stress analysis of composite material embedded with optical fiber ...

Asia-Pacific Conference on FRP in Structures (APFIS 2007)
S.T. Smith (ed)
© 2007 International Institute for FRP in Construction
STRESS ANALYSIS OF COMPOSITE MATERIAL EMBEDDED WITH OPTICAL
FIBER SENSOR SUBJECTED TO ANTI-PLANE SHEAR
S.C. Her * and B.R. Yao
Department of Mechanical Engineering, Yuan Ze University, Taiwan.
Email: mesch@saturn.yzu.edu.tw
ABSTRACT
The use of smart materials has grown considerably over the past two decades in applications ranging from civil
structure to aerospace vehicle. Fiber optic sensor with small size, light weight and immunity to electromagnetic
interference can be embedded and integrated into the host material to form an ideally smart structure system.
One must recognize that optical fibers are foreign entities to the host structure, therefore will induce high stress
state in the vicinity of the embedded sensor irrespective of the small size of the fiber. This is a result of the
material and geometric discontinuity introduced by the embedded optical fiber. To address this concern, present
paper focuses the attention on constitutent interaction between the optical fiber, coating, matrix and host material.
An analytical model to predict the stress fields in the vicinity of the embedded optical fiber are derived. The
theoretical development is based on the four concentric cylinders model which represents the optical fiber,
protective coating, matrix and host material, respectively. The host material is considered to be a composite with
reinforced fiber parallel to the optical fiber. In this investigation, the host structure is subjected to anti-plane
shear. The effects of the coating and host material on the stress distribution in the vicinity of the embedded
optical fiber are presented through a parametric study.
KEYWORDS
Embedded optical fiber, smart structure, concentric cylinder, in-plane shear, anti-plane shear.
INTRODUCTION
Structures embedded with fiber optic sensor are considered as primary sensing candidates for smart structures to
monitor a wide range of physical quantities such as strain, temperature. Although optical fibers have small
diameter, we must realize that they are foreign entities to the host structure. Therefore, the optical fiber can be
considered as cylindrical elastic inclusion. The discontinuity of material and geometry induce stress and strain
concentration in the vicinity of the embedded optical fiber. Benventiste et al. 1989 predicted the stress field in
composite with coated inclusion based on the average stress in the matrix. Dasgupta et al. 1992 investigated the
resin pocket geometry for stress analysis of optical fiber embedded in laminates composites. Dasgupta and Sirkis
1992 showed that strain concentrations near an embedded optical fiber are highly dependent on the thickness and
material properties of the coating. Case and Carman 1994 studied the compression strength of composites
containing embedded sensors. Eskandari and Carman 1996 predicted the compression strength of a
unidirectional polymer composite embedded with circular fibers. Melin et al. 1997 used moiré inerferometry to
measure the deformation fields around the optical fibers embedded in carbon/epoxy composites. In the present
study, the objective is to calculate the stress field around the embedded optical fiber subjected to anti-plane shear
loading. The effects of the coating and host material on the stress distribution in the vicinity of the embedded
optical fiber are presented through a parametric study.
ANTI-PLANE SHEAR ANALYTICAL MODEL
The optical fiber embedded composites are modeled as four concentric cylinders as shown in Fig.1, represent
optical fiber, coating, matrix and host material, respectively. This geometry is applicable when the optical fiber
is embedded parallel to the reinforcing fibers in the surrounding host composite. The host material is assumed to
be transversely isotropic, while the optical fiber, coating and matrix are considered as isotropic materials. The
interfaces between each constituent are assumed to be perfect bonds, such that the tractions and displacements
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are continuous across the each interface. Fig.2 shows optical fiber embedded composite subjected to anti-plane
shear stressσ xz
= τ
0
in the far field.
Figure 1. Four concentric cylinders model
Figure 2. Anti-plane shear loading
The equilibrium equations for the anti-plane problem can be expressed in terms of displacements as follows
1
u
r u 1
z, rr+ z, r+ 2
uz,
θθ
= 0
r
(1)
Solutions for the partial differential equations are
n An
uz ( r,
θ,
z)
= [ + Bnr]cosθ
r
n = f , i , m ,c
(2)
Where the superscript and subscript n = f , i , m, c refer to optical fiber, coating, matrix and composite,
respectively, throughout the paper. A n
, B
n
are unknown constants associated with each material, can be
determined by the far field and interface continuity conditions. To avoid the displacement singularity at r = 0 ,
let the constant A = 0 . The shear stresses in each material can be written as
f
A
σ
rz
= C
n n
55
[ − + B
n
]cosθ
2
r
(3)
A
σ
θz
= C
55
[ − −B
n
]sinθ
2
r
n
where C 55
represents the constant of stiffness matrix (σ
ij
= C
ijε
j
)associate with each material, in the case
n
of isotropic material, C 55
equals to the shear modulus.
The far field conditions
σ
xz
= τ 0
σ
yz
= 0 r → ∞
can be rewritten in polar coordinate system as follow
σ
rz
= σ
yz
sinθ
+ σ
xz
cosθ
= τ
0
cosθ
(4a)
σ θ z
= σ
yz
cosθ
−σ
xz
sinθ
= −τ
0
sinθ
(4b)
Substituting eq(3) into eq(4) yields
C c 55Bc
= τ 0
(5)
The interface continuity conditions can be expressed as
c
m
c
m
u
z
( rm
) = u
z
( rm
) ; σ
rz
( rm
) = σ
rz
( rm
)
m
i
m
i
u
z
( ri
) = u
z
( ri
) ; σ
rz
( ri
) = σ
rz
( ri
)
(6)
i
u
f
r ) = u ( r ) ;
i
σ
f
r ) = σ ( r )
z
(
f z f
rz
(
i rz f
Where r
m
, r
i
, rf
are the radii of matrix, coating and optic fiber, respectively.
