The finite-element method is a powerful tool to stimulate the creativity of engineers, scientists, and applied mathematicians. Therefore it is not surprising that many elements have been presented in the literature and are available for a range of applications.

A finite element is defined not only by a geometric subdomain, or even the explicit mathematical expression of the shape functions. A finite element also includes the specification of whether it is an isoparametric element (as we shall essentially consider here) and the definition of the space integration formulas. In a broader sense we can also consider that the space discretization procedure for each variable, the time integration procedure when required, and the mathematical formulation of the physical problem also contribute to the FE definition.

Evaluation of the behavior of an element type will depend on the kind of problem that is considered, the required accuracy, and the time and effort allocated toward code development. In the realm of metal forming, the subject of this book, the choice of element formulation depends primarily on the following considerations.

1. Mode of deformation: the chosen element must approximate real material stiffness in all modes of deformation important to the physical problem being simulated. An element can be too stiff in some modes, producing “locking” for example, particularly when incompressibility constraints are considered; or it can be too compliant, producing “hourglass” and other spurious low-energy deformation modes.

This book assumes a background in the fundamentals of solid mechanics and the mechanical behavior of materials, including elasticity, plasticity, and friction. A previous book by the same authors covers these topics in detail, including derivation or explanation of the most important concepts. It is beyond the scope of the current book to reproduce all of this important information.

In this chapter, the essential equations from this background are reproduced. This serves two purposes: to introduce the notation that will be used throughout the remaining chapters, and to list the principal background equations in one place. Frequent reference to the equations presented in this chapter will be made. However, it should be kept in mind that the full context for these equations is found in Fundamentals of Metal Forming.

Notation

There are many alternate forms of notation used in solid mechanics and finite-element modeling. In some cases, it is clearer to use a form that has become a de facto standard in the area, even though such usage might not be rigorous. In other cases, there is no consensus on notation, so it is less confusing to be consistent with other equations.

The finite-element formulation of a given physical problem depends primarily on the nature of the problem and, to some extent, on several numerical choices the scientist decides to make. In this chapter we shall review five different topics, linked to plastic forming, which have a prominent influence on the way the problem will be solved.

First, the explicit and implicit formulations originally corresponded to different problems, distinguished by the deformation rate level, but they are now used for a much larger range of processes.

Second, historically the first attempts to model metal-forming processes by the finite-element method were based on the flow formulation, or the rigid-plastic material model, in which elastic deformation is neglected. Despite an additional cost and a more complex treatment, the elastoplastic approach is more and more often applied to forming problems in engineering.

Third, the Eulerian formulation, when applicable, is very efficient to treat steady-state processes, whereas the updated Lagrangian formulation is the most popular method for the other cases involving large strain.

Fourth, the displacement or velocity approaches were mostly used in the past for practical applications, as they are more economical. However, mixed methods provide an additional flexibility for the finite-element problem formulations, which can result in more satisfactory solutions, both from the mathematical and from the numerical point of view.

Fifth, the problem of time integration of constitutive equations has received considerable attention during the past 20 years, and some progress is still necessary to achieve complete accuracy for any real process.

Metal Forming Analysis has two purposes: (a) to acquaint the advanced graduate student with numerical principles and procedures used in the modern analysis of industrial forming operations, and (b) to provide reference material for those performing such an analysis in industrial settings, government laboratories, and academia. In both cases, an understanding of the most important methods and their respective characteristics is the goal.

The first seven chapters focus on principles and procedures, which are derived and presented in an intuitive, informal manner. Exercises appear throughout these chapters, proposing and then solving illuminating problems related to the subject. Extensive problems are provided in three categories at the end of each chapter: proficiency, depth, and numerical, to solidify the information presented.

The last five chapters focus on applications of the numerical analysis to specific forming operations in order to illustrate the lessons learned from these simulations. Most of these results are drawn from the authors' research in this area, using programs developed over many years at their laboratories. Exercises are presented where appropriate and practical, and a limited number of problems are provided at the end of some chapters.

It should be noted that this advanced text and reference volume does not provide a detailed treatment of the underlying physical equations or principles necessary to understand metal deformation itself. This material is limited to Chapter 1, which is a very brief review of the physical descriptions and equations.

There are many books devoted to a general introduction to the finite-element method. It is impossible to list all of them here, but those referenced in the footnotes appear in increasing order of difficulty. Interested readers could select one or two for study, depending on their backgrounds. In this chapter a very simple introduction is given, mainly in view of applications to the numerical modeling of metal-forming processes.

A first introduction is given in Section 2.1, with a comparison with another popular numerical method: the finite-difference method. The selected thermal problem allows a complete “hand calculation” by the two methods, and a comparison of the results with an analytical solution. The aim of this first overview is to summarize very briefly the main aspects (and vocabulary) of the method. It shows that a quasi-intuitive approach is conceivable and could help to demythologize the finite-element method.

