Ideal Flow Design of Plane-Strain Bending Driven by Springback

Abstract

The present paper is concerned with the design of the plane-strain bending of wide sheets driven by the sheet’s geometric shape after unloading. The solution is based on the ideal flow theory. It determines the loading path (the dependence between the dimensionless bending moment and the dimensionless tensile force) that produces the desirable shape. The ideal flow theory is valid for the rigid perfectly plastic material model. A method proposed in the literature combines a rigid plastic solution at loading and an elastic solution for springback. The final design solution is practically analytical. A numerical method is only necessary to solve a system of transcendental equations. An illustrative example is provided. This example shows that, in some cases, it is sufficient to use the solution at loading to determine an accurate loading path that produces the desirable sheet’s shape after springback.

1. Introduction

All sheet metal forming processes incorporate some bending. There is a vast amount of literature on springback for such processes. A comprehensive review of several issues related to springback was provided in [1]. Most theoretical studies have focused on the springback predictions using analytical and numerical methods. Several works have proposed procedures for reducing springback [2,3,4,5,6]. Paper [2] recommended reheating before post-stretching to reduce the elastic recovery. Optimal temperature, punch speed, holding time, and sheet orientation relative to the rolling direction were found in [3] for anisotropic sheets of Inconel 625 alloy. The effect of different heating methods on springback in the dieless bending of mild steel sheets has been studied in [4,5]. Paper [6] has employed regression analysis to develop an analytical model for optimizing the V-bending process of JSC-590 sheets. The present paper provides a design procedure that aims to achieve the desired shape after four-point bending followed by unloading. The design is based on the ideal flow theory [7].

The ideal flow theory was invented in [8], as the planar stationary flow of rigid perfectly plastic material was considered. The theory has been extended to three-dimensional stationary and non-stationary flows in [9]. It is used for the preliminary design of metal forming processes [7]. However, all available non-trivial bulk ideal flow theory applications are restricted to stationary flows. A plane strain strip drawing design was proposed in [8]. A more general solution for an arbitrary flow with a symmetry axis was derived in [10]. An axisymmetric drawing design was proposed in [11]. A review of ideal flow design solutions for drawing and extrusion has been provided in [12]. All these solutions determine a die shape that produces an ideal flow. Additionally, solution [11] supplies the die of minimum length.

The present paper combines the ideal flow theory for non-stationary planar processes and the method for analyzing plane-strain bending proposed in [13]. A review of the latter is presented in [14]. The design objective is to determine an ideal flow loading path that results in desirable geometric parameters after unloading. It is worthy of note that the ideal flow theory is based on a rigid plastic model. Therefore, the approach proposed in [15] is adopted for calculating springback. It has been shown in [16] that the effect of neglecting elasticity on loading on springback in bending is negligible. It is worthy of note that the new design disposes of the reversed loading for fibers in a central part of the sheet. Therefore, the Bauschinger effect, which may affect springback [17], is immaterial.

The solution at loading is purely analytical. Assuming that unloading is purely elastic, paper [18] derives a general solution for the final shape. This solution is used in conjunction with the new solution developed in the present paper.

2. Statement of the Problem

A wide sheet of initial thickness H and initial width 2L is subject to bending under tension. Its initial and intermediate/final configurations are shown in Figure 1a,b, respectively. The bending moment is denoted as M and the tensile force as F. Both are in per unit length. The current thickness, radius of AB, and radius of CD are denoted as h, r_AB, and r_CD, respectively. It is required to find a loading path in the M − F space such that

$$r_{CD}=r_f\hspace{1em}\mathrm{and}\hspace{1em}h=h_f$$

(1)

after unloading. Additionally, the process must be an ideal flow process. It is assumed that the material obeys the von Mises yield criterion and its associated flow rule.

