A hybrid analytical / extended finite element method for directevaluation of stress intensity factorsJulien Rethore1, Stephane Roux2, Francois Hild2

1LaMCoS, INSA-Lyon 2LMT Cachan

Introduction

Stress intensity factors KI andKII are key parameters forfracture mechanicsNeed for a robust evaluationNo post-processing, directevaluationMesh independenceOptimal convergence...

Displacement field in m

PU enrichment [2, 3]

10−2

10−1

100

10−3

10−2

10−1

100

h/w

|KI-K

Io|/K

Io

nmax=1nmax=3nmax=5nmax=7

convergence rate < 0.5

singular enrichment • discontinuous enrichment

u =∑i∈N

Niudi+∑

i∈Ncut

NiHbi+∑

i∈Ntip

∑j=I,II

NiFjKj

Crack displacement fields

Let us consider a homogeneous body with isotropic elastic behavior, anda 2D setting, the displacement field u is conventionally represented by itscomplex writing, u = ux + iuy. It was expanded by Williams [4] for astraight crack as a double series

u(r , θ) =∑i=I,II

∑n

cni φ

ni (r , θ)

with

φnI (r , θ) = rn/2

(κeinθ/2 −

n2

ei(4−n)θ/2 + (n2

+ (−1)n)e−inθ/2)

φnII(r , θ) = irn/2

(κeinθ/2 +

n2

ei(4−n)θ/2 − (n2− (−1)n)e−inθ/2

)where κ is Kolossov’s constant, namely, κ = (3 − ν)/(1 + ν) for planestress or κ = (3−4ν) for plane strain conditions, ν being Poisson’s ratio.

n = 0: translationn = 1: usual asymptotic fields

n = 2: rotation and T -stressn ≥ 3: sub-singular fields

Hybrid model

Ω1

Ω2

Ω12

Model 1:u1(x) =

∑i∈N1

N i(x)di +∑

i∈Ncut

N i(x)H(x)bi

Model 2:

u2(x) =1

2µ√

2π

∑n∈[0;nmax]

(φn

I (x) pn + φnII(x)qn

)Partition of energy:

α1(x) + α2(x) = 1Weighted bilinear forms:

ai(ui, v∗i

)=

∫Ωi

αi ε (ui) : C : ε(v∗i)

dΩ

Coupling:

Π (u1 − u2, µ∗) =

∫Ω12

λ · (u1 − u2) dΩ = 0

Arlequin method [1]:

a1(u1, v∗1

)+ a2

(u2, v∗2

)+ Π (u1 − u2, µ

∗) + Π(v∗1 − v∗2, λ

)= l1

(v∗1)

Results

Rinner R

outer

Displacement field in m

KIo = f(

aw

)σ√πa = 2.98 MPa

√m

Region of analytical model Router = routerh = 13h Region of ‘inactive’ elements Rinner = rinnerh = 10h

Coupling region Router − Rinner = `overlaph = 3h Number of terms in the analytical model nmax = 5

0 5 10 150.8

0.85

0.9

0.95

1

1.05

rinner

KI/K

Io

nmax=2nmax=3nmax=5nmax=7

`overlap = 1⇒ large rinner needs higher nmax to

accomodate boundary effects

10−2

10−1

100

10−3

10−2

10−1

100

h/w

|KI-K

Io|/K

Io

nmax=2nmax=3nmax=5nmax=7

rinner = 4,`overlap = 1⇒ convergence rate ≈ 1.5

Bibliography[1] H. Ben Dhia and G. Rateau.

The Arlequin method as a flexible engineering design tool.International Journal for Numerical Methods in Engineering, 62:1442–1462, 2005.

[2] N. Moes, J. Dolbow, and T. Belytschko.A finite element method for crack growth without remeshing.International Journal for Numerical Methods in Engineering, 46(1):133–150, 1999.

[3] S. Nicaise, Y. Renard, and E. Chahine.Optimal convergence analysis for the extended finite element method.http://hal.archives-ouvertes.fr/hal-00339853 2009.

[4] ML. Williams.On the stress distribution at the base of a stationary crack.ASME Journal Applied Mechanics, 24:109–114, 1957.

Conclusions and perspectives

Hydrid analitycal / extended finite element methodAccuracy and robustness wrt. geometrical parametersQuasi-optimal convergence of SIFs

Crack propagationAnalytical solutions for cohesive cracksDigital image correlation

julien.rethore@insa-lyon.fr

LaMCoS, Universite de Lyon, CNRS, INSA-Lyon UMR5259, 18-20 rue des Sciences - F69621 Villeurbanne Cedex