Incremental forming consists of introducing the rapid prototyping technology known as “layered manufacturing”. In the incremental forming process (Figure 1), the sheet metal is fixed along the edge by a simple fixture, the forming tool is driven by a special NC device to press the sheet metal according to the pre-prepared NC program command, and the forming tool is then driven by the forming device to continuously move along the sheet metal surface according to the pre-prepared NC program. In the forming process, the forming tool locally contacts the sheet metal. A tiny area around the contact point is under high pressure due to its force, and local plastic deformation occurs. Thus, the continuous movement of the forming tool is accompanied by the continuous local plastic deformation of the sheet metal. The required part shape can be obtained by accumulating this continuous local plastic deformation [29].

2.2. Mechanical Analysis of the Deformation Zone

During the forming process, the forming tool starts to contact the metal sheet from point D and leaves the sheet at point C, and the sheet metal has contact deformation in the CD area, as shown in Figure 1. That is, under the action of the forming tool, the sheet metal has local contact plastic deformation, starting from point D and ending at point C. Therefore, the CD area can be considered as the deformation area in the forming process.

As a result of the local effect of the forming tool, the metal material in the deformation area undergoes plastic deformation under the action of radial and tangential tensile stresses. In other words, the material is in a state of two-way tensile stress, the billet is radially stretched, the thickness direction is thinner, and the tangential direction has not changed. Figure 2 shows the stress-strain relationship. The constitutive equation reveals the following relationships [30,31]:

$$\{\begin{array}{c}{\epsilon }_r=\frac{\overline{\epsilon }}{\overline{\sigma }}[{\sigma }_r-\frac{1}{2}({\sigma }_t+{\sigma }_{\theta })]\\ {\epsilon }_t=\frac{\overline{\epsilon }}{\overline{\sigma }}[{\sigma }_t-\frac{1}{2}({\sigma }_r+{\sigma }_{\theta })]\\ {\epsilon }_{\theta }=\frac{\overline{\epsilon }}{\overline{\sigma }}[{\sigma }_{\theta }-\frac{1}{2}({\sigma }_r+{\sigma }_t)]\end{array}$$

(1)

According to the yield criterion, there are:

$${\sigma }_r-{\sigma }_t=\beta {\sigma }_s$$

(2)

According to the constant volume law, it is:

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

(3)

As the stress value ${\sigma }_t$ in the thickness direction is much smaller than ${\sigma }_r$ and ${\sigma }_{\theta }$ during the contact deformation process, it can be approximated that it is null. In addition, during the forming process, there is no tangential deformation. That is, ${\epsilon }_{\theta }$ is null. If $k=1.1 \mathrm{a}\mathrm{n}\mathrm{d} {\sigma }_s={\sigma }_{sm}$ (the average deformation resistance of the studied instantaneous deformation area), Equations (1)–(3) can be simultaneously obtained:

$$\{\begin{array}{c}{\sigma }_r=1.1{\sigma }_{sm}\\ {\sigma }_{\theta }=0.55{\sigma }_{sm}\end{array}$$

(4)

3. Calculation of the Deformation Force

The deformation force is a crucial parameter that affects the forming quality in the sheet metal incremental forming process. The amount of deformation force is one of the criteria for selecting the forming equipment. It is also one of the parameters used for controlling the forming accuracy. The calculation of the deformation force is difficult and it depends on several parameters [16].

