Boris Desmorat

Rodrigue Desmorat

Comptes Rendus Mécanique (CRM)

C. R. Mecanique 336 (2008) 448–453http://france.elsevier.com/direct/CRAS2B/Topology optimization in damage governed low cycle fatigueBoris Desmorat a,b,∗, Rodrigue Desmorat ca IJLRdA, Université Paris 6 – Pierre-et-Marie-Curie – CNRS, 4, place Jussieu, case 161, 75252 Paris cedex 05, Franceb Université Paris-Sud 11, 91405 Orsay cedex, Francec LMT, ENS Cachan – CNRS – Université Paris 6, 61, avenue du Président-Wilson, 94235 Cachan cedex, FranceReceived 12 October 2007; accepted after revision 18 December 2007Available online 4 March 2008Presented by Jean-Baptiste LeblondAbstractTopology optimization is applied here to discuss an optimization problem of fatigue resistance. Fatigue lifetime is maximized byoptimizing the shape of a structure in cyclic plasticity combined with Lemaitre damage law. The topology optimization algorithmis detailed. A 3D numerical example is given. To cite this article: B. Desmorat, R. Desmorat, C. R. Mecanique 336 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméOptimisation topologique en fatigue oligo-cyclique. L’optimisation topologique est utilisée pour traiter un problème d’opti-misation de résistance en fatigue. La durée de vie en fatigue est maximisée en optimisant la forme d’une structure en plasticitécyclique combinée à la loi d’endommagement de Lemaitre. L’algorithme d’optimisation est détaillé. Un exemple numérique 3Dest présenté. Pour citer cet article : B. Desmorat, R. Desmorat, C. R. Mecanique 336 (2008).© 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Keywords: Fatigue; Topology optimization; Low cycle fatigueMots-clés : Fatigue ; Optimisation topologique ; Fatigue oligocyclique1. IntroductionTopology optimization is often restricted to elastic behavior and monotonic loading. Classical algorithms are nowefficient to handle the complex 3D cases [1]. In order to maximize the global rigidity of an elastic structure, onepossible formulation is the minimization of the complementary elastic energy at both the local and global scales(alternate directions algorithm [2]). The local minimization is an important feature to keep in fatigue as, if a damagelaw with an energetic basis is used, it can lead to the minimization of local plasticity and damage. Ensuring this localminimization over a cycle in fatigue leads to the maximization of the lifetime. These reasons make us consider andextend the algorithm of alternate directions to low cycle fatigue in three steps: define a cyclic elasto-plasticity law that* Corresponding author.E-mail addresses: borisdes@ccr.jussieu.fr (B. Desmorat), rodrigue.desmorat@lmt.ens-cachan.fr (R. Desmorat).1631-0721/$ – see front matter © 2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crme.2008.01.001B. Desmorat, R. Desmorat / C. R. Mecanique 336 (2008) 448–453 449derives from a state potential, relate this potential to fatigue life and define a fatigue life maximization problem (onwhich a generalized alternate directions algorithm is used). The design problem is then stated as follows: determinethe lightest structure that can achieve a given lifetime.2. Cyclic elasto-plasticity lawLet us consider an isotropic and homogeneous material in the framework of generalized standard materials underthe assumption of small strains and displacements. The behavior is elasto-plastic with linear kinematic hardening, sothat the initial set of constitutive equations may be integrated in cyclic plasticity where the assumptions of proportionalloading and of a symmetric stabilized cycle have been made (see for instance [3]).The stress amplitude Δσ and the total strain amplitude Δε are then related by the nonlinear relation Δε = ∂ψcyclic∂Δσwithψcyclic(Δσ) = 12[Δσ 2eqRνE+ 〈Δσeq − 2σY 〉2+C](1)with Δσeq =√32ΔσDij ΔσDij , ΔσH = 13Δσkk , Rν = 23 (1 + ν) + 3(1 − 2ν)[ΔσHΔσeq ]2 (triaxiality function), where E and νare the elasticity parameters, C is the plastic modulus