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A flexible surface tooling for sheet-forming processes: conceptual studies and numerical simulation
Journal of Materials Processing Technology 124 (2002) 133–143
A flexible surface tooling for sheet-forming processes: conceptual studies and numerical simulation P.V.M. Raoa,*, Sanjay G. Dhandeb a
Department of Mechanical Engineering, Indian Institute of Technology, New Delhi 110016, India b Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208016, India Received 31 July 2000; accepted 5 March 2002
Abstract All sheet-forming processes suffer from a limitation that tooling in the form of molds/dies is a precursor for realizing any simple or complex shape. In order to overcome this limitation, discrete surface tooling has been proposed in the past. This consists of a large number of closely spaced surface tool elements arranged in a matrix form whose heights can be adjusted to approximate the contour of desired surface shapes. Such a tooling has been successfully demonstrated experimentally. In this paper, a variation of discrete surface tooling, called as flexible surface tooling, has been proposed. The proposed tooling consists of discrete surface tool elements draped with flexible sheet of rubber-like material to provide a continuous surface required for tooling applications. Computational simulation of the proposed tooling is carried out for feasibility studies. Computational simulation involves the deformation analysis of rubber-like membranes in multiple contact by FEM. The results of the analysis indicate that the flexible surface tooling is an option for sheet-forming processes in general and for processes like composite layup in particular. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Sheet-forming processes; Flexible surface tooling; Rubber-like material; Computational simulation
1. Shape-conforming processes In a class of manufacturing processes, shape is realized by making the original material in the form of a deformable sheet to conform to a relatively rigid surface of the tool. Such shape conformance involves an elastic/plastic deformation of the material by application of heat/pressure. As the shape realization here is due to close conformance of one surface to another, these processes can be called as shape-conforming processes. Press brake bending of sheet metal components, stretch forming, deep drawing, composite layup, compression molding of sheet molding compound (SMC) and thermoforming are some of the examples in this category (Fig. 1). In all shape-conforming processes, tool is a precursor for fabrication of every simple or complex shape. Tooling in the form of molds and dies provide a single or multiple base surfaces required for shaping the metallic sheet or the combination of reinforcing and matrix materials. If we neglect the shape variations due to springback and thermal effects, the shape of the product obtained in these processes is completely dictated by the shape of the mold or die used. It
*
Corresponding author.
can said that, in an ideal situation, the realizability of a shape in all shape-conforming processes is mainly governed by the tool used and is independent of the process. This reduces the problem of shape realization in these processes to that of realization of a tool surface. The realization of a tool surface for shape-conforming processes is a time consuming and often a costly affair. Typically, one spends more time on design and fabrication of a tooling than that involved in the fabrication of component itself. For example, the mold design and fabrication for a free-from composite product often involves a slow, laborintensive stage of master preparation [1]. In the case of sheet metal forming, design and fabrication of a die is an experimentally iterative process. A number of forming experiments with improved die shape at each stage is required to coverage to a final die shape which can be used for realizing the desired component shapes [2,3]. In a prototyping environment, where the design changes are frequent and the number of components required are often limited, such forming experiments are both costly and time consuming. Moreover, in such an environment the accuracy requirements of the product shape are not very critical. In such cases, the hard tooling is generally expensive and can be substituted by a flexible tooling for a class of component shapes. One such tooling which has been
0924-0136/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 1 4 1 - 3
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Fig. 2. A discrete approximation to a continuous surface. Fig. 1. Shape-conforming processes.
proposed in this work for shape-conforming processes is flexible surface tooling.
2. Flexible surface tooling The flexible surface tooling is based on the concept of discrete approximation to a continuous surface of a mold or a die (Fig. 2). It consists of a number of closely spaced multiple rigid surface tool elements, known as indentors, each of which is a surface element of an expected contour. The height of the indentors can be adjusted to approximate the desired surface shapes. The positioning of each indentor can be carried out either manually or using a computer control. The discrete nature of the tool can be overcome by placing a deformable elastomeric sheet over the indentors. The flexible sheet draped over the discrete surface provides with a continuous surface required for the tooling application (Fig. 3). The advantage of such a tooling is that it is adjustable. A variety of surface shapes can be realized by properly adjusting the heights of surface tool elements. The total time involved in the tool set up is considerably less than that involved in the development of a hard tool. The successful development of a flexible surface tooling will result in considerable reduction of the time involved in prototyping certain class of shapes. Moreover, automated nature of the process may relegate the need of some of the experts/skilled labors involved in tool design and development. Experiments with new shapes and incorporating design changes will be much easier once such a tooling is made available.
3. Earlier work The concept of adjustable discrete surface tooling was first given by Nakajima [4]. The concept was used to fabricate complicated dies for presswork and electrodes for electrochemical machining. An extensive research work on the concept of discrete surface tooling has been carried out by Hardt et al. [2,3,5–9] at the Laboratory for Manufacturing and Productivity at Massachusetts Institute of Technology. Unlike the forming system proposed by Nakajima, they propose the use of a variable configuration discrete element die in a closed loop shape control system (Fig. 4). More recently, Walczyk et al. [10,11] have proposed a reconfigurable tooling system along the lines of Hardt et al. for forming sheet metal parts and composites. The tooling system is hydraulically actuated and pin configuration used is 4 4. They also studied relative merits and demerits of using open and closed loop control for forming process using their die. Bogolyubov et al. [12] have proposed to use removable glass plastic shells resting on point supports as a universal tooling for forming airframe structures. Deformation analysis of shell resting on point supports and carrying uniformly distributed loading is carried out to demonstrate the concept. The discrete element die proposed by Nakajima and improvements proposed by Hardt et al. and Walczyk et al. is primarily meant for forming sheet metal components. The work carried out by both these groups is based on the forming experiments. Another approach to study the realizability of shapes by a discrete element die is using numerical methods such as FEM. A computational study will be
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Fig. 3. Flexible surface tooling.
Fig. 4. Open and closed loop shape control systems for sheet metal forming with variable configuration die.
