The Smoothed Finite Element Method For Solving Elastic And Hyperelastic Contact Problems

This paper presents a cell-based smoothed finite element method(CS-FEM)for solving elastic and hyperelastic contact problems with the bi-potential formulation.Three contact states are investigated accurately,which can be stated as sticking,separating and sliding respectively.Using the Skyline method can effectively save calculation time and reduce computer memory consumption for the storage of equations.Combined with the strain smoothing technique,the weak form of smoothed Galerkin is derived for the elasticity problem.The smoothed deformation gradient tensor is derived for the hyperelastic problem.At the same time,the smoothed Green-Lagrange strain is used to describe the nonlinear relationship between strain and displacement,and the second smoothed Piola-Kirchhoff stress tensor is used to establish the strain energy density function of the hyperelastic material.The Total Lagrange method is used to discretize the hyperelastic constitutive laws of Blatz-Ko and Yeoh based on the incremental form.The nonlinear equations are solved by the Newton-Raphson method.For contact problems,the contact force and the relative displacements on the contact surface are coupled with each other.The classic Coulomb friction criterion is used to describe the tangential friction phenomenon of contact;the Signorini condition is used to describe the normal criterion of unilateral contact,which has the characteristics of geometric non-penetration,static non-adhesion and mechanical complementarity.The Uzawa algorithm,which is a local iterative technique,is used to solve the contact force.There is no need to select any user-defined parameter in the whole process.In this paper,six different types of smoothing domains are constructed,only boundary integrations instead of domain integral are required in the calculation,and no coordinate mapping is needed.Some numerical examples are presented to verify the effectiveness and reliability of the method.The effect of the friction coefficient for the contact is also investigated.Some conclusions are shown below:The strain energy solutions calculated by the CS-FEM converge to the reference solution with the increase of the number of degrees of freedom.The results produced by the CS-FEM are more accurate than those of the traditional FEM.All the obtained numerical solutions agree well with the reference values.For the different constraints and driving conditions of the problems,moreover,the CS-FEM can provide both an upper bound and a lower bound of the strain energy solutions while using different smoothing domains.For the hyperelastic problems,CS-FEM-1SD is not suitable for large deformation analysis,because of large displacement and large rotation,and the nature of CS-FEM-1SD is similar like to FEM-Q4-1GP,it is unstable.Regarding the convergence rate of the strain energy solution,the convergence rates of their strain energy solutions all show linear convergence for the elastic contact problems.However,the convergence rate of the strain energy solution shows nonlinear convergence for the hyperelastic contact problem,the convergence rate gradually decreases with the number of smoothing domain increases.At the same time,the convergence rate of the FEM is the smallest.The smoothing domain division of CS-FEM-4SD is regular,and the calculation of stress is uniform.Although it is not the most accurate solution,its stability is very well.When using the same number of smoothing domains,the accuracy of the strain energy solution of CS-FEM-4SD is more than 20%higher than that of the FEM.So,CS-FEM-4SD can be better applied to linear and nonlinear analysis.On the one hand,penetration is an important condition to measure the quality of a contact algorithm.For the bi-potential contact algorithm,the penetration can reach the order of10^-15,which is almost negligible.On the other hand,the accuracy of the algorithm can be highlighted by comparison with Hertz's analytical solution.