Optimal structures made of two elastic materials and void

Transcription

1 Optimal structures made of two elastic materials and void Grzegorz Dzierżanowski in collaboration with Nathan Briggs and Andrej Cherkaev Introduction In this talk, some novel, recently obtained, exact results in optimal design of three-phase elastic structures are discussed. The problem is formulated as follows: Two isotropic materials, the strong and the weak one, are laid out with void in a given two-dimensional domain so that the compliance plus weight of a structure is minimized. As in the classical two-phase problem, the optimal layout of three phases is also determined on two levels: macro- and microscopic. On the macrolevel, the design domain is divided into several subdomains. Some are filled with pure phases, and others with their mixtures (composites). The main aim of the talk is to discuss the non-uniqueness of the optimal macroscopic multiphase distribution. This phenomenon does not occur in the twophase problem, and in the three-phase design it arises only when the moduli of material isotropy of strong and weak phases are in certain relation. Statement of the problem Let K 1, K 2, K 1 < K 2 denote the inverses of the bulk moduli of respective elastic materials and assume that K 3 = + in void. Similarly, write L 1, L 2, L 1 < L 2 and L 3 = + for the inverses of the shear moduli. Equilibrium conditions of linearized elasticity are τ = 0 in Ω, τ n = f on Ω f, τ = τ T, where τ stands for the symmetric 2nd order tensor field (elastic stress field), f denotes the vector field on Ω f (loading of the structure) and n is a vector field normal to Ω. Stress energy accumulated in e-th phase (e = 1, 2) is given by 8W e (τ) = K e (τ I + τ II ) 2 + L e (τ I τ II ) 2, where τ I, τ II stand for the eigenvalues of τ. In void it is assumed that { 0 if τ = 0, W 3 (τ) = + otherwise. 1

2 Compliance + weight of a structure (for fixed division of Ω) is calculated according to { ) J(χ 1, χ 2 ) = min (Φ 1 (τ)χ 1 + Φ 2 (τ)χ 2 dx }, τ Σ Ω with Φ 1 (τ) = 2W 1 (τ) + γ 1, Φ 2 (τ) = 2W 2 (τ) + γ 2, where γ 1, γ 2, γ 2 < γ 1, are the coefficients associated to the areas occupied by phases 1 and 2. The problem of optimal phase distribution in Ω { } (P 1 ) : J 0 = inf J(χ 1, χ 2 ) χ 1 i : Ω Ω χ i dx V i, V 1 + V 2 1 is equivalent to a non-convex variational problem { { } (P 2 ) : J 0 = inf F (τ) }, τ Σ F (τ) = min Φ 1 (τ), Φ 2 (τ), 0. and γ 1, γ 2 in (P 2 ) denote the Lagrange multipliers for restrictions in (P 1 ). Results Introduce α = γ 2 γ 1, α (0, 1). Results discussed in the talk shows that for certain values of α the macroscopic distribution of phases cannot be uniquely defined. It turns out that optimal volume fractions of phases are not unique in two following cases: Case A: For α = α A, α A = K 1 + L 1 K 2 + L 1 and Case B: For and Acknowledgement det τ > 0. α = α B, α B = K 1 + L 1 K 1 + L 2 det τ < 0. Grzegorz Dzierżanowski acknowledges the support through the Research Grant no 2013/11/B/ST8/04436 financed by the National Science Centre (Poland), entitled: Topology optimization of engineering structures. An approach synthesizing the methods of: free material design, composite design and Michell-like trusses. 2

3 1/17 Optimal structures madeoftwomaterialsandvoid Grzegorz DZIERŻANOWSKI Faculty of Civil Engineering, Warsaw University of Technology(Poland) Nathan BRIGGS, Andrej CHERKAEV Department of Mathematics, University of Utah Continuum Models Discrete Systems 13 SaltLakeCity,UT,USA,21-25July,2014

4 2/17 Summary Equations of 2D three-phase elasticity Variational problem of optimal phase distribution Discussion on the uniqueness of solution Example of optimal design

5 3/17 Equations of 2D three-phase elasticity Design domain Ω=Ω 1 Ω 2 Ω 3 2Ddomain, Ω boundaryofω. Statically admissible elastic stresses(tensor fields) Σ= { τ : divτ=0inω, τn=fon Ω f Ω }, τ symmetric2ndordertensorfield, f vectorfieldon Ω f (loadingofthestructure), n vectorfieldnormalto Ω. Kinematically admissible displacements(vector fields) V= { u : u=0on Ω u Ω } Small strains in the elastic body(tensor fields) ε= 1 2 ( u+( u) T)

6 Equations of 2D three-phase elasticity Constitutive equation ε=e:τ, E 4thordertensorfield:E ijkl =E klij =E jikl,i,j,k,l=1,2, E 1111 =E 2222 = K+L 4, E 1122 =E 2211 = K L isotropy 4, E 1212 = L 2, E 1211 =E 1222 =0. Mechanical properties of phases K 1 ifx Ω 1, L 1 ifx Ω 1, K(x)= K 2 ifx Ω 2, L(x)= L 2 ifx Ω 2, + ifx Ω 3, + ifx Ω 3, 4/17 suchthatk 1 <K 2 andl 1 <L 2.

