High strain-rate mechanical properties of SnAgCu leadfree alloys

High Strain-Rate Mechanical Properties of SnAgCu Leadfree Alloys Pradeep Lall, Sandeep Shantaram, Mandar Kulkarni, Geeta Limaye, Jeff Suhling Auburn University Department of Mechanical Engineering NSF Center for Advanced Vehicle and Extreme Environment Electronics (CAVE3) Auburn, AL 36849 Tele: (334) 844-3424 E-mail: [email protected] Abstract Electronic products are subjected to high G-levels during mechanical shock and vibration. Failure-modes include solder-joint failures, pad cratering, chip-cracking, copper trace fracture, and underfill fillet failures. The second-level interconnects may be experience high-strain rates and accrue damage during repetitive exposure to mechanical shock. Industry migration to leadfree solders has resulted in proliferation of a wide variety of solder alloy compositions. Few of the popular tin-silver-copper alloys include Sn1Ag0.5Cu and Sn3Ag0.5Cu. The high strain rate properties of leadfree solder alloys are scarce. Typical material tests systems are not well suited for measurement of high strain rates typical of mechanical shock. Previously, high strain rates techniques such as the Split Hopkinson Pressure Bar (SHPB) can be used for strain rates of 1000 per sec. However, measurement of materials at strain rates of 1100 per sec which are typical of mechanical shock is difficult to address. In this paper, a new test-technique developed by the authors has been presented for measurement of material constitutive behavior. The instrument enables attaining strain rates in the neighborhood of 1 to 100 per sec. High speed cameras operating at 300,000 fps have been used in conjunction with digital image correlation for the measurement of full-field strain during the test. Constancy of cross-head velocity has been demonstrated during the test from the unloaded state to the specimen failure. Solder alloy constitutive behavior has been measured for SAC105, and SAC305 solders. Constitutive model has been fit to the material data. Samples have been tested at various time under thermal aging at 25C and 125C. The constitutive model has been embedded into an explicit finite element framework for the purpose of life-prediction of leadfree interconnects. Test assemblies has been fabricated and tested under JEDEC JESD22-B111 specified condition for mechanical shock. Model predictions have been correlated with experimental data. Introduction Electronic products are generally subjected to high G-loads in shock and vibration environments. The second-level solder interconnects bear a considerable portion of the deformation load subjected on the printed circuit board during mechanical shock and vibration and are susceptible to damage and eventual failure. Previously, eutectic or near eutectic tin-lead based solder joints were widely used in the electronics industry because of their ease of solderability and long term 978-1-61284-498-5/11/$26.00 ©2011 IEEE reliability under a variety of commonly used environmental conditions. In the recent past, the electronics industry has migrated to leadfree solder alloy compositions or so called “green” products under the ROHS initiative. Tin-SilverCopper (SnAgCu or SAC) alloys are being widely used as replacements for the standard 63Sn-37Pb eutectic solder. Properties of leadfree solder alloys at strain rates typically experienced by the solder joint during typical mechanical shock events are scarce. Previously, constitutive material behavior of solder has been studied at high strain rates by using Split Hopkinson Pressure Bar test [Chan 2009, Siviour 2005] high strain rate impact tester [Wong 2008] dynamic Impact tester [Meier 2009]. In this paper, an impact hammer has been used in conjunction with digital image correlation and high-speed video for measurement of material constitutive behavior of leadfree SAC alloys. Previously, researchers have studied the microstructure, mechanical response and failure behavior of leadfree solder alloys when subjected to elevated isothermal aging and/or thermal cycling [Darveaux 2005, Ding 2007, Hsuan 2007, Pang 2004, Xiao 2004] and effects of room temperature aging on lead-free solder alloys properties [Chuang 2002, Coyle 2000, Darveaux 2005, Lee 2002, Pang 2004, Tsui 2002, Zhang 2009] at low strain rate events (<1 per sec). In the past, Digital Image Correlation (DIC) has been used in the electronic industry for various applications. DIC has been used to measure full field displacement and deformation gradient in electronic assemblies subjected to drop and shock [Lall 2007b, 2008a,b, 2009, 2010a,b, Miller 2007, Park 2007a,b, 2008], damping ratio on the surface of the board [Peterson 2008] examination of velocity, rotation, bending on portable products subjected to impact test [Scheijgrond 2005], stresses in solder interconnects of BGA packages under thermal loading [Bieler 2006, Rajendra 2002, Sun 2006, Xu 2006, Yogel 2001, Zhang 2005, Zhou 2001]. Previously, prediction of transient dynamics has been investigated using equivalent layer models [Gu 2005], smeared property models [Lall 2004, 2005], Conventional shell with Timoshenko-beam Element Model and the Continuum Shell with Timoshenko-Beam Element Model [Lall 2006a,b, 2007a-d, 2008a-d], implicit global models [Irving2004, Pitaressi 2004], and global-local submodels [Tee 2003,Wong 2005, Zhu 2001, 2003, 2004]. Explicit sub models [Lall 2009, 2010]. However, the high strain rate properties of leadfree alloys at elevated isothermal aging and room temperature aging are scarce. In this paper, a motion-controlled impact-hammer with a slip-joint has been used for stress-strain measurement of the 684 2011 Electronic Components and Technology Conference