Plasticating instabilities

Y.L. Bai, Thermo-Plastic Instability in Simple Shear, J. Mech. Phys. Solids 30, 195-207 (1982). 

We now turn to the other end of the stress-strain curve and explain why, in tensile straining, materials eventually start to neck, a name for plastic instability. It means that flow becomes localised across one section of the specimen or component, as shown in Fig. 11.5, and (if straining continues) the material fractures there. Plasticine necks readily chewing gum is very resistant to necking. 

Plastic instability is very important in processes like deep drawing sheet metal to form car bodies, cans, etc. Obviously we must ensure that the materials and press designs are chosen carefully to avoid instability. 

The plastic deformation of a member terminates with its rupture which normally occurs at the smallest section of the neck formed due to plastic instability. After being loaded into the plastic range, if the member is unloaded before plastic instability occurs then the elastic component of the strain can be recovered. This is a consequence of the atoms returning to... 

Figure 1.6 A load versus elongation diagram for a typical ductile metal, showing plastic instability.

It has been seen earlier that the condition for plastic instability can be expressed as... 

Assuming that the onset of plastic instability occurs at a fairly large value of plastic strain, one may consider the volume to remain essentially constant, so that A L = Aq L0. Differentiation gives... 

This relationship indicates that for plastic instability to occur the modulus of strain hardening, i.e., the slope of the true stress-true strain plot, should be equal to the true stress. This result is independent of any assumed functional relationship between o and s. If the relationship o = kt s " assumed to hold then one obtains... 

This relationship implies that the value of the true strain at which plastic instability sets in, i.e., necking starts to occur, is equal to the strain hardening exponent. 

The ultimate tensile strength (UTS) of a material refers to the maximum nominal stress that can be sustained by it and corresponds to the maximum load in a tension test. It is given by the stress associated with the highest point in a nominal stress-nominal stress plot. The ultimate tensile strengths of a ductile and of a brittle material are schematically illustrated in Figure 1.11. In the case of the ductile material the nominal stress decreases after reaching its maximum value because of necking. For such materials the UTS defines the onset of plastic instability. 

It is always very useful to be able to predict at what level of external stress and in which directions the macroscopic yielding will occur under different loading geometry. Mathematically, the aim is to find functions of all stress components which reach their critical values equal to some material properties for all different test geometries. This is mathematically equivalent to derivation of some plastic instability conditions commonly termed as the yield criterion. Historically, the yield criteria derived for metals were appHed to polymers and, later, these criteria have been modified as the knowledge of the differences in deformation behavior of polymers compared to metals has been acquired [20,25,114,115]. 

It seems reasonable to assume that crazing is a process which can occur quite naturally in any orientation hardening material, which exhibits plastic instability at moderate strains and in which the yield stress is much higher than the stress required for the nucleation of voids (cavitations). 

Since then the research activity may even have increased. Central themes were the molecular mechanisms in craze initiation, the influence of molecular weight and presence of entanglements, the nature of plastic instabilities, the role of crazes as precursors to cracks, and last but certainly not least, the formation of crazes in semicrystalline and multiphase polymers. Although considerable progress has been made in the above mentioned fields, some important questions are still open today. 

In most polymers, a marked necking phenomenon occurs very early after the yield point. This is the reason why it is not possible, in the range of large deformations, to determine strains in a large representative volume element (RVE). Consequently none of the dilatometers utilized to date can be used, except in a very restricted strain range. The latter statement concerns dual clip gage extensometers (axial + transversal) and also hquid displacement dilatometers (23). Once plastic instability has... 

Plastic instability—Incremental collapse incremental collapse is cyclic strain accumulation or cumulative cyclic deformation. Cumulative damage leads to instability of vessel by plastic deformation. 

Keeler SP (1961) Plastic instability and fractine in sheet stretched ovct rigid punches. PhD Thesis. Massachusetts Institute of Technology, Boston... 

SP Keeler. Plastic Instability and Fracture in Sheet Stretched over Rigid Punches. ScD thesis, Massachusetts Institute of Technology, Cambridge, MA, 1961. 

Although Y ions present only weak obstacles to dislocation motion [58], they are present in high concentrations and could, in theory, yield a large contribution to the flow stress. However, the crystals presently available apparently contain very small precipitates of ZrN [71] which provide stronger obstacles to slip than do unassociated Y ions [58, 72, 73]. Nonetheless, these unassociated Y ions do cause plastic instabilities, such as dynamic strain aging or the Portevin-Le Chatelier effect in... 

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