Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 41, 3, pp. 459-472, Warsaw 2003 LAYOUT OPTIMIZATION OF TWO ISOTROPIC MATERIALS IN ELASTIC SHELLS Grzegorz Dzierżanowski Tomasz Lewiński Institute of Structural Mechanics, Warsaw University of Technology e-mail: G.Dzierzanowski@il.pw.edu.pl; T.Lewinski@il.pw.edu.pl The two-phase layoutproblemwithin the thinplate theorywas solvedby Gibiansky and Cherkaev in 1984. The same problem in the plane-stress formulation was solved by the same authors in 1987 and eventually cle- ared up byAllaire andKohn in 1993. In the thin shell theory both these formulations are coupled, which is clearly seen in the homogenization formulae found by Lewiński and Telega in 1988, Telega and Lewiński in 1998,and inageneral settingof the layoutproblempresented in thebook by the same authors.The aimof the present paper is to set this problem within the Mushtari-Donnell-Vlasov approximation. The main result of the present examination is the lowerbound of the complementary energy found by using the translationmethod. The translationmatrix involves off-diagonal components,which leads to the effective complementarypo- tential of a specific coupled form, expressible in terms of invariants of the stress and couple resultants. Key words: homogenization, minimum compliance problem, relaxation by homogenization 1. Introduction Let us imagine a spatial curve γ1 alongwhich a loading is prescribed and a supporting curve γ2. Let two isotropicmaterials (1) and (2) of given amounts are at ourdisposal.Ouraim is formingof the stiffest transversely homogeneous shell (of a given constant thickness h) transmitting the loading on γ1 to the support γ2. We note that two different sets of design variables are present in the formulation: the design variables defining the middle surface of the shell and the characteristic function of the domain occupied by material (2). Both sets of the design variables are of different nature. Necessity of handling the 460 G.Dzierżanowski, T.Lewiński middle surface as a compound variable invokes the methods of the minimal surface problem, see e.g. Nitsche (1975). Although challenging, this geometric aspect of the problemwill not be dealt with. In the present paper, themiddle surface will be held fixed, while the distribution of material (2) will play the role of the design variable. To pose the problemcorrectly, we drawupon the known results concerning the plane elasticity, see Allaire and Kohn (1993), and the thin plate theory, see Gibiansky and Cherkaev (1984, 1987) and Lipton (1994). They teach us that the problemmust be reformulated to a relaxed form by admittingmicro- structures of properties governed by the homogenization formulae. A general scheme for relaxing the layout optimization problem for thin shells was given in Lewiński and Telega (2000, Sec. 28.2). This scheme will be specified here for Mushtari-Donnell-Vlasov shells. In this shell model, the measures καβ of change of curvature are approximate. It is assumed that the tangent displacements have a negligible influence on καβ. Consequently, the first two local equilibrium equations do not involve the couple resultants. This approximation is sometimes recommended for shallow shells. In the optimal layout problem in its relaxed form this approximation is essential, because it simplifies the homogenization formulae for effective stif- fnesses of shells with highly oscillating material properties, see