Application of homogenization FEM analysis to regular and re-entrant honeycomb structures

Honeycomb structures are widely used in structural applications because of their high strength per density. Re-entrant honeycomb structures with negative Poisson's ratios may be envisaged to have many potential applications. In this study, an homogenization finite element method (FEM) technique developed for the analysis of spatially periodic materials is applied for the analysis of linear elastic responses of the regular and re-entrant honeycomb structures. Young's modulus of the regular honeycomb increased with volume fraction. Poisson's ratio of the regular honeycomb structure decreased from unity as volume fraction increased. The re-entrant honeycomb structure had a negative Poisson's ratio, its value dependent upon the inverted angle of cell ribs. Young's modulus of the re-entrant honeycomb structure decreased as the inverted angle of cell ribs increased. The results are in good agreement with previous analytical results. This homogenization theory is also applicable to three-dimensional foam materials — conventional and re-entrant.Nomenclature b _i Body force - E, E _ijkl Young's modulus, elasticity tensor - E _e Effective Young's modulus - E _ijkl ^H Homogenized elasticity tensor - t _i Traction - u _i , u Displacement - v _i , v Virtual displacement - x _i , x Macroscale coordinate - y _i , y microscale coordinate - Microscopic/macroscopic ratio - Volume fraction - v Poisson's ratio - v_e Effective Poisson's ratio - _ij Stress - _P ^KL Microscale parameter of separation of variables