solder sample at strain rates in the range of 1-100 per sec. All the SAC solder specimens were prepared in-house. Effect of aging on mechanical behavior of lead free solder have been examined using tensile tester for SAC105 and SAC305 alloys that were aged for various durations (0-1 month) at room temperature (25oC) and at elevated temperature (125oC). All the events were monitored using two high-speed cameras. DIC has been used to measure full field strain contour on each specimen subjected for tensile test. FE model for high-strain rate pull test has been developed and analyzed using both dynamic explicit as well as explicit integration schemes. In addition, test assemblies has been fabricated and tested under high-G level loading condition for mechanical shock. Node based explicit sub-models and finite-element based peridynamic models for test assemblies under high G-level loading have been developed incorporating material properties extracted from high strain rate tensile testing for solder interconnect life prediction. Specimen Preparation Set-Up In the current study, mechanical measurements of aging as well as elevated temperature effects on lead free solders have been performed at high strain rates. Figure 1 shows specimen preparation setup where solder test specimens were formed in high precision rectangular cross-section glass tubes using a vacuum suction process. The solder was first melted in a quartz crucible using a solder pot maintained around 250C. One end of the high precision glass tube is placed inside the rubber tube connected to suction pump. The other end is inserted into the quartz crucible. The molten solder rises in the glass tube by suction pressure applied from the pump. The suction force is regulated through a control knob on the vacuum line so that only a desired amount of solder is drawn into the tube. The glass tube has then been removed from the crucible and room temperature air cooling is employed to cool the specimen. Temperature profile obtained during air cooling is shown in Figure 2. The specimen are examined by X-ray imaging to ensure that the sample is free from void or premature crack in the gage length as shown in Figure 3 and Figure 4. -2 Room Temperature 150 Recorded Temperature 100 -4 -6 Rate of cooling 50 o 200 ( C/sec) 0 Rate of Cooling 250 o Temperature ( C) Cooling Profile (Air Cooling) -8 0 -10 0 50 100 Time (second) 150 Figure 2: Cooling Profile implemented for Specimen Preparation Figure 3: Specimen inside Glass Tube Figure 4: X-ray inspection Experimental Set-Up A motion-control impact-hammer has been used to conduct a high-strain rate test at high velocity. Slip-joint has been incorporated in the load train to allow the cross-head to attain a constant high-velocity prior to loading the specimen. The load frame incorporates a piezoelectric load-cell and a linear voltage displacement transducer. In addition, the specimen is speckle coated and deformation captured with two high-speed video cameras. The force data is captured with high-speed data acquisition system at 5 million samples per sec . The cross-head deformation history has been captured with image tracking software for computation of cross-head displacement and cross-head velocity. Force sensor Speckled solder specimen Specimen length 0.040m Full rod extension 0.0254m T0 = 0 sec at rest Figure 1: Specimen Preparation Setup T1 > T0 zero specimen elongation up to complete slip joint extension T2> T1 specimen pulled at constant high velocity Figure 5: Specimen Configuration with a Slip-Joint 685 900 after the slip-joint reaches the maximum length and engages the specimen. 900 Description of Crosshead Based Strain Rate Measurement Three points have been used to monitor the complete tensile testing event. Point 1 is fixed to the test frame and serves as a reference. Point 2 is attached to the impact hammer and point 3 is attached to the crosshead through the slip-joint as shown in Figure 7. All the 3 targets are mounted such that they are in the same plane with respect to the lens of the camera 2. The motion of the impact hammer causes the downward motion of point 2. Crosshead motion causes the motion of point 3 through the slip-joint mechanism. The initial 0.005m displacement of the cross head is considered for strain and strain rate computation since all the specimens have shown almost or complete failure within this range. 0.500m High-speed CAM 1 3.7m High-speed CAM 2 Displacement vs. Time 0.15m Specimen Target (a) Figure 6: Camera configuration for capturing specimen deformation at high strain-rate. Displacement (m) 0 0 0.05 0.1 0.15 0.2 Contact 0.25 0.3 -0.2 -0.4 Point 2 - Point 1 Point 3 - Point 1 -0.6 -0.8 (b ) Displacement (m) -1 Time (sec) 0 -0.005 0 -0.01 -0.015 -0.02 -0.025 -0.03 -0.035 -0.04 Displacement vs Time Specimen Slip Joint Deformation Elongation 0.005 0.01 0.015 0.02 point 3 Time (sec) Displacement vs Time The high mass of the impact hammer along with the slip-joint enables the cross-head to maintain a constant high velocity during the entire pull-test. The constant velocity during the pull testing provides evidence of elimination of the possible inertial effects experienced in high-speed pull testers under dynamic loading conditions. The slip-joint enables the crosshead to attain the desired velocity prior to any specimen deformation during the test. At time t > 0, the slip-joint undergoes downward direction due to externally applied load. The slip-joint continues to move keeping the specimen stationary till the slip-joint reaches the maximum extended length. The crosshead pulls the specimen at constant velocity (c) Displacement (m) 0 Figure 7: High speed camera (CAM 2) monitoring targets during tensile testing event. -0.001 0 0.002 0.004 0.006 0.008 0.01 point 3 (Fc=1000Hz) -0.002 -0.003 -0.004 -0.005 -0.006 Time (sec) y = -0.8367x Figure 8: Crosshead motion time-history and specimen deformation. 