Interfaces are often believed to play a role in the mechanical behavior of mineralized biological and biomimetic materials. This motivates the micromechanical description of the elasticity and brittle failure of interfaces between crystals in a (dense) polycrystal, which serves as the skeleton of a porous material defined one observation scale above. Equilibrium and compatibility conditions, together with a suitable matrix-inclusion problem with a compliant interface, yield the homogenized elastic properties of the polycrystal, and of the porous material with polycrystalline solid phase. Incompressibility of single crystals guarantees finite shear stiffness of the polycrystal, even for vanishing interface stiffness, while increasing the latter generally leads to an increase of polycrystal shear stiffness. Corresponding elastic energy expressions give access to effective stresses representing the stress heterogeneities in the microstructures, which induce brittle failure. Thereby, Coulomb-type brittle failure of the crystalline interfaces implies Drucker-Prager-type (brittle, elastic limit-type) failure properties at the scale of the polycrystal. At the even higher scale of the porous material, high interfacial rigidities or low interfacial friction angles may result in closed elastic domains, indicating material failure even under hydrostatic pressure. This micromechanics model can satisfactorily reproduce the experimental strength data of different (brittle) hydroxyapatite biomaterials, across largely variable porosities. Thereby, the brittle failure criteria can be well approximated by micromechanically derived criteria referring to ductile solid matrices, both criteria being even identical if the solid matrix is incompressible.