Composites Science and Technology 103 (2014) 127–133
Contents lists available at ScienceDirect
Composites Science and Technology
journal homepage: www.elsevier.com/locate/compscitech
Generalized curvature tailoring of bistable CFRP laminates by curing
on a cylindrical tool-plate with misalignment
Junghyun Ryu a, Jong-Gu Lee a, Seung-Won Kim b, Je-Sung Koh b, Kyu-Jin Cho b, Maenghyo Cho a,⇑
a
Smart Structures & Design Laboratory, School of Mechanical and Aerospace Engineering/IAMD, Seoul National University, Gwanak 599 Gwanak-ro, Gwanak-gu,
Seoul 151-742, Republic of Korea
b
Biorobotics Laboratory, School of Mechanical and Aerospace Engineering/IAMD, Seoul National University, Gwanak 599 Gwanak-ro, Gwanak-gu, Seoul 151-742, Republic of Korea
a r t i c l e
i n f o
Article history:
Received 28 February 2014
Received in revised form 18 August 2014
Accepted 24 August 2014
Available online 29 August 2014
Keywords:
A. Carbon fibers
B. Non-linear behavior
C. Laminate theory
C. Residual stress
a b s t r a c t
Inducing an initial curvature is advantageous for tailoring the curvature of bistable Carbon Fiber
Reinforced Prepreg (CFRP) laminate because the final curvature of the laminate can be tailored without
changing its mechanical properties, i.e., bending rigidity, thickness, and weight. However, curvature
tailoring with initial curvature has been limited to tailoring the curvature of only one state of the two
equilibrium states. In this study, we propose a curvature tailoring scheme which can tailor the curvatures
of both equilibrium states by misaligning the laminate and the cylindrical tool-plate for curing. This
method was verified by analysis with the Rayleigh–Ritz method and experiments. In addition, explicit
equations to determine the misalignment angle and tool-plate curvature are derived for curvature tailoring of the bistable CFRP cross-ply laminates. These equations provide a simple engineering guideline for
designers of bistable CFRP cross-ply laminate. The proposed curvature tailoring method gives the
designer the ability to select the curvature of both equilibrium states without changing its mechanical
properties, and increasing the functionality of bistable CFRP laminates.
Ó 2014 Published by Elsevier Ltd.
1. Introduction
Unsymmetric lay-up sequence causes the curvature of laminate
after curing due to the difference of the thermal expansion coefficient between layers. If the side length of the laminate is short, it
deforms to the saddle shape, which is predicted by the classical
lamination theory. If the side length of the laminate exceeds
certain threshold value, however, it shows bistable phenomenon.
There are two stable modes after curing the bistable CFRP
laminates. Two different cylindrical shapes can be obtained by
inducing external force to generate snap-through action from one
equilibrium state to the other [1–3].
Rayleigh–Ritz method with von Karman nonlinearity is proposed to handle this phenomenon. Originally, Hyer [1,3] proposed
the model for cross-ply laminate which is based on the polynomial
displacement field and von-Karman non-linearity. This analytical
modeling technique proposed by Hyer has been extended to various bistable panel problems, such as approximation of bifurcation
temperature/side length [4–6], deformation behavior of angle-ply
laminate [6,7], slippage effect between laminate and tool plate
⇑ Corresponding author.
http://dx.doi.org/10.1016/j.compscitech.2014.08.024
0266-3538/Ó 2014 Published by Elsevier Ltd.
[8,9], identification of snap-though force [10,11], and initial
curvature effect [12–14].
Bistable structure is an attractive design component for morphing structures which requires a compact, lightweight, and energy
efficient shape-changing mechanism. Bistable structures do not
require an energy supply to maintain their deformed shapes in
their stable states. Moreover, shape transition from one stable state
to another propagates automatically by residual stress if the deformation reaches a certain threshold point. Slap bracelets, selfretracting tape measures, and the Venus-flytrap robot in [15] are
examples of the application of bistable structures.
The main objective of this study is the general curvature tailoring of bistable CFRP laminate using a cylindrical tool-plate. The
curvature of a bistable CFRP laminate is a key feature as a design
component of a morphing structure because it is closely related
with efficiency and functionality of the application. The lay-up
sequence is the crucial factor determining the curvature because
the curvature is a function of the thermal strain difference along
the thickness of the laminate. However, the lay-up sequence is
closely related with the mechanical properties of laminate such
as bending rigidity, thickness, and weight. Curvature tailoring
schemes, which minimize the influence on these properties, is an
essential research topic because these properties are important
design requirements of various applications.
