Size-dependent nonlinear dynamics and 1:3 internal resonance of thermally graded Osteon micro-beams under nonlocal strain gradient theory

This study investigates the nonlinear dynamic stability and internal resonance of osteon micro-beams under localized thermal gradients. Utilizing Nonlocal Strain Gradient Theory (NSGT), the size-dependent behavior of the Haversian system is modeled to bridge microscopic nuances and macroscopic thermal-mechanical responses. The governing equations are derived and solved via the Method of Multiple Scales (MMS) to determine nonlinear frequency-response characteristics. The results indicate that thermal gradients cause frequency reduction and convergence, while the nonlinear response retains a persistent hardening-type behavior at large amplitudes. A critical highlight is the identification of a 1:3 internal resonance threshold. At specific parametric configurations, the system exhibits sophisticated modal coupling, a ‘double-peak’ resonance profile, and the saturation phenomenon, indicating nonlinear energy transfer between the fundamental and second modes. Furthermore, 3D coupled parametric maps illustrate a ‘stability canyon’, highlighting osteon dynamic sensitivity under combined thermal and nonlocal influences. The obtained nonlinear vibration characteristics and internal resonance behavior may provide useful insight into how dynamic loading conditions influence stress redistribution and mechanical response at the osteon scale.