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Terasaki et al.: Use of Modified Accumulated Damage Model (1/11)

[Technical Paper]

Use of Modified Accumulated Damage Model to Predict Fatigue

Failure Lives of Sn-Ag-Cu-based Solder Joints in

Ball-Grid-Array-Type PackagesTakeshi Terasaki*, Hisashi Tanie*, Nobuhiko Chiwata**, Motoki Wakano***, and Masaru Fujiyoshi****

*Hitachi Research Laboratory, Hitachi, Ltd., 832-2, Horiguchi, Hitachinaka, Ibaraki 312-0034, Japan

**Corpolate Development Center, Hitachi Metals, Ltd., 1-2-1 Shibaura, Minato-ku, Tokyo 105-8614, Japan

***Technical Department, NEOMAX KAGOSHIMA Co., Ltd., 50-8 Midori, Izumi, Kagoshima 899-0201, Japan

****Metallurgical Research Laboratory, Hitachi Metals, Ltd., 2107-2 Yasugi, Yasugi, Shimane 692-8601, Japan

(Received December 27, 2011; accepted July 12, 2012)

Abstract

We have developed a modified accumulated damage model that can be used to predict fatigue failure lives of solder joints

in electronic devices. Our model calculates the fatigue failure life of solder on the basis of the damage that accumulates

during crack propagation by using a finite element method and corrects for the dependence of element size on the cal-

culated life by using the Hutchinson-Rice-Rosengren singularity theory. We predicted the fatigue lives of conventional and

copper-core solder bump joints in ball-grid-array packages in thermal cycling tests. The good agreement between these

predictions and experimental results indicates that our model can effectively predict fatigue failure life in solder joints.

Keywords: Solder Joint, Fatigue Life, Fatigue Crack Propagation, Modified Accumulated Damage Model, Finite

Element Method, HRR Singularity Theory, Sn-Ag-Cu, Ball-Grid-Array-Type Package

Fig. 0 •••

1. IntroductionThe trend in recent years toward greater compactness

and component mounting density in electronic equipment

has led to solder joints of electronic components that are

smaller and more diverse in shape. The thermal expansion

properties of the solder joints differ from those of the elec-

tronic components and the printed circuit board, so there

is repeated deformation during a change in temperature

owing to the difference in the thermal deformations of the

materials. This repeated deformation results in fatigue

cracking that originates on the surface of the solder and

propagates slowly within the solder until the joint fails.

The fatigue failure life of solder joints has conventionally

been investigated by using the correlation between the

fatigue failure life determined through durability testing

and fatigue-crack-initiation lives predicted by simulation.

[1, 2] However, that approach requires durability testing

for each solder joint size and shape, and the lifetime mea-

surements are time-consuming and expensive. To shorten

the development time and expedite reliability improve-

ments, there is a need for simulation techniques that

enable quantitative prediction of the fatigue failure lives of

various solder joint shapes.

Fatigue crack propagation in solder is measured at the

test-piece level on the basis of nonlinear fracture mechan-

ics,[3, 4] but there are few reports on applying simulation

methods based on nonlinear fracture mechanics to solder

joints because of the difficulty in creating the analysis

model.[5] In recent years, simulation methods for repro-

ducing the fatigue cracking path by evaluating the occur-

rence and development of fatigue cracking on the basis of

the accumulated damage at each location in the solder

joint have been proposed.[6–8]

Although the accumulated damage model that we pro-

posed[7] has the issue that the calculated fatigue crack

propagation life is dependent on the crack tip element size,

room-temperature fatigue crack propagation experiments

with solder center-cracked plate (CCP) test pieces[12, 13]

This paper was published in Journal of Japan Institute of Electronics Packaging, Vol. 14, No. 4 (2011), pp. 287–295 (in Japanese).

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Transactions of The Japan Institute of Electronics Packaging Vol. 5, No. 1, 2012

show that this problem can be solved by applying a correc-

tion factor calculated according to the Hutchinson-Rice-

Rosengren (HRR) singularity theory.[9–11] We refer to

this as the modified accumulated damage model.

In the work reported here, we tested the validity of the

modified accumulated damage model by comparing the

results for solder joint temperature cycle testing per-

formed on two types of ball grid arrays (BGAs) with simu-

lation results for fatigue failure life, crack propagation

path, and dispersion in the fatigue failure life.