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J. Ryu et al. / Composites Science and Technology 103 (2014) 127–133
Inducing initial curvature in bistable CFRP laminates by curing
on a curved tool-plate satisfies the requirement for a curvature
tailoring scheme described above because it allows tailoring of
the curvature without changing the lay-up sequence of the laminate. The effect of initial curvature on bistable structures has been
discussed by Ren et al. [12], Pirrera et al. [13], and Ryu et al. [14].
Ryu et al. [14] shows that the final curvature of a bistable CFRP
cross-ply laminate can be easily predicted because the final curvature of the laminate with initial curvature can be expressed as the
sum of the tool-plate curvature and final curvature of the laminate
without initial curvature. However, initial curvature was limited to
tailoring the curvature of only one state of the two equilibrium
states [12–14]. Therefore, the curvature of the other state mode
cannot be tailored in the bistable CFRP cross-ply laminates as
shown in Fig. 1.
Inducing the initial curvature using saddle-shaped or spherical
tool-plates appears to be a plausible generalization of the previous
study, because both of the normal curvatures in these cases are
non-zero without twisting curvature. However, putting the
laminate on such tool-plates is impossible. It requires large values
of extension and compression because such tool-plates are
non-developable surfaces, whereas the prepreg is the developable
surface. As a result, saddle or sphere shape tool-plates for curvature tailoring are impractical solutions.
A misalignment angle between the laminate and tool-plate is
proposed as a general curvature tailoring scheme. Putting the laminate on a cylindrical tool-plate does not induce large values of
extension and compression because the cylindrical tool-plate and
flat prepreg are developable surface while the non-zero initial normal curvatures are imposed to the laminate by the misalignment
angle. Detail formulation of this generalization is described in the
following paragraphs.
In this paper, a curvature tailoring scheme using a misalignment angle between the tool-plate and laminate, illustrated in
Fig. 2, is proposed to overcome the limitation described above.
The curvature of both stable states can be tailored while minimizing the effects on other processing and design properties of the
bistable CFRP laminate, such as bending rigidity, thickness and
density of the lamina. Detailed formulation of the generalized initial curvature effect is presented in Section 2. The analysis by the
proposed model and its verification with experiments is presented
in Section 3. The simulations are focused on the cross-ply laminate
in which we derive simple equation to predict final curvatures of
Fig. 2. Method of laying bistable CFRP laminate on a cylindrical tool-plate. (a) Fiber
direction coincides with the principal curvature direction of the tool-plate. (b) Fiber
direction is misaligned with the principal curvature direction of the tool-plate.
the bistable CFRP structure because initial bending curvature can
be decoupled from twisting curvature.
2. Analytical model development
To describe the effect of initial curvature with misalignment,
the Green-Lagrangian strain field proposed in a previous study
[14] must be generalized. The key feature required to describe
the initial curvature effect in the previous study is the rearrangement of Green-Lagrangian strain field, which is described in
Eq. (1); a flat reference state is introduced to define the final strain
field and define the initial strain, i.e. a kind of inelastic strain. This
concept is illustrated in Fig. 3.
Eij ¼
1
1
1
ðg Gij Þ ¼ ðg ij dij Þ ðGij dij Þ ¼ Efianl
Einitial
ij
ij
2 ij
2
2
ð1Þ
where gij is the metric at final state and Gij is the metric at initial
state.
A trigonometric displacement field, the basis of the GreenLagrangian strain field for constant curvature assumption, provides
more accurate results than a quadratic displacement field. [2] The
Green-Lagrangian strain field to describe initial curvature without
the misalignment angle between the bistable CFRP cross-ply
laminate and tool-plate is described by:
2
Exx
6
4 Eyy
2Exy
82
>
<
7
6
5¼ 4
>
:
3
nxx þ kxx y2
nyy þ kyy x2
ðjxx jyy þ 2kxx þ 2kyy Þxy
3
3 9
2
>
jxx jinitial
xx
=
7 6
7
5 4 jyy jinitial
5z
yy
>
0
;
ð2Þ
where nxx, nyy, nxy, means constant strains, kxx, kyy, means constants
for quadratic normal strain variation, jxx, jyy, means principal
Fig. 1. Initial curvature effect with cylindrical tool-plate.
