finite element methods in fracture mechanics
DESCRIPTION
FEA in FMTRANSCRIPT
Naoto Sakakibara
Finite Elements Methodin Fracture Mechanics
outlineIntroductionCollapsed Quadrilateral QPE ElementEnriched ElementDemo – NS-FFEM1.0Result Extended Finite Element MethodSummary
FEM in Fracture MechanicsEarly Application for Fracture Mechanics
> 5-10% error for simple problem *1 > solutions around tip cannot guaranteed*2
r
ru
1~
~
a. Crack tip element
– Quarter Point Elementb. Enriched Element
– Add another DOF
Collapsed Quarter Point Element
jiji uxNxu )()(
4
1 5
6
3
2
7
8
1 5 2
8 6
4 7 3
H/4
3H/4
•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.
Ex)
Transition ElementLynn and Ingraffea, 1978Combined with QPE elementImproving the accuracy of SIF, under special
configuration Located between Normal Element & QPE
4
12
LLL
(L,0)
(βL,0)
(1,0)
Collapsed QPE
Meshing tips
a
L -QPE
L -Tra.
Quarter Point Element
Transitional Element
Isoparametric Element
Suggestion•L-QPE/ 4a ~ 0.05-0.2•L-QPE/L-Tra. ~ 1.5244•Number of QPE ~ 6 – 12
Note:No optimal element size!
Enrich Element
General FEM
Singular field term
k
ikkiIIk
kkiIk
ikki QNQKQNQKuNu )()( 2211
'2221
1211
F
F
K
K
u
KK
KK
II
I
•Adding the analytic expression of the crack tip field to the conventional FEM
2sin
2
1
2cos
1 21
GQI
Part of the solution of displacement field
Drawbacks•Additional DOF Not able to use general FEM•Higher order more integration point•Incompatibility in displacement Transition element
NS-FFEM ver1.0
B,D
Method•Gaussian Elimination•Algebraic BC
Input•CPE4,CPE8,QPE8+Transitional•Mesh number•Geometry•Material Property
Output•SIF (QPDT)•σ, ε•u, v
Fem.exe
Deformed Configuration
ABAQUS QPE with CPE8 NS-FFEM with QPE
Result-1
)''(2
1
2DBI vv
L
GK
SIF QPDT method
SIF DCT method
)''()''((2
1
2ECDBI vvvv
L
GK
B
D
C
E
Result - 2
Enriched by singular function around tip.
Extended FEM-1
A
B
C
D
FI
FII
EIEII
III
n
j
n
hhh
mt
k
mf
lklkj HNFNuNu
j1 11 1
))(()())()(()()( axxbxxxx
F - Singular field function
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
n
j
n
hhh
mt
k
mf
lklkj HNFNuNu
j1 11 1
))(()())()(()()( axxbxxxx
ReferenceChona, 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.