How can we theoretically calculate the dependence of torsion constant of a wire on it's length and radius?

Question

Is there a way to calculate how the torsion constant depends on the length and radius of a uniform wire without doing the experiments? Also do mention if we require value of any kind of property of the material like density, elastisity etc.

Answer 1

For a wire or rod (shaft) of uniform cross sectional area the torsion constant is the same as the polar moment of inertia and relates to torsional stress. For a solid circular cross section it only depends on the radius.

The polar moment of inertia of a solid circular section is given by

$$J=\frac{πr^4}{2}$$

Further details follow.

ADDENDUM

Twisting involves torsional stress and torsional strain.

Torsional stress, in circular solid or thick-walled ($t>0.1r$) shafts, is given by:

$$τ=\frac{Tr}{J}$$

Where $τ$ is the torsional stress, $T$ is the applied torque, $r$ is the radius of the shaft and $J$ is the polar moment of inertia, or torsional constant.

The shaft’s response to the torsional stress is its torsional strain, which can be expressed as the total angle of twist $ϕ$:

$$ϕ=\frac{TL}{GJ}$$

Where $T$ and $J$ are as before, $L$ is the length of the shaft from the fixed end to where the angle of twist is measured, and $G$ is the shear modulus, a material property. It is related to the modulus of elasticity, $E$ (Young’s modulus) and Poisson’s ratio, $ν$. Material properties are determined by applicable material tests.

Hope this helps.