DYNAMIC CRACK PROPAGATION BETWEEN
TWO BONDED ORTHOTROPIC PLATES
M. S. MATBULY
Received 30 June 2003 and in revised form 30 August 2003
The problem of crack propagation along the interface of two bonded dissimilar
orthotropic plates is considered. Using Galilean transformation, the problem is reduced
to a quasistatic one. Then, using Fourier transforms and asymptotic analysis, the problem is reduced to a pair of singular integral equations with Cauchy-type singularity.
These equations are solved using Gauss-Chebyshev quadrature formulae. The dynamic
stress intensity factors are obtained in closed form expressions. Furthermore, a parametric study is introduced to investigate the effect of crack growth rate and geometric and
elastic characteristics of the plates on values of dynamic stress intensity factors.
1. Introduction
Composite materials have been extensively employed in many engineering fields such as
mechanical and aerospace structures. When the material used as member of such structures contains a crack, it is seriously necessary to study the stress field distribution at the
immediate vicinity of crack tips. The inertia action must be considered when the applied
loads or crack length depend on time. Also, the most frequently observed phenomenon
in the experiments shows that the crack growth rate is constant during the extending history except in the final unstable or arresting stage [11]. So, the elastodynamic analysis of
a moving crack, with constant velocity, is one of the most important problems in fracture
mechanics. The dynamic stress intensity factors (DSIF) play a key role in characterizing the fracture behaviour of such problems. Thus, analytical determination of DSIF in
predicting the fracture cannot be overemphasized.
In general, there are two approaches for analytical determination of DSIF. The first one
employs the integral transforms and asymptotic analysis to reduce the problem to that of
a system of singular integral equations [1, 3, 10, 15, 20, 22, 23]. The second approach
employs complex analysis to reduce the problem to that of a system of Riemann-Hilbert
problems [12, 13, 14, 19].
The present work is concerned with elastodynamic stress disturbance problem of a
moving Griffith crack with constant velocity. The crack is located at the interface of two
Copyright © 2004 Hindawi Publishing Corporation
Journal of Applied Mathematics 2004:1 (2004) 55–68
2000 Mathematics Subject Classification: 42C20
URL: http://dx.doi.org/10.1155/S1110757X04306170
56
Dynamic crack propagation between two bonded plates
X2
1
1
σ12
= σ22
=0
=1
Plate 1
1
σ22
= F1 (X1 )
H1
1
= F2 (X1 )
σ12
X1
Interfacial line
2
σ12
= F2 (X1 )
2
= F1 (X1 )
σ22
H2
=2
a0 +ct
a0 +ct
Plate 2
2
2
σ12
= σ22
=0
Figure 1.1. Two bonded dissimilar plates containing moving interfacial crack.
bonded dissimilar orthotropic plates, as shown in Figure 1.1. Each plate possesses a finite
width and is subjected to a static stress distribution along crack surfaces. This is the main
difference between the present work and previous ones [1, 3, 4, 6, 7, 8, 10, 12, 13, 14, 15,
19, 20, 22, 23], which were concerned only with plates of infinite widths. The governing
equations of the problem are described. Then, Fourier transforms and asymptotic analysis are employed to reduce the solution of the problem to that of a system of first-kind
singular integral equations with Cauchy-type singularity. These are solved numerically
according to the algorithm in [17]. Then, closed-form expressions for the asymptotic
stress field distribution at the immediate vicinity of crack tips are obtained.
2. Governing equations
Assuming that the Cartesian coordinate axes are the axes of symmetry of the elastic materials, the displacement equations of motion for orthotropic plates are [3]
2 ∂2 U2
∂2 U1
∂2 U1
∂ U1
= m
,
2 + C66
2 + C12 + C66
∂X1 ∂X2
∂t 2
∂X1
∂X2
2 2 ∂2 U1
∂2 U2
∂ U2
∂ U2
= m
,
C66
2 + C22
2 + C12 + C66
∂X1 ∂X2
∂t 2
∂X1
∂X2
C11
(2.1)
where is a superscript ( = 1 for orthotropic material in X2 > 0, while = 2 for orthotropic material in X2 < 0), as shown in Figure 1.1; U j ( j = 1,2) are the displacement
components in direction of X1 and X2 , respectively; Cij (i, j = 1,2) and C66
are the elastic
constants of orthotropic plate materials; and m and t are the material mass density and
time, respectively.