2. Temperature Cycle Testing Method[14]The dimensions and material properties of the BGA

package used in this research are shown in Fig. 1. The sol-

der bumps are arranged with a pitch of 0.5 mm. The pack-

age has a 9 × 9 array of bumps at the center and three rows

of bumps around the outside for a total of 333 bumps. The

BGA package is mounted on one side of an FR-4 board.

The lands on the BGA package have a solder mask defined

(SMD) structure and the lands on the FR-4 board have a

non-SMD (NSMD) structure. The package’s land surfaces

are covered with electrolytic Ni/Au. For the board’s lands,

the combination of Cu and organic solderability preserva-

tive (OSP) is generally used.

Two types of solder joints were evaluated: ordinary sol-

der bumps (conventional bumps) and solder bumps that

have a copper core and excellent resistance to thermal

fatigue (copper-core bumps). The solder composition was

Sn, 3.0 mass% Ag, and 0.5 mass% Cu (Sn-3Ag-0.5Cu) for

both types. The copper cores were coated with a 2-µm Ni

base layer.

The temperature cycle testing was performed in the

temperature range from –40°C to 125°C, with a 20-minute

cycle; 4000 cycles were performed for a set of 15 packages.

The number of cycles to fatigue failure lifetime was defined

as the number of cycles at which the electrical resistance

becomes twice the initial value under continuous monitor-

ing of the resistance.

3. Fatigue Failure Life Prediction Method3.1 BGA solder joint analysis model

To obtain a quantitative fatigue failure lifetime value for

BGA solder joints by using the modified accumulated dam-

age model, it is necessary to obtain the equivalent elastic

strain range in the BGA solder joints. Because the BGA

package used in the temperature cycle testing has 333 sol-

der bumps, a two-step analysis method[15] was used to

both reduce the analysis time and increase the analysis

accuracy. Specifically, we used a global model that has rel-

atively coarse elements for the solder bump to obtain the

solder bumps where the strain is maximum, and then

input the displacement calculated in the vicinity of those

locations to a sub-model for crack propagation analysis.

For the finite element analysis, we used general-purpose

software that was developed in house.

The global finite element method (FEM) model that we

used for conventional bumps is shown in Fig. 2. The con-

ventional bump sub-model is shown in Fig. 3 (a) and the

copper-core bump sub-model is shown in Fig. 3 (b). The

analysis region of the global model was restricted to 1/4 of

the entire model by utilizing symmetry, and 20-node hexa-

hedron elements were used for good accuracy. The sym-

Fig. 1 Cross-sectional diagrams of thermal cycle test pack-age.[14]

Fig. 2 FEM global model.

Fig. 3 FEM sub-model.

3


metry boundary plane is constrained in the direction nor-

mal to the plane, and the nodes for which the plane of

symmetry and the bottom face of the board cross are con-

strained in the z direction. In the sub-models, the solder

and the copper core are approximated by sets of 10-µm

cubes divided into eight-node hexahedron elements. The

element division of cubes in the crack propagation region

has the major advantage of not requiring the time-consum-

ing redivision of elements as the crack shape changes, but

there is the disadvantage of errors in strain distribution

due to surface roughness. Here, our priority was on com-

putational time rather than accuracy, so we chose the divi-

sion of elements by cubes. The node counts of the sub-

models were 29,781 for the conventional bump and 31,529

for the copper-core bump.

To test the effect of correction for the dependence of

crack propagation life on the element size, crack propaga-

tion life was calculated using a sub-model with 5-µm cube

element division. The sub-model node counts were 205,461

for conventional bumps and 219,456 for copper-core

bumps, about seven times as many nodes as for sub-mod-

els using cubic elements with 10-µm sides.

3.2 Material constants and temperature load con-ditions

The material constants used in the analysis are listed in

Table 1. Materials other than the copper core and the sol-

der were assumed to have isotropic elasticity. The plastic

deformation behavior of the copper core was determined

by a reverse analysis so that the load-displacement rela-

tionship of ball compression testing can be reproduced.

The material constants of the solder used were the values

for Sn-3.5Ag-0.75Cu[2] because there are no measured

values for Sn-3Ag-0.5Cu stress-strain behavior.

