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When you apply a shear force onto a solid piece of material (say a block on a surface or a cantilever beam with a load) that creates shear stress in the elastic regime, there is a restoring force that opposes the applied force and tries to restore the object's shape.

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Since the shear strain deforms the block "sideways" in the above image, the associated restoring force by the lower half of the block onto the upper half of the block is to the left and the restoring force by the upper half of the block onto the lower half of the block is to the left. So if the block is sheared "horizontally" I'd expect the forces to be horizontal as well.

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These restoring forces, as I depicted them, certainly seem non-central. My question is, do these restoring forces arise from central forces between the molecules of the object? Or are the forces between the molecules themselves non-central? Which of the two sketches below is more accurate? A, B, or is there something else happening such that neither sketch is helpful?

enter image description here

If B is accurate (and the molecular forces really are central), then I have a contention of incredulity to bring up: Wouldn't this mean the vertical components of the force have to be ludicrously large just for the horizontal components (the restoring shear forces) to be the values that they are? For example, if the restoring shear force is $3\,\textrm{N}$ horizontally, and the block is deformed by a small angle $\theta = 10^{-6}$ (this is the angle the side of the block makes with the vertical direction), would we have to have a force of $3\,\textrm{N} / \sin\theta \approx 3\cdot 10^{6}\,\textrm{N}$. In other words, even the most mundane shear force on an object would create unimaginably large forces. Is this correct?! If not, what is wrong with this reasoning?

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  • $\begingroup$ You also need to draw the forces that keep the block from rotating. Maybe the question would be easier to answer if all the forces are shown. $\endgroup$
    – Bill Watts
    Jul 9, 2023 at 4:49
  • $\begingroup$ Was writing the same comment. Try drawing a representative volume element with all four necessary traction forces for a pure shear stress state. This may clear up the issue. $\endgroup$ Jul 9, 2023 at 4:52
  • $\begingroup$ @Chemomechanics What if I consider a cantilever beam with a load at the exposed end? That would introduce a similar shear stress, but it's not clear (to me) how all four traction forces come into play or if they are necessary. (Alternatively perhaps the block could be fixed to the ground but I acknowledge that I'd have to consider normal forces then.) $\endgroup$ Jul 9, 2023 at 7:58
  • $\begingroup$ If you don’t see why all four forces are always required to maintain a shear state on any infinitesimal element, I recommend reviewing free-body diagrams, Newton’s second law, and the equations of equilibrium. It tends to be more convincing to prove something to oneself rather than just reading it. $\endgroup$ Jul 9, 2023 at 17:31

2 Answers 2

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My question is, do these restoring forces arise from central forces between the molecules of the object?

It is more than a little strange to draw the restoring forces on the block of material. The restoring forces would be the Newton’s 3rd law pairs from the shear forces. So they do not act on the block at all.

The forces that act on the block are the shear forces and they point in the opposite direction as what you have shown. This holds even if you subdivide the block into two half blocks.

Regarding the centrality of the forces, there is no spontaneous generation of angular momentum, but the shear forces themselves can produce a net external torque. I don’t know if it is useful to take those “macroscopic” facts and explain them as a lattice of semi-classical point particles interacting with central forces, but it is certainly possible to do so.

Actual molecules are not point particles, and it is better to think directly in terms of stress tensors and Lagrangian/Hamiltonians than in terms of forces at that level. Forces are a macroscopic simplification used to approximate the integral of a stress tensor.

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An example is to make a torsion on a mass suspended by a rope. The resulting oscillations must come from some restoring force.

There is a pure shear stress condition at a small square on the rope surface, with sides parallel and perpendicular to the rope. However, if we draw another square twisted $45^{\circ}$ in the same surface, there is a condition of compression in one direction and tension in the other, both equal in magnitude. It can be understood by imagining a line inclined $45^{\circ}$ in the rope surface. After twisting a small angle, this line is elongated or shortened, depending on the twist direction.

The restoring forces comes from that tensile and compressive stresses.

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