The boundary conditions along the interface of plates are
1
2
X1 ,X2 ,t = lim− σ22
X1 ,X2 ,t = F1 X1 ,
lim+ σ22
X2 →0
1
lim σ12
X2 →0+
X1 ,X2 ,t
X2 →0
2
= lim− σ12
X2 →0
X1 ,X2 ,t = F2 X1 ,
X1 < a0 + d,
M. S. Matbuly 57
lim U11 X1 ,X2 ,t = lim− U12 X1 ,X2 ,t ,
X2 →0+
lim U21
X2 →0+
X1 ,X2 ,t
X2 →0
= lim− U22
X2 →0
X1 > a0 + d
X1 ,X2 ,t ,
1
2
lim+ σ22
X1 ,X2 ,t = lim− σ22
X1 ,X2 ,t ,
X2 →0
1
lim σ12
X2 →0+
X1 ,X2 ,t
X2 →0
2
= lim− σ12
X2 →0
X1 > a0 + d,
X1 ,X2 ,t ,
(2.2)
where a0 is the initial half crack length, and d = |c|t, where c is the magnitude of crack
propagation velocity. Moreover, Fi (X1 ) (i = 1,2) are known functions. They represent the
applied static stress along crack surfaces; F1 (−X1 ) = F1 (X1 ) and F2 (−X1 ) = − f2 (−X1 ).
The boundary conditions along the external boundaries are
1
1
2
σ12
X1 ,H1 ,t = σ22
X1 ,H1 ,t = σ12
X1 , −H2 ,t
2
X1 , −H2 ,t = 0,
= σ22
(2.3)
X1 < ∞,
where H1 and H2 are as shown in Figure 1.1.
Using Galilean transformation: x = (X1 − ct)/a0 , y = X2 /a0 , and t = t, the governing
equations (2.1), (2.2), and (2.3) can be reduced to a quasistatic form as follows:
2 2 ∂2 u ∂ ν
∂ u
+
a
= 0,
+
a
1
2
∂x2
∂x∂y
∂y 2
( = 1,2)
2 2 ∂2 ν
∂ u
∂ ν
+
b
+
b
= 0,
1
2
∂x2
∂x∂y
∂y 2
(2.4)
where
u (x, y) =
U1 X1 ,X2 ,t
,
a0
ν (x, y) =
a1 =
C12
+ C66
2 ,
C11
1 − M1
a2 =
b1 =
C12
+ C66
2 ,
C66
1 − M2
b2 =
c
M j = Vj
( j = 1,2),
C11
C66
V1 =
U2 X1 ,X2 ,t
,
a0
(2.5a)
C66
2 ,
1 − M1
(2.5b)
C22
2 ,
1 − M2
(2.5c)
C11
,
m
V2 =
C66
.
m
(2.5d)
Furthermore, the values of Mach numbers M j (, j = 1,2) are assumed to be less than
unity for subsonic crack propagation.
58
Dynamic crack propagation between two bonded plates
The boundary conditions for the reduced quasistatic problem can be expressed as follows:
lim σ y1y (x, y) = lim− σ y2y (x, y) = f1 (x),
y →0
y →0+
|x| < 1,
1
2
lim+ σxy
(x, y) = lim− σxy
(x, y) = f2 (x),
(2.6)
y →0
y →0
lim u1 (x, y) = lim− u2 (x, y),
y →0
y →0+
|x| > 1,
lim+ ν (x, y) = lim− ν2 (x, y),
1
(2.7)
y →0
y →0
lim+ σ y1y (x, y) = lim− σ y2y (x, y),
y →0
y →0
1
2
lim σxy
(x, y) = lim− σxy
(x, y),
y →0
y →0+
σxy
=
σ12
,
Q
σ y y =
|x| > 1,
(2.8)
1
2
σ y1y x,h1 = σxy
x,h1 = σ y2y x, −h2 = σxy
x, −h2 = 0,
|x| < ∞,
(2.9)
where
H
h = a0
σ22
,
Q
1 2
C66 ,
Q = C66
(2.10)
( = 1,2).
