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Measurement of the Shear Properties of Extruded Polystyrene Foam by In-Plane Shear and Asymmetric Four-Point Bending Tests

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The shear modulus and shear strength of extruded polystyrene foam were obtained by the in-plane shear and asymmetric four-point bending tests. In addition, the test data were numerically analysed, and the effectiveness of these tests was examined. The numerical and experimental results suggest that the shear modulus and shear strength obtained from the in-plane shear test are significantly smaller than those obtained from the asymmetric four-point bending test because the influence of the stress concentration was less significant. Although the in-plane shear test is standardised in ASTM C273/C273M-11, it is considerable to adopt the asymmetric four-point bending test as another candidate for obtaining the shear properties of extruded polystyrene foam.
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polymers
Technical Note
Measurement of the Shear Properties of Extruded
Polystyrene Foam by In-Plane Shear and Asymmetric
Four-Point Bending Tests
Hiroshi Yoshihara 1,* and Makoto Maruta 2
1Faculty of Science and Engineering, Shimane University, Nishikawazu-cho 1060, Matsue,
Shimane 690-8504, Japan
2
Faculty of Science and Technology, Shizuoka Institute of Science and Technology, Toyosawa 2200-2, Fukuroi,
Shizuoka 437-8555, Japan; maruta.makoto@sist.ac.jp
*Correspondence: yosihara@riko.shimane-u.ac.jp; Tel.: +81-852-32-6508
Received: 4 December 2019; Accepted: 22 December 2019; Published: 30 December 2019


Abstract:
The shear modulus and shear strength of extruded polystyrene foam were obtained by the
in-plane shear and asymmetric four-point bending tests. In addition, the test data were numerically
analysed, and the eectiveness of these tests was examined. The numerical and experimental
results suggest that the shear modulus and shear strength obtained from the in-plane shear test
are significantly smaller than those obtained from the asymmetric four-point bending test because
the influence of the stress concentration was less significant. Although the in-plane shear test is
standardised in ASTM C273/C273M-11, it is considerable to adopt the asymmetric four-point bending
test as another candidate for obtaining the shear properties of extruded polystyrene foam.
Keywords:
asymmetric four-point bending (AFPB) test; in-plane shear (IPS) test; extruded polystyrene
foam (XPS); shear modulus; shear strength; stress concentration
1. Introduction
At present, extruded polystyrene foam (XPS) is used in the construction of sandwich panels [
1
4
],
flooring materials, such as tatami mats, used in traditional rooms in Japan [
5
8
], and heat-storage
tanks [
9
] and geofoams [
10
12
], because the lightweight nature of XPS is eective to attenuate the
seismic forces. To ensure that such construction designs are reliable and cost-eective, it is important
to accurately characterize the mechanical properties of XPS, including the shear properties, such as the
shear modulus and shear strength.
In a previous study, flexural vibration (FV) tests were conducted to measure the Young’s modulus
and shear modulus values of XPS, and its eectiveness was discussed based on numerical and
experimental results [
13
]. In another previous study, torsional vibration (TV) and square-plate twist
(SPT) tests were conducted to measure the shear modulus values of XPS [
14
]. Nevertheless, the tests
conducted in these previous studies enable the determination of the shear modulus value alone, and it is
impossible to determine the shear strength value. Several methods are considered to measure both the
shear modulus and the shear strength values of foam materials. Among them, the in-plane shear (IPS),
Arcan, and Iosipescu shear tests are often conducted for several foam materials [
15
20
]. Nevertheless,
these methods are not always convenient, because a pair of metallic plates must be bonded or connected
to the facings of the foam sample to apply a shearing force. Therefore, the preparation of the specimen is
often time-consuming. In particular, there is concern that the stress concentration between the metallic
plate and foam sample seriously influences the shear modulus and shear strength values in the IPS
test, although the IPS test is standardised in ASTM C273/C273M-11 [
19
]. Considering these drawbacks,
Polymers 2020,12, 47; doi:10.3390/polym12010047 www.mdpi.com/journal/polymers
Polymers 2020,12, 47 2 of 15
alternate methods should be used to measure the shear modulus and shear strength values of XPS.
The asymmetric four-point bending (AFPB) test, which is regarded as an application of the Iosipescu
shear test, could help to overcome these drawbacks [
21
26
]. The AFPB test is more advantageous
than the aforementioned shear tests in that it does not require an apparatus specially designed for the
test but that for the universal four-point bending test; therefore, the test can be conducted easily and
conveniently. Despite this advantage, however, there are few examples examining the shear properties
of foam materials with the AFPB test [
23
]. In particular, it is extremely dicult to find any examples
conducting the AFPB test using XPS, including the experiment and numerical analyses.
In this study, IPS and AFPB tests were performed on XPS specimens to measure the shear modulus
and shear strength values. The validity of the test methods was examined by comparing their results
with finite element (FE) calculations. The aim of the study was to use the static test to determine the
shear properties of XPS, including the shear modulus and shear strength values.
2. Materials and Methods
2.1. Specimens
Table 1shows the XPS panels (STYROFOAM
TM
series fabricated in The Dow Chemical Company,
Tokyo, Japan) used to obtain the test specimens for this study and the nominal densities of the panels [
5
].
These panels were also used in previous studies [
13
,
14
]. The directions along the length, width,
and thickness of the XPS panel were defined as the L, T, and Z directions, respectively. The L direction
coincided with the extruded direction of the panel. As shown in Figure 1, the initial dimensions of
the panels were 910
×
910
×
25 mm in the L, T, and Z directions, respectively. Ten specimens with
initial dimensions of 300
×
25
×
25 mm and 170
×
25
×
25 mm were cut from the XPS for the IPS
and AFPB tests, respectively. A specimen with its longest dimension coinciding with the L direction
was defined as an L-type specimen, whereas one with its longest dimension coinciding with the T
direction was defined as a T-type specimen. Therefore, the shear properties in the LT and LZ planes
were obtained from the L-type specimen, whereas those in the TL and TZ planes were obtained from
the T-type specimen.
Table 1. Extruded polystyrene foam (XPS) panels used in this study and their nominal densities.
Material Code Density (kg/m3)
STYROFOAM IB IB 26
STYROFOAM B2 B2 29
STYROACE-II ACE 32
STYROFOAM RB-GK-II RB-GK 36
Polymers 2020, 12, x FOR PEER REVIEW 2 of 15
modulus and shear strength values of XPS. The asymmetric four-point bending (AFPB) test, which is
regarded as an application of the Iosipescu shear test, could help to overcome these drawbacks [21–
26]. The AFPB test is more advantageous than the aforementioned shear tests in that it does not
require an apparatus specially designed for the test but that for the universal four-point bending test;
therefore, the test can be conducted easily and conveniently. Despite this advantage, however, there
are few examples examining the shear properties of foam materials with the AFPB test [23]. In
particular, it is extremely difficult to find any examples conducting the AFPB test using XPS,
including the experiment and numerical analyses.
