"Think about a 1" diameter round bar cantilever beam fixed at one end and shear force on the other. The finer mesh will better capture the true cross-sectional area than the course mesh."
This would only result in a meaningful difference if the order of the element was low enough such that it couldn't represent the section of the curve that it divides. Since Ronan stated that he was using a non-draft quality mesh, the addition of the mid-side node should be more than enough to capture the curves of the geometry to remove any meaningful error due to geometry misrepresentation.
"The finer mesh also better captures any stress concentrations. Hence less displacement and higher stress."
The first part is true, but not the second part. I think I get your reasoning: finer mesh better captures geometry (cross-sectional area) => more area = stiffer geometry = lower displacements, and finer mesh = higher stresses. However, this line of thinking isn't correct because the decrease in stiffness do to inaccurate geometry representation does not compare to the increase in stiffness from too few elements. This can easily be confirmed by setting up a model like you described (a cantilever rod) and looking at how the maximum stress and displacement change as your progressively refine a draft quality mesh (I use draft quality because it will give the worse geometric representation of the circular cross-section of the rod).
We can see with the lowest mesh refinement level that the rod appears almost hexagonal:
And with the finest mesh that it appears very close to circular:
But yet we see both the maximum displacement and stress increase as we refine the mesh (and get a better approximation of the cross-sectional area):
In fact, if we think about the assumptions in a linear static analysis, then it becomes apparent that the stress should increase if the displacement increases. A linear static analysis assumes Hooke's Law:
Stress = Young's Modulus * Strain
and
Strain = d(Displacement)
where d() is the derivative.
In other words, larger displacement = larger strains = larger stresses. It should be noted that taking the derivative of the displacement fields to get the strain field introduces error, hence displacement results being more accurate than strain/stress results for a given mesh.
"It looks like you have a uniform load on the ball. Most likely that is the reason for inaccuracy in results. In real world, it is not loaded like that."
This is possible if the statically equivalent point of the load is offset further out on the uniform load when compared to the non-uniform load.
