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Etude de l'endommagement pendant la mise en forme à froid de tôles d'aluminium

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7 pages
Annexe A Homogenisation procedures for non linear heterogeneous materials Cette annexe est extraite d’un rapport du projet européen VIRFORM D3. Homogenisation procedures such as the self-consistent scheme or distribution-based models [Pon95][Bor96] for microheterogeneous materials have been originally formulated in the field of linear elasticity theory, for constant moduli given to the constitutive phases. The methods formally hold for any linear phase behaviour law, and have also been considered (since [Hil65]) as incrementally applicable to "linearizable behaviour laws". Many works concerning elastic-plastic materials have thus been modelized using a linear rate (or incremental) formulation for the phase behaviour laws, say: φ φ φ Q ' = t q ' ij ijkl kl φ φ φ(for the appropriate local conjugate stress and strain rate tensors Q ' and q ', or dQ φ φand dq increments), using the current (instantaneous) tangent (t ) phase moduli to approximate the overall effective related ones. These approaches are uncorrect when considering that the current moduli distribution in each phase, which is not uniform since strain-dependant (and the strain field is in general not uniform) can be represented by an average current value. In the mean time, theoretical considerations, not to be discussed here, have yield to consider as a "better" linearized formulation, the one using current ...
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Annexe A Homogenisation procedures for non
linear heterogeneous materials
Cette annexe est extraite d’un rapport du projet européen VIRFORM D3. Homogenisation procedures such as the selfconsistent scheme or distributionbased models [Pon95][Bor96] for microheterogeneous materials have been originally formulated in the field of linear elasticity theory, for constant moduli given to the constitutive phases. The methods formally hold for any linear phase behaviour law, and have also been considered (since [Hil65]) as incrementally applicable to "linearizable behaviour laws". Many works concerning elasticplastic materials have thus been modelized using a linear rate (or incremental) formulation for the phase behaviour laws, say: φ φφ  Q' =t q' ij ijklkl φ φφ (for the appropriate local conjugate stress and strain rate tensorsQ' andq', or dQφ φ and dq increments),using the current (instantaneous) tangent (t) phase moduli to approximate the overall effective related ones. These approaches are uncorrect when considering that the current moduli distribution in each phase, which is not uniform since straindependant (and the strain field is in general not uniform) can be represented by an average current value. In the mean time, theoretical considerations, not to be discussed here, have yield to consider as a "better" linearized formulation, the one using current instantaneous φ φφ φ (s= sq ,) secant phase moduli, writting Qwith applications to damageelastic ij ijklkl plastic materials [EST99] while more recently, an "affine" formulation has been stressed as an even better approximate of the overall material response [MAS00]. In the field of homogeneization procedures developed for non linear materials, for a better account of the stress and strainfluctuations within the phases, it is worth also mentioning approaches using second order moments [SUQ95], yielding an improved secant approach, and "second order procedures", from potential expansions, [PON96] which generally are more accurate but are of more difficult use thandirect stressstrain approaches. Note in particular that all (tangent, secant, affine) procedures fail to account for the non linearity of the response of porous materials under hydrostatic loading, while second order procedures catch it.
