doi: 10.1111/j.1460-2695.2006.01065.x
Review of methods to predict solder joint reliability under
thermo-mechanical cycling
S. RIDOUT ∗ and C. BAILEY
School of Computing and Mathematical Sciences, University of Greenwich, Park Row, Greenwich, London, SE10 9LS, UK
Received in final form 20 July 2006
A B S T R A C T Solder joints are often the cause of failure in electronic devices, failing due to cyclic
creep induced ductile fatigue. This paper will review the modelling methods available to
predict the lifetime of SnPb and SnAgCu solder joints under thermo-mechanical cycling
conditions such as power cycling, accelerated thermal cycling and isothermal testing, the
methods do not apply to other damage mechanisms such as vibration or drop-testing.
Analytical methods such as recommended by the IPC are covered, which are simple to
use but limited in capability. Finite element modelling methods are reviewed, along with
the necessary constitutive laws and fatigue laws for solder, these offer the most accurate
predictions at the current time. Research on state-of-the-art damage mechanics methods
is also presented, although these have not undergone enough experimental validation to
be recommended at present.
Keywords constitutive law; fatigue; finite element; life prediction; solder joint; thermal
cycling.
N O M E N C L A T U R E A = crack area
a = joint diameter
da
dN = crack propagation rate
c i = material constants (Unless stated otherwise, similarly numbered constants in
different equations are not the same, e.g. c2 in Eq. (2) is not equal to c2 in Eq.
(13))
D = damage
E = Young’s modulus (elastic modulus)
g = cohesive zone stiffness
h = solder joint height
L = distance to neutral point
N f = mean number of cycles to failure
N ff = number of failure free cycles
N 0 = number of cycles to crack initiation
Q = activation energy
R = universal gas constant
T = temperature
T = difference in temperature between the hot and cold extremes during cycling
W = accumulated strain energy density per cycle
Y = cohesive zone traction
α = CTE (coefficient of thermal expansion)
α = difference in CTE between the circuit board and the component
∗
Correspondence: S. Ridout. E-mail: s.w.ridout@gre.ac.uk
400 c© 2006 The Authors. Journal Compilation c© 2007 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 30, 400–412

REVIEW OF METHODS TO PREDICT SOLDER JOINT REL IABIL ITY 401
γ = shear strain range
ε = normal strain
εacc = accumulated effective plastic strain per cycle
εcr = total creep strain
ε in = instantaneous plastic strain
εp = total plastic strain
εs = steady-state creep strain
ε th = thermal strain
η = characteristic life
μ = cohesive zone relative opening displacement
σ = normal stress
I N T R O D U C T I O N
Most electronic devices are expected to last for many years
of use. Therefore it is unfeasible for companies to perform
field tests due to the time and cost involved. Instead, ac-
celerated tests are used which impose much harsher con-
ditions on the joint, causing it to fail in a shorter time and
allowing a judgement to be made on the products reliabil-
ity. The most popular accelerated test is thermal cycling,
in which a soldered assembly is placed in an oven and sub-
jected to a cyclic temperature profile representing condi-
tions considerably harsher than expected field use condi-
tions. During cycling or at periodic intervals the samples
will be monitored for electrical resistance/continuity, if
the resistance shows a very large increase this indicates
that a crack has grown completely through the solder joint
and the sample is considered to have failed. The average
number of cycles to failure in such a test is used as a mea-
sure of the reliability of a chip assembly.
Another accelerated test is the power cycling test, during
which heat is periodically generated from within a com-
ponent. This results in an anisothermal temperature field
more closely matching the conditions experienced in ac-
tual field use.
Two of the more common types of chip assemblies tested
are surface mount passives and flip-chips shown in Fig. 1.
The circuit board is typically FR4 but alumina or other
materials may be used. Two of the more common solder
alloys are SnPb, which due to its Pb content is banned
from use in most applications throughout Europe from
July 2006 under the RoHS directive,1 and SnAgCu which
is the most popular Pb-free replacement. Throughout the
paper SnPb refers to eutectic Sn37Pb and SnAgCu refers
to Sn3.5Ag0.7Cu or a slight variation on this composition.
Although many of the methods reported in this paper were
developed for a particular solder alloy, in most cases the
same methods could be used for a different alloy if the
material constants are changed.
The primary failure mechanism of solder joints is duc-
tile fatigue fracture. During operation or during testing,
an electronic component will be subject to changes in tem-
perature. This causes the materials to expand and contract
at different rates depending on their coefficients of ther-
mal expansion. This cycling results in internal stresses
developing which, in turn, causes the solder to creep.
Creep is a time-dependent plasticity, which occurs in met-
als when they are close to their melting points. Over many
cycles, the creep leads to ductile fatigue damage, which
manifests itself as microvoids in the solder, which grow and
coalesce into macro-cracks that slowly propagate through
the joint over the course of its lifetime (Fig. 2), ultimately
resulting in failure of the joint.
It should be noted that other failure mechanisms are pos-
sible depending on the external conditions applied. For
instance, if the device was subject to vibration or dropped
onto the floor then the sudden acceleration may cause a
brittle failure.
M O D E L L I N G M E T H O D S
Modelling is a useful tool used to supplement or replace
accelerated tests, particularly in the early design stages.
The modelling discussed in this paper applies to creep
induced ductile fracture only, and so will be suitable for
modelling the damage that occurs due to typical thermal
or power cycling of joints, but not to vibration or drop
testing, or shear-strength testing. Most of the methods
described in this paper can be applied to any solder alloy
provided the correct material constants are known. Many
methods exist, and there is no clear-cut answer as to which
is best. Figure 3 shows the classes of modelling methods
which will be discussed in this paper.
The methods in Fig. 3 increase in complexity going from
top to bottom, the analytical method proposed by Engel-
maier2,3 being simple to implement but with many caveats
restricting its use in certain situations. The constitutive law
+ fatigue law class of methods (encompassing FEA and
other alternatives) are very popular, providing more ac-
curate predictions with fewer restrictions than analytical
methods, however, with increased set up time and com-
putational cost. The damage mechanics based methods re-
quire considerably more effort both in implementation
and computational cost and their predictive capability is
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402 S. R IDOUT and C. BA ILEY
Fig. 1 Solder joints in surface mount
resistor and flip chip packages.
currently unproven, however, they promise to provide the
most accurate predictions once they are developed further.
