Multiscale homogenization

Abstract We here treat a diffusion problem coupled with water flow in bentonite. The remarkable behavior originates from molecular characteristics of its constituent clay mineral, namely montmorillonite, and we show the behavior based on a unified simulation procedure starting with the molecular dynamic (MD) method and extending the obtained local characteristics to a macroscale behavior by the multiscale homogenization analysis (HA Sanchez-Palencia. 1980). Not only the macroscale effective diffusion property but also the adsorption behavior is well defined based on this method. 

For characterizing the microstructure we use a confocal laser scanning microscope (CLSM). By CLSM we can specify a 3-D configuration under atmospheric condition. Smectite minerals are extremely fine and poorly crystallized, so it is difficult to determine the properties by experiment. We inquire into the physicochemical properties by a molecular dynamics (MD) simulation method. Then, we develop a multiscale homogenization analysis (HA) method to extend the microscopic characteristics to the macroscopic behavior. We show numerical examples of a coupled water-flow and diffusion problem. 

We here showed that for bentonite clay, we can determine the nano-scale material properties such as diffusion coefficient and viscosity by molecular dynamics (MD) simulation and extend the microscale characteristics to the macroscale behavior by the multiscale homogenization analysis (HA) method. A seepage flow and diffusion problem is treated. The micro/macro problem can be simulated well by this procedure if we know the microscale geometrical characteristics. 

We outline the essential features of a multiscale homogenization analysis. A problem of a one-dimensional elastic bar is given as an example. 

Ichikawa Y, Kawamura K, Fujii N, Theramast N (2002) Molecular dynamics and multiscale homogenization analysis of seepage/diffusion problem in bentonite clay. Int J Numerical Methods Eng 55 1 10... 

Ichikawa Y, Kawamura K, Theramast N, Kitayama K (2(X)4) Secondary and tertial consolidation of bentonite clay consolidation test, molecular dynamics simulation and multiscale homogenization analysis. Mech Mater 36 487-513... 

Luo et al. [80] have used multiscale homogenization (MH) and FEM for wavy and straight SWCNTs, they have compare their results with Mori-Tanaka, Cox, Halpin-Tsai, Fu, et al., [81] Lauke [82]. Trespass et al. [83] used 3D elastic beam for C-C bond and 3D space frame for CNT and progressive fracture model for prediction of elastic modulus, they used rule of mixture for compression of their results. Their assumption was embedded a single SWCNT in polymer with perfect bonding. The multiscale modeling, MC, FEM and equivalent-continuum method was used by Spanos and Kontsos [84] and compared with the results of Zhu et al. [85] and Paiva et al. [86]... 

Hughes, T. J. R., L. Mazzei, A. A. Oberai, and A. A. Wray (2001b). The multiscale formulation of large eddy simulation Decay of homogeneous isotropic turbulence. 

Concurrent multiscale methods have also been employed to address fatigue. Oskay and Fish [82] and Fish and Oskay [83] introduced a nonlocal temporal multiscale model for fatigue based upon homogenization theory. Although these formulations were focused on metals, Fish and Yu [84] and Gal et al. [85] used a similar concurrent multiscale method for analyzing fatigue of composite materials. 

Next to metals, probably the synthetic polymer-based composites have been modeled most by hierarchical multiscale methods. Different multiscale formulations have been approached top-down internal state variable approaches, self-consistent (or homogenization) theories, and nanoscale quantum-molecular scale methods. 

Q. Yu, J. Multiscale asymptotic homogenization for multiphysics problems with multiple spatial and temporal scales a coupled thermo-viscoelastic example problem. Int. J. Solids Struct. 39, 6429-6452 (2002)... 

Berthon-Fabry et al. have studied multiscale structure formation as a function of the concentration of monomers and of acid. The % mass ratio (percent of monomers to the total mass of the sol) was used as a controlling parameter for the effect of monomer concentration. Formulations with % mass ratio >35 and relatively low acid catalyst concentration (R/C > 50) gave homogenous monoliths. With a higher monomer % mass ratio (>55) and acid catalyst concentrations RjC < 1), a very viscous resin-like material was obtained. Varying the concentration of monomers and acid is an effective method to produce carbon aerogels with different morphologies [11, 37]. 

---