Porosity distribution effect on stress, electric field and nonlinear vibration of functionally graded nanostructures with direct and inverse flexoelectric phenomenon

In the present paper, the effect of the diverse distribution of porosity on the static and nonlinear dynamic responses of a sandwich functionally graded nanostructure with thermo-electro-elastic coupling is presented based on the modified flexoelectric theory. The rectangular nanoplate satisfying the classical Kirchhoff’s assumptions is taken into consideration as example of a nanostructure. It is assumed that sandwich functionally graded porous nanoplate is embedded on an elastic foundation and subjected to a thermo-electro-mechanical load. The Winkler-Pasternak model was considered to present the effect of elastic foundation. Components of the Green-Lagrange strain tensor are taken as linear and infinitesimal. The appeared source of nonlinearity in nanostructure is caused by using the properties of the flexoelectric effect. Therefore, change a volume element due to the elastic deformation cannot be negligible. The constitutive relations for functionally graded porous material are expressed by a power-law variation of material parameters in conjunction with cosine functions to create a possibility to investigate the effect of distribution of diverse types of porosity on mechanics of structures. The equilibrium and governing equations of the sandwich nanoplate resting on the Winkler-Pasternak foundation are derived based on Hamilton’s principle. The obtained differential equations for the static and dynamic problems of nanoplate were solved by Galerkin’s and multiple scale methods. The influence of some important material and geometrical parameters as well as diverse boundary conditions on mechanical responses of the nanostructure were comprehensively investigated and discussed.