Kobe University Repository: Kernel
Kobe University Repository: Kernel
Kobe University Repository: Kernel
Introduction
To accomplish optimum package, cushion curve (combination of static stress–peak acceleration
curve and static stress–strain curve) is often used in packaging field [1, 2]. Using the cushion curve,
we can calculate the minimum thickness of the cushioning materials swiftly. If there is not a
ready-made cushion curve (Fig. 1) on hand for a cushioning material, the packaging designer should
plot it by himself. However, traditional plotting method is dependent on massive repeated dynamic
compression tests. Furthermore, using self-made cushion curve based on the traditional method, we
may not accomplish desired cushioning effect. Therefore, to quantitatively evaluate the performance
of the cushion materials, it is necessary to search a new simplified method that not depends on the
large number dynamic compression tests. According to literature, many simplified methods [3-8]
for the cushion curve were proposed so far, nevertheless, precision evaluation of those methods
were not completely carried out.
Drop height: 60 cm. Average of 2~5 dropping
Fig90 Dynamic cushion performence curve Fig91 Instant maximum strain curve
Plastic cushion and paper cushion are extensively applied in packaging field [10]. Their
representative cushioning materials are expanded polyethylene (EPE) and corrugated fiberboard.
Therefore, EPE and laminated-board cushion (LBC) was used as test materials to represent plastic
and paper cushions in this study (Fig. 2).
(a) (b)
Fig. 2 Two kinds of cushioning materials. (a) Plastic cushion, (b) Paper cushion.
(a) (b)
Fig. 3 Dynamic compression test with shearing. (a) Dynamic compression test, (b) Illustration of
the effect of shearing
60
Cushion curve based on
dynamic compression test without shearing
50
Peak acceleration (G)
10
0
0.001 0.01 0.1
Static stress (MPa)
Fig. 4 Comparison between experimental results and cushion curves with and without shearing
EPE (25 times expanded rate, 178×178×80 mm) as shown in Fig. 2(a) was used as the test
material. A dynamic compression test with the effect of shearing was carried out (Fig. 3(a)) [11]. A
test configuration following Fig. 3(b) was set to yield the effect of shearing. An iron plate (Base2:
100×100×63 mm) was fixed on the table of the test equipment (Base1) and EPE was placed on the
iron plate. Test conditions were set as: the drop height h was 0.6 m, 5 consecutive droppings were
performed, 5 points of the static stress was plotted, 3 tests under the same conditions were carried
out for each point.
Based on the experimental data, we plot the cushion curve with shearing as shown in Fig. 4.
Moreover, a cushion curve without shearing based on a test (h: 0.6 m, the thickness of the test
material t: 80 mm) and the result of dummy package (content: 20 kg, 400×400×300 mm;
cushioning material: 8 corner pads, 80×80×80 mm) equivalent drop test [12-17] based on the test
conditions above are also provided.
According to Fig. 4, the results of dummy package equivalent drop test and the cushion curve
with shearing are almost identical. Hence, it can be say that the shearing should be a major reason
that the desired cushion effect is not available. Therefore, to get more accurate packaging design,
the effect of shearing should not be ignored when we use the cushion curve.
0.8 10
Dynamic cushion factor
Dynamic stress (MPa)
8
0.6
6
0.4
4
0.2
2
s2
0 0
0 0.2 s1 0.4 0.6 0.8 1.0 0.01 0.1 1 10
Strain Dynamic stress (MPa)
(a) (b)
Fig. 5 Strain–dynamic stress and C curves of EPE. (a) Strain–dynamic stress curve, (b) C curve
90
T=2
Acceleration (G)
4
c
50 a
b d 5
6
8
10
a critical acceleration
0 0.02 0.05 0.1 0.2 0.3
Static stress (kg/cm2)
Fig. 6 Example of calculating thickness of cushioning material using cushion curve
Generally, there are two methods to calculate the minimum thickness of the cushioning material
tmin for a certain drop height and acceleration when packaging design. One is a method using the
lowest point of the cushion curve; the other is a method using the lowest point of the C curve. An
example of calculating tmin using the cushion curve shows in Fig. 6. Using the peak acceleration–
static stress curve of cushion curve, we draw a line ‘a’ from a critical acceleration, and then connect
the lowest points of two adjacent cushion curves of line ‘a’ and get line ‘b’. Line ‘b’ is broken up
into two parts by an intersection: lines ‘c’ and ‘d’. According to proportion of lines ‘c’ and ‘d’, we
can calculate tmin [1]. Using these two methods, we calculated tmin for equal free fall heights and
accelerations, as shown in Fig. 7. Horizontal axis shows the thickness t0 calculated by the
ready-made cushion curve [9], vertical axis shows the thickness tc calculated by the C curve. It can
be known that the two results are almost the same roughly. However, the error between the two
results becomes large gradually with respect to the increase in the thickness of the cushioning
material.
