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Large-Scaled 3-D Area Array Electronic Packaging Analysis Kuo-Ning Chiang1 Department of Power Mechanical Engineering, National Tsing Hua University HsinChu, Taiwan 300, R.O.C. E-Mail: [email protected] Hsien-Chie Cheng2 National Center for High-Performance Computing, Taiwan 300, R.O.C. Wen-Hwa Chen3 Department of Power Mechanical Engineering, National Tsing Hua University HsinChu, Taiwan 300, R.O.C. Abstract As integrated circuit functionality, performance and density continue to increase, innovative next generation packaging approaches are in great demand. The ball grid array (BGA) type packaging technology such as flip chip (FC), plastic/ceramic BGA (PBGA/CBGA, see Figure 1), and chip scale package (CSP) has been gaining world-wide interest and commitment as the potentially lowest-cost package for high-I/O devices and even for lower-pincount (e.g., Tessera’s BGA) applications. Drivers include the I/O density advantages of an area array, as well as the potential for excellent electrical and thermal performance, etc. However, some of the reliability issues of the BGA type package are not eradicated, research is critically needed in the area of next generation electronic package analysis and design, e.g., improvement of the package reliability, increase of the assembly yields rate and reduction of the solder bump pitches, etc. A substructuring like local-global finite element method with multi-point constraints boundary condition is developed for prediction of the fatigue life of the solder joint, this methodology could significantly reduce the analysis CPU time and make the 3-D large-scaled electronic packaging analysis possible. Keyword: Ball Grid Array (BGA), Flip Chip (FC), Chip Scale Packaging (CSP), Solder bump pitch. 1 2 3 Associate Professor, ***Corresponding Author Associate Research Scientist Professor 1 Large-Scaled 3-D Area Array Electronic Packaging Analysis Kuo-Ning Chiang1 Department of Power Mechanical Engineering, National Tsing Hua Universit Hsien-Chie Cheng2 National Center for High-Performance Computing, Taiwan 300, R.O.C. Wen-Hwa Chen3 Department of Power Mechanical Engineering, National Tsing Hua University Taiwan 300, R.O.C. Summary As integrated circuit functionality, performance and density continue to increase, innovative next generation packaging approaches are in great demand. The ball grid array (BGA) type packaging technology such as flip chip (FC), plastic/ceramic BGA (PBGA/CBGA, see Figure 1), and chip scale package (CSP) has been gaining world-wide interest and commitment as the potentially lowest-cost package for high-I/O devices and even for lower-pincount (e.g., Tessera’s BGA) applications. Drivers include the I/O density advantages of an area array, as well as the potential for excellent electrical and thermal performance, etc. However, some of the reliability issues of the BGA type package are not eradicated, research is critically needed in the area of next generation electronic package analysis and design, e.g., improvement of the package reliability, increase of the assembly yields rate and reduction of the solder bump pitches, etc. A substructuring like local-global finite element method with multi-point constraints boundary condition is developed for prediction of the fatigue life of the solder joint, this methodology could significantly reduce the analysis CPU time and make the 3-D large-scaled electronic packaging analysis possible. Introduction The low cycle fatigue-induced failure of solder balls in surface mounted electronic devices has become one of the most critical reliability issues in the ball grid array type packages. Solder ball reliability performance was found to be highly dependent on the configuration of the package, such as the combination of substrates and geometry/material properties of die, etc., which in turn, is governed by bond pad geometry, solder ball configuration, thermal behaviors of each component, moisture condition, as well as the solder reflow characteristics, etc. Using finite element methods with multiphysics capabilities to predict the package reliability and to reduce the time-to-market is imperative in the area of electronic packaging industry. One of the dominant failure mechanism of the electronic packages is the low cycle fatigue when subjected to thermal/power cycling loading. In general, predicting the fatigue life of the solder joint requires the detailed thermal-mechanics and package geometry information. Application of a simplified, approximation model, such as a “slice” or “2-D plain strain/stress” model provides a very effective alternative due to its ease of modeling and efficiency of computation. However, Pao et. al. [1] has pointed 1 2 3 Associate Professor Associate Research Scientist Professor 2 out that the “slice” model could originate a significant error in comparison with the three-dimensional, full-scaled models