INTRODUCTION

Plasticity theory deals with yielding of materials, often under complex states of stress. Plastic deformation, unlike elastic deformation, is permanent in the sense that after stresses are removed the shape change remains. Plastic deformations usually occur almost instantaneously, but creep can be regarded as time-dependent deformation plastic deformation.

There are three approaches to plasticity theory. The approach most widely used is continuum theory. It depends on yield criteria, most of which are simply postulated without regard to how the deformation occurs. Continuum plasticity theory allows predictions of the stress states that cause yielding and the resulting strains. The amount of work hardening under different loading conditions can be compared.

A second approach focuses on the crystallographic mechanisms of slip (and twinning), and uses understanding of these to explain continuum behavior. This approach has been quite successful in predicting anisotropic behavior and how it depends on crystallographic texture. Ever since the 1930s, there has been increasing work bridging the connection between this crystallographic approach and continuum theory.

Plastic deformation of crystalline materials usually occurs by slip, which is when the sliding of planes of atoms slide over one another (Figure 8.1). The planes on which slip occurs are called slip planes and the directions of the shear are the slip directions. These are crystallographic planes and directions, and are characteristic of the crystal structure. The magnitude of the shear displacement by slip is an integral number of inter-atomic distances, so that the lattice is left unaltered by slip. If slip occurs on only a part of a plane, there remains a boundary between the slipped and unslipped portions of the plane, which is called a dislocation. It is the motion of these dislocations that cause slip.

Slip lines can be seen on the surface of deformed crystals. The fact that we can see these indicates that slip is inhomogeneous on an atomic scale. Displacements of thousands of atomic diameters must occur on discrete or closely spaced planes to create steps on the surface that are large enough to be visible. Furthermore, the planes of active slip are widely separated on an atomic scale. Yet the scale of the slip displacements and distances between slip lines are small compared to most grain sizes so slip usually can be considered as homogeneous on a macroscopic scale.

The effects of pressure on the yield locus can be confused with the effects of the sign of the stress. For example, twinning is sensitive to the sign of the applied stresses and causes the yield behavior under compression to be different from that under tension.

S-D EFFECT

With a so-called strength differential (SD) effect in high strength steels [1], yield strengths under tension are lower than under compression. The fractional magnitude of the effect, 2(|σc| – σT)∕[(|σc| + σT] is between 0.10 and 0.20. Figure 14.1 shows the effect in an AISI 4330 steel and Figure 14.2 indicates that it is a pressure effect.

Although the flow rules, equation 4.18, predict a volume increase with yielding, none has been observed.

POLYMERS

For polymers, the stress-strain curves in compression and tension can be quite different. Figures 14.3 and 14.4 are stress strain curves for epoxy and PMMA in tension and compression. Figure 14.5 compares the yield strengths of polycarbonate as tested in tension, shear and compression. The effect of pressure on the yield strength of PMMA is plotted in Figure 14.6.

In 1950, R. Hill wrote an authoritative book, Mathematical Theory of Plasticity that presented a comprehensive treatment of continuum plasticity theory as known at that time. Much of the treatment in this book covers some of the same ground but there is no attempt to treat the same topics treated by Hill. This book, however, includes more recent developments in continuum theory, including a treatment of anisotropy that has resulted from calculations of yielding based on crystallography, analysis of the role of defects and forming limit diagrams. There is a much greater emphasis on deformation mechanisms, including chapters on slip and dislocation theory and twinning.

This book should provide a useful resource to those involved with designing processes for sheet metal forming. Knowledge of plasticity is essential to those involved in computer simulation of metal forming processes. Knowledge of the advances in plasticity theory are essential in formulating sound analyses.

INTRODUCTION

For pencil glide, the five independent slip variable necessary to produce an arbitrary shape change can be the amount of slip in a given direction and the orientation of the plane (angle of rotation about the direction). There are two possibilities for five systems: Either three or four active slip directions can be active. Chin and Mammel [1] used a Taylor type analysis for combined slip on {110}, {123}, and {112} planes, finding that Mav for axially symmetric flow = 2.748 (Figure 10.1). Hutchinson [2] approximated pencil glide by assuming slip on a large, but finite number of slip planes. Both of these analyses used the least work approach of Taylor. Penning [3] described a least-work solution considering the possibility of both three and four active slip directions. Parniere and Sauzay [4] described a least work solution.

METHOD OF CALCULATION

Piehler et al [5, 7, 8] used a Bishop and Hill-type approach, by considering the stress states capable of activating enough slip systems. Explicit expressions were derived for the stress states in the case of four active slip directions. Instead of explicit solutions for the case of three active slip directions, a limited number of specific cases were considered. The stress states are:

Calculation of exact forces to cause plastic deformation in metal forming processes is often difficult. Exact solutions must be both statically and kinematically admissible. This means they must be geometrically self-consistent as well as satisfying stress equilibrium everywhere in the deforming body. Slip-line field analysis for plane strain deformation satisfies both and are therefore exact solutions. This topic is treated in Chapter 15. Upper and lower bounds are based on well-established principles [1, 2].

Frequently, it is difficult to make exact solutions and it is simpler to use limit theorems, which allows 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.

UPPERBOUNDS

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.

Of concern in plasticity theory is the yield strength, which is the level of stress that causes appreciable plastic deformation. It is tempting to define yielding as occurring at an elastic limit (the stress that causes the first plastic deformation) or at a proportional limit (the first departure from linearity). However, neither definition is very useful because they both depend on accuracy of strain measurement. The more accurately the strain is measured, the lower is the stress at which plastic deformation and non-linearity can be detected.

To avoid this problem, the onset of plasticity is usually described by an offset yield strength that can be measured with more reproducibility. It is found by constructing a straight line parallel to the initial linear portion of the stress strain curve, but offset from it by a strain of Δe = 0.002 (0.2%). The yield strength is taken as the stress level at which this straight line intersects the stress strain curve (Figure 2.1). The rationale is that if the material had been loaded to this stress and then unloaded, the unloading path would have been along this offset line resulting in a plastic strain of e = 0.002 (0.2%). This method of defining yielding is easily reproduced.

INTRODUCTION

Slip-line field analysis involves plane-strain deformation fields that are both geometrically self-consistent and statically admissible. Therefore, the results are exact solutions. Slip lines are really planes of maximum shear stress and are oriented at 45 degrees to the axes of principal stress. The basic assumptions are that the material is isotropic and homogeneous and rigid-ideally plastic (that is, no strain hardening and that shear stresses at interfaces are constant). Effects of temperature and strain rate are ignored.

Figure 6.1 shows a very simple slip-line field for indentation. In this case, the thickness, t, equals the width of the indenter, b and both are very much smaller than w. The maximum shear stress occurs on lines DEB and CEA. The material in triangles DEA and CEB is rigid. Although the field must change as the indenters move closer together, the force can be calculated for the geometry 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 6.2 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 are considered throughout this chapter.