Uncertainty quantification of mechanical properties for three-dimensional orthogonal woven composites. Part I: Stochastic reinforcement geometry reconstruction

Journal Pre-proofs Uncertainty quantification of mechanical properties for three-dimensional orthogonal woven composites. Part I: Stochastic reinforcement geometry reconstruction Wei Tao, Ping Zhu, Can Xu, Zhao Liu PII: DOI: Reference:

S0263-8223(19)32494-8 https://doi.org/10.1016/j.compstruct.2019.111763 COST 111763

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

1 July 2019 18 October 2019 28 November 2019

Please cite this article as: Tao, W., Zhu, P., Xu, C., Liu, Z., Uncertainty quantification of mechanical properties for three-dimensional orthogonal woven composites. Part I: Stochastic reinforcement geometry reconstruction, Composite Structures (2019), doi: https://doi.org/10.1016/j.compstruct.2019.111763

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Uncertainty quantification of mechanical properties for three-dimensional orthogonal woven composites. Part I: Stochastic reinforcement geometry reconstruction Wei Tao a,b, Ping Zhu a,b,*, Can Xu a,b, Zhao Liu a,c * Corresponding author, [email protected] aState

key laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, P.R. China

bShanghai

Key Laboratory of Digital Manufacture for Thin-walled Structures, Shanghai Jiao Tong University, Shanghai, 200240, P.R. China

cSchool

of Design, Shanghai Jiao Tong University, Shanghai, 200240, P.R. China

Corresponding author: Ping Zhu (Tel)+86-21-34206787 (Fax)+86-21-34206787 Email: [email protected]

Abstract For woven composites, the stochasticity of mechanical properties is mainly dependent on the reinforcement variability. Represent Volume Elements with realistic reinforcement architecture can obtain accurate predictions and identify the variability of mechanical properties. This work presents a data-driven modeling framework to generate statistically equivalent RVEs for three-dimensional orthogonal woven composites, in which retaining the knowledge of reinforcement variability from experimental observation. The reinforcement geometry is characterized by Micro CT

in terms of fiber tow centroid coordinates and cross-sectional dimensions. A comprehensive slicing and data correction method, which transforms the intact 3D geometry into a four-dimensional dataset, is established for all tow genera. A quantitative understanding of the variability of reinforcement architecture is presented via various statistical descriptors. In order to preserve the mainly statistical characteristics of tow feature parameters, D-vine copula functions are adopted to address their irregular marginal distributions and joint dependence. The reconstructed data is verified by the marginal probability density functions, the bivariate distributions and the correlation matrices of feature parameters. An inverse design from data to textile geometry is achieved by dedicated codes in TexGen. The whole framework is driven by acquired data automatically and can generate any number of statistically equivalent RVEs for further simulations. Keywords: Reinforcement variability, 3D orthogonal woven composites, D-vine copula function, Geometry reconstruction 1.

Introduction Owning to their outstanding physical and mechanical properties, woven

composites have gained more attention in light-weight structural applications [1-3]. However, woven composites also show significant stochasticity of mechanical performances [4]. In order to achieve more reliable and robust design, the uncertainty of mechanical properties must be quantified. To author’s knowledge, the variability of mechanical properties is mainly dependent on the uncertainty of defects in the composites, which come from manufacturing process including weaving, moulding,

curing, and machining [5, 6]. The defects include reinforcement geometry variability, voids, residual thermal stress and so on [7-9]. Among them, the reinforcement geometry variability plays a vital role on the variability of mechanical properties [10]. Due to the periodic nature of woven composites, the mesoscale Represent Volume Element (RVE) is generally adopted to represent the repetitive geometry of reinforcement [11]. As finite element approaches become more and more popular, it has become an emerging paradigm to predict mechanical properties of woven composites via mesoscale RVEs [12]. In a mesoscale RVE, the reinforcement geometry is usually constructed based on deterministic morphology parameters, e.g., constant tow dimensions and tow spacing. However, those morphology parameters of realistic reinforcement exhibit significant variations not only at different locations of the composite but also inside the RVE [13]. Compared with idealized RVE model, the RVE with realistic internal reinforcement architecture can obtain an improved prediction and identify variability of mechanical properties. It has been implemented to investigate the relationship between reinforcement variability and mechanical properties, such as elastic properties [14], strength [15, 16] and damage [17, 18]. The knowledge of spatial fluctuation for fiber tows is the key issue leading to accurate predictions of mechanical properties. The stochastic reinforcement geometry can be experimentally investigated by various imaging techniques. As a non-destructive way, X-ray Micro Computed Tomography (CT) has been found to be an effective and reliable means to acquire the reinforcement architecture of woven composites [19, 20]. To some extent, the

stochasticity of the tow architecture has been studied for 2D laminates. Bale et al [21] captured the characteristics of fiber tows of ceramic-matrix textile composites by X-ray micro CT, which consist of the coordinates of centroids, the cross-sectional dimensions and orientations. Then, a statistical analysis of the shape and positioning of the fiber tows was performed, based on a decomposition of the spatial variations of characteristics of the tows into non-stochastic periodic trends and non-periodic stochastic deviations. Wang et al [22] analyzed the 3D images of C/Epoxy with real yarn normal cross-section information obtained by slice data correction. Systematic trends, standard deviations and correlation lengths of stochastic deviations were evaluated respectively by the application of reference period method. For Three-dimensional Orthogonal Woven Composites (3DOWC), the binder tow interlaces the warp and weft tows in the thickness direction. Therefore, it encounters more challenges than 2D laminates in variability characterization. Firstly, warp/weft tows of 3DOWC belonging to different layers may present distinguish tow feature parameters, which makes it consist of more types of tow genera than its counterpart [23]; Secondly, since the binder tow exhibits a complicated spatial centroid path, a comprehensive slicing and data correction method needs to be established for binder tows [24, 25]. A correct identification of the spatial geometrical variations of textile reinforcement, such as tow path and cross-sectional dimension, permits to reconstruct a high-fidelity statistical equivalent RVE geometry [26, 27]. Moreover, variability characterization enables a quantitative understanding of reinforcement architecture via various statistical descriptors, which can be further utilized for establishing the

