mlr session 10 ductile brittle

7/30/2019 MLR Session 10 Ductile Brittle

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Ductile fractureExamples of applications

Benot Tanguy

Department of Nuclear MaterialsCommisariat lEnergie Atomique

[email protected]

With the acknowledged contribution ofJ. BessonW. BrocksS. Marie


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Motivations


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Motivations

Residual

strength

?!


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Ductile fracture

Ductile fracture mechanisms


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Ductile fracture features

Fracture surface : presence of dimples

[Luu, PhD, 2006]

inclusions


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Ductile micromechanisms

Industrial materials : presence of inclusions, precipitates

Several scales


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Nucleation : cavities creation (mechanisms to be defined!) related to plasticity

Mechanisms:

Voids issued form particules:

Carbides, precipitates, (micron to nanometer scale)

Iron carbides (Fe3C)[Al-Li] (Turck, 2007)


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Dislocations pile-ups

Deformation localisation (involved in coalescence process)

[Zener mechanim]

[Tanguy, 2005][Luu, PhD, 2007]

Void nucleation : Energetic criteria + stress criteria(Surface creation) (interface separation

Particle frcature)


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Void Coalescence : final stage of the failure process

Internal necking

Coalescence in shear (Void sheet : 2nd particules population at lower scale

involved)

Internal necking


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Void sheet


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Short introduction to ductile fracture

Main features of ductile fracture

The three main steps of ductile fracture :

1. Formation of microvoids through decohesion of the metallic matrix around

inclusions or fracture of the particle,2. Growth of these voids through plastic flow of the matrix

3. Colaescence of the voids and/or formation of the shear bands

The plasticity is in general, not confined to the vicinity of the crack tip front

Focus on macroscopic monotonic loading


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Approaches for simulating crack extension

1. Morphological (Global approach)

no analysis of microscopic mechanisms mere transposition of the methods andreasonings of LEFM (SSY assumption)

Node release + fracture mechanics criterion (R-curve)

No splitting of dissipation into global plasticity and local separation

Approach widely used in the industry because it is simple, but thetransposition of the results from one situation to another is somewhat

problematic (size effect, material heterogeneities, anisotropic behavior)

2. Cohesive surface

Interface elements with traction-separation law responsible for local separation

3. Continuum damage mechanics (Local approach)

Unified constitutive equations for deformation and damage, e.g. porous metalplasticity

Relies on micro-macro analyses more reliable but still relatively little used

because of its complexity

Simplic

ity

Complexca

sesdescrip

tion


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Fracture of brittle material

el

c( ) 2a = =G

elel

el rel

(2 ) (2 )

UU

B a B a = =

G

sep

c2

(2 )

U

B a

= =

( )elrel sepcrack

0U UA

crack extends if

energy release rate

separation energy (SE)

(energy per area)

fracture criterion

fracture stress c2E

a

=

A.A. Griffith (1920)

(to create surfaces)

(more details in session 1)


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Short introduction to ductile fracture

Typical load-displacement curve of a cracked specimen :

Small scale yielding

Large small yielding

Linear fracturemechanics

Non-linear fracturemechanics

Rp


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Non linear elasticity: the J-integral

Assumptions:

Homogeneous, isotropic material

Small strains

Non linear elastic behavior:n

B 0=Radial loading : no variation of the load trajectory

No direct extension to cyclic loadings

x

y

n

T(n)

ds

==== ds

x

u).n(TWdyJ

W : elastic energy density

T(n) : stress vector (stress acting on the contour)

x

u

: partial derivative of the displacement vector

ds : path increment on

J-integral independent on

J-integral: link between local parameters (near to crack tip) and global loading ->fracture

parameterJ-integral = G, energy release rate (surface energy: unit J/m2) (Budiansky and Rice,73)

Path-independant integral(Cherapanov, 1967), (Rice, 1968)

Crack tip

= ijij dw

(more details in session 5)

