Continuum methods solids

In literature, some researchers regarded that the continuum mechanic ceases to be valid to describe the lubrication behavior when clearance decreases down to such a limit. Reasons cited for the inadequacy of continuum methods applied to the lubrication confined between two solid walls in relative motion are that the problem is so complex that any theoretical approach is doomed to failure, and that the film is so thin, being inherently of molecular scale, that modeling the material as a continuum ceases to be valid. Due to the molecular orientation, the lubricant has an underlying microstructure. They turned to molecular dynamic simulation for help, from which macroscopic flow equations are drawn. This is also validated through molecular dynamic simulation by Hu et al. [6,7] and Mark et al. [8]. To date, experimental research had "got a little too far forward on its skis however, theoretical approaches have not had such rosy prospects as the experimental ones have. Theoretical modeling of the lubrication features associated with TFL is then urgently necessary. 

Therefore, when Jenike developed his methods to mathematically model the flow of bulk solids, he concluded that a bulk solid must be modeled as a plastic, and not a visco-elastic, continuum of solid particles (1). This approach included the postulation of a flow-no-flow criterion that states the bulk solid would flow from a bin when the stresses applied to the bulk solid exceed the strength of the bulk solid. The terms stress and strength are further discussed in this section on cohesive strength tests below. The flow properties test methods discussed are used to obtain the equipment parameters required to provide consistent, reliable flow. 

Macroscopic phenomena are described by systems of integro-partial differential algebraic equations (IPDAEs) that are simulated by continuum methods such as finite difference, finite volume and finite element methods ([65] and references dted therein [66, 67]). The commonality of these methods is their use of a mesh or grid over the spatial dimensions [68-71]. Such methods form the basis of many common software packages such as Fluent for simulating fluid dynamics and ABAQUS for simulating solid mechanics problems. 

ZoandaAy lubAd.catd.on is a familiar term in the vocabulary of the tribologist. In the general concept of the boundary lubricated condition, the lubricant film between the two surfaces is no longer a liquid layer instead the surfaces are separated by films of only molecular dimensions. Friction is influenced by the nature of the underlying surface and by the chemical constitution of the lubricant films. This view of lubricating action at the solid surface was introduced by Sir W. B. Hardy [1] as an extension of Osborne Reynolds concept that hydrodynamic action within the liquid film is a process treated by continuum methods which are not applicable at the discontinuity or "boundary" between liquid and solid. 

R. Lakes (1995). Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. In Continuum Methods for Materials with Microstructures (Ed. H. Muhlhaus), pp. 1-25. John Wiley Chichester. 

For the reasons given above, a number of authors [23-28] have applied MD to shock wave studies in an effort to obtain details of the various shock compression processes that are not easily available from the conventional continuum method. We have carried out calculations of the shock compression of one-, two- and three-dimensional systems in both solid and liquid phases [29,30], using essentially the same model as in Fig. 1. Here I shall first summarize the general features of the shock profile from our studies, then I shall discuss one representative case, with special reference to the thermal relaxation problem, as an illustration of some of the general results. 

However, the length and time scales that molecular-based simulations can probe are still very limited (tens of nanosecond and a few nanometers), due to computer memory and CPU power limitations. On the other hand, nanoscale flows are often a part of larger scale devices that could contain both nanochannels and microfluidic domains. The dynamics of these systems depends on the intimate connection of different scales from nanoscale to microscale and beyond. MD simulation cannot simulate the whole systems due to its prohibitive computational cost, whereas continuum Navier-Stokes simulation cannot elucidate the details in the small scales. These limitations and the practical needs arising from the study of multiscale problems have motivated research on multiscale (or hybrid) simulation techniques that bridge a wider range of time and length scales with the minimum loss of information. A hybrid molecular-continuum scheme can make such multiscale computation feasible. A molecular-based method, such as MD for liquid or DSMC for gas, is used to describe the molecular details within the desired, localized subdomain of the large system. A continuum method, such as finite element or finite volume based Navier-Stokes/Stokes simulation, is used to describe the continuum flow in the remainder of the system Such hybrid method can be applied to solve the multiscale phenomena in gas, liquid, or solid. 

As noted before, thin film lubrication (TFL) is a transition lubrication state between the elastohydrodynamic lubrication (EHL) and the boundary lubrication (BL). It is widely accepted that in addition to piezo-viscous effect and solid elastic deformation, EHL is featured with viscous fluid films and it is based upon a continuum mechanism. Boundary lubrication, however, featured with adsorption films, is either due to physisorption or chemisorption, and it is based on surface physical/chemical properties [14]. It will be of great importance to bridge the gap between EHL and BL regarding the work mechanism and study methods, by considering TFL as a specihc lubrication state. In TFL modeling, the microstructure of the fluids and the surface effects are two major factors to be taken into consideration. 

Using as the background continuum the short-lived spontaneous fluorescence of rhodamine B or 6 G, McLaren and Stoicheff 233) developed this method further to obtain inverse Raman spectra over the range of frequency shifts 300-3500 cm" in liquids and solids in a time of 40 nsec The stimulating monochromatic radiation at 6940 A is provided by a giant-pulse ruby laser. A small part of the main laser beam is frequency-doubled in a KDP-crystal and serves to excite the rhodamine fluorescence, thus ensuring simultaneous irradiation of the sample by both beams. 

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