A Damage-Mechanics-Based Constitutive Model ... - Semantic Scholar
C. Basaran1 Professor, and Director Electronic Packaging Laboratory, University at Buffalo, Buffalo, NY e-mail: [email protected]
Y. Zhao Senior Engineer Analog Devices, Boston, MA
H. Tang Senior Engineer NEC Honda Electronics, Detroit, MI
J. Gomez Graduate Student University at Buffalo, Buffalo, NY
A Damage-Mechanics-Based Constitutive Model for Solder Joints Sn-Pb eutectic solder alloy is extensively used in microelectronics packaging interconnects. Due to the high homologous temperature, eutectic Sn-Pb solder exhibits creepfatigue interaction and significant time-, temperature-, stress-, and rate-dependent material characteristics. The microstructure is often unstable, having significant effects on the flow behavior of solder joints at high homologous temperatures. Such complex behavior makes constitutive modeling an extremely difficult task. A viscoplasticity model unified with a thermodynamics-based damage concept is presented. The proposed model takes into account isotropic and kinematic hardening, and grain size coarsening evolution. The model is verified against various test data, and shows strong application potential for modeling thermal viscoplastic behavior and fatigue life of solder joints in microelectronics packaging. 关DOI: 10.1115/1.1939822兴
Introduction As the complexity of electronic packages grows, high reliability of assembled components is critical to maintaining final product quality, especially in light of trends toward miniaturization and higher-level integration 关1兴. It is well known that the coefficient of thermal expansion mismatch between the different materials in the package is the major cause of fatigue failure. Thermal cycling causes progressive damage in the package, especially in the Sn-Pb solder joints used to attach the surface-mounted devices to the printed-circuit boards or a chip to a substrate. Eventually, this damage accumulation beyond a critical point leads to the electrical failure of the assembly. In order to predict the fatigue life of the solder joints, a unified constitutive model of the solder material is essential. Implementing this constitutive model in a generalpurpose finite element code enables an engineer to simulate the thermomechanical response of a new package design on a computer screen. Computer simulations during early design stages significantly lower development cost and reduce design-to-market time cycles. Eutectic solder alloys are routinely used at high homologous temperatures. The melting point of the eutectic Sn-Pb solder alloy is 183° C, and it is at 0.65 Tm at room temperature, where Tm is the melting point. Therefore, solder joints exhibit time-, temperature-, and stress-dependent deformation behavior. Reversed plasticity, creep, and creep-plasticity interactions dominate the behavior of solder joints during thermomechanical cycling. The microstructure also has significant effects on the flow behavior of the eutectic solder alloy, as shown by Kashyap and Murty 关2兴, Morris and Reynolds 关3兴, and Basaran and Chandaroy 关4兴. Material models ranging from purely elastic to elastoplastic using various stress-strain relations have been proposed for Sn-Pb solder material, such as in Kitano et al. 关5兴, Lau and Rice 关6兴, Basaran et al. 关7兴, and many others. Adams 关8兴 proposed a simple viscoplastic model without hardening. Wilcox et al. 关9兴 proposed a rheological model to represent the inelastic behavior of the material; however, it is applicable to a limited range of strain rates. The purely phenomenological models proposed by Knecht and Fox 关10兴, Darveaux and Banerji 关11兴, and Hong and Burrell 关12兴 decouple the creep and plasticity effects artificially. This decoupling 1 Corresponding author. Contributed by the Electronic and Photonic Packaging Division for publication in the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received: June 28, 2004; final manuscript received: July 23, 2004. Review conducted by: Bahgat Sammakia.
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does not have any physical basis and is more a mathematical convenience. One of the drawbacks of the Sn-Pb solder material constitutive models available in the literature is the large number of material parameters required. A large number of material parameters are difficult to determine in practice rendering the model impractical for industrial use. An extensive literature survey on Sn-Pb constitutive models was presented in Basaran et al. 关7兴. On the other hand, classical forms of decoupled plasticity and creep theories have been shown to be quite inferior for modeling cyclic plasticity, creep, and interaction effects 关13兴. The tests on stainless steels showed that the combination of plasticity and creep equations yielded unsatisfactory modeling of certain deformation features, including cyclic creep and creep-plasticity interactions. The development of inelastic constitutive equations has shown that, at the present time, the plasticity-creep approaches are gradually being replaced by “unified” methods, as pointed out by Krempl 关14兴. A number of researchers recently attempted to develop unified creep-plasticity models in order to model the solder material behavior more precisely. Some are Busso et al. 关15兴, Mcdowell et al. 关13兴, Tachibana and Krempl 关16兴, Skipor et al. 关17兴, Wei et al. 关18兴, Basaran and Chandaroy 关4兴, Basaran and Tang 关26,27兴, and others. In this paper, a constitutive model for the eutectic Sn-Pb solder is formulated within a unified framework, where the plastic rateindependent and rate-dependent creep strains are assumed coupled to yield total strain. Both isotropic and kinematic hardenings are incorporated to model cyclic behavior. The influence of microstructure is represented in the constitutive equations by incorporating the grain size in the viscoplastic flow rule. A thermodynamics-based damage evolution equation is incorporated to account for the material damage accumulation and fatigue life predictions. Damage is introduced by means of the train equivalence principle. Furthermore, damage is described in terms of entropy, based on the second law of thermodynamics. Uniaxial tension and shear cycling loading conditions are simulated. The model is validated against uniaxial experimental data available from the literature, and against mechanical shear cycling experiments performed at the University at Buffalo Electronic Packaging Laboratory. The material properties for the constitutive model were taken out of experimental data available in the literature, and from our own testing.
