A Thermo-chemo-mechanically Coupled Continuum Theory Based On Non-equilibrium Thermodynamics And Its Applications

Multi-field coupling problems concerning mass transport,heat exchange,substance changes and mechanical deformation widely exist in such cases as curing of polymers,swelling of hydrogels,oxidation of metals,lithiation or de-lithiation of Li ion batteries and growth of biological tissues.Thermo-chemo-mechanically coupled dynamical process are the common features of these problems and a theoretical study on them is of great significance.The first and second laws of thermodynamics play an important role in studying the constitutive behaviors of a system.In literatures,there exist many versions of thermodynamic laws for an open system who exchanges mass with its surroundings,bringing great difficulties to choose one reasonable version for the thermodynamic analysis of an open system.To analyze their reasonability and validity,we develop a theoretical continuum framework for the thermodynamics of multi-component systems based on Biot's view that a deformable solid is chosen as the diffusing media,which distinguishes from traditional mixture theory considering constituents to be open and the entire mixture as a closed system.In doing so,the difference between common versions of thermodynamic laws can be accounted for.Based on the proposed continuum thermodynamics laws for an open system,a theoretical model for thermo-chemo-mechanically coupled problems considering plastic flow at large deformation is proposed,which can be applied to predict mechanical behavior of materials under thermal and chemical environments.Different from other models in literatures,the present model derives the driving forces in the case of large deformation for reaction and diffusion: the chemical affinity and the chemical potential,by employing the extent of reaction and the diffusion concentration as two kinds of independent variables so that diffusion and reaction can be distinguished.Then,a modified chemical kinetics is developed to satisfy the dissipation inequality,which depends not only on species concentrations,but also on deformation.This modified chemical kinetics is constructed from the most popular chemical kinetics in chemistry,initially expressed as a power function of the concentrations of reactants and products,by incorporating the Eshelby stress into the chemical affinity to reflect the influence of deformation on the chemical kinetics.A case study of metal oxidation is provided to demonstrate the present model.Next,the fully coupling model for large deformation cases is also applied to simulate the lithiation process of silicon electrode in Li ion batteries.Silicon is extensively used as an electrode in Li ion batteries for its superiority in the charging capacity.However,the lithiation of silicon would induce large deformation and significant stress jump across a two-phase interface in silicon electrodes and might eventually lead to structure failure of batteries.To improve the performance and life of Li ion batteries,it is of great importance to exactly model the lithiation process.In the past decade,although many models have been developed for the lithiation,most of them just relate the deformation with the diffusion,without considering the effect of electrochemical reactions,so that they cannot simulate the formation of the two-phase interface.In this paper,we aim to employ the thermo-chemo-mechanically coupled model to simulate the lithiation process,where a reaction barrier effect is proposed to describe the formation of the two-phase interface due to fast reaction and the mechanical behaviors of Si electrodes such as large plastic flow during lithiation process are predicted.Furthermore,we propose a thermodynamically consistent continuum model for chemo-active soft materials,by coupling large deformation,chemical reaction,species transport and heat conduction.Although many efforts have been devoted to studying large deformation coupled with diffusion in soft materials,a full thermo-chemo-mechanically coupled model for soft materials is still necessary especially when diffusion and reaction are coexistent.In this paper,to consider the coupling between diffusion and reaction,we introduce two kinds of independent state variables into the Helmholtz free energy function,i.e.,the diffusion concentration of a diffusive species absorbed by the host solid and the extent of a reaction.We also establish the evolution rule of the reaction extent based on conventional reaction kinetics and thermodynamics,different from the traditional linear phenomenological description.To illustrate this proposed model,a gel with moisture absorption and hydrolysis reaction is studied by using a hyperelastic constitutive model and its chemo-mechanical responses are predicted at the transient and steady states.Finally,a diffusion-reaction-deformation coupled model is employed and implemented as a user-defined element(UEL)subroutine in the commercial finite element software package ABAQUS.Chemical reaction and diffusion are treated as two distinct processes by introducing the extent of reaction and the diffusion concentration as two kinds of independent variables,for which the independent governing equations for these two processes are obtained.An exponential form of chemical kinetics,instead of the linearly phenomenological relation,between the reaction rate and the chemical affinity is used to describe reaction process.As a result,complex chemical reaction can be simulated,no matter it is around or away from equilibrium.Two numerical examples are presented,one for validation of the model and another for the modelling of the deflection of a plane caused by a chemical reaction.