Continuum behavior

Apart from obvious features such as laminarity, there are speculations that flows in micro channels exhibit a behavior deviating from predictions of macroscopic continuum theory. In the case of gas flows, these deviations, manifesting themselves as, e.g., velocity slip at solid surfaces, are comparatively well understood (for an overview, see [130]). However, for liquid flows on a length scale above 1 pm, there is no clear theoretical foundation for deviations from continuum behavior. Nevertheless, various unexpected phenomena such as friction factors deviating from the continuum prediction [131-133] have been reported. A more detailed discussion of this still unsettled matter is given in Section 2.2. At any rate, one has to be careful here since it may be that measurements in small systems lack precision, essentially because of the incompatibility of analysis in a confined space and with large measuring equipment... 

Reduction in Nusselt number is observed as the flow deviates from the continuum behavior, or as Kn takes higher values. 

A second difficulty that attends the use of continuum models is the fact that such models are indifferent to the presence of a minimum length scale as dictated by the typical separation between atoms. This absence of small scale resolution within the continuum framework can have a souring influence on the success of such theories in predicting features in the immediate vicinity of defects. One of our main objectives will be to show how, via the device of incorporating nonlinear information into our models, many of these difficulties may be sidestepped. As a result of the evident appeal of microscopic insights into the ultimate origins of continuum behavior, in the next chapter we undertake a review of the theoretical foundations of microscopic modeling, the partnership of quantum and statistical mechanics. 

The question has been discussed extensively in technical journals and in complicated ways. The stimulus for this book is the belief that the topic need not be so complicated. There are two equations that describe the stresses in the cylinder that have up to now not been used using these neglected equations provides a point of view not taken by other writers, and it is the fresh point of view that permits certain simplicities to be seen (the key equations are 6.3 and 8.10). Of course, we make headway only to a limited extent not all problems are answered, not all complications are resolved. The existence of a central and unresolvable complication is recognized toward the end of this overview, in the section on Continuum Behavior and Atoms. 

Figure 11.5 Homogeneous constriction, (a) A homogeneous deformation, with uniform radial shortening in yz planes and elongation parallel to x. (b) An imaginary wafer and a migration that would contribute to the overall deformation in (a). For continuum behavior, such wafers would need to be infinitely numerous and infinitely small, but the length of each migration path would remain finite, (c) Diagrams to suggest the concept of a dense swarm of such paths.

The description in terms of a substrate that self-diffuses plus components that interdiffuse permits a further distinction it is the substrate that has continuum properties, to which the reasoning in Chapter 11 applies, and for which we use eqn. (12.7) specifically along one direction or another. The interdiffusive effects, here mimicked by the motion of the additive a, do not resemble continuum behavior in isotropic materials, the additive a affects only the volume of a sample-element and cannot affect its shape the additive responds directly to the mean stress and produces only an isotropic change in mean strain. In the cylinder problem treated above, the symmetry and uniformity assumed are such that this distinction leaves the mathematical solution unchanged in form. But if a less regular physical situation were to be treated, the distinction between the behavior of BX and the behavior of the additive would have more noticeable consequences. 

As discussed in Chapter 15, the interdiffusion or exchange of atoms of A and B is not a continuum behavior it is driven strictly by variation in mean stress and results only in isotropic change of volume. From eqn. (13.5a) we have ... 

Equations 3.3 show the interesting result that the bed resistivity takes on a pseudo-continuum behavior for particles smaller than 65 pm. 

Most adhesives are polymer-based materials and exhibit viscoelastic behavior. Some adhesives are elastomer materials and also exhibit full or partial rubberlike properties. The word elastic refers to the ability of a material to return to its original dimensions when unloaded, and the term mer refers to the polymeric molecular makeup in the word elastomer. In cases where brittle material behavior prevails, and especially, when inherent material flaws such as cracks, voids, and disbonds exist in such materials, the use of the methods of fracture mechanics are called for. For continuum behavior, however, the use of damage models is considered appropriate in order to be able to model the progression of distributed and non-catastrophic failures and/or irreversible changes in material s microstructure, which are sometimes described as elastic Hmit, yield, plastic flow, stress whitening, and strain hardening. Many adhesive materials are composite materials due to the presence of secondary phases such as fillers and carriers. Consequently, accurate analysis and modeling of such composite adhesives require the use of the methods of composite materials. 

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