Yield Surface In order to separate elastic behavior from inelastic deformation region, a von Mises type yield surface is used
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F = ¯eff − 共1 − D兲共R + k0兲 艋 0
共1兲
Table 1 Strain dependent material properties at various temperatures
where k0 is the initial size of the yield surface, R is the evolution of the size of the yield surface, D is the internal state variable of damage, and the ¯eff denotes the reduced effective stress or the von Mises distance in the deviatoric stress space, which is given by ¯eff = 冑3/2共S − ␣⬘兲:共S − ␣⬘兲
共2兲
where S is the deviatoric stress tensor and ␣⬘ is the deviator of back stress tensor ␣.
Nonlinear-Kinematic Hardening (NLK) Rule The NLK rule is taken from Chaboche 关19兴, and was originally proposed by Armstrong and Frederic 关20兴. Nonlinearities are introduced as a recall term to the Prager linear rule d␣ = 3 Cd p − ␥␣dp 2
共3兲
where dp = 冑2 / 3d p : d p is the increment of the plastic strain trajectory, and C and ␥ are material constants. The first term represents the linear kinematic hardening as defined by Prager. The second term is a recall term, often called a dynamic recovery term, which introduces the nonlinearity between the back stress ␣ and the actual plastic strain. When ␥ = 0, Eq. 共3兲 reduces to the linear kinematic rule. The NLK equation describes the rapid changes due to the plastic flow during cyclic loadings, and plays an important role, even under stabilized conditions 共after saturation of cyclic hardening兲. In other words, these equations take into account the transient hardening effects in each stress-strain loop. After unloading, dislocation remobilization is implicitly described due to the back stress effect and the larger plastic modulus at the beginning of the reverse plastic flow 关19兴.
Isotropic Hardening The isotropic hardening rule follows the formulation proposed by Chaboche 关19兴. Contrary to the kinematic effect, the isotropic hardening takes many successive cycles to be saturated, giving rise to the stabilized behavior. The evolution of the size of the yield surface is represented as R = Q⬁共1 − e−bp兲
共4兲
The associated internal variable is the accumulated plastic strain p. R is the evolution of the size of the yield surface, and Q⬁ is the asymptotic value of the size of the yield surface. The material constant b determines how fast the yield stress evolves.
Viscoplastic Flow Rule The temperature-dependent viscoplastic flow rule incorporated in this paper is based on the work by Kashyap and Murty 关2兴. A series of isothermal creep tests on Sn60-Pb40 were performed in their work, with different temperatures, different strain rates and different grain sizes. The creep behavior at three regimes was studied. Sn-Pb solder alloy has a very small primary creep region, and it reaches the steady-state creep region almost immediately 关13兴. On the other hand, the solder alloy, when used as an interconnect material in electronic packaging applications, never goes into the tertiary region during its normal service life. Therefore, a viscoplastic flow rule, based on a steady-state creep behavior, is a valid representation of the alloy material. The concept of the viscous overstress is adopted in the formulation. Grain size effect on creep rate in the flow rule is included, as in Kashyap and Murty 关2兴. Here, the grain size effect is incorporated through a measure of the phase size. As viscous flow evolves, there is coarsening of the phase, and this phase coarsening is accompanied by a corresponding increase in the grain size. The viscoplastic flow rule is given by Journal of Electronic Packaging
˙ vp =
冉 冊冉
AD0Eb b kT d
p
具⌬¯eff典 E
冊 冉 冊 n
exp −
Q N RT
共5兲
where, N is the direction of inelastic strain rate, defined by N 3 1/2 = F / ; ⌬¯eff is the effective viscous overstress; ⌬¯eff = 共 2 兲 储s − ␣储 − 共1 − D兲共R + k0兲, where the angular brackets are Macauley brackets. The other variables are material parameters, with the same definition as in Kashyap and Murty 关2兴: A ⫽ a dimensionless material parameter D = D0 exp关−共Q / RT兲兴 ⫽ diffusion coefficient D0 ⫽ frequency factor= 100 mm2 / s 关21兴 Q ⫽ creep activation energy for plastic flow R ⫽ universal gas constant= 8.314 J / K mol= 8.314 N mm/ K mol T ⫽ absolute temperature in Kelvin E ⫽ Young’s modulus b ⫽ characteristic length of crystal dislocation 共magnitude of Burger’s vector兲 taken as 3.2⫻ 10−7 mm for Pb-Sn alloy 关21兴 k ⫽ Boltzmann constant= 1.38⫻ 10−23 J / K = 1.38⫻ 10−20 N mm/ K d ⫽ average phase size p ⫽ grain size exponent SEPTEMBER 2005, Vol. 127 / 209
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n ⫽ stress exponent for plastic deformation rate, where 1 / n indicates strain rate sensitivity The grain size exponent p is taken as 3.34, and the stress exponent n = 1.67. As shown by Kashyap and Murty 关2兴, the stress exponent n is not significantly influenced by the test temperature or grain size. Also based on the Arrhenius plot, Kashyap and Murty 关2兴 have shown that activation energy is Q = 44.7 kJ/ mol for temperatures lower then 408 K, and Q = 81.1kJ/ mol for tempratures higher than 408 K.
Damage Evolution Basaran and Yan 关22兴 developed a damage evolution function based on thermodynamics theory. Their damage equation is adopted in this model to account for the prediction of solder fatigue life. The damage function is given by Fig. 1 Constant A versus temperature „based on Adams test data…
D = 1 − e−共⌬e−⌬/N0kT/m¯s兲
共6兲
Fig. 2 Present model results versus Adams test data. „a… Strain rate= 1.67Ã 10−2 / s, variable temperature; „b… Strain rate= 1.67 Ã 10−3 / s, variable temperature; „c… Strain rate= 1.67Ã 10−1 / s, variable temperature; „d… Room temperature, variable strain rate = 1.67Ã 10−3 / s.
