Numerical Solution of the Tri-harmonic Kirchhoff Plate Equation Resulting from a Strain Gradient Theory

Abstract

A second gradient continuum theory is formulated based on second gradients of displacements. For a reduction of additional material parameters, the modified strain gradient model is used and a partial dierential equation of rank six is developed using the Kirchhoff plate assumptions. The solutions of the governing tri-harmonic plate bending equation incoorperate size-effects. Balance equations are presented and higher-order stress-strain relations are derived. In order to account for second gradients of displacements, which manifest themselves in the higher-order terms of a strain energy density, a C¹–continuous displacement field is preferable. So-called Hermite finite element formulations allow for merging gradients between elements and are used to achieve global C¹–continuity of the solution. Element stiffness matrices as well as the global stiffness matrix are developed for a lexicographical order of nodes and for equidistantly distributed elements. The convergence, the C¹–continuity, and the size effect are demonstrated.