Lewiński and Telega (2000, Sec. 17.1). Thus, Mushtari-Donnell-Vlasov approximation con- cernsboth the levels:macro- andmicroscopic. Its role at themicro-level is even more important, since it results in decoupling of the homogenized constitutive relations, see Lewiński and Telega (2000, Sec. 17.4). Having these relations, one can apply the translation method to rearrange the general formula of the effective potential of the relaxed problem to an explicit form, expressible in terms of invariants of the stress and couple resultants. Finding the explicit form of this potential is the main objective of the present paper. The result obtained is somewhat paradoxical. In the case of shape design, the effective potential turns out to involve the terms coupling N and M, namely tr(NM) and trNtrM.Moreover, the definitions of regimes, see (5.2) and Table 1, couple these tensors. Therelaxationbyhomogenizationhasat least twoaims.First, it rearranges the initial problem to thewell-posed form. Secondly, it admits considering the degenerated case of shape design, which is understood now as mixing a given material with voids. Passing to zerowith the values of the elasticmoduli of the weaker material can be performed, which leads to a non-degenerated relaxed form of the shape design problem. In the present paper we show that this passage can also be done in the context of the shell theory. Layout optimization of two isotropic materials... 461 2. Mushtari-Donnell-Vlasov equations Consider a shell of constant thickness hwith amiddle surface S being an image of a plane domain Ω; a point ξ=(ξ1,ξ2)∈Ω is mapped on S and ξ1, ξ2 parametrize the surface S. Deformation of the shell is fully determined by the displacement fields (u1,u2,w) of its middle surface; uα is a displacement tangent to the ξα coordinate, while w is normal to S. The strain measures are assumed according to theMushtari-Donnell-Vlasov approximation ǫαβ(u,w)= 1 2 (uα‖β +uβ‖α)− bαβw (2.1) καβ(w) =−w‖αβ here (·)‖α represents the αth covariant derivative and (bαβ) is the curvature tensor. The shell is considered as elastic and the constitutive relations are given by Nαβ =Aαβλµǫλµ(u,w) M αβ =Dαβλµκλµ(w) (2.2) where A and D are tensors of membrane and bending stiffnesses, respecti- vely. The stress resultants satisfy the equilibrium equations whose variational form is ∫ S [Nαβǫαβ(v,v)+M αβκαβ(v)] dS= f(v,v) (2.3) for all kinematically admissible (v,v). Here v=(vα) are trial tangent displacements, and v represents a trial nor- mal displacement.The linear form f(v,v) expresses thework of the loading on the trial displacements. The displacements are viewed as kinematically admis- sible if they are sufficiently regular and fulfil kinematic boundary conditions. 3. Compliance minimization of a two-phase shell Theshell is consideredasmadeof two isotropicmaterials ofmoduli (k̃1, µ̃1) and (k̃2, µ̃2) such that k̃2 > k̃1, µ̃2 > µ̃1. The tensors of membrane and bending stiffnesses have the following representations Aα =2kαI1+2µαI2 Dα = h2 12 Aα (3.1) 462 G.Dzierżanowski, T.Lewiński where kα =hk̃α, µα =hµ̃α and I αβλµ 1 = 1 2 gαβgλµ (3.2) I αβλµ 2 = 1 2 (gαλgβµ+gαµgβλ−gαβgλµ) Here (gαβ) are components of the metric tensor on S. Let S be the set of statically admissible stress resultants (N,M). Accor- ding to Castigliano’s theorem the compliance C, defined as f(u,w), can be represented by C = min (N,M)∈S ∫ S [N : (A−1N)+M : (D−1M)] dS (3.3) where A and D