Vibration analysis of nonlocal advanced nanobeams in hygro-thermal environment using a new two-unknown trigonometric shear deformation beam theory

In this work, the effects of moisture and temperature on free vibration characteristics of functionally graded (FG) nanobeams resting on elastic foundation is studied by proposing a novel simple trigonometric shear deformation theory. The main advantage of this theory is that, in addition to including the shear deformation influence, the displacement field is modeled with only 2 unknowns as the case of the classical beam theory (CBT) and which is even less than the Timoshenko beam theory (TBT). Three types of environmental condition namely uniform, linear, and sinusoidal hygrothermal loading are studied. Material properties of FG beams are assumed to vary according to a power law distribution of the volume fraction of the constituents. Equations of motion are derived from Hamilton's principle. Numerical examples are presented to show the validity and accuracy of present shear deformation theories. The effects of hygro-thermal environments, power law index, nonlocality and elastic foundation on the free vibration responses of FG beams under hygro-thermal effect are investigated.