desmorat r damage and fatigue
TRANSCRIPT
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DAMAGE AND FATIGUE
Continuum Damage Mechanics modeling
for fatigue of materials and structures
Rodrigue Desmorat
LMT CachanENS Cachan, 61 av. du Pt Wilson
94235 Cachan Cedex
ALERT School 2006
Revue Europenne de Gnie Civil, Vol. 10, n6/7, pp. 849-877, 2006
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Fatigue issuesFatigue issues
Fatigue = failure under repeated (initially cyclic) loading
1-10 cycles: material behavior coupled with damage
10-100 cycles: very low cycle fatigue100-104cycles: low cycle fatigue
105-107cycles: high cycle fatigue
>108cycles: gigacycle fatigue
86400s/day so that 105cycles at 1Hz takes around 1 day
Objectives of the courseGive background on Damage Mechanics applied to fatigue problems
Give background to build tools able to handle complex loadings (3D,
random, seismic, with temperature variations, with coupling with
other physics for instance by use of poromechanics effective
stress or by multiscale analyses)
Thermodynamics framework should allow more finalized extension
to geomaterials (rocks, soils)
Modeling s till in progress in the Mechanical / Civil Engineering
communities
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Example: thermo-mechanical random fatigue
A thermo-hydraulic computation gives the temperatureand stresseshistory.
A DAMAGE computation must gives the damage D(t), the locationof wheredamage is maximum (where a crack will initiate) and the time to mesocrack
initiation.
Loading
sequence made
of 1000 points
3D stresses
Temperature
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Damage Mechanics becoming an engineering tool ?
on damage models
on engineering applications:ductile, creep, fatigue, creep-fatigue and
brittle failures
on parameters identification on numerical topics
on damage threshold
on damage anisotropy
on micro-defects closure effect
(Springer 2005)
A book
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OutlineOutline
I- Elasto-plastic ity / Continuum Damage Mechanics
1. Plasticity in thermodynamics framework
2. Damage and effective stress concept3. Elasticity coupled with damage
4. Von Mises plasticity coupled with damage
II- Ampli tude damage laws
III- Damage evolution laws for fatigue
1. Lemaitre's damage law2. Quasi-brittle materials
3. Rocks or soils
4. Micro-defects closure effect - Mean stress effect
5. Damage post-processing
6. Jump-in-cycles procedure
IV- Toward an unified approach for damage and fatigue ?1. Basis of a thermodynamics framework
2. Application to metals, concrete, elastomers and rocks
V- High Cycle Fatigue 5
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I- ELASTO-PLASTICITY /CONTINUUM DAMAGE MECHANICS
!
"E E
"p "e Damage = scalar variable D
D = 1!E
E
!
"-
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OutlineOutline
I- Elasto-plastic ity / Continuum Damage Mechanics
1. Plasticity in thermodynamics framework
2. Damage and effective stress concept3. Elasticity coupled with damage
4. Von Mises plasticity coupled with damage
II- Ampli tude damage laws
III- Damage evolution laws for fatigue
1. Lemaitre's damage law2. Quasi-brittle materials
3. Rocks or soils
4. Micro-defects closure effect - Mean stress effect
5. Damage post-processing
6. Jump-in-cycles procedure
IV- Toward an unified approach for damage and fatigue ?1. Basis of a thermodynamics framework
2. Application to metals, concrete, elastomers and rocks
V- High Cycle Fatigue 7
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PlasticityPlasticity 1D1D
Strain partition
Elasticity
Criterion function
Hardening
Accumulated plast ic st rain
! = !e + ! p
! = E"e
f = ! "
X "
R" !
y
R =R(p)
X = C!p " #Xp
p = !p dt"
( )
!
