Experimental and numerical study on size and constraining effects in lead-free
solder joints
Dr. Joël Cugnoni (*), Prof. John Botsis (*), V. Sivasubramaniam (*, #), Dr. Jolanta Janczak-Rusch
(#)
* Ecole Polytechnique Fédérale de Lausanne, LMAF-STI, Switzerland, [email protected]
# Swiss Federal Laboratories for Materials testing and Research, EMPA, Switzerland
Abstract
The reliability and durability of lead-free solder joints depends on a large number of factors, like
geometry, service temperature, processing parameters and size effects. In this work, the nature and
influence of the plastic constraints in the solder due to elastic joining partners have been studied by
parametric finite element simulation of solder joints with variable dimensions. The apparent
hardening due to plastic constraints has been shown to strongly depend on the gap to thickness ratio
with an inversely proportional evolution. Due to the interdependence of the geometrical, processing
and size effects, the macroscopic stress-strain constitutive law of lead-free solder materials should
be determined in the most realistic conditions.
In order to identify the elasto-plastic constitutive law of Sn-Ag-Cu solders, a sub-micron resolution
Digital Image Correlation technique has been developed to measure the evolution of strain in solder
joints during a tensile test. Experimental results of the stress-strain response of Sn-Ag-Cu solder
joints have been determined for several gaps. The measured load-displacement curves have been
used in an inverse numerical identification procedure to determine the constitutive elasto-plastic
behaviour of the solder material. The effects of geometrical constraints in a real solder joint with
heterogeneous stress and strain fields are then studied by comparing the apparent (constrained) and
constitutive (non-constrained) stress-strain relationships.
Once the size dependant constraining effects have been removed from the stress-strain relationship,
the size effects can be studied separately by comparing the constitutive elasto-plastic parameters of
joints with a variable thickness. Experimental stress-strain curves (constrained and unconstrained)
for Sn-4.0Ag-0.5Cu solder in joints of 0.25 to 2.4 mm gap are presented and the constraining and
the size effects are discussed.
1. Introduction
In the last decade, the evolution of microelectronic devices, dictated by the well known “Moore’s
law”, has continued to follow its exponential progression with an increase of transistor density of
more than 60 times resulting in one order of magnitude in dissipated power density. Moreover, the
evolution of the markets, with an incredible expansion of the mobile communication &
entertainment sectors, requires an always higher level of integration and has lead to new concepts
like System-On-Chip or stacked 3D packaging of electronic devices. The transportation, mobile or
automotive microelectronic applications require a high survivability of electronic devices under
complex thermo-mechanical dynamic loads, like power-cycling, impacts or vibrations.
Unfortunately, the increased complexity of electronic packages, higher transistor and power
densities are responsible for higher and higher temperature gradients and thermal-induced
mechanical loads in electronic packages which, combined with temperature-dependant material
properties, can lead to severe reliability issues. Moreover, in recent electronic designs like BGA, the
solder joints do not only act as electrical interconnections but also play a significant role in the
mechanical stability of the package. In this sense the joint and soldered zone are in close interaction
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which introduce very complex three-dimensional stress states in the solder.. It has also been
observed that different joint geometries, but from thesame materials, can show totally different
mechanical behaviours, with, for example, a change from ductile failure to quasi-brittle fracture and
large differences in the ultimate stresses. The effects of joint geometry on the mechanical
interaction of the solder with the substrates are not completely understood yet and should therefore
be studied.
The need for miniaturisation of electronic components induces also a large reduction of the typical
volume of the interconnects, until a point where the characteristic size of solder joints is comparable
to the characteristic size of the microstructure of the solder material itself. In this case, the
mechanical properties of the solder may change significantly due to scale effects. Moreover, the
mechanical properties of lead-free solder joints have shown to be considerably dependent on the
manufacturing process which can strongly affect their micro-structure. To ensure the reliability of
future products within the requirements of reduced cost and development times, numerical
simulation and optimization methods appears to be the most efficient tools today, but, in turn, these
techniques require a good knowledge of the thermo-mechanical loads and accurate material models
to predict and optimize the lifetime of solder joints. From this point of view, the recent progress of
the numerical simulation capabilities has lead to an important increase of the demand for highly
detailed 3D constitutive material models that can accurately represent the behaviour of complex
materials in a large range of thermo-mechanical loads. Due to several parameters that affect the
durability and reliability of solder joints and the necessity to identify complex multi-parameter
constitutive models, there is now a strong need for versatile characterization techniques that could
be used in-situ to determine accurately various mechanical properties of solder materials from
industrial electronic packages.
