On the Nonlocal Theory of Microstructures and it's Implementation of Finite Element Methods

This paper reviews the recent development of several nonlocal theories as well as corresponding finite element methods. Discussion focuses on size effects on microscale and the mesoscale material parameters. We can show that both number and role of mesoscale material parameters l are different with the nonlocal theories. In this paper we use the finite element method to model Stolken's microbend test, where three nonlocal theories with one l are adopted. Numerical results show that the existing nonlocal theories with one mesoscale material parameter l are preferred to the theories with more parameters l in engineering. The governing equations of nonlocal theories are inhomogeneous differential equations for which the conventional patch test is not robust and complete. In this paper, based on nonlocal theories of C1 continuity, the variational principles with relaxed inter-element continuity requirements of the nonconforming element and the test function for the patch test are given. The element RCT9+RT9 satisfied requirement of both C0 and C1 displacement continuities simultaneously, is adopted herein. In addition, this paper also includes some information of recent patents on the applications of nonlocal theories.