Finite Elements Method
in Fracture Mechanics
Naoto Sakakibara
outline
 Introduction
 Collapsed Quadrilateral QPE Element
 Enriched Element
 Demo – NS-FFEM1.0
 Result
 Extended Finite Element Method
 Summary
FEM in Fracture Mechanics
 Early Application for Fracture Mechanics
> 5-10% error for simple problem *1
> solutions around tip cannot guaranteed*2
u~
 ~1
r
r
a. Crack tip element
– Quarter Point Element
b. Enriched Element
– Add another DOF
Collapsed Quarter Point Element
•Henshell and Shaw,1975
•1/√r variation for strain can be achieved
•Same shape function N,
•Standard FEM can be used
•Collapsed Element, more accuracy than other QPEs.
ui ( x)  N ij ( x)u j
7
4
3
Ex)
473
H/4
8
6
6
8
3H/4
1
5
2
1
5
2
0
0.2
0.4
0.6
0.8
1
Transition Element
 Lynn and Ingraffea, 1978
 Combined with QPE element
 Improving the accuracy of SIF, under special configuration
 Located between Normal Element & QPE
L  2 L 1
L 
4
Collapsed QPE
(1,0) (βL,0)
(L,0)
Meshing tips
Suggestion
•L-QPE/ 4a
~ 0.05-0.2
•L-QPE/L-Tra.
~ 1.5244
•Number of QPE
~ 6 – 12
a
L -Tra.
L -QPE
Quarter Point
Element
Transitional
Element
Isoparametric
Element
Note:
No optimal element size!
Enrich Element
•Adding the analytic expression of the crack tip field to the
conventional FEM


ui   N k uik  K I (Q1i   N k Q1k )  K II (Q2i   N k Q2ik )
k
k
k


Singular field term
General FEM
QI 1 
Drawbacks
•Additional DOF
 Not able to use general FEM
•Higher order
 more integration point
•Incompatibility in displacement
 Transition element
1 
   1

cos 
 sin 2 
G 
2 2
2
Part of the solution of displacement field
K
 21
K
11
 u 
K 
 F 
KI    
22  
K 
  F '
K
 II 
12
NS-FFEM ver1.0
Method
•Gaussian Elimination
•Algebraic BC
B,D
Input
•CPE4,CPE8,QPE8+Transitional
•Mesh number
•Geometry
•Material Property
Output
•SIF (QPDT)
•σ, ε
•u, v
Deformed Configuration
ABAQUS QPE with CPE8
NS-FFEM with QPE
Result-1
SIF QPDT method
KI 
2G 2
(v ' B v ' D )
 1 L
SIF DCT method
2G 2
KI 
((v'B v'D )  (v'C v'E )
 1 L
E
C
D
B
Result - 2
Enriched by singular function around
tip.
Extended FEM-1
n
mt
mf
n
j 1
k 1
l 1
h 1
u (x)   N j (x)u j   N k (x)( Fl (x)b k )   N h (x) H ( (x))a h
F - Singular field function
EII
EI
D
A
II
I
B
FI
FII
C
H – Discontinuous function
•H – step, sign, etc.
•εI(x), εII(x) – different function
•a – associated with displacements at E & F
•Mesh – independent from crack
Extended FEM-2
Discontinuous
Function H
Singular field
Function
Summary
1.QPE
Transitional
• DOF ~ # of nodes
• Mesh size, no optimal size
• Mesh, depend on crack
2.Singular Field
• Additional DOF
• More Integration point at crack tip element
• Mesh, depend on crack
3.Singular Field
Discontinuous
• Additional DOF
• Mesh, independent from crack
• No remeshing for crack growth
n
mt
mf
n
j 1
k 1
l 1
h 1
u (x)   N j (x)u j   N k (x)( Fl (x)b k )   N h (x) H ( (x))a h
Reference
Chona, R., Irein, G., and Sanford, R.J. (1983). The influence of specimen size and shape
on the singurarity-dominated zone. Proceedings, 14th National Symposium on Fracture
Mechanics, STP791,Vol.1, American Soc. for Testing and Materials, (pp. I1-I23). Philadelphia.
I.L.Lim, I.W.Jhonston and S.K.Choi. (1993). Application of singular quadratic distorted
isoparametric elements in linear fracture mechanics. International journal for numerical
methods in engineering , Vol.36, 2473-2499.
I.L.Lim, I.W.Johnston and S.K.Choi. (1992). On stress intensity factor computation from
the quater-point element displacements. Communications in applied numerical methods , Vol.8,
291-300.
Mohammad, S. (2008). Extendet finite element. Blackwell Publishing.
Nicolas Moes, John Dolbow and Ted Belystschko. (1999). A finite element method for
crack growth withiout remeshing. International jounarl for numerical methods in engineering ,
131-150.
Sanford, R. (2002). Principle of Fracture Mechanics. Upper Saddle River, NJ 07458: Pearson
Education, Inc.