2. Phenomenological investigations and methodology
2.1 Constraining effects
A typical solder joint can be considered as a structure consisting of two elastic substrates joined by
an elasto-visco-plastic solder material. Although considered as a simple uniaxial structure, solder
joints can show very complex stress states due to the proximity of interfaces between the substrate
and solder If plastic or creep flow occurs in service conditions, the solder material will tend to keep
a constant volume and will shrink in the lateral directions. Due to the presence of the elastic
substrates that prevents this lateral shrinkage near the solder-substrate interfaces, an inhomogeneous
three dimensional stress state will then develop in the solder, which in turn may introduce large
hydrostatic pressure in the centre of the solder joint. When decreasing the solder gap relative to the
other dimensions, the triaxiality of the stress field, defined by the ratio R=p/m of the hydrostatic
pressure p and the Mises stress m, can completely modify the overall load-displacement response
of the solder joint by introducing a significant apparent hardening. This apparent hardening due to
triaxial stress state is usually called the constraining effect and it strongly depends on the geometry
of the solder joint.
Figure 1: Geometry and dimensions of the joint
2/18
In order to demonstrate the effects of plastic constraints on the behaviour of typical solder joints, a
parametric finite element study of Cu-SnAgCu-Cu joints with variable dimensions (Figure 1) has
been carried out and the hardening introduced by the plastic constraints has been monitored by
defining the constraining effect ratio Q = (ujoint-usolder)/usolder that compares the ultimate
engineering stress ujoint of the solder joint (applied load / area of the solder joint) with the ultimate
Mises equivalent stress of the bulk solder material usolder. The solder gap g and the length L of the
joint have been kept constant throughout the parametric study with values of 1 mm and 60 mm. The
width w and thickness t of the solder joint have been varied according to the following ratios: W =
w/t = {1, 2, 5, 10, 20} and G = g/t = {0.05, 0.1, 0.2, 0.5, 1, 2}. The combination of all the possible
cases has lead to the evaluation of thirty different joint geometries covering a broad range of aspect
ratios. The copper substrates have been considered as purely elastic and the solder has been
modelled as an elastoplastic material with exponential and linear isotropic hardening terms defined
by the following hardening function:

 y ( p )   y0  Q 1  e
b  p
 K 
(1)
p
where the material constants  y0 , Q  , b and K represent respectively the initial yield stress, the
asymptotic exponential hardening stress, the exponential hardening rate and the linear hardening
modulus. The values of the mechanical properties used in the model are given in Table 1.
The average Mises stress m and average hydrostatic pressure p in the solder as well as the apparent
engineering stress-strain response of the joint have been extracted from tensile loading conditions in
order to evaluate the constraining effect and triaxiality ratios Q and R. The correlation between the
triaxiality ratio R and the constraining effects (Figure 2) proves that the hydrostatic pressure
induced by the plastic constraints is the dominant cause of the apparent hardening that can be
observed in very thin joints. If we analyze the effects of plastic constraints as a function of the
geometry (Figure 3), we observe that the constraining effect ratio Q shows a hyperbolic dependency
on the ratio G = solder gap / joint thickness with Q being approximately proportional to 1/G. From
these results, we can see that the effects of plastic constraints may, in an ideal case, increase the
apparent ultimate stress of very thin joints by a factor of up to 6 for G = 1/20 and that the
constraining effects begins to play a significant role when G < 1/2. In comparison, the width to
thickness ratio W has a relatively small influence on the constraining effects (Figure 4) and we can
consider that the width does not play any significant role when W>5. When W is less than 2, the
effects of the edges are much more pronounced and the constraining effects decreases of up to 20%.
These results, showing the considerable impact of constraining effects on the overall response of
solder joints, prove that constraining effects have to be taken into account in the characterization
and design stages of new electronic packages. Another consequence of the strong dependency of the
behaviour of the joint on its geometry is that results from different mechanical tests can lead to
radically different apparent mechanical properties of the joint and we should clearly distinguish
between the stress-strain response of the joint and the constitutive behaviour of the solder. For
example, this implies that simple mechanical test data of solder joints should not be directly used in
a finite element as constitutive properties for the design of solder joints, but we should use instead
test data from bulk solder specimens (no effects of constraints) or by taking into account the
constraining effects in the characterization process of solder joints.