The general concept of ideal flow design was described in detail in [7]. The present section applies this concept to the process of plane-strain bending under tension. The general structure of the rigid/plastic solution for this process is illustrated in Figure 2a. All fibers in Region 1 are extended, and all fibers in Region 2 are compressed in the circumferential direction throughout the deformation process. All fibers in Region 3 are first compressed and afterward extended in the circumferential direction. Region 3 does not exist at the initial instant. The ideal flow condition is satisfied in Regions 1 and 2. The boundary between Regions 2 and 3 propagates to surface CD and moves through the material. For this reason, the ideal flow condition is not satisfied in Region 3. Therefore, the ideal flow design should determine such a dependence between the bending moment and the tensile force that the initial boundary between Regions 1 and 2 does not move through the material. Thus, Region 3 does not exist in the ideal flow process of bending under tension throughout the deformation process (Figure 2b).

It is worthy of note that Region 2 does not exist at the initial instant if the tensile force is large enough. Such processes will be considered separately.

The ideal plastic solutions are rigid/plastic. Nevertheless, the determination of residual stresses and springback is possible using the method proposed in [15].

3. Solution at Loading

3.1. Kinematics

With no loss of generality, one can choose the origin of the Cartesian coordinate system (x, y) shown in Figure 2b at the intersection of the axis of symmetry of the process and the outer surface of the sheet. Then, the process is symmetric relative to the x-axis. The kinematics of plane strain bending of incompressible material was described in [13]. This description is independent of the constitutive equations. It is also valid for plane strain bending under tension [18]. It follows from [13] that

$$\frac{x}{H}=\sqrt{\frac{\zeta }{a}+\frac{s}{a^2}}\cos \left(2a\eta \right)-\frac{\sqrt{s}}{a}\hspace{1em}\mathrm{and}\hspace{1em}\frac{y}{H}=\sqrt{\frac{\zeta }{a}+\frac{s}{a^2}}\sin \left(2a\eta \right).$$

(2)

Here, (ζ, η) is the Lagrangian coordinate system satisfying the condition $\zeta =x/H$ and $\eta =y/H$ at the initial instant, a is a time-like parameter, and s is a function of a. This function should be found from the solution. At the initial instant:

$$a=0\hspace{1em}\mathrm{and}\hspace{1em}s=\frac{1}{4}.$$

(3)

It is also convenient to introduce a polar coordinate system $\left(r, \theta \right)$ by the transformation equations:

$$\frac{r\cos \theta }{H}=\frac{x}{H}+\frac{\sqrt{s}}{a}\hspace{1em}\mathrm{and}\hspace{1em}\frac{r\sin \theta }{H}=\frac{y}{H}.$$

(4)

Then, $r=r_{AB}$ on AB, $r=r_{CD}$ on CD, $\theta ={\theta }_0$ on BC, and $\theta =-{\theta }_0$ on AD (Figure 2b). Equations (2) and (4) combine to give

$$\frac{r}{H}=\sqrt{\frac{\zeta }{a}+\frac{s}{a^2}}\hspace{1em}\mathrm{and}\hspace{1em}\theta =2a\eta .$$

(5)

It follows from the definition for the Lagrangian coordinates that $\zeta =0$ on AB, $\zeta =-1$ on CD, $\eta =L/H$ on BC, and $\eta =-L/H$ on AD (Figure 2b). Then, Equation (5) supplies

$$\frac{r_{AB}}{H}=\frac{\sqrt{s}}{a},\hspace{1em}\frac{r_{CD}}{H}=\sqrt{\frac{s}{a^2}-\frac{1}{a}}\hspace{1em}\mathrm{and}\hspace{1em}\frac{h}{H}=\frac{r_{AB}-r_{CD}}{H}=\frac{\sqrt{s}-\sqrt{s-a}}{a}.$$

(6)

The equations in (2) show that the principal strain rate trajectories coincide with the coordinate curves of the Lagrangian coordinate system. Moreover, the quadratic invariant of the strain rate tensor is determined from these equations as

$${\xi }_{eq}=\frac{\left|\zeta +ds/da\right|}{\sqrt{3}\left(\zeta a+s\right)}\frac{da}{dt}.$$

(7)

Then, ${\xi }_{eq}=0$ at

$$\zeta ={\zeta }_0=-\frac{ds}{da}.$$

(8)

This equation is valid at the boundary between Regions 1 and 2 at the initial instant (Figure 2b).