3. Solution of the problem
Decoupling (2.4) then employing Fourier sine (cosine) transforms with respect to x, one
can find that
4 ∞
1
Aj (α)eαr j y sinαx dα ,
u (x, y) =
π j =1 0
4 ∞
1
ν (x, y) =
kj Aj (α)eαr j y cosαx dα ,
π j =1 0
( = 1,2)
(3.1)
where α is the transform variable, Aj (α) ( = 1,2 and j = 1,4) are unknown functions,
and r j ( = 1,2 and j = 1,4) are the real distinct roots of
f1 r 4 − 2 f2 r 2 + 1 = 0 ( = 1,2),
(3.2)
provided that
f1 = a2 b2 ,
kj =
2 f2 = a2 + b2 − a1 b1 ,
2
a2 r j − 1
a1 r j
f2 > f1 , f1 > 0,
( = 1,2, j = 1,4).
(3.3)
(3.4)
M. S. Matbuly 59
From the stress-displacement relationship of orthotropic materials [9], one can get
(x, y) =
σxy
C66
∂ν ∂u
+
Q ∂x
∂y
4 ∞
1
αpj Aj (α)eαr j y sinαx dα
=
π j =1 0
σ y y (x, y) =
=
1 ∂u
∂ν
C12
+ C22
Q
∂x
∂y
( = 1,2),
4 ∞
1
αoj Aj (α)eαr j y cosαx dα
π j =1 0
(3.5)
( = 1,2),
where
pj
C66
r j − kj
=
,
Q
oj
C12
+ C22
rj kj
=
.
Q
(3.6)
On suitable substitution from (2.9) into (3.5), one can find that
Am (α) =
4
Ljm eαβ jm Aj (α) (m, = 1,2),
(3.7)
j =3
where
 h1 r − r for = 1
j
m
(m = 1,2, j = 3,4),
=
h2 r − r for = 2
j
m
2
n
n=1 (−1) 1 − δnm pn o j − p j on
L jm =
(,m = 1,2, j = 3,4),
p1 o2 − p2 o1
βjm
(3.8)
(3.9)
δnm is the Kronecker delta.
Substituting (3.5) and (3.7) into (2.8), one can find that
E2 D1 − D42 E41 1
E42 D31 − D42 E31 1
A3 (α) + 4 4
A4 (α),
∆
∆
D2 E1 − E32 D41 1
D2 E1 − E32 D31 1
A3 (α) + 3 4
A4 (α),
A24 (α) = 3 3
∆
∆
A23 (α) =
(3.10)
where
∆ = D32 E42 − D42 E32 ,
Dj = pj +
2
m=1
αβ jm
pm
L jm e ,
Ej = oj +
2
m=1
om Ljm eαβ jm
( = 1,2, j = 3,4).