relationship among manufacturing, material and property [28]. However, obtaining realistic RVEs considering reinforcement variability by repeating three-dimensional imaging, such as X-ray Micro CT technique, is a time-consuming and costly process. Therefore, many scholars try to characterize the variability of reinforcement architecture within limited 3D imaging samples, then reconstruct abundant statistical equivalent RVEs that possess the same statistical characteristics of reinforcement variability with the experimental observation. Blacklock and Rinaldi et al. [29, 30] presented a Monte Carlo algorithm based on Markov Chain operators for generating replicas of textile composite specimens. Three-dimensional tows were generated to provide geometrical models that possess the same statistical characteristics imaged using X-ray CT, which had resolved interpenetrations and ordering errors among tows. Vanaerschot et al. [23, 31-33] provided a roadmap for generating realistic virtual textile specimens for high-fidelity simulations. The geometrical variability of the reinforcement was experimentally quantified in terms of average trends, standard deviations and correlation lengths. Then, Monte Carlo Markov Chain method, K-L Series Expansion technique and Fourier Transform method were utilized to generate the average trend and zero-mean deviations of reinforcement parameter. The spatial variability of tow feature parameters can be commonly simulated and reconstructed by multivariate cross-correlated random fields [21, 33-35]. However, descriptions of random fields in terms of standard deviations on slices and correlation coefficients between slices are incomplete. The marginal distributions and

multidimensional correlations of the tow geometrical fluctuations should be integrally identified. An effective mathematical tool named copula functions has been developed to process the correlations of random variables [36]. The copula function disassembles the random variables’ joint distribution construction problem independently into the marginal probability distributions and a dependence structure among marginal distributions [37]. Moreover, the development of vine copula provides reliable and efficient means for constructing joint probability models with multidimensional correlations [38]. The copula functions, which have been widely applied in the uncertainty analysis field, are suitable for more generalized distribution conditions with tail dependence and nonlinear correlation. In this work, the copula functions have been introduced into the variability characterization and the reconstruction of reinforcement geometry. This work presents a data-driven modeling framework to generate statistical equivalent RVEs of three-dimensional orthogonal woven composites, in which retaining details of the variability of tow geometry. This article starts with the detailed roadmap of proposed framework. Section 3 focuses on variability characterization and data processing for specific 3DOWC. An overall statistical description and analyzation for the reinforcement data is carried out in section 4. Then, section 5 introduces the adopted Vine copula method and discusses the variability reconstruction results. At last, the way of statistical equivalent RVE generation is present in section 6. 2. A data-driven modeling framework

In this work, a data-driven modeling framework is proposed for generating mesoscale

statistical

equivalent

Represent

Volume

Element

(RVE)

of

Three-Dimensional Orthogonal Woven Composites (3DOWC), which is illustrated in Fig.1. The modeling framework consists of four main steps: (1) Data collection; (2) Variability characterization; (3) Variability reconstruction; (4) RVE generation. In the first step, the reinforcement geometry of 3DOWC samples is adopted as the data source to drive the following steps. The intact 3D architecture of composite samples is characterized by Micro CT, which is discretized into two-dimensional (2D) slices along the global axes of the 3D image. The interested feature parameters of each fiber tow (i.e. data) are extracted from the 2D slices, which consists of centroid coordinates and tow cross-sectional dimensions. After data correction and translation, all tow feature parameters are stored in a dataset. The second step is to analyze the tow feature parameters by statistical descriptors, which provides the statistical information required for realistic description of the textile reinforcement. In the next step, the simulated stochastic centroid coordinates and corrected cross-sectional dimensions are reconstructed by D-vine copula method, which replicates the measured statistics, e.g., probability density function and correlation information. At last, the reconstructed feature parameters constitute a new dataset, which is utilized to generate the geometry of statistical equivalent RVEs in TexGen. [Insert Fig. 1] 3. Data collection 3.1 Micro CT experiment for the material

The material studied in this work is a three-dimensional orthogonal woven composite. The reinforcement of the composites was fabricated by 6k carbon fibers of type T700s from Toray, which consists of 4 warp layers and 5 weft layers. The nominal spacing between neighboring warp tows was set at 2.0mm, while it was 2.5mm for weft tows during fabrication. The reinforcement was impregnated by Araldite LY 1572 CI epoxy resin from HUNTSMAN. Vacuum Infusion Process (VIP) was implemented to manufacture the 3DOWC plate. Then, the curing process was carried out at 80℃ for 9 h. After curing, the final thickness of composite plate was 3.0mm. The internal reinforcement architecture of the composites was inspected via the X-ray Micro CT technology in nanoVoxel 2000. This test set-up, which can get a minimum voxel size of 0.5 μ m, had been utilized to reconstruct high-resolution 3D images of C/Epoxy plain weave composite in Wang’s work [22]. In order to enhance the representativeness of the collected data, four Micro CT samples were cut from different locations of the composite plates. Considering the periodic length in warp and weft directions, a length of 20mm was selected for each square sample. A voxel size of 16.3μm was found to be optimal for micro CT scanning to provide a high-resolution image of the reinforcement architecture. A typical three-dimensional Micro CT image for a composite sample is shown in Fig. 2, in which warp tows and weft tows lie along the X-axis and the Y-axis, respectively. [Insert Fig. 2] 3.2 Data acquisition

