Composites Part B 173 (2019) 106870
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Composites Part B
journal homepage: www.elsevier.com/locate/compositesb
Failure of brittle micro-spherical shells embedded in elastomer matrix
under indentation
Yinan Lu a, Jay D. Carroll b, Kevin N. Long b, *, Rong Long a, **
a
b
Department of Mechanical Engineering, University of Colorado Boulder, Boulder, CO, 80309, USA
Sandia National Laboratories, Albuquerque, NM, 87123, USA
A R T I C L E I N F O
A B S T R A C T
Keywords:
Glass micro balloons
Elastomer matrix
Indentation
Damage
Sylgard
Syntactic foam
Silicone elastomer filled with glass micro balloons (GMB) is an elastomeric syntactic foam used in electronics and
component packaging for encapsulation, potting, stress-relief layer, and electrical insulation purposes. Under
mechanical loading, the reinforcing phase, namely the GMBs embedded in the elastomer matrix, may break or
delaminate, leading to internal damage and macroscale stiffness degradation, which can alter the material’s
protective capacity against mechanical shock and vibration. The degree of damage is controlled by the loading
history, delamination, and failure behavior of the GMBs. We investigate the GMB failure behavior in this work
wherein we present an indentation experiment to measure the force required to fail individual GMBs that are
either embedded in the elastomer matrix or adhered to the surface of an elastomer layer. The indentation
apparatus is augmented with an inverted optical microscope to enable in situ imaging of the GMB. Failure modes
for the embedded or non-embedded GMBs are discussed based on the morphology of the broken GMBs and the
measured failure forces. We also measure the adhesion energy between the glass balloon and the elastomer,
based on which the possibility of delamination between the GMB and the surrounding elastomer matrix during
the failure process is evaluated. Our results can facilitate the development of a failure criterion of GMBs which is
necessary for establishing a physics-based constitutive model to describe the macroscopic damage mechanics of
elastomeric syntactic foams.
1. Introduction
microspheres. Such glass microspheres, referred to as glass micro bal­
loons (GMBs), have a shell thickness (~1 μm) that is much smaller than
the diameter (~10–100 μm). Recently uniaxial compression tests were
performed on the GMB-elastomer composite [4,8], where it was found
that the Young’s modulus of the composite can be increased to 5 times
that of the PDMS matrix modulus if the GMB volume fraction reaches
40%. More interestingly, the composite exhibits significant hysteresis
accompanied by irrecoverable softening during cyclic loadings, which
resembles the Mullins effect typically observed in filled rubber [9–11].
This softening phenomenon implies an internal damage mechanism in
the GMB-elastomer composite, which was attributed to the fracture of
GMBs—rather than GMB-matrix delamination, matrix cracking, or other
damage mechanisms—through postmortem microscopic imaging and in
situ X-Ray computed tomography micrographs [8,12]. Fracture of GMBs
was also observed as an important damage mechanism in syntactic
foams with stiff matrix (i.e., epoxy resin) under compression [13–17].
Furthermore, recent modeling efforts showed that failure of GMBs has a
Elastomeric syntactic foam is a class of composites that consist of a
soft elastomer matrix filled with hollow microspheres [1–3]. In com­
parison to the pure elastomer, syntactic foam is endowed with increased
modulus, lowered density, lowered coefficient of thermal expansion,
improved dielectric properties, and substantially enhanced capability of
mechanical energy absorption [4,5]. As a result, this material is
deployed for encapsulation, potting, stress relief layer and electrical
insulation applications [4–7]. For example, elastomeric syntactic foam is
used in electronic packaging to protect electronics and delicate com­
ponents from stray voltage and mechanical shock or vibration.
Since applications of syntactic foams often involve mechanical shock
and vibration, fundamental understandings on their mechanical be­
haviors are of vital importance for reliability analysis. This paper focuses
on a representative material system of elastomeric syntactic foams: a
polydimethylsiloxane (PDMS) silicone elastomer filled with hollow glass
* Corresponding author.
** Corresponding author.
E-mail addresses: knlong@sandia.gov (K.N. Long), rong.long@colorado.edu (R. Long).
https://doi.org/10.1016/j.compositesb.2019.05.081
Received 20 September 2018; Accepted 6 May 2019
Available online 11 May 2019
1359-8368/© 2019 Elsevier Ltd. All rights reserved.
Y. Lu et al.
Composites Part B 173 (2019) 106870
much stronger degradation effect on the small deformation elastic bulk
modulus of such composites than the delamination mechanism [4].
Constitutive models capable of relating the macroscopic damage me­
chanics to the microscopic GMB failure events and predicting the frac­
tion of surviving GMBs under given mechanical loadings are highly
desirable for the design of elastomeric syntactic foams. To develop such
a physics-based constitutive model, a thorough understanding on the
physical nature of the GMB failure process must be established.
A well-known failure mode of thin spherical shells under compres­
sion is buckling. Due to the brittleness of glass, the large displacement
resulting from buckling will immediately cause a GMB to fracture. Sig­
nificant efforts have been devoted to the theoretical analysis of buckling
instability of spherical shells in the literature. For example, the critical
buckling loads for a non-embedded spherical shell under point load or
uniform pressure have been determined [18]. In addition, the buckling
of a spherical shell between two rigid plates has been analyzed in Audoly
and Pomeau [19]. More recently, the post-buckling deformation of
spherical shells or caps have been studied through numerical simula­
tions [20–23]. On the experimental side, the failure of GMBs or other
types of brittle micro-spherical shells was tested under either isostatic
pressure [24] or compression between two rigid plates (e.g., through
nano-indentation [25,26]). However, for the damage mechanics of
elastomeric syntactic foams, it is more relevant to consider the failure of
GMBs that are embedded in an elastic matrix. In this case, the buckling
instability may be affected by the constraint provided by the sur­
rounding matrix material. Analytical or numerical solutions have been
developed to determine the critical load for the buckling of an embedded
spherical shell under hydrostatic pressure [27,28] or uniaxial
compression [28–30]. In contrast, experimental studies on the failure of
embedded GMBs are much more limited. As a result, many important
questions regarding GMB failure still remain unanswered. For example,
is buckling the main failure mode for embedded GMBs, or is there
another failure mode? How does the onset of GMB failure depend on its
size? Can delamination occur on the GMB/matrix interface before
failure?
Here we present an experimental study to elucidate the failure pro­
cess of individual GMBs. Specifically, we use indentation to apply
compression to individual GMBs in two configurations: 1) GMBs
embedded in a PDMS layer with a dilute concentration to isolate a GMB
from the neighboring ones; 2) standalone GMBs adhered to the top
surface of a PDMS layer. Comparison of these two configurations will
shed light on the effect of matrix constraint on GMB failure. Although
slightly different in geometry, the second configuration is similar to the
nano-compression test [25] where individual GMBs were crushed be­
tween two rigid surfaces by a nano-indenter. Serendipitously, the second
configuration also allows us to quantify the adhesion between the GMB
shells and the PDMS matrix. The paper is organized as follows. In Section
2, we describe the experimental methods used for sample preparation
and indentation tests as well as the associated finite element analysis.
