Mécanique linéaire élastique de la rupture (rupture ...lazarus/Telecharger/SoutenanceHdRVLazarus.pdf · Mécanique linéaire élastique de la rupture (rupture fragile) tridimensionnelle:

Mécanique linéaire élastique de la rupture (rupture fragile)tridimensionnelle: de la théorie à la pratique

Three-dimensional linear elastic fracture mechanics (brittlefracture): from theory to practiceHabilitation à Diriger des Recherches

Véronique Lazarus

Laboratoire FAST, UPMC Univ Paris 6

Tuesday 6th of July 2010

1

CV

1991-1994 Engineering degree.Ecole Nationale Supérieure de Techniques Avancées.

1994 DEA (Postgraduate degree)in Mechanical Engineering (University of Paris 6).

1994-1997 PhD in Mechanical engineering at the Laboratoire deModélisation en Mécanique (Paris 6).Advisor: J.-B. Leblond.“Some three-dimensional problems of brittle fracturemechanics”.

1997–98 ATER at University of Paris 6.1998– Associate professor at University Paris 6.

2

Teaching

Almost 2200 teaching hours.60% practical (TD), 20% numerical work (TP num), 20% Lab work (TP exp),20% courses.

I Continuum thermomechanics.I Solid mechanics.I Fluid mechanics.I Mathematics.I History of sciencesI Computer science.

3

Research

1994–2006 Laboratoire de Modélisation en Mécanique (LMM),incorporated in 2007 into the Institut Jean Le Rondd’Alembert (IJLRDA, UMR 7190).Mechanics of Solids and Structures Team.

2006–2008 Institut Jean Le Rond d’Alembert and FASTlaboratoryduring teaching sabbatical leave (3 X 6 months, CRCT,Delegation CNRS).

2008– Laboratoire Fluides, Automatique et SystèmesThermiques (FAST, UMR 7108).Porous and Fractured Media Team.

4

Publications

I 21 articles in peer-reviewed journals(7 J Mech Phys Solids, 4 Int J Solids Struct, 3 Cr Acad Sci II B, 2 Int JFracture, 1 Phys Rev E, 1 EPL-Europhys Lett, 1 Langmuir, 1 J ApplMech-T ASME, 1 Reflets de la Physique).

I 13 articles in International Conference proceedings.I 2 successful ANR and 2 successful Triangle de la Physique funding

requests.

5

PhD students

I On 3D LEFM theory with J.B. Leblond:09/2001-10/2005 Elie FAVIER.09/2005-07/2009 Nadjime PINDRA .09/2008– Laurène LEGRAND.

I On drying of colloidal suspensions with L. Pauchard:2007 Cécile BOUSQUET. Industrial PhD, CEA

Marcoule.10/2007– Mourad CHEKCHAKI.

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Brittle Fracture mechanics (LEFM): History

I Starting point: Cracked Liberty ship (second World War).I Griffith 20’s→ Irwin 50’s (Naval research Laboratory).I Failure occurs for small strains and negligible plasticity.

(glass, metal at low temperatures, rocks...).

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Brittle Fracture mechanics: the aims

Take an elastic loaded body:

1. Will a crack appear?

2. If yes, can we predict its shape? Can we predict the number of radialcracks?

3. If cracks are present, will they propagate, over which distance? until thetotal breakdown of the body?

4. Can we predict the crack front shape?

8


JL Prensier

Take an elastic loaded body:1. Will a crack appear?2. If yes, can we predict its shape? Can we predict the number of radial

cracks?3. If cracks are present, will they propagate, over which distance? until the

total breakdown of the body?4. Can we predict the crack front shape?

8


JL Prensier




8


J Tignon




8

Outline

1. Bases (Irwin 1950-)

If cracks are present, will they propagate, over which distance? until thetotal breakdown of the body?

2. Deformation of the crack front (Rice 1985-)

Can we predict the crack front shape?

