Accepted Manuscript The effective elastic properties analysis of periodic microstructure with hybrid uncertain parameters Wenqing Zhu , Ning Chen , Jian Liu , Siyuan Xia PII: DOI: Reference: S0020-7403(18)31036-1 https://doi.org/10.1016/j.ijmecsci.2018.08.018 MS 4476 To appear in: International Journal of Mechanical Sciences Received date: Revised date: Accepted date: 31 March 2018 16 August 2018 18 August 2018 Please cite this article as: Wenqing Zhu , Ning Chen , Jian Liu , Siyuan Xia , The effective elastic properties analysis of periodic microstructure with hybrid uncertain parameters, International Journal of Mechanical Sciences (2018), doi: https://doi.org/10.1016/j.ijmecsci.2018.08.018 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT Highlights Hybrid uncertain-but-bounded model is used to treat the uncertainty of microscopic materials in homogenization analysis. Homogenization-based hybrid uncertain analysis method (HHUAM) is with bounded hybrid uncertainty. CR IP T presented to estimate the effective elastic tensor of microscopic materials The influence of microscopic bounded hybrid material uncertainties on the AC CE PT ED M AN US homogenized macroscopic elastic property is investigated. ACCEPTED MANUSCRIPT The effective elastic properties analysis of periodic microstructure with hybrid uncertain parameters Wenqing Zhu, Ning Chen*, Jian Liu, Siyuan Xia CR IP T (State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan, 410082, People’s Republic of China) Abstract: This paper presents a homogenization-based hybrid uncertain analysis material properties with AN US method (HHUAM) for the prediction of the effective elastic tensor for microscopic uncertain-but-bounded parameters. For those uncertain-but-bounded parameters related to the microscopic material properties, the M ones with sufficient statistic information are modelled as bounded random variables, ED and those without enough statistics to build the probability density functions are defined as interval variables. Based on the finite element framework for PT homogenization method, the effective elastic tensor with bounded hybrid uncertain CE parameters can be expanded by using Gegenbauer series expansion. The variation ranges of the expectation and variance of the effective elastic tensor can be obtained AC due to the orthogonality relationship of Gegenbauer polynomials. Two numerical cases are carried out to verify the effectiveness and the efficiency of the HHUAM. The influence of the bounded hybrid uncertainties in microstructures on homogenized macroscopic elastic properties of heterogeneous materials is also investigated. ACCEPTED MANUSCRIPT 1. Introduction Composite materials are extensively used in various fields due to its more tunable and versatile material properties. The macroscopic performance of a composite CR IP T material is generally determined by the complex microstructures [1,2]. By using homogenization method, the homogenized parameters of a periodic composite material can be obtained, and the heterogeneous media can be transformed into an energetically equivalent material model [3-5]. So far, the most commonly used method for AN US homogenization problems is the finite element method [6-9]. In the manufacturing process of composite materials, the physical parameters of its constituent materials and the morphological parameters of its microstructure are M inevitably affected by uncertain factors. Therefore, these uncertainties existed in the ED microstructure or the microscopic material properties must be taken into consideration during the process of homogenization. For this reason, many researchers focused on PT the uncertain analysis of homogenization problems. Koishi et al. presented a new CE finite element approach based on the homogenization method for composite materials with stochastic uncertainties [10]. Kaminski et al. reported a stochastic analysis AC method to solve the homogenization problems by using perturbation technique [11]. Sakata et al. discussed how the homogenized elastic property influenced by some microscopic uncertainty through the Monte-Carlo method [12]. Besides, Sakata et al. used a perturbation-based method to predict the equivalent elastic properties of composite materials [13]. Ostoja-Starzewski investigated the microstructural randomness in thermomechanics using homogenization method [14]. Xu and Brady ACCEPTED MANUSCRIPT used Fourier Galerkin method to solve the stochastic homogenization of random media elliptic problems [15]. Ma et al. investigated the homogenization analysis of heterogeneous materials based on random factor method [16]. Kaminski reported the homogenization for polymers with rubber particles considering probabilistic CR IP T uncertainty in Poisson ratio [17]. Recently, Chen et al. proposed an interval homogenization-based method for solving the homogenization problems when the probabilistic information of the microscopic uncertain parameter cannot be obtained AN US unambiguously [18]. From the overall perspective, research of homogenization problems for composite materials with microscopic uncertainties are focused on either random M parameters or interval parameters. But in some cases, random parameters and interval parameters may exist at the meantime. Under this circumstance, a hybrid uncertain ED model was presented by Elishakoff and Colombi [19]. Up to now, the hybrid uncertain PT model was extensively used in structural reliability analysis and structural-acoustic response analysis [20-23]. Hybrid uncertain methods were generally on the basis of CE the perturbation method (PM) or Monte-Carlo method (MCM) [24, 25]. It is well AC known that the MCM is the most straightforward approach for uncertainty analysis [26]. However, the excessive computational burden of MCM limits its applications in engineering problems. The perturbation method has a much higher efficiency than the MCM, but it can only be applied to uncertain problems with small uncertainty level because of its inherent drawback. To overcome the defects of the PM and MCM, the polynomial approximation method has been introduced to handle the hybrid uncertain ACCEPTED