Experimental results are presented in Section 3, while interpretation and
discussion of the experimental results are included in Section 4. Con­
clusions are made in Section 5.
glass slide and surrounded by a thin plastic frame which acted as a
spacer. After degassing in a vacuum chamber for 20 min, we put another
glass slide on top of the mixture and the plastic frame, and clamped the
two glass slides together. The assembly was then placed in an oven for
curing (16 h at 71 � C). The plastic frame spacer was chosen such that the
resulting thickness of the PDMS layer is around 200 μm. This thickness is
larger than the diameter of most GMBs (30–115 μm [31]), and thus is
sufficient for keeping the GMBs embedded in the PDMS layer. After
curing, one of the two glass slides was peeled from the PDMS film,
leaving the film attached to the other glass slide. A schematic of the
GMB-embedded PDMS layer and the glass slide underneath is shown in
Fig. 1a.
For indentation tests of non-embedded GMBs, the material sample
was prepared by first creating a pure PDMS thin layer attached to a glass
slide. The PDMS layer served as an adhesive substrate to fix individual
GMBs in place during indentation. To retain adhesion of the top surface
of the PDMS layer, we used spin coating to deposit the precursor mixture
on the glass slide instead of using the approach with two glass slides and
a spacer as described above. Specifically, we prepared mixture of Syl­
gard 184 base resin (100 PBW) and curing agent (10 PBW). After mixing
and vacuum degassing for 20 min, the mixture was deposited on a glass
slide using a spin coater (WS-400-6NPP-Lite Spin processer, Laurell
Technologies, North Wales, PA) with 2000 revolutions per minute for
3 min. This procedure can result in a PDMS layer with a thickness of
80–100 μm. After curing in an oven at 71 � C for 16 h, GMBs were gently
blown on the top of the PDMS layer. Adhesion between the GMBs and
the PDMS surface was sufficiently strong to immobilize the GMBs (see
Section 3.3). A schematic of the non-embedded GMB on top of the PDMS
layer is shown in Fig. 1d.
2.2. Indentation testing
2.2.1. Indentation set-up
Indention testing was performed using a custom developed set-up.
The set-up included an XYZ linear stage (Optosigma, Santa Ana, CA)
with a range of �6.5 mm for the XY directions and �5 mm for the Z
direction, and the resolution for all three directions is 0.01 mm. The XYZ
stage was used as a base to support an aluminum beam. A load cell
(Interface Inc., Scottsdale, AZ) with a capacity of 1 N and a resolution of
10 4 N was attached to the end of the aluminum beam. A 0.5 mmdiameter steel ball was used as a spherical indenter, and was mounted to
the load cell using epoxy glue. During an indentation experiment, this
device was situated next to an inverted microscope (TE2000, Nikon,
Melville, NY) to enable in situ imaging. The microscope is equipped with
a piezoelectric stage to which the sample is attached during indentation.
This stage allowed accurate positioning of the sample so that a GMB
could be placed in the center of the indenting region and imaging
window. A schematic and a photograph of the indentation set-up are
shown in Fig.A1 of Appendix A.
To ensure accuracy of our measurement, the load cell’s calibration
was verified before the indentation experiments. In addition, the applied
vertical displacement measured from the XYZ linear stage (see Appendix
A) included a contribution from compliance of the loading system
consisting of the load cell, the aluminum beam and connection between
them. This compliance may vary slightly when we detach and reattach
the load cell to the aluminum beam. Therefore, before each set of
indentation experiments, we calibrated the loading system compliance
by indenting on a glass plate supported by a rigid surface. As shown in
Fig.A2 of Appendix A, the applied displacement was found to be a linear
function of the measured force with a slope of 0.21–0.24 mm/N. Theo­
retically speaking, the elastic contact between the steel ball and the glass
plate may also contribute to the compliance measured in the calibration
experiments. We ruled out this possibility by using the Hertz contact
theory [32] to show that the glass plate and steel ball can be considered
as rigid relative to the measured compliance (see Appendix A for de­
tails). Therefore, the measured compliance was solely due to the loading
2. Materials and method
2.1. Sample preparation
For the indentation tests of embedded GMBs, the material sample
consisted of a PDMS layer with a dilute concentration of GMBs. The
dilute concentration not only allowed us to indent on individual GMBs
that are isolated from neighboring GMBs, but also made the polymer
film transparent, thereby allowing in situ microscopic imaging. Specif­
ically, we prepared mixture consisting of three components: i) Sylgard
184 (Dow Corning, Midland, MI) base resin with 100 parts by weight
(PBW), ii) Sylgard 184 curing agent with 10 PBW, and iii) A16/500 GMB
(3 M, St Paul, MN [31]) with 0.01 PBW. The mixture was deposited on a
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Composites Part B 173 (2019) 106870
Fig. 1. (a) Schematic of indentation tests on individual embedded GMBs. (b) Image of an embedded GMB. (c) An embedded GMB subjected to indentation. The
shadowed bounded by the dashed line was casted by the indenter. (d) Schematic of indentation tests on individual non-embedded GMBs. (e–f) Image of a nonembedded GMB. The images in (b, c, e) were taken using the reflection mode of the microscope, while the image in (f) was taken using the transmission mode.
system, which was applied to correct the displacement data in subse­
quent indentation experiments.
larger than 80 μm, while a larger displacement increment of 30 μm for
smaller GMBs.
Procedures for the indentation experiments on non-embedded GMBs
(see Fig. 1d) were similar to those described above. However, since the
GMBs were sitting on top of the PDMS layer, it was difficult to obtain
images with adequate resolution using the reflection mode (see Fig. 1e
for example). Therefore, we used the transmission mode of the micro­
scope, in which the light source was above GMB and the transmitted
light was collected by the objective lens below, to image the GMB before
and after the indentation experiment. As shown in Fig. 1f, the trans­
mission mode can significantly improve the image quality, which
allowed us to observe morphology of the failed GMB in a postmortem
manner.
2.2.2. Indentation tests of individual GMBs
Procedures of the indentation experiments on embedded GMBs, as
schematically illustrated in Fig. 1a, are briefly summarized below.
a) First, the indentation device was set up next to the microscope. The
sample of GMB-embedded PDMS layer was placed on the piezo­
electric stage of the microscope. Using a 20X objective lens, we were
able to image individual GMBs in situ using the reflection mode of the
microscope, i.e., both the light source and objective lens were below
the sample so that the indenter above would not block the light.
b) An isolated GMB was selected to minimize the interference from
neighboring GMBs, and its diameter was measured from the micro­
scopic image. An example of the selected GMB is shown in Fig. 1b.
c) The piezoelectric stage and XYZ linear stage were tuned such that the
steel ball indenter was right above the selected GMB. This would
ensure the GMB to be positioned at the center of the shadowed region
casted by the indenter (marked by the dashed line in Fig. 1c).
d) The XYZ stage was adjusted so that the indenter was just in contact
with the PDMS film, which was determined by first zeroing the load
cell and observing the emergence of non-zero indentation force. This
was taken as the initial position of indentation.
e) Displacement of the indenter was applied in increments through the
XYZ linear stage until the GMB failed. After each increment, the force
measured from the load cell and the applied displacement were
recorded, and an image of the GMB was taken using the microscope.
