Lecture 4: Fracture of metals and polymers II. Crack

Institute of Mechanics, Materials and Civil Engineer ingEcole Polytechnique de Louvain& Research Center on Architectured and Composite Materials (ARCOMAT)

Lecture 4: Fracture of metals and polymers II. Crack propagation

10 mars 2016Institut de Mathématiques, Quartier Polytech 1, Allée de la Découverte12, Bâtiment B37, 4000 Liège - Auditoire 02

Chaire Francqui 2016, ULg

Thomas Pardoen

Outline

1. Experimental characterization of ductile tearinga. The traditional JR curve approach & cob. The essential work of fracturec. A few words on the mechanisms

2. Modelling strategies

3. Example 1 : Thick components - prediction of JR curves with Gurson type model

4. Example 2 : Thin sheet fracture

5. Example 3 : Application of « stress at a distance » model in adhesive joints

6. Example 4 : Fracture in Al with PF

y

x

z

rθ

σx

y στxy

zσ

σrθσ

xy

z

Mode I Mode II

x

z

y x

z

Mode III

y

( ) ( ) ( )∑∞

=

−+=

,...3,2

12

1 ,,2 n

n

nijij fraKCfr

K θθπ

σ II

IIII dimensionsother nij

2a

σ

σ ∞

∞

KI = σ ∞ πa

1.5

1.

0.5

0.0.0001 0.001 0.01 0.1 1. 10.

r/a

σ yyapprox./σ yyexact

1st term only

1st and 2nd terms

1st, 2nd and 3nd terms

exact

( ) ( ) ( )θπ

θπ

θπ

σ III1

IIIII1

III1

I

r2r2r2ijijij f

Kf

Kf

Kij ++=

aYK ∞= σIIIor IIor I

Back to linear elastic staticcrack analysis

JR curves

2

strain plane

3

1

=

0

I

σπK

rY

2

stress plane 1

=

0

I

σπK

rY

log σyslope -1/2

log r/L Finite strain zone

Plastic zone

K-dominated zone

K-field validity

Small Scale Yielding

Validity of LEFM

JR curves

F

F

hB

a

h

A

CF

∂∂

∂∂

2

2

=A

-=P

G

32

2212

hEB

aF=G

−++=

ν1*

1 222 III

III

KKK

EG

Energetic approach (energy release rate)Linear elastic

P is the potential energy of the systemC is the compliance of the structureA is the crack area

JR curves

G ≥ Gc (J/m2) or KI ≥ KIc (MPa.m1/2)

K ≥ KR(∆a) (or G ≥ GR(∆a))

∆a

R

K Ic

K (∆a)

∆a

K ss

∆aRef

K eng

(b)

a

RG , G

ainitial

G G R

Stable

Unstable

Fracture resistance curves(linear elastic)

Cracking initiation (fracture toughness) :

Crack propagation

Stability

JR curves

( )

∫

−=

−=

∂∂

σ

∂∂

dsx

unnW

AJ

jijixV

2J/m P

( ) ∫−=

F

FduawB

J0

η

F

Ψ

uw-a

S

B

wa

( ) duFawB

JF

∫−=

0

2SENBdeep

Non linear fracture mechanics: J integral

valid for radial loadingsWv is the strain energy density

JR curves

J.R. Rice, 1968

0

1

2

3

4

5

6

0 2 4 6 8 10

σyy

/σ0

rσ0/J

Material with N=10 αε0=0.001

εplan

σplan

HRR fields

J2 deformation theory(non linear elastic response)

See HHR tables by Fong Shih, 1983 (Brown University )

JR curves

Hutchinson, JMPS1968; Rice and Rosengren, JMPS 196 8

N

=

00 σσα

εε

( )NrI

Jij

N

Nij ,~1

1

000 θσ

εασσσ

+

=

( )NrI

J N

N

,~1

1

000 θσ

εασσσ

+

=

( )NrI

Jij

N

N

Nij ,~1

000 θε

εασαεε

+

=

JR curve and definition of fracture toughness

JR curves

From B. Tanguy, CEA

Fatigue precracking of CT or SENB specimens,

then load with partial unloading sequences


JR curves

From B. Tanguy, CEA


Ji

∆a0.2 mm

J

loss of constraint

Steady state regime

active plasticzone

crack wake

JIc ΓSS

Γp

JR curves

J = Γ0 + Γp

Ji

y

x

δ

Initial crackr

u

uy

x

45°

( )NurI

Jru i

N

N

Ni ,~1

000 θ

εασαε

+

=

Fracture toughness can also bedefined as critical CTOD δc

JR curves

( )0

0,σ

αεδ JNd=

Crack tip opening displacement

Within the finite strain zone

3 41 5 62

0.05-0.1

εp

0.577

Stress triaxiality

loss of constraint

increasingn

constraint changes

r/δFracture process zone

this is where damage develops !