In reality the behaviour of solder joints is very com-
plex and all the modelling methods discussed must make
many simplifications and omit certain aspects of the prob-
lem, for example the behaviour of the intermetallic par-
ticles in SnAgCu, or the differences between the Pb-rich
and Sn-rich phases of SnPb. The constitutive law + fa-
tigue law methods make fewer simplifications than the an-
alytical methods, and the damage mechanics methods make
fewer simplifications still but they all make simplifications.
Many phenomena will not be explicitly modelled, and any
changes in the effect of these phenomena when using a
different geometry, temperature range or other condition
will not be captured. (e.g. The intermetallic layer is usually
not explicitly modelled, this is fine for most conditions but
will yield inaccurate predictions if modelling extremely
thin joints where the intermetallic layer forms the bulk of
the joint.)
The methods discussed are all intended to predict the re-
sults of accelerated tests and not field-use reliability. They
cannot accurately predict field-use reliability as they have
not been validated against field-use data, because gener-
ally field-use data do not exist. The assumption is that the
relative reliability of various components under acceler-
ated conditions will be the same, or at least similar, under
field-use conditions. The models have been extrapolated
to simulate field-use conditions,4 but this is potentially
misleading.
A N A LY T I C A L M E T H O D S
Clech5 has reported the use of an analytical method where
the strain range in the solder is calculated assuming the
solder joint is totally compliant:
γ =
LαT
h
. (1)
This strain range is then used to predict the characteristic
lifetime per unit crack area of a single joint using a Coffin–
Manson fatigue law with an added crack area adjustment:
Nf
A
= c 1(γ )−c 2 . (2)
Predictions made using this method were correlated with
a data set of 27 experimental data points covering differ-
ent kinds of assemblies under different thermal profiles
for SnAgCu solder. Despite not taking into account the
creep behaviour of the solder at all, the accuracy of the
predictions was in the range of ±2×.
A slightly different analytical approach is Engelmaier’s
model for predicting the lifetime of Sn37Pb joints recom-
mended in the IPC-D-279 standard.3 There are versions
for both leaded and non-leaded components (leaded as in
gull-wing lead, not to be confused with Pb) and they are
very simple to use when compared with FEA modelling.
The way it works is to assume the solder deforms to
its maximum amount based on the CTE (coefficient
of thermal expansion) of the substrate and component.
This strain range is calculated for non-leaded joints using
Eq. (3), this is very similar to the Eq. (1) used by Clech
but with an added empirical factor.
γ = c 1
L
h
(αT). (3)
The strain range γ , calculated using Eq. (3) is not ac-
curate as it does not take into account the stiffness and
creep properties of the solder. The solder stiffness would
prevent the strain range from ever reaching this value. To
compensate for this, the standard Coffin–Manson fatigue
law for predicting N f (number of cycles to fail) has been
modified to include temperature and frequency effects:
Nf =
1
2
[
γ
2c 1
]1/c
(4)
c (T, f ) = −0.442 − 6 × 10−4T¯SJ + 1.74
×10−2 ln
(
1 +
360
tD
)
. (5)
Where T¯SJ is the mean cyclic joint temperature for which
a formula is provided.3 This approach has been shown
to predict lifetimes with an accuracy of ±2× under the
appropriate conditions.2 However, there are many con-
ditions in which this law does not apply. A list of caveats
from the IPC standard3 are stated below along with a brief
discussion of whether they also apply to the constitutive
law + fatigue law methods.
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REVIEW OF METHODS TO PREDICT SOLDER JOINT REL IABIL ITY 403
Fig. 2 SEM image of a crack through a
SnAgCu solder joint.
Fig. 3 Modelling methods to predict cycles
to fail in thermo-mechanical cycling.
1 Solder joint quality: This caveat is to ensure that the failure
mechanism is actually ductile fatigue within the solder joint
and not brittle failure at the interface which could be caused
by poor materials choice (e.g. alloy 42 leads) or very small
solder joint gaps (<75 nm). This caveat could be applied
all the modelling methods covered in this paper, which are
intended to predict the lifetime of joints which fail due to
creep induced ductile fracture, not brittle failure.
2 Large temperature excursions: The damage mechanism
changes in solder joints experiencing large temperature
excursions (−50 ◦C to +80 ◦C). Using a constitutive
law + fatigue law method could capture the difference in
solder behaviour within this temperature range. For ex-
ample, Darveaux et al.4 validated a fatigue law for SnPb
for temperature ranges up to −55 ◦C to +125 ◦C, and
Syed6 validated a similar law for SnAgCu up to the same
range.
3 High frequency/Low temperature: For frequencies
>0.5Hz and/or temperatures <0 ◦C the direct applica-
tion of Coffin–Manson may be more appropriate (c = ∼
−0.6). Under these conditions, creep induced fatigue may
not be the primary failure mechanism, in which case none
of the modelling methods discussed in this paper would be
suitable.
4 Local expansion mismatch. The local CTE mismatch be-
tween the solder joint and the board or component is not
taken into account. For instance, an alumina component on
an alumina substrate would generate no global strain in the
solder and would, according to this model, experience no
fatigue. However in reality, local strains will be generated
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404 S. R IDOUT and C. BA ILEY
at the solder/component and solder/substrate interfaces,
which would lead to damage. This damage mechanism is
captured perfectly well by using FEA modelling with an
appropriate constitutive law and fatigue model.
5 Very stiff leads/very large expansion mismatches: The an-
alytical method does not make use of the Young’s modulus
of any of the materials so it cannot predict accurately the
amount of strain which will occur in the joints, leads, com-
ponent and substrate. When the package geometries and
materials are similar this may not matter, but deviations
from the norm could result in very inaccurate results. For
instance, if we consider two assemblies, similar in all re-
spects except that the component Young’s modulus is high
in one case and low in the other. The component with the
high Young’s modulus will cause a greater strain in the sol-
der joints, but this model will predict the same lifetime for
both. Using FEA will predict the strains occurring in the
solder accurately based on the Young’s modulus and other
mechanical properties of the materials.
So of the five caveats, three would no longer be neces-
sary if a Constitutive law + Fatigue law method were used
instead. The Analytical methods, while simple to use, are
limited in their applicability compared to the other meth-
ods.