3.1. Material Deformation Analysis

According to the above analysis, the stress state at any position on the contact deformation zone can be obtained:

$${\sigma }_{ij}=\left[\begin{array}{ccc}1.1{\sigma }_{sm}& 0& 0\\ 0& 0.55{\sigma }_{sm}& 0\\ 0& 0& 0\end{array}\right]$$

(5)

The stress tensors of Equation (5) are rewritten into the form of the stress deviator tensor and stress sphere tensor as:

$${\sigma }_{ij}={\sigma }^{\text{'}}{}_{ij}+{\delta }_{ij}{\sigma }_m$$

(6)

where ${\sigma }_{ij}$ is the stress deviator and ${\sigma }_m$ is the Average stress

Therefore, the deviatoric stress tensor is expressed as:

$$\sigma {\text{'}}_{ij}=\left[\begin{array}{ccc}0.55{\sigma }_{sm}& 0& 0\\ 0& 0& 0\\ 0& 0& -0.55{\sigma }_{sm}\end{array}\right]$$

(7)

The above analysis shows that, during the forming process, the contact denatured zone can be considered as a hydrostatic pressure superimposed by deviatoric stress. During forming, the hydrostatic pressure can only affect the volume of the metal sheet, and not its shape (plastic deformation). The deviatoric stress can only change the shape (plastic deformation) of the metal sheet, but it cannot change its volume. Therefore, the plastic deformation of the material is caused by the deviatoric stress. It can be seen from Equation (7) that the deformation of the contact zone meets the condition of plane pure shear deformation. That is, the principal stresses in two opposite directions have the same size, and those in the other direction are null. Consequently, the material deformation in the contact zone during sheet metal incremental forming can be classified as pure shear deformation.

3.2. Tangential Deformation Force

As the material deformation in the process of sheet metal incremental forming conforms to the condition of plane pure shear deformation, it can be assumed that the material has only shear deformation along the axis of the forming tool during the deformation process. That is, the incremental forming process is assumed to be an ideal deformation diagram (Figure 3), and the material slides along the axis to the inclined plane with the forming angle.

According to this assumption, only the shear strain on the $R_Z$ plane occurs in the deformed material. It can be seen from Figure 3 that R is the distance from the center of the blank to the axis of rotation, and s is the unit having a thickness of $dR$ and sliding along the axis. Therefore, the shear strain is equal to the distance s of the axial slip of the element divided by its thickness $dR$:

$$\gamma =\frac{s}{dR}=\mathrm{tg}\alpha$$

(8)

where $\alpha$ is the forming angle.

The value of the tangential force $P_{\theta }$ can be derived according to the equilibrium condition of the work that is conducted and the deformation work. If the shear stress on the material is τ and the shear strain is γ, the plastic deformation work per unit volume of the material is expressed as:

$$w={\displaystyle \underset{0}{\overset{\gamma }{\int }}\tau \cdot d\gamma }={\displaystyle \underset{0}{\overset{tg\alpha }{\int }}\tau \cdot d\gamma }$$

(9)

Due to the deformation hardening of the material, τ will be a function of γ. In addition, because the τ-γ curve is more difficult to obtain than the σ-ε (-) stretching curve, Equation (9) can be rewritten as:

$$w={\displaystyle \underset{0}{\overset{\overline{\epsilon }}{\int }}\sigma \cdot d\overline{\epsilon }}$$

(10)

where $\sigma$ is the tensile stress of the material (N/mm).

According to the “deformation energy constant condition”, the relationship between γ and ε shows that the actual equivalent effect in the incremental forming process becomes:

$$\overline{\epsilon }=\frac{1}{\sqrt{3}}\gamma =\frac{1}{\sqrt{3}}\mathrm{tg}\alpha$$

(11)

Assuming that, during the forming process, the radius of the part at the forming position is R, the rotational speed of the forming tool is n, the feed of the tool along the axis of the workpiece is f, the initial thickness of the sheet is $t_0$, and the wall thickness of the part at the forming position is $t=t_0\cos \alpha$, the volume change rate of the material per unit time in incremental forming is then given by:

$$\begin{array}{l}\frac{\mathrm{d}V}{\mathrm{d}T}=2\pi n\cdot R\cdot t_0\frac{dR}{dT}\\ =2\cdot \pi n\cdot R\cdot t_0\cdot f\cdot \mathrm{c}tg\alpha \end{array}$$

(12)