JR curves

σyy/σ0

3

Evolution of stress fields duringcrack propagation

JR curves

Slight stress elevation during crack propagation

3 41 5 62

HRR ~ 0.6

Stress triaxiality

increasingsheet thickness

r/δ

Fracture process zone = necking zone

0.6

1.

mid-planesurface

Additional complexitycoming from 3D effects

JR curves

log σy

slope -1/(N+1)

slope -1/2

log r/L

log σ y

slope -1/(N+1)

log r/L

log σ y

slope -1/(N+1)

log r/L

slope -1/2

Finite strain zoneHRR validity

J-dominated zone

Plastic zone

K-dominated zone

K-field validity

Finite strain zone

HRR validity

J-dominated zone

Plastic zone

Finite strain zonePlastic zone

Small Scale Yielding Large Scale Yielding

No single parameter approach

slope -1/2

(a) (b)

The big picture on the validity

*

2

E

KJ =≡G only J valid

JR curves

Outline

1. Experimental characterization of ductile tearinga. The traditional JR curve approachb. The essential work of fracturec. A few words on the mechanisms






pef wlww α0+=

00tl

Ww f

f =

DENT geometry

pepef wltwltWWW 20000 α+=+=

Essential Work of Fracture (J)

Plastic Work (J)

Total work of Fracture (J)

Thickness samples (m)Ligament length (m)

Essential specific Work of Fracture (J/m²)

Plastic work density (J/m 3)

(Double EdgeNotch Tension)

The essential of fracture method

Essential work of fracture

Cotterell and Reddel, IJF, 1977

l0


Example EWF method

0

5 104

1 105

1.5 105

2 105

2.5 105

3 105

3.5 105

4 105

0 5 10 15 20 25

t0 = 1 mm

t0 = 2 mm

t0 = 3 mm

t0 = 4 mm

t0 = 5 mm

t0 = 6 mm

wf (J/m2)

l0 (mm)

Al 6082 O temper– 6 thicknesses

Pardoen, Marchal, Delannay, JMPS 1999


EWF method widely applied to polymers

Outline







Classical ductile crack propagation mechanisms

Initial sharp crack (e.g. fatigue precrack)

Crack tip blunting

Physical initiation of cracking

Fracture mechanisms at crack tip


Crack tip just after initiation in Cu from Pardoen & De lannay, EFM 2000

Fatigue zoneBlunting zone =

stretch zone width(≈ δc/2) Ductile tearing


Fracture surface near the original crack tip

Propagating crack remains sharp (opening on the order of voidsize and not much larger like at cracking initiation)

Crack tip opening angle remains relatively constant af ter the initiation transient

CTOA


Classical ductile crack propagation mechanisms

Additional complexity

Mechanisms at intermediate scale change with plate thickness

and also with loading rate, environment and temperature …


Recent result : localization leading to slant fracture can occur before damage

Recent progress in the characterization of slant fracture


Morgeneyer et al., Acta Mater 2014


Recent progress in the characterization of slant fracture

Localization leading to slant fracture after damage (preexisting or grown) - major effect of second

population and specimen orientation Ueda et al., Acta Mater 2014

… but the transition or not to slant fracture dependi ngon strain hardening capacity, texture, rate dependenc y

etc is still an open issue ….

G or J (kJ/m )

c c2

Specimen thickness (mm)

20

200

100

2 10

Sometimes, even more complex : flip-flap fracture


see recent work by K.L. Nielsen, J.W. Hutchinson

In the absence of slantfracture, major thickness effect

Ductile tearing in bulk materials with undefinedfracture process zone vs plastic zone


Du et al., Acta Mater 2000

« extreme case » : perfectly ductile flow with no fracture

Can we still talk about fracture ? Does it matter ?


