Uncertainty quantification of dry woven fabrics: A sensitivity analysis on material properties

Composite Structures 116 (2014) 1–17

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Uncertainty quantification of dry woven fabrics: A sensitivity analysis on material properties A.B. Ilyani Akmar a,b,⇑, T. Lahmer b, S.P.A. Bordas d, L.A.A. Beex d, T. Rabczuk b,c,* a

Faculty of Civil Engineering, MARA University of Technology, 40450 Shah Alam, Selangor, Malaysia Institute of Structural Mechanics, Bauhaus University of Weimar, Marienstrasse 15, 99423 Weimar, Germany c School of Civil, Environmental and Architectural Engineering, Korea University, Republic of Korea d Institute of Mechanics & Advanced Materials (IMAM), School of Engineering Cardiff University, Queen’s Buildings, The Parade, Cardiff CF24 3AA, Wales, UK b

a r t i c l e

i n f o

Article history: Available online 17 May 2014 Keywords: Uncertainty criteria MataBerkait-dry woven fabric unit cells Global sensitivity analysis Latin hypercube sampling

a b s t r a c t Based on sensitivity analysis, we determine the key meso-scale uncertain input variables that influence the macro-scale mechanical response of a dry textile subjected to uni-axial and biaxial deformation. We assume a transversely isotropic fashion at the macro-scale of dry woven fabric. This paper focuses on global sensitivity analysis; i.e. regression- and variance-based methods. The sensitivity of four meso-scale uncertain input parameters on the macro-scale response are investigated; i.e. the yarn height, the yarn spacing, the yarn width and the friction coefficient. The Pearson coefficients are adopted to measure the effect of each uncertain input variable on the structural response. Due to computational effectiveness, the sensitivity analysis is based on response surface models. The Sobol’s variance-based method which consists of first-order and total-effect sensitivity indices are presented. The sensitivity analysis utilizes linear and quadratic correlation matrices, its corresponding correlation coefficients and the coefficients of determination of the response uncertainty criteria. The correlation analysis, the response surface model and Sobol’s indices are presented and compared by means of uncertainty criteria influences on MataBerkait-dry woven fabric material properties. To anticipate, it is observed that the friction coefficient and yarn height are the most influential factors with respect to the specified macro-scale mechanical responses. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The mechanical behavior of dry fabrics is important for the manufacturing process of textile composites. The first step of the manufacturing process of a textile composite is the placement of a dry woven fabric in a preform. Subsequently, resin is poured over it in order to form the solid shape of the textile composite. Any deformations and uncertainty factors during the placement possibly affect the elastic behavior of the final textile composite. For instance, wrinkles, folds, and tearing possibly lead to unexpected mechanical behavior of the final textile composites. Using numerical simulation methods [1], the placement of a dry fabric in a preform can be investigated. Since not all individual yarns can be incorporated at the macro-scale due to the computational costs, meso-scale models are mostly used for this. The

⇑ Corresponding authors at: Institute of Structural Mechanics, Bauhaus University of Weimar, Marienstrasse 15, 99423 Weimar, Germany. E-mail addresses: [email protected] (A.B. Ilyani Akmar), [email protected] (T. Rabczuk). http://dx.doi.org/10.1016/j.compstruct.2014.04.014 0263-8223/Ó 2014 Elsevier Ltd. All rights reserved.

material properties of the meso-scale descriptions in turn depend directly on the geometry of meso-scale unit cells of the fabric and the yarn materials used. Several studies therefore focused on fitting a macro-scale continuum description on the response of meso-scale unit cell models of fabrics. Some of the earliest studies are those of Clulow and Taylor [2], Hearle et al. [3], Kawabata et al. [4,5], Testa and Yu [6] and Pan [7]. More recent studies are those of Beex et al. [8], Buet-Gautier and Boisse [9], Lomov et al. [10], Tabiei and Yi [11], Cavallaro et al. [12], Kumazawa et al. [13] and Komeili and Milani [14]. The meso-scale modeling of dry fabrics implies that all of the mechanical properties of the meso-scale constituents must be considered. These properties can be determined based on the microscale modeling of an individual yarn or by defining an appropriate constitutive model for the yarn material [14]. A study by Komeili and Milani [14] presented two sets of geometrical and materialrelated meso-level uncertainty criteria on a glass fibre plain weave fabric using two-level factorial designs. For the geometrical uncertainty factors, the yarn spacing, yarn width, yarn thickness and misalignment of the yarns angle were investigated. For material

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uncertainty criteria, the longitudinal Young’s modulus, transverse Young’s modulus and friction coefficient were adopted in the analysis. Gasser et al. [15] used an inverse characterization method on experimental results (experiments performed on large pieces of fabrics) to obtain the material properties of yarns. This approach uses special material constitutive models for yarns to account for the effects of the discrete fibres at the micro-level as utilized by Sherbun [16], which was adopted by [17] as well. Conversely, Peng and Cao [18] utilized classical elastic material properties and finite element procedures in defining the material properties. They also implemented homogenization to predict the effective non-linear elastic moduli of textile composites at the macro-scale similar to Takano et al. [19,20], Rabczuk et al. [21] and Bakhvalov and Panasenko [22]. Uncertainty analysis of moderate to complex computational models is costly due to the high number of uncertainty criteria that might be considered in the design process. In many cases, a structural response is dominated by only a few uncertainty criteria [23]. Correlation analysis (regression-based) and response surface models (variance-based) are the methods of global sensitivity analysis. In this paper, the influence of four meso-scale uncertainty criteria on the macro-scale response of a MataBerkait-dry woven fabric are investigated via these methods. The MataBerkait-dry woven fabric is previously generated by Ilyani Akmar et al. [24], and the meso-scale uncertainty criteria of interest are the yarn spacing, yarn width, yarn height and the friction coefficient between the yarns. According to previous studies, these properties have been the most important criteria in yarn designation. The geometry of the meso-scale unit cell is generated in TexGen and ABAQUS is used to discretise the unit cell with finite elements and analyse its response as a function of applied uni-axial and biaxial deformations under periodic boundary conditions, as explained in Section 2. The sensitivity analysis is considered in Section 3. The numerical results are presented in Section 4. Finally, conclusions are detailed in Section 5. 2. Dry woven fabric unit cell 2.1. Geometry and discretisation The dry woven fabric unit cells used in this study are based on the unit cell models introduced by Ilyani Akmar et al. [24] and inspired by M.H.D.C. [25] who adopted Mata Berkait as the pattern arrangement of the yarns, see Fig. 1. Three basic dimensional values of the yarns specified in this paper are s ¼ 5:13 mm; w ¼ 4:44 mm and h ¼ 0:5 mm for initial yarn spacing, yarn width and yarn height, respectively. The cross-sections are assumed to have an elliptical shape. The assumption of uniform fabric deformation at the meso-scale is applied in fabric unit cell modeling and periodic boundary conditions are applied to replicate its repetitive nature. Periodic boundary conditions is used to simulate a large system by modeling a small system that in order to consider the effects of adjacent unit cells. The periodicity conditions can be determined by imposing the force nodes in opposite corners to have the same transverse displacement and the same rotation over global opposite edges.

Once a unit cell model has been specified as described above, the unit cell is imported in ABAQUS. Fig. 2 contains a flowchart that explains the procedures for uncertainty quantification on Mata Berkait-dry fabric unit cell. 2.2. Material properties of the yarns It is essential to have a detailed study of the fabric behavior at the meso-scale to determine its equivalent material properties for macro-scale models. The meso-scale modeling of dry fabrics implies that an appropriate constitutive description is formulated for the meso-scale constituents (the yarns and the interactions between them in this study). The material properties of the yarns are obtained from Peng and Cao [20], see Table 1 in which the materials used are pure E-Glass and Polypropylene. 2.3. Macro-scale response At the macro-scale, the out-of-plane response is assumed to be negligible compared to the in-plane responses. For this reason, we focus on the in-plane responses and no loading will be imposed in the z-direction. This condition is crucial in the absence of a resin or matrix. Furthermore, yarns in dry fabrics can easily slide over each other which results in a substantial decrease of the shear modulus and transverse Young’s modulus. It has been assumed that the longitudinal stiffness and shear moduli are given by Hooke’s law

rxx exx ryz Gyz ¼ 2eyz rxz Gxz ¼ 2exz rxy Gxy ¼ : 2exy Exx ¼

ð1Þ

A postulation is made on the stiffness functions due to the lack of a comprehensive and consistent source of data/benchmark for the material properties of fabric yarns at meso-level. Constitutive models of yarns can also be estimated based on physical observations of yarn characteristics. For instance, due to the absence of a matrix, it is apparent that there are zero or quasi zero stiffness values in the constitutive model of yarn materials due to the fibrous nature in the fabric. The negligible longitudinal compressive response of yarns is in agreement with this observation. For the same reason, Gasser et al. [15] has introduced the crushing law in modeling the transverse behavior of fibres. As mentioned by Komeili and Milani [14], this law is developed based on the observations that the more compression is applied to the fibres, the stiffer it becomes. With respect to the crushing law, the transverse modulus of yarns is a function of contact conditions between yarns. Conversely, Kawabata et al. [4]526 defined the transverse modulus as a function of contact force. Other crushing functions in obtaining the transversal behavior can be retrieved in Badel et al. [27]. In this study, Gasser’s approach is adopted as similar to Komeili and Milani’s implementation. Gasser et al. [15] defined that the transversal stiffness which related to transversal behavior

Fig. 1. The unit cell geometry used by Ilyani Akmar et al. [24]: (left) an elementary pattern; (right) the half of elementary pattern.

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Fig. 2. Uncertainty quantification procedures on Mata Berkait-dry fabric unit cell.

