Stochastic concepts in the study of size effects in the mechanical strength of highly oriented polymeric materials

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Abstract
A study of size effects in the ultimate mechanical properties of crystalline and semicrystalline polymeric materials (fibers, single crystals) is conducted. The concept of size effect and its importance are discussed. A statistical/stochastic approach is adopted and is shown to yield analytical predictions which are at least as accurate as other modeling schemes (mainly based on fracture mechanics theory) previously proposed in the literature. Within this framework, we propose a possible scheme for simultaneous interpretation of both longitudinal (gauge length) and transversal (diameter) size effects, which provides some interesting information regarding the type of flaw population present in a given polymeric material. The statistical scheme used is based on the Poisson/Weibull model, and variants of it, since this model is relatively well established from both physical and experimental viewpoints. Some unsolved problems inherent to the Weibull/weakest link model for failure are discussed, and a new distribution function for the strength of solids is derived. This is illustrated through an analysis of available experimental data for ultrahigh‐molecular‐weight polyethylene, poly‐p‐phenylene terephthalamide (Kevlar), polydiacetylene, and polyoxymethylene. A maximum likelihood approach is used for the first time for interpreting the effect of diameter variability on the mechanical strength of polymeric fibers. We propose a procedure by which the Weibull shape parameter is maximized over a continuous range of values of the size exponent. Finally, we conjecture on the relative roles of defect populations distributed over the boundary and in the bulk.