Can stress be generalised to points?

Question

I know we can talk about a 'stress' in an area for which a 'uniform' force acts, can we generalise this to the stress at each point in the area, by making smaller areas around a point such that:

Over the area $S=\frac{F}{A}$ we can determine that $s=\frac{F}{A_s}$ and then take the limit as $A_s$ approaches $0$ yielding $s=\frac{F}{dA}$ over all of $A$?

Is stress evenly distrubuted across all points such that $S=\frac{F}{A}=\frac{F}{dA}$ so we can talk about stress over infinitesimal areas as well? Even areas under uniform force?

Answer 1

IMHO, stress is primarily defined for a point. Also keep in mind that stress is a vector quantity (not scalar). So the stress, on the same point can be different in different direction.

However, Even in the same cross-section adjacent points can have different stress (even in the same direction). e.g. have a look at the stress distribution of the following tensile specimen.

You can see that even in the same cross section on the curved region the stresses at adjacent points are significantly different. (although in the picture here a Von Mises stress is used, similar patters can be observed for the $\sigma_{xx}, \sigma_{yy}, \sigma_{zz}, \tau_{xy}, \tau_{yz},\tau_{zx}$ )

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Its just that it is easier and more intuitive for a first year student to explain it in larger crosssection/surfaces, rather than using a 3d infinitesimal cube.

Answer 2

$S = \dfrac{F}{A} = \dfrac{Mc}{I}$. The stress in the latter case is non-uniform but in the form of linear varying.

Is the stress derived from $S = \dfrac{F}{A}$ really uniform? No, depending on the flexibility of the element in contact, and the support rigidity, the true stresses may vary. But for the force applied over a small, limited area, the variation is insignificantly small, thus the average is conveniently taken as a uniform stress over the entire surface area.

Yes, $S = \dfrac{F}{A}$, when $A$ approaches zero, the force goes to infinite, which is considered a point load; whether it is uniform or non-uniform is inconsequential, as the stress variation will be negligibly small.