J. Ryu et al. / Composites Science and Technology 103 (2014) 127–133
129
Fig. 3. Reference state definitions to describe initial and final strain field.
curvature values after curing, and jinitial
, jinitial
means initial curvaxx
yy
ture values.
The Green-Lagrangian strain field in Eq. (2) does not contain
constant shear strain and twisting curvature because it is applied
to bistable CFRP cross-ply laminate and the non-zero curvature
direction of the tool-plate is aligned with the fiber direction. However, as shown in Eq. (3), the misalignment angle u has an additive
effect the both normal curvatures that have non-zero values,
although the tool-plate has only one non-zero principal curvature
value. A graphical illustration of the coordinate system is shown in
Fig. 4. It should be noted that the reference of the coordinate system is the fiber direction of the laminate; a fiber angle of 0° means
that the fiber direction is aligned to the x-direction and a fiber
angle of 90° means that the fiber direction is aligned to y-direction.
Positive curvature indicates convex shape. As a result, the non-zero
tool plate curvature, j, in Fig. 4(a) has a negative value. Initial
curvature tensor with misalignment angle u is described by:
"
initial
j
j sin2 u
j sin u cos u
¼
j sin u cos u
j cos2 u
#
ð3Þ
where misalignment angle between principal curvature directions
of tool plate and material axis of the laminate, u, is illustrated in
Fig. 4(b).
Constant shear strain and twisting curvature should be considered in the strain field because the initial twisting curvature can
affect not only the final twisting curvature but also shear strain.
However, it should be noted that the constant curvature assumption is still valid because the tool plate curvature, j, is constant
in the overall domain of the laminate. As a result, the final shape
of the bistable CFRP laminate is cylindrical because this is the only
surface that can satisfy the requirement for a constant curvature
surface and developable surface.
In this sense, using a strain field for an unsymmetric angle-ply
laminate [7] is an efficient scheme to accomplish this generalization because it can describe the degrees of freedoms – constant
shear strain and twisting curvature – with a minimum of
unknowns. Combining this scheme with the initial curvature
formulation, Eq. (1), describes the general deformation behavior
of a bistable CFRP laminate with sufficiently long side length. The
generalized Green-Lagrangian strain field is described by:
2
3 2 2
3
nxx
m
n2
mn
6
7 6 7 6
7
m2
mn 5
4 Eyy 5 ¼ 4 nyy 5 þ 4 n2
nxy
2Exy
2mn 2mn m2 n2
02
3 2 3 1 2 initial 3
jxx
j1
kxx y02
B6
7 6 7 C 6 jinitial 7
@4
kyy x02
5 4 j2 5zA þ 4 yy 5z
0
ðj1 j2 þ 2kxx þ 2kyy Þx0 y0
2jinitial
xy
Exx
3
2
ð4Þ
where nxx, nyy, nxy, means constant strains kxx, kyy, constants for quadratic normal strain variation, j1, j2, principal curvature values,
and h means angle difference between principal curvature direction
and material coordinate. In addition, m = cos h, n = sin h, x0 = mx + ny,
and y0 = nx + my.
Calculation of the unknown variables in Eq. (4), nxx, nyy, nxy, kxx,
kyy, j1, j2, and h is the same as in previous study [2]. Minimizing
Fig. 4. Initial curvature change by misalignment. (a) Fiber direction coincides with the principal curvature direction of the tool-plate. (b) Fiber direction is misaligned with the
principal curvature direction of the tool-plate.
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J. Ryu et al. / Composites Science and Technology 103 (2014) 127–133
the strain energy function using the Newton–Raphson iteration
procedure, the equilibrium states can be determined by obtaining
the unknown variables in Eq. (4). The Strain energy function with
proposed strain field and thermal strain is described by:
U¼
1
2
Z
C ijkl ðEij aij DTÞðEkl akl DTÞdV
2
B11
0
0
0
6B
6 11
6
6 0
6
6 0
6
6
4 0
D11
0
0
0
0
A22
B22
0
0
B22
D22
0
0
0
0
A66
6
0 7
76 jxx
76
0 76 nyy
76
6
0 7
76 jyy
76
B66 54 nxy
0
0
0
B66
D66
0
V
¼ U normal þ U shear þ U coupling
ð5Þ
where Unormal: (i = j & k = l), Ushear: (i – j & k – l), Ucoupling: other
indices, detail expressions of Unormal, Ushear, and Ucoupling is
described in Appendix A.