Table 1: constitutive properties used in the parametric FEM
Cu substrate
E

(GPa)
(-)
112
0.33
E
(GPa)
48.4
SnAgCu lead-free solder
 y0
Q
b

(-)
(MPa)
(MPa)
(-)
0.35
35.1
16.6
675
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K
(MPa)
85.6
Correlation between Constraining Effect ratio & Triaxiality ratio of stress field
7
Constr. effect ratio, Q
6
5
4
3
y = 0.9686x - 0.4707
R2 = 0.9938
2
1
0
0
1
2
3
4
5
6
7
8
Triaxiality ratio, R
Figure 2: Correlation between constraining effect ratio Q and triaxility ratio R in the solder joint
Constraining effect ratio in function of Gap / Thickness ratio
8
Constraining effect ratio, Q
7
6
5
4
3
-1.3
Q = 0.151G
2
R2 = 0.988
1
0
-
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.80
Gap / Thickness ratio, G
Figure 3: Constraining effect ratio Q as a function of gap to thickness ratio G
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2.00
Constraining effects as a function of geometry
Constraining effect ratio, Q
6
5
4
3
2
1
20
0
10
0.05
5
0.1
0.2
G= g
/ t
2
0.5
1
1
2
W
=
w
/t
Figure 4: Evolution of the constraining effects as a function of the geometry of the solder joints
Constraining effects may also affect considerably the damage mechanisms by changing the threedimensional stress distribution in the solder joint. For example, we have observed that, for different
joint geometries, the equivalent von Mises stress distribution can be radically different (Figure 5
and Figure 6). From the results of the present parametric finite element simulations, we have
noticed that a large plastic deformation zone in the middle of the joint appears for large gap to
thickness ratio (G>1). On the contrary, a clear concentration of plastic deformation can be observed
near the interfaces for thin joints (G<0.5), which is responsible for crack initiation and propagation
along the interface.
Figure 5: Concentration of Mises stress and plastic damage
in the center of thick joints (G>1) due to the effects of
substrates.
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Figure 6: Stress concentration at the interface in
thin joints (G < 0.5). due to the induced
constraining effects.
2.2 Size effects
When decreasing the characteristic size of solder joints, changes in the global mechanical behaviour
of the solder material might be observed due to multiple phenomena. First of all, the microstructure
and also the composition of the solder may change due to diffusion processes during production.
The change in microstructure mainly depends on the manufacturing parameters (local temperature
profiles and gradients) and on the change of the heat conduction and diffusion processes with the
size of the solder. On the other hand, a change in the statistics and impact of defects due to a
reduced volume of material may also play a significant role (weakest link assumption) in the same
way as what has already been observed in other materials. For example, the number and influence
of pores or the effects of grain boundaries, which can be both considered as weak points in the
solder material, may change when the characteristic volume of the tested solder decreases up to the
characteristic size of the microstructure (grain size). Due to several possible causes, studying the
size effects is a complex experimental task that can be seen from different point of view. But the
most reasonable approach to this problem is a systematic study of the phenomena by carefully
designing experiments and checking the consistency of the observations for each specimen size. In
this present work, we have tried to isolate the purely mechanical size effects by keeping the same
processing parameters and controlling the microstructure from all solder gap sizes.
2.3 In-situ characterization method
Besides size and constraining effects, lead-free solder materials have shown very complex elastovisco-plastic behaviours [1-19] with strong dependencies on time, strain history, temperature and
processing parameters. Due to these multiple influences, the macroscopic stress-strain constitutive
law of lead-free solder materials should therefore be determined in the most realistic conditions in
order to maximise the consistency of the identified mechanical properties.
For an in situ characterisation of the mechanical properties of solder, an adequate strain
measurement technique for small scale joint specimen must be developed. In most cases, solder
joints consist of only a thin layer or a small ball of solder alloy, with a typical size of less than 100
m, confined between two relatively rigid substrates. Testing material properties at this length scale
is a difficult task as it requires very sensitive and accurate measurements of both applied loads and
displacements with typical values in the order of 10 N and 0.1 m respectively. Optical strain
measurement techniques, like Electronic Speckle Pattern Interferometry (ESPI), Moiré
Interferometry or Digital Image Correlation (DIC) [31-34], are the best candidates to achieve the
required displacement and spatial resolutions. Despite their high spatial resolution and full field
capabilities, the ESPI and Moiré interferometry techniques are usually very difficult to use in
practice because of their stability requirements and inability to accurately measure strain fields
when the specimen is loaded with a rate-controlled displacement ramp. But, by combining DIC
image post-processing, optical microscopy observation and high-resolution CCD image recording, a
versatile high resolution strain measurement technique can be developed to characterize small scale
solder joints in a convenient way.
As seen before, the load-displacement curve of the joint does not directly represent the constitutive
behaviour of the solder material and constraining effects must be taken into account in the
characterization of the solder. Since no analytically invertible model is available for the plastic
response of constrained solder material, the constitutive stress-strain relation of the confined solder
material cannot be determined directly from the global load-displacement curve of the whole joint.