140
t min based on C curve (mm)
120
100
80
60
40
20
0
0 20 40 60 80 100 120 140
t min based on cushion curve (mm)
Fig. 7 Comparison of two calculating methods for tmin
To quantitively discuss the error between tc and t0, Fig. 8 is plotted. Horizontal axis is the set-up
acceleration and vertical axis is the difference of thicknesses tc – t0. Considering that the thickness
of the cushioning material increases or decreases in 10 mm increments when packaging design, we
highlight the part that tc – t0 is larger than 10 mm, and know tc – t0 becomes large when the set-up
acceleration is smaller than 50 G. To plot the ready-made cushion curve, test conditions of the
dynamic compression test are usually set as: h = 0.6 m, t = 40 mm. Peak acceleration of the
dynamic compression test using these test conditions is still larger than 65 G even in small
acceleration region. Hence, the lowest point of the C curve based on the dynamic compression test
under above conditions cannot express the cushioning performance in small acceleration region
correctly. Moreover, tc – t0 becomes larger in small acceleration region. Therefore, we should make
careful judgment on the test condition of the dynamic compression test when the cushioning
characteristic of the C curve is used.
20
tc – t0 (mm) 10
-10
-20
-30
0 25 50 75 100
Set-up acceleration (G)
Fig. 8 Difference of thicknesses in different acceleration regions
0.20
Dynamic stress (MPa)
0.16
0.12
0.08
0.04
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Strain
Fig. 9 Strain–dynamic stress curves of LBC
It can be seen that strain–dynamic stress curves of LBC considerably differ. Hence, it is not
possible to use one curve as same as the plastic cushion to represent others. Therefore, we proposed
a method that using an average stress–strain curve to calculate the strain and the acceleration. To
ensure that the packaging design has sufficient safety in terms of acceleration, we also calculated an
average+3σ strain–dynamic stress curve (σ: standard deviation). The definite integral of the strain–
dynamic stress curve is ε. Hence, if let ε=Ep (Ep: absorbable energy calculated by the potential
energy that based on m and h), we can calculate the peak acceleration and the strain. The calculating
process is shown in Fig. 10: First, based on the average strain–dynamic stress curve, the strain
corresponding to ε is derived. Second, according to average and average+3σ strain–dynamic stress
curves, the peak acceleration until the strain becomes the calculating value is calculated.
According to Fig. 10, comparing the calculating strain and acceleration with the experimental
data, it is known that although the calculating strain matches the experimental data, the calculating
acceleration is small than the experimental data when the average strain–dynamic stress curve is
used and large than the experimental data when the average+3σ strain–dynamic stress curve is used.
Hence, we must determine that based on which of these curves to let the calculating results
approach the experimental data. The investigate results show that the average+σ strain–dynamic
stress curve is optimum. Therefore, if the average strain–dynamic stress curve and the standard
deviation can be available, we can evaluate the cushioning performance of LBC swiftly.
0.20
Dynamic stress (MPa)
0.16
0.12
0.08
0.04
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Strain
Fig. 10 Estimation of the strain and the peak acceleration of LBC
Fig. 11 shows acceleration and the strain comparisons of LBC between calculating results and
experimental data based on 20 dynamic compression tests. It is known that Gc = 1.0041 Ge, Sc =
0.0249 Se. For both the acceleration and the strain, there is approximate linear relationship between
calculating results and experimental data. Thus, it can be say that the proposed method has enough
precision to evaluate the cushioning performance of LBC.
Calculating peak acceleration Gc (G)
100 1.0
Calculating strain Sc
80 0.8
60 0.6
40 0.4
20 0.2
Gc = 1.0041 Ge Sc = 1.0249 Se
0 0
0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1.0
Experimental peak acceleration Ge (G) Experimental strain Se
(a) (b)
Fig. 11 Comparison between calculating results and experimental data. (a) Comparison of peak
acceleration, (b) Comparison of strain
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