Characteristic dimensionless numbers in multi-scale and rate-dependent processes--颗粒学报PARTICUOLOGY

Bai, Y., Xia, M., Wang, H., & Ke, F. (2003). Characteristic dimensionless numbers in multi-scale and rate-dependent processes. China Particuology, 1(1), 7-12. https://doi.org/10.1016/S1672-2515(07)60093-1

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

Yilong Bai ^*, Mengfen Xia, Haiying Wang, Fujiu Ke

State Key Laboratory of Non-Linear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, P. R. China

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

Volume 1, Issue 1, April 2003, Pages 7-12

Received 26 February 2003, Accepted 26 March 2003, Available online 30 November 2007.

E-mail: Baiyl@lnm.imech.ac.cn

Highlights

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.

Graphical abstract

Keywords

multi-scale; rate-dependent; Deborah number

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