On the other hand, if all the material properties were known, the elasto-plastic response of an
arbitrary solder joint could be easily simulated with standard non-linear finite element models
(FEM). Therefore, given the geometry and the load-displacement response of the joint, the problem
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of identifying the constitutive elasto-plastic properties of the solder from the constrained response
of the solder joint can be reduced to an inverse numerical procedure where the unknown material
constants of a finite element model must be determined so that the difference between predicted and
measured load-displacement curves is minimal in the least squares sense. This inverse numerical
procedure can be simply understood as a fitting method of a numerical model to experimentally
determined data. This class of mixed numerical-experimental identification techniques provide the
unique capability of identifying any unknown constitutive parameter for any specimen geometry as
long as the measured quantities are sufficiently sensitive to the identification parameters [24-30].
When the constitutive parameters have all been determined, the constraining effects can be studied
by comparing the constitutive stress-strain curve of the solder material and the stress - strain
response of the solder joint. By varying the solder gap of the test specimens, the same procedure
can also lead to a better understanding of the size effects by simply studying the evolution of the
identified constitutive properties as a function of the characteristic size of the specimen. In the
present study, the main advantage of this identification approach is that it can separate the effects of
constraints which depend on size (accounted for in the FEM) and the purely constitutive size effects
of the material. The flowchart presented in Figure 7 illustrates the major steps of the proposed
characterization method.
Figure 7: Flowchart of the inverse identification procedure
3. Experimental approach
The objectives of the present experimental work is to isolate and study the scale effects in lead free
solder materials and to develop a versatile in-situ characterization procedure that takes the
constraining effects into account. A set of specimens with a variable solder gap have been tested in
tension to study the size and constraining effects while the manufacturing parameters have been
kept the same in order to minimize the effects of the microstructure on the different gaps. An
inverse identification technique is used to extract the constitutive properties of the solder from the
7/18
measured load-displacement curve of the joints. Finally the constraining and size effects are
evaluated from the identified data and compared as a function of the solder gap.
3.1 Materials and specimens
The chosen test specimen geometry (Figure 8) consists in a 20x1 mm joint of Sn-4.0Ag-0.5Cu
solder alloy confined between two 60 mm long copper plates, with solder gaps of 0.25, 0.5 mm, 0.7
mm, 1.2 mm and 2.4 mm. The tensile specimens were produced as follows: two 60 x 20 x 1 mm Cu
sheet are clamped into a solder jig with a spacer to maintain the nominal solder gap and an
industrial solder paste (Alpha OM-338) is applied in the solder gap. Then the solder jig is placed on
a hotplate and the specimen is heated to melting point of the solder at 234 °C (heating rate of
approx. 3-4°C/min). The solder jig is then transferred to a second hotplate in which the solder is
maintained at its liquid state at 234 deg for 60 seconds. Finally the jig is quenched in water on a
steel block to ensure rapid cooling. For 0.25 and 0.5 mm gap joints, an additional vacuum is applied
during 1 minute while the solder is held liquid in order to reduce the porosity. Finally, the two sides
of the tensile specimens are roughly polished to create a sufficient amount of optical details for
accurate DIC strain measurements.
3.2 Mechanical testing
Tensile tests have been performed on an Instron 5848 MicroTester with special test grips designed
to ensure pure tension loading while minimizing out of plane displacements. The test profile
consists in a single displacement ramp up to rupture at a constant 0.5 m/s speed. A small region in
the centre of the joint is observed through an standard optical microscope (magnification of 24x to
48x) and a sequence of high resolution images is recorded during the test by a 1.3 MPixels CCD
video camera. A second video camera is also placed on the opposite side of the sample in order to
monitor the global damage mechanisms of the joint during testing. Containing about 300 images,
the recorded sequences are then used in a custom Digital Image Correlation software in order to
calculate the evolution of the average strain in the solder during the whole test.
Figure 8: Geometry of the test specimens and Digital Image Correlation measurements
8/18
Based on a bi-cubic image interpolation scheme, the present DIC software has a displacement and
strain resolution of approximately 0.1 m and respectively 0.02% with the microscope set at 48x
magnification. The DIC strain measurement is performed on two regions on the copper plates
adjacent to the interfaces (Figure 8), which were meshed with a correlation grid of 10x1 cells of
100x50 pixels. The equivalent gauge length of this strain measurement technique is approximately
equal to 1.5 times the solder gap of the tested joint, and gives a very good representation of the
average strain in the solder even for small gaps. Finally, the stress –strain responses of the solder
joints have been calculated from the load displacement curves. In this present work, the stress-strain
or load-displacement curves measured with the DIC optical measurement technique will always be
called the response of the solder joint to distinguish it from the constitutive behaviour of the solder
material itself.