Substituting eq(2) and eq(3) into eq(6), results
Am
Ac
+ B
m
− − B
c
= 0
(7a)
2 2
rm
rm
A i
Am
+ B − − = 0
2 i
B
(7b)
2 m
r r
i
i
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The constants
n
Ai
B
f
− − B
i
= 0 (7c)
2
r
f
− C
A + C B + C A m m m
c c c
55 m
− C B =
2 55 55 2 55 c
0 (7d)
r
r
m
m
− C
A A
i i i
m m m
55
+ C B
i
+ C − C B =
2 55 55 2 55 m
0 (7e)
r
r
i
i
C B C A f
i i i
55 f
+
55 2
− C
55Bi
= 0
(7f)
r
f
A , B can be obtained by solving the system equations (5) and (7). Back substituting these
n
constants into eq(2) and eq(3), yields to the displacements and stresses.
NUMERICAL EXAMPLES
The material properties using in the numerical examples for the optical fiber are E
f
= 431 GPa , ν
f
= 0.25 ; for
matrix are E m
= 2. 89GPa
, ν
m
= 0. 35 . The material properties of coating and composite (host structure) are
varying to investigate the influence of coating and host structure on the stress distribution in the vicinity of
embedded optical fiber. The ratios of radii between optical fiber, coating and matrix are
r
f
: ri
: rm
= 0.768 : 0.847 :1. The radius of the composite is assumed to be infinite i.e. r
c
→ ∞ . The optical
fiber embedded composites is subjected to far field anti-plane shear σ
xz
= τ
0
. The shear stresses of σ
rz
and
0 0 0
σ
θz
along the radial direction for θ = 0 , 45 , 90 are calculated.
To investigate the effect of coating and host material on the stress distribution, two different coatings named
polymide and acrylate, three different host materials named graphite/epoxy, gelion/epoxy and T300/epoxy are
considered as shown in table 1. Thus, total of six different combinations of host material and coating are
obtained as shown in table 2. The stress distributions for these six different combinations are shown in Figs.3 - 8,
0
respectively. The numerical results show that the shear stress σ
rz
is decreasing as angle θ increasing from 0
0
0
0
to 90 , the shear stress σ
θz
is increasing as angle θ increasing from 0 to 90 . Both the maximum shear
stresses σ
rz
and σ
θz
are occurred in the optical fiber. The shear stresses are also increased with the increase of
shear modulus of coating. The shear stresses exhibit the opposite trend as varying the host material, i.e.
decreased with the increase of shear modulus of host material.
Table 1. Material properties of coating and composite
polyimide E = 3 .1GPa
ν = 0. 32
coating
acrylate E = 5 .5 GPa ν = 0. 35
composite
Graphite/epoxy
Gelion70/Epoxy
T300/epoxy
E L
= 149GPa
E T
= 11. 4GPa
G LT
= 4. 5GPa
G TT
= 4. 385GPa ν = 0. 3
LT
E L
= 298. 8GPa
E T
= 4. 3GPa
G LT
= 3. 5GPa
G TT
= 1. 6285GPa ν = 0. 32
LT
E L
= 52. 4GPa
E T
= 50. 6GPa
G LT
= 3. 5GPa
G TT
= 19. 5GPa ν = 0. 06
LT
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Table 2. Six different material combinations
combination Coating composite
1 Polyimide Graphite/epoxy
2 Polyimide Gelion/epoxy
3 Polyimide T300/epoxy
4 Acrylate Graphite/epoxy
5 Acrylate Gelion/epoxy
6 Acrylate T300/epoxy
Figure 3. Stress variation with radius
Figure4. Stress variation with radius
for material combination 1 for material combination 2
Figure 5. Stress variation with radius
Figure 6. Stress variation with radius
for material combination 3 for material combination 4
Figure 7. Stress variation with radius
Figure 8. Stress variation with radius
for material combination 5 for material combination 6
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CONCLUSIONS
In this study, the stress distribution of optical fiber embedded composite subjected to anti-plane shear loadings is
investigated basing on the four concentric cylinders model. The influence of the host material and coating on the
stress concentration is presented through numerical examples.
.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the financial support provided by National Science Council of R.O.C. under
grant no. NSC-93-2212-E-155-007 for this work.
REFERENCES
Benveniste, Y., Dvorak, G.J. and Chen, T. (1989). “Stress fields in composite with coated inclusions”,
Mechanics of Materials, 7, 305-307.
Case, S.W. and Carman, G.P. (1994). “Compression strength of composites containing embedded sensors or
actuator”, J. of Intelligent Material Systems and Structures, 5, 4-11.
Dasgupta ,A. and Sirkis, J.S. (1992). “Importance of coating to optical fiber sensors embedded in smart
structures”, AIAA, 30, 1337-1343.
Dasgupta, A., Wan, Y. and Sirkis, J.S. (1992). “Prediction of resin pocket geometry for stress analysis of optical
fibers embedded in laminated composites”, Smart Mater. Struct., 1, 101-107.
Eskandari, S. and Carman, G. (1996). “Evaluating the influence of fiber coating on the compression strength of a
unidirectional polymer composite”, J. of Composite Material, 30, 1958-1976.
Melin, L.G., Levin K., Nilsson, S., Palmer, S.J. and Rae, P. (1999). “A study of the displacement field around
embedded fibre optic sensors”, Composites: Part A, 30, 1267-1275.
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