Compaction and Forging of Porous Metals

Powder forming, once considered a laboratory curiosity, has evolved into a manufacturing technique for producing high-performance components economically in the metal-working industry because of its low manufacturing cost compared with conventional metal-forming processes. Generally, the powder-forming process consists of three steps: (1) compacting a precise weight of metal powder into a “green” preform with 10–30% porosity (defined by the ratio of void volume to total volume of the preform); (2) sintering the preform to reduce the metal oxides and form strong metallurgical structures; (3) forming the preform by repressing or upsetting in a closed die to less than 1% residual porosity. Powder forming has disadvantages in that the preform exhibits porosity. Because of this porosity, the ductility of the sintered preform is low in comparison with wrought materials. In forging compacted and sintered powdered-metal (P/M) preforms, where large amount of deformation and shear is involved, pores collapse and align in the direction perpendicular to that of forging and result in anisotropy. However, repressing-type deformation, where very little deformation and shear are present, does not lead to marked anisotropy. A low-density preform will result in more local flow and a higher degree of anisotropy than will a preform of high initial density. These anisotropic structures can lead to nonuniform impact resistances of the forged P/M parts. Also, in forming of sintered preforms, materials are more susceptible to fracture than in forming of solid materials, and the analysis is of particular importance in producing defect-free components by determining the effect of various parameters (preform and die geometries, sintering conditions, and the friction conditions) on the detailed metal flow. In this chapter, the plasticity theory for solid materials is extended to porous materials, applicable to the deformation analysis of sintered powdered-metal preforms. In characterizing the mechanical response of porous materials, a phenomenological approach (introducing a homogeneous continuum model) is employed. For the finite-element formulations of the equilibrium and energy equations based on the infinitesimal theory, the following assumptions are made: the elastic portion of deformation is neglected because the practical forming process involves very large amounts of plastic deformation; the normality of the plastic strain-rates to the yield surface holds; anisotropy that occurs during deformation is negligible; and thermal properties of the porous materials are independent of the temperatures.

Axisymmetric Isothermal Forging

According to Spies, the majority of forgings can be classified into three main groups. The first group consists of compact shapes that have approximately the same dimensions in all three directions. The second group consists of disk shapes that have two of the three dimensions (length and width) approximately equal and larger than the height. The third group consists of the long shapes that have one main dimension significantly larger than the two others. All axially symmetric forgings belong to the second group, which includes approximately 30% of all commonly used forgings. A basic axisymmetric forging process is compression of cylinders. It is a relatively simple operation and thus it is often used as a property test and as a preforming operation in hot and cold forging. The apparent simplicity, however, turns into a complex deformation when friction is present at the die–workpiece interface. With the finite-element method, this complex deformation mode can be examined in detail. In this chapter, compression of cylinders and related forming operations are discussed. Since friction at the tool–workpiece interface is an important factor in the analysis of metal-forming processes, this aspect is also given particular consideration. Further, applications of the FEM method for complex-shaped dies are shown in the examples of forging and cabbaging. Finite-element discretization with a quadrilateral element is similar to that given in Chap. 8. The cylindrical coordinate system (r, ϑ, z) is used instead of the rectangular coordinate system. The element is a ring element with a quadrilateral cross-section, as shown in Fig. 9.1. The ξ and η of the natural coordinate system vary from −1 to 1 within each element.

