Relationship between angle of twist and temperature gradient

[Sud89]

Homework Statement

To derive a formula to find the angle of twist when a torque T is applied and the temperature gradient is ΔT

Homework Equations

The Attempt at a Solution

Angle of twist Φ=TL/JG
I believe as temperature decreases, torque will increase which means angle of twist is inversely proportional to temperature.
Thermal strain is α.ΔT.. Should I relate torque and strain so I will be able to get T and ΔT in my equation?

 

[KataKoniK]

Yes, you can relate torque and strain to incorporate the effect of temperature in your equation. The formula for thermal strain is αΔT, where α is the coefficient of thermal expansion and ΔT is the temperature gradient. This strain will contribute to the overall deformation of the material, which can be represented by the angle of twist Φ.

To incorporate this in your equation, you can modify the formula to include the thermal strain as:

Φ = (TL/JG) + (αΔT)

Where Φ is the angle of twist, T is the applied torque, L is the length of the material, J is the polar moment of inertia, G is the shear modulus, and α is the coefficient of thermal expansion.

This modified formula takes into account the effect of temperature on the material, and as you mentioned, the angle of twist will be inversely proportional to the temperature gradient ΔT. As the temperature decreases, the thermal strain will decrease, resulting in a smaller angle of twist for the same applied torque.