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Annexe A
The affine formulation derives from the tangent moduli rate (or incremental) formulation, in writting, for any appropriate conjugate (Q,q) stress and strain tensors such that Q=C(q),q=D(Q):  Q'~ dQ = Qo+dQ  Qo = Q  Qo (resp q' ~ dq = q  qo) and Q= toq (Qo toqo )= toq ij ijklkl +ij ijklkl ijklkl ij or q= yoQ (qo yoQo )= yoQ ij ijklkl +ij ijklkl ijklkl ij 1 with to= C(qo),q (respyo =D(Qo) ,Q) and where Q°= toqo  ijkl ijkl ijklij klij ijklkl C(qas a «o) actsstress, (resp q°polarization »» or «non mechanicalQo = yo ij ijijkl kl D(Qo) ,as a non mechanical, or eigen, strain), as dilatation stresses (or strains). Both stress ij 1 and strain formulations proove to be dual, thanks toyo =tThe finaly obtained generalo . formulation is the one of thermoelasticity, with evolutive non mechanical stresses (resp strains) at each calculation step. So describing each phase of a composite non linear material provides a heterogeneous linear thermoelastic material of comparison, for which a homogeneous stepwise equivalent linear thermoelastic material can be determined. The Eshelby homogeneous equivalent inclusion solution is next used. For a φ φφ heterogeneoustellipsoidal inclusion supporting thee(resp° eigenstrainp° polarization m mstress), embedded in an infinitetmatrix, also possibly supporting aeigenstrain (resp m stress), the problem first corresponds to the one of the heterogeneity polarization φ φm supporting de° =e°  withno eigenstrain (no polarization stress) in the matrix m [MUR79b] Anystrain needs be uniformly substracted for the calculations, to be m uniformly redistributed afterwhile. The Eshelby thus provides, for dε=εstrain fields, in the strain approach, for eachφphase in an aggregate or in a matrix/inclusion inclusional structure, and with regard to a reference (m) medium (the matrix phase if any), the relation: φ φφ φφ φφmφ φ  σ=t:(εe° ) =t:(dε de° ) =t:(dεe* ) φ φφ φmφφ φ =>t:(dΕ+S:e* de° ) =t:(dΕ+S:e* e* ) φ φφ φ with dε= dΕ+S:e* andSthe Eshelby tensor (resulting from theΓGreen strain operator integral related to the reference medium) for theφphase, and d inclusionalΕ=Ε  m a uniform strain to be specified (Εthe applied isEat infinity for an isolated one inclusion in m). Except for a few cases of high symmetry having analytical solutions for the 1 The commas indicate partial derivatives.
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Eshelby tensors, calculations need be performed by numerical integration procedures. A dual stress approach can be derived similarily, using the dualΔstress Green operator integral. The strain approach yields, for the eigenstrains in each inclusional phase: φm mφ φ1 mφ φφ e[* =t (tt):S] :[(tt):dE+t):de° ] φ φφ φmφ φ Conversely, in terms of polarization stresses, from:σ=t:dεdp° =t:dεp* φmφ1φφ φφ φm1 mφ φφ dε=(tt) :(pd* p° )= dΕ+S:e* =dΕ+ (S:t):(t:e= d* )Ε+T:p* mφ1φφ φ1φmφ1φ => [(tt) T]:p* =G:pd* =Ε+ (tt) :dp° φ φmφ1φ =>p* =G:[dΕ+ (tt) :dp° ] φ The advantage of theT« influence » tensors, compared to the Eshelby ones, is to be φ ij/kl symmetric. Note that the initialp° polarizationspossibly present in the phases do not modify the materialt effectivemoduli to be estimated, which write, whatever are the eff φ φ p° polarizations,or the relatede° eigenstrains: m mφ φm mφ φmφ  t =<t:A> = fmt:A +<t:A> =t <(tt):A>, sincefmA+ <A> eff =I. φ "<.>" denotes the ensemble average over all phases, and "<.>" overtheφinclusionalφ phases only (ie all phases but the matrix), andAare the mean strain localization tensors for φ the inclusional phases. When no matrix, fm=0, and "<.>" <=> "<.>". From the dual stress m mφ φ approch, effective compliances will be obtained as y< =y:B> = fmy:B +<y:B> = eff m mφ φφ φφ φ1 y <(yy):B>, withB thestress concentration tensors (B=t:A:tFor ). eff t t symmetry reasons,<t:A<> =A:t> and <y:B<> =B:y>. Thus, for a macroscopic behaviour law expressed as: Σ=t:E =t: (Eeff effeff ) eff one has : Σ=t: (dE d) =t: dE d, eff effeff eff m =>=t: = d+t:. eff eff effeff eff
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Annexe A
