Stiffness experimental data

To conclude, the concept of bond stiffness, based on the energy/distance curves for the various bond types, goes a long way towards explaining the origin of the elastic modulus. But we need to find out how individual atom bonds build up to form whole pieces of material before we can fully explain experimental data for the modulus. The... 

The mechanics of materials approach to the micromechanics of material stiffnesses is discussed in Section 3.2. There, simple approximations to the engineering constants E., E2, arid orthotropic material are introduced. In Section 3.3, the elasticity approach to the micromechanics of material stiffnesses is addressed. Bounding techniques, exact solutions, the concept of contiguity, and the Halpin-Tsai approximate equations are all examined. Next, the various approaches to prediction of stiffness are compared in Section 3.4 with experimental data for both particulate composite materials and fiber-reinforced composite materials. Parallel to the study of the micromechanics of material stiffnesses is the micromechanics of material strengths which is introduced in Section 3.5. There, mechanics of materials predictions of tensile and compressive strengths are described. 

The mechanics of materials approach to the estimation of stiffness of a composite material has been shown to be an upper bound on the actual stiffness. Paul [3-4] compared the upper and lower bound stiffness predictions with experimental data [3-24 and 3-25] for an alloy of tungsten carbide in cobalt. Tungsten carbide (WC) has a Young s modulus of 102 X 10 psi (703 GPa) and a Poisson s ratio of. 22. Cobalt (Co) has a Young s modulus of 30x 10 psi (207 GPa) and a Poisson s ratio of. 3. 

Another analysis method was based on the local wave vector estimation (LFE) approach applied on a field of coupled harmonic oscillators.39 Propagating media were assumed to be homogeneous and incompressible. MRE images of an agar gel with two different stiffnesses excited at 200 Hz were successfully simulated and compared very well to the experimental data. Shear stiffnesses of 19.5 and 1.2 kPa were found for the two parts of the gel. LFE-derived wave patterns in two dimensions were also calculated on a simulated brain phantom bearing a tumour-like zone and virtually excited at 100-400 Hz. Shear-stiffnesses ranging from 5.8 to 16 kPa were assumed. The tumour was better detected from the reconstructed elasticity images for an input excitation frequency of 0.4 kHz. 

The dynamic behavior of liquid-crystalline polymers in concentrated solution is strongly affected by the collision of polymer chains. We treat the interchain collision effect by modelling the stiff polymer chain by what we refer to as the fuzzy cylinder [19]. This model allows the translational and rotational (self-)diffusion coefficients as well as the stress of the solution to be formulated without resort to the hypothetical tube model (Sect. 6). The results of formulation are compared with experimental data in Sects. 7-9. 

In this article, we have surveyed typical properties of isotropic and liquid crystal solutions of liquid-crystalline stiff-chain polymers. It had already been shown that dilute solution properties of these polymers can be successfully described by the wormlike chain (or wormlike cylinder) model. We have here concerned ourselves with the properties of their concentrated solutions, with the main interest in the applicability of two molecular theories to them. They are the scaled particle theory for static properties and the fuzzy cylinder model theory for dynamical properties, both formulated on the wormlike cylinder model. In most cases, the calculated results were shown to describe representative experimental data successfully in terms of the parameters equal or close to those derived from dilute solution data. 

Recently, the stiff-chain polyelectrolytes termed PPP-1 (Schemel) and PPP-2 (Scheme2) have been the subject of a number of investigations that are reviewed in this chapter. The central question to be discussed here is the correlation of the counterions with the highly charged macroion. These correlations can be detected directly by experiments that probe the activity of the counterions and their spatial distribution around the macroion. Due to the cylindrical symmetry and the well-defined conformation these polyelectrolytes present the most simple system for which the correlation of the counterions to the macroion can be treated by analytical approaches. As a consequence, a comparison of theoretical predictions with experimental results obtained in solution will provide a stringent test of our current model of polyelectrolytes. Moreover, the results obtained on PPP-1 and PPP-2 allow a refined discussion of the concept of counterion condensation introduced more than thirty years ago by Manning and Oosawa [22, 23]. In particular, we can compare the predictions of the Poisson-Boltzmann mean-field theory applied to the cylindrical cell model and the results of Molecular dynamics (MD) simulations of the cell model obtained within the restricted primitive model (RPM) of electrolytes very accurately with experimental data. This allows an estimate when and in which frame this simple theory is applicable, and in which directions the theory needs to be improved. 

In reality the interactions between polymer and solvent molecules, which determine the solution viscosity, are very complicated and dependent on a great number of parameters. The literature mentions the solubility parameters of polymer and solvent, polymer chain stiffness, free volume of the solution, etc. In principle, all these factors should be taken into account in predicting the viscosity of a polymer solution. However, the available experimental data are insufficient for this purpose. 

Numerieal finite element models of Astley et al. (1998) take aeeount of ehemieal eomposition, mierofibril angles and their dispersion within all the eell wall layers (not simply the S2) and eell geometry (shape, size, wall thiekness). Where adapted to realistie models of earlywood/latewood entities, these simulations reduee the disparity between stiffness as the MFA ehanges from 45 to 10° to a faetor of about 3 as opposed to about 5 in the original model (although that work was supported by experimental data) by Cave (1968). 

Figure 2 7. Comparison of fit of power law, Bingham plastic and Robertson-Stiff rheological models to experimental data from bentonite drilling fluid. (Data from reference 106.)...

The mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation software used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over RI1/AI2O3 (Figure 8.8) demonstrates how adequate the mechanistic model is. 

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