What is the meaning behind stress distribution in a material, physically?

Why do we even have pressure/stress distributions, why not simply force distributions? Yes, pressure and stress (depending on if we are talking about a fluid, or a solid material's internal forces) is the measure of how force is spread out. But take this image for example. What does it actually mean to have a stress distribution? Why don't we simply have a force distribution instead, since if we want to calculate the force at some arbitrary point, we cannot actually do that, since a point doesn't have an area. Having a stress/pressure defined at a single point is unintuitive (yes, I do understand it's a limit of ratio of force/area while area is approaching zero, and it is a definition). But still, wouldn't it be simpler to have a force distribution? How does it make sense to have a varying pressure anyways? Why not just represent them using different force magnitudes?



The point is that the total force at any point(atom) in a body is is zero --- unless that part of the body is accelerating. What we care about is what force one part of a body exerts on an adjacent part, and that force (per unit area) is what we call stress.

You might talk about a force distribution in a freely falling object. Gravity acts on every atom independently. They all have the same acceleration.

Things are different for an object under compression or tension. The total force on each atom is $0$ if the object is at rest. Two opposing external surface forces squeeze atoms together or pull them apart. Two neighboring atoms are pressed together or stretched apart. The atomic bond between them keeps them still. If the bond isn't strong enough, the object will break. This is different from an object in free fall.

The diagram shows a shear stress. This is much the same. An external force is pushing two surfaces in opposite directions. Atoms in adjacent layer exert sideways forces on each other. The bonds prevent movement.

We have both things. It is interesting an example where forces are net forces, resulting in acceleration.

In a rotating body, a small volume around each point have an instantaneous momentum $$\Delta \vec p = \Delta m\vec v = \rho \Delta V (\vec \omega \times \vec R)$$ where $\vec R$ is the distance vector from the rotation axis.

Making the derivative with respect to time, and supposing a rotation around a principal inertia axis ($\omega$ constant) $$\Delta \vec F_c = \rho \Delta V \left(\vec \omega \times \frac {d\vec R}{dt}\right) = \rho \Delta V (\vec \omega \times \vec v)$$ The magnitude is:$$\Delta F_c = \rho \Delta V \omega^2 R \implies f_c = \rho \omega^2 R$$ where $f_c$ is the centripetal force per unit of volume. So, we have a force distribution along the body indeed.

However, thinking of the momenta as a field that follows local conservation, it is not possible a change of momentum with time without some kind of flow to the surroundings. What flows? $$\frac{d\vec p}{dt} = -\nabla \cdot (???)$$ The pressure is exactly the unknown, and its units allow us to think of it as a kind of a density of current: momentum per unit of area and time. In this example there is no shear stresses and the diagonal elements of the stress tensor are $\sigma_{jj} = -p_{jj}$ $$\frac{d\vec p}{dt} = -\nabla \cdot \boldsymbol p = \nabla \cdot \boldsymbol \sigma$$

The relation also holds in a static situation, as a body on the ground for example. In this case we can use the equivalence principle, and pretend that the body is being accelerated upward. The change of momentum with time is also the divergence of the stress tensor at each point of the body.

Stress is force per area (N/m$^2$) or equivalently, stored mechanical energy per volume (J/m$^3$).

The values of forces depend on the size and geometry of the object in question, while the values of stresses do not. The strength of a material is given in terms of stress, e.g. 100,000 psi or 690 MPa, and is the maximum that the atomic or grain structure can take before failing. It may take 1 million pounds of force to produce this stress in a large bridge beam, or only 5 pounds to produce it in a thin wire. But in either case the stress is what determines the behavior.

The stress state of an object is essentially the totality of its internal forces in an infinitesimal volume, and in the context of solid materials is more fundamental than forces. Rather than being a scalar or vector quantity, the full stress state is a rank-2 tensor quantity.