We consider an estimate of the lower bound for the critical load in a monotonic dead loading process for an isotropic compressible elastic solid. Within a constitutive class that includes many commonly used strain energy functions, we seek a lower bound on the Hadamard stability functional and develop an estimate of the lower bound for the critical load. As an example, we apply our estimate to the case of a simple shear loading process for a generalized Blatz – Ko material. Our result, albeit for a special (but reasonably broad) constitutive class, shows a significant improvement of the estimate of the lower bound as compared to other proposals contained in the literature.