1. Fatigue damage modeling in solder interconnects
using a cohesive zone approach
Adnan Abdul-Baqi, Piet Schreurs, Marc Geers
AIO-Meeting: 03-06-2003
Supported by Philips

2. Outline
• Introduction
• Geometry and loading
• Cohesive zone method:
– Cohesive zone formulation
– Cohesive tractions
– Damage evolution law
– One dimensional example
• Results:
– Damage distribution
– Corresponding total effective damage and reaction force
– Life-time prediction in comparison with empirical models
• Conclusions

3. Printed circuit board (PCB)
• Solder joints provide mechanical & electrical connection between the silicon
chip and the printed circuit board.
• Repeated switching of the device → temperature fluctuations → fatigue of the
solder joints → device failure.

4. Solder bump
• Interconnects failure contributes by up to 20 % to device failure.

5. Tin-Lead solder
Typical Tin-Lead microstructure (A. Matin).
• Simplified microstructure is chosen for the simulations:
– Physically: rapid coarsening → continuous change.
– Numerically: Large number of degrees of freedom → time consuming.

6. Geometry and loading: solder bump
Ux
0.1 mm
Lead
y
Tin
x 0.1 mm
• Plane strain formulation, thickness = 1 mm.
• Elastic properties: Tin (E = 50 GPa, ν = 0.36), Lead (E = 16 GPa, ν = 0.44) .
max
• Loading: cyclic mechanical with Ux = 1 µm.

7. Cohesive zone method: cohesive zone?
continuum element
3 n 4
t ∆ cohesive zone
1 2
continuum element
• Cohesive zones are embedded between continuum elements.
• Constitutive behavior: specified through a relation between
the separation ∆ (initially = 0) and a corresponding traction T(∆).

8. Cohesive zone method: stiffness matrix and nodal force vector
• The cohesive zone nodal displacement vector is constructed in the local frame
of reference (t,n):
uT = {u1, u1 , u2, u2 , u3, u3 , u4, u4 }.
t n t n t n t n
• The relative displacement vector ∆ is then calculated as:
 

 ∆ 
t 
∆= = Au
 ∆n 

where A is a matrix of the shape functions:
 
−h1 0 −h2 0 h1 0 h2 0
A= 
0 −h1 0 −h2 0 h1 0 h2
 
and
1 1
h1 = (1 − η), h2 = (1 + η).
2 2
The parameter η is defined at the cohesive zone mid plane and varies between
−1 at nodes (1,3) and 1 at nodes (2,4).

9. • The cohesive zone internal nodal force vector and stiffness matrix are now writ-
ten as:
l +1
f = S ATT dS = −1 ATT dη
2
l +1
K = S ATBA dS = −1 ATBA dη
2
where S is the cohesive zone area, l is the cohesive zone length and B is the
cohesive zone constitutive tangent operator given by:
∂Tt ∂Tt 
 

∂∆t ∂∆n 
 

B= 
.

∂Tn ∂Tn 


 
 
∂∆t ∂∆n
• Finally, K and f are transformed to the global frame of reference (x,y).

10. Cohesive tractions: monotonic loading
1 1
Tn/σmax
0
Tt/τmax
0
−1
−2 (a) −1 (b)
−1 0 1 2 3 4 5 6 −3 −2 −1 0 1 2 3
∆ /δ ∆ /δ
n n t t
Cohesive zone monotonic normal (a) and shear (b) tractions.
• Characteristics: peak traction and cohesive energy.
• The softening branch is the energy dissipation source.

11. Cohesive tractions: cyclic loading
• A linear relation is assumed between the cohesive traction and the corresponding
cohesive opening:
Tα = kα (1 − Dα )∆α
where kα is the initial stiffness and α is either the local normal (n) or tangential
(t) direction in the cohesive zone plane.
• Energy dissipation is accounted for by the damage variable D.
• The damage variable is supplemented with an evolution law:
˙ ˙
D = f (∆, ∆, T, D, ...).

12. Cyclic loading: damage evolution
• Evolution law (motivated by Roe and Siegmund, 2003):
 
|Tα |
Dα = cα |∆α | (1 − Dα + r)m 
˙ ˙  − σf 

1 − Dα
where cα , r, m are constants and σf is the cohesive zone endurance limit.
• Satisfies main experimental observations on cyclic damage:
– Damage increases with the number of cycles.
– The larger the load, the larger the induced damage.
– Damage is larger in the presence of mean stress/strain.
– Load sequencing: cycling at a high stress level followed by a lower level
(H–L) causes more damage than when the order is reversed (L–H).
σf = 0 −→ linear damage accumulation (Miner’s law).

