Sinusoidal shear deformable beam theory for analytic nonlocal elasticity

This paper presents a unified nonlocal sinusoidal shear deformation theory to comprehensively analyze nanobeam bending, buckling, and free vibration. The proposed model effectively distinguishes bending and shear components, accurately capturing small-scale effects and transverse shear deformation without shear correction factors. The energy and governing equations were derived using Hamilton's principle and solved analytically through the Laplace Transformation method. This approach led to the exact expressions for key mechanical responses, including the displacement equation for bending, the buckling load for stability, and the frequency equation for vibration analysis. The results are extensively presented in table and graphical formats, offering a detailed study of the effects of various parameters on the behavior of nanobeams. Furthermore, the model's predictions were validated against existing beam theories, demonstrating its enhanced accuracy and robustness. This study significantly advances the understanding of nanobeam mechanics by providing a powerful and versatile framework for designing and analyzing nanoscale structures. The findings are particularly relevant for applications where precise control over mechanical properties is crucial, making this work a valuable contribution to the field of nanotechnology and advanced material engineering.