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helpful in reducing the number of forming experiments required to arrive at the correct die shape. However, in the case of sheet metal-forming processes a numerical study accounting for large strains, material properties of the sheet, springback and the tribological aspects associated with the process is difficult. In the case of composite fabrication processes such as layup, the complexities associated with the sheet metal forming are absent. In this case, the forming forces are also considerably less. However, the thermal effects are likely to be present in all composite fabrication processes. But, in the case of wet or prepreg layup with room temperature curing these effects can be neglected. These simplifications make the numerical study of the flexible surface tooling feasible for applications like composite layup. This work deals with a computational study of the realizability of simple and complex shapes by the proposed flexible surface tool. A computational procedure, which deals with the deformation analysis of nonlinear elastic membranes in multiple contact, has been developed and used for the tool simulation. The effect of geometric and material properties of the flexible sheet on the shape accuracies has been studied with an objective to identify the range of these parameters for a given application. Unlike the tooling proposed by others [2–12], which are primarily meant for sheet metal-forming processes, the present tooling is meant for composite fabrication using layup. Composite layup being a process where forming forces are considerably lower, a relatively small number of tool elements can be used for forming free-form shapes, which reduces set up time considerably. Further, the effect of various tooling parameters on shape realizability has been studied in the present work, which was not dealt in detail by earlier researchers.
of the undeformed body [13]. For rubber-like materials, the most widely used form of w is the Moony–Rivlin material description. The strain energy function for this case is given by w ¼ C1 ðI1 3Þ þ C2 ðI2 3Þ;
I3 ¼ 1
(1)
where C1 and C2 are the material constants. The contact problems involving hyper-elastic membranes have been attempted by many researchers in the past. The incremental-iterative strategies based on the total or updated Lagrangean concepts are commonly used to solve this nonlinear problem [14–18]. The material incompressibility and contact constraints are dealt either by mixed formulation or penalty-based approach [19]. However, when the problem involves frictionless contacts and plane or membrane state of stress, it can also be attempted by a mathematical programming approach. This approach is commonly known as energy search and has been followed by a number of investigators for the deformational analysis of rubber-like membranes with or without contact constraints [20–26]. In this work, the problem of deformation analysis of flexible sheet has been attempted by finite element discretization and potential energy minimization in the presence of contact constraints. The approach is the same as that of Becker [21] except for the contact constraints. Three-noded triangular elements are used for the finite element discretization. Moony–Rivlin model is used for the strain energy calculations with incompressibility condition. The incompressibility condition has been dealt by analytical treatment which is possible in the case of plane or membrane state of stress [19,27]. The conjugate gradient method with Polak– Ribiere correction is used for energy minimization. The contact constraints are treated using a penalty approach.
5. Problem formulation 4. Deformation analysis In the proposed tooling system, the deformed configuration of the flexible elastomeric sheet is used as a base surface for tooling requirements. The elastomeric sheet undergoes deformation due to indentation by a matrix of indentors. Knowing the deformed configuration of the sheet for a given configuration of the indentors is an important aspect in the above-mentioned tooling. This involves solving of an elastostatic contact problem accounting for the large deformations of rubber-like materials. The flexible sheet used for the purpose of tooling here is of uniform thickness before deformation and its thickness is small compared to its other dimensions (less than 1%). As the state of stress in the sheet is essentially two-dimensional in this case, the sheet can be idealized as a thin incompressible isotropic hyper-elastic membrane. The hyper-elastic materials are characterized by the presence of an elastic potential, w, known as strain energy density, which represents the strain energy per unit volume
Consider the geometry of deformation of a thin homogeneous elastic membrane of uniform thickness h0. The undeformed and deformed configurations are represented by O0 and O, respectively (Fig. 5). The region O0 is bounded by planes X3 h0 =2, the middle surface being X3 ¼ 0. The position vectors x ¼ x1 i1 þ x2 i2 þ x3 i3 and X ¼ X1 i1 þ X2 i2 þ X3 i3 of the deformed and initial configurations, respectively, are related through expression given by x¼Xþu
(2)
where u is the displacement vector. The Lagrangean strain tensor is now given by the relation 1 @xk @xk gij ¼ dij (3) 2 @Xi @Xj where dij is the Kronecker delta. Thickness of the membrane after deformation, h, is given by h ¼ lh0
(4)
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Fig. 5. The geometry of deformation of a membrane.
where l is the extension ratio at the middle surface in a direction normal to the middle surface. Assuming the membrane to be thin, strain tensor can be written as 1 @ua @ub @ui @ui gab ¼ þ þ ; 2 @Xb @Xa @Xa @Xb 1 g23 ¼ 0; g33 ¼ ðl2 1Þ (5) g13 ¼ 0; 2 where a; b ¼ 1; 2 and i ¼ 1; 2; 3. The three strain invariants I1, I2, I3 are given by I1 ¼ l2 þ 2ð1 þ gaa Þ;
I2 ¼ l4 þ l2 I1 þ
1 I3 ; l2
I3 ¼ l2 ð1 þ 2gaa þ 2eab elm gal gbm Þ
(6)
where a; b; g; m ¼ 1; 2 and eab, elm are the two-dimensional permutational symbols ðe12 ¼ 1; e21 ¼ 1; e11 ¼ e22 ¼ 0Þ. Imposing the incompressibility constraints I3 ¼ 1, the above expression can be written as I1 ¼ l2 þ 2ð1 þ laa Þ;
I2 ¼ l4 þ l2 I1 þ
l2 ¼ ð1 þ 2gaa þ 2eab elm gal gbm Þ1
1 I3 ; l2 (7)
The potential energy of deformed membrane, P, is given by Z Z (8) P ¼ U W ¼ wðI1 ; I2 Þ dv t v ds v
s
where U is the strain energy of the deformed membrane and W the virtual work of external traction forces, tv the kinematically admissible displacement field and s the part of boundary on which tractions are specified.