When failure of the GMB was observed, the indenter was retracted
and a final image of the GMB was taken after the indenter was
completely removed.
2.2.3. Sample characterization
All indentation tests on the embedded GMBs were performed using
one sample of GMB-embedded PDMS layer, which will be referred as
Sample 1 hereafter. For the non-embedded GMBs, we used two samples
due to the insufficient number of GMBs adhering on each sample. These
two samples will be named as Sample 2 and 3, respectively. To interpret
the indentation data, additional experiments and analysis were per­
formed to determine several parameters of the indentation sample,
specifically thickness of the PDMS layer, Young’s modulus of the PDMS
elastomer, and wall thickness of the GMBs, as described in detail below.
We measured thickness of the PDMS layer in the three samples using
the indentation device. Briefly, we first zeroed the load cell, then
adjusted the XYZ stage until the indenter just contacted the glass plate,
which was determined as the point where the load cell started to pick up
a force signal. The Z-displacement of the XYZ stage at the initial contact
with the glass plate was recorded. The Z-displacement at which the
indenter just contacted the PDMS surface was determined in a similar
manner. The difference between the two displacements was taken as the
thickness of the PDMS layer. Since the surface of the PDMS layer may
not be exactly flat, the thickness may vary slightly across the entire
layer. Therefore, multiple measurements were conducted and the
average thickness was used. For the GMB-embedded PDMS layer
(Sample 1), the average thickness measured by indentation device is
226 μm. To verify this result, we also used an optical profilometer
The above procedures were repeated for individual GMBs with
different diameters. Interestingly, we observed that larger GMBs tend to
fail at a smaller indentation displacement, which is attributed to the fact
that larger GMBs can make the substrate stiffer and hence experienced
larger force at a given indentation displacement. As a result, a smaller
displacement increment of 10 μm was used for GMBs with diameter
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(Wyko, Model: NT2000) to scan the same sample, and the average
thickness was found to be 224 μm (varied from 215 μm to 232 μm),
which is consistent with the value obtained by indentation. For the two
samples with non-embedded GMBs on top of the PDMS layer, the
average thickness of the PDMS substrate was measured to be 82 μm
(Sample 2) and 100 μm (Sample 3) using the indentation device.
The PDMS elastomer in our indentation samples was assumed to be
incompressible (i.e., Poisson’s ratio ¼ 0.5) while its Young’s modulus
was measured from indentation tests. Specifically, we first identified an
area on the surface of PDMS layer that was free of any GMBs, and fol­
lowed the procedures outlined in Section 2.2.2 to measure the inden­
tation force F when the applied displacement was increased. After that
the calibrated loading device compliance was applied to convert the
applied displacement to the indentation depth δ. The Young’s modulus
of the PDMS layer Ep can be obtained by fitting the F-δ data using the
Hertz contact theory [32]. However, a correction factor must be
included to account for the rigid backing surface (i.e., the glass slide)
underneath the thin PDMS layer [33,34]. We adopt the correction factor
formula given in Long et al. [34], as detailed in Appendix B. This for­
mula was obtained from results of finite element simulations assuming
no-slip condition on the interface between the indenter and the PDMS
substrate. For Sample 1, 2, and 3, the average PDMS modulus was found
to be 8.6 MPa, 7.4 MPa and 6.0 MPa, respectively. These values are
larger than typical Young’s modulus of bulk PDMS elastomer with 10:1
ratio of base resin versus curing agent [35–38]. For example, we per­
formed uniaxial compression tests using bulk PDMS samples, i.e., cyl­
inders that are 28 mm in length and 14 mm in diameter following Brown
et al. [4], and the Young’s modulus was found to be about 2.9 MPa. To
understand this discrepancy, we note that the modulus of PDMS is
known to be sensitive to the curing conditions (e.g., temperature [37,
39]). In particular, Liu et al. [40] prepared PDMS thin layers with
different thicknesses (30μm-1.8 mm) by spin coating and found that the
modulus increases significantly (up to a two-fold increase) for thinner
layers. This was attributed to the shear stress induced by the spin coating
process. The thickness-dependent modulus of PDMS thin layers was also
observed in Gao et al. [41]. In our case, the PDMS layers in Sample 2 and
3 were first spin coated and then cured with the top surface remaining
free, consistent with the process in Liu et al. [40] and Gao et al. [41], and
exhibited a similar trend of increasing modulus as the layer thickness
decreases. In contrast, Sample 1 was prepared differently: it was
confined between two glass slides during the entire curing process,
which might explain the higher modulus despite that it is thicker than
the PDMS layers of Sample 2 and 3.
The wall thickness t of GMBs is also an important parameter for
understanding the experimental data of GMB failure. Direct measure­
ment of the wall thickness from the microscope images is challenging
due to the limited optical contrast between the GMB shell and the PDMS
matrix. Previous studies suggest that the wall thickness of 3 M A16/500
GMBs is around 1 μm based on scanning electronic microscope (SEM)
images of broken GMB shells [8]. In Appendix C, we present an analysis
to estimate the wall thickness based on the true density of GMB powder
and the size distribution of individual GMBs. This analysis suggests that
the GMB wall thickness is approximately 1 μm, consistent with the SEM
data. Motivated by the fact that the wall thickness t may vary slightly for
different GMBs [25] and the results in Section 4.1.1 (see Fig. 8), we will
assume the range of t ¼ 1~2 μm in our analysis.
2.3. Finite element modeling
We built finite element (FE) models for the indentation tests of in­
dividual GMBs using a commercial software ABAQUS (version 6.13,
Simulia Inc, Providence, RI). The purpose is to enable comparison of the
failure mechanisms between the embedded and non-embedded GMBs.
For the embedded GMBs, the indentation force at failure is dominated by
the response of the PDMS matrix. In contrast, for the non-embedded
GMBs, the indentation force is sustained by the GMB. As a result, it is
not appropriate to directly compare the failure force measured in the
two types of tests. With the FE models, our main objective is to convert
the measured failure force to the maximum normal stresses in the GMB
shell at failure, which can facilitate further understanding of the GMB
failure mechanism.
Examples of the FE model for the indentation of embedded and nonembedded GMB are shown in Fig. 2a and b, respectively. In both cases,
axisymmetry was assumed since the centers of the GMB and the spherical
indenter were aligned in the experiments. The indenter (i.e., a steel ball
with a diameter of 0.5 mm) was assumed to be rigid, since its modulus is
several orders of magnitude higher than that of the PDMS substrate. The
GMB was modeled as an elastic shell with a Young’s modulus of 61 GPa
and a Poisson’s ratio of 0.19 following Brown et al. [4], and was meshed
by axisymmetric shell elements SAX1. The PDMS substrate was modeled
as an incompressible neo-Hookean solid and was meshed with continuum
elements CAX4. The Young’s modulus and thickness of the PDMS sub­
strate have been characterized in Section 2.2.3 and were incorporated
into the FE model. For the embedded GMB, we assumed the GMB was
perfectly bonded to the PDMS matrix (enforced by the “tie” constraint).