and (.)D denotes the deviatoric part of a tensor.The stress-strain cyclic law reads:Δεij = 1 + νEΔσij − νEΔσkkδij + 32CΔσDijΔσeq〈Δσeq − 2σY 〉+ (2)This cyclic elasto-plasticity law can be interpreted as a nonlinear elasticity law relating the amplitude of stress Δσ(during one cycle) to the amplitude of total strain Δε. In the following sections, we will use the notation ψcyclic(Δσ) =ψcyclice (Δσ) + ψcyclicp (Δσ) with (e and p standing for elastic and plastic contributions):ψcyclice (Δσ) =Δσ 2eqRν2E, ψcyclicp (Δσ) = 〈Δσeq − 2σY 〉2+2C(3)3. Damage governed low cycle fatigueDamage is next classically described by the state variable D which models at mesoscale a loss of resisting areadue to micro-cracks or micro-cavities (0  D < 1) [4]. Failure of the representative volume element occurs whenD reaches its critical value Dc (of the order of magnitude of 0.2–0.3 for metals). Isotropic damage is coupled toelasticity and plasticity by means of the effective stress tensor σ̃ij = σij1−D , i.e. the stress tensor σij is replaced by σ̃ijin the elasticity law as well as in the yield criterion. Lemaitre’s damage evolution law is used:Ḋ =(YS)sṗ with Y = σ2eqRν2E(4)in which Y is the strain energy density release rate, S, s and Dc are material parameters, and σeq is the von Misesstress.In order to evaluate the number of cycles to crack initiation, a reliable damage increment over one cycleδDδN= ∫1 cycle Ḋ dt can be determined with the assumption of uncoupled elasto-plasticity and damage (i.e. of dam-age estimated by post-processing an elasto-plastic computation) [5].Assuming that the strain energy release rate does not vary much between the applied load inducing the reach of theyield stress and the minimum and maximum loads inducing the maximum von Mises stress, the damage incrementover one cycle δDδNreads (for a symmetric periodic loading):⎧⎪⎪⎨⎪⎪⎩δDδN=(Δσ 2eqRν8ES)sδpδNif Δσeq > 2σYδD = 0 otherwise(5)δN450 B. Desmorat, R. Desmorat / C. R. Mecanique 336 (2008) 448–453in which δpδN= ∫1 cycle ṗ dt is the increase of accumulated plastic strain during one cycle, Rν is the triaxiality functionand where the yield stress σY acts as the asymptotic fatigue limit in Wöhler diagram. For cyclic loading, the number ofcycles to crack initiation NR is obtained for D = Dc after integration over the number of cycles. Considering Eq. (3)and with δpδN= 23/2C1/2(ψcyclicp )1/2, one obtains:NR = Dc( δDδN)= A(ψcyclice (Δσ))s(ψcyclicp (Δσ))1/2with A = 22s−3/2√C SsDc (6)4. Fatigue life maximization problem4.1. Cyclic elasto-plasticity problemConsider a 3D medium Ω . The external boundary is split into two surfaces: Γ0 and Γ1. On Γ0 is imposed azero displacement, on Γ1 a periodic surface load F(t) which vary symmetrically with an amplitude ΔF . The cyclicelasto-plasticity problem is defined as a nonlinear elasticity problem (P ):(P){Δσij,j = 0 in Ω, Δεij = ∂ψcyclic(Δσ)∂Δσij , Δεij = 12 (Δui,j + Δuj,i)Δσijnj = ΔFi on Γ1, Δui = 0 on Γ0(7)This problem is put into a variational form, in which Ec =∫Ωψcyclic(Δσ)dV is the complementary energy.4.2. Optimization problemIn order to increase the number of cycles to crack initiation in the structure, consider the following minimizationproblem, with respect to material and void distribution:min[ ∫Ωψcyclice (Δσ)dV +∫Ωψcyclicp (Δσ)dV]This optimization problem is the classical compliance optimization in the case of linear elasticity (i.e. if ψcyclicp (Δσ) =0 everywhere in the structure) which corresponds to the maximization of the global rigidity of an elastic structure [6,1].In its general form, this optimization problem is similar to the problem of minimizing the complementary energy of astructure made of nonlinear elastic material [7,8]. This optimization problem has to be formulated in an integral form(and not in a local form) because the distribution of material is not a priori known, and then the location of the failurepoint in which damage will reach its critical value is also not a