Therefore, the machining power N, the corresponding torque $M_t,$ and the tangential force $P_{\theta }$ can be computed as:

$$N=w·\frac{dV}{dT}=2\pi ·t_0·f·n·R·{\displaystyle {\int }_0^{\overline{\epsilon }}\sigma }d\overline{\epsilon }$$

(13)

$$M_t=\frac{N}{2·\pi ·n}=t_0·f·R\mathrm{ctg}\alpha ·{\int }_0^{\overline{\epsilon }}\sigma d\overline{\epsilon }$$

(14)

$$Pt=\frac{Mt}{R}=t0·f·ctg\alpha ·{\displaystyle {\int }_0^{\overline{\epsilon }}\sigma }d\overline{\epsilon }$$

(15)

$$P_{\theta }=\frac{M_t}{R}=t_0\cdot f\cdot ctg\alpha \cdot {\displaystyle \underset{0}{\overset{\overline{\epsilon }}{\int }}\sigma \cdot d\overline{\epsilon }}$$

(16)

When the forming process parameters are determined, the values of N, $M_t,$ and $P_t$ can be obtained. However, as σ is a function of ε, the average stress can be used to simplify the calculation:

$$\overline{\sigma 0}=({\displaystyle {\int }_0^{\epsilon }\sigma }d\epsilon )/\overline{\epsilon }$$

(17)

This can be approximated by the arithmetic mean of ${\sigma }_{0.2}$ and ${\sigma }_n$ (the actual stress corresponding to the tensile actual stress-strain curve). By substituting Equation (11) into Equation (16), the tangential force $P_{\theta }$ can be calculated as:

$$P_{\theta }=t_0·f\mathrm{ctg}\alpha ·\overline{{\sigma }_0}·\overline{\epsilon }=\frac{1}{\sqrt{3}}·t_0·f\mathrm{ctg}\alpha ·\overline{{\sigma }_0}·\tan \alpha$$

(18)

3.3. Radial and Axial Deformation Forces

It is assumed that the average pressure p is the contact surface between the forming tool and the workpiece during the incremental forming process. The projected areas of the contact surface in the tangential, radial, and axial directions are $F_{\theta }$, $F_r,$ and $F_z,$ respectively. The components of the force in the three directions are then given by:

$$\{\begin{array}{c}{P}_{\theta }=p\cdot F_{\theta }\\ P_r=p\cdot F_r\\ P_z=p\cdot F_z\end{array}$$

(19)

Therefore, the radial forces $P_r$ and $P_z$ can be expressed as:

$$\{\begin{array}{c}{P}_r=P_t\frac{F_r}{F_{\theta }}\\ P_z=P_t\frac{F_z}{F_{\theta }}\end{array}$$

(20)

The projected areas $F_{\theta }$, $F_r,$ and $F_z$ in the three directions can be determined using the method of drawing and analysis.

The drawing method projects the contact area between the tool and the workpiece in three directions. It is challenging to accurately define these three projected figures’ outlines by a formula. According to the results of the drawing, the following approximate calculations can be made:

$$\{\begin{array}{c}a\approx r_r·\sin \alpha \\ b=R·\theta \\ c=f\\ d\approx r_r(1-\cos \alpha )\end{array}$$

(21)

where $r_r$ is the forming tool ball nose radius and $\theta$ is the angle of the deformation zone.

The included angle of the deformation zone is given by:

$$\theta =\frac{\pi }{180}\arccos \left[\frac{c-\sqrt{1+\frac{b^2}{R^2}\cdot \left(c-1\right)}}{\left(c-1\right)}\right]$$

(22)

a, b, c, and d have the following relationship:

$$\{\begin{array}{c}{a}^2=r_r^2-r_1^2\\ r_1=r_r-f\\ b^2=2H\cdot a+a^2\\ c=\frac{b^2}{a^2}\end{array}$$

(23)

where $H$ is the forming tool height.