A correct and complete identification of all feature parameters of the textile reinforcement is the key issue to reconstruct a high-fidelity statistical equivalent RVE geometry. The interested data is extracted from the obtained 3D Micro CT images. The classification for tow genera is an essential step for data acquisition of reinforcement variability. The tows in the same genus should share the same characteristics of tow feature parameters. For the present 3D orthogonal woven composites, the morphology and spatial location for tows are dependent on the fabrication and manufacturing process. Owing to its periodic nature, the tows in the same layer nominally bear the same periodic contact from the top layer and the bottom layer. Therefore, in this work, the tows are categorized by tow layers, which include four warp genera, five weft genera and one binder genus. As shown in Fig. 3, the warp genera and the weft genera are named from the top of the sample to the bottom. [Insert Fig.3] The obtained 3D Micro CT fields of the composite samples were discretized by a series of uniformly-spaced slices perpendicular to three coordinate axes. Since the quantifying process is operated manually, a trade-off has been made between the precision of geometry description and the efficiency for labeling each slice. The interval between parallel slices is set at 0.179mm for discretizing the Micro CT volumes of each sample. Then, all related data are recorded by identifying fiber tow cross-sections on each slice. The acquisition of tow feature parameters is implemented by a public Java image processing program named ImageJ, which can measure

distances of user-defined selections on images. The adopted feature parameters of the fiber tows in this work consist of tow path centroid coordinates and cross-sectional dimensions. Since the upper and lower bounds of all cross-sectional shapes are perpendicular to the global Z-axis, the rotation angle of tow sections is neglected in this study. The dimensions for warp tows and weft tows are characterized on the 2D slices along the Y-axis and the X-axis, respectively. Since the binder tow exhibits a square-wave path, the slices perpendicular to the X-axis and the Z-axis are consociated to record the complete information of binder tows. Typical slices along three axes are illustrated in Fig. 3. As shown in Fig. 3, the cross-sections of Weft1 tows and Weft5 tows present a semi-elliptical shape, while the others are close to a rectangle shape. The parameters in terms of width and height are sufficient to describe the geometric figures of tow cross-sections. 3.3 Data processing Feature parameters for each tow genus are quantified from parallel slices of the discretized 3D Micro CT fields. However, these parallel slices are not perpendicular to the tow path, which results in the discrepancy between obtained feature parameters and the ones in normal cross-sections. In order to present the actual statistical characteristics, slice data should be converted into normal cross-sectional feature information, which has a significant impact on the dimensions of the binder genus. In this study, a projection method is adopted to correct the slice data. The proposed projection method has two assumptions: (1) tow centroids on slices coincide with the ones on the corresponding normal planes; (2) the distortion of tow figure is

neglected after projection. In the proposed projection method, the coordinates of centroid O for one specific tow at i th slice is assumed to be （x i , y i , z i）. When the adjacent four slices are taken into consideration, the tangent vector k i of this tow at i th slice can be calculated by “Five-Point Scheme” [39]. Qi , Qi  (1   i )V i   i V i 1 Qi

ki 

i =

V i -1  V i V i -1  V i + V i +1  V i +2

V i =[x i , y i , z i ]-[x i -1 , y i -1 , z i -1 ]

(1)

(2) (3)

As illustrated in Fig.4, Point P （x P , y P , z P） is on the tow edge of i th slice, point A ( x A , y A , z A ) is the projection of point P on the normal cross-sectional plane S. Since vector k i is perpendicular to the plane S, for given k i  [ x k , y k , z k ] , the analytical equation for plane S can be written by:

x k ( x  xi )  y k ( y  y i )  z k ( z  z i )  0

(4)

Point A is also on the plane S:

x k ( x A  xi )  y k ( y A  y i )  z k ( z A  z i )  0

(5)

 Since vector AP is parallel with vector k i , the coordinates of point A can be obtained as following equations: ( xi  x P ) x k  ( y i  y P ) y k  ( z i  z P ) z k k x  xP k 2 k 2 k 2 (x )  ( y )  (z )

(6)

( xi  x P ) x k  ( y i  y P ) y k  ( z i  z P ) z k k y = y  yP k 2 k 2 k 2 (x )  ( y )  (z )

(7)

xA =

A

zA=

( xi  x P ) x k  ( y i  y P ) y k  ( z i  z P ) z k k z  zP k 2 k 2 k 2 (x )  ( y )  (z )

After projection, the corrected tow dimensions can be obtained.