6. Computational procedure The computational procedure used here consists of minimizing the total potential energy of the system subjected to given geometric constraints. For this purpose, the domain O0
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is discretized into three-noded triangular finite elements. The total number of nodes and elements being N and E, respectively. The number of variables in the minimization problem are 3N0 , N0 being the total number of unconstrained nodes. The objective function is the total potential energy of the system summed over E elements. Potential energy of the individual element consist of the strain energy due to deformation and potential of loads, if any, on the element. In order to calculate the strain energy due to stretching of the membrane, undeformed element is placed in the plane of deformed one. Such a transformation does not affect the strain energy calculation as it involves only rigid body translation and rotation. The external forces in this problem are in the form of pressure loads acting on the surface element. As the membrane deforms, the changing direction of this pressure and the changing area of the element on which it acts is accounted. The constraints in this problem are in the form of contacts. The deformed configuration of the flexible sheet due to indentation is such that no interpenetration of the indentor and membrane material is allowed. This condition on the kinematically admissible displacement field of the flexible sheet can be imposed by penalty approach. The penalty is being proportional to the amount of violation. To solve this constrained optimization problem efficiently, the sequential unconstrained minimization technique (SUMT) has been used. SUMT transforms the constrained optimization problem to an unconstrained one by penalizing the constraint violation if any. The conjugate gradient algorithm with Polak–Ribiere correction is adopted as the minimization technique. The method of multipliers (MOM) type of penalty which is equally effective in both feasible and non-feasible solution domains is used for transforming constrained optimization problem into an unconstrained one [28]. The computer program developed for this purpose is written in C is implemented on DEC-3000 platform. The correctness of the formulation and the numerical scheme used here has been verified by running the computer program developed for three standard test cases taken from the works of Becker [21], Feng et al. [23] and Feng and Huang [25].
7. Results Based on the theoretical formulation and the computational procedure presented earlier, numerical experiments were carried out with an objective to study the shape realization by the proposed tooling. The software developed for this purpose is in three modules: a preprocessor, a deformational analysis program and a post-processor. The input to shape realization software consist of the following:
The shape to be realized or the desired surface shape;
Dimensions of the flexible sheet;
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Sheet thickness; Material properties of the sheet; Geometry of the individuals indentors; Layout or the arrangement of indentors.
Knowing the heights by which indentors are to be raised to approximate a given continuous surface is the first step in the process of shape realization. The preprocessor part of the software deals with this aspect. For this purpose, a common coordinate system is used to define the desired shape and the indentor positions. Each of the indentors is now raised incrementally till it makes a contact with the desired shape. After all the indentors are raised in a similar manner, it forms a discrete approximation to the given continuous surface. The height of the indentors decided as part of the preprocessor together with various tooling parameters mentioned above, from an input to analysis module. The deformational analysis program is based on the theoretical formulation and the computational procedure described earlier in this paper. The analysis program uses the middle surface of the flexible sheet as a reference surface, whereas the upper surface of the flexible sheet is the desired one. This offset can be easily accounted using Eq. (4) as the deformation of the flexible sheet proceeds. As a part of the post-processor, the deformed configuration of the flexible sheet obtained as a result of analysis is compared with the desired surface shape to evaluate the shape error. The realizability of a given shape here is measured in terms of root mean square (RMS) error and the average absolute (AVAB) error between the desired and realized surface shapes. Two numerical examples are presented in this section. The results of an extensive study in this direction can be found in Ref. [29]. The first example deals with the realization of an elliptic cylindrical surface patch. The parametric representation of the patch is given by x ¼ 1:51625 cosðuÞ;
y ¼ v 0:5;
z ¼ 0:758125 sinðuÞ 0:658125
(9)
where 1:0534 u 2:0882 and 0:0 v 1:0: In order to realize the above surface, a rectangular sheet of size 1:6 m 1:2 m and thickness 5.0 mm is indented with a total of 121 indentors. Two of the edges of the flexible sheet of 1.2 m length are constrained. The circular indentors of 49.5 mm radius with spherical end (radius ¼ 100 mm) are used. The indentors are placed in an area of 1:1 m 1:1 m forming 11 rows; each row consisting of 11 equally spaced indentors (Fig. 6). The ratio of the first and second Mooney constants, C1/C2 is taken as 4.0. The analysis has been carried out with a total of 3072 triangular elements. Fig. 7 shows the desired and deformed surface patches. The figure shows only the region of the deformed surface, which is in contact with indentors. It also depicts the error between the desired and deformed surface patches at higher magnification.
Boundary condition
AVAB error (mm)
RMS error (mm)
Two sides constrained Four sides constrained
0.6936 0.7591
0.8611 0.9350
In order to realize the above surface patch, the analysis is also carried using a square sheet of size 1.6 m2 with all the four sides constrained. The AVAB error and the RMS error between the desired and deformed surfaces for the two cases are indicated in Table 1. The second numerical example deals with the realization of a doubly curved surface patch. The equation of the surface patch expressed in an explicit form is given as
5x 5x 5y 5y z ¼ 0:25 1 1 4 4 4 4
(10)
where 0:5 < x < 0:5 and 0:5 < y < 0:5: In order to realize this surface patch, a square membrane of size 1:6 m 1:6 m and thickness 6 mm is constrained at all the four edges and indented with a total of 121 indentors. Square indentors of 99 mm side are used in this case (Fig. 8). The other tooling parameters are same as that of previous example. The deformed configuration and the error profile are shown in Fig. 9. The AVAB and RMS errors are given in Table 2. The numerical examples presented in this section assume that there is no external loading on the flexible sheet. The numerical experiments were carried out for uniformly distributed load (UDL) acting on the flexible sheet. The loads are assumed to be acting on the flexible sheet in an area where indentors are placed. The effect of UDL on the accuracy of a typical realized shape is shown in Fig. 10. The thickness of the flexible sheet used here is 4 mm. As can be seen from figure, the effect of dead loading is to improve the accuracy of the shape at small loads. This is because, in the absence of any loading, the flexible sheet does not make a contact with all the indentors. The application of UDL is to bridge the gaps left between the sheet and indentors, if any. With further increase in the load, the accuracy deteriorates. The values of the AVAB and RMS errors presented above are only typical values and not necessary the best values obtained. Selecting the optimum values of the tooling parameters can further reduce the error between the desired and deformed surface patches. The dimension of the flexible sheet, the sheet thickness and the material of the flexible sheet form some of the important variables in present study of shape Table 2 AVAB and RMS errors (numerical example II) AVAB error (mm)
RMS error (mm)
1.1949
1.4755
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Fig. 6. Indentor layout (numerical example I).
realization. The effect of these parameters on the accuracy of realized shape is important and helps in knowing the optimum values of these parameters for a given application. The dimensions of the sheet are important as it affect the realizability of a given shape as well as the size of the surface tooling machine. The flexible sheet used here is of rectangular shape. The size of the sheet is larger than the area in which the indentors are placed (Fig. 11). The extra area of the sheet that does not come in contact with indentors is an important variable and can be measured in terms of margins. The effect of the sheet margins on the surface accuracy for a typical case is given in Table 3. The indentors were placed in an area of 1:0 m 1:0 m in this case. Table 3 Effect of sheet margin on the surface shape accuracy
Fig. 7. Realization of an elliptic cylindrical surface patch (numerical example I).
