Gamma distribution

For each 𝑥, 𝐹(𝑥) is the area under the density curve to the left of 𝑥.

2.1.5.4. Normal distribution. A continuous random variable 𝑋 is said to have a normal distribution with parameters 𝜇 and 𝜎, where −∞ < 𝜇 < ∞ and 𝜎 > 0, if the pdf of 𝑋 is,

𝑓(𝑥; 𝜇, 𝜎) = 1 √2𝜋𝜎𝑒

−(𝑥−𝜇)2

2𝜎2 _{, −∞ < 𝑥 < ∞} (2.18)

It can be shown that the expected value 𝐸(𝑋) = 𝑚𝑒𝑎𝑛 = 𝜇 and the variance 𝑉(𝑋) = 𝜎2_{, so the} parameters are the mean and the standard deviation of 𝑋.

The normal distribution with parameter values 𝜇 = 0 and 𝜎 = 1 is called a standard normal distribution. A random variable which has a standard normal distribution is called a standard normal random variable and is denoted by 𝑧.The pdf of 𝑧 is

𝑓(𝑧; 0,1) = 1 √2𝜋𝑒

−(𝑧)2

2 _{, −∞ < 𝑧 < ∞} (2.19)

2.1.5.5. Gamma distribution. The graph of any normal distribution is bell shaped and

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about the mean. To describe this type of skewed phenomena, a set of non-symmetric pdf is used. They are called gamma family. For 𝛼 > 0, the gamma function Γ(𝛼) is defined as,

Γ(𝛼) = ∫ 𝑥𝛼−1𝑒−𝑥𝑑𝑥 ∞

0

(2.20)

A continuous random variable 𝑋 is said to have a gamma distribution if the pdf of 𝑋 is

𝑓(𝑥; 𝛼, 𝛽) = { 1 𝛽𝛼_Γ(𝛼)𝑥𝛼−1𝑒 −_𝛽𝑥 𝑥 ≥ 0 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑤 (2.21)

where the parameters 𝛼 and 𝛽 satisfy 𝛼 > 0 and 𝛽 > 0.

The standard gamma distribution has 𝛽 = 1, so the pdf of a standard gamma random variable is given by, 𝑓(𝑥; 𝛼) = { 1 Γ(𝛼)𝑥 𝛼−1_𝑒−𝑥 _{𝑥 ≥ 0} 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑤 (2.22)

The mean and the variance of a random variable 𝑋 having the gamma distribution 𝑓(𝑥; 𝛼, 𝛽) are

𝐸(𝑥) = 𝛼𝛽 𝑎𝑛𝑑 𝑉(𝑥) = 𝛼𝛽2 (2.23)

When 𝑋 is a standard gamma random variable, the cumulative distribution function (cdf) of 𝑋 is,

𝐹(𝑥; 𝛼) = ∫ 1 Γ(𝛼)𝑦 𝛼−1_𝑒−𝑦_{𝑑𝑦 , 𝑥 > 0} 𝑥 0 (2.24)

which is called the incomplete gamma function.

2.1.5.6. Exponential distribution. 𝑋 is said to have an exponential distribution if the pdf of 𝑋 is,

𝑓(𝑥; 𝜆) = {𝜆𝑒−𝜆𝑥 𝑥 ≥ 0, 𝜆 > 0

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(2.25)

The exponential pdf is a special case of the gamma pdf where 𝛼 = 1 and 𝛽 =1 𝜆 The mean and the variance are as follows:

43 𝐸(𝑋) = 𝛼𝛽 =1 𝜆 𝑉(𝑋) = 1 𝜆2 (2.26)

2.1.5.7. Weibull distribution. A random variable 𝑋 is said to have a Weibull distribution with parameters 𝛼 and 𝛽 if the of 𝑋 is

𝑓(𝑥; 𝛼, 𝛽) = { 𝛼 𝛽𝛼𝑥 𝛼−1_𝑒−(𝛽𝑥)𝛼 _{𝑥 ≥ 0} 0 𝑥 < 0 (2.27)

The cdf of a Weibull random variable having parameters 𝛼 and 𝛽 is,

𝐹(𝑥; 𝛼, 𝛽) = {1 − 𝑒−( 𝑥

𝛽)𝛼 _{𝑥 ≥ 0} 0 𝑥 < 0

(2.28)

2.1.5.8. Lognormal distribution. A non-negative random variable 𝑋 is said to have a lognormal distribution if the random variable 𝑌 = ln (𝑋) has a normal distribution. The resulting pdf of a lognormal random variable when ln (𝑋) is normally distributed with parameters 𝜇 and 𝜎 is, 𝑓(𝑥; 𝜇, 𝜎) = { 1 √2𝜋𝜎𝑒 −(ln (𝑥)−𝜇)2 2𝜎2 _{𝑥 ≥ 0} 0 𝑥 < 0 (2.29)

