Yield Functions and Plastic Potentials for BCC Metals and Possibly Other Materials

Yield functions and plastic potentials are expressed in terms of the invariants of the stress tensor for polycrystalline metals and other isotropic materials. The plastic volume change data of Richmond is used to evaluate the embedded materials properties for some bcc metals and one polymer. A general form for the plastic potential is found that is intended to represent and cover a wide range of materials types. Introduction The present work is concerned with the yield functions describing the departure from ideal, linear elastic conditions, and with the plastic potentials which are used to describe the ensuing plastic flow which occurs after the yield functions have been traversed. The definitive theoretical work in this area was formalized by Hill in his early and insightful book, Hill (1950), and his many later contributions such as Hill (1959, 1968) and Hill and Rice (1972). The definitive experimental work was given by Richmond and colleagues (to be cited later), based mainly upon body centered cubic (bcc) metals. The present work follows the lead of these two valuable sources in pursuing these matters. In the time since these two contributions, most efforts on using yield functions and plastic potentials have proceeded by taking whatever forms were expedient for the particular application of immediate interest. A main objective here is to deduce general representations for yield functions and plastic potentials that have a minimum number of embedded parameters (properties) in order to have the most reasonably useful forms for application to a wide range of full density materials. The resulting forms will be evaluated for various materials types. We begin with the consideration of very ductile metals. Face centered cubic (fcc) metals provide the backbone of ideal elastic-plastic behavior. Such metals as copper, nickel, aluminum, silver, gold and lead constitute the basis for ideal plastic flow, whether described at the dislocation level or the continuum level using so-called J2 plasticity theory. The first significant evidence for the non-ideal behavior not adequately described by J2 theory is the class of bcc metals: chromium, molybdenum, tantalum, tungsten, vanadium, iron and most steels. These bcc metals provide the perfect test bed for studying the inception of non-ideal plastic effects, with the ultimate aim to generalize beyond this class to much broader classes of materials such as polymers and ceramics. The present work is at the macroscopic level, but it is helpful to rationalize controlling effects at a more basic level. There are at least two possible sources for the departure of most bcc metals from the ideal behavior exhibited by most fcc metals. One is the far from ideal form of grain boundaries on the atomic scale. The state of disorder quite naturally implies a state of non-uniformity and heterogeneity in the strength properties of grain boundaries. The other possible source of non-ideal behavior for bcc metals is the fact that the core structure of dislocations spreads over many atomic layers of glide planes, Hirsch (1960), Christian (1983) and Vitek (1975). This greatly decreases the mobility of the dislocations. A consequence of this is a greater sensitivity to temperature (and pressure) dependent behavior. Other explanations are certainly possible for the non-ideal behavior of bcc metals. Dislocation dynamics studies related to these matters are rapidly evolving and likely will ultimately provide new insights. Until that time however only the two sources just mentioned will be further considered here. Concerning the non-uniformity of strength of grain boundaries in fcc materials, this of little importance because the great mobility of the dislocation structures implies that the loads on the grain boundaries are insufficient to cause any disruption of the grain boundary. However, in bcc metals the grain boundaries are much more highly stressed than in fcc metals. Interest here is with initially isotropic materials so only polycrystalline aggregates of bcc crystals will be considered. The actual behavior on the grain scale involves variability from grain to grain, and progressive and accumulating degrees of irreversible damage. Because of this variability, the slip on the grain boundaries and the slip systems within the grains may coordinate and interact in some grains. A macroscopic description is necessarily an average over all grains. Probably the grain boundary behavior is much more variable than that of the grain-to-grain form. The grain failure itself and the grain boundary failure are not necessarily independent and competing physical events. They can be interactive with the grain boundaries operating to some extent as slip systems in conjunction with those within the crystal. In the macroscopic view, sufficiently general descriptors must be used to cover these possibilities. Even if only shear stresses are needed for the individual crystals, both shear and normal stresses are needed for the grain boundary failure. Macroscopically this then requires both shear and normal stresses. The behavior of the polycrystalline aggregate thus depends not only upon the shear stress on the slip planes in the individual grains but also upon normal stresses acting within the grains and upon the grain boundaries. The corresponding macroscopic characteristics involved are the shear stresses and the mean normal stress. For the failure of isotropic materials, we will then use the invariants that involve the shear stresses and the mean normal stress. The formal statement of the yield function is given by f(!ij) " 1 (1) The plastic potential G(!ij) describes the plastic flow through the standard flow form