7 Equations of 2D three-phase elasticity StressenergyW e accumulatedine-thphase(e=1,2) 8W e (τ)=k e (τ I +τ II ) 2 +L e (τ I τ II ) 2, τ I,τ II eigenvaluesofτ. Invoiditisassumedthat { 0 ifτ=0, W 3 (τ)= + otherwise. Compliance+weightofastructure(forfixeddivisionofΩ) { ) J(χ 1,χ 2 )=min (Φ 1 (τ)χ 1 +Φ 2 (τ)χ 2 dx Ω }, τ Σ 5/17 Φ 1 (τ)=2w 1 (τ)+γ 1, Φ 2 (τ)=2w 2 (τ)+γ 2, γ 1,γ 2 arbitrarycoefficientsassociatedtoareas occupiedbyphases1and2.

8 6/17 Variational problem of optimal phase distribution TheproblemofoptimalphasedistributioninΩ { } (P 1 ): J 0 =inf J(χ 1,χ 2 ) χ 1 i: Ω Ω χ idx V i,v 1 +V 2 1 is equivalent to a non-convex variational problem { { } (P 2 ): J 0 =inf F(τ)dx },F(τ)=min τ Σ Φ 1 (τ),φ 2 (τ),0. Ω γ 1,γ 2 in(p 2 ) Lagrangemultipliersforrestrictionsin(P 1 ).

9 Variational problem of optimal phase distribution Solution exists if the problem is relaxed by allowing mixtures of pure phases(limits of classical designs). Relaxation meansquasiconvexificationoff(τ)in(p 2 ) { (P 3 ): QJ 0 =min QF(τ)dx }, τ Σ { Ω QF(τ)=min 2W (τ,m 1,m 2,m 3 )+γ 1 m 1 +γ 2 m 2 } 0 m e 1,e=1,2,3,m 1 +m 2 +m 3 =1. 7/17 τ averagestress, m 1,m 2,m 3 volumefractions, W minimalstressenergy,i.e.energystoredinamixture composed of optimally stressed phases. Sufficient optimality conditions for microstresses are expressedintermsofτ I andτ II. 1 1 A.Cherkaev,GD(2013),Int.J.SolidsStruct.,50,

10 Variational problem of optimal phase distribution Composite region R(α)= { } τ:qf(τ)<f(τ) forgiven α= γ 2 γ 1. 8/17 QF(τ)issupportedbyΦ 1,Φ 2 andw 3 (0)=0. QF(τ) F(τ) but QJ 0 =J 0.

11 Discussion on the uniqueness of solution In a locally optimal microstructure, volume fractions ofphases:m 1,m 2,m 3,m 1 +m 2 +m 3 =1,depend onthequotientαandaveragestressτ, Optimalstressfieldsinphases:τ (1),τ (2),τ (3) =0, have to be statically admissible. Thus it is necessary that they are in rank-1 connection on phase interfaces: det(τ (i) τ (j) )=0, i,j=1,2,3. Optimal stress fields in phases and the average stress are univalent: detτ 0 detτ (i) 0 (worksalsofor ). 9/17 QUESTION: Are the volume fractions and microstresses uniquely determinedforallpossiblechoicesofαandτ?

12 10/17 Discussion on the uniqueness of solution. Large values of α Assumethatαisgreaterthanacertainthreshold,α>α A, whichresultsinm 2 =0. QuasiconvexenvelopeissupportedbyΦ 1 andw 3 (0)=0. ThusQF=QF 13,suchthat Forτ R(α): QF 13 (τ)= ) K 1 τ I τ II ifτ I τ II <0, (K 1 +L 1 )γ 1 ( τ I + τ II L 1 τ I τ II ifτ I τ II >0. Forτ/ R(α): QF 13 (τ)=φ 1 (τ). R(α)= } γ1 {τ: τ I + τ II ξ 0, ξ 0 =2. K 1 +L 1

13 11/17 Discussion on the uniqueness of solution: Large values of α Optimal volume fractions of phases are uniquely defined m 1 = τ I + τ II ξ 0, m 2 =0, m 3 =1 m 1.

14 12/17 Discussion on the uniqueness of solution: Special values of α Letα=α A α A = K 1+L 1 K 2 +L 1. Thenγ 2 =α A γ 1 andenergywellφ 2 touchesqf 13,i.e. at two points Φ 2 (τ)=qf 13 (τ), τ 1 =(ξ,ξ), τ 2 =( ξ, ξ), ξ= 1 2 α Aξ 0. Both tensors above correspond to pure spherical stress.

15 Discussion on the uniqueness of solution: Special values of α 13/17

16 14/17 Discussion on the uniqueness of solution: Special values of α Optimal volume fractions of phases are not uniquely defined forα=α A ξ 0 τ I τ II ξ 0 2ξ if τ R 1, ξ 0 τ I τ II τ II m max ξ 0 ξ τ II ξ if τ R 2.1, 2 = ξ 0 τ I τ II ξ 0 τ I ξ τ I ξ if τ R 2.2, τ I τ II ξ 2 if τ R 3, 0 m 2 m max 2, m 1 +m 3 =1 m 2.

17 15/17 Discussion on the uniqueness of solution: Special values of α Letα=α B α B = K 1+L 1 K 1 +L 2. Thenγ 2 =α B γ 1 andenergywellφ 2 touchesqf 13,i.e. at two points Φ 2 (τ)=qf 13 (τ), τ 3 =( η,η), τ 4 =(η, η), η= 1 2 α Bξ 0. Both tensors above correspond to pure deviators. Optimal volume fractions of phases are not uniquely defined forα=α B.

18 Discussion on the uniqueness of solution: Special values of α 16/17

19 Exampleofoptimaldesign 2 Thankyou! 17/17 2 N.Briggs,A.Cherkaev,GD(2014),acceptedforpublicationinStruct. Multidiscip. Optimiz., see also