We present a new method to study the free surface dissolution of stressed solids. The model couples a thermodynamic/kinetic approach with an elastic discrete element model. In the model we mimic experiments where we stress a crystal vertically and let it dissolve. The reaction is either progressing on a free surface of the crystal in contact with fluid or from an initial hole in the crystal. Once the elastic solid is stressed it develops an instability in the form of a roughness on its free surface. This so called Asaro-Tiller-Grinfeld instability is well known in Physics from thin-film overgrowth studies. It was also recently reproduced on stressed crystals of brittle elastic salts. The simulations show that the surface roughness progresses towards cusp instabilities that can develop into grooves and into crack-like structures or anti-cracks. The roughness has a characteristic size or wavelength that depends on the elastic and surface energies in the model. A comparison with linear stability analysis, experiments and other numerical approaches shows that our model reproduces reasonable results. The localization of dissolution in anti-cracks that propagate into the crystal can introduce the growth of secondary mode I cracks along the anti-crack. These fractures show a similar characteristic scaling behavior than the structures that are produced by the Asaro-Tiller-Grinfeld instability. Combinations of anti-cracking and cracking may cause grain-size reduction and failure.