Power Law Multi-Scaling of Material Strength

TL;DR

This paper introduces a flexible 'knee' function to model transitions between different power law regimes in physical systems, simplifying the analysis of complex scaling behaviors in material strength and mechanics.

Contribution

The paper proposes a general merging function to accurately describe regime transitions in power law scaling laws, reducing the need for complex detailed solutions.

Findings

Successfully applied to various known scaling laws

Effectively models sharpness of regime transitions

Simplifies analysis of power law behavior in materials

Abstract

Power law is one of the the simplest forms of the relationship between different variables of a system. It leads naturally to the introduction of compound parameters describing physical properties of the system. Often one of the variables of interest is the object dimension, or time. The prevalence of a simple power law over the entire range of dimensions or times can be helpfully interpreted as size or time independence of the corresponding compound physical parameter of the system. However, it is also often found that a simple power law only persists for some extreme values, e.g. for very large and/or small sizes, or very short or long times. Transitions between regimes of different power law asymptotic behaviour are encountered frequently in the description of a wide variety of physical systems. While asymptotic power law behaviour may often be readily predicted, e.g. on dimensional grounds, the evaluation of the relationship between system parameters in the transition range usually requires laborious detailed solution. To obviate this difficulty we introduce, on rather general basis, something we refer to as the merging, or 'knee' function. The function possesses sufficient flexibility to describe transitions of various sharpness. To demonstrate its usefuness, the merging function is applied to a variety of well-known scaling laws in the mechanics and strength of materials and structures.