The contour values of the three projection diagrams are then obtained, as shown in Figure 4. They are approximately processed according to the area of the triangle:

$$\{\begin{array}{c}{F}_{\theta }\approx \frac{1}{2}f·r_r·\sin \alpha \\ F_r\approx \frac{1}{2}f·r_r·(1-\cos \alpha )·R·\theta \\ F_z\approx \frac{1}{2}{r}_r·\sin \alpha ·R·\theta \end{array}$$

(24)

By substituting Equation (22) into Equation (20), the following can be obtained:

$$\{\begin{array}{c}{P}_r=P_{\theta }\cdot \frac{\left(1-\cos \alpha \right)\cdot R\cdot \theta }{f\cdot \sin \alpha }\\ P_z=P_{\theta }\cdot \frac{R\cdot \theta }{f}\end{array}$$

(25)

The tangential, radial, and axial forming forces, as well as the forming force, can be obtained from the above analysis:

$$\{\begin{array}{c}\begin{array}{l}{P}_{\theta }=\frac{1}{\sqrt{3}}{t}_0·f·\overline{{\sigma }_0}·\tan \alpha \\ P_r=P_{\theta }\cdot \frac{\left(1-\cos \alpha \right)\cdot R\cdot \theta }{f\cdot \sin \alpha }\end{array}\\ \begin{array}{l}{P}_z=P_{\theta }\cdot \frac{R\cdot \theta }{f}\\ P=\sqrt{P_{\theta }^2+P_r^2+P_z^2}\end{array}\end{array}$$

(26)

As $P_{\theta }$ << [$P_r,P_z$], the deformation force can be simplified as:

$$P=\sqrt{P_r^2+P_z^2}$$

(27)

4. Materials and Methods

4.1. Experimental Materials

To verify the validity of the previous deformation force calculation formula, two conical parts, with forming angles of 45° and 58°, heights of 50 mm and 25 mm, and opening radii of 55.4 mm and 33.6 mm, are designed, as shown in Figure 5 and Figure 6. The material of the designed punches is tool steel, and the diamond film is sputtered after quenching. The surface of the processed punch is intact and can meet the requirements of high-performance forming applications.

4.2. Experimental Method

In this study, a three-dimensional theoretical model of the test component was developed using the CAD/CAM software. Numerical control programming based on the model was then conducted, creating the processing path and outputting the processing instructions of the numerical control forming equipment. The forming test conditions are shown in

Table 1. Conditions of the forming experiment.

Project	Specimen 1	Specimen 2
Forming equipment	Incremental forming machine	Incremental forming machine
Forming tool diameter	Φ8 mm	Φ8 mm
Sheet material	LY12M	LY12M
Blank size	Φ80 mm	Φ130 mm
Blank thickness	1.4 mm	1.4 mm
Lubricating oil	L--ECC 30	L--ECC 30

.

4.3. Deformation Force Measurement Solution

The deformation force was measured using a Kistler 9443B three-way piezoelectric dynamometer. The fixture for securing the metal sheet was placed above the dynamometer (Figure 7), so that the forming tool operates on the metal plate during the forming process. The material’s force will be completely transmitted to the dynamometer.

5. Analysis of Experimental Results

5.1. Results of the Deformation Force Measurement

Deformation forces in three directions can be obtained by measuring using a Kistler 9443B three-way piezoelectric force gauge, as shown in Figure 8 and Figure 9. The relationship between the deformation forces $F_x$,$F_y$, and the radial forming $F_r$ at any position P on the forming plane is given by (Figure 10):

$$\{\begin{array}{c}{F}_x=F_r\cos (\phi )\\ F_y=F_r\cos (\phi +\pi /2)\end{array}$$

(28)

The deformation force in the $xy$ direction is periodic. During the forming process, each contour layer can be considered as a cycle with two peaks. The two peaks have essentially the same size and opposite directions. The radial deformation force on the modified forming plane is defined as the average of the absolute value of the peak value, while the axial deformation force is defined as the average value of the z-direction force at any ten points on the contour layer, as shown in Figure 11 and Figure 12.