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Fig. 3 Present model results versus Skipor’s test data. „a… Strain rate= 1.0Ã 10−1 / s, variable temperature; „b… Strain rate= 1.0 Ã 10−2 / s, variable temperature; „c… Strain rate= 1.0Ã 10−3 / s, variable temperature; „d… Strain rate= 1.0Ã 10−4 / s, variable temperature.
⌬e − ⌬ =
1
冉冕 冊 冕
t
ijdijp −
o
to
1 qi dt + xi
冕
t
␥˙ dt
共7兲
to
where D denotes a monotonically increasing scalar variable of damage at current state, T denotes absolute temperature, N0 is ¯ s is the average Avogadro’s constant, k is Boltzman’s constant, m molecule quantity/mol, is specific mass, ij is the stress tensor, dijp is the increment of plastic strain tensor, ␥˙ is the distributed internal heat production rate per unit mass, and qi is the heat flux tensor. The incremental stress-strain relationship modified by damage can be obtained from the strain equivalence principle el dij = 共1 − D兲Cijkldkl
共8兲
where dij is the increment of the stress tensor, Cijkl is the elastic constitutive matrix, and del ij is the increment of the elastic strain tensor.
Model Verification Against Test Data The model is verified against the monotonic tensile test data from Adams 关8兴, Skipor 关17兴, McDowell 关13兴, and cyclic shear test data performed at the University at Buffalo Electronic Packaging Laboratory. For this verification, material constants were Journal of Electronic Packaging
inferred from their experimental data or obtained from our own testing. As shown by Adams 关8兴, Young’s modulus varies significantly for the same Pb–Sn composition from one specimen to another. Furthermore, there is a big scatter on the values of Young’s module reported in the literature. Sn–Pb solder alloy is a highly temperature-dependent and rate-dependent material. In addition, its material properties are also very sensitive to its microstructure. To compare the model with the test results, material properties are obtained for every test. Adams 关8兴 performed a series of tensile tests on Sn60-Pb40 alloy at isothermal conditions, with various temperature levels. The author also obtained Young’s modulus E and Poisson’s ratio from ultrasonic testing. The variations of E and G with temperature are as follows: E共GPa兲 = 62.0 − 0.067 T
共9兲
G共GPa兲 = 24.3 − 0.0029 T
共10兲
where G is the shear modulus and T is the temperature in Kelvin. The phase size data corresponding to Adams 关8兴 is obtained after comparison of Adams’ results with Kashyap and Murty’s 关2兴 creep test data. Following this approach, Chandaroy 关23兴 found d = 15 m. The optimum value of the kinematic hardening modulus to fit the experimental data was found to be C = 1600 MPa. The SEPTEMBER 2005, Vol. 127 / 211
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Fig. 4 Present model results versus McDowell’s test data. „a… Strain rate= 1.0Ã 10−4 / s, variable temperature and „b… strain rate = 1.0Ã 10−2 / s, variable temperature.
recovery coefficient ␥ was determined by Busso et al. 关15兴 from Adams’ 关8兴 test data for a strain rate ˙ = 1.67⫻ 10−3 s−1. This relation is used here: ␥ = 159+ 0.89共°C兲. The initial yield strength k0 is obtained by observation of the test data at different temperatures and different strain rates. It is obvious from Adams’ 关8兴 test data that the initial yield strength depends on both temperature and applied strain rate. Therefore, k0 has to be determined for each temperature at every given strain rate. This consideration also applies to the determination of the material constants Q and b used in the isotropic hardening rule. The values of k0, Q, and b for different temperature levels at different strain rates are given in Table 1 for Adams’ 关8兴 tests. The constant A is dependent on temperature, and the value of A is fitted based on Adams’ data. A linear regression was performed 212 / Vol. 127, SEPTEMBER 2005
to obtain the relationship between A and temperature, as shown in Fig. 1. The regressed equation yielded the following relationship: A = b1 ⫻ Tb2 or log A = log b1 + b2 log T where log b1 = 41.368,b2 = − 13.692. The stress-strain Adams 关8兴 test data versus model simulation results are plotted in Figs. 2共a兲–2共d兲. Good predictions are generally obtained with the material model. The stress-strain Skipor 关17兴 and McDowell 关13兴 test data versus model simulation results are plotted in Figs. 3共a兲–3共d兲, 4共a兲, and 4共b兲, respectively. Some differences in the hardening behavior between the model and Skipor’s experimental result are present. This may be due to the lack of better estimates of the hardening material parameters. Transactions of the ASME
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Table 2 Solder joint material properties
Fig. 6
Thin-layer solder joint tested in cyclic shear
An experimental study was conducted at the University at Buffalo Electronic Packaging Laboratory on homemade Pb–Sn thinlayer solder joint specimens in order to verify the capability of the model to describe cycling loading conditions. The homemade thin-layer solder joints were 38 mm long and 0.46 mm thick, with an out-of-plane thickness of 6.35 mm, as shown in Fig. 6. The testing and numerical simulations were performed under displacement-controlled conditions at room temperature, and with constant strain rate using an MTS 858 material testing system. Details about the testing set up and determination of material parameters can be found in Tang 关24兴. The testing was performed at a constant strain rate of 1.67⫻ 10−3 / s. The material parameters are reported in Table 2. The stress-strain test data for shear-cycling conditions is compared with the model simulations in Figs. 5共a兲–5共d兲. In general, a good prediction is obtained with the constitutive model for the case of cycling loads. In this analysis, the material parameters were directly obtained from testing, and the differences shown are more likely due to the differences in the
Fig. 5 Thin-layer solder. Present model versus test results. Cyclic shear „a… strain rate= 1.67Ã 10−3 / s, ISR= 0.005; „b… strain rate= 1.67Ã 10−3 / s, ISR= 0.012; „c… strain rate= 1.67Ã 10−3 / s, ISR= 0.02; and „d… Strain rate= 1.67Ã 10−3 / s, variable ISR.
