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MECHANICAL PROPERTY MEASUREMENT OF FLEXIBLE MULTI-LAYERED
MATERIALS USING POSTBUCKLING BEHAVIOR
Atsumi OHTSUKI
Department of Mechanical Engineering
Meijo University
1-501 Shiogamaguchi, Tempaku-ku, Nagoya, 468-8502 Japan
ABSTRACT
In application of flexible materials it is very important to evaluate mechanical properties of these materials in both analytical
and technological interests. This report deals with an innovative method (:Compression Column Method) of measuring
Young’s modulus under large deformations of a flexible multi-layered specimen. Exact analytical solutions based on the nonlinear large deformation theory are derived in terms of elliptic integrals. By just measuring the applied load and a vertical
displacement and/or a deflection angle, each Young’s modulus can be easily obtained for thin and long multi-layered flexible
materials. Measurements were carried out on a two-layered material (PVC: a high-polymer material and SUS: a steel material).
The results confirm that the new method is suitable for flexible multi-layered thin plates or thin rods. Based on the
assessments made the method can be further applied to multi-layered thin sheets and multi-layered fiber materials (e.g., steel
belts, glass fibers, carbon fibers, optical fibers, etc.). In the meantime, the Circular Ring Method [1], [2], the Cantilever Method
[3] and the Compression Column Method [4] have already been developed and reported for single-layered thin/slender
beam/column specimens.
Introduction
In recent years, flexible multi-layered materials with very high performance are used in a wide ranging and diverse applications.
Therefore, the importance of large deformation analyses has been increasingly recognized in both analytical and technological
interests of structural design of mechanical springs, fabrics and various multi-layered thin walled structures (aerospace
structures, ship, car, rail structures, etc.). Consequently, evaluation of mechanical property such as Young’s modulus is
needed to predict large deformations occurring in flexible multi-layered materials. Many of the current material testing methods
to examine mechanical properties for flexible materials are duplicates of the conventional testing methods for metallic
materials and these methods are based on the small deformation theory
With the large deformation theory a new bending test method (:Compression Column Method) which is much more practical
than the conventional bending/tensile test is proposed in the present paper. Exact analytical solutions for large deformation are
obtained in terms of elliptic integrals under the assumptions that the geometrical nonlinearity arises as a result of large
deformation, while the material remains linearly elastic. In using this method, Young’s modulus of thin and long flexible
materials such as plastics and advanced composites can be easily obtained by just measuring the applied load and a vertical
displacement and/or a deflection angle.
In order to assess the applicability of the proposed method, several experiments were carried out using a two-layered material
(PVC: a high-polymer material and SUS: a steel material). As a result, it becomes clear that the new method is suitable for
flexible multi-layered materials. The measured modulus is the secant modulus of elasticity.
Besides the Compression Column Method for multi-layered materials, the Circular Ring Method [1, 2], the Cantilever Method
[3] and the Compression Column Method for a single-layered specimen have already been developed and reported, based on
the large deformation theory.
Theory
The material testing methods for metal or plastics is used to examine a mechanical property (Proportional limit, Elastic limit,
Yield point, Ultimate strength, Elastic modulus, Elongation, Contraction, etc.) of a material. The three/four-point bending
L
λ 2
s
A
x
E1 I 1
C' C
Q ( x, y )
θ
z
E2
I2
δ
y
y
yZ
P
P
D
RA ( − )
R( − )
neutral axis
θB
B
arbitrary axis
y
y
Figure 1. Schematic configuration of multi-layered column
subjected to axial compressive forces at both
hinged supports.
Figure 2. Illustration of cross-section of
two-layered material..
methods or tensile method are applied in general. Although these methods are very simple, they also have several
disadvantages (e.g., a stress concentration around a loading nose, a gripping problem of specimen).
From this point of view, a new testing method (Compression Column Method) is devised considering large deformation
behaviors of a test specimen. The new method can be applied to various multi-layered wires, long fiber materials (Glass fibers,
Carbon fibers, Optical fibers, etc.) and multi-layered thin sheet materials.
