INTRODUCTION

Slip-line field theory is based on analysis of a deformation field that is both geometrically self-consistent and statically admissible. Slip lines are planes of maximum shear stress and are therefore oriented at 45° to the axes of principal stress. It is assumed that

1. the material is isotropic and homogeneous,

2. the material is rigid – ideally plastic (i.e., no strain hardening),

3. effects of temperature and strain rate are ignored,

4. plane-strain deformation prevails, and

5. the shear stresses at interfaces are constant: usually frictionless or sticking friction.

Figure 9.1 shows the very simple slip line for indentation where the thickness, t, equals the width of the indenter, b. The maximum shear stress occurs on line DEB and CEA. The material in triangles DAE and CEB is rigid. As the indenters move closer together the field must change. However, for now, we are concerned with calculating the force when the geometry is as shown. The stress ςy must be zero because there is no restraint to lateral movement. The stress ςz must be intermediate between ςx and ςy. Figure 9.2 shows the Mohr's circle for this condition. The compressive stress necessary for this indentation is ςx= − 2k. Few slip-line fields are composed of only straight lines. More complicated fields will be considered.

An important concern in forming is whether a desired process can be accomplished without failure of the work material. Forming limits vary with material for any given process and deformation-zone shape. As indicated in Chapter 10, central bursts may occur at a given level of Δ in some materials and not in others. Failure strains for a given process depend on the material.

DUCTILITY

In most bulk forming operations, formability is limited by ductile fracture. Forming limits correlate quite well with the reduction of area as measured in a tension test. Figure 11.1 shows the strains at which edge cracking occurs in rolling as a function of the tensile reduction in area. The fact that the limiting strains for strips with square edges are much higher than those with rounded edges indicates that process variables are also important. Similar results are reported for other processes.

METALLURGY

The ductility of a metal is strongly influenced both by the properties of the matrix and by the presence of inclusions. Factors that increase the strength generally decrease ductility. Solid solution strengthening, precipitation, cold work, and decreased temperatures all lower fracture strains. The reason is that with higher strengths, the stresses necessary for fracture will be encountered sooner.

Inclusions play a dominant role in ductile fracture. The volume fraction, nature, shape, and distribution of inclusions are important. In Figure 11.2, the tensile ductility is seen to decrease with increased amounts of artificial inclusions.

Calculation of exact forces to cause plastic deformation in metal forming processes is often difficult. Exact solutions must be both statically and kinematically admissible. That means they must be geometrically self-consistent as well as satisfying the required stress equilibrium everywhere in the deforming body. Frequently it is simpler to use limit theorems that allow one to make analyses that result in calculated forces that are known to be either correct or higher or lower than the exact solution.

Lower bounds are based on satisfying stress equilibrium, while ignoring geometric self-consistency. They give forces that are known to be either too low or correct. As such they can assure that a structure is “safe.” Conditions in which η = 0 are lower bounds. Upper-bound analyses, on the other hand, predict stress or forces that are known to be too large. These are usually more important in metal forming. Upper bounds are based on satisfying yield criteria and geometric self-consistency. No attention is paid to satisfying equilibrium.

UPPER BOUNDS

The upper-bound theorem states that any estimate of the forces to deform a body made by equating the rate of internal energy dissipation to the external forces will equal or be greater than the correct force. The method of analysis is to

1. Assume an internal flow field that will produce the shape change;

2. Calculate the rate at which energy is consumed by this flow field; and

3. Calculate the external force by equating the rate of external work with the rate of internal energy consumption.

When metals are deformed plastically at temperatures lower than would cause recrystallization, they are said to be cold worked. Cold working increases the strength and hardness. The terms work hardening and strain hardening are used to describe this. Cold working usually decreases the ductility.

Tension tests are used to measure the effect of strain on strength. Sometimes other tests, such as torsion, compression, and bulge testing are used, but the tension test is simpler and most commonly used. The major emphasis in this chapter is the dependence of yield (or flow) stress on strain.

