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
1
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
2
Printed circuit board (PCB)
• Solder joints providemechanical& electricalconnection between the siliconchip and the printed circuit board.
• Repeated switching of the device→ temperature fluctuations→ fatigue of thesolder joints→ device failure.
3
Solder bump
• Interconnects failure contributes by up to20 % to device failure.
4
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.
5
Geometry and loading: solder bump
x
0.1
mm
0.1 mmx
y
U
Lead
Tin
• Plane strain formulation, thickness= 1 mm.
• Elastic properties:Tin (E = 50 GPa,ν = 0.36), Lead(E = 16 GPa,ν = 0.44) .
• Loading: cyclic mechanical withUmaxx = 1µm.
6
Cohesive zone method: cohesive zone?
continuum element
1 2
3 4
∆ cohesive zone
continuum element
t
n
• Cohesive zones are embedded between continuum elements.
• Constitutive behavior: specified through a relation betweentheseparation∆ (initially = 0) and a correspondingtractionT(∆).
7
Cohesive zone method: stiffness matrix and nodal force vector
• The cohesive zone nodal displacement vector is constructed in the local frameof reference (t,n):
uT = {u1t , u
1n, u
2t , u
2n, u
3t , u
3n, u
4t , u
4n}.
• The relative displacement vector∆ is then calculated as:
∆ =
∆t
∆n
= Au
whereA is a matrix of the shape functions:
A =
−h1 0 −h2 0 h1 0 h2 0
0 −h1 0 −h2 0 h1 0 h2
and
h1 =1
2(1− η), h2 =
1
2(1 + η).
The parameterη is defined at the cohesive zone mid plane and varies between−1 at nodes (1,3) and1 at nodes (2,4).
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• The cohesive zone internal nodal force vector and stiffness matrix are now writ-ten as:
f =∫S
ATT dS =l
2
∫ +1
−1ATT dη
K =∫S
ATBA dS =l
2
∫ +1
−1ATBA dη
whereS is the cohesive zone area,l is the cohesive zone length andB is thecohesive zone constitutive tangent operator given by:
B =
∂Tt
∂∆t
∂Tt
∂∆n
∂Tn
∂∆t
∂Tn
∂∆n
.
• Finally, K andf are transformed to the global frame of reference (x,y).
9
Cohesive tractions: monotonic loading
−1 0 1 2 3 4 5 6
−2
−1
0
1
(a)
∆n/δ
n
Tn/σ
max
−3 −2 −1 0 1 2 3
−1
0
1
(b)
∆t/δ
tT
t/τm
axCohesive zone monotonic normal(a) and shear(b) tractions.
• Characteristics:peak tractionandcohesive energy.
• The softening branch is the energy dissipation source.
10
Cohesive tractions: cyclic loading
• A linear relation is assumed between the cohesive traction and the correspondingcohesive opening:
Tα = kα(1−Dα)∆α
wherekα is the initial stiffness andα is either the local normal (n) or tangential(t) direction in the cohesive zone plane.
• Energy dissipation is accounted for by thedamage variableD.
• The damage variable is supplemented with anevolution law:
D = f (∆,∆, T,D, ...).
11
Cyclic loading: damage evolution
• Evolution law (motivated by Roe and Siegmund, 2003):
Dα = cα|∆α| (1−Dα + r)m
|Tα|1−Dα
− σf
wherecα, r,m are constants andσf 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).
12
Uniaxial cyclic tension-compression example
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Geometry:L = 20µm,R = 10µm.Loading: axial sinusoidal displacementU with amplitude of0.2µm.Continuum:E = 30 GPa,ν = 0.25.Cohesive zone:k = 106 GPa/mm, c = 100 mm/N, σf = 150 MPa, r = 10−3,m = 3.
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Initial cohesive stiffness
High initial stiffness→ minimize artificial enhancement of the overall compliance.
For a bar containingn equally spaced cohesive zones:
σ =(U − n∆)
LE,
T = k(1−D)∆.