One should distinguish two cases:${\zeta }_0\ge -1$ and ${\zeta }_0<-1$ at the initial instant. Since the coordinate $\zeta$ is Lagrangian, the ideal flow condition formulated in Section 2 requires that ${\zeta }_0$ is independent of a if ${\zeta }_0\ge -1$ at the initial instant. The line $\zeta ={\zeta }_0$ lies outside the domain occupied by the material if ${\zeta }_0<-1$. Therefore, it is not necessary that ${\zeta }_0$ is independent of a, though it is important to verify that the condition ${\zeta }_0\le -1$ is satisfied throughout the deformation process. However, the condition that ${\zeta }_0$ is independent of a allows one to determine an ideal flow process if ${\zeta }_0<-1$. This process is considered below. Then, independently of the value of ${\zeta }_0$ at the initial instant, solving (8) together with the initial condition (3) yields

$$s=\frac{1}{4}-{\zeta }_{0}a.$$

(9)

Let $a_l$, $h_l$, $r_{AB}^{\left(l\right)}$ and $r_{CD}^{\left(l\right)}$ be the values of a, h, $r_{AB}$ and $r_{CD}$ at the end of loading, respectively. Then, it follows from (6) and (9) that

$$\frac{h_l}{H}=\frac{\sqrt{1-4{\zeta }_0{a}_l}-\sqrt{1-4a_l\left({\zeta }_0+1\right)}}{2a_l},\hspace{1em}\frac{r_{AB}^{\left(l\right)}}{H}=\frac{\sqrt{1-4a_l\left({\zeta }_0+1\right)}}{2a_l},\hspace{1em}\frac{r_{CD}^{\left(l\right)}}{H}=\frac{\sqrt{1-4a_l\left({\zeta }_0+1\right)}}{2a_l}.$$

(10)

3.2. Stress Solution

Let ${\sigma }_r$ and ${\sigma }_{\theta }$ be the radial and circumferential stresses, respectively, referred to the $\left(r\theta \right)-$ system introduced in (5). It has been shown above that the principal strain rate trajectories coincide with the coordinate curves of the Lagrangian coordinate system. The material model is coaxial. Therefore, the principal stress trajectories also coincide with the coordinate curves of the Lagrangian coordinate system. Then, Equation (5) implies that the radial and circumferential stresses are the principal stresses.

3.2.1. Case ${\zeta }_0\ge -1$

The stress solution is discontinuous across the curve $\zeta ={\zeta }_0$. The yield criterion is

$${\sigma }_r-{\sigma }_{\theta }=\frac{2{\sigma }_0}{\sqrt{3}}$$

(11)

in the region $-1\le \zeta \le {\zeta }_0$ and

$${\sigma }_r-{\sigma }_{\theta }=-\frac{2{\sigma }_0}{\sqrt{3}}$$

(12)

in the region ${\zeta }_0\le \zeta \le 0$. Here, ${\sigma }_0$ is the yield stress in tension, a material constant. The only stress equilibrium equation that is not identically satisfied is

$$\frac{\partial {\sigma }_r}{\partial r}+\frac{{\sigma }_r-{\sigma }_{\theta }}{r}=0.$$

(13)

Using (5), (9), (11), and (12), one transforms this equation into

$$\frac{\partial {\sigma }_r}{\partial \zeta }=-\frac{4a{\sigma }_0}{\sqrt{3}\left[1+4a\left(\zeta -{\zeta }_0\right)\right]}$$

(14)

in the region $-1\le \zeta \le {\zeta }_0$ and

$$\frac{\partial {\sigma }_r}{\partial \zeta }=\frac{4a{\sigma }_0}{\sqrt{3}\left[1+4a\left(\zeta -{\zeta }_0\right)\right]}$$

(15)

in the region ${\zeta }_0\le \zeta \le 0$.

Surface AB is traction free (Figure 2b). Therefore,

$${\sigma }_r=0$$

(16)

for $\zeta =0$. The tensile force is assumed to be balanced by a uniform pressure P applied over surface CD. For equilibrium, it is evident that $P{r}_{CD}=F$. Using (6) and (9), one can rewrite this equation as

$$p=\frac{P}{{\sigma }_0}=\frac{2fa}{\sqrt{1-4a\left({\zeta }_0+1\right)}},\hspace{1em}f=\frac{F}{{\sigma }_{0}H}.$$

(17)

Then,

$$\frac{{\sigma }_r}{{\sigma }_0}=-p$$

(18)

for $\zeta =-1$. Moreover, the radial stress must be continuous across the curve $\zeta ={\zeta }_0$:

$$\left[{\sigma }_r\right]=0$$

(19)

for $\zeta ={\zeta }_0$. Here, the square brackets denote the amount of jump in an enclosed quantity.