Table 1 Material Properties of E-Glass/Polypropylene fibrous composite predicted by [20]. Mechanical Inputs

Values

Longitudinal stiffness, El Transverse stiffness, Et Longitudinal Poisson’s ratio, mlt Transverse Poisson’s ratio, mtt Longitudinal shear stiffness, Glt Transverse shear stiffness, Gtt

51.92 GPa 21.97 GPa 0.2489 0.2143 8.856 GPa 6.250 GPa

of yarns is a function of both longitudinal and transverse strain using Eq. (2) by taking direction 1(x-direction for the present work) as the longitudinal axis

Ei ¼ E0 j enii j em xx

½i ¼ y; z;

ð2Þ

where E0 ; n and m are constants that can be determined using an inverse identification method; exx is the strain in x-direction. Eventually, the elastic moduli, E and shear moduli, G, are summarized in Eq. (3) wherein the other material properties due to the Poisson’s ratios’ effect are zero leading to the following stiffness matrix

2 6 6 6 6 C¼6 6 6 6 4

3

Ex

0

0

0

0

0

0

Ey

0

0

0

0

0

Ez

0

0

0

0

0

Gyz

0

0

0

0

0

Gzx

0 7 7 7 0 7 7: 0 7 7 7 0 5

0

0

0

0

0

Gxy

ð3Þ

3. Sensitivity analysis: material properties of woven fabric unit cells A sensitivity analysis is performed to determine the influence of uncertain input variables on the elastic macro-scale mechanical response of MataBerkait-dry woven fabrics. The correlation analysis (regression-based), the response surface model (variancebased) and Sobol’s indices (variance-based) [28] are used. The Anthill-Plots are presented to describe the correlation between the uncertainty criteria and their influences on the material properties of MataBerkait-dry woven fabrics. 3.1. Uncertainty criteria The four uncertainty criteria of interest in this study are: (a) (b) (c) (d)

Yarn spacing, sy Yarn width, wy Yarn height, ty Friction coefficient,

l.

Yarn spacing primarily relates to the fabric arrangement and has a major influence on fabric macro-scale mechanical properties. Here, the yarn spacing is defined as the distance between the righthand edges of the picks in woven fabric. The geometrical designation of basic yarn is given in Fig. 3. The yarn width wy , has an inter-related correlation with the yarn height, t y . Both of these uncertainty criteria are commonly

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A.B. Ilyani Akmar et al. / Composite Structures 116 (2014) 1–17

Fig. 3. Geometrical designation of basic yarn.

used to identify the yarn cross-sectional aspect ratio, ARy , as given in Eq. (4) [24]. Here, ARy is not necessarily constant because the t y and wy are the uncertainty criteria

wy ¼ ARy t y :

ð4Þ

In textile manufacturing, the yarn height, t y or thickness of yarns refers to the yarn weight of fabrics. The identification of the fabric height is classified as from super fine to super bulky. Varying the yarn weights can have a significant impact on the finished fabrics. The higher the yarn weight, the heavier the yarn. Frictional properties are dependent on the composition of the material, the surface of the yarns, pressure between the yarn surfaces, temperature and relative humidity. Yarns experience friction either between themselves or against metallic surfaces where both yarn-to-yarn and yarn-to-metal friction play a significant role in textile manufacturing. The friction response appears to be significantly sensitive to the relative positioning and orientation of the dry woven fabrics during manufacturing due to the highly non-linear behavior of woven fabrics. It can also lead to manufacturing defaults such as unweaving or wrinkles [29]. Further research on the friction coefficient of woven fabrics can be found in studies such as [30–35]. Table 2 shows the tolerances of the four uncertainty criteria used in the analysis. 3.2. Correlation analysis (regression-based method) A correlation analysis is commonly measured in terms of a correlation coefficient q. It quantifies the strength and direction of a linear/quadratic correlation between the output (the macro-scale mechanical response) and an uncertain variable, i.e. the input (e.g. the friction coefficient). The best known correlation coefficient is the Pearson correlation coefficient. It is derived by dividing the covariance of the two uncertainty criteria by the product of their standard deviations. The format of the correlation analysis is to ascertain a set of variables, and to vary each variable independently while keeping the others at their original values. The correlation coefficient, q can be expressed according to Eq. (5) [23] where EðÞ denotes the expectation of a random variable

EðXYÞ  EðXÞEðYÞ

q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2

2

2

ð5Þ

2

EðX Þ  E ðXÞ EðY Þ  E ðYÞ

Since the correlation coefficient q can be written as the Pearson correlation coefficient in estimating the correlation of X and Y, Eq. (5) can be expressed in terms of the Pearson correlation coefficient as follows

PN

Þ ðx  xÞðy  y

i i¼1 i ffi; q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P N i¼1 ðxi

 xÞ2

N i¼1 ðyi

ð6Þ

Þ2 y

 are the mean of the xi and yi valwhere N is the sample size, x and y ues; respectively. The coefficients resulting from Eq. (5) and Eq. (6) are located at the interval [-1,1] where the values between the investigated uncertainty criteria X and output response, Y are categorized as follows: 0:8 and above (strong correlation), 0:5  0:79 (moderate correlation) and 0:49 and below (weak correlation). The þ1 and 1 indicate the perfect positive and perfect negative correlations, respectively. Perfect positive correlation defines that as one variable moves, either up or down, the other variable will move in lockstep, in the same direction. Alternatively, perfect negative correlation implies that if one variable moves in either direction the variable that is perfectly negatively correlated will move in the opposite direction. The Pearson correlation coefficient is associated with the regression coefficient derived by linear regression analysis. A linear regression as shown in Eq. (7) is modeled for n data inputs with ^ and ^ and b one independent variable xi , two uncertainty criteria a one error term, i

^ þe; ^ þ bx yi ¼ a i i

ð7Þ

^ are estimated via least ^ and b where the uncertainty criteria a squares fittings

^¼q b

PN



i¼1 ðxi  xÞðyi  PN 2 i¼1 ðxi  xÞ

Þ y

ð8Þ

;

^x: ^¼y b a

ð9Þ

The interrelation between linear regression and the Pearson correlation coefficient is given by [23]

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EðY 2 Þ  E2 ðYÞ ^ ¼ q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : b EðX 2 Þ  E2 ðXÞ

ð10Þ

The proportion of variability in the investigated data through the linear regression is defined by the coefficient of determination, R2 . The variability of the data is measured via the residuals as follows

^ Þ: ^ þ bx ^ i ¼ yi  ða u i

ð11Þ 2

In the case of linear regression R is the square of the Pearson correlation coefficient. The coefficient of determination R2 is given by

PN 2 ^ u R2 ¼ 1  PN i¼1 i 2  i¼1 ðyi  yÞ

ð12Þ

and interprets the accuracy of the regression fit. 3.3. Response surface model Response surface modeling is a method used in the empirical study of the correlation between the output response and a number of input uncertainty criteria. It is typically used to find the optimal setting for the input uncertainty criteria that maximize (or minimize) the predicted response [37, Chapter 4]. This method is based on the development of a response surface approximation of the unit cell model response. This approximation is then used as a surrogate for the original model in uncertainty and sensitivity analysis. Sensitivity measures for the uncertainty criteria are

Table 2 Uncertainty criteria. Uncertainty criteria

Lower bound

Upper bound

Data sources

Yarn spacing, sy Yarn width, wy Yarn height, t y Friction coefficient,

2.5% 3.55% 4.0% 0

2.5% 3.55% 4.0% 0.5

Peng and Cao [20] Peng and Cao [20] Upper estimation Komeili and Milani [14] Modified estimation of Lin et al. [36]

l

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A.B. Ilyani Akmar et al. / Composite Structures 116 (2014) 1–17

derived from the constructed response surface. This surface plays the same role in a response surface methodology as the Taylor series in a differential analysis. Due to its advantage in minimizing the numerical effort, a response surface model [38] is implemented to evaluate the effects of uncertainty criteria on material properties of MataBerkait-dry woven fabric. The response surface model is constructed using Latin Hypercube Sampling (LHS) in order to quantify the regression coefficient, coefficient of determination, R2 and adjusted coefficient of determination, Radj . R2 is often interpreted as the proportion of response variation as explained by the regression mode in the model. Radj is derived as an alternative approach of regression evaluation in which it penalizes the statistic as extra variables are included in the model. In comparison to Monte-Carlo sampling ðMCSÞ e.g., using LHS can significantly reduce the number of required dry woven fabric unit cell samples because LHS is not depending on the dimension of the random vectors as well as the number of random variable [39,40]. By using LHS, the computational cost can be reduced due to the dense stratification across the range of each dry woven fabric unit cell samples [41]. Based on a linear polynomial, the response of the meso-scale unit cell model (here, in terms of the elastic meso-scale properties), YðN  1Þ, as a function of the uncertain input uncertainty criteria, X ¼ ðX1 ; X2 ; . . . ; Xk ÞðN  kÞ is approximated as

^ ¼ b þ b X1 þ b X2 þ    þ b Xk þ e ¼ Xb þ e; Y 0 1 2 k

ð13Þ

where N is the number of samples, k is the input uncertainty variables, b is the regression coefficient vector ðk  1Þ and the error ^ and Y is e as Eq. (14). Consequently, between the real models, Y the sum squared errors, SSE can be calculated as Eq. (15)

^ e ¼ Y  Y;

ð14Þ

SSE ¼ eT e:

ð15Þ

By substituting Eq. (14) into Eq. (15), SSE can be written as N X ^ T ðY  YÞ: ^ SSE ¼ ðY  YÞ

ð16Þ

i¼1

^ Minimizing SSE with respect to b yields the regression coefficient, b as

^ ¼ ðXT XÞ1 XT Y: b

ð17Þ

Hence, the coefficient of determination, R2 and the total variation, SST can be approximated by

SSE R ¼1 ; SST 2

2

06R 61 T

SST ¼ VðYÞ ¼ ðY  YÞ ðY  YÞ;

ð18Þ ð19Þ

where Y is the mean value of Y i expressed by Eq. (20) N 1X Y¼ ðY i Þ: N i¼1

ð20Þ

By substituting Eqs. (16), (19) and (20) into Eq. (18), R2 can be simplified by

PN ðY i  Y^i Þ R2 ¼ 1  Pi¼1 : N i¼1 ðY i  Y i Þ

ð21Þ

N1 ð1  R2 Þ: Nk

þ bkN X 2kN þ e;

