The mathematical theory of elasticity for planar pentagonal quasicrystals is developed and some analytic solutions for a class of mixed boundary-value problems (corresponding to a Griffith crack) of the theory are offered.An alternate procedure and a direct integral approach are proposed.Some analytical solutions are constructed and the stress and displacement fields of a Griffith crack in the quasicrystals are determined.A basis for further studying the mechanical behavior of the material related to planar defects is provided.