1. Basic Equation
A typical illustration of a load-deflection shape for a column compressed between freely pivoted ends is given in Fig.1. While
an ideal straight column is subjected to a small compressive load P, it produces only a deformation slightly shrunk in the
longitudinal direction. However, the load P reaches the critical load, the so-called Euler buckling will take place. As an example,
a cross section of two-layered plate (n=2) is shown in Fig.2.
Due to the symmetry of the deformed shape, the analysis is carried out for the region AB (only 1/2 of the whole arc length 2L).
The horizontal displacement is denoted by x, vertical displacement by y, and θ is the deflection angle. Moreover, an arc length
is denoted by s, the radius of curvature by R and the bending moment by M. The relationship among R, M, s, x, y and θ are
given by:
1
dθ
=−
=
R
ds
⎫
⎪⎪
⎬
⎪
⎪⎭
M
n
∑ (E I )
i
i
i =1
dx = ds ⋅cos θ , dy = ds ⋅sin θ
where
(1)
EiIi= flexural rigidity of each layer
The moment applied at an arbitrary position Q(x, y) is expressed as
M = P⋅ y + MA
(2)
Introducing the following non-dimensional variables
x
L
y
L
s
R
L
L
MAL
ξ = , η = , ζ = ，ρ =
γ=
2
PL
n
∑ (E I )
i
i =1
i
,α=
n
∑ (E I )
i
i =1
i
⎫
⎪
⎪
⎬
⎪
⎪
⎭
(3)
the basic equation of large deformation (: postbuckling behavior) is derived from Eqs.(1) - (3) in the form of :
dθ
+ γ ⋅ sin θ ＝0
2
dζ
2
Considering the boundary condition
(4)
(dθ dζ ) θ =θ = 0 (M θ =θ ＝0 )
B
at the point B, Eq.(4) can be integrated to yield the
B
following.
dθ
＝ ± 2γ (cos θ－cos θ B )
dζ
(0 ≦θ ≦ θ B )
(5)
Equation (5) is the basic equation that determines large deformation behaviors of a compressed column. The double sign (±)
on the right hand side of Eq.(5) means that the positive or negative sign is adopted when the deflection angle θ is increasing or
decreasing, respectively, with the increase of the non-dimensional arc length ζ.
In analysis of the compressed column, only the plus sign (+) is adopted.
In analyzing the differential equation (5) the following formula may be used to transform the variables with respect to angle θ
cosθ B = 1 − 2k 2
⎛ 0 ≦φ ≦ π ⎞
cosθ = 1 − 2k 2 ⋅ sin 2 φ　⎜
⎟
2⎠
⎝
⎫
⎪
⎬
⎪⎭
(6)
Denoting the half length of a specimen by L and taking into the conditions ζB (=sB/L) =1, ηB=δ/L, ξB=(L-λ/2)/L, the maximum
non-dimensional arc length ζB, the maximum non-dimensional vertical displacement ηB and the maximum non-dimensional
horizontal displacement ξB are obtained as follows.
ζB =1=
ηB =
ξB =
where
δ
=
L
π
F ⎛⎜ k , ⎞⎟
γ ⎝ 2⎠
1
(where, κ =
(1 − cos θ B ) / 2
)
2k ⎛
π ⎞ 2k
⎜1 − cos ⎟ =
2⎠
γ⎝
γ
L−λ 2
1
=
L
γ
⎧2 E ⎛ k , π ⎞ − F ⎛ k , π ⎞ ⎫
⎟⎬
⎜
⎟
⎨ ⎜
⎩ ⎝ 2⎠
⎝ 2 ⎠⎭
(7)
(8)
(9)
F (k , π 2 ) = Legendre − Jacobi’s complete elliptic integrals of the first kind
E (k , π 2 ) = Legendre − Jacobi’s complete elliptic integrals of the second kind
On the other hand, the bending moment MA at the point A becomes as follows;
M A＝－Pδ
(10)
Therefore, the relationship of the non-dimensional curvature ρA at the point A and the vertical displacement δ at the point B is
derived as follows;
n
∑ (E I
i
ρA =
i =1
MAL
i
n
)
∑ (E I
i
=-
i
)
i =1
PLδ
From Eq.(11), each Young’s modulus Ei can be calculated as an indirect process in the form of;
(11)
n
∑ ( E I ) + PL
⋅η B ⋅ ρ A = 0
2
i
i
(12)
i =1
Moreover, Young’s modulus Ei can be calculated as a direct process in the form of;
ηB
⎫
⎧
( Ei I i ) − PL ⋅ ⎨
⎬ =0
∑
i =1
⎩ 2 sin(θ B 2) ⎭
2
n
2
(13)
On the other hand, the following formula based on Eq.(3) is useful in calculating Young’s modulus Ei as an indirect process.
