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Yilong Bai, Mengfen Xia, Haiying Wang and Fujiu Ke State Key Laboratory of Non-Linear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, P. R. China Author to whom correspondence should be addressed. E-mail: Baiyl@lnm.imech.ac.cn  
Multi-scale modeling of materials properties and chemical processes has drawn great attention from sci-ence and engineering. For these multi-scale and rate-dependent processes, how to characterize their trans-scale for-mulation 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 transforma-tions 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 inde-pendent dimensionless numbers: the imposed Deborah number De=(ac)/(LV) and the intrinsic Deborah num- ber D = (nNc5)/V* ,where a. L, c, V and nN are wave speed, sample size, microcrack size, the rate of micro- crack grovvth and the rate of microcrack nucleation density, respectively. Clearly, the dimensionless number De=(ac)l(LV) includes length and time scales on both meso- and macro- Ievels and governs the progressive process. Whereas, the intrinsic Deborah number D' indicates the characteristic transition of microdamage to macroscopic rup-ture 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 stili open now. The comparison of various mechanisms governing the enhancement of meso-size effects re-minds 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.
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