for some particular package’s analyses. As a consequence, a 3-D, full-scaled model is favored for obtaining a more reliable, robust solution. On the other hand, due to that a BGA typed package often consists of 300-700 solder balls and additionally, the aspect ratio of some components is extremely exclusive, its finite element model will likely come out with enormously huge degree of freedoms. For such a large-scaled model undergoing a nonlinear/transient analysis is in fact impractical for most computing resources currently available. In order to resolve this difficulty, the most common approach in the literature is to apply the substructuring based techniques. One typical option is to adopt a fine mesh to the most reliability-concerned solder ball and apply a coarser mesh elsewhither. However, in most cases, the most reliability concerned solder ball is not known a priori. The other way that is preferred for the current study is to apply a substructuring typed local-global (namely, micro-macro) analysis technique (see, e.g., Voleti [2]). This technique has been significantly applied in the past to deal with various engineering problems, particularly in the applications of composite materials (e.g., Griffin and Vidussoni [3]). In the area array packaging analysis, the local-global finite element approach as well has played an important role (see, Corbin [4]; Ju [5]). In their study, an equivalent beam model is adopted to elastically/plastically simulate the specific 3-D local model, as shown in Figure 1, such that similar stiffness characteristics will be yielded when subjected to the same force. However, their approaches may present some disadvantages. For example, their 3-D local models contain one single solder joint, sandwiched by a half thickness of substrate (i.e., BT or ceramic) on top and PCB on bottom as shown in Figure 1, and more importantly, in the global analysis a plate model is respectively used to simulate the substrate and PCB. Obviously, this modeling is likely oversimplified and less accurate, and lacks of feasibility for expansion to include other components of the package for analysis, such as molding compound etc. The main purpose of this study is to propose an improved equivalent beam model using a FEM based method together with an optimization approach. In addition, the solution to a realistic 3-D PBGA structure made up of multi-materials system will be also investigated to substantiate the proposed methodology. The computational time as well as the accuracy of the equivalent model will be considerably compared with those of the full-scaled model. Heat Sink Gold Wire Die Cap C-4 Solder Ceramic 90Pb/10Sn Ceramic Ball Molding Compound Eutectic Solder Joint Die Attach BT Solder Mask Ball Pad Solder Ball FR-4 FR-4 Module mid-plane Module mid-plane Module eutectic solder fillet Ceramic module 90%Pb/10%Sn solder ball Card eutectic Solder fillet 60Sn/40Pb eutectic solder Molybdenum pad BT module Solder pad FR-4 card Copper pad FR-4 card Card mid-plane Card mid-plane (a) CBGA/Corbin’s Local Model [4] Fig. 1 (b) PBGA/Ju’s Local Model [5] Existing Local Models of Ball Grid Array Typed Packages 3 Furthermore, if anyone of the solder joints is fatigued, the package can be considered as failure. Hence, the most stressed/strained one shall be located from the global analysis, and the corresponding net displacements of the beam will be extracted. These displacements will be again applied to the local model to further examine the stress/strain information. However, a significant geometry change involved in the local model will result in a stress/strain concentration field that is very sensitive to the mesh size. In the literature, there are several methods that can be applied to investigate the problem. A nonlinear analysis with nonlinear material properties or detailed geometry descriptions, such as a recommended corner fillet, may ease the stress concentration but the strain concentration. It is important to note that the maximum cyclic effective strain range is extremely crucial to the low cycle fatigue life of the solder ball as well as overall reliability based on an empirical relationship, such as Coffin [6], Manson, Cheng et. al. [7] or Darveaux et. al. [9] techniques. It is, therefore, vital to effectively characterize the strain concentration. In this study, an engineering approximation approach using a volume-weighted averaging strain in a finite local zone, proposed by Clark and Mcgregor [10], will be applied for characterizing the strain concentration field. To the end, a finite local zone with a specific dimension will be characterized. Based on the average plastic strain in the specific local zone, the fatigue life of the most susceptible ball can be then rationally predicted. Local/Global Analysis In general, the solution procedure of local/global analysis used in this study can be briefly characterized into the following main steps. A