Materials Sciences and Application, 2011, 2, 1027-1032
doi:10.4236/msa.2011.28139 Published Online August 2011 (http://www.SciRP.org/journal/msa)
1027
Crack-Tip Stress Analysis at a Bi-Material
Interface by Photoelastic, Isopachic and FEA
George A. Papadopoulos1*, Elen B. Bouloukou1, Elen G. Papadopoulou2
1National Technical University of Athens, Department of Engineering Science, Section of Mechanics, Zografou campus, Athens,
Greece; 2National Technical University of Athens, Department of Mining Engineering and Metallurgy, Laboratory of physical Met-
allurgy, Athens, Greece.
Received April 30th, 2011; revised May 25th, 2011; accepted June 2nd, 2011.
ABSTRACT
The paper investigates the stress state at the bi-material interface crack-tip by the Photoelastic and Isopachic methods
and the Finite Element Analysis (FEA). The principal stresses at the bi-material interface crack-tip are theoretically
determined using the combination photoelastic and isopachic fringes. The size and the shape of crack-tip isochromatic
and isopachic fringes, at a bi-material interface under static load, are studied. When the crack-tip, which is perpen-
dicular to interface, is placed at the interface of the bi-material, the isochromatic and the isopachic fringes depend on
the properties of the two materials. Thus, the isochromatic and the isopachic fringes are divided into two branches,
which present a jump of values at the interface. The size of the two branches mainly depends on the elastic modulus and
the Poisson’s ratio of the two materials. From the combination of the isochromatic and the isopachic fringes, the prin-
cipal stresses σ1 and σ2 can be estimated and the contour curves around the crack-tip can be plotted. For the FEA
analysis, the program ANSYS 11.0 was used. The bi-material cracked plates were made from Lexan (BCBA) and Plexi-
glas (PMMA).
Keywords: Crack, Bi-Material, Photoelasticity, Isochromatic, Isopachic, FEA
1. Introduction
The study of the behaviour of a transverse crack propa-
gating through the mesophase of composites has become
a subject of great interest. The problem of crack propaga-
tion in a duplex plate was studied by Williams et al. [1-5]
and later was approached by Dally and Kobayashi [6] by
means of dynamic photoelasticity. Theocaris et al. [7-9]
have studied the influence of both the mesophase and the
material characteristics of either phase, in biphase plates
consisting of different materials, on the stress distribution
around the crack tip. They have extended their study to
the magnitude and the variation of the crack propagation
velocities during fracture in duplex plates under dynamic
loading [10]. Also, Theocaris et al. [11,12] have studied
the influence of the hard or soft fiber and the mesophase
layers in a soft-hard-soft or hard-soft-hard combination of
biphase plate subjected to a dynamic tensile load, on the
fracture mode and bifurcation process in both phases.
Also, theoretical studies on this subject were carried out
by Gdouto s et al. [13- 15] and Theo tokoglou et al. [16,17].
The study of the size and the shape of crack-tip caustics
at a bi-material interface was carried out by Papadopou-
los [18]. Sadowski et al. [19,20] have studied the bi-material
interface problem for ceramic materials by FEA. In the
present paper, we estimate the principal stresses and its
contour curves at the bi-material interface crack-tip from
the combination of the isochromatic and the isopachic
fringes and the FEA.
2. Crack-Tip Stresses at the Interface
Two plates of module E1 and E2 and Poisson’s ratio ν1
and ν2 are perfectly bonded along their common interface
(Figure 1). In plate (1) there is a crack perpendicular to
the interface. The crack-tip is placed exactly at the inter-
face of the two plates. The Airy stress function X(r,θ) for
this problem, used by Zak and Williams [5], is:
1
(, )( )Xrr F




 (1)
with:




 
sin1cos 1
sin 1cos 1
Fb
cd

 



 
 (2)

Crack-Tip Stress Analysis at a Bi-Material Interface by Photoelastic, Isopachic and FEA 1028
where ,,,bc d

are constants and

takes values be-
tween 0 and 1, which is depended on the ratio E12 = E1/E2
of the two plates moduli [18].
From the stress function (1) the polar stresses at the
crack-tip are taken:




1
1sin 1
1cos1
3sin 1
3cos1
r
b
r
c
d

 

 





 
 
 











(3)





1
sin 1
cos 1
1sin 1
cos 1
b
r
c
d




 





 
 

 




(4)




1
1cos1
1sin 1
1cos1
1sin 1
r
b
r
c
d


 

 







 














(5)
3. Specimens
Specimens were made from Plexiglas (PMMA) with
Poisson ratio ν = 0.34 and Elasticity modulus E = 3.4
GPa and Lexan (BCBA) with ν = 0.36 and E = 2.8 GPa,
subjected to static uniaxial tension. The specimens thick-
ness was t = 0.003 m. The crack width was 0.3mm with a
sharp tip at the interface (Figure 1).
4. Theory of Photoelasticity
Isochromatic fringes are loci of points with the same
Figure 1. Geometry of bi-material plate
value fors or the the difference of the principal stresse
maximum shear stress. According to the stress optical
law, the difference in the principal stresses is given by
Frocht [21] :
max
12
2cc
Nf
t
   (6)
where is the isochromatic fringe order, is the
obtained:
c
N
ss otgthickne f the plate and c
f is the material frine value
or stress-optical constant.
From equations (3)-(6) is

 
1
1
1,2
1/2
22
22
(1)()( )4
cc
Nf
r
FFF t

 













  







(7)
The relation between the stress-optical constants of
m the aterials 1 and 2 is:
 
1
2
12 2
1
cc
1
f
f
E


 (8)
where 121 2
/EEE

is the ratio of the elastic modulii
and 1

, 2

are the Poisson’ s ratio of the two materials.
5. Theory of Isopachic Fringes
Isopachics fringes are loci of points with the same value
for the sum of the principal stresses. The fringe order
p
N of isopachic is related to the sum of the principal
sses by: stre
12
p
p
r
Nf
t

   (9)
where
p
N
late is the order of isopachics, t is the thickness of the p and
p
f
is the isopachic fringe constant.
The sum of theresses (equations (3) and (4)) is: st
12
(1)() ()rFF

 

12 r


 
 

 (10)
From equations (9) and (10) is obtained:
1
1
1,2
2
()
1
(1)() ()
pp
Nf
rt
FF











(11)
with:
1
21
12 2
1
() ()
p
p
f
f
E


 (12)
where are the isopachic fringe constants of the
Thystem
1,2
()
p
f
materials 1 and 2 of the bi-material plate, respectively.
6. Principal Stresses Estimation from the
Isochromatic and Isopachic Fringes
e principal stresses can be estimated from the s

Crack-Tip Stress Analysis at a Bi-Material Interface by Photoelastic, Isopachic and FEA1029
(6) and (9) system,
th
of isochromatic and isopachic fringes (Eqs (6), (9)). The
solution of the system is valid at the cross points of the
isochromatic and isopachic fringes (Figure 2). Figure 2
presents the overlapping of isochromatic and isopachic
fringes by Eqs (7) and (11), for Lexan 1-Plexiglas 2
bi-material plate with E12 = 0.82353, ν1 = 0.36, ν2 = 0.34,
λ = 0.5192, t = 0.003 and d2 = 1 (d2 is an arbitrary con-
stant which represents the ten sile load of the plate. At the
cross points of the fringes the principal stresses can be
calculated by the Eqs (6) and (9).
From the solution of the equations
e principal stresses are obtained:



22
1
1,2
1,2
1
r 
 4
22
2
rr
pp cc
Nf Nf
t


  
 (13)

22
2
1,21,2
14
22
() ()
2
r
rr
pp cc
Nf Nf
t







 
(14)
By substituting the stresses from the Eqs (3)-(5) into
Eqs (13), (14) the contour curves of the principal stresses,
around the crack-tip, are obtained:
1
1
1
1,2 1,2
2
2
2
22
() ()
(1)( )()
(1)( )( )4(())
pp cc
Nf Nf
t
r
FF
FF
F


 
 

1 /2








 


  




 















(15)
2
1

1
1,2 1,2
2
2
21/2
22
() ()
(1)()( )
(1)( )()4(())
pp cc
Nf Nf
t
r
FF
FF
F

 
 