THE TENSION TEST

The temperature and strain rate influence test results. Generally the strain rate is in the order of 10–2 to 10–3/s and the temperature is between 18 and 25°C. These effects are discussed in Chapter 5. Measurements are made in a gauge section that is under uniaxial tension during the test.

Initially the deformation is elastic and the tensile force is proportional to the elongation. Elastic deformation is recoverable. It disappears when the tensile force is removed. At higher forces the deformation is plastic, or nonrecoverable. In a ductile material, the force reaches a maximum and then decreases until fracture. Figure 3.1 is a schematic tensile load-extension curve.

There are many tests of formability for sheet materials. A given test may correlate well with behavior in one type of forming process and poorly with behavior in another. This is because the relative amounts of drawing and stretching vary from test to test and process to process.

CUPPING TESTS

The Swift cup test is the determination of the limiting drawing ratio for flat-bottom cups. In the Erichsen and Olsen tests, cups are formed by stretching over a hemispherical tool. The flanges are very large so little drawing occurs. The results depend on stretchability rather than drawability. The Olsen test is used in America and the Erichsen in Europe. Figure 18.1 shows the setup.

The Fukui conical cup test involves both stretching and drawing over a ball. The opening is much larger than the ball so a conical cup is developed. The flanges are allowed to draw in. Figure 18.2 shows the setup. A failed Fukui cup is shown in Figure 18.3.

Figure 18.4 shows comparison of the relative amounts of stretch and draw in these tests.

LDH TEST

The cupping tests discussed above are losing favor because of irreproducibility. Hecker attributed this to “insufficient size of the penetrator, inability to prevent inadvertent drawing in of the flange, and inconsistent lubrication.” He proposed the limiting dome height test (LDH) which uses the same tooling (4 in. diameter punch) as used to determine forming limit diagrams.

My coauthor Robert Caddell died in 1990. I have greatly missed interacting with him.

The biggest changes from the second edition are an enlargement and reorganization of the last third of the book, which deals with sheet metal forming. Changes have been made to the chapters on bending, plastic anisotropy, and cup drawing. An entire chapter has been devoted to forming limit diagrams. There is one chapter on various aspects of stampings, including the use of tailor-welded blanks, and another on other sheet-forming operations, including hydroforming of tubes. Sheet testing is covered in a separate chapter. The chapter on sheet metal properties has been expanded to include newer materials and more depth on aluminum alloys.

In addition, some changes have been made to the chapter on strain-rate sensitivity. A treatment of friction and lubrication has been added. A short treatment of swaging has been added. End-of-chapter notes have been added for interest and additional end-of-chapter references have been added.

No attempt has been made in this book to introduce numerical methods such as finite element analyses. The book Metal Forming Analysis by R. H. Wagoner and J. L. Chenot (Cambridge University Press, 2001) covers the latest numerical techniques. We feel that one should have a thorough understanding of a process before attempting numerical techniques. It is vital to understand what constitutive relations are imbedded in a program before using it. For example, the use of Hill's 1948 anisotropic yield criterion can lead to significant errors.

The right lines in a moving body

01. I began to explain in chapter 3 the importance of and the ubiquitous nature of that special kind of line in a moving body which I called there, for the want of a better name, right line. There and in other places the n-line and the contact normal are also mentioned (§ 3.01, § 3.05, § 3.06) and a distinction made between a line which is obliged by virtue of some constraint to be a right line in a body as soon as the body moves and other lines in the body which may or may not be right lines according to how the body moves. Because I am speaking here in chapter 9 about the kinematics of a body, I am speaking about the motional (not the actional) right lines within the body; the distinction between the two is explained at § 3.43.

02. In any event it is taken for granted here that the reader is aware of the practical significance of the right line in the kinematics of mechanism. The objective of this chapter is not to study the significance of right lines but to plot the overall distribution of them in a moving body or in a body which has the capacity to move with one degree of freedom only (§ 3.24). We are trying to discover, not only where the right lines in a body are, but also how many of them there are.