A more systematic approach follows in Sections 2.2 to 2.5, in order to separate as clearly as possible the steps that lead to the rational development of the method. In most occasions the basic concepts are first explained in detail by using one-dimensional examples, and the generalization is only mentioned, as more detail will be given in Chapters 3 and 4 for two- and three-dimensional approaches.

The basic idea is to follow four steps:

1. Establish the physical equations of the model (see Chapter 1, or an earlier work on this subject).

This chapter is devoted to the analysis of the interactions between thermal phenomena and deformation processes. In hot forming, the effect of temperature is especially important, as in general the tools are used at relatively low temperature and thus produce a highly heterogeneous temperature field. For the workpiece itself, there is always a competition between the cooling effect that is due to the contact with tools and radiation plus convection on free surfaces on one hand, and heating by plastic deformating in the core of the part or by the friction dissipation on the surface in contact on the other hand. In fact, it may happen for forming of complex parts that the final result is cooling in some regions and heating in other regions. For a complete analysis of a forming sequence, different situations must be considered:

1. The preform can be heated in an oven and then transferred to the forming machine: in this case the strain and stress evolution can be neglected as a first approach, and a purely thermal analysis may be sufficient.

2. During the forming process, large plastic or elastoplastic deformation (respectively viscoplastic or elastic viscoplastic) is imposed on the workpiece, while the local slipping distance at the tool-part interface can also be important. A coupled thermal and mechanical analysis with large strain theory is therefore necessary.[…]

The aim of this chapter is to provide the necessary numerical concepts for the practical implementation of the finite-element method. First, we review the notion of isoparametric elements; such elements are important when the elements are distorted and the shape functions cannot be easily expressed in term of the physical coordinates. This is always the case in the numerical simulation of metal-forming processes, where significant deformation of the initial mesh cannot be avoided. Second, we consider in some detail the procedure for numerical integration, which is essential for many applications, particularly when the constitutive equation is nonlinear. We resolve the resulting finite-element equation into two types: linear and nonlinear. Finally, we introduce a simple, one-dimensional mechanical example with linear material behavior. This example is treated completely and a nonlinear material is also used to illustrate the differences in the methods.

Isoparametric Elements

The physical problem under consideration is posed in a domain and is discretized into small subdomains, called elements, as introduced in Chapter 2. Elements are physical in the sense that they are defined themselves in real space. So far we have used the physical domains of those elements directly, by defining shape functions inside an element in terms of nodal values and positions. It is often more convenient to map the physical element domains to even simpler shapes and thus to formulate the discretized problem on these shapes. The expression of the shape functions is much simpler (for nonlinear ones).

In this chapter we gathered the three main problems one faces in metal-forming simulation once the element and the constitutive formulation are chosen:

1. The contact problem is still among the most challenging ones, as it introduces a very stiff constraint, which results in a discontinuity of the velocity field at the onset of the contact.

2. The appropriate introduction of the friction law is also crucial in forming processes, as the material flow can be more sensitive to a small variation of the friction parameters than to a change of rheology.

3. The introduction in the finite-element formulation of the incompressibility requirement for dense materials, when elasticity is neglected. This topic is not straightforward from the engineering point of view, so it has motivated very theoretical works in the field of functional analysis; we shall not review these here as it is far beyond the scope of this book.

The Contact Problem

The contact problem with tools is probably the most challenging issue in modeling metal-forming processes. It is intrinsically a difficult mathematical problem, as it corresponds to a time discontinuity of the velocity when contact is established and, even in the stationary case, it involves unilateral conditions. Moreover if contact is not taken into account properly, or if the approximation is too crude, the final results of the computation are greatly affected and can be completely wrong. For simplicity we shall restrict ourselves here to rigid tools.

Within the name of forging, a lot of different processes are referred to, so that a classification must be endeavored. As for other metal-forming processes, the deformation temperature can be the first criterion, and we shall therefore distinguish between the usual three cases.

In hot forging the part is heated above half the fusion absolute temperature to reduce the forging force, take advantage of the higher material ductility, and allow large deformation. However, the tools are submitted to high thermal stresses as the temperature difference between the part and the tool is high (even when the tools are preheated). For example, steel is forged at about 1100–1200 °C, while the tools have a temperature ranging from room temperature to 250 °C. For simplicity we shall introduce here free forging, in which the shape of the tools is rather simple, and which is generally used for a small series of large or very large parts; open-die forging, in which the preform is squeezed between two dies to give a prescribed shape; and closed-die forging (see Fig. 10.1).

Cold forging is performed at room temperature, but the dissipated plastic work may raise the temperature of the part up to about 250 °C. Cold forging is used to achieve better tolerance, higher mechanical properties, and a better surface aspect that can avoid further machining. But as cold metals are less ductile in general than hot ones, cold forging allows moderate deformation, unless a heat treatment is introduced to restore a deformation capability.

