Plasticity, dislocation, yield stresses

At room temperature, NiAl deforms almost exclusively by (100) dislocations [4, 9, 10] and the availability of only 3 independent slip systems is thought to be responsible for the limited ductility of polycrystalline NiAl. Only when single crystals are compressed along the (100) direction ( hard orientation), secondary (111) dislocations can be activated [3, 5]. Their mobility appears to be limited by the screw orientation [5] and yield stresses as high as 2 GPa are reported below 50K [5]. However, (110) dislocations are responsible for the increased plasticity in hard oriented crystals above 600K [3, 7]. The competition between (111) and (110) dislocations as secondary slip systems therefore appears to be one of the key issues to explain the observed deformation behaviour of NiAl. 

In textbooks, plastic deformation is often described as a two-dimensional process. However, it is intrinsically three-dimensional, and cannot be adequately described in terms of two-dimensions. Hardness indentation is a case in point. For many years this process was described in terms of two-dimensional slip-line fields (Tabor, 1951). This approach, developed by Hill (1950) and others, indicated that the hardness number should be about three times the yield stress. Various shortcomings of this theory were discussed by Shaw (1973). He showed that the experimental flow pattern under a spherical indenter bears little resemblance to the prediction of slip-line theory. He attributes this discrepancy to the neglect of elastic strains in slip-line theory. However, the cause of the discrepancy has a different source as will be discussed here. Slip-lines arise from deformation-softening which is related to the principal mechanism of dislocation multiplication a three-dimensional process. The plastic zone determined by Shaw, and his colleagues is determined by strain-hardening. This is a good example of the confusion that results from inadequate understanding of the physics of a process such as plasticity. 

Dislocations multiply in a facile manner during a plastic deformation process, and several mechanisms for this have been observed by electron miscroscopy. Dislocations are destroyed by the processes of recovery and recrystallization during annealing after plastic deformation. Since dislocations cause low-yield stresses in metals and other solids, solid strengthening is accomplished either by eliminating dislocations or by immobilizing them. 

PLASTIC DEFORMATION. When a metal or other solid is plastically deformed it suffers a permanent change of shape. The theory of plastic deformation in crystalline solids such as metals is complicated but well advanced. Metals are unique among solids in their ability to undergo severe plastic deformation. The observed yield stresses of single crystals are often 10 4 times smaller than the theoretical strengths of perfect crystals. The fact that actual metal crystals are so easily deformed has been attributed to the presence of lattice defects inside the crystals. The most important type of defect is the dislocation. See also Creep (Metals) Crystal and Hot Working. 

The transition metal carbides do have a notable drawback relative to engineering applications low ductility at room temperature. Below 1070 K, these materials fail in a brittle manner, while above this temperature they become ductile and deform plastically on multiple slip systems much like fee (face-centered-cubic) metals. This transition from brittle to ductile behavior is analogous to that of bee (body-centered-cubic) metals such as iron, and arises from the combination of the bee metals strongly temperature-dependent yield stress (oy) and relatively temperature-insensitive fracture stress.1 Brittle fracture is promoted below the ductile-to-brittle transition temperature because the stress required to fracture is lower than that required to move dislocations, oy. The opposite is true, however, above the transition temperature. 

Investigation of the deformation relief occurring on the surface of samples additionally subjected to by 15% strain after different number of compression steps have shown that plateau on the initial portion of strain curves is result of strain localization (Fig. 2a) in macro shear bands (MSB). Its appearance is result of scattering some dislocation boundaries onto individual dislocations (Baushinger effect) and formation of avalanche of mobile dislocations (Fig. 2b). So, in this case yield of titanium is controlled by substructure that, probably, leads to weak dependence of yield stress on strain. Macrobands formed at the beginning of the cycle of loading remain until the end of loading. So, plastic flow of titanium is localized. 

Fatigue cracks are initiated at stresses below the conventional yield stress at which bulk plastic flow occurs. In this stress range, surface inhomogeneities and favorably oriented grains allow slip by movement of dislocations, which produce surface offsets that, due to localized work hardening (slip interference), are not reversed as the stress is reversed. As a result, surface intrusions and extrusions of the form proposed in Fig. 7.118 are produced (Ref 162). The intrusions and extrusions,... 

