Using the result from Problem 2.10

a. Find the value of the angle \(\theta\) (other than \(0^{\circ}\) or \(90^{\circ}\) ), where the curve of \(G_{x y}\) versus \(\theta\) has a possible maximum, minimum, or inflection point.

b. For the value of \(\theta\) found in part (a), find the bounds on \(G_{12}\), which must be satisfied if \(G_{x y}\) is to have a maximum or minimum.

c. Qualitatively sketch the variation of \(G_{x y}\) versus \(\theta\) for the different cases and identify each curve by the corresponding bounds on \(G_{12}\), which give that curve.

d. Using the bounds on \(G_{12}\) from part (b), find which conditions apply for E-glass/epoxy composites. The bounds on \(G_{12}\) in part (b) should be expressed in terms of \(E_{1}, E_{2}\), and \(v_{12}\).

Problem 2.10

Derive the third of Equation 2.40 for the off-axis shear modulus, \(G_{x y}\).

2V12 E-[1 + (1-2)+1] Ex = Ey E E 1 2V12 + 1 1 G12 E2 1 E2 2V12 - E 2G12 1 1 sc E E1 E2 E2 G12 Gxy = 1 G12 Vxy = Ex ($4 V12 +c c). +4 1 1 1 + -c)- + 1 2 + 1 (2.40)