Spherical Indentation

Vickers Hardness. The Vickers or diamond pyramid hardness (DPH) developed in 1924 was an improvement over the Brinell test. The Vickers test used a pyramidal diamond as the indenter. This permitted the hardness testing of much harder materials, and the constant 136° angle of the indenter eliminated the problem of variable indentation shape encountered using spherical indenters (1). 

Figure 2.7 Plastic flow stresses of from Brinell spherical indentation-hardnesses versus elastic shear moduli. Nominally pure fee metals at 200K (Gilman, 1960).

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. 

Fig. 3.12. Schematic drawings of indentation (or liber push-out) techniques using (a) a spherical indenter (b) a Vickers microhardness indenter (c) on a thin slice. After Grande ct al. (1988).

Indentation hardness determinations were performed in dynamic mode ( 1500 mm/sec impact speed) using a pendulum impact device and in quasistatic mode ( 0.008 mm/sec impact speed) with a custom-built indentation tester. The spherical indenters were of 2.54 cm diameter and 65.6 g mass, and the pendulum length was 92.3 cm with a release angle of 30°. Quasistatic indentation forces were selected to produce indentations of a similar size to the dynamic indentation test (1.5 to 2.0 mm radius). The compact indentations were measured using a white light interferometer (Zygo Corporation, Middlefield, Connecticut, U.S.A.) and the dent depth, dent diameter, apparent radius of curvature, and pendulum initial and rebound heights were used to calculate the indentation hardness of the compacts. 

Fig. 6.2.7 (a). Diagram of parameters of chevron cracks associated with spherical indenter action and (b) crack mechanism seen in cross-section of Hertz s cone (C—C crack plane). Crack growth tends to enlarge in (r, 0) in relation to the initiation of the crack peak, which optimizes the conditions for energy liberation. (After Lawn et at., 1975)... 

The equation, unlike that for a spherical indenter, contains no terms relating to the state of the sample surface (flaw population, etc.). Instead, it considers the cone angle as a variable through its influence on the penetration field. 

Mattice et al. (2006) extended Hertz analysis to spherical indentation of a viscoelastic material ... 

It should be pointed out that Equation (11) should be modified to Equation (1) when compression of a single particle between two surfaces is modelled, that is, spherical indentation is replaced by compression between two surfaces. Applications in which these models were applied to experimental data from compression testing are described later. 

In the Brinell test (Brinell, 1900 Meyer, 1908) the indenter consists of a hard steel ball, though in examining very hard metals the spherical indenter may be made of tungsten carbide or even of diamond. Another type of indenter which has been widely used is the conical or pyramidal indenter as used in the Ludwik (1908) and Vickers (see Smith Sandland (1925)) hardness tests, respectively. These indenters are now usually made of diamond. The hardness behaviour is different from that observed with spherical indenters. Other types of indenters have, at various times, been described, but they are not in wide use and do not involve new principles. 

Staines M, Robinson WH, Hood JAA Spherical indentation of tooth enamel. J Mater Sci 1981 16 2551-2556. 

Chatzis and Dullien (1977) were the first to use tubular bonds and spherical sites to simulate pore throats and bodies respectively, in network models of porous media. Previously, intersections (sites) were assumed not to have any volume. In their model, individual elements are represented as cylindrical tubes with spherical indentations in the middle (Fig. 3-17C). Bond lengths and bond and site radii can be drawn from independent distribution functions, or as is more common, correlated with each other so that only one distribution function is required (Ioanni-dis Chatzis, 1993). 

Figure 12-18. Degree of contact for a hard spherical indenter pressed into the flat, rough surface of an unconstrained deformable body. The data divide into two classes. Homogeneous solids drawn rod aluminum, bead blasted A cold rolled aluminum, bead blasted cold rolled aluminum, bead blasted and then annealed Ygold, bead blasted and then annealed work-hardened turned copper. Degree of 0.007. Solids with hardened surface layers aluminum, bead blasted. Degree of contact ranged Data by Williamson and Hunt [16].

Fig. 2. Schematic of crack formation in oxide scales due to spherical indentation (R = 200 pm, h l pm,a = 30°, load F= 20 to 100N).

The arrangement of the radial and circumferential-like cracks around spherical indents depends strongly on the surface orientation [33], A schematic representation of the crack pattern on (111) NiAl is shown in Fig.9 together with surface profiles in different crystal directionfMatfiin the surface plane. 

Fig. 9. Arrangement of cracks around a spherical indent on (111) NiAl toghether with surface profiles measured by LSM.

Fig. 10. Results of FliM calculations of the spherical indentation surface profile (a), radial (b) and circumferential (c) surface strains versus radial distance from the centre of the indent for a piecewise linear work hardening (tangent modulus Et - 0.1 E for , < 0.02 and Er -- 0.0044 E for e, >0.02 (full line) T = 0.1 E (dashed line) E - 180 GPa, v =- 0.3,...

Spherical indentation is easily to perform. However, a determination of the resulting stress state in the scale seems to be too difficult.Thus, this test can be used mainly for comparative studies. The bend test is presumably more appropriate for determining fracture-mechanical properties. Other specimen deformation can be more suitable for special purposes. 

More recent is the University of Twente mixing ring (or TMR). This device is made olf three parts smooth wall barrel extension, a torpedo with semi-spherical indentations (similar to these in CTM) and a perforated sleeve that goes between them. The sleeve provides shallow indentations. It slowly rotates between the torpedo and the barrel by the virtue of a drag flow. The flow in TMR is both dispersive and distributive. The advantage of TMR is its suitability to improve mixing in injection or blow molding machines [Housz, 1989]. 

Some attempts to evaluate gel strength as a function of the solvent were done using a spherical indenter. Unfortunately, nothing was reported on crack appearance. However, it was reported that gels become stronger as the amount of fumed silica increases [48],... 

Scott, O.N., Begley, M.R., Komaragiri, U., and Mackin, T.J. Indentation of freestanding elastomer films using spherical indenters, Acta Materialia 2005,52,4877-4885. 

Stress distribution in indentation is largely affected by the indenter tip geometry, which is a vital factor in determining the boundary conditions for the field. The major types of indenter tips shown schematically in Figure 3 may be separated into two groups, viz. point-force (pyramidal and conical) and spherical indenters. Correspondingly, Boussinesq and Hertzian stress fields will describe point-force and spherical indentation in the case of purely elastic loading (Fig. 4). To account for possible elastic compliance of the indenter, a reduced elastic modulus Er is... 

The quantification of contact pressures in spherical indentation is based on the indentation model of Field and Swain [43]. The procedure is similar to the point-force indentation and includes the evaluation of the contact radius at a given load (fle) from its value at peak load (ac.max)- The mean contact pressure is then found as [43]... 

Fig. 54. Schematic of indentation damage for (a) Vickers and (b) spherical indentations in zirconia-based ceramics. Dark areas comespond to higher microcrack density. Dashed lines show the elastic-plastic boundary. After Reference [251].

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