Fundamental Micro–Macro Concepts

Abstract

As stated in the introduction,itisclear that for the relation between averages to be useful it must be computed over as ample containing as tatistically rep-resentative amount of material.This requirement can be formulatedinavery precisem athematical way,w hichh as ac lear physical meaning. Ac ommonly acceptedm acro/microc riterion usedi ne ffectivep ropertyc alculationsi st he so-called Hill's condition, σ : Ω = σ Ω : Ω .H ill's condition (Hill [[79]], 1952)dictates the size requirements on the RV E. The classical argument is as follows. Forany perfectlybonded heterogeneousbody, in theabsence of body forces, two physicallyi mportantl oadings tates satisfy Hill's condition. They are (1) pure linear displacements of the form: u | ∂Ω = E·x ⇒⇒ Ω = E (4.1) (2) pure tractions in the form: t | ∂Ω = L·n ⇒⇒σ Ω = L ;(4.2) where E and L areconstantstrain and stress tensors,respectively.Clearly,for Hill's conditions to be satisfied within am acroscopic body under nonuniform externall oading, thes ample must be large enough to possess small boundary fieldfluctuationsrelativetoits size. Therefore applying (1)-or (2)-typebound-aryc onditions to al arge sample is aw ay of reproducing approximately what maybeoccurringinastatistically representativemicroscopic sample of mate-rial in am acroscopic body.T hus, therei sac lear interpretation to these test boundary conditions. Ourr equirement that thesamplemust be large enoughto have relatively smallb oundaryfi eld fluctuations relativet oi ts size and small enough relative to them acroscopic engineering structure,f orcesu st oc hoose boundary conditions that areu niform.T his is not optional. Next we will deriveatesting procedure for the computation of the consti-tutivet ensor IE *