Ideal elastic-plastic deformation

Let us consider two limiting cases of the adduced in Fig. 4.15 dependence at = 0 and 1.0, both at d = 3. In the first case (d = 2) the value dU = 0 or, as it follows from dU definition (the Eq. (4.31)), dW = dQ and polymer possesses an ideal elastic-plastic deformation. Within the frameworks of the fractal analysis d =2 means, that (p, = 1.0, that is, amorphous glassy polymer structure represents itself one gigantic cluster. However, as it has been shown above, the condition d =2 achievement for polymers is impossible in virtue of entropic tightness of chains, joining clusters, and therefore, d > 2 for real amorphous glassy polymers. This explains the experimental observation for the indicated polymers dU 0 or dW dQ [57], At Vg, =... 

An ideal elastic-plastic solid is one that deforms elastically until the yield stress is reached and thereafter deforms plastically under that stress without hardening. One such material is isotropic, with Young s modulus 80 X 10 Pa and yield stress 12 X 10 Pa. Aright circular cylinder of this material, initially 0.2 m long and of radius... 

The resistance to plastic flow can be schematically illustrated by dashpots with characteristic viscosities. The resistance to deformations within the elastic regions can be characterized by elastic springs and spring force constants. In real fibers, in contrast to ideal fibers, the mechanical behavior is best characterized by simultaneous elastic and plastic deformations. Materials that undergo simultaneous elastic and plastic effects are said to be viscoelastic. Several models describing viscoelasticity in terms of springs and dashpots in various series and parallel combinations have been proposed. The concepts of elasticity, plasticity, and viscoelasticity have been the subjects of several excellent reviews (21,22). 

It is instructive to describe elastic-plastic responses in terms of idealized behaviors. Generally, elastic-deformation models describe the solid as either linearly or nonlinearly elastic. The plastic deformation material models describe rate-independent behaviors in terms of either ideal plasticity, strainhardening plasticity, strain-softening plasticity, or as stress-history dependent, e.g. the Bauschinger effect [64J01, 91S01]. Rate-dependent descriptions are more physically realistic and are the basis for viscoplastic models. The degree of flexibility afforded elastic-plastic model development has typically led to descriptions of materials response that contain more adjustable parameters than can be independently verified. 

Rheology is the science that deals with the deformation and flow of matter under various conditions. The rheology of plastics, particularly of TPs, is complex but understandable and manageable. These materials exhibit properties that combine those of an ideal viscous liquid (with pure shear deformations) with those of an ideal elastic solid (with pure elastic deformation). Thus, plastics are said to be viscoelastic. 

The present review shows how the microhardness technique can be used to elucidate the dependence of a variety of local deformational processes upon polymer texture and morphology. Microhardness is a rather elusive quantity, that is really a combination of other mechanical properties. It is most suitably defined in terms of the pyramid indentation test. Hardness is primarily taken as a measure of the irreversible deformation mechanisms which characterize a polymeric material, though it also involves elastic and time dependent effects which depend on microstructural details. In isotropic lamellar polymers a hardness depression from ideal values, due to the finite crystal thickness, occurs. The interlamellar non-crystalline layer introduces an additional weak component which contributes further to a lowering of the hardness value. Annealing effects and chemical etching are shown to produce, on the contrary, a significant hardening of the material. The prevalent mechanisms for plastic deformation are proposed. Anisotropy behaviour for several oriented materials is critically discussed. 

The term elastic limit is mainly a definition. It describes a stress which, if exceeded, will influence plastic deformation. Experimentally, the elastic limit is practically unattainable because it is a limit. Either it has not been reached or it is overreached. Ideally, the elastic limit and proportional limit are the same. 

Figure 5.8 Illustration of (a) ideal elastic deformation followed by ideal plastic deformation and (b) typical elastic and plastic deformation in rigid bodies. From Z. Jastrzebski, The Nature and Properties of Engineering Materials, 2nd ed.. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc.

The type of deformation that occurs will depend upon the material s inherent properties and the amount of force being applied. Deformation can be described in three main ways, elastic, plastic, and brittle fragmentation, but it is important to realize that these are idealized deformation mechanisms—most real materials are some combination of two or all three mechanisms. 

The liquid metal mercury-solution interface presents the advantage that it approaches closest to an ideal polarizable interface and, therefore, it adopts the potential difference applied between it and a non-polarizable interface. For this reason, the mercury-solution interface has been extensively selected to carry out measurements of the surface tension dependence on the applied potential. In the case of other metal-solution interfaces, the thermodynamic study is much more complex since the changes in the interfacial area are determined by the increase of the number of surface atoms (plastic deformation) or by the increase of the interatomic lattice spacing (elastic deformation) [2, 4]. 

Note that the preceding equation is for ideal cases, in which the particles are monodis-persed, spherical, and totally elastic, and the contact surface is clean. In practice, the particles are usually nonspherical and polydispersed the collision could have involved some heat loss, plastic deformation, or even breakup and the contact surface may have impurities or contaminants. In these cases, a correction factor tj is introduced to account for the effects of these nonideal factors. The applicable form of the electric current through the ball probe is, thus, given by... 

