An iteration method to study nonlinear wave propagation for a non-Green elastic 1D bar.

This article focuses on the propagation of nonlinear waves in a one-dimensional (1D) bar using constitutive equations where the linearized strain tensor depends on the Cauchy stress tensor, specifically for non-Green elastic solids. Two nonlinear constitutive models are examined: one for rock mechanics and another exhibiting strain-limiting behavior. The authors develop a novel iterative numerical method that integrates the equation of motion with these constitutive relations to approximate solutions, starting from the linear elastic case as an initial guess. Numerical results demonstrate significant differences in displacement predictions between the nonlinear models and the classical linearized elastic model, particularly for the strain-limiting constitutive equation, under sinusoidal and linearly increasing external loads. This methodology provides a versatile computational approach for studying nonlinear wave propagation in 1D elastic bars with complex constitutive behaviors.