Cell Periodicity
Use the Cell Periodicity node to model a unit cell (representative volume element; RVE) of a larger repetitive structure. Periodic boundary conditions will be used on the outer boundaries of this unit cell.
From the Cell Periodicity node, is possible to generate the elasticity matrix for the equivalent homogenized material.
If more than one Cell Periodicity node is used, they must have disjoint selections. It is thus possible to evaluate more than one RVE in the same study.
The Cell Periodicity node is only available with some COMSOL products (see https://www.comsol.com/products/specifications/). Cell Periodicity is available for 3D and 2D.
Periodicity Type
Select a Periodicity type — Free expansion, Average strain, Average stress, or Mixed.
With the Free expansion periodicity type, the unit cell is allowed to expand freely in a periodic manner. This option is useful to determine the thermal expansion or hygroscopic swelling coefficients of a heterogeneous unit cell.
Select Calculate average properties — None, Coefficient of thermal expansion, or Coefficient of hygroscopic swelling.
In 2D, use either a Plane stress or a Generalized plane strain approximation to calculate averaged properties with the Free expansion periodicity type.
The Average strain periodicity type makes it possible to derive homogenized elastic properties of media with periodic structures, such as a perforated plates, porous media or composites structures. In this case, you have to examine six load cases in 3D. Enter values or expressions for the components of the Average strain tensor εavg. In a geometrically nonlinear analysis, the average strains are interpreted as Green–Lagrange strains.
Select Calculate average properties — None, Elasticity matrix, Standard (XX, YY, ZZ, XY, YZ, XZ), or Elasticity matrix, Voigt (XX, YY, ZZ, YZ, XZ, XY).
The Average stress periodicity type makes it possible to derive the homogenized compliance matrix of media with periodic structures. Enter values or expressions for the components of the Average stress tensor σavg. In a geometrically nonlinear analysis, the stresses are interpreted as Second Piola–Kirchhoff stresses.
Select Calculate average properties — None, Compliance matrix, Standard (XX, YY, ZZ, XY, YZ, XZ), or Compliance matrix, Voigt (XX, YY, ZZ, YZ, XZ, XY).
In 2D, use a Generalized plane strain approximation to calculate averaged properties with the Average strain or Average stress periodicity types.
With the Mixed periodicity type, enter either the average strain or average stress tensor components.
When using a Plane stress or a Plane strain 2D approximation with the Mixed periodicity type, you only need to enter three components (XX, YY, XY) of the average strain or average stress tensors. For the Generalized plane strain approximation, you also need to enter the ZZ component.
For all values of Periodicity type, select the Cell volume — Solid, Void volume fraction, or User defined.
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When Solid is selected, the volume of the RVE is computed from the domains selected in the Cell Periodicity node.
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The Void volume fraction option can be used to scale the RVE volume when there are voids inside the RVE that are not selected as domains. Enter the void volume fraction, f. This is the fraction of the total RVE that is occupied by voids. The volume of the RVE is computed as
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By using the User defined option, you can enter the volume of the RVE, V, explicitly.
The variable <item>.vol contains the volume of the RVE used when computing average strains and stresses.
Load Group, Material and Study Generation
You can create the elasticity matrix for a homogenized material, and make it accessible as a material to be used in other components. In order to do this, a number of fundamental load cases must be analyzed. The necessary definitions and steps required for such an analysis can be automated as described below.
On the Periodicity Type section toolbar, there is an icon Study and Material Generation (). It has a list with three entries:
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Create Load Groups and Study . This option can be selected when Periodicity Type is Average strain or Average stress. When you select it, the following changes will be made to the model:
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A number of load groups will be created under Global Definitions. They are collected in a group named Load Groups for Cell Periodicity. The load groups correspond to unit loads along different axes.
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The Average strain or Average stress tensor in the Periodicity Type section will be populated using the load group variables.
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A new study, named Cell Periodicity Study, will be created. This study contains a stationary study step. In the Study Extensions section of the new Cell Periodicity Study, one load case is added for each load group.
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Create Material by Reference (). This option can be selected when Periodicity Type is Average strain or Average stress, and Calculate average properties not set to None. When you select it, a new material will be created under Global>Materials. It contains the elasticity matrix or the compliance matrix. The name of this material is Homogeneous Material.
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Create Material by Value (). This option is similar to Create Material by Reference, with the difference that the material properties are numerical values, and not references to variables in the node. The benefits of this option is faster computation when the material is used by other physics interfaces. Also, if the Cell Periodicity node is deleted, the material created from it can still be used in sequential computations.
For this type of analysis to work correctly, it is important that you do not edit the generated nodes manually. By clicking the Create button again, you can reset all settings in the generated nodes to their default values.
See also
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Load Cases and Effective Properties of Periodic Structures in the Structural Mechanics Modeling chapter.
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Periodic Cell Theory in the Structural Mechanics Theory chapter.
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Materials and Solid Mechanics Material Properties in the Materials chapter in the COMSOL Multiphysics Reference Manual.
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Micromechanical Model of a Fiber Composite: Application Library path Structural_Mechanics_Module/Material_Models/micromechanical_model_of_a_fiber_composite
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Micromechanics and Stress Analysis of a Composite Cylinder: Application Library path Composite_Materials_Module/Tutorials/composite_cylinder_micromechanics_and_stress_analysis
Advanced
To display this section, click the Show More Options button () and select Advanced Physics Options in the Show More Options dialog box. Furthermore, this section is shown only if elasticity or compliance matrices are requested through Calculate average properties.
In that case, you can modify the behavior of how the studies for computing the homogenized matrices are generated by selecting a Study tag — Automatic or Legacy. The Legacy option is mainly for maintaining compatibility with versions prior to 6.0. With the default Automatic option, it is, for example, possible to set up nested homogenizations. A unique tag for the study will be composed by combining the physics interface tag and the tag of the Cell Periodicity node, for example solid1cp1std.
Constraint Settings
To display this section, click the Show More Options button () and select Advanced Physics Options in the Show More Options dialog box.
In the COMSOL Multiphysics Reference Manual:
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Constraint Reaction Terms
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Weak Constraints
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Constraint Settings
Location in User Interface
Context Menus
Solid Mechanics>More>Cell Periodicity
Ribbon
Physics tab with Solid Mechanics selected:
Domains>More>Cell Periodicity