VI. Approaches to Understanding Metal Forming

  • ANALYTICAL VERSUS EXPERIMENTAL APPROACH
  • When developmental work is conducted with production equipment in the plant, it disrupts operation of the plant and causes intolerable expenses. It is common practice to establish separate research and development activities. The full spectrum of such an activity and the alternative approaches, complementing one another, are presented in Fig. 35.

    Short of doing developmental work on a production scale, all other methods can be classified as modeling or simulation. In Fig. 35, the pie is divided into formulation and a physical lab of simulation methods or into mathematical and physical modeling sections. Mathematical modeling is further divided into analytical and numerical modeling. The old, but still in existence, method of developing and using empirical expressions is overlooked here.

  • PHYSICAL MODELING; SCALED-DOWN MODELS AND VISIOPLASTICITY
  • The most common physical modeling procedures are modeled by scaling down of size or by experimenting with modeling materials. In either case the equipment may be highly instrumented with sensors and recording and control capabilities that are not used on production equipment. When the proper tooling

    FIG. 35. Classification for planned activities in metal forming.

    and procedures to produce the desired product have been identified by physical modeling, the transition to the production mode is usually smooth with no surprises.

    Scaling down of the size of a component with which the experimental work proceeds brings savings in materials, tooling, equipment size, space requirements, and operating costs as well as expediting the development stage. The discussion of mathematical modeling will show that for metal forming at room temperature the scaling up is automatic, and conditions, for example, that prevent failure on small-scale models are identical to those required for the production size.

    The loads employed to impose plastic flow in the scaled-down model are proportionally lower than those required for the production size. Thus, the measured loads from the experiment provide the information regarding loads to be needed for production runs.

    Production equipment can be replaced in the laboratory by less expensive means when the actual workpiece is modeled by a softer material. Lead was used to study flow patterns for many years. Plasticine and wax are very popular today. Since the loads are smaller, expensive die materials can be replaced by aluminum, wood, or Plexiglas. The equipment itself need not have a strong frame or a large power supply. It can be built at a fraction of the cost of production equipment and is inexpensive to operate. Several laboratories have been built for modeling material simulation by wax.

    To make the model realistic, waxes can be mixed with additives so that their properties may resemble those of the workpiece material. Strain-hardening properties and strain-rate behavior can be controlled by those additives. Even tool materials can be chosen so that tool rigidity can be made to match the strength of the wax in the same manner that the real tool will relate to the workpiece. Friction conditions can be controlled by means of lubrication to best simulate the friction conditions during actual production (Wanheim, 1978-1979). The main purpose of studying through modeling materials is to observe the flow patterns and thus to develop dies that produce a sound product.

    A picture of a cross section along the axis of symmetry of a multilayer wax forging is reproduced here as Fig. 36. The original cylindrical workpiece was constructed from two colors of wax, checkered in coaxial cylinders and flat planes normal to the axis of symmetry. Each rectangle represents a ring of rectangular cross

    FIG. 36. Wax models. (From Wanheim, 1978-1979).

     

    section. The boundaries of the rectangles form a rectangular network of grid lines. These billets can be forged, extruded, rolled, or formed by any other conceivable metal-forming process. A full billet can be formed by using a complete set of dies. On completion of the forming process, the billet can be sliced and the deformation pattern studied. On the other hand, half billets can be formed and the flat surface may be exposed to view through a Plexiglas window. Pictures of the deforming specimen can then be taken at predetermined intervals. The position coordinates of each corner of the checkered grid can be determined at each interval. This procedure is defined as "visioplasticity." The data is then fed into a computer with a program that will determine the following: (1) a displacement field, (2) a velocity vector for each point at each moment. Thus, when the velocity field is fully determined, (3) a strain-rate field is calculated from the velocity field; (4) a stress field is calculated from the strain-rate field, the history of deformation, and the characteristic stress-strain relationship for the workpiece; (5) the stress load on the surfaces of the die is calculated from the stress field, and (6) the measurement of the total load during forming is determined. By integration of surface stresses on the die over its entire surface, the load is calculated and compared with the measured total force.

    In parallel with the study of the distorted grid in the workpiece, the deformations of the die itself are occasionally measured by the light photoelastic method. In those cases, it is possible to compare the stresses in the die, including sites of stress concentration and their levels, with those derived from the workpiece.

    The far-reaching program described here is still in its developmental stages. Nevertheless, the results achieved so far leave no doubt about the contributions already made and the potential value of this approach. Preliminary tooling for new products and even for new processes can be designed reliably by using a modeling materials laboratory at a fraction of the cost of running developmental work on production equipment.

