23rd International Congress of Theoretical and Applied Mechanics,
August, 19-24, 2012, Beijing, China
Interfacial cracks in bi-material solids:
Stroh formalism and skew-symmetric
weight functions
Lorenzo Morini
,
Enrico Radi
,
Dipartimento di Scienze e Metodi dell’ Ingegneria,
Università di Modena e Reggio Emilia,
Via Amendola 2, 42100, Reggio Emilia, Italy
Alexander B. Movchan
,
Natalia V. Movchan
,
Department of Mathematical Sciences, University of Liverpool,
Liverpool L69 3BX, U.K.
Outline
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Conclusions•
References•
Interface cracks in anisotropic materials:
Stroh formalism
;
•
Riemann-Hilbert
formulation;
•
Symmetric and skew-symmetric
weight functions;
•
Stress intensity factors
evaluation;
•
Application: point forces applied at crack faces;
Interfacial cracks: Stroh formalism
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Conclusions•
Quasi-static semi-infinite
plane interfacial crack with
general
loading acting on the faces;
•
Displacements and tractions in terms of functions of
complex
variable
z
j
= x
1
+ µ
j
x
2
:
u
,
1
(x
1
, x
2
) = 2
Re
[Ag(z)],
T
(x
1
, x
2
) = 2
Re
[Bg(z)],
Assuming
Stroh representation
:
[Q
ik
+ (R
ik
+ R
ki
)µ
j
+ T
ik
µ
2
j
]A
kj
= 0
Riemann-Hilbert formulation
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Conclusions•
ReferencesTraction-free
crack problem:
-Free traction condition at
x
1
< 0
;
-Tractions and displacements continuity at
x
1
> 0
;
-
Boundary conditions
at the interface yields to a
R-H problem
:
h
+
(x
1
) + H
−1
Hh
−
(x
1
) = T (x
1
)
for
x
1
> 0
h
+
(x
1
) + H
−1
Hh
−
(x
1
) = 0
for
x
1
< 0
Mirror traction-free problem
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vsskew-symmetric SIF
•
ConclusionsMirror
traction-free crack problem:
-Free traction at
x
1
> 0
;
-Tractions and displacements continuity at
x
1
< 0
;
-At the interface:
w
+
(x
1
) + H
−1
Hw
−
(x
1
) = 0
for
x
1
> 0
w
+
(x
1
) + H
−1
Hw
−
(x
1
) = Σ(x
1
)
for
x
1
< 0
Weight functions
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Conclusions•
References•
Symmetric
weight functions:
[U](x
1
) = U(x
1
, x
2
= 0
+
) − U(x
1
, x
2
= 0
−
)
•
Skew-symmetric
weight functions:
hUi(x
1
) =
1
2
(U(x
1
, x
2
= 0
+
) + U(x
1
, x
2
= 0
−
))
•
Mirror traction-free problem is solved in
Fourier space
;
•
A
Wiener-Hopf
-like equation is derived:
[ ˆ
U
]
+
(ξ) = −
1
|ξ|
n
Re
H
− i
sign
(ξ)
Im
H
o ˆ
Σ
−
(ξ),
•
The skew-symmetric weight function become:
h ˆ
U
i(ξ) = −
1
2|ξ|
n
Re
W
−i
sign
(ξ)
Im
W
o ˆ
Σ
−
(ξ),
ξ ∈
R
.
Decoupling plane and antiplane strain and stress
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Conclusions•
Materials where
A
,
B
and
Y
and then
H
and
W
have the
following structure are considered:
∗ ∗ 0
∗ ∗ 0
0 0 ∗
•
Uncoupled inplane
and
antiplane
strain and stresses;
•
Monoclinic
and
orthotropic
materials have this property;
•
Physical tractions:
T
(x
1
) =
√
1
2πx
1
Re
Kx
iε
1
w
⇒
Mode I and II
T
3
(x
1
) =
√
K
3
2πx
1
⇒
Mode III
•
K = K
I
+ iK
II
, and
w
= (w
1
, w
2
)
is a
complex vector
;
•
K
3
is a
real scalar
;