MANUSCRIPT problems. Lately, Wu et al. proposed a Polynomial-Chaos-Chebyshev-Interval (PCCI) method for vehicle dynamics with hybrid uncertain parameters [27]. In practical engineering practice, uncertain parameters are always bounded because of the tolerance design. Naturally, a bounded hybrid uncertain model is emerged, in which CR IP T the range of both the random variables and interval variables are strictly defined. Relatively speaking, the bounded hybrid uncertain model has more practical application value. Yin et al. proposed an orthogonal polynomial approximation for structural-acoustic system and acoustic field with hybrid AN US method uncertain-but-bounded parameters, in which a derivative λ-PDF model based on Gegenbauer polynomial expansion is employed[28, 29]. Compared with the traditional orthogonal polynomial approximation methods in probabilistic analysis, M the advantage of the derivative λ-PDF model is that it can approximate arbitrary ED mono-valley and mono-peak PDF. Furthermore, the accuracy of Gegenbauer PT polynomial expansion method is better compared with the perturbation method in interval analysis, and the efficiency of the Gegenbauer polynomial expansion method CE is greater than the MCM. AC From these backgrounds mentioned above, research on the homogenization problems with bounded hybrid uncertain parameters is promising and interesting, which is still unreported. Thus, a homogenization-based hybrid uncertain analysis method (HHUAM) for estimating the effective elastic properties of periodic microstructure with bounded hybrid uncertain parameters is presented in this paper. Based on the homogenization method and the Gegenbauer series expansion method, ACCEPTED MANUSCRIPT the formulation of HHUAM is deduced. In HHUAM, the effective elasticity tensor with bounded hybrid uncertain parameters is approximated through a N-order Gegenbauer series expansion. The expectation and variance of the effective elasticity tensor with bounded hybrid uncertain parameters can be calculated due to the CR IP T orthogonality of the Gegenbauer polynomials. The influence of bounded hybrid uncertainties in microstructures on homogenized macroscopic elastic properties of heterogeneous materials is also investigated. AN US This paper is organized as follows. The homogenization method is introduced in Section 2. The theory of Gegenbauer series expansion method is introduced in Section 3. The HHUAM is proposed for the homogenization problems with bounded hybrid M uncertain parameters in Section 4. Two numerical cases are investigated in order to drawn in Section 6. ED verify the accuracy and the efficiency of HHUAM in Section 5. The conclusions are CE PT 2. Homogenization method The homogenization method can be used to calculate the effective properties of AC periodic composite materials from their unit cell topology [30-32]. In this section, the homogenization method will be introduced briefly. Two levels of coordinate system in the homogenization method includes the macro-level x and the micro-level y = x / ε, where ε denotes the small positive scaling parameter. Here, the representative volume element is defined as Ω, and a body subject to the body force fi satisfies the following equation ACCEPTED MANUSCRIPT ij xj where fi 0 in (1) ij represents the stress, xj denotes the macro generalized coordinate system. ui The prescribed displacement boundary F on the boundary u and the traction Fi on the are CR IP T ui 0 on u (2) ij n j Fi on F (3) where nj denotes the normal vector. expressed as AN US The relationship between the stress-strain and the strain-displacement can be 1 ui u j eij (u ) 2 x j xi (4) M ij Dijkl ekl (u ) (5) ED where eij represents the strain, Dijkl denotes the constitutive matrix. PT By the use of asymptotic expansion, the displacement ui can be approximated CE with respect to ε and expressed as follow 0 0 1 (6) 1 is the macroscopic displacement, ui denotes the first order variation AC in which, ui ui (x) ui (x, y) ui (x, y) of average displacement and is defined as follow [33]. where ik l ui ik l ekl ui 1 0 (7) denotes the characteristic displacement function, and can be obtained by the auxiliary equation as follow kmn D y j ijkl yl d y j Dijmn d (8) ACCEPTED MANUSCRIPT After solving the ik l through Eq. (8), the effective elasticity tensor can be calculated by H ijmn D pkl Dijkl Dijpq yq 1 d (9) expressed as Kχ F 1 D I-bχ d AN US DH = CR IP T By the use of finite element method, Eqs. (8) and (9) can be rewritten and (10) (11) in which, b denotes the strain matrix at the microscale, and the stiffness matrix K and the force vector F can be represented as K bT Dbd (12) F bT Dd (13) M ED PT 3. Gegenbauer series expansion method (GSEM) CE 3.1. Gegenbauer polynomials AC The Gegenbauer polynomials of n degree denoted by Gn (ξ) can be defined as below [34]: G0 ( ) 1 G1 ( ) 2 (n 1)Gn 1 ( ) 2(n ) Gn ( ) (n 2 1)Gn 1 ( ), n 2 (14) where λ is a polynomial parameter and λ > 0. The orthogonality relationship of Gegenbauer polynomials on ξ [-1, 1] related to ACCEPTED MANUSCRIPT the weight function ρλ (ξ) can be expressed as 1 1 hi , i j ( )Gi ( )G j ( )dx 0, i j (15) where ( 1) (1/ 2)( (1/ 2)) 21 2 ( 1) hi i !(i ) 2 ( ) in which, () represents the Gamma function. (16) CR IP T ( ) k (1 2 ) (1/2) , k AN US 3.2. Gegenbauer series expansion (GSE) for the approximation of a function On the basis of the orthogonal relationships of the Gegenbauer polynomials, a M continuous function f (ξ) defined on ξ [-1, 1] can be expressed as N f ( ) N ( ) fiGi ( ) (17) ED i 0 where fi represents the i th (i = 0, 1,…, N) expansion coefficient, N is the retained order PT of GSE. N1 NL i1 0 iL 0 f ( ) fi1 ,...,iL Gi1 1 (1 ) GiLL ( L ). AC CE As for the L-dimensional problem, f (ξ) can be expanded as where Nl (l = 1, 2,…, L) denotes the retained order of GSE in relation to ξl, (18) fi1 ,...,iL denotes the expansion coefficient. 