Cylindrical void in a rigid-ideally plastic single crystal II: Experiments and simulations

Experimental results and finite element simulations of plastic deformation around a cylindrical void in single crystals are presented to compare with the analytical solutions in a companion paper: Cylindrical void in a rigid-ideally plastic single crystal I: Anisotropic slip line theory solution for face-centered cubic crystals [Kysar, J.W., Gan, Y.X., Mendez-Arzuza, G., 2005. Cylindrical void in a rigid-ideally plastic single crystal I: Anisotropic slip line theory solution for face-centered cubic crystals, International Journal of Plasticity, 21, 1481–1520]. In the first part of the present paper, the theoretical predictions of the stress and deformation field around a cylindrical void in face-centered cubic (FCC) single crystals are briefly reviewed. Secondly, electron backscatter diffraction results are presented to show the lattice rotation discontinuities at boundaries between regions of single slip around the void as predicted in the companion paper. In the third part of the paper, the finite element method has been employed to simulate the anisotropic plastic deformation behavior of FCC single crystals which contain cylindrical voids under plane strain condition. The results of the simulation are in good agreement with the prediction by the anisotropic slip line theory.     

A coupled kinetic-constitutive approach to the study of high temperature crack initiation in single crystal nickel-base superalloys

A rate-dependent crystallographic constitutive theory coupled with a mass diffusion model has been used to study crack initiation in single crystal nickel-base superalloys, exposed to an oxidising environment and subjected to mechanical loading. The time to crack initiation under constant load has been predicted using a strain-based failure criterion. A notched compact tension (CT) specimen containing a single casting defect, idealised as a cylindrical void close to the notch surface, has been studied. Finite element analysis of the CT specimen revealed that, due to the strong localisation of inelastic strain at the void, a microcrack will initiate in the vicinity of the void rather than at the notch surface. The numerical results have also shown that the time to crack initiation depends strongly on the void location. The coupled diffusion-deformation studies have revealed that environmental effects reduce the time to crack initiation due to the oxidation-induced material softening in the vicinity of the notch and void. The applicability of a failure assessment approach, based on the linear elastic stress intensity factor, K, to predict the crack initiation time under creep loading is examined and a probabilistic framework for prediction of component lifetime is proposed.     

Creep flow, diffusion, and electromigration in small scale interconnects

This paper proposes a three-dimensional electromigration model for void evolution in small scale interconnects. Concurrent kinetics of creep flow and surface diffusion as well as the effect of the surrounding material is considered to provide better understanding of the evolution process. The multiple kinetics and energetics are incorporated into a diffusive interface model. A semi-implicit Fourier spectral method and the preconditioned biconjugate-gradient method are proposed for the computations to achieve high efficiency and numerical stability. We systematically studied kinetic processes in diffusion dominated to creep dominated regime. Which process dominates, as revealed by the analysis, is determined by a combination of viscosity, mobility, interconnect thickness, and void radius. Previous studies on electromigration suggest that a circular void subjected to an electron wind force and surface diffusion is always stable against any small shape perturbation. Our simulations show that a shape that is stable in surface diffusion can become unstable in a creep dominated process, which leads to a quite different void morphology. A spherical void can evolve into a bowl shape and further break into smaller voids. It is also shown that the interconnect geometry has an important effect.     

金属互连结构的电迁移失效分析新算法

综合考虑了电子风、温度梯度、应力梯度和原子密度梯度四种电迁移驱动机制, 基于ANSYS软件平台和FORTRAN程序提出一种新的电迁移失效分析算法．通过ANSYS电－热－结构耦合分析获得模型的电流密度分布、温度分布和应力分布, 基于FORTRAN编写的原子密度重分布算法获得不同时刻的原子密度, 依据空洞生成和扩展失效准则进行电迁移动态空洞演化模拟并得到失效寿命．最后, SWEAT结构与CSP结构的应用算例验证了算法的精度．     

