First, the conditions for uniqueness of the stress boundary value problem for infinite cracks in elastic–plastic bodies are discussed on the basis of the Laurent series representation of the plastic boundary in an elastic perfectly plastic body in anti-plane strain (mode III). Next, for two cases, exact closed-form solutions of the shape of the elastic–plastic boundary are found in terms of elementary functions. A crack under shear stress acting on its surfaces, and a crack under constant remote shear stress σ ∞ 13=− p, are considered. In the first case, also the complete stress distribution is obtained, for the second the physical coordinates as functions of stresses are found. The new elastic–plastic solutions are compared with ones predicted by linear elastic fracture mechanics.