In this study, IPS and AFPB tests were performed on XPS specimens to measure the shear
modulus and shear strength values. The validity of the test methods was examined by comparing
their results with finite element (FE) calculations. The aim of the study was to use the static test to
determine the shear properties of XPS, including the shear modulus and shear strength values.
2. Materials and Methods
2.1. Specimens
Table 1 shows the XPS panels (STYROFOAMTM series fabricated in The Dow Chemical
Company, Tokyo, Japan) used to obtain the test specimens for this study and the nominal densities
of the panels [5]. These panels were also used in previous studies [13,14]. The directions along the
length, width, and thickness of the XPS panel were defined as the L, T, and Z directions, respectively.
The L direction coincided with the extruded direction of the panel. As shown in Figure 1, the initial
dimensions of the panels were 910 × 910 × 25 mm in the L, T, and Z directions, respectively. Ten
specimens with initial dimensions of 300 × 25 × 25 mm and 170 × 25 × 25 mm were cut from the XPS
for the IPS and AFPB tests, respectively. A specimen with its longest dimension coinciding with the
L direction was defined as an L-type specimen, whereas one with its longest dimension coinciding
with the T direction was defined as a T-type specimen. Therefore, the shear properties in the LT and
LZ planes were obtained from the L-type specimen, whereas those in the TL and TZ planes were
obtained from the T-type specimen.
Table 1. Extruded polystyrene foam (XPS) panels used in this study and their nominal densities.
Material Code Density (kg/m3)
STYROFOAM IB IB 26
STYROFOAM B2 B2 29
STYROACE-II ACE 32
STYROFOAM RB-GK-II RB-GK 36
Figure 1. Diagram of the XPS panel. Unit = mm.
910
910
25
L
T
Z
T-type specimen
L-type specimen
Figure 1. Diagram of the XPS panel. Unit =mm.
Polymers 2020,12, 47 3 of 15
The shear modulus values in the LT, LZ, TL, and TZ planes were defined as G
LT
,G
LZ
,G
TL
,
and G
TZ
, respectively. Additionally, the shear strength values corresponding to these planes were
defined as S
LT
,S
LZ
,S
TL
, and S
TZ
. To conduct the IPS and AFPB tests, the density of the specimen
was measured. After measuring the density, the IPS and AFPB specimens were fabricated by the
procedure described below.
2.2. In-Plane Shear (IPS) Tests
As described in previous studies [
13
,
14
], IPS tests were conducted according to the method based
on ASTM C273/C273M-11 [
19
]. As shown in Figure 2, the specimen was rigidly supported with
aluminium plates bonded to the facings using epoxy resin (LOCTITE Easy Mix, cure time =24 h,
Henkel Japan, Yokohama, Japan). The load plate was tapered to a knife-edge and fitted into V-notch
loading blocks. A load P(N) was applied at the crosshead speed of 1 mm/min until it reached the
maximum. The relative displacement between the loading plates
δ
(mm) was measured using a linear
variable dierential transducer, LVDT (CDP-5M, capacity =10 mm, Tokyo Sokki Kenkyujo, Tokyo,
Japan). In the IPS loading, the shear stress is supposed to be homogeneously distributed. Therefore,
the shear stress τIPS was obtained from the following equation:
τIPS =P
TL (1)
and shear strain γIPS were obtained from the following equation:
γIPS =δ
B(2)
where B,L, and Tare the width, length, and thickness of the specimen, respectively. The shear modulus
value was measured from the initial slope of the
τIPS
γIPS
diagram [
13
,
14
], whereas the shear strength
value was obtained by substituting the maximum load Pmax into Equation (1).
Polymers 2020, 12, x FOR PEER REVIEW 3 of 15
The shear modulus values in the LT, LZ, TL, and TZ planes were defined as GLT, GLZ, GTL, and
GTZ, respectively. Additionally, the shear strength values corresponding to these planes were defined
as SLT, SLZ, STL, and STZ. To conduct the IPS and AFPB tests, the density of the specimen was measured.
After measuring the density, the IPS and AFPB specimens were fabricated by the procedure described
below.
2.2. In-Plane Shear (IPS) Tests
As described in previous studies [13,14], IPS tests were conducted according to the method
based on ASTM C273/C273M-11 [19]. As shown in Figure 2, the specimen was rigidly supported with
aluminium plates bonded to the facings using epoxy resin (LOCTITE Easy Mix, cure time = 24 h,
Henkel Japan, Yokohama, Japan). The load plate was tapered to a knife-edge and fitted into V-notch
loading blocks. A load P (N) was applied at the crosshead speed of 1 mm/min until it reached the
maximum. The relative displacement between the loading plates δ (mm) was measured using a linear
variable differential transducer, LVDT (CDP-5M, capacity = 10 mm, Tokyo Sokki Kenkyujo, Tokyo,
Japan). In the IPS loading, the shear stress is supposed to be homogeneously distributed. Therefore,
the shear stress τIPS was obtained from the following equation:
𝜏=𝑃
𝑇𝐿 (1)
and shear strain γIPS were obtained from the following equation:
𝛾=𝛿
𝐵 (2)
where B, L, and T are the width, length, and thickness of the specimen, respectively. The shear
modulus value was measured from the initial slope of the τIPSγIPS diagram [13,14], whereas the shear
strength value was obtained by substituting the maximum load Pmax into Equation (1).
Figure 2. Diagram of the in-plane shear (IPS) test.
2.3. Asymmetric Four-Point Bending (AFPB) Tests
Figure 3a shows the diagram of the AFPB test. As shown in Figure 3b, the shear force is
maximum between the inner spans, whereas the bending moment is zero at the mid-span. Therefore,
the AFPB test is advantageous to characterize the shear properties in that the shear force is dominant
when the failure is induced at the mid-span. This method is regarded as an application of the
Iosipescu shear test, which was originally proposed for the measurement of the shear properties of
metals [27].
P
Aluminium plate
XPS specimen
V-notched attachmen
t
V-notched attachmen
t
LVDT
Figure 2. Diagram of the in-plane shear (IPS) test.
2.3. Asymmetric Four-Point Bending (AFPB) Tests
Figure 3a shows the diagram of the AFPB test. As shown in Figure 3b, the shear force is maximum
between the inner spans, whereas the bending moment is zero at the mid-span. Therefore, the AFPB
test is advantageous to characterize the shear properties in that the shear force is dominant when the
Polymers 2020,12, 47 4 of 15
failure is induced at the mid-span. This method is regarded as an application of the Iosipescu shear
test, which was originally proposed for the measurement of the shear properties of metals [27].
Figure 3. Diagram of the asymmetric four-point bending (AFPB) test.
A rectangular bar with the aforementioned dimensions was sandwiched between a pair of moulds
with the V-notched configuration: then, it was cut into the shape shown in Figure 3a using a heat wire.