Experimentally it was difficult to measure the vertical position of the
GMB through the thickness of the PDMS layer. Therefore, we assumed the
GMB was located at the mid-height of the PDMS layer. Moreover, we
assumed the GMB wall thickness to be 1.5 μm in all FE models, motivated
by the range of 1~2 μm of the GMB wall thickness (see Section 2.2.3).
Mesh convergence tests were performed to ensure that the mesh density
was adequate. The effect of the GMB thickness and the vertical position
of the embedded GMB will be further discussed in Section 4.1 and
Appendix E.
Fig. 2. Geometry of the axisymmetric finite element model. (a) Indentation on a PDMS layer with an embedded GMB. (b) Indentation on a non-embedded GMB on
top of a PDMS layer. The inset illustrates the directions of normal stress components σθθ (meridional) and σ ϕϕ (hoop) within the GMB shell.
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Composites Part B 173 (2019) 106870
It should be emphasized that no failure models for the GMBs were
implemented in the FE simulations. Instead, we used the FE model to
extract the maximum tensile and compressive stresses in the GMB shell
when the experimentally measured failure force was achieved in the
simulation. Specifically, we increased the indentation displacement
until the experimental failure force was reached. After that, the merid­
ional stress σθθ and hoop stress σ ϕϕ (see inset of Fig. 2b) were extracted at
both the top and bottom section point of every shell element across the
GMB, from which the maximum tensile and compressive stresses were
determined and recorded.
3. Results
3.1. Failure of embedded GMB
We performed indentation tests on 47 embedded GMBs (diameter
ranging from 40 to 130 μm). Two different types of failure modes were
observed, and examples are shown in Fig. 3. In one mode (see Fig. 3b),
the GMB exhibited signs of partial fracture at the early stage of inden­
tation, marked by the point A2 in Fig. 3a. The fracture pattern became
more severe as the indentation force was increased (see A3 in Fig. 3b).
Therefore, we name this mode as the “progressive failure”. In the other
mode, the GMB remained intact (see B2 in Fig. 3c) until a relatively large
indentation force was achieved and then suddenly fractured (see B3 in
Fig. 3c). This mode is named as the “sudden failure” mode. It is worth
noting that we did not observe any unloading events in Fig. 3a due to
fracture of the GMB. This is attributed to the fact that the indentation
force was mainly contributed by the elastic deformation of PDMS ma­
trix. More interestingly, even though the GMB clearly fractured during
indentation, after retraction of the indenter the GMB recovered its
original shape, as shown in A4 of Fig. 3b and B4 of Fig. 3c. Such recovery
implies that fragments of the GMB were still adhered to the PDMS matrix
after fracture, which further indicates that the adhesion between GMB
and PDMS matrix is strong and delamination did not occur during GMB
fracture. This behavior was observed for all the embedded GMBs tested.
In each indentation experiment, we identified the indentation force
at which GMB fracture was first observed, e.g., A2 in Fig. 3b and B3 in
Fig. 3c. This force will be referred to as the failure force of embedded
GMBs. The failure force is plotted against the GMB diameter in Fig. 4.
Different symbols are used to distinguish the progressive and sudden
failure modes. For the sudden failure mode, the failure force exhibits a
decreasing trend with the diameter, i.e., larger GMBs failed at a smaller
force, while for progressive failure mode the failure force was roughly
insensitive to the diameter. In addition, the failure force for sudden
failure (0.3~1 N) was slightly larger than that for progressive failure
Fig. 4. Failure force of embedded GMBs versus the GMB diameter. Different
symbols were used to distinguish the progressive failure (circles) and the sud­
den failure (squares).
(0.1–0.6 N), but overall the range of failure forces for the two modes
overlaps significantly.
3.2. Failure of non-embedded GMB
The failure behaviors of 40 non-embedded GMBs (diameter ranging
from 40 to 110 μm) were tested in the indentation experiments. Unlike
the embedded GMBs, the indentation force was fully sustained by the
non-embedded GMBs. As a result, an abrupt drop in force would occur
even if the GMB partially failed (see Fig. 5a). Therefore, the progressive
failure mode was not observed for the non-embedded GMBs; the failure
always occurred in a sudden manner. However, we observed two
different types of failure modes, as illustrated in Fig. 5. In one mode, a
large part of the GMB shell remained intact after the failure event (see
A4 in Fig. 5b), while in the other mode, the GMB shell shattered into
pieces after failure (see B4 in Fig. 5c). These two modes are named as the
partial failure and complete failure modes, respectively. In addition to
the postmortem morphology, one can also see in Fig. 5a that the failure
force of the complete mode is much larger than that of the partial mode.
This large difference in failure force can also be used to distinguish the
two modes.
Because of the abrupt force drop upon failure, the failure force for
non-embedded GMBs can be clearly identified as the peak indentation
Fig. 3. (a) Indentation force versus depth for two
embedded GMBs that experienced progressive failure
and sudden failure, respectively. (b) Images showing
the progressive failure of an embedded GMB (diam­
eter ¼ 107 μm). (c) Images showing the sudden failure
of an embedded GMB (diameter ¼ 92 μm). The data
points associated the images are marked in (a) except
A4 and B4 which were taken after the indenter was
fully retracted. All images were taken using the
reflection mode of the microscope.
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Composites Part B 173 (2019) 106870
Fig. 5. (a) Indentation force versus depth for two
non-embedded GMBs that experienced partial
failure and complete failure, respectively. (b)
Images showing the partial failure of a nonembedded GMB (diameter ¼ 88 μm). (c) Images
showing the complete failure of a non-embedded
GMB (diameter ¼ 96 μm). The data points asso­
ciated the images are marked in (a). The images
A1, B1, A4, and B4 were taken using the trans­
mission mode of the microscope while the rest
were taken using the reflection mode.
force (e.g. A3 and B3 in Fig. 5a). The failure forces for non-embedded
GMBs with different diameters are plotted in Fig. 6. Different symbols
are used to distinguish the partial and complete failure modes. It is
worth noting that the failure forces measured for non-embedded GMBs
were in the range of 1 mN to 0.06 N, much smaller than those for
embedded GMBs (0.1~1 N). This is expected since for non-embedded
GMBs the indentation force was completely applied on GMB, while for
embedded GMBs the indentation force was mainly contributed by the
PDMS layer. Furthermore, the failure force data in Fig. 6 are clearly
separated into two groups according to the failure mode. Specifically,
the failure force for complete failure shows a strong decay with GMB
diameter, and are about one order of magnitude larger than those for
partial failure. In contrast, the failure force for partial failure was found
to be insensitive to GMB diameter. These features will be discussed in
Section 4.1.
and the PDMS substrate could be identified (see the plane B–B0 ), from
which we were able to measure the contact radius a. By moving up the
focal plane further, we could focus on the equator of the GMB (see A-A’
in Fig. 7a) from which the GMB radius r was measured. Apart from these
two planes, the image of the GMB was blurry and no sharp boundary
could be identified. Weight of the GMB can be ruled out from the factors
causing the contact. Using the density of borosilicate glass (2.23 g/cm3),
the weight of GMBs with 1 mm thickness and 40–120 μm diameter is
estimated to be on the order of 0.1 nN–1 nN. The contact radius caused
by this range of weight is estimated to be below 0.3 μm using the Hertz
contact theory and thus can be neglected.