  

  




 


(16)
Figure 3 presents the contour curves of the prin
st
)
finite element
roperties for Lexan
cipal
resses σ1 and σ2 around the crack-tip for Lexan
1-Plexiglas 2 bi-material plate with E12=0.82353, ν1 =
0.36, ν2 = 0.34, λ = 0.5192, d2 = 1 and t = 0.003 for (a)
isopachic fringe order Np=1 and isochromatic fringe or-
ders Nc = 3,4 and (b) Np=5 and Nc = 3,4.
7. Finite Element Analysis (FEA
The program ANSYS 11 is used in the
analysis (FEA) modeling. The mesh is generated using
triangular elements 8-node 82 for the crack and the
crack-tip of the model (Figure 4(a)).
The elements have the material p
Figure 2. Theoretically overlapping of Isochromatic (Nc)
= 2.8 GPa and ν = 0.36 and for Plexiglas E=3.4 GPa
and Isopachic (Np) fringes.
E
12
( 0.82353)E

and ν = 0.34. Figure 4(b) shows the
k-tip that had been used in the analysis.
Figure 4(c) shows the FEA model mesh for the crack and
the interface of bi-material plate. For the calculation of
the stresses and its contour curves at the bi-material in-
terface crack-tip, with Plexiglas as material 1 and Lexan
as material 2 or with Lexan as material 1 and Plexiglas as
material 2 (Figure 1), a tensile load of 100 N/m was ap-
plied.
Figu
mesh at the crac
re 5 shows the contour curves of principal stress
σ1
r curves of principal stress
σ2
r curves of principal stress
σ1
of principal stress
σ2
he direction of 45˚ relative to
cr
cussion
the principal stresses σ1
(Pa) at the bi-material interface crack-tip for Plexi-
glas-Lexan bi-material plate.
Figure 6 shows the contou
(Pa) at the bi-material interface crack-tip for Plexi-
glas-Lexan bi-material plate.
Figure 7 shows the contou
(Pa) at the bi-material interface crack-tip for
Lexan-Plexiglas bi-material plate.
Figure 8 shows the contour curves
(Pa) at the bi-material interface crack-tip for Lexan-
Plexiglas bi-material plate.
The stresses variation in t
ack direction are presented in next figures. Figure 9
shows the variation of principal stress σ1 (Pa) and Figure
10 shows the variation of principal stress σ2 (Pa) at the
bi-material crack-tip.
8. Results and Dis
From Figure 3 it is observed that
and σ2 are rapidly increased approaching to the

Crack-Tip Stress Analysis at a Bi-Material Interface by Photoelastic, Isopachic and FEA 1030
(a)
(b)
Figure 3. Contour curves oincipal stresses around the f pr
crack-tip for (a)
p
N
= 1 and c
N
= 3,4 and (b)
p
N
= 5 and
c
N
= 3,4.
rack-tip. achic fringe order
cFor isop
p
N
Nc =
(a)
(b) (c)
Figure 4. (a) Elemee crack nt 8-node 82, (b) FEA mesh at th
tip, (c) Model FEA mesh.
Figure 5. Contour of principal stress σ1 (Pa) at the
bi-material interface crack-tip tip for Plexiglas-Lexan
bi-material plate.
and and
1928.6Pa 24 28.6Pa


rial 1 (Lexan
11714.3Pa


and
214.3Pa , respe
principal ctivelyin mate) the stresses are 11333.3Pa
. While,


and 1500Pa and
2333.3Pa


and 166 ively. ter-
f principal stsses values is observed be-
cause of the difference constants of two materials. So,
from the experimental combination of the isochromatic
and isopachic fringes the contour curves of principal
stresses around the crack-tip can taken.
Applying the FEA, the contour curv
.7Pa , resp
re ectAt the in face a jump o
= 1 (Figure 3(a)) and
for isochromatic fringe orders 3 and 4 the principal
stresses in material 2 (Plexiglas) are 1857.1Pa

 and
1071.4Pa and 2428.6Pa

 and 6pec-
ile, in man) th
are 1666.7 Pa
42.9Pa , res
e principal stresses tively. Whaterial 1 (Lex

 and 833.3Pa and 2333.3Pa


and opachi
500Pa , respectively. For isc fringe order
p
N= 5 (ure 3(b)), closer to the crack-tip than in state
igure 3(a), and for isochromatic fringe orders Nc = 3
and 4 the principal stresses in material 2 (Plexiglas) are
Fi
of F
g
es of principal
stresses around the crack-tip can theoreticall y tak e n.

