Equations Narkis

Equations, similar to Hatschek s empirical equation (11) can be found in calculations by Ishai and Cohen [54] or Narkis [55] for describing the dependence of relative modulus on concentration for disperse composites ... 

In the Einstein equation, b = 1 for spherical particles at low concentration and a depends on the adhesion between the matrix and the filler. This equation predicts that the addition of filler increases tensile strength which was found to be not always the case, so this equation has been modified by various researchers. The Nicolais and Narkis equation is a common modification in which a=1.21 and... 

A modified Nielsen modef is another frequently used equation, cially in the form proposed by Nicolais and Narkis ... 

Good agreement with equation (12.5) was also obtained by Narkis and Nicolais (1971), who studied the effect of glass bead concentration on styrene-acrylonitrile copolymers at 110°C, a temperature close to the glass temperature, the best lit being obtained with equation (12.6d) to represent ij/Vf. As expected, the filler shifted the relaxation curves to longer times, in proportion to the volume fraction of filler—an observation consistent with reported increases in 7 (see below). 

Applicability of equation (12.23) (with K = 1.21) to several filled polymer systems has been reported (Nicolais and Narkis, 1971). 

Making use of equation (12.24), Nicolais and Narkis (1971) proposed the following master curve equation to take account of both temperature and filler effects oh yield stress as a function of strain rate ... 

Superposition techniques may also be used to correlate stress-strain behavior in the rubbery state. In their study of styrene-acrylonitrile copolymers filled with glass beads, Narkis and Nicolais (1971) obtained stress-strain curves at temperatures above 7. Stress-strain curves were plotted for different fractions of filler, and in terms of both the polymer and composite strain. At a given strain, the stress increased with increasing filler concentration, as expected. It was possible to shift curves of stress vs. polymer strain along the stress axis to produce a master curve (Figure 12.12). In addition to the empirical measurements, an attempt was made to calculate stress-strain curves from the strain-independent relaxation moduli (see Section 1.16 and Chapter 10) by integrating the following equation ... 

Figure 12.11. Yield stress vs. a-rB for styrene-acrylonitrile copolymers containing diiferent concentrations of glass beads (T r = 24°C the numbers on the curves give Vf, the volume fraction of filler). The upper curve corresponds to equation (12.25) with constants A and B equal to 1.0 x 10 and 3 x 10, respectively it also corresponds closely to estimated values of a,., the yield stress of unfilled polymer. (Nicolais and Narkis, 1971.)...

Table 2.5 contains a general equation, Eq. (2.35), for the effect of particulate fillers on the tensile strength of a polymer, a common modification of this equation by Nicolais and Narkis, (Eq. (2.36)), and an additional equation proposed by Piggott and Leidner,... 

A large number of equations have been proposed to predict a variety of mechanical properties of filled plastics. The principles underlying the derivations require detailed and lengthy study to be understood, and regrettably there is no room here to discuss the rational basis for the predictive equations mentioned we shall simply indicate the kind of predictions that have emerged and moreover we shall confine the discussion to one property the modulus. The original references can be found in any review of the effect of fillers on mechanical properties, such as the theses by Wainwright and Phipps, mentioned elsewhere in this article. Contributors include Nielsen, Paul, Narkis, Ishai, Bueche, Sato and Furukawa, Halpin, Chow etc. 

Many other equations have also been offered. One of the simplest was a semi-empirical one by Narkis ... 

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