Fatigue crack propagation life analysis of solder joints under thermal cyclic loading

Fatigue crack propagation life analysis of solder joints under thermal cyclic loadings was investigated by the strain energy release rate method using finite element analysis. A relationship between the crack-growth rate and the strain energy release rate was derived. Finite element simulations were carried out to investigate the effect of crack growth along the interface of solder and lead in a solder joint assembly. The crack propagation life of the solder joint with an interface crack was predicted from the derived relationship between crack growth rate and values of the strain energy release rate. It was found that crack propagation life is much higher than the crack initiation life.     

A thermodynamics based damage mechanics constitutive model for low cycle fatigue analysis of microelectronics solder joints incorporating size effects

Below certain length scales and in the presence of a non-uniform plastic strain field the mechanical behavior of many metals and its alloys is substantially different from that in bulk specimens. In particular, an increase in resistance with decreasing size has been observed in Pb/Sn eutectic solder alloys which are extensively used in microelectronics packaging interconnects. Due to the high homologous temperature, the Pb/Sn solder exhibits creep–fatigue interaction and significant time, temperature, stress and rate dependent material characteristics. The simultaneous consideration of all the above mentioned factors makes constitutive modeling an extremely difficult task. In this paper, a viscoplastic constitutive model unified with a thermodynamics based damage evolution model is embedded into a couple stress framework in order to simulate low cycle fatigue response coupled to size effects. The model is implemented into commercial finite element code ABAQUS. The microbending experiment on thin nickel foils is used to validate the model. Analyses are performed on a thin layer solder joint in bending under cyclic loading conditions.     

96.5Sn-3.5Ag钎料合金的时相关单轴应变循环变形行为及其本构关系研究

在室温下对96.5Sn-3.5Ag钎料合金进行了不同加载波形下的单轴应变循环实验。研究了在具有不同保持时间、不同应变率、不同应变幅值及其历史对材料的循环变形行为的影响。基于材料时相关变形行为,提出了统一粘塑性本构模型,并对该材料的变形行为进行本构模拟。实验结果表明:该钎料合金单轴变形行为具有应变率、保持时间以及应变幅值依赖性。本构关系的预言结果与实验结果吻合得一致性说明该种模型能够很好地描述材料的单轴应变循环变形行为。     

Cyclic viscoelastoplasticity and low-cycle fatigue of polymer composites

Observations are reported on a polymer composite (polyamide-6 reinforced with short glass fibers) in tensile relaxation tests with various strains, tensile creep tests with various stresses, and cyclic tests with a stress-controlled program (ratcheting with a fixed maximum stress and various minimum stresses). Constitutive equations are developed in cyclic viscoelastoplasticity of polymer composites. Adjustable parameters in the stress–strain relations are found by fitting observations in relaxation tests and cyclic tests (16 cycles of loading–unloading). It is demonstrated that the model correctly predicts experimental data in creep tests and dependencies of maximum and minimum strains per cycle on number of cycles up to fatigue fracture of specimens. The influence of strain rate and minimum stress on number of cycles to failure is studied numerically.     

Constitutive and damage model for 63Sn37Pb solder under uniaxial and torsional cyclic loading

A constitutive model with Ohno–Wang kinematic hardening rule is developed and employed to simulate the isothermal cyclic behavior of Sn–Pb solder under uniaxial and torsional loading. An implicit constitutive integration scheme is presented for inelastic flow of solder. Then a modified low cycle fatigue life prediction model is put forward in which the sum of maximum shear strain range and normal strain range based on the critical plane concept is adopted to replace the uniaxial strain range used by Stolkarts et al. [Stolkarts, V., Keer, L.M., Fine, M.E., 1999. Damage evolution governed by microcrack nucleation with application to the fatigue of 63Sn–37Pb solder. J. Mech. Phys. Solids 47, 2451–2468]. Comparison of the experimental results and simulation verifies that the stress strain hysteresis loops and peak stress decline curve of solder can be reasonably modeled over a wide range of loading conditions with implement of damage coupled constitutive model, and the lifetime estimations of 63Sn37Pb solder based on the assumption of microcrack nucleation governed damage is effective to provide a conservative prediction.     

