Robust parameter design approach and statistical modeling

As explained in the introduction section, the main objective of robust parameter design is to minimize variance of the results while the sources of uncertainty still exist. Main causes that can potentially impart uncertainty into the reinforcement behavior of this study include the fabrication processes, characterization methods and material uncertainties, including in- herent natural fiber properties’ variation and design parameters setting of the reinforcement. Uncertainties due to fabrication and characterization methods are addressed here through de- veloping consistent reinforcement and composite fabrication methods (explained in appendix C and section 3.7, respectively) as well as using reliable permeability mold and measurement procedures (section 3.6) while composite testing is performed according to ASTM standards. However, as will be explained in the results chapter, the IBS test method (section 3.4) shows a quite large variability in the results. Furthermore, natural fibers of this study have always been supplied from the same companies (Innofibre for Kraft pulp and Safilin for flax yarns) which favor the consistency of constituent material properties and subsequently of results. Principally in this thesis, studying inherent variation in physical/mechanical properties of different natural fiber species was not targeted and such an aspect is already studied in some other research works [102, 105].

For studying the variation of results due to parameters setting, the classical robust param- eter design method shown in Figure 3-31 is considered in this thesis, as a general approach. This algorithm is devised based on references [39, 106, 107] and its detailed description is

explained in appendix F (section F.4). Explanations about design of experiment (DOE), anal- ysis of variance (ANOVA) and regression modeling used in this approach are also given in the same appendix.

Taguchi method is reported less efficient than the classical method [106, 107], and with regard to the parameters of this study and their settings, it was hardly possible to implement the notion of outer array design of experiments used in Taguchi's method. Moreover, since the practical domain of each factor was not large and set at the beginning of experiments on the feasible experimental region, the heuristic portion of the algorithm (steps 5 to 10 in Figure 3-31) was not actually applicable and hence was not followed.

Throughout this thesis two DOE trials shown in Table 3-9 (called 1st _{DOE hereafter) and} Table 3-11 (called 2nd DOE hereafter) are used. The 1st DOE is a 24-1 fractional-factorial resolution IV design of experiment considered for step 3 of Figure 3-31 and the 2nd DOE is a full-factorial 32 experimental design used for step 11 of Figure 3-31. Because it was difficult to conduct a central composite design (CCD) for the parameter settings of this study (due to axial portion of the CCD), full factorial 2nd _{DOE is used for step 11 of Table 3-11.}

F igure 3 -31 . Algor it hm of c lassica l robust pa ra met er de sign [39, 106, 107] .

Firstly, IBS, permeability and tensile tests are conducted based on the 1st DOE in con- formance with step 3 of Figure 3-31 and their results are presented and analyzed in the sec- tions 4.1.2, 4.2.1 and 4.3.2, respectively. As will be explained in these sections, because IBS and tensile test results did not fulfill the criterion of step 4, other steps of Figure 3-31 are no more followed for IBS and tensile behavior evaluation. However, areal density of UD flax layer (factor B in the 1st DOE) is concluded influential on both mean and standard deviation of K1 permeability (fulfilling robust criterion of step 4), therefore a second series of permea- bility measurements is conducted based on the 2nd DOE to implement steps 11 to 14 of Figure 3-31 and its results are presented and analyzed in section 4.2.2.

The first two columns of Table 3-8 show the factors used in the 1st _{DOE and their defini-} tions, and the last two columns indicate their coded and actual settings. In this table, the 116 ± 4.3 and 172 ± 6.9 g/m2 surface densities correspond to 16 and 24 yarns/in. flax plies (tex 200 yarns) respectively. Likewise, the two studied paper surface densities are 29 ± 0.99 and 38 ± 0.64 g/m2. High and low limits of factors A were chosen low because a low paper surface density is better for the reinforcement permeability. For factor B, it was not possible to put more than 24 yarns/in. and 16 yarns/in. was considered a low limit in tem of yarn spacing. High and low levels for factors C and D are selected based on technical considera- tions (reasonable drying temperatures and compressing pressures).

Table 3-9 shows the 24-1 fractional-factorial resolution IV design of experiment along with the measured values using this table. In this 1st _{DOE, the fiber volume fraction was kept} constant at 35 % for permeability tests and composite fabrication. Each response indicated in Table 3-9 is measured at least four times. According to appendix M of [107] such a sample

size yields a confidence of 90 % to avoid type II statistical error (β) and 95 % to avoid type I statistical error (α) for standard deviation values. The confidence to avoid both types of error in the case of mean values are higher than this.

