Scalability and homogenization of transitional functions: Effects of non-equilibrium and non-homogeneity

Abstract

Mono-scale functions refer to the individual scale range of the SI system of measurement. Coarseness of the segmented scale was dictated by state-of-the-art of technology at that time. By to-day’s standard, non-equilibrium and non-homogeneity (NENH) are first order considerations whereby scales must be refined to include micro, nano and pico effects. The conventional technology applies to monoscaling, confined to equilibrium and homogeneity (E&M). Their conversion to multiscaling requires the use of transitional functions. The ultra high strength and light weight structural materials rely on the absorption of energy at more than one scale. For considerations are effects at microscopic, nanoscopic and picoscopic scales.The irony is that NENH are subject to eventual homogenization for otherwise the multiscale effects could not be transferred to improve and modify the monoscale rules in practices. To this end, additional Postulate and Corollary are needed to account for scale directionality of energy transfer, prevalent to NENH. Direction-dependency differentiates the transition of macro→micro and micro→macro. Transitional functions are not the same when they traverse up and down the scale. Homogenization averages out NENH effects such that corrections may be applied to monoscaling.Transitional functions can lock-in the load, material and geometry effects of the macro–micro test data to produce the nano-pico data. This is related to the obtainment of small crack data from large crack test data. To this end, the volume energy density factor (VEDF) or the volume energy density (VED) can be used as the transitional functions as a form-invariant criterion such that multiscale effects can be used to correct and modify monoscale results. When NENH effects are highly localized and cannot be averaged out, the time rate of VEDF and VED or the equivalent of the power energy density must be used.The scalar correction for NENH is derived for the macro–micro cracking of a line crack subject to the combined effects of loading, material and geometry. Two correction factors Λ and Ω are used, one for macro→micro and another for micro→macro. The outcome can be checked by results for crack length and/or crack growth rate.