Time-dependent thermoelastic creep analysis of rotating disk made of Al–SiC composite

Time-dependent creep stress redistribution analysis of rotating disk made of Al–SiC composite is investigated using Mendelson’s method of successive elastic solution. All mechanical and thermal properties except Poisson’s ratio are radial dependent based on volume fraction percent of SiC reinforcement. The material creep behavior is described by Sherby’s constitutive model using Pandey’s experimental results on Al–SiC composite. Loading is an inertia body force due to rotation and a distributed temperature field due to steady-state heat conduction from inner to outer surface of the disk. Using equations of equilibrium, stress strain, and strain displacement, a differential equation, containing creep strains, for displacement is obtained. History of stresses and deformations are calculated using method of successive elastic solution. It is concluded that the uniform distribution of SiC reinforcement does not considerably influence on stresses. However, the minimum and most uniform distribution of circumferential and effective thermoelastic stresses belongs to composite disk of aluminum with 0% SiC at inner surface and 40% SiC at outer surface. It has also been found that the stresses, displacement, and creep strains are changing with time at a decreasing rate so that after almost 50 years the solution approaches the steady-state condition.     

Mathematical and numerical modelling of large creep deformations for annular rotating disks

A simulation model is presented for the creep process of the rotating disks under the radial pressure in the presence of body forces. The finite strain theory is applied. The material is described by the Norton-Bailey law generalized for true stresses and logarithmic strains. A mathematical model is formulated in the form of a set of four partial differential equations with respect to the radial coordinate and time. Necessary initial and boundary conditions are also given. To make the model complete, a numerical procedure is proposed. The given example shows the effectiveness of this procedure. The results show that the classical finite element method cannot be used here because both the geometry and the loading (body forces) change with the time in the creep process, and the finite elements need to be redefined at each time step.     

Micromechanical model of the high temperature cyclic behavior of 9–12%Cr martensitic steels

9–12%Cr quenched and tempered martensitic steels are known to soften under cyclic loadings at high temperature. The present article proposes a model based on physical mechanisms described at the scale of slip systems. This model describes explicitly the microstructural recovery (corresponding to a decrease of the dislocation density and subgrain coarsening) observed experimentally. The scale transition is carried out in the framework of self-consistent homogenization schemes. The model assumptions and its physical basis are explicitly discussed. The parameters are identified on a very limited amount of experimental data. The model turns out to give very good predictions and extrapolations for the cyclic softening effect observed in uniaxial tension–compression loadings for strain ranges larger than 0.3%. Stress–relaxation and creep behavior can also be simulated for high stresses. In addition the cyclic softening effect is reproduced for multiaxial tension–torsion loadings.     

An application of a damage constitutive model to concrete at high temperature and prediction of spalling

A characteristic feature of concrete under uniaxial compression is the development of cracks parallel to the loading direction. A damage constitutive model proposed by Ortiz [Ortiz, M., 1985. A constitutive theory for the inelastic behaviour of concrete. Mech. Mater. 4, 67–93] can predict the transverse tensile stress responsible for these cracks by considering the interaction between the aggregate and the mortar and the development of damage in the latter. When concrete is exposed to high temperature, as is the case during fire, the failure mode is thermal spalling. In order to improve the prediction of the stresses involved in this failure Ortiz’s model is extended to account for the effects of high temperature. Published experimental results for uniaxial and biaxial compression at high temperatures are used to calibrate the temperature dependence of some of the material properties. The transient creep strain is accounted for by modifying the constrained thermal strain. The stress analysis is coupled with hygro-thermal analysis of heat, mass transfer and pore pressure build-up. The effect of pore pressure on the damage evolution is modeled by applying a body force in the stress analysis module proportional to the pressure gradient. A numerical example of concrete under fire is solved and the computed results are discussed. Spalling is predicted when the damage variable reaches its maximum value of unity. The predicted depth and time of spalling for a range of variation of permeability and initial liquid water content are presented. They are in good agreement with published experimental results.     

