Two-scale Asymptotic Properties Analysis For One Class Of Chemo-mechanical Coupling Problem

The local structures and combinations of different materials are different,and the mechanism of composite materials has some coupling problem of span between multiple scales.In the process of chemical-force coupling,the materials of different scale models can be given different meanings.Due to the complexity of materials of local distribution,the analysis of the effective properties for these materials is an important research direction.In mathematics,the chemical-mechanical coupling problem can be characterized by some differential equations.In this paper,we study the two-scale asymptotic behavior of solutions for a class of coupled chemical-mechanical problem.By constructing proper cell functions,the two-scale asymptotic expansion of the static problem for chemo-mechanical coupling problem in periodic domain is constructed,the homogenization constant and homogenization solution is obtained.Also,using the general theory of two-scales method,the error estimates for the formal asymptotic solution are analyzed.The sections of this research are organized as follows:In the first section,the history research for the coupling problem of chemo-mechanical coupling problems in composite materials is introduced.The research review and progress at home and abroad are introduced,some fundamental knowledge,some known research results and methods to solve the chemo-mechanical coupling problem are presented.In section 2,the static problem of chemo-mechanical coupling problem in periodic domain with Dirichlet boundary condition is discussed.Firstly,the existence and uniqueness theorem of weak solution for problem are discussed.Then,according to the general framework of two-scale method,the two-scale asymptotic expansion of the solution for the problem is constructed.Lastly,the homogenization behavior of the solution is discussed and the asymptotic error of the two-scale expansion is estimated.In section 3,based on the proposed method in section 2,the two-scale asymptotic expansion for the solution of the model problem for the static problem of chemo-mechanical coupling problem in the periodic domain with mixed boundary conditions is obtained,and the asymptotic error of the two-scale expansion is analyzed.In section 4,the main results and method in this thesis are summarized,and someon-going researches in future are introduced.