The mean and the variance of the distribution is

𝐸(𝑋) = 𝑒𝜇+𝜎

2

2 _{𝑎𝑛𝑑 𝑉(𝑋) = 𝑒}2𝜇+𝜎2_(𝑒𝜎2_{− 1)} (2.30)

2.2. Experimental 2.2.1. Materials

Flax fibers were provided by Composite Innovation Center, Winnipeg, Canada. Flax plants of the variety Bethune and Prairie Grande were grown in two different locations in

Manitoba, Canada in 2015. Flax stems were harvested and collected before they were able to ret. Next measurements were made on these flax stems in order to separate them into two distinct categories. The criteria was that if the diameter of flax stems were less than 1.5 mm, they were

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categorized as small stems (small diameter). On the other hand, if the diameter of flax stems were larger than 1.65 mm, they were categorized as large stems (large diameter). Combining these different criteria such as stem diameter, location of growth, and variety, six batch of samples were prepared: Arborg Large (Arb L), Arborg Small (Arb S), Melita Large (Mel L), Melita Small (Mel S), Prairie Grande Large (PG L), and Prairie Grande Small (PG S). Table 2.1 shows this categorization more specifically.

Table 2.1: Formation of different samples

Sample Name Location Variety Stem Size % Medium Stems

Arborg Large Arborg Bethune Large 16%

Arborg Small Arborg Bethune Small 16%

Melita Large Melita Bethune Large 8%

Melita Small Melita Bethune Small 8%

PG Large Melita Prairie Grande Large 14%

PG Small Melita Prairie Grande Small 14%

Prior to enzyme retting, straw stems were gently crimped using a wooden tool so that the enzyme solution can penetrate easily into the stems. This facilitates proper enzyme retting. Straw stems were enzyme retted using a pectinase enzyme (Bioprep 3000L), EDTA and buffer solution. The straws were soaked in the enzyme solution for 5 minutes and then removed from the soaking tub. The unrinsed straws were placed in sealed bags and incubated at 55° C for two hours. Next the straws were soaked in an EDTA and buffer solution for 30 minutes and

incubated again at 55° C for 22 hours. The straw was then rinsed thoroughly with tap water to remove the remaining enzyme/EDTA solution. The straws were spread out flat to dry in a fume hood at room temperature for 12 hours. Next the fibers were decorticated from the stem using the same wooden crimping tool used in the pre-crimp step.

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2.2.2. Tensile testing

Tensile tests were performed on technical flax fibers having a diameter range of 35-150 μm. A gauge length of 20 mm were selected for proper handling of the fiber specimen. A Deben micro-tensile tester, which is more generally used for in-situ observation of a specimen under tension in electron microscope, was selected for the testing. The maximum load that can be measured with the machine is 180 N with a load cell linearity of 1.5%. Force and position resolution of the machine were 0.001 N and 0.0001 mm respectively. Displacement rate of the crosshead were selected to be 0.75 mm/min. ASTM standard D3822 was followed for

determining the mechanical properties of the fibers.

On the other hand, the instrument used in Composite Innovation Center (CIC) was a load cell and a screw based linear actuator to measure and produce stress, respectively. The

combination is sold as a LEX810 tensile tester from Diastron. The equipment has a load cell capacity of 10N with a load cell linearity of 0.08%. The speed range for the CIC tensile tester is 0.01-60 mm/min. The positional accuracy is 10 micron and the maximum elongation is 50 mm. The gauge length used for flax fiber in CIC was 4 mm. For the diameter measurement, a

rotational laser scanner was used.

At least 50 fibers were tested from each of the six samples mentioned above. Before performing the tensile test, diameter of each fiber was measured using a Zeiss Axiovert 40 MAT inverted optical microscope with an attached camera of Progress C10 Plus. The diameter of each fiber specimen was measured in three different locations and then averaged for each sample. Figure 2.2 shows a typical diameter measurement of the technical fiber. A circular cross-section was assumed and the average diameter was used for calculating the area. The effect of lumen was neglected. Thomason et.al [11] maintained that the assumption of a circular cross-section

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overestimates the true cross-section of flax fibers and proposed a common scale factor in order to correct the circular area to a more elliptical shape. In this study, a factor of 1.5 was used in order to correct for the circular area assumptions. All the testing of flax fibers were performed in a temperature and humidity controlled laboratory (21°𝐶 ± 1°𝐶, 18%RH±3%).

Figure 2.2: Diameter measurement with optical microscope