Those material constants were obtained by approximat-

ing measurements from torsion testing of thin cylindrical

test pieces (cycling frequency: 0.01 Hz) using a multiple-

step increasing amplitude method[17] (Fig. 4) with an

elasto-plastic linear hardening model. The measured cyclic

stress-strain behavior does not exhibit remarkable work

hardening and softening over the entire temperature

range, so the kinematic hardening rule was used. The

approximate material constant, noted in Fig. 4, does not

have good accuracy for the strain range of 0.5% or less, but

from the constraints of the finite element analysis software

that we used and previous experience,[2, 7, 13, 15, 16] we

Fig. 4 Cyclic stress-strain curves for Sn-Ag-Cu at 0.01Hz.[2]

Table 1 Material properties.

ComponentTempe rature (°C)

Young’s modulus

(GPa)

Poisson’s ratio

Coefficient of thermal expansion (×10–6/°C)

Yield stress (MPa)

Strain hardening coefficient

(MPa)

Chip — 170 0.07 3 — —

DAF — 0.28 0.3 115 — —

Interposer — 31 0.2 13 — —

Resin — 29 0.3 8 — —

Pad — 127 0.3 16 — —

PCB — 25 0.2 14 — —

Solder

–50 41.3

0.35 23

48 1100

–15 40.8 36 900

26 38.2 30 700

75 35.5 22 450

125 33.8 17 250

Cu core — 127 0.3 16 60 900

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decided to use it in this research.

Because the solder exhibits strain rate dependence of

the stress-strain behavior and stress relaxation character-

istics, the elasto-plastic creep model[1] or the visco-plastic

model[18] describes the stress-strain behavior with higher

accuracy than the elasto-plastic model does, but the com-

putation time is longer. For that reason, the strain rate

dependence was approximated by using data from the low-

est possible cycling frequency (0.01 Hz) in the testing

range as data that is close to the strain rate that occurs

during the temperature cycle testing. The stress relaxation

characteristic was ignored.

The range of plastic strain that occurs in the solder

bump is obtained as half of the accumulated equivalent

plastic strain that arises in one cycle, given 1.5 cycles of

temperature loading and excluding the initial 0.5 cycles.[1]

To reduce the computation time in the work reported

here, however, we took the equivalent plastic strain that

develops in the solder bump when a change in tempera-

ture from 125°C to –40°C is applied uniformly to the entire

model as the temperature load to be the approximately

equivalent plastic strain range. That approximation pro-

duces error, but we put priority on the computation time.

3.3 Accumulated damage model[7]An overview of the accumulated damage model previ-

ously proposed[7] is presented in Fig. 5. The region of the

analysis object in which cracking is permitted to occur is

divided into elements of equal dimensions and each ele-

ment is assigned a number. Then, function fi, which repre-

sents the accumulated damage to each element, is initial-

ized to 0. This function increases as damage to each

element accumulates, and when it reaches a value of 1,

that element has reached the end of its life (is removed).

Here, the subscript i is the element number.

Next, an elasto-plastic stress analysis that models tem-

perature cycle testing or mechanical load testing is per-

formed, and the equivalent plastic strain that arises in each

element for one cycle of testing, Δεipeq, is computed. Next,

the amount of damage for one cycle, Δ fi, is calculated from

the plastic strain value for each element.

∆

∆f

Cipeqi

p

p

=

ε α

1

(1)

Here, Cp and αp are coefficients of the Coffin-Manson

rule,[19, 20] which shows the relation between the plastic

strain range Δεp and the low cycle fatigue life Nf.

∆ε α

p p fC N p= − (2)

From the obtained Δfi and the accumulated damage

already incurred by each element fi, the number of cycles

needed to reach the end of each element’s life, ΔNi, is cal-

culated using the following equation.

∆

∆N

ffi

i

i= −1

(3)

In the analysis, ΔN is the minimum value of ΔNi because

the finite element analysis should be performed whenever

the crack propagates and the object shape changes. Given

that, the additional damage to each element caused by

increasing the number of cycles by ΔN is given by the fol-

lowing formula.

f f N fi i i→ + ⋅∆ ∆ (4)

The function fi excludes elements that are equal to or

exceed (1-e), and the stress analysis is performed again for

the changed shape. The term e is a constant for eliminat-

ing elements whose accumulated damage value is close to

1 and is introduced to reduce the number of repetitions of

the stress analysis. In the work reported here, an e value of

0.1 was used.