ð22Þ 2

It is desirable that the coefficient of determination (COD), R is greater than 0.8 reflecting a strong correlation. For large N values, R2 is equivalent to R2adj . Higher order approaches are also carried out in the analysis as such quadratic without mixed terms

ð23Þ

and fully quadratic terms as follows

^ ¼ b þ b X 1 þ b X 2 þ    þ b X kN þ b X 2 þ b X 2 þ    Y 0 1 2 kN 11 1 22 2 þ bkNkN X 2kN þ b12 X 1 X 2 þ    þ bkN1kN1 X kN1 X kN þ e:

ð24Þ

3.4. Sensitivity indices/Sobol’s indices (variance-based method) We apply the variance-based sensitivity methods for arbitrary complex computational models. The estimation of the sensitivity indices is based on the effect of the random input variables on the model output (‘the macro-scale material parameters’). The ‘‘first-order sensitivity index’’, Si also known as ‘‘main effect index’’ is defined as a direct variance-based measure of sensitivity. However, when the number of variables becomes large, the ‘‘total-effect index’’ or ‘‘total-order index’’, STi gives an exclusive and residual influence [28,42]. 10000 sampling points are generated to quantify the sensitivity indices of uncertainty criteria where the regression coefficient adopted are based on the values obtained in the response surface model. This subsection provides a short elaboration for computing the full set of first-order and total-effect indices for a model of k-factors [43]. (a) A (N,2k) matrix of random numbers is generated where k is the number of inputs and two matrices of data (A and B) are defined which contain half of the sample. N is the number of sample which varies from few hundreds to few thousands.

2

ð1Þ

x1 6 ð2Þ 6 x 6 1 6 A ¼ 6 ... 6 ðN1Þ 6x 4 1 ðNÞ x1 2

ð1Þ

xkþ1

6 ð2Þ 6 x 6 kþ1 6 B ¼ 6 ... 6 ðN1Þ 6x 4 kþ1 ðNÞ xkþ1

ð1Þ

...

xi

x2

ð2Þ

...

xi

...

...

ðN1Þ

. . . xi

x2

x2

ð1Þ ð2Þ

... ðN1Þ

ðNÞ

...

xi

xkþ2

ð1Þ

...

xkþi

xkþ2

ð2Þ

...

xkþi

...

...

ðN1Þ

. . . xkþi

x2

xkþ2

ðNÞ

xkþ2

ðNÞ

ð1Þ ð2Þ

... ðN1Þ

...

ðNÞ

xkþi

ð1Þ

...

xk

...

xk

3

7 7 7 7 ... ... 7 7 ðN1Þ 7 . . . xk 5 ðNÞ . . . xk ...

ð2Þ

ð1Þ

x2k

ð25Þ

3

7 ð2Þ x2k 7 7 7 ... ... 7 7 ðN1Þ . . . x2k 7 5 ðNÞ . . . x2k ...

ð26Þ

(b) A matrix Ci is defined by taking all columns of B except the ith column which taken from A as

2

ð1Þ

xkþ1

6 ð2Þ 6 x 6 kþ1 6 Ci ¼ 6 . . . 6 ðN1Þ 6x 4 kþ1 ðNÞ xkþ1

xkþ2

ð1Þ

...

xi

xkþ2

ð2Þ

...

xi

...

...

...

...

ðN1Þ xi ðNÞ xi

ðN1Þ xkþ2 ðNÞ xkþ2

...

ð1Þ ð2Þ

...

ð1Þ

x2k

3

7 ð2Þ x2k 7 7 7 ... . . . 7: 7 ðN1Þ . . . x2k 7 5 ðNÞ . . . x2k ...

ð27Þ

(c) The model output is computed for all the input values in the sample matrices A; B and Ci which yields three vectors of model outputs of dimension N  1

yA ¼ f ðAÞ;

The adjusted coefficient of determination R2adj is given by

R2adj ¼ 1 

^ ¼ b þ b X 1 þ b X 2 þ    þ b X kN þ b X 2 þ b X 2 þ    Y 0 1 2 kN 11 1 22 2

yB ¼ f ðBÞ;

yC ¼ f ðCi Þ:

ð28Þ

These vectors are needed in computing the first-order and totaleffect indices, Si and STi , for a given factor X i . Since there are the kfactors, the cost of this approach is N þ N runs of the model for matrices A; B, plus k  N to estimate k-times the output vector corresponding to matrix C i [43]. The first-order sensitivity index Si can be written as

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A.B. Ilyani Akmar et al. / Composite Structures 116 (2014) 1–17

Si ¼

PN ðjÞ ðjÞ 2 VðEðYjX i ÞÞ ð1=NÞ j¼1 yA yC i  f0 ¼ ; P ðjÞ 2 VðYÞ ð1=NÞ N ðy Þ  f 2 j¼1

ð29Þ

0

A

where VðEðYjX i ÞÞ is the variance of the expected value of Y under the condition that X i is kept fixed. X i and VðYÞ is the unconditional variance of Y. The f02 is given by

f02 ¼

N 1X ðjÞ y N j¼1 A

criterium, X i to the output, i.e. first order effects in addition to all higher order effects and the expression of STi is defined by

!2 ð30Þ

:

The high and low values of yA and yCi are randomly associated if X i is non-influential parameter. Contradictorily, if X i is influential, then high (or low) values of yA will be preferentially multiplied by high (or low) values of yCi increasing the value of the resulting scalar product. Note that the accuracy of both f0 and VðYÞ can be improved by using both yA and yB points rather than yA only. The accuracy of the estimation for Si and STi will be obtained with respect to the above condition, although the factors ranking will remain unchanged [43]. The index Si is a measure for the exclusive influence of an uncertainty criterium, X i . If the sum of all Si is close to one, the model is additive, and no interaction of the uncertainty criteria exists. In complex engineering problems, interactions between input variables may exist, thus, higher order sensitivity indices are needed [44]. Each of Sobol’s [28] indices represents a sensitivity measure that describes which amount of each variance is caused due to the randomness of the single random input variables and its mapping onto the output variables [23]. The total-effect index STi is used to present the total contribution of the uncertainty

STi ¼ 1 

VðEðY j X i ÞÞ : VðYÞ

Due to the mutual interaction of the uncertainty criteria, the totaleffect index of a parameter in uncertainty criteria increases, therefore RSTi will always be constant (on top of RSTi P 1). The difference of STi  Si measures the numbers of interactions that X i has with other input variables in the uncertainty criteria [44]. In addition, Si and STi demand a large number of samples which leads to an expensive computational analysis. The sensitivity indices Si and STi shown in Eqs. (29) and (31) are reduced by the coefficient of determination as follows

^Si ¼ R2 Si ^STi ¼ R2 STi :

ð32Þ

4. Numerical results In this section, we will quantify the main parameters and their interactions that play an important role in influencing the fabric material properties. It is obligatory for a significantly statistic analysis to represent the design space equally which corresponds to an equal representation of the upper and lower bounds of the variables. In this case, the Latin Hypercube Sampling (LHS) was used as a random sample parameter distribution into equal probability intervals [45]. This method represents the design space equally by minimizing the variation vectors. Each variable is assumed to

Exx values as a function of yarn height variation 51.906

51.905

51.905

51.904

51.904 Exx (GPa)

Exx (GPa)

Exx values as a function of friction coefficient variation 51.906

51.903

51.903

51.902

51.902

51.901

51.901

51.900 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

ð31Þ

0.4

0.45

51.900

0.5

−4

−3

−2

−1

E values as a function of yarn spacing variation

51.905

51.904

51.904 Exx (GPa)

Exx (GPa)

51.905

51.903

51.902

51.901

51.901

0 Yarn spacing (%)

3

4

51.903

51.902

−1

2

xx

51.906

−2

1

E values as a function of yarn width variation

xx

51.906

51.900 −3

0 Yarn height (%)

Friction coefficient, (µ)

1

2

3

51.900 −4

−3

−2

−1

0

1

2

3

4

Yarn width (%)

Fig. 4. Effects of friction coefficient, yarn height, yarn spacing and yarn width variations on Exx under uni-axial loading in X-direction for MataBerkait-dry woven fabric.

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A.B. Ilyani Akmar et al. / Composite Structures 116 (2014) 1–17

be uniformly distributed due to insufficient information with respect to the meso-scale uncertain input parameters. 4.1. Correlation analysis The scatter plots in Figs. 4–6 show the influence of different input variables on the mechanical properties for uni-axial loading. The scatter plots for biaxial loading are depicted in Figs. 7–9. The ranking of the uncertainty criteria can be gained from the Ant-hill plots where 100 sampling points for each pair-wise uncertainty criteria are used. Based on the scatter plots, we can preliminarily predict the influences of uncertainty criteria (friction coefficient, yarn height, yarn width and yarn spacing) on material properties (Exx ; Eyy ; Ezz ; Gxy ; Gxz and Gyz ). The changes of the Young’s moduli with respects to the input parameters are very small due to the sample size of the unit cells (meso-scale analysis). It is shown that Exx and Ezz either under uni-axial or biaxial loadings, will gradually increase with an increasing friction coefficient. In addition, Eyy will decrease with an increase of friction coefficient under both loading cases. This behavior is expected since there is no loading imposed in z-direction in all cases and the direction of loading in x- and ydirections in the biaxial case. Consequently, the behavior of Ezz is mainly affected by the weave arrangement and its undulation. The decrement in Eyy is also correlated to the yarn height which is significantly influenced by the ratio of height and width of the yarns. The friction shows an important role through the external forces imposed at the contact surfaces on moving yarns. As for the yarn height, a decrease of elastic moduli can be observed as the yarn height variation becomes higher.