single solder ball (i.e., the local model) is first defined, and moreover, its elastic-plastic characteristics is extensively simulated using an equivalent beam model. The second step is to perform the global analysis, in which the solder joints will be substituted by the equivalent beams. Finally, the detailed stress/strain information of the solder joint subjected to the net displacements extracted from the most susceptible equivalent beam in the global analysis will be calculated. The success of this approach relies on reliable calculation of the effective properties of the prescribed local model. In the literature, several approaches can be employed, such as the rule of mixture, the self-consistent method (Willis [11]), the homogenization method (Sanchez-Palencia [12]; Guedes and Kikuchi [13]), or the FEM based method (Cheng et. al. [7]). However, the first three methods are not appropriate for the current study due to that solder joints are discontinuous. As a result, the FEM based method becomes a preferred choice. The other implied advantages of the FEM based method are in that an existing finite element code is simply required for obtaining the effective properties without the necessity of conducting a complicated modeling. Effective Properties and Geometry Data of the Equivalent Beam The equivalent beam theory has been fully discussed by Cheng et. al. [7], and this research will apply their approach and formulations to a realistic 3-D plastic BGA package reliability analyses. Furthermore, to maintain the readability of this work, some equations from Cheng et. al. [7] are included. Figure 2 presents the local model used in this study, simply including a single near eutectic solder joint (60%Sn/40%Pb) without comprising of any BT substrate and FR-4 PWB. The major advantage of this model in comparison with Corbin’s and Ju’s stems from its flexibility and generality in 3-D modeling the entire assembly associated with the area array electronic packaging. The geometry profile of the 3-D solder ball is obtained from the simulation result of the Surface Evolver (Racz and Szekely [14]; Brakke [15]). The Evolver is for the simulation of surfaces whose geometry is determined by surface tension and other energies, such as gravity, and external forces, etc. It evolves a surface towards its equilibrium configuration by minimizing total system free energy. The surface is represented as a set of interconnected triangular facets and then iterating this initial surface combined with geometry constraints toward a minimal energy configuration by using a gradient descent method. The Surface Evolver has been successfully applied in predicting the final shape of the BGA joint after reflow according to various pad sizes and shape, solder volume, specific solder height, and surface tension, etc. The advantages of using the Evolver for BGA reflow shape predictions are 4 several and clearly, it is more economical and less time consuming than laboratory experiments. Do nL/2 mDo (1-n)L/2 L nL/2 Fig. 2 A Typical Cantilever Beam with Variable Cross Sections First of all, it should be noticed that the following derivations are based on the fact that the effect of local CTE mismatch between solder joints and substrate/PCB is negligible in contrast to the global effect between silicon chip and PWB. Furthermore, in order to derive the corresponding equivalent beam using a FEM based technique, the following relations are required: shear resultant forces (P) vs. shear displacements ( ws ), axial resultant forces (F) vs. axial displacements ( wa ), and moments (M) vs. shear displacements ( ws ) of the solder joint. Once they are obtained, the elastic and plastic properties as well as the corresponding geometry of the equivalent beam can be derived in the following. Consider an equivalent beam, as shown in Figure 2, with Young's modulus E , shear modulus G , and Poisson's ratio . This cantilever beam is fixed at the left-hand side and subjected to a concentrated force, either a tension force F or a shear force P, at the right-hand side. Note that n and m are both the geometry parameters of the equivalent beam used in the design optimization process. The shear correction factor (see Cowper [16]) corresponding to a solid circular beam can be deduced as: 61 7 6 (1) Based on the governing equations of the beam theory as well as the shear correction factor, see Eq. (1), the elastic shear displacement s (or the axial displacement a ) at the most right-hand side of the cantilever beam subjected to a shear (or axial) concentrated force P (or F) can be obtained: 7 6 n1 12 12 PL m m s 3EA 1 1 n1 m 2 m 2 FL a AE 1 1 4 1 4 m 3m 2 n 3 3 n n PL 2 4 12 EI (2) (3) 5 where A and I, respectively, represent the area and moment of inertia of the base beam. On the other hand, based on Eq. (2), with a given P, L, n, m, and s at the tip of the right-hand side of the cantilever beam, the effective elastic material