Download - MLR Session 10 Ductile Brittle
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Ductile fractureExamples of applications
Benot Tanguy
Department of Nuclear MaterialsCommisariat lEnergie Atomique
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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Ductile micromechanisms
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Ductile micromechanisms
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Ductile micromechanisms
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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Ductile micromechanisms
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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Ductile micromechanisms
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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Ductile micromechanisms
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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Application : Charpy test modelling
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Application : Charpy test modelling
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Application : Charpy test modelling
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Ductile failure model
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Parameters adjustement
)]2
1ln(1[22
aR
ra
zz
++=
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Parameters adjustement
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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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Mesh size dependance
Compact tension specimen
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Mesh size dependance
Compact tension specimen
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Mesh size dependance
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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Charpy test modelling
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Charpy test modelling
DYNVIQS
Loading rate effectExp.
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Towards ductile to brittle transition
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Towards ductile to brittle transition
Constant value of the peak of the maximal principal stress with
increasing loading
CVN and CT
without damage
modelling:
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Towards ductile to brittle transition
i i i i
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Towards ductile to brittle transition
T d d il b i l i i
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Towards ductile to brittle transition
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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Modelling plastic anisotropy
New model for plastic anisotropy
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New model for plastic anisotropy
New model for plastic anisotropy
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New model for plastic anisotropy
Yield surface shape
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Yield surface shape
(biaxial loading)
1E
Yield surface shape
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Yield surface shape
New model for plastic anisotropy
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New model for plastic anisotropy
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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Modelling plastic anisotropy
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Modelling 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
Parameters adjustement
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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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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
Application: cracked pipe
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
Application: cracked pipe
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
Application: cracked pipe
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
Application: cracked pipe
Local approach : Rousselier model
Geometry definition : 1 quarter of the pipe, straight crack front
Initial crack size : =50 (correspond to the fatigue defect)
27 elements along the crack front : average length = 0.55 mm
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Application: cracked pipe
Local approach : 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
Application: cracked pipe
Local approach : Rousselier model
Early convergence problems, located at the inner surface crack front element due tonecking
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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
Application: cracked pipe
Local approach : 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
Application: cracked pipe
Local approach : Rousselier model
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
Application: cracked pipe
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
Application: cracked pipe
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
Application: cracked pipe
Pipe test analysis
Geometry definition : 1 quarter of the pipe, straight crack front
Initial crack size : =50 (correspond to the fatigue defect)
27 elements along the crack front : average length = 0.55 mm
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Application: cracked pipe
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
Application: cracked pipe
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
Application: cracked pipe
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 - - - -