Cell Periodicity
The heterogeneous properties of a material at the microscopic scale are often impractical for direct application at the macroscopic scale, which is typically the focus of structural analysis. In such cases, it becomes necessary to employ a homogeneous material model that incorporates appropriately averaged properties.
The Cell Periodicity feature facilitates the evaluation of such average properties. It is based on the idea of a repeating unit cell (RUC) or a representative volume element (RVE). The cell is a microscopic domain that is representative for the material on a macroscopic length scale.
The terms RVE (Representative Volume Element) and RUC (Representative Unit Cell) are frequently used interchangeably in the academic literature. Generally, an RVE describes a subvolume that characterizes heterogeneous materials with a statistically homogeneous microstructure. Conversely, the RUC is employed to represent subvolumes characterized by periodic microstructure patterns.
When dealing with materials characterized by a random distribution of particles, fibers, or pores, the appropriate subvolume to consider would be an RVE. To be numerically efficient, this RVE should be as small as possible, but sufficiently large to provide a representative sample for the determination of average material properties at a macroscopic scale.
If the material or lattice exhibits true periodicity, the suitable choice for representing a material subvolume would be an RUC, which repeats itself to uniformly fill the space with a repetitive pattern.
The distinction between RVEs and RUCs necessitates the application of different sets of boundary conditions. The Cell Periodicity node offers two distinct sets of boundary conditions, namely Periodic and Homogeneous, designed to suit these subvolumes.
To model a microscopic structure, 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 cell, as long as the basic assumptions about homogenization 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.
Homogenization Method
The homogenization method introduces two scales: a macroscale and a microscale. The macroscale usually refers to the homogenized continuous medium, and the microscale to the heterogeneous cell. The macroscopic stress tensor and the macroscopic strain tensor are derived by averaging the stresses and strains in the cell
(3-181) and
where V is the volume of the cell. The macroscopic elasticity tensor of the homogenized continuum is then defined by
(3-182)
Periodic boundary conditions
For a periodic structure that consists of an array of RUC, the displacement field is written as
where is the global average strain tensor, X is the position, and u* is a function that is periodic from one cell to another. Given that the array of cells forms a continuous structure, it is essential to ensure continuity of displacements across the boundaries between these cells. The boundaries in a Boundary Pair subnode must therefore 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) boundaries can be written as
and
where Xdst and udst are the position and displacement on the destination side, and Xsrc and usrc are the position and displacement on the source side.
Hence, the displacement boundary condition is established by
(3-183)
The continuity of tractions should also be satisfied for RVEs together with the continuity of displacements. This means that periodic conditions for tractions are equivalent to the periodic conditions for displacements.
Homogeneous boundary conditions
For a statically homogeneous microstructure represented by an RVE, the displacement boundary condition is written as
where is a global average strain.
The displacements on a pair of parallel (and opposite) boundaries is written as
and
where Xdst and udst are the position and displacement on the destination side, and Xsrc and usrc are the position and displacement on the source side.
For a statistically homogeneous microstructures, RVE, the traction boundary condition is written as
where is a global average stress.
The tractions on a pair of parallel (and opposite) boundaries are written as
and
where Tdst and Tsrc are the tractions on the destination and source sides, respectively.
There are different options available to enforce the Homogeneous or Periodic boundary conditions:
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Free Expansion
•
Prescribed Average Strain
•
Prescribed Average Stress
•
Mixed
Free Expansion
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 3-183 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 3-183 as a constraint.
It is possible to apply either periodic or homogeneous boundary conditions.
Prescribed Average Strain
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 3-183 is a user input. The average stress is computed from Equation 3-181, and the elasticity tensor is obtained from Equation 3-182. It is possible to apply either periodic or homogeneous boundary conditions.
Prescribed Average Stress
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 3-183 is a global degree of freedom, which is defined by a global weak equation
(3-184)
The compliance tensor is then obtained from the following equation
It is possible to apply either periodic or homogeneous boundary conditions.
Mixed
This option studies the response of the unit cell when subjected to a combination of prescribed stress and strain. You can choose which components of the average stress and strain tensors to enter as user inputs; the remaining components are computed from Equation 3-181 and Equation 3-184 by enforcing Equation 3-183 as a constraint. It is only possible to apply periodic boundary conditions.