13. Cohesive zone: k = 106 GPa/mm, c = 100 mm/N, σf = 150 MPa, r = 10−3, m = 3.
Continuum: E = 30 GPa, ν = 0.25.
Loading: axial sinusoidal displacement U with amplitude of 0.2 µm.
Geometry: L = 20 µm, R = 10 µm.
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Uniaxial cyclic tension-compression example

14. Initial cohesive stiffness
High initial stiffness → minimize artificial enhancement of the overall compliance.
For a bar containing n equally spaced cohesive zones:
(U − n∆)
σ= E,
L
T = k(1 − D)∆.
Stress continuity → σ = (U/L)E ∗, where E ∗ is given as:
 
1
E ∗ = 1 − kL
  E.

nE (1 − D) + 1
nE
To ensure a negligible enhancement of the overall compliance → kL << 1.
E
In a two-dimensional model the condition is estimated by kl << 1, where l ≈ L/n
is the average cohesive zone length.

15. 400
0.15
(a)
0.1 (b) 200
T (MPa)
0.05
F (N)
0 0
−0.05 −200
−0.1
−0.15 −400
0 200 400 600 800 1000 −0.05 0 0.05 0.1 0.15 0.2
N (cycles) ∆ (µ m)
(a) Reaction force vs. cycles to failure. (b) Cohesive traction vs. opening.
• Assumption: damage does not occur under compression:
– Physically: infinite compressive strength.
– Numerically: minimizes inter-penetration (overlapping) of neighboring con-
tinuum elements under compression.

16. F versus N : experimental (Erik de Kluizenaar: Philips).

17. 0.15 0.15
0.1 (a) 0.1 (b)
0.05 0.05
F (N)
F (N)
0 0
−0.05 −0.05
−0.1 −0.1
−0.15 −0.15
0 20 40 60 80 100 0 500 1000 1500 2000
N (cycles) N (cycles)
Different damage parameters: (a) r = 10−3, m = 1. (b) r = 0, m = 3.

18. 1 1
H−L
0.8 0.8 L−H
0.6 εmean = 0 0.6
D
εmean = 0.5 %
D
0.4 0.4
0.2 (a)
0.2 (b)
0 0
0 200 400 600 800 1000 0 100 200 300 400
N (cycles) N (cycles)
(a) Mean strain effect. (b) Load sequencing effect.
H–L: 200 cycles at max = 1 % followd by 200 cycles at max = 0.5 %
L–H: 200 cycles at max = 0.5 % followd by 200 cycles at max = 1 %

19. Cohesive parameters: solder bump
czg1
czg2
czg3
czg4
• Initial cohesive zone stiffness kα = 106 GPa/mm.
– Sufficiently high compared to continuum stiffness. Identical for all cohesive
zone groups.
• Damage coefficient cα in [mm/N]: czg1 : 0, czg2 : 25, czg3 : 100, czg4 : 0.
• σα = 0 MPa, r = 10−3.
f

20. Computational time reduction
• Loading is applied incrementally.
• For large number of cycles → time consuming.
• Computational time reduction: only selected cycles are simulated.
• Time reduction of more than 90 % in some cases.

21. Results: damage distribution
N = 500; Deff = 0.14 N = 1000; Deff = 0.22
Damage distribution in the solder bump at different cycles. Red lines indicate
i
damaged cohesive zones (Deff ≥ 0.5).
2 2
Deff = (Dn + Dt − DnDt )1/2
i i i i i

22. N = 2000; Deff = 0.31 N = 8000; Deff = 0.4

23. 0.5
0.4
0.3
eff
D
0.2
0.1
0
0 2000 4000 6000 8000
N (cycles)
The total effective damage versus the number of cycles.
The total effective damage is calculated by averaging over all cohesive zones:
1 N
Deff = Deff S i
i
S i
i
where Deff is the effective damage at cohesive zone (i).

24. 8
6
4
F (N) 2
0
x
−2
−4
−6
−8
0 2000 4000 6000 8000
N (cycles)
The reaction force versus the number of cycles.
• Slow softening followed by rapid softening (Kanchanomai et al., 2002)

25. S-N curve
−0.5
FEM
) −1 linear fit
max
−1.5
log(ε
−2
−2.5
−3
1 2 3 4 5 6
log(2N )
f
Applied strain max versus the number of reversals to failure 2Nf .

26. • Finite element data can be fitted with the Coffin-Manson model:
max = a(2Nf )b
a: fatigue ductility coefficient
b: fatigue ductility exponent
• Failure criteria: 50% reduction in the reaction force
−→ a = 0.83, b = −0.49.
• Reduction of 25% or 75% → same value of b.
• Change by ±50 % in the Young’s modulii → same value of b.

27. Effect of the elastic parameters
4
3
r
Nf/Nf
2
1
0
0.5 0.75 1 1.25 1.5
E/Er
Variation of Nf with E at max = 1%. Fitting curve: Nf /Nfr = (E/E r)−1.83.

28. Conclusions
• Evolution law captures main cyclic damage characteristics.
• The model’s prediction of the solder bump life-time agrees with the Coffin-
Manson model.
• More efficient computational time reduction scheme:
−→ simulation of larger number of cycles.
−→ more realistic microstructure.

29. Movie ...

30. Thank you
Questions?