Stationnary crack


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The J-integral

x

y

n

T(n)

ds

==== ds

x

u).n(TWdyJ

HRR stress field

(non linear elastic material) )n,(~rIB

J

)n,(~rIB

J

ij1n

n

n0ij

ij1n

1

n0ij

====

====

++++

++++

++++

==== rd

x

uTcosWJ

r

1~dx

uTcosW

++++

lorsque r 0

r

1~klij

1n

n

ij1n

1

ijn

0r

1~

r

1~B ++++++++

====

Stress singularity controlled by J-integral

nB 0= plastic behavior only valid for monotonic loading

J plays the role of an intensity factor like K in the case of LEFM

Small strain deformation theory

Elastic case: n=1


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The J-integral

Fracture criterion : J=Jc, Jc material characteristic

Practical determination: ASTM standard E1820

0

1000

2000

3000

4000

5000

6000

7000

8000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Ouverture (microns)

Force(

304L-1

Size criterion reached for little propagation

Value of J at initiation: Ji

Value of J after 0.2 mm crack propagation: JIc


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The J-integral: limitation

For extending crack: -path dependence of J-geometry dependence of J-Da curves

( )el plex sepcrack crack

WU U U

A A

= + +

pl sep pldissc

crack crack crack

UU UR R

A A A = = + = +

energy balance

dissipation rate

commonly: Rpl causes geometry dependence

Problem: separation of Rpl and c

c

plR >>


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Global approach: the J-integral

Size dependance of J

J increases due to the plasticdissipation taking place in thecrack wake due to theprogressive elastic unloading,due to the nonradial loadingsin the active PZ of apropagating crack, and due to

change in the crack tipgeometry

JR very sentitive to sligth

changes of constraintwhich affect a lot theextrinsic plasticdissipation -> Quantifiedby the so-called Q stress(2 parameters approach,

needs FE calculations)


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Global approach: the J-integral

Size dependance of J

E36 Steel (-50C)

0

50

100

150

200

250

300

0.00 0.20 0.40 0.60 0.80

a/W

JIC(KJ/m)

SENB 0.03 < a/W < 0.77

CCP 0.63 < a/W < 0.77

Sumpter


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Ductile fracture

Continuum modelling of ductile damage

The so-called local approach


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Application : fracture toughness from Charpy tests

Tanguy and al.


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Application : Charpy test modelling

Objective : simulation of the Charpy transition curve


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Ductile failure model


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Parameters adjustement

)]2

1ln(1[22

aR

ra

zz

++=


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Mesh size dependance

In classical approach, mesh size is a compromise:-description of the stress-strain gradients-size of the FE calculations-fracture energy related to mesh size

Notched specimen

[Mealor, 2004]


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Compact tension specimen


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Compact tension specimen


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0

abBVW RR == 00 baBabBSW RR 000 )/()(/ ===

Interpretation Failure is a consequence of localisation in a strip of thickness b Estimation of the apparent surface energy

Dissipated energy to fracture

Characteristic length: b=In standard FE : b~~hSo that

h00 =h has to be fixed related to microstructure


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Modelling of Charpy test

FE Mesh


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Modelling of Charpy test

Ductile fracture simulation : Load-displacement curve


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Charpy test modelling


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DYNVIQS

Loading rate effectExp.


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Towards ductile to brittle transition


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Constant value of the peak of the maximal principal stress with

increasing loading

CVN and CT

without damage

modelling:


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i i i i


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T d d il b i l i i


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On-going work

L l h li ti 2


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Local approach- application 2

Anisotropic plastic models for sheets materials

Besson and al.


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T t i


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Test specimens

Test specimens


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Test specimens

Plastic anisotropy


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Plastic anisotropy

Modelling plastic anisotropy


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New model for plastic anisotropy


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Yield surface shape


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Yield surface shape

(biaxial loading)

1E

Yield surface shape


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Yield surface shape



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Introducing plastic anisotropy in the GTN model


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g p py

Ductile tearing of pipeline steel


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g p p

Motivations


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Motivations


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Ductile tearing of pipeline steel


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g p p

Outline


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Sheet materials


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Test specimens


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Large facility at Arcelor-Mittal


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Plastic anisotropy


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Example : Fit for X100 steel


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Application : limit load of a pressurized pipe under bending


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compression

buckling

Ductile rupture modified GTN model


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parameters

X70 steel


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Wide plate test


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Wide plate test: measurements


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sudden propagation



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Characteristic length, mesh size


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Simulation


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Plate thickness

Crack tunneling


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Plate thickness effect on Z


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Through-thickness hardness gradient


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X100 Europipe -- Plate


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X100 ?