Crack-Tip Stress Analysis at a Bi-Material Interface by Photoelastic, Isopachic and FEA1031
Figure 6. Contour of principal stress σ2 (Pa) at the
bi-material interface crack-tip tip for Plexiglas-Lexan
bi-material plate.
Figure 7. Contour of principal stress σ1 (Pa) at the
bi-material interface crack-tip tip for Lexan-Plexiglas
bi-material plate.
Figure 8. Contour of principal stress σ2 (Pa) the
rom Figures 5-8 it is observed that the contour curves
study is concluded that the dis-
at
bi-material interface crack-tip for Lexan-Plexiglas
bi-material plate.
F
of principal stresses are zones with a average values of
principal stresses. Approaching the crack-tip, the princi-
pal stresses σ1 and σ2 are rapidly increased. The variation
of the average values of principal stresses σ1 and σ2 in
front of the crack-tip in th e direction of 45˚ relative to the
crack-axis are presented in Figures 9 and 10, respectively.
Figure 11 presents the experimentally overlapping of
isochromatic and isopachic fring es for E12 = 1 (one mate-
rial, Lexan). At the cross points of the fringes the princi-
pal stresses can be calculated by the Eqs (6) and (9). Also,
at crack tip the caustic was taken from which the stress
intensity factor KI can be calculated [22].
9. Conclusions
According to the above
tribution of the principal stresses close and far from the
crack-tip can be experimentally considered by the meth-
ods of photoelasticity and isopachics. Also, the contour
curves of principal stresses can be experimentally plotted
Figure 9. The variation of principal stress σ1 (Pa) in the di-
rection of 45˚ relative to crack direction (inset photo) at the
bi-material interface crack-tip for Plexiglas-Lexan
bi-material plate.
Figure 10. The variation of principal stress σ2 (Pa) in the
bi-material plate.
direction of 45˚ relative to crack direction (inset photo) at
the bi-material interface crack-tip for Plexiglas-Lexan

Crack-Tip Stress Analysis at a Bi-Material Interface by Photoelastic, Isopachic and FEA 1032
[10] P. S. Theocaris and J. Milios, “Crack Arrest at a Bimate-
rial Interface,” International Journal of Solids and Struc-
tures, Vol. 17, No. 2, 1981, pp. 217-230.
doi:10.1016/0020-7683(81)90077-9
[11] P. S. Theo caris, M. Siar ova and G . A . Papadopo
Propagation and Bifurcation in Fibbeulos, “Crack r-Composite Models:
I Soft-Hard-Soft Sequence of Phases,” Journal of Rein-
forced Plastics and Composites, Vol. 5, No. 1, 1986, pp.
23-50. doi:10.1177/073168448600500104
[12] P. S. Theocaris and G. A. Papadopoulos, “Crack Propaga-
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[13] E. E. Gdoutos, “Failure of a Bimaterial Plate with a Crack
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Figure 11. Experimentally overlapping Isochromatic
Isopachic fringes at the crack tip for one material (Lexan).
[1] M. L. Williamularities Resulting
from Various BAngular Corners of
oundary Conditions in Angular Corners of
eismological Societ
of Applied
olids and Struc-
and
around the crack-tip at the interface of the bi- material fromber Science
and Technology, Vol. 15, No. 1, 1981, pp. 27-40.
doi:10.1016/0015-0568(81)90029-4
[14] E. E. Gdoutos and A. Giannakopoulou, “Stress an
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the overlapping of the isochrom ati c and is opachic fringes.
From the FEA analysis is concluded that: (a) The ex-
actly contour curves of principal stresses can be theoreti-d Fail-
art I: Stress
cally estimated. (b) From the contour curves of stresses
the variation of the principal stresses can be calculated in
difference directions in front of bi-material crack-tip.
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