Stress intensity factors evaluation
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Conclusions•
ReferencesBetti integral’s theorem
relates
(u, T
(+)
)
to
(U, Σ
(−)
)
:
•
For
plane strain
:
[ ˆ
U
]
+T
R ˆ
T
+
− ˆ
Σ
−T
R
[ˆ
u
]
−
= −[ ˆ
U
]
+T
R
hˆpi − h ˆ
U
i
T
R
[ˆ
p
]
Where
R
is the rotation matrix:
R
=
−1 0
0 1
•
For
antiplane strain
:
[ ˆ
U
3
] ˆ
T
3
+
− ˆ
Σ
3
[ˆ
u
3
]
−
= −[ ˆ
U
3
] hˆ
p
3
i − h ˆ
U
3
i[ˆ
p
3
]
•
Integral formulas for
stress intensity factors
:
K
=
M
−1
1
2πi
Z
∞
−∞
n
[ ˆ
U
]
+T
(τ )Rhˆpi(τ) + h ˆ
U
i
T
(τ )R[ˆ
p
](τ )
o
dτ
K
3
=
1
2πiK
33
Z
∞
−∞
n
[ ˆ
U
3
]
+
(τ )h ˆ
p
3
i(τ) + h ˆ
U
3
i(τ)[ ˆ
p
3
](τ )
o
dτ
Where
K
= (K, K)
T
;
Plane strain in orthotropic bimaterials I
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Conclusions•
2D vector problem in
orthotropic bimaterials
;
•
Symmetric
bimaterial matrix:
H
=
H
11
−iβ
√
H
11
H
22
iβ
√
H
11
H
22
H
22
•
Bimaterial parameters:
H
11
= [2nλ
1
4
(˜
s
11
s
˜
22
)
1
2
]
(1)
+ [2nλ
1
4
(˜
s
11
s
˜
22
)
1
2
]
(2)
,
H
22
= [2nλ
−
1
4
(˜
s
11
s
˜
22
)
1
2
]
(1)
+ [2nλ
−
1
4
(˜
s
11
s
˜
22
)
1
2
]
(2)
,
β
pH
11
H
22
= [((˜
s
11
s
˜
22
)
1
2
+ ˜
s
12
)]
(2)
− [((˜s
11
s
˜
22
)
1
2
+ ˜
s
12
)]
(1)
,
Where:
λ =
s
˜
11
˜
s
22
,
ρ =
1
2
2˜
s
12
+˜
s
66
√
˜
s
11
s
˜
22
,
n =
1
2
(1 + ρ)
1
2
,
•
Generalized Dundurs parameter
connected to oscillatory index:
ε =
1
2π
ln
1 − β
1 + β
Plane strain in orthotropic bimaterials II
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Conclusions•
References•
Skew-symmetric
bimaterial matrix:
W
= Y
(1)
− Y
(2)
=
δ
1
H
11
iγ
√
H
11
H
22
−iγ
√
H
11
H
22
δ
2
H
22
•
Bimaterial parameters:
δ
1
=
[2nλ
1
4
(˜
s
11
s
˜
22
)
1
2
]
(1)
− [2nλ
1
4
(˜
s
11
s
˜
22
)
1
2
]
(2)
H
11
,
δ
2
=
[2nλ
−
1
4
(˜
s
11
s
˜
22
)
1
2
]
(1)
− [2nλ
−
1
4
(˜
s
11
s
˜
22
)
1
2
]
(2)
H
22
,
γ =
[((˜
s
11
s
˜
22
)
1
2
+ ˜
s
12
)]
(1)
+ [((˜
s
11
s
˜
22
)
1
2
+ ˜
s
12
)]
(2)
√
H
11
H
22
,
•
Homogeneous
material
⇒ δ
1
, δ
2
= 0
, but
γ 6= 0
then even in
homogeneous case we have
non-zero skew-symmetric
weight
functions;
Plane strain in orthotropic bimaterials: weight functions
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Conclusions•
Fourier transform
symmetric and skew-symmetric
weight
functions:
[ ˆ
U
]
+
= −
√
H
11
H
22
|ξ|
q
H
11
H
22
iβ
sign
(ξ)
−iβ
sign
(ξ)
q
H
22
H
11
ˆ
Σ
−
(ξ);
h ˆ
U
i = −
√
H
11
H
22
2|ξ|
δ
1
q
H
11
H
22
−iγ
sign
(ξ)
+iγ
sign
(ξ) δ
2
q
H
22
H
11
ˆ
Σ
−
(ξ);
•
Inverting these expressions we get
[U]
and
hUi
.
•
Since Mode I and II are
coupled
:
U
=
U
1
1
U
1
2
U
2
1
U
2
2
, Σ =
Σ
1
1
Σ
2
1
Σ
1
2
Σ
2
2
•
[ ˆ
U
] ⇒
Wiener-Hopf equation
;
Asymmetric loading
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Conclusions•
ReferencesAsymmetric
point forces acting on the crack faces:
hp
2
i(x
1
) = −
F
2
δ(x
1
+ a) −
F
4
δ(x
1
+ a + b) −
F
4
δ(x
1
+ a − b)
[p
2
] (x
1
) = −F δ(x
1
+ a) +
F
2
δ(x
1
+ a + b) +
F
2
δ(x
1
+ a − b)
Plane strain: SIF
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vs skew-symmetric SIF•
Conclusions!"
b/a
0
"!#
0.4
1
$!"
0.6
0.8
!"
"!"
%
a
K
I S/F
#!"
& = '1/2
& = '1/4
& = 0
a
b/a
0
"!#
0.4
0.6
0.8
1
b
#!(
& = )1/2
#!"