Plane-Strain Problems

This chapter is concerned with the formulations and solutions for plane plastic flow. In plane plastic flow, velocities of all points occur in planes parallel to a certain plane, say the (x, y) plane, and are independent of the distance from that plane. The Cartesian components of the velocity vector u are ux(x, y), uy(x, y), and uz = 0. For analyzing the deformation of rigid-perfectly plastic and rate-insensitive materials, a mathematically sound slip-line field theory was established (see the books on metal forming listed in Chap. 1). The solution techniques have been well developed, and the collection of slip-line solutions now available is large. Although these slip-line solutions provide valuable insight into deformation modes and forming loads, slip-line field analysis becomes unwieldy for nonsteady-state problems where the field has to be updated as deformation proceeds to account for changes in material boundaries. Furthermore, the neglect of work-hardening, strain-rate, and temperature effects is inappropriate for certain types of problems. Many investigators, notably Oxley and his co-workers, have attempted to account for some of these effects in the construction of slip-line fields. However, by so doing, the problem becomes analytically difficult, and recourse is made to experimental determination of velocity fields, similarly to the visioplasticity method. Some of this work is summarized in Reference [2]. The applications of the finite-element method are particularly effective to the problems for which the slip-line solutions are difficult to obtain. The finite-element formulation specific to plane flow is recapitulated here.

The Finite-Element Method—Part I

The concept of the finite-element procedure may be dated back to 1943 when Courant approximated the warping function linearly in each of an assemblage of triangular elements to the St. Venant torsion problem and proceeded to formulate the problem using the principle of minimum potential energy. Similar ideas were used later by several investigators to obtain the approximate solutions to certain boundary-value problems. It was Clough who first introduced the term “finite elements” in the study of plane elasticity problems. The equivalence of this method with the well-known Ritz method was established at a later date, which made it possible to extend the applications to a broad spectrum of problems for which a variational formulation is possible. Since then numerous studies have been reported on the theory and applications of the finite-element method. In this and next chapters the finite-element formulations necessary for the deformation analysis of metal-forming processes are presented. For hot forming processes, heat transfer analysis should also be carried out as well as deformation analysis. Discretization for temperature calculations and coupling of heat transfer and deformation are discussed in Chap. 12. More detailed descriptions of the method in general and the solution techniques can be found in References [3-5], in addition to the books on the finite-element method listed in Chap. 1. The path to the solution of a problem formulated in finite-element form is described in Chap. 1 (Section 1.2). Discretization of a problem consists of the following steps: (1) describing the element, (2) setting up the element equation, and (3) assembling the element equations. Numerical analysis techniques are then applied for obtaining the solution of the global equations. The basis of the element equations and the assembling into global equations is derived in Chap. 5. The solution satisfying eq. (5.20) is obtained from the admissible velocity fields that are constructed by introducing the shape function in such a way that a continuous velocity field over each element can be denned uniquely in terms of velocities of associated nodal points.

Plasticity and Viscoplasticity

The theory of plasticity describes the mechanics of deformation in plastically deforming solids, and, as applied to metals and alloys, it is based on experimental studies of the relations between stresses and strains under simple loading conditions. The theory described here assumes the ideal plastic body for which the Bauschinger effect and size effects are neglected. The theory also is valid only at temperatures for which recovery, creep, and thermal phenomena can be neglected. The basic theory of classical plasticity is described by Hill, and also in References, in addition to the books listed in Chap. 1. A concise description of the general plasticity theory necessary for metal forming is given in the book by Johnson et al.. In this chapter, certain important aspects of the theory are presented in order to elucidate the developments of the finite-element solutions of metal-forming problems discussed in this book. First, various measures of stress and strain are introduced. Then, the governing equations for plastic deformation and principles that are the foundations for the analysis are described. The extension of the theory of plasticity to time-dependent theory of viscoplasticity is outlined in Section 4.8. Particular references are made, in Sections 4.3 through 4.7, to the books by Hill and by Johnson and Mellor, and to the section on general plasticity theory in the book by Johnson et al.. The basic quantities that may be used to describe the mechanics of deformation when a body deforms from one configuration to another under an external load are the stress, strain, and strain-rate. Various measures of these quantities are defined, depending upon how closely formulations represent actual situations. Although it is not possible to provide the complete mathematical formulations in one-dimensional deformation, these measures are introduced for the case of simple uniaxial tension. Consider the uniaxial tension test of a round specimen whose initial length is l0 and cross-sectional area is A0. The specimen is stretched in the axial direction by the force P to the length l and the cross-sectional area A at time t, as shown in Fig. 4.1. The response of the material is recorded as the load-displacement curve, and converted to the stress-strain curve as shown in the figure. The deformation is assumed to be homogeneous until necking begins.

