| Constitutive model is applied to describe the relationship between stress and strain,it is essential for the modeling of materials and structures.The majority of macroscopic phenomenological constitutive models are based on the analysis of experimental results and the framework of thermodynamics,some parameters are lack of physical meanings.For crystalline materials,plasticity origins from crystal slip and twining,macroscopic phenomenological constitutive models cannot reflect the physical mechanism during deformation.Therefore,crystal plasticity model has been widely applied to the modeling of crystalline materials.Polycrystalline materials are composed of grains with different orientations microscopically,the grains behave differently even under the same kind of loading.The traditional phenomenological constitutive models can hardly simulate the inhomogeneous deformation due to different orientations,they cannot relate the microscopical deformation of every single grain and the macroscopic deformation of overall materials.Mean-field and full-field model are made up for these deficiencies.These two kinds of models are both investigated in the present thesis for polycrystalline materials under thermo-mechanical loading.With the size decreasing of solders,they may only contain one or few grains,the anisotropy of the deformation cannot be ignored.Thus,in the present study,multi-scale models are proposed for polycrystalline materials.The crystal plasticity model is combined with the micromechanical based methods to describe the constitutive behavior more physically and accurately.In this work,a multi-scale constitutive model is proposed for Sn-based solder alloys.The mechanical behavior of every single crystal is described by crystal plasticity model,and every single crystal is assumed to be an ellipsoidal inclusion embedded in a uniform infinite matrix,namely homogeneous equivalent medium（HEM）.By self-consistent method,the mechanical behavior of HEM can be obtained,which is regarded as the same as the polycrystalline material.The elastic-plastic self-consistent model was developed for elastic-plastic deformed materials.The model is modified to make it suitable for visco-plastic deformed conditions.In addition,the hardening rate decreases with the increasing of strain,the Voce hardening model is improved to characterize this phenomenon.To verify the developed model,the calculated Taylor factor is compared with the theoretical results.On the other hand,the stress-strain curves of Sn-3.0Ag-0.5Cu and Sn-0.7Cu alloy at different temperatures and strain rates are simulated and compared with the experimental results.The model is also applied for cyclic deformed Sn-0.7Cu alloy by introducing the Armstrong-Frederick kinematic hardening model.The results show that the proposed model is effective for Sn-based solder alloys both under uniaxial tension and cyclic loading conditions.The above Voce hardening model is a phenomenological one in essence,which is developed and improved based on the experimental observation,the parameters are lack of physical meanings.In addition,intermetallic compound（IMC）,Sn-Cu and Sn-Ag compounds exist in the polycrystallineb-Sn matrix of Sn-Ag-Cu eutectic solder alloys.With different content of IMC,Sn-3.0Ag-0.5Cu and Sn-1.0Ag-0.5Cu alloy usually behave differently even under the same kind of load.Thus,the influence of IMC cannot be neglected in the modeling.Microscopically,the existence of IMC impedes the movement of dislocations.Macroscopically,it changes the overall stiffness of materials.A multi-scale model is proposed in the current study to capture these two characteristics.At grain scale,a dislocation density based hardening law considering non-shearable precipitates is adopted locally for singleb-Sn crystal;the mechanical behavior of polycrystallineb-Sn matrix is obtained by self-consistent method;the Mori-Tanaka scheme is applied to describe the overall mechanical behavior of solder alloys.By comparing with the experimental results,it can be concluded that the proposed model can predict Young’s modulus of Sn-1.0Ag-0.5Cu alloy based on the Young’s modulus of Sn-3.0Ag-0.5Cu with reasonable accuracy.The uniaxial tension stress-strain curve of Sn-1.0Ag-0.5Cu and Sn-3.0Ag-0.5Cu alloys at different temperatures and strain rates can be predicted accurately by the proposed model.Solders in electronic packaging usually work under complex loading.Therefore,research on the influence of temperature and strain rate is essential for the prediction of solders cyclic behavior under different working conditions.In the present work,the strain controlled tension-compression cyclic experiments are performed at 15℃,50℃,100℃,150℃,200℃,and three strain rates of 10-4s-1,10-3s-1 and 10-2s-1,respectively.The temperature and strain rate dependence of apparent Young’s modulus,maximum true stress are analyzed based on the experimental results.On the other hand,the above dislocation density based model is not applicable for cyclic loading,to take the dislocation storage and annihilation mechanisms during strain path changes into consideration,the total dislocation density is assumed to be the sum of the forward dislocation density and reversible dislocation density.Based on Eshelby’s inclusion analysis,a dislocation density related slip system level intra-granular backstress model is developed to simulate the Bauschinger effect.The proposed intra-granular backstress model is compared with the Armstrong-Frederick backstress model to show the similarities and differences between them.The proposed model is applicable for Sn-3.0Ag-0.5Cu alloy and modified 9Cr-1Mo steel under cyclic loading.Predicting of fatigue crack nucleation is important in preventing catastrophic failure of structures.A method is proposed by combining the crystal plasticity finite element（CPFE）model and theoretical analysis to predict the fatigue crack nucleation of polycrystalline materials.The models presented above can be regarded as the mean-field model.In this thesis,the full-field model is also investigated.According to the topological structure of polycrystalline materials,a 2D finite element model contains 150 different orientated grains is generated by 2D Voronoi tessellation.The overall stress strain can be obtained by the micromechanical homogenization method.It is assumed that the softening of material before fatigue crack nucleation is caused by the moving and gathering of dislocation dipoles,therefore,a dislocation density based damage model is developed and coupled with the crystal plasticity based model to describe the softening behavior of Sn-3.0Ag-0.5Cu alloy and P91 steel under cyclic loading.In addition,the phase transformation based fatigue crack nucleation prediction model is improved to adapt the constitutive model better.Some parameters in the prediction model is calibrated by the simulation results of CPFE model.By comparing with the experimental results,it can be concluded that the proposed method can predict the fatigue crack nucleation cycle with reasonable accuracy. |