(8)

Owning to the periodicity of the reinforcement, the continuous tows can be segmented successively with a nominal periodic length. Before tow segmentation, reference slices should be selected as the starting positions of the repeating periods of each genus. As shown in Fig.5, for warp tows and binder tows, the reference slices are located on the slices with the minimum Z-coordinates of the binder tows. For weft tows, the corresponding reference slices are the ones showing the top edge of binder tows. According to the preset tow spacing, the nominal periodic length of warp yarns and binder yarns is 5mm, while it is 4mm for weft yarns. The periodic length for each genus is consistent with the dimensions of a RVE. In a RVE, there are 29, 24 and 18 slices along X-axis, Y-axis and Z-axis, respectively, decided by the discretized 3D Micro CT volume. Along each axis, the feature parameters of all fiber tows on the last slice are set to be consistent with the ones on the first slice to guarantee the periodic boundary condition of a RVE. As shown in Fig.5, the tows of the same genus can be translated into one RVE via the following translation vector:

l T  mx e x  n y e y  (x e x   y e y ) 2 where m, n and l are integers,

  0,1

(9) ,

x and  y are the nominal periodic length

for warp and weft tows. m, n and l are decided by the distance between two reference slices of tows. After data collection, the composite reinforcement geometry information in four composite samples has been extracted and combined into a four-dimensional dataset :

 =  t ,  , s, j 

(10)

where t is the genus type, t  (Warp1, Warp2, Warp3, Warp4, Weft1, Weft2, Weft3, Weft4, Weft5, Binder); ε represents the feature parameters of each genus: X, Y, Z, W and H; X, Y and Z describe centroid coordinates, W and H represent width and height of the normal cross-sections; s is the slice number of each genus, s=1…Ns; j indicate j th tow, j=1…Nj. The value of Nj and Ns for different genera are listed in Table1, which indicate the number of measurements for each genus. There are totally 50 feature parameters and more than 2200 sample sizes for each feature parameter in the dataset. Every slice of each feature parameter has sufficient data to implement statistical analysis for reinforcement variability. [Insert Fig.4] [Insert Fig.5] [Insert Table1] 4.

Variability characterization

4.1 Centroid coordinates Fig.6 illustrates the distribution of tow centroids, where each dot represents the coordinates of one tow centroid on a slice. As shown in Fig.6, the coordinates of centroids present significant stochasticity. For quantitatively understanding the variability, the statistical characteristics of centroid coordinates for all genera are displayed in Table 2 and Table 3. Since the slices are perpendicular to the axes, samples on the same slice possess an identical normal coordinate, which is not listed in Table 2 and Table 3.

In Table 2, the mean values of Y-coordinate for warp genera increase progressively from top to bottom, which demonstrates a tilting arrangement along the thickness direction. According to the mean coordinates of warp genus, the warp genera and the binder genus are tilted along Y-axis approximately at an angle of 15°, as illustrated in Fig.6a. The tilting arrangement results from the deformability of the dry fabric in the manufacturing process. However, the weft genera generally share the same mean values of X-coordinate, which indicates that the weft genera stack vertically. In most cases, the standard deviations of centroid coordinates for weft and warp genera are less than 0.05 mm. Therefore, the weft and warp genera possess approximately straight paths, which demonstrates that the data correction process plays a slight influence on them. As shown in Table 2, the fluctuation of weft tows is more obvious than warp tow. Table 3 shows the mean values and the standard deviations of centroid coordinates for a quarter of binder genus. Slice 1-5 are perpendicular to X-axis and Slice 6-11are along Z-axis, highlighted in the frame of Fig. 6b. Although the slices are chosen along two different axes, the mean values of centroid location are almost uniformly spaced, which can record the spatial centroid path appropriately. The typical path of binder tows can be observed in Fig. 6b, verifying the effectiveness of slicing methods for binder genus. [Insert Fig.6] [Insert Table 2] [Insert Table 3] 4.2 Tow dimensions

Fig. 7 shows the statistical characteristics of tow dimensions for warp genera and weft genera in all four samples. In Fig.7, the histogram of each tow genus stands for the mean value of corresponding tow dimension in all slices, while the error bar represents the standard deviation in all slices. As shown in Fig.7, the tow dimensions of inner warp genera (Warp2 and Warp3) distinct from the ones of outer warp genera (Warp1 and Warp 4). The inner warp genera possess larger widths and smaller heights than its counterpart. The same circumstance is also observed for weft genera. Besides, the mean values of tow width and height also present a negative correlation for warp and weft genera. The standard deviations of weft genera are significantly greater than the ones of warp genera, which indicates that weft genera sustain more drastic undulation of tow dimensions. Fig.8 illustrates the mean values of tow width and height for weft genera on each slice. As shown in Fig.8, the tow width and height present disciplinary fluctuations, which are inversely correlated with each other on these slices. For all slices of weft genera, the difference of mean values of tow height is in the range of 8-10%, while it is 2-4% for tow width. Fig.9 illustrates the mean values of width and height of warp tows on each slice. Although the fluctuations of warp tow dimensions still exist, they are less fluctuating in each slice than weft genera. For all slices of warp genera, the difference of mean values of tow height is in the range of 3-5%, while it is 2-3% for tow width. The mean values of tow width and height of binder genus on each slice is demonstrated in Fig.10. The data correction plays a significant impact on the height of a binder tow, owing to the dramatic variation of centroid coordinates. After data

correction, the tow dimensions present roughly sinusoidal patterns, where the peak values are approximately twice the valley values for tow height and width. As shown in Fig. 10, the binder genus possesses the minimum height and the maximum width on the surface of the composite, while presents a square figure at the interior of the composite. [Insert Fig.7] [Insert Fig.8] [Insert Fig.9] [Insert Fig.10] 5. Variability reconstruction Challenges that confront quantifying the uncertainty in feature parameters include describing their irregular marginal distributions on each slice and joint dependence between slices. As a robust and tractable method, statistical copula functions are often used to address these challenges by the practicing engineer. For preserving the mainly statistical characteristics of feature parameters, no decompositions or assumptions are adopted in variability reconstruction. 5.1 D-vine copula method The word copula means ‘link’ or ‘bond’ in Latin, and a copula is a connection function between the marginal distributions of multiple random variables and their joint distribution. Let F1 ( x1 ) , F2 ( x2 ) ,…, Fn ( xn )

denote the marginal Cumulative

Distribution Functions (CDF) of random variables x1, x2,…, xn, respectively. Assuming that F ( x1 , x2 ,..., xn ) is the joint CDF of F1 ( x1 ) , F2 ( x2 ) ,…, Fn ( xn ) , based

on Sklar’s theorem [36], there exists a unique copula function C satisfying the following expression: (11)