3.3 Experimental results
The present procedure has been used to test 26 solder joint specimens with solder gap width equal
to 0.25, 0.5, 0.7 1.2 and 2.4 mm which corresponds to the following gap to thickness ratios G =
{0.29, 0.58, 0.77, 1.33, 2.71}. An important scatter of the measured stress-strain curves has been
observed with a typical standard deviation of 5%. This scatter appeared to be clearly due to a visible
variability in the quality of the produced joints, especially for 0.25 and 0.5 mm gap joints where a
few macroscopic voids could be observed. A lot of effort has been devoted to optimize the quality
of the produced joints and the final results shown here have required the use of vacuum during the
joining process in order to reduce the visible porosity of 0.25 and 0.5 mm solder joints. Finally, the
stress-strain curves are averaged for each gap in order to be used for the inverse numerical
identification of the constitutive properties (Figure 9). We have also observed that the optical strain
measurement of the elastic part of the curves can be perturbed by small out of plane displacements
that slightly change the optical magnification of the microscope. Despite our efforts to avoid such
displacements by carefully designing the specimen fixtures and alignment tools, the Young
modulus can not always be measured accurately with the present test setup, but, as our interest is
focused on the plastic stage of the joint response, this problem is not considered important in the
present study. Taking the standard deviation into account, we can observe that the average stressstrain curves of 0.5 mm, 0.7 mm, 1.2 mm and 2.4 mm joints appears to be similar, even if a slight
trend can be observed. However, the stress-strain response of 0.25 mm joints shows a more
pronounced hardening as well as a reduced ultimate strain compared to the other joints. The
ultimate stress of 0.25 mm joints (G=0.29) increases by 20% compared to 0.5 mm joints (G=0.58)
and as discussed before, this trend can be explained by the effects of plastic constraints that start
playing a significant role for G<0.5. For comparison, a similar tensile test has been carried out on a
cast solder specimen produced in the same conditions (rapid cooling in water). Even with the same
processing parameters, the experimental stress-strain curve of the cast specimen is completely
different from the response of the solder joints, but we cannot really compare these curves directly
because of the possible constraining effects. To compare the actual constitutive properties for
different gaps, the experimental load-displacement response of these joints should be used in an
identification procedure that takes into account the constraining effects.
9/18
Experimental stress-strain curves of the tested joints
90
80
70
stress (MPa)
60
50
40
30
20
0.25 mm
0.7 mm
2.4 mm
10
0.5 mm
1.2 mm
Cast specimen
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
strain (-)
Figure 9: Constrained stress-strain curves of Sn-4.0Ag-0.5Cu solder in 0.25 to 2.4 mm gap joints
3.4 Microstructure & damage analysis
Even if the processing parameters have been kept constant for the production of all specimens, the
microstructure has been checked in order to ensure the consistency of the comparisons for the
different joint size. Moreover, as mentioned before, damage mechanisms can differ depending on
the joint geometry and a fractography analysis has been carried out to study the failure processes.
Microstructure:
The microstructure has been characterised by optical microscopy of polished specimens from
untested joints. We notice in Figures 9 to 11 that the produced Sn-4.0Ag-0.5Cu solder has a very
fine microstructure due to rapid cooling after soldering. The microstructure of 0.7 to 2.4 mm gap
joints does not change significantly but for the joints produced under vaccum (0.25 mm & 0.5 mm),
few long Sn dendrites were observed near the interfaces. Fortunately, these dendrites are rather
seldom and randomly oriented and their effect should be very small on the average stress-strain
response. The thickness of the interfacial layer is constant for each solder gap with a typical size in
the range of 3 to 5 microns.
Porosity:
Two types of porosity have been observed in the produced joints. First, macro porosity that is
visible by eye consists usually of very few spherical pores with typical diameter of 50 to 100
microns. The diameter of macro pores does not seem to vary with the gap, but their number is
increasing for smaller gaps even with the vacuum joining process (Figure 12). Second, when the
joint is observed at high magnification, micro pores can be observed with typical diameter in the
order of 5 microns. The number of micro pores is also increasing for smaller joints and they appear
to be concentrated near the interfaces. Due to the combination of these two types of porosity, the
overall void volume fraction is difficult to quantify precisely, but, the trend is a clear increase of
porosity fraction with smaller gaps. The evaporation of the flux seems to be responsible for the
observed porosity and it seems that bubbles of flux are captured between the interfaces due to
surface tension effects during the soldering process. The capillary effects and interaction between
10/18
bubbles and interface can also explain why more pores are kept inside the solder alloy with smaller
gaps widths. The vacuum used in the soldering process clearly decreases both macro and micro
porosity but does not remove all the voids, especially near the interfaces.