3.3. Computation of the coefficients of the Gegenbauer series expansion The coefficient of the GSE can be obtained by the use of the Gauss-Gegenbauer ACCEPTED MANUSCRIPT integration method and the weighted least squares method [35]. The weighted least squares error E can be expressed as 1 E ( )(rN )2 d (19) 1 where rN is the residual error of the Gegenbauer series expansion, which can be N CR IP T expressed as rN f ( ) N ( ) f ( ) f j G j ( ) . j 0 (20) To minimize E, the necessary condition of the coefficient fi (i = 0, 1,…, N) is i 0,1,..., N AN US E 0, fi (21) Substituting Eqs. (19) and (20) into Eq. (21), we can obtain N j 0 1 ( ) f ( )Gi d 1 f jG j ( ) Gi ( ) ( )d , Gi ( ) simplified as i 0,1,..., N . (22) is orthogonal to each G j ( ) for j = 0, 1,…, N, Eq. (22) can be ED Since 1 M 1 1 hi PT fi 1 1 ( ) f ( )Gi d , i 0,1,..., N (23) CE Through the Gauss-Gegenbauer integration formula [35], Eq(23) can be rewritten AC as follow fi 1 hi 1 1 ( ) f ( )Gi d 1 m f ( j )Gi ( j ) Aj hi j 1 (24) In the above equation, j ( j 1, 2,..., m) represents the interpolation point; Aj ( j 1,2,..., m) denotes the weight which is defined by Aj 222 ( ) 2 2 (2 m) (1 x 2j )1 Gm ( x j ) , (1 m) j 0,1,..., m (25) ACCEPTED MANUSCRIPT where Gm ( x j ) 2Gm 11 (26) Likewise, through the least-squares approximation and the Gauss-Gegenbauer integration formula, the expansion coefficient fi1 ,...,iL can be calculated and fi1 ,...,iL 1 1 h1 hLL 1 1 h1 hLL 1 1 1 CR IP T represented as i1 ,...,iL ( ) f ( )Gi1 ,...,iL ( )d1 d 2 1 m1 mL j1 1 jL 1 f ( (27) j1 ,..., jL )Gi1 ,...,iL ( j1 ,..., jL ) Aj1 ,..., jL Aj1 ,..., jL can be expressed as AN US in which, j1 ,..., jL are the interpolation points, Aj1 ,..., jL Aj11 Aj22 AjLL . (28) Generally, when the number of the integration points related to ξi increases to Ni + M 1, the accuracy of the integration method can be acceptable. Therefore, the integration ED points are set as mi = Ni + 1 in this paper. Through Eq. (27), the total number of PT integration points is Ntotol = (N1 + 1) (NL + 1). CE 4.HHUAM for homogenization problems with hybrid AC uncertain-but-bounded parameters In this part, by introducing the GSEM to the FEM of homogenization, a hybrid uncertain homogenization-based method (HHUAM) is proposed for homogenization problems with hybrid uncertain-but-bounded parameters. ACCEPTED MANUSCRIPT 4.1. Definition of bounded hybrid uncertain model. In this section, the uncertainties in constituent material properties are treated as bounded hybrid variables. On the one hand, when the PDFs of the bounded uncertain parameters cannot be constructed because of the insufficient information, the bounded CR IP T uncertain parameters are defined by an interval vector a = [a1, a2, …, ai]. The interval variable ai satisfies ai aiI ai , ai , i 1, 2,..., L1 (29) AN US where L1 denotes the total number of the interval variables. On the other hand, when the PDFs of the bounded uncertain parameters can be structured unambiguously, the bounded uncertain parameters are defined by a bounded M random vector b = [b1, b2, …, bi]. The bounded random variable bi (i = 1, 2, …, L2) satisfies ED Pb (bi ) 0, bi bi , bi Pb (bi ) 0, else PT (30) where Pb (bi) denotes the PDF of bi. CE Thus, the hybrid uncertain vector x = [x1, x2, …, xL] can be expressed as L1 L2 L (31) AC x [a, b] a1 , a2 ,..., aL1 , b1 , b2 ,..., bL2 , where L denotes the total number of the hybrid uncertain-but-bounded parameters; L1 and L2 denote the interval and the bounded random variables, respectively. 4.2 Transformation of bounded hybrid uncertain variables The key point of random polynomial expansion method is that the PDFs of ACCEPTED MANUSCRIPT random variables can be represented by the weight function of orthogonal polynomials. The Gegenbauer polynomial can be applied to stand for a family of bounded PDFs, termed as λ-PDF, because it is a parametric polynomial [34]. Thus, uncertain problems with bounded random variables following λ-PDFs can be solved through the CR IP T Gegenbauer series expansion. An arbitrary bounded random variable β with λ-PDF is defined as k (1 x 2 ) (1/2) x 1,1 P ( x) ( x) else 0 (32) AN US where ρλ () denotes the weight function as shown in Eq. (16). Gegenbauer series expansion can be directly employed for random problems with bounded random parameters following λ-PDFs. However, when the bounded random M variables are following arbitrary probability distribution, we should transform the ED bounded random variables into a function of β [36]. By using the transformation process, an arbitrary random variable b is represented as 1 , 2 ,..., L PT b b( ) , bi bi ,k ( i ) bi ,0 bi ,1i bi ,2 i 2 bi , B i B , i 1, 2,..., L2 (33) CE where βi is the i th bounded random variable with λ-PDF; B denotes the polynomial AC order; bi,k (k = 0, 1,…, B) is the transformation coefficient of the function bi,k (βi), which can be obtained by To find , bi ,k (k 0,1,..., B) J Min Pori ( xi ) P ( xi ) i 1 b 2 (34) In the abovementioned equation, xi are discrete points in the domain of b, Pori (xi) is the original PDF of b, Pb ( xi ) denotes the PDF of the polynomial function bi,k (βi) . ACCEPTED MANUSCRIPT Eq. (34) can be solved by numerous optimization methods. When the parameter λ and the transformation coefficients bi,k are chosen properly, the PDF of the function bi,k (βi) can approximate the mono-valley and mono-peak PDFs in any interval [36]. Similarly, the interval variable bi in the arbitrary interval vector a can be CR IP T transformed as a function of interval variable i [-1, 1]. The arbitrary interval vector a is represented as: 1 ,2 ,...,i a a( ), where ai ,0 AN US ai ai (i ) ai ,0 ai ,1i , i 1, 2,..., L1 ai ai 2 , ai ,1 ai ai 2 (35) (36) M As with GSEM for the random problem, the polynomial parameter λ in GSEM for interval analysis is also determined by Eq. (34). Ref. [28] shows that the GSE can ED achieve high precision for interval analysis with a very small value of λ. Therefore, the PT parameter λ of the Gegenbauer series related to interval variables is taken as 0.001 and named as λ0 in this paper. CE Thus, the hybrid uncertain vector x = [x1, x2 ,…, xL] can be transformed as AC x [a( ), b( )] a1 (1 ), a2 (2 ),..., aL1 (L1 ), b1,k ( 1 ), b2,k ( 2 ),..., bL2 ,k ( L ) , L1 L2 L 2 (37) 4.3 HHUAM for hybrid uncertain analysis of homogenization problem Based on the theory of GSEM and the transformation formula of bounded hybrid variables in Eq. (37), the effective elasticity matrix related to bounded hybrid uncertain ACCEPTED MANUSCRIPT variable x can be expressed as DH DH x DH a( ), b( ) N1 N L1 L2 i1 0 iL1 L2 0 = fi1 ,...,iL L Gi1 ,...,iL ( )GiL 1 ,...,iL L ( ), 0 1 2 1 1 1 k 1, 2,..., Ntotal 2 (38) where the constant coefficients of the expansion fi1 ,...,iL L can be obtained through Eq. 1 2 fi1 ,...,iL L 1 2 m1 1 L1 L2 L1 L2 j1 1 h11 h mL1 L2 jL1 L2 1 CR IP T (27) and expressed as 0 0 DH a(ˆ ), b( ˆ ) Gi1 ,...,iL ( )GiL 1 ,...,iL L ( ) A A 1 1 1 2 (39) denotes the value of the AN US In the abovementioned equation, DH a(ˆ ), b( ˆ ) effective elasticity matrix at the interpolation point. L1 and L2 are the number of interval variables and bounded random variables, respectively. ˆ and A ˆ and represent the A represent M integration points and weights with respect to interval variable. 