assume the form A1 and D1 or A2 and D2. Let χα be the characteristic function of the domain occupied by the αth material. The materials are distributed transversely homogeneous, hence the integrals ∫ S χα dS =Aα (3.4) determine the volumes occupied by the materials. The layout optimization problem can be put in the following form inf {∫ Ω [N : (A−1N)+M : (D−1M)] √ g dξ ∣∣∣ (N,M)∈ S, (3.5) χ2 ∈L∞(Ω,{0,1}), ∫ Ω χ2 √ g dξ=A2 } where g=det(gαβ).Weknowthat thisproblemrequires relaxation, seeLewiń- ski andTelega (2000, Sec. 26.1), which, roughly speaking, introduces anunder- lyingmicrostructure. To each point ξ∈Ωwe assign a cell Y =(0, l1)×(0, l2) in which the distribution of the αthmaterial is determined by the characteri- stic functionχYα(y), y=(y1,y2)∈Y , χY1 =1−χY2 . Let the averaging over Y be denoted by 〈·〉 or 〈f〉= 1 |Y | ∫ Y f(y) dy (3.6) Layout optimization of two isotropic materials... 463 Letus recall thedefinitions of the sets of statically admissible stress and couple resultants defined for the cells Y S per 1 (Y ) = { n∈L2(Y,Es2) ∣∣∣〈n(y)〉=0, ∂nαβ ∂yβ =0 in Y, nαβνβ take opposite values at the opposite sides of Y } (3.7) S per 2 (Y ) = { m∈L2(Y,Es2) ∣∣∣〈m(y)〉=0, ∂2mαβ ∂yα∂yβ =0 inY, mν =m αβνανβ take equal values at the opposite sides of Y, q= να ∂mαβ ∂yβ + ∂(mαβνατβ) ∂s take opposite values at the opposite sides of Y } Here ν = (να) represents a unit outward vector normal to ∂Y , and E s 2 is a set of symmetric 2×2 matrices. The distribution of stiffnesses within Y is given by A(y)=A1χ Y 1 (y)+A2χ Y 2 (y) (3.8) D(y)=D1χ Y 1 (y)+D2χ Y 2 (y) and, consequently, A −1(y)=A−11 χ Y 1 (y)+A −1 2 χ Y 2 (y) (3.9) D −1(y)=D−11 χ Y 1 (y)+D −1 2 χ Y 2 (y) We introduce the notation σ=(n,m)⊤ a = [ A −1(y) 0 0 D−1(y) ] (3.10) and define the effective complementary potential 2W∗(N,M,ρ) = min { 〈σ : (aσ)〉 ∣∣∣σ∈ Sper1 (Y )× S per 2 (Y ), (3.11) 〈σ〉=(N,M)⊤, 〈χY2 〉= ρ } for ρ∈ [0,1]. 464 G.Dzierżanowski, T.Lewiński The relaxation rearranges problem (3.5) to the form min { 2 ∫ Ω W∗(N,M,m2) √ g dξ ∣∣∣m2 ∈L∞(Ω; [0,1]), (3.12)∫ Ω m2 √ g dξ=A2, (N,M)∈ S } 4. The lower bound of W∗ The expression for W∗ is not explicit. It is the translation method (see Cherkaev, 2000) which makes it possible to express W∗ in terms of the inva- riants of N and M. This method consists of two stages. The first one finds a lower bound of W∗. The final stage is to prove that the bound is sharp. This means showing amicrostructure realizing this lower bound.The present paper concerns the first stage of the method. To make the further computations possibly simple, we introduce the vec- torial basis a1 = 1√ 2 (e1⊗e1+e2⊗e2) a2 = 1√ 2 (e1⊗e1−e2⊗e2) (4.1) a3 = 1√ 2 (e1⊗e2+e2⊗e1) where ei, i=1,2,3, are unit vectors of the Cartesian basis; ei ·ej = δij. We introduce representations N= 3∑ i=1 Niai M= 3∑ i=1 Miai (4.2) Note that detN=− 1 2 N ⊤ TN (4.3) with T= diag(−1,1,1) (4.4) or T= I2− I1, see Lewiński and Telega (2000). Layout optimization of two isotropic materials... 