Analyze a smart structure formed using a SMA wire with a return spring.

Parts a$-$d below refer to SMA NiTiNOL wire with diameter $d_0$ and unstretched length $L_0.$

$1.$ Assume the stress$-$strain law for the SMA in the austenitic state is the linear equation: ${}_A=E_A{}_A.$

$2.$ Assume the stress$-$strain law for the SMA in the martensitic state is the linear equation: ${}_M=E_M{}_M.$

$3.$ Assume at room temperature the SMA is in the martensitic state.

$4.$Assume when heated the SMA is in the austenitic state.

Parts a$-$e below are independent of each other. Solve using symbols. Refer to class notes on Canvas for a diagram of the SMA$-$spring

system configuration.

$($a$)$ A SMA wire in the un$-$stretched condition is then given an initial strain of ${}_0($to preload the wire$)$ at room temperature $($RT$).$ Its

ends are then rigidly fixed. What force is developed in the wire?

$($b$)$ A SMA wire in the un$-$stretched condition at RT is given an initial strain of ${}_0,$ the ends are rigidly fixed, and the wire is then

heated to the austenite finish temperature, $A_f.$ What force is developed in the wire?

$($c$)$ A SMA wire initially in the un$-$stretched condition at room temperature is connected in series to a spring of stiffness $K\frac{N}{m}.$ The

spring and wire are pulled and stretched until the SMA wire is preloaded with an initial strain ${}_0.$ The free end of the spring and the

free end of the wire are now fixed. Find: $($i$)$ the stretch $($displacement$)$ of the spring, $($ii$)$ the stretch $($displacement$)$ of the wire, $($iii$)$ the

strain in the wire, $($iv$)$ the force in the wire, $($v$)$ the force in the spring. Find these parameters both before and after heating the SMA

wire.

$($d$)$ Are the assumptions $(1-4)$ above accurate or could there be error in any of the assumptions and why?

$($e$)$ Write a simple equation that tells how much energy can be generated every time the SMA and spring actuate.