0 ED the integration points and weights related to the bounded random variable. d kH of the DH is written as follow The k th element PT d kH d kH x d kH a( ), b( ) N1 N L1 L2 i1 0 iL1 L2 0 CE = fi1k,...,iL L Gi1 ,...,iL ( )GiL 1 ,...,iL L ( ), 0 1 2 1 1 1 k 1, 2,..., Ntotal 2 (40) AC where Gi1 ,...,iL ( ) 0 1 iL11 ,...,iL1 L2 G 0 i0 0 Gi1 i1 Gi2i2 GiL L1 1 i i i 1 1 1 ( ) GiL 1 GiL 2 GiL L L11 L1 2 (41) L1 L2 2 It needs two steps to obtain the bounds of the expectation and variance of the effective elasticity matrix related to bounded hybrid uncertain variables [28]. First of all, the interval parameters are treated as constant parameters, then Eq. (40) can be rewritten as ACCEPTED MANUSCRIPT N1 N L1 k 0 d fi1 ,...,iL L Gi1 ,...,iL ( ) GiL 1 ,...,iL L ( ) 1 2 1 1 1 2 iL11 0 iL1 L2 0 i1 0 iL1 =0 N L11 N L1 L2 H k N L11 = iL11 0 N L1 L2 iL1 L2 0 cikL 1 ,...,iL L GiL 1 ,...,iL L ( ) 1 1 2 1 1 (42) 2 where c Thus, the expectation of H k N L1 i1 0 iL1 =0 fi1k,...,iL L Gi1 ,...,iL ( ) d kH 0 1 2 1 can be expressed as N L11 N L1L2 k E ciL 1 ,...,iL L GiL 1 ,...,iL L ( ) 1 1 2 1 1 2 iL11 0 iL1L2 0 (43) AN US d N1 CR IP T k iL11 ,...,iL1 L2 N N L1 L2 L11 cikL 1 ,...,iL L GiL 1 ,...,iL L ( ) P ( L1 1 ) P ( L1 L2 ) d L1 1 d L1 L2 1 1 2 1 1 2 L11 L1 L2 iL11 0 iL1L2 0 M (44) Since the orthogonality of the Gegenbauer polynomial basis, the expectation of PT ED d kH can be calculated analytically [36] and expressed as follow For calculating the variance of k d c0,...,0 (45) H k d kH , the expectation of the mean square of d kH AC CE need to be calculated and can be expressed as [36] 2 N L11 N L1L2 k E (d ) E ciL 1 ,...,iL L GiL 1 ,...,iL L ( ) iL11 0 iL1L2 0 1 1 2 1 1 2 H 2 k N L11 N L1 L2 iL11 0 Accordingly, the variance of iL1 L2 0 d kH c k iL11 ,...,iL1 L2 2 h L11 iL11 hiLL1LL2 . 1 2 can be calculated and expressed as (46) ACCEPTED MANUSCRIPT d2 E (d kH )2 ( d ) 2 H k H k N L11 N L1 L2 iL11 0 iL1 L2 0 2 c h k iL11 ,...,iL1 L2 L11 iL11 hiL L c L1 L2 1 (47) 2 k 0,...,0 2 To substitute Eq. (43) into Eq. (44) and Eq. (47), we can obtain N1 N L1 i1 0 iL1 =0 d H d H ( ) fi1k,...,iL1 ,0,...,0Gi1 ,...,iL1 ( ) 2 d kH k (48) CR IP T k 0 2 N1 N L1 k 0 fi1 ,...,iL L Gi1 ,...,iL ( ) hiLL111 hiLL1LL2 1 2 1 1 1 2 iL11 0 iL1 L2 0 i1 0 iL1 =0 N L11 N L1 L2 2 (49) AN US N1 N L1 0 fi1k,...,iL ,0,...,0Gi1 ,...,iL ( ) 1 1 i 0 i =0 L1 1 Secondly, the bounds of the expectation and variance of d kH can be obtained through Monte-Carlo simulation and expressed as I H k I ED 2 d kH , minI d H ( η) , maxI d H ( η) ηη k k ηη minI d2H ( η) , maxI d2H ( η) ηη k k ηη M d (50) PT Various methods can be used to calculate the extremum in Eq. (50). Interval arithmetic is efficient to obtain the bounds of interval functions. Nonetheless, the CE wrapping effect of interval arithmetic may lead overestimation [37]. Up to now, the AC Monte-Carlo simulation (MCS) is regarded as the most robust method for the interval problems [38]. Therefore, in this paper, the MCS is adopted to calculate the extreme value of the functions in Eq. (50). It can be noted that the interval functions in Eq. (50) are simple functions. Thus, the computational cost of the MCS for the interval functions in Eq. (50) is acceptable. ACCEPTED MANUSCRIPT 4.4 The procedure of HHUAM for homogenization problem with hybrid uncertain-but-bounded parameters Step 1. Calculating the polynomials parameter λ and the transformation coefficient related to each bounded uncertain parameter. For the bounded random variable λ0 = 0.001 [28], ai,k is determined by Eq. (36). CR IP T bi, λ and bi,k can be obtained by Eq. (34). For the interval variable ai, λ is set as integration through Eq. (25). AN US Step 2. Calculating the weights and the interpolation points of Gauss-Gegenbauer Step 3. Calculating the effective elasticity matrix at the interpolation points based on Eq. (11). M Step 4. Calculating the expansion coefficients of the GSE by Eq. (39). ED Step 5. Calculating the bounds of the expectation and variance of the effective elasticity matrix with bounded hybrid uncertain parameters according to Eq. (50). PT It is worth mentioning that the accuracy of GSE can be improved via employing CE higher retained order [36]. Thus, the proposed HHUAM can be conveniently applied to the bounded hybrid uncertain problem with large uncertain level. Considering the AC computational accuracy and the computational cost, the relative improvement criterion is used to estimate the retained order of HHUAM [28]. 