(3.11)
60
Dynamic crack propagation between two bonded plates
The boundary conditions (2.6) and (2.7) in conjunction with (3.7), (3.8), (3.9), (3.10),
and (3.11) lead to
1 ∞ 1 1
α E3 A3 (α) + E41 A14 (α) cosαx dα = f1 (x),
π 0
|x| < 1,
1 ∞ 1 1
1 1
α D3 A3 (α) + D4 A4 (α) sinαx dα = f2 (x),
π 0
∞
1
α τ1 (α)A13 (α) + τ2 (α)A14 (α) sinαx dα = 0,
π 0
|x| > 1,
1 ∞ α ξ1 (α)A13 (α) + ξ2 (α)A14 (α) cosαx dα = 0,
π 0
(3.12)
(3.13)
where
2
τ1 (α) = 1 +
2
τ2 (α) = 1 +
ξ1 (α) =
− 1+
m=1
k31 +
2
m=1
ξ2 (α) = k41 +
2
m=1
− k42 +
2
2
L23m eαβ3m
1 1 αβ13m
km
L3m e
2
k32 +
−
m=1
∆3 − 1 +
2 2 αβ23m
km
L3m e
2
L24m eαβ4m
∆2 ,
∆4 ,
m=1
∆1
2
1
m=1
2 1
E4 D3 − D42 E31
2
2 2 αβ4m
km
L4m e
∆2 ,
1 1 αβ4m
km
L4m e
− k32 +
2
2
2
L24m eαβ4m
m=1
2
L23m eαβ3m
2
∆1 − 1 +
m=1
m=1
2
m=1
1
L14m eαβ4m
− k42 +
∆1 =
− 1+
m=1
1
L13m eαβ3m
2
m=1
(3.14)
2
2 2 αβ3m
km
L3m e
∆3
2
2 2 αβ4m
km
L4m e
∆4 ,
,
∆
E42 D41 − D42 E41
,
∆3 =
∆
D32 E31 − E32 D31
,
∆
D2 E1 − E32 D41
∆4 = 3 4
.
∆
∆2 =
The unknown functions A1j (α) ( j = 3,4) can be determined through solving the integral
equations (3.12) and (3.13) as follows [21].
Let
∂ν1 (x, y)
∂ν2 (x, y)
− lim−
= φ1 (x) 1 − H(x − 1) ,
y →0
y →0
∂x
∂x
∂u1 (x, y)
∂u2 (x, y)
lim+
− lim−
= φ2 (x) 1 − H(x − 1) ,
y →0
y →0
∂x
∂x
lim+
(3.15)
where H(x − 1) is the unit step function [2], while φ j (x) ( j = 1,2) are unknown odd and
even functions of x, respectively.
M. S. Matbuly 61
From (3.13) and (3.15), one can deduce that
τ2 Φ1 (α) + ξ2 Φ2 (α)
,
(αΨ)
− τ1 Φ1 (α) + ξ1 Φ2 (α)
A14 (α) =
,
(αΨ)
A13 (α) =
(3.16)
where
Φ1 (α) = 2
Φ2 (α) = 2
1
0
1
0
φ1 (x)sinαx dx,
(3.17)
φ2 (x)cosαx dx,
Ψ = τ1 ξ2 − τ2 ξ1 .
Substituting (3.16) into (3.12), the problem is reduced to the following system of integral
equations:
2 1
j =1 −1
H̆i j (x,t)φ j (t)dt = fi (x),
|x| < 1, i = 1,2,
(3.18)
where the kernels H̆i j (x,t) are
1 ∞ 1 1
E3 τ2 − E41 τ1 sinα(t − x)dα,
π 0 Ψ
1 ∞ 1 1
H̆12 (x,t) =
E3 ξ2 − E41 ξ1 cosα(t − x)dα,
π 0 Ψ
1 ∞ 1 1
D3 τ2 − D41 τ1 cosα(t − x)dα,
H̆21 (x,t) =
π 0 Ψ
1 ∞ 1 1
H̆22 (x,t) = −
D ξ2 − D41 ξ1 sinα(t − x)dα.