priori known.To solve this optimization problem (known to be ill-posed even in the linear elasticity setting), we will considera distributed fictitious density β allowed to vary between βmin > 0 and 1 (βmin = 0 is set for numerical reasons).Introduce β in the state potential ψβ as following:ψβ = 1βnψcyclic (8)This leads to the definition of an equivalent Young’s modulus E∗ = βnE, an equivalent plastic modulus C∗ = βnCand to an unchanged elastic limit. Introduce a cost term in the objective function in order to limit the total quantity ofmaterial in the optimal design. The optimization problem then reads:minβ∈[βmin,1][ ∫Ωψβ(Δσ)dV +∫Ωcost(β)dV]This problem is put into the form of a double minimization with the use of the variational formulation of the localproblem (P):minβ∈[βmin,1]minΔτ∈ad[ ∫Ωψβ(Δτ)dV +∫Ωcost(β)dV]B. Desmorat, R. Desmorat / C. R. Mecanique 336 (2008) 448–453 451Remark 1. β has no specific physical meaning and is not related to a damaged configuration. We are looking forwardto optimizing the distribution of material in a fixed domain and thus want to obtain values of β mostly close to 0 and 1(black and white design), which explains the use of the power n (n ≈ 3).4.3. Optimization algorithmThe optimization procedure (initially introduced in the framework of linear elasticity in [2]) consists in:– an initialization of iteration (q = 0): β(0) ⇒ Δσ(0),– local minimizations minβ∈[βmin,1]( 1βn ψcyclic(Δσ (q)) + cost(β)) with respect to β with fixed stress Δσ(q) whichgive β(q+1) for iteration q + 1,• a global minimization with respect to Δσ with fixed optimization parameters β(q+1) which gives Δσ(q+1)(finite element computation and use of complementary energy theorem),• iterations on local and global minimizations (q ← q + 1).After one set of local minimizations followed by a global minimization, the criterion decreases. The criterion being apositive quantity, the algorithm is convergent.4.4. Local minimizations with fixed stress fieldThe cost functional is chosen proportional to the distributed fictitious density: cost(β) = kβ , where the parameterk is a constant chosen by the user. The local minimization problem reads then:minβ∈[βmin,1][Bβn+ kβ]with the β-independent constant B = 12(Δσ 2eqRνE+ 〈Δσeq − 2σY 〉2+C)Let β0 = ( nBk )1/(n+1). The principle of the minimization is identical to the classical Solid Isotropic Material withPenalization (SIMP) case:{β = βmin if β0  βminβ = β0 if βmin  β0  1β = 1 if 1  β04.5. Fatigue life maximization problemDuring the local minimizations steps (with fixed stress amplitude), the locally heavily loaded points in the structurehave an optimal fictitious density equal to 1. Locally, the maximization of lifetime for a fixed stress amplitude (i.e.minimization of (ψcyclice )s(ψcyclicp )1/2) is achieved for a fictitious density equal to 1. Thus, with the optimizationprocedure, the heavily loaded points where the crack initiation will occur have a fictitious density optimal with respectto fatigue life.The design problem of finding the lightest structure that achieves a given lifetime is solved using the followingprocedure: determine the lightest structure that can achieve a given lifetime. The numerical procedure is: (i) choosean initial value for the cost parameter k, (ii) apply the optimization algorithm, (iii) if the number of cycles to ruptureis lower than the targeted one, allow for more material in the design domain by decreasing the value of k and runthe optimization procedure again (increase k otherwise). When the targeted number of cycles to crack initiation isobtained, the optimal distribution of material is given by the last optimization procedure.5. Numerical exampleConsider as an example a beam clamped on both lateral sides. A surface load is imposed on the upper surface atits center on 20% of the length of the bar (Fig. 1). The load varies symmetrically between 100 and −100 MPa. Thematerial parameters are: Young’s modulus E = 200 000 