C. R. Acad. Sci. Paris, t. 328, Série II b, p. 713–718, 2000Mécanique des solides et des structures/Mechanics of solids and structures

An existence theorem for the limit analysis problemStefano LENCI a,b

a Laboratoire de modélisation en mécanique, Université Pierre-et-Marie-Curie, CNRS, case 162, 4, placeJussieu, 75252 Paris cedex 05, France

b Istituto di Scienza e Tecnica delle Costruzioni, Università di Ancona, via Brecce Bianche, Monte d’Ago, 60131Ancona, ItalyE-mail: lenci@popsi.unian.it

(Reçu le 17 juillet 2000, accepté le 25 juillet 2000)

Abstract. The limit analysis problem has been considered. The statical and kinematical formulationare summarised, and it is noted that while an existence theorem for the statical problem canobtained by the results of convex analysis, similar theorem for the kinematical problem mustbe obtained independently. We illustrate the main reasons that lead to a Suquet’s relaxedversion of the original kinematical problem, and we prove an existence theorem showingthat the functional of dissipation is lower semicontinuous. This theorem is compared withan analogous theorem existing in the literature. 2000 Académie des sciences/Éditionsscientifiques et médicales Elsevier SAS

limit analysis / convex analysis / existence theorem

Un théorème d’existence pour le problème de l’analyse limite

Résumé. On considère ici un problème d’analyse limite. L’approche statique et cinématique sontrésumés et il est remarqué qu’alors qu’un théorème d’existence pour le problème statiquepeut être obtenu à l’aide de résultats d’analyse convexe, un résultat similaire peutêtre obtenu de façon indépendante pour le problème cinématique. Nous explicitons lesprincipales raisons qui ont conduit à la version relaxée du problème cinématique proposépar Suquet et nous obtenons un résultat d’existence en démontrant que la fonctionnellede dissipation est s.c.i. Ce théorème est comparé à un théorème analogue existant dans lalittérature. 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

analyse limite / analyse convexe / théorème d’existence

Version française abrégée

Soit une structure déformable constituée d’un matériau occupant un domaine bornéΩ deRn de bord∂Ωde classeC1. La structure est fixée sur la partieuΓ de∂Ω , et on applique une tensionλg ∈C0

c (gΓ ,Rn) surla partie complémentairegΓ = ∂Ω\uΓ . De plus la structure est soumise à un champ de force volumiqueλf ∈ L∞(Ω ,Rn). La capacité de résistance du matériau est définie par le convexe ferméP (x) ⊂ Symqui contient toutes les contraintes admissibles. On suppose queB(0, r)⊂ P (x)⊂B(0,R) et que la non-homogénéité deP (x) est telle que la fonctionπ(x,e) définie par (3) est continue par rapport à toutes sesvariables. Le problème d’analyse limite(1) consiste à déterminer le plus grand multiplicateurλs pouvantêtre équilibré par un champ de contrainte admissible. (1) définit l’approche statique et de façon classique [1]on peut définir l’approche cinématique (2), (3). Les formulations (1) et (2) peuvent être traitées à l’aide de ladualité entre fonctions convexes [2,3]. Cela permet de montrer queλs = λk. L’analyse convexe permet aussi

Note présentée par Pierre SUQUET.

S1620-7742(00)01250-2/FLA 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés. 713

S. Lenci

de démontrer un théorème d’existence pour l’approche statique mais ne fournit pas un résultat similaire pourl’approche cinématique.

Dans sa formulation (2), (3), le problème cinématique n’a pas de solution, car on ne peut pas extraireune sous suite convergeante d’une suite minimisante [4]. D’un point de vue physique, ceci correspondà la possibilité que les champs cinématiquement admissibles soient discontinus. Les espacesBD(Ω)ont été introduits [5,6] pour surmonter cette difficulté. Dans ces espaces les champs de déplacementspeuvent être discontinus et les suites minimisantes de (2) sont compactes pour la topologie faible étoile.L’introduction de ces espaces permet de résoudre le problème mais introduit de nouvelles difficultés, carV = u∈ BD(Ω), u= 0 in uΓ et l’ensembleL(u) = 1 ne sont pas fermés pour la nouvelle topologie.Cette question a été résolue par Suquet qui considère indépendamment les déplacementsu dansΩ etu+

sur le bord∂Ω , l’idée de relaxer le terme de bord étant due à Temam et Strang. Dans [2] (voir aussi [7])Suquet suggère la forme relaxée (4) de (2). Les termes contenantπb traduisent la possible plastification de∂Ω . L’expressionΠ (u,u+) est une fonctionnelle convexe sur les mesures, qui a été définie, par exemple,dans [4]. Une définition équivalent est (6).

