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<p>In recent years, nonlocal continuum theories have gained increasing attention for modeling micro- and nanoscale structures, metamaterials, and heterogeneous composites such as laminated glass. Peridynamics (PD) offers a robust nonlocal framework by incorporating long-range mechanical interactions between material points. This study presents an analytical investigation of peridynamic first-order shear deformation (FSDT) beam equations under both static and dynamic loading conditions, considering various boundary constraints. While classical beam theories allow numerous closed-form solutions, analytical results for peridynamic formulations remain scarce. For static cases, the derived bending moment and shear flux solutions coincide with classical FSDT stress resultants. Polynomial solutions satisfying peridynamic equilibrium and pointwise boundary conditions are also obtained, but they exhibit nonzero power fluxes through the constraints, highlighting the need for more general formulations. To address this, series solutions based on trigonometric functions are developed, ensuring exact satisfaction of boundary conditions and zero power flux at simply supported edges. For dynamic scenarios, solutions for free vibrations, forced vibrations, and lateral impact of simply supported beams are presented, showing excellent agreement with classical FSDT solutions when the horizon is sufficiently small. These findings demonstrate the accuracy of the peridynamic FSDT framework and provide a foundation for exploring general nonlocal boundary conditions and complex loading scenarios.</p>

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Analytical solutions to first-order shear deformation beam equations in a peridynamic framework

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月9日

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  • Zhenghao Yang,
  • Konstantin Naumenko,
  • Holm Altenbach

摘要

In recent years, nonlocal continuum theories have gained increasing attention for modeling micro- and nanoscale structures, metamaterials, and heterogeneous composites such as laminated glass. Peridynamics (PD) offers a robust nonlocal framework by incorporating long-range mechanical interactions between material points. This study presents an analytical investigation of peridynamic first-order shear deformation (FSDT) beam equations under both static and dynamic loading conditions, considering various boundary constraints. While classical beam theories allow numerous closed-form solutions, analytical results for peridynamic formulations remain scarce. For static cases, the derived bending moment and shear flux solutions coincide with classical FSDT stress resultants. Polynomial solutions satisfying peridynamic equilibrium and pointwise boundary conditions are also obtained, but they exhibit nonzero power fluxes through the constraints, highlighting the need for more general formulations. To address this, series solutions based on trigonometric functions are developed, ensuring exact satisfaction of boundary conditions and zero power flux at simply supported edges. For dynamic scenarios, solutions for free vibrations, forced vibrations, and lateral impact of simply supported beams are presented, showing excellent agreement with classical FSDT solutions when the horizon is sufficiently small. These findings demonstrate the accuracy of the peridynamic FSDT framework and provide a foundation for exploring general nonlocal boundary conditions and complex loading scenarios.