To measure the “apparent” shear strain, a biaxial strain gauge (nominal gauge factor =2.1, gauge length
=1 mm; FCA-1-11, Tokyo Sokki Kenkyujo Co., Ltd., Tokyo, Japan) was bonded at the centre of a side
surface. The gauge axes were in directions inclined at
±
45
with respect to the x-direction. The normal
strains in directions inclined at +45
and
45
were defined as
εI
and
εII
, respectively. In bonding the
strain gauge on XPS, there were the following two obstacles:
1. Since the XPS hardly accepted adhesives, the adhesive strength was not often high.
2. Several adhesives often melted the XPS.
Previously to the AFPB tests, several adhesives, including cyanoacrylate, epoxy, and vinyl acetate,
were examined to address these obstacles, and, finally, a cyanoacrylate adhesive (CC-35, cure time
=1 h, Kyowa Dengyo, Co., Ltd., Tokyo, Japan) was selected. To promote the adhesive strength,
a surface-preparing agent (S-9B, Kyowa Dengyo, Co., Ltd., Tokyo, Japan) was coated on the specimen
before using the adhesive. The specimen was eccentrically supported at two trisected points, and the
loads were applied at the remaining two points at a crosshead speed of 1.0 mm/min. The distance
between the left and right loading points was 150 mm. The shear stress
τAFPB
is supposed to be
homogeneously distributed between the notch roots, and therefore it was obtained from the following
equation [22,25,26]:
τAFPB =V
bt =P
2bt (3)
Polymers 2020,12, 47 5 of 15
where bis the distance between the notch roots, and tis the thickness of the specimen. The normal
strains in the length and depth directions were defined as
εx
and
εy
, respectively, and the shear strain
in the length/depth plane was defined as
γxy
. Then, the normal strains in directions inclined at +45
and 45,εIand εII, respectively, were derived as follows:
(εI=εx
2+εy
2+γxy
2
εII =εx
2+εy
2γxy
2
(4)
Therefore, the “apparent” shear strain measured using strain gauge
γg
was obtained from the
following equations:
γg=γxy =εIεII (5)
The method for obtaining the shear modulus value is described below. In contrast, the shear
strength value was obtained by substituting the maximum load into Equation (3).
2.4. Tension and Compression Tests for the Strain Gauge Calibration
In the IPS test, the shear strain can be easily measured using an LVDT as described above.
In contrast, because it is dicult to set the LVDT in the AFPB test to measure the shear strain in the
gauge region, which corresponds to the region between the notch roots, an alternative method for
measuring shear strain was required. Optical methods, such as digital image correlation (DIC) and
virtual fields method (VFM), are promising in measuring the strain induced in a material with low
stiness, such as foam and paper materials [
15
18
,
20
,
28
]. In the preliminary tests, it was examined
whether the shear strain could be accurately measured using a high-speed digital image sensor
(Keyence CV-5000SO, Keyence Corporation, Osaka, Japan), which was eective for measuring the
elongation induced during a tensile test of the paper material [
29
]. On the gauge region of an AFPB
specimen, a pair of straight lines, inclined at 45
with respect to the length direction of the AFPB
specimen surface, were drawn. Then, the elongation between the lines was photographed using a CCD
camera at intervals of 0.5 s and analysed using the high-speed digital image sensor. The shear strain
was calculated from dividing the elongation by the initial distance between the lines. In this method,
however, the rotation of the lines, which was induced from the large deflection of the AFPB specimen,
was significant in the field of view. Additionally, it was dicult to conduct the aforementioned DIC
and VFM methods because of the lack of the equipment. Therefore, the optical methods should have
been abandoned in this study.
Instead of the optical method, the shear strain was measured using a strain gauge bonded to the
gauge region, despite being a classical and provisional method, and the relevancy for using the strain
gauge was examined. However, when bonding a strain gauge on a material with a small stiness
such as XPS, the sensitivity of the gauge significantly decreases. As a result, the strain obtained from
the strain gauge output is often smaller than the actual strain. Several examples have examined
the possibility of using the strain gauge for cellular plastics [
30
,
31
]. In this study, the tension and
compression tests were conducted for the strain calibration, and it was examined whether the strain
obtained from the strain gauge could be transformed into the actual strain. The dimensions of the
tension test specimen were 170
×
25
×
25 mm, whereas those of the compression test specimen were
100
×
25
×
25 mm. The length direction coincided with the direction 45
inclined with respect to
the L direction because the shear strain in the AFPB test was detected as the normal strains in the
±
45
directions, as described above. A strain gauge similar to that used in the AFPB test was bonded
to the centre of both LT surfaces. Additionally, a displacement gauge (capacity =50 mm; PI-5-50,
Tokyo Sokki Kenkyujo Co., Ltd., Tokyo, Japan) was attached to the same surfaces. Ten specimens were
used for each calibration test. In the tension test, a tensile load was applied to the specimen with a grip
the length of which was 35 mm. In the compression test, a compressive load was applied to the end
surface of the specimen. The crosshead speed was 1 mm/min, and the strain output from the strain
and displacement gauges, defined as
εg
and
εd
, respectively, were obtained by averaging the strains
Polymers 2020,12, 47 6 of 15
measured at both LT surfaces. Using the calibrated
εd
εg
relationship, the calibrated shear strain
γc
was obtained from the shear strain measured from the strain gauge
γg
. The detailed method of the
transformation is described below.
3. Finite Element Analysis
Two-dimensional finite element analyses (2D-FEAs) were independently conducted on the actual
IPS and AFPB tests. The ANSYS 18.2 program was used for the FE analyses. Table 2presents the
elastic properties used in this study. The values of Young’s modulus and shear modulus are similar to
those measured in previous studies [
13
,
14
]. In these studies, however, the values of the Poisson’s ratios
were not measured and but derived as 0.35. Since the values of the Poisson’s ratio were measured
from the tension tests in another study [
32
], they are also used in this study. The model consisted of
four-node plane elements. The horizontal and vertical axes of the model were defined as the xand y
directions, respectively.
Table 2. Elastic properties of the XPS and aluminium models used for the FEAs [13,14,32].
Code
Young’s Modulus (MPa) Shear Modulus (MPa) Poisson’s Ratio
ELETEZGLT GZL GTZ vLT vZL vTZ
IB 17.7 15.7 16.7 6.93 9.42 8.95 0.46 0.53 0.40
B2 24.4 15.9 20.2 7.26 10.3 8.55 0.58 0.44 0.46
ACE 29.0 19.8 24.4 9.18 12.8 10.5 0.53 0.40 0.37
RB-GK 37.5 23.5 30.5 12.1 15.5 13.5 0.43 0.36 0.51
Young’s Modulus (GPa) Shear Modulus (GPa) Poisson’s Ratio
Aluminium
69.0 27.0 0.28
As shown in Figure 4, the IPS test model consisted of the XPS and aluminium portions.