π 0 Ψ 3
H̆11 (x,t) =
(3.19)
Since the integrands of (3.19) are continuous functions of α, then it is clear that any
possible singularity of the kernels must result from the asymptotic analysis of the integrands as t → x and α → ∞ [16, 21]. Then, by adding and subtracting the asymptotic
expressions of these integrands under the integral sign, the problem can be reduced to
the following pair of singular integral equations with Cauchy-type singularity:
1
1
2 φ j (t)
1
δi j G i j
Hi j (x,t)φ j (t)dt = fi (x),
dt +
π j =1
−1 t − x
−1
|x| < 1, i = 1,2,
(3.20)
62
Dynamic crack propagation between two bonded plates
where
H11 (x,t) =
H12 (x,t) =
H21 (x,t) =
H22 (x,t) =
∞
0
1 1
E3 τ2 − E41 τ1 − G11 sinα(t − x)dα,
Ψ
0
1 1
E3 ξ2 − E41 ξ1 − G12 cosα(t − x)dα,
Ψ
0
1 1
D τ2 − D41 τ1 − G21 cosα(t − x)dα,
Ψ 3
∞
∞
∞
0
1
−1 Ψ
D31 ξ2 − D41 ξ1
− G22 sinα(t − x)dα,
o3 B − s2 /B − o14 B − s1 /B
,
Z
G11 =
1
G12 =
p1 B − s2 /B − p41 B − s1 /B
G21 = 3
,
Z
o3 k41 B − s4 /B − o14 k31 B − s3 /B
,
Z
p1 k1 B − s3 /B − p31 k41 B − s4 /B
G22 = 4 3
,
Z
(3.21)
1
s − s + k1 s − k41 s1 s1 s4 − s2 s3
,
+
Z = k41 − k3 + 3 4 3 2
B
B2
B = L231 L242 − L241 L232 p12 o22 − p22 o21 ,
s1 = B13 + B24 L231 L242 + B12 + B27 L241 L232 ,
s2 = B33 + B44 L231 L242 + B32 + B47 L241 L232 ,
s3 = k12 B13 + k22 B24 L231 L242 + k22 B12 + k12 B27 L241 L232 ,
s4 = k12 B33 + k22 B44 L231 L242 + k22 B32 + k12 B47 L241 L232 ,
B13 = p31 o22 − p22 o13 ,
B24 = p12 o13 − p31 o21 ,
B12 = p31 o21 − p12 o13 ,
B27 = p22 o13 − p31 o22 ,
B33 = p41 o22 − p22 o14 ,
B44 = p12 o14 − p41 o21 ,
B32 = p41 o21 − p12 o14 ,
B47 = p22 o14 − p41 o22 .
From (3.15) and (2.5d), one can deduce the single-valuedness conditions ensuring the
uniqueness of φi (α) (i = 1,2) as follows:
1
−1
φ1 (t)dt = 0,
1
2
φ2 (t)dt =
c
−1
2
C11
−
m2
Equations (3.20) and (3.22) can be solved as follows [17].
1
C11
.
m1
(3.22)
M. S. Matbuly 63
Assume that
qi (t)
φi (t) = √
1 − t2
(i = 1,2),
(3.23)
where qi (t) (i = 1,2) are bounded continuous functions for all t ∈ [−1,1].
By substituting from (3.23) into (3.20) and (3.22) then employing Gauss-Chebyshev
integration formulae, the solution of the problem can be reduced to the following system
of linear algebraic equations [17]:
n1 2 wk
j =1 k=1
δi j G i j
+ Hi j xl ,tk q j tk = fi xl
tk − xl
n1
wk q2 tk
k =1
wk q1 tk = 0,
k =1
n1
(i = 1,2) xl = 1,n1 − 1 ,
2
=
πc
(3.24)
2
C11
−
m2
1
C11
,
m1
where n1 is the number of collocation points in the interval [−1,1],
w1 = wn1 =
1
,
2n1 − 2
tk = cos
xl = cos
wk =
π(k − 1)
n1 − 1
π(2l − 1)
2n1 − 2
1
n1 − 1
for k = 2,n1 − 1 ,
k = 1,n1 ,
(3.25)
l = 1,n1 − 1 .
