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wikifemfuchde2019:lez_2019-03-26

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: $T=2\iint_A \phi dA$

wikifemfuchde2019/lez_2019-03-26.txt · Ultima modifica: 2019/03/27 11:38 da ebertocchi