La topologie, notéeτ , du problème (4) est la topologie faible étoile dansBD(Ω)×M1(gΓ ,Rn). Puisquel’ensembleL(u,u+) = 1 est fermé pourτ , afin de montrer la validité de la formulation relaxée (4), ilreste à montrer que la fonctionnelleΠ (u,u+) est s.c.i. pour la topologieτ . Ceci est démontré dans [2,p. 298] pour le cas particuliergΓ =∅. Dans cette note on montre que ceci reste vrai dans le cas général :

THÉORÈME 1. – Π (u,u+) est s.c.i. pour la topologieτ .

Ce résultat permet alors d’employer des arguments classiques de calcul variationnel afin de démontrer lethéorème d’existence suivant :

THÉORÈME 2. –La formulation cinématique relaxée (4) admet au moins une solution.

Le théorème 2donne un résultat d’existence du minimiseur deΠ (u,u+) pour (u,u+) ∈ BD(Ω) ×M1(gΓ ,Rn), et ceci ne semble pas avoir été remarqué dans la littérature. En effet la conditionu+ ∈M1(gΓ ,Rn) ne peut pas être améliorée comme le montre l’exemple de lafigure 1qui est résolu de façonexplicite.

Le théorème 2peut être comparé avec celui obtenu par Demengel [9,10] qui est légèrement différentpuisque nous considérons ici des domaines admissibles bornés alors que dans [9,10] la résistance dumatériau est déterminée uniquement par le déviateur des contraintes. Les deux théorèmes sont obtenusdans le cadre du calcul variationnel mais dans la note présentée ici, on utilise des arguments classiquesd’analyse convexe alors que dans [10], les résultats sont démontrés à l’aide de théorèmes de compacité pourBD(Ω). Les résultats de Demengel permettent de discerner si le bord deΩ plastifie ou non. Toutefois ilssont obtenus au prix du calcul d’un chargement limite fictif¯λ, qui en général est aussi difficile à déterminerqueλ= λs = λk. Si λ < ¯λ le bord ne plastifie pas. Dans le cas contraire le bord plastifie.

Let Ω be a bounded domain inRn with the boundary∂Ω of classC1. On the partuΓ of ∂Ω the bodyis fixed, while on the remaining partgΓ = ∂Ω\uΓ a tensionλg ∈ C0

c (gΓ ,Rn) is applied. Note that thisimpliesg = 0 on the boundary ofgΓ . This regularity is necessary to guarantee the continuity ofL(u,u+)with respect tou+ for the topologyτ (see equations (5)). Other choices are possible: for example, it ispossible to assume thatgΓ is closed and thatλg ∈ C0(gΓ ,Rn). Ω is further submitted to body forcesλf ∈L∞(Ω ,Rn), while its strength in each point is defined by the closed convex setP (x)⊂ Sym, whichcontains all admissible stress tensors. We assume thatB(0, r) ⊂ P (x) ⊂ B(0,R) (B(0, ρ) is the ballcentered in0 and of radiusρ) and thatP (x) can vary in such a way that the functionπ(x,e) defined in (3)is continuous with respect to both variables. Thelimit analysis(or yield design) problem consists in finding

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An existence theorem for the limit analysis problem

the maximum multiplier of loads which can be equilibrated by an admissible stress field (n is the outernormal of∂Ω ):

λs = supλ ∈R such that there existsσ ∈ L2(Ω ,Sym), such that

divσ+ λf = 0 in Ω , σn= λg in gΓ , σ(x) ∈ P (x) a.e. inΩ

(1)