5. Numerical examples In this section, two cases are used to investigate the accuracy and efficiency of the ACCEPTED MANUSCRIPT HHUAM. On account of the convergence of the Monte-Carlo simulation (MCS) [39], the results calculated by the MCS with enough sampling points for homogenization with bounded hybrid uncertain parameters are used as reference solution. In MCS, the interval variables are given random sampling values within the intervals, and the CR IP T expectation and variance can be obtained by sampling the bounded random variables. With time repeating, we can obtain the bounds of the expectation and variance. Simulations in numerical cases are carried out by using MATLAB R2015b on a 5.1. Rectangular microscale voids AN US 3.30GHz Intel(R) Xeon(R) CPU E3-1230 V2. M In this case, a unit square cell contains a 0.4×0.6 rectangular hole as shown in Fig. 1 is used to demonstrate the HHUAM. The finite element model consists of 304 ED elements and 364 nodes. The Young’s modulus E and the Poisson’s ratio ν of the solid PT phase material are 4.5 GPa and 0.39, respectively. Considering the microscopic uncertainty, the Young’s modulus E and the Poisson’s ratio ν are treated as bounded CE uncertain parameters. The uncertain level of E and ν are defined as E and ν, AC respectively. ACCEPTED MANUSCRIPT 0.6 CR IP T 0.4 Fig. 1 Unit cell with rectangular hole For simplicity, the PDF of the bounded random Young’s modulus E is assumed to AN US be a linear function of E [36], Em represents the mean value, ∆E represents the deviation amplitude, E is a random variable which is defined on [-1, 1] and follows λE-PDF. The polynomial parameter λ is assumed to be 3. The information of M uncertain-but-bounded parameters is listed in Table 1. ED Table 1 The information of uncertain-but-bounded parameters. Uncertain parameters PT E (GPa) PDF EI = 4.5 [1- E , 1+ E ] E=Em + E ∆E, λ=3 νI = 0.39 [1- ν , 1+ ν ] Unknown CE ν Range AC In this numerical case, the uncertain level of input is assumed as E =ν =0. By using the relative improvement criterion shown in the appendix, the retained order vector N req estimated [ N Ereq , Nreq ] of HHUAM with respect to the uncertain parameter can be [28]. When Nreq [ N Ereq , Nreq ] 0 =0.2, respectively. setting the tolerance to 10-2, the estimation of are [2, 2], [2, 3] and [2, 5] at uncertain level 0 =0.05, 0 =0.1 and ACCEPTED MANUSCRIPT The effective elastic tensor of the unit cell is calculated by the MCS and the HHUAM. In the MCS, when the sample points are up to 5106, the results calculated by MCS almost no longer change. It worth mentioning that the interval and random sample points are respectively set as 1000 and 5000. Therefore, the total number of CR IP T sample points is 5106. The bounds of the expectation and variance of the effective elastic tensor are listed in Tables 2a, 2b and 2c when 0 = 0.05, 0 = 0.1 and 0 = 0.2, respectively. AN US Tables 2a-2c show that results obtained by HHUAM are very close to the referenced results obtained by MCS. It can be found out that the errors of the expectation of the effective tensor are much less than that of the variance of the M effective tensor. The reason for this phenomenon is that the variance is related to the square of the expectation. On the basis of the error propagation rule, the relative error of ED variance is larger than that of expectation. Besides, note that the statistics information PT of the effective elastic tensor is not deterministic. It is owing to the PDF information of the interval parameter is missing. Therefore, the statistics information of the effective CE elastic tensor is an interval. Furthermore, it can be found that the error may be larger AC than the tolerance tol =10-2. Nevertheless, the largest error is 1.92E-02, which is acceptable. It is indicated that the accuracy of HHUAM for predicting the effective elastic tensor is good. ACCEPTED MANUSCRIPT Table 2a Bounds of the expectation and variance of the effective elastic tensor (0 = 0.05). Bounds H D12 H D22 H D66 Variance MCS HHUAM Error MCS HHUAM Error Lower 2.7315 2.7316 3.66E-05 2.30E-03 2.33E-03 1.30E-02 Upper 2.9945 2.9940 1.67E-04 2.76E-03 2.80E-03 1.45E-02 Lower 1.1517 1.1519 1.74E-04 4.08E-04 4.15E-04 1.72E-02 Upper 1.4571 1.4567 2.75E-04 6.54E-04 6.63E-04 1.38E-02 Lower 3.6968 3.6968 1.62E-05 4.21E-03 4.27E-03 1.43E-02 Upper 4.0461 4.0454 1.73E-04 5.04E-03 5.11E-03 1.39E-02 Lower 0.5211 0.5211 5.76E-05 8.36E-05 8.49E-05 1.56E-02 Upper 0.5334 0.5334 9.37E-05 8.76E-05 8.89E-05 1.48E-02 CR IP T H D11 Expectation Bounds H D22 H MCS HHUAM Error MCS HHUAM Error Lower 2.6307 2.6301 2.28E-04 8.52E-03 8.65E-03 1.53E-02 Upper 3.1705 3.1698 2.21E-04 1.24E-02 1.26E-04 1.61E-02 Lower 1.0318 1.0315 2.91E-04 1.31E-03 1.33E-03 1.53E-02 Upper 1.6574 1.6568 3.62E-04 3.38E-03 3.43E-03 1.48E-02 Lower 3.5621 3.5614 1.97E-04 1.56E-02 1.59E-02 1.92E-02 Upper 4.2780 4.2770 2.34E-04 2.25E-02 2.29E-02 1.78E-02 Lower 0.5164 0.5163 1.94E-04 3.28E-04 3.33E-04 1.52E-02 Upper 0.5417 0.5416 1.85E-04 3.61E-04 3.67E-04 1.66E-02 PT D66 Variance M D12H Expectation ED H D11 AN US Table 2b Bounds of the expectation and variance of the effective elastic tensor (0 = 0.1). CE Table 2c Bounds of the expectation and variance of the effective elastic tensor (0 = 0.2). Bounds H D12 H D22 H D66 Variance MCS HHUAM Error MCS HHUAM Error Lower 2.4705 2.4707 8.10E-05 3.01E-02 3.05E-02 1.33E-02 Upper 3.6954 3.6903 1.38E-03 6.72E-02 6.81E-02 1.34E-02 Lower 0.8362 0.8369 8.37E-04 3.44E-03 3.50E-03 1.74E-02 Upper 2.2387 2.2342 2.01E-03 2.47E-02 2.50E-02 1.21E-02 Lower 3.3474 3.3475 2.99E-05 5.52E-02 5.60E-02 1.45E-02 Upper 4.9549 4.9492 1.15E-03 1.21E-01 1.22E-01 8.26E-03 Lower 0.5089 0.5089 7.86E-05 1.28E-03 1.29E-03 7.81E-03 Upper 0.5697 0.5691 1.05E-03 1.60E-03 1.62E-03 1.25E-02 AC H D11 Expectation ACCEPTED MANUSCRIPT It can be seen from Fig. 2 that the uncertainty levels of the interval of the expectation and the interval of variance of the effective elastic tensor are influenced by different uncertain levels of input. When 0 = 0.15, the retained order vector -2 Nreq [ N Ereq , Nreq ] is Nreq 4 [2,4] with the prescribed tolerance tol = 10 . Fig. 2 CR IP T shows that with the uncertain level of input increasing, the uncertainty levels of the interval of the expectation and the interval of variance are monotonously increasing. H Besides, note that the D12 is the most influenced by the hybrid uncertain-but-bounded H H H AN US parameters in microscopic material properties, followed by D11 and D22 , and D66 is the least influenced. In the meantime, it can be found that the uncertainty levels of the H H H interval of the expectation and the interval of variance of the D11 , D12 and D22 are H M much larger than the uncertain levels of input, whereas those of D66 are smaller than the uncertain levels of input. This phenomenon is usually caused by the combined ED influences of Young’s modulus and Poison’s ratio. As an example, the D11 and D12 PT terms in stiffness matrix for plain strain state are E(1-ν) / (1+ν)(1-2ν) and Eν / (1+ν)(1-2ν), respectively. Comparing D11 with D12, the influence decreases for D11 CE because of the negative correlation between E and -Eν in some degree. In detail, when AC the uncertain level of input increases to 20%, the uncertainty levels of the interval of the H expectation and the interval of variance of D12 are more than 45% and 75%, respectively, whereas the uncertainty levels of the interval of the expectation and the H interval of variance of D66 are less than 6% and 12%, respectively. The uncertainty H H levels of the expectation and the interval of variance of D11 and D22 are almost the same as those of input. ACCEPTED MANUSCRIPT CR IP T a PT ED M AN US b AC CE Fig. 2 The uncertainty levels of the interval of expectation (a) and the interval of variance (b) result from different uncertain levels of input Except the accuracy, another important factor in the evaluation of a numerical method is the computational efficiency, which is crucial for its application in practical engineering. Hence, the computational time of MCS and HHUAM for predicting the effective elastic tensor with respect to three uncertain levels of input is investigated and listed in Table 3. We can observe form Table 3 that the execution time of MCS at ACCEPTED MANUSCRIPT different uncertain level of input are very close to each other. This is because the sample points of MCS with respect to three uncertain levels of input are the same. Besides, Table 3 shows that more execution time is spent for predicting the effective elastic tensor through HHUAM with the uncertain level of input increasing. The CR IP T reason of this is that higher retained order of HUHAM is needed for reducing the estimation error for homogenization problem with larger uncertain level. Nevertheless, it can be observed from Table 3 that the computational cost of HHUAM for predicting AN US the effective elastic tensor is much less than that of MCS under these three uncertain levels of input, even if with a higher retained order. It can be indicated that the computation efficiency of HHUAM for predicting the effective elastic tensor is M desirable. Uncertainty of inputs 0 =0.1 Time of HHUAM (s) Order of HHUAM 1.47410 6 6.727 [2, 2] 1.47410 6 7.899 [2, 3] 10.333 [2, 5] 1.474106 CE 0 =0.2 Time of MCS (s) PT 0 =0.05 ED Table 3 Execution time of MCS and HHUAM for predicting the effective elastic tensor. AC 5.2 Particle filled polymers In this case, a unit Representative Volume Element (RVE) of a unidirectional fiber reinforced composite is investigated, as shown in Fig. 3. The similar finite element mesh model has been studied in Refs. [40-43]. The radius of the fiber in the center of the matrix is 0.2. The finite element model consists of 361 elements and 340 ACCEPTED MANUSCRIPT AN US CR IP T nodes. Fig. 3 RVE of a unidirectional fiber reinforced composite M The Young’s modulus E1 and the Poisson’s ratio ν1 of the matrix are 4.5 GPa and ED 0.39, respectively. The Young’s modulus E2 and the Poisson’s ratio ν2 of the fiber are 73 GPa and 0.21, respectively. Considering the microscopic uncertainty, the Young’s PT modulus E1, E2 and the Poisson’s ratio ν1, ν2 are treated as bounded uncertain parameters. The uncertainty level of E1, E2 and ν1, ν2 are defined as E , E and , 1 2 1 2 CE , respectively. The information of these uncertain-but-bounded parameters is listed AC in Table 4. For brevity but without loss of generality, the PDF of the bounded random Young’s modulus E is assumed to be a linear function of E [36]. E1 and E2 m m denotes I I the mean value of E1 and E2 , respectively. E1 and E2 denotes the deviation I I amplitude of the E1 and E2 , respectively. E1 and E2 are random variables which are defined on [-1, 1] and follows E PDF and E PDF , respectively. 1 2 ACCEPTED MANUSCRIPT The values of polynomial parameter E1 and E2 are assumed to be 1 and 2, respectively. The information of uncertain-but-bounded parameters is listed in Table 4. Table 4 The information of the uncertain-but-bounded parameters. Range PDF E1 (GPa) E1 4.5[1 E , 1 E ] I 1 1 1 1 0.39 [1 , 1 ] E2 (GPa) E2 73[1 E , 1 E ] I 1 1 I 2 2 2 0.21[1 , 1 ] I 2 2 2 E1 E1 1 E1 ， 1 m E E1 CR IP T Uncertain parameters Unknown E2 E2m E E2 ， E 2 2 2 Unknown AN US The uncertain level of input is assumed as E1 E2 1 2 0 , The retained order of HHUAM can be estimated by the relative improvement criterion [28]. req req req req req When setting the tolerance to 10-2, the estimation of N [ N E1 , N1 , N E2 , N 2 ] are M [2, 2, 2, 2], [2, 2, 2, 3] and [3, 2, 2, 6] at uncertain level 0 = 0.05, 0 = 0.1 and 0 = 0.2, respectively. ED In this case, the sample points of the MCS are 8106, the interval and random PT sample points are 1000 and 8000, respectively. The results obtained by MCS and CE HHUAM are listed in Table 5a-5c. Tables 5a-5c show that the bounds of expectation and variance of the effective AC elastic tensor obtained by HHUAM match the referenced results obtained by MCS perfectly. Note that even if the largest error of the bound of the effective elastic tensor is less than 2%, which can certainly be accepted. It can be observed that when 0 increases from 0.05 to 0.1, the variations of the lower and upper bounds of the expectation are small, whereas when 0 increases from 0.1 to 0.2, the variations of the lower and upper bounds of the expectation are much larger and become ACCEPTED MANUSCRIPT disproportionate. This indicates that the bounds of the expectation are more sensitive to large uncertainty. Besides, it can be noted that when 0 increases from 0.05 to 0.2, the variations of the bounds of the variance are larger than those of the expectation throughout. This phenomenon may be caused by that the variance is related to the CR IP T square of the expectation. Fig. 4 shows the uncertainty levels of the interval of the expectation and the interval of variance influenced by different uncertain levels of input. By using the AN US req req req req req relative improvement criterion, here the estimation of N [ N E1 , N1 , N E2 , N 2 ] is N4 [3, 2, 2,5] when the uncertain level of input 0 = 0.15 and the prescribed req tolerance is tol = 10-2. From Fig. 4, it can be observed that the uncertainty