C O N S T I T U T I V E L A W P L U S FAT I G U E
L A W M E T H O D S
These methods work by running a transient simulation to
predict the solder’s stress strain behaviour during a ther-
mal cycle. From this simulation, either the accumulated
effective plastic strain per cycle (εacc) or the accumulated
strain energy density per cycle (W ) is extracted to be
used in a fatigue law. First, the methods of modelling the
mechanical response of an assembly under thermal cycling
will be discussed (FEA and alternatives), followed by the
various constitutive laws available, and finally the fatigue
laws.
5. FEA
FEA (finite element analysis) is a powerful and widely used
numerical method, which can be used to accurately pre-
dict the mechanical response of the solder joints and sur-
rounding assembly under thermo-mechanical cycling. A
detailed description of the FEA method is provided by
Zienkiewicz.7
To use FEA requires considerably more investment in
time than the analytical approaches mentioned so far, and
detailed knowledge of the test being modelled. In order
to use FEA to determine the fatigue life of solder joints
under thermal cycling, the following steps are required:
1 Create geometry
The geometry of the model should strike an acceptable
balance between accuracy and simplicity. Common sim-
plifications to the geometry include:
• The intermetallic layer formed between the solder joint
and the copper pads is usually ignored, due partly to its
size and partly to the lack of material property data.
• Good judgement is needed in deciding what details of the
actual geometry are important to include in the model
and which are superfluous.
• Symmetry. Usually only a quarter of a surface mount
resistor needs to be modelled, and only an eighth of a
BGA or flip-chip.
2 Create mesh
It is important that the mesh is fine enough to make the
solution mesh independent. This can be verified by per-
forming simulations using progressively finer meshes until
the results no longer change. It is also important that the
aspect ratio of elements is not too high as this can introduce
numerical errors.
3 Specify material properties
It is crucial to use accurate material properties to obtain
accurate results. When available, temperature-dependent
material properties should be used. The solder properties
are discussed later in the paper, because the focus is on
solder joint modelling the properties of the other materials
involved are not discussed.
4 Specify boundary conditions
During typical thermal cycling regimes the temperature
may be regarded as uniform throughout the sample. For
very rapid thermal shock, or when there is heat genera-
tion from within a chip (e.g. during power cycling) then
the temperature field will not be uniform and should be
calculated throughout the mesh on each time step.
5 Apply fatigue law
To apply the fatigue law, a εacc or W value needs to be
extracted from the FEA simulation results. Because the
variation in strain across a joint is captured, this raises
the question of whether to average over the whole joint
or only a portion of it. Two sources have been found
which address this,6,8 and both use an average value over a
25 μm layer at the top of the solder bump in a BGA pack-
age. Altering the size of this volume averaged region may
alter the value of εacc or W significantly, so if the size
needs to be changed, for example when modelling a dif-
ferent sized or shaped joint, then the fatigue law constants
should ideally be re-calibrated with experimental data to
ensure accurate predictions.
Alternative numerical methods
Alternatives to FEA are usually more simplified meth-
ods that capture the creep behaviour of the solder using
a constitutive law but do not require as much work or
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REVIEW OF METHODS TO PREDICT SOLDER JOINT REL IABIL ITY 405
Fig. 4 Diagram showing the response of
solder during a creep test.
Fig. 5 Experimental and simulated
hysteresis curves using the McDowell UCP
model.
expertise to set up. The simplest example of this kind of
approach would model only the prominent shear compo-
nent of strain in the solder joint, this is an approximation
as in reality a BGA solder bump would experience signif-
icant normal and shear strains. Furthermore, the strain is
assumed to be uniform throughout the solder joint thus
making it suitable to model global strains only.
A simulation is run where the temperature cycles be-
tween two extremes and at each time step the stress and
creep strain rate in the solder are calculated. After two or
three thermal cycles the response of the solder will have
stabilized and the simulation can finish. The output of the
simulation is a hysteresis loop – a graph of stress versus
strain, and values of εacc and W . Due to its simplicity it
will provide results with less effort and less computation
time than FEA, and because it uses a constitutive law to
model the solder’s creep behaviour it should provide more
accurate results over a wider range of conditions than an
analytical method, although no validation against experi-
ment has been found to back this up.
A more advanced example of this kind of approach
is provided by Clech9 which provides the capability of
modelling local as well as global thermal expansion mis-
matches, and the effect of underfill. It has been calibrated
against 19 experimental results with an error of about
±2.5×, and validation was performed against a further
14 experimental results.
A hybrid method is suggested by Darveaux8 where two
purely elastic analyses are conducted to provide (1) the
stiffness of the assembly surrounding the joint and (2) the
displacement between the top and bottom pads assuming
the solder joint is not present. These are then used in a 1D
numerical model, which uses a constitutive law to predict
the average response of the solder joint over a thermal
c© 2006 The Authors. Journal Compilation c© 2007 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 30, 400–412

406 S. R IDOUT and C. BA ILEY
cycle. This strikes a compromise between the accuracy of
FEA and the computational ease of a simpler numerical
method. However, it will not predict the damage caused by
the local thermal expansion mismatch between the solder
and the component or substrate.
5. Solder constitutive laws
A constitutive law for a material will govern its mechani-
cal behaviour when subject to different stresses at differ-
ent temperatures. The importance of using the correct
constants for the constitutive laws used in modelling is
highlighted by a sensitivity study10 which shows that the
differences in SnPb Young’s modulus, CTE and creep ac-
tivation energies within the range of variation reported in
the literature can lead to inaccurate lifetime predictions
when used in FEA modelling.
Elasticity
The elastic properties of solder are usually determined
from a strain rate controlled tensile test. From this test the
Young’s modulus can be determined along with the ulti-
mate tensile/shear strength of the solder. There is a large
strain rate and temperature effect and the Young’s modulus
will appear lower at lower strain rates or higher tempera-
tures.11 The temperature dependence is usually modelled
but no work was found which modelled the strain rate
dependence.
The elastic deformation of the solder is governed by the
Young’s modulus and Poisson’s ratio. In one dimension the
relationship between stress and strain is given by Hooke’s
law:
σ = Eε. (6)
In 3D the relationship is more complex12 and takes into
account the Possion’s ratio (the proportion by which the
cross-section area shrinks for each unit strain applied).
