Bulk modulus, definition

What we would like to do is use these thermodynamic properties to calculate an equilibrium elastic moduli. The bulk modulus is by definition the constant of proportionality that links the infinitesimal pressure change resulting from a fractional change in volume (Section 2.2.1). In colloidal terms this becomes... 

Such expressions can be derived from the general treatment by Kynch.34 The definition of a bulk modulus is... 

One of the basic mechanical properties of the foam is its compressibility (elasticity). By definition the bulk modulus of elasticity Ev is expressed by... 

This number should be contrasted with the experimental lattice constant of a typical free electron metal such as sodium, for which the lattice parameter is 4.23 A. In addition to the lattice parameter, this model may also be used to estimate the bulk modulus of free electron metals by resorting to the definition B = V d E/dV ). This is left as an exercise for the reader. 

When S 1, bulging can only occur near the sides of the block. Near the centre, the rubber compresses in volume with zero lateral strains, in uniform stress conditions. Therefore, the volume strain dV/V is approximately equal to the vertical compressive strain, except at the edges of the plates. From the definition of the bulk modulus K in Eq. (3.30), we find that, in the limit as 5 tends to infinity... 

Most polycrystalline solids are considered to be isotropic, where, by definition, the material properties are independent of direction. Such materials have only two independent variables (that is elastic constants) in matrix (7.3), as opposed to the 21 elastic constants in the general anisotropic case. The two elastic constants are the Young modulus E and the Poisson ratio v. The alternative elastic constants bulk modulus B and shear modulus /< can also be used. For isotropic materials, n and B can be found from E and t by a set of equations, and on the contrary. 

Bulk modulus n. The modulus of volume elasticity, i.e., the resistance of a solid or liquid to change in volume with change in pressure, at constant temperature. The thermodynamic definition is ... 

Eiastic Materiai Functions. The material functions that are used here for an isotropic, elastic material are the shear modulus G, the extensional modulus E, the bulk modulus K, and the Poisson s ratio v. However, any two of these provides the full set of information needed to describe such a material, as they are not all independent. For simple deformations, the definitions of these moduli are... 

Returning to the static moduH, the subject of this section. Fig. 12.1 serves as a reminder of the definitions of the engineering quantities bulk modulus (K), shear modulus (G), Young s modulus E), and Poisson s ratio (v), which in an isotopi-cally elastic soHd (fine, randomly textured grains) are simply related to each other according to ... 

Fig. 11.5. (a,b) The definition of the elements for the calculation of Young s modulus, (c) The subdivision of a stress per unit area into its perpendicular and tangential components, (d) The elements for the description of bulk modulus arrows are forces acting isotropically on a piece of material. 

Multi-bond crystals. The bulk modulus of a multi-component material results from the superposition of that of the individual component. From the definition of bulk modulus, Tian et al. derived... 

More definitive evidence of enzymatic attack was obtained with 1 1 copolymers of e-caprolactone and 6-valerolactone crosslinked with varying amounts of a dilactone (98,99). The use of a 1 1 mixture of comonomers suppressed crystallization and, together with the crosslinks, resulted in a low-modulus elastomer. Under in vitro conditions, random hydrolytic chain cleavage, measured by the change in tensile properties, occurred throughout the bulk of the samples at a rate comparable to that experienced by the other polyesters no weight loss was observed. However, when these elastomers were implanted in rabbits, the bulk hydrolytic process was accompanied by very rapid surface erosion. Weight loss was continuous, confined to the... 

Thus, E of the bulk solution can be used to predict the elasticity modulus of films at a definite thickness and bulk concentration. 

Kj / y, where K is the surface dilationcil modulus, defined in 13.6.19). An alter-native Marangoni number was introduced by Edwards et al., who considered creep flow around an emulsion droplet. Their definition is Ma s K° /kar/, where a is the radius of the droplet, rj the bulk viscosity and fc (in s ) a rate constant, characteristic of the rate of supply of surfactants to the interface by transport from the bulk. The second definition rather applies to Gibbs monolayers it is a measure of the extent to which surface tension gradients can develop against the counteracting replenishment of the surface. 

Another condition for regime 2 is that the amount of A that reacts in the film before reaching the bulk be negligible. For the very slow and slow reactions, the kinetic term resides along with the differential equation describing the transport of the species in the film, such that the definition of Hatta modulus is necessary as was derived in Chapter 6 for a pseudo-first-order reaction. In Chapter 6, we defined the Hatta modulus as... 

One should be cautious in reviewing this work. The definition of the aspect ratio for the oblate spheroids (the thickness divided by the long dimension) is the inverse of what is described in the bulk of this chapter. There is a difference in assigning the Young s modulus in the first article (178 GPa) and the second article (167 GPa) for montmoriiionite. 178 GPa is generally accepted as the Young s modulus for montmoriiionite in the bulk of the work featured in this chapter. The bulk and the shear moduli are calculated from the model and utilized to calculate the Young s modulus. 

Eurther work needs to be done with WAXS, a higher magnification TEM, and tensile modulus evaluations of injected molded test samples before a definite judgment can be made concerning the degree of exfoliation of this organomontmorillonite in a styrene composite prepared by bulk polymerization. 

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