Isostrain

Figure 6-22 Distribution of the Moduli of Components x, y, upper (Gy) and Lower Bound (G ) Moduli from the Takayanagi Isostress and Isostrain Blending Laws the Continuous Phase is Indicated in Parenthesis (plotted from data in Morris, 1992).

Calculate (a) the modulus of elasticity, (b) the tensile strength, and (c) the fraction of the load carried by the fibers for a continuous glass fiber-reinforced epoxy resin, with 60% by volume E-glass fiber, stressed under isostrain conditions. The tensile strength and modulus of the fibers are 1800 MPa and 76 GPa, respectively, and the values of these quantities for the matrix are 60 MPa and 2.4 GPa, respectively. 

The membrane conductivity in Equation 19.4 can be estimated using the following three models [48] isostrain or parallel model... 

Therefore, in the classic limit, thermal averages in the grand mixed isostress isostrain ensemble may be cast as... 

Equations (2.116)-(2.118) can be rewritten in a slightly different way, which permits to derive a general relation between partition functions in various mixed is(xstre s isostrain ensembles. Notice, for example, that we may define... 

Figure 5.20 Shear modulus C44 as a function of shear strain asxo- (O) MC simulations in grand mixed isostress isostrain ensemble (—) representation of small-strain approximation C44 (asxo) = 00 + 02 (asxo) (see Eqs. (5.113) and (5.114)].

To obtain a more concise picture of thermodynamic stability of different film morphologies, we plot grand mixed isostress isostrain ensemble, we calculate (j> directly from Eq. (5.119) using the molecular expression for Tyy [see Eq. (5.85)], which does not contain any fluid substrate contribution between the fluid substrate... 

Figure 5.22 (a) Normal compressional stress (see Appendix E.3 for molecular expressions) as a function of substrate separation from GCEMC simulations (O) (o.Sxo = 0.0). Solid lines aie intended to guide the eye. (b) As (a) but for [see Eq. 5.119]. Intersections between the latter and the vertical lines demarcate (meta- or thermodynamically) stable states in the grand mixed isostress isostrain ensemble for = 0.0 (see text). 

Figure 5.25 (a) as a fimction of shear strain qSxO for mono- (O), bi- (A), and trilayer (-I-) morphologies calculated in grand mixed isostress isostrain ensemble... 

As we showed in Section 2.5.4, thermal averages in isostress isostrain ensembles can be related through a Laplace transformation. Hence, for the conjugate stress r and strain A, wc may employ Echange variables according to Tja — t and —> A giving... 

According to the above discussion the distribution of microstates in the current mixed isastross isostrain ensemble is governed by the probability density given in Exj. (5.193). The similarity between the present probability density and the one relevant in the closely related ensemble discussed in Section 5.2.4 suggests we should design an adapted Metropolis algorithm closely related to the one described in that section. In fact, from the detailed discussion in Section 5.2.4, it turns out that we just need to replace the substrate separation s by the area A. More specifically, we need to replace Ex s. (5.46)-(5.48)... 

Table 5.5 Overview of fluctuation-related response coeflScients in the mixed isostress isostrain ensemble at constant T, N, T, and s o (see text for details of the derivations).

Table 5.6 Compaiison of results for isostrain heat capacity from consistency relation Eq. (5.140) with directly computed values from canonical ensemble (CE) [see eq. (5.206)] for model A (see text). ...

In accord with the mean-field theory developed in Section 5.7.5, the inversion temperature, however, does depend on the density of the fluid. This can be seen from plots of Ta — 1 in Fig. 5.29 based on isostress isostrain ensemble simulations. Regardless of T, Ta /ke — 1 turns out to be a nonmonotonic function of density. It has a maximum that increases and shifts to lower densities with decreasing temperature. In the limit 7> — 0, one exj)ccts all curves to approach zero according to [see Ekj. (5.155)]... 

The discussion of the Joule-Thonison effect in the previous section clearly showed that it is advantageous in theoretical treatments of confined fluids to tackle a given physical problem by a combination of different methods. This was illustrated in Section 5.7 whore wo employed a virial expansion of the equation of state, a van der Waals type of equation of state, and MC simulations in the specialized mixed isostress isostrain ensemble to investigate various aspects of the impact of confinement on the Joule-Thomson effect. The mean-field approach was particularly useful because it could predict certain trends on the basis of analytic equations. However, the mean-field treatment developed in Sections 4.2.2 and 5.7.5 is hampered by the assump-... 

This being the primary goal of the subsequent discussion we would also like to emphasize two other, perhaps more practical, aspects. On account of the rigidity of the underl3dng lattice it seems inconceivable to develop mixed isostress isostrain ensembles suitable for lattice MC simulations. On the other hand, lattice simulations are computationally much less demanding because molecules can occupy only discrete positions in. space. Hence, the number of configurations possible on a lattice is greatly reduced compared with simulations of continuous model systems. 

To this end we performed MC simulations in the mixed isostress isostrain ensemble introduced in Section 5.7 where a thermodynamic state is specified... 

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