Elasticity/Warping functions

Warping Function and Torsion of Non-Circular Cylinders

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

${\mathrm{\nabla }}^2\psi =0\mathrm{\forall }(x_1,x_2)\in \text{S}$

Traction-free lateral BCs are satisfied if

$({\psi }_{,1}-x_2)\frac{d{x}_2}{ds}-({\psi }_{,2}+x_1)\frac{d{x}_1}{ds}=0\mathrm{\forall }(x_1,x_2)\in \partial \text{S}$

or,

$({\psi }_{,1}-x_2){\hat{n}}_1+({\psi }_{,2}+x_1){\hat{n}}_2=0\mathrm{\forall }(x_1,x_2)\in \partial \text{S}$

The twist per unit length is given by

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

where the torsion constant

$\tilde{J}=\underset{S}{\int }(x_1^2+x_2^2+x_1{\psi }_{,2}-x_2{\psi }_{,1})dA$