Figure 1: Deformation of a circle into an ellipse.
1.) On the ellipse in figure one, precisely draw and label, the principal stretch directions, the lines of no finite longitudinal strain, and the angles y, and q'. (5 points)
Red line is S1; Green line is S3; Blue lines are the lines of no finite longitudinal strain (both lengths are 4.0 cm so S = 1 for these lines). q' is the acute angle between S1 and the shear direction. y the angular shear is the deflection between two lines that were originally perpendicular.
2.) Calculate the percent elongation (%e) for line A which becomes line A'. (5 points)
%e = [(Lf-Lo)/Lo]
x 100
Lo = diameter of undeformed circle
4.0 cm;
Lf = length of A' = 2.0 cm
[(2.0 cm - 4.0 cm) / 4.0 cm] x 100 = -50%
3.) Calculate the stretch for S1 and S3. (5 points)
radius of the undeformed circle = 2.0 cm (Lo);
1/2 the length of the major semi axis of the
ellipse = L1f = 4.35 cm;
1/2 the length of the minor semi axis of the
ellipse = L2f = 0.925 cm;
S1 = L1f / Lo = 4.35 cm / 2.0 cm = 2.18; S3 = L2f / Lo = 0.925 cm/ 2.0 cm = 0.46
4.) Calculate the shear strain (g) from the angle y. (5 points)
g = Tan y;
y = 60°;
Tan 60°=1.732;
g =1.732
5.) Calculate the shear strain (g) from the principal stretch directions. (5 points)
g = S1
- S3;
S1 = 2.18;
S3 = 0.46
(2.18 - 0.46) = 1.72
g = 1.72