Solid Formulation, Comparison of Two Formulations, and Concluding Remarks

In the previous chapters we have discussed only the applications of flow formulation to the analysis of metal-forming processes. Lately, elastic-plastic (solid) formulations have evolved to produce techniques suitable for metal-forming analysis. This evolution is the result of developments achieved in large-strain formulation, beginning from the infinitesimal approach based on the Prandtl–Reuss equation. A question always arises as to the selection of the approach—“flow” approach or “solid” approach. A significant contribution to the solution of this question was made through a project in 1978, coordinated by Kudo, in which an attempt was made to examine the comparative merits of various numerical methods. The results were compiled for upsetting of circular solid cylinders under specific conditions, and revealed the importance of certain parameters used in computation, such as mesh systems and the size of an increment in displacement. This project also showed that the solid formulation needed improvement, particularly in terms of predicting the phenomenon of folding. For elastic-plastic materials, the constitutive equations relate strain–rate to stress–rates, instead of to stresses. Consequently, it is convenient to write the field equation in the boundary-value problem for elastic-plastic materials in terms of the equilibrium of stress rates. In this chapter, the basic equations for the finite-element discretization involved in solid formulations are outlined both for the infinitesimal approach and for large-strain theory. Further, the solutions obtained by the solid formulation are compared with those obtained by the flow formulation for the problems of plate bending and ring compression. A discussion is also given concerning the selection of the approach for the analysis. In conclusion, significant recent developments in the role of the finite-element method in metal-forming technology are summarized. The field equation for the boundary-value problem associated with the deformation of elastic-plastic materials is the equilibrium equation of stress rates. As stated in Chap. 1 (Section 1.3), the internal distribution of stress, in addition to the current states of the body, is supposed to be known, and the boundary conditions are prescribed in terms of velocity and traction-rate.

The Finite-Element Method—Part II

Numerical integration is an important part of the finite-element technique. As seen in Section 6.5 of Chap. 6, volume integrations as well as surface integrations should be carried out in order to represent the elemental stiffness equations in a simple matrix form. In deriving the variational principle, it is implicitly assumed that these integrations are exact. However, exact integrations of the terms included in the element matrices are not always possible. In the finite-element method, further approximations are made in the procedure for integration, which is summarized in this section. Numerical integration requires, in general, that the integrand be evaluated at a finite number of points, called Integration points, within the integration limits. The number of integration points can be reduced, while achieving the same degree of accuracy, by determining the locations of integration points selectively. In evaluating integration in the stiffness matrices, it is necessary to use an integration formula that requires the least number of integrand evaluations. Since the Gaussian quadrature is known to require the minimum number of integration points, we use the Gaussian quadrature formula almost exclusively to carry out the numerical integrations.

Introduction

In the late 1970s and early 1980s the use of computer-aided techniques (computer-aided engineering, design, and manufacturing) in the metal-forming industry increased considerably. The trend seems to be toward ever wider application of this technology for process simulation and process design. A goal in manufacturing research and development is to determine the optimum means of producing sound products. The optimization criteria may vary, depending on product requirements, but establishing an appropriate criterion requires thorough understanding of manufacturing processes. In metal-forming technology, proper design and control requires, among other things, the determination of deformation mechanics involved in the processes. Without the knowledge of the influences of variables such as friction conditions, material properties, and workpiece geometry on the process mechanics, it would not be possible to design the dies and the equipment adequately, or to predict and prevent the occurrence of defects. Thus, process modeling for computer simulation has been a major concern in modern metal-forming technology. Figure 1.1. indicates the role of process modeling in some detail. In the past a number of approximate methods of analysis have been developed and applied to various forming processes. The methods most well known are the slab method, the slip-line field method, the visioplasticity method, upper- (and lower-) bound techniques, Hill’s general method, and, more recently, the finite-element method (FEM). In the slab method, the workpiece being deformed is decomposed in several slabs. For each slab, simplifying assumptions are made mainly with respect to stress distributions. The resulting approximate equilibrium equations are solved with imposition of stress compatibility between slabs and boundary tractions. The final result is a reasonable load prediction with an approximate stress distribution. The slip-line field method is used in plane strain for perfectly plastic materials (constant yield stress) and uses the hyperbolic properties that the stress equations have in such cases. The construction of slip-line fields, although producing an “exact” stress distribution, is still quite limited in predicting results that give good correlations with experimental work. From the stress distributions, velocity fields can be calculated through plasticity equations.