多物理场下FCBGA焊点电迁移失效预测的数值模拟研究

随着微电子封装技术的快速发展, 焊点的电迁移失效问题日益受到关注. 基于有限元法并结合子模型技术对倒装芯片球栅阵列封装(flip chip ball grid array, FCBGA)进行电-热-结构多物理场耦合分析, 详细介绍了封装模型的简化处理方法, 重点分析了易失效关键焊点的电流密度分布、温度分布和应力分布, 发现电子流入口处易产生电流拥挤效应, 而整个焊点的温度梯度较小. 基于综合考虑“电子风力”、温度梯度、应力梯度和原子密度梯度四种电迁移驱动机制的原子密度积分法, 并结合空洞形成/扩散准则及失效判据, 分析FCBGA焊点在不同网格密度下的电迁移空洞演化过程, 发现原子密度积分算法稳定, 不依赖网格密度. 采用原子密度积分法模拟真实 工况下FCBGA关键焊点电迁移空洞形成位置和失效寿命, 重点研究了焊点材料和铜金属层结构对电迁移失效的影响. 结果表明, 电迁移失效寿命随激活能的增加呈指数级增加, 因此Sn3.5Ag焊点的电迁移失效寿命约为63Sn37Pb的2.5倍, 有效电荷数对电迁移寿命也有一定的影响;铜金属层结构的调整会改变电流的流向和焊点的应力分布, 进而影响焊点的电迁移失效寿命.      

A phase field model for failure in interconnect lines due to coupled diffusion mechanisms

We describe a diffuse interface, or phase field model for simulating electromigration and stress-induced void evolution and growth in interconnect lines. Microstructural evolution is tracked by defining an order parameter, which takes on distinct uniform values within solid material and voids, and varying rapidly from one to the other over narrow interfacial layers associated with the void surfaces. The order parameter is governed by a form of the Cahn-Hilliard equation. An asymptotic analysis demonstrates that the zero contour of order parameter tracks the motion of a void evolving by coupled surface and lattice diffusion, driven by stress, electron wind and vacancy concentration gradients. Efficient finite element schemes are described to solve the modified Cahn-Hilliard equation, as well as the equations associated with the accompanying mechanical, electrical and bulk diffusion problems. The accuracy and convergence of the numerical scheme is investigated by comparing results to known analytical solutions. The method is applied to solve various problems involving void growth and evolution in representative interconnect geometries.     

Numerical modeling and experimental investigation of gas–liquid slug formation in a microchannel T-junction

Gas–liquid two-phase flow in a microfluidic T-junction with nearly square microchannels of 113 μm hydraulic diameter was investigated experimentally and numerically. Air and water superficial velocities were 0.018–0.791 m/s and 0.042–0.757 m/s, respectively. Three-dimensional modeling was performed with computational fluid dynamics (CFD) software FLUENT and the volume of fluid (VOF) model. Slug flow (snapping/breaking/jetting) and stratified flow were observed experimentally. Numerically predicted void fraction followed a linear relationship with the homogeneous void fraction, while experimental values depended on the superficial velocity ratio U_G/U_L. Higher experimental velocity slip caused by gas inlet pressure build-up and oscillation caused deviation from numerical predictions. Velocity slip was found to depend on the cross-sectional area coverage of the gas slug, the formation of a liquid film and the presence of liquid at the channel corners. Numerical modeling was found to require improvement to treat the contact angle and contact line slip, and could benefit from the use of a dynamic boundary condition to simulate the compressible gas phase inlet reservoir.     

Measurement and modeling of two-phase flow parameters in scaled 8 × 8 BWR rod bundle

The behavior of reactor systems is predicted using advanced computational codes in order to determine the safety characteristics of the system during various accidents and to determine the performance characteristics of the reactor. These codes generally utilize the two-fluid model for predictions of two-phase flows, as this model is the most accurate and detailed model which is currently practical for predicting large-scale systems. One of the weaknesses of this approach however is the need to develop constitutive models for various quantities. Of specific interest are the models used in the prediction of void fraction and pressure drop across the rod bundle due to their importance in new Natural Circulation Boiling Water Reactor (NCBWR) designs, where these quantities determine the coolant flow rate through the core. To verify the performance of these models and expand the existing experimental database, data has been collected in an 8 × 8 rod bundle which is carefully scaled from actual BWR geometry and includes grid spacers to maintain rod spacing. While these spacer grids are ’generic’, their inclusion does provide valuable data for analysis of the effect of grid spacers on the flow. In addition to pressure drop measurements the area-averaged void fraction has been measured by impedance void meters and local conductivity probes have been used to measure the local void fraction and interfacial area concentration in the bundle subchannels. Experimental conditions covered a wide range of flow rates and void fractions up to 80%.     