Theoretically, the results from yarn height variation can be correlated with the results obtained in yarn width variation because they are inter-related by yarn aspect ratio as derived in Eq. (4). Nevertheless, Exx decreases gradually with the varying of yarn height. The decrement in Exx is due to greater yarn surfaces as the yarn height varies. A contradiction also occurred for the variation of yarn width since Eyy and Ezz show a significant increase whilst Exx is gradually decreased with an increasing yarn width. However, yarn spacing variation increases the values of Exx and Eyy for both loading cases. The shear moduli components are presented in the correlation analysis summary. A summary of correlation analysis is carried out by identifying the strong/poor correlation between the inputs (uncertainty criteria) and the MataBerkait-dry woven fabric response (material properties) to give a clear observation based on the scatter plots. Table 3 shows the summary of the correlation analysis under both loading conditions. According to Table 3, it is observed that friction coefficient and yarn height have the strongest correlations on elastic moduli whilst a weak correlation on shear moduli of MataBerkait-dry woven fabric under both loadings. It can be concluded that more than 90% of the correlation coefficient reflected by the change of the friction coefficient and yarn height for the distribution of Emoduli (Exx ; Eyy and Ezz ). Due to the low value of R2 for all shear moduli, it can be concluded that the shear modulus is insignificant. This is because we observed the effect of dry-woven fabric loading in a direction that is neither parallel nor perpendicular to the fibres. If an external force is applied perpendicular to the yarns then the E-Glass/Polypropylene fibrous composite will stretch in

E values as a function of friction coefficient variation

E values as a function of yarn height variation

yy

yy

26.0

25.8

25.8

25.6

25.6

Eyy (GPa)

Eyy (GPa)

26.0

25.4

25.4

25.2

25.2

25.0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

25.0

0.5

−4

−3

−2

−1

0

1

2

3

4

Yarn height (%)

Friction coefficient, (µ) Eyy values as a function of yarn spacing variation

Eyy values as a function of yarn width variation

26.0

26.0

25.8

25.6

25.6 Eyy (GPa)

Eyy (GPa)

25.8

25.4

25.4

25.2

25.2

25.0 −3

−2

−1

0 Yarn spacing (%)

1

2

3

25.0 −4

−3

−2

−1

0 Yarn width (%)

1

2

3

4

Fig. 5. Effects of friction coefficient, yarn height, yarn spacing and yarn width variations on Eyy under uni-axial loading in X-direction for MataBerkait-dry woven fabric.

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A.B. Ilyani Akmar et al. / Composite Structures 116 (2014) 1–17 Ezz values as a function of friction coefficient variation

Ezz values as a function of yarn height variation

25.8

25.8 Ezz (GPa)

26.0

Ezz (GPa)

26.0

25.6

25.6

25.4

25.4

25.2 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

25.2

0.5

−4

−3

−2

−1

0

1

2

Friction coefficient, (µ)

Yarn height (%)

Ezz values as a function of yarn spacing variation

Ezz values as a function of yarn width variation

26.0

25.8

25.8

4

Ezz (GPa)

Ezz (GPa)

26.0

3

25.6

25.6

25.4

25.4

25.2 −3

−2

−1

0

1

2

3

Yarn spacing (%)

25.2 −4

−3

−2

−1

0

1

2

3

4

Yarn width (%)

Fig. 6. Effects of friction coefficient, yarn height, yarn spacing and yarn width variations on Ezz under uni-axial loading in X-direction for MataBerkait-dry woven fabric.

the same direction. Note that the stiffness of the E-Glass/Polypropylene fibrous composite, measured perpendicular to the fibres increases much more slowly than stiffness measured parallel to the fibres as the volume fraction of fibres is increased. Moreover, in a fibrous composite (for instance E-Glass/Polypropylene) with the applied stress aligned perpendicular to the fibres, the stress is transferred to the fibres through the fibre matrix interface and both the fibre and the matrix experience the same stress. While the rule of mixtures has proved adequate for tensile modulus (E) in the axial direction, the iso-strain rule of mixtures does not work for either the shear (G) moduli. Instead, these are dependent on the phase morphology. Since the dry-woven fabric is used in the analysis, the strain in the yarn will be the volume average of the strain in E-Glass/Polypropylene fibrous composite. In this study, the volume fraction of E-Glass is given by 0.70. The type of weave affects dimensional stability and the drapability of the fabric over complex surfaces. MataBerkait weaves, for example, exhibit good drapability. Unfortunately, good drapability and resistance to shear are mutually exclusive. Most of 2D-weaves involve two orthogonal directions of yarn, implying weak in-plane shear resistance within a single ply. Obviously, these are the reasons on why a poor shear stiffness is obtained in the analysis. Table 4 summarizes the coefficient of determination between the inputs (uncertainty criteria) and the MataBerkait-dry woven fabric response (material properties). Again, the shear stiffness contributes a lower R2 with less than 0.49. In addition to the previous elaboration, this scenario also relates to the effect of transverse pressure on the individual yarns caused by the tension of the yarn. When a yarn is subjected to such compression, the indi-

vidual fibres (fibres that construct a yarn) lose contact with each other. This results in a considerable reduction in the axial stiffness of the yarn, consequently to the shear stiffness. Besides, the properties of the yarns, the properties of the fibrous composite, the orientation of the yarns, and the size and shape of the yarns also reflect to the result of R2 in stiffness properties. Further uncertain variables with relevant correlations are yarn spacing and yarn width variations. Yarn spacing and width are found to be important criteria in almost every elastic moduli under both loading cases. As for yarn spacing, it is noticeable that its contribution is marginal. This is because the yarn spacing is related to the undulation of yarns along the loading directions. This can be proven with the results of Ezz where in uni-axial case, the q is almost perfectly positive correlation whilst in biaxial case, the q is moderately negative correlation. In uni-axial loading case, the yarn spacing becomes one of the most dominant uncertain variable influencing the elastic moduli. Theoretically, the increase in yarn height or width relates to the increase of yarn cross sectional area which acquires a greater reaction force corresponding to the stretching of the larger yarn cross-sectional area. A wider yarn will let the side contact between the yarns becomes closer. The undulation of the yarns is also influenced by these two uncertainty criteria (height and width variations) even though the variations given for these uncertainty criteria are considerably small. This is clearly seen in the results of R2 for Exx due to yarn width and Ezz with respect to yarn spacing variations. Both show a poor R2 value due to the discussed matter. A greater yarn height contributes to a curvier path line of the yarn. Subsequently, a yarn with greater height acquires larger strain to enforce elonga-

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A.B. Ilyani Akmar et al. / Composite Structures 116 (2014) 1–17 E values as a function of friction coefficient variation

E values as a function of yarn height variation xx

51.904

51.904

51.902

51.902

51.900

51.900 E (GPa)

51.906

51.898

xx

51.898

xx

E (GPa)

xx

51.906

51.896

51.896

51.894

51.894

51.892

51.892

51.890 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

51.890

0.5

−4

−3

−2

−1

E values as a function of yarn spacing variation

1

2

3

4

E values as a function of yarn width variation

xx

xx

51.906

51.906

51.904

51.904

51.902

51.902

51.900

51.900 E (GPa)

51.898

51.898

xx

Exx (GPa)

0 Yarn height (%)

Friction coefficient, (µ)

51.896

51.896

51.894

51.894

51.892

51.892

51.890 −2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Yarn spacing (%)

51.890 −4

−3

−2

−1

0

1

2

3

4

Yarn width (%)

Fig. 7. Effects of friction coefficient, yarn height, yarn spacing and yarn width variations on Exx under biaxial loading in X and Y-directions for MataBerkait-dry woven fabric.

tion in the yarn and eventually affect the reaction force of the unit cell. At certain strain level, where yarns are not in fully-tensioned, undulation and relative movements will be the two dominant factors. This is when the yarns will elongate correspondingly to the imposed load and friction becomes significant through the reaction forces at the contact surfaces. It is clearly shows that in any loading cases, the cross-sectional area of the yarns, yarn undulations and the friction coefficients between the moving yarns will certainly influence the strains in each axes. Consequently, the strains are associated to the performance of elastic moduli. In practical, the cross-sectional aspect ratios of yarns are not uniform along their axes. Each yarn is subjected to irregular pressures imposed by neighboring yarn during both manufacture of the textile preform and the consolidation process. Each yarn is pinched in different directions at different points. Indeed all four uncertainty criteria show a reliable result that relates to the theoretical and practical issues. However, a correlation analysis is only a preliminary analysis for a sensitivity analysis. For a more significant result, a response surface model and sensitivity indices are presented in Sections 4.2 and 4.3. The influences of each uncertainty criteria on the material properties of MataBerkait-dry woven fabric are clearly presented in both models. 4.2. Response surface model The results are illustrated in Tables 5 and 6 for both loading conditions with respect to the linear, quadratic without mixed terms and fully quadratic polynomial regression, respectively.

