Prob 2 22 Line rotation from uniaxial extension Consider a line inclined at an angle p from the vertical with a slope y0 x0 After stretching by an amount y we have 0 y x z tan x x x0 y y y0 yx0 1 FG 1 IJ x H K y0 x0 0 1 tan 0 3 2 Prob 2 23 Orientation function Fraction of sigments in range d f phi 2 Pi r 2 sin phi 2 Pi r 2 f sin Segment orientation after stretching from previous problem phi 2 arctan 1 lambda 3 2 tan phi tan phi 2 arctan 3 2 Integrate over sphere to obtain mean square orientation angle phi avg simplify int cos phi 2 2 f phi phi 0 Pi 2 3 1 3 2 1 arctan 1 arctan 2 3 1 3 1 phi avg 3 2 3 1 Evaluate for 3 Digits 4 evalf subs lambda 3 phi avg Digits 4 7581 Denote the Herrman orientation parameter as ff ff 1 2 3 phi avg 1 3 3 1 3 2 1 arctan 1 arctan 2 3 1 3 1 3 ff 3 2 2 3 1 Evaluate for 3 evalf subs lambda 3 ff 6370 Plot orientation function versus extension ratio plot ff lambda 1 5 0 8 0 6 0 4 f 0 2 0 0 2 0 4 1 2 3 4 5 3 1 2
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