The proposed strain field which is described in Eq. (4) can handle the initial curvature effect with misaligned fiber angle configuration for angle-ply as well as cross-ply laminate. The final
curvature prediction from the strain field given in Eq. (4) cannot
be made in simple explicit manner because final curvature direction can be changed when lay-up sequence changes or initial curvature in Eq. (3) changes.
In the case of cross-ply laminates, however, explicit prediction
of final curvature can be made even including the effect of initial
curvature with misalignment angle. Final curvature direction of
unsymmetric cross-ply laminate should be aligned with fiber
directions because the difference of the thermal expansion coefficients between layers is maximized along that directions and they
are orthogonal to each other. As a result, strain field in Eq. (4) can
be reduced to the following compact equation for the cross-ply
laminate configuration because m = 1 and n = 0. The reduced
Green-Lagrangian strain filed for cross-ply laminate is described
by:
3 2
3 2
3 2
3
j1 jinitial
nxx
Exx
kxx y2
xx
6
7 6n 7 6
7
6
initial 7
kyy x2
4 Eyy 5 ¼ 4 yy 5 þ 4
5 4 j2 jyy 5z
nxy
2Exy
ðj1 j2 þ 2kxx þ 2kyy Þxy
2jinitial
xy
0
32
A11
nxx
3
2
N1
3
7 6M 7
7 6 17
7 6
7
7 6 N2 7
7¼6
7
7 6M 7
7 6 27
7 6
7
5 4 N6 5
2jxy
ð8Þ
M6
Reduced stain field in Eq. (6) can be reduced further as the side
length of the laminate goes to infinite because the deformed shape
yield to a perfect cylinder, i.e. one of jxx or jyy vanishes. As a result,
the number of variable is reduced to five, nxx, nyy, nxy, jxy, and either
of jxx or jyy. The simplified strain field for cross-ply laminate with
infinite side length yield to:
82
3 2
3
>
j1 jinitial
xx
>
> nxx
>
6
7 6 jinitial 7
>
>
5z for Mode 1
>
yy
> 4 nyy 5 4
>
>
< nxy
2jinitial
xy
E¼ 2
3 2
3
>
jinitial
nxx
>
xx
>
>
>6
7 6
initial 7
>
>
4 nyy 5 4 j2 jyy 5z for Mode 2
>
>
>
: n
initial
2jxy
xy
ð9Þ
If the laminate shows bistability, either one of normal moment
balance equation cannot be satisfied because the deformed shape
is cylinder. As a result, explicit approximation equation for final
curvature and constant strains for each stable states are derived
and expressed by:
2
ð6Þ
Final curvatures, j1 and j2, can be predicted in explicit manner
for the following reasons. First, minimizing strain energy in Eq. (5)
for the cross-ply laminates is the sum of minimal normal strain
energy and minimal shear strain energy, i.e. coupling strain energy,
Ucoupling, is zero. Second, the principal curvature values, j1 and j2,
become independent of the shear strain. It is because second terms
of in-plane shear strain in Eq. (6) yields to zero as the side length of
the laminate increases [5,6]. Therefore, principal curvature values
of the bistable laminate, j1 and j2, can be determined by minimizing the normal strain energy only. It should be noted that normal
strain field in Eq. (6) is same as that of Eq. (2).
In this sense, explicit approximation scheme in previous study
[14] is still valid. The detailed derivation is given below. First,
force/moment balance equations for classical lamination theory
are given by:
2
A11
6B
6 11
6
6 A12
6
6B
6 12
6
4 0
0
B11
D11
A12
B12
B12
D12
0
0
B12
A22
B22
0
D12
B22
D12
0
0
0
0
A66
0
0
0
B66
32
3 2
3
nxx
0
N1
7
7
6
6
0 76 jxx 7 6 M 1 7
7
76
7 6
7
0 76 nyy 7 6 N2 7
76
7¼6
7
7 6
7
6
0 7
76 jyy 7 6 M 2 7
76
7 6
7
5
5
4
4
nxy
B66
N6 5
2jxy
D66
M6
ð7Þ
where the stiffness expression, Aij, Bij, Dij, and force and moment
resultant Ni, Mi, are described in Appendix B.
