From Hertzian contact to Pochhammer-Chree dynamics: Solitons, chaos, and bifurcation in granular metamaterials

Mst. Ifat Zahan Soma, Tarikul Islam, Tobibur Rahman and Shahariar Ryehan

Chaos, Solitons & Fractals, 2026, vol. 208, issue P1

Abstract: This study is concerned with the complex propagation patterns of solitary waves within one-dimensional metamaterials consisting of discrete granular elements. Granular mechanics relies heavily on the Hertzian contact theory, a concept which has not been adequately captured by linearised models of particulate contact. In this work, the Hertz theory of contacts is extended to the nonlinear Pochhammer-Chree equation, incorporating spatiotemporal dispersion effects. We present exact solitary wave solutions using the improved auxiliary equation and the improved tan-hyperbolic methods. These methods provide solutions which include the Kink, Anti-kink, Dipole, and Bell-shaped solitary waves, as well as compactons, which have characteristic localisation of energy. The progression of these wave patterns with differing roughness parameters is depicted through three-dimensional surface plots and contour profiles. A comprehensive bifurcation analysis has been carried out to establish the system's stability and to examine the onset of chaos, which is induced by external forcing, by applying the Melnikov theory. Analytical predictions are validated against Discrete Element Method (DEM) simulations, demonstrating excellent agreement with an Error Value (RMSE) of approximately 1.2%. The computed Lyapunov exponent was positive (1.9515), which suggests that the system is sensitive to initial conditions. This new approach is the most advanced that has been developed to date for calculating the nonlinear propagation of stress waves and is essential for designing materials and structures that are resistant to earthquake damage.

Keywords: Pochhammer-Chree equation; Solitary wave solution; Bifurcation analysis; Chaos analysis; Granular chain; Seismic wave modeling (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:208:y:2026:i:p1:s0960077926002742

DOI: 10.1016/j.chaos.2026.118133

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