Integrated numerical–experimental analysis of interfacial fatigue fracture in SnAgCu solder joints

In ball grid array (BGA) packages, solder balls are exposed to cyclic thermo-mechanical strains arising from the thermal mismatch between package components. Thermo-mechanical fatigue crack propagation in solder balls is almost always observed at the chip side of the bump/pad junction. The objective of the experimental part of this study is to characterize the bump/pad interface under fatigue loading. Fatigue specimens are prepared by reflowing Sn3.8Ag0.5Cu lead-free solder alloy on Ni/Au substrates. Obtained results show that fatigue damage evolution strongly depends on the microstructure. Applied strain and solder volume both have an influence on the fatigue damage mechanism. In the numerical part of the study, fatigue experiments are modeled using the finite element technique. A cohesive zone approach is used to predict the fatigue damage evolution in soldered connections. Crack propagation is simulated by an irreversible linear traction–separation cohesive zone law accompanied by a non-linear damage parameter. Cohesive zone elements are placed where failure is experimentally observed. Damage evolution parameters for normal and tangential interaction are scrutinized through dedicated fatigue tests in pure tensile and shear directions. The proposed cohesive zone model is quantitatively capable of describing fatigue failure in soldered joints, which can be further extended to a numerical life-time prediction tool in microelectronic packages.     

Fully plastic crack problems: The center-cracked strip under plane strain

Crack problems are formulated for solids characterized by a pure power hardening relation between the stresses and the strains. For such problems there are simple functional relationships between the amplitude of the dominant crack-tip singularity, as measured by the path-independent J-integral, and the applied load, the load point displacement, and the crack opening displacement. The solutions are valid for both incremental and deformation theories of plasticity; they also apply to problems involving steady-state creep. Numerical results are presented for the center-cracked strip of finite width under plane strain conditions. A preliminary discussion is given of the applicability of the solutions to large scale yielding fracture mechanics.     

温度循环应力剖面对QFP焊点热疲劳寿命的影响

采用统一型粘塑性Anand本构方程描述了QFP(四方扁平封装)焊点的粘塑性力学行为,利用有限元分析软件建立组装在印制电路板上QFP的有限元模型,通过研究焊点内部总应变范围的变化进而研究温度循环应力剖面各个参数对焊点热疲劳寿命的影响,为设计合理的温度循环应力剖面提供了理论依据。     

96.5Sn-3.5Ag 钎料合金的含非比例度的多轴时相关本构模型研究

基于96.5Sn-3.5Ag钎料合金的多轴时相关变形行为,提出了一个考虑其多轴变形特性的本构模型.在该模型背应力演化方程中,引入了非比例度对背应力演化率的影响,并提出了模型参数的确定方法.在室温下对96.5Sn-3.5Ag钎料合金进行了十字形、双三角形及椭圆形等拉扭组合非比例循环应变路径下的变形行为的本构模拟,并将预言结果与实验结果进行了比较.预言结果表明,该模型对于96.5Sn-3.5Ag钎料合金的多轴应变循环变形行为具有很好的预言能力.     

低成本基板倒装焊底充胶分层裂缝扩展研究

采用MIL-STD-883C热循环疲劳加载标准，通过电学检测方法测定了B型和D型两种倒装焊封装焊点寿命。并使用无损声学C-SAM高频超声显微镜技术观测这两种倒装焊封装在焊点有无断裂两种情况时芯片／底充胶界面的分层和扩展，计算得到分层裂缝扩展速率。在有限元模拟中采用粘塑性和时间相关模量描述了SnPb焊点和底充胶的力学行为。使用裂缝尖端附近小矩形路径J积分方法作为断裂力学参量得到不同情况下的界面分层裂缝顶端附近的能量释放率。然后由实验裂缝扩展速率和有限元模拟给出的能量释放率得到可作为倒装焊封装可靠性设计依据的Par—is半经验方程。     

A constitutive model in finite viscoelasticity

A new constitutive model is suggested for the viscoelastic behavior of rubber-like materials at finite strains. The model treats a viscoelastic medium as a system with a variable number of purely elastic links, which can arise and collapse due to micro-Brownian motion of molecules.Assuming that the processes of birth and death for elastic links are independent of stresses, we obtain operator linear constitutive equations in finite viscoelasticity. According to this model, elastic and viscous effects may be distinguished and described independently of each other by a relaxation measure and a strain energy density.The potential energy of deformations is assumed to depend on the principal invariants of the relative Finger tensor of strains. Unlike the standard approach, we do not suggest any expression for the strain energy densitya priori, but suppose that this function is presented as a sum of two functions of one variable which are found by fitting experimental data.The proposed approach allows results of several experiments (uniaxial tension, biaxial tension, and torsion) for styrene butadiene rubber and butyl rubber to be predicted correctly.     

