International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Stochastic multi-scale finite element analysis for laminated composite plates Zhou, Xiao-Yi; Gosling, Peter D. 2015-07

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Stochastic Multi-scale Finite Element Analysis for LaminatedComposite PlatesXiao-Yi ZhouResearch Associate, School of Civil Engineering and Geosciences, Newcastle University,Newcastle upon Tyne, UKPeter D. GoslingProfessor, School of Civil Engineering and Geosciences, Newcastle University,Newcastle upon Tyne, UKABSTRACT: This paper proposes a dual stochastic finite element method for conducting stochastic anal-ysis for laminated composite plates with consideration of microscopic material property uncertainties. Astochastic multiscale finite element method, which couples the multiscale computation homogenizationmethod with the second order perturbation technique, is developed first to propagate variability in the ef-fective elasticity property of composite arising from microscopic uncertainties such as Young’s modulusand Poisson’s ratio of constituent materials. Then a standard stochastic finite element analysis is con-ducted to consider structural level uncertainties. The performance of the proposed approach is evaluatedby comparing the estimates of mean values and coefficients of variation for the effective elastic propertiesand structural response for a laminated composite plate with corresponding results from the Monte Carlosimulation method.1. INTRODUCTIONComposite materials are becoming increasingly im-portant in civil engineering applications where thetailoring opportunities offered by composite fi-bre reinforcement can achieve engineering require-ments not attainable by conventional steel and con-crete options. Enormous efforts have been devotedto predict the effective mechanical properties ofcomposite materials to advance the structural anal-ysis for composite structures in deterministic con-text. However, it has been widely accepted and ex-tensively reported that uncertainties exist in all as-pects of a composite structure, e.g. constituent ma-terial properties, that result in uncertainties in struc-tural responses. Due to the heterogeneous natureof composites, the uncertain behaviour in perfor-mance of a composite structure such as laminatedplate is often more higher than the conventionalisotropic material structures. An efficient and ac-curate quantification of the uncertainty in structuralperformance is thus desirable to analyse a compos-ite structure, and it is also a crucial part of the devel-opment of reliability-based design method for com-posite structures.The objectives to quantitative modelling of un-certainty of systems with random properties canbe divided into three groups: the description ofthe random properties of the system (Jeong andShenoi, 2000; Chuang, 2006), the calculation ofstatistical information of the response from the sys-tem properties (Motley and Young, 2011; Sasiku-mar et al., 2014; Talha and Singh, 2014), and theinterpretation and use of this statistical response in-formation for design, maintenance, repair, and soon (Murotsu et al., 1994; Frangopol and Recek,2003; Soares, 1997). The calculation of the statis-tical response characteristics requires an extensionof the traditional deterministic analysis that leadsto stochastic analysis. Monte Carlo simulationmethod, perturbation method and spectral method112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015are the available solutions for this type of problem(Matthies, 2007; Schuëller and Pradlwarter, 2009;Stefanou, 2009). Although stochastic finite elementbased methods have been used for a long time, itis only recently that attempts are being made toextend these methods for uncertainty quantifica-tion in composite materials and composite struc-tures (Chen and Guedes Soares, 2008; Ngah andYoung, 2007; Talha and Singh, 2014). However,most of previous work on stochastic analysis forlaminated composite plates focused on meso-scaleuncertainties such as ply or lamina material proper-ties. For instance, Chen and Guedes Soares (2008)consider the uncertainty of Young’s modulus. Ngahand