In this study a response surface model is used to determine the uncertain input variables to the fabric response outputs. We established two types of second-order regression models by separating the mixed terms (quadratic without mixed terms) and fully quadratic terms (main effect, quadratic terms and interactions terms). The response surface models are well-replicated under uni-axial loading for all output parameters while only the response surface model of Exx is replicated well under biaxial loading. This could be seen through the comparison of R2 values obtained wherein the R2 values are approximately greater than 0.80. Theoretically, a higher order of the polynomials will lead to the higher values of coefficient of determination and adjusted coefficient of determination. This has been proven by the results in Tables 5 and 6, where as the polynomial order becomes higher, the R2 value for each elastic moduli becomes larger. For example, Eyy in all regression modes under uni-axial loading contributed R2 ¼ 0:82 for linear regression, R2 ¼ 0:83 for quadratic without mixed terms and R2 ¼ 0:84 for full quadratic analysis. Under biaxial loading, it can be seen that the R2 values for Exx in each regression modes are higher than the R2 values for Exx under uni-axial loading. We should recall, that when loaded in the transverse direction, the fibrous composite (fibre and matrix) experience the same stress. This relates to the effect of yarn materials, the volume fraction of yarns, orientation of the yarns and loading cases. For instance, we found that Eyy and Ezz results in all regression modes under biaxial loading are low. Eyy contributes 0.51, 0.54 and 0.57 whilst Ezz with 0.17, 0.18 and 0.23 with respect to the linear, quadratic without mixed term and full quadratic regressions, respectively. These are caused by the yarn orientation effect that

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A.B. Ilyani Akmar et al. / Composite Structures 116 (2014) 1–17 Eyy values as a function of friction coefficient variation

Eyy values as a function of yarn height variation

21.3 21.3 21.2 21.2 21.1 21.1

Eyy (GPa)

21.0

20.9

yy

E (GPa)

21.0

20.8

20.8

20.7

20.7

20.6

20.6

20.5

20.5

20.4 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

20.4

0.5

−1

0

1

2

Eyy values as a function of yarn width variation

21.2

21.1

21.1

21.0

21.0 Eyy (GPa)

21.3

20.9

20.8

20.7

20.7

20.6

20.6

20.5

20.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Yarn spacing (%)

3

4

20.9

20.8

−1.5

−2

Eyy values as a function of yarn spacing variation

21.2

−2

−3

Yarn height (%)

21.3

20.4 −2.5

−4

Friction coefficient, (µ)

yy

E (GPa)

20.9

20.4 −4

−3

−2

−1

0

1

2

3

4

Yarn width (%)

Fig. 8. Effects of friction coefficient, yarn height, yarn spacing and yarn width variations on Eyy under biaxial loading in X and Y-directions for MataBerkait-dry woven fabric.

behave towards the load given and other related factors as discussed previously. Contour plot is a helpful visualization of the response surface when the measured input variables are no more than three. Since the uncertain input variables are more than three, it is impossible to visualize the response surface. For that reason, in order to locate the optimum value, finding the stationary point could be the best approach. Concerning the shear moduli, the values of R2 are too low in order to derive a conclusion from the response surface model. Moreover, as only axial modes (uni-axial and biaxial) are considered in our studies, shear effects seem to be negligible. It is favorable to highlight that the analysis is focused on a 2D-weave only and the samples are assumed to be perfectly aligned in the arrangements. As mentioned earlier, the Radj takes into account only the independent variables that assist in explaining the variation of the dependent variable. It means that the R2 will increase if more independent variables (uncertainty criteria) are added to the regression, however the Radj will only increase if the independent variables that are added to the regression affect the overall explanatory power of the regression. It is found that the values of R2 and Radj are approximately similar which reflects that a sufficient number of samples have evaluated. Unfortunately, a high R2 does not give any guarantee for the accuracy of the fit of the model. The reason is that a high R2 can occur in the presence of misspecification of the functional form of a correlation. With regards to this, the evaluation of R2 values cannot be fully taken as an evaluation of sensitivity analysis for the uncertainty criteria in which the approach does not quantify

the dominant factor significantly. It is important to note that each coefficient is influenced by the other variables in a regression model. Due to the uncertainty criteria that are nearly always inter-related, two or more variables may explain the same variation in material properties. Therefore, each coefficient does not explain the total-effect on material properties of its corresponding variable, as it would if it was the only variable in the model. Rather, each coefficient represents the additional effect of adding that variable into the model, if the effects of all other variables in the model are already accounted for. Therefore, each coefficient will change when other variables are added to or deleted from the model. As a conclusion, the response surface model is an empirical statistical modeling that is able to develop an appropriate approximating relationship between the fabric response (elastic properties) and the input variables. 4.3. Sensitivity indices (Sobol’s indices) Tables 7 and 8 depict the first-order and total-effect sensitivity indices that demonstrate the influences of uncertainty criteria on material properties. Based on the results in Table 7, it can clearly be observed that a strong interaction between the uncertain input variables lies in Exx . A relative difference can be measured as such Exx under uniaxial loading where the difference, ^ STi  ^ Si are 0.07, 0.05 and 0.06 for linear, quadratic without mixed term and full quadratic regressions, respectively. The difference between ^ Si and ^ STi values for Eyy can be concluded as 0.01:0.03:0.03 whilst Ezz with 0.01:0.02:0.03. It

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A.B. Ilyani Akmar et al. / Composite Structures 116 (2014) 1–17 Ezz values as a function of friction coefficient variation

Ezz values as a function of yarn height variation

22.0 22.0 21.9 21.8 21.8 21.7 21.6 Ezz (GPa)

Ezz (GPa)

21.6

21.4

21.5 21.4 21.3

21.2

21.2 21.1

21.0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

21.0

0.5

−4

−3

−2

−1

Friction coefficient, (µ)

21.9

21.9

21.8

21.8

21.7

21.7

21.6

21.6

21.5

21.4

21.3

21.3

21.2

21.2

21.1

21.1

−1.5

−1

−0.5

0

0.5

2

3

4

21.5

21.4

−2

1

Ezz values as a function of yarn width variation 22.0

Ezz (GPa)

Ezz (GPa)

Ezz values as a function of yarn spacing variation 22.0

21.0 −2.5

0 Yarn height (%)

1

1.5

2

2.5

21.0 −4

−3

−2

−1

Yarn spacing (%)

0

1

2

3

4

Yarn width (%)

Fig. 9. Effects of friction coefficient, yarn height, yarn spacing and yarn width variations on Ezz under biaxial loading in X and Y-directions for MataBerkait-dry woven fabric.

Table 3 Correlation coefficient for four uncertainty criteria under uni-axial and biaxial loadings (0:8 and above: strong correlation, 0:5  0:79: moderate correlation and 0:49 and below: weak correlation). Variables

Correlation coefficient, q Exx

Eyy

Ezz

Gxy

Gxz

Gyz

Uni-axial mode C. friction Yarn height Yarn width Yarn spacing

0.97 (strong) 0.99 (strong) 0.77 (moderate) 0.96 (strong)

0.96 (strong) 0.99 (strong) 0.97 (strong) 0.95 (strong)

0.95 (strong) 0.99 (strong) 0.99 (strong) 0.95 (strong)

0.07 (weak) 0.3 (weak) 0.15 (weak) 0.04 (weak)

0.09 (weak) 0.01 (weak) 0.11 (weak) 0.27 (weak)

0.02 (weak) 0.07 (weak) 0.18 (weak) 0.27 (weak)

Biaxial mode C. friction Yarn height Yarn width Yarn spacing

0.99 (strong) 0.99 (strong) 0.09 (weak) 0.99 (strong)

0.99 (strong) 0.99 (strong) 0.95 (strong) 0.95 (strong)

0.96 (strong) 0.97 (strong) 0.99 (strong) 0.59 (moderate)

0.05 (weak) 0.53 (moderate) 0.25 (weak) 0.31 (weak)

0.06 (weak) 0.01 (weak) 0.12 (weak) 0.04 (weak)

0.1 (weak) 0.18 (weak) 0.20 (weak) 0.12 (weak)

seems that the differences are considered small for transverse directions. Furthermore, it is found that the friction coefficient is the most influential uncertain variable in Exx values. The ^ S4 (friction coefficient) in each regression modes are obtained as 0.39 for Exx . The yarn height ^ S3 followed the rank with 0.33 for all regression modes. The least influential variable in Exx values is yarn width variations. This is agreeable due to the relationship between yarn height and yarn width ratio. This is because the yarn width and yarn height are correlated with the yarn cross-sectional aspect ratio, ARy . Concurrently, the decreases of yarn width indices will increase the yarn height indices or vice versa. In contrast, the most dominant variable in Eyy and Ezz values is yarn height variation.

Again, the yarn width is the least contributor to the Eyy values however shows a better performance in Ezz for all regression modes. It is also discovered that the yarn spacing presented itself as a variable with marginal importance in all regression modes under uni-axial loading case. Under biaxial loading case, it is observed that the friction coefficient behaves dominantly in Exx and Eyy with more than 80% and 40% respectively. Contradictorily, the dominant variable in Ezz is given by yarn width variable with 10% of the total contribution. It is also discovered that the value of ^ STi is greater than ^ Si for all mode of regression in each stiffness properties. The presence of interactions in the model can be calculated by 1  Ri Si . With the

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A.B. Ilyani Akmar et al. / Composite Structures 116 (2014) 1–17

Table 4 Coefficient of determination, R2 of correlation analysis. Variables

Coefficient of determination, R2 Exx

Eyy

Ezz

Gxy

Gxz

Gyz

Uniaxial mode C. friction Yarn height Yarn width Yarn spacing

0.94 0.98 0.59 0.92

0.92 0.98 0.94 0.90

0.90 0.98 0.98 0.90

0.005 0.09 0.02 0.002

0.008 0.0001 0.01 0.07

0.0004 0.005 0.03 0.07

Biaxial mode C. friction Yarn height Yarn width Yarn spacing

0.98 0.98 0.008 0.98

0.98 0.98 0.90 0.90

0.92 0.94 0.98 0.35

0.003 0.28 0.06 0.10

0.004 0.0001 0.01 0.002

0.01 0.03 0.04 0.01

presence of interaction between the uncertainty variable and the response output (stiffness properties), significantly the value of STi will be greater than Si . The difference, STi  Si is a measure of how much uncertainty variable is involved in interactions with any other uncertainty variable. In addition, it is observed that most of the total-effect indices are less than 1 except for Exx under biaxial loading. This shows that the uncertain input variable is non-additive to all stiffness properties under uni-axial loading and similarly to Eyy and Ezz for biaxial loading. Overall, the dominance of each uncertain input variable also affected by the loading modes and the yarn properties. Again, the yarn properties that include the orientation of the yarns, size and shape of the yarns and the fibrous composite may affect the stiffness values. For instance, the friction

Table 5 Regression coefficient, coefficient of determination, R2 and adjusted coefficient of determination, Radj under uniaxial loading. Response surface model Regression mode Linear regression