The Young’s modulus is both temperature and strain rate
dependent, and a formula for calculating the Young’s mod-
ulus of SnAgCu provided by Pang:13
E(T, ε˙)SnAgCu = (−0.0005T + 6.4625) log ε˙
+ (−0.2512T + 71.123). (7)
The temperature dependence is usually modelled but no
modelling work was found which included the strain rate
dependence. More temperature dependent Young’s mod-
ulus values of SnAgCu have been reviewed by Syed,6 there
is some scatter in the values reported which may be as a
result of differences in the strain rate used. It is pointed
out by Basaran and Jiang14 that when modelling SnPb, the
use of different values of Young’s modulus (literature val-
ues range from 9GPa to 48GPa) can adversely affect the
results of a simulation and that ideally a modulus measure-
ment should be made on an actual manufactured package
using a nano-indentation technique.
The CTE of the solder governs the amount by which it
expands under changing temperatures:
α =
dεth
d T
. (8)
A CTE of 25 ppm/◦C has been reported for SnPb,15 and
a CTE of 20 ppm/◦C has been reported for SnAgCu.16
Creep
Creep is a time-dependent plastic deformation, which oc-
curs to metals under stress at high homologous tempera-
tures, because solder has a melting point of 183 ◦C (SnPb)
or 217 ◦C (SnAgCu) then it creeps at room temperature.
During a creep test a constant load is applied to a sol-
der specimen at a constant temperature. The character of
the response of the solder is illustrated in Fig. 4. At the
instant the force is applied, the solder will experience a
strain, which is part elastic and part plastic. The elastic
part can be predicted given the Young’s Modulus of the
solder as discussed earlier. The plastic part is the instanta-
neous plasticity, Darveaux et al.4 and Wiese and Rzepka17
offer laws to predict this, although the distinction between
this instantaneous plasticity and the following primary
creep region is not well defined. As the test continues the
strain increases, first rapidly (primary region) and grad-
ually slowing to a steady strain rate (secondary region).
This additional strain is due to creep. After the secondary
region comes the tertiary region during which the strain
rate increases until rupture. This occurs due to both neck-
ing (shrinking of the cross section resulting in increased
stress) and damage (cracking) occurring in the solder.
The standard method for FEA modelling of solder joints
involves modelling only the steady-state creep and it
is possible to get reasonable predictions using this ap-
proach.4,6 However, it is possible that better accuracy may
be achieved by incorporating primary creep and instanta-
neous plasticity into the model. (If the goal of a simula-
tion is to accurately predict the amount of deformation
occurring in the solder, then including primary creep is
essential.)
Steady-state creep
The secondary creep, also called steady-state creep, is
commonly the only kind of creep to be modelled. The
simplest steady-state creep law is the Norton law:
ε˙cr = c 1σ c 2 exp
(
−Q
kT
)
. (9)
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REVIEW OF METHODS TO PREDICT SOLDER JOINT REL IABIL ITY 407
Fig. 6 Diagram showing predicted damage of surface mount
resistor fillet joint after thermal cycling.
This law has been used to model SnPb10 but the law does
not capture the change in creep mechanism between low
stresses and high stresses. To capture this change Wiese18
has used a double-power model to model SnAgCu:
ε˙cr = c 1
(
σ
σn
)c 2
exp
(
−Q1
kT
)
+ c 3
(
σ
σn
)c 4
exp
(
−Q2
kT
)
.
(10)
The first term on the RHS of this formula repre-
sents climb controlled creep (low stress) and the second
term represents glide/climb controlled creep (high stress).
Constants are provided for this law to model SnAgCu.18
An alternative and widely used method for capturing the
change in creep mechanism is to use a sinh law:
ε˙s = c 1[sinh(c 2σ )]c 3 exp
(
−Q
kT
)
. (11)
Many authors have used this approach8,17,19,20 and
Darveaux et al. have published constants for four differ-
ent alloys including SnPb and Sn3.5Ag.4 There is spread
in the published experimental creep data, differences in
strain rates in the order of 10 to 100 times at a given
stress and temperature are common. This can be partly at-
tributed to different scales of sample being used by differ-
ent researchers (bulk samples behave differently to joint-
scale samples) as well as the fact that the strain rate is very
sensitive to small changes in stress.
Primary creep/Kinematic hardening
Good lifetime predictions are made using only steady state
creep laws,4 probably because although the absolute val-
ues of strain predicted are inaccurate, the fatigue laws have
been calibrated to these results and thus provide reason-
able lifetime predictions. Even better predictions should
be possible by modelling primary creep. Three differ-
ent approaches have been found to model primary creep,
the use of a time-variable, isotropic hardening and kinematic
hardening. Of these, only kinematic hardening is suitable
to model the solder behaviour during thermo-mechanical
cycling.
Darveaux et al.4 and Schubert et al.19 have reported laws
that make use of a time variable in the creep strain rate
function. Darveuax’s law is represented by Eq. (12), while
Schubert uses a slightly more complex formula.
ε˙cr = ε˙s(1 + c 1c 2 exp(−c 2ε˙s t)). (12)
This may be adequate for modelling a monotonic creep
test, but modelling of cyclic temperature cycling or fa-
tigue cycling where the stress and therefore ε˙s is constantly
changing this law is not appropriate.
Cheng et al.15 used the Anand model to model the pri-
mary and secondary regions of the creep curve using
isotropic hardening:
ε˙cr = c 1
[
sinh
(
c 2
σ
s
)]1/c 3
exp
(
−
Q
RT
)
. (13)
s is a scalar internal variable, which represents the averaged
isotropic resistance to plastic flow. It changes according to
the following formula:
s˙ =
{
c 4
∣
∣
∣
1 −
s
s ∗
∣
∣
∣
c 5
.sign
(
1 −
s
s ∗
)}
.ε˙cr. (14)
Where s∗ is the saturation value of s given by:
s ∗ = c 6
[
ε˙cr
c 1
exp
(
Q
kT
)]c 7
(15)
Where c1 in Eq. (15) is the same as c1 in Eq. (13).