The thickness of the model, T, was 25 mm. The horizontal and vertical lengths of the XPS portion, Band
L, respectively, were 25 and 300 mm, respectively. The mesh of the XPS portion was homogeneously
divided with dimensions of 2.5 and 5 mm in the xand ydirections, respectively. The number of elements
was 1950. Prior to the FEAs using this finite element mesh, other FEAs had been examined using a
coarser mesh. Nevertheless, the obtained results were similar to each other, and therefore the finite
element mesh shown in Figure 3was confirmed to be fine enough. In contrast, the aluminium portion
had dimensions of 5 and 30 mm in the horizontal and vertical directions, respectively. The elastic
moduli of the aluminium were derived as listed in Table 1. The displacement of the bottom edges
was restricted in the xand ydirections, whereas the displacement in the ydirection, defined as u
y
,
was applied downward to the nodes at the top of the aluminium portion as u
y
=1.0 mm, as shown
in Figure 4. Under this boundary condition, the stress components corresponding to each node,
defined as
σx
,
σy
, and
τxy
, were obtained. Additionally, the nominal shear stress
τIPS
and shear strain
γIPS
were calculated using Equations (1) and (2), respectively. In the FEA, the applied load Pwas
obtained from the sum of the reaction forces at the loading points, whereas the relative displacement
between the loading plates
δ
was obtained as the value of u
y
(=1 mm). The shear modulus value was
derived as τIPS/γIPS, as determined in a previous study [13].
Figure 5shows the finite element mesh for the AFPB test simulation. The horizontal length of the
model was 170 mm, and the model thickness, t, was 25 mm. The depth of the model, H, was 25 mm,
and the distance between the notch roots, b, was 7 mm. The mesh was constructed to be finer at the
region between the circular notches, as shown in Figure 5b. The number of the elements was 2000.
Prior to the FEAs using the IPS simulations, other FEAs had been conducted using a coarser mesh for
the AFPB analysis. Nevertheless, the obtained results were similar to each other; therefore, the finite
element mesh shown in Figure 5was also confirmed to be fine enough. The finite element mesh was
confirmed to be fine enough as well as that of the IPS model. The nodes corresponding to the locations
Polymers 2020,12, 47 7 of 15
at x=10 and 110 mm at the bottom of the model (y=0 mm) were restricted, whereas a displacement of
1 mm was applied downward to the nodes corresponding to the locations at x=60 and 160 mm at
the top of the model (y=250 mm). The asymmetric loading condition was realised by this boundary
condition. Similar to the IPS test simulation, the stress components corresponding to each node,
σx
,
σy
,
and
τxy
, were obtained. Additionally, the nominal shear stress
τAFPB
was calculated using Equation (3),
whereas the shear strain
γAFPB
was obtained from that of the node located at the centre of the model.
The shear modulus value was derived as τAFPB/γAFPB.
Polymers 2020, 12, x FOR PEER REVIEW 7 of 15
(3), whereas the shear strain γAFPB was obtained from that of the node located at the centre of the
model. The shear modulus value was derived as τAFPB/γAFPB.
Figure 4. Finite element meshes used in the IPS test simulations and boundary condition. Unit = mm.
Figure 5. Finite element meshes used in the AFPB test simulations and boundary condition. Unit =
mm.
10
300
2510
XPS
Aluminium
5
5555
2
5.5
BC
B'C'
u
y
= 1 u
y
= 1
x
y
x
y
(a) Whole mesh of AFPB model
10
A
u
y
= 1
u
y
= 1
50 50 50 10
25
(b) Detail of zone A
R = 2.5
CC'
B
B'
7
Figure 4.
Finite element meshes used in the IPS test simulations and boundary condition. Unit =mm.
Polymers 2020, 12, x FOR PEER REVIEW 7 of 15
(3), whereas the shear strain γAFPB was obtained from that of the node located at the centre of the
model. The shear modulus value was derived as τAFPB/γAFPB.
Figure 4. Finite element meshes used in the IPS test simulations and boundary condition. Unit = mm.
Figure 5. Finite element meshes used in the AFPB test simulations and boundary condition. Unit =
mm.
10
300
2510
XPS
Aluminium
5
5555
2
5.5
BC
B'C'
u
y
= 1 u
y
= 1
x
y
x
y
(a) Whole mesh of AFPB model
10
A
u
y
= 1
u
y
= 1
50 50 50 10
25
(b) Detail of zone A
R = 2.5
CC'
B
B'
7
Figure 5.
Finite element meshes used in the AFPB test simulations and boundary condition.
Unit =mm
.
Polymers 2020,12, 47 8 of 15
4. Results and Discussion
4.1. Finite Element Analysis
In the IPS test simulation, the stress components corresponding to each node
σx
,
σy
, and
τxy
were normalised by the nominal shear stress
τIPS
calculated from Equation (1). Figure 6shows the
distribution of the normalised stresses
σx
/
τIPS
,
σy
/
τIPS
and
τxy
/
τIPS
at the mid-width and boundary
between the XPS and aluminium plate, which correspond to BB’ and CC’ in Figure 4, respectively.
At the mid-width, the shear stress component is more significant than the normal stresses, and its
distribution is relatively uniform. However, at the boundary between the XPS and the aluminium
plate, the compressive stresses in the xand ydirections are markedly enhanced at point C’ (
σx
/
τIPS
=
4.47 and
σy
/
τIPS
=
1.64), because of the rectangular edge between the XPS and aluminium plate.
When the shear modulus is measured using an LVDT as determined in the ASTM C273/C273M-11,
these stress concentrations and combined stress condition enhance the displacement measured by the
LVDT, and the shear modulus value is estimated as low. Additionally, the stress concentrations also
enhance the failure. Therefore, there is a concern that the shear strength value is also estimated as low.
Polymers 2020, 12, x FOR PEER REVIEW 8 of 15
4. Results and Discussion
4.1. Finite Element Analysis
In the IPS test simulation, the stress components corresponding to each node σ
x
, σ
y
, and τ
xy
were
normalised by the nominal shear stress τ
IPS
calculated from Equation (1). Figure 6 shows the
distribution of the normalised stresses σ
x
/τ
IPS
, σ
y
/τ
IPS
and τ
xy
/τ
IPS
at the mid-width and boundary
between the XPS and aluminium plate, which correspond to BB’ and CC’ in Figure 4, respectively. At
the mid-width, the shear stress component is more significant than the normal stresses, and its
distribution is relatively uniform. However, at the boundary between the XPS and the aluminium
plate, the compressive stresses in the x and y directions are markedly enhanced at point C’ (σ
x
/τ
IPS
=
4.47 and σ
y
/τ
IPS
= 1.64), because of the rectangular edge between the XPS and aluminium plate.
