Applied Plasticity

In a single crystal of many metals, the main mechanism of plastic deformation is simple shear parallel to preferred planes and directions, which at ordinary temperatures coincide with those of the highest atomic density. Slip is initiated along a particular plane and in a given direction when the associated component of the shear stress attains a critical value under increasing external load. The amount of plastic deformation in a single crystal is specified by the glide strain, which is the relative displacement of two parallel slip planes at a unit distance apart. When there are several possible slip directions in a crystal lattice, the displacement of any point in the crystal due to simultaneous shears in the appropriate directions can be found from simple geometry. The mechanism of slip-induced plasticity in single crystals, governed by the glide motion of dislocations along corresponding slip planes, has been the subject of numerous investigations in the past.

In many problems of practical interest, it is a reasonable approximation to disregard the elastic component of strain in the theoretical analysis even when the body is only partially plastic. In effect, we are then dealing with a hypothetical material which is rigid when stressed below the elastic limit, the modulus of elasticity being considered as infinitely large. If the plastically stressed material has the freedom to flow in some direction, the distribution of stress in the deforming zone of the assumed rigid/plastic body would approximate that in an elastic/plastic body, except in a transition region near the elastic/plastic interface where the deformation is restricted to elastic order of magnitude. The assumption of rigid/plastic material is generally adequate not only for the analysis of technological forming processes, where the plastic part of the strain dominates over the elastic part, but also for the estimation of the yield point load when the rate of work-hardening is sufficiently small (Section 1.2). In the present chapter, we shall be concerned with problems in plane stress involving rigid/plastic bodies which are loaded beyond the range of contained plastic deformation.

Many physically important problems of plasticity involve solids of revolution which are loaded symmetrically about their geometrical axes. Typical examples of axially symmetrical distribution of stress and strain are provided by the expansion of circular cylindrical tubes, upsetting of cylindrical blocks, extrusion of cylindrical billets, and the axial drawing of wires and tubes. In the theoretical analysis of such problems, which involve large plastic strains, it is customary to assume the material to be rigid/plastic, for which no deformation can occur until the load attains the yield point value. The rigid/plastic assumption is adequate for the estimation of the yield point load itself, and also for the determination of stresses and strains in the plastically deforming zone during the continued loading. In this chapter, we shall deal with several examples in which the stress distribution is axially symmetrical, together with a few related problems involving three-dimensional states of stress.

In this chapter, we shall be concerned with the yield point state of perfectly plastic plates whose thickness is small compared to the dimensions of its plane faces. The load acting on the plate is normal to its surface, and is regarded as positive if it is pointing vertically downward. The vertical displacement of the middle surface is assumed to be generally small compared to the plate thickness, and plane vertical sections are assumed to remain plane during the bending. The deformation of the plate is therefore entirely defined by the vertical displacement of its middle surface, which remains effectively unstrained during the bending. A theory based on this model is found to be satisfactory not only in the elastic range but also in the plastic range of deflections. However, when the deflection of the plate exceeds the thickness, significant membrane forces are induced by the bending, the effect of which is to enhance the load carrying capacity of the plate.

A shell is a thin-walled structure in which the material fills the space between two parallel or nearly parallel surfaces, the mean surface that halves the shell thickness being known as the middle surface of the shell. When the shell is loaded to the point of plastic collapse, the deformation proceeds in an unrestricted manner under constant load, provided geometry changes are disregarded and the material is considered as ideally plastic. As in the case of thin plates, the incipient deformation mode of the shell will be described in terms of that of the middle surface, and the associated stress field will be specified in terms of the stress resultants, acting across the shell thickness. The usual assumptions of rigid/plastic material will be made for the estimation of the collapse load.

During continued plastic deformation of a polycrystalline metal, certain crystallographic planes tend to rotate with respect to the direction of the greatest principal strain, and the texture of the metal assumes a fibrous appearance. The phenomenon is similar to that of a single crystal in which the slip planes tend to rotate in such a way as to become parallel to the direction of the maximum principal strain. Since the individual crystal grains in a polycrystalline aggregate cannot rotate freely due to their mutual constraints, the development of a preferred orientation in a polycrystalline metal is much more complex. As a result of progressive cold work, an initially isotropic metal therefore becomes anisotropic, and its mechanical properties vary with the direction. The yield strength of the metal in the direction of mechanical working may be greater or less than that in the transverse direction, depending on the type of preferred orientation that is produced by cold work. In a cold rolled sheet of brass, for instance, the tensile yield stress transverse to the direction of rolling can be considerably higher than that in the rolling direction. Although anisotropy and the Bauschinger effect always occur together, the latter can be largely removed by mild annealing, while the former can be altered only by carrying out the heat treatment above the recrystallization temperature.

In a typical boundary value problem, involving prescribed nominal traction rates on a part S _F of the boundary surface, and prescribed velocities on the remainder S _v , more than one mode of deformation may be possible when the applied load reaches a critical value. The lack of uniqueness of the deformation mode under given boundary conditions is commonly referred to as bifurcation, the current shape and mechanical state of the body being supposed to be given or previously determined. For a linear solid, in which the strain rate is a unique linear function of the stress rate during both loading and unloading, a bifurcation mode corresponds to an eigensolution of the field equations, and represents a mode quasi-statically possible under constant loads on S _F and rigid constraints on S _v In dealing with the conventional elastic/plastic solid, which is bilinear in the sense that the strain rate is related to the stress rate by separate linear functions for loading and unloading, it is convenient to introduce a linear comparison solid with identical boundary conditions (Section 1.5). While bifurcation in the linearized solid can occur under any given traction rates on S _f and velocities on S _v when the load becomes critical, bifurcation in the actual elastic/plastic solid would occur only under those traction rates for which there is no instantaneous unloading of the material that is currently plastic. The incremental theory of plasticity will be almost exclusively used in this chapter for the estimation of the critical load.

In this chapter we shall be concerned with the class of problems in which the plastic deformation is so rapid that the inertia effects cannot be disregarded. Problems of dynamic plasticity arise in the high-velocity forming of metals, penetration of high-speed projectiles into fixed targets, enlargement of cavities by underground explosion, and the design of crash barriers related to collisions, to name only a few. The rate of loading and the size of the components are usually such that the deformation process can be described in terms of the propagation of elastic/plastic waves. However, simplified theories which disregard the wave propagation phenomenon are generally capable of providing useful information for practical purposes. In the case of structural members subjected to impact loading, the mode of plastic deformation can be most conveniently represented by the existence of discrete yield hinges that rapidly move away from the point of loading. The concept of moving yield hinges is a useful device for the dynamic analysis of structures.