levels of the M interval of the expectation and the interval of variance increase monotonously with the uncertain level of input increasing. It can be noted that the uncertainty levels of the H H H ED interval of the expectation and the interval of variance of the D11 , D12 and D22 PT obtained by MCS and HUHAM are much larger than the uncertain levels of input, H while those of D66 are smaller than the uncertain levels of input. In detail, when the CE uncertain level of input increases to 20%, the uncertainty levels of the interval of the H AC expectation and the interval of variance of D12 are more than 70% and 90%, respectively, whereas the uncertainty levels of the interval of the expectation and the H interval of variance of D66 are less than 5% and 10%, respectively. The uncertainty H H levels of the interval of the expectation and the interval of variance of D11 and D22 are almost the same as those of input. Because of the combined influences of Young’s H modulus and Poison’s ratio, it can also be concluded that the D12 is the most ACCEPTED MANUSCRIPT influenced by the hybrid uncertain-but-bounded parameters in microscopic material H H H properties, followed by D11 and D22 , and D66 is the least influenced. Table 5a Bounds of the expectation and variance of the effective elastic tensor (0 = 0.05). Bounds H D22 H D66 MCS HHUAM Error MCS HHUAM Error Lower 8.8823 8.8915 1.04E-03 0.04740 0.04700 8.44E-03 Upper 11.2192 11.2119 6.51E-04 0.07510 0.07422 1.17E-02 Lower 4.8010 4.8091 1.69E-03 0.01387 0.01377 7.21E-03 Upper 7.1754 7.1672 1.14E-03 0.03075 0.03036 1.27E-02 Lower 8.8809 8.8901 1.04E-03 0.04739 0.04698 8.65E-03 Upper 11.2175 11.2102 6.51E-04 0.07508 0.07420 1.17E-02 Lower 1.8873 1.8877 2.12E-04 0.00218 0.00216 9.17E-03 Upper 1.9229 1.9233 2.08E-04 0.00227 0.00224 1.32E-02 CR IP T H D12 Variance AN US H D11 Expectation Table 5b Bounds of the expectation and variance of the effective elastic tensor (0 = 0.1). Bounds H D66 MCS HHUAM Error Lower 8.1605 8.1553 6.37E-04 0.1604 0.1585 1.18E-02 Upper 13.2819 13.2695 9.34E-04 0.4184 0.4132 1.24E-02 Lower 4.0530 4.0459 1.75E-03 0.0396 0.0391 1.26E-02 Upper 9.2449 9.2309 5.72E-02 0.2027 0.1999 1.38E-02 Lower 8.1592 8.1539 6.50E-04 0.1604 0.1585 1.18E-02 Upper Lower Upper M Error ED H D22 HHUAM PT H D12 Variance MCS 13.2799 13.2676 9.26E-04 0.4183 0.4131 1.24E-02 1.8729 1.8737 4.27E-04 0.0086 0.0085 1.16E-02 1.9428 1.9437 4.63E-04 0.0093 0.0092 1.08E-02 CE H D11 Expectation AC Table 5c Bounds of the expectation and variance of the effective elastic tensor (0 = 0.2). Bounds H D11 H D12 H D22 Expectation Variance MCS HHUAM Error MCS HHUAM Error Lower 7.1687 7.1863 2.46E-03 0.4969 0.4932 7.45E-03 Upper 23.8403 23.8030 1.56E-03 5.2016 5.1276 1.42E-02 Lower 2.9967 3.0102 4.50E-03 0.0869 0.0865 4.60E-03 Upper 19.7493 19.7086 2.06E-03 3.5551 3.5014 1.51E-02 Lower 7.1675 7.1851 2.46E-03 0.4968 0.4931 7.45E-03 Upper 23.8377 23.8003 1.57E-03 5.2006 5.1266 1.42E-02 ACCEPTED MANUSCRIPT H D66 Lower 1.8638 1.8627 5.90E-04 0.0337 0.0333 1.19E-02 Upper 1.9867 1.9886 9.56E-04 0.0388 0.0385 7.73E-03 AN US CR IP T a AC CE PT ED M b Fig. 4 The uncertainty levels of the interval of expectation (a) and the interval of variance (b) result from different uncertain levels of input It is well known that the uncertain level of the material properties may be different in practical engineering problems. Thus, a simple case is used to demonstrate the HHUAM for bounded hybrid uncertain homogenization problems with ACCEPTED MANUSCRIPT non-uniform uncertain level of input. In this case, the uncertain level of input is assumed as E 0.05, 0.2, E 0.15, 0.1 . The Young’s modulus E1 and E2 are 1 1 2 2 1 1 2 2 assumed to be the non-linear function of E ( E E m E E E E ( E ) 2 ), in which Em represents the mean value of the interval of the Young’s modules, ∆E represents the CR IP T deviation amplitude, E is a random variable defined on [-1, 1] with λE-PDF. The assumed information is listed in Table 6. When setting the tolerance to 10-2, the req req req req req estimation of N [ N E1 , N1 , N E2 , N 2 ] is N5 [2,2,2,5] in the HHUAM. The req AN US MCS is also used to calculate the referenced results. The samples are 8106, the interval and random sample points are 1000 and 8000, respectively. The bounds of the expectation and variance of the effective elastic tensor of the RVE obtained by both M HHUAM and MCS is listed in Table 7. Table 7 shows that the bounds of expectation ED and variance of the effective elastic tensor obtained by HHUAM match the referenced results obtained by MCS perfectly. Furthermore, the execution time of HHUAM and PT MCS for predicting the effective elastic tensor of the RVE in this case are 83 s and CE 2.6106 s, respectively. From these analysis results, it can be concluded that both the accuracy and efficiency of HUHAM are desirable for uncertain homogenization AC problems with uncertain-but-bounded parameters. Table 6 The information of the uncertain-but-bounded parameters. Uncertain parameters Range PDF E1 (GPa) [4.275,4.725] E1 = 4.3875+0.225 +0.1125( 1 [0.312,0.468] Unknown E2 (GPa) [62.05,83.95] E2 = 67.525+10.95 +0.1125( 2 [0.189,0.231] Unknown E1 E2 E1 )2 ， E 6 1 E2 )2 ， E 4 2 ACCEPTED MANUSCRIPT Table 7 Bounds of the expectation and variance of the effective elastic tensor of the RVE ( E1 0.15, 1 0.2, E2 0.05, 2 0.1). Bounds H D12 H D22 H MCS HHUAM Error MCS HHUAM Error Lower 7.0080 7.0186 1.51E-03 0.0092 0.0091 1.09E-02 Upper 23.1949 23.1373 2.48E-03 0.0978 0.0961 1.74E-02 Lower 2.934 2.9445 3.58E-08 0.0016 0.0016 9.38E-03 Upper 19.1979 19.1409 2.97E-03 0.0669 0.0657 1.79E-02 Lower 7.0068 7.0174 1.51E-03 0.0092 0.0091 1.09E-02 Upper 23.1923 23.1348 2.48E-03 0.0978 0.0961 1.74E-02 Lower 1.8215 1.8189 1.43E-03 6.22E-04 6.14E-04 1.30E-02 Upper 1.9409 1.9410 5.15E-05 7.14E-02 7.06E-04 1.12E-02 AN US D66 Variance CR IP T H D11 Expectation It is worth mentioning that interval homogenization-based method (IHM) and M subinterval homogenization-based method (Sub-IHM) are only developed for estimating the effective elastic tensor of periodic microstructure with interval ED uncertainty [19], while the proposed homogenization-based hybrid uncertain