Bulk modulus also

The equations of state across the series are similar to Fig. 11, the principal change being that the f-contribution is progressively reduced. The f-pressure for USb is a httle more than half the value for UN and the d-pressure is relatively more important. The sp repulsive pressure is similarly reduced as the anion size is increased. The bulk modulus also decreases approximately inversely proportional to the lattice parameter. 

That this form does indeed hold had been demonstrated for the cavitation of glassy polypropylene (Mott et al. 1993) in a computational study furnishing the validity of this extension of the universal binding-energy relation to symmetrical bulk response. The additional attraction of this expression is that it points out directly that application of a pressure produces symmetrical elastic compaction in an isotropic solid. However, more interestingly, one notes that, when dilatation is imposed, the bulk modulus monotonically decreases and eventually, at a dilatation of 1 /p, vanishes. This also leads to the observation that, if dilatation results from thermal expansion in response to a temperature increase, the bulk modulus also decreases. This simple observation represents the essence of the temperature dependence of all other elastic constants in anisotropic solids, beyond the mere effect on the bulk... 

The bulk modulus also can be expressed as the reciprocal of compressibility in the elastic range. Therefore, a low bulk modulus of 90 GPa (13 x 10 ... 

The derived functions of the volume with respect to temperature and pressure are expansivity, a, and compressibility, p, written as Eqs. (9) and (10), respectively. Both refer to unit volume and both are usually positive. The best known substance with a negative expansivity up to about 277 K is water. The negative sign for compressibility in Eq. (10) takes care of the fact that increasing pressure always decreases the volume. Note also that the inverse of the compressibility is the bulk modulus, also called the isothermal elasticity coefficient. 

The foam bulk modulus also can be predicted by using the Kelvin model [62]... 

For crystalline polymers, the bulk modulus can be obtained from band-structure calculations. Molecular mechanics calculations can also be used, provided that the crystal structure was optimized with the same method. 

Another commonly used elastic constant is the Poisson s ratio V, which relates the lateral contraction to longitudinal extension in uniaxial tension. Typical Poisson s ratios are also given in Table 1. Other less commonly used elastic moduH include the shear modulus G, which describes the amount of strain induced by a shear stress, and the bulk modulus K, which is a proportionaHty constant between hydrostatic pressure and the negative of the volume... 

A guide to tire stabilities of inter-metallic compounds can be obtained from the semi-empirical model of Miedema et al. (loc. cit.), in which the heat of interaction between two elements is determined by a contribution arising from the difference in work functions, A0, of tire elements, which leads to an exothermic contribution, and tire difference in the electron concentration at tire periphery of the atoms, A w, which leads to an endothermic contribution. The latter term is referred to in metal physics as the concentration of electrons at the periphery of the Wigner-Seitz cell which contains the nucleus and elecUonic structure of each metal atom within the atomic volume in the metallic state. This term is also closely related to tire bulk modulus of each element. The work function difference is very similar to the electronegativity difference. The equation which is used in tire Miedema treatment to... 

When a craze occurs around a rubber droplet the droplet is stressed not only in a direction parallel to the applied stress but also in the plane of the craze perpendicular to the applied stress (see Figure 3.9). Such a triaxial stress leading to dilation of the particle would be resisted by the high bulk modulus of the rubber, which would thus become load bearing. The fracture initiation stress of a polyblend should not therefore be substantially different from that of a glass. 

The bulk modulus of an ideal SWNT crystal in the plane perpendicular to the axis of the tubes can also be calculated as shown by Tersoff and Ruoff and is proportional to for tubes of less than 1.0 nm diameter[17]. For larger diameters, where tube deformation is important, the bulk modulus becomes independent of D and is quite low. Since modulus is independent of D, close-packed large D tubes will provide a very low density material without change of the bulk modulus. However, since the modulus is highly nonlinear, the modulus rapidly increases with increasing pressure. These quantities need to be measured in the near future. 

Physical hardness can be defined to be proportional, and sometimes equal, to the chemical hardness (Parr and Yang, 1989). The relationship between the two types of hardness depends on the type of chemical bonding. For simple metals, where the bonding is nonlocal, the bulk modulus is proportional to the chemical hardness density. The same is true for non-local ionic bonding. However, for covalent crystals, where the bonding is local, the bulk moduli may be less appropriate measures of stability than the octahedral shear moduli. In this case, it is also found that the indentation hardness—and therefore the Mohs scratch hardness—are monotonic functions of the chemical hardness density. 

A brief review is made of the methods that are currently being used to determine the density (p) and compressibility (6) of electrolyte solutions as a function of pressure. The high pressure equations of state used to represent these properties are also discussed. The linear secant bulk modulus [K = Ppp/(pP - p )] equation of state... 

Second-order equations of state (i.e., equations involving the second derivative of the bulk modulus in P ) have also been proposed (see for this purpose the synoptic table in Kim et ah, 1976). However these high-order equations are of limited application in geochemistry because of the still large incertitude involved in high-P compressional studies. 

Fig. 3 Experimental heat capacities of benzene [11], Cv is obtained from observed Cp after subtracting the expansion work, computed using the experimentally determined bulk modulus. The Cv estimated from molecular translational and librational lattice modes (obtained from neutron diffraction ADP s) is also plotted. Note that these external modes well reproduce the observed Cv up to ca. 100 K. Above this temperature the internal modes are active and Cv exceeds the classical limit of 3 k T...

The trends in several ground state properties of transition metals have been shown in Figs. 2, 3 and 15 of Chap. A and Fig. 7 of Chap. C. The parabolic trend in the atomic volume for the 3-6 periods of the periodic table plus the actinides is shown in Fig. 3 of Chap. A. We note that the trend for the actinides is regular only as far as plutonium and that it is also broken by several 3 d metals, all of which are magnetic. Similar anomalies for the actinides would probably be found in Fig. 15 of Chap. A - the bulk modulus - and Fig. 7 of Chap. C - the cohesive energy if more measurements had been made for the heavy actinides. 

In addition to the tensile and shear moduli, a compressive modulus, or modulus of compressibility, K, exists to describe the elastic response to compressive stresses (see Fignre 5.7). The compressive modulus is also sometimes called the bulk modulus. It is the proportionality constant between the compressive stress, CTc, and the bulk strain, represented by the relative change in bulk volume, AV/Vo-... 

When used in load-bearing applications, isotropic polymers may also fail because of low modulus. The moduli that must be considered in the design of functional polymers are Young s modulus E, shear modulus G, and bulk modulus K. Poisson s ratio P should also be considered. 

The explicit formulae given by Rosen55 are also of value. They are derived from a model consisting of a random assemblage of composite cylinders (Hashin and Rosen56 ) and expressed in terms of the axial Young modulus E, the Poisson ratio for uniaxial stress in the fibre direction v, the transverse plane strain bulk modulus k, the axial shear modulus G and the transverse shear modulus G. ... 

---