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

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.

• Are you trying to define the torsional deflection formula from the Hook's law of linear stress-strain? – John Alexiou Nov 27 '18 at 20:18
• See relevent engineering notes on the subject. – John Alexiou Nov 27 '18 at 20:23
• Did I answer your question? – Bob D Dec 4 '18 at 12:29

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.

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.

• But shouldn't the force that is generated by twisting be different for different materials? I mean same twist will produce more torque in case of steel than, say, silk. – NiRVANA Nov 30 '18 at 4:32
• For a solid circular section, the torsional stress depends on the applied torque, radius, and polar moment of inertia, or torsional constant. The torsional strain (the materials response to the torsional stress) does depend on the specific materials shear modulus, a property of the material determined by applicable material tests. I have revised my answer to provide further details. Hopefully this will acceptably answer your question. – Bob D Nov 30 '18 at 13:30