n
∑ (E I ) −
i
PL
=0
γ
i
i =1
where
2
(14)
Ii = second moment of area of the each layer cross section
Therefore, it is possible to calculate Young’s modulus Ei using Eq.(12) or (13) or (14).
In the meantime, it is necessary to determine the neutral axis multi-layered plates (In case of multi-layered rods, the cross
section is symmetry).
The distance y to the neutral axis (Fig.2) and the relation of
y - Ii are obtained respectively as follows;
n
∑ E (S )
i
y=
i
z
i =1
n
(15)
∑ (E A )
i
i
i =1
i
bhi3
hi
⎛
+ bhi ⎜ y − − ∑ hk −1 ⎞⎟
Ii =
12
2 k =1
⎝
⎠
where
2
(16)
Si = first moment of area of the each layer cross section
2
i
bh
( S i ) z = i + bhi ⎛⎜ ∑ hk −1 ⎞⎟
2
⎝ k =1
⎠
2. Measuring Techniques
Though there exist some methods in order to measure Young’s modulus, representative three methods are introduced in this
paper.
Two quantities, i.e., a vertical displacement δ and a deflection angleθB are needed for the direct process based on Eq.(13). On
the other hand, the indirect process, based on Eq.(12) or (14), needs one quantity δ or θB. In case of Eq. (12), a
chart(:Nomograph) of δ -ρA relation is presented, which was computed previously by using Eq.(7). In case of Eq.(14), a chart
(:Nonograph) of γ -δ relation or γ -θB relation is presented.
Here, for the sake of simplicity, the usage of the chart is recommend by the author.
2.1 Method 1 [Direct Process]: (Measurement of δ and θB)
The usage of this method is shown below. Each Young’s modulus Ei is obtained for a PVC thin plate(first layer) with
length:2L1=300.0[mm], width: b1=27.0[mm], thickness: h=0.515 [mm] and a SUS thin plate(second layer) with length:2L2=300.0
[mm], width: b1=27.0[mm], thickness: h=0.100 [mm].
When P=0.066 [kgf] (1kgf=9.8N), δ =53.2[mm](i.e., ηB=δ/L=0.354) and θB = 33.33 [deg] are measured for a two-layered
condition. From Eq.(13), a combined flexural rigidity is as follows.
ηB
⎧
⎫
E1 I 1 + E 2 I 2 = PL ⋅ ⎨
⎬
⎩ 2 sin (θ B 2) ⎭
2
2
0.354 ⎞
= 0.066 × 9.8 × (0.1500) 2 × ⎛⎜
⎟
⎝ 2 × 0.2867 ⎠
= 5.547 × 10 −3
2
(17)
Similarly, δ and θB are measured for a single-layered condition after removing of second (or first) layer. Therefore, a flexural
rigidity is as follows from Eq.(13)
B
ηB
⎧
⎫
E1 I 1 = PL ⋅ ⎨
⎬
⎩ 2 sin (θ B 2 ) ⎭
2
2
(18)
Using Eqs.(15),(16) and the simultaneous equations (17),(18) each Young’s modulus E1, E2 is calculated as E1=3.35[GPa] for
PVC material, E2=205.64[GPa] for SUS material.
2.2 Method 2 [Indirect Process]: (Measurement of δ only)
If the value of the non-dimensional load γ is given by using a certain means, the computation of each Young’s modulus Ei can
be made by using Eq.(14). The usage of the chart is recommended by the author.
2.55
2
n
[= PL /Σ(E
iIi)]
i=1
2
2.65
γ
2.55
n
[= PL /Σ
(EiIi)]
i=1
2.65
γ
A chart (:Nomograph) is given in Fig.3, illustrating the relationship of γ and δ/L [shown in Eq.(8)] in order to facilitate the
calculation of γ. This chart (γ is computed previously by using Eq.(7)) is prepared considering user-friendliness. Using this
chart, each Young’s modulus Ei is calculated from the relational expression given in Eq.(14).