F ( x1 , x2 ,..., xn )  C  F1 ( x1 ), F2 ( x2 ),..., Fn ( xn ); 

where C is the corresponding copula function of the joint CDF F ( x1 , x2 ,..., xn ) ; θ is the parameter of copula function C. Then the joint Probability Density Function (PDF)

f ( x1 , x2 ,..., xn ) can be

obtained by: f ( x1 , x2 ,..., xn )  f1 ( x1 ) f 2 ( x2 )... f n ( xn )

 nC  F1 ( x1 ), F2 ( x2 ),..., Fn ( xn );  x1x2 ...xn

=f1 ( x1 ) f 2 ( x2 )... f n ( xn )c  F1 ( x1 ), F2 ( x2 ),..., Fn ( xn ); 

(12)

where c denotes the PDF of C, and fi ( xi ) is the PDF of xi . The joint PDF f ( x1 , x2 ,..., xn ) also can be decomposed as f ( x1 , x2 ,..., xn )=f1 ( x1 ) f 21 ( x2 x1 )... f n 1,2,...,n 1 ( xn x1 , x2 ,..., xn 1 )

(13)

where f k 1,2,...,k 1 ( xk x1 , x2 ,..., xk 1 ) , k=2,3,…,n, is the conditional PDF. For high-dimensional correlation problems, there are a large number of possible decomposition forms of Eq. (13). To help depict them, the regular vine has been proposed by Bedford and Cooke [38, 40]. Each class of the regular vine provides a specified method of decomposition on the joint PDF of multidimensional random variables. D-vine is a type of the regular vine, and Fig. 11 shows a five-dimensional D-vine by a form of a nested set of trees. In a n-dimensional D-vine model, there are n-1 trees Tj, j=1,2,…,n-1, where tree Tj has n–j+1 nodes and n-j edges. Each edge stands for a copula density, e.g. edge13|2 corresponds to the copula density function c13|2. The nodes on the tree Tj+1 are determined by edges in the tree Tj. The joint

PDF f ( x1 , x2 ,..., xn ) of D-vine can be expressed as n

n 1 n  j

k 1

j 1 i 1

f ( x1 , x2 ,..., xn )= fk ( xk )   ci ,i  j i 1,...,i  j 1 ( F ( xi xi 1 ,..., xi  j 1 ), F ( xi  j xi 1 ,..., xi  j 1 )) (14)

where index j stands for the tree Tj, while subscript i runs over the edges in Tree Tj. Referring to Eq. (14), though D-vine structures, the joint PDF can be determined by the marginal PDF of each random variable and multiple two-dimensional copula functions. Since the marginal PDFs are decided, the construction of joint PDF is transformed into a problem of establishing multiple bivariate copula functions. [Insert Fig.11] The commonly-used copula functions consist of elliptical copulas (e.g. Gaussian copula and Student t copula) and Archimedean copulas (e.g. Clayton copula and Frank copula). The Gaussian copula has a multivariate standard normal distribution, while the t copula is a multivariate t distribution. The Archimedean copulas enable the use of an explicit form of C to model dependence in high dimensions using only one copula parameter [41]. The bivariate copula functions considered in this study are summarized in Table 4 [42]. The h function is the binary conditional distribution expressed as

hij (ui , u j )=F ( xi x j ) 

C (ui , u j ) u j

(15)

where ui =Fi ( xi ) and u j =Fj ( x j ) . For two random variables in this study, their best-fit copula function and copula parameters should be determined by observed sampling results. In order to ensure the accuracy of the correlation analysis obtained by copula functions, maximum

likelihood estimation (MLE) [43] is adopted to estimate the corresponding parameters of a copula function. After deciding copula parameters for all candidate copula functions in Table 4, Akaike information criterion (AIC) [44] is implemented to select the optimum copula function. The copula function with the smallest AIC value gives the best fit for the samples. Then the constructed D-vine tree structure can be utilized to sample on multidimensional random variables[42]. [Insert Table 4] 5.2 Verifying the sampling results The statistical results of collected data exhibit two basic characteristics: Firstly, the probability distribution of feature parameters on each slice varies; Secondly, for a specific feature parameter of a genus, the mean value fluctuates regularly along slices, which indicates that the feature parameter may have multidimensional dependency among slices. Therefore, if a genus has Ns slices in a tow, the feature parameter on Ns slices are regarded as Ns-dimensional random variables during reconstruction. To summarize, the adopted D-vine copula algorithm involves the following steps: 1.

Decomposing the joint PDF into two-dimensional copula functions and marginal PDFs of Ns-dimensional random variables of each feature parameter.

2.

Based on the collected data, choosing optimal copula functions and copula parameters by AIC criterion and MLE method, respectively.

3.

Obtaining the multidimensional joint PDF and generating new sampling sets for Ns-dimensional random variables.

