Indentation hardness equation

Because the indentation varies with time, the modulus must be specified for a certain indentation time, eg, a 10-s modulus. The Hertz equation holds only for purely elastic materials. However, it has been appHed to viscoelastic materials, including polymers and coatings, with excellent results (249—256). Indentation hardness vs temperature curves are shown in Figure 40 (249,251). 

The Bib and BI are defined as shown below in Equations (2) and (3), respectively. Both indices describe the bonding ability of the tablets by using the ratio of tensile strength and indentation hardness. However, Bib is determined at low-indentation speed (Hq) and BIw is determined at high speed (Hio). Hence, BI is considered more indicative of actual tableting conditions, whereas Bib relates to slower compactions speeds typical of development or roller compaction processes. 

The experimental data for the temperature dependence of the strain rate is shown in Figure 8. Table I lists the ductile-brittle temperatures derived from Figure 8 by equating expressions for Eq and Ep, the measured values of Z, the Vickers indentation hardness of the materials, and other relevant parameters (defined earlier) [59]. 

The development of indentation fracture mechanics has also allowed fracture toughness to be determined using indentation cracks. Indeed, the ease with which these cracks can be introduced and the simple specimen preparation involved has popularized this approach. Moreover, it has allowed crack behavior to be studied for cracks in a size range that is close to that found in practice. There are two main approaches for determining fracture toughness from indentation cracks. In the first approach, the size of the radial cracks that emanate from the hardness impression are measured. It is recognized that the parameter x depends on the elastic properties of the indented material. It has been proposed that x=P E H), where )8 is a constant that depends only on the indenter geometry and H is hardness. Equation (8.64) can, therefore, be written as... 

Other penetrometer-indentometers include transducers to sense the position and movement of the probe and microprocessors for temperature control and data collection and reduction. These instruments are used mainly to measure softening points, which are not glass transitions but are usually close to those values. Because a softening point is indicative of behavior under load, it is often more useful for predicting performance than the Tg. Penetrometer-indentometers can also be used to measure indentation hardness, creep, creep recovery, and modulus. Examples of such instruments include the TA Instruments, Mettler, Perkin-Elmer, Seiko, and Shimadzu thermomechanical analyzers (TMAs). They can be used to generate modulus and modulus-temperature data from indentation-time plots by applying the Hertz equation (eq. 36) (170,296), where E is the elastic or Young s modulus, jx the Poisson s ratio, r the radius of the hemispherical indentor, P the force on the indentor (mass load x g), h the indentation, and ifk the indentation hardness. 

Now, substituting equation (1.39) and the area of indent into equation (1.33), we have an expression for theoretical hardness ... 

Because WC is oxidized to WO3 at high cutting speeds and this oxide is volatile, in order to cut hard steels, substantial amounts of TiC are added to some WC-Co grades. Some hardness data for cemented carbides is given in Table 6.23 along with other carbide data, and equations (5.51), (5.52), and (5.55) can be used to determine fracture toughness parameters from indentation hardness measurements. 

An estimate of the difference caused by measuring hardness at instantaneous load application and at infinitely slow application of load can be made as follows for the commonly encountered Vickers indenter. From equation (1.6) the Vickers hardness is defined as Hy = 0.4636 P/d where P is the final load in newtons, or... 

As equation (5.91) shows, the threshold load can be estimated from a knowledge of Hy, K, and all of which are found by the indentation hardness technique. In the case just considered, with Kc = 0.75 MN Hy = 6 GN m, ( c = 130 MN m , and A = 0.076, an estimate for P is 0.43 N. This small load over the area represented by a, is of course a large local stress, but the estimate does emphasize that small particles can generate critical flaws beneath the surface. The effect of the surface compression layer can be seen by removing the -1.6critical load decreases to 0.068 N for this glass. 

The indentation hardness decreases with increasing loading time (1 to 100 s) at constant temperature (>.3r i) in the plastic region. A series of straight lines were obtained. The activation energy, Q, for creep and the constant, m, expressed by the following equations are calculated for various carbides (see Table 7) ... 

Here M is the moment and Mp the fully-plastic moment of, for instance, a beam P/A is the indentation pressure and H the hardness of, for example, armour plating.) The left-hand side of each of these equations describes the loading conditions the right-hand side is a material property. When the left-hand side (which increases with load) equals the right-hand side (which is fixed), failure occurs. 

Proceeding from equation (4.3.19), we obtain, as for the Vickers indenter (4.3.22) and (4.3.23), the following hardness formulae for commonly used indenters ... 

The Vickers hardness of the ceramic was measured in air on a polished section at indentation loads of 5 and 10 Newtons. Indentation crack length measurements were made in air and water on the same section. The crack lengths introduced in air were used to calculate KIC by Equation (1). 

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