Application of a non-conforming tetrahedral element in the context of the three-dimensional strain gradient elasticity

Abstract

In the size-dependent continuum theories such as the strain gradient theory, the higher-order derivatives of displacement field appear in the energy functional of the structure which leads to the employment of C1 continuous shape functions within the finite element discretization procedure. Although a wide range of one- and two-dimensional small-scale finite elements were developed to analyze the structural behavior of micro- and nano-structures, a few studies can be found on the development of size-dependent three-dimensional (3D) finite elements. Hence, the main purpose of this work is the introduction of a four-node tetrahedral element to analyze the size-dependent mechanical behavior of micro- and nano-structures based on the three-dimensional strain gradient theory (SGT). In the proposed element, the values of displacement components and their related first-order derivatives are considered as degrees of freedom at each node. To present the governing equations, the matrix form of kinetic and strain energies and the work of external forces are derived based on the 3D elasticity theory and strain gradient model. To show the efficiency of the proposed model, the size-dependent linear vibration analysis of circular and elliptical micro- and nano-plates is presented. Various numerical results including comparative and convergence studies are reported to check the accuracy and performance of the introduced finite element.