Existence of Weak Solutions for the Cahn-Hilliard Reaction Model Including Elastic Effects and Damage

摘要

In this paper,we introduce and study analytically a vectorial Cahn-Hilliard reaction model coupled with rate-dependent damage processes.The recently proposed Cahn-Hilliard reaction model can e.g.be used to describe the behavior of electrodes of lithium-ion batteries as it includes both the intercalation reactions at the surfaces and the separation into different phases.The coupling with the damage process allows considering simultaneously the evolution of a damage field,a second important physical effect occurring during the charging or discharging of batteries.Mathematically,this is realized by a Cahn-Larch system with a non-linear Newton boundary condition for the chemical potential and a doubly non-linear differential inclusion for the damage evolution.We show that this system possesses an underlying generalized gradient structure which incorporates the non-linear Newton boundary condition.Using this gradient structure and techniques from the field of convex analysis we are able to prove constructively the existence of weak solutions.