All feature parameters from various genus types have been reconstructed independently by the proposed D-vine copula algorithm. Any number of samples can be generated by D-vine copula for each feature parameter. In this work, 1000 samples are reconstructed for feature parameters on each slice, which is adequate for statistical equivalent RVE generation. Given that displaying all 50 reconstructed feature parameters might be redundant, the width of Weft2 genus (Weft2W) and the Z-coordinate of Weft5 genus (Weft5Z) are selected to verify the reconstructed data set. Table 5 demonstrates comparison of mean values and standard deviations for original data from experiment and simulated data from variability reconstruction. Without loss of generality, partial slices selected with regular intervals are shown in Table 5, where Weft2W1 stands for width of Weft2 genus on slice 1. Although the mean values and standard deviations for original data are discrepant on different slices in Table 5，the simulated data presents a relatively small error for each slice，which are less than 0.6% for mean values and 1.7% for standard deviations. The histograms of Weft2W and Weft5Z on these slices are demonstrated in Fig. 12 and Fig. 13, respectively, where diverse and irregular distributions of two feature parameters are observed. Owning to its nonparametric characteristics, the Kernel Density Estimation (KDE) [45, 46] method with Gaussian Kernel is employed to fit the marginal Probability Density Functions (PDF) without assuming prior distribution pattern. The obtained smoothing PDFs are also depicted in Fig.12 and Fig.13. The PDFs of feature parameters of simulated data are consistent with the ones of original data for Weft2W and Weft5Z, which demonstrates the high accuracy of proposed method in reconstructing the

marginal distributions on each slice. [Insert Table 5] [Insert Fig.12] [Insert Fig.13] The joint dependences of feature parameters are verified by the bivariate distributions and the correlation matrices in this work. Corresponding correlation information of D-vine copulas between neighboring slices is presented in Table 6. In Table 6, Weft2W1-2 presents the correlation between the width of Weft2 genus on slice 1 and slice 2. Distinct differences of copula functions and parameters are observed for Weft2W and Weft5Z. All four copula functions contribute to constructing the correlation of each feature parameter on neighboring slices, where the Frank copula function is the most frequently selected function. In order to further reveal the joint distribution, the distributions of feature parameters on neighboring slices from original data and simulated data are illustrated in Fig.14. As shown in Fig. 14, the original samples and simulated samples possess similar density distributions with the corresponding copula contour. Although the marginal distributions of feature parameters are irregular and diverse, the adopted D-vine copula method presents strong ability to characterize and reconstruct the nonlinear correlations between variables. In order to quantify the correlation of different slices, the Kendall correlation coefficient is adopted as the measurement for correlation. The Kendall correlation coefficient is a type of nonlinear correlation coefficient and does not rely on the marginal distributions, which makes it suitable for the inconsistent and

irregular marginal distributions in this work. Fig. 15 and Fig. 16 demonstrate the comparison of correlation matrices between original data and simulated data obtained by the proposed D-vine copula method. The labels in the diagonal of correlation matrix stand for the slice number of the row or the column. For instance, the circles in blue frame represent the Kendall correlation coefficients between slice 1 and slice 3 in Fig. 15. The correlation coefficients in the lower triangular matrix are obtained from the simulated data, while the ones in the upper triangular matrix come from the original data. The color and size of each circle represent the degree of correlation for two slices: red color stands for positive correlation, while blue color stands for negative correlation; larger is the size, stronger is the correlation. As shown in Fig. 15 and Fig. 16, the correlation coefficients are mostly positive. Moreover, the correlation coefficients in Fig. 15 grow larger along with decreasing the distance between two slices. Consequently, the strongest correlations locate near by the diagonal of matrix. Nevertheless, for slice 1 and slice 2 in Fig.16, the correlation coefficients increase when they meet the last six slices. As illustrated in these two figures, the size and color of circles in the lower triangular matrix and upper triangular are basically symmetric along the diagonal, which demonstrates the consistence of correlation coefficients from original data and simulated data. It also proves the accuracy of the proposed method for joint dependence of each feature parameter between slices. [Insert Table 6] [Insert Fig.14]

[Insert Fig.15] [Insert Fig.16] 6.

RVE generation After variability reconstruction, a new four-dimensional dataset Ω’, which

consists of simulated feature parameters, is generated by the proposed D-vine copula method. The last step of this work is achieving an inverse design from the simulated data to the reinforcement geometry of statistically equivalent RVEs. In this work, the RVE geometry generation of 3D orthogonal woven composites is exemplified in TexGen software, which mainly contains the following steps: 1.

Assuming that j1 is a sample tow for a specific genus t in Ω’, the centroid path can be easily established in TexGen with the simulated centroid coordinates of j1. The normal cross-section on each centroid is built by tow dimensions of j1 with the observed cross-sectional shape by Micro CT. The cross-sections of top weft tows and bottom weft tows are constructed with approximately semi-elliptical shape, while the cross-sections for the other tows are close to rectangle in TexGen.

2.

Since there are two tows of the same genus in a RVE, another sample tow j2 is also chosen from genus t in Ω’. Considering the periodic nature of the composite, tow j2 needs to be duplicated and translated along the following vector T’:

i j T '  x e x   y e y , i  1, j  1 2 2

(16)

where x and  y are the nominal periodic length for warp and weft tows; ex and e y are the vectors of X-axis and Y-axis, respectively. After the duplication

and translation, the j2 tow section outside the RVE will be cut off, and only the inside part of j2 is remained. 3.

Repeatedly conducting step (1)-(2) for all genera. The statistically equivalent RVE considering reinforcement variability is reconstructed, as shown in Fig.17.

4.

