Theoretical Calculation of Elastic Constants

In recent years there has been increased interest in the theoretical calculation of the elastic constants for ideal and fully oriented polymers based on knowledge of their crystal structures. This increased interest arises from the major developments in computational methods and from the success achieved in producing very highly oriented polymers with reasonably high stiffness. 

The theoretical calculations require knowledge of the force constants for deformation of the structure. These fall into two categories, those relating to intramolecular deformation of the polymer chains, which are primarily bond stretching and bond bending, and those concerned with intermolecular deforma- 

The results of such calculations for semi-crystalline polyethylene have been reviewed elsewhere [37]. A rather wide range of predicted values is obtained, due to the choice of force constants and also to sensitivity to detailed assumptions on the unit cell structure. In spite of these limitations the principal predictions for the elastic anisotropy are clear. These include the anticipated high values for C33 and the very low values for the shear stiffnesses C44, C55 and cee, which reflect the major differences between bond stretching and bond bending forces that control C33 and the intermolecular dispersion forces that determine the shear stiffnesses. It is therefore of value to compare such theoretical results with those obtained experimentally. Table 7.3 shows results for polyethylene where data for the orthorhombic unit cell at 300 K are used to calculate these constants for an equivalent fibre (Voigt averaging procedure see Section 7.5.2 below) compared with ultrasonic data for a solid sheet made by hot compaction. It can be seen that 

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Theoretical calculations of elastic constants of pol5aner systems with a particulate filler enable us to find the analytical expressions for effective values of Lame coefficients x and A, m at varying filler loading ... 

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