When the shear modulus is measured using an LVDT as determined in the ASTM C273/C273M-11,
these stress concentrations and combined stress condition enhance the displacement measured by
the LVDT, and the shear modulus value is estimated as low. Additionally, the stress concentrations
also enhance the failure. Therefore, there is a concern that the shear strength value is also estimated
as low.
Figure 6. Distribution of the normalised stress σ
x
/τ
IPS
, σ
y
/τ
PS
, and τ
xy
/τ
IPS
at (a) the mid-width and (b)
the boundary between the XPS and aluminium plate obtained from the IPS test simulations.
In the AFPB test simulation, the σ
x
, σ
y
, and τ
xy
values were normalised by the nominal shear
stress τ
AFPB
calculated from Equation (3). Figure 7 shows the distribution of the normalised stresses
σ
x
/τ
AFPB
, σ
y
/τ
AFPB
and τ
xy
/τ
AFPB
at the mid-span and along the bottom notch edge, which correspond to
BB’ and CC’ in Figure 5, respectively. The shear stress component is more significant than the normal
stresses, but it distributes more uniformly than that in the IPS test. Therefore, when the shear strain
is measured at the mid-span, it is expected that the shear modulus value obtained from the actual
AFPB test is more precise than that obtained from the actual IPS test. At the notch edge, however, the
tensile and compressive stresses in the x direction are markedly enhanced as σ
x
/τ
AFPB
= 2.40 and 2.36
at the points of x = 1.31 and 1.31 mm, respectively. These values are smaller than that at the
rectangular edge in the IPS test. However, there is also a concern that the shear strength value is
estimated as low because of the combined stress condition. Further research should be conducted on
the specimen configuration to measure the shear strength value with reducing the effect of the stress
concentration and combined stress condition.
Figure 6.
Distribution of the normalised stress
σx
/
τIPS
,
σy
/
τIPS
, and
τxy
/
τIPS
at (
a
) the mid-width and (
b
)
the boundary between the XPS and aluminium plate obtained from the IPS test simulations.
In the AFPB test simulation, the
σx
,
σy
, and
τxy
values were normalised by the nominal shear
stress
τAFPB
calculated from Equation (3). Figure 7shows the distribution of the normalised stresses
σx
/
τAFPB
,
σy
/
τAFPB
and
τxy
/
τAFPB
at the mid-span and along the bottom notch edge, which correspond
to BB’ and CC’ in Figure 5, respectively. The shear stress component is more significant than the normal
stresses, but it distributes more uniformly than that in the IPS test. Therefore, when the shear strain is
measured at the mid-span, it is expected that the shear modulus value obtained from the actual AFPB
test is more precise than that obtained from the actual IPS test. At the notch edge, however, the tensile
and compressive stresses in the xdirection are markedly enhanced as
σx
/
τAFPB
=
2.40 and 2.36 at the
points of x=
1.31 and 1.31 mm, respectively. These values are smaller than that at the rectangular
edge in the IPS test. However, there is also a concern that the shear strength value is estimated as
low because of the combined stress condition. Further research should be conducted on the specimen
configuration to measure the shear strength value with reducing the eect of the stress concentration
and combined stress condition.
Polymers 2020,12, 47 9 of 15
Polymers 2020, 12, x FOR PEER REVIEW 9 of 15
Figure 7. Distribution of the normalised stress σ
x
/τ
AFPB
, σ
y
/τ
AFPB
, and τ
xy
/τ
AFPB
at (a) the mid-width and
(b) the boundary between the XPS and aluminium plate obtained from the AFPB test simulations.
Table 3 shows the shear modulus values obtained from the IPS and AFPB test simulations.
Because of the stress concentration described above, the shear modulus values obtained from the IPS
test simulations are significantly smaller than those input into the FEM program. In contrast, the
shear modulus values obtained from the AFPB simulations are closer to the input values than those
obtained from the IPS test simulations. The concentration of the normal stress components is also
found in the AFPB test simulation at the notch edge. However, as described above, the stress
concentration in the AFPB test is less significant than that in the IPS test. Because the IPS test is
standardised as the ASTM C273/C273M-11 [19], it is conducted more frequently than the AFPB test.
However, based on the FEA results, it is preferable to measure the shear modulus and shear strength
values of XPS from the AFPB test rather than the IPS test.
Table 3. Shear moduli obtained from the IPS and AFPB test simulations by FEM.
Code G
LT
(MPa) G
LZ
(MPa) G
TL
(MPa) G
TZ
(MPa)
IPS AFPB IPS AFPB IPS AFPB IPS AFPB
IB 5.54 6.16 7.08 8.30 5.58 6.05 6.80 7.77
B2 5.75 6.62 7.65 10.6 5.87 6.21 6.63 7.60
ACE 7.00 9.18 9.08 13.2 7.13 7.87 7.84 8.95
RB-GK 8.69 11.1 10.5 15.5 8.91 10.3 10.4 11.5
Input data of shear modulus: See Table 2.
4.2. IPS and AFPB Tests
Figure 8 shows the typical examples of the relationships between the strain measured from the
displacement gauge ε
d
and that measured using the strain gauge ε
g
in the tension and compression
tests for strain calibration. As shown in this figure, the ε
d
ε
g
relationships obtained from the
specimens cut from the same panel coincide well with each other in the tension tests. In contrast, the
relationships obtained from the compression tests vary. In the compression test, the load was applied
directly to the end surface of the specimen. Therefore, there was a concern that the load was often
applied eccentrically to the specimen due to the distortion of the end surface. Because of the variation,
it is often difficult to calibrate the strain using the results obtained from the compression tests. In
contrast, the load was applied via the grips in the tension tests and the condition of the end surface
did not influence the ε
d
ε
g
relationship. Therefore, the ε
d
ε
g
relationship was obtained stably in the
tension test. Based on these testing results, the ε
d
ε
g
relationships obtained from the tension tests
were used for the calibration in this study.
Figure 7.
Distribution of the normalised stress
σx
/
τAFPB
,
σy
/
τAFPB
, and
τxy
/
τAFPB
at (
a
) the mid-width
and (
b
) the boundary between the XPS and aluminium plate obtained from the AFPB test simulations.
Table 3shows the shear modulus values obtained from the IPS and AFPB test simulations.
Because of the stress concentration described above, the shear modulus values obtained from the IPS
test simulations are significantly smaller than those input into the FEM program. In contrast, the shear
modulus values obtained from the AFPB simulations are closer to the input values than those obtained
from the IPS test simulations. The concentration of the normal stress components is also found in the
AFPB test simulation at the notch edge. However, as described above, the stress concentration in the
AFPB test is less significant than that in the IPS test. Because the IPS test is standardised as the ASTM
C273/C273M-11 [
19
], it is conducted more frequently than the AFPB test. However, based on the FEA
results, it is preferable to measure the shear modulus and shear strength values of XPS from the AFPB
test rather than the IPS test.
