Study On Meshless Radial Basis Reproducing Kernel Particle Method For Functionally Graded Materials

Functionally graded materials(FGMs)are new type of intelligent composite materials,which has more excellent comprehensive mechanical properties than traditional homogeneous materials,and has a very broad application prospect in the field of science and engineering.Due to the non-uniformity of FGMs,it is difficult and limited to obtain the exact solution of the problem theoretically,especially for complex problems such as nonlinear large deformation.Consequently,in most cases,people look for the solution of the problem from the numerical aspect.The meshless method is a new numerical simulation method,which effectively avoids the dependence on meshes and elements,and shows unique advantages in the analysis of mechanical problems for FGMs.At present,the reproducing kernel particle method(RKPM)is one of the meshless methods,which is more mature and widely used.Nevertheless,the computational accuracy of traditional RKPM is easily affected by different kernel functions.Aiming at the existing problems of the RKPM,the meshless radial basis reproducing kernel particle method(RRKPM)is presented by combining the RKPM and the radial basis functions.Moreover,the meshless RRKPM is utilized to analyze the elasticity,geometrically nonlinear and elasto-plasticity problems of FGMs,and an efficient numerical simulation program is developed based on MATLAB.Finally,the obtained results are compared to the exact solutions or the reference solutions of the finite element method,the correctness and reliability of the meshless RRKPM for solving mechanical problems of FGMs are demonstrated.Combined with the numerical examples,the influence of different functional gradient functions and exponents on displacement and stress distribution is studied.The results show that with the increase of gradient index,the displacement of FGMs structure decreases and the variation range of stress value increases.Compared with the linear function,the deformation of FGMs structure is more obvious when the material parameters change exponentially.Different parts of FGMs have different mechanical properties,so FGMs can be selected according to the requirements of different working environment.Furthermore,the penalty factor,shaped parameters of radial basis function,control parameters of the radius of influencing domain,node distribution and the number of loading steps are discussed in detail,and the optimal parameters for solving mechanical problems of FGMs are determined.The proposed method is easy to solve the elasticity,geometrically nonlinear and elasto-plasticity problems of FGMs.In particular,the FGMs parameters can be changed continuously according to the function.In other words,there are different FGMs parameters at different nodes.Compared with the traditional RKPM,the proposed method can reduce the negative influence of choosing different kernel functions on the calculating accuracy,and show higher computational accuracy and numerical stability for solving mechanical problems of FGMs.Additionally,the proposed method does not require meshes,and avoid mesh reconstruction when dealing with large deformation problems such as geometrically nonlinear of FGMs.This work provides a new method for the calculation and analysis of FGMs.