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Plastic anisotropy


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Fracture surface : two void populations


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Same observations on Charpy specimens


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Delamination on CT specimens


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Results of mechanicals tests


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Local approach


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Local approach


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Model


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FE meshes


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Elastoplastic anisotropic behevior


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FE results for notched bars


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FE results for plane strain specimens


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FE results for Charpy specimens (static and dynamic)


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FE results for CT specimens


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Parametric study : yield surface


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X100 NSC Plate & Prestrain


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Prestrain at NSC


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Application : toughness after UOE forming or ground movement


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Similar characterization program as for the Europipe material


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Cyclic behavior


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Rupture anisotropy : notched bars


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Rupture anisotropy : CT specimens


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Prestraining reduces ductility


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and crack growth resistance


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CODLoad for 0% and 6% prestrain


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Damage model with kinematic hardening


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FE simulation of crack advance


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Simulation of rupture anisotropy


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Conclusions : main characteristics for ductile rupture of pipeline steels


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Local approach


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On-going work

Cohesive model


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113W. Brocks and al.

Use of cohesive elements for thesimulation of ductile tearing

Cohesive model

W. Brocks and al.


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a

crackLigament

Separation of thecohesive elements

Phenomenological representationof various failure mechanisms

by cohesive interfaces

Implementation:

structure is divided into

material with elastic-plastic

properties(continuum elements)

interface with damageproperties(cohesive elements)

Cohesive law

two material parameters:

TractionTraction--Separation Law (TSL)Separation Law (TSL)


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CrackLigament

c

c n n n

0

( )d

=

Cohesive stress, c (MPa) (maximumstress carrying capacity of the

interface)

Critical separation, c (mm)

Separation energy, c (J/mm2), defined by

( ) ( )

( ) ( )

n n

1 1

n 2 n 2

0 2 0 2

2

n 1

n n c n c 1 n 2

3 2

2 n c

2

( ) ( ) 1

2 3 1

set A

SET A will be used for pipe tests FE calculations

Application: cracked pipe

Common assumptions to F.E. analyses

Geometry definition : 1 quarter of the pipe, straight crack front

Initial crack size : =50 (correspond to the fatigue defect)

Modelling


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( p g )

At the end of the pipe, thin rigid ring to take into account the embedded section


40

60

80

100

120

140

M(

kN.m

)

Test 1

J-Da approach

J-a approach :

&&

Maximum moment well predicted but

conservative prediction of the global

behaviour :


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0

20

40

60

80

100

120

140

0 1 2 3 4 5 6CMOD (mm)

M(

kN.m

)

Test 1

J-Da approach

020

0 1 2 3 4 5 6rotation ()

expmax,MaJmax,M

expmax,MaJmax,M

expaJ

rotationrotation

CMODCMODaa

>>

>>>

&&

0

20

40

60

80

100

120

140

0 5 10 15 20 25 30 a (mm)

M(

kN.m

)

Test 1

J-Da approachc


40

60

80

100

120

140

M(

kN.m

)

Test 1

J-Da approach

Ji-Gfr approach

Energetic approach :

Ji-Gfrapproach


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020

40

60

80

100

120140

0 2 4 6 8 10CMOD (mm)

M(

kN.m

)

Test 1

J-Da approach

Ji-Gfr approach

0

20

40

0 1 2 3 4 5rotation ()

Ji Gfr approach

020

40

60

80

100

120

140

0 5 10 15 20 25 30 a (mm)

M(

kN.m

)

Test 1

J-Da approach

Ji-Gfr approachc


120

140

120

140

Energetic approach :

Ji-Gfrapproach


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0

20

40

60

80

100

120

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0strain gage J2 (%)

M(

kN.m

)

Test 1

Ji-Gfr approach

0

20

40

60

80

100

120

0 2 4 6 8ovalisation ov2 (mm)

M(

kN.m

)