"
"p "e
!y
R+X
f=0
f
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Thermodynamics frameworkThermodynamics framework
Thermodynamics variables
Thermodynamics potential
!" =1
2(# $ #p ) : E : (# $ #p ) + G(p)
State laws
!" = #$%
$& p! ="
#$
#%=E : (% & %p ) =E : %e
R ="#$#p
= %G (p)=
Kp linear
Kp1/M power
R&(1' e'bp ) exponential
(
)*
+*
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Thermodynamics frameworkThermodynamics framework
Thermodynamics variables
Thermodynamics potential
!" =1
2(# $ #p ) : E : (# $ #p ) + G(p)
State laws
!" = #$%
$& p! ="
#$
#%=E : (% & %p ) =E : %e
R ="#$#p
= %G (p)=
Kp linear
Kp1/M power
R&(1' e'bp ) exponential
(
)*
+*
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Stored (blocked)
energy density ws
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Criterion function
Dissipation potential
Evolution laws
f = !eq "R " !y
F = f
(associated model for single isotropic hardening)
Determination of the plastic multip lier
plasticity
visco-plasticity
!p= "
#F
#$= "
3
2
$D
$eq
p = !"#F
#R= "=
2
3$p : $p
f= 0, f = 0 ! "
f= !v , !v = KNp1/ N
" #= p =f
KN
N
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CaseCase of tension from von Mises plasticityof tension from von Mises plasticity
!=
! 0 0
0 0 0
0 0 0
#
$
$
$ &
'
'
' !
D=
! " 1
3
tr
!1 =
23 ! 0 0
0
"1
3 ! 0
0 0 " 13 !
#
$
%
%
%
&
'
(
(
(
!eq =3
2!
D :!D = !
!p = p3
2
"D
"eq=
p 0 00 # 12 p 0
0 0 # 12 p
$
%
&&&
'
(
)))
! =!e + !p = "E+p
" =R(p) + "y
#
$%
&%' "(!)
!Plastic
incompressibiliy
! "11p =p = " p
Tension
curve :
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OutlineOutline
I- Elasto-plastic ity / Continuum Damage Mechanics
1. Plasticity in thermodynamics framework
2. Damage and effective stress concept3. Elasticity coupled with damage
4. Von Mises plasticity coupled with damage
II- Ampli tude damage laws
III- Damage evolution laws for fatigue
1. Lemaitre's damage law2. Quasi-brittle materials
3. Rocks or soils
4. Micro-defects closure effect - Mean stress effect
5. Damage post-processing
6. Jump-in-cycles procedure
IV- Toward an unified approach for damage and fatigue ?1. Basis of a thermodynamics framework
2. Application to metals, concrete, elastomers and rocks
V- High Cycle Fatigue 13
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Damage and effective stress concept
D
Effective stress
! =
F
S
VER
S
SD
!=
F
S=
F
S" SD
=
F
S 1"S
SD
( )
!=
!
1"D
!= E"e # ! = E"eE
=
E(1$D)
%&
'
Principle of
strain equivalence
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OutlineOutline
I- Elasto-plastic ity / Continuum Damage Mechanics
1. Plasticity in thermodynamics framework
2. Damage and effective stress concept3. Elasticity coupled with damage
4. Von Mises plasticity coupled with damage
II- Ampli tude damage laws
III- Damage evolution laws for fatigue
1. Lemaitre's damage law2. Quasi-brittle materials
3. Rocks or soils
4. Micro-defects closure effect - Mean stress effect
5. Damage post-processing
6. Jump-in-cycles procedure
IV- Toward an unified approach for damage and fatigue ?1. Basis of a thermodynamics framework
2. Application to metals, concrete, elastomers and rocks
V- High Cycle Fatigue 15
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Elasticity coupled with damage
D=D(Y) ou D=D( )!+
!
"
E
E(1-D)
f=0
f
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Thermodynamics potential
State laws
Thermodynamics variables
!" =1
2(1#D)$ : E : $
! ="#$
#%=E(1& D) : %
!Y ="#$
#D ! Y =
1
2
": E : "
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Damage criterion function
Marigo damage modelMarigo damage model
f =Y !"(D)
Damage potential F = f (associated model)
Damage evolution law
D = !"F
"Y= ! dtermined from the consistency condition
f= 0, f = 0 ! "
g(Y) =
Y!YD
S
s
Y! YDS
"
#$$
%$$
D = !"1(YMax ) =g(YMax ) Ex:
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Mazars damage modelMazars damage model
Damage criterion function
f= !" # != !+
: !+
-4
-3
-2
-1
0
1
2
-5 -4 -3 -2 -1 1 2
!1
!u
!2
!u
! eq
" (# = 0.2)
" (# = 0.3)
^
^
Critre de Mazars
r re e von ses
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Damage evolution law in tension
Damage evolution law in compression
Different damage evolution in tension and in compression
D = ! tDtraction +!cDcompression
D traction=
1!"D(1!A t )
"Max!