X and Y margin (mm)
AVAB error (mm)
RMS error (mm)
200.0 300.0 400.0 500.0
0.21750 0.23716 1.70882 4.58881
0.26821 0.27479 2.00238 4.98410
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Fig. 8. Indentor layout (numerical example II).
Fig. 9. Realization of a doubly curved surface patch (numerical example II).
As can be inferred from the table, better shape accuracies can be obtained at smaller margins of the flexible sheet. This is due to higher tensions of the flexible sheet associated with smaller margin values. At higher marginal values at the flexible sheet looses contact with more indentors giving rise to poor realized shapes. Fig. 12 shows the effect of sheet thickness on the surface shape accuracy in terms of AVAB error. In the present analysis, the flexible sheet is idealized as a membrane. Because of this, the sheet thickness has no effect on the deformed configuration of the membrane when there is no external loading. However, the effect of sheet thickness in the presence of an external loading is significant. This is evident from the figures which shows the variation of shape error with sheet thickness in the presence of an UDL of 0.25 kPa. The variation of the error between the desired and deformed surface shapes with the change in material properties of the flexible sheet is presented in Fig. 13. The ratio of
P.V.M. Rao, S.G. Dhande / Journal of Materials Processing Technology 124 (2002) 133–143
Fig. 10. Variation of RMS error with UDL.
Fig. 11. Sheet margins.
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Fig. 12. Variation of AVAB error with sheet thickness.
Fig. 13. Effect of sheet material on the surface shape accuracy.
first and second Moony constants C1/C2 is generally used to characterize rubber-like materials. The figure presents the variation of AVAB and RMS errors with increasing values of C1/C2. As the material of the flexible sheet tends to be more elastic with increasing values of C1/C2, better results were obtained at these values.
8. Conclusions Based on the computational results presented above, it can be concluded that a flexible surface tooling is an option for realizing a class of shapes by processes like composite layup. The proposed tooling is a variation of the already
P.V.M. Rao, S.G. Dhande / Journal of Materials Processing Technology 124 (2002) 133–143
existing discrete surface tooling which uses large number of discrete tool elements and is meant for those processes where the forming forces are considerably high. The proposed tooling is likely to be effective for realizing free-from shapes with small curvature values. Though the results presented here are quantitative in nature, the design inferences are more qualitative in nature. It has been felt that the numerical study of the proposed flexible surface tool should be extended to compare the results of computational analysis with the physical experiments. This is necessary to get a clear picture of the realizability of a shape by the proposed tooling. The efforts to build a flexible surface tooling system are already in progress and some of the lessons learnt from the computational analysis are being incorporated in this endeavor.
Acknowledgements The authors gratefully acknowledge the support rendered by Aeronautical Development Agency, Bangalore, India, by sponsoring the present research work through a project. References [1] N.M. Ham, M.S. Molitor, Thermal analysis of composite tooling materials, in: Proceedings of the Fifth Annual ASM/ESD Advanced Composites Conference, Dearborn, MI, 1989, pp. 161–165. [2] D.E. Hardt, R.D. Webb, Sheet metal die forming using closed-loop shape control, Ann. CIRP 31 (1) (1982) 165–169. [3] D.E. Hardt, Closed-loop sheet metal-forming processes, in: Proceedings of the IFAC Workshop on Intelligent Manufacturing System, Dearborn, MI, 1992, pp. 187–192. [4] N. Nakajima, A newly developed technique to fabricate complicated dies and electrodes with wires, Bull. JSME 12 (54) (1969) 1546–1554. [5] D.E. Hardt, D.C. Gossard, A variable geometry die for sheet metal forming, in: Machine Design and Control, Proceedings of the Joint Automatic Control Conference, San Francisco, CA, 1980. [6] R.D. Webb, Spatial frequency-based closed loop control of sheet metal forming, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1987. [7] R.D. Webb, D.E. Hardt, A transfer function description of sheet metal forming for process control, Trans. ASME J. Eng. Ind. 113 (1) (1991) 44–52. [8] D.C. Ousterhout, Design and control of a flexible process for threedimensional sheet forming, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1991. [9] D.E. Hardt, M.C. Boyce, K.B. Ousterhout, A. Karafillis, G.M. Eigen, CAD-driven flexible forming system for three-dimensional metal parts, in: Proceedings of the Sheet Metal and Stamping Symposium, SAE Special Publication No. 994, SAE, Warrendale, PA, 1993, pp. 69–76.