The contact area was due to the competition between adhesion en­
ergy on the interface and strain energy in the PDMS substrate, which is
captured by the Johnson-Kendall-Roberts (JKR) theory [42]. Assuming
the GMB is rigid relative to the flat PDMS substrate, the contact radius a
under zero applied force is given by:
�
�
1 ν2p 9πr2
3
a ¼
(1)
Wad ;
2
Ep
3.3. Adhesion between GMB and PDMS
When imaging the non-embedded GMBs on top of the PDMS layer,
we found that a contact region between the GMB and the PDMS sub­
strate could be established even when no external force was applied. As
shown in Fig. 7a, by adjusting the height of the focal plane, we found
that a sharp boundary of the triple contact line between the air, GMB,
where r is the GMB radius, Ep and νp are the Young’s modulus and
Poisson’s ratio of PDMS substrate, and Wad is the work of adhesion be­
tween the GMB shell surface and the PDMS substrate. We set νp ¼ 0.5
and use the value of Ep characterized in Section 2.2.3. The work of
adhesion Wad quantifies the work required to separate a unit area of
contact. If the two contacting materials are elastic, Wad can also be
written in terms of the surface energy of each of the two contact surfaces
[43]. Note that Eq. (1) is developed for infinitely thick substrates (i.e., an
elastic half space), while the PDMS substrate in our samples was thin (e.
g., 82 μm thickness for Sample 2). The effect of finite substrate thickness
can be accounted for using a correction factor given in Shull [44] that
depends on a/h where h is the PDMS substrate thickness. We measured
the contact radius a for 10 GMBs with different radius r, and it is shown
in Appendix D that the effect of correction factor is insignificant (�5%
difference). Therefore, the correction factor was neglected for
simplicity, and Eq. (1) was used to calculate work of adhesion Wad,
which was found to range from 4 to 15 J/m2 as shown in Fig. 7b.
4. Discussions
4.1. Failure mechanism of GMBs
As shown in Section 3, the indentation experiments enabled us to
acquire quantitative data on the failure force of individual GMBs. Im­
plications of these data on the failure mechanisms of the embedded or
Fig. 6. Failure force of non-embedded GMBs versus the GMB diameter.
Different symbols were used to distinguish the partial failure (circles) and the
complete failure (squares).
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Composites Part B 173 (2019) 106870
Fig. 7. (a) Schematic showing the adhesion between GMB and PDMS substrate. The inset shows an image of the GMB equator (A-A0 ) and the contact area (B–B0 ). (b)
The measured work of adhesion Wad for 10 GMBs with different diameters.
non-embedded GMBs are discussed in this section.
To put our results into perspective, we first note the failure of a
standalone GMB can also be characterized by measuring the isostatic
pressure required to crush it. For example, it was reported that the 3 M
A16/500 GMBs used in our experiments has a minimum survival frac­
tion of 80% under an isostatic pressure of 3.4 MPa (or 500psi) [31]. This
pressure can be considered as a conservative estimate of the GMB’s
strength under isostatic pressure. However, the physical nature of failure
was not reported for this type of GMBs [31]. On the other hand, Bratt
et al. [24] performed isostatic compression experiments on a class of
GMBs, and found that the failure pressure Pc can be well captured by
critical buckling of thin spherical shells [18]:
Pc ¼
8Eg t2
ffiffi ;
rffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�
D2 3 1 ν2g
where r is the radius of the spherical shell and t is the shell thickness.
Although in our experiments, the GMB was not compressed between two
rigid flat plates, but between a rigid curved indenter (steel ball) and a
compliant substrate (PDMS layer), we make two approximations that
allow us to apply Eq. (3). First, since the indenter diameter (¼ 500 μm) is
much larger than the GMB diameter (�110 μm), we neglect the curva­
ture of the indenter. Second, since Eq. (3) is associated with a buckling
instability occurring at the apex of the spherical shell, we neglect the
compliance of the PDMS layer underneath the GMB. The critical force Fc
given by Eq. (3) is compared with the experimental data of failure force
for the complete failure mode in Fig. 8, where we have set Eg ¼ 61 GPa,
νg ¼ 0.19 following Brown et al. [4]. The shell thickness is not known
exactly but is between 1 and 2 μm. Therefore, we plotted two curves
with t ¼ 1.35 μm or 1.65 μm to bound the experimental data. No addi­
tional fitting parameters were used. The data appear to scale well with
the 1.65 μm thickness at smaller diameters, but later scale with 1.35 μm
thickness predictions when the GMB diameter increases. It is possible
that the thickness of the balloons is correlated with the diameter.
Indeed, Koopman et al. [25] suggested that GMBs were blown into the
hollow form from glass particles by gas expansion within each particle
under high temperature, and thus larger GMBs should have thinner
walls, which is consistent with the bounding behavior in Fig. 8. Further
work would need to confirm the correlation between the wall thickness
and GMB size.
Unlike the complete failure mode where the failure force is clearly
dependent on the GMB size, the failure force for the partial failure mode
is insensitive to the GMB diameter D, but exhibits an upturn for large D
(see Fig. 6). Interestingly, the upturn occurs in a regime where failure
force of the two modes start to cross. Indeed, the failure force data for
D > 100 μm can also be captured by the buckling force in Eq. (3) with a
shell thickness t ~1 μm. This behavior is qualitatively similar to the
nano-indentation data reported in Koopman et al. [25], where the up­
turn was attributed to the lower probability of defects in larger GMBs
due to the thinner shell.
(2)
where D and t are the GMB’s diameter and thickness, respectively, and Eg
and νg are Young’s modulus and Poisson’s ratio of glass. A key feature of
the experiments by Bratt et al. [24] is that they screened out GMBs with
defects (e.g., non-uniform shell thickness) using interference micro­
scopy, and only defect-free GMBs were tested. For the 3 M A16/500
GMBs used in our experiments, Pc is found to be in the range of
24–180 MPa by setting Eg ¼ 61 GPa, νg ¼ 0.19, t � 1 μm and
D ¼ 40–110 μm, which is about one order of magnitude larger than the
strength (3.4 MPa) reported by 3 M [31]. This large discrepancy implies
that failure of many GMBs may be dominated by defects such as struc­
tural flaws or geometrical imperfections. For example, a void in the GMB
shell can result in stress concentration and subsequent fracture events,
causing the GMB to fail before the buckling load is achieved. In addition,
it is known that geometrical imperfections can significantly reduce the
critical isostatic buckling pressure of a spherical shell [20]. We will bear
in mind the potential effects of defects on GMB failure in the discussions
below.
4.1.1. Non-embedded GMBs
For the non-embedded GMBs, Fig. 6 shows two distinct failure modes
and the associated clear separation of failure force data. Based on the
discussions above, we hypothesize that the complete failure mode
observed for non-embedded GMBs is due to a buckling instability, while
the partial failure mode is associated with pre-existing defects within the
GMB shell.
