We investigate auxetic isotropic elastic materials with the Poisson's ratio of exactly −1 and isotropic materials with the Poisson's ratio of near 0.5 (incompressible). In both cases the energy is not positive definite, which can lead to the material instability. In incompressible materials, the instability manifests itself in the emergence of non-uniform displacement distribution. Investigation of the behavior of such materials with respect to damage or crack accumulation is based on the notion that during the instantaneous process of fracture formation, the fracture acts as a negative stiffness element; after the fracture is formed, it immediately turns into a conventional fracture with usual positive stiffness. We demonstrate that the cracks formed due to tensile or combined tensile and shear fracturing of the material are capable of momentarily bringing the Poisson's ratio of nearly incompressible material to 0.5, which instantaneously turns the material into unstable. On the other hand, the formation of shear cracks leads to reduction in the Poisson's ratio bringing the material away from the point of instability. Opposite to this, cracks of all kinds do not change the Poisson's ratio in extreme auxetics (the Poisson's ratio equal to −1). Thus the auxetic materials remain stable with respect to fracture formation.