A Second Look at the Higher-Order Theory for Periodic Multiphase Materials

In this communication, we present a reformulation, based on the local/global stiffness matrix approach, of the recently developed higher-order theory for periodic multiphase materials, Aboudi et al. [“Linear Thermoelastic Higher-Order Theory for Periodic Multiphase Materials,” J. Appl. Mech., 68(5), pp. 697–707]. This reformulation reveals that the higher-order theory employs an approximate, and standard, elasticity approach to the solution of the unit cell problem of periodic multiphase materials based on direct volume-averaging of the local field equations and satisfaction of the local continuity conditions in a surface-averge sense. This contrasts with the original formulation in which different moments of the local equilibrium equations were employed, suggesting that the theory is a variant of a micropolar, continuum-based model. The reformulation simplifies the derivation of the global system of equations governing the unit cell response, whose size is substantially reduced through elimination of redundant continuity equations employed in the original formulation, allowing one to test the theory’s predictive capability in most demanding situations. Herein, we do so by estimating the elastic moduli of periodic composites characterized by repeating unit cells obtained by rotation of an infinite square fiber array through an angle about the fiber axis. Such unit cells possess no planes of material symmetry in the rotated coordinate system, and may contain a few or many fibers, depending on the rotation angle, which the reformulated theory can easily accommodate. The excellent agreement with the corresponding results obtained from the standard transformation equations confirms the new model’s previously untested predictive capability for a class of periodic composites characterized by nonstandard, multi-inclusion repeating unit cells lacking planes of material symmetry. Comparison of the effective moduli and local stress fields with the corresponding results obtained from the original Generalized Method of Cells, which the higher-order theory supersedes, confirms the need for this new model, and dramatically highlights the original model’s shortcomings for a certain class of unidirectional composites.