4.2 Traction vector
- In a generalized view, stress is a resultant force
(\(\vec{F}\)) over a given area (\(\vec{A}\)).
- However it needs to be recognized that different planes can be considered within a body.
- Given that the loads are passed through the body without concern for the plane of observation, these planes consequently “feel” different intensities of stress.
- Further, the area of any cutting plane must be viewed as being
associated with the vector of the plane, and hence have vector
properties \[\vec{A} = A \, {\vec{n}}\]
- Lastly, since the magnitudes of stress vary as a function of position within a body, To determine the component of stress on different planes (at any point) requires an infinitesimal area (\(d A\)) to be examined.
- The differential forces passed through the differential areas of the planes are ultimately what permit a complete description of stress.
- Consider a differential area \(\Delta A\) of the internal surface, and a corresponding differential force \(\Delta \vec{F}\).
- Normal stress magnitude \[ \sigma = \lim_{\Delta A \rightarrow 0} \frac{\Delta F_n}{\Delta A} \approxeq \frac{d F_n}{d \vec{A}} = T_n \]
- Shear stress magnitude \[ \tau = \lim_{\Delta A \rightarrow 0} \frac{\Delta F_s}{\Delta A} \approxeq \frac{d F_s}{d \vec{A}} = T_s \]
- The combination of the above stress components constitute a vector, called a traction vector \[\begin{align} \vec{T} =&\, \vec{T}_n + \vec{T}_s \\ \end{align}\]
- The tractions, once integrated over an area, becomes the forces on (or passing through) that plane \[\vec{F} = A \, \vec{T}\]
- The traction on each surface depends on the orientation of the surface.
- A complete description of the stress state requires that the traction be known for all \(\vec{n}\)