Transcript Diapositive 1
Computational Fracture Mechanics
Anderson’s book, third ed. , chap 12
Elements of Theory
Energy domain integral method: - Formulated by Shih et al. (1986): CF Shih, B. Moran and T. Nakamura, “Energy release rate along a three-dimensional crack front in a thermally stressed body”, International Journal of Fracture 30 (1986), pp. 79-102 - Generalized definition of the
J
- integral (nonlinear materials, thermal strain, dynamic effects).
- Relatively simple to implement numerically, very efficient.
Finite element (FE) code
ABAQUS
version 6.5
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For details see the Getting Started Manual of Abaqus 6.5
Energy Domain Integral
: In 2D, under quasistatic conditions,
J
may be expressed by
J
G
lim
0 G
ds
The contour G surrounds the crack tip.
The limit indicates that G shrinks onto the crack tip.
n
: unit outward normal to G.
q
: unit vector in the virtual crack extension direction.
x 1
,
x 2
Cartesian system and,
H
w
I
w
s : strain energy density : Cauchy stress tensor displacement gradient tensor
H
: Eshelby’s elastic energy-momentum tensor (for a non-linear elastic solid)
For details see the Theory Manual of Abaqus 6.5, section 2.16
With
q
along
x 1
and the field quantities expressed in Cartesian components, i.e.
q n
n
1
n
2
H
T
w
0 0
w
s s 11 21 s 12 s 22
u
1
x
1
u
2
x
1
u
1
x
2
u
2
x
2 Thus,
n
1
n
2
w
0 0
w
n
1
n
2 s s 11 21 s 12 s 22
u
1
x
1
u
2
x
1
u
1
x
2
u
2
x
2 In indexed form, we obtain
J
G
lim
0 G
wn
1 s
u x
1
j
ds
The expression of
J
(
see eq. 6.45)
is recovered with
dx
2
dy
The previous equation is
not suitable
for a numerical analysis of
J.
Transformation into a domain integral
Following Shih et al. (1986),
J
G
lim
0 G
ds
q
1
C
m
T
H q
ds
C
C
t
u q
ds
(*)
m
: outward normal on the closed contour
C
C
G
C
t
σ m
: the surface traction on the crack faces.
q q
on
G
0
on C
is a sufficiently smooth weighting function in the domain
A
.
Note that,
q
q
with 1 0
on
G
on C otherwise arbitrary A
m
= -
n
on G
A
includes the crack-tip region as G 0
(*) Derivation of the integral expression
J
C
m
T
H q
ds
C
m 0
T
H q
ds
G
m
T
H q
ds
C
m q
T
H q
ds
C
C
m
T
H q
C
G
C
A.
ds
Noting that,
C
C
m
T
H q
ds
C
C
m
w
I
C
C
m
ds
w
I
T
u σ
T
q
ds
since
σ
T
σ
C
C
w
m q
ds
0
C
C
m σ
u q
ds
C
C
m σ
u q
ds
C
C
u q
ds
C
C
t
u q
ds
since
t
σ m
Using the divergence theorem, the contour integral is converted into the domain integral
J
A
div
T
H q
dA
C
C
t
u q
ds
Under certain circumstances,
H
is divergence free, i.e.
H
0 indicates the path independence of the J-integral.
In the general case of thermo-mechanical loading and with body forces and crack face tractions:
H
0 the J-integral is
only
defined by the limiting contour G 0 Introducing then the vector,
h =
div
in
A
or
h k
H
Using next the relationship,
div J
A
h q
dA
C
C
t
div
u q
ds
Contributions due to crack face tractions.
In Abaqus: - This integral is evaluated using ring elements surrounding the crack tip.
- Different contours are created: First contour (1) = elements directly connected to crack-tip nodes.
The second contour (2) are elements sharing nodes with the first, … etc Crack Refined mesh Contour (i)
q
1 nodes inside 2 1
q q
0 nodes outside 8-node quadratic plane strain element (CPE8)
Exception:
0
q
1 on midside nodes (if they exist) in the outer ring of elements
J-integral in three dimensions
Local orthogonal Cartesian coordinates at the point
s
on the crack front:
J
defined in the x 1 - x 2 plane crack front at
s L
G
lim
0 G
d
G Point-wise value For a virtual crack advance l (
s
) in the plane of a 3D crack,
T
L
: length of the crack front under consideration.
dA
ds d
G : surface element on a
vanishingly small
tubular surface enclosing the crack front along the length
L.
Numerical application (bi-material interface):
• SEN specimen geometry (see annex III.1): s
a
= 40 mm Material 1
b
= 100 mm
a/b
= 0.4
and
h/b
= 1
h
= 100 mm
2h a b y x
Material 2 s Remote loading: s 1 MPa.
Materials properties (Young’s modulus, Poisson’s ratio):
Material 1
:
E 1
n
1
= 3 GPa = 0.35
Material 2
:
E 2
n
2
= 70 GPa = 0.2
Plane strain conditions.
• Typical mesh: Material 1= Material 2 Material 1 Refined mesh around the crack tip Material 2 Number of elements used: 1376 Type: CPE8 (plane strain)
Simulation of the stress evolution s 22 (isotropic case)
Simulation of the stress evolution s 22 (bi-material)
Results:
Abaqus Material 1
J
(N/mm)
0.1641
Material 2
J
(N/mm)
0.0077
(*) same values on the contours 2-8
Bi-material
J
(N/mm)
0.0837
Annex III Abaqus Isotropic K
I 0.746
0.748
K
II 0.
0.
Bi-material K
I / 0.752
K
II / 0.072
(*) SIF given in
MPa m
• Ones checks that:
J
1
K I
2 n
i
2
)
for the isotropic case (
i
=1,2).
• Relationship between J and the SIF’s for the bi-material configuration: - For an interfacial crack between two dissimilar isotropic materials (plane strain), where and
G i
E i
n
i
i
n
i E i
1
E i
n
i
2 plane strain,
i =
1,2
K I
and
K II
are defined here from a complex intensity factor, such that with - Extracted from the Theory Manual of Abaqus 6.5, section 2.16.2.
H. Gao, M.Abbudi and D.M. Barnett, “Interfacial Crack-tip fields in anisotropic elastic solids thermally stressed body”, Journal of the Mechanics and Physics of Solids 40 (1992), pp. 393-416 Disagreement with the results of Smelser et al.