Table 3. Shear moduli obtained from the IPS and AFPB test simulations by FEM.
Code
GLT (MPa) GLZ (MPa) GTL (MPa) GTZ (MPa)
IPS AFPB IPS AFPB IPS AFPB IPS AFPB
IB 5.54 6.16 7.08 8.30 5.58 6.05 6.80 7.77
B2 5.75 6.62 7.65 10.6 5.87 6.21 6.63 7.60
ACE 7.00 9.18 9.08 13.2 7.13 7.87 7.84 8.95
RB-GK 8.69 11.1 10.5 15.5 8.91 10.3 10.4 11.5
Input data of shear modulus: See Table 2.
4.2. IPS and AFPB Tests
Figure 8shows the typical examples of the relationships between the strain measured from the
displacement gauge
εd
and that measured using the strain gauge
εg
in the tension and compression
tests for strain calibration. As shown in this figure, the
εd
εg
relationships obtained from the specimens
cut from the same panel coincide well with each other in the tension tests. In contrast, the relationships
obtained from the compression tests vary. In the compression test, the load was applied directly
to the end surface of the specimen. Therefore, there was a concern that the load was often applied
eccentrically to the specimen due to the distortion of the end surface. Because of the variation, it is
often dicult to calibrate the strain using the results obtained from the compression tests. In contrast,
the load was applied via the grips in the tension tests and the condition of the end surface did not
Polymers 2020,12, 47 10 of 15
influence the
εd
εg
relationship. Therefore, the
εd
εg
relationship was obtained stably in the tension
test. Based on these testing results, the
εd
εg
relationships obtained from the tension tests were used
for the calibration in this study.
Polymers 2020, 12, x FOR PEER REVIEW 10 of 15
Figure 8. Relationships between the normal strains obtained from the displacement gauge εd and
strain gauge εg in the calibration by (a) tension and (b) compression tests.
Because of the insensitivity of the strain gauge, εg is much smaller than εd, as shown in Figure
7a. The εdεg relationship is initially linear and gradually becomes concave, and these tendencies were
commonly found in every XPS material. Considering these tendencies, the εdεg relationship was
formulated using a power function as follows:
𝜀=𝛼𝜀+𝛽𝜀 (6)
where α, β, and c are the parameters obtained by the method of least squares. The a, b, and c values
corresponding to each XPS were obtained from the following procedure:
1. The εdεg relationship corresponding to each specimen was regressed into Equation (6) by the
method of least squares.
2. The εg values were virtually determined in the range from 0 to 0.005 at intervals of 0.0001, and
the εd values were obtained by substituting the εg values into the regressed equation.
3. The εd values obtained from the same εg value were averaged among the same XPS material.
The averaged εdεg relationship was regressed into Equation (6) again, and the a, b, and c values
were determined.
The α, β, and c values obtained from this procedure are listed in Table 4. The calibrated shear
strain γc was calculated from the shear strain γg, which was calculated from the strain gauge output
using Equation (6), as follows: 𝛾=𝛼𝛾+𝛽𝛾 (7)
Table 4. α, β, and c values obtained from the regression of εdεg relationship of the tension test data
into Equation (6).
Code α β c
IB 3.97 135 5.25
B2 5.20 216 12.6
ACE 2.83 131 8.60
RB-GK 2.18 130 8.95
Figure 9 shows the typical examples of the shear stress–shear strain relationships obtained from
the IPS test τIPSγIPS, and AFPB test τAFPBγg, and τAFPBγc. In the AFPB test, the shear strain obtained
from the strain gauge γg is much smaller than that in the IPS test γIPS because of the insensitivity of
the strain gauge. Additionally, the nonlinearity in the τAFPBγg relationship is not often significant,
such that the stress–strain relationship is extremely discrepant from that obtained from the IPS test.
Strain obtained from strain gauge ε
g
Strain obtained from displacement gauge ε
d
STYROACE-II
(a) Tension test
Strain obtained from strain gauge ε
g
STYROACE-II
(b) Compression test
Figure 8.
Relationships between the normal strains obtained from the displacement gauge
εd
and
strain gauge εgin the calibration by (a) tension and (b) compression tests.
Because of the insensitivity of the strain gauge,
εg
is much smaller than
εd
, as shown in Figure 7a.
The
εd
εg
relationship is initially linear and gradually becomes concave, and these tendencies were
commonly found in every XPS material. Considering these tendencies, the
εd
εg
relationship was
formulated using a power function as follows:
εd=αεg+βεgc(6)
where
α
,
β
, and care the parameters obtained by the method of least squares. The a,b, and cvalues
corresponding to each XPS were obtained from the following procedure:
1.
The
εd
εg
relationship corresponding to each specimen was regressed into Equation (6) by the
method of least squares.
2.
The
εg
values were virtually determined in the range from 0 to 0.005 at intervals of 0.0001, and the
εdvalues were obtained by substituting the εgvalues into the regressed equation.
3.
The
εd
values obtained from the same
εg
value were averaged among the same XPS material.
The averaged
εd
εg
relationship was regressed into Equation (6) again, and the a,b, and cvalues
were determined.
The
α
,
β
, and cvalues obtained from this procedure are listed in Table 4. The calibrated shear
strain
γc
was calculated from the shear strain
γg
, which was calculated from the strain gauge output
using Equation (6), as follows:
γc=αγg+βγgc(7)
Figure 9shows the typical examples of the shear stress–shear strain relationships obtained from
the IPS test
τIPS
γIPS
, and AFPB test
τAFPB
γg
, and
τAFPB
γc
. In the AFPB test, the shear strain obtained
from the strain gauge
γg
is much smaller than that in the IPS test
γIPS
because of the insensitivity of
the strain gauge. Additionally, the nonlinearity in the
τAFPB
γg
relationship is not often significant,
such that the stress–strain relationship is extremely discrepant from that obtained from the IPS test.
When conducting the strain calibration using Equation (6), however, the linear region is significant in
the linear portion of the τAFPBγcrelationship, and the nonlinear strain region is more pronounced.
Polymers 2020,12, 47 11 of 15
Table 4. α
,
β
, and cvalues obtained from the regression of
εd
εg
relationship of the tension test data
into Equation (6).
Code α β c
IB 3.97 135 5.25
B2 5.20 216 12.6
ACE 2.83 131 8.60
RB-GK 2.18 130 8.95
Polymers 2020, 12, x FOR PEER REVIEW 11 of 15
When conducting the strain calibration using Equation (6), however, the linear region is significant
in the linear portion of the τAFPBγc relationship, and the nonlinear strain region is more pronounced.
Figure 9. Typical examples of the shear stress–shear strain relationships obtained from the IPS and
AFPB test.
