Constitutive and Damage Accumulation Modeling

Transcription

1 Workshop on Modeling and Data needs for Lead-Free Solders Sponsored by NEMI, NIST, NSF, and TMS February 15, 001 New Orleans, LA Constitutive and Damage Accumulation Modeling Leon M. Keer Northwestern University 1

2 Acknowledgement Collaborator: Morris E. Fine Semyon Vaynman Graduate Students: Shengmin Wen Vladimir Stolkarts (Now with Motorola) Motorola Sponsor: Semiconductor Research Corporation Advance Micro Devices, Inc.

3 Outline Objective Constitutive and damage model Preliminary application results Data needed for lead free solders 3

4 Objectives Build constitutive model with damage so that numerical simulation of conditions encountered in use becomes possible (since experimental simulation is not a choice in some circumstances) Develop materials science based damage formation and evolution law to overcome size limitation Develop theory that gives realistic life prediction and metallurgical directions 4

5 Inelastic Strain Due to presence of both creep and plasticity in solders, Unified Creep and Plasticity strain rate is used: & in = 3 A S V d n e B S V d n + 1 ( T )N Q RT T m Θ = e for T 3 S v = S R with & = & in N N = S S Hooke s Law: ( tot in T & & & ) & = C : TI where S is the deviatoric stress tensor, and Θ is a diffusivity term, α is back-stress, R is the radius of the yield surface. 5

6 Mechanical Performance: 96.5Sn-3.5Ag Damage along with cyclic loading shown in UNIAXIAL test Sn-3.5Ag isothermal fatigue test at 5 o C Ramp: 1sec, No hold, ε = A 96.5Sn-3.5Ag isothermal fatigue test at 5 o C Ramp: 1sec, No hold, ε = Peak tensile stress Peak compressive stress Cycle number B C True Strain True Stress (MPa) True Stress (MPa) Cycle No 40 Cycle No 7940 Hysteresis loops at A and B (Data from H. Mavoori)

7 Mechanical Performance: 95Sn-5Ag 95Sn-5Ag isothermal fatigue test at 5 o C Ramp: 0sec, No hold, γ = 0.1 in shear A B Maximum shear stress Minimum shear stress C Damage along with cyclic loading shown in SHEAR straincontrolled test (Data from J. Liang, et al, Fatigue Fract. Engng. Mater. Struct., 19:1401 (1991)) Cycle number 7 True Stress (MPa)

8 Damage mechanics & D(ω) Strain equivalence is used, so that: = 1 M : and = *( ) ε σ For isotropic damage: M=(1-D)II = 1 ε ε 1 Corresponding Changes: S 3 1 = S R v 1 D ( ) in &α = 1 D & N & = (1 D)[ C ( θ ) : ( & & in χθ & I ) + C : : C θ θ 1 & ] (σ,ε) (σ,ε) D 8

9 Micro- and Macro-cracks Microcracks and macrocracks (96.5Sn-3.5Ag at ε=0.003, ramp:1sec, no hold, N/N f =1.06, at 5 o C) 9 Photo from H. Mavoori Ph.D. Dissertation (NU1996), Fig.6.3 Load At the end of the test, specimen is full of microcracks. Some of the microcracks have coalesced into macrocracks.

10 Damage Model: Percolation Damage accumulation: Percolation or a serial of independent local events N = 0 N 0 N i N f, Fracture mechanics approach N = 0 (Initial Crack) N i N f 10

11 Nucleation-Extension τ τ t Mura and Nakasone: (J. Appl. Mech., 57:1(1990)) ( G = W 1 W + γ G) n Dislocation Energy Density Fine (Fine: Scripta mater. 4:1007 (000)) W = γ d Cracks formation cycle n cr (Assume pennyshape shear cracks dominate): A = 0 A at n cr W 1 64(1 ν ) τ = 3( ν )E a 3 Dislocations of vacancy type form a crack n cr = 3(1 ( ν ν ) ) π ( d( γ γ τ d )E τ f ) τ 11

12 Physical damage metric ω = N 0 f (n cr ) dn cr f(n cr ) is a probability distribution function for n cr, or the probability of a grain or cell that has a crack formation cycle number of n cr. f(ncr) ω 1.0 ωc ncr Critical cycle distribution Physical damage metric Nf N 1

13 Calculation of D from ω = ) D 0 = g(0) = 0 D g( ω If, then ω= Thus: D = g ( ω ) ω + g ( ω ) ω + L 1 A special power law form: For bulk specimen, f(n cr ) is a constant. Let f(n cr ) = C, then D( ω ) = D c ω ω c η N ω = f (n cr )dn cr = Cdn cr = 0 N 0 CN ω / ω c = N / N f D = D c N N f η 13