Fractography:
In order to understand the damage and failure processes, fracture surfaces of broken specimens have
been observed with SEM and in-plane metallographic views of tested joints have been taken by
standard optical microscopy. Moreover, the image sequence used for the DIC strain measurement
contains also important information about the evolution of damage during the test. From these
observations, we have seen that the growth of pores and their interaction with the interfaces has a
large impact on the hardening and rupture behaviour of the solder. During plastic deformation and,
more precisely, in the softening region of the stress-strain curve, the size of the macro pores
contained in the solder increases radically and usually cracks tend to propagate through the pores
(Figure 13) and/or the interface. Moreover, interfacial porosity can act as a starting point for
interfacial cracks between intermetallic phases and solder, which can even lead to quasi-brittle
rupture of the joint (Figure 13).
When studying the fracture surface, we have noticed the presence of both “dimple” structures that
reveals ductile fracture and smooth rupture surfaces near the edges which are related to crack
propagations (Figure 15 and Figure 16). The macro pores after rupture appears to have grown
through most of the joint thickness and a close look has revealed the presence of pure Sn in these
macro pores (Figure 17) which shows that they were created during the manufacturing process and
are not due to void nucleation. In general, porosity seems to be the dominant parameter as it can
completely modify the mechanical response of solder joints and cannot be easily controlled in the
manufacturing process.
Global damage mechanisms:
As mentioned before, constraining effects can change the damage behaviour by modifying the
three-dimensional stress state in the solder joint. In our experiments, we have observed that thin
joints (0.25 mm) mainly break at the interface (brittle crack propagation), thick joints (1.2 & 2.4
mm) usually fails by accumulation of plastic damage and void growth in the joint centre and
moderately thick joints (0.5 & 0.7 mm) show both types of failures. These observed tendencies
corroborates the results of the numerical simulations of the plastic deformation in solder joints with
variable gap (Figure 5 and Figure 6) in which we clearly see stress concentrations near the interface
for G<0.5 and in the middle of the joint for G>1.
Figure 10: Typical microstructure of 2.4 mm joints
with a fine structure due to rapid cooling after
soldering
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Figure 11: Microstructure of 0.7 mm joints with some
small voids near interfaces
Figure 12: Typical microstructure of 0.5 mm joints
produced with vacuum showing the presence of pure
Sn dendrites and pores
Figure 14: Metallographic view showing the growth
of the macro porosity and crack propagation
through large voids
Figure 16: Typical “dimple” structure around
macro pores
Figure 13: Metallographic view of a broken interface
showing the effects of localized porosity on the crack
propagation
Figure 15: Fractography of a broken joint which shows
three typical features: macro voids, “dimple” structure
revealing ductile fracture and smooth rupture surfaces
due to crack propagation
Figure 17: Sn dendrites in the center of large voids
reveals that macro pores were created during
manufacturing.
4. Identification of constitutive properties
To identify the constitutive properties of the solder from the measured load-displacement response
of the joint, a numerical FE model corresponding to the experimental set-up must be built. The
interest of this type of numerical modelling is that thanks to its flexibility it can account for all the
geometrical effects which is particularly important here because of the constraining effects.
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4.1 Finite element modelling
The geometry of each sample has been measured precisely before testing and average dimensions
have been calculated for each solder gap to create a corresponding 3D finite element model of the
joints for the commercial solver AbaqusTM. In order to reduce the total number of degrees of
freedom, symmetries have been taken into account and only one eighth of the solder joint has been
meshed with 3D wedges and hexahedral elements. As shown in Figure 18, the solder alloy volume
has been divided in approximately 50x5x5 linear hexahedral elements with reduced integration to
prevent locking problems under large plastic deformation, while the copper part has been
discretized in approximately 5000 linear 3D wedges elements. Because of the iterative nature of the
proposed inverse numerical method, the mesh size has been chosen as an optimum between the
accuracy of the simulated load-displacement response of the joint and the finite element solution
time.
Typical elastic material properties (Young’s modulus ECu = 112 GPa , Poisson’s ratio  Cu = 0.33)
have been applied to the copper part of the joint. Among the various models that exist in the
literature, we have chosen an isotropic elasto-plastic law with linear and exponential hardening
components because of its relative simplicity and its ability to describe the plastic behaviour of bulk
lead free solder materials (experimentally verified by the authors). The hardening law
corresponding to this model is described relation (1) and has been implemented in Abaqus
through a UHARD Fortran user subroutine. A large deformation FE formulation has been selected
in order to capture the non-linear geometric effects.