The material properties for 60Sn40Pb, 62Sn36Pb2Ag and
96.5Sn3.5Ag were fitted from the conventional Darveaux
model parameters. The use of an isotropic hardening law
to capture the hardening during a creep test is ques-
tionable as a large part of the hardening is due to kine-
matic hardening, Stolkarts et al.21 have even reported that
isotropic hardening does not occur in SnPb solder. So al-
though this model may capture the behaviour of a mono-
tonic creep test it will not capture the true behaviour in
cyclic testing where the load is reversed. Despite this the
model has been used in a FEA analysis of chip assemblies
by many researchers, often with good agreement to ex-
perimental results.8,11,15
Kinematic hardening
To understand how the solder behaves under cyclic load-
ing, mechanical fatigue tests are used. During this test a
cyclic strain, either shear or tensile, is imposed on a solder
joint or bulk sample. The response of the solder is depen-
dent on the elastic, creep and fatigue properties making it
a useful test to validate the accuracy of constitutive laws.
c© 2006 The Authors. Journal Compilation c© 2007 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 30, 400–412

408 S. R IDOUT and C. BA ILEY
Kinematic hardening laws with material parameters for
SnPb have been found22–24 which can accurately pre-
dict the behaviour shown in isothermal fatigue tests, but
no laws with material parameters have been found for
SnAgCu. shows experimental and simulated hysteresis
loops of isothermal fatigue tests from Neu22 predicted us-
ing the McDowell UCP (unified creep plasticity) model25
which includes both isotropic and kinematic hardening
terms.
Instantaneous plasticity
Darveaux used the following law to capture the instan-
taneous plasticity, which occurs in creep tests in many
solders including SnPb and Sn3.5Ag:
εin = c 1
(
σ
E
)c 2
. (16)
Unfortunately, this law cannot be applied to other load-
ing conditions such as temperature cycling where the
stress would be constantly changing.
Wiese has used the multilinear elastic–plastic model of
ANSYS to simulate the time-independent behaviour of
Sn37Pb, Sn3.5Ag and Sn4Ag0.5Cu. This was shown to
accurately predict the behaviour of the solder in isother-
mal fatigue testing with cycle periods of 1 s to 3600 s.
Wiese shows that for fatigue tests with a frequency of
1 Hz, the instantaneous plasticity dominates but for cy-
cles with a period of 3600 s the creep dominates and the
instantaneous plasticity is insignificant. Because most test-
ing is in the order of 3600 s cycles then this would suggest
that the common practice of ignoring instantaneous plas-
ticity is justified.
It is difficult to say exactly how much of the strain is
instantaneous plasticity and how much is primary creep,
and even if it is worth distinguishing between the two.
An alternative is to use a model such as the McDowell
unified creep–plasticity model22,25 which appears to accu-
rately capture the behaviour of SnPb solder using isotropic
and kinematic hardening and no instantaneous plasticity.
This approach is preferable as it avoids the difficult-to-
measure distinction of how much strain is due to instan-
taneous plasticity and how much is due to creep.
Although going beyond steady-state creep modelling to
include instantaneous plasticity and/or hardening results
in more accurate predictions of the solder behaviour, it
remains unproven as to whether this will yield significantly
improved lifetime predictions.
Fatigue laws
Fatigue laws are required to predict the number of cycles
to fail when provided with either the accumulated effec-
tive creep strain or the strain energy from a simulation.
A review of fatigue laws for solder is provided by Lee
et al.26 The simplest fatigue laws are the Coffin–Manson
law Eq. (17) and the strain energy based law Eq. (18) used
by Akay.27
γ = c 1 N
c 2
f (17)
Nf =
(
W
c 1
)1/c 2
. (18)
If a creep law is used which outputs more than one kind
of plastic strain then the fatigue law may make use of
this added data to predict the contributions to lifetime
from different damage mechanisms. The following equa-
tion can then be used to calculate the overall N f:
1
Nf
=
1
N1f
+
1
N2f
+ ... +
1
Nnf
. (19)
Where N1f , N
2
f , . . . N
n
f are the number of cycles to fail pre-
dicted separately for each of the n damage mechanisms. An
example of this is provided by Syed6 where the low-stress
creep and high-stress creep contributions predicted using
the Wiese double power law (10) were used to predict the
fatigue life:
Nf =
(
0.013εIacc + 0.036ε
II
acc
)
−1
. (20)
This law along with strain and energy based laws of the
form of (2) and (18), respectively, were calibrated with a
set of experimental data on assembles with four different
ball pitches and sizes, three different substrate materials
and three different accelerated temperature profiles using
SnAgCu solder. All of the models were found to predict
the characteristic lifetimes to within 25% in most cases.
Darveaux et al. published a methodology4 using fatigue
laws which predict the time to crack initiation and the
growth rate of the primary and secondary cracks in an
SnPb solder bump joint. The crack growth rates are as-
sumed to be constant making the prediction of overall
lifetime possible. This has since been updated and sim-
plified8 by combining the primary and secondary crack
propagation rates.
Darveaux used FEA and calculated W by averaging
over a layer of solder 25 μm thick adjacent to the package
interface where the cracks are expected to develop. This
was then used in the law for predicting time to crack ini-
tiation Eq. (21), and the law for crack propagation rate
Eq. (22).
No = c 1W−c 2 (21)
da
dN
= c 3Wc 4 . (22)
c© 2006 The Authors. Journal Compilation c© 2007 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 30, 400–412

REVIEW OF METHODS TO PREDICT SOLDER JOINT REL IABIL ITY 409
The characteristic life and failure-free life can then be
calculated from Eq. (23) and Eq. (24), respectively.
η = No +
a
da/dN
(23)
Nff =
η
2
. (24)
Also provided8 is a formula to calculate the overall char-
acteristic life based on the number of worst-case joints
present in the whole system (typically only the four or
eight highest-stressed joints contribute in a BGA package
as this is where the vast majority of failures occur). The
method predicted lifetimes with an accuracy of ±2× or
better.
When using any fatigue law it is important to under-
stand that it will work best under conditions similar for
that which the constants were validated. Changing condi-
tions such as chip geometry and temperature profile, and
also the volume over which averaging of εacc or W is per-
formed will affect the accuracy of the lifetime predictions.
The accuracy of predictions was found to be in error by
a factor of up to 7× when using a modelling procedure
not consistent with that used in crack growth correla-
tion.8 The use of relative predictions is recommended8
when there is at least one data set of measured fatigue
lifetime for the package in question. The procedure is
to first calculate the lifetime for the known case, then
any further calculations on different cases can be ‘cali-
brated’ using the experimental data. The accuracy of rela-
tive predictions was found to be in the range of ±25% or
better.