686 Displacement (m) Figure 8a shows the relative motion of point 2 and point 3 with respect to point 1 for complete pull test event. Figure 8b - blue circled region shows the specimen elongation after when point 3 crosses 0.0254m or 1 inch (rigid body motion). Figure 8c shows the displacement as a function of time during specimen deformation. In this paper, uniaxial high strain rate tensile testing has been done under various conditions of thermal-aging and strain rates to determine the constitutive behavior of lead free solder alloys. 0.006 0.005 0.004 0.003 Cross head Velocity =0.8367m/sec 0.002 0.001 0 0.002 0.004 0.006 0.008 Time (sec) Figure 9: Displacement time-history for crosshead velocity of 0.84 m/sec. 0.006 0.005 0.004 0.003 Cross Head Velocity =2.262m/sec 0.002 0.001 0.000 0 0.002 0.004 0.006 0.008 Displacement (m) 0 Figure 12: Strain time-history and strain rate for crosshead velocity of 2.26 m/sec. The cross head velocity was found to be constant during the specimen deformation history, indicated by a straight line deformation time history of displacement. Data has been collected at two different crosshead velocities of 0.84 m-sec1 and 2.26 m-sec-1 with the corresponding strain rates being 20 sec-1 and 55 sec-1 respectively (Figure 9 to Figure 11). The strain rate based on crosshead displacement rate was also found to be constant during the deformation history indicated by the straight line time-history of strain. Measurement of deformation on the loaded test specimen Digital image correlation (DIC) technique is used to measure full-field deformation and the derivative of deformation on the surface of the loaded structure. DIC technique is used in dynamic testing to study deformation for flexible bodies [Reu 2006], material characterization at high strain rate [Tiwari 2005], characterization of the thermal property and fracture behavior of the plastic ball grid array assembly and underfill thin film [Shi 2004], characterization of materials used in electronic packages [Srinivasan 2005], charactering the mechanics of trabecular bone [Bay 1999]. The technique involves the application of speckle pattern on the surface of test structure and track a geometric point on the speckle patterned surface before and after loading in order to compute both in-plane as well as out-of-plane deformation in the structure. Time (sec) Strain Figure 10: Displacement time-history for crosshead velocity of 2.26 m/sec. Cross-Head Strain Rate 20/sec 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 Average specimen failure time 0.004 0.006 0.008 Time (sec) Figure 11: Strain time-history and strain rate for crosshead velocity of 0.84 m/sec. Strain 0 0.002 Cross-Head Strain Rate 55/sec 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 Average specimen failure time 0 0.002 0.004 0.006 Time (sec) Figure 13: 3D-Digital Image Correlation Measurement for a truss member Displacement field quantities are obtained by tracking a geometric point before and after deformation [Zhou 2001, Amodio 2003, Srinivasan 2005, Kehoe 2006, Lall 2007c, Lall 2008c-d, Lall 2009]. The tracking is achieved using digital image processing of speckle pattern on the specimen surface. Figure 13 shows the principle of DIC in 3 dimension case. The sub image at time t = 0 is referred as I1 (r) and at time t > 0 is referred as I2 (r) respectively, which are related as follows: (1) I (r )  I [r  U(r )] I1 (r )  I 2 [r  U(r )] 2 0.008 1 (2) where U(r) is the displacement vector at pixel r = (x,y,z)T . Correlation Criteria used for high-speed tensile testing is Normalized Sum-Squared Difference (NSSD) to correlate the 687 change in a reference pixel in the original image and corresponding reference pixel in the deformed image. Figure 14 shows the images captured by high speed cameras of a speckle patterned test specimen subjected to high speed uniaxial tensile test from time t = 0 to time t > failure time. From these images we note that the rupture occurs closer to centre of the test specimen along cone shaped edge (thickness direction) which forms an angle of approximately 45° to 50° with the original surface of the specimen which in turn indicates that shear is primarily responsible for the failure of SAC305 material. Figure 14: Images captured by the high speed cameras from time t = 0 to time t > failure time of the speckle patterned test specimen subjected to high speed uniaxial tensile test. 250 Figure 16: Stress (σ22) Vs Strain (E22) curve across failure location (Location A) Figure 15 shows the force (F2) Vs displacement (U2) curve for test specimen under high strain-rate at failure location (Location A). Force data is acquired using high-speed force sensors and displacement is computed using DIC technique in conjunction with high speed cameras. Stress component (σ22) in the test specimen along loaded direction is computed based on the acquired force (F2) data and original cross sectional area (5mm  0.5mm). Lagrangian strain components Eij (i, j =1, 2, 3) are computed using DIC technique. Figure 16 shows the stress (σ22) Vs strain (E22) curve for the corresponding uniaxial tensile test. The value of Elastic modulus (E) of the bulk SAC305 material at strain rate approximately 41.5 sec-1 is determined to be 8.56GPa which is critical values while simulating this test. Corresponding average Poisson’s ratio (υ) is determined to be 0.285 Test Repeatability Repeatability of the test has been quantified at different strain rates. Random variation in test conditions can produce variations in the measured material response and reduce the accuracy of the measured data. Figure 17 shows the measured repeatability of the measured displacement of the impact hammer, and specimen deformation after slip-joint engagement for the strain rates of 20 per sec and 55 per sec. In both cases three separate tests show good repeatability indicated by overlapping deformation histories of the impact hammer and the specimen deformation. Relative Displacement vs. Time Contact -1 time = Strain rate 20sec 0.24sec 0 150 Experimental Stiffness K=163.373E+04 N/m 100 Displacement (m) Force (F2) N 200 50 0 0 0.0005 0.001 0.0015 0.002 -0.2 0 -0.6 -1 Figure 15: Force (F2) Vs Displacement (U2) for test specimen under high strain-rate at failure location (Location A). 