As part of a wide-ranging research programme, whose objective is to extend sharp crack fracture mechanics to notches, so that credit can be taken for their non-sharpness, the author has recently analysed a general two-dimensional model of a blunt notch with radius of curvature ρ. For Mode III deformation, it has been shown that the elastic stress at a distance x≪ ρ ahead of the notch root only depends on x, ρ and the peak stress σ p, irrespective of the notch shape and the loading characteristics. However, a similar approach for Mode I deformation shows that the stress, besides depending on x, ρ and σ p, also depends on the notch shape and the loading characteristics. In this paper, the behaviour of a circular cylindrical notch in an infinite solid is analysed so as to provide support for these general conclusions. Thus with Mode III deformation, it is shown that the local stress is uniquely defined by x, a= ρ (the notch radius) and σ p, but is independent of the loading characteristics while, with Mode I deformation, the local stress does depend on the loading characteristics.