User:Eas4200c.f08.wiki.f/10-31-08

Neglect h.o.t.


 * → $$\frac{d\sigma}{dx} + \frac{f(x)}{A} = 0$$


 * Notes:
 * A is the applied load
 * Body force = force/volume
 * f(x) = force/length

Now, non-uniform stress field is in 3-D, but without an applied load, focusing on the x-direction only (i.e. without the other stress component to avoid cluttering the figure)

Below is an image of shear on a 3-D cube :



Note: the first part of the superscript refers to the normal direction of the cube face (facet), and the second part refers to the direction of shear.

$$\sum{F_x} = {\color{red}dydz[-\sigma_{xx}(x,y,z) + \sigma_{xx}(x+dx,y,z)]} + {\color{blue}dzdx[-\sigma_{yx}(x,y,z) + \sigma_{yx}(x,y+dy,z)]} + {\color{green}dxdy[-\sigma_{zx}(x,y,z) + \sigma{zx}(x,y,z+dz)]}$$
 * The red term has facets normal to the x-axis, the blue has facets normal to the y-axis, and the green has facets normal to the z-axis.

$$0 = $$( dxdydz )$$[\frac{d\sigma_{xx}}{dx} + \frac{d\sigma_{yx}}{dy} + \frac{d\sigma_{zx}}{dz}]$$

This ends Step C of the Road Map.