Summary of the mechanisms


Benzerga, Pardoen, Pineau, Acta Mater 2016

Outline







Starting point : to go beyondfracture mechanics approach

From X.K. Zhu, S.K. Jang, EFM 68 (2001) 285-301

JQ/JT/JA 2 theories … doubts …

Modelling strategies

Modelling strategies

A few references


Hutchinson and Evans, Acta Mater 2000

The big pictureModelling strategies


Outline







( )( )

31

0

0

43

=

−f

Tnucl

x

πεε

ε

HRRHRR

HRR

23

exp0.43exp22

exp

Analytical « first order » qualitative model

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2 0.25 0.3

σm/σ

e HRR

n

Plane stress

Plane strain

Assume ε0nucl ≈ 0

( )0

HRRN

ln

2

30.exp

3

20.43-

f3

156.0

~~1

0000

+≈

+

TXI

Jxx

NN

εεσαε

αε

( )N

N

xx T

XIJ

1

0

000~~

f31

56.0

+

+≈

HRR

0

NIc

2

3exp

3

20.43-

ln

εεαεσαε

≈ thickness,,, 00

00 fE

NFXJσσIc

Example 1

See models R&T + Brown&E in lecture 3

Simple uncoupled model basedon elastoplastic solution

X0/2

δ0(a)

See earlier works by McMeeking

Example 1

FE model with discrete void in elastoplastic matrix

X0

(b)

Tvergaard & Hutchinson, IJSS, 2002

Two “slightly different” ductile tearing mechanisms

X0

Z0

2R0

2Rz0

δ0

I. Multiple void interaction

Finite strain zone

II. void by void growth

Plastic localization = coalescence = finite strain zone

Plastic localization = coalescence

Example 1


Small scale yielding analysis (infinite medium) of ductile fracture with advanced “Gurson” model

K field

Model implemented in "ABAQUS Standard" through a User defined MATerial (UMAT), finite strain setting, fully implicit – or home code (Ph . D. Florence Scheyvaerts)

Pardoen, T., Hutchinson, J.W., 2003 Acta Mater . 51, 133-148.

Example 1

X0

(c)

0

0.5

1

1.5

2

0 0.01 0.02 0.03 0.04

Tvergaard and Hutchinson, 2001

Present work - 1 row of voids

Present work - voids everywhere

1/3

1/2

0

R=

σ0/E=.003, n=0.1, W

0=1, λ

0=1

JIc

/σ0X

0

f0

+ extended Gurson

Dotted lines

ValidationExample 1

Pardoen & Hutchinson, Acta Mater 2003


Plane strain fracture toughness of ductile metallic alloys

0

2

4

6

8

10-6 10-5 10-4 10-3 10-2 10-1

σ0/E = 0.003

σ0/E = 0.01

σ0/E = 0.001

f0

JIc

/σ0X

0

n = 0.1, W0 = 1, λ

0 = 1

Negligible effect of σσσσ0/E on JIc/σσσσ0X0

= ... ,,,,, 0000

00 λσσ WfE

nFXJIc

Example 1


0

2

4

6

8

10

10-5 10-4 10-3 10-2

f0

0.2

n=

σ0/E = .003, W

0 = 1, λ

0 = 1

JIc

/σ0X

00.1

0.01

Very strong effect of the strain hardening exponent n on JIc/σσσσ0X

Example 1


= ... ,,,,, 0000

00 λσσ WfE

nFXJIc


Large effect of strain hardening exponent partly ex plains why fracture toughness usually decreases with increasin g strength –

not a direct effect of the strength


σ

ε

↑ σu often means ↓ n

+ also an effect of second population of voids !

Example 1


= ... ,,,,, 0000

00 λσσ WfE

nFXJIc


σ0/E = .003, n = 0.1, λ

0 = 1

0

2

4

6

8

10-5 10-4 10-3 10-2

1/3 1

W0=

1/10

3

10

JIc

/σ0X

0

f0

Very strong effect of the initial void shape on JIc/σσσσ0X0

Example 1


= ... ,,,,, 0000

00 λσσ WfE

nFXJIc


Plane strain ductile tearing resistanceof ductile metallic alloys

0

10

20

30

40

50

60

70

0 20 40 60 80 100

∆a (coalescence)

MODE I, f0=10-3, W0=1, λ0=0, θ0=0σ0/E=3.10-3, n=0.1, non homogeneous

θc=0, θ

c=qcq

NLG=1

NLG=10

NLG=18

The number of « Gurson » rows has an impact on the predicted tearing resistance

Example 1

Scheyvaerts, PhD UCL 2008


Relatively small impact of void nucleationstress on tearing modulus

0

10

20

30

40

50

60

70

0 20 40 60 80 100

JR

/(σ0 X

0)