The constitutive law + fatigue law approach is an empirical
method tailored to a specific experimental data set, one of
the reasons that many researchers are investigating the use
of damage mechanics methods is that these will hopefully
remain accurate over a wider data set without having to
be re-calibrated.
D A M A G E M E C H A N I C S M E T H O D
In order to overcome the geometry dependence inherent
in the fatigue law approach to predicting lifetime, or to
allow crack paths to be predicted, damage-based constitu-
tive laws can be used. A number of researchers have pub-
lished results from such laws for SnPb solder although the
approach is still in the experimental stage. A review cov-
ering several methods is provided by Desai and White-
nack.28 Detailed validation against a large set of experi-
mental data has not been found for any of these models
and the computational cost involved is high. Given time,
with the increase in computational power and continued
research including more validation against experimental
lifetime data, this approach may prove useful in predicting
reliability.
J-integral
Ghavifekr and Michel29 determine the relationship be-
tween the J-integral of a notched solder tensile test spec-
imen and the crack growth rate and suggest that it can be
applied to the prediction of growth rate in solder joints.
Gu and Nakamura30 use the method to determine the
direction (but not rate) of crack propagation in a solder
bump joint.
A simplistic use of the method may be to predict the
crack growth rate at the small notch and assume a con-
stant propagation speed. Darveaux has shown that in BGA
bump joints the crack propagation speed is roughly con-
stant, making this viable, however, other kinds of joints
do not show constant crack speed (e.g. resistor joints31).
A more advanced approach would be to advance the crack
through the mesh by small amounts, refining the mesh to
achieve a high mesh density around the crack tip, and pre-
dict the crack propagation speed at each point in the cracks
life. One drawback to the use of the J-integral method for
predicting reliability is the necessity to manually intro-
duce small cracks (notches). It cannot predict the time or
location of crack initiation. No work has been found which
validates lifetime predictions using the J-integral method
against experimental data.
Continuum damage mechanics
In this approach, the damage builds up within the bulk
of the material, allowing for the modelling of regions of
cracking rather than one sharp crack as in the J-integral
method. This captures the phenomena seen in solder of
small micro-cracks gradually forming within a region and
coalescing to form macro-cracks.32 An example of this ap-
proach is shown in Fig. 6 from modelling work on surface
mount resistors by the authors of this paper.
A framework for implementing a continuum damage
model is the DSC (disturbed state concept) which is de-
scribed by Desai and Whitenack.28 This regards a ma-
terial as a composite, containing two different parts – the
intact part and the adjusted/disturbed/damaged part, each
of which has its own set of material properties. In the case
of modelling fatigue damage, the adjusted (or damaged)
part would have no resistance to shear stresses and possibly
no resistance to hydrostatic stresses. The implementation
of this approach is simpler than the J-integral methods in
that the mesh does not need to be split (although refining
could aid accuracy). Another big advantage is that no prior
information on the location of the crack is required.
c© 2006 The Authors. Journal Compilation c© 2007 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 30, 400–412

410 S. R IDOUT and C. BA ILEY
A formula based on accumulated effective strain in the
solder can be used to calculate damage:
D = 1 − exp
(
−c 1εaccc 2
)
. (25)
However, it is stated by Basaran et al.33 that although
εacc is often used to calculate damage, entropy is a better
damage metric.
Volume averaging is necessary to avoid mesh depen-
dence,34 this raises the issue of the size of the volume
to use. In reality, the influence of micro-crack interaction
occurs over a fairly small length scale, however, if the scale
is made smaller than the mesh element size then volume
averaging will have no effect.
Cohesive zone
This technique models the gradual degradation in the ad-
hesion between surfaces, making it ideal for modelling
delamination at solder/pad interfaces. The method uses
flat, 2D elements, called cohesive zone elements, at an in-
terface along which a crack is expected to develop (e.g.
the solder to copper pad interface, or the region of inter-
metallics near the top of a BGA joint). This element has
a stiffness, which governs how far it separates (in normal
and/or shear directions) depending on the applied stress.
As the normal and/or shear separation increases or cycles
then the damage present in the element increases and this
in turn reduces its stiffness. This is very similar to the con-
tinuum damage approach except using 2D rather than 3D
elements.
Abdul-Baqi et al.35 have described a method whereby the
bulk of the solder is modelled using a linear elastic law, but
with cohesive zones at the solder-pad interface, the inter-
faces between the phases of SnPb (Pb-rich islands within
a Sn-rich matrix) and even in the grain boundaries within
each phase. The cohesive zones each have an associated
damage parameter, which evolves according to:
D˙ = c 1|μ˙|(1 − D + c 2)c 3
(
|Y|
1 − D
− c 4
)
. (26)
An increase in the damage leads to a decrease of stiffness
as governed by:
Y = h(1 − D)μ. (27)
By using cohesive zones to model four kinds of interface
(solder to pad, Sn-rich phase to Pb-rich phase, Sn-rich
grain boundary, Pb-rich grain boundary) the method of-
fers a potentially very accurate description of the solder
behaviour. However, a model is useless without accurate
material constants and it is not mentioned how the ma-
terial properties for these different cohesive zones should
be found. Finding the parameters of just one kind of co-
hesive zone could be done reasonably by using an inverse
analysis/optimisation approach, but though the approach
could be applied in this case, it is likely that the relative
proportion of damage occurring in each kind of cohesive
zone (solder to pad, grain boundaries, etc.) will be very
difficult to predict.
An interesting consequence of the cohesive zone ap-
proach is that the compliance of the cohesive zone will
affect the compliance of the bulk of solder. This is an un-
wanted side effect and Abdul-Baqi et al.35 have in their
work ensured that the contribution on the joint compli-
ance from the cohesive zones in their initial, undamaged
state is negligible. The cohesive zones remain stiff under
compression, thus preventing the unrealistic occurrence
of overlapping crack surfaces.
A model is proposed by Yang et al.36 in which a single
cohesive zone element is used to model SnPb solder in a
pure shear cyclic fatigue test. Strangely, the cohesive zone
in this model has a thickness, therefore strains are dis-
cussed rather than separations or displacements, making
it fit the description of a continuum damage model better
than the cohesive zone model they describe it as. Sepa-
rate laws are used to describe the damage evolution under
monotonic strain Eq. (28) and cyclic strain Eq. (29).