电致失效力学

电致失效力学研究单调或交变电场载荷下由应力引起的失效行为,它包含了电致断裂、电致疲劳、电致迁移与电致损伤等新研究课题．本文概述了电致失效力学的领域与课题,并深入讨论了电致应变诱导断裂疲劳的机理及电迁移损伤的力电耦合过程．研究结果表明,电致失效力学可提供铁电陶瓷致动器和集成电路的若干关键设计参数．对铁电陶瓷多层共烧致动器,该分析提供其层厚、外加电场强度和交变电场循环周数．对集成电路内导线,该分析提供其允许电流密度和临界线长．      

Coupling effects of void size and void shape on the growth of prolate ellipsoidal microvoid

The combined effects of void size and void shape on the void growth are studied by using the classical spectrum method. An infinite solid containing an isolated prolate spheroidal void is considered to depict the void shape effect and the Fleck-Hutchinson phenomenological strain gradient plasticity theory is employed to capture the size effects. It is found that the combined effects of void size and void shape are mainly controlled by the remote stress triaxiality. Based on this, a new size-dependent void growth model similar to the Rice-Tracey model is proposed and an important conclusion about the size-dependent void growth is drawn: the growth rate of the void with radius smaller than a critical radius r_c may be ignored. It is interesting that r_c is a material constant independent of the initial void shape and the remote stress triaxiality.The project supported by the National Natural Science Foundation of China (A10102006) and the New Century Excellent Talents in Universities of China. The English text was polished by Keren Wang.     

Prediction of void fraction for gas–liquid flow in horizontal,upward and downward inclined pipes using artificial neural network

In this work, the ability of artificial neural networks (ANNs) to predict void fraction of gas–liquid two–phase flow in horizontal and inclined pipes was investigated. For this purpose, an ANN model was designed and trained using a total of 301 experimental data points reported in the literature for inclination angles between –20° and +20°. Pipe inclination angle as well as superficial Reynolds number of gas (Re_sg) and liquid (Re_sl) were chosen as input parameters of different structures of multilayer perceptron (MLP) neural networks, while the corresponding void fraction was selected as their output parameter. A hyperbolic tangent sigmoid and a linear function were employed as transfer functions of hidden and output layers, respectively, and Levenberg–Marquardt back propagation algorithm was used to train the networks. By trial–and–error method, a three–layer network with 10 neurons in the hidden layer was achieved as optimal structure of the ANN which made it possible to predict the void fraction with a high accuracy. Mean absolute percent error (MAPE) of 1.81% and coefficient of determination (R²) of 0.9976 for training data and MAPE of 1.52% and R² value of 0.9948 for testing data were obtained. Also for all data, MAPE of 1.95% and R² value of 0.9972 were calculated, and 96% data were within ±5% error band. In addition, the accuracy of the proposed ANN model was compared with the predictions from 17 void fraction correlations available in the literature for different flow patterns and horizontal and inclined flows. For all cases, the proposed ANN model gave better performance than all of the studied correlations. The results confirm the very good capability of the ANNs to predict the void fractions of gas–liquid flow in inclined pipes, regardless of flow pattern. Finally, by performing interpolation using the trained network, the void fraction values for some other conditions were predicted.     

