Smooth Finite Element For Sloving 3D Nonlinear Solid Mechanic Problem

With the development of science and technology and the need of engineering for production and construction,more and more nonlinear mechanical problems have emerged in practical projects,especially in the areas of infrastructure construction and aerospace.Many structures exhibit large displacements and large deformations under large loads and high pressures,which present a series of nonlinear phenomena.On the other hand,with the needs of social development and the development of materials disciplines,more and more new materials are being used in medical,aerospace,and mechanical fields,and the mechanical properties of those material may present physical nonlinearities.If theoretical design and simulation are performed according to the linear theory,great difficulties will be encountered.In fact,these problems can only be reasonably analyzed using the viewpoints and methods of nonlinear continuum mechanics in order to be effectively solved.For nonlinear solid mechanics problems,only a few of them can obtain analytical solutions.Using numerical methods to solve their numerical solutions has become an important approach to solve engineering problems.In this paper,the author uses the high-efficiency numerical method Smoothed Finite Element Methods(S-FEM)proposed in recent years to solve the three-dimensional nonlinear solid mechanics problem,and establishes a general solver to solve the three-dimensional nonlinear mechanical problem by S-FEM.The smoothed finite element combines the robustness of the finite element and the efficiency of the meshless method.In addition,many excellent properties have been obtained in linear elastic solid mechanics problems.For example,strain energy upper and lower bounds can be obtained under certain norms.In this paper,the S-FEM is used to solve the three-dimensional nonlinear solid problem,and it is verified whether the S-FEM can obtain the upper and lower bounds of the strain energy in nonlinear problems.In order to ensure the validity of the conclusion,the article not only studied the large geometric deformation of the structure under large loads,but also used typical nonlinear superelastic materials,and used two hyperelastic models—Mooney-Rivlin and Ogden models.Mooney-Rivlin's constitutive equation is expressed as strain invariants,and Ogden's constitutive equation is expressed as the principal stretches.Through a large number of numerical examples,the authors found that: Node-based and face-based S-FEM using automatically generable four-noded tetrahedral elements(NS-FEM-Te4 and FS-FEMTe4)are adopted to find the solution bounds in strain energy.The lower bound solutions are obtained using FEM-Te4 and FS-FEM-Te4,while the upper bound solutions are obtained using NS-FEM-Te4.A combined ?S-FEM-Te4 with a scaling factor ? that controls the combination is constructed to find nearly exact solutions for the nonlinear solids mechanics problems through adjusting ?.This is achieved using the property that a successive change of scaling factor ? can make the model transform from "overly-stiff" to "overly-soft".Considering the properties of FS-FEM and NS-FEM,a selective FS/NS-FEM-TE4 also is used to solve 3D nonlinear large deformation problems,which produces a lower bound in strain energy.