Viscoplastic Constitutive Behavior of Metallic Materials under Thermo-mechanical Loading

Author:He Xu

Supervisor:yao yao


Degree Year:2018





Finite element method has been widely applied in the designing and developing of new products,a properly defined numercial model can predict the weak point and analyze the damage failure mechanisms during deformation.To descibe the mechanical behavior of material,constitutive model is essential to ensure the accuracy of numerical analysis.A constitutive model can describe the stress strain relationship of the material under different loading conditions.Due to the differences in the microstructure,even the same material obtained by different processes will have different macroscopic mechanical properties.For most of the solid materials under high temperature(relative to the melting point),the evolution of stress is related not only to the loading condition but also the loading rate.Higher loading rate usually leads to greater saturation stress.The saturation stress decreases with the increasing of temperature,which is the softening phenomenon caused by heat.The influence of material,microstructure,temperature and loading rate on the mechanical properties has brought great challenges to the development of constitutive model.Developing an accurate constitutive model under coupled thermal-mechancial loading has always been an improtant issue in the research of solid mechanics.The research object of the thesis is to describe constitutive behavior of metals and alloys under thermo-mechanical conditions.The solder materials used in microelectronic packaging is investigated.Firstly,a macroscopic constitutive model for solder material has been developed in the present work.Generally,the macroscopic constitutive model can be divided into two classes,namely the macroscopic phenomenological constitutive model and the one based on specific physical mechanism.A phenomenological model is usually based on the experimental results.It has the advantages of relatively simple form and less parameters.However,it is not the only task for a constitutive model to characterize the stress strain relationship of materials,it also describes the physical mechanisms during the deformation process.At the same time,the scientificity and robustness of the developed model can be enhanced based on the physical mechanism.The irrecoverable deformation of metal materials is mainly caused by dislocation motion.Therefore,a constitutive model based on dislocation theory has been developed,the viscoplastic behavior of materials is investigated by the phenomenological model based on dislocation density theory.Based on the phenomenological constitutive model,an unified creep and plasticity model is developed to describe the mechanical behavior of solder alloys.The model is coupled with a amage model to predict the fatigue life of solder.By combining the Chaboche isotropic hardening and Prager kinematic hardening model,the modified unified creep and plasticity model is applied to simulate the stress strain behavior of lead-rich solder alloys(Pb-3.5Sn)and lead-free solder alloys(Sn-3.5Ag and Sn-3.9Ag-0.6Cu)under specific loading conditions.The developed model can predict the mechanical behavior of the solder alloys with reasonable accuracy.A new damage evolution equation is obtained based on the continuum damage mechanics.The effect of temperature is considered,the influence of temperature on damage nucleation and evolution is studied.The entire process of fatigue is simulated by the new developed damage unnified creep and plasticity model.Comparision between simulation result and the experimental data shows that the developed model can simulate the damage evolution of solder alloys,describe the characteristics and trends of damage accumulation,and predict the fatigue life of solder accurately.The traditional unnified creep and plasticity developed by Mc Dowell only contains one rate sensitivity parameter,it cannot descibe the temperature dependent behavior of rate sensitivity.The relationship between rate sensitivity and temperature is studied in the thesis.By analyzing the experimental stress-strain curve of solder alloys under different temperatures and loading rates,it is found that the influence of temperature on the evolution of rate sensitivity can be divided into three stages.At the first stage,with the increasing of temperature,the rate sensitivity increases.At the second stage,the rate sensitivity remains unchanged.After the temperature continues to rise into the third stage,rate sensitivity decreases with increasing temperature.For comparison,the effect of temperature on rate sensitivity of high strength alloy steels is also studied and the similar three stages are observed.At the first and second stage,the behavior is the same as that of solder alloys.While at the third stage,the trend of rate sensitivity evolution is opposite to that of solder alloys,which shows the rate sensitivity keeps increasing with elevated temperatures.Based on the experimental observations,a new unified creep plastic constitutive model is developed,which incorporates the evolution of rate sensitivity and temperature.The mechanical behavior of Sn-3.0Ag-0.5Cu and high strength alloy steels is investigated,the results shows that the proposed model can describe the viscoplastic deformation of the above materials with reseaonable accuracy.The phenomenological constitutive model can describe the viscoplastic denformation well,but it does not have a solid physical beackground,and the parameters adopted in the model are lack of clear physical meanings.In order to ensure the developed constitutive model can not only describe the experimental phenonmen,but also characterize the physical mechanisms during eformation,a 3D constitutive model has been developed based on the unified creep plasticity model combining the J2 flow rule.By considering the influence of strain hardening and dynamic recovery on the dislocation density of material,the traditional dislocation based constitutive model is expanded to multiaxial conditions and the dislocation density is regarded as an internal variable.The correpsonding implicit algorithm is provided.The model is applied to describe the stress-strain behavior of Sn-3.0Ag-0.5Cu at high strain rate,which shows good accuracy compared with experimental results.Then the model is applied to simulate the dynamic impact of 3D electronic package.The vulnerable points of the structure are sucessfully found based on the numercial analysis.By considering the influence of static recovery and stress state on dislocation recovery,the developed 3d dislocation density based constitutive model is modified to simulate the cyclic behavior.In addition to the dynamic recovery,the static recovery mechanism can reduce the dislocation density in the deformation process of metalic materials.Besides,the change of the stress state,especially the reverse change of stress,will promote the dislocation recovery mechanism.The effect of different recovery mechanisms on the dislocation density is considered in four stages: the initial loading without yielding,yielding,unloading,and reverse loading without yielding stage,respectively.The dislocation density is dynamically evolved with loading conditions,accordingly the cyclic behavior can be well described.To show the accuracy of the developed model,the experimental data of P91 steel is compared with the nuermcial analysis.A series of tension-compression cyclic experiments for Sn-3.0Ag-0.5Cu alloy at 20℃ and 50℃ were conducted and simulated by the proposed model.The strain rates of the experiments were set as 0.02/s,0.002/s and 0.0002/s.The results show that the proposed model can describe the cyclic behavior of P91 steel with reseanable accuracy.