STUDY THE CAUSTICS, ISOCHROMATIC AND ISOPACHIC FRINGES AT
A BI-MATERIAL INTERFACE CRACK
G.A. PAPADOPOULOS
Department of Engineering Science, Section of Mechanics, National Technical University of Athens,
GR-157 73, Zografou campus, Athens, Greece.
E-mail: [email protected]
Abstract. The shape of crack-tip caustics, isochromatic and isopachic fringes, at a bi-material interface under static load, are studied.
When the crack-tip, which is perpendicular to interface, is at the interface of the bi-material, the caustics, the isochromatic and the
isopachic fringes depend on the properties of the two materials. Thus, the caustics, the isochromatic and the isopachic fringes are
divided into two branches. The size of the two branches mainly depends on the elastic modulus and the Poisson’ s ratio of the two
materials. From the caustics, the stress intensity factor KI can be calculated while, from the comparison of the isochromatic and the
isopachic fringes, the principal stresses s 1, s 2 can be calculated.
1.
INTRODUCTION
The problems of crack propagation through the mesophase of composites has become a subject of great interest. This problem was
studied by Williams [1-4], Zak and Williams [5] and later by Dally and Kobayashi [6] by means of dynamic photoelasticity. The
influence of both the mesophase and the material characteristics of either phase, on the stress distribution around the crack-tip in biphase plate consisting of different materials have experimentally studied by Theocaris et. al. [7-10]. Also, theoretical studies on this
subject were carried out by Gdoutos [11-13], Theotokoglou et. al. [14] and Spyropoulos et. al. [15]. The study of the size and the shape
of crack-tip caustics at a bi-material interface was carried out by Papadopoulos [16,17].
In this work an attempt to study the shape of crack-tip caustics, isochromatic and isopachic fringes at a bi-material interface and the
evaluation of the principal stresses is studied.
2.
STRESS-FIELD AROUND THE CRACK-TIP
Two plates of modulii E1 and E2 and Poisson’s ratios ? 1 and ? 2 are perfectly bonded along their common interface (Figure 1). In plate
(1) there is a crack perpendicular to the interface. The crack-tip is exactly at the interface of the two plates.
Figure 1. Geometry of bi-material plate.
The Airy stress function X(r,?) for this problem, used by Zak and Williams (1963), is
X ( r ,θ ) = r λ +1 F (θ )
(1)
F (θ ) = α sin(λ + 1)θ + b cos(λ + 1)θ + c sin(λ − 1)θ + d cos(λ − 1)θ
(2)
with
where α , b, c, d are constants and
moduli [17].
λ takes values between 0 and 1, which is depended on the ratio E12=E1/E2 of the two plates
From the stress function (1) the polar stresses at the crack-tip are taken
1 ∂X (r , θ ) 1 ∂ 2 X ( r , θ )
+
= ( λ + 1) r λ −1 F (θ ) + r λ −1 F ′ (θ )
2
2
r
∂r
r
∂θ
α ( λ + 1) sin(λ + 1)θ + b( λ + 1) cos(λ + 1)θ + c( λ − 3) sin(λ − 1)θ
= − λ r λ −1 
 + d (λ − 3) cos(λ − 1)θ
σr =
σθ =
∂ 2 X ( r ,θ )
∂r
2
(3)



= λ ( λ + 1) r λ−1 F (θ )
+ b cos(λ + 1)θ + c sin(λ − 1)θ + d cos(λ − 1)θ
]
1 ∂X ( r , θ ) 1 ∂ 2 X (r , θ )
−
= −λ r λ −1 F ′(θ )
2
∂θ
r
r
∂ r∂θ
α (λ + 1) cos(λ + 1)θ − b( λ + 1) sin(λ + 1)θ + c( λ − 1) cos(λ − 1)θ
= −λ r λ −1 
− d ( λ − 1) sin(λ − 1)θ



