Using the results from Problem 10.10, derive the equations for both parts of the off-axis complex modulus, \(E_{x}^{*}=E_{x}^{\prime}\left(1+i \eta_{x}\right)\), for an arbitrary angle \(\theta\); then find numerical values of both parts for an angle of \(\theta=30^{\circ}\).

Problem 10.10

Extensional vibration experiments are conducted on longitudinal, transverse, and \(45^{\circ}\) off-axis unidirectional composite specimens, and the complex moduli results for a particular vibration frequency are, respectively:

Using the above data derive the equations for both parts of the complex shear modulus, \(G_{12}^{*}=G_{12}^{\prime}\left(1+i \eta_{12}\right)\), then find numerical values for both parts. Assume that all loss factors are very small ( \(\ll 1)\), and that the major Poisson's ratio \(v_{12}=0.3\) is a real constant.

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E=E(1+in)=35.6(1+0.0041) GPa, for 0=0 E2 E2(1+in2) = 10.8(1+0.009i) GPa, for 0 = 90 = Ex Ex(1+inx) = 11.6(1+0.011) GPa, for 0 = 45 =