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An error in the beam model proposed by Lam, Yang, Chong, Wang, & Tong (2003) for the modified couple stress theory is found and corrected, and then the corrected model is verified by the finite element analysis. In addition, various beam models based on the modified couple stress theory are classified according to their characteristics, and evaluated numerically by the finite element analysis using 2D and 3D continuum elements through elastic cantilever beam problems. The results show that the corrected beam model of Lam et al. is the best fit for plane strain beam problems, while none of the beam models are appropriate for general beams of similar width and thickness dimensions. The reason is confirmed by the energy term analysis which reveals that the kinematic assumptions applied to the beam models are inappropriate when the beam shape is general. [ABSTRACT FROM AUTHOR]

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 C 1 -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. • Mixed finite elements are developed for strain gradient theory. • A displacement gradient field is defined using the superconvergent patch recovery. • Only displacement degrees of freedom are used for the continuous gradient field. • The superconvergent characteristics of the gradient field is validated. • The developed finite elements capture size effect observed in experiments. [ABSTRACT FROM AUTHOR]

• We developed a novel 10-node tetrahedral element for the modified couple stress theory. • Higher-order rotation modes are introduced to condense out the Lagrange multiplier DOFs. • Since the Lagrange multipliers are condensed out, the computational efficiency is dramatically increased. • The structural behavior with size effect can be predicted efficiently using the developed element. In this paper, a mixed tetrahedral element based on the Lagrange multiplier is newly developed for the modified couple stress theory. The limitations of the Lagrange multiplier method are that the total number of degrees of freedom is increased and that there are zeros in the diagonal term of the stiffness matrix. We developed a method to overcome the limitations by condensing out the Lagrange multipliers in the calculation of the element stiffness. Higher-order rotation modes are introduced. As condensing out the rotation modes, the Lagrange multiplier is also condensed. The total number of degrees of freedom can be greatly reduced, maximizing computational efficiency. Through numerical examples, the convergence and efficiency of the proposed elements were validated. The developed element is expected to be used for the analysis of micro/nano-scale structures. [ABSTRACT FROM AUTHOR]