Elastic properties

Regression coefficient

Standardized regression coefficient

Exx Coefficient of determination, R2 ¼ 0.89

b0 ¼ 51.898 b1 ¼ 0.012 b2 ¼ 0.003

bx1 ¼ 0:452 bx2 ¼ 0.146

Adjusted coefficient of determination, R2adj ¼ 0.89

b3 ¼ 0.089

Adjusted coefficient of determination, R2adj ¼ 0.82

b3 ¼ 4.604 ¼ ¼ ¼ ¼

0.150 25.595 0.201 0.223

bx3 ¼ 0.543 bx4 ¼ 0.554 bx1 ¼ 0.324 bx2 ¼ 0.240 bx3 ¼ 0.778 bx4 ¼ 0.298

Coefficient of determination, R2 ¼ 0.79

b4 b0 b1 b2

Adjusted coefficient of determination, R2adj ¼ 0.79

b3 ¼ 3.806

bx3 ¼ 0.746

b4 ¼ 0.047

bx4 ¼ 0.108

Ezz

¼ ¼ ¼ ¼ ¼

51.50 0.101 0.037 0.230 0.014

Coefficient of determination, R2 ¼ 0.91

b0 b1 b2 b3 b4

Adjusted coefficient of determination, R2adj ¼ 0.90

b5 ¼ 0.009

Exx

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

0.005 0.319 0.013 22.235 2.952 1.587 3.200 0.256

Coefficient of determination, R2 ¼ 0.83

b6 b7 b8 b0 b1 b2 b3 b4

Adjusted coefficient of determination, R2adj ¼ 0.82

b5 ¼ 0.257

Eyy

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

0.200 1.423 0.212 24.701 1.657 1.269 1.939 0.112

bx1 ¼ 0.243 bx2 ¼ 0.331

bx1 bx2 bx3 bx4 bx5

¼ ¼ ¼ ¼ ¼

3.814 1.727 1.405 1.008 3.370

bx6 ¼ 1.875 bx7 ¼ 1.945 bx8 ¼ 0.464 bx1 bx2 bx3 bx4 bx5

¼ ¼ ¼ ¼ ¼

3.077 2.034 0.540 0.509 2.749

bx6 ¼ 2.273 bx7 ¼ 0.240 bx8 ¼ 0.218

Coefficient of determination, R2 ¼ 0.80

b6 b7 b8 b0 b1 b2 b3 b4

Adjusted coefficient of determination, R2adj ¼ 0.79

b5 ¼ 0.142

bx6 ¼ 2.219

b6 ¼ 0.168 b7 ¼ 1.860 b8 ¼ 0.130

bx7 ¼ 0.365 bx8 ¼ 0.155

Ezz

Full quadratic

0.008 25.420 0.311 0.187

Coefficient of determination, R2 ¼ 0.82

Eyy

Quadratic without mixed terms

¼ ¼ ¼ ¼

b4 b0 b1 b2

¼ ¼ ¼ ¼ ¼

51.633 0.082 0.020 0.064 0.008

bx1 bx2 bx3 bx4 bx5

¼ ¼ ¼ ¼ ¼

2.005 1.889 0.380 0.259 1.764

Coefficient of determination, R2 ¼ 0.91

b0 b1 b2 b3 b4

Adjusted coefficient of determination, R2adj ¼ 0.90

b5 ¼ 0.009

bx5 ¼ 3.429

b6 ¼ 0.004

bx6 ¼ 1.774

Exx

bx1 bx2 bx3 bx4

¼ ¼ ¼ ¼

3.074 0.912 0.392 0.555

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A.B. Ilyani Akmar et al. / Composite Structures 116 (2014) 1–17 Table 5 (continued) Response surface model

Coefficient of determination, R2 ¼ 0.84

b7 ¼ 0.267 b8 ¼ 0.012 b9 ¼ 0.003 b10 ¼ 0.018 b11 ¼ 0.000 b12 ¼ 0.003 b13 ¼ 0.000 b14 ¼ 0.039 b0 ¼ 13.547 b1 ¼ 6.114 b2 ¼ 0.497 b3 ¼ 19.259 b4 ¼ 0.427

bx7 ¼ 1.630 bx8 ¼ 0.460 bx9 ¼ 0.845 bx10 ¼ 0.671 bx11 ¼ 0.051 bx12 ¼ 0.089 bx13 ¼ 0.121 bx14 ¼ 1.383 bx1 ¼ 6.371 bx2 ¼ 0.637 bx3 ¼ 3.253 bx4 ¼ 0.850 bx5 ¼ 1.627

Adjusted coefficient of determination, R2adj ¼ 0.83

b5 ¼ 0.152

bx6 ¼ 3.108 bx7 ¼ 1.447 bx8 ¼ 0.212 bx9 ¼ 6.903 bx10 ¼ 1.441 bx11 ¼ 1.926 bx12 ¼ 2.845 bx13 ¼ 2.294 bx14 ¼ 0.723

Coefficient of determination, R2 ¼ 0.81

b6 ¼ 0.273 b7 ¼ 8.567 b8 ¼ 0.206 b9 ¼ 0.809 b10 ¼ 1.408 b11 ¼ 0.188 b12 ¼ 2.957 b13 ¼ 0.259 b14 ¼ 0.727 b0 ¼ 15.085 b1 ¼ 4.798 b2 ¼ 0.709 b3 ¼ 12.871 b4 ¼ 0.600

Adjusted coefficient of determination, R2adj ¼ 0.80

b5 ¼ 0.018

bx6 ¼ 3.079

b6 ¼ 0.233 b7 ¼ 8.934 b8 ¼ 0.132 b9 ¼ 0.743 b10 ¼ 2.308 b11 ¼ 0.131 b12 ¼ 2.607 b13 ¼ 0.180 b14 ¼ 1.685

bx7 ¼ 1.752 bx8 ¼ 0.158 bx9 ¼ 7.365 bx10 ¼ 2.742 bx11 ¼ 1.556 bx12 ¼ 2.911 bx13 ¼ 1.846 bx14 ¼ 1.943

Eyy

Ezz

bx1 bx2 bx3 bx4 bx5

¼ ¼ ¼ ¼ ¼

5.804 1.054 2.524 1.388 0.225

Table 6 Regression coefficient, coefficient of determination, R2 and adjusted coefficient of determination, Radj under biaxial loading. Response surface model Regression mode

Elastic properties

Regression coefficient

Standardized regression coefficient

Linear regression

Exx Coefficient of determination, R2 ¼ 0.94

b0 ¼ 51.881 b1 ¼ 0.009 b2 ¼ 0.002

bx1 ¼ 0.151 bx2 ¼ 0.045 bx3 ¼ 0.247

Adjusted coefficient of determination, R2adj ¼ 0.94

b3 ¼ 0.093

bx4 ¼ 0.887

Coefficient of determination, R2 ¼ 0.51

b4 b0 b1 b2

0.028 20.395 0.274 0.088

bx1 ¼ 0.202 bx2 ¼ 0.080 bx3 ¼ 0.304

Adjusted coefficient of determination, R2adj ¼ 0.51

b3 ¼ 2.545

bx4 ¼ 0.657

Eyy

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

0.467 20.738 0.014 0.240

Coefficient of determination, R2 ¼ 0.17

b4 b0 b1 b2

Adjusted coefficient of determination, R2adj ¼ 0.15

b3 ¼ 1.034

Ezz

bx1 ¼ 0.014 bx2 ¼ 0.297 bx3 ¼ 0.168 bx4 ¼ 0.196

b4 ¼ 0.102

Quadratic without mixed terms

¼ ¼ ¼ ¼ ¼

51.867 0.113 0.096 0.225 0.048

bx1 bx2 bx3 bx4 bx5

¼ ¼ ¼ ¼ ¼

1.850 1.944 0.599 1.509 1.708

Coefficient of determination, R2 ¼ 0.96

b0 b1 b2 b3 b4

Adjusted coefficient of determination, R2adj ¼ 0.96

b5 ¼ 0.010

bx6 ¼ 1.987

b6 ¼ 0.011 b7 ¼ 0.134 b8 ¼ 0.040

bx7 ¼ 0.358 bx8 ¼ 0.641

Exx

(continued on next page)

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Table 6 (continued) Response surface model Regression mode

Elastic properties

¼ ¼ ¼ ¼ ¼

35.022 13.668 9.185 1.146 0.854

Coefficient of determination, R2 ¼ 0.54 Adjusted coefficient of determination, R2adj ¼ 0.52

b5 ¼ 1.305

Eyy

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

1.025 3.799 0.785 24.273 12.630 6.346 4.497 0.108

Standardized regression coefficient bx1 bx2 bx3 bx4 bx5

¼ ¼ ¼ ¼ ¼

10.066 8.320 0.137 1.201 9.857

bx6 ¼ 8.243 bx7 ¼ 0.483 bx8 ¼ 0.570

Coefficient of determination, R2 ¼ 0.18

b6 b7 b8 b0 b1 b2 b3 b4

Adjusted coefficient of determination, R2adj ¼ 0.15

b5 ¼ 1.230

bx6 ¼ 7.546

b6 ¼ 0.688 b7 ¼ 3.416 b8 ¼ 0.001

bx7 ¼ 0.556 bx8 ¼ 0.001

Ezz

Full quadratic

Regression coefficient b0 b1 b2 b3 b4

¼ ¼ ¼ ¼ ¼

51.99 0.072 0.061 0.597 0.018

bx1 bx2 bx3 bx4 bx5

¼ ¼ ¼ ¼ ¼

12.682 7.837 0.732 0.208 12.670

Coefficient of determination, R2 ¼ 0.97

b0 b1 b2 b3 b4

Adjusted coefficient of determination, R2adj ¼ 0.97

b5 ¼ 0.003

bx5 ¼ 0.474

Coefficient of determination, R2 ¼ 0.57

b6 ¼ 0.017 b7 ¼ 0.185 b8 ¼ 0.040 b9 ¼ 0.022 b10 ¼ 0.015 b11 ¼ 0.018 b12 ¼ 0.052 b13 ¼ 0.021 b14 ¼ 0.054 b0 ¼ 32.387 b1 ¼ 16.047 b2 ¼ 11.041 b3 ¼ 51.919 b4 ¼ 2.279

bx6 ¼ 2.886 bx7 ¼ 0.492 bx8 ¼ 0.653 bx9 ¼ 2.905 bx10 ¼ 0.246 bx11 ¼ 2.882 bx12 ¼ 0.791 bx13 ¼ 2.988 bx14 ¼ 0.844 bx1 ¼ 11.818 bx2 ¼ 10.002 bx3 ¼ 6.197 bx4 ¼ 3.207 bx5 ¼ 9.186

