Noname manuscript No. (will be inserted by the editor) Do solids flow?

Abstract Are solids intrinsically different from liquids? Must a finite stress be applied in order to induce flow? Or, instead, do all solids only look rigid on some finite timescales and eventually flow if an infinitesimal shear stress is applied? Surprisingly, these simple ques-tions are a matter of debate and definite answers are still lacking. Here we show that solidity is only a time-scale dependent notion: equilibrium states of matter that break spontaneously translation invariance, e.g. crystals, flow if even an infinitesimal stress is applied. However, they do so in a way inherently different from ordinary liquids since their viscosity diverges for vanishing shear stress with an essential singularity. We find an ultra-slow decrease of the shear stress as a function of the shear rate, which explains the apparent yield stress identified in rheological flow curves. Furthermore, we suggest that an alternating shear of frequency ω and amplitude γ should lead to a dynamic phase transition line in the (ω,γ) plane, from a ’flowing ’ to a ’non-flowing ’ phase. Finally, we apply our results to crystals, show the corresponding microscopic process leading to flow and discuss possible experimental inves-tigations. We clearly know how to make the difference between a solid and a liquid in everyday life. However, the question of whether solid and liquid are distinct states of matter is far from being conclusively answered in the literature. A difficulty in addressing this issue is that even ordinary liquids may look solid, as one learns when diving into the water from an height of, say, 10 meters without being a pro. Actually, supercooled liquids and soft matter, such as emulsions, foams, and pastes behave like bona-fide solids, if probed on timescales that range from microseconds to hours. This phenomenon, called visco-elasticity, is a propert