Complex bulk modulus

Figure 8b. Frequency dependence of the complex bulk modulus of a viscoelastic polymer.

K, p p, denote the complex bulk modulus and the complex shear modulus of the first phase of the inhomogeneous medium, and K% = K2 and p = p2- denote the complex bulk modulus and the complex shear modulus of the second phase, respectively K, K2, pj, p2 are elastic properties for nonhomogeneous media phases). For nonbonded configurations the viscoelastic modulii and pi"11 are described by the formulae which result from the... 

B Complex bulk modulus, with real (elastic) and imaginary (viscous) components. 

Equations 11.8 and 11.9 are isomorphous to equations 9.9 and 9.10 which define the storage and loss components of the complex dielectric constant . Similar equations are also used to define the complex bulk modulus B, the complex shear modulus G, and the complex Poisson s ratio v, in terms of their elastic and viscous components. The physical mechanism giving rise to the viscous portion of the mechanical properties is often called "damping" or "internal friction". It has important implications for the performance of materials [8-15],... 

Further Applicatioiis. Detailed evaluations and numerical emulations of the theory are found in Vleeshouwers thesis. (31,38) One examine concerns the frequency and temperature dependence of the complex bulk modulus. Another is the simulation of cooling the melt over a transition region into the glassy state. To proceed the continuous temperature profile is approximated by a series of discrete steps depending on the cooling rate. The relaxing free volume and volume following each T-jump are then computed. 

The proposed solution is the calculation of the bubble motion of bubbles with different sizes using the Kirkwood-Bethe-Gilmore equations. Knowing the bubble-size distribution at a given sound pressure by calculating cavitation thresholds and using this information in an equation for the local total bubble number, the calculation of the complex bulk modulus of the bubbly mixture is possible. 

In principle, the frequency dependence of the complex bulk modulus could be obtained from the data for E, E", G, and G", In Fig. 15-6, the shear and extension data have been combined to calculate Poisson s ratio /it as a function of frequency. The values indicated show that /it passes through a minimum between the inflections in the dispersion of G and of E. ... 

Lu et al. [7] extended the mass-spring model of the interface to include a dashpot, modeling the interface as viscoelastic, as shown in Fig. 3. The continuous boundary conditions for displacement and shear stress were replaced by the equations of motion of contacting molecules. The interaction forces between the contacting molecules are modeled as a viscoelastic fluid, which results in a complex shear modulus for the interface, G = G + mG", where G is the storage modulus and G" is the loss modulus. G is a continuum molecular interaction between liquid and surface particles, representing the force between particles for a unit shear displacement. The authors also determined a relationship for the slip parameter Eq. (18) in terms of bulk and molecular parameters [7, 43] ... 

Spiering et al. (1982) have developed a model where the high-spin and low-spin states of the complex are treated as hard spheres of volume and respectively and the crystal is taken as an isotropic elastic medium characterized by bulk modulus and Poisson constant. The complex is regarded as an inelastic inclusion embedded in spherical volume V. The decrease in the elastic self-energy of the incompressible sphere in an expanding crystal leads to a deviation of the high-spin fraction from the Boltzmann population. Pressure and temperature effects on spin-state transitions in Fe(II) complexes have been explained based on such models (Usha et al., 1985). 

The mechanical properties of a linear, isotropic material can be specified by a bulk modulus, K, and a shear modulus, G. For an ideal elastic solid, these moduli are real-valued. For real solids undergoing sinusoidal deformation, these are best represented as complex quantities [49] K = K jA and G = G -I- jG". The real parts of K and G represent the component of stress in-phase with strain, giving rise to energy storage in the film (consequently K and G are referred to as storage moduli) the imaginary parts represent the component of stress 90° out of phase with strain, giving rise to power dissipation in the film (thus, K" and G" are called loss moduli). 

Any two complex, freguency dependent, elastic moduli are sufficient to describe the mechanical response of a viscoelastic polymer. The two moduli which are most freguently measured are the bulk modulus [41], and the Young s modulus E (o)) = E (o)) + iE"(o)) [42]. Poisson s... 

This equation relates the tensile complex relaxation modulus to the bulk and shear complex relaxation moduli. In the same way, Eq. (5.95) leads to the relationship... 

In the same way, the response to a sinusoidal change of volume yields the complex bulk relaxation modulus. 

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]. 

Mixing, solution/latex Modulus, bulk Modulus, complex, G Modulus, loss, G Modulus, measurements Modulus, plateau Modulus, storage, G Modulus, tensile/flexural... 

The effective complex Biot constants H, C, and M are defined in terms of the effective drained modulus of the composite, the effective undrained bulk-modulus K, and the effective Skempton s coefficient B as... 

Table 2 also contains data for one non-oxide host lattice and one Cr complex. The much larger shifts observed in these three systems is a consequence of the highly compressible nature of halide lattices and molecular complexes relative to oxide lattices. The bulk modulus of Cs2NaYClg ( 495 kbar [142]) is about five times smaller than that of ruby (2530 kbar [149,150]) and is primarily responsible for the difference in shift rate. A similar explanation holds for the Cr + complex. When normalized to compressibility, therefore, the variation in B for all of the systems in Table 2 is comparable. 

The SC solutions appear to run into difficulties when there is a large elastic mismatch between the constituent phases, for example, at high concentrations of a rigid phase in a compliant matrix or of a porous phase in a stiff matrix. The latter situation will be discussed in Section 3.6. One approach to this problem is known as the Generalized Self-Consistent Approach, the concept behind which is illustrated in Fig. 3.14. Instead of a single inclusion in an effective medium, a composite sphere is introduced into the medium. As in the composite sphere assemblage discussed in the last section, the relative size of the spheres reflects the volume fraction, i.e., V = a bf as before. Interestingly, this approach leads to the HS bounds for the bulk modulus. The solution for the shear modulus is complex but can be written in a closed form. 

For both magnesite and calcite, the elastic bulk modulus Bq was computed straightforwardly by the Murnaghan interpolation formula, while of the elasticity tensor only the C33 component and the C + C 2 linear combination could be calculated in a simple way. The relations used are C = (l/Vo)c (d L /crystal structure. To derive other elastic constants, the symmetry must be lowered with a consequent need of complex calculations for structural relaxation. A detailed account of how to compute the Ml tensor of crystal elasticity by use of simple lattice strains and structure relaxation was given previously[10, 11]. For the present deformations only the c-o ( ) relaxation need be considered. The results are reported in Table 6, together with the corresponding values extrapolated to 0 K from experimental data (Table 2). For calcite, the mea-... 

The greatest value of bulk modulus, B = 222.1 GPa, has been foimd for a complex-aUoyed phase Ni2.5Coo.5(Alo.5Reo.i25Uuo.i2sTao.i25Wo.i25)- Thus, bulk modulus increases by 19.5% as compared with pure NisAl. Elastic properties of this phase are presented in the fifth line of Table 10.2. The greatest elastic moduli were... 

---