Sheet-metal forming is one of the most important manufacturing processes for mass production in industry. It is inexpensive in large quantity production, but a great deal of time and expense are devoted to the design and production of reliable stamping dies. Since the mid-1970s, important advances have been made in the development of computer simulation codes for deformation modeling during sheet-metal forming.

Typical sheet-forming operations carried out in standard presses are characterized by quasi-static loading, principal loading by in-plane tension (with some bending influence), irregularly shaped and continuously changing contact surfaces, and large relative movements of material on the contacting surfaces. Nearly all FEM nodes in the sheet mesh will come into contact with the tooling at some stage of the simulation. (This contrasts with forming or bulk forming, in which the majority of mesh nodes are internal.) Therefore, sheet forming is in general much more dependent on friction and contact than most other forming operations. It is essential to handle contact and friction accurately and consistently in sheet-forming analysis; otherwise, large errors can accumulate. Furthermore, contact algorithms must be sufficiently stable that small perturbations do not induce numerical divergence and instability into the techniques.

Current numerical research focuses on whole-part analysis by FEM. The principal obstacles remain stability, time, knowledge of physical boundary conditions and friction, and limited experimental verification. While contact is more complex for sheet forming, the process offers some simulation advantages in terms of the possibility of neglecting through-thickness stresses and, in some cases, bending stresses.

Once the physical problem is taken into account properly, the major concern of the engineer is reliability of the numerical code, and cost and delay to run an application. The growing need of very complicated problems, for which meshing “by hand” is less and less thinkable, urged the development of friendly automatic meshing and remeshing tools.

In contrast, it is now possible to evaluate quantitatively the confidence we can put into an actual computation with the a posteriori error estimation. The combination of both approaches has been achieved: after a first trial computation, the most advanced methodology permits us to generate a new mesh automatically, which is locally refined in order that the error is always below an acceptable value for practical purpose.

Finally, when very large problems are in three dimensions, both computer time and memory allocation may become prohibitive. This concern can be taken into account by utilizing iterative methods for solving large linear systems, in order to save computer time, reduce the memory requirement, and facilitate the optimal use of parallel computers.

Meshing and Remeshing

The meshing capability of a FE code is now very important, as the manual generation of the complex meshes that are treated is almost impossible and always dissuading. When the incremental simulation of a forming sequence is considered, automatic remeshing is still more important. We shall briefly review the main methods for meshing and discuss their possible generalization to remeshing.

In this chapter, sheet-metal formability tests, such as the tensile test, the plane-strain test, and the in-plane stretching test are analyzed. The FEM and experimental methods were used in order to demonstrate how two-dimensional in-plane simulation can help interpret and develop such tests, as well as to understand the nature of material behavior and governing mechanics.

A successful sheet-metal forming process can convert an initially flat sheet into a useful part of the desired shape. The major failures that may be encountered are splitting, wrinkling, and shape distortion. The deformed part can be considered unusable if any one of these failures occurs. Formability tests may be used to assess the capacity of a sheet to be deformed into a useful part.

Since sheet-metal forming operations are diverse in type, extent, and rate, many formability tests have been proposed. No single one can provide an accurate indication of the formability for all situations. Formability tests can be divided into two types: intrinsic and simulative. The intrinsic tests measure the basic material properties under certain stress—strain states, for example, the uniaxial tensile test and the plane-strain tensile test. Simulative tests subject the material to deformation that closely resembles a particular press-forming operation. A simulative test can provide limited and specific information that may be sensitive to factors other than the material properties, such as the thickness, surface condition, lubrication, and geometry and type of tooling.

Gas cutting of iron and steel

Iron and steel can be cut by the oxy-hydrogen, oxy-propane, oxy-natural gas and oxy-acetylene cutting blowpipes with ease, speed and a cleanness of cut.

Principle of cutting operation

There are two operations in gas cutting. A heating flame is directed on the metal to be cut and raises it to bright red heat or ignition point. Then a stream of high-pressure oxygen is directed on to the hot metal. The iron is immediately oxidized to magnetic oxide of iron (Fe3O4) and, since the melting point of this oxide is well below that of the iron, it is melted immediately and blown away by the oxygen stream.

It will be noted that the metal is cut entirely by the exothermic chemical action and the iron or steel itself is not melted. Because of the rapid rate at which the oxide is produced, melted and blown away, the conduction of the metal is not sufficiently high to conduct the heat away too rapidly and prevent the edge of the cut from being kept at ignition point.

The heat to keep the cut going once it has started is provided partly by the heating jet, and partly by the heat of the chemical action.

The cutting torch or blowpipe (Fig. 7.1)

Cutting blowpipes may be either high or low pressure. The high pressure pipe, using cylinder acetylene or propane as the fuel gas,* can have the mixer in the head (Fig. 7.2), or in the shank, while the low pressure pipe with injector mixing can be used with natural gas at low pressure.