The nonlinear wave equation ∂ 2T ∂X 2 = ∂ ∂t ∂T ∂t (1+T 2+X 2) 2 provides a Lagrangian description of one-dimensional stress propagation in a class of model inhomogeneous ideally hard elastic materials. The equation is privileged in that it is associated with pseudospherical surfaces with constant Gaussian curvature K=−1 . Here, exact representations for the stress distribution evolution in model elastic materials are obtained corresponding to classical Beltrami and Dini surfaces as well as a two-soliton pseudospherical surface generated via the classical Bäcklund transformation.