= λ ( λ + 1) r
λ−1
[α sin(λ + 1)θ
(4)
τ rθ =
(5)
The boundary conditions of the problem are written as
τ rθ1 = 0 and σ θ1 = 0
τ rθ2 = 0 and υ 2 = 0
π
2
π
for θ 2 =
2
for θ 1 =
(6)
σ θ1 = σ θ2 , τ rθ1 = τ rθ 2 , u1 = u2 , υ1 = υ 2
for θ 1 = 0 and θ 2 =
where u1, 2 , υ 1, 2 are the displacement components.
From the boundary conditions (6) the values of the
the modulii of the plates.
Figure 2. Variation of
3.
λ
π
2
can be determined. Figure 2 presents the variation of
λ versus the ratio E12 of
λ versus ratio E12=E1/E2 for poisson’s ratios ?1=?2=0.30.
THEORY OF CAUSTICS
The deflection of light, either reflected from, or passing through a generic point P of the plate in the vicinity of the crack-tip, is given
by the deviation vector w, which, for a optically isotropic material, is expressed by [16]
Wr,t , f = X r ,t , f i + Y r,t , f j = r + w r ,t, f
(7)
w r ,t , f = −εz0 tc r,t , f grad x, y (σ r + σ θ ), r = r cosθ i + r sin θ j
(8)
with
where
ε
is a multiplying factor, which is equal to unity for reflected from the front (f) face or transmitted (t) light rays and equal to 2
for light rays reflected from the rear (r) face of the plate,
c r , t, f
are the stress-optical constants of the material, t is the thickness of
the plate and
z 0 is the distance between plate and reference screen. The stress-optical constant c f = ν 1,2 / E1,2
for the materials 1
and 2 respectively.
From relations (7) and (8) the parametric equations of the caustics (r) and (t) ((r) is the caustic which is formed by the reflected light
rays from the rear face of the plate and (t) is the caustic which is formed by the transmitted light rays through the plate) are obtained
X r ,t

1

= λ m r01,2 cosθ 1,2 −
c 2 sin(λ − 2)θ1,2 + d 2 cos(λ − 2)θ 1,2

(λ − 2) c22 + d 22
[

1

Yr ,t = λ m r01,2 sin θ1,2 −
c2 cos(λ − 2)θ 1,2 − d 2 sin(λ − 2)θ1,2

(λ − 2) c22 + d 22
[
where
λ m is the magnification factor of the optical set-up, which is given by
λm =
where
rays.
zi




]
(9)

]
(10)

z0 ± zi
zi
(11)
is the distance between plate and focus of the light beam, (+) for the reflected and (-) for the transmitted through the plate
The radius of the initial curve of the caustics is
1 /(3−λ )


