Formulation of three-dimensional finite element models for gradient elasticity analysis

Abstract

In the recent decades, gradient elasticity theories which are sometimes known as nonlocal elasticity theories have drawn considered attention for describing the non-local behaviour of the materials in micro- and nano-scale. The strain energy of the materials depends not only on the strain but also on its derivatives. This thesis will restrict to strain gradient elasticity which is often termed as gradient elasticity nowadays. In gradient elastic materials, the energy conjugate of the strain is stress and that of the strain gradient is known as double stress. Gradient elasticity theories usually impose the first-order continuity, i.e. C1 requirement on the interpolation of the displacement. Constructing C1 interpolation functions is not a trivial work. In this thesis, the discrete Kirchhoff (DK) method and the refined direct stiffness (RDS) method for the thin plate analysis are revisited and generalized to devise 3D finite element (FE) models for the gradient elasticity analysis. In the generalized DK method, the relations between the displacement and the displacement gradient are enforced at discrete points which are the corner nodes and midpoints of the element edges, respectively. Then, the displacement and displacement gradient are separately interpolated. When computing the element stiffness, the strain and the strain gradient are derived from the interpolated displacement and displacement gradient, respectively. In the relaxed-hybrid (RH) method, a generalized hybrid functional is employed. The assumed constant double stress is used to enforce the continuity of the normal derivative of the displacement in a relaxed-hybrid functional. By employing the basis of the double stress derived from displacement as that of the assumed double stress, element stiffness can be computed simply by augmenting the matrix relating the strain gradient and the element vector of degrees of freedom with a refinement matrix. In this thesis, 4-node tetrahedral, 14-node hexahedral and 8-node hexahedral elements are proposed by the generalized DK and RH methods. The individual element test (IET) is conducted for all models to verify that they can fulfil the constant double stress patch test. Numerical tests are presented to show the accuracy and convergence of the proposed element models, and compare the proposed hexahedral elements with an existing 3D C1 element. The proposed elements are more accurate in predicting the stress when the mesh is irregular and consume less computational time.