Any number of RVEs can be generated by choosing different sampling tows in Ω’. A dedicated MATLAB code, which modifies the input file of TexGen according

to the randomly selected sampling tows in Ω’, is programmed to construct the geometry of RVE automatically. Since the tow feature parameters on the last slice along each axis is equal to the ones on the first slice in the data set Ω’, the generated statistically equivalent RVEs satisfy the boundary periodic condition, which can be directly implemented for further finite element analysis. [Insert Fig.17] 7. Conclusion This work presents a data-driven modeling framework to generate statistically equivalent Represent Volume Elements for three-dimensional orthogonal woven composites. The reinforcement architecture of the composites is inspected by X-ray Micro CT technology. The tows of the studied composite are categorized by tow layers, including four warp genera, five weft genera and one binder genus. The interested feature parameters of the fiber tows consist of centroid coordinates and cross-sectional dimensions. A projection method is adopted to correct the slice data into normal cross-sectional feature information. After data collection, the composite

reinforcement geometry is transformed into a four-dimensional dataset, which has 50 feature parameters and more than 2200 sample sizes for each feature parameter. The D-vine copula functions are adopted to address irregular marginal distributions on each slice and joint dependence between slices. For warp and weft genera, tow width and height have a significant negative correlation on each slice. Inner genera of in-plane tows are found to be broader and shorter than outer warp genera. The warp genera are less fluctuating in tow dimensions and centroid coordinates than its counterpart. The rule of fluctuations for width and height of weft tows are closely related to the intersected locations, where the tows have maximum width and minimum height among all slices. A new method is proposed to record complete information the square-wave path of binder tow. The binder genus is observed to be tilted along axis Y. The data correction plays a significant impact on the dimensions of binder genus. The simulated data obtained by D-vine copula algorithm presents a relatively low error of statistical characteristics for feature parameters. The simulated data not only owns close PDFs with original data, but also shares similar distributions on a bivariate copula contour. Moreover, the correlation coefficients between any pair of slice are consistent with each other in the correlation matrix. It proves the accuracy of the proposed method for marginal distributions and joint dependence of each feature parameter. The whole framework is driven by acquired data automatically and can generate any number of statistically equivalent represent volume elements, which can be directly implemented for mechanical properties prediction. In the future work, the

correlation between feature parameters and debarring the overlap between tows will be considered into the variability reconstruction and geometry generation to obtain a more realistic RVE.

Acknowledgment The authors would like to acknowledge the support from Key National Natural Science Foundation of China (Grant No. U1864211), National Natural Science Foundation of China (Grant No.11772191), and National Science Foundation for Young Scientists of China (Grant No.51705312). Reference [1] Fu X, Ricci S, Bisagni C. Minimum-weight design for three dimensional woven composite stiffened panels using neural networks and genetic algorithms. Compos Struct 2015;134:708-15. [2] Liu Z, Lu J, Zhu P. Lightweight design of automotive composite bumper system using modified particle swarm optimizer. Compos Struct. 2016;140:630-43. [3] Tao W, Liu Z, Zhu P, Zhu C, Chen W. Multi-scale design of three dimensional woven composite automobile fender using modified particle swarm optimization algorithm. Compos Struct 2017;181:73-83. [4] Zhu C, Zhu P, Liu Z. Uncertainty analysis of mechanical properties of plain woven carbon fiber reinforced composite via stochastic constitutive modeling. Compos Struct 2019;207:684-700. [5] Mesogitis TS, Skordos AA, Long AC. Uncertainty in the manufacturing of fibrous

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Figure Captions Fig. 1. The data-driven modeling framework for generating statistical equivalent RVE Fig. 2. A typical three-dimensional Micro CT image for a composite sample Fig. 3. Tow feature parameter characterization on the 2D slice perpendicular to: (a) X-axis, (b) Y-axis, (c) Z-axis. Fig. 4. The schematic illustration of data correction Fig. 5. The schematic figure of translating data into a RVE Fig. 6. Overview of centroid coordinates for each genus Fig. 7. Statistical results of tow dimensions for warp genera and weft genera Fig. 8. The mean values of tow dimensions for weft genera on each slice Fig. 9. The mean values of tow dimensions for warp genera on each slice Fig. 10. The mean values of tow dimensions of binder genus on each slice Fig. 11. A five-dimensional D-vine model Fig. 12. The probability distribution functions of Weft2W of original data and simulated data: (a) on slice 1; (b) on slice 5; (c) on slice 9; (d) on slice 13; e) on slice 17. Fig. 13. The probability distribution functions of Weft5Z of original data and simulated data: (a) on slice 1; (b) on slice 5; (c) on slice 9; (d) on slice 13; e) on slice 17. Fig.14. The distributions of feature parameters on neighboring slices from original data and simulated data: (a) Weft2W1-2; (b) Weft2W10-11; (c) Weft5Z1-2; (d) Weft5Z6-7.

Fig. 15. The comparison of correlation matrices for Weft2W between original data and simulated data Fig. 16. The comparison of correlation matrices for Weft5Z between original data and simulated data Fig. 17. A generated statistical equivalent RVE in TexGen

Table Captions Table 1 Sample sizes for tows and slices of each genus Table 2 The statistical characteristics of centroid coordinates for warp genera and weft genera Table 3 The statistical characteristics of centroid coordinates for binder genus Table 4 Bivariate copula functions adopted in this study Table 5 Comparison of the statistical characteristics of original data and simulated data Table 6 Correlation information of constructed D-vine copulas

Uncertainty quantification of mechanical properties for three-dimensional orthogonal woven composites. PartⅠ: Stochastic reinforcement geometry reconstruction (Figures)

Fig. 1. The data-driven modeling framework for generating statistical equivalent RVE

Fig. 2. A typical three-dimensional Micro CT image for a composite sample

Fig. 3. Tow feature parameter characterization on the 2D slice perpendicular to: (a) X-axis, (b) Y-axis, (c) Z-axis.