r01,2 =  4λ ( λ − 1)( λ − 2) Cr ,t , f c12,2 + d12,2 


(12)
with
C r ,t , f = −
ε z 0 tcr ,t , f
λm
(13)
where the indices 1,2 the materials 1 and 2 of the bi-material plate are represented. The polar coordinate
[p/2, p]U[-p, -p/2] (material 1) and the
θ2
θ1 takes values in the region
takes values in the region [-p/2, +p/2] (material 2).
The constants α 1, 2 , b1, 2 , c1, 2 , d 1, 2 , which are calculated by the boundary conditions (6), are given by the relations
α 2 = c2 = 0,
d 2 = arbitrary constan t
α 1 = {E12 ( λ − 1)[ b2 (ν 2 + 1)( λ + 1) − d 2 ( λ (ν 2 + 1) + 3 − ν 2 )] − [b2 ( λ + 1) + d 2 (1 − λ )]
[ν 1 (λ − 1) + λ + 3]} cos(λπ / 2) / 4(λ + 1)
b1 = −{E12 [b2 ( λ (ν 2 + 1) + ν 2 + 1) − d2 (λ (ν 2 + 1) + ν 2 − 3)] − ν 1 (λ + 1)(b2 − d 2 )
+ b2 ( λ − 3) + d 2 (3 − λ )}0.25 sin(λπ / 2)
c1 = −{ E12 [b2 (ν 2 + 1)( λ + 1) − d 2 ( λ (ν 2 + 1) + 3 − ν 2 )] − [b2 (λ + 1)
+ d 2 (1 − λ )](ν 1 + 1)}0.25 cos(λπ / 2)
d1 = {E12 [b2 (ν 2 + 1)( λ + 1) − d 2 ( λ (ν 2 + 1) + ν 2 − 3)]
− ( λ + 1)(b2 − d 2 )(ν1 + 1)}0.25 sin(λπ / 2)
(14)
(15)
(16)
(17)
(18)
b2 = {d 2 [[ E12 [λ (3ν 2 − 5) + ν 2 − 3] − [ν 1 ( 3λ + 1) + 1 − λ ]] cos(λπ ) + [ν1 ( 2λ 2 + λ + 1)
+ ( λ − 1)( 2λ + 3)] − E12 [2λ 2 (ν 2 + 1) + λ (ν 2 + 1) + ν 2 − 3]]} /{[[ E12 (ν 2 + 1)
(19)
+ 3 − ν 1 ] cos(λπ ) − ( 2λ + 1)[ E12 (ν 2 + 1) − (ν1 + 1)]]( λ + 1)}
The variables E12, ? 1, ? 2 are depended on the materials while, the variable d2 is depended on the conditions of the experiment and
mainly on stress of loding.
For Plexiglas (PMMA), with E1=3.4 GPa and ? 1 =0.34, as material 1 and Lexan (PCBA), with E2=3.8 GPa and ? 2 =0.36, as material 2
(Figure 1) the theoretical caustics at the crack-tip at the bi-material interface were plotted according to equations (9)-(19). Figure 3
presents the plotted caustics, caustic (r) and caustic (f), for Plexiglas 1-Lexan 2 bi-material plate with E12=1.2142, ? 1=0.34, ? 2=0.36,
?=0.48, b2=0.353d2, c1=0.826d2, d1=0.78d2, c2=0 and d2=1.
Figure 3. Theoretical caustics for E12=1.2142, ?1=0.34, ?2=0.36 and ?=0.48
(Plexiglas 1-Lexan 2).
The relative sizes of the caustics (r) and (f) are depended on the stress-optical constants cr and cf of the two materials of the bi-material
plate.
4.
THEORY OF PHOTOELASTICITY
Isochromatic fringes are loci of points with the same value for the difference of the principal stresses or the maximum shear stress.
According to the stress optical law, the difference in the principal stresses is given by [18]
σ 1 − σ 2 = 2τ max =
Nc fc
t
where N c is the isochromatic fringe order,
constant, which is given by the relation
fc =
(20)
t
is the thickness of the specimens and
fc
is the material fringe value or stress-optical
Eλ l
2ν
(21)
where E is the elastic modulus, λl is the wave length of the used unicoloured light and ν is the Poisson’s ratio of the plate material.
From equations (3)-(5) and (20) is obtained

1

r =
 [ (1− λ 2 ) F (θ ) + F ′′(θ ) ] 2 + 4λ 2 ( F ′(θ )) 2
[
with
1
]
1/ 2
N c ( f c ) 1,2  λ −1

t

(22)
F (θ ) = α 2 sin(λ + 1)θ + b2 cos(λ + 1)θ + c 2 sin(λ − 1)θ + d 2 cos(λ − 1)θ
(23)
The relation between the stress-optical constants of the materials 1 and 2 is
( fc ) 2 =
1 ν1
( fc ) 1
E12 ν 2
where E12 = E1 / E 2 is the ratio of the elastic modulii of the two materials,
materials.
(24)
ν1
and
ν2
are the Poisson’ s ratios of the two
The isochromatic fringes are plotted around the crack-tip by the equation (22) for ( fc ) 1 = 1, t = 0.003 m , d 2 = 1 . Figure 4 presents
the plotted isochromatic fringes of order N c = 1÷ 10 , for Plexiglas 1- Lexan 2 bi-material plate, with E12 = 1.2142, ? 1= 0.34, ? 2= 0.36
and ?=0.48.
Figure 4: Isochromatic fringes of order
N c = 1÷ 10
for E12 = 1.2142, ? 1= 0.34, ? 2= 0.36 and ?=0.48
(Plexiglas 1- Lexan 2).
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
Np
of isopachic is
related to the sum of the principal stresses by
σ 1 + σ 2 = σ r + σθ =
Np fp
t
(25)
where N p is the order of isopachics, t is the thickness of the plate and
fp
is the isopachic fringe constant, which is given by the
relation
fp =
Eλ l
2ν
(26)
where E is the elastic modulus,
λl
is the wave length of the used unicoloured light and ν is the Poisson’s ratio of the plate material.
The sum of the stresses (from equations (3) and (4)) are
[
σ 1 + σ 2 = σ r + σ θ = r λ−1 (λ + 1) 2 F (θ ) + F ′′(θ )
]
(27)
with
F (θ ) = α 2 sin(λ + 1)θ + b2 cos(λ + 1)θ + c 2 sin(λ − 1)θ + d 2 cos(λ − 1)θ
F ′ (θ ) = −α 2 ( λ + 1) 2 sin(λ + 1)θ − b2 ( λ + 1) 2 cos(λ + 1)θ
− c2 ( λ − 1) 2 sin(λ − 1)θ − d 2 ( λ − 1) 2 cos(λ − 1)θ
(28)
(29)
By substituting the equations (31)-(33) into equation (29) is obtained
1

