Incompressible materials

As previously noted, the equilibrium constant is independent of pressure as is AG. Equation (7.33) applies to ideal solutions of incompressible materials and has no pressure dependence. Equation (7.31) applies to ideal gas mixtures and has the explicit pressure dependence of the F/Fq term when there is a change in the number of moles upon reaction, v / 0. The temperature dependence of the thermodynamic equilibrium constant is given by... 

Reconciliation of Equilibrium Constants. The two approaches to determining equilibrium constants are consistent for ideal gases and ideal solutions of incompressible materials. For a reaction involving ideal gases, Equation (7.29) becomes... 

So for an incompressible material v = 0.5 and Equation 2.4 is recovered. The value of Poisson s ratio for rubber is usually close to 0.5 but for many other solids the value is lower and we find 0.25 < v < 0.33. We may also describe a Bulk rigidity modulus, K, such as we would measure when we compress a material with hydrostatic pressure, in terms of Young s modulus ... 

Kalyanasundaram, Kumar, and Kuloor (K2) found the influence of dispersed phase viscosity on drop formation to be quite appreciable at high rates of flow. The increase in pd results in an increase in drop volume. To account for this, the earlier model was modified by adding an extra resisting force due to the tensile viscosity of the dispersed phase. The tensile viscosity is taken as thrice the shear viscosity of the dispersed phase, in analogy with the extension of an elastic strip where the tensile elastic modulus is represented by thrice the shear elastic modulus for an incompressible material. The actual force resulting from the above is given by 3nRpd v. 

Note 3 For an isotropic, incompressible material, // = 0.5. It should be noted that, in materials referred to as incompressible, volume changes do in fact occur in deformation, but they may be neglected. 

When a material is subjected to a tensile or compressive stress, Eqs. (5.63) through (5.74) should be developed with the shear modulus, G, replaced by the elastic modulus, E, the viscosity, rj, replaced by a quantity known as Trouton s coefficient of viscous traction, k, and shear stress, r, replaced by the tensile or compressive stress, a. It can be shown that for incompressible materials, k = 3r], because the flow under tensile or compressive stress occurs in the direction of stress as well as in the two other directions perpendicular to the axis of stress. Recall from Section 5.1.1.3 that for incompressible solids, E = 3G therefore the relaxation or retardation times are k/E. 

For incompressible materials, Trouton s coefficient can be related to X = 3tj, so that... 

The Poisson ratio is typically between 0.3 and 0.4, approaching 0.5 for incompressible materials. 

Figure 16 illustrates several test specimens which have been used (46) in the multiaxial characterization of solid propellants. The arrows indicate the direction of load application. The strip tension or strip biaxial test has been used extensively in failure studies. It can be seen that the propellant is constrained by the long bonded edge so that lateral contraction is prevented and tension is produced in two axes simultaneously. The sample is free to contract normal to these axes. The ratio of the two principal tensile stresses may be varied from 0 to 0.5 by varying the bonded length of incompressible materials. 

The stress-strain relations for some special cases of biaxial defonnation are derived from Eqs. (13) to (15) in the following way. Strip biaxial extension of incompressible material is defined as the mode of deformation in which one of the Xj, say X2, is kept at unity, while the other, Xt, varies. This deformation is also called pure shear . We have for it ... 

Finally, for simple uniaxial extension of incompressible material we obtain with °2 = °3 = 0... 

For incompressible material, the stress-strain relations for biaxial extension are given by Eqs. (13) and (14), which may be solved for bW/bli and bWjbI2 to give... 

Thus from the thin-walled data for Comp B (at a 10-fold expansion) one obtainsy E = 2.96km/sec and E /Q = 0.87. As shown in Section IV (and in particular in Table 5), a tangentially impinging detonation is less effective in propelling incompressible material than a head-on detonation. When this is taken into account (approx 0.87/0.93) E Q if E is based on measurements made at large expansion of the test cylinders... 

Unspecified isotropic pressure term in stress tensor p for incompressible materials. 

When a material is stretched there is also contraction in the direction perpendicular to the direction of stretching. The ratio of the lateral contraction to the longitudinal extension is Poisson s ratio. For incompressible materials, Poisson s ratio is 0.5 and as rubbers are very nearly incompressible they have values close to this. 

Quantities C and C2 are functions of the two invariants of the stress tensor /1 and I2 for incompressible material. 

Nevertheless, some conclusions may be drawn from the set of results presented here. First, with the notable exception of InN, the group III nitrides form a family of hard and incompressible materials. Their elastic moduli and bulk modulus are of the same order of magnitude as those of diamond. In diamond, the elastic constants are [49] Cu = 1076 GPa, Cn = 125 GPa and Cm = 577 GPa, and therefore, B = (Cn + 2Ci2)/3 = 442 GPa. In order to make the comparison with the wurtzite structured compounds, we will use the average compressional modulus as Cp = (Cu + C33)/2 and the average shear modulus as Cs = (Cu + Ci3)/2. The result of this comparison is shown in TABLE 8. 

PARAMETER RANGE COMPARISON. Table I summarizes the parameter ranges of the torqued cylinder apparatus, the resonance apparatus and the DMTA. Since the bulk moduli of the materials under consideration in this paper are much larger than the Young s or shear moduli, the materials are considered incompressible. For incompressible materials, the shear modulus is one third of Young s modulus. Comparisons are then made by converting Young s modulus to shear modulus for the data measured by the resonance apparatus and the DMTA. 

Tensors that are proportional to 8 are sometimes called isotropic tensors. For an incompressible material, gradients of p, but not p itself, can affect fluid motion. Thus, a uniform isotropic tensor of arbitrary magnitude can be added to T (or or) without consequence to the flow behavior. Adding such an isotropic tensor is equivalent to adding a constant to each diagonal component of the stress tensor. Thus, if the fluid is incompressible, a is determined only up to an additive isotropic tensor, and the stress-free state is synonymous with the state of isotropic stress. 

Consider the momentum and mass balance equations in the absence of inertia for an incompressible material ... 

Since for an incompressible material E = 3G (see Chapter 4), it can be asserted that the quantity NkgT of Eq. (3.33) coincides with the shear modulus of the elastomer. 

The empirical constitutive parameters for five materials with different compactibilities are shown in Table 2. The 9-pm spheres with n = d = 0, and K = Kq = constant correspond to an incompressible material. The Kaolin flat is moderately com-pactible and the Mierlo biosolidsj water treatment residue,activated sludge with n and S... 

As s goes to 0 for incompressible materials with definite rigid crystalline structures, a becomes a constant. 

The cake compactibility parameters for an incompressible material carbonyl iron, a moderately compactible kaolin flat D, and a super-compactible activated sludge are given in Table 22.6. 

The value of h can readily be calculated for other values of g/X and ggR/g. Figure 2 shows the solution of the enhancement of h (i.e. h/hincompressibie) for the two values of ggR/g considered by Love, but over a range of values of g/X. Recall that g/X = 0 for an incompressible material. Figure 2 confirms that as g/X —> 0, the solution (3) approaches the incompressible solution (3), i.e. enhancement of h goes to 1. The two cases evaluated by Love are indicated. Figure 2 shows that the solution seems well behaved as the ratios g/X and ggR/g vary. Enhancement of h increases over this range as the body becomes more compressible. 

Figure 6. The loci of the singularities in the (ggR/g,p,/A) plane are shown. No singularities are observed for small values of ggR/g. or for g/A = 0 (rigid and/or incompressible material). The Earth and Venus, represented by the Earth symbol, occupy a region of this space just below the lowest locus line.

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