0.1 Point Point Point Point Point Point -0.4 -0.8 Displacem ent (U2) m eter 0.05 0.15 2 Event 3 Event 2 Event 3 Event 2 Event 3 Event 0.2 0.25 0.3 1 1 2 2 3 3 Time (sec) Strain rate 50sec 100 90 80 70 60 50 40 30 20 10 0 -1 0 Displacement (m) Stress (Sigma 22) MPa Relative Displacement vs. Time E = 8.56 GPa Strain rate = 41.5/sec -0.2 -0.4 -0.6 -0.8 -1 0 0.025 0.05 0.075 0.1 0 0.05 0.1 0.15 Point 2 Event A Point 3 Event A Point 2 Event B Point 3 Event B Point 2 Event C Point 3 Event C 0.2 Contact time = 0.24sec 0.25 Time (sec) Figure 17: Repeatability of the Test Method. Strain (E22) 688 0.3 Stress (Sigma yy) MPa 20 SAC105 30days @25c + 20/sec @25C 10 SAC105 30days @25c + 55/sec @25C 0 Stress (Sigma yy) MPa 0.02 Stress (Sigma yy) MPa SAC105 1day @25c + 55/sec @ 25C 30 0.08 SAC305 30days @25c + 55/sec @ 25C 10 0 0.02 50 SAC105 1day @125c + 20/sec @25C 40 SAC105 1day @125c + 55/sec @25C 30 20 10 0 0.02 0.1 50 40 30 SAC305 1day @25c + 20/sec @ 25C 20 SAC305 1day @25c + 55/sec @ 25C 10 0.02 0.04 0.06 Strain (Eyy) 0.08 0.1 (b) Figure 18: (a) Stress-Strain for 1-day aged SAC105 (b) Stress vs. Strain for 1-day aged SAC305. (  = 20sec-1 and 55 sec-1). 0.04 0.06 Strain (Eyy) (a) 0.08 0.1 50 SAC305 1day @125c + 20/sec @25C 40 SAC305 1day @125c + 55/sec @25C 30 20 10 0 0 0 0 0.1 SAC305 30days @25c + 20/sec @25C 20 0 0.04 0.06 Strain (Eyy) (a) 0.08 40 0 0.02 0.04 0.06 Strain (Eyy) (a) 50 10 0 Stress (Sigma yy) MPa 30 0.04 0.06 0.08 0.1 Strain (Eyy) (b) Figure 19: (a) Stress vs. Strain for SAC105 aged 30 days @25C (b) Stress vs. Strain for SAC305 aged 30 days @RT. (  = 20sec-1 and 55 sec-1). SAC105 1day @25c + 20/sec @ 25C 20 40 0 40 30 50 0 Stress (Sigma yy) MPa Stress (Sigma yy) MPa High Strain Rate Material Data Uniaxial specimen of dimensions length 40mm width 5mm and thickness 0.4mm were formed for the SAC solder alloys using the methods described earlier in the paper. The samples were aged temperatures of T = 25oC, 125oC for periods of 1 days and 30 days. Cooling rates of the solder specimen were kept similar to actual solder interconnects and the sample thickness chosen to be similar to the height of typical solder joint, with similar microstructure in each planar cross-section Specimens for each alloy were prepared in batches followed by subjecting them to a specific set of aging conditions and high strain-rate test (aging temperature and aging time). The procedure was followed to reduce variability in aging conditions due to wait-time after completion of specimen formation, aging and prior to high strain rate test. Data measured from multiple samples was then averaged to obtain the nominal material response at the test condition. Figure 18 to Figure 21 illustrates the recorded stress vs. strain curves for the SAC105 and SAC305 solder alloys after aging for 1 day and 30 days at 25C and 125C. Data has been presented at two strain rates of 20 per sec and 55 per sec. Figure 18 shows the ultimate tensile strength increases with the increase in strain rate for both SAC105 and SAC305 after 1-day of thermal aging at 25C. In addition, the elastic modulus of the material also shows an increase for both SAC105 and SAC305 solders after 1-day of thermal aging at 25C. Figure 19 shows a similar trend of increase in UTS and Elastic Modulus with increase in strain rate after 30 days of aging at 25C for both SAC105 and SAC305. Figure 20 and Figure 21 show a trend of increase in UTS and Elastic Modulus with increase in strain rate after 1 days and 30 days of aging at 125C for both SAC105 and SAC305 50 0.02 0.04 0.06 Strain (Eyy) 0.08 0.1 (b) Figure 20: (a) Stress vs. Strain for SAC105 aged 1-day @125C (b) Stress vs. Strain for SAC305 aged 1-day @125C. (  = 20sec-1 and 55 sec-1). 689 Stress (Sigma yy) MPa Table 4: SAC305 Elastic modulus (GPa) at   55 sec Aging Period Elastic Modulus (GPa) . 50 SAC105 30days @125c + 20/sec @25C 40 SAC105 30days @125c + 55/sec @25C 30 Aging Temperature 1 Day 30 Day 20 25°C 7.54 (Pristine) 2.63 10 125°C 2.15 1.99 Table 5: SAC105 UTS (MPa) at   20 sec Aging Period UTS (MPa) . 0 0 0.02 0.04 0.06 0.08 0.1 Strain (Eyy) (a) Stress (Sigma yy) MPa 1 50 SAC305 30days @125c + 20/sec @25C 40 SAC305 30days @125c + 55/sec @25C Aging Temperature 1 Day 30 Day 25°C 33.73 (Pristine) 31.34 125°C 18.55 17.56 Table 6: SAC105 UTS (MPa) at   55 sec Aging Period UTS (MPa) . 30 20 1 1 Aging Temperature 1 Day 30 Day 10 25°C 37.6 (Pristine) 33.53 0 125°C 23.6 21.6 0 0.02 0.04 0.06 Strain (Eyy) 0.08 Table 7: SAC305 UTS (MPa) at   20 sec Aging Period UTS (MPa) 0.1 . (b) Figure 21: (a) Stress vs. Strain for SAC105 aged 30-days @125C (b) Stress vs. Strain for SAC305 aged 30-days @125C. (  = 20sec-1 and 55 sec-1). Table 1 to Table 4 shows numerical values of the elastic modulus for each of the tested cases. Table 5 to Table 8 provides the ultimate tensile strength (UTS) for each tested cases. Pristine samples exhibit a higher ultimate tensile strength compared to the aged samples in each case. Samples exposed to 30 days of isothermal aging exhibited the lowest ultimate tensile strength of all the cases tested. The SAC105 specimens exhibited longer elongation to failure compared to SAC305 specimen for all aging conditions studied. Table 1: SAC105 Elastic modulus (GPa) at   20 sec Aging Period Elastic Modulus (GPa) . 