4.2 Traction vector

Body under stress, inner plane exposed by equilibrium cut

  • 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.

Body under stress, inner plane exposed by equilibrium cut

  • 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.

Body under stress, inner plane exposed by equilibrium cut

  • 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 \[\vec{F} = A \, \vec{T}\]

Tractions on different planes

  • 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 \(\vec{n}\)