# Why do the concepts representative volume element and characteristic scale come apart?

Does every characteristic scale have an associated representative volume element? Are they unrelated concepts?

They are not unrelated concepts, because both are used to describe complex materials.

In physics, a characteristic length is an important dimension that defines the scale of a physical system. Often, such a length is used as an input to a formula in order to predict some characteristics of the system.

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In computational mechanics, a characteristic length is defined to force localization of a stress softening constitutive equation. The length is associated with an integration point. For 2D analysis, it is calculated by taking square root of the area. For 3D analysis, it is calculated by taking cubic root of the volume associated to the integration point

Length is one dimension, volume is three, and one can define a volume with three orthogonal lengths, but the word 'characteristic" defines a physical property whose manifestation depends on this length. The examples make this clear.

The Reynolds number (Re) is an important dimensionless quantity in fluid mechanics used to help predict flow patterns in different fluid flow situations.

The dimensionless numbers depend on the characteristic length, if one checks further in the links.

In the theory of composite materials, the representative elementary volume (REV) (also called the representative volume element (RVE) or the unit cell) is the smallest volume over which a measurement can be made that will yield a value representative of the whole. In the case of periodic materials, one simply chooses a periodic unit cell (which, however, may be non-unique), but in random media, the situation is much more complicated.

It defines where the continuum assumed for the material at hand for calculations breaks down to individual atoms and molecules .

They are not unrelated concepts, as both deal with the complexity of the break down of a material to its constituent atoms and molecules, but not directly derivable from each other, as each is used in a different context, in the examples above. In searches one finds specific models for specific materials relating the two concepts, not general statements. See the introduction in this paper.