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geometry between actual specimens and the numerical model. For instance, there are voids in the actual specimens that are not explicitly represented in the numerical model; therefore, some differences should be expected. In order to compute the stress-strain response with the testing results, the proper corrections were made to account for the voids in the solder joint and the stiffness, of the load train. In the numerical simulation the applied displacement profile was imposed by the shear strain rate.
Conclusions A damage-mechanics-based unified thermal viscoplasticity constitutive model with isotropic hardening and kinematic hardening for Pb–Sn solder alloy has been proposed and verified against various tensile- and shear-cycling loading test data. The model has a thermodynamics-based damage evolution function embedded in it. Hence, contrary to other constitutive models, such as Darveux 关11兴 where the constitutive model is used to predict the stressstrain behavior and a Coffin–Manson-type empirical curve is used to predict the fatigue life, the model proposed in this paper can accomplish these two steps in one step. The most important contribution of this model is that it has only four material parameters 共not including E and G兲. This model is capable of predicting a broad range of deformation patterns in monotonic, cyclic, and creep conditions. Material constants involved in the model have been evaluated based on different levels of temperatures, strain rates, and microstructures. Phenomenological in nature, the effect of microstructure is taken into account by incorporating grain size in the flow rule. Material properties of solder alloy with the same chemical composition vary in a wide range, as stated by Frear 关25兴, “There is no such a thing as the mechanical behavior of a particular solder, it depends on the microstructure, which in turn depends on how the microstructure is processed.” Furthermore, this material is proved to be highly temperature and rate sensitive. When the constitutive model is to be used to predict the thermomechanical behavior and fatigue life of a particular package, material properties must be obtained from a manufactured package rather than from a sample of the same composition 关26–28兴.
Acknowledgments This research project is sponsored by the Department of Defense Office of Naval Research Advanced Electrical Power Systems. Helpful discussions with Terry Ericsen at ONR are gratefully acknowledged.
关6兴 关7兴 关8兴 关9兴 关10兴 关11兴 关12兴
关13兴 关14兴 关15兴 关16兴
关17兴 关18兴 关19兴 关20兴 关21兴 关22兴 关23兴 关24兴
References 关1兴 Viswanadham, P., and Singh, P., 1998, Failure Modes and Mechanisms in Electronic Packaging, Chapman & Hall, New York, NY. 关2兴 Kashyap, B. P., and Murty, G. S., 1981, “Experimental Constitutive Relations for the High Temperature Deformation of a Pb-Sn Eutectic Alloy,” Mater. Sci. Eng., 50, pp. 205–213. 关3兴 Morris, J. W., Jr., and Reynolds, H. L., 1992, “The Influence of Microstructure on the Mechanics of Eutectic Solders,” Advances in Electronic Packaging, ASME, New York, EEP-Vol. 19共2兲, pp. 1592–1598. 关4兴 Basaran, C., and Chandaroy, R., 1998, “Mechanics of Pb40/ Sn60 Near Eutectic Solder Alloys Subjected to Vibrations,” Appl. Math. Model., 22, pp. 601– 627. 关5兴 Kitano, B. P., Kawai, S., and Shimizu, L., 1988, “Thermal Fatigue Strength
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关25兴 关26兴 关27兴 关28兴
Estimation of Solder Joints of Surface Mount IC Packages,” Proc. of 8th Annual Int. Elec. Packaging Conf., IEPS, Dallas, pp. 4–8. Lau, J., and Rice, J. R., 1990, “Thermal Stress/Strain Analyses of Ceramic Quad Flat Pack Packages and Interconnects,” Elect. Components and Technology 40th Conf., Las Vegas, IEEE, New York, Vol. 1, pp. 824–934. Basaran, C., Desai, C., and Kundu, T., 1998, “Thermomechanical Finite Element Analysis of Problems in Electronic Packaging Using the Disturbed State Concept-Parts 1 and 2,” ASME J. Electron. Packag., 120共1兲, pp. 41–54. Adams, P. J., 1986, “Thermal Fatigue of Solder Joints in Micro-Electronic Devices,” M.S. thesis, Department of Mechanical Engineering, MIT, Cambridge, MA. Wilcox, J. R., Subrahmanyan, R., and Li, C., 1989, “Thermal Stress Cycles and Inelastic Deformation in Solder Joints,” Proc. of 2nd ASME Int. Electronic Mat. and Processing Congr., Philadelphis, ASME, New York, pp. 208–211. Knecht, S., and Fox., L. R., 1990, “Constitutive Relation and Creep-Fatigue Life Model for Eutectic Tin-Lead Solder,” IEEE Trans. Compon., Hybrids, Manuf. Technol., 13, pp. 424–433. Darveaux, R., and Banerji, K., 1992, “Constitutive Relations for Tin-Based Solder Joints,” IEEE Trans. Compon., Hybrids, Manuf. Technol., 15共6兲, pp. 1013–1024. Hong, B. Z., and Burrell, L. G., 1995, “Nonlinear Finite Element Simulation of Thermoviscoplastic Deformation of C4 Solder Joints in High Density Packaging Under Thermal Cycling,” IEEE Trans. Compon., Packag. Manuf. Technol., Part A, 18共3, pp. 585–590. McDowell, D. L., Miller, M. P., and Brooks, D. C., 1994, “A Unified CreepPlasticity Theory for Solder Alloys,” Fatigue Testing of Electronic Materials, ASTM STP 1153, pp. 42–59. Krempl, E., 1999, “Viscoplastic Models for High Temperature Applications,” Int. J. Solids Struct., 37, pp. 279–291. Busso, E. P., Kitano, M., and Kumazawa, M., 1992, “A Visco-Plastic Constitutive Model for 60/ 40 Tin-Lead Solder Used in IC Package Joints,” ASME J. Eng. Mater. Technol., 114, pp. 331–337. Tachibana, Y., and Krempl, E., 1995, “Modeling the High Homologous Temperature Deformation Behavior Using the Viscoplasticity Theory Based on Overstress 共VBO兲: Part I Creep and Tensile Behavior,” ASME J. Eng. Mater. Technol., 117, pp. 456–461. Skipor, A. F., Harren, S. V., and Botsis, J., 1996, “On the Constitutive Response of 63/ 37 Sn/ Pb Eutectic Solder,” ASME