5.2. Comparison of the Deformation Forces

Studying the forming force is the most direct starting point for the study of the incremental forming process mechanism. Many researchers have started from the mechanism to calculate the forming force after analyzing the stress and strain in the deformation area. Two main methods exist. The first consists of predicting the forming force through finite element analysis, while the second one predicts it in the forming process through empirical models obtained by experiments.

Table 2. Comparison between different methods for predicting the forming force.

Methods	Results		Ref.
	Theoretical Result	Experimental Result
Model method	897.7 N	822.0 N	[27]
	1040.0 N	1080.0 N	[32]
	750.0 N	About 802.0 N	[33]
Finite element analysis	800.0 N	About 750.0 N	[34]
Proposed method	873.6 N	932.3 N	This paper
	850.8 N	948.5 N

presents a comparison between the recent trends and predictions of the forming force.

It can be seen from the above analysis that when the forming enters the stable stage, the radial deformation force F_r and the axial deformation force $F_Z$ are basically stable. Therefore, they can be calculated as statistical measurement values. The resultant force is approximately equal to the deformation force:

$$\{\begin{array}{c}{F}_r=\frac{{\displaystyle \sum _{i=1}^n{F}_{x{p}_{{}_i}}}}{n}=\frac{{\displaystyle \sum _{i=1}^n{F}_{y{p}_{{}_i}}}}{n}\\ \begin{array}{l}{F}_z=\frac{{\displaystyle \sum _{i=1}^n{F}_{z_{{}_i}}}}{n}\\ F=\sqrt{F_r^2+F_z{}^2}\end{array}\end{array}$$

(29)

The theoretical deformation force can be obtained by integrating the conditions of the two groups of forming experiments into Equations (26) and (27).

Table 3. Comparison between the theoretically calculated value of the deformation force and its actual measured value.

	Specimen 1 (45°)			Specimen 2 (58°)
Deformation Force	Calculated Value	Measured Value	Deviation	Calculated Value	Measured Value	Deviation
$P_r$	334.3	260.0	28.5%	412.4	336.7	22.5%
$P_z$	807.1	895.3	9.8%	774.1	887.1	16.1%
$P$	873.6	932.3	6.3%	850.8	948.8	10.3%

shows a comparison between the theoretically calculated value of the deformation force and its actual measured value.

The findings of this study were compared with those of other studies to demonstrate the efficiency of the proposed theoretical prediction model. Veera [35] studied the impact of the forming depth, feed speed, and tool rotation speed on the forming force, and provided a model for predicting the forming force. His experimental results showed that the model’s ability to predict the response value of the forming force is 95.0%. Li et al. [36] proposed a tangential force prediction and analysis model, which has an average error of 6.0% and 11.0%, respectively. This model greatly improved the prediction efficiency of the forming force. Bansal et al. [37] proposed an analytical model to predict the forming force of two different materials, reaching errors of 7.9% and 19.7%. According to the experimental results obtained in this study, the deviation of the forming resultant force is 6.3% and 10.3%, respectively, which is very close to the above experimental results, and therefore the deviation is acceptable.

6. Conclusions

This paper tackled the deformation forces of the sheet metal incremental forming process. The main conclusions are summarized as follows:

(1) The graphic approach was used to analyze the stress state of the deformation area during the forming process and the deformation of the contact region. It was also used to derive the deformation force calculation formula.

(2) The designed experiment scheme validated the accuracy of the proposed deformation force calculation method. Through experimental comparison, it was shown that the error between the theoretical analysis value and the measured value is within an acceptable range, which demonstrated the efficiency of the derived deformation force calculation method. Finally, statistical methods for measuring the radial deformation force, axial deformation force, and forming resultant force were proposed.