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. Usin g 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 epsilo n terms in these expansions depend on the standard (first order gradie nt) macroscopic moduli tensor, while all the higher order epsilon term s require the determination of higher order gradient macroscopic modul i. These macroscopic moduli, which are calculated by solving appropria te boundary value problems on the unit cell, relate the macroscopic (u nit cell average) stress rate increment to the macroscopic displacemen t rate gradients. The proposed general theory is subsequently applied to the investigation of the failure surfaces in periodic solids of inf inite extent. For these solids one can define in macroscopic strain sp ace a microscopic (local) failure surface, which corresponds to the on set of the first bulking-type instability in the solid, and a macrosco pic (global) failure surface, which corresponds to the onset of the fi rst 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 stan dard macroscopic moduli tenser. In addition, the regions where the two surfaces coincide is of significant practical interest, for a macrosc opic localized mode of deformation (e.g. in the form of a shear band o r a kink band) appears in the postbifurcation regime. The prediction o f these coincidence zones is based on a necessary criterion that depen ds on the higher order gradient macroscopic moduli. A detailed example is given for the case of layered composites, in view of the possibili ty of obtaining closed form expressions for all the required macroscop ic moduli and in view of the existence of an analytical solution to th e microscopic failure problem. Two applications are presented, one for a foam rubber composite and another for a graphite-epoxy composite wh ose properties have been determined experimentally. Following the veri fication of the above mentioned necessary criterion for the coincidenc e of the micro- and macrofailure surfaces in the two examples, the pre sentation is concluded by a discussion and suggestions for further wor k. Copyright (C) 1996 Elsevier Science Ltd