1 Aging Temperature 1 Day 30 Day 25°C 2.23 1.21 125°C 0.73 Aging Temperature 1 Day 30 Day 25°c 35.74 (Pristine) 33.7 125°c 23.22 22.2 Table 8: SAC305 UTS (MPa) at   55 sec Aging Period UTS (MPa) . Aging Temperature 1 Day 30 Day 25°C 5.02 1.55 125°C 1.019 0.856 Table 3: SAC305 Elastic modulus (GPa) at   20 sec Aging Period Elastic Modulus (GPa) . 1 Aging Temperature 1 Day 30 Day 25°C 2.94 (Pristine) 1.793 125°C 1.3 1.094 Aging Temperature 1 Day 30 Day 25°C 44.48 (Pristine) 38.26 125°C 29.9 23.41 50 1 Stress (Sigma yy) MPa Table 2: SAC105 Elastic modulus (GPa) at   55 sec Aging Period Elastic Modulus (GPa) 1 Effect of Thermal Aging SAC105 and SAC305 alloys properties including elastic modulus and UTS exhibited a reduction with increase in aging time from 1-day to 30-days for both the aging temperatures of 25C and 125C. Figure 22 to Figure 25 show similar trend of reduction in the elastic modulus and the UTS for both strain rates of 20 per sec and 55 per sec. 0.568 . 1 1 day @25c + 20/sec @ 25c 45 40 30 days @25c + 20/sec @ 25c 35 30 1 day @125c + 20/sec @ 25c 25 20 30 days @125c + 20/sec @ 25c 15 10 5 0 0 0.02 0.04 0.06 0.08 Strain (Eyy) Figure 22: Effect of thermal aging on stress-strain behavior of SAC105 at strain rate of 20 per sec. 690 0.1 Stress (Sigma yy) MPa 50 1 day @25c + 55/sec @ 25c 45 40 30 days @25c + 55/sec @ 25c 35 30 1 day @125c + 55/sec @ 25c 25 20 30 days @125c + 55/sec @ 25c 15 10 5 0 0 0.02 0.04 0.06 0.08 0.1 Strain (Eyy) Figure 23:Effect of thermal aging on stress-strain behavior of SAC105 at strain rate of 55 per sec. where f is scalar bond force,    represents current relative position vector connecting particles. Peridynamics concept has been implemented in FE code by creating truss elements as mesh with appropriate stiffness properties which represents the peridynamic bonds [Macek 07]. The discretized form of the equation of motion replaces the integral by finite sum as follows: .. n (7)  u i   f u np  u in , x p  x i Vp  b in Stress (Sigma yy) MPa 50 45 1 day @25c + 20/sec @ 25c 40 35 30 days @25c + 20/sec @ 25c 30 25 1 day @125c + 20/sec @ 25c 20 15 30 days @125c + 20/sec @ 25c 10 5 0 0 0.02 0.04 0.06 0.08 p 0.1 Strain (Eyy) where, f is given by 4, n is the time step number, and subscript denote the node number, therefore: (8) u in  u ( x i , t n ) Figure 24:Effect of thermal aging on stress-strain behavior of SAC305 at strain rate of 20 per sec. Where, Vp represents volume of the node p, which for a uniform rectangular grid is simply x where  is user specified value. The summation is taken over all the nodes p such that x p  x i   . The value of horizon  may depend 50 Stress (Sigma yy) MPa density in the reference configuration, and f is the Pairwise force function which is equal to force density (per unit volume squared) that x’ exerts on x. Relative position of these two particles in reference configuration is given by Equation (4) with their relative displacements as Equation (5).   x ' x (4)   u (x ' , t )  u ( x, t ) (5) In classical theory, bond extends over finite distance based on the idea of contact forces. But in the case of peridynamic theory, bonds for any given particle do not extend beyond the envelope i.e. particle only interact within the envelope. General form of the bond force for this basic theory is given by:  (6) ,  f (, )  f ( y( t ), , t )  45  1 day @25c + 55/sec @ 25c 40 35 30 days @25c + 55/sec @ 25c 30 25 1 day @125c + 55/sec @ 25c 20 15 30 days @125c + 55/sec @ 25c 10 5 0 0 0.02 0.04 0.06 0.08 0.1 Strain (Eyy) Figure 25:Effect of thermal aging on stress-strain behavior of SAC305 at strain rate of 55 per sec. Peridynamics In Finite Element Framework Peridynamic theory applies integration scheme rather than differentiation to compute the forces on a material particle. The main objective of this theory is to reformulate mathematical description of solid mechanics so that the same equation is valid on or off a discontinuity such as crack and voids [Silling 2000, 2003, 2005]. According to peridynamic theory, acceleration of any particle at x in the reference configuration at time t is given by .. (3)  u ( x , t )   f ( u ( x ' , t )  u ( x , t ), x ' x )dVx '  b( x , t ) upon the many factors such as nature of the physical problem to be modeled, maximum number of elements which could be possibly created by commercial codes, importance of capturing the complicated cross-sections along boundary in FE model representing the actual physical structure. Multiplying equation (7) by Vi leads to equation of motion identical in the form to that of finite element analysis: .. n (9) Vi u i   f (u np  u in , x p  x i )Vp Vi  b in Vi p M u  FTn  Fen .. n n where M is the lumped mass matrix, Fe is the external force n vector, and FT is the internal force vector. Each diagonal term of M is Vi and each component of Fe is bi Vi . n n n component of FT is n n  f (u p  u i , x p  x i )Vp Vi , which is the sum of all the forces Similarly, each p Hx where Hx is a peridynamic envelope of radius , u is the displacement field vector, b is the prescribed body force (10) from trusses connected to node i. Therefore, creating a truss assembly and providing appropriate stiffness properties for truss elements according to peridynamics theory are the 2 691 fundamental aspects in order to implement peridynamics in FE code. Peridynamics theory requires creating nodes on concentric circles as shown in Figure 26(a) but for simplicity only uniform rectangular meshes