MPa, Poisson’s ratio ν = 0.3, plastic modulus C = 6000 MPa,yield stress σY = 180 MPa, damage parameters S = 2.8 MPa, s = 2 and Dc = 0.2 (those of a 2-1/4 CrMo steel [5]).Table 1 shows for different optimized geometries the results of the proposed design procedure:452 B. Desmorat, R. Desmorat / C. R. Mecanique 336 (2008) 448–453Table 1Numerical resultsTableau 1Résultats numériquesCost parameter k 1.0 0.6 0.2Material volume 48 % 52 % 69 %Number of cycles to crack initiation 1580 5603 83 620(Normalized) Optimized energy∫Ω ψβ(Δσ)dx 1.0 0.75 0.53Fig. 1. Sketch of the 3D medium and boundary conditions.Fig. 1. Description du milieu 3D et des conditions aux limites considérés.Fig. 2. Von Mises stress-amplitude (MPa) on the optimized deformed geometry at crack initiation (material volume equal to 48%).Fig. 2. Amplitude de contrainte de von Mises (MPa) sur la géométrie déformée à initiation de fissure (pour un volume de matière de 48 %).Fig. 3. Damage field on half of the optimized geometry at crack initiation for a material volume equal to 48% (middle cutting plane).Fig. 3. Endommagement sur la moitié de la structure à initiation de fissure pour un volume de matière de 48 % (plan de coupe central).– the increase of the number of cycles to rupture with an increasing total material volume,– the increase of the total material volume with an decreasing cost parameter k,– the increase of the energy term∫Ωψβ(Δσ)dx (normalized with respect to the case k = 1) of the optimizedcriterion with a decreasing total material volume.B. Desmorat, R. Desmorat / C. R. Mecanique 336 (2008) 448–453 453Figs. 2 and 3 show the von Mises stress amplitude (more than 90% of the optimized structure is plastified with amaximum plastic strain amplitude of 1.3%) and the damage field on the optimized geometry for the number of cyclesto crack initiation in the case of a material volume equal to 48%. The critical damage is obtained at the lower clampedparts of the structure.6. ConclusionContinuum Damage Mechanics can be used for the low cycle fatigue design of a structure by considering Lemaitredamage evolution law: damage is governed by the plasticity (through the accumulated plastic strain rate) and isenhanced by the stress level (through the elastic energy density). The damage increment per cycle in fatigue is thenexpressed as a function of the associated complementary energy density. Both the state potential and the local damageincrement are minimized by the proposed topology optimization procedure. An optimized structure shape is gained infatigue under symmetric cyclic loading, as illustrated in a three-dimensional cantilever beam.References[1] M.P. Bendsøe, O. Sigmund, Topology Optimization, Springer, 2003.[2] G. Allaire, R. Khon, Optimal design for minimum weight and compliance in plane stress using extremal micro structures, European Journal ofMechanics A/Solids 12 (6) (1993) 839–878.[3] R. Desmorat, Fast estimation of localized plasticity and damage by energetic methods, International Journal of Solids and Structures 39 (12)(2002) 3289–3310.[4] J. Lemaitre, J.-L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, 1991.[5] J. Lemaitre, R. Desmorat, Engineering Damage Mechanics, Ductile, Creep, Fatigue and Brittle Failures, Springer, 2005.[6] G. Allaire, Shape Optimization by the Homogenization Method, Springer-Verlag, New York, 2002.[7] B. Desmorat, G. Duvaut, Compliance optimization with nonlinear elastic materials: Application to constitutive laws dissymmetric in tension–compression, European Journal of Mechanics A/Solids 22 (2) (2003) 179–192.[8] B. Desmorat, Structural rigidity optimization with frictionless unilateral contact, International Journal of Solids and Structures 44 (3) (2007).

Topology optimization in damage governed low cycle fatigue

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Topology optimization in damage governed low cycle fatigue

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