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[10] D.F. Walczyk, J. Lakshmikanthan, D. Kirk, Development of a reconfigurable tool for forming aircraft body panels, J. Manuf. Syst. 17 (4) (1998) 287–296. [11] D.F. Walczyk, Y.T. Im, A hydraulically actuated reconfigurable tool for flexible fabrication: implementation and control, Trans. ASME J. Manuf. Sci. Eng. 122 (3) (2000) 562–568. [12] A.S. Bogolyubov, A.G. Bratukhin, A.V. Karpov, V.V. Sadkov, Universal tooling with composite shells in aircraft industry, in: P.J. Hogg, D.H. Woolstencraft (Eds.), Composites Tooling II Proceedings, Elsevier, Oxford, 1992, pp. 123–141. [13] J.T. Oden, Finite Element of Nonlinear Continua, McGraw-Hill, New York, 1972. [14] J.M. Charrier, S. Shrivastava, R. Wu, Free and constrained inflation of elastic membrane in relation to thermoforming-axisymmetric problems, J. Strain Anal. Eng. Des. 22 (2) (1987) 115–125. [15] J.M. Charrier, S. Shrivastava, R. Wu, Free and constrained inflation of elastic membrane in relation to thermoforming-non-axisymmetric problems, J. Strain Anal. Eng. Des. 24 (1989) 55–74. [16] H.G. DeLorenzi, H.F. Nied, Blow modeling and thermoforming of plastics: finite element modelling, Comput. Struct. 26 (1987) 197– 206. [17] F. Gruttman, R.L. Taylor, Theory and finite element formulation of rubber-like membrane shell using principal stretches, Int. J. Numer. Meth. Eng. 35 (1992) 1111–1126. [18] N.G. Zamani, D.F. Watt, M. Esteghamatian, Status of the finite element method in the thermoforming process, Int. J. Numer. Meth. Eng. 28 (1989) 2681–2693. [19] M.H. Gadala, Numerical solution of nonlinear problem continua. II. Survey of incompressibility constraints and software aspects, Comput. Struct. 22 (1986) 841–855. [20] M. Levinson, The application of the principle of the stationary potential energy to some problems in finite elasticity, Trans. ASME J. Appl. Mech. 32 (3) (1965) 656–660. [21] E.B. Becker, Numerical solution of a class of problem of finite elastic deformation, Ph.D. Thesis, University of California, Berkeley, CA, 1966. [22] W.W. Feng, W.H. Yang, On the contact problem of an inflated spherical nonlinear membrane, Trans. ASME J. Appl. Mech. 40 (1973) 209–214. [23] W.W. Feng, J.T. Tielking, P. Huang, The inflation and contact constraints of a rectangular Moony membrane, Trans. ASME J. Appl. Mech. 41 (1974) 979–984. [24] J.T. Teelking, W.W. Feng, The application of the minimum potential energy principle to nonlinear axisymmetric membrane problems, Transactions of the ASME Journal of Applied Mechanics 41 (2) (1974) 491–496. [25] W.W. Feng, P. Haung, On the inflation of a plane nonlinear membrane, Trans. ASME J. Appl. Mech. 41 (3) (1974) 767–771. [26] W.W. Feng, P. Huang, On the general contact problem of an inflated nonlinear plane membrane, Int. J. Solids Struct. 11 (1975) 437– 448. [27] J.T. Oden, T. Sato, Finite strain and displacement of elastic membranes by the finite element method, Int. J. Solids Struct. 3 (1967) 471–488. [28] G.V. Reklaitis, A. Ravindran, K.M. Ragsdell, Engineering Optimization Methods and Application, Wiley, New York, 1883. [29] P.V.M. Rao, Shape realization using a flexible surface tooling system, Ph.D. Thesis, Indian Institute of Technology, Kanpur, 1996.
A flexible surface tooling for sheet-forming processes: conceptual studies and numerical simulation P.V.M. Raoa,*, Sanjay G. Dhandeb a
Department of Mechanical Engineering, Indian Institute of Technology, New Delhi 110016, India b Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208016, India Received 31 July 2000; accepted 5 March 2002
Abstract All sheet-forming processes suffer from a limitation that tooling in the form of molds/dies is a precursor for realizing any simple or complex shape. In order to overcome this limitation, discrete surface tooling has been proposed in the past. This consists of a large number of closely spaced surface tool elements arranged in a matrix form whose heights can be adjusted to approximate the contour of desired surface shapes. Such a tooling has been successfully demonstrated experimentally. In this paper, a variation of discrete surface tooling, called as flexible surface tooling, has been proposed. The proposed tooling consists of discrete surface tool elements draped with flexible sheet of rubber-like material to provide a continuous surface required for tooling applications. Computational simulation of the proposed tooling is carried out for feasibility studies. Computational simulation involves the deformation analysis of rubber-like membranes in multiple contact by FEM. The results of the analysis indicate that the flexible surface tooling is an option for sheet-forming processes in general and for processes like composite layup in particular. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Sheet-forming processes; Flexible surface tooling; Rubber-like material; Computational simulation
1. Shape-conforming processes In a class of manufacturing processes, shape is realized by making the original material in the form of a deformable sheet to conform to a relatively rigid surface of the tool. Such shape conformance involves an elastic/plastic deformation of the material by application of heat/pressure. As the shape realization here is due to close conformance of one surface to another, these processes can be called as shape-conforming processes. Press brake bending of sheet metal components, stretch forming, deep drawing, composite layup, compression molding of sheet molding compound (SMC) and thermoforming are some of the examples in this category (Fig. 1). In all shape-conforming processes, tool is a precursor for fabrication of every simple or complex shape. Tooling in the form of molds and dies provide a single or multiple base surfaces required for shaping the metallic sheet or the combination of reinforcing and matrix materials. If we neglect the shape variations due to springback and thermal effects, the shape of the product obtained in these processes is completely dictated by the shape of the mold or die used. It
*
Corresponding author.
can said that, in an ideal situation, the realizability of a shape in all shape-conforming processes is mainly governed by the tool used and is independent of the process. This reduces the problem of shape realization in these processes to that of realization of a tool surface. The realization of a tool surface for shape-conforming processes is a time consuming and often a costly affair. Typically, one spends more time on design and fabrication of a tooling than that involved in the fabrication of component itself. For example, the mold design and fabrication for a free-from composite product often involves a slow, laborintensive stage of master preparation [1]. In the case of sheet metal forming, design and fabrication of a die is an experimentally iterative process. A number of forming experiments with improved die shape at each stage is required to coverage to a final die shape which can be used for realizing the desired component shapes [2,3]. In a prototyping environment, where the design changes are frequent and the number of components required are often limited, such forming experiments are both costly and time consuming. Moreover, in such an environment the accuracy requirements of the product shape are not very critical. In such cases, the hard tooling is generally expensive and can be substituted by a flexible tooling for a class of component shapes. One such tooling which has been
0924-0136/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 1 4 1 - 3
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Fig. 2. A discrete approximation to a continuous surface. Fig. 1. Shape-conforming processes.
proposed in this work for shape-conforming processes is flexible surface tooling.
2. Flexible surface tooling The flexible surface tooling is based on the concept of discrete approximation to a continuous surface of a mold or a die (Fig. 2). It consists of a number of closely spaced multiple rigid surface tool elements, known as indentors, each of which is a surface element of an expected contour. The height of the indentors can be adjusted to approximate the desired surface shapes. The positioning of each indentor can be carried out either manually or using a computer control. The discrete nature of the tool can be overcome by placing a deformable elastomeric sheet over the indentors. The flexible sheet draped over the discrete surface provides with a continuous surface required for the tooling application (Fig. 3). The advantage of such a tooling is that it is adjustable. A variety of surface shapes can be realized by properly adjusting the heights of surface tool elements. The total time involved in the tool set up is considerably less than that involved in the development of a hard tool. The successful development of a flexible surface tooling will result in considerable reduction of the time involved in prototyping certain class of shapes. Moreover, automated nature of the process may relegate the need of some of the experts/skilled labors involved in tool design and development. Experiments with new shapes and incorporating design changes will be much easier once such a tooling is made available.
