The hypoexponential form for the fiber-length distribution in thermoplastic composites revisited: finite initial length, cascade fracture, and its mechanistic status

Abstract

In a previous work [Masubuchi et al., Compos. Sci. Technol. 134, 43 (2016)], a fiber-length distribution function was proposed as the hypoexponential, or Erlang-type, convolution of two exponential waiting lengths, motivated by two independent Poisson processes for breakage and for blocking of adjacent breaks. The present paper further evaluates this hypoexponential form by introducing a finite initial length L_{0}, and by analyzing the mechanistic basis of the buckling-induced fiber breakage process. The obtained finite-L_{0} closed-form solution was compared to a glass fiber dataset to retroactively justify the infinite-length idealization in the earlier work for such systems while providing the complete form for processes in which fragments remain comparable to L_{0}. Concerning the mechanics of fiber fragmentation, a Mellin transform analysis of the fragmentation equation showed that scale-invariant cascades yield power laws and therefore cannot generate characteristic lengths; those lengths must instead arise from scale-breaking ingredients, particularly the cutoff region and the arrested processing history. Monte Carlo simulations of a phenomenological finite-opportunity cascade showed that the hypoexponential form does not reproduce the peak of the generated distribution but captures its exponential-like tail, implying that it serves as a useful two-parameter fitting model, although not an exact generative distribution.