Specification of energy-based criterion of elastic limit states for cellular materials

The aim of the paper is to apply the energy-based criterion of limit states in anisotropic elastic solids proposed by Rychlewski [5] for prediction of elastic limit states in cellular materials. The analysis is based on elastic model of a skeleton and an idealized description of topological arrangement of cell structure for cellular materials. The considered unit cells have, respectively, the form of a cube, a cuboid, a simple prism with the base of equilateral triangle, and a simple prism with the base in the form of regular hexagon. The morphology of the skeleton in a particular unit cell modeled by means of the struts joined in a rigid node determines the elastic stiffness and its symmetry: cubic symmetry, orthotropy and transversal symmetry. An analytical formulation of force-displacement relations for the skeleton struts is found by considering the affinity of node displacements in tensile, bending, and shear deformation. The elements of the stiffness matrix for a single cell are expressed as functions of the compliance coefficients for stretching and bending of struts. The analytical formulae for the elastic Kelvin moduli and the critical energy densities as well as the graphical presentation of the results were obtained with application of symbolic operations provided by Mathcad program. The distributions of critical energy density of particular elastic eigen states with respect to the change of the stiffness of the skeleton were studied.