Adjusted coefficient of determination, R2adj ¼ 0.54

b5 ¼ 1.216

bx6 ¼ 6.317 bx7 ¼ 1.519 bx8 ¼ 0.568 bx9 ¼ 6.831 bx10 ¼ 2.516 bx11 ¼ 0.069 bx12 ¼ 2.794 bx13 ¼ 5.727 bx14 ¼ 1.251

Coefficient of determination, R2 ¼ 0.23

b6 ¼ 0.785 b7 ¼ 12.721 b8 ¼ 0.781 b9 ¼ 1.132 b10 ¼ 3.478 b11 ¼ 0.010 b12 ¼ 4.109 b13 ¼ 0.917 b14 ¼ 1.781 b0 ¼ 25.267 b1 ¼ 14.938 b2 ¼ 9.513 b3 ¼ 53.766 b4 ¼ 2.824

Adjusted coefficient of determination, R2adj ¼ 0.17

b5 ¼ 0.841

bx6 ¼ 3.955

b6 ¼ 0.361 b7 ¼ 20.150 b8 ¼ 0.019 b9 ¼ 1.629 b10 ¼ 2.092 b11 ¼ 0.430 b12 ¼ 4.772 b13 ¼ 0.399 b14 ¼ 2.549

bx7 ¼ 3.280 bx8 ¼ 0.019 bx9 ¼ 13.402 bx10 ¼ 2.063 bx11 ¼ 4.243 bx12 ¼ 4.423 bx13 ¼ 3.402 bx14 ¼ 2.440

Exx

Eyy

Ezz

coefficient variable becomes dominant in both loading cases for Exx and only for Eyy under biaxial loading. By definition, STi is greater than Si or equal to Si in the case that one uncertainty variable is not involved in any interaction with other uncertainty variables. However, if STi ¼ 0, it reflects that uncertainty variable is non-

bx1 bx2 bx3 bx4

bx1 bx2 bx3 bx4 bx5

¼ ¼ ¼ ¼

¼ ¼ ¼ ¼ ¼

1.188 1.239 1.591 0.563

14.999 11.749 8.750 5.419 8.666

influential and can be fixed anywhere in its distribution without affecting the variance of the response output. Furthermore, the sum of all Si is equal to 1 for additive models whilst less than 1 for non-additive models. Conclusively, a reliable result has been obtained through Sobol’s indices compared to other approaches.

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Table 7 First-order and total-effects sensitivity indices of uncertainty criteria on MataBerkait-dry woven fabric elastic properties under uni-axial loading: 1 – yarn spacing; 2 – yarn width; 3 – yarn height; 4 – friction coefficient. Material properties

Linear regression First-order sensitivity indices, ^ Si

Quadratic without mixed term ^i First-order sensitivity indices, S

Full quadratic First-order sensitivity indices, ^ Si

Exx

^ S1 ¼ 0.15 ^ S2 ¼ 0.04 ^ S3 ¼ 0.33

^ S1 ¼ 0.14 ^ S2 ¼ 0.03 ^ S3 ¼ 0.31

^ S1 ¼ 0.14 ^ S2 ¼ 0.03 ^ S3 ¼ 0.32

^ S4 ¼ 0.37 P4 ^ i¼1 Si ¼ 0.89

^ S4 ¼ 0.43 P4 ^ i¼1 Si ¼ 0.91

^ S4 ¼ 0.42 P4 ^ i¼1 Si ¼ 0.91

Total effect indices, ^ STi ^ ST1 ¼ 0.20 ^ ST2 ¼ 0.04

Total-effect indices, ^ STi ^ ST1 ¼ 0.20 ^ ST2 ¼ 0.04

Total-effect indices, ^ STi ^ ST1 ¼ 0.21 ^ ST2 ¼ 0.04

^ ST3 ¼ 0.33 ^ ST4 ¼ 0.39 P4 ^ i¼1 STi ¼ 0.96

^ ST3 ¼ 0.33 ^ ST4 ¼ 0.39 P4 ^ i¼1 STi ¼ 0.96

^ ST3 ¼ 0.33 ^ ST4 ¼ 0.39 P4 ^ i¼1 STi ¼ 0.97

First-order sensitivity indices, ^ Si ^ S1 ¼ 0.07 ^ S2 ¼ 0.06

First-order sensitivity indices, ^ Si ^ S1 ¼ 0.08 ^ S2 ¼ 0.06

First-order sensitivity indices, ^ Si ^ S1 ¼ 0.08 ^ S2 ¼ 0.06

^ S3 ¼ 0.59 ^ S4 ¼ 0.10 P4 ^ i¼1 Si ¼ 0.82

^ S3 ¼ 0.59 ^ S4 ¼ 0.10 P4 ^ i¼1 Si ¼ 0.83

^ S3 ¼ 0.60 ^ S4 ¼ 0.10 P4 ^ i¼1 Si ¼ 0.84

Total-effect indices, ^ STi ^ ST1 ¼ 0.08 ^ ST2 ¼ 0.06

Total-effect indices, ^ STi ^ ST1 ¼ 0.09 ^ ST2 ¼ 0.07

Total-effect indices, ^ STi ^T1 ¼ 0.09 S ^ ST2 ¼ 0.08

^ ST3 ¼ 0.59 ^ ST4 ¼ 0.10 P4 ^ i¼1 STi ¼ 0.83

^ ST3 ¼ 0.60 ^ ST4 ¼ 0.10 P4 ^ i¼1 STi ¼ 0.86

^ ST3 ¼ 0.60 ^ ST4 ¼ 0.10 P4 ^ i¼1 STi ¼ 0.87

First-order sensitivity indices, ^ Si ^ S1 ¼ 0.05 ^ S2 ¼ 0.13

First-order sensitivity indices, ^ Si ^ S1 ¼ 0.05 ^ S2 ¼ 0.13

First-order sensitivity indices, ^ Si ^1 ¼ 0.05 S ^ S2 ¼ 0.13

^ S3 ¼ 0.60 ^ S4 ¼ 0.01 P4 ^ i¼1 Si ¼ 0.79

^ S3 ¼ 0.60 ^ S4 ¼ 0.02 P4 ^ i¼1 Si ¼ 0.80

^ S3 ¼ 0.61 ^ S4 ¼ 0.02 P4 ^ i¼1 Si ¼ 0.81

Total-effect indices, ^ STi ^ ST1 ¼ 0.05 ^ ST2 ¼ 0.13

Total-effect indices, ^ STi ^ ST1 ¼ 0.06 ^ ST2 ¼ 0.13

Total-effect indices, ^ STi ^ ST1 ¼ 0.07 ^ ST2 ¼ 0.13

^ ST3 ¼ 0.61 ^ ST4 ¼ 0.01 P4 ^ i¼1 STi ¼ 0.80

^ ST3 ¼ 0.61 ^ ST4 ¼ 0.02 P4 ^ i¼1 STi ¼ 0.82

^ ST3 ¼ 0.61 ^ ST4 ¼ 0.03 P4 ^ i¼1 STi ¼ 0.84

Eyy

Ezz

5. Conclusion In this paper, two global sensitivity analyses are presented with particular consideration of the influences of four uncertainty criteria on MataBerkait-dry woven fabric material properties. They are presented in order to quantify the significant factor that influenced MataBerkait-dry woven fabric material properties under uni-axial and biaxial loadings. Latin Hypercube Sampling (LHS) was used as random sampling plan in the analysis. Furthermore, it is highlighted that the sensitivity analyses conducted are based on surrogate model predictions. The four uncertainty criteria of interest are defined as the yarn width, spacing, height and friction coefficient. These four uncertainty criteria are highlighted due to the importance in geometrical factors and the mechanical responses of fabric unit cells. The most significant uncertainty criterium that affects the material properties of the fabric was identified by the global sensitivity analyses. Overall, the reasons that influenced the material properties results based on uncertainty criteria are varies. To conclude, the type of weave, the properties of the yarns, the properties of the fibrous composite, the orientation of the yarns, and the size and shape of the yarns affect the stiffness values. The effect of loading also plays an important role in predicting the stiffness values.

A regression-based method helps in determining on how the output of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are held fixed. Correlation analysis is conducted by varying one uncertainty variable at a time. The scatter plots are beneficial for quick overviews and for showing patterns between the uncertainty criteria and responses (material properties). It shows large quantities of data and easy to interpret the correlation between variables and clustering effects. However, scatter plots confront with difficulties in discovering relationships which span more than two dimensions. On the other hand, the correlation analysis (scatter plot) measures the linear relation between the uncertainty criteria which shows that the friction coefficient and yarn height are dominantly influenced the material properties of the MataBerkaitdry woven fabric. The contribution of correlation ratio obtained by these two factors are extremely high (more than 0.90) in both loading cases evaluated. Shear moduli are found to have the weakest correlation compared to elastic moduli results. Moreover, the observations from the results of scatter plots provided valuable insight in the effects of uncertainty criteria on the response of the fabric. Indeed, the correlation analysis is a preliminary evaluation of the uncertainty criteria.