In the case of cross-ply CFRP laminate, Eq. (7) can be approximated to Eq. (8) by neglecting the normal strain coupling terms
because fiber stiffness, E1, is much higher than matrix stiffness,
E2. The simplified force/moment balance equations for cross-ply
laminate are given by:
2
A11 B11
0
0
0
32
nxx
3
2
0
3
2
N1
3
N1
3
76
7 6
7
6
6
initial 7
6 B11 D11 0 0 0 76 j1 jxx 7
6 0 7 6 M1 7
76
7
7 6
7
6
6
initial 6
7
7
7
6 0 0 A22 0 0 76
6
nyy
76
7 þ jyy 6 B22 7 ¼ 6 N 2 7
6
76
7
7
7
6
6
6
nxy
5
4 0 0 0 A66 B66 54
4 0 5 4 N6 5
initial
2jxy
0
0 0 0 B66 D66
M6
for Mode 1ðj2 ¼ 0Þ
2
A11 0 0 0
6
6 0 A22 B22 0
6
6 0 B22 D22 0
6
6
4 0 0 0 A66
0 0 0 B66
0
32
nxx
nyy
3
2
76
7
6
76
7
6
76
7
6
initial
initial
76 j2 jyy 7 þ j
6
xx
76
7
6
76
7
6
nxy
B66 54
5
4
2jinitial
D66
xy
0
0
B11
3
2
7 6
7
7 6 N2 7
7 6
7
7 ¼ 6 M2 7
7 6
7
7 6
7
0 5 4 N6 5
0
M6
0
0
for Mode 2ðj1 ¼ 0Þ
ð10Þ
Based on this, we can generalize the conclusion of the previous
study on the effect of initial curvature to the following statement:
final non-zero normal curvature of a bistable CFRP cross-ply laminate cured on a curved tool-plate is the sum of the tool-plate normal curvature and final non-zero normal curvature of the laminate
cured on a flat tool-plate. As a result, initial curvature effect to the
final curvature with misalignment angle can be summarized to:
j1 ¼ jflat þ jtool cos2 u j2 ¼ 0
Mode 2 : j1 ¼ 0
j2 ¼ jflat þ jtool sin2 u
Mode 1 :
ð11Þ
As a result, a manufacturer of bistable CFRP laminate can easily
tailor the final curvature of a bistable CFRP cross-ply laminate if he
or she knows the final non-zero normal curvature of the laminate
when cured on a flat tool-plate, jflat.
J. Ryu et al. / Composites Science and Technology 103 (2014) 127–133
131
3. Numerical simulations and verification
3.1. Numerical simulations
In order to verify the proposed strain field described in the previous section, numerical simulations for CFRP cross-ply laminate
with initial curvature are performed. CFRP from HANKUK CARBON
CO. LTD. was used for bitable CFRP cross-ply laminate; the material
properties are listed in Table 1. Final curvature of a bistable CFRP
cross-ply laminate with and without misalignment angle as a function of side length change is illustrated in Fig. 5(a) and (b). The final
curvatures of both stable states are changed when an initial curvature with misalignment angle is induced while only one final curvature of stable state is changed when misalignment angle is zero.
Fig. 6 shows the final curvature changes in bistable CFRP crossply laminates when the side lengths of the laminates were fixed to
150 mm 150 mm and the radius of curvature of the tool-plate is
200 mm. It shows that Eq. (11) is valid, i.e. the final curvature
change is proportional to the tool plate non-zero curvature when
the misalignment angle is fixed.
3.2. Experiments
To validate the Eq. (6), principal curvature direction is aligned
with fiber direction although initial twisting curvature is applied,
with experiment, bistable CFRP cross-ply laminates are cured with
Table 1
Material properties of CFRP.
Quantity
Axial tensile modulus
Transverse tensile modulus
Shear modulus
Major Poisson’s ratio
Mass per unit length
Density
Carbon fiber weight
Resin weight
Scrim weight
Thickness
Thermal expansion coefficient
parallel to fiber direction
Thermal expansion coefficient
perpendicular to fiber direction
Temperature change
Unit
Value
E1
E2
G12
GPa
GPa
GPa
a1
g/cm3
g/mm2
g/mm2
g/mm2
g/mm2
mm
°C1
160
12
8
0.3
800
1.8
54
38
34
0.085
0.19 106
a2
°C1
0.32 104
DT
°C
145
Fig. 6. Predicted final curvature of a bistable CFRP cross-ply laminate as a function
of misaligned angle (Rinitial = 150 mm).
two different misalignment angles, 0° and 45°. The shape of the
composites before curing was perfectly square with a length of
150 mm. Two square CFRP plies were perfectly aligned [0°/90°]
and cured at 170C with a curing time of two hours and bagging
pressure of 1 atm. Fig. 7 shows that twisting curvature and inplane shear strain are negligibly small and the equilibrium shape
is very close to cylindrical even with the misalignment angle.
