Failures of isotropic brittle materials were revisited to find out whether there are unified failure criteria, which can be applied to a load-carrying structural component made of a brittle material regardless of whether the component has a crack, a circular notch, or neither of these. To this end, a set of failure criteria were considered to determine failure of a local point which may be located at the crack tip, a notch tip, or any other location. The proposed criteria consist of two conditions, of which both must be satisfied for failure to occur. The first condition is that the local stress must not be lower than the failure strength of the material. The second condition is the stress-gradient condition. Even if the stress at a local point far exceeds the failure strength of the material, failure will not initiate until the stress gradient condition is satisfied. Four different examples cases were presented to explain the proposed failure concept.