## Warping Function and Torsion of Non-Circular Cylinders
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Warping functions are quite useful in the solution of problems involving the torsion of cylinders with non-circular cross sections.

For such problems, the displacements are given by

- $u_{1}=-\alpha x_{2}x_{3}~;~~u_{2}=\alpha x_{1}x_{3}~;~~u_{3}=\alpha \psi (x_{1},x_{2})$

where $\alpha \,$ is the twist per unit length, and $\psi \,$ is the warping function.

The stresses are given by

- $\sigma _{13}=\mu \alpha (\psi _{,1}-x_{2})~;~~\sigma _{23}=\mu \alpha (\psi _{,2}+x_{1})$

where $\mu \,$ is the shear modulus.

The projected shear traction is

- $\tau ={\sqrt {(\sigma _{13}^{2}+\sigma _{23}^{2})}}$

Equilibrium is satisfied if

- $\nabla ^{2}{\psi }=0~~~~\forall (x_{1},x_{2})\in {\text{S}}$

Traction-free lateral BCs are satisfied if

- $(\psi _{,1}-x_{2}){\frac {dx_{2}}{ds}}-(\psi _{,2}+x_{1}){\frac {dx_{1}}{ds}}=0~~~~\forall (x_{1},x_{2})\in \partial {\text{S}}$

or,

- $(\psi _{,1}-x_{2}){\hat {n}}_{1}+(\psi _{,2}+x_{1}){\hat {n}}_{2}=0~~~~\forall (x_{1},x_{2})\in \partial {\text{S}}$

The twist per unit length is given by

- $\alpha ={\frac {T}{\mu {\tilde {J}}}}$

where the torsion constant

- ${\tilde {J}}=\int _{S}(x_{1}^{2}+x_{2}^{2}+x_{1}\psi _{,2}-x_{2}\psi _{,1})dA$