J. Eng. Mater. Technol., 118, pp. 1–11. Wei, Y., Chow., C. L., Fang, H. E., and Neilsen, M. K., 1999, “Characteristics of Creep Damage for 60Sn-40Pb Solder Material, ASME 99-IMECE/EEP-15. Chaboche, J. L., 1989, “Constitutive Equations for Cyclic Plasticity and Viscoplasticity,” Int. J. Plast., 3, pp. 247–302. Armstrong, P. J., and Frederick, C. O., 1966, “A Mathematical Representation of the Multiaxial Bauschinger Effect,” CEGB Report RD/B/N731. Paydar, N., Tong, Y., and Akay, H. U., 1994, “A Finite Element Study of Factors Affecting Fatigue Life of Solder Joints,” ASME J. Electron. Packag., 116, pp. 265–273. Basaran, C., and Yan, C., 1998, “A Thermodynamic Framework for Damage Mechanics of Solder Joints,” ASME J. Electron. Packag., 120, pp. 379–383. Chandaroy, R., 1998, “Damage Mechanics of Microelectronic Packaging Under Combined Dynamic and Thermal Loading,” Ph.D. dissertation, State University of New York at Buffalo. Tang, H., 2002, “A Thermodynamic Damage Mechanics Theory and Experimental Verification for Thermomechanical Fatigue Life Prediction of Microelectronics Solder Joints,” Ph.D. thesis., University at Buffalo, State University of New York. Frear, D., Morgan, H., Steven, B., and Lau, J., 1994, The Mechanics of Solder Alloy Interconnects, International Thomson Publishing, Florence, KY. Basaran, C., and Tang, H., 2001, “Influence of Microstructure Coarsening on Thermomechanical Fatigue Behaviour of Pb/ Sn Eutectic Solder Joints,” Int. J. Damage Mech., 10共3兲, pp. 235–255. Basaran, C., and Tang, H., 2002, “Implementationof a Thermodynamic Framework for Damage Mechanics of Solder Interconnects in Microelectronic Packaging,” Int. J. Damage Mech., 11共1兲, pp. 87–108. Morris, J. W., Goldstein, F. J. L., and Mei, Z., 1994, “Microstructural Influences on Mechanical Properties of Solder,” Chapter 2 in The Mechanics of Solder Alloy Interconnects, D. Frear et al., eds., International Thomson Publishing, Florence, KY.
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Y. Zhao Senior Engineer Analog Devices, Boston, MA
H. Tang Senior Engineer NEC Honda Electronics, Detroit, MI
J. Gomez Graduate Student University at Buffalo, Buffalo, NY
A Damage-Mechanics-Based Constitutive Model for Solder Joints Sn-Pb eutectic solder alloy is extensively used in microelectronics packaging interconnects. Due to the high homologous temperature, eutectic Sn-Pb solder exhibits creepfatigue interaction and significant time-, temperature-, stress-, and rate-dependent material characteristics. The microstructure is often unstable, having significant effects on the flow behavior of solder joints at high homologous temperatures. Such complex behavior makes constitutive modeling an extremely difficult task. A viscoplasticity model unified with a thermodynamics-based damage concept is presented. The proposed model takes into account isotropic and kinematic hardening, and grain size coarsening evolution. The model is verified against various test data, and shows strong application potential for modeling thermal viscoplastic behavior and fatigue life of solder joints in microelectronics packaging. 关DOI: 10.1115/1.1939822兴
Introduction As the complexity of electronic packages grows, high reliability of assembled components is critical to maintaining final product quality, especially in light of trends toward miniaturization and higher-level integration 关1兴. It is well known that the coefficient of thermal expansion mismatch between the different materials in the package is the major cause of fatigue failure. Thermal cycling causes progressive damage in the package, especially in the Sn-Pb solder joints used to attach the surface-mounted devices to the printed-circuit boards or a chip to a substrate. Eventually, this damage accumulation beyond a critical point leads to the electrical failure of the assembly. In order to predict the fatigue life of the solder joints, a unified constitutive model of the solder material is essential. Implementing this constitutive model in a generalpurpose finite element code enables an engineer to simulate the thermomechanical response of a new package design on a computer screen. Computer simulations during early design stages significantly lower development cost and reduce design-to-market time cycles. Eutectic solder alloys are routinely used at high homologous temperatures. The melting point of the eutectic Sn-Pb solder alloy is 183° C, and it is at 0.65 Tm at room temperature, where Tm is the melting point. Therefore, solder joints exhibit time-, temperature-, and stress-dependent deformation behavior. Reversed plasticity, creep, and creep-plasticity interactions dominate the behavior of solder joints during thermomechanical cycling. The microstructure also has significant effects on the flow behavior of the eutectic solder alloy, as shown by Kashyap and Murty 关2兴, Morris and Reynolds 关3兴, and Basaran and Chandaroy 关4兴. Material models ranging from purely elastic to elastoplastic using various stress-strain relations have been proposed for Sn-Pb solder material, such as in Kitano et al. 关5兴, Lau and Rice 关6兴, Basaran et al. 关7兴, and many others. Adams 关8兴 proposed a simple viscoplastic model without hardening. Wilcox et al. 关9兴 proposed a rheological model to represent the inelastic behavior of the material; however, it is applicable to a limited range of strain rates. The purely phenomenological models proposed by Knecht and Fox 关10兴, Darveaux and Banerji 关11兴, and Hong and Burrell 关12兴 decouple the creep and plasticity effects artificially. This decoupling 1 Corresponding author. Contributed by the Electronic and Photonic Packaging Division for publication in the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received: June 28, 2004; final manuscript received: July 23, 2004. Review conducted by: Bahgat Sammakia.