Young (2007) considered the components inthe constitutive stiffness matrix as random field.Many studies have shown that mechanical proper-ties of multi-phase composite materials are stronglyaffected by their microscopic heterogeneous na-ture (Sriramula and Chryssanthopoulos, 2009; Pot-ter et al., 2008). Various of homogenization meth-ods are available in the literature to predict effec-tive material properties of composites. Although ithas been pointed out that potential benefits can beobtained (Charmpis et al., 2007), considerations onmicroscopic material property uncertainties, espe-cially through micromechanics method, are seldomreported in stochastic analysis. Chamis (2004) andWelemane and Dehmous (2011) are a few that in-vestigated microscopic uncertainties for stochasticanalysis of composite structures.In order to taking microscopic uncertainties intoaccount in stochastic analysis of composite struc-tures, this paper presents a dual stochastic finite-element based procedure. Among the many inher-ent uncertainties, we mainly focus on microscopicconstituent material properties in present study.The analytical development starts with the propa-gation of uncertainties at the microscopic materialproperties to determine the effective material prop-erties at macro or ply scale by using the stochastichomogenization method (Zhou et al., 2014). First-order shear deformation laminate theory is thenused as basis to propagate the uncertainty in the ma-terial behaviour at the laminate or structural scale.Next, the stochastic finite element analysis is per-formed to determine the stochastic structural re-sponses corresponding to the uncertain basic vari-ables. A case study of laminated composite plate isperformed to illustrate the proposed method for theanalysis of composite structures.2. STOCHASTIC VARIATIONAL FORMULATIONOF MULTISCALE APPROACH2.1. Stochastic homogenization method for com-posite materialsFirst, we will summarize the basic assumptions andthe final formulae of the stochastic homogeniza-tion method for the estimation of effective elas-tic moduli. For a detailed discussion and numer-ous references for this and related methods, thereader is referred to Zhou et al. (2014). The classof homogenization-based multi-scale constitutivemodels employed in the present study is charac-terised by the assumption that the strain and stresstensors at a point of the so-called macro-continuumare volume average of their respective microscopiccounterpart fields over a pre-specified Representa-tive Volume Element (RVE). That is, by definingthe deformation gradient ε¯ and the stress tensor σ¯as volume averages of their counterpart fields overthe RVE, we haveε¯ ≡ 1Vµ∫ΩµεµdV =1Vµ∫Ωµ∇uµdV (1)andσ¯ = 1Vµ∫ΩµσµdV (2)where εµ and σµ denote, respectively, the deforma-tion gradient and the stress fields of the RVE, Vµ isthe volume of the RVE and ∇ denotes the gradientoperator.By defining the field u˜µ of displacement fluctua-tions of the RVE asu˜µ ≡ uµ −u∗ with u∗ = ε¯y (3)The deformation gradient field of the RVE can bewritten as a sum of a uniform deformation gradi-ent coinciding with the macroscopic deformationgradient, ε¯ , and a displacement fluctuation gradientfield, ε˜ ≡ ∇u˜µ ,εµ = ε¯+ ε˜ (4)212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015To comply with the averaging strain assumption,the displacement fluctuation field must satisfy∫Ωµε˜µdV = 0. (5)To make upscaling transition, a second assump-tion also known as the Hill-Mandel principle is in-troduced, and it requiresσ¯ : ε¯ = 1Vµ∫Ωµσµ : εµdV (6)Considering the RVE equilibrium and the secondassumption, it introduces an addition constraint forthe RVE that requires:∫∂Ωµte ·ηdA = 0 η ∈ V , (7)in terms of the RVE boundary traction te.Hence, the RVE equilibrium problem for alinear-elastic material case consists of finding, for agiven macroscopic deformation ε¯ , a kinematicallyadmissible displacement fluctuation field u˜µ ∈ Vunder constraints Eqs.(5) and (7), such that∫Ωσ : ∇ηdV =∫Ω(Cµεµ): ∇ηdV = 0 ∀η ∈ V ,(8)where Cµ is material constitutive tensor and V isthe (as yet not defined) space of virtual kinemati-cally admissible displacement of the RVE.Considering the uncertainties in material prop-erties, b, constitutive matrix Cµ and displacementfluctuation u˜µ are stochastic function of b. Us-ing perturbation method, the variational functionEq.