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Table 8 First-order and total-effects sensitivity indices of uncertainty criteria on MataBerkait-dry woven fabric elastic properties under biaxial loading: 1 – yarn spacing; 2 – yarn width; 3 – yarn height; 4 – friction coefficient. Material properties

Linear regression

Quadratic without mixed term

Full quadratic

Exx

First-order sensitivity indices, ^ Si ^ S1 ¼ 0.02 ^ S2 ¼ 0.00

First-order sensitivity indices, ^ Si ^ S1 ¼ 0.02 ^ S2 ¼ 0.00

First-order sensitivity indices, ^ Si ^ S1 ¼ 0.05 ^ S2 ¼ 0.00

^ S3 ¼ 0.06 ^ S4 ¼ 0.86 P4 ^ i¼1 Si ¼ 0.94

^ S3 ¼ 0.05 ^ S4 ¼ 0.89 P4 ^ i¼1 Si ¼ 0.96

^ S3 ¼ 0.06 ^ S4 ¼ 0.86 P4 ^ i¼1 Si ¼ 0.97

Total-effect indices, ^ STi ^ ST1 ¼ 0.03 ^ ST2 ¼ 0.02

Total-effect indices, ^ STi ^ ST1 ¼ 0.03 ^ ST2 ¼ 0.02

Total-effect indices, ^ STi ^ ST1 ¼ 0.03 ^ ST2 ¼ 0.01

^ ST3 ¼ 0.08 ^ ST4 ¼ 0.86 P4 ^ i¼1 STi ¼ 0.99

^ ST3 ¼ 0.08 ^ ST4 ¼ 0.89 P4 ^ i¼1 STi ¼ 1.02

^ ST3 ¼ 0.09 ^ ST4 ¼ 0.94 P4 ^ i¼1 STi ¼ 1.07

First-order sensitivity indices, ^ Si ^ S1 ¼ 0.03 ^ S2 ¼ 0.01

First-order sensitivity indices, ^ Si ^1 ¼ 0.03 S ^ S2 ¼ 0.01

First-order sensitivity indices, ^ Si ^1 ¼ 0.01 S ^ S2 ¼ 0.00

^ S3 ¼ 0.07 ^ S4 ¼ 0.40 P4 ^ i¼1 Si ¼ 0.51

^ S3 ¼ 0.09 ^ S4 ¼ 0.41 P4 ^ i¼1 Si ¼ 0.54

^ S3 ¼ 0.08 ^ S4 ¼ 0.48 P4 ^ i¼1 Si ¼ 0.57

Total-effect indices, ^ STi ^ ST1 ¼ 0.03 ^ ST2 ¼ 0.01

Total-effect indices, ^ STi ^T1 ¼ 0.04 S ^ ST2 ¼ 0.02

Total-effect indices, ^ STi ^T1 ¼ 0.01 S ^ ST2 ¼ 0.00

^ ST3 ¼ 0.08 ^ ST4 ¼ 0.42 P4 ^ i¼1 STi ¼ 0.54

^ ST3 ¼ 0.10 ^ ST4 ¼ 0.42 P4 ^ i¼1 STi ¼ 0.58

^ ST3 ¼ 0.10 ^ ST4 ¼ 0.49 P4 ^ i¼1 STi ¼ 0.60

First-order sensitivity indices, ^ Si ^ S1 ¼ 0.00 ^ S2 ¼ 0.09

First-order sensitivity indices, ^ Si ^1 ¼ 0.01 S ^ S2 ¼ 0.10

First-order sensitivity indices, ^ Si ^1 ¼ 0.03 S ^ S2 ¼ 0.10

^ S3 ¼ 0.03 ^ S4 ¼ 0.04 P4 ^ i¼1 Si ¼ 0.17

^ S3 ¼ 0.03 ^ S4 ¼ 0.04 P4 ^ i¼1 Si ¼ 0.18

^ S3 ¼ 0.04 ^ S4 ¼ 0.06 P4 ^ i¼1 Si ¼ 0.23

Total-effect indices, ^ STi ^ ST1 ¼ 0.04 ^ ST2 ¼ 0.14

Total-effect indices, ^ STi ^ ST1 ¼ 0.01 ^ ST2 ¼ 0.10

Total-effect indices, ^ STi ^ ST1 ¼ 0.03 ^ ST2 ¼ 0.10

^ ST3 ¼ 0.07 ^ ST4 ¼ 0.00 P4 ^ i¼1 STi ¼ 0.25

^ ST3 ¼ 0.03 ^ ST4 ¼ 0.05 P4 ^ i¼1 STi ¼ 0.19

^ ST3 ¼ 0.04 ^ ST4 ¼ 0.07 P4 ^ i¼1 STi ¼ 0.24

Eyy

Ezz

Unlike, regression-based, variance-based methods allow full exploration of the input space, which account for interactions, and non-linear responses. Response surface model is an iterative process and involves only the main effects and interactions or may also have quadratic and possibly cubic terms to visualize the curvature. It is also known as surrogate models which commonly employed for design optimization and sensitivity analyses. Three types of regression modes (Linear, quadratic without mixed term and full quadratic regressions) are presented in the response surface model. Again, it is impossible to visualize the response shape for more than 3 dimensional cases. R2 values from response surface model showing an increase as the order of regression is increased. However, the high values of R2 could not guarantee the fitness of the model. Therefore, Radj is used to help the overall explanatory power of the regression. Radj value will be higher as more extra variables are added. However, the small difference between Radj and R2 reflects that the material properties response are not affected with any extra variables in the analysis. The evaluation of R2 values cannot be fully-taken as an evaluation of sensitivity analysis for the uncertainty criteria because the approach does not quantify the dominant factor significantly. The response

surface model results on shear properties might be misleading but also indicate some effects due to the variation of uncertainty criteria. Another approach of a variance-based method utilized is the Sobol’s indices. The Sobol’s sensitivity indices are ratios of partial variances to total variance, and none of the sensitivity indices may be negative or exceed 1. Importantly, first-order sensitivity index does not measure the uncertainty caused by interactions with other variables. A first-order sensitivity index is estimated directly because it measures the effect of one varying variable. The total-effect index places an upper bound on the importance of a given input by crediting the full effect of all relevant interactions to the given input. Eventually, the sensitivity indices (Sobol’s indices) provide a clear idea of the effect of each uncertain variable onto the variance of the result responses. Indeed, the sum of all total-effect sensitivity indices is always greater than 1 and the model is perfectly additive if its equal to 1. However, a more general study can be conducted to consider the effect of combined loading, as such combination of shear and biaxial loadings. It is also be worthwhile to investigate a series of loading magnitudes, yarn shapes and material properties of fibrous

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composite on the sensitivity analysis results as it was observed here some factors can either loosen or strengthen their significance by the increment of the loading magnitudes (only displacement is applied here). As for a large structure application, the combined loadings are imposed. Lenticular or power ellipse can be chosen as an option to the yarn shape selection. Other fibrous composite like E-Glass/Polyester also contributes some difference in the results. Most of the observations and conclusions made are limited to the 2D-weave of MataBerkait-dry woven fabric. Adoption of several plies of MataBerkait-dry woven fabric also a worthwhile trial to be considered as here we only used a single ply dry-fabric. It would be of interest to study on how the same uncertainty criteria reflect the response in 3D-weave of various fabric architectures. In conjunction to that, the uncertainty in yarn materials may be used to quantify the significance of material properties with the similar approaches. Acknowledgment The author would like to give an endless gratitude to the ‘‘Ministry of Higher Educational of Malaysia’’ for the supports given in completing this study. References [1] V Sagar T, Potluri P, Hearle JWS. Mesoscale modelling of interlaced fibre assemblies using energy method. Comput Mater Sci 2003;28:49–62. [2] Clulow EE, Taylor HM. An experimental and theoretical investigation of biaxial stress–strain relations in a plain weave cloth. J Text Inst Trans 1963;54:323–47. [3] Hearle JWS, Grosberg P, Backer S. Structural mechanics of fibers, yarns and fabrics, vol. 1. Wiley-Interscience; 1969. [4] Kawabata S, Niwa M, Kawai H. The finite deformation theory of plain weave fabrics part i: the biaxial deformation theory. J Text Inst 1973;64:21–46. [5] Kawabata S, Niwa M, Kawai H. The finite deformation theory of plain weave fabrics part ii: the uni-axial deformation theory. J Text Inst 1973;64:47–61. [6] Testa R, Yu L. Stress–strain relation for coated fabrics. J Eng Mech 1987;113:1631–46. [7] Pan N. Analysis of woven fabric strengths: prediction of fabric strength under uniaxial and biaxial extensions. Compos Sci Technol 1996;56:311–27. [8] Beex LAA, Verberne CW, Peerlings RHJ. Experimental identification of a lattice model for woven fabrics: application to electronic textile. Compos Part A 2013;48:82–92. [9] Buet-Gautier K, Boisse P. Experimental analysis and modeling of biaxial mechanical behavior of woven composite reinforcements. Exp Mech 2001;41:260–9. [10] Lomov SV, Huysmans G, Luo Y, Parnas RS, Prodromou A, Verpoest I, et al. Textile composites: modelling strategies. Compos Part A Appl Sci Manuf 2001;32:1379–94. [11] Tabiei A, Yi W. Comparative study of predictive methods for woven fabric composite elastic properties. Compos Struct 2002;58:149–64. [12] Cavallaro PV, Sadegh AM, Quigley CJ. Decrimping behavior of uncoated plainwoven fabrics subjected to combined biaxial tension and shear stresses. Text Res J 2007;77:403–16. [13] Kumazawa H, Susuki I, Morita T, Kuwabara T. Mechanical properties of coated plain weave fabrics under biaxial loads. Trans Jpn Soc Aeronaut Space Sci 2005;48:117–23. [14] Komeili M, Milani AS. The effect of meso-level uncertainties on the mechanical response of woven fabric composites under axial loading. Comput Struct 2012;90-91:163–71. [15] Gasser A, Boisse P, Hanklar S. Mechanical behaviour of dry fabric reinforcements. 3d simulations versus biaxial tests. Comput Mater Sci 2000;17:7–20.

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