Mixed finite elements based on superconvergent patch recovery for strain gradient theory

Strain gradient theory is one of the most powerful candidates to simulate micro/nano-sized structures considering size effect. However, too many degrees of freedom (DOFs) have been placed to satisfy the C1-continuity condition for the development of finite elements based on the strain gradient theory, greatly reducing the practicality of these elements. In this work, four-node quadrilateral and eight-node hexahedral elements are developed to simulate size effect based on strain gradient theory. Since only displacement DOFs are placed on each node without using displacement gradient DOFs, there is dramatic reduction in computational cost. The displacements are approximated with shape functions of standard finite elements. The displacement gradient field is approximated with linear polynomials and the coefficients are determined using the superconvergent patch recovery. It was found that the superconvergent properties of Barlow points are still valid even though the gradient recovery is performed during the calculation of the element stiffness matrix. The preconditioned conjugate gradient method is utilized to solve the system equations. Numerical examples demonstrate the accuracy and efficiency of the proposed elements. Size effect observed in experiments is captured by using the developed finite elements.