(8) can be transformed into its zeroth-, first- andsecond-order approximations as:• The zeroth order∫Ωsµ∇sη : Cµ(b¯) : ∇su˜µ(b¯)dV+∫Ωsµ∇sη : Cµ(b¯) : εdV = 0 (9)• The first-ordern∑p=1{∫Ωsµ∇sη :(Cµ(b¯) :[Dbp∇su˜µ(b¯)]+[DbpCµ(b¯)]: ∇su˜µ(b¯))dV+∫Ωsµ∇sη :[DbpCµ(b¯)]: εdV}δbp = 0(10)• The second-ordern∑p=1n∑q=1{∫Ωsµ∇sη :(Cµ(b¯) :[Hbpbq∇su˜µ(b¯)]+[HbpbqCµ(b¯)]: ∇su˜µ(b¯)+2[DbpCµ(b¯)]:[Dbq∇su˜µ(b¯)])dV+∫Ωsµ∇sη :[HbpbqCµ(b¯)]: εdV}δbpδbq = 0(11)2.2. Stochastic formulation for laminated com-posite platesIn this section, the stochastic variational formula-tion for probabilistic analysis of laminated compos-ite plates is deduced by the first-order or Reissner-Mindlin shear deformation theory (FSDT) and theperturbation method. Using FSDT the displace-ment components u, v and w can be expressed interms of the mid-plane displacements u0, v0, w0,and the rotations of transverse normal about y- andx-axes of θx and θy, respectively, as (details inOñate (2013) for example)u(x,y,z) = u0(x,y)− zθx(x,y) (12)v(x,y,z) = v0(x,y)− zθy(x,y)w(x,y,z) = w0(x,y)The strain components are computed using theabove displacement field with ε = ∇u, and it is ex-pressed asε =(εm 0)T+(−zεˆb εˆs)T= Sεˆ (13)whereεˆ =(εˆm εˆb εˆs)T(14)withεˆm =[∂u0∂x ,∂v0∂y ,(∂u0∂y +∂v0∂x)]T(15)εˆb =[∂θx∂x ,∂θy∂y ,(∂θx∂y +∂θy∂x)]Tεˆs =[∂w0∂x −θx,∂w0∂y −θy]Tare the generalized strain vectors due to membrane,bending and transverse shear deformation effects,respectively, andS=[I3 −zI3 03×202×3 02×3 I2](16)312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015is the generalized strain εˆ to the actual strain εtransformation matrix.Now, let us consider a composite laminated plateformed by a stacking of nl orthotropic layers withorthotropic axes 1, 2, 3 and isotropy in the 1 axis(i.e. in the plane 23). The 1 axis defines the di-rection of the longitudinal fibres which are em-bedded in a matrix of polymeric or metallic mate-rial. The relationships between the in-plane stressesσp = [σx,σy,τxy]T and the transverse shear stressesσs = [τxz,τyz]T with their conjugate strains for eachlayer k can be written asσ =(σpσs)=[Dp 00 Ds](εpεs)= Dε (17)The constitutive matrices Dp and Ds are symmet-rical and their terms are a function of independentmaterial parameters and the angle βk, and they canbe obtained throughDp = TT1 D1T1, Ds = TT2 D2T2 (18)with T1 and T2 are coordinate transformation ma-trix from material principal coordinate system toglobal system, and D1 and D2 are in-plane constitu-tive matrix and transverse shear constitutive matrixin material principal coordinate system.Now we are ready to establish the stress-strainrelation between laminate stress and strain by inte-grating through the thickness of the laminate of thestress, we haveσˆ = Dˆεˆ (19)withσˆ =σˆmσˆbσˆs , Dˆ=Dˆm Dˆmb 03×2Dˆmb Dˆb 03×202 02 Dˆsandσˆm =∫ t/2−t/2σpdt σˆb =−∫ t/2−t/2zσpdt σˆs =∫ t/2−t/2σsdtwhere −t/2 and t/2 are the z-coordinates of thelamina’s upper and lower surfaces, respectively.For a laminate with nl orthotropic layers and ho-mogeneous material within each layer we can writeDˆm =nl∑k=1tkDpk Dˆmb =nl∑k=1tkz¯kDpkDˆb =nl∑k=113[z3k+1− z3k]Dpk (20)where tk = zk+1− zk, z¯k =12 (zk+1 + zk) and Dpk isthe in-plane constitutive matrix for the kth layer.The principle of virtual work (PVW) is writtenin terms of the generalized strains, the resultantstresses and the external distributed load t as∫Aδ εˆT σˆdA =∫AδuT tdA (21)where A is the mid-plane surface of theplate, δu = [δu0,δv0,δw0,δθx,δθy]T andt= [ fx, fy, fz,mx,my]T .Considering the uncertainties in material proper-ties, b, constitutive matrix Dˆ and the structural re-sponse u are stochastic function of b. Using Taylorseries expansion based perturbation approach to ap-proximate the stochastic terms Dˆ, u and εˆ , and re-taining the second-order accuracy, then the zeroth-,first- and second-order equations of Eq.