3. Crack initiation (2000’s)

Will a crack appear?If yes, can we predict its shape? Can we predict the number of radialcracks?

9

Traditional LEFM approach

~up

∂Ωt

∂Ωu

~T p

s~e2(s)

~e1

F

~e3(s)

E , ν

Linear Elastic material. Condition of crack propagation?

10

Definition of the Stress Intensity Factors

~e12

~e2

31

process zone

rM

Westergaard (1938), Williams (1952), Leblond and Torlai (1992):

σ1p(M) ∝ Kp(s)√r

for r → 0.

Stress Intensity Factors (SIFs): K1(s), K2(s), K3(s)

Energy release rate G:

G(s) ≡ −dEelast

dS

=1− ν2

E(K1(s)2 + K2(s)2) +

1 + ν

EK3(s)2 Irwin’s formula

11

Crack propagation direction criterions

Whatever the loading, in an homogeneous brittle material, the crackpropagates in order to reach a situation of pure tension loading (Hull, 1993).

Mode 2: local kink ϕ

PLS: Goldstein and Salganik (1974)MTS: Erdogan and Sih (1963)

Review: Qian and Fatemi (EFM, 1996)

Mode 3: rotation along x1

Lazarus et al. (JMPS 2001-I,II, IJF2008)

Lin et al. (IJF 2010)Pons et Karma (Nature 2010)

12

Crack propagation direction criterionsIn memory of F. Buchholz (univ. Paderborn, Germany) who made thefollowing experiments in PMMA:

Mode 1+2: in-plane 4PS Mode 1+2+3: in-plane 3PB

Now, coplanar propagation (weak interface) even in mode 1+2+3

13

Crack advance versus loading criterions

I Brittle fracture: Griffith (1920)’ criterion

G < Gc ⇒ no propagation,

G = Gc ⇒ propagation.

In mode 1, it is equivalent to Irwin (1958)’s criterion:

K1 < Kc ⇒ no propagation,

K1 = Kc ⇒ propagation.

I Fatigue, subcritical fracture: Paris (1961)’ type law

∂a(t)∂t

= CGβ

For β 1, regularization of Griffith (1920)’ threshold criterion.

14

Determination of the SIF

I Engineering: Finite Elements simulations, XFEM (Belytschko et al.2000’s).

From Lazarus, Buchholz, Fulland, Wiebesiek (IJF, 2008).

I Research analytical approach:Crack front perturbation approaches initiated by Rice (1985).

15

Outline

Bases of the LEFM approach

Deformation of the crack front shape

Crack initiation

Conclusion

16

Crack front perturbation approach:δK(s) knowing their initial values K(s)

δ(s)~e2(s)

s0

τ

s

σ

K + δK

KI Mode 1: Rice (1989)I Mode 2+3 : Favier, Lazarus and

Leblond (IJSS, 2006)

δKi (s0) = Nij (ν) · Kj (s0)δ′(s0)

+1

2πPVZF

Wij (s0, s)

D2(s0, s)· Kj (s) [δ(s)− δ(s0)~e2(s0).~e2(s)] ds.

+Similar, but more complex, formula for δWij (s0, s1).

Initialisation: geometry for which Wij (s, s0) are known...

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Initialisation: Circular cracks

x

y

z

θ

σ∞zy

σ∞xy

σ∞yy

a

An internal circularcrack

x

y

z

θ

Oa

F∞ or U∞0

M∞0 or Ω∞

An external circular crack

I Internal: Kassir and Sih (1975), Tada et al. (1973), Gao et Rice (1987),Gao (1988).

I External: Stallybrass (1981), Gao et Rice (1987), Rice (1989)

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Initialisation: Half-plane cracks

z

x

y

I Homogeneous case: Meade and Keer (1984), Bueckner (1987), Rice(1985), Gao and Rice (1986).

I Interfacial crack: Lazarus et Leblond (1998), Bercial-Velez et al. (2005),Piccolroaz et al. (2007).