For the concerned problem, one can deduce that the DSIF at the left and right crack
tips are equal. Then, by making use of the following asymptotic relations, as α → ∞, [5]:
Φ1 (α) =
1
−1
q (t)
sinαt dt ≈ q1 (1)
1 − t2
1
q (t)
√2
cosαt dt ≈ q2 (1)
Φ2 (α) =
1 − t2
−1
∞
0
1
α
√ e−bα
√1
2π
π
1
+ϑ
,
sin α −
α
4
α
2π
π
1
+ϑ
,
cos α −
α
4
α
√
sin h
π
dα = 0.5tan−1
,
0.25
b
coshα
cos
b 2 + h2
sinhα b > 0,
(3.26)
64
Dynamic crack propagation between two bonded plates
the leading terms of the asymptotic stress field distribution (3.5) at the immediate vicinity
of the right crack tip (x → 1 and y → 0± ) can be obtained as follows:
ζ
ζ
θ
θ
σxy (x, y) ≈ q1 (1) 1 sin 1 − 2 sin 2
2ρ1
2
2ρ2
2
θ
θ
ζ
ζ
+ q2 (1) 3 cos 1 − 4 cos 2 ,
2ρ1
2
2ρ2
2
(3.27)
γ1
γ2
θ
θ
σ y y (x, y) ≈ q1 (1) cos 1 − cos 2
2ρ1
2
2ρ2
2
γ3
γ4
θ
θ
+ q2 (1) sin 1 − sin 2 ,
2ρ1
2
2ρ2
2
where
2
ρ1 = (x − 1)2 + r31 y ,
θ1 = tan−1
r31 y
,
x−1
ξ1 =
θ2 = tan−1
ξ2 =
2
r41 y
,
x−1
p41 1 − s1 /B
,
Z
p1 k1 − s4 /B
ξ3 = 3 4
,
Z
p31 1 − s2 /B
,
Z
ρ2 = (x − 1)2 + r41 y ,
γ1 =
o13 1 − s2 /B
,
Z
γ2 =
o14 1 − s1 /B
,
Z
γ3 =
o13 k41 − s4 /B
,
Z
γ4 =
o14 k31 − s3 /B
.
Z
(3.28)
p1 k1 − s3 /B
ξ4 = 4 3
,
Z
Therefore, the DSIF, KI , and KII can be determined as follows [18]:
KI + iKII = Q 2a0
lim
x→1,
√
y →0+
x − 1 σ y y (x, y) + iσxy (x, y)
(3.29)
such that by substituting from (3.27) and (3.28) into (3.29) then evaluating the limits,
one can find that
√ √ KI = Q a0 γ1 − γ2 q1 (1),
KII = Q a0 ξ3 − ξ4 q2 (1).
(3.30)
M. S. Matbuly 65
3.4
a0 = 0.75
Normalized DSIF (KI )
3.0
2.6
a0 = 0.5
2.2
a0 = 0.25
1.8
1.4
1.0
a0 = 0.10
0.6
0.2
−0.2
0.0
0.2
0.8
0.4
0.6
Crack growth rate (mm/h)
Figure 4.1. Variation of the normalized KI with the crack growth rate and initial crack length.
4. Numerical results
A parametric study is introduced to investigate the effects of crack growth rate, initial
crack length, and the plate width on the values of DSIF. Consider that a structure of two
bonded orthotropic plates as in Figure 1.1 possesses the following elastic characteristics:
m1 = m2 = 1g/cm3 ,
1
2
C66
= C66
= 1MPa,
1
= 2.5MPa,
C22
2
C11
= 1 + r1 C66
− r1 r2 C22
=
C12
2
r1 r2
(4.1)
2
C22
= 4MPa,
2
+ 1 − r1
2 + r2
( = 1,2),
− 1 C66
( = 1,2),
where (4.1) are derived from (2.5), (3.2), and (3.3). Also, for the concerned numerical
results, it is assumed that 1 > r1 > r2 > 0, r3 = −r1 , and r4 = −r2 . The initial crack length
2a0 ranges from 0.2 to 1.5 cm, while the width of plates ranges from 5 to 10 m. For simplicity, the numerical results are obtained for constant uniform stress distribution along
the crack surfaces, |x| < 1 and y → ±0.