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does not have any physical basis and is more a mathematical convenience. One of the drawbacks of the Sn-Pb solder material constitutive models available in the literature is the large number of material parameters required. A large number of material parameters are difficult to determine in practice rendering the model impractical for industrial use. An extensive literature survey on Sn-Pb constitutive models was presented in Basaran et al. 关7兴. On the other hand, classical forms of decoupled plasticity and creep theories have been shown to be quite inferior for modeling cyclic plasticity, creep, and interaction effects 关13兴. The tests on stainless steels showed that the combination of plasticity and creep equations yielded unsatisfactory modeling of certain deformation features, including cyclic creep and creep-plasticity interactions. The development of inelastic constitutive equations has shown that, at the present time, the plasticity-creep approaches are gradually being replaced by “unified” methods, as pointed out by Krempl 关14兴. A number of researchers recently attempted to develop unified creep-plasticity models in order to model the solder material behavior more precisely. Some are Busso et al. 关15兴, Mcdowell et al. 关13兴, Tachibana and Krempl 关16兴, Skipor et al. 关17兴, Wei et al. 关18兴, Basaran and Chandaroy 关4兴, Basaran and Tang 关26,27兴, and others. In this paper, a constitutive model for the eutectic Sn-Pb solder is formulated within a unified framework, where the plastic rateindependent and rate-dependent creep strains are assumed coupled to yield total strain. Both isotropic and kinematic hardenings are incorporated to model cyclic behavior. The influence of microstructure is represented in the constitutive equations by incorporating the grain size in the viscoplastic flow rule. A thermodynamics-based damage evolution equation is incorporated to account for the material damage accumulation and fatigue life predictions. Damage is introduced by means of the train equivalence principle. Furthermore, damage is described in terms of entropy, based on the second law of thermodynamics. Uniaxial tension and shear cycling loading conditions are simulated. The model is validated against uniaxial experimental data available from the literature, and against mechanical shear cycling experiments performed at the University at Buffalo Electronic Packaging Laboratory. The material properties for the constitutive model were taken out of experimental data available in the literature, and from our own testing.
Yield Surface In order to separate elastic behavior from inelastic deformation region, a von Mises type yield surface is used
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F = ¯eff − 共1 − D兲共R + k0兲 艋 0
共1兲
Table 1 Strain dependent material properties at various temperatures
where k0 is the initial size of the yield surface, R is the evolution of the size of the yield surface, D is the internal state variable of damage, and the ¯eff denotes the reduced effective stress or the von Mises distance in the deviatoric stress space, which is given by ¯eff = 冑3/2共S − ␣⬘兲:共S − ␣⬘兲
共2兲
where S is the deviatoric stress tensor and ␣⬘ is the deviator of back stress tensor ␣.
Nonlinear-Kinematic Hardening (NLK) Rule The NLK rule is taken from Chaboche 关19兴, and was originally proposed by Armstrong and Frederic 关20兴. Nonlinearities are introduced as a recall term to the Prager linear rule d␣ = 3 Cd p − ␥␣dp 2
共3兲
where dp = 冑2 / 3d p : d p is the increment of the plastic strain trajectory, and C and ␥ are material constants. The first term represents the linear kinematic hardening as defined by Prager. The second term is a recall term, often called a dynamic recovery term, which introduces the nonlinearity between the back stress ␣ and the actual plastic strain. When ␥ = 0, Eq. 共3兲 reduces to the linear kinematic rule. The NLK equation describes the rapid changes due to the plastic flow during cyclic loadings, and plays an important role, even under stabilized conditions 共after saturation of cyclic hardening兲. In other words, these equations take into account the transient hardening effects in each stress-strain loop. After unloading, dislocation remobilization is implicitly described due to the back stress effect and the larger plastic modulus at the beginning of the reverse plastic flow 关19兴.
Isotropic Hardening The isotropic hardening rule follows the formulation proposed by Chaboche 关19兴. Contrary to the kinematic effect, the isotropic hardening takes many successive cycles to be saturated, giving rise to the stabilized behavior. The evolution of the size of the yield surface is represented as R = Q⬁共1 − e−bp兲
共4兲
The associated internal variable is the accumulated plastic strain p. R is the evolution of the size of the yield surface, and Q⬁ is the asymptotic value of the size of the yield surface. The material constant b determines how fast the yield stress evolves.
