Study on nonlinear vibrations of temperature- and size-dependent FG porous arches on elastic foundation using nonlocal strain gradient theory

A nonlocal strain gradient theory is developed in this paper to study the large amplitude vibrations of arches made of functionally graded (FG) porous material. The case of shallow arches resting on nonlinear elastic foundation is modeled via a general higher-order shear deformation theory. The third-order model of Reddy, the first-order model of Timoshenko, and the classical model of Euler–Bernoulli are analyzed. Thermomechanical properties of the arch exposed to the uniform thermal field are assumed to be temperature dependent. The nonlinear motion equations of the arch are established by employing Hamilton’s principle and the von Kármán type of geometric nonlinearity. The two-step perturbation technique and the Galerkin method are utilized to solve the nonlinear governing equations. The size-dependent linear and nonlinear frequencies of the arch are obtained for the immovable pinned-pinned boundary conditions. The comparison studies are performed to verify the present solution method with the provided data in the literature, and a good agreement is observed. The novel parametric studies covered in this research include the effects of several parameters such as elastic foundation, nonlocal and length scale parameters, porosity, temperature field, power law index, and geometrical parameters on the frequencies of FG porous arches in detail.