A hybrid analytical / extended finite element method for direct evaluation of stress intensity factors Julien R´ ethor ´ e 1 , St ´ ephane Roux 2 , Franc ¸ois Hild 2 1 LaMCoS, INSA-Lyon 2 LMT Cachan Introduction Stress intensity factors K I and K II are key parameters for fracture mechanics Need for a robust evaluation No post-processing, direct evaluation Mesh independence Optimal convergence ... Displacement field in m PU enrichment [2, 3] 10 -2 10 -1 10 0 10 -3 10 -2 10 -1 10 0 h/w |K I -K Io |/K Io n max =1 n max =3 n max =5 n max =7 convergence rate < 0.5 singular enrichment • discontinuous enrichment u = X i ∈N N i ¯ ud i + X i ∈N cut N i Hb i + X i ∈N tip X j =I ,II N i F j K j Crack displacement fields Let us consider a homogeneous body with isotropic elastic behavior, and a 2D setting, the displacement field u is conventionally represented by its complex writing, u = u x + iu y . It was expanded by Williams [4] for a straight crack as a double series u (r ,θ )= X i =I ,II X n c n i φ n i (r ,θ ) with φ n I (r ,θ )= r n/2 κe inθ/2 - n 2 e i (4-n)θ/2 +( n 2 +(-1) n )e -inθ/2 φ n II (r ,θ )= ir n/2 κe inθ/2 + n 2 e i (4-n)θ/2 - ( n 2 - (-1) n )e -inθ/2 where κ is Kolossov’s constant, namely, κ =(3 - ν )/(1 + ν ) for plane stress or κ =(3 - 4ν ) for plane strain conditions, ν being Poisson’s ratio. n = 0: translation n = 1: usual asymptotic fields n = 2: rotation and T -stress n ≥ 3: sub-singular fields Hybrid model Ω 1 Ω 2 Ω 12 Model 1: u 1 (x )= X i ∈N 1 N i (x )d i + X i ∈N cut N i (x )H(x )b i Model 2: u 2 (x )= 1 2μ √ 2π X n∈[0;n max ] ( φ n I (x ) p n + φ n II (x )q n ) Partition of energy: α 1 (x )+ α 2 (x )= 1 Weighted bilinear forms: a i ( u i , v * i ) = Z Ω i α i (u i ): C : ( v * i ) dΩ Coupling: Π(u 1 - u 2 ,μ * )= Z Ω 12 λ · (u 1 - u 2 ) dΩ= 0 Arlequin method [1]: a 1 ( u 1 , v * 1 ) + a 2 ( u 2 , v * 2 ) +Π(u 1 - u 2 ,μ * )+Π ( v * 1 - v * 2 ,λ ) = l 1 ( v * 1 ) Results R inner R outer Displacement field in m K Io = f a w σ √ π a = 2.98 MPa √ m Region of analytical model R outer = r outer h = 13h Region of ‘inactive’ elements R inner = r inner h = 10h Coupling region R outer - R inner = ‘ overlap h = 3h Number of terms in the analytical model n max = 5 0 5 10 15 0.8 0.85 0.9 0.95 1 1.05 r inner K I /K Io n max =2 n max =3 n max =5 n max =7 ‘ overlap = 1 ⇒ large r inner needs higher n max to accomodate boundary effects 10 -2 10 -1 10 0 10 -3 10 -2 10 -1 10 0 h/w |K I - K Io |/K Io n max =2 n max =3 n max =5 n max =7 r inner = 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. Mo ¨ es, 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 method Accuracy and robustness wrt. geometrical parameters Quasi-optimal convergence of SIFs Crack propagation Analytical solutions for cohesive cracks Digital image correlation [email protected] LaMCoS, Universit´ e de Lyon, CNRS, INSA-Lyon UMR5259, 18-20 rue des Sciences - F69621 Villeurbanne Cedex
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
LaMCoS, Universite de Lyon, CNRS, INSA-Lyon UMR5259, 18-20 rue des Sciences - F69621 Villeurbanne Cedex
Related Documents
