The concept of stress is very useful to describe the effect of external loads on structures. However, as a basis for the prediction of failure the concept of stress becomes meaningless when the structure encompasses singularities as a result of discrete stiffness steps or geometric anomalies such as cracks. In this article it is argued that the concept of failure stress is incorrect and should be replaced by a generalized concept based on stress intensity factors and singularity orders. It appears that material failure stress is the critical stress intensity factor for a zero-order singularity stress field. By plotting the critical stress intensity factor as a function of singularity order, the strength of a material can be characterized in a general fashion that integrates tensile strength, fracture toughness and critical singularities in adhesive joints. It is also shown that plasticity does not eliminate the stress singularity in an adhesive joint but changes the order of the singularity due to the induced change in interface corner angle between the dissimilar materials in the joint.