were used for the simulations presented in the paper. When we compare rectangular grid generation with respect to circular grid as shown in Figure 26, certain percent of nodes get eliminated due to rectangular grid formations which are termed as missing nodes as a result certain degree of accuracy will be lost in the simulation. Firstly, to generate the geometry of the structure a uniform rectangular array of nodes is defined and then trusses are generated by connecting any given node i to any other node p that lies within a sphere of radius  centered at node i. Pictorial representation of computation grid with peridynamic envelope is as shown in Figure 27. According to the Figure 27 trusses of only 2 different lengths can be created, this is done to minimize the no. of truss elements suppose to be generated in the commercial FE software. Since, once we expand this element generating idea to develop electronic package FE model with more than 2 different truss lengths, no. of trusses to be generated will exceed the limit of the commercial FE platform capability to generate large no. of truss elements. thickness 2 x containing conventional continuum elements is defined to overlap the peridynamic trusses. Then these trusses are specified to be embedded in the continuum elements as shown in Figure 28. Since the overlap region will become too stiff due to embedded truss elements, elastic modulus and density for the host continuum elements are set to a very small value. Figure 28: Peridynamics based finite element model (Hybrid model) Figure 26: Comparison of circular and rectangular grid formation In the peridynamics based FE model of the uniaxial tensile test specimen, peridynamic trusses are modeled across experimental failure region by actually modeling failure region to reside in the upper part of peridynamic truss region as shown in Figure 29. This is done to verify the model prediction of the experimental failure region. Boundary conditions are defined via nodes located at the top end of the FE model. Both the translation and rotation d.o.f. of all the nodes located at the top side of the model are constrained in X1, X2 and X3 directions. Figure 27: Computation grid Inorder to model the test specimen subjected to uniaxial tensile test ‘  ’ is chosen as 1.4142  x where x represents the rectangular grid spacing as shown in Figure 28. Computation time is minimized through coupling peridynamics truss meshes with conventional FEA mesh using embedded nodes and elements which are available in ABAQUS/explicit commercial code [Macek 07]. Element types C3D8R are used as conventional elements. A band of 692 Figure 29: Peridynamics truss region in FE model Definition of Section and Material Properties Consistent with Peridynamics Theory For a uniform rectangular mesh, definition of cross sectional area A and elastic modulus E for trusses are of the form 10 and 11 respectively [Macek 2007, Silling 2005]: (11) A  x 2 m 2 simulation since there is no penetration of the one material type into another material type. Table 10: Stiffness of each truss sets based on peridynamics. Truss Set 1 Set 2 Elastic E 2  8.1745GPa E 1  8.1745GPa Modulus (E) N/m2 Table 9 indicates the geometric details of truss elements for modeling uniaxial tensile specimen based on peridynamic EA E A approach K1  1 1 K2  2 2 Stiffness L1 L2 Table 9: Geometric details of truss elements EA 4 K  N / m SET Truss 408.7253  10 N / m 289.0149  10 4 N / m SET1 x1 L 2 x 2 Finite element model generation will be complete by Length (L) 5E-04 m 7.071E-04 m modifying the elastic modulus of trusses that are within a Cross (5e-4)*(5e-4) (5e-4)*(5e-4) distance of free surface by a normalization factor derived sectional areas = 25e-08m2 = 25e-08m2 from Equation (15). Elastic modulus Eb(x) of an element near (A) a boundary is [Macek 2007]: (17) p1 E b (X)  1  E Peridynamic (12) 4 N E Peridynami cs  cx  2 p  (X) m (18) E b (X )  E Peridynamic where, 18 k 18 EClassicContinuum 1 N (13) Where Eb(X) is the Elastic modulus near boundary. c    4 6 4 31 2    m Therefore, equation (17) can be re-written as (19) K at ..a ..po int Where, K is the bulk modulus, is the Peridynamic envelope E b (X)   Ei or horizon and has a value of 7.071E-04 m. From Figure 16 K TRUSSi for bulk SAC305, at strain rate ≈ 41.5 sec-1 we have Where i = 1, 2 (trusses based on their initial length). EClassicContinuum =8.56 GPa and υ = 0.25 (inherent Poisson's Hence, the Elastic modulus has been normalized for trusses of ratio for peridynamic theory). Therefore, different length. For defining mass for truss element, truss 9 (14)  18  density of bulk-sac305 is used in the simulation instead of  8 . 56 10 1   c    31  20.25 7.071  10 4 4  setting truss density as zero [Macek 2007] and defining mass   of the node via lumped masses as Vi . Failure criteria N  13.07921  10 22 6 definition for trusses elements, based upon stress-strain m Normalization of spring constant ‘c’ is required if different response from Figure 16 of the bulk SAC305 test specimen is material particles lie within the peridynamic envelope as assumed to be elastic-plastic [Macek 2007]. In-order to implement failure criteria in FEA, truss element yield strain follows: ( y) is set to be equal to experimental yield strain = 0.01 1 (15) c p which can be referred from Figure 16 c( X)  1  p (X) Node-based modeling approach has been implemented to Where, define boundary condition in loaded direction. Previously,  (16) various tensile test simulation approach have been carried P(X )  f ( U, X  x ' )dVx '  out such as a finite displacement rate was applied to one end U H X of the model while the other end was fixed to compute thus, -P(X)Δu = restoring force (per unit volume) that the dependence