A unified enrichment scheme for fracture · PDF fileA uni ed enrichment scheme for fracture...

A unified enrichment scheme for fractureproblems
Safdar AbbasThomas-Peter Fries
AICES, RWTH Aachen University, Aachen, Germany
9th World Congress on Computational Mechanics and 4thAsian Pacific Congress on Computational Mechanics,
Sydney, Australia, July 19-23, 2010
1

Outline
Outline
Motivation
XFEM in fracture mechanics
Numerical examples (cohesionless cracks)
Numerical examples (cohesive cracks)
Conclusions
Future outlook
2

Motivation
Outline
Motivation
Conclusions
Future outlook
3

Motivation Types of fracture
Types of fracture
Strain
Ductile Fracture
Brittle FractureStress Brittel Ductile
From: Fracture mechanics: fundamentals andapplications by Ted L. Anderson
4

Types of fracture
Fracture Behavior
Linear ElasticFracture Mechanics
Fracture Process ZoneCohesive
Plastic Collapse
5

Cohesionless fracture
Negligible Plastic Zone. Infinite Stresses at Crack Tip. Different Criteria for Crack Growth.
6

Negligible Plastic Zone.
Infinite Stresses at Crack Tip. Different Criteria for Crack Growth.
6

Negligible Plastic Zone. Infinite Stresses at Crack Tip.
Different Criteria for Crack Growth.
6

Negligible Plastic Zone. Infinite Stresses at Crack Tip. Different Criteria for Crack Growth.
6

Cohesive fracture
Presence of Damaged zone:1. Plastic Zone for metals.2. Fracture Process Zone for cementitious materials and ceramics.
Finite Stresses at Crack Tip. Different size of the Plastic or Process zone. Different criteria for crack growth.
7

Cohesive fracture
Finite Stresses at Crack Tip.
Different size of the Plastic or Process zone. Different criteria for crack growth.
7

Cohesive fracture
Finite Stresses at Crack Tip. Different size of the Plastic or Process zone.
Different criteria for crack growth.
7

Cohesive fracture
Finite Stresses at Crack Tip. Different size of the Plastic or Process zone. Different criteria for crack growth.
7

Linear elastic fractureu
x
x
Cohesive fractureu
x
x
8

Motivation High-gradient enrichment functions
High-gradient enrichment functions
= r x ;d
dx= xr x1
=>d
dx=
x < 1,0 x > 1 & x < 2,
0 x > 2.
9

= r x ;d
dx= xr x1 =>
d
dx=
x < 1,
0 x > 1 & x < 2,
0 x > 2.
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
r
d! /
dr
9

= r x ;d
dx= xr x1 =>
d
dx=
x < 1,0 x > 1 & x < 2,
0 x > 2.
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
r
d! /
dr
9

Interpolated function f Interpolation functions = [1, 2, 3] Find
uh =
f , for ,
where uh =iui =
Tu
10

Interpolated function f
Interpolation functions = [1, 2, 3] Find
uh =
f , for ,
where uh =iui =
Tu
10

Interpolated function f Interpolation functions = [1, 2, 3]
Finduh =
f , for ,
where uh =iui =
Tu
10

Interpolated function f Interpolation functions = [1, 2, 3] Find
uh =
f , for ,
where uh =iui =
Tu
10

Optimal set of functions
= {r1.1, r1.3, r1.5, r1.7}
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r
!
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
r
d! /
dr
11

= {r1.1, r1.3, r1.5, r1.7}
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r
!
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
r
d! /
dr
11

= {r1.1, r1.3, r1.5, r1.7}
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r
!
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
r
d! /
dr
11

Outline
Motivation
Conclusions
Future outlook
12

XFEM Formulation in Fracture Mechanics
uh(x) =iI
Ni (x)ui Continuous
+jI?1
N?j (x) H(x)aj +
kI?2N?k (x)
(4
m=1
Bm(x)bmk
)
Discontinuous
Set of nodes whose support is cut by the interface. Set of crack tip nodes. Partition-of-unity functions. Enrichment functions. Additional degrees of freedom.
13

uh(x) =iI
Ni (x)ui Continuous
+jI?1
N?j (x) H(x)aj +
kI?2N?k (x)
(4
m=1
Bm(x)bmk
)
Discontinuous
Set of nodes whose support is cut by the interface.
Set of crack tip nodes. Partition-of-unity functions. Enrichment functions. Additional degrees of freedom.
135 140 145 150 15592
94
96
98
100
102
104
106
108
13

uh(x) =iI
Ni (x)ui Continuous
+jI?1
N?j (x) H(x)aj +
kI?2N?k (x)
(4
m=1
Bm(x)bmk
)
Discontinuous
Set of nodes whose support is cut by the interface. Set of crack tip nodes.
Partition-of-unity functions. Enrichment functions. Additional degrees of freedom.
135 140 145 150 15592
94
96
98
100
102
104
106
108
13

uh(x) =iI
Ni (x)ui Continuous
+jI?1
N?j (x) H(x)aj +
kI?2N?k (x)
(4
m=1
Bm(x)bmk
)
Discontinuous
Set of nodes whose support is cut by the interface. Set of crack tip nodes. Partition-of-unity functions.
Enrichment functions. Additional degrees of freedom.
135 140 145 150 15592
94
96
98
100
102
104
106
108
13

XFEM Formulation in

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