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Initialisation: Tunnel cracks

2a

x

y

z

σ

A tunnel-crack loaded by:I remote tensile: Leblond, Mouchrif et Perrin, 1996;I shear stresses: Lazarus and Leblond, 2002

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Examples of application of the perturbation approach

1. Largescale propagation simulations

2. Stability of the straight crack front shape in an homogeneous media

3. Crack propagation in heterogeneous media

21

Largescale propagation simulations: PlaneCracks

τ

σ

F

I Initialisation of K, W: internal circle under remote tensile or shear loadingI Step 1: Determination of K and W along the front F

by successive small perturbations of the circleI Step 2: Determination of the crack advance by Paris’ law

∂a(t)∂t

= CGβ

Rice (1989), Bower and Ortiz (1990), Lazarus (2003), Favier, Lazarus,Leblond (2006)

22

Examples in mode 1

Brittle fracture β = 50

The stationnary shape is circular.

Lazarus, 2003

23

Examples in mode 2+3 (coplanar propagation case)

τ a

b

Brittle fracture β = 50

The stationnary shape is nearly elliptical with:ab

= (1− ν)ββ+1

ab

= (1− ν) if β 1 (G = Gc)

Favier, Lazarus, Leblond (IJSS, 2006)

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The configuration stability problem

x

y

z

I Problem: circular, straight cracks are often used in engineering, is itsafe?

I If the crack front is perturbed, will the perturbation increases (instable) ordecreases (stable) in time?

This stability problem has been studied byI Rice and Gao (1985-1990) for circular and half-plane cracksI Lazarus and Leblond (1998) for the interfacial half-plane crackI Leblond, Favier, Pindra, Lazarus, Mouchrif, Perrin (1996-) for

tunnel-cracks

25

Straight front stability in mode 1: unperturbed problem

x

y

z

K = cst , α = 0

z

x

y

a

−σ

σ

K = 2q

2πσa1/2, α = 1/2

z

x

y

aP

−P

K =q

2πPa−1/2, α = −1/2

K (a) = kaα

I if α > 0,dK (a)

da> 0 : instable propagation at constant loading.

I if α < 0,dK (a)

da< 0 : stable propagation at constant loading.

26

Model problems: SIF along the perturbed configuration

x

y

z

a

δ(z)

δK (z)

K (a)= α

δ(z)

a+

12π

PVZ ∞−∞

δ(z′)− δ(z)

(z′ − z)2 dz′

In Fourier transform along z−axis:

δbK (k)

K (a)=

„α− ka

2

« bδ(k)

ak wavenumber

27

Stability of the crack shape. Case α < 0,dK (a)

da< 0.

Then α− ka2< 0 whatever the value of k .

Since,δbK (k)

K (a)=

„α− ka

2

« bδ(k)

a

it implies that any perturbation disappears.

B

Aλ

a

λ aK (A) < K (B)

a λ

A

B

λ aK (A) < K (B)

28

Stability of the crack shape. Case α > 0,dK (a)

da> 0.

λ aK (A) < K (B)

B

Aλ

a

λ aK (A) > K (B)

a λ

A

B

29

Stability of the crack shape. Case α > 0.

Stabilityλ < λc

K (A) < K (B)

B

Aλ

a

Bifurcationλ = λc = λ∗c aK (A) = K (B)

where

λ∗c =π

α

Instabilityλ > λc

K (A) > K (B)

a λ

A

B

Since λc = λ∗c a with a, ultimately any perturbation tends to disappear.If :

I The first order approach remains validI The fracture properties are homogeneous...

30

Crack propagation in heterogeneous media

Crack front shape such as K (x , z) = Kc(x , z), ∀(x , z) ∈ F?For slightly toughness heterogeneities:Kc(z, x) = Kc(1 + ∆Kc(z, x)), |∆Kc | 1:

δ(k , a) = −ad∆Kc(k , a)ka2− α

Meaningless if α ≥ 0 due to the existence of a bifurcation

⇒ α < 0 in the sequel.