Taking the calculations described above as one step and

repeating that step, we can evaluate the accumulated dam-

age for each element by the linear cumulative damage rule

of Eq. (4),[21] and we can remove elements in order as

they reach the ends of their lives. In this way, the crack

path is calculated automatically and the crack propagation

life for each the crack state is obtained at the same time.

By repeating this procedure from the initial shape up to

failure, we can obtain the failure life.

In this research, the values obtained as the least-squares Fig. 5 Steps in method for predicting solder fatigue life.[7]

5


approximation from the low-cycle fatigue data for Sn-3.5Ag-

0.75Cu at the upper-limit temperature of the temperature

cycle testing (125°C)[2] (Fig. 6) were used as the coeffi-

cients of the Coffin-Manson relationship of Eq. (2), (Cp =

0.316 and ap = 0.5). The reason for using that approxima-

tion is that there is no low-cycle fatigue data for Sn-3Ag-

0.5Cu. The low-cycle fatigue life was measured by torsion

testing of a thin cylindrical test piece with a cycling fre-

quency of 0.1 Hz in the same way as for the stress-strain

behavior.

The least-squares approximated lifetime curve for 125°C

is shown as the solid line in Fig. 6, accompanied by dotted

lines that indicate twice and half the fatigue lifetimes in

each plastic strain range. The data points for –50°C, room

temperature, and 125°C lie mostly in the region between

the dotted lines, and no remarkable temperature depen-

dence is seen. The Coffin-Manson coefficients used in this

research were calculated from this dispersed measure-

ment data, so the lives predicted below exhibit the same

dispersion.

3.4 Crack propagation life correction method[12, 13]

What requires attention in the accumulated damage

model is that the calculated crack propagation life is

dependent on the element size. The reason for that depen-

dence is a strain singularity in which the strain becomes

abruptly large in the vicinity of the crack tip.

The strain singularity at the crack tip in solder and other

materials that undergo plastic deformation was clarified

theoretically by Hutchinson,[9, 10] Rice, and Rosengren[11]

in the HRR singularity model. That model expresses the

crack tip strain component εij for two-dimensional cracking

of a plane stress state in an infinite region by the following

equation.

ε αεασ ε

θij

nn

nn

ijJ

Ir E=

⋅ ⋅ ( )+ −+

00 0

11 (5)

Here, α, σ0, ε0, and n are material constants that approxi-

mate the nonlinear relationship of the equivalent plastic

strain εpeq and the equivalent stress σeq of the material by

the power law of Eq. (6), J is the J integral value, which is a

nonlinear fracture mechanics parameter, I is the integral

constant, r is the distance from the crack tip in a cylindri-

cal coordinate system, and Eij(θ) is a dimensionless func-

tion that depends on the angle θ in the cylindrical coordi-

nate system.

εε

ασσ

peq eqn

0 0=

(6)

Because the J integral is the intensity factor of the strain

distribution, we know from Eq. (5) that crack tip strain is

proportional to the -n/(n+1) power of the distance r from

the crack tip. Equation (5) holds for the vicinity of the two-

dimensional crack tip in an infinite region and is thus appli-

cable for asymptotic solutions of crack tips for general

crack shapes that include three-dimensional cracking.

Nevertheless, it does hold approximately for any crack

shape when the material deformation behavior is modeled

by Eq. (6).

Our proposed crack propagation life correction method

using the HRR singularity model[13] is shown in Fig. 7.

First, we determine the life NUL taking into account appro-

priate limits for the Coffin-Manson relationship[14, 15] of

Eq. (2) and take the plastic strain range as a suitable value

for the equivalent plastic strain range of the crack tip ele-

ments ε ELcorr (Fig. 7 (a)).

ε αcorrEL

p ULC N p= − (7)

The value of NUL is 25: the same as has been reported pre-

viously.[13]

Because we know from Eq. (5) that the mean equivalent

plastic strain range of the crack tip element ε EL is propor-

tional to the (-n/(n+1)) power of the element size LEL, the

appropriate element size LELcorr for calculating the quantita-

tive crack propagation life can be obtained by using Eq.

(8), as shown in Fig. 7 (b).