  • ANALYTICAL AND NUMERICAL METHODS OF MODELING
  • Exact solutions are not available for problems in metal forming. Approximations and simplifying assumptions are inevitable, and many approaches-slug equilibrium, slip-line techniques, and others-have been partially successful.

    Limit analysis (Avitzur, 1980) is a promising approach and is being used with increasing frequency. In this approach, as applied to the study of drawing or extrusion force, two approximate solutions are developed. One, the upper-bound solution (Kudo, 1960, 1961) provides a value that is known to be higher than or equal to the actual force; the other, the lower-bound solution, provides a value that is known to be equal to or lower than the actual force; the actual force thus lies between the two solutions. For example, in Fig. 37 with the drawing stress as ordinate and the semicone angle of the die as abscissa, upper- and lower-bound solutions are plotted for several reductions together with corresponding measured values of the actual stress. Even when experimental results are not available, it is expected that the actual stress and the exact solution, if these were available, would lie between the upper and lower bounds as obtained analytically. Thus, by limit analysis, an approximate solution is given with an estimate of the maximum possible error. The gap between upper- and lower-bound solutions may be narrowed by providing several upper bounds, choosing the lowest, and by providing several lower bounds, choosing the highest. Upper- and Lower-bound solutions are obtained only by following strict rules (including the requirement of the proper description of friction behavior and

    material characteristics). A full illustration of limit analysis is given by Avitzur (1983).

    A large number of analytical approaches complement the still existing empirical expressions. Each approach has some advantages and shortcomings, and until a perfect approach is available, they all complement and assist each

    FIG. 37. Relative drawing stress versus semicone angle and percentage reduction in area during wire drawing.

     

    other. With further progress, and as each approach gets closer to the exact solution, they also get closer to each other to the extent that some initially thought to be conflicting rival approaches turn out to be identical. The upper- and lower-bound solutions for some metal forming processes, for example, disc and strip forging (Avitzur, 1983) are so close that when plotted together, both curves are almost overlapping. The expressions for drawing in plane strain, obtained by the upper bound are identical to those obtained by a force equilibrium on a rigid triangular element (Westwood, 1960). The same similarities can be observed between the upper bound and slipline solutions to other problems in plane strain. The popular slipline technique identifies surfaces within the workpiece, along which shear stress is at its peak. Thus, the material along these surfaces shears producing plastic flow and the shaping of the workpiece. The slip line technique, based on the stress equilibrium approach, identifies surfaces along which maximum shear stress prevails. In a comparable manner the upper bound approach, based on deformation patterns, offers the stream line technique and the associated stream function. See Talbert and Avitzur (1996). Here a flow line along the path of a particle through the deformation region determines a function f (x, y, z and e) = Constant, where x, y, and z are coordinates in space, and e identifies each specific flow line. When this function and its derivatives are set to conform to the geometrical boundary conditions imposed by the tooling, a stream function is derived. Such a stream function leads to the determination of the strains and strain rate components and to the derivation of an upper bound solution for the energy required. In practice this method is applied to two-dimensional conditions. Although this method is applicable to non-steady state flow, it is more easily applied to steady state flow.

    In the free slug equilibrium approach bulk elements of the workpiece are considered and the forces acting on them are set in equilibrium. As has been shown, with proper adaptation, the free slug approach provides the following expression for the relative drawing stress in wire drawing, which is very close to the upper bound solution presented in Fig. 38.

    (Avitzur, 1980).

    With the computer revolution, several numerical methods emerged as computational tools to study metalforming. In the procedures described next the workpiece is divided into a finite number of elements. Each element experiences a continuous plastic deformation pattern, which may also be a rigid body motion. The elements also experience sliding motion with respect to each other and along their contact surfaces with the tools. The elements are bordered (restricted) by each other, and by the surfaces of the tools, or they are free, exposed surfaces. The total energy required to impose plastic flow of the entire workpiece is calculated piece by piece as the sum of energies expended in each of the elements and along their surfaces.

    The numerical methods mentioned can be compiled into three major groups, i.e., the "Upper Bound Elemental Technique" (UBET), its offspring, the "Spatial Elementary Rigid Region" (SERR) Method, and the "Finite Element Method" (FEM). UBET was introduced by Prof. Kudo (1960, 1961) as an extension to his analytical upper bound solutions to fundamental elementary shapes. Any workpiece of complex shape undergoing plastic deformations can be constructed as an assembly of a number of the elementary shapes. The UBET computerized procedure then determines the total deformation energy. The initial UBET procedures were restricted to axisymmetric and plain strain deformations. The basic elements were rings and cylinders of rectangular and triangular cross-sections.