To support this hypothesis, we note that the buckling of a spherical
shell compressed between two rigid plates has been studied in Audoly
and Pomeau [19]. It was found that buckling occurs at the apex of the
shell, and the critical buckling force Fc is given by Ref. [19]:
Fc ¼ 6:41 �
1
Eg t 3
� ;
ν2g r
4.1.2. Embedded GMBs
For the embedded GMBs, the failure force decays with GMB size in
the sudden failure mode, and remains size-insensitive in the progressive
failure mode. Therefore, it is tempting to also associate the sudden mode
with a buckling instability and the progressive failure mode with a
defect-dominated mechanism. If this is true, we would expect that in the
sudden mode a GMB should be able to sustain much larger stresses than
the progressive mode, which would lead to larger failure forces in the
sudden mode. As demonstrated by Shams et al. [30], constraints from
the matrix can significantly increase the critical stress to buckle an
embedded GMB. In addition, the compressive strength of a syntactic
foam based on the assumption of GMB buckling (see Fig. 4a of Shams
et al. [30]) is an order of magnitude larger than experimental
(3)
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Composites Part B 173 (2019) 106870
1.5 μm), while in practice t may vary from 1 to 2 μm. In addition, our FE
models for the embedded GMBs assumed the GMB was located at the
mid-height of the PDMS layer. We performed additional simulations to
see how the failure stresses are sensitive to these two assumptions. The
results are summarized in Appendix E. For the embedded GMBs, we find
that the median failure stresses (tensile or compressive) can change by
no more than 34% when the GMB thickness is changed to 1 μm or 2 μm,
or the location of the GMB is moved up or down from the mid-height by
40 μm. Given the large variations of the failure stress, the relative
change (<34%) is not substantial and the two assumptions made in our
FE model are acceptable. However, the failure stresses for the nonembedded GMBs is more sensitive to the assumed wall thickness,
which implies that the large scattering seen in Fig. 9a and c may be
associated with, at least partially, the variation in wall thickness of the
tested GMBs.
Several remarks are made below for the data in Fig. 9.
� The failure stresses of non-embedded GMBs (partial failure) do not
show a clear trend of size dependence and are highly scattered (see
Fig. 9a and c). The max tensile stress at failure typically occurs at the
apex of the GMB while the max compressive stress occurs at the
equator.
� The failure stresses of the two failure modes of embedded GMBs are
close to each other. More importantly, the failure stresses are more
concentrated around the median value (roughly 200 MPa for both
tensile and compressive stresses). In particular, the range of
compressive failure stresses for the embedded GMBs (100–500 MPa)
is consistent with the compressive strengths measured for solid glass
spheres [47].
� When comparing the non-embedded and embedded GMBs, we can
see that the max tensile stress of non-embedded GMBs are consid­
erably larger than that of embedded GMB. The compressive failure
stress of the two types of GMBs are on the similar level but the values
for non-embedded GMBs are much more scattered.
Fig. 8. Comparison between the failure forces of non-embedded GMBs in the
complete failure mode with the critical buckling force given by Eq. (3).
measurements [45] where GMB failure is presumably dominated by
defects. However, this was not the case for the embedded GMBs in our
experiments: no separation in failure force was observed, i.e., failure
forces of the two modes are in the same range (see Fig. 4). This obser­
vation leads us to conclude that both modes are associated with defect
growth in the GMB shell. It is possible that the defect growth in the
sudden mode is catastrophic (e.g., unstable crack propagation), whereas
in the progressive mode the defect growth may be stabilized by the
surrounding matrix.
4.1.3. Failure stresses
Experimentally it would be difficult to resolve the physical nature of
the defect-dominated failure in GMB using our set-up. Imaging tech­
niques with much higher spatial and temporal resolution are needed to
capture how defects eventually lead to GMB failure, which is not pur­
sued in this work. Instead, here we show that the effect of defects can be
approximately quantified by a reduction of maximum stresses in the
GMB shell at failure. Take isostatic compression as an example. In this
case, the stress in the shell is compressive and its magnitude can be
calculated using [46].
σ¼
PD
;
4t
4.2. Effect of adhesion
Delamination between GMB surface and PDMS matrix, if it occurs,
may significantly affect the failure process of embedded GMBs.
Although it was difficult to directly observe delamination during
indentation due to the limited optical contrast between the PDMS matrix
and the GMB shell, the GMB images after failure (e.g. see Fig. 3) indicate
that no delamination has occurred during the failure process. Moreover,
in Section 3.3, we measured the work of adhesion Wad between the
PDMS matrix and GMB shell, which ranges from 4 to 15 J/m2. To test
whether this level of adhesion is sufficient to retain the PDMS/GMB
interface during indentation, we implemented a cohesive zone model
[48] between the GMB shell and the PDMS matrix in the FE simulations
for embedded GMBs. In particular, we used the triangular
traction-separation law along the direction normal to the interface, and
set the work of adhesion Wad to be 2.5 J/m2 which is well below the
experimentally measured range. Three different maximum separation
displacements for the cohesive zone model were attempted: 0.1 μm,
0.05 μm and 0.01 μm, all of which are much smaller than the GMB shell
thickness ~1 μm, and Wad remained fixed. A smaller maximum sepa­
ration implies that the cohesive zone can sustain a larger peak stress but
a smaller range of interface separation. Our simulation results suggested
that delamination should not occur before failure for all the embedded
GMBs tested in our indentation experiments. This is consistent with
other experimental observations in recent literature [4,8,12].
(4)
where P is the isostatic pressure applied to the spherical shell.
Combining Eq. (3) and Eq. (4) and assuming t � 1 μm and
D ¼ 40–110 μm, we can calculate the compressive stress at the onset of
buckling, which is found to be 650 MPa to 1.8 GPa. In contrast, using the
isostatic failure pressure P ¼ 3.4 MPa reported by 3 M [31] and the same
values of D and t, the compressive stress at failure is from 34 to 94 MPa,
which is significantly reduced in comparison to that of the buckling
mode, presumably due to defects in the GMB shell.
Following the method outlined in Section 2.3, we calculated the
maximum tensile and compressive stresses within the GMB shell when
the experimentally measured failure force is achieved, which will be
referred to as the failure stresses hereafter. Since we have identified that
the partial failure mode of non-embedded GMBs and both the sudden
and progressive failure modes of embedded GMBs are associated with
pre-existing defects, the failure stresses (tensile and compressive) are
plotted in Fig. 9 for comparison. The median, 25% percentile and 75%
percentile values for the failure stresses of non-embedded GMBs (partial
failure) and embedded GMBs (sudden and progressive failure) are
summarized in Table 1. The complete failure mode of non-embedded
GMBs is not included in Fig. 9, since it is due to buckling and the fail­
ure stresses would be dependent on the GMB size. It should be noted that
our FE models assumed a constant thickness t of the GMB shell (i.e.,
5. Conclusions
Our indentation experiments revealed several different failure modes
for GMBs. Phenomenologically, sudden and progressive failure modes
were observed for the embedded GMBs, while complete and partial
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Composites Part B 173 (2019) 106870
Fig. 9. Failure stresses of the GMBs (a, c): Maximum (a) tensile and (c) compressive stress at failure for non-embedded GMBs in the partial failure mode. (b, d):
Maximum (b) tensile and (d) compressive stresses at failure for embedded GMBs. The dashed lines represent the median value in each plot while the shadow areas
represent a range spanned by the 25% and 75% percentile values.