L LELcorr

ELcalc

ELcorr

ELcalc

nn

= ⋅

− +

ε

ε

1

(8)

Here, LELcalc is the element dimension used in the calcula-

tions, and ε ELcalc is the mean equivalent plastic strain range

of the crack tip element at the crack length taken to be the

crack life. On the other hand, if we substitute Eq. (5) into

Fig. 6 Fatigue life of Sn-3.5Ag-0.75Cu subjected to torsion testing.[2]

6


Eq. (2), the crack propagation life N is proportional to the

µ-th power of the element size.[13] The term µ is

expressed by Eq. (9), where the terms µCS and µAD correct

for the effects of crack shape and accumulated damage,

respectively, and are given values of 0.05 and 0.1, respec-

tively.[12]

µα

µ µ=+

⋅ − + +nn p

CS AD11

1 (9)

Accordingly, the quantitative crack propagation life Ncorr

can be calculated from life Ncalc, obtained by calculation,

using Eq. (10) (Fig. 7(c)).

N NLL

corr calc

ELcorr

ELcalc

= ⋅

µ

(10)

In the crack propagation analysis of actual solder joints,

ε ELcalc changes in each step, so at step j of the repeated

damage calculations, the life increase ΔN jcalc is converted

to quantitative crack propagation life increase ΔN jcorr,

using Eq. (11), which summarizes Eqs. (7), (8), and (10).

∆ ∆N NC N

corrj

calcj p UL

ELcalc

nnp

=⋅

− − +

⋅α µ

ε

1

(11)

The quantitative crack propagation life Ncorr is obtained by

summing ΔN jcorr up to the number of steps m that reach

the crack length taken to be the life (Eq. (12)).

N Ncorr corrj

j

m

==∑∆

1 (12)

The material constant n of the power law of Eq. (6) is

obtained by approximating the stress-strain behavior[2] of

Sn-3.5Ag-0.75Cu for each temperature (Table 2). In the

work reported here, the mean of the n values for all tem-

peratures (8.6) was used to correct the calculated life.

Thus, the correction value µ is 0.942.

4. Results of Temperature Cycling Testing[14]The Weibull plot of the fatigue failure life is shown in

Fig. 8. There were 14 packages failures in the conventional

bump type between 1350 and 3600 cycles and 8 packages

failure in the copper core bump type between 2000 and

3300 cycles. The 50% failure probability life was 2460

cycles for the conventional bumps and 3300 cycles for the

copper-core bumps. The failed bumps were all located in

the corners of the BGA package.

5. Results of Fatigue Failure Life Prediction5.1 Crack propagation behavior

The equivalent plastic strain distribution of the solder

joints of each global model for conventional and copper-

Fig. 7 Procedure for fatigue life adjustment.[13]

(a) Decision on upper limit of plastic strain range

(b) Calculation of appropriate element size

(c) Fatigue life correction

Table 2 Values of power-law index n.

Temperature (°C)

–50 –15 26 75 125 Average

n 4.5 4.5 9.1 11.7 13.3 8.6

Fig. 8 Thermal cycle test results.[14]

7


core bumps is shown in Fig. 9. For both, the maximum

equivalent plastic strain occurred in the solder bumps in

the BGA package corners, which is consistent with the

test results. Therefore, we input the corner solder bump

displacement to the sub-model and calculated the fatigue

crack propagation behavior. The crack propagation behav-

ior is shown in Fig. 10 (a) for the conventional bump and in

Fig. 10 (b) for the copper-core bump; the corrected crack

propagation life is also shown. The black areas in the fig-

ure are surfaces of equal accumulated damage of 0.5,

which is the equivalent of a fatigue crack. The number of

computation steps to failure was 77 for conventional bumps

and 121 for copper-core bumps.

The conventional bumps developed fatigue cracking

near the BGA-side lands that were coated with solder

resist around the margins. Almost no fatigue cracking

developed near the board-side lands, which were not

coated with the solder resist. The fatigue cracking surface

area near the BGA lands occupied about 2/3 of the joint

surface area at about 1300 cycles, and failure was pre-

dicted to occur at about 1600 cycles. The test results show

failures between 1350 and 3600 cycles, confirming that

fatigue failure life can be predicted roughly.

Fatigue cracking in copper-core bumps, on the other

hand, was seen near the BGA lands in addition to the main

fatigue cracking near the board-side lands. The fatigue

cracking surface area near the board lands covered about

half of the joint surface area at about 1300 cycles and

cracking in about 1/3 of the joint surface area was seen

near the BGA lands. After that, the cracking developed

mainly along the copper core interface, with failure pre-

dicted at 4000 cycles. In the testing, 8 out of 15 packages

failed before 4000 cycles, confirming that fatigue failure

life can be predicted roughly for the copper-core bumps as

well. Thus, fatigue cracking in the copper core solder

bumps was not concentrated at a particular joint interface,

but developed at multiple joint interfaces, so we think that

copper core bumps have superior thermal fatigue charac-

teristics.