(1) is thestatical definition of the limit load [1]. By classicalformal arguments [1], it is also possible togive akinematicaldefinition of the limit load:

λk = inf

Π (u), u ∈ V, L(u) = 1, (2)

where: Π (u) =

∫Ω

π[x,ε(u(x))

]dΩ , π(x,e) = sup

σ · e, σ ∈ P (x)

V =

u ∈H1(Ω ,Rn), u= 0 in uΓ

, L(u) =

∫Ω

f ·udΩ +

∫gΓ

g ·udΓ (3)

The hypothesis onP (x) guarantees thatr|ε(u)|L1(Ω ,Sym) <Π (u)<R|ε(u)|L1(Ω ,Sym). In the frameworkof yield design (i.e., without referring to the elasto-plastic properties of the material but only to its strengthresistance [1]), the kinematical definition is less intuitive, but it is very useful in numerical computations[2]. The statical and kinematical formulations (1) and (2) can be considered in the general framework of theduality between convex functions [2,3], and this permits us to prove the equality betweenλs andλk, whichguarantees the equivalence of the two approaches. The general theory of convex analysis also furnishesan existence theorem for the statical problem, but it does not provide a similar result for the kinematicalformulation, and this question should be addressed independently.

In the form (2), (3), the kinematical problem has no solution, basically because we cannot extract froma minimising sequence a converging subsequence [4], even if in (3) one considersLD(Ω) instead ofH1(Ω ,Rn) (see [4] for the definition ofLD(Ω)). Furthermore, non-existence is confirmed by physicalcommon sense, which suggests that the solution can admit discontinuities, and therefore cannot belong toLD(Ω). To overcome these difficulties, the functional spaceBD(Ω) has been introduced [5,6], whichensures that minimising sequences of (2) are compact with respect to its weak∗ topology and thatdiscontinuities of the displacement field are allowed for. The introduction of this space, however, solvesthe previous problems but creates new difficulties, becauseV = u ∈ BD(Ω), u = 0 in uΓ and theconstraintL(u) = 1 are no longer closed for the weak∗ topology of BD(Ω) (because the trace on theboundary ofΩ is not continuous with respect to this topology). This question has been solved by Suquet,which, to the author’s knowledge, was the first to realize that the displacement fieldsu onΩ andu+ on thewholeboundary∂Ω must be considered independently, although the idea of relaxing the boundary terms isdue to Temam and Strang (but in [3] only the fixed partuΓ is relaxed). In [2] (see also [7]) Suquet suggeststhe following relaxed form of (2) (M1 is the space of bounded measures [8]):

λk = inf

Π (u,u+), u ∈BD(Ω), u+ ∈M1(gΓ ,Rn

), L(u,u+

)= 1, (4)

where Π (u,u+) =

∫Ω

π[x,ε

(u(x)

)]dΩ +

∫gΓ

πb(x,u+ − γ(u)

)dΓ +

∫uΓ

πb(x,−γ(u)

)dΓ

πb(x,v) = π[x,Sym(n(x)⊗ v)

], L(u,u+) =

∫Ω

f ·udΩ +

∫gΓ

g ·u+ dΩ dΓ (5)

and he proves thatλk = λk. The terms containingπb account for the possible strain localization on theboundary∂Ω . The expressionΠ (u,u+) is a convex functional of measures, which has been defined, e.g.,in [4] (γ(u) is the trace ofu on the boundary and belongs toL1(∂Ω ,Rn) [2,4]). It must be consideredwith care, because the three terms in the definition ofΠ (u,u+) are not independent, as it may appear. Analternative definition ofΠ (u,u+) which clarifies this point is (see [2], p. 291):

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S. Lenci

Π (u,u+) = sup

∫Ω

σ · ε(u) dΩ +

∫gΓ

σn ·(u+ − γ(u)

)dΓ +

∫uΓ

σn ·(− γ(u)

)dΓ (6)

for everyσ ∈Σ such thatσ(x) ∈ P (x) in Ω

whereΣ =C(Ω ,Sym)/(S ∩C(Ω ,Sym)) andS = σ ∈ L2(Ω ,Sym), such thatdivσ = 0 in Ω , σn= 0in gΓ. BD(Ω) is the dual space ofΣ [2]

In other words,σ insideΩ andσn on the boundary used in the definition of the support functionsπ andπb are not completely independent on each other, but the latter is just the trace of the former. The equalitybetween the two definitions is substantially contained in Theorem 2.3 of [2] and can be proved by showingthat the problems (4)–(5) and (4)–(6) have the same dual. From a mechanical point of view, this fact followsfrom the equilibrium conditions on the boundary of the domain.