(21) can bewritten as follows (Kleiber and Hien, 1992):The zeroth order∫Aδ εˆT Dˆ(b¯)εˆ(b¯)dA =∫AδuT t(b¯)dA (22)The first ordern∑i=1{∫Aδ εˆT Dˆ(b¯)[Dbi εˆ(b¯)]dA−∫AδuT[Dbit(b¯)]dA+∫Aδ εˆT[DbiDˆ(b¯)]εˆ(b¯)dA}δbi = 0 (23)The second ordern∑i=1n∑j=1{∫Aδ εˆT Dˆ(b¯)[Hbib j εˆ(b¯)]dA−∫Aδ εˆT[Hbib jDˆ(b¯)]εˆ(b¯)dA−2∫Aδ εˆT[DbiDˆ(b¯)][Db j εˆ(b¯)]dA−∫AδuT[Hbib jt(b¯)]dA}δbiδb j = 0 (24)412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20153. FINITE ELEMENT IMPLEMENTATION3.1. Finite element approximation of C¯Using standard notations as follows,u˜µ = Na˜µ , ∇su˜µ = Ba˜µ , η = Nδa,∇sη = Bδa and ε = Ba∗,where N denotes shape function, B is the strain-displacement matrix, a˜µ is nodal displacement fluc-tuation vector, δa is virtual nodal displacementfluctuation vector, a∗ denotes the given nodal dis-placement vector. The finite element approxima-tion to the zeroth-, first- and second-order varia-tional principles in Eqs.(9-11), respectively are ob-tained as:The zeroth-order{Ka˜µ +Ka∗}·δa= 0 (25)The first-order{n∑p=1{K[Dbp a˜µ]+[DbpK]a˜µ+[DbpK,p]a∗}δbp}·δa= 0 (26)The second-order{n∑p=1n∑q=1{K[Hbpbq a˜µ]+[HbpbqK]a˜µ+2[DbpK][Dbp a˜µ]+[HbpbqK]a∗}δbpδbq}·δa= 0 (27)with K =∫ΩsµBTCµBdV ,[DbpK]=∫ΩsµBT[DbpCµ]BdV , and[HbpbqK]=∫ΩsµBT[HbpbqCµ]BdV denoted as the stiff-ness matrix and its first- and second-order partialderivatives, respectively. The solutions for mi-crostructure displacement fluctuation field a˜µ andits derivatives[Dbp a˜µ]and[Hbpbq a˜µ]can be foundby introducing appropriate boundary conditionsfor the microstructure such as the three classicboundary conditions of linear displacement, peri-odic displacement and anti-periodic traction, andconstant tractions. Details can be found in Zhouet al. (2014). With these at hand, the constitutiverelation between the applied macrostrain ε¯ and thehomogenized macrostress σ¯ can be calculated:C¯=σ¯ε¯ (28)and similarly for its first- and second-order partialderivatives[DbiC¯]and[Hbib jC¯], respectively.3.2. Finite element formulation for laminatedcomposite platesAgain, the finite element implementation to ana-lyze the laminated composite plates can be obtainedby introducing a conventional finite element dis-cretization in Eqs.(22-24). Then their finite elementapproximations can be written as:The zeroth orderKu= F (29)The first ordern∑i=1{K[Dbiu(b¯)]+[DbiK]u−[DbiF(b¯)]}δbi = 0 (30)The second ordern∑i=1n∑j=1{[Hbib jK(b¯)]−[Hbib jF(b¯)]−2[DbiK(b¯)][Db ju(b¯)]−[Hbib jK(b¯)]u}δbiδb j = 0 (31)with K=∫ABT DˆBdA, [DbiK] =∫ABT[DbiDˆ]BdA,and[Hbib jK]=∫ABT[Hbib jDˆ]BdA denoted as thestiffness matrix of laminated plate elements and itsfirst- and second-order partial derivatives, respec-tively. Solving the Eqs.(29-31) consecutively, thenodal displacements u and its first- and second-order partial derivatives, [Dbiu] and[Hbib ju]can beobtained, and other structural response terms can becalculated straightforwardly.It should be noted that the stiffness matrix Dˆand its derivatives[DbiDˆ]and[Hbib jDˆ]in theabove Eqs.(29-31) are obtained by Eq.(28) throughstochastic homogenization method. According tothe first-order shear deformation theory, the through512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015thickness direction stress of the k-th ply σ3 is as-sumed to be zero. A reduced stress-strain relationcan be obtained from the stiffness matrix Eq.(28) asσ =Qε withQi j = C¯i j−C¯i3C¯ j3C¯33, i, j = 1,2,6 (32)Hence, the in-plane and transverse shear stiffnessmatrix in Eq.(18) areD1 =Q11 Q12 Q16Q21 Q22 Q26Q61 Q62 Q66 D2 =[C¯44 C¯45C¯54 C¯55].4. NUMERICAL EXAMPLEFor the demonstration of adequacy of the proposedapproach, we analysed a square laminated compos-ite plate. The dimension of the structure is given inFig.1a. The plate is made of 4 graphite/epoxy uni-directional lamina with material properties listed inTable 1. The stacking sequence for respective lam-ina is (30/− 30)4 as an angle ply laminate, wherethe subscript 4 denotes the number of repetitions.The thickness of the lamina is assumed to be 5 mm.The structure is subject to a uniformly distributedload of 1000.0N/m2 or 1× 10−3N/mm2, and theedges are simply supported.Table 1: Statistical properties of random variablesRandom variable Symbol ValueFibreAxial modulus Ez 233 GPaTransverse modulus Ep 23.1 GPaAxial Poisson ratio νz 0.2Transverse Poisson ratio νp 0.4Axial shear modulus