Finite Element Method Of Elastic-Plastic Frictional Contact Analysis For Cosserat Materials

Microstructures of materials (including material heterogeneity, microscopic geometry configuration) play an important role in determining their macroscopic mechanical behaviors. It has been found that classical continuum theory is deficient both on physical and numerical aspects for some special materials (e.g., foams, truss structures, granular structures and trabecular bone, etc.) and physical phenomenon (such as stress concentration and strain localization, etc.). The reasons are the lack of an intrinsic length and the local change of the character of the governing equations, which result in that the corresponding numerical finite element solutions show pathological mesh dependence, and cause some incorrect numerical results. In order to study effects of microstructures of materials, researchers have proposed a series of generalized continuum constitutive model, such as Cosserat model. The main characteristic of Cosserat model is that kinematics of Cosserat model includes independent rotation degrees freedom, and internal length scale is introduced in the constitutive formulation.From a mechanics point of view, many cellular materials can be regarded as a structure of interconnected beams. In these materials, bending is often a prominent deformation mechanism, so at the micro-scale both displacements and rotations are present. This makes Cosserat theory a suitable candidate for continuum modeling of cellular materials, because it contains rotations as degrees of freedom and new material parameters. In this paper, based on the parametric variational principle, a quadratic programming method is presented for3D modeling of strain localization phenomena via Cosserat continuum model; a quadratic programming method is developed to solve the elastic-plastic contact problem of Cosserat materials, and the mechanical properties of contact surface for Cosserat materials was demonstrated qualitatively. In addition, a new multiscale finite element method is developed for mechanical analysis of periodic heterogeneous Cosserat materials, which avoid the estimation of the macroscopical equivalent material parameters for heterogeneous Cosserat materials and guarantee high accuracy of the numerical results.Firstly, based on the parametric variational principle, a quadratic programming method is developed for elastic-plastic finite element analysis of3D Cosserat continuum model. Since the classical continuum model which is lack of internal scale parameter suffers from pathological mesh dependence in the strain localization analysis, the governing equations of Cosserat continuum model are regularized by adding rotational degrees-of-freedom and internal scale parameters to the conventional continuum model. Numerical examples are calculated to demonstrate the efficiency and stability of the proposed computational algorithm for numerical simulation of strain localization problems. Particularly, the mesh independent results are ensured.Secondly, based on the parametric virtual work principle, a quadratic programming method is developed to solve the elastic-plastic contact problems of Cosserat materials. Both the elastic-plastic problem and the contact problem with friction between two Cosserat materials are studied by using the parametric variational principle, which can greatly simplify the solution process of the original nonlinear problem. The penalty factors, which are introduced into the algorithm for contact problem, are eliminated by using a definite numerical technique, resulting in the high accuracy of the numerical results. Numerical examples are performed to demonstrate the validity and feasibility of the proposed algorithm. It can be found that the contact forces are strongly dependent on the material parameters of Cosserat materials.Finally, a new multiscale finite element method is developed for mechanical analysis of periodic heterogeneous Cosserat materials. The main idea of the method is to numerically construct the multiscale base functions to capture the small-scale features of the coarse elements. Considering the existence of rotation in the Cosserat materials, specified boundary conditions of the base functions for extended multiscale finite element method (EMsFEM) are developed based on the relationship between transverse displacement and rotation (slope) of the two-node beam element, and the corresponding periodic boundary conditions are developed. By adopting both two kinds of boundary conditions, the numerical base functions for displacement and rotation fields of Cosserat materials are constructed respectively to establish the relationship between the macroscopic deformation and the microscopic stress and strain. It can be observed that the proposed method does not require the estimation of the macroscopical equivalent material parameters of the heterogeneous Cosserat materials like the general homogenization methods, and the downscaling computation could be realized easily.