The solution of Equations (14) and (15) satisfying conditions (16) and (18) is

$$\frac{{\sigma }_r}{{\sigma }_0}=\frac{1}{\sqrt{3}}\ln \left[\frac{1+4a\left(\zeta -{\zeta }_0\right)}{1-4a{\zeta }_0}\right]$$

(20)

in the region ${\zeta }_0\le \zeta \le 0$ and

$$\frac{{\sigma }_r}{{\sigma }_0}=\frac{1}{\sqrt{3}}\ln \left[\frac{1-4a\left(1+{\zeta }_0\right)}{1+4a\left(\zeta -{\zeta }_0\right)}\right]-p$$

(21)

in the region $-1\le \zeta \le {\zeta }_0$. Substituting (20) and (21) into (19) and using (17) determines f as a function of a at a given value of ${\zeta }_0$:

$$f=\frac{\sqrt{1-4a\left({\zeta }_0+1\right)}}{2\sqrt{3}a}\ln \left\{\left[1-4a\left(1+{\zeta }_0\right)\right]\left(1-4a{\zeta }_0\right)\right\}.$$

(22)

The right-hand side of this equation reduces to the expression $0/0$ as $a\to 0$. Applying l’Hospital’s rule, one can find that

$$f=-\frac{2\left(1+2{\zeta }_0\right)}{\sqrt{3}}$$

(23)

at $a=0$.

Equations (11), (12), (20), and (21) combine to give

$$\frac{{\sigma }_{\theta }}{{\sigma }_0}=\frac{1}{\sqrt{3}}\ln \left[\frac{1+4a\left(\zeta -{\zeta }_0\right)}{1-4a{\zeta }_0}\right]+\frac{2}{\sqrt{3}}$$

(24)

in the region ${\zeta }_0\le \zeta \le 0$ and

$$\frac{{\sigma }_{\theta }}{{\sigma }_0}=\frac{1}{\sqrt{3}}\ln \left[\frac{1-4a\left(1+{\zeta }_0\right)}{1+4a\left(\zeta -{\zeta }_0\right)}\right]-p-\frac{2}{\sqrt{3}}$$

(25)

in the region $-1\le \zeta \le {\zeta }_0$. The bending moment referred to the central fiber is determined as

$$M={\displaystyle \underset{r_{CD}}{\overset{r_{AB}}{\int }}\left({\sigma }_{\theta }-\frac{F}{h}\right)rdr}.$$

(26)

Using (5), (6), and (17), one can transform this equation to

$$m=\frac{2\sqrt{3}M}{{\sigma }_0{H}^2}=\frac{\sqrt{3}}{a}{\displaystyle \underset{-1}{\overset{0}{\int }}\frac{{\sigma }_{\theta }}{{\sigma }_0}d\zeta }-\frac{\sqrt{3}f}{\left(\sqrt{s}-\sqrt{s-a}\right)}.$$

(27)

The last term is a known function of a due to (9) and (22). One can evaluate the integral in (27) using (24) and (25). As a result,

$${\displaystyle \underset{-1}{\overset{0}{\int }}\frac{{\sigma }_{\theta }}{{\sigma }_0}d\zeta }=\frac{\sqrt{3}\ln \left\{\left[1-4a\left(1+{\zeta }_0\right)\right]\left(1-4a{\zeta }_0\right)\right\}-4\sqrt{3}a\left[1+2{\zeta }_0+\sqrt{3}p\left(1+{\zeta }_0\right)\right]}{12a}.$$

(28)

Here, p is a function of a due to (17) and (22). Then, substituting (28) into (27) determines m as a function of a:

$$m=-\frac{4a+8a{\zeta }_0+\sqrt{\left(1-4a{\zeta }_0\right)\left[1-4a\left(1+{\zeta }_0\right)\right]}\ln \left\{\left(1-4a{\zeta }_0\right)\left[1-4a\left(1+{\zeta }_0\right)\right]\right\}}{4a^2}.$$

(29)

The right-hand side of this equation reduces to the expression $0/0$ as $a\to 0$. Applying l’Hospital’s rule, one can find that

$$m=-4{\zeta }_0\left(1+{\zeta }_0\right)$$

(30)

at $a=0$.