� For the embedded GMBs, the small range of failure stresses implies
that both the sudden and progressive failure can be correlated with a
strength (either tensile or compressive) of the GMB shell. In addition,
no delamination was observed during the failure of the embedded
GMBs. The two findings point towards a possible failure mechanism:
the sudden and progressive failure modes may be both associated
with fracture processes in the GMB shell initiated by pre-existing
defects. The difference between the sudden and progressive modes
can be attributed to the stability of crack propagation in the GMB
shell, which is sensitive to the location and shape of initial defects.
Table 1
Maximum tensile and compressive (magnitudes) stresses within the GMB shell at
the experimentally measured failure force.
Max tensile stress (GPa)
Max compressive stress (GPa)
75% percentile
median
25% percentile
75% percentile
median
25% percentile
Non-embedded
Embedded
0.943
0.665
0.342
0.561
0.286
0.128
0.315
0.227
0.199
0.326
0.253
0.214
In addition to implications on the failure mechanisms, our data are
also useful for modeling the damage mechanics of elastomeric syntactic
foams by providing guidance at the individual GMB level. For example,
our results showed that there exists a statistical distribution in the failure
behaviors of individual GMBs, which needs to be understood and
modeled appropriately for accurate micromechanical models of
macroscopic damage. In addition, the maximum tensile and compressive
stresses at failure in embedded GMBs fall in a small range around
200 MPa. Future work can explore the potential of using such stresses as
a failure criterion for embedded GMBs. Finally, the work of adhesion
between GMB shell and PDMS was measured to be 4–15 J/m2. Our FE
simulations suggested that with such a strong adhesion, delamination is
not likely to occur during the failure of GMBs in the PDMS syntactic
foams studies here.
failure modes were observed for the non-embedded GMBs. Quantitative
analysis of the measured failure force suggested that the complete fail­
ure mode of non-embedded GMBs is due to a buckling instability. The
physical mechanisms underlying the other three failure modes are less
clear, and further experimental studies are needed to establish a com­
plete understanding. However, we can still infer the following conclu­
sions using the failure stress data in Fig. 9.
� The highly scattered failure stress for the partial failure mode of nonembedded GMBs suggests that this mode is not controlled by a crit­
ical stress in the GMBs. Physically a possible explanation is that the
partial failure mode may still be associated with a buckling insta­
bility, but the critical buckling load is knocked down by geometrical
imperfections or other defects in the GMBs shell [20]. In this case, the
buckling load would be highly sensitive to the detailed geometrical
features of imperfections, which corresponds to the larger scattering
of failure stresses.
Acknowledgement
Y.L. and R.L. thank the Sandia National Laboratories for supporting
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Composites Part B 173 (2019) 106870
this work. Sandia National Laboratories is a multi-mission laboratory
managed and operated by National Technology and Engineering Solu­
tions of Sandia, LLC., a wholly owned subsidiary of Honeywell
International, Inc., for the U.S. Department of Energy’s National Nuclear
Security Administration Under Contract DE- NA0003525.
Appendix A. Indentation apparatus
A photo of the indentation set-up is shown Fig.A1a, and a schematic drawing of the set-up is shown in Fig.A1b for better clarity. The aluminum
beam is based on the XYZ stage which is fixed to the optical table during indentation testing. The right end of the aluminum beam is attached with a
load cell and a steel ball indenter (0.5 mm in diameter). This set-up allows in situ imaging of the GMB by an optical microscope during indention.
Alignment of the indenter and the GMB can be achieved by using the X and Y components of the XYZ stage and the piezoelectric stage on which the
sample is place. The indentation displacement can be applied using the Z component of the XYZ stage.
Fig. A1. (a) Photo of the indentation set-up to probe failure of individual GMBs. (b) Schematic drawing of the indentation set-up.
As described in Section 2.2.1, the compliance of the loading device was calibrated by indenting on a glass plate supported by a rigid surface. An
example of the applied displacement versus the measured force F is shown in Fig.A2, where the experimental data exhibit a linear relationship. Next
we use the Hertz theory [32] to estimate how much the elastic contact between the steel ball and the glass plate can contribute to the total compliance.
According to the Hertz theory, the displacement δ* due to the elastic contact is related to the indentation force F through:
!
#2=3
"
3 1 ν2s 1 ν2g
F
δ* ¼
þ
;
(A1)
4
R1=2
Es
Eg
where R (¼250 μm) is the indenter radius, Es and Eg are the Young’s modulus of steel and glass respectively, and νs, νg are the corresponding Poisson’s
ratio. Using Es ¼ 200 GPa, νs ¼ 0.3, Eg ¼ 61 GPa and νg ¼ 0.19, we plot the displacement δ* given by Eq. (A1) in Fig.A2. Clearly the displacement δ* due
to the elastic contact between the steel ball and glass plate is negligible in comparison to total displacement. Therefore, we conclude that the steel ball
and glass plate can be considered as rigid in the calibration experiments.
Fig. A2. Displacement versus force data (symbols) measured in a compliance calibration experiment. The linear fit has a slope of 0.22 mm/N which was taken as the
compliance of the loading device. The dashed line is given by Eq. (A1) and shows that the compliance of the elastic contact between the steel ball and the glass plate is
negligible in comparison to that of the loading device.
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Composites Part B 173 (2019) 106870
Appendix B. Measurement of PDMS modulus
We performed indentation tests to measure the Young’s modulus of the PDMS elastomer in our samples, which is denoted as Ep. The data of
indentation force F and depth δ were fitted by the following equation developed by Long et al. [34]:
F¼
16Ep 1=2 3=2
1
�
�;
R δ
9
ψ ω; Rh
(B1)
where R is the indenter radius, h is the PDMS layer thickness, ω is a dimensionless parameter defined as ω ¼(Rδ/h2)3/2 and ψ (0< ψ � 1) is a correction
factor to account for the finite thickness of the PDMS layers. Note that we have assumed that the PDMS layer is incompressible and is perfectly bonded
to the glass plate underneath it. The steel ball is much stiffer than the PDMS layer and thus is assumed to be rigid. The correction factor ψ is sensitive to
the friction condition on the interface between the steel ball and the PDMS surface. Based on finite element simulations, Long et al. [34] developed
closed-form expressions for ψ under two limiting interface friction conditions: frictionless or no-slip. It is known in the literature that the dry friction
between steel and PDMS elastomer is strong (e.g. the coefficient of friction is between 0.8 and 1.2 [49]). Therefore, we assume the no-slip condition on
the indenter/PDMS interface, and use the following correction factor (valid for δ/h � 0.6):
�
�
R
1 þ 2:3ω
� �
¼
ψ ω;
;
h
1 þ 1:15ω1=3 þ 8:94 0:89h2 R2 ω þ 9:288ω2
(B2)
� �3=2
Rδ
ω¼ 2
:
h
Examples of the fitting for the three samples are shown in Fig.B1. As a comparison, we also used a different correction factor ψ given by an
analytical solution in Dimitriadis et al. [33] to fit the indentation data:
�
�
R
1
¼
ψ ω;
:
(B3)
h
1 þ 1:133ω1=3 þ 1:283ω2=3 þ 0:769ω þ 0:0975ω4=3
Note that this correction factor was based on the assumption of frictionless interface between the steel ball and PDMS substrate. As shown in Fig.