∆a/X0

k=1.5

k=3.8

k=3.5

k=2.5

k=4

k=0

k=4.25

f0=10-3, W

0=1, λλλλ

0=1, PZ=108x18

k=0, PZ=108x30

σnucleation = k σ0

Example 1


Tearing modulus increases with decreasing initial porosi ty

0

5

10

15

20

25

30

35

40

0 20 40 60 80

JR/(σ

0 X

0)

∆a / X0

n=0.1, W0 = 1, λλλλ

0 = 1.0, θθθθ

0 = 0.0

f0 = 10-2

f0 = 10-3

f0 = 10-4

Example 1



Plastic zone size increases with decreasing initial poro sityand with crack propagation

0

1000

2000

3000

4000

5000

0 20 40 60 80

f0 = 10-2

f0 = 10-3

f0 = 10-4

rp / X0

∆a / X0

n=0.1, W0 = 1.0, λλλλ0 = 1.0, θθθθ0 = 0.0

predictedanalytical

smooth fit of the original curve

ry

Example 1



Effect of initial void shape on tearing modulus issignificant mainly for oblate voids

0

10

20

30

40

50

0 20 40 60 80

JR/(σ

0 X

0)

∆a / X0

n=0.1, f0 = 10-3, λλλλ

0 = 1.0, θθθθ

0 = 0.0

W0 = 1/6

W0 = 1

W0 = 6

Example 1



Significant effect of void distribution

0

5

10

15

20

25

30

35

40

0 20 40 60 80 100

λ0 = 1/2

λ0 = 1

λ0 = 2

JR

/(σ0 X

0)

∆a / X0

n=0.1, f0 = 10-3, W

0 = 1, θθθθ

0 = 0.0

Example 1



CTOA

5 X0

Crack tip opening angle predictions

0°

5°

10°

15°

20°

25°

30°

35°

10 20 30 40 50 60 70 80 90

CTOA / 2

∆a / X0

n=0.1, W0 = 1, λλλλ

0 = 1.0, θθθθ

0 = 0.0

f0 = 10-2

f0 = 10-3

f0 = 10-4

Example 1



The main success of the micromechanics of ductile fracture : predicting constraint effects

in thick components

Example in 3D

Gao, Faleskog, Shih, Dodds, EFM 68 (2001) 285-301

Example 1

See also many studies in the group of A. Pineau & J. B esson from the Ecole des Mines de Paris

Outline







Gc

t

?

Mixed mode I-III crackingMode I cracking or

?

?G or J (kJ/m )

c c2


20

200

100

2 10

Ductile cracking in thin components –many open questions

Al alloy result

Example 2

( )( )

31

0

0

4

3

=

−f

Tnucl

x

πεε

ε

HRRHRR

HRR

2

3exp0.43exp22

exp

Analytical « first order » model

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2 0.25 0.3

σm/σ

e HRR

n

Plane stress

Plane strain

Assume ε0nucl ≈ 0

( )0

HRRN

ln

2

30.exp

3

20.43-

f3

156.0

~~1

0000

+≈

+

TXI

Jxx

NN

εεσαε

αε

( )N

N

xx T

XIJ

1

0

000~~

f3

156.0

+

+≈

HRR

0

NIc

2

3exp

3

20.43-

ln

εεαεσαε

≈ thickness,,, 00

00 fE

NFXJσσIc

X0 X0

cracking initiation

Example 2

0

0.5

1

1.5

2

0 1 2 3 4 5

Γ0/σ

0X

0

Stress triaxiality T

f0 = 0.01; n = 0.1; σ

0/E = 500

Typical stress triaxiality for "thick" cracked specimens with possible constraint effects

Analytical « first order » model explainswhy fracture toughness is larger in plane

stress compared to plane strain

But, it does not explain why fracture toughness would start first increasing

with increasing thickness

G or J (kJ/m )

c c2


20

200

100

2 10

Example 2

Based on FE void cellsimulations or extended

Gurson model + coalescence

L 0

t0

X0

Diffuse plastic zone

Localized necking zone

Final micro-zone of damage-induced localization

Additional effect in thinplates: crack tip necking

Example 2

L 0

t0


L0

Γ = Wtot

L0t0

t0

Γ0

wnwp

)()()( 0 aaa p ∆Γ+∆Γ=∆Γ

In thin plates

In thick plates

Wt = Γ0L0t0 + wnαt02L0 + wpβt0L0

2

In DENT panels

Γ = Γ0 + wnαt0 + wpβL0

Example 2

What are the energy contributions to the total work of facture ?

