A composite bone implant for a long bone, such as the femur, is designed with two materials to balance strength, flexibility, and biocompatibility. The two sections simulate bone's natural properties in different regions. A femoral implant is subject to a torque of $T=12Nm$ when the patient is engaged in light physical activity. The implant has two sections: Section $1$ is titanium with $L_1=100mm,$ outer diameter $D=12mm,$ and inner diameter $d=6$ mm $.$ The shear modulus of titanium is $41$ GPa $.$ Section $2$ is a biocompatible polymer $L_2=150mm,$ and the same outer and inner diameters. The shear modulus of the polymer is $2$ GPa $.$

a$)$ Determine the angle of twist for the entire implant.

b$)$ Perform a failure analysis to see if the applied torque will cause shear failure in either section. The implant should not exceed a maximum allowable shear stress $T_{\text{m}\text{a}\text{x}}=120$MPa for titanium and $T_{\text{m}\text{a}\text{x}}=20$MPa for the polymer.