465 Application of the translation method for the similar thin plate problem is described with all details in Lewiński and Telega (2000, Sec. 26). Thus the technique of the translation method will not be explained here. To take into account themembrane-bending couplingwe choose the trans- lation matrix in the form T = [ αT γT γT βT ] 6×6 (4.5) with α,γ,β ∈ R, β­ 0. Gibiansky and Cherkaev (1987, 1997) proved that 〈n :Tn〉= 〈n〉 :T〈n〉 ∀n∈ Sper1 (Y ) 〈m :Tm〉­ 〈m〉 :T〈m〉 ∀m∈ Sper2 (Y ) (4.6) In this paper we note that 〈n :Tm〉= 〈n〉 :T〈m〉 ∀n∈ Sper1 (Y ) ∀m∈ S per 2 (Y ) (4.7) The proof of (4.7) will be published elsewhere. Let us stress only that the differential conditions concealed in the definitions of sets (3.7) are crucial for properties (4.6) and (4.7). Equality (4.7) links nwith m,whichcanbeviewedas curious: twofields n and m turn out to be linked, although they seem to be independent, see (3.7). In the theory considered the stress couples do not intervene to the membrane equilibriumequations.None the less n and m are linkedby (4.7) onlybecause they obey some differential constraints. Tomake the dimensions of n and m equal we introduce m̃=( √ 12/h)m and M̃=( √ 12/h)M and put γ̃= γ(h/ √ 12), β̃=β(h2/12). We have σ : aσ= σ̃ : ãσ̃ (4.8) where σ̃=(n,m̃)⊤, ã = diag[A−1(y),A−1(y)] or ã = diag (1 2 K(y), 1 2 L(y), 1 2 L(y), 1 2 K(y), 1 2 L(y), 1 2 L(y) ) (4.9) and K(y)=K1χ Y 1 (y)+K2χ Y 2 (y) (4.10) L(y)=L1χ Y 1 (y)+L2χ Y 2 (y) 466 G.Dzierżanowski, T.Lewiński where Kα = 1 kα Lα = 1 µα (4.11) Similarly: T̃ is given by (4.5) with replacing γ̃ for γ and β̃ for β. The translation method requires positive definiteness of (ã − T̃ ) or the positive definiteness of the matrices   1 2 Kσ +α γ̃ 1 2 Lσ−α −γ̃ 1 2 Lσ−α −γ̃ γ̃ 1 2 Kσ+ β̃ −γ̃ 1 2 Lσ− β̃ −γ̃ 1 2 Lσ− β̃   (4.12) where σ=1,2. We remember that K1 > K2 and L1 > L2, see Sec. 3. One can prove thatmatrices (4.12) are positive definite provided that (α,β̃, γ̃) belong to the set Z Z = { (α,β,γ) ∣∣∣ − 1 2 Kσ ¬α¬ 1 2 Lσ, 0¬β¬ 1 2 Lσ, σ=1,2, (4.13) 4γ2 ¬ (Kσ+2α)(Kσ +2β), 4γ2 ¬ (Lσ−2β)(Lσ −2α) } The translation method with the shifting matrix given by (4.5) results in the following estimate W∗(N,M̃,ρ)­W(N,M̃,ρ) (4.14) 2W(N,M̃,ρ)= max (α,β,γ)∈Z [ σ : ( â(α,β,γ)σ )] with σ=(N,M̃)⊤ and â(α,β,γ) = 〈 (ã − T̃ )−1 〉−1 + T̃ (4.15) Layout optimization of two isotropic materials... 467 Tofind â we twice come across the algebraic task of inverting the 6×6matrix of a form X=   a −c1 b c2 b c2 −c1 d c2 e c2 e   (4.16) Thematrix X can be easily inverted. The result is surprisingly simple X −1 =   a′ −c′1 b′ c′2 b′ c′2 −c′1 d′ c′2 e ′ c′2 e ′   (4.17) with a′ = d f c′1 =− c1 f f = ad− (c1)2 b′ = e g c′2 =− c2 g g= be− (c2)2 d′ = a f e′ = b g (4.18) Thus, both thematrices X and X−1 have the same structure. Consequen- tly, the lower bound W has the form of (4.14)2 with σ : [â(α,β,γ)σ] = â(N1)2+ b̂[(N2)2+(N3)2]− (4.19) −2ĉ1M̃1N1+2ĉ2(M̃2N2+M̃3N3)+ d̂(M̃1)2+ ê[(M̃2)2+(M̃3)2] where â, b̂, ĉ1, ĉ2, d̂, ê depend on ρ, α, β, γ, Kσ, Lσ. These relations are closed-form but complicated, and they will not be reported here. Potential (4.19) corresponds to the linear shell theory with a membrane- bending coupling. Let us note, however, that performing maximization in (4.14)2 rearranges the potential W to a non-linear hyperelastic form, depen- ding upon the invariants (trN)2 ‖devN‖2 (trM̃)2 ‖devM̃‖2 trNtrM̃ tr(NM̃) (4.20) 468 G.Dzierżanowski, T.Lewiński where devmeans deviator and ‖devN‖= [ (N2)2+(N3)2 ]1 2 (4.21) is its norm. Performing themaximization in (4.14)2 is not an easy task, becausemany parameters enter the formulae. Further consideration will be confined to a specific case of shape design. 