Complex Variable Element-free Galerkin Method For Large Deformation Analysis Of Hyperelastic Materials Under Non-conservative Loads

Meshless method based on the information of the nodes,avoiding the difficulities which are broughted by creating the meshes.Therefore,the meshless method is flexible and adaptive when it comes to mesh deformity,mesh movement,etc.It shows obvious advantage as a powerful numerical analysis tool.To date,element-free Galerkin method(EFG)is the most widely used method of the meshless methods.By introducing the complex variable moving least-square(CVMLS)approximation into the EFG method,the complex variable element-free Galerkin(CVEFG)method is developed to overcome the problem of excessive computation.Compared with the moving least-squares(MLS)approximation,the dimension of the base function in the CVMLS approximation is reduced.So that the coefficients in the trial function are reduced,and the number of nodes required in the problem domain is reduced accordingly.Thus the computational efficiency is improved.The external loads often change with the deformation of the stress surface in the process of large deformation of engineering structures and materials.In this case,the load is a non-conservative force which depends on the state of deformation,and the numerical process can be relatively complex.Compared with the large deformation analysis of elastic materials,the superelastic materials can produce larger deformation when these materials are pressed.Besides,volume locking and pressure oscillation are easy to occur when numerical method is used to solve the problem because the near incompressibility of hyperelastic materials,which troublesome in analysis.In conclusion,it is necessary to discuss the large deformation of hyperelastic materials under non-conservative loads,which is prone to mesh distortion when using finite element method(FEM).Unlike the FEM,meshless method has an enormous advantage in dealing with such problems because it does not need to divide grids.The CVEFG method for large deformation analysis of elastic and hyperelastic meterials under non-conservative loads is proposed in this paper.The penalty method is used to apply the essential boundary conditions.The Galerkin integral weak form which is of incremental form and total Lagrange formulation for large deformation problems is obtained.The mixed variable method is used to solve the problem of incompressibility of hyperelastic materials.The CVMLS method is used to establish the approximation function of the displacement field,and the corresponding tangent modulus of hyperelasticity,strain-displacement-transformation matrix and the stiffness matrix are derived.The discrete equation of large deformation analysis by meshless method is established.The Newton-Raphson method is used for iterative solution.The algorithm steps for solving large deformation problems under non-conservative loads are established.The calculation program of MATLAB is compiled.A series of numerical examples are calculated by the corresponding program,such as classical cantilever beam example,honeycomb structure and pure bending beam example,etc.We come to the following meaningful conclusions.First,compared with the results obtained by EFG method,the CVEFG method is more efficient.Second,great deformation can be obtained without large error due to mesh distortion when CVEFG method is used to analyze large rotation problems.Third,the three-dimensional hyperelastic material is simulated and analyzed in this paper and the stress-strain relationship of hyperelastic material under basic load is analyzed.Fourth,the feasibility of solving negative Poisson's ratio structure by CVEFG method is analyzed,which lays a foundation for studying the physical and mechanical properties of negative Poisson's ratio structure.In this paper,the feasibility of CVEFG method for calculating large deformation of elastic and hyperelastic materials under non-conservative loads is discussed.By comparing the numerical results with the analytical solutions,the validity of the CVEFG method for large deformation problems under non-conservative loads is verified.