The topologyτ of the problem (4) is the weak∗ convergence ofBD(Ω) for u and the weak∗ convergenceof M1(gΓ ,Rn) for u+. SinceBD(Ω) andM1(gΓ ,Rn) are weakly sequentially compact, and since

c[∣∣ε(u)

∣∣M1(Ω ,Sym)

+∣∣u+ − γ(u)

∣∣M1(gΓ ,Rn)

+∣∣γ(u)

∣∣L1(gΓ ,Rn)

]< Π (u,u+) (7)

we can extract a converging subsequence from any minimising sequence. Furthermore, the constraintL(u,u+) = 1 is closed becauseL(u,u+) is continuous with respect toτ .

To prove that the relaxation (4) is the good one, it remains to show that the functionalΠ (u,u+) is lowersemicontinuous with respect toτ . This fact was proved in [2, p. 298] for the special casegΓ = ∅, but itholds in general.

THEOREM 1. – Π (u,u+) is lower semicontinuous with respect to theτ convergence.

Proof. –Let (un,u+n )→ (u,u+) with respect toτ . By definition (6) ofΠ (u,u+), the inequality:

Π (un,u+n )>

∫Ω

σ · ε(un) dΩ +

∫gΓ

σn ·(u+n − γ(un)

)dΓ +

∫uΓ

σn ·(− γ(un)

)dΓ (8)

holds for everyσ ∈Σ . Transforming the right hand side by means of Green’s formula [4], yields:

Π (un,u+n )>−

∫Ω

divσ ·un dΩ +

∫gΓ

σn ·u+n dΓ (9)

By taking the lim inf in (8) and remembering that, if(un,u+n )→ (u,u+) with respect toτ , thenun→ u

strongly inL1(Ω ,Rn) andu+n → u+ weakly∗ in M1(gΓ ,Rn), we obtain:

lim infn→∞

Π (un,u+n )>−

∫Ω

divσ ·udΩ +

∫gΓ

σn ·u+ dΓ (10)

or, by applying again Green’s formula:

lim infn→∞

Π (un,u+n )>

∫Ω

σ · ε(u) dΩ +

∫gΓ

σn · (u+ − γ(u)) dΓ +

∫uΓ

σn ·(− γ(u)

)dΓ (11)

By taking the supremum with respect to all the possibleσ in C1(Ω ,Sym), such thatσ(x) ∈ P (x), we seethat the right hand side is justΠ (u,u+), and this proves the theorem.

The previous result permit employing classical arguments of calculus of variations to prove the followingexistence theorem.

THEOREM 2. –The relaxed kinematical formulation (4) admits at least one solution.

Theorem 2gives the existence of the minimizer ofΠ (u,u+) for (u,u+) ∈ BD(Ω)×M1(gΓ ,Rn), andthis fact seems to be not well remarked in literature (but see [7]). Indeed, the requirementu+ ∈M1(gΓ ,Rn)cannot be improved, as shown by the example reported infigure 1. A circular, planar disc of unitary

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An existence theorem for the limit analysis problem

Figure. An example with solution inM1(gΓ ,Rn).

width is fixed on the lower partα ∈ ]π,2π[, while on the upper partα ∈ [0, π] a nominal tensiongx = 0,gy = p(α) = sin(α) is applied. There are no body forces, and the limit convex is the ball of radiusk,namely,P = σ, |σ|6 k. The limit multiplier isλ= λs = λk = k. A minimum of Π (u,u+) is attainedat u1 = 0 and onu+

