Homogeneous bulk modulus

An attempt has been made by Spiering et al. [39,40] to relate the magnitude of the interaction parameter F(x) as derived from experiment to the elastic interaction between HS and LS ions via an image pressure [47]. To this end, the metal atoms, inclusive of their immediate environments, in the HS and LS state are considered as incompressible spheres of radius /"h and Tl, respectively. The spheres are embedded in an homogeneous isotropic elastic medium, representing the crystal, which is characterized by specific values of the bulk modulus K and Poisson ratio a where 0 < a < 0.5. The change of molecular volume A Fas determined by X-ray diffraction may be related to the volume difference Ar = Ph — of the hard spheres by ... 

Thus a measurement of the ultrasonic velocity and density can be used to determine the adiabatic compressibility (or bulk modulus) of the material. For homogeneous solids measurements of the compression and shear velocities can be used to determine the bulk and shear moduli (see section 2.4). The Young s modulus of rod-like materials (e.g. spaghetti) can be determined by measuring the velocity of ultrasound. 

The relationship between the different state variables of a system subjected to no external forces other than a constant hydrostatic pressure can generally be described by an equation of state (EOS). In physical chemistry, several semiempirical equations (gas laws) have been formulated that describe how the density of a gas changes with pressure and temperature. Such equations contain experimentally derived constants characteristic of the particular gas. In a similar manner, the density of a sohd also changes with temperature or pressure, although to a considerably lesser extent than a gas does. Equations of state describing the pressure, volume, and temperature behavior of a homogeneous solid utilize thermophysical parameters analogous to the constants used in the various gas laws, such as the bulk modulus, B (the inverse of compressibUity), and the volume coefficient of thermal expansion, /3. 

Cribb [107] has adopted an approach in which no limitations are made on the shape and size of the fillers. The phases are supposed to be homogeneous, isotropic and linearly elastic. The simplicity of this approach is attractive but it converts the problem of calculating (x[ to the related question of calculating the bulk modulus of a composite. The Cribb equation is given as... 

Brown et al. (196 ) determined the room temperature Young s and shear moduli of scandium ( 7250 ppm impurity including 5800 oxygen). Bulk modulus and Poisson s ratio were computed from the data. Calculation of the bulk modulus and Poisson s ratio from Young s and shear moduli implies that the material is homogeneous and isotropic. Despite the use of polycrystalline scandium made by swaging and annealing a rod that had solidified in a water-... 

In solving viscoelastic stress analysis problems, assumptions on the material properties are often essential as gathering accurate time dependent data for viscoelastic properties is difficult and time consuming. Thus, one often only has properties for shear modulus, G(t) or Young s modulus, E(t), but not both. Yet of course for even the simplest assumption of a homogeneous, isotropic viscoelastic material, two independent material properties are required for solution of two or three dimensional stress analysis problems. Consequently, three assumptions relative to material properties are frequently encountered in viscoelastic stress analysis. These are incompressibility, elastic behavior in dilatation and synchronous shear and bulk moduli. Each of the common assumptions defines a particular value for either the bulk modulus or Poisson s ratio as follows. 

For this problem it is also easier to replace the stability of gels with the mechanical stability of networks. The requirement for mechanical stability of a homogeneous, isotropic material is given by bulk modulus > 0, and the corresponding stability requirement of networks is given... 

The first models describing the elastic behaviour of fractal structures used, as a rule, simulation within the frameworks of percolation theory [21-25]. Anon-homogeneous statistical mixture of solid and liquid displays solid properties (for instance, shear modulus G not equal to zero) only, when the solid component forms a percolation cluster at gelation in polymer solutions. If the liquid component is replaced by a vacuum then the bulk modulus K. will also be equal to zero below the percolation threshold [21]. This model gives the following relationship for elastic constants [21] ... 

Thus a measurement of the ultrasonic properties can provide valuable information about the bulk physical properties of a material. The elastic modulus and density of a material measured in an ultrasonic experiment are generally complex and frequency dependent and may have values which are significantly different from the same quantities measured in a static experiment. For materials where the attenuation is not large (i.e., a ca/c) the difference is negligible and can usually be ignored. This is true for most homogeneous materials encountered in the food industry, e.g., water, oils, solutions. 

A convenient basis for all that follows is obtained by looking at the modulus of a polymer material as a function of temperature. To obtain a basic representation, we consider the ideal case of a linear (unbranched) polymer chain of homogeneous chemical composition in a single phase bulk material which is amorphous (non-crystaUine). For all such materials, when modulus is plotted... 

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