A t
exp Bt ("Max ! "D )[ ]
Dcompression = 1!"D (1! Ac )
"Max
!
Ac
exp Bc ("Max ! "D )[ ]
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TensionTension// compression for concretecompression for concrete
Mazars model : 1 set of damage parameters for tension
1 set of damage parameters for compression
!
" (MPa)
E = E(1# D)
-0.003 -0.002 -0.001 0-0.004 0.001
-10
0
10
-20
-30
-40
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Anisotropic damage modeling
Local FE Non local FE
D22field D22field
classical mesh dependency
Nooru-Mohamed test (1992)
Desmorat, Gatuingt, Ragueneau (2004-2006)
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OutlineOutline
I- Elasto-plastic ity / Continuum Damage Mechanics
1. Plasticity in thermodynamics framework
2. Damage and effective stress concept3. Elasticity coupled with damage
4. Von Mises plasticity coupled with damage
II- Ampli tude damage laws
III- Damage evolution laws for fatigue
1. Lemaitre's damage law2. Quasi-brittle materials
3. Rocks or soils
4. Micro-defects closure effect - Mean stress effect
5. Damage post-processing
6. Jump-in-cycles procedure
IV- Toward an unified approach for damage and fatigue ?1. Basis of a thermodynamics framework
2. Application to metals, concrete, elastomers and rocks
V- High Cycle Fatigue 23
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Plasticity coupled with damage
!
"E E(1-D)
"p "e
E
f=0plasticit et endomagement
f
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Thermodynamics potential
Thermodynamics variables
!" =
1
2(# $ #
p
):E(1$D):(# $ #
p
)+
G(r)
Strain partition
! = !e+ !
p
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Stored (blocked)
energy density ws
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State laws
!Y ="#$
#D
! ="
#$
#%=E(1&D) : %
e
R =!"#
"r= $G (r) =
Kr linaire
Kr1/ M
puissance
R% (1&e&br
) exponentiel
'
()
*)
R! = 23 (1+!) + 3(1"2!) #
H
#eq$
%&& '
())
2
Strain energy density
release rate Y =
1
2!e : E : !e =
"eq2R
#
2E(1$D)2=
"eq2R
#
2E
Triaxialityfunction
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Dissipation potential
Evolution laws
(non associated model)
Criterion function
F = f+ FD
FD =S
(s +1)(1!D)
Y
S
"#$
%&'s+1
f= !eq "R " !y =!eq
1"D"R " !y
!p= "
#F
#$=
"
1%D
3
2
$D
$eq
r =!"#F
#R= "= p(1!D)
Damage
evolution law
(Lemaitre)D = !
"F
"Y=
Y
S
#$%
&'(s
p
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Determination of the plastic multip lier
plasticity
visco-plasticity
f= 0, f = 0 ! "
f = !v , !v = KNp1/ N
" p =f
KN
N
Norton law
Mesocrack initiation when D=Dc
Damage parameters (to be identif ied)!pD, S, s, Dc
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Rupture in monotonic loadingRupture in monotonic loading
pR =!pD +2ES
"u2R#
$
%&&
'
())
s
DcAccumulated plastic strainto rupture
Sensitivity analysis
stress triaxiality
ultimate stress
!pR
pR=STX
pR !TX
TX+S"u
pR !" u
"u
+SSpR !S
S+SE
pR !E
E+S#
pR !#
#+Ss
pR !s
s+S$pD
pR!$ pD
$pD
+SDcpR !Dc
D c
2.9 2.5 2.51.94
10.5 0.5
TX =!H
!eq
!u = !y +R"
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Stress triaxiality effectStress triaxiality effect on pon pRR
0 1 2 3 4 5
0.5
1
1.5
2
pR ! " pD
" pR ! "pD
1
5
s
#H
# eq
1
3
"A high stress triaxiality makes materials brittle"
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Time toTime torupturerupture inincreepcreep
p =!eq
KN0(1"D)
#
$%%
&
'((
N0
Initial Norton law
Time to rupture
Time to damage
initiation
tR =tD +1! (1!Dc )
2s+N0+1
2s+
N0+
1
2ES
"eq2
R#
$
%
&&
'
(
))
s
KN0
"eq
$
%
&&
'
(
))
N0
tD =! pDKN
0
"eq
#
$%%
&
'((
N0
D =Y
S
!"#
$%&s
p ='eq2R(
2ES(1)D)2!
