On nonlinear vibration and snap -through stability of porous FG curved micro -tubes using two-step perturbation technique

Considering the modi fied couple stress theory, nonlinear vibration and snap -buckling analyses of porous FG curved micro -tubes are performed in this research. Curved micro -tubes with shallow curvature resting on non- linear elastic foundation are analysed. Properties of the micro -tube with uniform distributed porosity are graded across the radius of cross-section by means of a power law function. The equations of motion are derived based on the high -order shear deformation theory by applying the Hamilton principle. Governing equations are obtained using the von K?rm?n assumptions as a system of nonlinear differential equations. For curved micro -tubes which are simply supported in flexure and axially immovable, governing equations are solved using an analytical approach. A two-step perturbation technique is used to obtain the closed -form solutions for nonlinear free vibration and snap -buckling problems. Since the porous FG curved micro -tube is not analysed in literature, the results are compared with the case of FG tubes. Parametric studies are provided to explore the effect of geometrical characteristics of the curved micro -tube on the static and dynamic responses of these structures. The in fluences of material length scale parameter, functionally graded patterns, porosity parameter and nonlinear elastic foundation are also investigated.