3.2.2. Case ${\zeta }_0<-1$

In this case, Equation (20) is valid in the range $0\ge {\zeta }_0\ge -1$. Equation (21) is immaterial. It follows from (18) and (20) that

$$p=\frac{1}{\sqrt{3}}\ln \left[\frac{1-4a{\zeta }_0}{1-4a\left(1+{\zeta }_0\right)}\right].$$

(31)

This equation and (17) combine to give

$$f=\frac{\sqrt{1-4a\left({\zeta }_0+1\right)}}{2\sqrt{3}a}\ln \left[\frac{1-4a{\zeta }_0}{1-4a\left(1+{\zeta }_0\right)}\right].$$

(32)

Applying l’Hospital’s rule, one can find that $f=2/\sqrt{3}$ at $a=0$.

Equation (27) is valid, and Equation (24) is valid in the range $0\ge {\zeta }_0\ge -1$. Then,

${\displaystyle \underset{-1}{\overset{0}{\int }}\frac{{\sigma }_{\theta }}{{\sigma }_0}d\zeta }=\frac{1}{\sqrt{3}}-\frac{\left[1-4a\left(1+{\zeta }_0\right)\right]}{4\sqrt{3}a}\ln \left[\frac{1-4a\left(1+{\zeta }_0\right)}{1-4a{\zeta }_0}\right].$

Substituting the right-hand side of this equation into (27) yields

$$m=\frac{1}{a}+\frac{\sqrt{\left(1-4a{\zeta }_0\right)\left[1-4a\left(1+{\zeta }_0\right)\right]}}{4a^2}\ln \left[\frac{1-4a\left(1+{\zeta }_0\right)}{1-4a{\zeta }_0}\right].$$

(33)

Applying l’Hospital’s rule, one can find that $f=0$ at $a=0$.

4. Design Procedure at Loading

At a given value of ${\zeta }_0$, the solution in Section 3 determines the path in the (f, m)−space in parametric form, with a being the parameter. This path ensures that the process is an ideal flow process. A restriction imposed on the solution is that $p\ge 0$. It is seen from (17) that this inequality is equivalent to

$$f\ge 0$$

(34)

if $a\ne 0$. Equation (32) shows that the inequality in (34) is always satisfied if ${\zeta }_0<-1$.

Consider the case $0\ge {\zeta }_0\ge -1$. It follows from (22) and (31) that

$$\frac{\sqrt{1-4a\left({\zeta }_0+1\right)}}{2\sqrt{3}a}\ln \left\{\left[1-4a\left(1+{\zeta }_0\right)\right]\left(1-4a{\zeta }_0\right)\right\}\ge 0.$$

(35)

Consider the equation

$$\frac{\sqrt{1-4a\left({\zeta }_0+1\right)}}{2\sqrt{3}a}\ln \left\{\left[1-4a\left(1+{\zeta }_0\right)\right]\left(1-4a{\zeta }_0\right)\right\}=0.$$

(36)

It is seen from (10) that $1-4a\left({\zeta }_0+1\right)>0$. Then, solving Equation (36) for a and using (35) leads to

$$a\le a_c\left({\zeta }_0\right)=\frac{1+2{\zeta }_0}{4{\zeta }_0\left({\zeta }_0+1\right)}.$$

(37)

Let ${\rho }_{CD}$ and $\beta$ be desirable values of ${\left(r_{CD}/H\right)}^2$ and $h/H$ at the end of loading. The inequality in (37) restricts these values. It follows from Equation (10) that

$${\rho }_{CD}=\frac{1-4a\left({\zeta }_0+1\right)}{4a^2}\hspace{1em}\mathrm{and}\hspace{1em}\beta =a^{-1}\left(\sqrt{\frac{1}{4}-{\zeta }_{0}a}-\sqrt{\frac{1}{4}-a\left({\zeta }_0+1\right)}\right).$$