Test 1

Ji-Gfr approach

O-2J-3

J-2


Local approach : Rousselier model



27 elements along the crack front : average length = 0.55 mm


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Early convergence problems, located at the inner surface crack front element due tonecking

140


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none influence of 1) the element length in the trough-thickness direction 2) a pre-damaged crack front elements

020

40

60

80

100120

0 1 2 3 4CMOD (mm)

M(

kN.m)

Test 1

Rousselier model



Early convergence problems, located at the inner surface crack front element due tonecking

140


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none influence of 1) the element length in the trough-thickness direction 2) a pre-damaged crack front elements

020

40

60

80

100120

0 1 2 3 4CMOD (mm)

M(

kN.m)

Test 1

Rousselier model



mesh size influence of the convergence study : meshes with = 0.3 and 0.45 mm


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convergence problems appears later when the mesh size increase.

0

2040

60

80

100

120140

0 1 2 3 4CMOD (mm)

M(

kN.m)

Test 1

l = 0.45mm

l = 0.3 mml = 0.15 mm



crack initiation prediction : which criterion ?1) initiation when one integration point reaches the critical voids fraction fc

2) initiation when at least one element is fully failed


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Test1 (from DDP) : Mini = 74.8 kN.m & Jini = 97.5 kJ/m

From CT12.5 tests : Jini = 100 kJ/m

criterion 1 criterion 2

M (kN.m) J (kJ/m) M (kN.m) J (kJ/m)

= 0.15 mm 61.0 58.3 75.7 100.7

= 0.3 mm 60.4 56.8 86.9 146.2

= 0.45 mm 101.8 230.4 109.1 284.3

= 0.3 mm + notch 102.4 234.2 106.3 262.4

Remark : J values are deduced from the F.E. calculation conducted for the J-a analysis


Choice of the new element shape :

The convergence of the calculation could be improved by decreasing the height h of the

elements in the crack growth area :


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h = Lc / 2


New calibration of the parameters :

pre-identification :D = 2 f 0 = 1,4.10

-3 ( Franklin relation)

fc = 0,05 1 = flow = 450 MPa


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CT test calculation : Lc = 0,4 mm

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5

Da (mm)

CMOD

(mm)

CT2 test

Rousselier model

0

5000

10000

15000

20000

25000

30000

0 1 2 3 4 5

CMOD (mm)

F(N)

CT2 test

Rousselier model


Pipe test analysis



27 elements along the crack front : average length = 0.55 mm


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Pipe test analysis

60

80

100

120

140

(kN.m

)

Test 1

J-Da approach 40

60

80

100

120

140

M(

kN.m

)

Test 1

Ji-Gfr approach


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0

20

40

0 2 4 6 8 10

CMOD (mm)

M Ji-Gfr approach

Rousselier model

0

20

40

60

80

100

120

140

-1 0 1 2 3 4 5 6 7

ov2 (mm)

M(

kN.m

)

Test 1

Ji-Gfr approach

Rousselier model

0

20

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0

J2 (%)

Rousselier model

O-2J-3

J-2


Pipe test analysis

Crack initiation : 1 element fully damaged (all integration points closed to the crack plan have reached fc)

Crack growth calculation : a = Area of fully damaged elements / Thickness


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0

20

4060

80

100

120

140

0 5 10 15 20 25 30

a (mm)

M(

kN.m

)

Test 1

J-Da approach

Ji-Gfr approach

Rousselier model


Conclusions

3 approaches have been considered : J-a, Ji-Gfr and Rousselier model

initiation Values at the maximum bending moment

M (kN.m) J (kJ/m) M (kN.m) Rotation () CMOD (mm) ( )


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The 3 approaches give good initiation prediction

J-a approach leads to a good estimation of the maximum endurable moment but underestimate displacements (rotation,

CMOD,.)

Ji-Gfr approach gives a good prediction of the complete behaviour of the pipe

a (mm)Test 1 74.8 97.5 124.6 2.47 >4 6

J-a 75.5 100 119.5 1.08 2.87 6

Ji-Gfr 75.5 100 126.8 2.25 6 11

Rousselier 75.7 100.7 - - - -

mlr session 10 ductile brittle

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