Research On C~1 Natural Element Method For Strain Gradient Elasticity And Its Application

Many microscale experiments have shown that materials display strong size effects when the characteristic length scale is on the order of microns or submicrons. Traditional theories are unable to explain the size effects which have been observed in the micro-scales and submicro-scales experiments. One remedy against the deficiency of the traditional theories is to use the strain gradient theories whereby the strain energy density function depends on both the strain tensors and the strain gradient tensors. As the characteristic length scales of materials are introduced in the constitutive equations, strain gradient theories can predict the size effects. The governing equations and the boundary conditions of strain gradient theories are more complicated than those in traditional theories; therefore, numerical methods are usually effective ways to solve these boundary value problems (BVPs). There are some shortcomings in FEM for strain gradient theories, for example, it is difficult to construct the higher-order continuous elements, some C0 continuous mixed elements exist extra nodal degrees of fromdom (DOFs), and the computational effiency is low. It is easy to construct higher-order continuous shape functions in meshless methods for strain gradient theories. However, for meshless methods based on Moving Least-square Method (MLS), there are many inverse matrixes are included, so that, the construction of shape functions is complicated and the computational cost is high. Moreover, the shape functions lack the Kronecker delta properties, so that it is not easy to deal with the essential boundary conditions (EBCs). In order to model the size effets of microstructures effectively, it is necessay to construct the meshless method possessing high accuracy, high effiency and interpolating property.In order to compare the discrepancies in computational accuracy, computational efficiency and interpolating property, C1 natural neighbor interpolant and shape function constructed by MLS are applied to surface fitting.C1 natural element method for the solution of couple-stress elasticity is proposed. The shape functions have the interpolation to nodal function and nodal gradient values, so that EBCs can be imposed directly. In order to examine the convergence and computational accuracy, simple shear problem of the block and the infinite plate with a central circular hole subjected to the unaxial tension, biaxial tension and pure shear are analyzed. For the simple shear problem, with the increase of nodes, the numerical solutions converge quickly to the exact solutions. For the irregular and regular nodes, there is little change in the solutions because Voronoi digram can adjust discrepancy caused by the nodes automatically. For the stress concentration problem of the hole, the effects of couple stresses on the distribution of stresses are related with the loads. In fact, the effect of couple stresses in the case of pure shear is greater than that in the case of simple tension; couple stresses have no effect at all on the stressses in a field of isotropic tension. Couple stresses influence on small area around the hole, when far from the hole; the solution in the couple-stress elasticity is close to that in the traditional theory.C1 natural element method for the solution of strain gradient elasticity is proposed. The shape functions have the interpolation to nodal function and nodal gradient values, so that it is easy to deal with EBCs. In order to examine the convergence and computational accuracy, boundary layer analysis of a bimaterial system and stress concentration due to a hole are analyzed. For the bimaterial system, with increase of nodes, the numerical solutions converge quickly to the exact solutions. For the non-uniform and uniform nodes, there is little change in the solutions because Voronoi digram can adjust discrepancy caused by the nodes automatically. For stress concentration due to a hole, the double stresses minish the stress concentration around the hole.The validity and accuracy of the proposed methods are investigated through the numerical examples, and the numerical solutions are in good agreement with the analytical ones, which show that C1 natural element method can be used to analyze the couple-stress elasticity and strain gradient elasticity problems.In the application of the proposed methods to MEMS, some pratical problems in engineering are analyzed. In these problems, typical components such as microgripper and microspeciem are taken as research objects, the impact of microspring's size on the bending stiffness, the impact of thickness of microbeam on voltage and the impact of the shape and size of hole on stress concentration factor are studied when considering the strain gradient effects. The dependent relations of these parameters on the characteristic length scales of materials are discussed. Numerical results can be used to provide some value evidence for design and experimental research.