property E can be eventually derived as: 7 6 n1 12 12 PL m m E 3 s A 1 1 n n 2 n 3 3 4 1 4 PL 3m m 2 4 12 s I (4) Note that in the case of m equal to 1, it becomes a uniform equivalent beam. Furthermore, in order to define the configuration of the equivalent beam, a , s , F, P, and M corresponding to the solder joint should be obtained in advance. Based on the shear resultant force (P) and the resultant moment (M) associated with the solder joint, the length of the equivalent beam can be first defined: L M /P (5) Subsequently, with a given, particular geometry parameter (i.e., m and n), the base diameter Do of the equivalent beam can be derived as follows: 1 1 n n 2 4 1 4 3m m 2 4 Do 4 L F s P a n 3 12 1 1 n 1 m 2 m 2 (7 6 ) 3 (6) The effective properties of the equivalent beam, including the elastic and plastic material properties as well as the geometry configuration, can be fully developed using an iterative method. In addition, an optimization technique, the design of experiment method, incorporating with a proper tolerance prescribed is applied for attaining accuracy to a great extend using the simple beam model. The tolerances defined in Corbin’s work will be applied in this study, in which 6% of discrepancy for the shear displacement and 16% for the axial displacement due to that the axial displacement of the entire module is estimated to be considerably smaller than the shear displacement. For simplicity, this study selects only five representative sets (i.e., forces vs. displacements) to determine the properties of the equivalent beam and to simulate the linear/nonlinear stiffness characteristics of the solder joint: one from the elastic region and the other four from the plastic zone. The elastic pair is used to determine the elastic properties, and the other four pairs are used to define the equivalent beam's plastic properties. For optimization, the objective is to minimize the least square mean error of the five representative stiffness characteristic sets between the solder joint and the equivalent beam model. In addition, the length of the equivalent beam L and the geometry data n and m are considered as the design variables in the optimization process, each of which is subjected to an appropriate side constraint: 1 L / Lo 1.2 , 0 n 1, 1 m 2, (7) 6 Concerning application of the design of experiment method, an appropriate number of trial tests based on variable discrete combinations of these design variables in the feasible domain will be defined and performed. Once all these trials are tested, a regression model can be constructed and analyzed. Note that the definition of the test trials is completely case-dependent. From the regression analysis, the trial comprising the minimum objective and satisfying all the specified tolerances will be chosen as the solution. The above approaches can, in general, lead to a considerably optimal solution. Furthermore, if other temperatures are considered, (e.g., the temperature-dependent material properties used in this study), the whole iterative procedure can be completely reduced to one time of estimation of the corresponding material properties using the geometry data derived formerly. Application Once the effective characteristics of the equivalent beam are derived, analysis of the 3-D equivalent package would be very efficient and present no difficulties. In order to verify the previously proposed methodology, a realistic PBGA package is substantially studied. In this application, the characteristics of the equivalent beam will be first derived. Based on the characterized equivalent beam model, a global solution to a typical package subjected to a given thermal loading can be then achieved, and more importantly, the performance and accuracy of the 3-D equivalent model using the equivalent beam will be extensively compared with those of the original 3-D full-scaled model. Furthermore, the beam that is the most reliability-concerned will be sought and the corresponding net displacements will be extracted and placed in the 3-D solder joint as enforced boundary conditions to further investigate the detailed stress/strain information. Due to that a significant geometry change is involved in the solder joint, a stress/strain concentration field that is critical to the mesh size will be resulted, in which it is crucial to the prediction of the solder joint’s fatigue life. Hence, in this application, an engineering approach based on the volume-weighted averaging strain in a finite zone will be applied for characterizing the strain field neighboring the singularity point. Stress (MPa) 30.0 o 20 C 20.0 10.0 100 oC 0.0 0.000 0.002 0.004 0.006 0.008 0.010 Strain Fig. 3 Tensile Stress-Strain Relationships. Linear/Nonlinear Characteristics of the Equivalent Beam Consider that the a typical eutectic 60Sn/40Pb solder joint holds a tensile stress-strain relationship, as shown in Figure 3. The Sn/Pb solder joint is considered