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 11, 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 12.1 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 strip 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 12.2, the tensile ductility is seen to decrease with increased amounts of artificial inclusions.

Sheet forming differs from bulk forming in several respects. In sheet forming, tension predominates whereas bulk forming operations are predominately compressive. In sheet forming operations at least one of the surfaces is free from contact with the tools. Useful formability is normally limited by localized necking, rather than by fracture as in bulk forming. There are instances of failure by fracture but these are unusual.

Sheet forming processes may be roughly classified by the state of stress. At one end of the spectrum is the deep drawing of flat-bottom cups. In this case, one of the principal stresses in the flange is tensile and the other is compressive. There is little thinning but wrinkling is of concern. At the other end of the spectrum are processes, usually called stamping, in which both of the principal stresses are tensile so thinning must occur. Rarely does the formability in sheet forming processes correlate well with the tensile ductility (either reduction in area or elongation at fracture).

CUP DRAWING

The deep drawing of flat-bottom cups is a relatively simple process. It is used to produce such items as cartridge cases, zinc dry cells, flashlights, aluminum and steel cans, and steel pressure vessels. The process is illustrated by Figure 15.1. There are two important regions: the flange where most of the deformation occurs and the wall, which must support the force necessary to cause the deformation in the flange.

My coauthor, Robert Caddell, died in 1990, and I have greatly missed working with him.

The most significant changes from the third edition are a new chapter on friction and lubrication and a major rearrangement of the last third of the book dealing with sheet forming. Most of the chapters in the last part of the book have been modified, with one whole chapter devoted to hydroforming. A new section is devoted to incremental forming. No attempt has been made to introduce numerical methods. Other books treat numerical methods. We feel that a thorough understanding of a process and the constitutive relations that are embedded in a computer program to analyze it are necessary. For example, the use of Hill's 1948 anisotropic yield criterion leads to significant errors.

I wish to acknowledge my membership in the North American Deep Drawing Research Group from whom I have learned so much about sheet forming. Particular thanks are due to Alejandro Graf of ALCAN, Robert Wagoner of the Ohio State University, John Duncan formerly with the University of Auckland, and Thomas Stoughton of General Motors.

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, in a tension test, 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.

Stress and strain are computed from measurements in a tension test of the tensile force, F, and the elongation, Δℓ.

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. Basic assumptions are:

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.

5. The shear stresses at interfaces are constant, usually frictionless or sticking friction.

Figure 10.1 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 restrain to lateral movement. The stress, σz, must be intermediate between σx and σy. Figure 10.2 shows the Mohr's circle for this condition. The compressive stress necessary for this indentation, σx = –2k. Few slip-line fields are composed of only straight lines. More complicated fields will be considered.

GOVERNING STRESS EQUATIONS

With plane-strain, all of the flow is in the x–y plane.

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 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 too high or too low 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 analysis involves:

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

2. Calculating the rate at which energy is consumed by this flow field.

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