)()( 0 aa ∆Γ=∆Γ )( an ∆Γ+ )( ap ∆Γ+

0

5 104

1 105

1.5 105

2 105

2.5 105

0 1 2 3 4 5 6 7

Jc (J/m2)

we (J/m2)

t0 (mm)

24300 J/m2

28800 J/m2

33MJ/m3

29MJ/m3

0

100

200

300

400

500

600

0 2000 4000 6000 8000

JR (kJ/m2)

∆atot

(µm)

0.6 mm

6 mm4 mm

2 mm

1 mm

blunting line

Ductile tearing of thin 6082-O aluminium plates

Pardoen, T., Marchal, Y., Delannay, F., 1999. J. Me ch. Phys. Solids 47, 2093-2123.

Example 2

Materials Thickness(mm)

Homog. E(GPa)

σ0(MPa)

n(Swift)

k(Swift)

Stainless steel A316L 0.65 to 3 yes 210 310 0.48 25

Al 6082-O 0.6 to 6 yes 70 50 0.26 265

Brass annealed 0.9 to 2 yes 110 100 0.6 33

Al NS4 // RD 0.57 to 1.5 yes 70 140 0.17 159

Zinc // RD 0.6 to 1.3 yes 61 100 0.15 118

Lead 0.8 to 1.8 yes 16 7 0.25 290

Bronze annealed 0.54 to 1.2 yes 100 120 0.51 38

Bronze ⊥ RD 0.54 to 1.2 yes 100 400 0.01 ? ?

Bronze // RD 0.54 to 1.2 yes 100 410 0.015 ? ?

Brass ⊥ RD 0.9 to 2 no 110 (240) (0.25)

Brass // RD 0.9 to 2 no 110 (210) (0.32)

Al NS4 annealed 0.57 to 1.5 no 70 (80) (0.2)

Al NS 4 ⊥ RD 0.57 to 1.5 no 70 (150) (0.14)

Mild steel ⊥ RD 0.87 to 1.5 no 210 (240) (0.17)

Mild steel // RD 0.79 to 1.5 no 210 (220) (0.17)

Zinc ⊥ RD 1.3 yes 86 140 0.08

Pardoen, T., et al. 2004 J. Mech. Phys. Solids 52, 423.

Same observations for a wide range of metals !!!

Example 2

A316L Brass Annealed Al NS4

Triax = T

TTsides= 0.6 centreε

f

Cup & cup fracture

Example 2

Model for the work of necking

=Γf

0

0

n ,,,, ενσσ E

knF

( )( )∫ ∫=

2

000

maxn

2un

u

ddn

ε ε

εεσ

h hhtlW

( )∫ ∫=Γ

2

0

maxn

2u0

u

ddn

h hh

ε

εεσ

ε u =2nk− 3

3k

Assumption of plane strain tension

active necking region

hnhu

tu

tnnecking region in the initial configuration

h

Example 2

Pardoen et al. J. Mech. Phys. Solids 2004

Model for the work of necking

0

0.1

0.2

0.3

0 0.5 1 1.5 2 2.5

wn/σ

0kn

εf-ε

u

n=0.1

n=0.25

n=0.4

n=0.5

Example 2


L 0

t0

X0


Localized necking zone

Final micro-zone of damage-induced localization

Model for the work of fracture

),,,,( 0000

00

0 λσσ

WfnE

FX

=Γ

Φgrowth ≡C

σ 2s + ησ hgX

2+ 2q g + 1( ) g + f( )cosh κ

σhg

σ

− g+ 1( )2 − q2 g+ f( )2 = 0

Φcoalescence≡σe

σ +

3

2

σ h

σ − F W,χ( )= 0

Example 2


Extended Gurson model (Gologanu + Thomason)

f0

0.1

1

10

100

1000

10-5 10-4 10-3 10-2 10-1 100

Γ0/σ

0X

0

Γ0/J

Ic

n = 0.1

n = 0.1

n = 0.3

n = 0.5

Example 2

Model for the work of fracture

Extended Gurson model (Gologanu + Thomason)


Combining the work of neckingand work of fracture

1t/ry

SSY

00Xc

σΓ

stressplaneX00

0

σΓ

Ic

strainplane

JX

=00

0

σΓ

A

B

C

C − Γ0

A Slant fracture (tentative)

B

C

Flat fracture with large σ0 small n

Flat fracture with small σ0 largen

α2

α1

nnn

k

wEk

0030

1

σσα =

C − Γn

Example 2


… but not the transition or not to slant fracture dep endingon strain hardening capacity, texture, rate dependenc y, etc

As a matter fact, nobody (I take a risk !) has everproduced a convincing quantitative prediction of this kind of curve for the full range of thicknesses !