as an elastic-plastic, temperature-dependent, and 7 time-dependent material. Before further constructing the equivalent beam, the elastic/plastic characteristics of the solder joint at temperature 20 o C and 100o C are first derived in Table 1 based on Figure 3. The effective properties of the equivalent beam can be optimally obtained using the previously proposed solution procedure. Table 1 also presents the obtained linear/nonlinear stiffness characteristics of the equivalent beam in comparison with those of the solder joint, including shear reaction forces vs. shear displacements, moments vs. shear displacements, as well as axial reaction forces vs. axial displacements at temperature 20o C and 100 o C . It is found that the maximum differences that are 6% for the shear displacements and 16% for axial displacements all occur in the elastic region (the first representative set). In other words, it implies the differences of the shear displacements and the axial displacements in the plastic region are all much less than the prescribed tolerances (i.e., 6% and 16% respectively). The mean error of these two models corresponding to shear resultant forces, shear resultant moment, and axial resultant forces are 3.0%, 6.6%, and 8.5% respectively, which are considerably satisfactory. Table 1: Resultant Forces/Moments versus Displacements. 20o C Shear Displacements (mm) -vs.- Shear Reaction Forces (N) Displacements Solder Joint Equivalent Beam Solder Joint 0.00015 0.962 0.907 0.435 0.0004 2.424 2.295 1.091 0.0010 4.171 4.099 1.678 0.0025 6.212 6.229 2.006 0.006 7.591 7.337 2.152 Shear Displacements (mm)-vs.-Shear Reaction Moments (N-mm) 0.00015 0.223 0.231 0.101 0.0004 0.562 0.585 0.253 0.0010 0.967 1.045 0.389 0.0025 1.441 1.589 0.466 0.006 1.764 1.871 0.502 Axial Displacements (mm)-vs.-Axial Reaction Forces (N) 0.0001 2.306 2.675 1.044 0.0003 6.180 7.031 2.764 0.0008 11.293 10.780 4.186 0.0015 14.661 13.560 4.700 0.003 17.423 17.110 5.008 100 o C Equivalent Beam 0.411 1.028 1.675 1.963 2.156 0.105 0.262 0.467 0.501 0.550 1.214 3.133 4.390 4.857 5.177 3-D Plastic BGA Analysis The solution to a realistic 3-D PBGA structure, as shown in Figure 4(a), made up of multi-materials system is then pursued. The full-scaled finite element model consists of a silicon chip, a piece of BT substrate, FR-4 printed circuit board, molding compound, and 72 eutectic solder bumps. The top view of a quarter of the package is also shown in Figure 4(b). The result obtained from this full-scaled model is extensively compared from that of the equivalent model, in which the eutectic solder joints with thousands of three-dimensional solid elements is replaced by a 10-linear-beam-element model. The material properties corresponding to these components are shown in Table 2. It should be noted that all these components are considered as elastic except of the eutectic solder joint or the equivalent beam, which is elastic-plastic. 8 SILICON Y EPOXY BT 5.5 4.0 2.5 1.0 0.0 (a) Finite Element Model Fig. 4 1.5 3.0 4.5 6.0 X (b) Top View A Typical PBGA Structure Table 2: The Material Properties and Geometry Data Used in the PBGA Package Silicon Chip BT FR4 EPOXY Young’s Modulus(MPa) 130000 19000 18200 8960 Possion Ratio 0.28 0.2 0.19 0.35 CTE(ppm) 2.62 15 16 19 Geometry Data(mm) 9 8 0.36 14 13 0.36 14 13 1.71 13.4 12.4 0.72 This entire package is subjected to a 100o C temperature change with the initial stress-free condition set at 25o C . Since the package is symmetric in the X- and Y- planes, only a quarter of the package is analyzed. The finite element approximation of the 1/4 full-scaled model comprises 53540 nodes and 52420 elements, and that for the equivalent model contains 22212 nodes and 18226 elements. The solution to the package can be eventually obtained using a commercial, finite element analysis code-- ABAQUS R . The net shear displacements of the equivalent beams and the solder joints associated with the equivalent model and the full-scaled model, respectively, are shown in Figure 5(a). It is observed that the equivalent approach can as well position the most critical joint, in which it locates at (X,Y)=(4.5, 4.0), right beneath the chip and in the farthest diagonal from the neutral point. Basically, the result matches closely with the experimental finding (see, Nagarajand and Mahalingam [17]). In addition, the differences between these two models are shown in Figure 5(b) in percentage. It is apparent that the accuracy for most of the joints is sufficient except of those in the first column/row, in which the discrepancy can be up to 12%. This is because ABAQUS R as well as the currently proposed equivalent beam model lack capabilities in handling a symmetric circular beam. An approximation is employed for this particular situation, and as a result, a larger error is induced. Furthermore, the total computational CPU time for the 3-D full-scaled model is 237,567 (second); on the other hand, it is 14,253 (second) for the equivalent model. Significantly, the performance increases up to 17 times. 