Indent plastic zone

An approximate model of the ISE can be developed with the aid of Figure 2.9. This figure shows a schematic cylindrical indenter with a conical tip being pushed into a specimen. The plastic zone is approximated by a segment of a sphere, and the diameter of the indent is 2r. The yield stress is Y, and the friction coefficient is a. 

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. 

Figure 2.1 illustrates the stress distribution on an amorphous PET sample at an indentation depth, h = 2 mi (Rikards et al, 1998). It can be seen that the depth of the plastic zone shown is here about five times the penetration distance of the Vickers indenter. 

Figure 2.1. Stress distribution, in MPa, for amorphous PET at an indentation depth h = 2 um (c is the depth of the plastic zone and a is one-half the projected length of the indentation diagonal). Stresses larger than 78 MPa are elastic. (After Rikards et ah, 1998.)...

The hemispherical core of radius a is immediately followed by a plastic zone. The plastic-elastic boundary lies here at a radius c, where c > a. The model allows the Pm/ T ratio to be related to a single non-dimensional variable ( tan 6)/ T, where is the contact angle between the sample and the indenter = 19.7° for a Vickers indenter). For v = 0.5, Johnson s analysis leads to ... 

An attractive higher-level framework within which to study plasticity during nanoindentation is provided by the dislocation dynamics methods described above. In particular, what makes such calculations especially attractive is the possibility of making a direct comparison between quantities observed experimentally and those computed on the basis of the nucleation and motion of dislocations. In particular, one can hope to evaluate the load-displacement curve as well as the size and shape of the plastic zone beneath the indenter, and possibly the distribution of dislocations of different character. While the... 

Fivel M. C., Robertson, C. F., Canova, G. R. and Boulanger, L., Three-Dimensional Modeling of Indent-Induced Plastic Zone at a Mesoscale, Acta Mater. 46, 6183 (1998). 

The hardness of a material is related to the plastic work performed in creating an indentation. Based on this concept, Burnett and Rickerby (1987) developed a model that takes into account the relative plastic zone size and the related amount of plastic work. Spherical cavity analysis using Marsh s relation (Marsh, 1964) applied to an indentation that creates a hemispherical plastic zone showed that the size of the latter varies with the size of the Vickers indentation according to Lawn, Evans and Marshall (1980)... 

Fig. 6. Schematic representation of the indentation model of Galanov et al. [37] in spherical coordinates (r < c—plastic core c < r < b—plastic zone r > b—elastic zone r < b —reversible phase transformations zone), (a) Phase transformation in the core and in the plastic zone, (b) Phase transformation within the plastic core.

The repeated measurement of the size of indents, and the interpretation of indent geometry for the purposes of calculation, may be tedious, and operator bias is almost unavoidable. The edge of the impression is not always well defined, and misleading edge effects may be associated with anisotropic plasticity or plastic recovery. Faceted and elongated grains, or other microstructural features, together with the limitations of contrast and resolution in the optical microscope, complicate the interpretation, while the shape of the indent may differ in different materials so-called pin-cushion or barreled indents, associated with different constitutive relations and frictional shear on the faces of the indentor in contact with the plastic zone [3]. Mismeasurement of indent size is a major source of scatter in the experimental data and the relative errors in the results of different operators. 

Four microcrack morphologies may develop in a brittle solid after formation of the plastic zone beneath the indent median cracks, Fig. 9a, radial cracks (a halfpenny shaped elliptical crack Fig. 9b, much larger than the median crack), lateral cracks (in a plane normal to the median and radial cracks). Fig. 9c, and Palmqvist... 

The median crack is a single, penny-shaped crack nucleated beneath the apex of the plastic zone created by the indentor. The diameter of the median crack is comparable with the indent size, and a median crack is not visible in a polycrystalline, opaque material. The driving force for nucleation of the median crack is the elastic tensile stress developed normal to the indentation direction at the elastic-plastic boundary when the external load is relaxed. Nucleation of a median crack depends on the presence of a suitable flaw. Once nucleated, the median crack will propagate spontaneously to a stable flaw size. The critical flaw size for growth is ... 

After nucleation, the median crack propagates away from the elastic-plastic zone boundary. Stable propagation occurs on increasing the external load, and the ratio of the diameter of the median crack, D, to the indent size is given by ... 

To be able to compare grain size and indentation size influences, the size of the plastically deformed zone must be known. This information comes from TEM investigations where an approximately constant ratio 2R /2a 4-5 between the plastic zone size 2/ pi and the length of the Vickers diagonals 2a can be derived for single as well as po/ycrystalline ceramics and for hard materials with fundamentally different bonding like (ionic) alumina [2] and (covalent) SiC [8]. 

With Eq. (3/3a), this deereasing influence of the load in more fine-grained microstructures is expressed by smaller values of the ratio (5e/5i)o- Table 2 displays fitting parameters for the experimental data of Fig. 4 for smaller grain sizes, increasing asymptotic hardness values H o are associated with decreasing parameters ( e/ i)o- la submicrometer alumina microstructures, the increase of the microplastic deformability (the decrease of the hardness) becomes smaller and smaller and approaches zero already at small indent sizes of 10-20 pm (see Fig. 4), and which characterizes the extension of the plastic zone approaches the initial deformability at rather small loads of about 1N. 

A difference appears only when, on increasing the load, the indentation size becomes much larger than the grains whereas in sapphire the deformability increases with the growing plastic zone, in the polycrystals this size effect is partly offset by the hindrance of dislocation activity due to the close spacing of the grain boundaries. 

Figure 8. Grain size effects in the hardness of alumina ceramics (HV-10, triangies), and indentation size effect (load influence) in the hardness of sapphire (squares). For HVIO vaiues, the lower x-axis gives the average grain size (data for sapphire indicated at 1000 pm). For the curves describing the size effect in sapphire, the upper x-axis represents the length of the Vickers diagonal (n = 1) or the plastic zone size ( = 4) [6,16]. See Fig. 2 (p. 166) for theoretical background.

The Estimated Temperature Rises in the Plastic Zone during Indentation Test of Several Brittle Materials... 

The indentation fracture mechanics approach likens abrasive-workpiece interactions for grinding of ceramics to small-scale indentation events. The deformation and fracture patterns observed for normal contact with a Vickers pyramidal indentor under an applied load P are illustrated in Figure 3.1. A zone of plastic deformation is foimd directly imder the indentor. Two principal crack systems emanate from the plastic zone med-ian/radial and lateral cracks. Median/radial cracks are usually associated with strength degradation and lateral cracks with material removal. 

Plastic zone, median/radial cracks (R), and lateral cracks (L) for Vickers indentation. (From Lawn, B.R. and Swain, M.V., /. Mater. Sci., 10, 113, 1975. With permission.)... 

Figure 7. A shallower indent showing a less pronounced plastic zone that did not reach the...

For most ceramics, the yield stress is not small, the indentation process is controlled by the dynamic process of the growth of a plastic zone around the indent, and the hardness will obviously be time dependent. In the context of this chapter the time dependency will be markedly influenced by mechanicochemical effects on the dynamic processes which involve bond rupture. 

B SiOg, 223 Bhat equation, 174 Bierbaum hardness, 1 Blunt punch, 12-13, 166-168 crack development, 166-168 equation for stress, 113-114 flow pattern, 12 indenter analysis, 12, 166 and plastic zone. 111 Bond breaking model. 132 rate equation, 132 Borazon, 231 Borides, 297-301 bonding in, 298-299 hardness anisotropy, 84, 93, 108-109 Knoop hardness, 87 slip systems, 108-109 structures, 299... 

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