    The extension to three-dimensional problems by the SERR method was made by the introduction of the Tetrahedral element that undergoes a linear rigid body motion. A workpiece of any shape can be constructed from an assembly of tetrahedral elements where the curved surfaces interfacing the tools are approximated by plane surfaces. The plastic deformation of the workpiece is then accommodated by the sliding of the elements, one with respect to each other, and along the surfaces of the tool. See Avitzur (1993), Azarkhin and Richmond (1992), and Prafulla Kumar Kar (1998). The tetrahedrons themselves act as rigid bodies and consume no energy of deformations. Energy is consumed only along the surfaces of the tetrahedral elements where sliding occur.

    The potential for the SERR method is unlimited. The linear rigid body motion of the elements may eventually be allowed to experience the more general rigid body rotational motion where the plane surfaces of the tetrahedral element are replaced by more complex curved surfaces. Any shape of the workpiece will be automatically divided to a minimal number of tetrahedral elements, and the motion of each element will be automatically determined. Better precision will be reached when each large element will be subdivided automatically to (twelve) smaller ones.

    Today the most popular of the numerical methods is the FEM, which proved itself in other fields such as electricity, heat transfer, and fluid flow. In the FEM the workpiece is divided into a finite number of elements that may undergo plastic deformations. In one approach, for each element the differential equations of equilibrium of the stresses are replaced by an equilibrium on a set of finite differences of the stresses.

    Material properties may be introduced in any complexity desired to reflect real behavior. The procedure to obtain flow patterns and tool loads is iterative in nature. The more complex the geometry and the material, the longer the time it takes; and larger computers are then required for a single solution. There is no doubt that with the advancement in computer technology and better finite element procedures more and more of the tooling design for metal forming will be assigned to the FEM method.

  • PROS AND CONS OF THE DIFFERENT MODELING PROCEDURES
  • The construction of production tools for large components may be rather expensive. Ideally, the first tool for a new product should be the best one possible. In practice, however, the first tool may fail to perform as expected, resulting in a defective product or a fractured tool. The possible modes of failure are numerous, and some of them are most common. Modeling by scaling down, by modeling materials, and by mathematical models makes the road to the choice of best production tooling shorter. Comparisons and guidelines for the choice of the most suitable modeling method for each circumstance will be attempted.

    In comparing experimental procedures of small-scale or material modeling on the one hand with mathematical modeling on the other hand, one notes that the development of a mathematical model is much more time-consuming. In the study of flow through conical converging dies (Fig. 39), many flow patterns are observed that lead to failure. The choices of die angle, reduction, and friction that lead to those failures are presented by the inserts. For example, the development of the criterion for the prevention of central burst took about 18 months. This derivation was based on years of work in the general area of flow through conical converging dies and the application of the upper bound approach to metal-forming processes in general. Such work requires a specialized expertise, which is quite scarce at present. Furthermore, the original criteria, although useful, may need further study.

    The specific problem that prompted the central-burst study (Avitzur, 1968) could have been solved by a trial-and-error, experimental procedure in a few weeks or months by personnel without an analytical background. As a matter of fact, experimental procedures were employed successfully for the elimination of central burst, when discovered, prior to the development of the criteria. Every new appearance of the phenomenon called for a new study, but the resolution of the problem did not add to the understanding of the causes of the defect or to a solution to the problem when it appeared again. The analytical criterion, on the other hand, is universal; it applies anywhere and to any material. When an analytical solution is applied successfully to a new problem, the technique itself advances further, making the next solution for another problem easier and more reliable. The potential for the application of analytical methods is unlimited.

    FIG. 38. Upper-bound solution to flow through conical converging dies.

    FIG. 39. Criteria for failure in flow through conical convergine dies.

     

    However, when a specific problem arises, the pressure of time may dictate experimental methods. For example, the study of flow patterns for complex forging may one day be conducted analytically or by numerical procedures. At present, however, experimental work is the only practical solution. For the study of die design to prevent unfilled corners and cracks due to folding, modeling materials are most helpful. On the other hand, complex deep-drawing studies cannot be made with plasticine, wax, or lead but can be conducted successfully on a scaled-down model using the real material. Also, studies to determine tool life have not yet been conducted mathematically or through modeling materials; even scaled-down models are not very useful. Die design, including the choice of die material, is only improved through practice on the production line with full-scale equipment.

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