Stress continuity→ σ = (U/L)E∗, whereE∗ is given as:
E∗ =
1− 1kLnE (1−D) + 1
E.To ensure a negligible enhancement of the overall compliance→ nE
kL << 1.
In a two-dimensional model the condition is estimated byEkl << 1, wherel ≈ L/n
is the average cohesive zone length.
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0 200 400 600 800 1000−0.15
−0.1
−0.05
0
0.05
0.1
0.15
N (cycles)
F (
N)
(b)
−0.05 0 0.05 0.1 0.15 0.2−400
−200
0
200
400
∆ (µ m)
T (
MPa
)
(a)
(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.
15
F versusN : experimental (Erik de Kluizenaar: Philips).
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0 20 40 60 80 100−0.15
−0.1
−0.05
0
0.05
0.1
0.15
N (cycles)
F (N
)
(a)
0 500 1000 1500 2000−0.15
−0.1
−0.05
0
0.05
0.1
0.15
N (cycles)
F (N
)
(b)
Different damage parameters:(a) r = 10−3,m = 1. (b) r = 0,m = 3.
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0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
N (cycles)
D
(a)
εmean
= 0 ε
mean = 0.5 %
0 100 200 300 4000
0.2
0.4
0.6
0.8
1
N (cycles)
D
(b)
H−LL−H
(a) Mean strain effect.(b) Load sequencing effect.
H–L: 200 cycles atεmax = 1 % followd by 200 cycles atεmax = 0.5 %
L–H: 200 cycles atεmax = 0.5 % followd by 200 cycles atεmax = 1 %
18
Cohesive parameters: solder bump
czg1
czg2
czg3
czg4
• Initial cohesive zone stiffnesskα = 106 GPa/mm.
– Sufficiently high compared to continuum stiffness. Identical for all cohesivezone groups.
• Damage coefficientcα in [mm/N]: czg1 : 0, czg2 : 25, czg3 : 100, czg4 : 0.
• σfα = 0 MPa, r = 10−3.
19
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 than90 % in some cases.
20
Results: damage distribution
N = 500; Deff
= 0.14 N = 1000; Deff
= 0.22
Damage distribution in the solder bump at different cycles.Red linesindicatedamaged cohesive zones (Di
eff ≥ 0.5).
Dieff = (Di
n2
+ Dit2 −Di
nDit)
1/2
21
N = 2000; Deff
= 0.31 N = 8000; Deff
= 0.4
22
0 2000 4000 6000 80000
0.1
0.2
0.3
0.4
0.5
N (cycles)
Def
f
The total effective damage versus the number of cycles.
Thetotal effective damageis calculated by averaging over all cohesive zones:
Deff =1
S
N∑iDi
eff Si
whereDieff is the effective damage at cohesive zone (i).
23
0 2000 4000 6000 8000−8−6−4−2
02468
N (cycles)
F x (N
)
The reaction force versus the number of cycles.
• Slow softening followed by rapid softening (Kanchanomai et al., 2002)
24
S-N curve
1 2 3 4 5 6−3
−2.5
−2
−1.5
−1
−0.5
log(2Nf)
log(
ε max
)FEM linear fit
Applied strainεmax versus the number of reversals to failure2Nf .
25
• Finite element data can be fitted with theCoffin-Mansonmodel:
εmax = a(2Nf)b
a: fatigue ductility coefficientb: fatigue ductility exponent
• Failure criteria:50% reduction in the reaction force−→ a = 0.83, b = −0.49.
• Reduction of25% or 75%→ same value ofb.
• Change by±50 % in the Young’s modulii→ same value ofb.
26
Effect of the elastic parameters
0.5 0.75 1 1.25 1.50
1
2
3
4
E/Er
Nf/N
fr
Variation ofNf withE at εmax = 1%. Fitting curve:Nf/Nrf = (E/Er)−1.83.
27
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.
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Movie ...
29
Thank you
Questions?
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