Chin-Gilman parameter

A measure of shear strength is the shear modulus. For covalent crystals this correlates quite well with hardness (Gilman, 1973). It also correlates with the hardnesses of metals (Pugh, 1954), as well as with ionic crystals (Chin, 1975). Chin has pointed out that the proportionality number (VHN/C44) depends on the bonding type. This parameter has become known as the Chin-Gilman parameter. 

The variation of the Chin-Gilman parameter with bonding type means that the mechanism underlying hardness numbers varies. As a result, this author has found that it is necessary to consider the work done by an applied shear stress during the shearing of a bond. This depends on the crystal structure, the direction of shear, and the chemical bond type. At constant crystal structure, it depends on the atomic (molecular volume). In the case of glasses, it depends on the average size of the disorder mesh. 

For interpreting indentation behavior, a useful parameter is the ratio of the hardness number, H to the shear modulus. For cubic crystals the latter is the elastic constant, C44. This ratio was used by Gilman (1973) and was used more generally by Chin (1975) who showed that it varies systematically with the type of chemical bonding in crystals. It has become known as the Chin-Gilman parameter (H/C44). Some average values for the three main classes of cubic crystals are given in Table 2.1. 

C44 measures the shear strengths of chemical bonds and the Chin-Gilman parameter indicates how directly they interact with dislocation motions which depends on how localized the bonding is. Thus it is relatively large for covalent bonding which is localized to pairs of atoms (electron pair bonding). 

The difference of the Chin-Gilman parameter for differing types of chemical bonding accounts for the Tabor constant not being three for non-metals. 

The Chin-Gilman parameters (H/G) are given in the figure captions. Note that the value for the bcc metals (0.02) is about five times greater than the value for the fee metals (0.0044). Thus the bcc metals deformation harden much more rapidly than the fee metals. 

Figure 7.2 Brinell Hardness Numbers (BHN) of the fee transition metals as a function of their average shear moduli (taken from Ledbetter, 2001).The hardness numbers are low temperature values measured at -200 °F. Note that this figure is similar to Figure 6.2 without A1 and Pb. Chin-Gilman parameter = H/G = 0.0044.

The Chin-Gilman parameter for these compounds is about 0.11. 

The most abundant of all minerals in the interior of the earth is (Mg,Fe)Si03 perovskite. It constitutes greater than seventy percent of the lower mantle, so it is of great importance to geophysics. At room temperature the hardness of MgSi03 is VHN = 1800kg/mm2 and its Chin-Gilman parameter is 0.01. 

The hardnesses of some perovskites are given in Table 11.1 (based on the data of Yamanaka et al., 2004). The table shows that these perovskites are moderately hard and the third column which lists their Chin-Gilman parameters indicates that they are predominately ionically bound. 

The hardness of A1203 is VHN = 2700kg/mm2 and its rms. shear stiffness is 366 GPa so its Chin-Gilman parameter is 0.074. This suggests that its chemical bonding is a combination of covalent and ionic bonding. 

A shear modulus of about 1 GPa has been measured for wet lysozyme. Thus its Chin-Gilman parameter is about 0.02 which is large compared with metals and small compared with covalent crystals. 

The Chin-Gilman parameter for PETN where the shear modulus is known is about 0.036 which is consistent with other molecular crystals. 

Molecular crystals come in too many varieties and mixtures of chemical binding for simple theories of their hardnesses to be feasible. This is aggravated by their relatively low symmetries, making them quite ansotropic. Rough estimates of their hardnesses can be made if their shear moduli are known using the Chin-Gilman parameter. However, the shear moduli have been measured in only a few cases. 

There is disagreement in the literature about the role of friction. Compare, for example, Cai (1993) with Ishikawa et al. (2000) This has arisen in various ways. In the case of metals, where the Chin-Gilman parameter is small, friction is not important for relatively large indents. However, as the C-G parameter becomes much larger for covalent crystals, and as the indent size decreases friction becomes more important. Also, environmental factors, such as humidity, affect friction coefficients. In the regime of superhardness with dry specimens and small indents friction becomes very important. 

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