Nonlinear in-plane buckling of small-curved and large-curved FG porous microbeams via strain gradient-based isogeometric collocation formulations

In the current investigation, for the first time, the changes in the limit loads and equilibrium branches associated with the nonlinear in-plane stability characteristics of curved microbeams are explored in the presence of different microstructural gradient tensors. In this regard, multiple microsize-dependent equilibria are analyzed relevant to thermomechanical loaded small-curved, medium-curved, and large-curved microbeams made of functionally graded porous (FGP) metal reinforced with nanofillers possessing clamped end supports. To this purpose, based upon the strain gradient elasticity within the framework of the third-order shear flexible curved beam model, the isogeometric collocation formulations incorporating Greville abscissae are constructed resulting in higher-continuity characters as well as remarkable accuracy for higher-order approximations. It is deduced that for the small-curved FGP reinforced microbeam, no limit load can be found due to the absence of the buckling phenomenon, but after rising the temperature by an enough amount, the initial instability mode appears. However, for the medium-curved FGP reinforced microbeam, the limit instability mode of buckling occurs which results in the normalized upper limit load equal to 0.7161 based on the classical theory and 0.7606 based on the strain gradient elasticity (6.21% enhancement). Also, it results in the normalized lower limit load equal to 0.3060 based on the classical theory and 0.3521 based on the strain gradient elasticity (15.07% enhancement). On the other hand, for the large-curved FGP reinforced microbeam, the bifurcation mode of buckling occurs which results in the normalized upper limit load equal to 1.0494 based on the classical theory and 1.1358 based on the strain gradient elasticity (8.23% enhancement). Also, it results in the normalized lower limit load equal to 0.2225 based on the classical theory and 0.2513 based on the strain gradient elasticity (12.94% enhancement).