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.
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
Further details follow.
Twisting involves torsional stress and torsional strain.
Torsional stress, in circular solid or thick-walled ($t>0.1r$) shafts, is given by:
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 $ϕ$:
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.