(38)

Solving these equations for ${\zeta }_0$ and a leads to

$$a_l=\frac{1}{\beta \left(\beta +2\sqrt{{\rho }_{CD}}\right)}\hspace{1em}\mathrm{and}\hspace{1em}{\zeta }_0=\frac{\beta \left(\beta +2\sqrt{{\rho }_{CD}}\right)}{4}-\frac{{\rho }_{CD}}{\beta \left(\beta +2\sqrt{{\rho }_{CD}}\right)}-1.$$

(39)

Substituting (37) into (38) yields

$$\begin{array}{l}{\rho }_{CD}\left({\zeta }_0\right)=\frac{1-4a_c\left({\zeta }_0\right)\left({\zeta }_0+1\right)}{4a_c^2\left({\zeta }_0\right)},\\ \beta \left({\zeta }_0\right)=\frac{1}{2a_c\left({\zeta }_0\right)}\left(\sqrt{1-4{\zeta }_0{a}_c\left({\zeta }_0\right)}-\sqrt{1-4a_c\left({\zeta }_0\right)\left({\zeta }_0+1\right)}\right).\end{array}$$

(40)

These equations determine a curve in the $\left({\rho }_{CD}, \beta \right)-$ space. This curve shown in Figure 3 corresponds to the condition $f=0$. The domains of the validity and invalidity of the solution are also indicated in this figure. Choosing ${\rho }_{CD}$ and $\beta$ in the domain of validity, one can find ${\zeta }_0$ and $a_l$ from (39). Equations (22) and (29) supply the loading path for this design if $0\ge {\zeta }_0\ge -1$, and Equations (32) and (33) if ${\zeta }_0<-1$.

5. Unloading and Final Design

Unloading is assumed to be purely elastic. This assumption is verified a posteriori. Let ${\rho }_{AB}$ be the value of ${\left(r_{AB}/H\right)}^2$ at the end of loading. The general solution for unloading is provided in [18]. Using this solution, one can find

$$\begin{array}{l}{R}_0^{\ast }=\sqrt{{\rho }_{CD}}\left\{1+C_1\left[1-\frac{\left(1-{\rho }_0^2\right)}{2\ln {\rho }_0}\right]-\frac{3kp}{2}\left(1+\frac{{\rho }_0^2}{2\ln {\rho }_0}\right)\right\},\\ R_1^{\ast }=\sqrt{{\rho }_{AB}}\left\{1+C_1\left[{\rho }_0^2-\frac{\left(1-{\rho }_0^2\right)}{2\ln {\rho }_0}\right]-\frac{3kp{\rho }_0^2}{2}\left(1+\frac{1}{2\ln {\rho }_0}\right)\right\}.\end{array}$$

(41)

Here, $R_0^{\ast }$ is the value of $r_{CD}/H$ after springback, $R_1^{\ast }$ is the value of $r_{AB}/H$ after springback, ${\rho }_0=\sqrt{{\rho }_{CD}/{\rho }_{AB}}$, $k={\sigma }_0/\left(3G\right)$ and $G$ is the shear modulus of elasticity. Additionally,

$$\begin{array}{l}{C}_1=-\frac{B_1}{A_1},\\ B_1=\frac{3kf\ln {\rho }_0}{\beta }-\frac{3kp{\rho }_0^4}{2}+\frac{k{\rho }_0^2}{2}\left[3p+2\ln {\rho }_0\left(\frac{\sqrt{3}m}{R_0^2}-\frac{3f}{\beta }-3p+6p\ln {\rho }_0\right)\right],\\ A_1={\left(1-{\rho }_0^2\right)}^2-4{\rho }_0^2{\ln }^2{\rho }_0.\end{array}$$

(42)

The quantities m, p, and f involved in (41) and (42) are understood to be calculated at the end of loading. The dimensionless strip’s thickness after springback is determined as

$${\beta }^{\ast }=R_1^{\ast }-R_0^{\ast }.$$

(43)

The solution in Section 3 supplies m, p and f as functions of ${\zeta }_0$ and $a_l$. Equation (10) provides ${\rho }_{CD}$ and ${\rho }_{AB}$ as functions of the same parameters. Since $R_0^{\ast }$ and ${\beta }^{\ast }$ are prescribed as the design objectives, Equations (41) and (43) constitute a system for determining ${\zeta }_0$ and $a_l$. This system should be solved numerically.

