There is no stress whatsoever trying to disrupt solid bodies – or bound states of any sort – caused by the expansion of the Universe.
The individual atoms or pieces of solid bodies are arranged to minimize the total energy and have stable relative positions for that reason. Equivalently, it is possible to parallel transport a body in the timelike direction, along the world line, and that's exactly what these bodies are doing as they evolve in time.
This becomes particularly clear in the de Sitter space – the exponentially accelerating expansion caused by the positive cosmological constant, a phase that we have been entering in recent billions of years. The isometry of $dS_4$ is $SO(5,1)$ which also includes a boost-like generator that plays exactly the same role as the time translations in the flat Minkowski space. So this isometry tells you how bound states evolve in time.
Your assumption reflects a widespread misconception about what is actually expanding. What is expanding is the Universe itself, not the size of the objects. The size of atoms, molecules, and even planets etc. stays the same which really means that the expanding Universe is able to harbor an increasing number of atoms, molecules, and/or planets. It is really expanding. It's not just some vacuous change of units that wouldn't change anything material.
Whether the distance between two objects is increasing as the result of the expansion of the Universe depends on what determines their location. If they're just "attached" to some regions of space, like galaxies, they will expand with the space itself. But bound objects' molecules or components have positions determined by the equilibrium of various forces, especially attractive forces, acting inside them. So they're not attached to "independent regions of space" which is why the distance between them isn't increasing, surely not by the same factor as the factor that stretches the distances between galaxies.
Some intermediate situations, like clusters of galaxies that are "partly/loosely bound", would deserve a special discussion. They may expand a bit and it's calculable how much. However, it's important that the systems dominated by the attractive binding forces, e.g. electromagnetic forces that keep solid matter connected, surely don't suffer from the same rate of expansion as the Universe itself. At the same moment, I have to emphasize that the "cosmological stretching" of atoms, molecules, solids, glass rods, and skeletons is exactly zero because all these proper distances are fully determined by local physics governed by non-gravitational forces. For example, there is nothing such as a hydrogen atom that is 1.00001 times the usual radius and the same observation holds for the other tightly bound states, too. This is elementary quantum mechanics. Some people and some papers may err about this basic point but they will never change the radius of the atom or other tight bound stats.