Bulk compressive compliance

Note 1 Also known as bulk compressive compliance. 

The bulk modulus K (= /H, the reciprocal of the bulk compliance) can be measured in compression with a very low height-to-thickness (h/i) ratio and unlubricatcd flat clamp surfaces. In pure compression with a high h/t ratio, and lubricated clamps, the compressive modulus (= /D, the reciprocal of the compressive compliance) will be measured. Any intermediate hjt ratios will measure part bulk and part compressive moduli. Hence it is vital for comparing samples to use the same dimensions in thermal scans and the same h/t ratio when accurately isotherming and controlling static and dynamic strains and frequencies. 

Once again the reader is reminded that, although all the equations in this chapter refer to deformation in shear, they are equally applicable to other types of deformation if the shear moduli, compliances, and spectra are replaced by the corresponding functions appropriate to extension or bulk compression, etc. 

The most extensive measurements of dynamic bulk compression have been those of McKinney and Belcher on poly(vinyl acetate), covering ranges of frequency from 50 to 1000 Hz, temperature from 0° to 100°C, and superposed hydrostatic pressure from 1 to about 1000 atm. Both storage and loss bulk compliances, B and B", were obtained. All data were successfully combined by the method of reduced variables taking into account the substantial dependence of both Bg and Bg (the equilibrium and glasslike bulk compliances) on both T and P, as follows ... 

Now, in rheological terminology, our compressibility JT, is our bulk compliance and the bulk elastic modulus K = 1 /Jr- This is not a surprise of course, as the difference in the heat capacities is the rate of change of the pV term with temperature, and pressure is the bulk stress and the relative volume change, the bulk strain. Immediately we can see the relationship between the thermodynamic and rheological expressions. If, for example, we use the equation of state for a perfect gas, substituting pV = RTinto a = /V(dV/dT)p yields a = R/pV = /Tand so for our perfect gas ... 

The quantity B is often expressed in terms of the compressibility, which is the reciprocal of the bulk modulus. Similarly, and G are known as the tensile and shear compliances, and given the symbols D and J, respectively. 

H6 and BCT4 both have a bulk modulus which, within the tight-binding model is comparable in magnitude to that of diamond. In accord with an empirical TB scheme [24] H6 seems to be even harder than diamond while SCF plane wave calculations [25,26] predict that the almost equal bulk moduli of BCT4 and H6 are 17% below the diamond value. The bulk moduli of the models have been determined by calculating the elastic compliances after applying suitable strains to the crystals and inversion of the volume compressibility [68]. 

Bulk modulus B is usually calculated from measurements of bulk compliance (1/B) or compressibility. The usual expression may be written... 

X10. The next three rows present the viscosity rj, the surface tension, and its tenqterature dependence, in the liquid state. The next properties are the coefficient of linear thermal expansion a and the sound velocity, both in the solid and in the liquid state. A number of quantities are tabulated for the presentation of the elastic properties. For isotropic materials, we list the volume compressihility k = —(l/V)(dV/dP), and in some cases also its reciprocal value, the bulk modulus (or compression modulus) the elastic modulus (or Young s modulus) E the shear modulus G and the Poisson number (or Poisson s ratio) fj,. Hooke s law, which expresses the linear relation between the strain s and the stress a in terms of Young s modulus, reads a = Ee. For monocrystalline materials, the components of the elastic compliance tensor s and the components of the elastic stiffness tensor c are given. The elastic compliance tensor s and the elastic stiffness tensor c are both defined by the generalized forms of Hooke s law, a = ce and e = sa. At the end of the list, the tensile strength, the Vickers hardness, and the Mohs hardness are given for some elements. 

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