3. Earlier work The concept of adjustable discrete surface tooling was first given by Nakajima [4]. The concept was used to fabricate complicated dies for presswork and electrodes for electrochemical machining. An extensive research work on the concept of discrete surface tooling has been carried out by Hardt et al. [2,3,5–9] at the Laboratory for Manufacturing and Productivity at Massachusetts Institute of Technology. Unlike the forming system proposed by Nakajima, they propose the use of a variable configuration discrete element die in a closed loop shape control system (Fig. 4). More recently, Walczyk et al. [10,11] have proposed a reconfigurable tooling system along the lines of Hardt et al. for forming sheet metal parts and composites. The tooling system is hydraulically actuated and pin configuration used is 4 4. They also studied relative merits and demerits of using open and closed loop control for forming process using their die. Bogolyubov et al. [12] have proposed to use removable glass plastic shells resting on point supports as a universal tooling for forming airframe structures. Deformation analysis of shell resting on point supports and carrying uniformly distributed loading is carried out to demonstrate the concept. The discrete element die proposed by Nakajima and improvements proposed by Hardt et al. and Walczyk et al. is primarily meant for forming sheet metal components. The work carried out by both these groups is based on the forming experiments. Another approach to study the realizability of shapes by a discrete element die is using numerical methods such as FEM. A computational study will be
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Fig. 3. Flexible surface tooling.
Fig. 4. Open and closed loop shape control systems for sheet metal forming with variable configuration die.
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helpful in reducing the number of forming experiments required to arrive at the correct die shape. However, in the case of sheet metal-forming processes a numerical study accounting for large strains, material properties of the sheet, springback and the tribological aspects associated with the process is difficult. In the case of composite fabrication processes such as layup, the complexities associated with the sheet metal forming are absent. In this case, the forming forces are also considerably less. However, the thermal effects are likely to be present in all composite fabrication processes. But, in the case of wet or prepreg layup with room temperature curing these effects can be neglected. These simplifications make the numerical study of the flexible surface tooling feasible for applications like composite layup. This work deals with a computational study of the realizability of simple and complex shapes by the proposed flexible surface tool. A computational procedure, which deals with the deformation analysis of nonlinear elastic membranes in multiple contact, has been developed and used for the tool simulation. The effect of geometric and material properties of the flexible sheet on the shape accuracies has been studied with an objective to identify the range of these parameters for a given application. Unlike the tooling proposed by others [2–12], which are primarily meant for sheet metal-forming processes, the present tooling is meant for composite fabrication using layup. Composite layup being a process where forming forces are considerably lower, a relatively small number of tool elements can be used for forming free-form shapes, which reduces set up time considerably. Further, the effect of various tooling parameters on shape realizability has been studied in the present work, which was not dealt in detail by earlier researchers.
of the undeformed body [13]. For rubber-like materials, the most widely used form of w is the Moony–Rivlin material description. The strain energy function for this case is given by w ¼ C1 ðI1 3Þ þ C2 ðI2 3Þ;
I3 ¼ 1
(1)
where C1 and C2 are the material constants. The contact problems involving hyper-elastic membranes have been attempted by many researchers in the past. The incremental-iterative strategies based on the total or updated Lagrangean concepts are commonly used to solve this nonlinear problem [14–18]. The material incompressibility and contact constraints are dealt either by mixed formulation or penalty-based approach [19]. However, when the problem involves frictionless contacts and plane or membrane state of stress, it can also be attempted by a mathematical programming approach. This approach is commonly known as energy search and has been followed by a number of investigators for the deformational analysis of rubber-like membranes with or without contact constraints [20–26]. In this work, the problem of deformation analysis of flexible sheet has been attempted by finite element discretization and potential energy minimization in the presence of contact constraints. The approach is the same as that of Becker [21] except for the contact constraints. Three-noded triangular elements are used for the finite element discretization. Moony–Rivlin model is used for the strain energy calculations with incompressibility condition. The incompressibility condition has been dealt by analytical treatment which is possible in the case of plane or membrane state of stress [19,27]. The conjugate gradient method with Polak– Ribiere correction is used for energy minimization. The contact constraints are treated using a penalty approach.
5. Problem formulation 4. Deformation analysis In the proposed tooling system, the deformed configuration of the flexible elastomeric sheet is used as a base surface for tooling requirements. The elastomeric sheet undergoes deformation due to indentation by a matrix of indentors. Knowing the deformed configuration of the sheet for a given configuration of the indentors is an important aspect in the above-mentioned tooling. This involves solving of an elastostatic contact problem accounting for the large deformations of rubber-like materials. The flexible sheet used for the purpose of tooling here is of uniform thickness before deformation and its thickness is small compared to its other dimensions (less than 1%). As the state of stress in the sheet is essentially two-dimensional in this case, the sheet can be idealized as a thin incompressible isotropic hyper-elastic membrane. The hyper-elastic materials are characterized by the presence of an elastic potential, w, known as strain energy density, which represents the strain energy per unit volume
Consider the geometry of deformation of a thin homogeneous elastic membrane of uniform thickness h0. The undeformed and deformed configurations are represented by O0 and O, respectively (Fig. 5). The region O0 is bounded by planes X3 h0 =2, the middle surface being X3 ¼ 0. The position vectors x ¼ x1 i1 þ x2 i2 þ x3 i3 and X ¼ X1 i1 þ X2 i2 þ X3 i3 of the deformed and initial configurations, respectively, are related through expression given by x¼Xþu
(2)
where u is the displacement vector. The Lagrangean strain tensor is now given by the relation 1 @xk @xk gij ¼ dij (3) 2 @Xi @Xj where dij is the Kronecker delta. Thickness of the membrane after deformation, h, is given by h ¼ lh0
(4)
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Fig. 5. The geometry of deformation of a membrane.