Preform Design in Metal Forming

Preform design in metal forming refers to the design of an initial shape of the workpiece that, when it has undergone an associated forming process, forms the required product shape with desired property successfully without formation of defects and without excessive waste of materials. A carefully selected preform can contribute significantly to the reduction of the production costs. Preform design problems are encountered in various metal-forming processes, such as closed-die forging, shell nosing, rolling, and sheet-metal forming. Design of an optimal preform shape requires simultaneous determination of optimal process conditions. However, we are here concerned with the determination of the best preform shape under a given set of process conditions. In this chapter, a new method of “backward tracing” is introduced as an alternative approach to the solution of preform design, and the applications of this method to some specific processes are discussed. Similarly to the forward simulation technique, the backward tracing method uses the finite-element method. The forward simulation technique has been discussed in the previous chapters. Backward tracing refers to the prediction of the part configuration at any stage in a deformation process, when the final part geometry and process conditions are given. The concept is illustrated in Fig. 15.1. At time t = t0, the geometrical configuration x0 of a deforming body is represented by a point Q. The point Q is arrived at from the point P, whose configuration is given as x0–1 at t = t0–1, through the displacement field during a time-step Δt, namely, x0 = x0–1 + u0–1 Δt, where u0–1 is the velocity field at t = t0–1. Therefore, the problem is to determine u0–1, based on the information (x0) at point Q. The solution scheme is as follows: taking a loading solution u0 (forward) at Q, a first estimate of P can be made according to P(1) = x0 – u0 Δt.

Analysis and Technology in Metal Forming

The design, control, and optimization of forming processes require (1) analytical knowledge regarding metal flow, stresses, and heat transfer, as well as (2) technological information related to lubrication, heating and cooling techniques, material handling, die design and manufacture, and forming equipment. The purpose of using analysis in metal forming is to investigate the mechanics of plastic deformation processes, with the following major objectives. • Establishing the kinematic relationships (shape, velocities, strain-rates, and strains) between the undeformed part (billet, blank, or preform) and the deformed part (product); i.e., predicting metal flow during the forming operation. This objective includes the prediction of temperatures and heat transfer, since these variables greatly influence local metal-flow conditions. • Establishing the limits of formability or producibility; i.e., determining whether it is possible to perform the forming operation without causing any surface or internal defects (cracks or folds) in the deforming material. • Predicting the stresses, the forces, and the energy necessary to carry out the forming operation. This information is necessary for tool design and for selecting the appropriate equipment, with adequate force and energy capabilities, to perform the forming operation. Thus, the mechanics of deformation provides the means for determining how the metal flows, how the desired geometry can be obtained by plastic deformation, and what the expected mechanical properties of the produced part are. For understanding the variables of a metal-forming process, it is best to consider the process as a system, as illustrated in Fig. 2.1 in Chap. 2. The interaction of most significant variables in metal forming are shown, in a simplified manner, in Fig. 3.1. It is seen that for a given billet or blank material and part geometry, the speed of deformation influences strain-rate and flow stress. Deformation speed, part geometry, and die temperature influence the temperature distribution in the formed part. Finally, flow stress, friction, and part geometry determine metal flow, forming load, and forming energy. In steady-state flow (kinematically), the velocity field remains unchanged, as is the case in the extrusion process; in nonsteadystate flow, the velocity field changes continuously with time, as is the case in upset forging.