2.60
2.573
2.50
2.60
2.575
2.50
33.33
0.354
0
0.2
ηB
0.4
[= δ L]
Figure 3. Non-dimensional chart for finding the parameter γ
when the parameter δ/L is given.
0
20
θB
40
[deg]
Figure 4. Non-dimensional chart for finding the parameter γ
when the parameter θB is given.
In order to demonstrate the usage of the non-dimensional chart (Ref. Fig.3) each Young’s modulus Ei is obtained for a PVC
thin plate(first layer) with length:2L1=300.0[mm], width: b1=27.0[mm], thickness: h=0.515 [mm] and a SUS thin plate(second
layer) with length:2L2=300.0[mm], width: b1=27.0[mm], thickness: h=0.100 [mm].
When P=0.066 [kgf] (1kgf=9.8N), δ =53.2[mm](i.e., ηB=δ/L=0.354) is measured for a two-layered condition. From Eq.(14), a
combined flexural rigidity is as follows and then, γ is read from Fig.3 (γ = 2.573).
E1 I 1 + E 2 I 2 =
P1 L2
γ
=
0.066 × 9.8 × 0.15 2
2.573
(19)
= 5.656 × 10 −3
Similarly, δ is measured for a single-layered condition after removing of second (or first) layer. Therefore, a flexural rigidity is
as follows from Eq.(14)
2
E1 I 1 =
P1 L
(20)
γ
Using Eqs.(15),(16) and the simultaneous equations (19),(20) each Young’s modulus E1, E2 is calculated as E1=3.42[GPa] for
PVC material, E2=209.35[GPa] for SUS material.
2.3 Method 3 [Indirect Process]: (Measurement of θB only)
A similar chart (:Nomograph) is given in Fig.4, illustrating the relationship of γ and θB [shown in Eq.(7)] in order to facilitate the
calculation of γ. Using this chart each Young’s modulus Ei is calculated from the relational expression given in Eq.(14).
In order to demonstrate the usage of the non-dimensional chart (Ref. Fig.4) each Young’s modulus Ei is obtained for a PVC
thin plate(first layer) with length:2L1=300.0[mm], width: b1=27.0[mm], thickness: h=0.515 [mm] and a SUS thin plate(second
layer) with length:2L2=300.0[mm], width: b1=27.0[mm], thickness: h=0.100 [mm].
When P=0.066 [kgf] (1kgf=9.8N), θB = 33.33 [deg] is measured for a two-layered condition. From Eq.(14), a combined flexural
rigidity is as follows and then, γ is read from Fig.4 (γ = 2.575).
B
Specimen(:column)
LM-Block
Indicator B
x
Pulley
y
Grid paper
Indicator A
LM-Rail
LM-Block
1mm
Base
Load pan
Figure 5. Experimental set-up (As an example, a multi-layered plate specimen is shown).
E1 I 1 + E 2 I 2 =
P1 L2
γ
0.066 × 9.8 × 0.15 2
=
2.575
= 5.654 × 10
(21)
−3
Similarly, θB is measured for a single-layered condition after removing of second (or first) layer. Therefore, a flexural rigidity is
as follows from Eq.(14)
2
E1 I 1 =
P1 L
(22)
γ
Average
150
S.D.=12.24
(S.D.:Standard Deviation)
64.0
[GPa]
200
Experiments
3.5
E
[GPa]
250
E
Using Eqs.(15),(16) and the simultaneous equations (21),(22) each Young’s modulus E1, E2 is calculated as E1=3.41[GPa] for
PVC material, E2=208.31[GPa] for SUS material.