The boundary conditions have been set to reflect the three symmetry planes and a displacement
ramp has been imposed on the surface corresponding to the loading fixtures of the real test. The
simulated total applied load and the calculated displacement of the nodes corresponding to the DIC
measurement area have been calculated at 25 equally spaced points in order to produce a precise
load-displacement curve that can be compared with the experimental measurements.
Figure 18: Finite element mesh of 1/8th of a 1mm gap joint.
4.2 Inverse numerical identification procedure
The goal of the proposed identification procedure is to determine the constitutive parameters  y0 ,
Q  , b and K of the linear/exponential plastic law in relation (2) from the measured load-
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displacement response of the solder joint. The constitutive parameters to determine are normalized
and combined to define the vector of identification parameters α :
~
~
~
~
(3)
α  [ E s Es ,  y0 ~ y0 , Q  Q  , log( b) log( b ) , K K ]
where the tilde represents the initially guessed values of the constitutive properties. Even though its
value was not of interest in this study, the Young’s modulus of the solder Es was also added to the
identification vector α in order to improve the accuracy of the identified plastic constitutive
parameters.
A non-linear least-squares Levenberg-Marquardt optimisation algorithm [35-37] has been selected
because of its robustness, its accuracy and its excellent convergence speed. The error function ε (α )
used by the minimisation procedure is defined as the vector of the normalized differences between
the experimental and simulated load-displacement curves P exp ( exp ) and P num ( num , α) .
Given this error norm ε (α ) , the identification of the constitutive properties leads to the following
minimization problem:
2
1
Find α k such that min
(4)
F (α k ) with F (α k )  ε(α k ) , k  1, 2, ....
k
α
2
where the k superscript indicates the iteration number and where F (α k ) represents the global leastsquares objective function (scalar) to be minimized. To start the iterative minimisation procedure,
an initial parameter vector 0 is supplied and is usually set equal to unity so that the initial
parameter vector corresponds to the initial guess of the constitutive properties Es,  y0 , Q  , b and K.
After several iterations, the optimization process is stopped when the global error residual was less
than a given tolerance and the identified constitutive parameters is computed from the parameter
vector α k at the last iteration from relation (3).
With 5 unknown parameters, the present identification algorithm requires approximately 9 finite
element solutions per iteration. In most cases, the convergence of the identification parameters is
reached after 4 to 5 Levenberg-Marquardt iterations, which corresponds to approximately 3 hours of
computations on a recent PC computer (Intel Pentium IV 3Ghz) with the present finite element
model.
4. Identified constitutive properties
The proposed inverse identification procedure has been applied to extract the unconstrained
constitutive properties from the measured load-displacement response of 0.25 mm to 2.4 mm joints.
In this study, the inverse method has shown excellent robustness and convergence properties, with a
maximum error of 4% between the identified and measured load-displacement curves even when
beginning the optimization with 300% error on the initial guess. As shown in Figure 19, all the
parameters converge very quickly and the global error norm decreases by more than 2 orders of
magnitude during the first iteration, which demonstrates the excellent conditioning of the chosen
inverse procedure. Table 2 summarizes the identified constitutive properties for the different gaps
and presents also the engineering yield stress y0.2% and the asymptotic exponential hardening stress
uy0+Q which represents approximately the ultimate stress.
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Convergence graph
10
1.2
1
1
Error norm
Relative parameter value
1.1
0.9
alpha1
alpha2
alpha3
alpha4
alpha5
F(alpha)
0.8
0.7
0.6
0.5
0.1
0.01
0
1
2
3
4
5
Iteration
Figure 19: Convergence graph of the identification procedure (1.2 mm gap joint )
Table 2: Initial and identified constitutive properties
Specimen
 y0
Initial guess
0.25 mm
0.5 mm
0.7 mm
1.2 mm
2.4 mm
(MPa)
25.0
36.5
38.4
25.6
35.1
38.3
Q
(MPa)
17.0
7.76
9.03
23.9
16.6
15.9
B
(-)
400
62.7
191
772
675
433
K
(MPa)
100
9.6
-7.7
58.5
85.6
56.3
y0.2%
uy0+Q
(MPa)
37.4
41.3
44.4
47.5
47.6
(MPa)
44.2
47.5
49.5
51.7
54.1
Identified constitutive stress-strain curves
70
60
stress (MPa)
50
40
30
0.25 mm
0.5 mm
0.7 mm
1.2 mm
2.4 mm
Cast specimen
20
10
0
0
0.005
0.01
0.015
0.02
0.025
strain (-)
0.03
0.035
0.04
0.045
0.05
Figure 20: Identified constitutive relations for Sn-4.0Ag-0.5Cu solder in 0.25 to 2.4mm joints
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4.4 Identified constraining and size effects
The identified constitutive stress-strain law for the different gaps are presented in Figure 20.
