The role of invariance in the finite similitude scaling theory

Abstract

A new scaling theory has appeared in the open literature called finite similitude, which provides an infinite number of new similitude rules that can in principle accommodate all scale effects. A difficulty with the practical application of the theory in experimentation is that (in the absence of supporting analysis) only lower-order similitude rules are feasible since the number of scaled experiments necessarily increases with rule order. Scaling analysis is not constrained to the same extent but nonetheless it is necessary to determine explicitly scaling functions that depend on a single parameter (the length scalar $\beta$). One approach to obtaining these scaling functions is by targeting invariants in the physical system under scrutiny. All manner of scale invariances can be targeted such as geometric measures of length, area, and volume, along with space–time measures of length-time, area-time, and volume-time. Important kinematics parameters such as speed (e.g., acoustic, light) and acceleration (e.g., gravitational) can also be targeted including material properties (e.g., viscosity) and important fields (e.g., velocity, stress). The focus of this paper is to examine the role and importance of invariances to the finite-similitude theory with the aim of providing insight into the possible options available. Although continuum mechanics under scaling is the focus here, invariances from microstructural considerations can often arise. A number of case studies in solid mechanics are presented involving both quasistatic and dynamic crack propagation to demonstrate the reach and benefits of the approach in scaling analysis.