A cross-section observation of a copper-core bump after

2200 cycles of temperature cycle testing[17] is presented

in Fig. 11 (a). Cracking has developed over most of the

bottom of the copper core (board side), but only half that

amount of cracking is seen at the top (BGA side). The

crack propagation prediction results shown in Fig. 11 (b)

are not perfectly consistent with the cross-section observa-

tion, but they do reproduce well the tendencies, such as

greater crack propagation at the bottom of the copper core

than at the top.

5.2 Effect of correcting for element size depen-dence

The crack propagation life and the crack shapes for each

element size and each bump shape before and after correc-

tion for element size dependence are shown in Fig. 12. The

cracks are represented by the surfaces of equal accumu-

lated damage of 0.5, as described in section 5.1 . The crack

Fig. 9 Equivalent plastic-strain distribution in solder joints.

(a) Conventional solder bump

(b) Cu-cored solder bump

Fig. 10 Crack propagation behavior examples.


(b) Cu-cored solder bump Fig. 11 Crack locations in Cu-cored solder bump after 2200 cycles.

(a) Thermal cycle test (b) Calculation

8


propagation lives after correction were about 800 cycles

and 1300 cycles.

For both the conventional and copper-core bumps, the

crack propagation was about the same for any element size

if we look at the number of cycles after correction. For the

same crack propagation state, the crack propagation life

before correction differed by a factor of about 1.5, confirm-

ing that the proposed correction method can correctly

compensate for the dependence of the solder bump crack

propagation life on element size.

For the conventional bump, the model with an element

size of 5 µm produced somewhat more crack propagation

on the board side and less crack propagation on BGA side

by the same amount. The balance in crack propagation at

the top and bottom of the solder bump had a large effect

on the fatigue failure life, as shown in the next section, so

using a smaller element should lead to somewhat longer

fatigue failure life predictions.

The corrected element size LELcorr at each step for con-

ventional bumps and copper-core bumps is shown in Fig.

13. The value of LELcorr is close to 1 µm regardless of the

bump or element size. Furthermore, LELcorr is small in the

initial step for either type of bump. It increases as the steps

advance and becomes more or less constant except for the

10-µm conventional bump. As we see from Eq. (8), LELcorr

is determined by ε ELcalc, the mean equivalent plastic strain

range of the crack tip element. Thus, the change in LELcorr

described above means that ε ELcalc is large in the first step

and decreases as the steps advance and thereafter has a

roughly constant value.

5.3 Evaluation of the effects of factors affecting life-time

In crack propagation analysis, a single value is obtained

as the fatigue failure life, but in temperature cycle testing,

Fig. 12 Correction of calculated crack-propagation life.

Fig. 13 Proper element size at each step.

(a) Conventional bump around 800 cycles

(b) Conventional bump around 1350 cycles

(c) Cu-cored bump around 750 cycles

(d) Cu-cored bump around 1350 cycles

(a) Conventional solder Bump

(b) Cu-cored solder Bump

9


there is much dispersion in the fatigue failure life values

(Fig. 8). The causes of that dispersion, in addition to the

dispersion in the fatigue life of the solder material shown

in Fig. 6, include voids in the solder bumps and dispersion

in the diameters of openings in the solder resist (SR open-

ing diameter) and land diameters, and variations in the vol-

ume of the solder bumps and the amount of soldering

paste supplied. The effects of voids on fatigue failure life

have already been investigated using crack propagation

analysis, and the results agree well with test results.[16]

The effects of SR opening diameter have been investi-

gated experimentally by Matsuzaki et al., who described

them as the main cause of dispersion in fatigue failure

life.[22] Although there are no quantitative measurements

of variations in SR opening diameter, it is empirically possi-

ble that the ordinary deviation from the design value of

φ0.25 mm is ±0.01 mm, or at most ±0.02 mm. Therefore,

we predicted the fatigue failure life using our crack propa-

gation analysis for the conditions described in section 3.2

for the case of ±0.02-mm variation from the design value

for the SR opening diameter of 0.25 mm. The sub-model

element size was 10 µm.