Similarly, Table 3-10 shows the factors used in the 2nd DOE with their corresponding actual and coded setting. In this table, the flax ply surface densities of 125 ± 1.8, 153 ± 1.3 and 175 ± 1.8 g/m2 respectively correspond to using 16, 20 and 24 yarns/inch flax layers. As can be noticed the middle-level surface density of flax ply (153 ± 1.9 g/m2) is not exactly in the middle of high and low level settings. This is because these values are measured experi- mentally with inherent variations in the values. Because of this, in the regression modeling of surface density according to Table 3-11, corresponding ‘number of yarns per inch’ (16, 20 and 24) values are used and once the optimum value of ‘number of yarns per inch’ is found, it has been converted to flax layer surface density measured in g/m2_.

Table 3-8. Reinforcement factors considered for the 1st DOE. Factor

name Description

Low and high level setting coded actual A Paper ply surface density

(g/m2₎

+1 38 ± 0.60 1 29 ± 1.0

B Flax ply surface density

(g/m2₎

+1 172 ± 6.9 1 116 ± 4.3

C Forming pressure (bar) +1 _1 3 ± 0.1 1 ± 0.1

D Drying temperature (°C) +1 _1 101.5±1.5 71.5±1.5

Table 3-11 shows the full-factorial 32 experimental design (2nd DOE). Other reinforcement factors are set at A = 1 (27±0.67 g/m2), C = +1 (3 ± 0.1 bar) and D = +1 (101.5±1.5) for the reinforcements fabricated according to this table of experiments. Each run of this table is repeated four times.

Table 3-9. 1st _{DOE with coded level setting of the factors and the evaluated responses.}

Run Factor Responses

A B C D Reinforcement Composite Surface density IBS Permeability E1 E2 σU K1 K2 1 1 1 1 1 2 1 1 +1 +1 3 1 +1 1 +1 4 1 +1 +1 1 5 +1 1 1 +1 6 +1 1 +1 1 7 +1 +1 1 1 8 +1 +1 +1 +1

Table 3-10. Reinforcement factors considered for the 2nd _DOE. Factor

name Description

level setting coded actual B Flax ply surface density

(g/m2₎

+1 175 ± 1.9 0 153 ± 1.3 1 125 ± 1.9

E Fiber volume fraction (%)

+1 45

0 40

Table 3-11. 2nd DOE with coded level setting of the factors and the evaluated response. Run Factors Responses B E Reinforcement Permeability K1 K2 1 −1 −1 2 0 −1 3 +1 −1 4 −1 0 5 0 0 6 +1 0 7 −1 +1 8 0 +1 9 +1 +1

Tables 3-12 and 3-13 show the surface densities of the flax/paper reinforcements for the 1st and the 2nd DOE, respectively. These values are used in Equation 3-5 to calculate the cavity height. The measured reinforcement surface densities are based on weighting at least 12 reinforcement layers of 140 mm × 140 mm (5.5 in. × 5.5 in.) which are cut using scissors and a precise metallic template of the same dimension (for 1st DOE samples) and using cutter and the metallic template for 2nd DOE samples. Prior to weighting, all samples are dried at 103°C for at least 18 hours and stored in a desiccator to prevent humidity absorption. All other reinforcements used for permeability tests and composites molding are conditioned the same way. This was done to promote consistency in the results as natural fibers are hydro- philic and easily affected by the humidity level. It is reported that drying of fibers before processing is important, because water on the fiber surface can weaken the interface strength and consequently the mechanical properties of composites [100].

Table 3-12. Final surface density (mr) of flax/paper reinforcements of 1st DOE. Run mr (g/m 2₎ mean STD 1 147 2.30 2 144 2.41 3 197 8.54 4 204 3.95 5 152 3.74 6 155 4.82 7 216 3.42 8 204 6.15

Table 3-13. Final surface density (mr) of flax/paper reinforcements of 2nd DOE. Run mr (g/m 2₎ mean STD 1 152 2.73 2 180 1.97 3 202 3.46 4 152 2.73 5 180 1.97 6 202 3.46 7 152 2.73 8 180 1.97 9 202 3.46