The purpose of this article is to present a simplified methodology for analysis of sandwich structures using the homogenization method. This methodology is based upon the strain energy criterion. Normally, sandwich structures are composed of hexagonal core and face sheets and a complete and complex hexagonal core is modeled for finite element (FE) structural analysis. In the present work, the hexagonal core is replaced by a simple equivalent volume for FE analysis. The properties of an equivalent volume were calculated by taking a single representative cell for the entire core structure and the analysis was performed to determine the effective elastic orthotropic modulus of the equivalent volume. Since each elemental cell of the hexagonal core repeats itself within the inplane direction, periodic boundary conditions were applied to the single cell to obtain the more realistic values of effective modulus. A sandwich beam was then modeled using determined effective properties. 3D FE analysis of Three and FourPoint Bend Tests (3PBT and 4PBT) for sandwich structures having an equivalent polypropylene honeycomb core and Glass Fiber Reinforced Plastic (GFRP) composite face sheets are performed in the present study. The authenticity of the proposed methodology has been verified by comparing the simulation results with the experimental bend test results on hexagonal core sandwich beams.
Applications of composite sandwich structures are continuously increasing in the recent times due to their excellent outofplane shear and compressive properties. Modern manufacturing aerospace, ship building, and automotive industries are the main users of sandwich structures. A typical sandwich structure consists of a central core material covered by top and bottom face sheets. Generally honeycomb, truss, and foam are used as the inner cores in sandwich structures. Nomex, Polypropylene, and Aluminium are regularly used core materials, whereas, Glass Fiber Reinforced Plastic (GFRP), Carbon Fiber Reinforced Plastic (CFRP), and Aluminium are commonly used face sheet materials.
Composite sandwiches are used as substantial structures in cars, ships, beams, and so forth. Complex and large hexagonal honeycomb core shapes are not only difficult to model but also are computationally expensive. Due to these limitations, an alternative strategy is desired in which a complex shaped core material may be replaced by a simple equivalent volume having elastic orthotropic properties.
Many authors attempted to determine the equivalent properties of central core by separately modeling the honeycomb core sheet with hexagonal unit cells [
Different authors like Kelsey et al. [
In the present study only a single hexagonal element with periodic boundary conditions has been modeled to determine the equivalent properties. This approach to determine the equivalent properties is based on the work of Gornet et al. [
This article is organized as follows: representative volume element (RVE) for honeycomb core material is defined and described in Section
Figure
Representative volume element (RVE).
Honeycomb core RVE.
By considering
One can express the average stress and strain tensors as
Here
The 21 components of the orthotropic compliance matrix
Equations (
In this section, FE analysis and results are presented and discussed in detail. Two different FE models were developed and analyzed. First, FEA of a single cell of hexagonal honeycomb representing the RVE is performed and the equivalent orthotropic properties are determined using the methodology explained in previous section. Then a sandwich beam is modeled and FEA (3PBT and 4PBT) is performed using the determined equivalent properties.
FEA to determine the equivalent properties was done using the commercially available software CAST3M [
The honeycomb core is modeled with threedimensional brick elements having 20 nodes (CU20) [
As adopted by Gornet et al. [
Comparison of FEA results with published [
Property  3D FEA results  Gibson and Ashby [ 
Chamis et al. [ 





Unit cell  

























0.81  0.99  0.42 

0.024  0.031  0.42 

0.028  0.0054  0.42 
Boundary conditions for five different loadings.
Von Mises stress profile for different loading conditions on hexagonal core.
Honeycomb cores are usually designed for outofplane normal (
In the present work, only one honeycomb core cell with actual double wall thickness is modeled and results are obtained by applying periodic displacement boundary conditions. The predicted outofplane elastic modulus
The sandwich structure usually consists of central core materials covered by the face sheets on both sides of the core. In the present study, two different types of analyses were performed, where the face sheets were modeled with the shell elements in the first and with the 3D solid elements in the second analysis. However, the core material is modeled by 3D solid elements in both the cases. The authenticity of the performed analysis has been established by comparing the FE results with the reported work [
The geometry and boundary conditions for 3PBT and 4PBT specimens are shown in Figures
Threepoint bend test (3PBT) specimen.
Fourpoint bend test (4PBT) specimen: (a) full length; (b) symmetric.
Figures
Force versus deflection curve (3PBT).
Force versus deflection curve (4PBT).
The above simulation results on 3PBT and 4PBT indicate that the adopted strategy to model the hexagonal honeycomb material by equivalent orthotropic material gives good results. Hence, instead of modeling a large number of hexagonal cells, one can determine the equivalent properties by modeling a single hexagonal cell as an RVE only.
Figures
Von Mises stress profile (3PBT).
Normal stress (
Similarly Figures
Von Mises stress profile (4PBT).
Normal stress (
Selected nodal path direction for 3PBT: (a) height; (b) width.
Selected nodal path direction for 4PBT: (a) height; (b) width.
Figure
Normal stress distribution along the height direction (3PBT).
Normal stress distribution along the width direction (3PBT).
Similarly, Figures
Normal stress distribution along the height direction (4PBT).
Normal stress distribution along the width direction (4PBT).
In this article, the strain energy based homogenization methodology is explained to determine the equivalent orthotropic properties of honeycomb sandwich structures. The obtained properties are further successfully employed to simulate the 3PBT and 4PBT. A 3D representative volume element (RVE) of one cell of honeycomb is selected to apply the homogenization technique rather than selecting a continuous structure having multiple cells, hence considerably reducing the computational time and effort. In the first part, homogenization is performed for honeycomb core material and determined orthotropic properties are compared with the properties based on available theories of different authors. In the second part, 3D FEA of 3PBT and 4PBT are performed based on the determined equivalent orthotropic properties. The bend test results for both analyses are compared with experimental data and are found in good agreement.
The constitutive equations representing the 3D stressstrain relation in terms of 6 × 6 compliance matrix,
The inverse relationship in terms of 6 × 6 stiffness matrix,
The HillMandel theorem equates the RVE energy to the average of the energies of its components:
Components of the 6 × 6 compliance matrix,
The orthotropic material parameters (
The authors declare that there are no conflicts of interest regarding the publication of this paper.