Our predictions for conventional and copper-core bump

fatigue failure lives for the SR opening diameter range

from 0.23 mm to 0.27 mm are presented in Fig. 14. The

error bars in the figure indicate ranges predicted on the

basis of the variation in the solder material fatigue life (the

number of cycles for half and double the fatigue life in

each plastic strain range). The range in which failures

occurred in the temperature cycle testing and the 50% fail-

ure probability life are also noted in the figure. It appears

that the dispersion in the temperature cycle testing results

can be mostly explained by the dispersion in the solder

material fatigue life, but confirmation requires a detailed

fatigue life distribution of the solder material, which is cur-

rently unavailable.

For the conventional bumps, an increase in the SR open-

ing diameter extends the fatigue failure life. In particular,

the fatigue failure life at φ0.27 mm is more than twice that

at φ0.26 mm (5000 cycles or more). The predicted fatigue

failure life obtained for conventional bumps roughly over-

laps the range of the temperature cycle testing results.

The opposite is true for the copper-core bumps. The

fatigue failure life increases as the SR opening diameter

decreases, reaching 5000 cycles or more at φ0.24 mm or

less. The fatigue failure life range obtained for the copper-

core bumps does not cover the interval between 2000 to

3000 cycles, which is often seen in the temperature cycle

testing. We think that this results from the actual bump

shape differing somewhat from the shape used in the anal-

ysis model (Fig. 11). The 50% failure probability life of con-

ventional bumps and copper-core bumps is roughly consis-

tent with the fatigue failure life for SR opening diameters

near φ0.26 mm. SR opening diameter deviations to about

φ0.26 mm were considered, but the effects of other shape

factors must also be studied and taken into account.

The crack shapes at the time of failure for conventional

bumps with diameters ranging from 0.24 mm to 0.26 mm

are shown in Fig. 15. As the SR opening diameter

increased, the amount of crack propagation on the board

NSMD pad side increased. That is the result of higher

bump rigidity on the BGA side for larger SR opening diam-

eters and distribution of the load on the NSMD side, where

rigidity was already high. That load distribution increased

fatigue failure life. The crack shapes at the time of failure

for copper-core bumps ranging from φ0.24 mm to φ0.26

mm are shown in Fig. 16. Copper-core bumps failed on the

board side, and as the SR opening diameter increased,

crack propagation decreased on the BGA side, and the

load concentrated on the board side.

The investigation described above shows that the SR

opening diameter has a large effect on fatigue failure life

and that the predicted fatigue failure life roughly repro-

duces the temperature cycle test results.


(b) Cu-cored solder bump

Fig. 14 Effect of diameters of solder resist opening on fatigue failure life.

10


6. ConclusionTo reduce the great amount of work required to test sol-

der joint fatigue failure life, we developed a simulation

method (accumulated damage model) that can predict

quantitative fatigue failure life in a short time and verified

the prediction accuracy for two types of BGA solder joints

with temperature cycle testing. This method places greater

priority on a large reduction in computation time than on

prediction accuracy. It uses (a) cube-shaped elements in

the crack propagation region, (b) an elasto-plastic model

that ignores the strain rate dependence of the solder

stress–strain behavior and the stress relaxation character-

istics, and (c) an approximation of the equivalent plastic

strain range; moreover, it assumes that the HRR stress sin-

gularity holds for three-dimensional crack tips. In this

report, we obtained the following conclusions.

(1) Fatigue failure lives predicted by the developed

simulation method agree with BGA solder joint

temperature cycle testing results for both the con-

ventional solder bumps and the copper core solder

bumps.

(2) The proposed life correction method eliminates

the dependence of calculated number of cycles on

element size. Accordingly, a quantitative fatigue

failure life prediction can be made in a short time

with a small analysis model.

(3) Crack-propagation-analysis of the degree of effect

of SR opening diameter, which is considered to be

a cause of dispersion in fatigue failure life results

obtained from temperature cycle testing, showed

that an SR opening diameter variation of ±0.02 mm

can reproduce most of the dispersion in the tem-

perature cycle test results.

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11


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Takeshi TerasakiHisashi Tanie Nobuhiko ChiwataMotoki Wakano Masaru Fujiyoshi

fig. 0 ••• use of modified accumulated damage model to ... · the accumulated damage at each...

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