9 Er ro r i n Pe r c e nt a g e S he a r Di s p l a c e m e n t 1.5e- 3 1.0e- 3 Y=1.0 Y=2.5 Y=4.0 Y=5.5 Y=1.0 Y=2.5 Y=4.0 Y=5.5 5.0e- 4 ( 3D F ul l- S c a le d) ( 3D F ul l- S c a le d) ( 3D F ul l- S c a le d) ( 3D F ul l- S c a le d) ( Eq uiva le nt ) ( Eq uiva le nt ) ( Eq uiva le nt ) ( Eq uiva le nt ) 14 .0 Y= 1.0 Y= 2.5 Y= 4.0 Y= 5.5 12 .0 10 .0 8.0 6.0 4.0 2.0 0.0 0.0e+0 0.0 2.0 4.0 6.0 0 (a) The Net Shear Displacement Fig. 5 1.5 3 4.5 6 X -d is t a nc e F r o m t h e S y mm e tr y X -d is t a nc e F r o m t h e S y mm e tr y (b) The Error Analysis The Net Shear Displacements and Error Analysis Using Two Various Approaches Cyclic Thermal Fatigue Analysis Using a Local/Global Technique Cyclic Thermal Fatigue Analysis Using a Local/Global Technique Since the low-cycle fatigue is one of the major failure mechanism of the electronic packages when subjected to thermal/power cycling loading, of great importance is to investigate the thermally induced complex viscoplastic deformation of the solder joints in a package. A typical accelerated thermal cycling curve is shown at figure 6. In this study, the local/global finite element analysis will be applied for characterizing the detailed stress/strain information of the solder joint. Assume that in the global analysis, only time-independent plastic deformation implies in the ramp time of the thermal cycling loading, and only time-dependent creep deformation is involved in the dwell time. The deformation of the most reliability-concerned solder joint in the dwell time (i.e., during the upper/lower temperature) can be extracted by performing a nonlinear finite element analysis of the global model with equivalent beam inside subjected to a net temperature swing loading. The magnitude of the net temperature swing depends on the range between the upper/lower temperature in the thermal cycling loading and the stress-free temperature. Once the thermal displacement boundary conditions of the most reliability-concerned solder joint at these two temperatures are obtained, detailed thermal cycling simulation can be performed in the local level. Temperature (oC) 125 -55 0 15 30 45 60 75 Time (Min.) Fig. 6 Accelerated Thermal Cycling Curve 10 Local Analysis Based on the net deformations of the most reliability-concerned beam in the global analysis, a detailed stress/strain analysis can be performed on the solder joint with as the highest mesh density as possible in order to achieve the most converged solution. Once the stress/strain information is obtained, the fatigue life of the solder joint as well as the overall package can be predicted by directly plugging them into the empirical Coffin-Manson relationship. However, an abrupt geometry change does occur in the circumference of both top and bottom surfaces, in which it significantly induces a stress/strain concentration field. To verify the great effect of the geometry and material singularity on the stress/strain field, a typical eutectic solder joint shown in figure 7 will be practiced. Four different finite element models, each with a different mesh density (i.e., 768, 2592, 6144, 14520 elements), are applied for modeling the solder joint. In addition, the solder joint is subjected to a prescribed displacement boundary condition in both top and bottom sides. Using a finite element analysis code-- ABAQUS R , the maximum Von Mises stress, plastic strain, and plastic dissipation can be derived, shown in figure 8 to 10, respectively, corresponding to these four finite element models. Max. Von Mises Stress (MPa) Based on figure 8 to 10, it can be detected that all these structural responses are mesh-dependent. Note that the maximum plastic strain and dissipation are much sensitive to the mesh density than the maximum Von Mises stress. This is due to the fact that a nonlinear material modeling is implemented in the analysis. There are two empirical formulations that are intensively used by the electronic packaging researchers for the solder joint reliability prediction, which are, the equivalent plastic strain based Coffin-Manson relation [6] for predicting the cycles to failure and the viscoplastic strain energy density based formulation that proposed by Darveaux et. al [9] for estimation of the number of cycles to crack initiation. The finite element mesh density effect for these two methods are shown at figure 11 and 12, it is clear to see that reliability results of the Coffin-Manson relation and the Darveaux model both are very sensitive to the mesh density. Furthermore, the prediction of the Coffin-Manson relation with different mesh density is illustrated in figure 11, it shown that the prediction of cycles to failure of plastic BGA package may vary from 48 cycles (14,520 elements) to 4997 cycles (768 elements). The similar tendency is also shown at figure 12, the