Modulus prediction

Predicted results for E2 are plotted in Figure 3-10 for three values of the fiber-to-matrix-modulus ratio. Note that if Vj = 1, the modulus predicted is that of the fibers. However, recognize that a perfect bond between fibers is then implied if a tensile <32 is applied. No such bond is implied if a compressive 02 is applied. Observe also that more than 50% by volume of fibers is required to raise the transverse modulus E2 to twice the matrix modulus even if E, = 10 x E ,l That is, the fibers do not contribute much to the transverse modulus unless the percentage of fibers is impractically high. Thus, the composite material property E2 is matrix-dominated. 

Figure 5.27 The storage modulus predicted by the reptation model...

In addition to the modulus prediction, the ultimate properties including elongation and ultimate tensile strength, assuming good adhesion between filler and polymer, can be modeled by the following equations ... 

The compressive modulus predicted for a triblock (/ = 0.5) in the strong-segregation limit is theoretically the same as for the diblock formed by cutting the triblock in the middle. The bend modulus for the triblock is predicted to be about twice as large as that of the diblock because of the presence of bridging configurations (Turner 1995). 

Stress relaxation modulus predicted by the Rouse model for a melt of unentangled chains with jV= 10 . The solid curve is the exact Rouse result [Eq. (8.55)] and the dotted curve is the approximate Rouse result [Eq. (8.48)]. 

In 0-solvents 1/2), the stress relaxation modulus decays as the - 2/3 power of time, while in good solvents (i/ 0.588) G(t) decays approximately as the - 0.57 power of time. Like the stress relaxation modulus of the Rouse model [Eq. (8.47)], Eq. (8.63) crosses over from kT per monomer at the monomer relaxation time tq to kT per chain at the relaxation time of the chain tz TqN [Eq. (8.25)]. Once again, an excellent approximation to the stress relaxation modulus predicted by the Zimm... 

The WLF predictions were in the wrong direction for 100% modulus of compounds K and W at 40 °C, probably because of the accelerated results changing direction with time. The same applies to compound D at 300%. The 300% modulus predictions were also in the wrong direction for compounds A, C, G and N. For compounds A and C changing direction in accelerated tests was again apparent but for compounds G and N the natural and accelerated results are simply different. 

A careful reading of the literature [coupled, in this case, with experimental work (Manson and Chiu, 1973a)] reveals that both small and large filler particles can induce significant changes in phenomena such as sorption, permeability, or relaxation behavior of the matrix (notably in the glass transition temperature Tg and in damping characteristics, E" or tan 5), in addition to the noninteractive increases in modulus predicted by relationships such as Kerner s equation (12.3). The fact that some exceptions and contradictions continue to exist serves to stimulate further study rather than to deny the tendency toward filler-matrix interaction. Typical evidence may now be summarized. 

Fisher, F. X, Bradshaw, R. D., and Brinson, L. C. Fiber waviness in nanotube-reinforced polymer composites— I modulus predictions using effective nanotube properties. Comp Sci and Tech., 63,1689-1703 (2003). 

A single experimentally obtained cooling curve of the elastic modulus, such as the one shown in Figure 2 (circular markers), can be used in conjunction with Equations (1) and (10) can be used to find the appropriate values of an for a particular porous ceramic system. The modulus predicted by Equation (10) is shown by the solid curve in Figure 2. Thus, the agreement between model and experiment is very good for this material. 

Al-Khateeb G., A. Shenoy, N. Gibson, and T. Harman. 2006. A new simplistic model for dynamic modulus predictions of asphalt paving mixmres. Journal of the Association of Asphalt Paving Technologists, Vol. 75E, pp. 1-40. 

Robbins M.M. and D. Timm. 2011. Evaluation of dynamic modulus predictive equations for NCAT test track asphalt mixtures. Proceedings of the Transportation Research Board 90th Annual Conference, January 23-27. 

The results of Young s modulus prediction, according to (15.5) and (15.7), for uncross-linked isotropic all-cellulosic based composites are presented in Figs. 15.7 and 15.8. The modeling parameters used for (15.5) are listed in Table 15.5. 