The size effect on void growth in ductile materials

We have extended the Rice-Tracey model (J. Mech. Phys. Solids 17 (1969) 201) of void growth to account for the void size effect based on the Taylor dislocation model, and have found that small voids tend to grow slower than large voids. For a perfectly plastic solid, the void size effect comes into play through the ratio εl/R₀, where l is the intrinsic material length on the order of microns, ε the remote effective strain, and R₀ the void size. For micron-sized voids and small remote effective strain such that εl/R₀?0.02, the void size influences the void growth rate only at high stress triaxialities. However, for sub-micron-sized voids and relatively large effective strain such that εl/R₀>0.2, the void size has a significant effect on the void growth rate at all levels of stress triaxiality. We have also obtained the asymptotic solutions of void growth rate at high stress triaxialities accounting for the void size effect. For εl/R₀>0.2, the void growth rate scales with the square of mean stress, rather than the exponential function in the Rice-Tracey model (1969). The void size effect in a power-law hardening solid has also been studied.     

Theoretical and experimental studies on the contact line motion of second-order fluid

In this study, we studied the contact line motion of second-order fluids theoretically and experimentally. The theoretical study showed that the positive first normal stress difference (N ₁) increases the contact line velocity while the second normal stress difference (N ₂) does not affect the contact line motion. The increased contact line velocity is caused by the hoop stress acting on the curved stream lines near the contact line. The hoop stress increases the liquid pressure near the contact line, and the increased pressure changes the surface profile to have the smaller curvature and smaller dynamic contact angle. The contribution of N ₁is 1 order of magnitude smaller than the contribution from the viscous component when the Deborah number remains O(1). For experiments, silicone oils of different kinematic viscosities (1,000–200,000 mm ²/s) were used while eliminating the drying problem and shear-thinning effect near the contact line. The silicone oils were well fitted to the second-order fluid model with the positive first normal stress difference. The spreading rate of a silicone oil drop on a solid surface was faster than the spreading rate predicted by the theory for Newtonian fluids. As the theory predicts that N ₁increases the contact line velocity and the experimental result confirms the theoretical prediction, the effect of N ₁is established.     

The use of Hugoniot analysis for the propagation of vapor explosion waves

Experimental studies have shown that a mixture of molten metal and water can support the propagation of a quasi-steady vapor explosion wave. Analysis of steadily propagating vapor explosion waves has been carried out by applying the one-dimensional conservation laws of mass, momentum and energy and appropriate equations of state to a homogeneous mixture of molten tin, water and steam. The effects of void fraction, melt/water volume fraction and melt temperature on the Hugoniot curves have been considered. For low temperature melts, the Hugoniot curve lies partially inside the saturation dome and a Chapman-Jouguet (CJ) detonation point occurs only for low void fractions. For high melt temperatures, the downstream states lie entirely outside the saturation region. Increasing the volatility of the coolant or the addition of chemical reactions increases the predicted CJ detonation pressure and velocity. CJ deflagration solutions were obtained in all cases. The existence of a CJ detonation or CJ deflagration for a multiphase fuel-coolant mixture has yet to be substantiated experimentally and nonequilibrium effects may play a role in the divergence between theory and experiments.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

Constitutive potential for void-containing nonlinear materials and void growth

Based on an analysis of the deformation of an isolated void in a finite nonlinear viscous material, we establish the constitutive potentials for voided nonlinearly viscous materials, from which the related curves of the macroscopic stress, the average flow stress of the matrix material and the void volume fraction f are derived. However, the theory applies equally well to small strain, rate-independent J₂ deformation theory solid. By considering the effects of the strain-hardening directly, a modifies Gurson equation are developed. Finally, we calculate the void relative growth-rates for the nonlinear materials, and in good agreement with existed numerical results.     

Modeling of ductile fracture: Significance of void coalescence

In this paper void coalescence is regarded as the result of localization of plastic flow between enlarged voids. We obtain the failure criterion for a representative material volume (RMV) in terms of the macroscopic equivalent strain (E_c) as a function of the stress triaxiality parameter (T) and the Lode angle (θ) by conducting systematic finite element analyses of the void-containing RMV subjected to different macroscopic stress states. A series of parameter studies are conducted to examine the effects of the initial shape and volume fraction of the primary void and nucleation, growth, and coalescence of secondary voids on the predicted failure surface E_c(T, θ). As an application, a numerical approach is proposed to predict ductile crack growth in thin panels of a 2024-T3 aluminum alloy, where a porous plasticity model is used to describe the void growth process and the expression for E_c is calibrated using experimental data. The calibrated computational model is applied to predict crack extension in fracture specimens having various initial crack configurations and the numerical predictions agree very well with experimental measurements.     