!∞
!4
Furthermore, we have found that | 0 Ii j (α,x,t)dα − 0 Ii j (α,x,t)dα| < 10−16 , where
Ii, j (α,x,t) (i, j = 1,2), are the integrands of (3.21). Therefore, the improper integrals of
(3.21) are approximated and evaluated numerically from α = 0 → 4 by using the trapezoidal rule. Then, the values of DSIF are normalized such that
NormalizedKI =
KI
1 √ ,
σ22
a0
Normalized KII =
KII
1 √ ,
σ12
a0
1
1
and σ12
represent the applied stress along the upper crack surface.
where σ22
(4.2)
Dynamic crack propagation between two bonded plates
Normalized DSIF (KII )
66
7.0
6.0
5.0
4.0
3.0
2.0
1.0
0.0
−1.0
−2.0
−3.0
−4.0
0.1
a0 = 0.5
a0 = 0.25
a0 = 0.1
a0 = 0.75
0.2
0.3
0.4
0.5
0.6
Crack growth rate (mm/h)
0.7
0.8
Figure 4.2. Variation of the normalized KII with the crack growth rate and initial crack length.
Normalized DSIF (KI )
2.2
H1 = 10, H2 = 5
1.8
1.4
H1 = 5, H2 = 5
1.0
H1 = 5, H2 = 10
0.6
0.1
0.2
0.3
0.4
0.5
0.6
Crack growth rate (mm/h)
0.7
0.8
Figure 4.3. Variation of the normalized KI with the crack growth rate for different plate widths.
Figure 4.1 shows that the value of normalized KI continuously decreases with increasing the crack growth rate, where 0 < c < 0.32 and 0.5 < c < 0.6, while it increases elsewhere. Also, it shows increasing the pathological oscillatory behaviour [3] for the normalized KI with increasing the initial crack length.
Figure 4.2 shows that the value of normalized KII continuously increases with increasing the crack growth rate except for 0.5 < c < 0.65. But this decreasing interval is shifted
to 0.4 < c < 0.5 for the case of a relatively long initial crack length 2a0 = 1.5 cm. One can
notice as well that the pathological oscillatory behaviour of the normalized KII increases
with increasing a0 .
For the prescribed elastic and geometric characteristics, Figure 4.3 shows that the value
of normalized KI continuously increases with increasing the crack growth rate except
when c < 0.5 and H1 > H2 , it is decreased. Also, Figure 4.4 shows that the value of normalized KII continuously decreases with increasing the crack growth rate, while it increases
with increasing c when c > 0.62 and H1 < H2 .
M. S. Matbuly 67
Normalized DSIF (KII )
1.7
1.6
1.5
H1 = 10, H2 = 5
1.4
1.3
H1 = 5, H2 = 10
H1 = 5, H2 = 5
1.2
1.1
1.0
0.1
0.2
0.3
0.4
0.5
0.6
Crack growth rate (mm/h)
0.7
0.8
Figure 4.4. Variation of the normalized KII with the crack growth rate for different plate widths.
5. Conclusion
The present work is concerned with elastodynamic analysis of crack propagation between
two bonded dissimilar orthotropic plates. The width of each plate is assumed to be finite.
This is the new trend and the main difference between this work and the previous ones
[1, 3, 4, 6, 7, 8, 10, 12, 13, 14, 15, 19, 20, 22, 23]. So, the present work can be considered
as an extension for the analysis of interfacial crack problems.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
R. Babaei and S. A. Lukasiewicz, Dynamic response of a crack in a functionally graded material
between two dissimilar half planes under antiplane shear impact, Engrg. Fracture Mech. 60
(1998), 479–487.
R. N. Bracewell, The Fourier Transform and Its Applications, McGraw-Hill Series in Electrical
Engineering, Circuits and Systems, McGraw-Hill, New York, 1986.
S. Das and B. Patra, Moving Griffith crack at the interface of two dissimilar orthotropic half planes,
Engrg. Fracture Mech. 54 (1996), 523–531.
, Stress intensity factors for moving interfacial crack between bonded dissimilar fixed orthotropic layers, Comput. & Structures 69 (1998), no. 4, 459–472.
F. Delale and F. Erdogan, Transverse shear effect in a circumferentially cracked cylindrical shell,
Quart. Appl. Math. 37 (1979), 239–258.