To validate Eq. (11), the change of the final curvatures of two
stable states as a function of misalignment angle was measured
experimentally. The radius of curvature of the tool-plate was fixed
at 300 mm. Three different misalignment angles, 15°, 30°, and 45°
were tested for the lay-up sequences [0°/90°] and [90°/0°], and
three specimens were cured for each misalignment angle. The final
curvatures of the specimens as a function of misalignment angle
and their scattering are given in Fig. 8.
Tool plate curvature is measured based on the assumption that
the final curvature of the laminate is almost constant. Curvature
equation as a function of chord length and height of the laminate
is given by:
j¼
8d
2
C 2 þ 4d
ð12Þ
Fig. 5. Final curvature of a bistable CFRP cross-ply laminate as a function of side length change. (a) The fiber direction coincides with the principal curvature direction of toolplate. (b) The fiber direction is misaligned with the principal curvature direction of the tool-plate (45°).
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J. Ryu et al. / Composites Science and Technology 103 (2014) 127–133
Fig. 7. Deformation behavior of bistable composite with and without misalignment angle (Rinitial = 200 mm, u = 45°).
Fig. 8. Final curvature
(Rinitial = 300 mm).
changes
as
a
function
of
misalignment
angle
where C is the length of chord and d is the height of laminate after
curing.
The model predictions agree well with the experimental
measurements. It should be noted the [90°/0°] lay-up sequence
with misalignment angles 15°, 30° and 45° is equivalent to the
[0°/90°] lay-up sequence with misalignment angles 75°, 60° and
45°, respectively.
3.3. Design guideline
For the designer, schematic diagram of the procedure to determine the misalignment angle and initial tool plate curvature in
order to meet the required final laminate curvatures j1 and j2 is
presented in Fig. 9. The key equations in Fig. 9 are given in Eq.
(13) which is the rearrangement of Eq. (11). These equations are
useful for designing the final curvatures of a bistable CFRP crossply laminates using misalignment angles. Misalignment angle as
a function of final curvatures j1, j2 and misalignment angle u is
given by:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
j2 þ jflat
j1 jflat
j1 jflat j2 þ jflat
¼
¼
2
cos2 u
sin u
final curvature of the bistable CFRP cross-ply laminate without
initial curvature, jflat, is 18 (1/m); The final curvature of the bistable CFRP cross-ply laminate without initial curvature can be
obtained by either simulation or the experiment. In this case, we
can calculate the misalignment angle and tool plate curvature in
an explicit manner by following the procedure shown in Fig. 9.
First, the misalignment angle is 35.2644°, from the first row of
the Eq. (13). Next, the amount the tool-plate curvature is 9 (1/m)
based on the second row of Eq. (13). Thus, we can satisfy the design
requirements for the final curvatures of a bistable CFRP laminate
with a cylindrical tool-plate alone.
4. Conclusion
u ¼ tan1
jtool
Fig. 9. Schematic diagram of the procedure to determine the misalignment angle
and initial tool plate curvature in order to meet the required final laminate
curvatures j1 and j2.
ð13Þ
Based on Eq. (13) and Fig. 9, the tool curvature and initial angle
between the fiber direction of the laminate and x-axis are predicted for the desired final curvature of a bistable CFRP cross-ply
laminate. These can be calculated systematically. For example,
suppose that the design requirements for the final curvatures of
a bistable CFRP cross-ply laminate are 24 (1/m) in the x-direction,
j1, and 15 (1/m) in the y-direction, j2, and the magnitude of the
To control the final curvatures of both stable states of a bistable
CFRP laminate with a cylindrical tool-plate, the effect of misalignment between fiber direction and principal curvature directions of
the tool-plate has been studied. Through experiment and a developed analytical model, we verified that the final curvature of a
bistable CFRP cross-ply laminate is the sum of the normal curvature of the tool-plate and final curvature of the laminates without
initial curvature. We expect that the results of this study will provide convenient guideline for designers seeking to tailor smart
structures with bistable CFRP laminates.
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J. Ryu et al. / Composites Science and Technology 103 (2014) 127–133
2
Acknowledgements
This work was supported by the National Research Foundation
of Korea (NRF) Grant funded by the Korea Government (MSIP) (No.