Application to

1. Crack trapping by obstacles

2. Crack propagation in disordered medium

31

Application to the crack trapping

y

z

Kc

Kc(1 + ∆Kc)

1. Gao and Rice (1989, 1991)

2. Dalmas, Barthel, Vandembroucq (2009)

3. ANR MEPHYSTARTheory: S. Patinet, postdoc with V. Lazarus D. Vandembroucq.Experiments: L. Alzate (PhD D. Dalmas, St Gobain)

32

Application to disordered medium.

δ(k , a) = −ad∆Kc(k , a)ka2− α

I Power spectrum of the crack front fluctuations:

|δ(k , a)|2 = a2 |d∆Kc(k)|2„|α|+ ka

2

«2

as a function of |d∆Kc(k)|2 power spectrum of the toughness fluctuations∆Kc .

I If |d∆Kc(k)|2 = cst = bK0 (white noise), one obtains:

|δ(k , a)|2bK0a2=

1„|α|+ ka

2

«2

which corresponds to a Family-Viscek (1985) scaling ζ = 0.5 and τ = 1.

33

Application to disordered media. Other results

I Interfacial/homogeneous: minor influence.

Pindra, Lazarus, Leblond, JMPS 2008.I Mode mixity: minor influence

Pindra, Lazarus, Leblond, JMPS 2010I Tunnel-crack/half-plane crack: minor influenceI Loading α: MAJOR influenceI Irwin/Paris law: MAJOR influence with a memory effect

bδ(k , a) =

Z a

a0

»exp(−ka/2)

exp(−ka′/2)

–2β “ aa′”βcδc(k , a′) da′,

Lazarus and Leblond (2002); Favier, Lazarus and Leblond (JMPS, 2006).

Statistical physics approach:Daguier, Bouchaud and Lapasset (1995), Schmittbuhl, Maloy et al.(1995-2010), Ramanatha, Fisher, Ertas (1997), Krauth and Rosso (2002),Hansen et al. (2003), Roux, Vandembroucq, Hild, (2003), Katzav etAdda-Bedia (2006), Ponson (2007), Bonamy (2009)...

34

Interaction between several cracks

I Interaction between two tunnel-cracks:

I Determination of Wij for two-tunnel cracks (Pindra, Lazarus, Leblond, 2010)

I Bifurcation wavelength λc = λ∗a when a (Legrand, Leblond, 2010).

2a2b 2b

σ

I Extension of the code PlaneCracks for two cracks (PhD of L. Legrand).

35

Comparison of PlaneCracks with experiments

Lazarus (2003) Dupeux et al. (1998)

Collaboration envisaged with Muriel Braccini (SIMAP, Grenoble).One crack, interaction of two cracks (L. Legrand).

36

Heterogeneous media: comparison with experiments

D. Dalmas and D. VandembroucqANR Mephystar J. Schmittbuhl et al.

37

Outline

Bases of the LEFM approach

Deformation of the crack front shape

Crack initiation

Conclusion

38

JL Prensier

1. Will a crack appear?

2. If yes, can we predict its shape?

39

Variational approach to fracture.Bourdin, Francfort and Marigo (1998-2008)

PrincipleF cracks such as Etot (F) ≡ Eelastic(F) + Gc length(F) is minimum.

I Identical to the traditional approach if a crack is still present.I Applicable only if an “idea” of the crack shape.

Regularized form: Non-local damage modelα(M) damage field such as Etot (α) ≡ Eelastic(α) + Gc f (α, `) is minimum.

where f (α, `) chosen such as lim`→0 f (α, `) = length(F).I Convergence toward the initial principle for `→ 0.I Suitable for numerical purposes.

40

Directional drying in capillary cells

Gauthier, Lazarus and Pauchard(Langmuir 2007, EPL 2010)

I Capillary tubes of diameter ∼ 1 mm.I Ludox R© colloidal aqueous suspension of

silica particles (diameter ∼ 10 nm).I Drying from the single bottom open edge.I Contraction prevent by adhesion⇒ tensile stresses⇒ vertical star-shaped cracks.

