Periodic Cell Theory
The heterogeneous micromechanics material properties cannot be directly used in a macroscopic scale in which the structure is to be analyzed. There the need to use a homogeneous material model with appropriate average properties.
The Cell Periodicity feature facilitates the evaluation of such average properties. It is based on the idea of a representative volume element (RVE). The RVE is a domain which represents the material microscopically.
The RVE typically is the smallest possible unit cell. If the material has a random distribution of for example porosity, it should be large enough to be representative for the average properties of the material on a macro scale.
The only requirement on the shape of the RVE is that it should be possible to fill space with a repetitive pattern of RVEs. This means that there are a set of matching boundary pairs, each of them having the same geometry, but offset by a given distance.
To model an RVE, you add the feature Cell Periodicity, and select the domains representing the unit cell. For each pair of matching boundaries, add a Boundary Pair subnode, and select the boundaries.
In principle, there are no limitations on the physics features you can use for modeling the RVE, as long as the basic assumptions about periodicity are not violated. You should however not add any displacement constraints, because the possible rigid body motions are automatically constrained by the Cell Periodicity node.
Homogenization method
The homogenization method introduces two scales, a macro scale and a micro scale. Macro scales are usually referred as homogenized continuous media and the micro scale refers to the heterogeneous cell/RVE. The macro stress tensor and the macro strain tensor are derived by averaging the stresses and strains in the periodic cell
and
where V is the volume of the cell. The macroscopic elasticity matrix of the homogenized continuum is then defined from