Contrary to the stress-strain response of the joints that could not be compared directly due to
constraining effects, the constitutive stress-strain curves identified by the inverse numerical
procedure represent the actual behaviour of the solder without any geometrical effects. The
identified constitutive relations exhibit here a clear trend: when decreasing the gap, the engineering
yield stress y0.2%, and the ultimate stress u of the as-soldered material are all decreasing
significantly. Due to the porosity content, we cannot assess the material scale effects from these
identified properties since they reflect a combination of both factors. If we compare the identified
constitutive relations of the solder inside the joints with the bulk solder specimen, we observe very
different mechanical properties with more than 50% difference between the ultimate stress of the
SnAgCu alloy inside the joint and the cast specimen. The large difference between the properties of
bulk and joint specimens as well as the significant effects of the manufacturing process
demonstrates the need for in-situ characterization of solder properties in the development of the
future generation of electronic packages.
From these results, the effects of the plastic constraints have also been studied by comparing the
engineering ultimate stress of the joint and the constitutive ultimate stress of the solder. The
constraining effect ratio Q has been calculated from the test and constitutive data with the following
formula:
ˆ   u
(5)
Q u
u
where ˆ u denotes the ultimate stress of the measured stress-strain curves of the joints.
The evolution of identified constraining effects ratio Q as a function of the gap to thickness ratio G
is represented in Figure 21. This figure shows once again the hyperbolic shape of constraining
effects as a function of the gap to thickness ratio G, with an evolution in the form of Q = 0.25 G-1.
1.40
constraining effect ratio, Q
1.20
1.00
Q = 0.25 G -
0.80
1
0.60
0.40
0.20
0
0.5
1
1.5
2
2.5
gap / thickness ratio, G
Figure 21: Constraining effect ratio Q as a function of gap to thickness ratio G
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3
The comparison of plastic constraining effects (represented here by  = ˆ u -  u ) and the size
effects (Figure 22) shows that, when changing the solder gap in the present joint geometry, the
constraining effects completely compensate the decrease of properties due to size effects and even
dominate the evolution of the plastic response of the joint for small gaps. Indeed, in the present
case, when the solder gap is reduced from 2.4 mm to 0.25 mm, the increase of apparent ultimate
stress of the joint due to plastic constraints (35MPa) dominates the decrease of the constitutive
properties due to the manufacturing quality (10MPa), which explains the clear improvement
observed on the measured response of the thinnest joints in Figure 9.
Constraining and size effects
60
50
Stress (MPa)
40
30
20
10
Ultimate stress
Eng. Yield stress
Effects of Constraints
0
0.25
0.5
0.7
1.2
2.4
Gap width (mm)
Figure 22: Constitutive properties and constraining effects as a function of solder gap
5. Conclusion
The influence of plastic constraints and size effects on the elasto-plastic response of lead-free solder
joints has been successfully studied with the use of a novel inverse numerical identification method
based on optical DIC strain field measurement and 3D finite element simulation. The proposed
inverse numerical technique has demonstrated excellent robustness and convergence properties,
with a typical identification error of less than 4% between measured and identified loaddisplacement curves. The measured constrained stress-strain response of 0.25 mm to 2.4 mm joints
showed only clear improvements of yield and ultimate stresses when decreasing the gap to
thickness ratio below 0.5. On the other side, a clear trend in the identified constitutive properties of
the solder was observed with a decrease of the constitutive yield and ultimate stresses for thinner
joints. In the present case, a metallographic study of the test specimen showed the consistency of
the microstructure of the joints for all gaps as well as an apparent increase of the porosity for the
smaller joints. Due to the presence of porosity, the scale effects of the solder material could not be
assessed from the identified results due to their dependency on a combination of those two factors.
The effects of plastic constraints were studied by comparing the apparent (constrained) and
constitutive (unconstrained) stress-strain curves. Important constraining effects were observed, with
a 60% increase of the ultimate stress of the thinner joints solely due to the plastic constraints and the
evolution of the constraining effects with the gap showed an inversely proportional dependency on
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the gap to thickness ratio. Finally, with the chosen joint geometry, the increase of yield and ultimate
stress due to plastic constraints was shown to completely dominate the decrease due to size effects
and manufacturing quality.
Acknowledgments
This research lies within the scope of the European Community COST 531 Action: Lead free solder
materials and was supported by the Swiss State Secretariat for Education and Research (SER).
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