Fracture toughness of two-dimensional cellular material with periodic microstructure

The brittle fracture behavior of periodic 2D cellular material weakened by a system of non-interacting cracks is investigated. The material is represented as a lattice consisting of rigidly connected Euler beams which can fail when the skin stress approaches some limiting value. The conventional Mode I and Mode II fracture toughness is calculated first and its dependence upon the relative density is examined. To this end the problem of a sufficiently long finite length crack in an infinite lattice produced by several broken beams is considered. It is solved analytically by means of the discrete Fourier transform reducing the initial problem for unbounded domain to the analysis of a finite repetitive module in the transform space. Four different layouts are considered: kagome, triangular, square and hexagon honeycombs. The results are obtained for different crack types dictated by the microstructure symmetry of the specific material. The obtained results allowed to define the directional fracture toughness characterizing the strength of a material with many cracks for the given tensile loading direction. This quantity is presented in the form of polar diagrams. For all considered layouts the diagrams are found to be close to circles thus emphasizing quasi-isotropic fracture behavior. The deviation from isotropy in the case of a square honeycomb is essentially less than for the corresponding published axial stiffness polar diagram.