The crystalline fibril model

Using the Cox model for a fibre composite, already discussed [9], and 

In the absence of information regarding the arrangement of the crystalline bridges, it is assumed that they are randomly placed so that the probability of a crystalline sequence traversing the disordered regions to link adjaeent crystalline blocks is given in terms of a single parameter p, defined as 

The first term in this expression, which corresponds to the crystalline bridge sequences, is next treated as an array of short fibres, so introducing the shear lag (efficiency) factor d , which is a function of the finite aspect ratio of the crystalline bridges. The analogous equation to Equation (8.20) is 

The aspect ratio of the fibre component, which is a measure of the width of the crystalline bridges, is not directly accessible but can be deduced from the value of the shear lag factor O required to give the best match between the predicted and observed patterns of mechanical behaviour as a function of temperature. This exercise yields a radius of 1.5 nm for the crystalline bridge sequences, which suggests that each bridge is comprised of several extended polymer chains. Detailed considerations of the way in which the modulus increases at temperatures below —50 °C suggest that the modulus of the matrix increases with increasing draw ratio due to an increase in E in Equation (8.20), which corresponds to an increase in the modulus of the non-crystalline material. 

Even for block copolymers, in which the phase separation can be distinguished in electron micrographs, there are problems in matching parameters such as Poisson s ratios of the two components nevertheless the simple Takayanagi models, particularly when extended by a treatment to account for the finite length of the reinforcing component, can describe numerous features of static and d3mamic elastic behaviour. 

Using the Cox model for a fibre composite, already discussed [12], and neglecting the very small contribution E V arising from the tensile modulus of the comphant matrix, the extensional modulus E of the highly oriented polymer becomes 

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