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 that is representative for the material on a microscopic scale.

To model an RVE, you add a Cell Periodicity node, 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, since the possible rigid body motions are automatically constrained by the Cell Periodicity node.

where V is the volume of the cell. The macroscopic elasticity tensor of the homogenized continuum is then defined by

where is global average strain tensor, and u* is a function that is periodic from one unit cell to another. As the array of cells is a continuous structure, displacement continuity must be satisfied across the boundaries between the cells. The boundaries in a Boundary Pair subnode must always appear in parallel pairs. One of them is labeled as source and the other as destination. The displacements on a pair of parallel and opposite boundary surfaces can be written as

Together with displacement continuity, the traction continuity should be satisfied for the RVE, which is done implicitly in the displacement based finite element method. There are different options available to enforce the displacement continuity constraint:

In order to determine the homogeneous coefficient of thermal expansion of a heterogeneous material, the material is subjected to unit rise in temperature, while it is allowed to expand freely. To model this behavior, the global average strain tensor in Equation 3-108 is considered as a global degree of freedom that varies freely. The averaged coefficient of thermal expansion α is computed as

Similarly, to determine the homogeneous coefficient of hygroscopic swelling, the material is subjected to a unit rise in concentration. The averaged coefficient of hygroscopic swelling β is then computed as

If no averaged properties are computed, the free expansion option only computes the average stress and strain tensors by enforcing Equation 3-108 as a constraint.

In order to determine the homogeneous elasticity tensor of a heterogeneous material, the material is subjected to a unit strain in each direction, one-by-one, while keeping the other strain components equal to zero. To model this behavior, the global average strain tensor in Equation 3-108 is a user input. The average stress is computed from Equation 3-106, and the elasticity tensor is obtained from Equation 3-107.

In order to determine the homogeneous compliance tensor of a heterogeneous material, the material is subjected to a unit stress in each direction, one-by-one, while keeping the other stress components equal to zero. The average stress is a user input, and the global average strain tensor in Equation 3-108 is a global degree of freedom, which is defined by a global weak equation

This option studies the response of the unit cell when subjected to a combination of prescribed stress and strain. You can chose which components of the average stress and strain tensors to enter as user inputs; the remaining components are computed from Equation 3-106 and Equation 3-109 by enforcing Equation 3-108 as a constraint.