Elasticity/Warping functions

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

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

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

The stresses are given by

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

where ${\displaystyle \mu \,}$ is the shear modulus.

The projected shear traction is

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

Equilibrium is satisfied if

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

Traction-free lateral BCs are satisfied if

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

or,

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

The twist per unit length is given by

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

where the torsion constant

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