I Variation of the drying conditions⇒various number n of radial cracks

Model: Linear elastic cross-section 2Dproblem:

σ =Eν

(1 + ν)(1− 2ν)trε1+

E(1 + ν)

ε+σ01

41

Use of the regularized approach.

VL, Gauthier, Pauchard, Maurini, Valdivia (ICF12, 2009):

Qualitative agreement.Idea of the crack shape.

42

Use of the direct approachFor dimensional reasons:

n = f (L)

where L =Lc

R∝ fracture energy

elastic energy

More precisely, Lc is the Griffith length defined by (Kc toughness):

Lc =EGc

σ20

=K 2

c

σ20

0

1

2

3

4

5

6

7

8

9

0.01 0.1 1

n

Lc/R43

Comparison theory/experiments:

Good agreement between theory and experiments.

44

Application: Geological shrinkage crack patterns

Septarias Giant’s causeway, Ireland Port Arthur, Tas-mania

Minimum principle⇒ Lc

R=

K 2c

σ20

= f−1(n)⇒ σ0 =Kcp

f−1(n)⇒ origin?

With A. Davaille (FAST), S. Morris (Univ Toronto), colleagues of IDES (Orsay)

45

Application to crack nucleation

Impact (Vandenberghe, Ver-morel, 2009)

Indentation (Rhee et al.,2001)

Drying of a thin colloidal sus-pension film (L. Pauchard)

46

Conclusion.

New theoretical developments in LEFM in the last 10-20 years:

I Traditional approachI Statistical physicsI Variational approachI Phase-fieldI XFEMI Experiments

Now:I CollaborationsI Application several fields:

I EngineeringI Soft MatterI Geophysics

47

Conclusion.





x

y

z

a

δ(z)

Gao, Rice, Leblond, Lazarus et al. (1985-)

Bower, Ortiz, Favier, Lazarus, Leblond (1990-)

47

Conclusion.





Bouchaud, Schmittbuhl, Maloy, Hansen,Roux, Vandembroucq, Katzav, Adda-Bedia,

Ponson, Bonamy ...(1995-)

47

Conclusion.





Marigo, Francfort, Bourdin, Maurini (1998-)

47

Conclusion.





Corson et al. (PRL 2009)

Karma, Hakim, Henry, Levine...(2001-)

47

Conclusion.





Belytschko et al. (2000):

Belytschko, Moes, Gravouil, Sukumar,...(1997-)

47

Conclusion.





Pauchard, Gauthier

Dalmas, Barthel, Alzate, Teisseire (St Gobain)

47

Conclusion.New theoretical developments in LEFM in the last 10-20 years:




x

y

z

a

δ(z)

Lazarus

Dalmas, Vandembroucq (ANR MEphystar)Schmittbuhl et al.

47





Bourdin, Maurini

Gauthier, Lazarus, Pauchard

47

Conclusion.





Marigo, Francfort, Bourdin, Maurini (1998-)

Karma, Hakim, Henry, Levine...(2001-)

47





Belytschko et al. (2000):

Belytschko, Moes, Gravouil, Sukumar,...(1997-)

Lazarus

47

Conclusion.





M. Braccini, SIMAP (Grenoble)

Lazarus

47

Conclusion.



Now:I CollaborationsI Application several

fields:I EngineeringI Soft MatterI Geophysics

Inverse method: determination of ∆Kc

More tough materials

47

Conclusion.





n = f (Lc/R)Shrinkage cracks in wood or concrete

47

Conclusion.





Better comprehension of the consolidationprocess

47

Conclusion.





Origin?

47

x

y

z

a

δ(z)

48

Mécanique linéaire élastique de la rupture (rupture ...lazarus/Telecharger/SoutenanceHdRVLazarus.pdf · Mécanique linéaire élastique de la rupture (rupture fragile) tridimensionnelle:

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