According to equivalency relations established in [LEV67], the effective polarization fullfills: tm mm tφ φφt t:fm =A:t:+ <A:t:> = <A:t:<> =:t:A<> =:A> eff eff tφ φφ φφ φ or t: d= d:t< =A:t:d> = <d:t:A> = deff effeff effeff φmφm and the related strain : =<():B> +< =:B> eff 1 tφ φφ φφ φ1φ φ or d =t :<A:t:d<d> =:t:A>:t =<d:B>. eff effeff This "affine" law differs from the « rate » approachwhich givesΣ' =t:E', sayΣ= eff t:E +(Σo t:Eo), with "o" referring to the iteration step previous to the current one, ie eff eff the "reactualisazed" configuration. But the main difference between the two approaches does not only come from the difference betweenand (Σo t:Eo) for same phase tangent eff eff moduli, and at same assignedEstrain evolution. It mainly comes from the fact that the strain localisation (and stress concentration) fields in each phase vary differently, such that the set of simultaneously involved current phase (tangent) moduli are not identical in both approaches. Further resolution requires to specify the considered material microstructure. In the case of an inclusion/matrix structure, considering a distributionbased modelling with ellipsoidaly distributed pair of (ellipsoidal) inclusions, it has been shown that theEstrain takes the form: m Ε=ΕT:<p*> m (Ponte Castaneda &Willis 95) withTa tensor defining the inclusion pair spatial φ distribution ellipsoidal symmetry, analogous to theTtensors defining the inclusion ellipsoidal shapes.Eis the applied strain at infinity, and <p*> the average polarization over the inclusional phases (now on omitting theφsuperscript). For all shapeidentical inclusions φm (T=T), distributed in the same ellipsoidal symmetry as their common shape (T=T), it comesΕ=Em, the mean matrix strain. When, in this latter case, the matrix volume fraction is decreased to an infinitesimal value, the matrix/inclusion structure turns into an aggregate one. An iterative procedure on the effective moduli estimate then provides a SelfConsistent (SC) approximation fort, what next provides the related (,) terms. Note that for the eff effeff φ φφ φ SC scheme applying to an aggregate, de° =e° e° (respdp° =p° p° ),since the eff eff reference "matrix" phase is the homogeneous equivalent material itself, quantities which are unknown a priori, ast. One still has< =:A> and= <:B>. eff effeff
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m The solution from the distributionbased model, for aT ellipsoidalinclusion pair φ distribution, is obtained in first averaging thepto derive <* expressionsp*> from the relation: φmφ1φ  <p*> =<G:[dΕT:<p*>+Δt:dp° ]>, φmφ say, withΔt=tt: m 1  <p*> =<G>:[dΕT:<p*>]+ <G:Δt:dp°> 1 m1 m1 1 => <p[<*> =G> +T] :dE+ [I+ <G>:T] :<G:Δt:dp°> φ Introducing next the <p*> expression into thepyields :* one φ φm 1m 1m 11φ φp* =G:[dE T:[[<G> +T] :dE[ +I +<G>:T] :<G:Δt:dp°>] +G:Δt 1φ :dp° φm 1φ1m1 1φ φ =G:[I +T:<G>] :dE G: [T< +G>] :<G:Δt:dp°> +G:Δt 1φ :dp° φ φ  =P:dE+ dP° such that finally<p<*> =P>:dE+ <dP°>. Then, the (mean) strains in eachφphase can be obtained from: φ φ1φ φ1φ φ1φ] φφφ φ dε=Δt:p* Δt:dp° =Δt:[P:dE+ dP° dp° =A:dE+ dφ φφ φ1φ φφ1φm 11 With :d=T:G:Δt:t:de° Δt:G T:<P>:<G> :<G:Δt:t:de°> φ φφm And := d+ (IA):
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Annexe A
φ φ This provides the strain localisation tensor,Afor the actual "mechanical" part, andfor the complementary one (analogous to the temperature, ie non mechanical, dilatation strain in thermoelasticity). The inclusion (mean) stresses can next be written: φ φφ φφ φφ φφ φφ  σ= t:[A:dE+ d de° ]= t:[A:E+e=° ]B:Σ+φ φφ φ  =B:[t:(Ee° )]+Bb° =:[t:(dE de° )]+eff effeff eff φ φφ1 yielding, as expected:B=t:A:t , eff φ φφ φφ φφ φφ and : =t:[A:de° +d de=° ]t:[A:e° +e° ] eff eff φ φφ φ and where bothAandderive from the calculation of theTtensors inG, or from the Eshelby tensors. These tensors fulfill<A> = <B> =Iand <a> = <b> = 0. φ φ The effective moduli, independent on thepare calculated for all° stressesp° =0 from: m m  t =t <Δt:A> =t <P>. eff Whentwritten under this last form, the iterative procedure to obtain a Self is eff φm Consistent approximation (for fm~0 andT=T=T) simply writes: (N+1) (N)(N+1) t =t <P> eff eff φm Note that for a cavity,Δt=t, and for a f cavity concentration, the mean matrix stress m m σ=Σ/(1f), thusσeq=Σeq/(1f).
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