User:Eas4200c.f08.wiki.c/homework report 4

The expression:

$$\theta = \frac{\partial \alpha }{\partial x}$$

can be used to express the rate of twist of a cross section.

Hooke's law can be represented as:

$$\tau = G\gamma = G\rho \theta$$

 $$\tau (s)= G\gamma = G\rho(s) \theta (x)$$

With variables as defined in the previous diagram.

Integrate along $$\textit{C}$$

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

where

$$ \tau (s)ds = \frac{q(s)}{t(s)}$$

$$\rho (s) = 2\bar{A}$$

To get an expression for $$\theta$$ as discussed in the previous lecture.

The question, 'What is ad-hoc about the above derivation of $$\theta$$ expression on p 20.2 and the derivation of $$T = 2q\bar{A}$$?' was posed in the discussion of the equation. The answer can be provided in two parts:

1) Strain ($$\gamma$$) must be obtained using the displacement of $$P$$ in the direction tangent to $$\textit{C}$$ at $$P$$, but $$PP''$$ in the previous diagram is not necessarily tangent to $$\textit{C}$$ (but is actually close, due to the small angles involved).

2) $$\tau = \frac{q}{t}$$ obtained from ad-hoc assumption that $$\tau$$ was uniform across wall thickness, this is assuming that the wall thickness is very thin, or else this assumption breaks down.

Now formal justification (derivation) of elasticity theory.

This section refers back to the road map that was previously discussed.

A. Kinematic Assumption

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

Where $$\theta$$ is considered constant with respect to x (due to it being a uniform bar).

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

To transform equations in Sun[2006] to those using our unified notation, we use cyclic permutation

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

$$\varepsilon _{xx} = \frac{\partial u_{x}}{\partial x}(y,z) = 0$$

$$\gamma _{yz} = \frac{\partial u_{y}}{\partial z}(x,z) + \frac{\partial u_{z}}{\partial y}(x,y) = -\theta x + \theta x = 0$$ HW:

$$\varepsilon _{yy} = \frac{\partial u_{y}}{\partial y}(x,z) = 0$$

$$\varepsilon _{zz} = \frac{\partial u_{z}}{\partial z}(x,y) = 0$$

These results are obtained due to the fact that both derivatives are taken with respect to a variable that does not show up in either function being differentiated.