Fig. 4. The schematic illustration of data correction

Fig. 5. The schematic figure of translating data into a RVE

Fig. 6. Overview of centroid coordinates for each genus

Fig. 7. Statistical results of tow dimensions for warp genera and weft genera

Fig. 8. The mean values of tow dimensions for weft genera on each slice

Fig. 9. The mean values of tow dimensions for warp genera on each slice

Fig. 10. The mean values of tow dimensions of binder genus on each slice

Fig. 11. A five-dimensional D-vine model

Fig. 12. The probability distribution functions of Weft2W of original data and simulated data: (a) on slice 1; (b) on slice 5; (c) on slice 9; (d) on slice 13; e) on slice 17.

Fig. 13. The probability distribution functions of Weft5Z of original data and simulated data: (a) on slice 1; (b) on slice 5; (c) on slice 9; (d) on slice 13; e) on slice 17.

Fig.14. The distributions of feature parameters on neighboring slices from original data and simulated data: (a) Weft2W1-2; (b) Weft2W10-11; (c) Weft5Z1-2; (d) Weft5Z6-7.

Fig. 15. The comparison of correlation matrices for Weft2W between original data and simulated data

Fig. 16. The comparison of correlation matrices for Weft5Z between original data and simulated data

Fig. 17. A generated statistical equivalent RVE in Texgen

Uncertainty quantification of mechanical properties for three-dimensional orthogonal woven composites. PartⅠ: Stochastic reinforcement geometry reconstruction （Tables） Table 1 Sample sizes for tows and slices of each genus Genus type Tows in a genus Nj Slices in a tow Ns Warp 88 29 Weft 93 24 Binder 82 43

Sample size 2552 2232 3526

Table 2 The statistical characteristics of centroid coordinates for warp genera and weft genera X(mm) Y(mm) Z(mm) Warp1 Warp2 Warp3 Warp4 Weft1 Weft2 Weft3 Weft4 Weft5

Mean

Std

Mean

Std

Mean

Std

-0.001 -0.002 -0.003 -0.001 0.006

0.043 0.050 0.035 0.030 0.043

0.001 0.131 0.271 0.428 -

0.037 0.020 0.023 0.026 -

2.280 1.759 1.260 0.710 2.663 2.032 1.511 0.998 0.354

0.033 0.020 0.018 0.022 0.041 0.036 0.037 0.022 0.049

Table 3 The statistical characteristics of centroid coordinates for binder genus X(mm) Y(mm) Z(mm) Binder Mean Std Mean Std Mean Std Slice1 Slice2 Slice3 Slice4 Slice5 Slice6 Slice7 Slice8 Slice9 Slice10 Slice11

-2.500 -2.321 -2.141 -1.962 -1.783 -1.509 -1.417 -1.363 -1.329 -1.305 -1.291

0.047 0.044 0.035 0.030 0.027 0.022

0.103 0.100 0.079 0.052 0.015 -0.060 -0.123 -0.158 -0.214 -0.276 -0.317

0.037 0.026 0.043 0.040 0.043 0.041 0.043 0.046 0.039 0.041 0.040

0.105 0.112 0.134 0.193 0.276 0.538 0.717 0.897 1.076 1.255 1.434

0.021 0.025 0.029 0.032 0.036 -

Table 4 Bivariate copula functions adopted in this study Copula

C  u, v |  

Range of θ

h function

Gaussian

   1 (u ),  1 (v) |  

 1,1

  1 (u )   1 (v)    1 2  

Student t

T , (T1 (u ), T1 ( v )| )

 1,1

  T1 (u )   T1 ( v )  T +1   ( v  (T 1 ( v )) 2 )(1   2 ) /   1    

u

Clayton Frank



 v   1

1/

  e u  1 e v  1    ln 1     e  1   1

 0,   ,   \ 0

v  1  u   v   1

1/

e v  1 1  e  e v  1  u 1 e

Table 5 Comparison of the statistical characteristics of original data and simulated data Mean value Standard deviation Original Simulated ERROR Original Simulated ERROR （%） （mm） （mm） （%） （mm） （mm） Weft2W1 1.751 1.746 -0.323 0.0532 0.0541 1.669 Weft2W5 1.729 1.727 -0.099 0.0566 0.0559 -1.165 Weft2W9 1.749 1.747 -0.139 0.0514 0.0508 -1.150 Weft2W13 1.749 1.747 -0.141 0.0486 0.0491 1.037 Weft2W17 1.733 1.730 -0.130 0.0548 0.0542 -1.207 Weft5Z1 0.355 0.354 -0.196 0.0244 0.0244 -0.020 Weft5Z5 0.361 0.359 -0.486 0.0264 0.0267 1.213 Weft5Z9 0.359 0.357 -0.461 0.0288 0.0292 1.306 Weft5Z13 0.349 0.348 -0.528 0.0262 0.0266 1.499 Weft5Z17 0.349 0.348 -0.535 0.0239 0.0237 -0.757 Table 6 Correlation information of constructed D-vine copulas Slice Copula Copula Slice Copula function parameter function Weft2W1-2 Frank 7.518 Weft5Z1-2 Frank Weft2W4-5 Student t 0.723/5.886 Weft5Z6-7 Frank Weft2W9-10 Clayton 1.211 Weft5Z16-17 Student t Weft2W10-11 Frank 4.970 Weft5Z17-18 Gaussian

Copula parameter 6.627 9.994 0.846/7.332 0.878