تعیین مقدار عدم قطعیت پارچه های بافته شده خشک: تجزیه و تحلیل حساسیت بر روی خواص مواد

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.

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 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] and [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 micro-scale 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 material-related 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 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] and [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.

In this paper, two global sensitivity analyses are presented with particular consideration of the influences of four uncertainty criteria on MataBerkaitMataBerkait-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 MataBerkait-dry 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. 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. R2R2 values from response surface model showing an increase as the order of regression is increased. However, the high values of R2R2 could not guarantee the fitness of the model. Therefore, RadjRadj is used to help the overall explanatory power of the regression. RadjRadj value will be higher as more extra variables are added. However, the small difference between RadjRadj and R2R2 reflects that the material properties response are not affected with any extra variables in the analysis. The evaluation of R2R2 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 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 MataBerkaitMataBerkait-dry woven fabric. Adoption of several plies of MataBerkaitMataBerkait-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.