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

Abstract

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.

Homogenization of stratified thermoviscoplastic materials

“…Homogenization 1 Ten years ago, a new generation of heterogeneous materials, called functionally graded materials, appeared in the mechanical literature (see e.g. [1], [2], [3], [4] and [11]). These materials, characterized by high resistance to loading and/or to temperature increase, have been studied intensively, and numerical homogenization formulas have been proposed under the assumption of smoothly varying fields, which means that the different components of these materials are supposed to be perfectly bonded by thermomechanical processing in order to exhibit continuously changing properties.…”

Biaxial Loading of Continuously Graded Thermoviscoplastic Materials

“…Then, using (81), (76), (78), (11), (77), we obtain (4), (5), (6), where g is defined by (7). We assume that the material is non-homogeneous, in the sense that all material parameters depend on the coordinates x and y.…”

“…Recently, Bansal and Pindera [6,7] have shown that the "higher-order theory", introduced in [2,3] for functionally graded materials, is in fact a simple finite-volume direct averaging micromechanics approach. These materials are expected to show superior properties for high temperature resistance.…”

Stability by Homogenization of Thermoviscoplastic Problems

“…This method can of course be used to "tailor" (design) new materials with enhanced properties by averaging in a clever (but very heterogeneous) way simple phases. It has been often described (and sometimes revisited) in the mechanical literature and here again giving a complete list is impossible; let us mention Suquet (1982), Francfort, Leguillon and Suquet (1983), Francfort, Nguyen andSuquet (1983), Hashin (1983), Suquet (1983), Maugin (1992), Aboudi et al (1999), Ghosh et al (2001), Bansal and Pindera (2003), Michel and Suquet (2004), Alshirts and Maugin (2005), Bansal and Pindera (2005), Bardzokas and Zobnin (2005), Guinovart-Diaz et al (2005), Suquet (2005), Idiart et al (2006)). In particular, the concept of "homogeneous equivalent continuum" (Maugin (1992)) has been used to predict the macroscopic response from microscopic analyses by replacing the macroscopic heterogeneous medium, represented by an element called "representative volume element", by a continuum model (see Van der Sluis et al (1999), where a numerical homogenization technique has been proposed, which offers the possibility to determine the effective properties of a viscoplastic constitutive model).…”