of ductile crack formation in tensile tests on particle X experiences if it is displaced incrementally through stress triaxiality, stress and strain ratios [Bao 2005] . In this small vector Δu while holding all other points fixed and paper, the global output of the DIC data has been used to c  c .In the current modeling approach, test specimen is develop boundary conditions for the high-speed uniaxial test assumed to be linear, homogeneous and isotropic and hence c specimen [Lall 2009]. The approach has been used to develop will also remain constant. P = Symmetric tensor and +ve correlation of uniaxial test specimen and corresponding FE definite for reasonably behaved materials. It has 3 Eigen model and to predict failure mode and the failure location at values {P1, P2, P3} and P1 is the largest. P = Analogous the equivalent time step. Digital image correlation has been used to extract displacement and velocity at the speckle tensor obtained for large homogeneous body and 3 Eigen patterned surface on test specimen. Non-linear velocity 1 2 3 1 values {P , P , P } with {P } is the largest of three. histories on the test specimen have been measured during the Therefore, we have Elastic modulus (E) and corresponding high speed tensile test. stiffness (K) for two sets of truss elements as shown in Table 10. The short range forces are not considered in this Peridynamics     693 FE Prediction For High Speed Uniaxial Tensile Test Figure 30 shows the finite element model prediction of the stress field for high-speed uniaxial tensile test specimen at various time steps. As expected, stress field is dominant in the upper part of the truss region in FE model. Though the dropping direction is 2, since the trusses has only axial component in ABAQUS it is represented as S11. Model predictions have been correlated with high strain rate tension tests. Time to failure and failure mode is accurately captured by FEA based on peridynamic theory and also failure region predicted by FEA is in the vicinity of true failure region. Figure 31 shows the model predictions of the failure mode and time to failure of 3.2 ms, which correlates well with experiment. Figure 32: PCB (L*B =132* 77mm2 and thickness 1.5mm) and one PBGA-324 package located at centre of the test board Figure 33 Test board showing unique 4 quadrants continuity design for PBGA324 package. Table 11: Package Architecture of 324 I/O PBGA 19mm, 324 I/O, PBGA Ball Count 324 Ball pitch (mm) 1 Die Size (mm) 7.5 Substrate Thickness 0.3 (mm) Substrate Pad Type SMD Ball Diameter (mm) 0.65 Figure 30: Stress field prediction for high-speed uniaxial tensile test at various time steps Figure 31: Time to failure and failure mode predicted by FEM based on peridynamic theory. Application of High Strain-Rate Properties to PBGA Test Assembly at High G-Level Peridynamic via FEM concept has been extended to model the crack phenomena for printed circuit board assembly subjected to drop impact at high G-levels. Test-board used for experimentation has the dimension 132mm  77mm  1.5mm with one PBGA package located at centre of the test board having I/O count of 324 and a pitch of 1mm as shown in Figure 32. Detail of the PBGA package architecture is mentioned in the Table 11. Test board used in this paper has the unique 4 quadrant continuity design for PBGA package as shown in Figure 33 . The purpose of this continuity design is to track the in situ failure location within the package with respect to time as the varying quantity. The stand-off used in this experiment is of 1 inch tall which is taller than the normal stand-off used during JEDEC 0° drop test. The tall stand-off ensure free oscillation of PCB without hitting the base during high G-level tests. Relative movement of the stand-off is also monitored during the actual drop test with single high speed camera to ensure the rigidity of the stand off as shown in Figure 34. Test board was dropped on the impact surface and the corresponding measured acceleration curve is as shown in the Figure 35. The peak G value is found to be 12500g’s. Figure 34: Test board with targets A, B, C to measure relative displacements 694 Repeatability of G-value at drop height 60" Peak G=12500 g's Shock event 1 Voltage (V) (m/sec 2) Failure time (Region B) 10 12000 Acceleration 12 Shock event 2 8000 Shock event 3 4000 8 Time to failure 0.749 millisec 6 4 2 0 -2 0 0 0.0000 0.0001 0.0002 -4 0.0005 0.001 0.0015 0.002 Time (sec) Time (Sec) Figure 35: Measured acceleration curve corresponding to drop height 60inch. Figure 38: Continuity time history in 0°-drop-shock indicating the failure time for various package sub-regions B. The entire drop event was monitored using 2 high speed cameras in order to extract the full field in-plane and out of plane transient strain histories. In the Figure 36 hatched red and blue regions indicates failure locations of the package during high-G level drop test. For this particular test it has been found that package failure time is below 1-ms. Figure 38 shows continuity time histories for this test indicating the failure time for various package sub-regions. The failed package has been cross-sectioned and optical microscope image (Figure 39) used to determine failure cites. Figure 36: Speckle patterned test board indicating failure locations Failure time (Region D) 12 Voltage (V) 10 8 Figure 39: Failure mode Digital Image Correlation for PCB subjected to 0° drop test 3D-DIC measurement concept for a truss member as shown in Figure 13 can be expanded to entire PCB assembly subjected to drop and shock [Lall 2009, 2007c, 2008d]. Fullfield in-plane 2D strain contour at different time steps has been extracted (within 1-ms of the drop event Figure 40, first cycle of the