where l is the extension ratio at the middle surface in a direction normal to the middle surface. Assuming the membrane to be thin, strain tensor can be written as 1 @ua @ub @ui @ui gab ¼ þ þ ; 2 @Xb @Xa @Xa @Xb 1 g23 ¼ 0; g33 ¼ ðl2 1Þ (5) g13 ¼ 0; 2 where a; b ¼ 1; 2 and i ¼ 1; 2; 3. The three strain invariants I1, I2, I3 are given by I1 ¼ l2 þ 2ð1 þ gaa Þ;
I2 ¼ l4 þ l2 I1 þ
1 I3 ; l2
I3 ¼ l2 ð1 þ 2gaa þ 2eab elm gal gbm Þ
(6)
where a; b; g; m ¼ 1; 2 and eab, elm are the two-dimensional permutational symbols ðe12 ¼ 1; e21 ¼ 1; e11 ¼ e22 ¼ 0Þ. Imposing the incompressibility constraints I3 ¼ 1, the above expression can be written as I1 ¼ l2 þ 2ð1 þ laa Þ;
I2 ¼ l4 þ l2 I1 þ
l2 ¼ ð1 þ 2gaa þ 2eab elm gal gbm Þ1
1 I3 ; l2 (7)
The potential energy of deformed membrane, P, is given by Z Z (8) P ¼ U W ¼ wðI1 ; I2 Þ dv t v ds v
s
where U is the strain energy of the deformed membrane and W the virtual work of external traction forces, tv the kinematically admissible displacement field and s the part of boundary on which tractions are specified.
6. Computational procedure The computational procedure used here consists of minimizing the total potential energy of the system subjected to given geometric constraints. For this purpose, the domain O0
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is discretized into three-noded triangular finite elements. The total number of nodes and elements being N and E, respectively. The number of variables in the minimization problem are 3N0 , N0 being the total number of unconstrained nodes. The objective function is the total potential energy of the system summed over E elements. Potential energy of the individual element consist of the strain energy due to deformation and potential of loads, if any, on the element. In order to calculate the strain energy due to stretching of the membrane, undeformed element is placed in the plane of deformed one. Such a transformation does not affect the strain energy calculation as it involves only rigid body translation and rotation. The external forces in this problem are in the form of pressure loads acting on the surface element. As the membrane deforms, the changing direction of this pressure and the changing area of the element on which it acts is accounted. The constraints in this problem are in the form of contacts. The deformed configuration of the flexible sheet due to indentation is such that no interpenetration of the indentor and membrane material is allowed. This condition on the kinematically admissible displacement field of the flexible sheet can be imposed by penalty approach. The penalty is being proportional to the amount of violation. To solve this constrained optimization problem efficiently, the sequential unconstrained minimization technique (SUMT) has been used. SUMT transforms the constrained optimization problem to an unconstrained one by penalizing the constraint violation if any. The conjugate gradient algorithm with Polak–Ribiere correction is adopted as the minimization technique. The method of multipliers (MOM) type of penalty which is equally effective in both feasible and non-feasible solution domains is used for transforming constrained optimization problem into an unconstrained one [28]. The computer program developed for this purpose is written in C is implemented on DEC-3000 platform. The correctness of the formulation and the numerical scheme used here has been verified by running the computer program developed for three standard test cases taken from the works of Becker [21], Feng et al. [23] and Feng and Huang [25].
7. Results Based on the theoretical formulation and the computational procedure presented earlier, numerical experiments were carried out with an objective to study the shape realization by the proposed tooling. The software developed for this purpose is in three modules: a preprocessor, a deformational analysis program and a post-processor. The input to shape realization software consist of the following:
The shape to be realized or the desired surface shape;
Dimensions of the flexible sheet;
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Sheet thickness; Material properties of the sheet; Geometry of the individuals indentors; Layout or the arrangement of indentors.
Knowing the heights by which indentors are to be raised to approximate a given continuous surface is the first step in the process of shape realization. The preprocessor part of the software deals with this aspect. For this purpose, a common coordinate system is used to define the desired shape and the indentor positions. Each of the indentors is now raised incrementally till it makes a contact with the desired shape. After all the indentors are raised in a similar manner, it forms a discrete approximation to the given continuous surface. The height of the indentors decided as part of the preprocessor together with various tooling parameters mentioned above, from an input to analysis module. The deformational analysis program is based on the theoretical formulation and the computational procedure described earlier in this paper. The analysis program uses the middle surface of the flexible sheet as a reference surface, whereas the upper surface of the flexible sheet is the desired one. This offset can be easily accounted using Eq. (4) as the deformation of the flexible sheet proceeds. As a part of the post-processor, the deformed configuration of the flexible sheet obtained as a result of analysis is compared with the desired surface shape to evaluate the shape error. The realizability of a given shape here is measured in terms of root mean square (RMS) error and the average absolute (AVAB) error between the desired and realized surface shapes. Two numerical examples are presented in this section. The results of an extensive study in this direction can be found in Ref. [29]. The first example deals with the realization of an elliptic cylindrical surface patch. The parametric representation of the patch is given by x ¼ 1:51625 cosðuÞ;
y ¼ v 0:5;
z ¼ 0:758125 sinðuÞ 0:658125
(9)
where 1:0534 u 2:0882 and 0:0 v 1:0: In order to realize the above surface, a rectangular sheet of size 1:6 m 1:2 m and thickness 5.0 mm is indented with a total of 121 indentors. Two of the edges of the flexible sheet of 1.2 m length are constrained. The circular indentors of 49.5 mm radius with spherical end (radius ¼ 100 mm) are used. The indentors are placed in an area of 1:1 m 1:1 m forming 11 rows; each row consisting of 11 equally spaced indentors (Fig. 6). The ratio of the first and second Mooney constants, C1/C2 is taken as 4.0. The analysis has been carried out with a total of 3072 triangular elements. Fig. 7 shows the desired and deformed surface patches. The figure shows only the region of the deformed surface, which is in contact with indentors. It also depicts the error between the desired and deformed surface patches at higher magnification.