Figure 2.4 schematically depicts the indentation geometry for a Vickers indenter penetrating a cylindrical surface with a radius r. In the case of an ideal plastic deformation (i.e. when elastic stresses are absent) after load removal, the square pyramidal indenter leaves a rhombic indentation with one of its diagonals parallel to the filament axis. Let 2BC be the measured indentation length, f x, normal to the filament axis and 2DE the indentation length, which would arise on a flat surface for the same penetration depth. For an isotropic material, = C. However, as a result of the existing curvature, i > (anisometric indentation). From Fig. 2.4, DE = BC + BD tan a/2) and since tan(a/2) —1/2 one has = li + IBD.

Miiser [25] examined yield of much larger tips modeled as incommensurate Lennard-Jones solids. The tips deformed elastically until the normal stress became comparable to the ideal yield stress and then deformed plastically. No static friction was observed between elastically deformed surfaces, while plastic deformation always led to pinning. Sliding led to mixing of the two materials like that found in larger two-dimensional simulations of copper discussed in Section IV.E. 

Our treatment thus far has centered on idealized geometries in which the dislocation is presumed to adopt highly symmetric configurations which allow for immediate insights from the linear elastic perspective. From the phenomenological standpoint, it is clear that we must go beyond such idealized geometries and our first such example will be the consideration of kinks. The formation of kink-antikink pairs plays a role in the interpretation of phenomena ranging from plastic deformation itself, to the analysis of internal friction. 

Nevertheless, machines with concave die rings and internal press rollers do have advantages. For example, if the feed material exhibits a certain elastic behavior, because the forces in the relatively long and slender nip increase slowly, a more complete conversion of temporary elastic into permanent plastic deformation takes place. Fig. 8.41b is another presentation of the forces at work. Feed, ideally deposited in a uniform layer on the die, is pulled into the space (nip) between roller and die and compressed. Friction between roller, die, and material as well as interparticle friction in the mass are responsible for the pull of the feed into the nip and for densification. Smooth surfaces of roller and/or die may result in slip. Axial grooves in the roller, which may also favor build-up of a thin layer of material, and the above mentioned residual layer of densified feed on the die effectively reduce slip. Low interparticle resistance to flow or a distinct plasticity result in a more or less pronounced tendency of the mass to avoid the squeeze" (back-flow), thus reducing densification and potentially choking the machine (see above). 

The creep and recovery of plastics can be simulated by an appropriate combination of elementary mechanical models for ideal elastic and ideal viscous deformations. Although there are no discrete molecular structures which behave like individual elements of the models, they nevertheless aid in understanding the response of plastic materials. 

In terms of rheology ceramic bodies hold a special position between ideal elastic and ideal plastic bodies, as they exhibit Bingham behaviour. Plotted on a shear stress/shearing speed graph, ceramic plastic bodies start to deform only after having reached a certain shear stress tq, the so-called yield point. 

Hardness is a measure of a material s ability to resist elastic and plastic deformation. The hardness of non-ideal material is determined by the intrinsic stiffness of the material, as well as by the nature of its defects, be they point defects, dislocations, or macroscopic defects such as microcracks etc. For ideal systems, the hardness of a material will scale with its bulk modulus. 

On the other hand materials deform plastically only when subjected to shear stress. According to Frenkel analysis, strength (yield stress) of an ideal crystalline solid is proportional to its elastic shear modulus [28,29]. The strength of a real crystal is controlled by lattice defects, such as dislocations or point defects, and is significantly smaller then that of an ideal crystal. Nevertheless, the shear stress needed for dislocation motion (Peierls stress) or multiplication (Frank-Read source) and thus for plastic deformation is also proportional to the elastic shear modulus of a deformed material. Recently Teter argued that in many hardness tests one measures plastic deformation which is closely linked to deformation of a shear character [17]. He compared Vickers hardness data to the bulk and shear... 

Figure 7. The graph on the left shows the ideal shear stress-deformation curves for small beech specimens (right) with an ideal elastic adhesive coimection or with an ideal plastic adhesive connection. The industrial partner prepared several adhesive layers which exhibited stress-deformation diagrams close to the theoretical requirements. Three of them are shown above 027-2,009-05 and 013-1.

The idealized laws just reviewed can, however, not describe the behavior of matter if the ratios of stress to strain or of stress to rate of strain is not constant, known as stress anomalies. Plastic deformation is a common example of such non-ideal behavior. It occurs for solids if the elastic limit is exceeded and irreversible deformation takes place. Another deviation from ideal behavior occurs if the stress depends simultaneously on both, strain and rate of strain, a property called a time anomaly. In case of time anomaly the substance shows both solid and liquid behavior at the same time. If only time anomalies are present, the behavior is called linear... 

That property of plastic materials because of which they tend to recover their original size and shape after removal of a force causing deformation. If the strain is proportional to the applied stress, the material is said to exhibit Hookean or ideal elasticity. 

Elasticity (1664) n. A property that defines the extent to which a material resist small deformations from which a material recovers completely when deforming force is removed. When the deformation is proportional to the applied load, the material is said to exhibit Hookean elasticity or ideal elasticity. Elasticity equals stress divided by strain. Shah V (1998) Handbook of plastics testing technology. John Wiley and Sons, New York. Elias HG (1977) Macromolecules, vols 1-2. Plenum Press, New York. Weast RC (ed) (1978) CRC handbook of chemistry and physics, 59th edn. The Chemical Rubber Co., Boca Raton, EL. 

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