User:Eas4200c.f08.blue.a/Lecture 20b

Hooke's Law shows that $$\tau = G\gamma =G\rho \theta $$

In this case: $$\tau (s)=G\rho (s)\theta (s)$$

and integration along the contour yields:

$$\oint_{}^{}{\tau (s)}=G\theta (x)\oint_{}^{}{\rho (s)}$$

Where $$\tau (s)=\frac{q(s)}{t(s)}$$ and$$\oint_{}^{}{\rho (s)}=2\bar{A}$$

Discussion

Why is this method ad hoc? *to get PP' we assume $$\alpha$$ is small *to get PP" we assume finite $$\alpha$$
 * 1) PP' is projected tangent to the surface but, it is not necessarally on the surface (it is only close)
 * 2) $$\tau$$ is assumed to be uniform across the wall thickness
 * 3) Inconsistancy in assuming the size of $$\alpha$$

Formal Justification of Elasticity Theory
$$u_{x}(y,z)=\theta \psi (y,z)$$

$$u_{y}(x,z)=-\theta \psi (x,z)$$

$$u_{z}(x,y)=\theta \psi (x,y)$$

We assume, given an applied torque abouyt the x-axis that:

$$\varepsilon _{xx}=\varepsilon _{yy}=\varepsilon_{zz} =\gamma _{yz}=0$$

Where each strain is computed as:

$$\varepsilon _{xx}=\frac{du_{x}}{dx}(y,z)$$

$$\varepsilon _{yy}=\frac{du_{y}}{dy}(x,z)$$

$$\varepsilon _{zz}=\frac{du_{z}}{dz}(x,y)$$

$$\gamma _{yz}=\frac{du_{y}}{dz}+\frac{du_{z}}{dy}$$