1x = 0, u+1y = δ(x = 0, y = 1) (δ(.) is the Dirac delta), which is a measure ongΓ ,

although in the interior of the domain the solution isC∞. Incidentally, we note that this solution is notunique, and another minimum ofΠ (u,u+) is u2x = 0, u2y = 0 if y < R sinβ, u2y = 1/(2R cosβ) ify > R sinβ, u+

2 = γ(u2). In this case the boundary solutionu+ is more regular, because it is the traceof u and therefore it belongs toL1(gΓ ,Rn). However, the interior solution is less regular, and it belongsto BD(Ω) but not toLD(Ω), because it has a jump. Note that(u1,u

+1 ) is the limit of (u2,u

+2 ) whenβ

tends toπ/2. A more regular solution isu3x = 0, u3y = 0 if y < 0, u3y = 2y/(πR2) if y > 0,u+3 = γ(u3),

which belongs toLD(Ω)×C∞(gΓ ,Rn).Theorem 2should be compared with those previously proved by Demengel [9,10], which are obtained

under slightly different hypotheses. Indeed, we consider bounded admissible domains whereas in [9,10]only the deviatoric part of the stress determines the strength of the material (as occurs in the case of von-Mises and Tresca criteria).

We remark that, although both theorems are obtained in the natural framework of calculus of variations,the two proofs are based on different arguments and ideas. In this work classical tools of convex analysisare employed, while in [10] compactness theorems forBD(Ω) have been utilised.

The results of Demengel provide more information on the solution of the problem, because they permitus to discern if there is strain localization on boundary ofΩ . However, this is obtained at the price ofcomputing a “fictitious” limit loadλ (using her notation), which in general is as difficult to determine as thetrue limit loadλ= λs = λk. The multiplier¯λ can be computed by supposing that there is no admissibilitycondition in theinterior of Ω , i.e., the admissibility is required only on the boundary, and ifλ < ¯λ there isno strain localization on the boundary, as occurs in the opposite case.

Acknowledgements.The work was done during the author permanence at Laboratoire de Modélisation enMécanique, Université Pierre et Marie Curie (Paris 6), as part of the Training and Mobility of Researchers (TMR)programme, contract ERBFMBI CT97 2458. The financial support of this programme is greatly acknowledged.

The author wishes to thank Prof. G. Geymonat and Prof. F. Krasucki for useful comments, and Prof. P. Suquet forhis suggestions which have contributed to improve the paper.

References

[1] Salençon J., Calcul à la rupture et analyse limite, Presses ENPC, Paris, 1983.[2] Suquet P., Discontinuities and plasticity, in: J.J. Moreau, P.D. Panagiotopoulos (Eds.), Non-Smooth Mechanics

and Applications, Springer-Verlag, Wien, 1988, pp. 280–340.[3] Temam R., Strang G., Duality and relaxation in the variational problems of plasticity, J. Mec. 19 (1980) 493–527.[4] Temam R., Mathematical Problems of Plasticity, Gauthier-Villars, Paris, 1985.

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[5] Suquet P., Sur un nouveau cadre functionnel pour les équations de la plasticité, C. R. Acad. Sci. Paris Sér. A 286(1978) 1129–1132.

[6] Strang G., A family of model problems in plasticity, in: R. Glowinski, J.L. Lions (Eds.), Proc. Symp. Comp. Meth.in Appl. Sci., Lecture Notes in Mathematics, Vol. 704, 1978, pp. 292–308.

[7] Bouchitté G., Suquet P., Plasticity, yield design and homogenization, in: G. Dal Maso, A. Dell’Antonio (Eds.),Proc. Workshop on Composite Media and Homogenization Theory, Trieste, 1990.

[8] Bourbaki N., Intégration, Hermann, Paris, 1954.[9] Demengel F., Espaces de fonctions à dérivées mesure et applications à l’analyse limite en plasticité, C. R. Acad.

Sci. Paris, Sér. I 302 (1986) 179–182.[10] Demengel F., Compactness theorems for spaces of functions with bounded derivatives and applications to limit

analysis problems in plasticity, Arc. Rat. Mech. Anal. 105 (1989) 123–161.

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