The Imperfection Sensitivity of Isotropic Two-Dimensional Elastic Lattices

Abstract

The imperfection sensitivity of the effective elastic properties is numerically explored for three planar isotropic lattices: fully triangulated, the Kagome grid, and the hexagonal honeycomb. Each lattice comprises rigid-jointed, elastic Euler–Bernoulli beams, which can both stretch and bend. The imperfections are in the form of missing bars, misplaced nodes, and wavy cell walls. Their effect on the macroscopic bulk and shear moduli is numerically investigated by considering a unit cell containing randomly distributed imperfections, and with periodic boundary conditions imposed. The triangulated and Kagome lattices have sufficiently high nodal connectivities that they are stiff, stretching dominated structures in their perfect state. In contrast, the perfect hexagonal honeycomb, with a low nodal connectivity of 3, is stretching dominated under pure hydrostatic loading but is bending dominated when the loading involves a deviatoric component. The high connectivity of the triangulated lattice confers imperfection insensitivity: Its stiffness is relatively insensitive to missing bars or to dispersed nodal positions. In contrast, the moduli of the Kagome lattice are degraded by these imperfections. The bulk modulus of the hexagonal lattice is extremely sensitive to imperfections, whereas the shear modulus is almost unaffected. At any given value of relative density and level of imperfection (in the form of missing bars or dispersed nodal positions), the Kagome lattice has a stiffness intermediate between that of the triangulated lattice and the hexagonal honeycomb. It is argued that the imperfections within the Kagome lattice switch the deformation mode from stretching to a combination of stretching and bending. Cell-wall waviness degrades the moduli of all three lattices where the behavior of the perfect structure is stretching dominated. Since the shear response of the perfect hexagonal honeycomb is by bar bending, the introduction of bar waviness has a negligible effect on the effective shear modulus.