Characteristic dimensionless numbers in multi-scale and rate-dependent processes

Abstract

Multi-scale modeling of materials properties and chemical processes has drawn great attention from science and engineering. For these multi-scale and rate-dependent processes, how to characterize their trans-scale formulation is a key point. Three questions should be addressed: • How do multi-sizes affect the problems? • How are length scales coupled with time scales? • How to identify emergence of new structure in process and its effect? For this sake, the macroscopic equations of mechanics and the kinetic equations of the microstructural transformations should form a unified set that be solved simultaneously. As a case study of coupling length and time scales, the trans-scale formulation of wave-induced damage evolution due to mesoscopic nucleation and growth is discussed. In this problem, the trans-scaling could be reduced to two independent dimensionless numbers: the imposed Deborah number De *= ( ac * )/( LV * ) and the intrinsic Deborah number D* = ( n N * c * 5 )/ V *, where a, L, c *, V * and n N * are wave speed, sample size, microcrack size, the rate of microcrack growth and the rate of microcrack nucleation density, respectively. Clearly, the dimensionless number De *= ( ac * )/( LV * ) includes length and time scales on both meso- and macro- levels and governs the progressive process. Whereas, the intrinsic Deborah number D * indicates the characteristic transition of microdamage to macroscopic rupture since D * is related to the criterion of damage localization, which is a precursor of macroscopic rupture. This case study may highlight the scaling in multi-scale and rate-dependent problems. Then, more generally, we compare some historical examples to see how trans-scale formulations were achieved and what are still open now. The comparison of various mechanisms governing the enhancement of meso-size effects reminds us of the importance of analyzing multi-scale and rate-dependent processes case by case. For multi-scale and rate-dependent processes with chemical reactions and diffusions, there seems to be a need of trans-scale formulation of coupling effect of multi-scales and corresponding rates. Perhaps, two trans-scale effects may need special attention. One is to clarify what dimensionless group is a proper trans-scale formulation in coupled multi-scale and rate-dependent processes with reactions and diffusion. The second is the effect of emergent structures and its length scale effect.