drop event Figure 41 ) Time to failure 0.689 millisec 6 4 2 0 -2 0 0.0005 0.001 Time (sec) 0.0015 0.002 Figure 37: Continuity time history in 0°-drop-shock indicating the failure time for package sub-regions D. Figure 40: DIC based 2D full field strain contour (E11) on board (Within 1-ms of the drop event) 695 implemented in the FE analysis as initial and boundary conditions [Lall 2009]. Relative velocity component (V3) in dropping direction OLT 5000 ORT Velocity (mm/sec) 10000 ILT IRT 0 -5000 0 0.001 0.002 ORB 0.003 IRB OLB -10000 ILB Tim e (sec) Figure 41: DIC based 2D full field strain contour (E11) on board (first cycle of the drop event) Figure 44: Velocity (V3) components along dropping directions of the board at 8 discrete locations using DIC technique. Peridynamics Based FEA of Electronic Package Subjected To Drop Test Simulation of the PCB assembly drop test has been carried out using peridynamic approach within FE code. Coupling of FE and peridynamic scheme is retained to reduce the computational time [Macek 2007].   Table 12: Truss lengths for solder interconnect SET Truss SET1 x1 2 x 2 Length (L) 0.045E-03 m   0.06366E03 m Figure 42: Strain (E11) along the length of the board at centre location and corresponding velocity component in dropping direction. PCB transient strain history (E11) at the centre location with the corresponding velocity time history for this drop test is shown in Figure 42. In the same plot green arrow indicates that the board is undergoing maximum compression at zero velocity as expected. Figure 45: Peridynamic based FE Modeling concept for electronic package across the solder interconnect interface. Figure 43: Speckle patterned test board indicating discrete locations where velocity components (V3) are being extracted using DIC technique Figure 44 shows relative velocity vectors extracted from DIC at discrete locations of the test board. Specific locations have been identified by OLT: outer left top; ILT: inner left top; ORT: outer right top; IRT: inner right top; ORB: outer right bottom; IRB: inner right bottom; OLB: outer left bottom; C: centre. These experimentally extracted data will be Hybridization of the conventional finite elements with peridynamics trusses modeling concept for uniaxial tensile test has been expanded to simulate the PCB assembly drop and shock event. Figure 45 shows peridynamic based FE modeling concept for electronic package across the solder interconnect interface. All corner solder interconnects region has been chosen to be the peridynamic truss region since they are the weakest link in the entire PCB assembly [Lall 2005]. Peridynamic truss region has been deployed on both PCBsolder and substrate-solder side inorder to simulate multiple 696 failure modes such as cracks between copper and solder on package side as well as board side, bulk solder failure, cracks between PCB and solder. This model contains two set of trusses similar to peridynamics based uniaxial tensile test simulation but with the different dimension. Truss lengths modeled for peridynamic region across solder interconnect is mentioned in the Table 12. Elastic modulus (E) for different layers of truss region has been derived similar to the peridynamic E calculation implemented for truss region under uniaxial tensile test. Figure 46 shows the 3D view of the peridynamics based truss elements across the solder interconnect interface with dotted line representing truss regions. Figure 46: 3D view of the peridynamics based truss elements across the solder interconnect interface (elements with in dotted ellipse represents peridynamic truss region) FE Model Correlation With Experiment Figure 47 shows the location of peridynamic based solder interconnects in the finite element model. All the four corners of PBGA is modeled using peridynamic truss elements. Figure 48 shows the damage initiation and damage progression across the left bottom (LB) solder interconnect at various time steps. For the LB solder interconnect which is residing in the red hatched region damage is predicted between the time interval 0.6ms to 1ms and in the actual drop event red hatched region failure time is 0.7ms. Figure 48: Damage initiation and damage progression across left bottom (LB) solder interconnect on board side. This shows the capability of the peridynamic FE modeling approach to predict the damage initiation and damage progression for extremely loaded structures. Mostly for all the corner solder interconnects damage initiates between solder and board side. This is expected since experimentally board has failed with in 1ms from the point of impact (i.e. before board reaching maximum compression state). Figure 47: corner solder balls locations represented as LT, RT, RB and LB. Summary In this paper, detailed procedure of a simple uniaxial highspeed tensile test simulation based on peridynamic approach has been provided. Peridynamic modeling concept has been extended to simulate damage phenomena for the electronic package subjected to controlled drop test. Material properties and failure thresholds have been derived from high-strain rate 697 uniaxial tension tests. In addition, high-speed imaging and DIC measurements on board assemblies have been used to measure the initial and boundary conditions for explicit time integration scheme. Peridynamic based FE analysis of the board assembly has been used to predict the failure location in the second-level solder interconnects. Measurements show that FE based peridynamics is a useful technique to predict damage initiation and damage progression for various test boards with ball-grid array package architecture under extreme high-G loading situations. 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