profile_in_torsion_a2019_v003.mfd
chapter_9_torsion_of_thin-walled_tubes.pdf
Curiosità: Griffith, A.A, Taylor, G.I., Use of Soap Films in Solving Torsion Problems, 1917
Prandtl stress function $\phi$
shear stresses: $\tau_{zx}=\frac{\partial \phi}{\partial y}$ , $\tau_{yz}=-\frac{\partial \phi}{\partial x}$
shear stresses are hence defined as the 90° clockwise rotated $\phi$ gradient.
equilibrium equation $\frac{\partial \tau_{zx}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y}=0$ satisfied due to Schwarz.
Compatibility relationship, for an homogeneous, isotropic hookean material: $\nabla^2\phi=-2G\beta$ where $\beta$ is the rate of twist.
Here, $\nabla^2\phi=\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2}$.
Origin of the compatibility equations: the twist rate defines the derivatives $\frac{\partial u}{\partial z}=-\beta y$ and $\frac{\partial v}{\partial z}=+\beta x$; the two terms $\frac{\partial w}{\partial x}$ and $\frac{\partial w}{\partial y}$ complete the $\gamma_{zx}$ and the $\gamma_{yz}$ shear strain component definition by introducing the $w$ warping displacement. Those shear strains are related to the shear stresses by the constitutive equations $G\gamma_{zx}=\tau_{zx}$ and $G\gamma_{yz}=\tau_{yz}$.
Let's consider a closed path along the section consisting of four segments defined as
| from: | to: | $\frac{d x}{d s}=\frac{d s}{d x}$ | $\frac{d y}{d s}=\frac{d s}{d y}$ |
|---|---|---|---|
| $x ,y $ | $x+dx,y $ | +1 | 0 |
| $x+dx,y $ | $x+dx,y+dy$ | 0 | +1 |
| $x+dx,y+dy$ | $x ,y+dy$ | -1 | 0 |
| $x ,y+dy$ | $x ,y $ | 0 | -1 |
Then, the circuital integral along such a loop $$ \oint \frac{\partial w}{\partial s} ds = 0 $$ must be zero to enforce a single valued $w$ displacement at the $x,y$ point, thus avoiding a Penrose stairs-like impossible behaviour.
Then we have $$ 0= \oint \left( \frac{\partial w}{\partial x}\frac{d x}{d s}+ \frac{\partial w}{\partial y}\frac{d y}{d s} \right) ds, $$ whose warping slope components may be expressed based on the twist rate and the shear stresses, thus obtaining $$ 0= \oint \left( \left(\frac{\tau_{zx}}{G}+\beta y\right)\frac{d x}{d s}+ \left(\frac{\tau_{yz}}{G}-\beta x\right)\frac{d y}{d s} \right) ds. $$ By accumulating the contributions on the four segments that constitute the loop we obtain $$ \left(-\frac{1}{G}\frac{\partial\tau_{zx}}{\partial y}dy-\beta dy\right)dx + \left(+\frac{1}{G}\frac{\partial\tau_{yz}}{\partial x}dx-\beta dx\right)dy =0 $$ and $$ +\frac{\partial\tau_{zx}}{\partial y} -\frac{\partial\tau_{yz}}{\partial x} =-2 G \beta. $$ By substituting the shear stress component definition based on $\phi$, the compatibility equation is finally obtained.
boundary conditions: $\phi=c_i$, where $c_i$ is a constant on the $i$-th boundary; such condition, along with the way the $\tau_{zx},\tau_{yz}$ components based are derived from the $\phi$ Prandtl stress function, enforces the normal shear component with respect to the boundary to be zero.
Torque: $T=2\iint_A \phi dA$