2012R1A3A2048841 and 2012-041247).
B22
B31
B32
D11
D12
6
4 D21
D22
D31
D32
unormal ¼ C xxxx ðExx axx DTÞðExx axx DTÞ þ 2C xxyy ðExx axx DTÞ
U normal
ðEyy ayy DTÞ þ C yyyy ðEyy ayy DTÞðEyy ayy DTÞ
R
¼ 12 V unormal dV
2
ðA-1Þ
where Eij is Green-Lagrangian strain tensor component, aij is thermal expansion coefficient component, DT is the temperature
change during curing process, and unormal, ushear, ucoupling are strain
energy density functions.
Appendix B. Detail expression of stiffness expression and force/
moment resultant
6
4 T 21
T 31
T 22
T 32
T 13
3
2
2
cos2 h
sin h
2 cos h sin h
3
7 6
7
T 23 5 ¼ 4 sin2 h
cos2 h
2 cos h sin h 5
2
2
T 33
cos h sin h cos h sin h cos h sin h
ðB-1Þ
where h is angle between fiber direction and x-axis.
2
ðiÞ
Q
6 11
6 Q ðiÞ
4 12
0
ðiÞ
Q 12
ðiÞ
Q 22
0
2
3 2 E1
T 11 T 12 T 13
1m12 m21
7 6
76
6 m12 E2
¼
T
T
T
4
5
0 7
21
22
23
5
4 1m12 m21
ðiÞ
T 31 T 32 T 33
Q 66
0
2
3
T 11 T 21 T 31
6
7
4 T 12 T 22 T 32 5
0
3
T 13
T 23
m12 E2
1m12 m21
E2
1m12 m21
0
0
3
7
0 7
5
G12
T 33
ðB-2Þ
2
A11
6
4 A21
A31
A12
A22
A32
3
2
ðiÞ
Q 11
n
7 X6
ðiÞ
6
A23 5 ¼
4 Q 12
i¼1
A33
0
A13
ðiÞ
Q 12
ðiÞ
Q 22
0
0
6 7
4 N2 5 ¼
2
ðA-3Þ
T 12
3
3
7 ztop
i
0 7
5zjzbtm
i
ðiÞ
Q 66
ðB-3Þ
M1
3
D13
n 6
X
3
ðiÞ
ðiÞ
Q 11
2
3
Q 12
i¼1
ðiÞ
Q 22
ðiÞ
Q 66
0
ðiÞ
ðiÞ
Q 11
Q 12
6 ðiÞ
6 Q 12
4
Q 22
n 6
X
0
ðiÞ
Q 12
ðiÞ
0
0
7 ztop
7 2
0 7z zibtm
5 i
ðiÞ
Q 66
0
ðiÞ
Q 12
0
0
ðB-4Þ
3
7 ztop
7 3
0 7z zibtm
5 i
ðiÞ
Q 66
ðiÞ
Q 22
ðiÞ
xy
3
0
ðiÞ
Q 22
32
0
aðiÞ
76 xx
76 ðiÞ
0 74 ayy
5
ðiÞ
Q 11
6 ðiÞ
6 Q 12
4
i¼1
0
M6
2
n 6
7 1 X6 ðiÞ
D23 5 ¼
6Q
3 i¼1 4 12
D33
0
2
6
7
6 M2 7 ¼ 1
4
5 2
ðiÞ
Q 11
n 6
7 1 X6 ðiÞ
B23 5 ¼
6Q
2 i¼1 4 12
B33
0
B13
ðA-2Þ
ucoupling ¼ 2C xxxy ðExx axx DTÞðExy axy DTÞ þ 2C yyxy ðEyy ayy DTÞðExy axy DTÞ
R
U coupling ¼ 12 V ucoupling dV
T 11
N1
N6
ushear ¼ 4C xyxy ðExy axy DTÞðExy axy DTÞ
R
U shear ¼ 12 V ushear dV
2
B12
6
4 B21
2
Appendix A. Detail expression of strain energy
B11
2
ðB-5Þ
3
7 ztop
7zj ibtm
5 z
ðB-6Þ
i
2a
32
ðiÞ
axx
3
76
7
76 ðiÞ 7 2 ztop
0 76 ayy 7z zibtm
54
5 i
ðiÞ
ðiÞ
2axy
Q 66
ðB-7Þ
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