无铅焊料合金压入蠕变性能参数表征研究

微互连焊点高温下的服役可靠性很大程度上决定了电子产品性能的优劣，本文通过微纳米压入法开展了典型无铅焊料合金Sn-3.0Ag-0.5Cu 的高温蠕变性能研究，采用“快速加载–保载–快速卸载”的测试方式获取不同温度下完备的蠕变行为，分析发现温度升高焊料软化加剧、蠕变变形更为明显；得到了焊料合金不同温度下的稳态蠕变应力指数n 与不同硬度下的蠕变激活能Q，分析了温度与硬度对二者的影响并给出激活能Q 与lnH之间的关系．     

A unified constitutive model for superalloys

A physically based unified constitutive model is presented for an aircraft engine nickelbase superalloy. The model accounts for deformation modes that can be activated under different stress, time, and temperature combinations. Two internal state variables and a flow function have been utilized to prdict strain rate sensitivity, stress hold creep, strain hold relaxation, monotonic loading, cyclic loading, and thermal mechanical cycling. In the model flow function, creep deformation and plasticity deformation modes have been incorporated over a wide range of temperatures (0.4 < T/T_melt < 0.75). The model is checked with independent isothermal and thermal mechanical experiments. Different temperature ranges are explored to assess model capabilities.     

A non-linear uniaxial integral constitutive equation incorporating rate effects,creep and relaxation

A previously proposed first order non-linear differential equation for uniaxial viscoplasticity, which is non-linear in stress and strain but linear in stress and strain rates, is transformed into an equivalent integral equation. The proposed equation employs total strain only and is symmetric with respect to the origin and applies for tension and compression. The limiting behavior for large strains and large times for monotonic, creep and relaxation loading is investigated and appropriate limits are obtained. When the equation is specialized to an overstress model it is qualitatively shown to reproduce key features of viscoplastic behavior. These include: initial linear elastic or linear viscoelastic response: immediate elastic slope for a large instantaneous change in strain rate normal strain rate sensitivity and non-linear spacing of the stress-strain curves obtained at various strain rates; and primary and secondary creep and relaxation such that the creep (relaxation) curves do not cross. Isochronous creep curves are also considered. Other specializations yield wavy stress-strain curves and inverse strain rate sensitivity. For cyclic loading the model must be modified to account for history dependence in the sense of plasticity.     

The creep deformation of vibrating structures

In a recent paper [1] it was shown that the evaluation of certain bounding solutions for a structure subjected to cyclic loading was equivalent to assuming that the cycle time Δt was short compared with a stress redistribution time. Comparisons between values which are likely to occur in creep design situations indicated that Δt may often be assumed to be small and the bounding solution may be expected to closely approximate the actual stress history. In this paper the solution for the limiting case when Δt → 0 is evaluated for a class of constitutive relationships which may be expressed in terms of a finite number of state variables. Strain-hardening viscous, visco-elastic and Bailey-Orowan equations are discussed and particular solutions for which the residual stresses remain constant in time are derived. The solution for a non-linear visco-elastic model indicates that, for the stationary cyclic state, the constitutive equation need only predict the creep strain over a discrete number of cycles and need not predict the strains during a cycle. This observation should considerably simplify creep analysis.The solution of a simple example demonstrates the similarity between the predicting of the various constitutive relationships for isothermal problems. In fact they provide virtually identical solutions when expressed in terms of reference stress histories. The finite element solution of a plate containing a hole and subjected to variable edge loading is also presented for a viscous material. The solutions show behaviour which is similar to that of the two bar structure.