Elasticity bulk modulus

Bulk elastic modulus, of binary compound semiconductors, 22 145, 146-147t Bulk enzymes, from genetically engineered microbes, 22 480 Bulk erosion, 9 78 Bulk fluid velocity method, 16 688 Bulk gallium nitride, supercritical ammonia solution growth of, 14 96-97 Bulk gases... 

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 ... 

Figure 47. Semilogarithmic dependence of the shear elasticity modulus (a) and the bulk elasticity modulus (b) on the iteration number n for p = 0.2088 (/), 0.2092 (2), and 0.2098 (2).

We will consider below isotropic media, for which, just as Eq. (361), the concept of a complex bulk elastic modulus K (co) can be introduced [131]. 

The elastic force constant Ke is set equal to the functional form of the bulk elastic modulus (27) and normalized as the fourth parameter of the model. The remaining parameters were estimated using molecular and quantum mechanical calculations (see the Glossary of Symbols). 

In contrast, the Enskog model results in nonmonotonous dependencies for fluidized bed particulate pressure and the bulk elasticity modulus. Furthermore, both quantities fall off to zero as the bed attains the state of close packing. The problem of choosing one of the utilized approximate statistical models therefore assumes a fundamental significance. As has been pointed out, this problem can be successfully resolved by considering the behavior of fluidized beds at concentrations differing little from that corresponding to the close-packed state [25]. 

If the bulk elasticity modulus for a fluidized bed is negative at large concentrations, as is required by the Enskog model, the particulate pseudo-gas would be absolutely unstable with respect to virtual concentrational perturbations. At a negative bulk elasticity modulus, any occasional perturbation will grow under action... 

This modulus is always smaller than the bulk elasticity modulus of the phase A Bef < Ba-When the gas is absent in the solution and the cell is rigid (small coefficient dVceii/dP) then the effective modulus is determined only by the solution bulk elasticity Bef Ba- When the cell is not rigid (deformational) or the volume of the solution Va is very small then the effective... 

The adiabatic and isothermic bulk elasticity modules for water only slightly differ. The adiabatic modulus for water is 2.2-lO Pa. The bulk elasticity modulus for gas can be obtained from the equation of state. For ideal gas the isothermic modulus is approximately equivalent to Pa, and the adiabatic modulus is equivalent to yPA, where Pa is the pressure inside the cell, y = 1.4 the adiabatic constant. The isothermic modulus can be used in the case of infinitely slow processes. In the case of pressure oscillations at sound frequencies the adiabatic modulus should be used. Gas is much more compressible than liquids. The bulk elasticity modulus for gas is four orders of magnitude smaller than for water. Therefore, even small amounts of gas much smaller than the volume of the solution (Voas Va), can mimic small values of the effective cell elasticity modulus Eq. (6), i.e. a strong decrease of the cell resistance to pressure variations. It is extremely important to avoid the presence of any small amounts of gas in the solution because it can lead to uncontrolled changes of the effective cell elasticity modulus. 

Io(x) and l2(x) are modified Bessel functions of zero and second order, Bb and pe are the bulk elasticity modulus and the density of phase B, / is the capillary length (/ ac). Substituting Eqs. (12) and (14) into (8) one can obtain the general expressions for the bubble (drop) volume variation 6V ,(ico), the pressure variations inside the bubble (drop) 5P, (ici)), and inside the cell 6P (ico) at the given bP, and 6Vp . The effect of both the flow mobility and the compressibility of the medium in the capillary on the oscillating bubble or drop measurements is analysed in details in [26]. Here only the simpler case of quasi-stationary flow of incompressible media in the capillary is described. 

When the bulk elasticity modulus Bb is large and the capillary is not very long and narrow, the incompressible medium approximation can be used. It is valid when the pressure variations are not very fast, and the corresponding frequency limit is obtained [26]... 

For more concentrated suspensions, other parameters should be taken into consideration, such as the bulk (elastic) modulus. Clearly, the stress exerted by the particles depends not only on the particle size but on the density difference between the partide and the medium. Many suspension concentrates have particles with radii up to 10 pm and a density difference of more than 1 g cm . However, the stress exerted by such partides will seldom exceed 10 Pa and most polymer solutions will reach their limiting viscosity value at higher stresses than this. Thus, in most cases the correlation between setfling velocity and zero shear viscosity is justified, at least for relatively dilute systems. For more concentrated suspensions, an elastic network is produced in the system which encompasses the suspension particles as well as the polymer chains. Here, settling of individual partides may be prevented. However, in this case the elastic network may collapse under its own weight and some liquid be squeezed out from between the partides. This is manifested in a dear liquid layer at the top of the suspension, a phenomenon usually... 

Force-Curves and Force-Modulation Calibration. In figure l(a b), typical force-indentation curves obtained respectively on a rigid ( = 610 MPa) and a soft (E = 27 MPa) polymer are presented. The elastic modulus derived from the analysis of the force-indentation curves is compared to the bulk elastic modulus measured by DMA in figure 1(c). For this analysis, the used tip geometry was adapted to the maximum indentation depth reached during the experiment, Smax- For Smax Rj the spherical geometry was considered while, for Smax the conical one was used. For intermediate values, the paraboloid model was used. 

Figure 2. (a) Variation of the dynamic elastic response, d /z, measured by local force modulation as a function of the bulk elastic modulus, (b) Comparison between the surface Young s modulus deduced from force modulation experiments and the volume modulus calculated using Hertz model. 

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