Vibration of cracked functionally graded microplates by the strain gradient theory and extended isogeometric analysis

In this study, the vibration behaviours of functionally graded microplates with cracks are investigated by means of a simple yet rigorous version of Mindlin's generialised continuum and the extended isogeometric analysis (XIGA). The simplified strain gradient theory which includes one material length parameter and an additional micro-inertia term is employed to capture the size effects. Meanwhile, the displacement field of the plates is described using the refined plate theory with four unknowns and the XIGA in which enrichment functions are involved to effectively predict the responses of microplates with cracks. In addition, the IGA approach with highly smooth basis functions of non-uniform rational B-spline (NURBS) ensures a clean and efficient treatment of higher continuity requirements in the strain gradient theory. The benchmark numerical results show significant departure from those analysed by the classical continuum elasticity. Indeed, they reveal strong influences of microstructural characteristics on the vibration responses of microplates which are not shown in the platform of the classical theory and the influences are more pronounced as the size of the plates becomes comparable with the material length parameter.