Kemer equation

Keratinous proteins, in wool, 26 378-379 Kermesite, 3 41 Kemer equation, 11 309 Kemite (rasortie), 4 133t, 243t, 245 ... 

It is interesting to consider an extreme case for very stiff spheres (a very high) at a low volume fraction. The Kemer equation can then be written as ... 

For the quantitative description of the stiffness of the composite, the Kemer equation of 9.1.5. and 9.2 can again be applied, though in a somewhat modified shape. Since a fibre brings about anisotropy of properties in its environment, we have to consider two different cases, viz. the E-modulus parallel to the fibre, Ep, and the one perpendicular to it, Et. For both cases Kemer s equation holds, in which now A, in the parallel case, Ev, equals 2-l/d, while for Et, A = j. 

In the case that the composite is filled with rigid particles featuring some adhesion, the so-called Kemer equation (Kemer 1956) can be used to estimate the modulus of the composite ... 

Figure 26.11 shows the stress-strain behavior of ES30 filled with various levels of ATH. It can be seen that the yield stress increases with increasing level of ATH while the ultimate elongation is in excess of several hundred percent even for materials with more than 50wt% ATH. The modulus of composite materials can be modeled by the generalized Kemer equation ... 

Many theories have been advanced for predicting the modulus of filled composites. The Kemer theory is often used for the G modulus in the case of filled systems containing spheres Halpin--Tsai modified the Kemer equation in a more general form Lewis and Nielsen suggested a further modification by taking into consideration the packing factor and obtained, in the case of E modulus, the following equation ... 

The moduli of PPO/glass bead composites are reported by Trachte and DiBenedetto (23) to follow the Kemer equation and by Wambach al (24), to follow the equation of Van der Poel. These may therefore be more appropriate choices in modeling the composite behavior of the PpCIS/PPO blends but considering the low moduli ratio of the PpCIS/PPO blends, EppQ] 3/EppQ = 1.3, the difference between values of moduli predicted by use of either of these equations or use of the simpler Eq. 3 would be slight. 

When spherical particles are included in the polymeric matrix the behavior of such material is isotropic and the elastic properties can be easUy jnedicted by using the Kemer equation (15) shown below or the Halpin-Tsai equations for G,2 and discussed earlier ... 

Figure 13 Relative moduli Ec/Em vs volume fraction of glass beads for various comijosites (24). The solid line is the plot of the Kemer equation.

Thermal Properties. Since polymers generally have a much larger thermal expansion coefficient than most rigid fillers, there is a significant mismatch in thermal expansion in a filled polymer. This mismatch could lead to generation of thermal stresses aroimd filler particles during fabrication and, most severely, induce microcracks at the filler interface that could lead to prematiu-e failure of the filled polymer. As for the thermal expansion coefficient of a filled polsrmer, it generally falls below the value calculated from the simple rule of mixtiu-es but follows the Kemer equation (40) for nearly spherical particles. 

FIG. 14-19. Plots of log E and log tan 5 against temperature at a frequency of 110 Hz for a two-phase blend of equal weight fractions of polyfmethyl methacrylate) and slightly cross-linked polyfbutyl acrylate). Curves calculated from modified Kemer equation for a polyfmethyl methacrylate) matrix with inclusions of polyfbutyl acrylate) containing 14.4% poly(methyl methacrylate) by volume. (Dickie and Cheung. ) Reproduced, by permission, from the Journal of Polymer Science. 

However, the calculation according to the Eqs. (15.11) and (15.12) does not give a good correspondence to the experiment, espeeially for the temperature range ofT= 373 13 K in PC case. As it is known [38], in empirical modifications of Kemer equation it is usually supposed, that nominal concentration scale differs from mechanically effective filler fraction, which can be written accounting for the designations used above for natural nanocomposites as follows [41]. 

Hence, the stated above results have shown the modified Kemer equation application correctness for natural nanocomposites elastic response description. Really this fact by itself confirms the possibility of amorphous glassy polymers treatment as nanocomposites. Microcomposite models usage gives the clear notion about factors, influencing polymers stiffness. 

Hence, at the indicated above conditions fulfillment within the temperature range of T < for PC perfect intercomponent adhesion can be obtained, corresponding to Kemer equation, and then the value E estimation should be carried out according to the Eq. (15.15). At T= 293 K (cp j= 0.56,... 

The interlayer model represents an extension of van der Pool s theory derived from works by Frohlich and Sack devoted to viscosity of suspension by a shell-model. Van der Poel obtained expressions for G and K (bulk modulus). In his model, the filler sphere of a radius, a, is supposed to be sturounded by the sphere of the matrix material with radius 1. The sphere in sphere obtained in this way is sturotmded by the great sphere of radius, R, consisting of material with macroscopic properties of heterogeneous composition. The residts of calculations according to the equations proposed by van der Poel are very close to those obtained using the Kemer equation. Detailed description of this approach can be found elsewhere. ... 

There are no adjustable parameters. Such models include the Kemer equation [68], Halpin-Tsai equation [69] and Chow equation [70]. The second group of equations, on the other hand, incorporate adjustable parameters to account for interactions between particles as well as between the matrix and the particles. Factors such as critical solid volume fraction, degree of agglomeration and powder-matrix adhesion are taken into account. Equations and models under this group would include the Nielsen generalized equation [71] and the modified Kemer equation [72,73]. 

The Kemer equation [68] can be used if the particles are near spherical and there is good adhesion between the particles and the matrix. For particles that are much more rigid than the polymer matrix, the equation can be expressed (up to moderate powder loading) as follows ... 

Mukhopadhyay et al. uses a modified Kemer equation to evaluate the dynamic mechanical properties of silica filled ethylene vinyl acetate copolymers. The expression for the relative storage modulus is as follows [72, 73] ... 

A plot of logarithm of (1versus logarithm of 4> would yield the exponent n as the slope. The parameter B can also be evaluated from the intercept. It was shown in the same report that the parameter B obtained from equation (21c) agreed well with those used to fit the modified Kemer equation (21a) to the experimental relative storage modulus data. 

Figure 24 shows a plot of relative storage modulus versus volume fraction of iron powder in the feedstock. The relative storage modulus obtained at various dynamic stress frequency are shown. The data obtained at various frequency were very close and seems to be independent of the frequency. Both the Nielsen equation and modified Kemer equation were used in an attempt to fit the experimental data. The modified Kemer equation fits the experimental data at low powder loading very well. However, it failed to predict the data at powder volmne fraction of 0.58. In contrast, the Nielsen equation fits the data for the entire range of volume fraction. 

Figure 24. Plot of Nielsen equation and modified Kemer equation with experimental relative storage modulus data obtained at various dynamic stress frequency.

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