A. H. England, A crack between dissimilar media, Trans. ASME J. Appl. Mech. 32 (1965), 400–
402.
F. Erdogan and G. Gupta, Layered composites with an interface flow, Internat. J. Solids Structures
7 (1971), 1089–1107.
F. Erdogan and B. Wu, Interface crack problems in layered orthotropic materials, J. Mech. Phys.
Solids 41 (1993), no. 5, 889–917.
R. F. Gibson, Principles of Composite Material Mechanics, McGraw-Hill, New York, 1994.
S. Itou, Dynamic stress intensity factors around two rectangular cracks in a half-space under impact load, Acta Mech. 121 (1997), no. 1–4, 153–164.
J. F. Kalthoff, J. Beinert, and S. Winkler, Measurements of dynamic stress intensity factors for fast
running cracks in double-cantilever-beam specimens, Fast Fracture and Crack Arrest (G. T.
Hahn and M. F. Kanninen, eds.), ASTM STP, vol. 627, American Society for Testing and
Materials, Pennsylvania, 1977, pp. 161–176.
68
Dynamic crack propagation between two bonded plates
[12]
K. H. Lee, Stress and displacement fields for propagating the crack along the interface of dissimilar
orthotropic materials under dynamic mode I and mode II load, Trans. ASME J. Appl. Mech.
67 (2000), 223–228.
K. H. Lee, J. S. Hawong, and S. H. Choi, Dynamic stress intensity factors KI , KII and dynamic
crack propagation characteristics of orthotropic material, Engrg. Fracture Mech. 53 (1996),
119–140.
J. Y. Liou and J. C. Sung, Singularities at the tip of a crack terminating normally at an interface
between two orthotropic media, Trans. ASME J. Appl. Mech. 63 (1996), 264–270.
C.-C. Ma and Y.-S. Ing, Dynamic crack propagation in a layered medium under antiplane shear,
Trans. ASME J. Appl. Mech. 64 (1997), no. 1, 66–72.
M. S. Matbuly, Mode II stress intensity factor for cracked layered material, The 3rd International
Conference on Engineering Mathematics and Physics (Cairo, Egypt), Faculty of Engineering, Cairo University, Cairo, 1997, pp. 801–808.
M. S. Matbuly, S. A. Mohamed, and M. Nassar, On the numerical solution of the first kind singular integral equations, The 3rd International Conference on Engineering Mathematics and
Physics (Cairo, Egypt), Faculty of Engineering, Cairo University, Cairo, 1997, pp. 11–17.
V. Z. Parton and E. M. Morozov, Elastic-Plastic Fracture Mechanics, Mir Publishers, Moscow,
1978.
J. R. Rice, Elastic fracture mechanics concepts for interfacial cracks, Trans. ASME J. Appl. Mech.
55 (1988), 98–103.
C. Rubio-Gonzalez and J. J. Mason, Response of finite cracks in orthotropic materials due to concentrated impact shear loads, Trans. ASME J. Appl. Mech. 66 (1999), no. 2, 485–491.
N. I. Shbeeb, W. K. Binienda, and K. L. Kreider, Analysis of the driving forces for multiple cracks
in an infinite non-homogeneous plate, part I: theoretical analysis, Trans. ASME J. Appl. Mech.
66 (1999), no. 2, 492–500.
M. Valentini, S. K. Serkov, D. Bigoni, and A. B. Movchan, Crack propagation in a brittle elastic
material with defects, Trans. ASME J. Appl. Mech. 66 (1999), 79–86.
B. L. Wang, J. C. Han, and S. Y. Du, Dynamic response for non-homogeneous piezoelectric medium
with multiple cracks, Engrg. Fracture Mech. 61 (1998), 607–617.
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
M. S. Matbuly: Department of Engineering Mathematics and Physics, Faculty of Engineering,
Zagazig University, P.O. 44519, Zagazig, Egypt
E-mail address: mohamedmatbuly@hotmail.com