Table 5 lists the shear modulus values obtained from the IPS and AFPB tests. Similar to the
results of the FEAs, the shear modulus values obtained from the IPS tests are significantly lower than
those obtained from the AFPB tests. In the IPS test, the stress concentration at the rectangular edge
between the XPS and aluminium plate induces a large shear deformation; therefore, the shear
modulus value is measured as lower in the IPS test. In contrast, the stress concentration at the notch
edge in the AFPB test is not more significant than that at the rectangular edge between the XPS and
aluminium plate in the IPS test.
Table 5 also lists the values of the in-plane shear modulus GLT (GTL) obtained from the static SPT
tests in a previous study [14] and the unpaired t-tests of the differences of the GLT and GTL values
obtained from these tests were conducted. The SPT tests were conducted using the specimens cut
from XPS panels similar to those used in this study. The GLT and GTL values obtained from the IPS
tests are significantly lower than those obtained from the SPT tests in the significance level of 0.01. In
contrast, the difference between the GLT and GTL values obtained from the AFPB and SPT tests are not
significant in the significance level of 0.05. Therefore, the AFPB test is promising in obtaining the
shear modulus, in a similar way to the SPT test. However, the measurement of the shear strain is
indirect, despite the effectiveness in using the strain gauge. To improve the accuracy in measuring
the shear modulus, the aforementioned alternative techniques, such as the digital image correlation
(DIC) and virtual field method (VFM), should be adopted for the AFPB test in place of the strain
gauge [15–18,20]. Further research should be performed on this topic.
Table 5. Shear moduli obtained from the IPS and AFPB tests.
Code GLT (MPa) GLZ (MPa) GTL (MPa) GTZ (MPa) GLT, GTL (MPa)
IPS AFPB IPS AFPB IPS AFPB IPS AFPB SPT
IB 4.47 8.00 7.04 9.60 4.64 7.36 6.06 7.47 6.32
(0.16) (2.70) (0.47) (1.62) (0.24) (1.68) (0.63) (1.72) (0.22)
B2 4.96 7.34 7.53 6.76 5.25 6.56 6.43 6.42 6.99
(0.60) (1.05) (0.63) (0.68) (0.71) (0.59) (0.37) (0.57) (0.09)
ACE 6.55 10.4 11.1 16.3 6.93 11.3 8.03 10.6 9.78
(0.60) (2.3) (0.2) (1.3) (0.33) (2.5) (0.44) (1.4) (0.36)
RB-GK 10.3 15.0 12.7 16.6 10.4 11.9 8.96 13.7 13.6
(0.8) (2.8) (0.6) (3.1) (0.6) (2.5) (1.05) (3.0) (0.5)
Shear strain γ
IPS
, γ
g
, and γ
c
: IPS,
τ
IPS
-γ
IPS
: AFPB,
τ
AFPB
-γ
g
: AFPB,
τ
AFPB
-γ
c
ACE
LT plane
50
0
100
150
200
250
300
350
Shear stress
τ
IPS
and
τ
AFPB
(kPa)
Figure 9.
Typical examples of the shear stress–shear strain relationships obtained from the IPS and
AFPB test.
Table 5lists the shear modulus values obtained from the IPS and AFPB tests. Similar to the results
of the FEAs, the shear modulus values obtained from the IPS tests are significantly lower than those
obtained from the AFPB tests. In the IPS test, the stress concentration at the rectangular edge between
the XPS and aluminium plate induces a large shear deformation; therefore, the shear modulus value is
measured as lower in the IPS test. In contrast, the stress concentration at the notch edge in the AFPB
test is not more significant than that at the rectangular edge between the XPS and aluminium plate in
the IPS test.
Table 5. Shear moduli obtained from the IPS and AFPB tests.
Code
GLT (MPa) GLZ (MPa) GTL (MPa) GTZ (MPa) GLT,GTL (MPa)
IPS AFPB IPS AFPB IPS AFPB IPS AFPB SPT
IB 4.47 8.00 7.04 9.60 4.64 7.36 6.06 7.47 6.32
(0.16) (2.70) (0.47) (1.62) (0.24) (1.68) (0.63) (1.72) (0.22)
B2 4.96 7.34 7.53 6.76 5.25 6.56 6.43 6.42 6.99
(0.60) (1.05) (0.63) (0.68) (0.71) (0.59) (0.37) (0.57) (0.09)
ACE 6.55 10.4 11.1 16.3 6.93 11.3 8.03 10.6 9.78
(0.60) (2.3) (0.2) (1.3) (0.33) (2.5) (0.44) (1.4) (0.36)
RB-GK
10.3 15.0 12.7 16.6 10.4 11.9 8.96 13.7 13.6
(0.8) (2.8) (0.6) (3.1) (0.6) (2.5) (1.05) (3.0) (0.5)
Results are given as the averages
±
(SD). The results of the IPS and static SPT tests are referredfrom [
13
,
14
], respectively.
Table 5also lists the values of the in-plane shear modulus G
LT
(G
TL
) obtained from the static SPT
tests in a previous study [
14
] and the unpaired t-tests of the dierences of the G
LT
and G
TL
values
obtained from these tests were conducted. The SPT tests were conducted using the specimens cut
from XPS panels similar to those used in this study. The G
LT
and G
TL
values obtained from the IPS
tests are significantly lower than those obtained from the SPT tests in the significance level of 0.01.
In contrast, the dierence between the G
LT
and G
TL
values obtained from the AFPB and SPT tests
Polymers 2020,12, 47 12 of 15
are not significant in the significance level of 0.05. Therefore, the AFPB test is promising in obtaining
the shear modulus, in a similar way to the SPT test. However, the measurement of the shear strain is
indirect, despite the eectiveness in using the strain gauge. To improve the accuracy in measuring
the shear modulus, the aforementioned alternative techniques, such as the digital image correlation
(DIC) and virtual field method (VFM), should be adopted for the AFPB test in place of the strain
gauge [1518,20]. Further research should be performed on this topic.
Table 6lists the shear strength values obtained from the IPS and AFPB tests. The shear strength
values obtained from the AFPB tests are significantly higher than those obtained from the IPS tests.
Because the eect of the stress concentration is less significant in the AFPB test than in the IPS test,
the AFPB test is preferable to the IPS test for measuring the shear strength value.
Table 6. Shear strengths obtained from the IPS and AFPB tests.
Code
SLT (kPa) SLZ (kPa) STL (kPa) STZ (kPa)
IPS AFPB IPS AFPB IPS AFPB IPS AFPB
IB 157 225 208 261 157 213 185 242
(7) (2) (13) (13) (12) (7) (33) (14)
B2 150 267 235 298 163 245 196 250
(14) (9) (16) (30) (7) (12) (16) (19)
ACE 212 363 289 344 228 271 231 260
(19) (45) (9) (9) (11) (12) (5) (27)
RB-GK 282 403 351 429 283 355 278 320
(36) (18) (18) (40) (33) (16) (31) (26)
Results are given as the averages ±(SD).
