The Mesh-free Method For The Higher-order Continuum Structures

This paper focuses on the application of mesh-free method in the numerical simulation of higher-order continuous structure by taking traditional bending beams, higher-order continuous beam and the layered rock mass as example, and considering the characteristics of higher-order continuous, which can be satisfied by moving least-square approximations. This paper is organized as follows.Firstly, traditional finite element method to construct the shape function with high-order continuous structure features takes heavy workload. Therefore, this paper proposed to study the high-order derivative of the node’s displacement was used to approximate by the displacement based on the characteristics of the shape function of mesh-free method And the possibility of setting up the frame of mesh-free method of numerical calculation, and the research of the corner of boundary conditions are applied and the like related problems.Secondly, traditional curvature in bending beam was the second order derivative of displacement, and can be simply seen as a one-dimensional high-order continuous issue. This paper started form the continuous shape functions and its first-order derivative and high-order derivative were build by the moving least squares approximation in this paper. The corner and the curvature can be approximated with the node deflection interpolation., then construct the framework based on mesh-free method. The efficiency and precision of this method can be tested using example.Thirdly, the special material of bending beam can be processed for high order continuous beam, but it requires the higher-order strain to be the basic variables. The high-order variable can also be directly used the node deflection to interpolation by establish a shape functions and its third-order derivative with the characteristics of high-order continuous. This paper further studied the application of the moving least-square approximation in the numerical simulation of the higher-order continuum problems, and also analyzes the size factor of the influencing rule.Finally,this paper introduced the Cosserat theory to build the constitutive relation of layered rock mass by considering the bending effect of thin layer. The strain and the curvature can be approximated with the node deflection interpolation. Based on these methods, we establish the frame of mesh-free method of numerical calculation. The mesh-free method is verified by concrete example in the application of two-dimensional higher-order continuum, and in the meanwhile studies the deformation effect of rock mass based on the Cosserat theory of the mesh-free method.The constitutive reflection of the special material is quite complicated; therefore, it is valuable to deal with it as high-order continuous structure. The study is expected to do much more contribution on the special material beam, the theoretical analysis and calculation of layered rock mass.