The influence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models

Of interest here is the scale size effect on the stability of finitely strained, rate-independent solids with periodic microstructures. Using a multiple scales asymptotic technique, we express the critical load at the onset of the first instability and the corresponding eigenmode in terms of the scale size parameter epsilon. The zeroth order epsilon terms in these expansions depend on the standard (first order gradient) macroscopic moduli tensor, while all the higher order epsilon terms require the determination of higher order gradient macroscopic moduli. These macroscopic moduli, which are calculated by solving appropriate boundary value problems on the unit cell, relate the macroscopic (unit cell average) stress rate increment to the macroscopic displacement rate gradients. The proposed general theory is subsequently applied to the investigation of the failure surfaces in periodic solids of infinite extent. For these solids one can define in macroscopic strain space a microscopic (local) failure surface, which corresponds to the onset of the first bulking-type instability in the solid, and a macroscopic (global) failure surface, which corresponds to the onset of the first long wavelength instability in the solid. The determination of the macrofailure surface is considerably easier than the determination of the microfailure surface, for it requires the calculation of the standard macroscopic moduli tenser. In addition, the regions where the two surfaces coincide is of significant practical interest, for a macroscopic localized mode of deformation (e.g. in the form of a shear band or a kink band) appears in the postbifurcation regime. The prediction of these coincidence zones is based on a necessary criterion that depends on the higher order gradient macroscopic moduli. A detailed example is given for the case of layered composites, in view of the possibility of obtaining closed form expressions for all the required macroscopic moduli and in view of the existence of an analytical solution to the microscopic failure problem. Two applications are presented, one for a foam rubber composite and another for a graphite-epoxy composite whose properties have been determined experimentally. Following the verification of the above mentioned necessary criterion for the coincidence of the micro- and macrofailure surfaces in the two examples, the presentation is concluded by a discussion and suggestions for further work. Copyright (C) 1996 Elsevier Science Ltd