Boundary condition
AVAB error (mm)
RMS error (mm)
Two sides constrained Four sides constrained
0.6936 0.7591
0.8611 0.9350
In order to realize the above surface patch, the analysis is also carried using a square sheet of size 1.6 m2 with all the four sides constrained. The AVAB error and the RMS error between the desired and deformed surfaces for the two cases are indicated in Table 1. The second numerical example deals with the realization of a doubly curved surface patch. The equation of the surface patch expressed in an explicit form is given as
5x 5x 5y 5y z ¼ 0:25 1 1 4 4 4 4
(10)
where 0:5 < x < 0:5 and 0:5 < y < 0:5: In order to realize this surface patch, a square membrane of size 1:6 m 1:6 m and thickness 6 mm is constrained at all the four edges and indented with a total of 121 indentors. Square indentors of 99 mm side are used in this case (Fig. 8). The other tooling parameters are same as that of previous example. The deformed configuration and the error profile are shown in Fig. 9. The AVAB and RMS errors are given in Table 2. The numerical examples presented in this section assume that there is no external loading on the flexible sheet. The numerical experiments were carried out for uniformly distributed load (UDL) acting on the flexible sheet. The loads are assumed to be acting on the flexible sheet in an area where indentors are placed. The effect of UDL on the accuracy of a typical realized shape is shown in Fig. 10. The thickness of the flexible sheet used here is 4 mm. As can be seen from figure, the effect of dead loading is to improve the accuracy of the shape at small loads. This is because, in the absence of any loading, the flexible sheet does not make a contact with all the indentors. The application of UDL is to bridge the gaps left between the sheet and indentors, if any. With further increase in the load, the accuracy deteriorates. The values of the AVAB and RMS errors presented above are only typical values and not necessary the best values obtained. Selecting the optimum values of the tooling parameters can further reduce the error between the desired and deformed surface patches. The dimension of the flexible sheet, the sheet thickness and the material of the flexible sheet form some of the important variables in present study of shape Table 2 AVAB and RMS errors (numerical example II) AVAB error (mm)
RMS error (mm)
1.1949
1.4755
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Fig. 6. Indentor layout (numerical example I).
realization. The effect of these parameters on the accuracy of realized shape is important and helps in knowing the optimum values of these parameters for a given application. The dimensions of the sheet are important as it affect the realizability of a given shape as well as the size of the surface tooling machine. The flexible sheet used here is of rectangular shape. The size of the sheet is larger than the area in which the indentors are placed (Fig. 11). The extra area of the sheet that does not come in contact with indentors is an important variable and can be measured in terms of margins. The effect of the sheet margins on the surface accuracy for a typical case is given in Table 3. The indentors were placed in an area of 1:0 m 1:0 m in this case. Table 3 Effect of sheet margin on the surface shape accuracy
Fig. 7. Realization of an elliptic cylindrical surface patch (numerical example I).
X and Y margin (mm)
AVAB error (mm)
RMS error (mm)
200.0 300.0 400.0 500.0
0.21750 0.23716 1.70882 4.58881
0.26821 0.27479 2.00238 4.98410
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Fig. 8. Indentor layout (numerical example II).
Fig. 9. Realization of a doubly curved surface patch (numerical example II).
As can be inferred from the table, better shape accuracies can be obtained at smaller margins of the flexible sheet. This is due to higher tensions of the flexible sheet associated with smaller margin values. At higher marginal values at the flexible sheet looses contact with more indentors giving rise to poor realized shapes. Fig. 12 shows the effect of sheet thickness on the surface shape accuracy in terms of AVAB error. In the present analysis, the flexible sheet is idealized as a membrane. Because of this, the sheet thickness has no effect on the deformed configuration of the membrane when there is no external loading. However, the effect of sheet thickness in the presence of an external loading is significant. This is evident from the figures which shows the variation of shape error with sheet thickness in the presence of an UDL of 0.25 kPa. The variation of the error between the desired and deformed surface shapes with the change in material properties of the flexible sheet is presented in Fig. 13. The ratio of
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Fig. 10. Variation of RMS error with UDL.
Fig. 11. Sheet margins.
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Fig. 12. Variation of AVAB error with sheet thickness.
Fig. 13. Effect of sheet material on the surface shape accuracy.
first and second Moony constants C1/C2 is generally used to characterize rubber-like materials. The figure presents the variation of AVAB and RMS errors with increasing values of C1/C2. As the material of the flexible sheet tends to be more elastic with increasing values of C1/C2, better results were obtained at these values.
8. Conclusions Based on the computational results presented above, it can be concluded that a flexible surface tooling is an option for realizing a class of shapes by processes like composite layup. The proposed tooling is a variation of the already
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existing discrete surface tooling which uses large number of discrete tool elements and is meant for those processes where the forming forces are considerably high. The proposed tooling is likely to be effective for realizing free-from shapes with small curvature values. Though the results presented here are quantitative in nature, the design inferences are more qualitative in nature. It has been felt that the numerical study of the proposed flexible surface tool should be extended to compare the results of computational analysis with the physical experiments. This is necessary to get a clear picture of the realizability of a shape by the proposed tooling. The efforts to build a flexible surface tooling system are already in progress and some of the lessons learnt from the computational analysis are being incorporated in this endeavor.
Acknowledgements The authors gratefully acknowledge the support rendered by Aeronautical Development Agency, Bangalore, India, by sponsoring the present research work through a project. References [1] N.M. Ham, M.S. Molitor, Thermal analysis of composite tooling materials, in: Proceedings of the Fifth Annual ASM/ESD Advanced Composites Conference, Dearborn, MI, 1989, pp. 161–165. [2] D.E. Hardt, R.D. Webb, Sheet metal die forming using closed-loop shape control, Ann. CIRP 31 (1) (1982) 165–169. [3] D.E. Hardt, Closed-loop sheet metal-forming processes, in: Proceedings of the IFAC Workshop on Intelligent Manufacturing System, Dearborn, MI, 1992, pp. 187–192. [4] N. Nakajima, A newly developed technique to fabricate complicated dies and electrodes with wires, Bull. JSME 12 (54) (1969) 1546–1554. [5] D.E. Hardt, D.C. Gossard, A variable geometry die for sheet metal forming, in: Machine Design and Control, Proceedings of the Joint Automatic Control Conference, San Francisco, CA, 1980. [6] R.D. Webb, Spatial frequency-based closed loop control of sheet metal forming, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1987. [7] R.D. Webb, D.E. Hardt, A transfer function description of sheet metal forming for process control, Trans. ASME J. Eng. Ind. 113 (1) (1991) 44–52. [8] D.C. Ousterhout, Design and control of a flexible process for threedimensional sheet forming, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1991. [9] D.E. Hardt, M.C. Boyce, K.B. Ousterhout, A. Karafillis, G.M. Eigen, CAD-driven flexible forming system for three-dimensional metal parts, in: Proceedings of the Sheet Metal and Stamping Symposium, SAE Special Publication No. 994, SAE, Warrendale, PA, 1993, pp. 69–76.
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