analysis PT method (HHUAM) can be used for predicting the effective elastic tensor for microscopic material properties with bounded hybrid uncertainty. This is to say that the CE proposed HHUAM can deal with the uncertain homogenization problems which the AC interval and subinterval homogenization-based method cannot deal with. Furthermore, with choosing the value of λ properly, the HHUAM can also be applied to determine the effective elastic properties of periodic microstructure with interval parameters and have a better performance than the IHM and Sub-IHM. Here, as a contrast, the bounds of the effective elastic tensor of the RVE are calculated by IHM (Sub-IHM) and HHUAM. In this case, all of the uncertain parameters are assumed to be interval variables and their ACCEPTED MANUSCRIPT uncertain level is assumed as E E . The results calculated by IHM 1 1 2 2 and HHUAM when uncertain level α =5% are shown in Table 8. The results calculated by Sub-IHM and HHUAM when uncertain level α =10% are shown in Table 9. The reference results calculated by MCS are also listed in Table 8 and Table 9. Table 8 and CR IP T Table 9 show that the bounds of the effective elastic tensor obtained by HHUAM match the referenced results obtained by MCS better than IHM and Sub-IHM. Table 8 H H D11 D12 Upper Lower MCS 8.4736 11.7364 4.5870 HHUAM 8.4404 11.7833 4.5621 Error 0.39% 0.40% 0.54% IHM 8.3627 11.3451 4.4377 Error 1.31% 3.33% 3.25% H D22 H D66 Upper Lower Upper Lower Upper 7.4985 8.4722 11.7346 1.7952 2.0180 7.5361 8.4390 11.7815 1.7934 2.0195 0.50% 0.39% 0.40% 0.09% 0.07% 7.1496 8.3613 11.3434 1.8406 1.9684 4.65% 1.31% 3.33% 2.53% 2.46% ED M Lower AN US Bounds of the effective elastic tensor ( = 0.05). Table 9 PT Bounds of the effective elastic tensor ( = 0.1). H D11 H H D22 D66 Upper Lower Upper Lower Upper Lower Upper MCS 7.3814 14.4631 3.6703 10.0364 7.3802 14.4610 1.6911 2.1335 HHUAM 7.3404 14.5983 3.6416 10.1552 7.3392 14.5962 1.6864 2.1382 Error 0.56% 0.93% 0.78% 1.18% 0.56% 0.93% 0.28% 0.22% AC CE Lower H D12 Sub-IHM 7.3172 13.8896 3.5766 9.5095 7.3159 13.8877 1.7287 2.0826 Error 0.87% 4.1% 1.77% 5.25% 0.87% 3.96% 2.22% 2.39% 6. Conclusions A homogenization-based hybrid uncertain analysis method, termed as HHUAM, is proposed for predicting the effective elastic properties of periodic microstructure ACCEPTED MANUSCRIPT with hybrid uncertain-but-bounded parameters in this paper. In the HHUAM, the bounded hybrid random and interval variables are transformed into the unitary uncertain-but-bounded variables related to the polynomial parameter λ. By choosing a suitable λ, the effective elastic tensor can be expanded by GSE. The expansion CR IP T coefficients of the Gegenbauer polynomials of the HHUAM can be calculated through the Gauss-Gegenbauer integration method. The influence of bounded hybrid uncertainties in microstructures on homogenized macroscopic elastic properties of AN US heterogeneous materials is also investigated through the MCS and HHUAM. The HHUAM and MCS are used to calculate the effective elastic tensor of a unit square cell with a rectangular hole and a unit RVE of a unidirectional fiber reinforced M composite with microscopic hybrid uncertain-but-bounded constituent material properties. By comparing the results calculated by the HHUAM and the MCS, the ED accuracy and the efficiency of the HHAUM are verified. It should be pointed out that PT with the uncertain level of input increasing, more computational time is spent for predicting the effective elastic tensor through HHUAM. This is because of that higher CE retained order of HUHAM is needed for reducing the estimation error for AC homogenization problem with larger uncertain level. But compared with MCS, the computational efficiency of HHUAM is still significantly improved. Besides, it can be observed from the numerical results that the uncertain levels of the interval of the H H H expectation and the interval of variance of the D11 , D12 and D22 are much larger than H the uncertain levels of the uncertain-but-bounded parameters, while those of D66 are H smaller than the uncertain levels of input. Moreover, the D12 is the most influenced by ACCEPTED MANUSCRIPT the hybrid uncertain-but-bounded parameters in microscopic material properties, H H H followed by D11 and D22 , and D66 is the least influenced. This phenomenon is due CR IP T to the combined influences of Young’s modulus and Poisson’s ratio. Acknowledgement The paper is supported by the Key Project of Science and Technology of AN US Changsha (Grant No. KQ1703028) and the Fundamental Research Funds for the Central Universities (Grant No. 531107051148). The author would also like to thank AC CE PT ED M reviewers for their valuable suggestions. ACCEPTED MANUSCRIPT Appendix For an uncertain system with multiple uncertain-but-bounded parameters, the relative improvement of response via increasing the retained order of HHUAM with respect to the j th uncertain parameter is defined as [28] Where U (k , j ) U (k , e j ) ; U (k , j ) U (k , j ) Ir(k , j ) (51) U (k , j ) U (k , e j ) . U (k , j ) U (k , j ) (52) AN US Ir(k , j ) k 1, 2,... CR IP T Ir(k , j ) max Ir(k , j ), Ir(k, j ) tol , In Eq(52), U (k, j ) and U (k, j ) denotes the lower and upper bounds of the response obtained by GSEM, k=[k1, k2,…, kL] is the retained order vector, kj denotes M the retained order related to the jth uncertain parameter. ej denotes the L-dimension vector, in which the j th element is equal to 1 while the other elements are zero. ED For each uncertain parameter, the least vector N req [ N1req , N 2req ,..., N Lreq ] which AC CE PT satisfies Eq.(51) can be obtained by an iteration procedure shown as Fig. 5. Fig. 5 The procedure of estimating the least order via the relative improvement criterion. ACCEPTED MANUSCRIPT References 1. Hashin, Z. and B.W. Rosen, The Elastic Moduli of Fiber-Reinforced Materials. Journal of Applied Mechanics, 1964. 31(2). Tanaka, S., et al., A probabilistic investigation of fatigue life and cumulative cycle CR IP T 2. ratio. Engineering Fracture Mechanics, 1984. 20(3): p. 501-513. 3. Hollister, S.J. and N. 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