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<p><?tk 3?>This paper presents a comprehensive investigation into the nonlinear flexural behavior of bi-functionally graded (BFG) microbeams resting on partially elastic foundations (PEFs) and subjected to elastic boundary conditions (EBCs). The spatial gradation of constituent materials is defined along both the axial and thickness directions via independent power-law indices, enabling precise tailoring of stiffness distributions. Size-dependent effects, inherent to micro-scale beams, are incorporated through the modified couple stress theory (MCST) combined with a refined beam theory (RBT), ensuring an accurate representation of transverse shear deformation without the need for shear correction factors. The partially elastic foundation (PEF) is modeled as a hybrid Winkler–Pasternak medium with spatially varying stiffness, while the elastic end restraints are represented by translational and rotational springs of finite rigidity. The governing nonlinear equations are derived using Hamilton’s principle, discretized through the finite element method (FEM), and solved iteratively via the Newton–Raphson scheme. Parametric analyses reveal the intricate coupling between power-law indices, length scale parameters, foundation stiffness variation, and EBCs. The results highlight the significant influence of length scale effects and EBCs on the nonlinear load–deflection characteristics of BFG microbeams, providing valuable guidelines for the optimal design of advanced micro-scale structural elements in MEMS/NEMS applications.</p>

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The Nonlinear Flexural Behavior of Bi-Functionally Graded Microbeams Resting on Partially Elastic Foundations and Subjected to Elastic Boundary Conditions

  • 期刊论文

  • 发表日期：

    2025年10月27日

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  • Ngoc-Tu Do,
  • Trung Thanh Tran,
  • The Van Tran,
  • Quoc Hoa Pham,
  • Nhan Thinh Hoang

摘要

This paper presents a comprehensive investigation into the nonlinear flexural behavior of bi-functionally graded (BFG) microbeams resting on partially elastic foundations (PEFs) and subjected to elastic boundary conditions (EBCs). The spatial gradation of constituent materials is defined along both the axial and thickness directions via independent power-law indices, enabling precise tailoring of stiffness distributions. Size-dependent effects, inherent to micro-scale beams, are incorporated through the modified couple stress theory (MCST) combined with a refined beam theory (RBT), ensuring an accurate representation of transverse shear deformation without the need for shear correction factors. The partially elastic foundation (PEF) is modeled as a hybrid Winkler–Pasternak medium with spatially varying stiffness, while the elastic end restraints are represented by translational and rotational springs of finite rigidity. The governing nonlinear equations are derived using Hamilton’s principle, discretized through the finite element method (FEM), and solved iteratively via the Newton–Raphson scheme. Parametric analyses reveal the intricate coupling between power-law indices, length scale parameters, foundation stiffness variation, and EBCs. The results highlight the significant influence of length scale effects and EBCs on the nonlinear load–deflection characteristics of BFG microbeams, providing valuable guidelines for the optimal design of advanced micro-scale structural elements in MEMS/NEMS applications.