Viscoplastic Flow Rule The temperature-dependent viscoplastic flow rule incorporated in this paper is based on the work by Kashyap and Murty 关2兴. A series of isothermal creep tests on Sn60-Pb40 were performed in their work, with different temperatures, different strain rates and different grain sizes. The creep behavior at three regimes was studied. Sn-Pb solder alloy has a very small primary creep region, and it reaches the steady-state creep region almost immediately 关13兴. On the other hand, the solder alloy, when used as an interconnect material in electronic packaging applications, never goes into the tertiary region during its normal service life. Therefore, a viscoplastic flow rule, based on a steady-state creep behavior, is a valid representation of the alloy material. The concept of the viscous overstress is adopted in the formulation. Grain size effect on creep rate in the flow rule is included, as in Kashyap and Murty 关2兴. Here, the grain size effect is incorporated through a measure of the phase size. As viscous flow evolves, there is coarsening of the phase, and this phase coarsening is accompanied by a corresponding increase in the grain size. The viscoplastic flow rule is given by Journal of Electronic Packaging
˙ vp =
冉 冊冉
AD0Eb b kT d
p
具⌬¯eff典 E
冊 冉 冊 n
exp −
Q N RT
共5兲
where, N is the direction of inelastic strain rate, defined by N 3 1/2 = F / ; ⌬¯eff is the effective viscous overstress; ⌬¯eff = 共 2 兲 储s − ␣储 − 共1 − D兲共R + k0兲, where the angular brackets are Macauley brackets. The other variables are material parameters, with the same definition as in Kashyap and Murty 关2兴: A ⫽ a dimensionless material parameter D = D0 exp关−共Q / RT兲兴 ⫽ diffusion coefficient D0 ⫽ frequency factor= 100 mm2 / s 关21兴 Q ⫽ creep activation energy for plastic flow R ⫽ universal gas constant= 8.314 J / K mol= 8.314 N mm/ K mol T ⫽ absolute temperature in Kelvin E ⫽ Young’s modulus b ⫽ characteristic length of crystal dislocation 共magnitude of Burger’s vector兲 taken as 3.2⫻ 10−7 mm for Pb-Sn alloy 关21兴 k ⫽ Boltzmann constant= 1.38⫻ 10−23 J / K = 1.38⫻ 10−20 N mm/ K d ⫽ average phase size p ⫽ grain size exponent SEPTEMBER 2005, Vol. 127 / 209
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n ⫽ stress exponent for plastic deformation rate, where 1 / n indicates strain rate sensitivity The grain size exponent p is taken as 3.34, and the stress exponent n = 1.67. As shown by Kashyap and Murty 关2兴, the stress exponent n is not significantly influenced by the test temperature or grain size. Also based on the Arrhenius plot, Kashyap and Murty 关2兴 have shown that activation energy is Q = 44.7 kJ/ mol for temperatures lower then 408 K, and Q = 81.1kJ/ mol for tempratures higher than 408 K.
Damage Evolution Basaran and Yan 关22兴 developed a damage evolution function based on thermodynamics theory. Their damage equation is adopted in this model to account for the prediction of solder fatigue life. The damage function is given by Fig. 1 Constant A versus temperature „based on Adams test data…
D = 1 − e−共⌬e−⌬/N0kT/m¯s兲
共6兲
Fig. 2 Present model results versus Adams test data. „a… Strain rate= 1.67Ã 10−2 / s, variable temperature; „b… Strain rate= 1.67 Ã 10−3 / s, variable temperature; „c… Strain rate= 1.67Ã 10−1 / s, variable temperature; „d… Room temperature, variable strain rate = 1.67Ã 10−3 / s.
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Fig. 3 Present model results versus Skipor’s test data. „a… Strain rate= 1.0Ã 10−1 / s, variable temperature; „b… Strain rate= 1.0 Ã 10−2 / s, variable temperature; „c… Strain rate= 1.0Ã 10−3 / s, variable temperature; „d… Strain rate= 1.0Ã 10−4 / s, variable temperature.
⌬e − ⌬ =
1
冉冕 冊 冕
t
ijdijp −
o
to
1 qi dt + xi
冕
t
␥˙ dt
共7兲
to
where D denotes a monotonically increasing scalar variable of damage at current state, T denotes absolute temperature, N0 is ¯ s is the average Avogadro’s constant, k is Boltzman’s constant, m molecule quantity/mol, is specific mass, ij is the stress tensor, dijp is the increment of plastic strain tensor, ␥˙ is the distributed internal heat production rate per unit mass, and qi is the heat flux tensor. The incremental stress-strain relationship modified by damage can be obtained from the strain equivalence principle el dij = 共1 − D兲Cijkldkl
共8兲
where dij is the increment of the stress tensor, Cijkl is the elastic constitutive matrix, and del ij is the increment of the elastic strain tensor.
Model Verification Against Test Data The model is verified against the monotonic tensile test data from Adams 关8兴, Skipor 关17兴, McDowell 关13兴, and cyclic shear test data performed at the University at Buffalo Electronic Packaging Laboratory. For this verification, material constants were Journal of Electronic Packaging
inferred from their experimental data or obtained from our own testing. As shown by Adams 关8兴, Young’s modulus varies significantly for the same Pb–Sn composition from one specimen to another. Furthermore, there is a big scatter on the values of Young’s module reported in the literature. Sn–Pb solder alloy is a highly temperature-dependent and rate-dependent material. In addition, its material properties are also very sensitive to its microstructure. To compare the model with the test results, material properties are obtained for every test. Adams 关8兴 performed a series of tensile tests on Sn60-Pb40 alloy at isothermal conditions, with various temperature levels. The author also obtained Young’s modulus E and Poisson’s ratio from ultrasonic testing. The variations of E and G with temperature are as follows: E共GPa兲 = 62.0 − 0.067 T
共9兲
G共GPa兲 = 24.3 − 0.0029 T
共10兲
where G is the shear modulus and T is the temperature in Kelvin. The phase size data corresponding to Adams 关8兴 is obtained after comparison of Adams’ results with Kashyap and Murty’s 关2兴 creep test data. Following this approach, Chandaroy 关23兴 found d = 15 m. The optimum value of the kinematic hardening modulus to fit the experimental data was found to be C = 1600 MPa. The SEPTEMBER 2005, Vol. 127 / 211
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Fig. 4 Present model results versus McDowell’s test data. „a… Strain rate= 1.0Ã 10−4 / s, variable temperature and „b… strain rate = 1.0Ã 10−2 / s, variable temperature.