3.4
3.3
Av.:194.93[GPa]
3.2
66.0
P
Experiments
Average
S.D.=0.032
64.0
S.D.=2.56
150
[GPa]
3.5
3.4
3.3
Av.:204.95[GPa]
64.0
3.2
66.0
P
[gf]
S.D.=0.005
64.0
Av.:205.46[GPa]
S.D.=2.38
150
64.0
[GPa]
3.5
3.4
3.3
3.2
66.0
P
66.0
[gf]
(b) Method 2
E
[GPa]
E
200
Av.:3.40[GPa]
P
(b) Method 2
250
[gf]
(a) Method 1
E
[GPa]
E
200
66.0
P
[gf]
(a) Method 1
250
Av.:3.38[GPa]
[gf]
(c) Method 3
Figure 6. Comparison of Young’s modulus
among the three measuring methods
for a steel material (SUS). [1Kgf=9.8N]
S.D.=0.006
64.0
Av.:3.40[GPa]
66.0
P
[gf]
(c) Method 3
Figure 7. Comparison of Young’s modulus
among the three measuring methods
for a high-polymer material (PVC).
Experimental Investigation
In order to assess the applicability of the proposed Compression Column Method, several experiments were carried out using
a two-layered specimen [PVC(Polyvinyl chloride) layer :a high-polymer material + SUS layer : a stainless steel material].
The experimental set-up is shown in Fig.5, which shows a multi-layered plate as an example. The hinge condition is
accomplished by using rotative supports, which are thin steel bars with a 5 mm diameter in this testing method.
Each Young’s modulus of SUS and PVC by applying the Method 1(Direct Process), Method 2(Indirect Process) and Method 3
(Indirect Process) are shown in Figs. 6 and 7, respectively.
In this experiment, a maximum vertical displacement δ at the midpoint of column and/or a deflection angle θB at the end of
column are measured under several compressive loads P by using a grid paper and a protractor.
In a SUS layer (Ref. Fig.6), although Method 1 has a little scattering, the measured values of Method 2 and 3 remain nearly
constant for a compressive load and the deviation is very small. On the whole, the mean Young’s moduli determined by the
three methods are reasonably in good agreement each other.
On the other hand, Trends similar to that of Fig.6 is observed herein for Young’s moduli of a PVC layer (Ref. Fig.7). The mean
values by the three methods agree well mutually.
Conclusions
Effective use of flexible multi-layered materials under a variety of loads requires an understanding of the material properties,
e.g., Young’s modulus of each member.
The Compression Column Method is analyzed theoretically and proposed as a new and simpler material testing method for
measuring each Young’s modulus of flexible multi-layered materials.
The method is based on bending due to postbuckling behavior of an axial compressed long column.
Since no loading device is attached at the mid-region of a specimen, there is no undesirable effect of loading nose comparing
with the three- or four-point bending test. Then, the new method overcomes a difficult gripping problem of a specimen in the
tensile test because of the compression method.
For the sake of convenience, two charts(:Nomographs) are drawn on the basis of the proposed theory.
On the new idea, a set of testing devices was designed, and two-layered materials (a steel material, SUS + a high-polymer
material, PVC) were tested. Theoretical and experimental results clarify that the new method is suitable for measuring each
Young’s modulus of flexible multi-layered materials.
Based on the assessments the proposed method can be further applied to multi-layered thin sheets and multi-layered fiber
materials (e.g., steel belts, glass fibers, carbon fibers, optical fibers, etc.).
Acknowledgements
The author thanks to Mr. M. Takada, Meijo University, Japan, for assistance and also is grateful for the Grant-in Aid for
Scientific Research (C), Japan Society for the Promotion of Science.
References
1.
2.
3.
4.
Ohtsuki, A. and Takada, H., “A New Measuring Method of Young’s Modulus for a Thin Plate / Thin Rod Using the
Compressive Circular Ring,” Transactions of Japan Society for Spring Research, 47, 27-31 (2002).
Ohtsuki, A., “A New Measuring Method of Young’s Modulus for Flexible Materials,” Proceedings of the 2005 SEM Annual
Conference & Exposition on Experimental and Applied Mechanics, Section 72, 113(1)-113(8) (2005)[CD-ROM].
Ohtsuki, A., “A New Method of Measuring Young’s Modulus for Flexible Thin Materials Using a Cantilever,” Proceedings
of the 4th International Conference on Advances in Experimental Mechanics, 3-4, 53-58 (2005).
Ohtsuki, A., “A New Young’s Modulus Measuring Method for Flexible Thin Materials Using Postbuckling Behaviour,”
Fourth International Conference on Thin-Walled Structures, 233-240 (2004).
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