Meshless Method Study Of Functionally Graded Beam

The functionally graded material (FGM) is a special kind of nonhomogeneous material. It refers to the material constituting the elements which has a continuous gradient change along one side to the other, so that the nature and function of the material also showed a new material gradient. FGM has caused great concern to the majority of scholars in the manufacturing, design and applications with a very broad application prospects in the aerospace, nuclear energy, optics, biology, and civil engineering and many other fields. The research of the mechanical behavior to functionally graded materials has both rich non.uniform material mechanics and also the need for application in practical engineering.In recent years, functionally graded materials applications continue to expand, but the non.uniformity of the functionally graded materials and some special application to the FGM structure of theory analysis increases the difficulty. Meshless method is a class of numerical methods which was been developed in recent20years, As the Meshless method is fully based on point information, no grid, avoiding the generate complex process of grid. Meshless method is very suitable for the analysis of crack propagation, hypervelocity impact and penetration, and other issues, and has broad application of prospects.In many kinds of meshless method, element.free Galerkin method draw more attentions for its stability, high accuracy, insensitivity to distribution of discrete points, and most suitable for structural analysis has been paid more attention. This paper attempts elementfree Galerkin method for handling mechanical problems in functionally graded materials, the work and achievements of this paper are as follows(1) Based on elasticity Control equation using the variational principle, The EFGM balance equation and mechanical boundary conditions equivalent integral weak form a more comprehensive derivation.(2) Had a depth research on the element.free Galerkin method accuracy with the cantilever basis example. Calculat the value of displacement by junction points and the type of arrangement, the comparison with the analytical solution, the obtained distribution influence on the calculation accuracy. Analyzed weight function radius of influence in a number of different nodes and obtained element.free Galerkin method convergence of Lagrange multiplier approaches to satisfy the boundary conditions. In the calculation of the actual problem, the penalty function method and the Lagrange multiplier approaches to satisfy the boundary conditions, obtained calculation results with the theoretical value, summed up the characteristics of the two methods of calculation.(3) Derived the analytical solution of orthotropic functionally graded cantilever which changes the thickness direction of any functional form under tensile loads, cited two examples to verify the analytical solution in line with the classical elasticity solution, suitable for functionally graded cantilever which changes the thickness direction of arbitrary gradient functional form under any tension load.(4) Use of general.purpose finite element software sub.layer method to simulate the deformation of functionally graded cantilever under tension, can be used as the reference of similar analysis.(5) Meshless method used to study the discontinuous mechanical response of functionally graded materials in the elastic modulus varies linearly and index changes, obtained different material elastic modulus ratio of the displacement curve and compared with the theoretical solution and finite element calculations, and analyzed error, the result is in line with the actual.(6) Finally, the problem which extended to the thickness of the material parameters along the beam in any form analytical solutions can not be calculated, the use of meshless method analysis of the different components of the mechanical response of functionally graded materials, to provide a basis for future study complex models, can be used as an analytical engineering method.In this article, The purpose is to make element.free Galerkin method for the calculation of the functionally graded materials, to provide a new method for the calculation of functionally graded materials, to provide a basis for future study complex models.