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<p>An exact analytical solution is presented for the static bending response of bidirectionally functionally graded (BDFG) Timoshenko nanobeams within the framework of nonlocal strain gradient theory (NSGT). The material properties are assumed to vary simultaneously along the beam axis and through the thickness, described by an exponential law in the axial direction and a power-law distribution across the thickness. Based on Timoshenko beam kinematics, the coupled governing equations accounting for both nonlocal elasticity and strain-gradient effects are derived via Hamilton’s principle, together with consistent higher-order boundary conditions. To overcome the mathematical complexity associated with variable coefficients and higher-order operators, the Laplace transform method is employed, and closed-form expressions for transverse displacement and rotation are obtained through explicit inverse transformation in physical space. The proposed solution is validated through comparisons with available results for homogeneous and unidirectionally graded nanobeams, demonstrating excellent agreement and correct recovery of limiting cases. A detailed parametric study is conducted to examine the effects of axial and thickness-wise material gradation, nonlocal parameter, and strain-gradient length scale under various classical boundary conditions. The results reveal that axial gradation effectively tailors stiffness along the span, whereas increasing the thickness-wise gradation index leads to a more metal-dominated core and increased flexibility. A direct comparison between NSGT and classical nonlocal elasticity theory highlights the crucial role of strain-gradient hardening in mitigating nonlocal softening, resulting in more balanced and physically realistic predictions. The availability of exact closed-form solutions provides reliable benchmark results and offers new insight into the coupled influence of bidirectional material gradation and size-dependent effects in nanobeams, supporting future analytical, numerical, and multi-physics investigations.</p>

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Exact analytical solution for static bending of bidirectionally functionally graded Timoshenko nanobeams based on nonlocal strain gradient theory

  • 期刊论文

  • 发表日期：

    2026年6月24日

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  • Minhaj Uddin Mahmood Siddique,
  • I. M. Nazmul,
  • Indronil Devnath

摘要

An exact analytical solution is presented for the static bending response of bidirectionally functionally graded (BDFG) Timoshenko nanobeams within the framework of nonlocal strain gradient theory (NSGT). The material properties are assumed to vary simultaneously along the beam axis and through the thickness, described by an exponential law in the axial direction and a power-law distribution across the thickness. Based on Timoshenko beam kinematics, the coupled governing equations accounting for both nonlocal elasticity and strain-gradient effects are derived via Hamilton’s principle, together with consistent higher-order boundary conditions. To overcome the mathematical complexity associated with variable coefficients and higher-order operators, the Laplace transform method is employed, and closed-form expressions for transverse displacement and rotation are obtained through explicit inverse transformation in physical space. The proposed solution is validated through comparisons with available results for homogeneous and unidirectionally graded nanobeams, demonstrating excellent agreement and correct recovery of limiting cases. A detailed parametric study is conducted to examine the effects of axial and thickness-wise material gradation, nonlocal parameter, and strain-gradient length scale under various classical boundary conditions. The results reveal that axial gradation effectively tailors stiffness along the span, whereas increasing the thickness-wise gradation index leads to a more metal-dominated core and increased flexibility. A direct comparison between NSGT and classical nonlocal elasticity theory highlights the crucial role of strain-gradient hardening in mitigating nonlocal softening, resulting in more balanced and physically realistic predictions. The availability of exact closed-form solutions provides reliable benchmark results and offers new insight into the coupled influence of bidirectional material gradation and size-dependent effects in nanobeams, supporting future analytical, numerical, and multi-physics investigations.