G or J (kJ/m )

c c2


20

200

100

2 10

Sometimes, even more complex : flip-flap fracture (see recent work by K.L. Nielsen, J.W. Hutchinson)

Example 2

Outline







Back to bond line thickness effect in adhesive bonds

0

1000

2000

3000

4000

5000

6000

7000

0 0,5 1 1,5 2 2,5 3

Kinloch and Shaw 1981

Adh

esiv

e jo

int t

ough

ness

(J/

m2 )

Bond line thickness (mm)

XD1493

XD4600

Betamate 73455

ESP110

Example 3

See lecture 1

Cohesive zone based fracture model

• 2D plane strain FE model• Steady-state formulation (see Material flow below)• Far-field plasticity accounted for with von Mises plasticity• Damage mechanisms condensed in a cohesive zone

with zero thickness and with constant parameters

Continuum elements with elastic-plastic behaviour

Continuum elements with elastic-plastic behavior

Cohesive zone elementswith zero thickness

σ

δn

δn

σ

σ

adherend adhesive

Γ0

σ

Example 3

See lecture 1

“3 active plastic zones” whose intensity depends on adhesive thickness

Example 3

See lecture 1

See Martiny et al., IJAA2010, IJF 2012

Yellow Grey Blue

σpeak 87 MPa 99 MPa ~ 100 MPa

σpeak/σ0 2.9 2.7 ~ 3

Γ0 112 J/m2 845 J/m2 ~3500J/m2

Γp ~ 100 J/m2 ~ 40 J/m2 ~1000J/m2

(for interm. h) (OK to capture h effect) (KO to capture h effect) (partly OK h effect)

n 0.13 0.47 0.1

See Martiny et al., IJAA2010, IJF 2012

Example 3

See lecture 1

Betamate 73455 ESP110 Betamate 1493

Important values

Thickness effect in ESP110 not captured

Example 3

See lecture 1

• Elastic-plastic, 2D plane-strain, finite element model• Crack imposed at a predefined location• Failure criterion: the maximum principal stress must reach

a value of σc at a distance rc ahead of the crack tipwith σσσσc and r c kept constant for a given adhesive!

Example 3

Solution : “stress at a distance” model

Martiny et al., EFM 2013

Motivation behind the failure criterion

• Crack propagation is dominated by the nucleationof voids from the second-phase particles

• A maximum principal stress equal to σc is required to nucleate voids by cleavage, debonding, cavitation,…

• The second-phase particles are distant from each other by an average distance equal to rc

• Criterion initially suggested for brittle fracture in metals by Ritchie et al. (1973) and later applied to adhesives a.o. by Clarke and McGregor (1993)

Example 3

“Stress at a distance” model

Identification of the material parameter valuesElastic-plastic properties of the adhesive/adherend:least-square fit of uniaxial tensile stress vs. strain curves with a proper equation for the hardening law

Aluminum alloy

Adhesive ESP 110

Example 3



Identification of the material parameter values• Elastic-plastic properties of the adhesive/adherend• Parameters, rc and σc, of the failure criterion:

inverse analysis, i.e. chosen to give the best possible agreement between numerical predictions and experimental results

Betamate 73455 ESP 110 Betamate 1493

rc = 18 µm, σc = 98 MPa rc = 49 µm, σc = 210 MPa rc = 6.9 µm, σc = 141 MPa

Example 3



Attempt to link to the physics of damage

X = 50 to 150 µm

SEM fracture surface ESP110

Example 3

Example 3

Predictions Betamate 73455


Example 3

Predictions ESP110


Example 3


Predictions Betamate 1493

Adhesive fracture energy values• Predicted by the model of the TDCB test

• Normalized by a reference value corresponding to an elastic material (with the same elastic properties) under small-scale yielding conditions

As a function of:• The thickness of the adhesive layer• The elastic-plastic properties of the adhesive assuming the

following stress vs. strain behavior in uniaxial tension:

= if ≤

otherwise

• The parameters, rc and σc, of the failure criterion

Example 3

Parametric study

• Ga varies (increases)non-linearly with hadh

• Ga increases with σc

• Ga increases with smaller values, i.e. lower hardening capabilities

Example 3

Parametric study – effect of thickness, σc and


Back to bond line thickness effect in adhesive bonds

0

1000

2000

3000

4000

5000

6000

7000

0 0,5 1 1,5 2 2,5 3

Kinloch and Shaw 1981

Adh

esiv

e jo

int t

ough

ness

(J/

m2 )

Bond line thickness (mm)

XD1493

XD4600

Betamate 73455

ESP110

Example 3

See lecture 1


• Ga increases rapidlywith σc at the onsetof plasticity

• Ga increases less rapidly with smaller values, i.e. lower hardening capabilities

Example 3

Parametric study – effect of n


See lecture 1

Tvergaard & Hutchinson, JMPS 1992

Outline






6. Example 4 : Fracture in Al with PFZ

Grain

PFZ

Ee

Se

Grain

PFZ

Se

Ee

Intergranular versus transgranularductile fracture in 7xxx Al alloys

Depending on the mismatch between soft and

hard zones, fracture is either intergranular with very low toughness or

transgranularinter trans

Hard grain core

Soft precipitate free grain

boundary layer

Example 4

Objective : micromechanics based multiscalemodelling to relate inter- versus transgranular

cracking resistance to microstructure

requires proper account of shear !

θ δΣδ

L g1 Dg1

Lp1

Dp1

Dp2

L g2

Dg2d

h

Example 4

Scheyvaerts et al., JMPS 2011

Damage model

Rz

Rx

Extension by Gologanu, Leblond et al. (1993-1997) of Gurson model to spheroidal voids

Φgrowthnew≡C

σ y2 ΣΣΣΣ' +ηΣhX

2 + 2q g +1( ) g + f( )coshκ Σh

σ y

− g +1( )2 − q2 g + f( )2 = 0

( ) pkk

pkkp hhS ε

ε&

&&&

21 :3

12

3 +

−+= Pδδδδεεεε

( ) = py f εεεεσσσσ && −1 ypεσ

( ) piigrowth ff ε&& −= 1nuclgrowth fff &&& +=( )xz RRWS lnln ==

with

Beremin void nucleation criterion Thomason void coalescence criterion

bulkc

erfintc

particleprinc or σσσ max =

( )0maxmax σσσσ −+= eprinc

particleprinc k

( ) ppnucl gf εε && =

( ) ( )

+

−−=χχ

χαχσσ 1

2.11

12

2

Wn

y

z

Pardoen & Hutchinson, JMPS 2000

Example 4

See lecture 3

Void rotation must be taken into accountespecially for inclined PFZ

Example 4


Fully periodic 3D FE cell calculations under constant stress triaxiality + shear

Kailasam and Ponte Castaneda, JMPS, 1998

22 nωn =& = Pω Ω -Ω

( )1221 nnnnΩPP ⊗+⊗−= ω

( ) ( ) P12212

P1 DAnnnneD:Ce ::

1

1

2

1-

2

2

⊗+⊗−+−=

W

WPω

2L10

2L20

e1

e2

2U12

2U12

U11U11

U22

U22

n2

n1

−θ

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6

Unit Cell

Model with unit cell data

Extended Gurson model

W0=1/3

W0=3

θ (rad)

γ

Simple shear, n1//e

2 (θ=π/2)

Example 4


Void rotation

Generalized void coalescence model

( ) ( )( ) ( )( ) ( ) ( )

+

−−=

δχδδχδχ

δχσ

δσeffeffeff

effeffloc

y

n

W

124.1

11.01

2

2

0

0.5

1

1.5

2

2.5

3

3.5

4

-100° -50° 0° 50° 100°

U12

/U22

=0

U12

/U22

=1

U12

/U22

=2

εc

eq

δ

W0=1, f

0=10-3

T=0.57

T=1A

B

Example 4


Most of the time, coalescence in the ligament oriented perp. to the main loading direction