%
a
K
I A/F
& = )1/4
!(
& = 0
& = 1/4
!"
& = 1/2
"!(
"!"
b/a
0
"!#
0.4
1
"!*
0.6
0.8
)"!+
)"!#
)"!*
%
a
K
I I S/F
"!"
"!+
"!#
& = )1/2
& = )1/4
& = 0
& = 1/4
& = 1/2
c
b/a
0
"!*
"!#
0.4
0.6
0.8
1
d
"!+
"!#
%
a
K
I I A/F
"!"
=
& )1/2
& = )1/4
& = 0
& = 1/4
& = 1/2
)"!#
)"!+
)"!*
Symmetric vs skew-symmetric SIF
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vs skew-symmetric SIF•
Conclusions•
ReferencesΒ = 1 2
Β = 1 4
Β = -1 2
Β = -1 4
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 b a KiA KiS•
As
b/a → 1
increase,
K
I
A
≈ 40% − 50%
of
K
I
S
;
•
Skew-symmetric part of the loading
is not negligible
and needs to
be taken into account;
Antiplane strain: weight functions
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Conclusions•
Anisotropic materials with
symmetry
plane at
x
3
= 0
are
considered;
•
Fourier transform of weight functions:
[ ˆ
U
3
](ξ) = −
H
33
|ξ|
Σ
ˆ
3
(ξ);
h ˆ
U
3
i(ξ) =
η
2
[ ˆ
U
3
](ξ);
•
Inverting we obtain:
[U
3
](x
1
) =
√
H
33
2π
x
1
2
1
;
hU
3
i(x
1
) =
η
2
√
2π
H
33
x
1
2
1
;
•
H
33
=
h
p˜s
44
s
˜
55
− ˜s
2
45
i
(1)
+
h
p˜s
44
s
˜
55
− ˜s
2
45
i
(2)
;
η =
h
p˜s
44
s
˜
55
− ˜s
2
45
i
(1)
−
h
p˜s
44
s
˜
55
− ˜s
2
45
i
(2)
/H
33
Antiplane strain: SIF
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vs skew-symmetric SIF•
Conclusions•
References0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
b
a
a K 3 S F ΒΗ = -1 4
Η = -1 2
Η = -1
Η = 1
Η = 1 2
Η = 1 4
0.0
0.2
0.4
0.6
0.8
1.0
-1.0
-0.5
0.0
0.5
1.0
b
a
a K 3 A F•
Same loading configuration directed along
x
3
:
hp
3
i(x
1
) = −
F
2
δ(x
1
+ a) −
F
4
δ(x
1
+ a + b) −
F
4
δ(x
1
+ a − b)
[p
3
] (x
1
) = −F δ(x
1
+ a) +
F
2
δ(x
1
+ a + b) +
F
2
δ(x
1
+ a − b)
Symmetric vs skew-symmetric SIF
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vs skew-symmetric SIF•
ConclusionsΗ = -1
Η = -1 2
Η = -1 4
Η = 1
Η = 1 4
Η = 1 2
0.0
0.2
0.4
0.6
0.8
1.0
-1.0
-0.5
0.0
0.5
1.0
b
a
K3 A K3 S•
As for plane strain,
K
3
A
increase with
b/a
, expecially for
|η| > 1/2
;
•
For
b/a > 0.5
, skew-symmetric part of the loading
is not
negligible
;
Conclusions
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Conclusions•
References•
A new general approach for deriving the weight functions for 2D
interfacial cracks in
anisotropic
bimaterials has been developed;
•
For
perfect interface
conditions, the new method
avoid
the use of
Wiener-Hopf technique and the challenging factorization problem
connected;
•
Both
symmetric and skew-symmetric
weight functions can be
derived by means of the new approach;
•
Weight functions can be used for deriving
singular integral
formulation
of interfacial cracks in anisotropic media;
•
The proposed method can be applied for studying interfacial
cracks problems in many materials:
monoclinic, orthotropic, cubic,
piezoelectrics, poroelastics, quasicrystals
;
Furter developments:
-Applications to steady state moving cracks and wavy cracks;
-Analysis of inclusions effects of interface cracks propagation;
-Extension to 3D case;
References
•
Outline•
Interfacial cracks: Stroh formalism•
Riemann-Hilbert formulation•
Mirror traction-free problem•
Weight functions•
Decoupling plane andantiplane strain and stress
•
Stress intensity factors evaluation•
Plane strain in orthotropic bimaterials I•
Plane strain in orthotropic bimaterials II•
Plane strain in orthotropic bimaterials: weight functions•
Asymmetric loading•
Plane strain: SIF•
Symmetric vsskew-symmetric SIF
•
Antiplane strain:weight functions
•
Antiplane strain: SIF•
Symmetric vsskew-symmetric SIF