recovery coefficient ␥ was determined by Busso et al. 关15兴 from Adams’ 关8兴 test data for a strain rate ˙ = 1.67⫻ 10−3 s−1. This relation is used here: ␥ = 159+ 0.89共°C兲. The initial yield strength k0 is obtained by observation of the test data at different temperatures and different strain rates. It is obvious from Adams’ 关8兴 test data that the initial yield strength depends on both temperature and applied strain rate. Therefore, k0 has to be determined for each temperature at every given strain rate. This consideration also applies to the determination of the material constants Q and b used in the isotropic hardening rule. The values of k0, Q, and b for different temperature levels at different strain rates are given in Table 1 for Adams’ 关8兴 tests. The constant A is dependent on temperature, and the value of A is fitted based on Adams’ data. A linear regression was performed 212 / Vol. 127, SEPTEMBER 2005
to obtain the relationship between A and temperature, as shown in Fig. 1. The regressed equation yielded the following relationship: A = b1 ⫻ Tb2 or log A = log b1 + b2 log T where log b1 = 41.368,b2 = − 13.692. The stress-strain Adams 关8兴 test data versus model simulation results are plotted in Figs. 2共a兲–2共d兲. Good predictions are generally obtained with the material model. The stress-strain Skipor 关17兴 and McDowell 关13兴 test data versus model simulation results are plotted in Figs. 3共a兲–3共d兲, 4共a兲, and 4共b兲, respectively. Some differences in the hardening behavior between the model and Skipor’s experimental result are present. This may be due to the lack of better estimates of the hardening material parameters. Transactions of the ASME
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Table 2 Solder joint material properties
Fig. 6
Thin-layer solder joint tested in cyclic shear
An experimental study was conducted at the University at Buffalo Electronic Packaging Laboratory on homemade Pb–Sn thinlayer solder joint specimens in order to verify the capability of the model to describe cycling loading conditions. The homemade thin-layer solder joints were 38 mm long and 0.46 mm thick, with an out-of-plane thickness of 6.35 mm, as shown in Fig. 6. The testing and numerical simulations were performed under displacement-controlled conditions at room temperature, and with constant strain rate using an MTS 858 material testing system. Details about the testing set up and determination of material parameters can be found in Tang 关24兴. The testing was performed at a constant strain rate of 1.67⫻ 10−3 / s. The material parameters are reported in Table 2. The stress-strain test data for shear-cycling conditions is compared with the model simulations in Figs. 5共a兲–5共d兲. In general, a good prediction is obtained with the constitutive model for the case of cycling loads. In this analysis, the material parameters were directly obtained from testing, and the differences shown are more likely due to the differences in the
Fig. 5 Thin-layer solder. Present model versus test results. Cyclic shear „a… strain rate= 1.67Ã 10−3 / s, ISR= 0.005; „b… strain rate= 1.67Ã 10−3 / s, ISR= 0.012; „c… strain rate= 1.67Ã 10−3 / s, ISR= 0.02; and „d… Strain rate= 1.67Ã 10−3 / s, variable ISR.
Journal of Electronic Packaging
SEPTEMBER 2005, Vol. 127 / 213
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geometry between actual specimens and the numerical model. For instance, there are voids in the actual specimens that are not explicitly represented in the numerical model; therefore, some differences should be expected. In order to compute the stress-strain response with the testing results, the proper corrections were made to account for the voids in the solder joint and the stiffness, of the load train. In the numerical simulation the applied displacement profile was imposed by the shear strain rate.
Conclusions A damage-mechanics-based unified thermal viscoplasticity constitutive model with isotropic hardening and kinematic hardening for Pb–Sn solder alloy has been proposed and verified against various tensile- and shear-cycling loading test data. The model has a thermodynamics-based damage evolution function embedded in it. Hence, contrary to other constitutive models, such as Darveux 关11兴 where the constitutive model is used to predict the stressstrain behavior and a Coffin–Manson-type empirical curve is used to predict the fatigue life, the model proposed in this paper can accomplish these two steps in one step. The most important contribution of this model is that it has only four material parameters 共not including E and G兲. This model is capable of predicting a broad range of deformation patterns in monotonic, cyclic, and creep conditions. Material constants involved in the model have been evaluated based on different levels of temperatures, strain rates, and microstructures. Phenomenological in nature, the effect of microstructure is taken into account by incorporating grain size in the flow rule. Material properties of solder alloy with the same chemical composition vary in a wide range, as stated by Frear 关25兴, “There is no such a thing as the mechanical behavior of a particular solder, it depends on the microstructure, which in turn depends on how the microstructure is processed.” Furthermore, this material is proved to be highly temperature and rate sensitive. When the constitutive model is to be used to predict the thermomechanical behavior and fatigue life of a particular package, material properties must be obtained from a manufactured package rather than from a sample of the same composition 关26–28兴.
Acknowledgments This research project is sponsored by the Department of Defense Office of Naval Research Advanced Electrical Power Systems. Helpful discussions with Terry Ericsen at ONR are gratefully acknowledged.
关6兴 关7兴 关8兴 关9兴 关10兴 关11兴 关12兴
关13兴 关14兴 关15兴 关16兴
关17兴 关18兴 关19兴 关20兴 关21兴 关22兴 关23兴 关24兴
References 关1兴 Viswanadham, P., and Singh, P., 1998, Failure Modes and Mechanisms in Electronic Packaging, Chapman & Hall, New York, NY. 关2兴 Kashyap, B. P., and Murty, G. S., 1981, “Experimental Constitutive Relations for the High Temperature Deformation of a Pb-Sn Eutectic Alloy,” Mater. Sci. Eng., 50, pp. 205–213. 关3兴 Morris, J. W., Jr., and Reynolds, H. L., 1992, “The Influence of Microstructure on the Mechanics of Eutectic Solders,” Advances in Electronic Packaging, ASME, New York, EEP-Vol. 19共2兲, pp. 1592–1598. 关4兴 Basaran, C., and Chandaroy, R., 1998, “Mechanics of Pb40/ Sn60 Near Eutectic Solder Alloys Subjected to Vibrations,” Appl. Math. Model., 22, pp. 601– 627. 关5兴 Kitano, B. P., Kawai, S., and Shimizu, L., 1988, “Thermal Fatigue Strength
214 / Vol. 127, SEPTEMBER 2005
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