Coalescence process

Weck et al.∆εn ∆εn

with

and

( ) statestrain ,, ccn WF χε =∆

Example 4

Scheyvaerts et al., IJDM 2010

See lecture 3

Meshing Voronoï-based grain discretization

1 2 3

4 5 6

Random grain dispersion Voronoi tessellation Elementary sub-cell division

Sub-cell meshing Grain boundary merging Material properties allocation

Example 4

Scheyvaerts PhD thesis UCL 2008

Uniaxial tension sample

Small scale yielding conditions

Single grain

Multi-grain box

Crack line

Crack tip

Example 4


Meshing

( )( )( ) ( )( )1expln

1exp

0

RVE deformedon

−∆=

−∆=∆

nnc

nncn

X

Xu

εε

ε

( ) statestrain ,, ccn WF χε =∆

ncσ

Extra strain from coalescence to final fracture of RVE is known

Corresponding extra displacement is known

nσ

nu∆

δ

Example 4


Introduction of the length scale

GRAINσog/Eg νg ng f0g W0g λog

10-3 0.35 0.05 5 10-3 1 1

PFZσog/ σop νp np f0p W0p λop

2 to 8 0.35 0.3 5.17 10-2 1/3 1

Material parameters

Example 4


Single grain

Grain

PFZ

Σe

Εe

Σe

Grain

PFZ

Εe

Σe

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

σ22

/σ0g

ε22

Σ11

/Σ22

=0, Wgrain

=1, (L/D)0p

=1.85,

np=0.3, W

0p=1/3, λ

0p=1, R

0=0.0525

σ0g

/σ0p=2

σ0g

/σ0p=4

σ0g

/σ0p=6

σ0g

/σ0p=8 Transgranular fracture

Intergranular fracture

σ0g

/σ0p=3

σ0g

/σ0p=5

σ0g

/σ0p=7

Example 4


Example 4


Uniaxial tension multigrain specimen

σ0g/ σ0p = 2 leads to transgranular fracture

Example 4



σ0g/ σ0p = 4 leads to intergranular fracture

0

0.1

0.2

0.3

0.4

0.5

0.6

0% 20% 40% 60% 80% 100%

Wg=1, (L/D)

p=1.844

Wg=1, (L/D)

p=1.5

Wg=1, (L/D)

p=3

Wg=3, θ

g=π/2, (L/D)

p=1.844

Wg=3, θ

g=0, (L/D)

p=1.844

Wg=3, θ

g=π/4, (L/D)

p=1.844

Hexagonal, (L/D)p=1.5

εf

22=ln(A

0/A)

% Intergranular Fracture

Example 4



COEXISTENCE OF BOTH FRACTURE

MODES

0

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6 8 10 12

σ0g

/σ0p

Wg=3, (L/D)

0p=1.85

Wg=3, θ

g=π/4

Wg=1

Wg=3, θ

g=0

Wg=3, θ

g=π/2

εf

22=ln(A

0/A)

Example 4


Strong anisotropy resulting from an elongated grain shape and orientation with respect to the loading direction.

Uniaxial tension multigrain specimen –effect of grain shape

Diapositive 104

FS1 scheyvaerts; 12/12/2008

R100RrY ≤

Crack opens in Mode I

Process zone = multi-grains mesh

Ifield KK =

ry

Example 4


Cracked configuration – SSY conditions

Grain aspect ratio=1

Example 4

SSY cracked multigrain configuration


σ0g/ σ0p = 2 leads to transgranular fracture

Grain aspect ratio=1

Example 4


σ0g/ σ0p = 4 leads to intergranular fracture


σ0g/σ0p=2 • important crack tip blunting • path tends to branch and deviate from the horizontal plane

σ0g / σ0p

σ0g/σ0p=3

σ0g/σ0p=4

σ0g/σ0p=6

TRANS

INTER

• small CTOA • path close to the horizontal plane.

Example 4



Crack advance = # of cracked material pointsCrack advance = crack projection on the crack plane

0

2

4

6

8

10

12

0 10 20 30 40 50 60 70 80

JR/(σ

0 X

0)

∆a/X0

σ0g

/σ0p

=2

Example 4


SSY cracked multigrain configuration JR curves

Loading in the transverse

direction TD decreases

significantly the tearing resistance,

whatever the fracture mode.

Chaire Francqui 2016

Lectures

04/02/2016: Fracture of interfaces, adhesive joints and welds

18/02/2016: Fracture of coatings and electronic devices

03/03/2016: Fracture of metals and polymers - I. Damage

10/03/2016: Fracture of metals and polymers - II. Crack propagation

17/03/2016: Fracture of composites

24/03/2016: Fracture of nanomaterials

Adresse des lectures

Institut de Mathématiques, Quartier Polytech 1, Allée de la Découverte 12, Bâtiment B37, 4000 Liège - Auditoire 02

Inscription via le site web: http://www.facsa.ulg.ac.be/chairefrancqui/2016

Program of the Chair

Documents

Lecture 4: Fracture of metals and polymers II. Crack