Venkatraman, S., Deformation-Behavior of Poly(dimethyl Siloxane) Networks 0.1. Applicability of Various Theories to Modulus Prediction. J.Appl. Polym. 

Fig. 6.4 (a) Width of the recovery window [63], and (b) relaxed creep modulus at 92.3°C. Also shown (dashed line) is the relaxed modulus predicted by the affine theory of rubber elasticity. Symbols are as in Fig. 6.3 [63, 377]... 

Figures 4.2,4.3, and 4.4 show the modulus predicted by equation (4.8) vs. Young s modulus, E (experiment). As with the swelling data, the network imperfections and the contributions of the physical crosslinks, if any, were minimized by determining the two crosslink levels required for E (theory) on the separate homopolymer networks. Unfortunately, Millar did not report modulus data for his polystyrene/polystyrene homo-IPNs.

In addition to experimental fiber modulus data. Figure 4 also illustrates two axial modulus predictions calculated using the rule of mixtures. A previous investigation (14) has indicated that the modulus of neat PP fibers has a... 

Figure 10.6a represents an upper bound model, meaning that the modulus predicted is the highest achievable for a two-phased mixture. 

Thus the Takayanagi longitudinal modulus prediction is seen to be nearly correct, but the Takayanagi transverse modulus prediction is seen to be low. 

EJE for PC is adduced. As one can see, both indicated equations give a good enough correspondence with the experiment their average discrepancy makes up 5.6% in the Eq. (15.7) case and 9.6% for the Eq. (15.10). In other words, in both cases the average discrepancy does not exceed an experimental error for mechanical tests. This means, that both considered methods can be used for PC elasticity modulus prediction. Besides, it necessary to note, that the percolation relationship (the Eq. (15.7)) qualitatively describes the dependence E better, than the empirical relationship... 

For a network with a crosslink functionality (i/ ) of 4, the phantom network model predicts a modulus which is 1/2 of the modulus predicted by the affine network model. 

In fig.7-9 four curves of compressibility for roll press are shown. One curve represents the measured values and other three curves are predictions according to the particular equations of compressibility where parameters K, ao, ai, a.2, m were determined from curves of compressibility measured in a die press. It can be seen that equation (8) derived from simple linear modulus of volume transformation predicts the experimental values best. Equation derived from power modulus predicts the measured values in roll press with substantial error, although the equation fits measured values in die press and roll press very accurately. This fact is due to dependencies between parameters 02 nd m or in other words the equation is overparameterized. Prediction based on equation derived from linear modulus is not shown but substantial deviation between measured and predicted values also occurs. 

In the smdy of mechanical properties of particulate filled polymers, numerous models were developed to predict the effect of the particles on tensile or shear modulus. Most of these were derived from rheological models such as Einstein s, Eilers and Mooney s equations. A strong relationship exists between rheology and mechanical properties measurements and such correlations were studied by Gahleitner et al [66], as well as by Pukansky and Tudos [67]. There seems to have a direct relation between viscosity and shear modulus [59]. However, compensation has to be taken for matrix s Poisson ratio which is lower than 0.5 as shown by Nielsen and Landel [59]. Nevertheless, these equations on modulus predictions can be broadly classified under two groups. 

In Chapter 6 it is shown that the Doi-Edwards model for an entangled, monodisperse, linear polymer gives a value of 6/5 for this product, which reflects the nearly-exponential relaxation modulus predicted by this model. 

The Phantom Network Model. The theory of James and Guth, which has subsequently been termed the phantom network theory, was first outlined in two papers (186,187), followed by a mathematically more rigorous treatment (188-190). More recent work has been carried out by Duiser and Staverman (191), Eichinger (192), Graessley (193,194), Flory (82), Pearson (195), and Kloczkowski and co-workers (196,197). The most important physical feature is the occurrence of junction fluctuations, which occur asymmetrically in an elongated network in such a manner that the network chains sense less of a deformation than that imposed macroscopically. As a result, the modulus predicted in this theory is substantially less than that predicted in the affine theory. 

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