5. Bounding W∗ for the shape design Relaxed formulation (3.11) comprises the case of shape design or the case of k1 → 0, µ1 → 0, K1 →+∞, L1 →+∞. For simplicity we put K2 =K, L2 =L, m2 = θ and find 2W(N,M̃,θ)= 1 2θ { K(N1)2+L[(N2)2+(N3)2] } + +α 1−θ θ [(N1)2− (N2)2− (N3)2]+ 1 2θ { K(M̃1)2+L[(M̃2)2+(M̃3)2] } + (5.1) +β̃ 1−θ θ [(M̃1)2− (M̃2)2− (M̃3)2]+2γ̃1−θ θ [M̃1N1−M̃2N2−M̃3N3] Theparameters α, β̃, γ̃ are chosen according toTable 1.The regimes 1÷6 are defined by (5.2). Table 1.Values of α, β̃, γ̃ for regimes 1÷6 defined by (5.2) Regimes 2α 2β̃ 2γ̃ 2 and 4 and 6 −K 0 − √ L(L+K) 2 and 3 and 6 −K L 0 1 and 4 and 6 L 0 − √ K(K+L) 1 and 3 and 6 L L −(K+L) 2 and 4 and 5 −K 0 √ L(L+K) 2 and 3 and 5 −K L 0 1 and 4 and 5 L 0 √ K(K+L) 1 and 3 and 5 L L K+L Layout optimization of two isotropic materials... 469 Regime number 1 : detN­ 0 4 : detM̃¬ 0 2 : detN¬ 0 5 : trN · trM̃­ tr(NM̃) 3 : detM̃­ 0 6 : trN · trM̃¬ tr(ÑM) (5.2) Let us extract the potential of a homogeneous shell 2W◦(N,M)= 1 2 [1 2 K(trN)2+L‖devN‖2 ] + 1 2 [1 2 K(trM̃)2+L‖devM̃‖2 ] (5.3) from (5.1), and rearrange (5.1) to the form W(N,M̃,θ)=W◦(N,M)+ 1−θ 2θ G(N,M̃) (5.4) where G(N,M̃)= 2W◦(N,M)+ ∣∣∣α [1 2 (trN)2−‖devN‖2 ]∣∣∣+ (5.5) + ∣∣∣β̃ [1 2 (trM̃)2−‖devM̃‖2 ]∣∣∣+2|γ̃[trNtrM̃− tr(NM̃)]| and α, β̃, γ̃ are piece-wise constant quantities, according toTable 1.Note that G(N,M̃) is non-negative. Let us replace W∗ in (3.12) by W to consider the shape design problem in the form min { 2 ∫ Ω W(N,M̃,θ) √ g dξ ∣∣∣θ∈L∞(Ω; [0,1]), (5.6)∫ Ω θ √ g dξ=A2, (N,M)∈ S } Now, we introduce the Lagrangian multiplier λ associated with the iso- perimetric condition and interchange the order of the minimum in (5.6) and maximum over λ. For a fixed λ we find min {∫ Ω Fλ(N,M) √ g dξ ∣∣∣(N,M)∈ S } (5.7) where Fλ(N,M)= min 0¬θ¬1 [2W(N,M̃,θ)+λθ] (5.8) 470 G.Dzierżanowski, T.Lewiński The result of Allaire andKohn (1993) concerning the plane elasticity case applies here. Thus, the function Fλ can be expressed as Fλ(N,M)=2W ◦(N,M)+    2 √ λG(N,M̃)−G(N,M̃) for G(N,M̃)¬λ λ otherwise (5.9) 6. Case of small volume If the quantity A2 is a small number, then λ is big and the main part of problem (5.7) reduces to min {∫ Ω √ G(N,M̃) √ g dξ ∣∣∣(N,M̃)∈ S } (6.1) Let us note that the function √ G(N,M̃) is homogeneous of rank 1. Thus, there exists a closed set B such that the problem dual to (6.1) assumes the form max{f(v,v) |(ǫ(v,v),κ(v))∈B} (6.2) where ǫ(v,v), κ(v) are defined by (2.1) and f(v,v) represents the work of the loading. Here, the loading must be applied to the boundary; the surface loadings are excluded. The passage from (6.1) to (6.2) has been thoroughly explained in Lewiński and Telega (2001) for the thin plate problem. Similar arguments link (6.2) with (6.1). The set B is called a locking locus, and problem (6.2) is, in fact, a locking problem. 7. Final remarks Although estimate (4.14)1 is correct, its sharpness has not been shown. It means that the underlying microstructure has not been disclosed and the method cannot be alternatively applied by controlling the properties of the microstructure. Two questions are crucial here: do the higher rank laminates suffice and do they possess oblique fibres or are they all mutually orthogonal? Layout optimization of two isotropic materials... 