Representative elementary volume
In the theory of composite materials, the representative elementary volume 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, but in random media, the situation is much more complicated. For volumes smaller than the RVE, a representative property cannot be defined and the continuum description of the material involves Statistical Volume Element and random fields. The property of interest can include mechanical properties such as elastic moduli, hydrogeological properties, electromagnetic properties, thermal properties, and other averaged quantities that are used to describe physical systems.
Definition
defined the RVE as a sample of a heterogeneous material that:- "is entirely typical of the whole mixture on average", and
- "contains a sufficient number of inclusions for the apparent properties to be independent of the surface values of traction and displacement, so long as these values are macroscopically uniform."
Both of these are issues of mesoscale of the domain of random microstructure over which smoothing is being done relative to the microscale. As L/d goes to infinity, the RVE is obtained, while any finite mesoscale involves statistical scatter and, therefore, describes an SVE. With these considerations one obtains bounds on effective response of elastic linear and inelastic random microstructures. In general, the stronger the mismatch in material properties, or the stronger the departure from elastic behavior, the larger is the RVE. The finite-size scaling of elastic material properties from SVE to RVE can be grasped in compact forms with the help of scaling functions universally based on stretched exponentials. Considering that the SVE may be placed anywhere in the material domain, one arrives at a technique for characterization of continuum random fields.
Another definition of the RVE was proposed by Drugan and Willis:
- "It is the smallest material volume element of the composite for which the usual spatially constant macroscopic constitutive representation is a sufficiently accurate model to represent mean constitutive response."
Examples
RVEs for mechanical properties
In continuum mechanics generally for a heterogeneous material, RVE can be considered as a volume V that represents a composite statistically, i.e., volume that effectively includes a sampling of all microstructural heterogeneities that occur in the composite. It must however remain small enough to be considered as a volume element of continuum mechanics. Several types of boundary conditions can be prescribed on V to impose a given mean strain or mean stress to the material element.One of the tools available to calculate the elastic properties of an RVE is the use of the open-source EasyPBC ABAQUS plugin tool.
Analytical or numerical micromechanical analysis of fiber reinforced composites involves the study of a representative volume element. Although fibers are distributed randomly in real composites, many micromechanical models assume periodic arrangement of fibers from which RVE can be isolated in a straightforward manner. The RVE has the same elastic constants and fiber volume fraction as the composite. In general RVE can be considered same as a differential element with a large number of crystals.
RVEs for porous media
Establishing a given porous medium's properties requires measuring samples of the porous medium. If the sample is too small, the readings tend to oscillate. With increasing sample size, the oscillations begin to dampen out. Eventually the sample size will become large enough that readings are consistent. This sample size is referred to as the representative elementary volume.If sample size is increased further, measurement will remain stable until the sample size gets large enough that it begins to include other hydrostratigraphic layers. This is referred to as the maximum elementary volume.
Groundwater flow equation has to be defined in an REV.