  Customarily, crystalline solids are defined to be {\em rigid} since they
resist changes of shape determined by their boundaries. However, rigid solids
cannot exist in the thermodynamic limit where boundaries become irrelevant.
Particles in the solid may rearrange to adjust to shape changes eliminating
stress without destroying crystalline order. Rigidity is therefore valid only
in the {\em metastable} state that emerges because these particle
rearrangements in response to a deformation, or strain, are associated with
slow collective processes. Here, we show that a thermodynamic collective
variable may be used to quantify particle rearrangements that occur as a solid
is deformed at zero strain rate. Advanced Monte Carlo simulation techniques are
then employed to obtain the equilibrium free energy as a function of this
variable. Our results lead to a new view on rigidity: While at zero strain a
rigid crystal coexists with one that responds to infinitesimal strain by
rearranging particles and expelling stress, at finite strain the rigid crystal
is metastable, associated with a free energy barrier that decreases with
increasing strain. The rigid phase becomes thermodynamically stable by
switching on an external field, which penalises particle rearrangements. This
produces a line of first-order phase transitions in the field - strain plane
that intersects the origin. Failure of a solid once strained beyond its elastic
limit is associated with kinetic decay processes of the metastable rigid
crystal deformed with a finite strain rate. These processes can be understood
in quantitative detail using our computed phase diagram as reference.