On the use of area-averaged void fraction and local bubble chord length entropies as two-phase flow regime indicators

In this work, the use of the area-averaged void fraction and bubble chord length entropies is introduced as flow regime indicators in two-phase flow systems. The entropy provides quantitative information about the disorder in the area-averaged void fraction or bubble chord length distributions. The CPDF (cumulative probability distribution function) of void fractions and bubble chord lengths obtained by means of impedance meters and conductivity probes are used to calculate both entropies. Entropy values for 242 flow conditions in upward two-phase flows in 25.4 and 50.8-mm pipes have been calculated. The measured conditions cover ranges from 0.13 to 5 m/s in the superficial liquid velocity j _f and ranges from 0.01 to 25 m/s in the superficial gas velocity j _g. The physical meaning of both entropies has been interpreted using the visual flow regime map information. The area-averaged void fraction and bubble chord length entropies capability as flow regime indicators have been checked with other statistical parameters and also with different input signals durations. The area-averaged void fraction and the bubble chord length entropies provide better or at least similar results than those obtained with other indicators that include more than one parameter. The entropy is capable to reduce the relevant information of the flow regimes in only one significant and useful parameter. In addition, the entropy computation time is shorter than the majority of the other indicators. The use of one parameter as input also represents faster predictions.     

Electromigration induced strain field simulations for nanoelectronics lead-free solder joints

Electromigration is a major road block on the way to realization of nanoelectronics. Determination of plastic deformation under high current density is critical for prediction of electromigration failure. A new displacement–diffusion coupled model is proposed and implemented using finite element method. The model takes into account viscoplastic behavior of solder alloys, as a result, vacancy concentration evolution and electromigration process are accurately simulated. Finite element simulations were performed for lead-free solder joints under high current density and compared with experimental moiré interferometry measurements. The comparison validates the model.     

Hydrodynamics of two-phase flow in vertical up and down-flow across a horizontal tube bundle

Flow patterns, void fraction and friction pressure drop measurements were made for an adiabatic, vertical up-and-down, two-phase flow of air–water mixtures across a horizontal in-line, 5×20 tube bundle with a pitch-to-diameter ratio of 1.28. The flow patterns in the cross-flow zones were obtained and flow pattern maps were constructed. The data of average void fraction were less than the values predicted by a homogenous flow model and showed a strong mass velocity effect, but were well-correlated in terms of the Martinelli parameter X_tt and liquid-only Froude number Fr_LO. The two-phase friction multiplier data could be well-correlated with the Martinelli parameter.     

Interface balance laws,phase growth and nucleation conditions for multiphase solids with inhomogeneous surface stress

In this paper, the thermodynamic configurational force associated with a moving interface is used to derive the conditions for phase growth and nucleation in bodies with multiple diffusing species and arbitrary surface stress at the phase interface. First, the mass, momentum and energy balances are derived on the evolving phase interface. The thermodynamic conditions that result from free energy inequality at the interface are derived leading to the analytical form of the configurational force for bodies subject to mechanical loads, heat and multiple diffusing species. The derived second law condition naturally extends the Eshelby energy–momentum tensor to include species diffusion terms. The above second law restriction is then used to derive the condition for the growth of new phases in a body undergoing finite deformation subject to inhomogeneous as well as anisotropic interface stress, and multiple diffusing species. The growth conditions are derived in both current and reference configurations. The statistical temperature-dependent growth velocity is next derived using the Boltzmann distribution. The derived finite deformation form of growth requirement is simplified to obtain the small deformation diffusive void growth condition. Next, a general, finite deformation, arbitrary surface stress form of phase nucleation condition is derived by considering uncertainty in growth of a small nucleus. The probability of nucleation is shown to naturally depend on a theoretical estimate of critical volumetric energy density, which is directly related to the surface stress. The classical nucleation theory is shown to result from a simplified special case of the general criterion. As an application of the developed theory, the classical Blech electromigration experiment is simulated to estimate the critical energy density corresponding to the onset of electromigration voids at Al–TiN interface.