Voigt and Reuss bounds

Before this page

Homogenization schemes, where the concentration tensors and the average relations they satisfy are set out.

The two bounds use no shape, no reference medium and no Eshelby problem: they assume a uniform field throughout the RVE, the strain for one and the stress for the other. That crude assumption is what makes them rigorous, since a uniform field is admissible in the corresponding energy principle (Hill, 1952).

1. Voigt: a uniform strain

Every phase is assumed to undergo the macroscopic strain,  , which is exact for bars loaded in parallel. The average relation of §2 of the overview then gives the arithmetic mean

Its bound property follows from the principle of minimum potential energy. Under the affine boundary condition   , the effective energy is the minimum of    over the kinematically admissible fields, and the uniform field   is one of them:

2. Reuss: a uniform stress

Every phase is assumed to carry the macroscopic stress, which is exact for layers loaded in series. The compliances then average,

and the principle of minimum complementary energy, applied to the statically admissible field  , gives the dual inequality      . Together,

in the sense of quadratic forms, whatever the microstructure. A phase without stiffness, a void, makes vanish: the lower bound is then uninformative, which is the price of using no geometry at all.

3. Cracks, conduction, laminates

A crack family has no volume, so it enters neither mean: both bounds ignore cracks, which is why Voigt and Reuss refuse an RVE holding cracks alone. In conduction the same arguments give    and   .

A laminate attains both, each on its own block: across the layers, the inverse of the acoustic tensor    is the Reuss mean of the layers' inverses, and in their plane, the Schur complement of is the Voigt mean of the layers' complements (Laminate, §6).

4. Concentration tensors

The two assumptions are statements on the concentration tensors themselves, in the sense of the overview. Under Voigt every phase undergoes the macroscopic strain, so that   and  ; under Reuss every phase carries the macroscopic stress,  , so that   and   . The two averages then return and . Neither bound singles out a matrix, so that a contribution must be given an explicit reference, and crack families are absent, as they are from the bounds.

Where to go next

The bounds know nothing of shapes. The next page, Dilute estimates, introduces the Eshelby problem into the estimate, with each inclusion alone in the matrix, under either the remote strain or the remote stress. The bounds are computed on a first RVE in the tutorial Bounds and classical schemes.