471 The results of the paper apply not only to shells but to a general plate problem as well. If a flat plate is subjected both to an in-plane and transverse loading, its optimal design reflects the simultaneous work in the two planes. The optimal layout becomes a compromise between the optimal in-plane and bending type designs. Acknowledgement The work was supported by the State Committee for Scientific Research (KBN) through the grant No. 7T07A04318. References 1. AllaireG.,KohnR.V., 1993,Optimal design forminimumweight and com- pliance inplane stress using extremalmicrostructures,Eur. J.Mech., A/Solids, 12, 839-878 2. Cherkaev A., 2000,Variational Methods for Structural Optimization, Sprin- ger, NewYork 3. GibianskyL.V.,CherkaevA.V., 1984,Designing composite plates of extre- mal rigidity, In: Fiziko-Tekhnichesk. Inst. Im. A. F. Ioffe. AN SSSR, preprint No. 914,Leningrad(inRussian).English translation in:Topics in theMathema- tical Modelling of Composite Materials, Cherkaev A.V. and Kohn R.V., Edit., Birkhäuser, Boston 1997 4. Gibiansky L.V., Cherkaev A.V., 1987, Microstructures of elastic compo- sites of extremal stiffness and exact estimates of the energy stored in them, In: Fiziko-Tekhnichesk. Inst. Im. A. F. Ioffe. AN SSSR, preprint No. 1115, Leningrad (in Russian), pp. 52. English translation in:Topics in theMathema- tical Modelling of Composite Materials, Cherkaev A.V. and Kohn R.V., Edit., Birkhäuser, Boston 1997 5. Lewiński T., Telega J.J., 1988, Asymptotic method of homogenization of twomodels of elastic shells,Arch. Mech., 40, 705-723 6. Lewiński T., Telega J.J., 2000, Plates, Laminates and Shells. Asymptotic Analysis and Homogenization, World Scientific, Series on Advances in Mathe- matics for Applied Sciences, 52, Singapore, New Jersey, London, Hong Kong 7. Lewiński T., Telega J.J., 2001, Michell-like grillages and structures with locking,Arch. Mech., 53, 303-331 8. Lipton R., 1994, On a saddle-point theorem with application to structural optimization, J. Optim. Theory. Appl., 81, 549-568 472 G.Dzierżanowski, T.Lewiński 9. Lurie K.A., Cherkaev A.V., 1986, Effective characteristics of composite materials and optimum design of structural members, Adv. Mech. (Uspekhi Mekhaniki), 9, 3-81 (in Russian) 10. Nitsche J.C.C., 1975, Vorlesungen über Minimalflächen, Springer, Berlin- Heidelberg-NewYork 11. Telega J.J., Lewiński T., 1998, Homogenization of linear elastic shells: Γ- convergence and duality. Part I. Formulation of the problem and the effective model,Bull. Polon. Acad. Sci., Ser. Tech. Sci., 46, 1, 1-9; Part II. Dual homo- genization, ibidem, 11-21 Optymalizacja rozmieszczenia dwu materiałów w powłokach sprężystych Streszczenie Zagadnienieoptymalnegorozkładudwóchmateriałówizotropowychwsprężystych płytach cienkich rozwiązali Gibianskij i Czerkajew w roku 1984. Minimalizacji pod- legała podatność płyty. Analogiczne zadanie dotyczące teorii tarcz rozwiązali ci sami autorzy w 1987r. Sformułowanie to uzupełnili i uściślili Allaire i Kohn w roku 1993. Wzadaniudotyczącympowłok cienkichoba te sformułowania są ze sobą sprzężone, co jasno jest widoczne w formułach homogenizacji znalezionych w pracach Lewińskiego i Telegi z roku 1988orazwpracachTelegi i Lewińskiego z roku 1998; ogólne, niejawne sformułowanie tego zadania optymalizacji omówionow książce tych samych autorów. Celem niniejszej pracy jest sformułowanie tego zadania w sposób jawny w zakresie technicznej teorii powłok Musztariego-Donnella-Własowa. W pracy wyprowadzamy w sposób jawny dolne oszacowanie energii komplementarnej z wykorzystaniem me- tody translacji. Macierz translacji zawiera tutaj składniki pozadiagonalne. Ta postać macierzy translacji prowadzi do zastępczego potencjału o specyficznej postaci sprzę- żonej, wyrażalnej za pomocą niezmienników sił wewnętrznych w powłoce. Manuscript received March 5, 2003; accepted for print March 11, 2003