Figure 10a shows the large deformation at the rectangular edge between the XPS and aluminium
plate during the loading. The failure was initiated and propagated from the rectangular edge. The
large deformation is induced due to the stress concentration, and, as described above, there is a concern
that the shear modulus and shear strength are estimated as low because of the stress concentration
at this point. Figure 10b shows the failures induced in the AFPB test. The failure is induced at the
top and/or bottom notch edges, and it propagates obliquely in the specimen. It is desirable that the
failure is exactly induced at the mid-length, which corresponds to the region between B and B’ in
Figure 5b, because the failure at this region is induced due to the pure shear stress, as shown in Figure 7.
However, Figure 10b indicates that the catastrophic failure was often induced at a point deviating from
the mid-length at the top and/or bottom notch edges. Therefore, there is a concern that failure due to
the combined stress condition is induced, as predicted from the FE calculation (Figure 7b). In strongly
orthotropic materials like solid wood, even if a failure was initiated at a point deviating from the
mid-length, the failure due to the shear stress was also induced at the mid-length when the load was
applied continuously [
22
]. Further research is required on the specimen configuration to reduce the
eect of stress concentration thoroughly, although the shear strengths obtained from the AFPB tests are
higher than those obtained from the IPS tests as listed in Table 6.
Polymers 2020,12, 47 13 of 15
Polymers 2020, 12, x FOR PEER REVIEW 13 of 15
Figure 10. Large deformation of the XPS specimen at the rectangular edge between the XPS and
aluminium plate during the loading (a), and failures induced at the top and bottom notch edges in
the AFPB test (b).
5. Conclusions
To measure the shear modulus and shear strength of extruded polystyrene foam (XPS), the in-
plane shear (IPS) and asymmetric four-point bending (AFPB) test methods were experimentally and
numerically analysed, and the following results were obtained:
1. The results of the FE analyses indicated that the shear modulus values obtained from the AFPB
test simulations were closer to the input ones than those obtained from the IPS test simulation.
Therefore, it was expected to measure the shear modulus value by the AFPB test accurately.
2. In the actual AFPB tests, the measurement of the shear strain using the strain gauge was
indirectly determined from the calibration tests. The in-plane shear modulus values obtained
from the AFPB tests were close to those obtained from the square plate twist (SPT) tests
conducted in a previous study [14]. Therefore, the AFPB test is effective in measuring the shear
modulus. To accurately determine the shear strain, however, alternative methods, such as digital
image correlation (DIC) and virtual fields method (VFM), should be adopted in place of the
strain gauge.
3. From the FE analyses, the normal stress components were concentrated in a particular region.
Therefore, there was a concern that the shear strength value would be estimated as low because
the stress concentration enhanced the failure of the specimen. However, the effect of the stress
concentration was less in the AFPB test than in the IPS test.
4. Similar to the results obtained from the FE analyses, the experimental results indicated that the
shear modulus and shear strength values obtained from the AFPB test were higher than those
obtained from the IPS test. Therefore, the effect of the stress concentration was less significant in
the AFPB test than in the IPS test.
Author Contributions: All authors contributed to the design and implementation of the research, to the analysis
of the results and to the writing of the manuscript.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
References
1. Vinson, J.R. Sandwich structures. Appl. Mech. Rev. 2001, 54, 201–214, doi:10.1115/1.3097295.
2. Hu, Y.; Nakao, T.; Nakai, T.; Gu, J.; Wang, F. Dynamic properties of three types of wood-based composites.
J. Wood Sci. 2005, 51, 7–12, doi:10.1007/s10086-003-0614-y.
3. Hu, Y.; Nakao, T.; Nakai, T.; Gu, J.; Wang, F. Vibrational properties of wood plastic plywood. J. Wood Sci.
2005, 51, 13–17, doi:10.1007/s10086-003-0624-9.
Failure induced at the notch edge
Large deformation during the loading
a b
Figure 10.
Large deformation of the XPS specimen at the rectangular edge between the XPS and
aluminium plate during the loading (
a
), and failures induced at the top and bottom notch edges in the
AFPB test (b).
5. Conclusions
To measure the shear modulus and shear strength of extruded polystyrene foam (XPS), the in-plane
shear (IPS) and asymmetric four-point bending (AFPB) test methods were experimentally and
numerically analysed, and the following results were obtained:
1.
The results of the FE analyses indicated that the shear modulus values obtained from the AFPB
test simulations were closer to the input ones than those obtained from the IPS test simulation.
Therefore, it was expected to measure the shear modulus value by the AFPB test accurately.
2.
In the actual AFPB tests, the measurement of the shear strain using the strain gauge was indirectly
determined from the calibration tests. The in-plane shear modulus values obtained from the
AFPB tests were close to those obtained from the square plate twist (SPT) tests conducted in
a previous study [
14
]. Therefore, the AFPB test is eective in measuring the shear modulus.
To accurately determine the shear strain, however, alternative methods, such as digital image
correlation (DIC) and virtual fields method (VFM), should be adopted in place of the strain gauge.
3.
From the FE analyses, the normal stress components were concentrated in a particular region.
Therefore, there was a concern that the shear strength value would be estimated as low because
the stress concentration enhanced the failure of the specimen. However, the eect of the stress
concentration was less in the AFPB test than in the IPS test.
4.
Similar to the results obtained from the FE analyses, the experimental results indicated that the
shear modulus and shear strength values obtained from the AFPB test were higher than those
obtained from the IPS test. Therefore, the eect of the stress concentration was less significant in
the AFPB test than in the IPS test.
Author Contributions:
All authors contributed to the design and implementation of the research, to the analysis
of the results and to the writing of the manuscript. All authors have read and agreed to the published version of
the manuscript.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
References
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... For this reason, XPS boards have an advantage over EPS boards in structural assemblies in constant contact with moisture (inverted roof, basement wall, foundation slab, etc.), as they preserve low thermal conductivity regardless of the presence of moisture. Research into EPS boards used as so-called geofoams for reducing the self-weight of the soil (Athanasopoulos et al. 1999;Vejelis et al. 2008;Gnip et al. 2007;Maleki and Ahmadi 2011;Forcellini 2020;Yoshihara et al. 2018;Yoshihara and Maruta 2020) has provided answers regarding their durability and preservation of good mechanical properties when in constant contact with the soil and moisture (strength and deformability). On the other hand, the foundation slab structural assembly must be suitably waterproofed if EPS boards are used as thermal insulation under it. ...
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... Although the abovementioned stress concentration can be reduced in this method, it is difficult to obtain the in-plane shear modulus directly from the test because of the effect of out-of-plane shear modulus, and finite element (FE) calculations were required to improve the accuracy. In addition to these standardized methods, asymmetric four-point bending (AFPB) tests were conducted [23]. As shown in Table 1, the shear modulus value obtained from the AFPB test is higher than those obtained from the EN 12090, ASTM C273, and ISO 15310 methods. ...
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