N p ( f p ) 1,2  λ −1
1
r =

2
t
 ( λ + 1) F (θ ) + F ′′(θ )

(30)
with
( f p )2 =
1 ν1
( f p )1
E12 ν 2
(31)
where ( f p ) 1,2 are the isopachic fringe constants of the materials 1 and 2 of the bi-material plate, respectively.
The isopachic fringes are plotted around the crack-tip by the equation (30) for ( f p ) 1 = 1, t = 0.003 m, d 2 = 1 . Figure 5 presents the
plotted isopachic fringes of order N p = 1÷ 10 , for Plexiglas 1- Lexan 2 bi-material plate, with E12 = 1.2142, ? 1= 0.34, ? 2= 0.36 and
?=0.48.
Figure 5: Isopachic fringes of order
N p = 1 ÷ 10
for E12 = 1.2142, ?1= 0.34, ?2= 0.36 and
?=0.48 (Plexiglas 1-
Lexan 2).
6.
PRINCIPAL STRESSES ESTIMATION FROM THE ISOCHROMATIC AND ISOPACHIC FRINGES
The principal stresses can be estimated from the system of isochromatic and isopachic fringes (Eqs (20), (27)). This solution of the
system is valid at the cutting points of the isochromatic and isopachic fringes (Fig. 6). Figure 6 presents the overlapping of
isochromatic and isopachic fringes, for Lexan 1-Plexiglas 2 bi-material plate with E12=0.82353, ? 1=0.36, ? 2=0.34, ?=0.5192, d2=1 and
t=0.003.
Figure 6: Isochromatic (Nc ) and Isopachic (Np) fringes.
From the solution of the equations (20) and (27) system, the principal stresses are obtained
N p ( f p )1, 2 + N c ( f c ) 1,2
σ r + σθ 1
+
(σ r − σ θ ) 2 + 4τ r2θ =
2
2
2t
N p ( f p )1,2 − N c ( f c )1,2
σ + σθ 1
σ2 = r
−
(σ r − σ θ ) 2 + 4τ r2θ =
2
2
2t
σ1 =
(32)
(33)
By substituting the stresses from the Eqs (3)-(5) into Eqs (32), (33) the contour curves of the principal stresses, around the crack-tip,
are obtained
1
rσ 1

2σ 1
=
2
2
 (1 + λ ) F (θ ) + F ′′(θ ) + [(1 − λ ) F (θ ) + F ′′(θ )] 2 + 4λ2 ( F ′ (θ )) 2
rσ 2

2σ 2
=
2
2
 (1 + λ ) F (θ ) + F ′′(θ ) − [(1 − λ ) F (θ ) + F ′′(θ )] 2 + 4λ2 ( F ′(θ )) 2
[
[
][
 λ−1
1/ 2 

]
][
(34)
1
 λ−1
1/ 2 

]
(35)
For one material, E12=1, ? 1= ? 2=? and ?=0.5, the contour curves are got from the equations
−2
rσ 1




