Directional dissimilarity of transitional functions: Volume energy density factor

Scale segmentation of the SI system of measurement has necessitated the use of transitional function for connecting the results of the segmented scales. The state-of-the-art of measurement in the 1800s is reflected by the coarseness of scale segmentation. By up-to-day standard of nanonization, scale refinement is necessary to account for the interactive benefits of multiscaling, the building block of which is dual scaling, say macro-micro or micro-nano. The directional dissimilarity between two adjacent scales, say macro → micro and micro → macro, arises naturally in physical processes. An index Λ is defined to measure the severity of directional dissimilarity that may prevail in the volume energy density factor (VEDF) for macro → micro. A similar index Ω applies to micro → macro. Both A and Ω reflect the combined effects of loading, material and geometry. The macro-micro transition of VEDFmacro can differ from the micro-macro transition of VEDFmicro. The relation between the macrostress intensity factor and VEDFmacro is incompatible with that for the microstress intensity factor and VEDFmicro. The stress intensity factor, a monoscale concept, will not hold for both the macro- and micro-scale. To begin with, the units may differ. The same holds for the volume energy density factor. The irreconcilability can be resolved by an argument of dimensional compatibility, provided that the agreement is to preserve the monoscale definitions. These considerations call for a scrutiny of the crack growth data whose scatter may be contributed by the directional effects of scale transition.