Modeling of crystallization in fiber-reinforced composites

To predict the conversion of melt into spherulites during isothermal crystallization of fiber-reinforced composites, the concept of extended volume was used 

The key problem in the probabilistic approach is to calculate the probability, Po(t), that an arbitrarily chosen point will remain in an uncrystallized fraction of material at a certain time t elapsed from the beginning of crystallization, that is, from the moment of nucleation of the first spherulite. In the case of instantaneous nucleation, the point A will remain at time t outside of growing spherulites, if none of them is nucleated within a sphere with the point A at the center and radius of R(t) equal to a spherulite radius at time t  

The probability, Po(t), that during instantaneous nucleation no spherulite is nucleated in a sphere of radius R = R(t) is a product of two probabilities the probability that no spherulite is nucleated in polymer bulk, denoted here as Rb(0. and the probability that no spherulite is nucleated on fibers, denoted as P t). P it) is given by the simple expression  

If all fibers are aligned in one direction the fibers crossing the considered sphere have to cross a circle around point A with radius R. The probability that no fiber crosses the circle reads  

The probability that a fiber crosses the considered circle in a distance Si from A equals exp(- rDf R ) InDt Si dsi). The length of the fiber section embedded in the sphere equals 2(R -Siy, whereas the probability that no nucleation event occurs on this section reads exp[-2T s(i -SiY ], where denotes nonzero nucleation density on fibers, equal to a number of nuclei per a fiber length unit. To calculate Pt, it is necessary to determine at first the probability that n fibers cross the 

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