<p>In this work, we propose two different generalizations of one-dimensional strain-limiting elasticity model where the linearized strain is given as a nonlinear function of the stress. These formulations are called stress gradient-type and strain gradient-type generalizations, and their constitutive relations are presented in both differential form and integral form. One important feature of this framework is that contrary to the theory of strain-limiting elasticity, the propagation of linear stress waves becomes dispersive as a consequence of inclusion of stress or strain gradients. We study traveling stress wave solutions to the governing equations of the nonlinear models proposed in this work. For a sample case of the constitutive relation belonging to the stress gradient-type formulation, we obtain explicit expressions of smooth solitary wave solutions when the stress is small but finite. Finally, we show that, for both the stress gradient-type and strain gradient-type formulations, the propagation of small amplitude long waves is described by the well-known KdV equation with the same coefficients.</p>