B1, Eq. (B3) together with Eq. (B1) can also fit the experimental data well, but it leads to values of Ep that are slightly larger than those using Eq. (B2).
This is because relative to the no-slip interface, the frictionless interface makes the PDMS substrate less constrained and thus more compliant, which
must be compensated by larger PDMS modulus when the frictionless solution is used to fit the same experimental data. As mentioned earlier, strong
friction between steel ball and PDMS elastomer is expected. Therefore, we adopted the no-slip solution in Eq. (B2) to fit the experimental data.
Fig. B1. Indentation force F versus depth δ to determine the Young’s modulus of PDMS layers. (a) Sample 1 (226 μm) thick), (b) Sample 2 (82 μm thick), (c) Sample 3
(100 μm thick). The symbols represent experimental data. The solid lines represent fittings based on Eq. (B2), which gave 8.2 MPa, 7.2 MPa, and 6.0 MPa for Sample
1, 2 and 3, respectively. The dashed lines represent fittings based on Eq. (B3), which gave 8.8 MPa, 11.4 MPa, and 8.0 MPa for Sample 1, 2 and 3, respectively.
Appendix C. Estimating the wall thickness of GMBs
Here we estimate the GMB wall thickness t based on its density and size. The true density of the 3 M A16/500 GMB powder [31] was reported to
vary from 0.14 to 0.18 g/cm3 with a typical value of 0.16 g/cm3. For a single GMB with radius r and wall thickness t, its true density ρi can be
calculated as follows:
V
Va
(C1)
ρi ¼ ρg i ;
where ρg is the density of borosilicate glass shell of the GMB, Vi is the volume of the GMB shell and Va is the apparent volume of the GMB. Since the
GMB is spherical in shape, Va ¼ 4πr3/3 and Vi ¼ 4πr2t (assuming r ≫ t). The size of GMBs in powder is not uniform, which can be described by a
probability distribution function f(r) in terms of the radius r. Note that the wall thickness t may be not uniform in a GMB or between different GMBs,
but the characteristics of variation are not known. Therefore, we assume a uniform t across all GMBs to obtain a first-order estimate of t. Incorporating
the size distribution f(r) in Eq. (C1), we have
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Composites Part B 173 (2019) 106870
Z ∞
f ðrÞr3 dr
ρ 0
Z ∞
t¼
ρg
3
(C2)
;
f ðrÞr2 dr
0
where ρ is the true density of GMB powder.
Next we determine the distribution function. A few data points for the cumulative distribution function c(r) defined below for the 3 M A16/500
GMBs were known [31] (in terms of diameters) and plotted in Fig.C1 as symbols.
Z r
0
0
cðrÞ ¼
f ðr Þdr :
(C3)
0
We used two types of distribution functions to fit the data in Fig.C1: Log-normal and Weibull. Using the Log-normal distribution, we obtained that
!
1
ðlnðr=rm Þ Þ2
(C4)
f ðrÞ ¼ pffiffiffiffiffi exp
2σ 2
σr 2π
where rm ¼ 30μm and σ ¼0.4108. Submitting Eq. (C4) into Eq. (C2) and using ρ ¼ 0.16 g/cm3 and ρg ¼ 2.23 g/cm3, we find the wall thickness t to be
1.1 μm. On the other hand, if the Weibull distribution is used, we have
� �k !
� �k 1
k r
r
f ðrÞ ¼
(C5)
exp
rw rw
rw
where rw ¼34.63μm and k ¼2.555. With this distribution, t is found to be 0.97 μm. The two distribution functions give consistent result that the wall
thickness of GMB is around 1 μm.
Fig. C1. Cumulative distribution function of the GMB radius. The symbols represent data reported by 3 M [31], while the solid line and dash-dotted line are fittings
based on the Log-normal and Weibull distribution, respectively.
Appendix D. Calculating adhesion between GMB and PDMS substrate
The JKR theory that underlies Eq. (1) in the main text assumes the substrate is infinitely thick. A correction factor to account for the finite thickness
of the substrate is provided in Shull [44]. Specifically, by combining Eqs.(8), (12) and (30) of Shull [44], we obtain
�2
1 νp 9π r2 Wad
a3 ¼
gða=hÞ;
(D1)
Ep
2
and
�
�2
0:75 þ ða=hÞ þ ða=hÞ3
gða=hÞ ¼ �
�2 �
�;
1 þ 0:25ða=hÞ3
0:56 þ 1:5ða=hÞ þ 3ða=hÞ3
(D2)
where the variables in Eq. (D1) are the same as those defined in Eq. (1) except that h is the thickness of the PDMS substrate, and g(a/h) is the correction
factor. It should be emphasized that the formula above is applicable only if the substrate is incompressible and no-slip boundary condition is assumed
on the GMB/substrate interface. Both conditions are satisfied in our case. All adhesion data were measured using Sample 2 where the PDMS substrate
thickness h ¼ 82 μm. The measured contact radius ranges from 15.7 to 37.2 μm. Using Eq. (D2), we find the correction factor g(a/h) ranges from 1.03 to
1.05. Therefore, the correction factor can only cause a relative change that is within 5%, and thus was neglected in our calculation of Wad.
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Composites Part B 173 (2019) 106870
Appendix E. Sensitivity of the failure stresses to model assumptions
To test the sensitivity of the failure stresses to the assumed GMB shell thickness (¼1.5 μm) in the FE models, we repeated the calculations of failure
stresses by setting the shell thickness to 1 μm or 2 μm, median values of the failure stresses for the non-embedded (partial failure) and embedded GMBs
(sudden and progressive failure) are summarized in Table E1.
Table E1
Median of the maximum tensile and compressive (magnitude) stresses at GMB failure with different assumptions of GMB shell stress (2, 1.5 and 1 μm).
Shell thickness (μm)
2
1.5
1
Non-embedded
Embedded
Max tensile (GPa, median)
Max compressive (GPa, median)
Max tensile (GPa, median)
Max compressive (GPa, median)
0.463
0.665
0.773
0.166
0.286
0.714
0.165
0.227
0.291
0.198
0.253
0.339
For the embedded GMBs, we also assumed that the GMB was located at the mid-height of the PDMS layer. To test the effect of this assumption, we
changed the vertical location of the GMB center by moving it up or down by 40 μm, the resulting median values of failure stresses for embedded GMBs
(sudden and progressive failure) are summarized in Table E2. It can be seen that the failure stress results are not sensitive to the vertical location of the
embedded GMB.
Table E2
Median of the maximum tensile and compressive (magnitude) stresses at GMB failure with different
assumed locations of the GMB center: “0” means the GMB is at the mid-height of the PDMS layer, while
“40 μm” or “ 40 μm” means that the GMB center is moved up or down by 40 μm, respectively.
GMB location
40 μm
0
40 μm
Embedded
Max tensile (GPa, median)
Max compressive (GPa, median)
0.177
0.227
0.174
0.237
0.253
0.209
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