Extraction of Vibration Characteristics of Cracked Structures Using Peridynamic and Bézier Numerical Modeling

Modeling the initiation and propagation of cracks in dynamic structures remains a persistent challenge within the Classical Continuum Mechanics (CCM) framework, primarily due to stress singularities at crack tips and the breakdown of spatial derivatives in the presence of displacement discontinuities. These difficulties are compounded by the strongly nonlinear nature of solid mechanics, which leads to stiff systems of time domain ordinary differential equations and places severe restrictions on numerical stability and efficiency. Peridynamic (PD) theory, formulated as a nonlocal continuum framework, provides a natural remedy to these limitations by replacing local differential operators with integral expressions, thereby eliminating the requirement for spatial differentiability and enabling a unified description of deformation, damage initiation, and crack growth. Despite these advantages, the practical application of PD to dynamic fracture problems is still constrained by the choice of time-integration scheme. Conventional explicit direct-integration methods are straightforward to implement but typically demand prohibitively small time steps to satisfy stability criteria, resulting in high computational cost. On the other hand, Adaptive Dynamic Relaxation (ADR) techniques are effective for accelerating convergence in quasi-static analyses but rely on artificial damping, which compromises their accuracy and physical fidelity in genuine dynamic simulations. In this work, a novel explicit time-integration strategy based on the Bézier numerical method is introduced for solving the peridynamic governing equations, referred to as the PD-Bézier approach. The method exploits the smooth approximation properties of Bézier polynomials to construct a stable and accurate explicit time update, enabling larger admissible time steps without sacrificing solution quality. The performance of the proposed PD-Bézier scheme is systematically assessed through comparative studies against conventional direct-integration and ADR methods. Numerical results demonstrate that the PD-Bézier formulation accurately captures the dynamic response of cracked structures, including natural frequencies and transient displacement fields, while maintaining computational efficiency and avoiding artificial damping effects. These findings indicate that the proposed approach offers a robust and effective alternative for dynamic peridynamic simulations and provides a reliable computational tool for the identification, analysis, and health monitoring of damaged mechanical and structural systems.

کلیدواژه ها:

Peridynamics ، Computational Mechanics ، Plane Vibration ، Bézier Numerical Method