The in-plane elastic-plastic response of an incompressible, filled hexagonal honeycomb

Abstract

Exact solutions are derived for the small-strain, in-plane, elasto-plastic response of a hexagonal honeycomb using slender beam theory; incompressibility of the honeycomb is enforced by filling its voids with an incompressible, inviscid fluid. The honeycomb has sides of equal length, but its inclined struts subtend an angle that can deviate from 120° with respect to the vertical side walls. The relative density is sufficiently small that the struts are slender and can be treated as Euler-Bernoulli beams. Exact solutions are obtained for the elastic moduli and macroscopic yield surface of the rigid, ideally plastic lattice under general in-plane loading: the solutions satisfy equilibrium, compatibility and the constitutive response of each elastic, ideally plastic beam. Prior to conducting an elastic analysis, and a rigid, ideally plastic analysis, initial insight is gained by exploring the vector space of inextensional collapse mechanisms of the pin-jointed, compressible version of the hexagonal truss. Two inextensional collapse mechanisms of the compressible honeycomb are identified from the null space of the kinematic matrix. The presence of an incompressible, inviscid fluid in the voids of the honeycomb locks-up one mechanism but the other mechanism survives and generates macroscopic shear strain. Consequently, the incompressible hexagonal honeycomb with rigid joints has a high shear compliance and a low shear strength, with values equal to that of the unfilled, compressible honeycomb. In contrast, macroscopic tensile straining of the incompressible honeycomb requires the stretching of bars in addition to bar-bending, and the tensile modulus and strength of the incompressible honeycomb are thereby elevated. Explicit analytical formulae are derived for the macroscopic tensile modulus and strength of the incompressible honeycomb.