The effective elastic-piezoelectric properties of nanostructures have been shown to be strongly size-dependent. In this paper, a nonlocal second-order shear deformation formulation is presented to study the size-dependent thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core. Temperature is considered as uniform and nonlinear distributions across plate’s thickness direction. Based on the developed nonlocal second-order shear deformation theory, the size-dependent equations of motion are derived. The nonlocal thermal buckling responses of simply supported nanoplates are solved via Navier method. The reliability of present approach is verified by comparing the existing results provided in the open literature. The influences of nonlocal parameter, gradient index, electric voltage, and Winkler–Pasternak parameters on the thermal buckling characteristics of functionally graded nanoplates are examined.

Flexural and longitudinal wave behaviors of nanobeams made of nanoporous-graded materials while surrounded by Winkler-Pasternak foundation, subjected to the longitudinal magnetic field and exposed to the hygrothermal environment are studied analytically. To this end, the governing equation derived by Euler–Bernoulli beam theory in conjunction with the nonlocal strain gradient theory is defined by employing Hamilton’s principle. By adopting an analytic model, the flexural and longitudinal dispersion relations between phase velocity and wave number are derived. The reliability of the present method is confirmed by comparing the obtained results with those stored in the literature. Finally, the effects of the power-law index, porosity volume fraction, nonlocal and material characteristic parameters, uniform temperature and moisture rise, elastic foundation parameters, magnetic field intensity, and wave number are also investigated in detail. It is found that the small-scale parameters are more influential in higher wave numbers where the wavelength is close to the length scale of nanostructures. However, foundation parameters, porosity volume fraction, and longitudinal magnetic field are more influential in lower wave numbers.

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