Statistical Analysis of the Mechanical Behavior of High-Performance Polymers: Weibull’s or Gaussian Distributions?

Abstract

This work addresses the following problem: which of the statistical approaches, Weibull’s or Gaussian, is more appropriate to correctly describe the statistical distributions of the mechanical properties of the high-performance polymer materials of different sample types (single or multifilament oriented fibers) and chain architectures (ultra-high-molecular-weight polyethylene, polyamide 6, or polypropylene)? Along with the routine mechanical properties such as strength, strain at break, and Young’s modulus, an apparent viscoelastic modulus and an apparent strain at break found when differentiating the stress–strain curves have been considered for the first time. For this purpose, a large sample number (50 in each series) has been tested. It has been shown that the values of the Weibull’s modulus (m) characterizing the data scatter were dependent both on the chain architecture and the sample type for the five elastic, viscoelastic and fracture characteristics analyzed. The Weibull’s model has been found to be more correct as compared to the Gaussian one. The different statistical approaches used for the analysis of the large arrays of the data are important for a better understanding of the deformation and fracture mechanisms of quasi-brittle and quasi-ductile high-performance polymer materials.

Impact of Crystallization on the Development of Statistical Self-Bonding Strength at Initially Amorphous Polymer–Polymer Interfaces

“…The best conformity of the lap shear strength distributions to the Gaussian model was found for non-crystallized interfaces, indicating that the interface self-bonding process in this phase state is controlled, on the one hand, by the sum of many independent and equally weighted factors, and, on the other hand, by the crack propagation across the quasi-brittle interface, as it follows from the Weibull-like behavior. These are the signs of statistical dualism observed for high-performance polymer materials earlier [ 17 , 23 ].…”

“…It should be noted that the comparable intervals of the m variation were reported for the lap shear strength of the weak (partially self-healed) PS–PS ( m = 1–4), PMMA–PMMA ( m = 1–5), and PS–PMMA ( m = 6–11) interfaces [ 18 ], and for completely different types of materials such as high-performance polymer fibers characterized by a very high tensile strength σ = 0.5–5 GPa ( m = 7–10) [ 17 , 23 ]. This behavior suggests similar fracture mechanisms of these two completely different polymer systems consisting in their brittleness.…”

“…However, for the purpose of comparison of the data scatter for the materials that drastically differ in their mechanical properties, it seems to be preferable to operate with the Weibull’s modulus than with SD since the former and the latter are expressed in arbitrary and absolute units, respectively. Therefore, the use of m is more convenient because it allows one to compare the distribution behaviors of materials differing drastically in strength (e.g., m ≈ 5–10 for both weak polymer–polymer interfaces with σ = 0.01–0.5 MPa [ 18 , 19 , 20 , 21 ] and ultra-high-strength fibers with σ = 2–6 GPa [ 17 , 23 ]). By contrast, such a comparison is not straightforward when using SD (±0.01 and ±100 MPa, respectively) since roughly the same difference in the σ values (by four orders of magnitude) is retained for the SD values as well.…”

“…Furthermore, it is still not clear as whether the σ distributions of weak polymer–polymer interfaces, along with the Weibull’s distribution, also follow the most widely used normal (or Gaussian) distribution representing a bell curve [ 22 ], as it does, for instance, for the tensile strength distributions of high-strength polymer fibers (a type of statistical dualism) [ 17 , 23 ]. To put it differently, the manifestation of such dualism for the σ distributions of weak polymer–polymer interfaces is still an open question and clarification is important for further elucidation of the interface self-healing and fracture mechanisms.…”

Uncertainty propagation, entropy, and relative entropy in the homogenization of some particulate composites

“…Some recent studies in biomechanics show that human tympanic membranes' Young modulus could be efficiently approximated using bimodal Gaussian distribution, which has been discussed in Reference 48. Some recent statistical studies exhibit that material characteristics of polymers and polymer‐based materials can be described by both Gaussian and Weibull distributions, compare 49 . Some other studies include next to these two PDFs the log‐normal probability distribution 50 nevertheless the Gaussian PDF seems to be the most efficient.…”