Kemer’s equation

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. 

Kemer s equation is based on the assumption that there is good adhesion between the granules and the gel around them, and that the granules are uniformly distributed. 

Hence, the data of Figs. 11 and 12 have shown, that the Eqs. (38) and (39) fulfilled in case of polymer microcomposites are incorrect for nanocomposites. In case of the Eq. (38) correctness and Kemer s equation application for G calculation the lower boundary of viscosity r can be obtained according to the equation [63] ... 

Considering the latter ratio, Kemer s equation can be transformed by introducing 4>e= B in which B = parameter characterizing the interaction. If F = 0.5 and Ep > fun the equation will become... 

But then the result is that the thidcness of the fixed layer decreases with the decrease of the thickness of the layer between the particles, whidi is difficult to exjdain. The dependence of B on 0 msdces it imposable to use the modified Kemer s equation, which does not account for this dependence. 

The PP-nylon system can be treated similarly. If the debonded region between the fiber and the matrix is viewed as an assembly of contiguous voids but having a volume fraction corresponding to a hypothetical cylinder (fiber) surrounding the voids, the modified form of Kemer s expression (7) for foams and polyblends can be shown to give a result in excellent agreement with Equation 2. 

The interlayer model was developed by Maurer et al. The model of the particulate-filled system is taken in which a representative volume element is assumed which contains a single particle with the interlayer surrounded by a shell of matrix material, which is itself surrounded by material with composite properties (almost the same as Kemer s model). The radii of the shell are chosen in accordance with the volume fraction of the fQler, interlayer, and matrix. Depending on the external field applied to the representative volmne element, the physical properties can be calculated on the basis of different boundary conditions. The equations for displacements and stresses in the system are derived for filler, interlayer, matrix, and composite, assuming the specific elastic constants for every phase. This theory enables one to calculate the elastic modulus of composite, depending on the properties of the matrix, interlayer, and filler. In... 

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. ... 

Thermal property evaluation took place with respect to thermal expansion, diffusivity, specific heat, and ultimately thermal conductivity. Instantaneous CTE data at 20°C is presented in Figure III. The thermal expansion of these materials is an important consideration to take account of, as many applications require matching CTE s to help reduce thermal mismatch stresses during cycling. Results are plotted with a rule of mixture model as well as the Turner and Kemer models for CTE. These are shown in Equations 5, 6, and 7 respectively. These predictions were also based on the SiCrSi system, with the property inputs provided in Table II. 

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