Finite element bending analysis of square microplates with circular hole based on the three-dimensional strain gradient elasticity theory

Experimental studies indicates that the mechanical behavior of materials at micro and nano scales are size-dependent. Since the classical continuum mechanics theories cannot capture the size effect, employment of different non-classical theories has received a considerable attention among researchers. In this study, the finite element formulation is presented to investigate the bending of square microplates with circular hole subjected to uniform pressure based on the three-dimensional strain gradient elasticity theory. For this account, the 8-node C^1 continuous hexahedral element is introduced in which, in addition to the values of displacement components, some related higher-order mix derivatives are further considered as nodal values. The governing equations are derived based on the strain gradient theory and three-dimensional elasticity model and the finite element formulation is presented using the introduced element. Note that by considering some specified values for coefficients of strain gradient theory, the numerical results can be obtained for modified strain gradient theory and modified couple stress theory. To demonstrate the efficiency of the proposed finite element, the convergence and accuracy of the results are firstly checked and then the impacts of geometrical parameters on the bending of microplates with circular hole are studied.