Nucleation of cracks in two-dimensional periodic cellular materials

Abstract

This paper describes the nucleation and propagation of cracks in brittle cellular material. Four basic patterns with triangular, square, hexagonal and kagome-type cells are considered. The cracks propagate by sequential failure of critical elements. The analysis technique hinges on the combined use of the structural variation method and the representative cell method. While the latter allows for the analysis of periodic structures under arbitrary loads, by means of the discrete Fourier transform, the former analyzes modified structures (the cracked lattices) on the basis of analysis of the pristine structure (the periodic lattices). Within the assumptions of Bernoulli-Euler beam theory the suggested method for the analysis of infinite cracked lattices is exact. Although most cracks follow intuitive paths it was found that the microstructure of cellular materials has a significant influence on the crack pattern.

Fracture toughness of two-dimensional cellular material with periodic microstructure

“…This solution, proportional to the remote loading σ r , can be found by implying the symmetry conditions at the boundary nodes of the cell. The interested reader is referred to Lipperman et al (2007) where the results for all the lattice types considered herein are presented. Using the expression for the displacements (6) it is easy to calculate the internal forces in all elements of the lattice in terms of coefficients C i .…”

“…As noted, the method employed here for evaluating the fracture toughness follows the technique presented in detail in Lipperman et al (2007). The material is viewed as an assemblage of identical parallelogram cells defined by the two vectors of the lattice translational symmetry.…”

“…The evaluation of the fracture toughness for individual cracks is carried out by a novel analytical method employed by Lipperman et al (2007) for the analysis of crack nucleation. It involves the application of the discrete Fourier transform and enables to determine the stress state of an unbounded infinite lattice with an embedded finite-length crack.…”

Design of crack-resistant two-dimensional periodic cellular materials

“…The location of the most loaded elements (Figure 9) may be counterintuitive however an earlier work on crack nucleation and propagation has shown that this type of crack under same loads at some early stage of the crack formation changed direction from the path perpendicular to the loading to a direction inclined by 60 • [Lipperman et al 2007b]. The most loaded elements at the crack-tip are probably the harbingers of the kink in the crack path.…”

“…Clearly, the crack length used in the simulations must be large enough such as to produce self-similar K -fields close to the crack tips. The issue of possible crack types was investigated by Lipperman et al [2007b], who found numerically that they are parallel to the microstructure symmetry planes. It should be noted that the optimization of the fracture properties was carried out for a specific loading direction.…”