σ
1

=
 KI
θ
θ 
cos 1 + sin  

2
2 
 2π
rσ 2




σ2

=
 KI
θ
θ 
cos 1 − sin  

2
2
 2π
(36)
−2
(37)
Figure 7 presents the contour curves of principal stresses s 1 and s 2 for one material. Respective contour curves can be plotted
according to the Eqs (36) and (37) for two materials.
Figure 7: Contour curves of principal stresses around the crack-tip, for one material.
7.
CONCLUSIONS
According to the above study is concluded that the stress state at the crack-tip can be considered by the method of caustics, while the
stress state far from the crack-tip can be considered by the methods of photoelasticity and isopachics. Also, the contour curves of
principal stresses can be plotted around the crack-tip at the interface of the bi-material from the overlapping of the isochromatic and
isopachic fringes.
References
[1] Williams, M.L. (1952a). Surface stress singularities resulting from various boundary conditions in angular corners of plates under
bending. Proceedings ,First U.S. Nat. Congress of Applied Mechanics, ASME, 325-329.
[2] Williams, M.L. (1952b). Stress singularities resulting from various boundary conditions in angular corners of plates in extension.
Journal of Applied Mechanics 19, Trans. ASME 74, 526-528.
[3] Williams, M.L. (1957). On the stress at the base of a stationary crack. Journal of Applied Mechanics 24, Trans ASME 79, 109-114.
[4] Williams, M.L. (1959). The stresses around a fault or crack in dissimilar media. Bulletin of the Seismological Society of America
49, 199-204.
[5] Zak, A.R. and Williams, M.L. (1963). Crack point stress singularities at a bimaterial interface. Journal of Applied Mechanics 30,
142-143.
[6] Dally , J.W. and kobayashi, T. (1978).Crack arrest in duplex specimens. International Journal of Solids
And Structures 14, 121-126.
[7] Theocaris, P.S. and Milios, J. (1979). Crack propagation velocities in bi-phase plates under static and dynamic loading.
Engineering Fracture Mechanics 13, 559-609.
[8] Theocaris, P.S. and Milios, J. (1981). Crack arrest at a bimaterial interface. International Journal of Solids and Structures 17,
217-230.
[9] Theocaris, P.S., Siarova, M. and Papadopoulos, G.A. (1986a). Crack propagation and bifurcation in fiber-composite models. I:
Soft-hard-soft sequence of phases. Journal of Reinforced Plastics and Composites 5, 23-50.
[10] Theocaris, P.S. and Papadopoulos, G.A. (1986b). Crack propagation and bifurcation in fiber-composite models. II: hard-soft-hard
sequence of phases. Journal of Reinforced Plastics and Composites 5, 120-140.
[11] Gdoutos, E.E. (1981). Failure of a bimaterial plate with a crack at an arbitrary angle to the interface. Fiber
Science and
Technology 15, 27-40.
[12] Gdoutos, E.E. and Giannakopoulou, A. (1999). Stress and failure analysis of brittle matrix composites. Part I: Stress analysis.
International Journal of Fracture 98, 263-277.
[13] Gdoutos, E.E., Giannakopoulou, A. and Zacharopoulos, D.A. (1999). Stress and failure analysis of brittle matrix composites. Part
II: failure analysis. International Journal of Fracture 98, 279-291.
[14] Theotokoglou, E.N. , Tsamasphyros, G.J. and Spyropoulos, C.P. (1989). Photoelastic study of a crack approaching the bonded
half-plates interface. Engineering Fracture Mechanics 34, 31-42.
[15] Spyropoulos, C.P., Theotokoglou, E.N. and Tsamasphyros, G.J. (1990). Evaluation of the stress intensity factors for a crack
approaching the bonded half-plates interface from isopachics. Acta Mechanica 81, 75-89.
[16] Papadopoulos, G.A. (1993). Fracture Mechanics. The Experimental Method of Caustics and the Det.-Criterion of Fracture.
Springer-Verlag, London Lim.
[17] Papadopoulos, G.A. (1999). Crack-tip caustics at a bi-material interface. International journal of Fracture 98, 329-342.
[18] Frocht, M.M. (1948). Photoelasticity, John Willey, New York.