6. Numerical Example

The numerical solutions were found for $k=0.0035$. This value of k is typical for ultra-high-strength steels [19]. The two design solutions shown in

Table 1. Design objectives.

Design (a)	$R_0^{\ast }=1$	${\beta }^{\ast }={\beta }_m,$$\mathrm{or} {\beta }^{\ast }=0.9,$$\mathrm{or} {\beta }^{\ast }=0.8,$$\mathrm{or} {\beta }^{\ast }=0.7$
Design (b)	${\rho }_{CD}=1$	$\beta ={\beta }_m,$$\mathrm{or} \beta =0.9,$$\mathrm{or} \beta =0.8,$$\mathrm{or} \beta =0.7$

are compared. Design (a) requires that the radius of CD (Figure 1) and strip thickness take prescribed values after springback. Design (b) requires that these quantities take prescribed values at the end of loading. For illustrative purposes, it is assumed that $R_0^{\ast }=1$ in Design (a) and ${\rho }_{CD}=1$ in Design (b).

Design (b) does not require numerical efforts other than solving the first equation in (40), from which the maximum value of ${\zeta }_0$ is determined. This maximum value is ${\zeta }_0\approx -0.661$. The corresponding value of $\beta$ is determined from the second equation in (40) as $\beta ={\beta }_m\approx 0.947$. The loading paths that require ${\zeta }_0>-0.661$ violate the inequality in (34). Several paths calculated using the procedure in Section 4 are depicted in Figure 4, Figure 5 and Figure 6. The curves in these figures are for $\beta ={\beta }_m$, $\beta =0.9$, $\beta =0.8$, and $\beta =0.7$. The corresponding values of ${\zeta }_0$ for the last three cases are ${\zeta }_0=-0.73,$ ${\zeta }_0=-0.886$, and ${\zeta }_0=-1.057$. Figure 4 and Figure 5 show the dependencies of f and m on a, respectively. Due to (5), these curves can be regarded as dependencies on ${\theta }_0$ at any given value of $\eta$. Figure 6 depicts the loading paths in the (f, m)−space.

Design (a) requires a numerical method for solving Equations (41) and (43). This solution has shown that both designs lead to practically the same dependence of f on a. Therefore, this dependence is not illustrated for Design (a). The variation of m with a slightly depends on the design objectives. Figure 7 compares the two designs (${\beta }^{\ast }={\beta }_m$ and $\beta ={\beta }_m$ in Figure 7a, ${\beta }^{\ast }=0.9$ and $\beta =0.9$ in Figure 7b, ${\beta }^{\ast }=0.8$ and $\beta =0.8$ in Figure 7c, and ${\beta }^{\ast }=0.7$ and $\beta =0.7$ in Figure 7d).

The solid curves correspond to Design (a) and the broken curves to Design (b). The difference decreases as the desired final thickness decreases and becomes invisible in Figure 7d.

7. Conclusions

A design solution for plane-strain bending was presented. The solution is based on the ideal flow theory and determines the loading path that ensures desirable geometric parameters after unloading. The solution is practically analytical. A numerical method is only necessary to solve a system of transcendental equations. Section 6 provides several loading paths that illustrate the general solution.

The two design objectives specified in Table 1 were compared. One of these designs requires that the process produces desirable geometric parameters of the sheet after springback. The other requires that these geometric parameters be obtained at the end of loading. Subsequent unloading changes the sheet’s geometry. The illustrative example in Section 6 showed that the difference between these two designs is very small. However, this conclusion is not universal and may depend on the design objectives.

The general structure of the found solution shows that it may be extended to strain-hardening materials, even though the general ideal flow theory is not available for this class of material models. This extension should allow for the ideal flow design of real sheet metal forming processes driven by springback to be provided.