number of cycles to crack initiation that predicted by Darveaux model could change from 757 cycles (768 element) to 143 cycles (14,520 elements) as the finite element mesh become denser. These results indicate the methods that can be applied to the characterization of the singularity are in critical demand. In the literature, several existing techniques are available to resolve the stress concentration problem; however, there still lacks an effective way to discharge or to characterize the strain concentration. Z Y X 33 32 31 30 29 28 27 26 25 24 768 2592 6144 14520 Mesh Density Fig. 7 A 3-D Eutectic Solder FE Model Fig. 8 The Maximum Von Mises Stress 11 Max. Plastic Dissipation (N/mm^2) Max. Pastic Strain 0.025 Nf = C * (eq)n 0.02 0.015 0.01 0.005 0 768 2592 6144 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 No = C * (W)n 768 14520 2592 6144 14520 Mesh Density Mesh Density Fig. 9 The Maximum Plastic Strain Fig. 10 The Maximum Plastic Dissipation 757 420 290 Mesh Density 143 (14,520) 2.1 (6,144) (768) (2,592) (2,592) 1.15 0.84 (6,144) 0.34 (14,520) Mesh (768) 1018.7 27.14 54.73 48 227 500 4,997 Density .4 Fig. 11 Coffin-Manson Relation Fig. 12 Darveaux et. al. Model An engineering approximation approach based on the volume-weighted averaging equivalent plastic ~ strain pl in a finite zone is used in the study: n ~ pl Ve epl e 1 n V e 1 (8) e in which Ve is the volume of an element and epl is the corresponding equivalent plastic strain. The radius of the finite zone is determined in an observed manner that it should be small enough to capture the maximal strain field; on the other hand, large enough to obtain a converging solution as the mesh density increases. In this research, the engineering approach defines a total of four various types of finite element approximations for the solder joint, each of which corresponds to a different element size. In addition, three different 12 semi-circular-ring zones (i.e., A, B, and C) around the circumference of the top surface of the solder joint, each of which comprises a particular radius (i.e., 0.02, 0.062, and 0.12 (mm), respectively), are defined. figure 13 presents the volume-weighted averaging equivalent plastic strains of these three different zones as well as the maximum equivalent plastic strains with respect to these four finite element models. It can be easily seen that the maximum equivalent plastic strain is singular and extremely mesh-sensitive, and moreover, zone A seems to be able to provide better agreement to the proposed criterion in the selection of the finite zone than the others. However, a further study on the dimension of the finite zone, such as use of a finer zone or verification of the predicted life using experimental data, should be conducted in the future work. Equivalent Plastic Strain 0.010 0.008 Average at Zone A 0.006 Average at Zone B Average at Zone C Maximum 0.004 0.002 0.000 0 4000 8000 12000 16000 Mesh Size Fig. 13 The Convergence of the Equivalent Plastic Strains in Various Zones Conclusions An effective local/global methodology is proposed for analysis of the thermally-induced, elastic-plastic deformations of three-dimensional, full-scaled area array typed electronic packages. In addition, an engineering approximation based on a volume-weighted averaging technique incorporating with an observed criterion is applied to discharge the stress/strain concentration problem. It is shown that the proposed equivalent approach can not only derive a considerable accurate global solution but also significantly reduce the computational CPU time. Note that the proposed methodology is based on the fact that the local CTE mismatch between solder and PCB/BT is negligible; however, as a matter of fact, for some applications, this by no means sustains. For these particular cases, the local CTE mismatch effect should be studied and incorporated into the modeling. In addition, only the temperature-dependent, elastic-plastic material property is considered in the modeling of the equivalent beam while the time-dependent viscous effect, such as creep mechanisms, can still be integrated into the model during the detailed stress/strain analysis of the most reliability-concerned solder joint. The typical thermal-cycling analysis of the package can be as well fully conducted using the proposed methodology as soon as the net displacements of the most critical solder joint in both low- and high-temperature ranges in the thermal cycling are derived. Finally, based on the obtained averaging viscoplastic equivalent strain, the low-cycle thermal fatigue life of the solder joint can then be predicted using a Coffin-Manson’s or Darveaux’s empirical relationship. Acknowledgment 13 The support of the research by the National Science Council of the Republic of China through the grants NSC-87-2212-E007-004 and NSC87-2212-E321-001 is gratefully acknowledged. 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