Modulus calculation

The homogenization was also found [200] to be incredibly fast for both longitudinal and transverse overall elastic moduli of the transversely-random [201] unidirectional composite of parallel fibers. It was almost impossible to distinguish [200] the overall ensemble-average moduli calculated with 9 and 289 fibers in a disordered unit cell. 

High polymer calculations can be performed on polysaccharides. Calculation of unit cell translation vectors (15), heats of polymerization (15), and elastic moduli (16) can readily be done. The accuracy of such calculations is the same as that of equivalent molecular species. A limitation of elastic moduli calculations is that the polymer is assumed to be 100% ordered, a state not commonly found in polysaccharides. 

Shpakov, V.P. Tse, J.S. Tulk, C.A. Kvamme, B. Belosludov, V.R. (1998). Elastic moduli calculation and instability in structure I methane clathrate hydrate. Chemical Physics Letter, 282, 107-114. 

De Boer and coworkers ° °" parameterized the shell model for silica polymorphs on the results of ab initio calculations of the potential energy surfaces, polarizabilities, and dipole moments of Si(OH)4 and (0H)3Si-0(H)-Si(OH)3 clusters. The structural characteristics and elastic moduli calculated with this set of parameters for three structures compared well with results computed with the use of both the rigid ion and the empirical shell models. ... 

Examples of complex moduli calculated for continuous strings corresponding to this model are shown in Fig. 4.1 and 4.2 Unfortunately, the hydrodynamic interaction has not yet been taken into account in this model so that the results in these figures are not satisfactory with respect to hydrodynamics. However, the result may qualitatively show the important characteristics erf the dynamics of this model Figure 4.1 is an example erf a very rigid case. Apparently, the relaxation times can be divided into two groups separated by more than three decades on the time scale. Thus, there is a region where G R does not depend on a>R... 

Figure 1.1 represents the elasticity moduli calculation for nanocomposite BSR/nanoshungite components (matrix, nanofdler particle and interfacial layers), derived from the interpolation process of the nano-indentation data. The processed SPIP polymer nanocomposite image with shungite nanoparticles allows experimental determination of the interfacial layer... 

Table 1. The Weibull moduli calculated from the data in Figs 8 and 9...

In panels (a)-(c), comparison of the blend moduli calculated from the model (curves Equations 3.69 through 3.73) with the moduli data (symbols) for various high-M polyisoprene/ poly(p-tert butyl styrene) (PI/PtBS) blends as indicated. The sample code nmnbers of the blends indicate 10 M of the components. The model considers the cooperative Rouse equilibration and successive constraint release (CR)/reptation relaxation of the component chains, and the model parameters summarized in Table 3.1 were determined experimentally. (Redrawn, with permission, from Watanabe, H., Q. Chen, Y. Kawasaki, Y. Matsumiya, T. Inoue, and O. Urakawa. 2011. Entanglement dynamics in miscible polyisoprene/poly(p-fert-butylstyrene) blends. Macromolecules 44 1570-1584). 

Table 5.1 Comparison of bulk moduli calculated forthe free-electron gas and experimental data for alkali metals.

A cubic crystal possesses two shear moduli, Cn — Cn and C44. The method of the elastic moduli calculation is described in Appendix D. 

In Fig. 6.1, the obtained according to the original methodics results of elasticity moduli calculation for nanocomposite butadiene-styrene rub-ber/nanoshungite components (matrix, nanofiller particle and interfacial layers), received in interpolation process of nanoindentation data, are presented. The processed in SPIP pol5mier nanocomposite image with shimgite nanoparticles allows experimental determination of interfacial layer thickness which is presented in Fig. 6.1 as steps on elastomeric... 

Table IV Effective elastic moduli calculated of actual reconstructed 3D models...

Figure 6.7 shows the force errors of three Gaussian Approximation Potential models for diamond with cutoffs of 2.0, 2.75 and 3.7 A. The difference between the latter two models is negligible. However, the elastic moduli calculated from... 

The elastic moduli calculated with our model, the Finnis-Sinclair potential [14] and Density Functional Theory are given in Table 6.5. The elastic properties and the phonon dispersion relations described by the GAP model show excellent agreement with the values calculated by Density Functional Theory. 

In Fig. 1 the obtained according to tiie original methodics results of elasticity moduli calculation for nanocomposite butadiene-stirene rubber/ nanoshungite components (matrix, nanofiller particle and interfacial lay-... 

It has been found for many cubic phases consisting of spherical particles that the moduli calculated with the hard-sphere model are often in good agreement with the ones determined experimentally. However, it must be stated here that also for the cubic phases a generally valid model for a theoretical description of their rheological behaviour is not available at present (mainly due to the uncertainties in the choice of fi). 

In a recent study, the elastic constants of pure SmS at ambient pressure and for compounds in the system Smi elastic constant data are... 

The numerical values of c,y at 300 and 4.2 K and the values of the basal plane Ea,b and c-axis Ec = IA33 Young s moduli calculated from the elastic constants data are shown in table 14. 

Figure 8. Relative bulk modulus of porous alumina (red) and zirconia (blue) modified exponential relation for spherical pores and matrix Poisson ratio 0.2 (green solid curve), predictions for alumina with intrinsic bulk modulus 2.139 (red solid curve) and for zirconia with intrinsic bulk modulus 2.724 (blue solid curve), dashed and dotted curves are relative bulk moduli calculated via elasticity standard relations using the differential and Mori-Tanaka prediction, respectively.

Chains in Networks. One of the first studies of chains in networks is by Gao and Weiner (235) where they performed extended simulations of short chains with fixed (affinely moving) end-to-end vectors. The first extensive molecular dynamics simulations of realistic networks were performed by Kremer and collaborators (236). These calculations were based on a molecular dynamics method that has been applied to study entanglement effects in polymer melts (237). The networks obtained by cross-linking the melts were then used to study the effect of entanglements on the motion of the cross-links and the moduU of the networks. The moduli calculated without any adjustable parameters were close to the phantom network model for short chains, and supported the Edwards tube model for long ones. Similar molecular dynamics analyses were used to understand the role of entanglements in deformed networks in subsequent studies (238-240). 

Figure 2 Effective Moduli calculated for the Filament Wound Tubing at varying Winding Angles...

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