Nielsen—Lewis equation

As discussed previously, a number of mathematical models for prediction of the mechanical reinforcement in polymer matrices exist. Halpin-Tsai (Halpin and Kardos 1976 Cadek et al. 2002) and Lewis-Nielsen (Lewis and Nielsen 1970) equations are based on simple approximations and provide reliable predictions of the modulus of reinforced matrices where the filler is unidirectional or randomly distributed. Therefore one can utilise these predictions to quantify the level of dispersion and the possible orientation of the filler particles in the matrix (see Fig. 8.10). 

Many equations have been proposed for the transport properties of two-phase systems and in-depth details of the existing models are discussed elsewhere [4]. Noticing that virtually all the early theories neglected the effects of the particle shape, their packing density, and the possible formation of anisotropic clusters, Lewis and Nielsen modified the Halpin-Tsai equation for the elastic modulus of composite materials by incorporating the maximum volume fraction of filler cpm while still maintaining a continuous matrix phase [33,34]. Transposed to thermal conductivity Lewis and Nielsen s equation becomes... 

We denote the dynamic rheological modulus of a composite as G, that of the amylose matrix as G, and of the starch granule as G, and the volume fraction of the granules as 0. The various parameters that affect the rigidity of a composite may be seen in Equation 6.42 of Kemer (1956) and its modification. Equation 6.45, by Lewis and Nielsen (1970) for reinforcement ((j /Gq) by spherical particles ... 

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 increase in modulus may also often be expressed in terms of the slightly different Mooney (1951) or Guth (1944)-Smallwood (1944) equations. For example, with glass-bead-filled epoxy resins, Lewis and Nielsen (1970) found agreement between predicted and observed values of modulus in the rubbery region using the Mooney (1951) equation ... 

Lewis-Nielsen equation An equation derived from the theory of mixtures that provides an estimate of the modulus of a short-fiber, thermoplastic composite. It is given below. 

The Lewis-Nielsen model considers the effect of the shape of the filler and the orientation or type of packing for a two-phase system or single-phase reinforcement composite, resulting in equation (11.8) for effective thermal conductivity (Tavman... 

Lewis-Nielsen or modified Kerner equation [17] is one of the most successful models ... 

Lewis and Nielsen [71] showed that the Halpin-Tsai equation could be further generalized as follows ... 

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