Some sigmoid and reverse-sigmoid response patterns emerging from high-power loading of solids

Abstract

The objective of the present review is twofold. First, it aims at highlighting some sigmoid and reverse-sigmoid response patterns observed recently in the course of simulations of the high-strain-rate loading of (mostly, quasibrittle) solids. Second, it aims at reviewing various properties of two models used frequently as curve fitting tools for nonlinear and saturable phenomena. These two models-inspired by the Hill and the Weibull cumulative distribution functions-are bounded by two horizontal asymptotes with a smooth transition between the baseline and the final saturation state, characterized by a non-negative (a non-positive) derivative at each point for the sigmoid (the reverse-sigmoid) shape. Although they were used primarily for data fitting because of their flexibility and effectiveness, these nonlinear models possess other properties useful for the analysis of the irreversible, nonlinear and far-from-equilibrium phenomena. The main features of these two models are systematically examined in this review. In spite of the fact that satisfactory curve-fitting of data could not be considered a proof of causality it could underline a pattern of behavior and, perhaps, provide an investigation guidance.

A scaling approach to size effect modeling of J CDF for 20MnMoNi55 reactor steel in transition temperature region

“…where η and β are, respectively, the scale parameter and the Weibull modulus (shape parameter) [36] and the symbol J is used for brevity hereinafter instead of Jc. The Weibull parameters η and β are material constants sensitive to the specimen preparation, surface condition and temperature [37].…”

Shattering impact fragmentation of slender nanoprojectiles

“…The dependence of the maximum fragment mass upon the aforementioned state variable Ξ for the entire (non-negligible, Ξ  Ξ 0 ) fragmentation range is schematically depicted in Fig. 3 by a characteristic reverse-sigmoid curve in the logarithmic space [41].…”

“…4 are obtained by rescaling the raw simulation results (shown in the inset of the same figure) by appropriate powers of the nanoprojectile width defined by scaling exponents υ = 0.77  0.01 and β = -0.25  0.02. It is important to note that the preceding scaling exponents are identified based on the requirement of the critical-point data (v 0 , m max0 ) collapse [41]. Therefore, unlike the m max vs. v scaling of Timar an coauthors [18] with emphasis on the linear-range data collapse, the main panel plot of…”

Chemical sensors based on photonic colloidal crystals

“…Кривые на рис. 9 представляют собой семейство сигмоидальных кривых, что свидетельствует об аномальных процессах, происходящих при диффузии органических неполярных растворителей в коллоидные микросферы [15,28]. При этом математические модели для нестационарных режимов диффузии в средах, содержащих поры, требуют особого решения набора дифференциальных уравнений [25,29].…”