This result gives the dependence of crystal yield strength on particle size and work hardening due to the creation and movement of N dislocations during plastic flow. In the limit of small particle size, < 20 microns, where typically dUo/dx (N d)r Uo and the third order correction term is small, the yield stress behaves as Ty Nj if. This prediction accounts for the observed increased yield strength of small crystals and forms the basis of the specialty iron and steel industry [26.27], It is also likely to be the reason why the shock and impact sensitivity decrease with crystal size as. ... 

For some metals, notably steels, there is an abrupt break in the stress-strain plot at the upper yield point (Figure 10.18). This is followed by continued deformation at a lower yield stress, at the lower yield point, before the curve rises again. Between the upper and lower yield points, deformation occurs in localised regions that have the form of bands, rather than across the specimen in a uniform manner. The reason for this is that the dislocations, which would move during plastic deformation, are pinned in the steel, mainly by the interstitial carbon atoms present. At the upper yield stress, these become mobile and, once released, they can move and multiply at a lower stress value. This is analogous to sticking and slipping when a body over-... 

The yield strength (Oy) is expressed in terms of the yield stress, Oo (Oo is related to the intrinsic stress, Oi, resisting dislocation motion) and the grain size, d. When this relationship was deduced, it applied to a situation in which the grains were deformed by plastic deformation and the GBs acted as barriers to dislocation motion. This model is unlikely to be valid in general for ceramics since deformation by dislocation glide is not common. However, the relationship between d and Oy does hold as we saw for polycrystalline MgO in Figure 1.2. 

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... 

It has been possible, for metals and ceramic materials, to demonstrate by direct observation the existence of lattice defects called dislocations, using the techniques of transmission electron microscopy. These studies have shown that it is often adequate to assume that dislocation motion is responsible for the observed plastic, or permanent, deformation, and that this motion is negligible at stresses below the yield stress. Although very refined microstrain measurements and internal friction experiments have failed to define a stress range in which dislocation motion is completely absent, there is still a clear distinction for these materials between elastic and plastic strain, both on a macroscopic level, in terms of permanency of deformation, and on a microscopic level in terms of large scale dislocation motion. 

To probe the models for nucleation-controlled plastic flow we compare the predicted temperature dependence of the tensile plastic resistance with the tensile-yield-stress experimental results of Brooks and Mukhtar (2000). For comparison the polyethylene PE3 of average molecular weight = 131000 with a crystallinity of only 0.673 and lamella thickness of 34.3 nm is chosen. For the predictions of the temperature dependence, eqs. (9.26)-(9.28) of Section 9.4.3 are used, where we take in the denominator of eq. (9.25) the factor (1 + K), since the experiments were performed in tension. Noting that the lamella thickness of this polymer type is 1 = 34.3 nm, which is thicker than that for mode A of monolithic-screw-dislocation nucleation, we consider only modes B and C involving nucleation of screw-dislocation half loops and edge-dislocation half loops, and, together with the results of Fig. 9.21, we state the expected tensile yield stress Oy to be... 

UO2 has a surprisingly low brittle-ductile transformation. The only observed slip system at low temperatures is lll (110), and this does not depend on stoichiometry. Sources of mobile dislocation are an issue, however, and in order to achieve deformation at temperatures below 600 °C the crystals must be pre-deformed at 600 °C. With such pre-deformed crystals, deformation to plastic strains >1% is possible with modest yield stresses, typically 80 MPa at 450°C, 110 MPa at 400 °C, and 120 M Pa at 250 °C. Attempts to deform these crystals at room temperature were not successful, although perhaps a more careful alignment of the load train might have allowed plastic deformation at temperatures below 250 °C. Microindentation at room temperature is always possible, however, and the Knoop hardness anisotropy at room temperature is also consistent with 111 slip [74]. The yield stress at 600 °C was variable, but surprisingly was not a function of the 0/ U ratio the plastic deformation... 

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