Gz 8.96 GpaMatrixYoung’s modulus Em 4.62 GPaPoisson ratio νm 0.36 GPaThe response of the material at microscale levelis analysed using representative volume element(RVE) with width l. In general, the width l is muchsmaller than the characteristic thickness of the lam-inae. The fibre-reinforced material is assumed tohave periodic arrangement of finescale fibres em-bedded in a polymer matrix as shown in Fig.1b. Acubic RVE sample is shown in Fig.1c for a compos-ite with hexagonal arrangements of unidirectionalfibres. A local Cartesian coordinate system (1-2-3)is introduced at the microscale and oriented such1-axis is aligned parallel to the axis of the fibres.yxza bt(a) Plate(b) Detail of fibres arrange-ment132(c) RVEFigure 1: Example of laminated fibre-reinforced com-posite plateTo calculate the effective stiffness matrix of therepresented fibre-reinforced composite, the RVEhas been meshed into 6263 four-node tetrahe-dral elements consisting of 4206 elements for fi-bres and 2057 elements for matrix, with a to-tal of 1441 nodes. Implementing Eqs.(25-27) onthe MoFEM (mesh-oriented finite element method)program (Kaczmarczyk et. al., 2014), the effectiveelastic tensor and its first- and second-order deriva-tives can be obtained through Eq.(28) for specifiedboundary condition. Here the effective constitutivemodel obtained under periodic displacement andanti-periodic traction boundary condition case arepresented.With the obtained effective stiffness matrix ofeach lamina, the reduced stiffness matrix and itsfirst- and second-order partial derivatives can beobtained through Eq.(32). Introducing a certaintype of element, the stochastic finite element anal-ysis for the laminated plate can be conducted basedon Eqs.(30-31). In the present study, the plate is612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015discretized into 12× 12 four-node rectangular ele-ments.To demonstrate the accuracy of the present dualSFEM method, a comparison between the presentapproach and Monte Carlo simulation method(MCS) with 5000 samples has been performed. Un-certainty in the axial modulus of fibre, Ez were con-sidered and various coefficients of variation rangedfrom 0.025 to 0.15 were examined. Results areshown in Fig 2 for the effective elastic propertieswith CoV of 0.15 for Ez, and Fig.3 for the deflec-tion at the centre of the plate.(a) Mean value(b) Coefficient of variationFigure 2: Statistics of the components of the effectiveelastic property due to variation in axial modulus offibre, Ez, with CoV of 0.15In general, Fig 2 and Fig.3 show good agreementbetween the present approach and MCS on the re-sults of mean value and coefficient of variation withrespect to various CoVs of Ez. This indicates thatthe numerical accuracy of the present dual SFEM issufficient.CoV of axial modulus of fibre, Ez0 0.025 0.05 0.075 0.1 0.125 0.15 0.175Mean value (mm)-0.337-0.336-0.335-0.334-0.333-0.332-0.331-0.33PSMFEMMCS(a) Mean valueCoV of axial modulus of fibre, Ez0 0.025 0.05 0.075 0.1 0.125 0.15 0.175Coefficient of variation00. Coefficient of variationFigure 3: Statistics of the deflection at the centre ofthe plate w.r.t. various coefficients of variation in axialmodulus of fibre, Ez5. CONCLUSIONSA dual stochastic finite element formulation tak-ing into account the multi-layer effect and the mi-croscopic variability of material properties in eachlamina was developed for stochastic analysis oflaminated composite plates. Stochastic homog-enization method was used to propagate micro-scopic uncertainties. A comparison with Monte-Carlo simulation on the numerical accuracy showsthat the proposed approach could provide reason-able probabilistic prediction.712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015ACKNOWLEGEMENTThe authors gratefully acknowledge the financialsupport provided for this study by the UK Engineer-ing and Physical Sciences Research Council (EP-SRC) under grant reference EP/K026925/1.REFERENCESChamis, C. C. (2004). “Probabilistic simulation ofmulti-scale composite behavior.” Theoretical and Ap-plied Fracture Mechanics, 41(1–3), 51–61.Charmpis, D. C., Schuëller, G. 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