∂ Dmon
∂γ
= 0.166γ −0.648 (28)
∂ Dcyc
∂γ
= 0.067γ −0.543. (29)
This reflects the fact that it takes considerably less accu-
mulated strain to destroy a joint under monotonic loading,
than it does under cyclic loading. The way that these are
used in a simulation is to use Eq. (28) to model the first
half cycle, and then use Eq. (29) for all the remaining cy-
cles. In fact the use of Eq. (28) in the first half cycle in this
way is probably insignificant, given that 1000 s of cycles
are typically necessary to cause failure.
Hybrid
Towashiraporn et al.37 describe a method in which a dam-
age parameter is calculated based on the accumulated ef-
fective strain according to Eq. (25), as might be done with
a continuum damage approach. But unlike a continuum
damage approach, the damage does not affect the stiff-
ness of the bulk of the material; instead, the damage is
monitored across the critical interface where the crack is
expected to develop. When the damage at a node in this
interface reaches a critical value, the connectivity at that
node is released, thus creating a crack.
c© 2006 The Authors. Journal Compilation c© 2007 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 30, 400–412

REVIEW OF METHODS TO PREDICT SOLDER JOINT REL IABIL ITY 411
In order to reduce computation time, a global–local sub-
modelling method was used. This involves first perform-
ing a simulation on a 1/8 symmetry mesh of the whole
BGA package. From this, the displacements at the top and
bottom interfaces of the most critical solder joint are used
as boundary conditions in a more detailed simulation of
the critical joint. This was found to be inadequate due to
the changing response of the solder as the crack grows. To
capture this, a more sophisticated method was used where
the refined submodel was coupled to the whole assem-
bly model by multiple constraints. However, this involved
considerably more computational effort.
To save on computational expense, an approximate solu-
tion procedure is used which utilizes a critical disturbance
range rather than a single value. The damage accumula-
tion rate is calculated from a simulation of two cycles. This
rate is assumed to remain constant and used to work out
the cycle before the most damaged node will increase be-
yond the upper bound of the critical disturbance range. At
this cycle, all nodes with damage falling within the crit-
ical range are disconnected, and the process is repeated
iteratively until 70% of the interface is cracked, at which
point the joint is considered to have failed. The 70% fig-
ure is chosen because it was found experimentally that only
70% of a BGA joint crack is caused by creep–fatigue and
the remaining 30% is caused from shear overload. This
is also supported by simulations continuing beyond 70%
cracking which predict a roughly constant rate of crack
growth up to 70%, followed by an exponential increase in
crack growth rate. It was found that increasing the critical
disturbance range from (0.75–0.85) to (0.5–0.9) did not
affect lifetime predictions adversely, but it did reduce the
accuracy of crack front predictions.
This is an interesting approach but as with all the dam-
age mechanics methods is in need of further experimental
validation. Also the fact that the critical surface must be
specified beforehand makes this method unsuitable for
modelling new or more complicated geometries.
C O N C L U S I O N S
For many engineers, an analytical method such is attrac-
tive due to its ease of use. Furthermore, both Engelmaier2
and Clech9 have achieved lifetime predictions with an er-
ror of less than ±2× with analytical approaches. How-
ever, these methods do not model the creep behaviour
of the solder and are therefore limited in their capabil-
ity to model a wide range of temperature conditions and
geometries. The caveat restricting its use under large tem-
perature ranges (−50–80 ◦C) makes Engelmaier’s method
unsuitable to model many accelerated thermal profiles of
interest.
A constitutive law + fatigue model will offer more accu-
rate results, with fewer caveats to its range of applicability.
Darveaux8 has shown that lifetimes can be predicted with
an accuracy of 25% for relative predictions, and ±2× for
absolute predictions. Syed6 has also shown a good cor-
relation with experimental results. The use of the Anand
isotropic hardening law15 is popular but arguably inap-
propriate because the kinematic hardening may have a
greater influence over solder behaviour. The inclusion of
kinematic hardening laws should improve prediction of
solder strains but the impact on reliability predictions is
at present unproven. The disadvantage of Finite Element
methods is the expertise and time required to set up the
analysis. This can be partly overcome by using simpler nu-
merical methods such as Clech’s SRS software,9 but these
approaches do not capture all the geometry details possi-
ble using FEA.
All of the above methods make use of an empirical fa-
tigue law whose constants are geometry dependent, the
more advanced damage mechanics based methods avoid the
geometry dependence to a large extent by explicitly mod-
elling the crack propagation. They offer the potential for
even better accuracy of results over an even wider range
of conditions, but are in need of further development and
experimental validation before they can be recommended
to an industry.
Acknowledgement
The work was carried out as part of a project in the Ma-
terials Processing Metrology Programme of the UK De-
partment of Trade and Industry. Other sponsors include
the EPSRC (Engineering and Physical Sciences Research
Council), Prime Faraday and the NPL (National Physical
Laboratory) in the UK.
R E F E R E N C E S
1 LaDou, J. Printed circuit board industry. Int. J. Hygiene
Environ. Health (in press).
2 Engelmaier, W. (1991) Solder attachment reliability,
accelerated testing, and result evaluation. In: Solder Joint
Reliability: Theory and Applications (Edited by J. H. Lau). Van
Nostrand Reinhold, New York.
3 IPC (1996) IPC-D-279: Design Guidelines for Reliable Surface
Mount Technology Printed Board Assemblies. IPC.
4 Darveaux, R., Banerji, K., Mawer, A. and Doddy, G. (1995)
Reliability of Plastic Ball Grid Array Assembly in Ball Grid Array
Technology (Edited by J. H. Lau). New York.
5 Clech, J. P. (2004) Lead-free and mixed assembly solder joint
reliability trends. In: APEX Designers Summit. IPC, Anaheim,
California.
6 Syed, A. (2004) Accumulated creep strain and energy density
based thermal fatigue life prediction models for SnAgCu solder
joints. In: ECTC. IEEE.
7 Zienkiewicz, O. C., Taylor, R. L. and Zhu, J. Z. (2005) The
Finite Element Method: Its Basis and Fundamentals.
