Finite Element Methods in Fracture Mechanics

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.