Equivalent-continuum approach

The establishment of mathematical models (including the governing equations of coupled processes and constitutive models) and numerical methods for the solutions of practical problems attracted extensive attention in this area since mid 1990 s. Both equivalent continuum approach and discrete fracture system approach are used, with focus on coupled T-H, H-M, M-H-C and T-H-M processes of fractured rocks, soils and general porous media. Summarized below are some of the representative works ... 

Models and methods for coupled thermo-hydro-mechanical processes of fractured rocks, in saturated conditions using crack-tensor approach (Liu et al., 2002), the equivalent continuum approach and FEM solution technique with consideration for two-phase flow of water/vapour due to phase change by evaporation for fully or partially saturated cases (Li et al., 2000). 

In terms of numerical methods, the dominance by FEM with equivalent continuum approach might not be most suitable for sparsely or moderately fractured hard rocks and more advanced methods and codes using discrete approach are needed. The issue of applicability of the equivalent media approach, the associated scale effects, and uncertainty evaluations need to be fully explored. The processes are dominated by coupled stress-flow problems and effects of thermo-chemical effects need more attention. More works for soils, clays, sands and other similar media, which are equally, if not more, important in the fields of geo-engineering and environments, seem also needed. 

The Compliant Joint Model (CJM) was chosen for the mechanical calculations for this analysis. The CJM uses an equivalent continuum approach to model the behavior of jointed media. An equivalent continuum approach captures the average response of a jointed rock mass by distributing the response of the individual joints throughout the rock mass. The CJM in JAS3D can model up to four joint sets of arbitrary orientation, with the fractures in each set assumed to be parallel and evenly spaced. The intact rock between joints is treated as an isotropic linear-elastic material. More detailed descriptions of the CJM model can be found in Chen (1991). 

The direct use of micromechanical models for nanocomposites is however doubtfid due to the significant scale difference between nanoparticles and macro-partides. As such, two methods have recently been proposed for modeling the mechanical behavior of polymer nanocomposites equivalent continuum approach and self-similar approach. In equivalent continuum approach, molecular dynamics (MD) simulation is first used to model the molecular interaction between nanopartide and polymer. Then, a homogeneous equivalent continuum reinforcing element (i.e., an effective nanopartide) is constmcted. Finally, micro-mechanical models are used to determine the effective bulk properties of a... 

Mori-Tanaka model Kalpin-Tsai model Lattice-spring model Finite element method Equivalent continuum approach Seif-similar approach... 

Moving now to QM/continuum approaches, we shall limit our exposition to the so-called apparent surface charges (ASC) version of such approaches, and in particular to the family known with the acronym PCM (polarizable continuum model) [11], In this family of methods, the reaction potential Vcont defined in Eq. (1-2) has a form completely equivalent to the Hel part of the Z/qm/mm operator defined in Eq. (1-4), namely ... 

Comparing (6) with (1), we find that the conditions of equivalence between the (one-dimensional) continuum approach and the discrete approach are ... 

Because of the approach uses the energy terms that are associated with molecular mechanics modeling, a brief description of molecular mechanics is given first followed by an outline of the equivalent-truss and equivalent-continuum model development. 

Many traditional simulation techniques (e.g., MC, MD, BD, LB, Ginzburg-Landau theory, micromechanics and FEM) have been employed, and some novel simulation techniques (e.g., DPD, equivalent-continuum and self-similar approaches) have been developed to study polymer nanocomposites. These techniques indeed represent approaches at various time and length scales from molecular scale (e.g., atoms), to microscale (e.g., coarse-grains, particles. 

We now consider in somewhat more detail a simplified approach based on the curve of growth . For this, we ignore fine details of the observed line profile and use the equivalent width (EW) defined in Fig. 3.4, WA = f RdX or Wv = f Rdv, where R(AX) or R(Av) is the relative depression below the continuum at some part of the line. The curve of growth is a relationship between the equivalent width of a line and some measure of the effective number of absorbing atoms. Equivalent... 

Much like the RISM method, the LD approach is intermediate between a continuum model and an explicit model. In the limit of an infinite dipole density, the uniform continuum model is recovered, but with a density equivalent to, say, the density of water molecules in liquid water, some character of the explicit solvent is present as well, since the magnitude of the dipoles and their polarizability are chosen to mimic the particular solvent (Papazyan and Warshel 1997). Since the QM/MM interaction in this case is purely electrostatic, other non-bonded interaction terms must be included in order to compute, say, solvation free energies. When the same surface-tension approach as that used in many continuum models is adopted (Section 11.3.2), the resulting solvation free energies are as accurate as those from pure continuum models (Florian and Warshel 1997). Unlike atomistic models, however, the use of a fixed grid does not permit any real information about solvent structure to be obtained, and indeed the fixed grid introduces issues of how best to place the solute into the grid, where to draw the solute boundary, etc. These latter limitations have curtailed the application of the LD model. 

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