Butterworth-Heinemann, Oxford, UK.
c© 2006 The Authors. Journal Compilation c© 2007 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 30, 400–412

412 S. R IDOUT and C. BA ILEY
8 Darveaux, R. Effect of Simulation Methodology on Solder Joint
Crack Growth Correlation (White Paper), Amkor.
9 Clech, J. P. (1997) Solder reliability solutions: a PC-based
design-for-reliability tool. Solder. Surf. Mount Technol. 9, 45–54.
10 Lee, S.-W. R. and Zhang, X. (1998) Sensitivity study on
material properties for fatigue life prediction of solder joints
under cyclic thermal loading. Circuit World. 24, 26–31.
11 Pang, J. H. L., Low, P. T. H. and Xiong, B. S. (2004) Lead-free
95.5Sn-3.8Ag-0.7Cu solder joint reliability analysis for
micro-BGA assembly. In: ITHERM. IEEE, Las Vegas.
12 Bathe, K.-J. (1996) Finite Element Procedures. Prentice Hall,
New Jersey.
13 Pang, H. L., Xiong, B. S., Neo, C. C., Mang, X. R. and Low,
T. H. (2003) Bulk solder and solder joint properties for lead free
95.5Sn-3.8Ag-0.7Cu solder alloy. In: ECTC 2003. IEEE.
14 Basaran, C. and Jiang, J. (2002) Measuring intrinsic elastic
modulus of Pb/Sn solder alloys. Mech. Mater. 34, 349–362.
15 Cheng, Z. N., Wang, G. Z., Chen, L., Wilde, J. and Becker, K.
(2000) Viscoplastic Anand model for solder alloys and its
application. Solder. Surf. Mount Technol. 12, 31–36.
16 Dudek, R., Walter, H., Doering, R. and Michel, B. (2004)
Thermal Fatigue Modelling for SnAgCu and SnPb Solder
Joints. In: EuroSimE. IEEE, Brussels.
17 Wiese, S. and Rzepka, S. (2004) Time-independent
elastic-plastic behaviour of solder materials. Microelectron.
Reliab. 44, 1893–1900.
18 Wiese, S. and Wolter, K. J. (2004) Microstructure and creep
behaviour of eutectic SnAg and SnAgCu solders. Microelectron.
Reliab. 44, 1923–1931.
19 Schubert, A., Dudek, R., Do¨ring, R., Walter, H., Auerswald, E.,
Gollhardt, A., Schuch, B., Sitzmann, H. and Michel, B. (2002)
Lead-free solder interconnects: Characterisation, testing and
reliability. In: EuroSimE 2002. IEEE, Paris.
20 Pang, J. H. L., Xiong, B. S. and Low, T. H. (2004) Creep and
fatigue characterization of lead free 95.5Sn-3.8Ag-0.7Cu solder.
In: ECTC 2004. IEEE.
21 Stolkarts, V., Keer, L. M. and Fine, M. E. (1999) Damage
evolution governed by microcrack nucleation with application
to the fatigue of 63Sn-37Pb solder. J. Mech. Phys. Solids 47,
2451–2468.
22 Neu, R. W., Scott, D. T. and Woodmansee, M. W. (2000)
Measurement and modeling of back stress at intermediate to
high homologous temperatures. Int. J. Plast. 16, 283–301.
23 Gomez, J. and Basaran, C. Damage mechanics constitutive
model for Pb/Sn solder joints incorporating nonlinear
kinematic hardening and rate dependent effects using a return
mapping integration algorithm. Mech. Mater. (in press).
24 Yang, X. and Nassar, S. (2005) Constitutive modeling of
time-dependent cyclic straining for solder alloy 63Sn-37Pb.
Mech. Mater. 37, 801–814.
25 McDowell, D. L. (1992) A Nonlinear kinematic hardening
theory for cyclic thermoplasticity and thermoviscoplasticity. Int.
J. Plast. 8, 695–728.
26 Lee, W. W., Nguyen, L. T. and Selvaduray, G. S. (2000) Solder
joint fatigue models: review and applicability to chip scale
packages. Microelectron. Reliab. 40, 231–244.
27 Akay, H., Zhang, H. and Paydar, N. (1997) Experimental
correlations of an energy-based fatigue life prediction method
for solder joints. In: International Intersociety Electronic and
Photonic Packaging Conference (INTERpack). ASME.
28 Desai, C. S. and Whitenack, R. (2001) Review of models and
the disturbed state concept for thermomechanical analysis in
electronic packaging. J. Electron. Packag. 123, 19–33.
29 Ghavifekr, H. B. and Michel, B. (2002) Generalized fracture
mechanical integral concept JG and its application in
microelectronic packaging technology. Sensors Actuators A:
Physical. 99, 183–187.
30 Gu, Y. and Nakamura, T. (2004) Interfacial delamination and
fatigue life estimation of 3D solder bumps in flip-chip packages.
Microelectron. Reliab. 44, 471–483.
31 Shangguan, D. (1999) Analysis of crack growth in solder joints..
Solder. Surf. Mount Technol. 11, 27–32.
32 Wen, S. (2004) Damage based fatigue criterion for solders in
electronic packaging. In: ITHERM. IEEE.
33 Basaran, C., Tang, H. and Nie, S. (2004) Experimental damage
mechanics of microelectronic solder joints under fatigue
loading. Mech. Mater. 36, 1111–1121.
34 Desai, C. S., Basaran, C. and Zhang, W. (1997) Numerical
algorithms and mesh dependence in the disturbed state concept.
Int. J. Numer. Methods Engng. 40, 3059–3083.
35 Abdul-Baqi, A., Schreurs, P. J. G. and Geers, M. G. D. (2005)
Fatigue damage modeling in solder interconnects using a
cohesive zone approach. Int. J. Solids Struct. 42, 927–942.
36 Yang, Q. D., Shim, D. J. and Spearing, S. M. (2004) A cohesive
zone model for low cycle fatigue life prediction of solder joints..
Microelectron. Engng. 75, 85–95.
37 Towashiraporn, P., Subbarayan, G. and Desai, C. S. (2005) A
hybrid model for computationally efficient fatigue fracture
simulations at microelectronic assembly interfaces. Int. J. Solids
Struct. 42, 4468–4483.
c© 2006 The Authors. Journal Compilation c© 2007 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 30, 400–412