14 63Sn-37Pb Eutectic Solder 14

15 Pb-3.5Sn high-lead solder 15 Temp:5~80 o C, ε:0~.16%, Cycle No.4 TMF:5~80 o C, ε=.16% True Strain Peak tensile stress at max strain Peak compressive stress at 0 strain 1 1 number of cycles 15 True peak stresses (MPa) True stresses (MPa) Data Model 15 Temp:5~80 o C, ε:0~.16% Cycle No.8 with damage True stresses (MPa) TMF data from Lawrence R. Lawson True Strain Data Model

16 96.5Sn-3.5Ag lead-free solder 96.5Sn-3.5Ag isothermal fatigue test at 5 o C True Strain True Stress (MPa) Sn-3.5Ag isothermal fatigue test at 5 o C True Strain Data Model Ramp: 1sec, No hold, ε = , No40 Cycle No 40, no damage Data Model Isotermal data from Hareesh Mavoori Ramp: 1sec, No hold, ε = , No Cycle No 7940, damage 16 True Stress (MPa)

17 Database and Model Mechanical properties: Young s Moduli Tensile and shear strengths Elongations to failure Stress-relaxation and Creep Activation energy Thermal coefficient Stress exponents Creep exponents Other material constants strain rate temperature Product design Fatigue: Isothermal fatigue data Thermo-mechanical fatigue Data base Model Fractography and Microstructure data 17

18 Current state Extensive databases exist for Pb-based solders, but almost none are available for lead-free solder (CINDAS collected some data but not enough) Tests not designed for investigation of damage formation and evolution Current modeling is without damage or with damage assumed Fatigue characterization is largely empirical and not materials science based 18

19 Mechanical Properties Properties: Young s Moduli, Shear moduli, Ultimate properties (such as yield, tensile and shear strengths, elongation to failure, etc.), Stress-relaxation and creep, stress and creep exponents, thermal expansion and capacity constants, and others. Current state: What are needed: Data are available but not sufficient Specimens, strain rates, temperatures, set-ups and procedures differ Modeling, validation and comparison not easy to determine Good correlation between bulk and joints (H. Solomon from GE and R. Gagliano, et al, from Northwestern) Creep and relaxation and stress-strain curves under uniaxial and shear Rate and temperature dependence Standardized experimental procedures. Data on actual interconnects (quantitative data in shear) Size effects may be needed from actual joints 19

20 Fatigue tests Tests: Isothermal Thermomechanical Shear Uniaxial (mostly) Ramp time, hold time Temperature range Current state: Different failure definitions, test procedures (most uniaxial) Phenomenological formula must be proved effective for smaller joints and/or under multi-axial stress conditions Intermediate damaged state is seldom investigated What are needed: Standardize definition of fatigue failure for data sharing Intermediate state investigation to monitor damage evolution Design various microstructure specimens from damage perspective to seek better metallurgical measures Test conditions to duplicate field operating conditions 0

21 Fractography Discover microcrack origin, nucleation, evolution, coalescence phenomena Relate the mechanical performance and the damage state and its evolution Understand the failure mechanism What is needed: Testing oriented to damage mechanism identification. 1

22 Microstructure f Microstructure is very important to understand and model the failure mechanisms and can lead to better design: cr 3(1 ) ( d( d )E ) n ( ν γ γ = ν Grain size and its distribution Orientation distribution d Microstructure evolution τ What are needed: Tests on various but wellcharacterized microstructures π τ τ n cr n cr ) τ

23 ( d ) f Grain size and distribution k j m l j k l excellent good fair m poor Use metallography to find out f ( d ) d 3

24 σ α σ τ α α σ Grain orientation Local shear stress range depends on the grain s slip plane orientation: π/4 n g n s π/ α τ α =0.5sinα σ Generally: τ=n s σ n g n g is the normal to the glide plane while n s is the slip direction. Remark: Stereology, numerical simulation, and others, to find grain orientation distribution function 4 m

25 Metallurgical aspects From damage accumulation perspective, metallurgical measures, such as reduced grain size, can improve fatigue life of solder structures. Knowledge from lifetime improvement by metallurgical measures can be used to improve the accuracy of the damage accumulation method. Collaboration between disciplines of materials science and mechanics are needed. 5

26 How accurate can it be? With the data input from prior discussion, a damage accumulation model can be constructed. A question is: Can this model make accurate predictions of reliability? Answer: The model may give statistically significant predictions. 6

27 Conclusion A database for lead-free solders in needed Testing oriented to damage mechanism should be conducted Numerical simulation and computational analysis should play a more important role in the modeling when direct measurement is not available 7