Mechanics of deformable solids Definition

Representative elementary volume

Definition

Let a heterogeneous material have a microstructure with characteristic length $\ell$ , such as the size of its grains. Let $L$ be the length on which the body or the applied loads vary. For a region $V$ of the material and a field $g$ , write $\langle g \rangle_V = \frac{1}{\abs{V}} \int_V g \, dx$ . A region $V$ of diameter $D$ is a representative elementary volume (RVE) for a property $P$ when:

$\ell \ll D \ll L$ ;
the apparent value of $P$ computed on $V$ changes by less than a prescribed tolerance when $V$ is moved or enlarged;
this apparent value changes by less than the same tolerance when the affine, uniform-traction or periodic boundary conditions on $V$ are exchanged.

Remarks

Example: for a periodic composite, a single period cell with periodic boundary conditions gives the exact homogenized tensor of homogenization, so it is an RVE for every linear property. Non-example: for a polycrystal, a cube containing three grains violates clause 2, since its apparent stiffness depends strongly on the orientations of those three grains. For a random material the definition depends on the property and on the tolerance; a larger volume is usually needed for nonlinear properties than for the elastic moduli. For an elastic modulus, the apparent value is computed from the averages $\langle \sigma \rangle_V$ and $\langle \varepsilon \rangle_V$ . The Hill–Mandel condition links the averages of stress and strain on an RVE to the macroscopic energy.

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Homogenization concept
Representative elementary volume definition

Tag	`G0D4RWPCS8`
URI	`m2://G0D4RWPCS8`
Kind	definition
Writing	W0 stub
MSC	74Q05, 74A60, 74E30
Defines	Representative elementary volume
Uses	Homogenization
Changed	3 November 2026

Representative elementary volume

Definition

Remarks

Reading path

Further reading

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