"##
$
%&&
s'eq
KN0(1)D)
!
"##
$
%&&
N0
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Sensitivity analysis
!tRtR
=STXtR !TX
TX+S"eq
tR!"
eq
" eq
+SNtR !NN
+SKNtR !KN
KN+SEt
R !EE
+SStR !SS+S#t
R !##
+SstR ! s
s+S$pD
tR!$pD
$pD
+SDctR !Dc
Dc
. 1 1 0.8
0.6 0.5 0.1
0
0
0
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Previous elasticity coupled with damage models (with no plasticity)
cannot reproduce neither material hysteresis nor fatigue damage
Plasticity coupled with damage model suitable for low cycle fatigue of
metals but difficulties encountered in damage threshold
measurements (loading dependency)
High temprerature fatigue of metals (creep-fatigue) represented
Absolute need of kinematic hardening in fatigue of metals (even if
only briefly presented): Bauschinger effect
Rate form constitutive equations possible for damage: facilities to
handle 3D, non proportional loadings, temperature variations, coupling
with other physics
Still a lot to do for application to (more complex) geomaterials
Partial conclusion for damage models and fatiguePartial conclusion for damage models and fatigue
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II- AMPLITUDE DAMAGE LAWS
Numbers of cycles N
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"
!
$!p $"
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II- AMPLITUDE DAMAGE LAWS
or "Numbers of cycles N
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$"=2"Min case of symmetric loading
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Damage from number of cycles measurementDamage from number of cycles measurement
Engineering damage for fatigue
with NRithe number of cycles to rupture at strain level i
Miner's l inear damage accumulation rule
Example on two level loading
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Amplitude damage law in terms of s tress
"D
"N= g(D)G# ($#,R# )
R"=
"min
"Max
load ratio
R"=
#"M
"M
=1symmetric
loading
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Amplitude damage law in terms of s tress
Does a nonl inear g(D) function leads to nonlinear damage
accumulation ? The answer is NO
Integrate over each level i
Sum over all the levels i
"D
"N= g(D)G# ($#,R# )
R"=
"min
"Max
load ratio
R"=
#"M
"M
=1symmetric
loading
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Amplitude damage law in terms of s trains
"D"N
= g(D)G#($%,R
#)
Limitations
Link between stress and strain amplitude laws not so clear, at least
as long as no rate form damage law allows to recover both
Non cyclic loading ? Needs of cycles counting methods (rainflow)
Extension to 3D ?
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II- DAMAGE EVOLUTION LAWS FOR FATIGUE
D = YS
"
#$ %
&'s
p Y = 12"e :E :"e = #
eq
2
R$2E
Lemaitre's law Strain energy release rate density
Paas law Generalized damage law
D = Y
S"#$ %
&'s
(D =Cg(D)"eq# "eq
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OutlineOutline
I- Elasto-plastic ity / Continuum Damage Mechanics
1. Plasticity in thermodynamics framework
2. Damage and effective stress concept3. Elasticity coupled with damage
4. Von Mises plasticity coupled with damage
II- Ampli tude damage laws
III- Damage evolution laws for fatigue
1. Lemaitre's damage law2. Quasi-brittle materials
3. Rocks or soils
4. Micro-defects closure effect - Mean stress effect
5. Damage post-processing
6. Jump-in-cycles procedure
IV- Toward an unified approach for damage and fatigue ?1. Basis of a thermodynamics framework
2. Application to metals, concrete, elastomers and rocks
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R! = 23 (1+!) + 3(1"2!) #
H#eq
$
%&&
'
())
2
Lemaitre's damage lawLemaitre's damage law
D =Y
S
!"#
$%&
s
p si p > pD
Elastic strain energy
Triaxiality function
Damage gouverned by plasticity
D = Dc
amorage dune fissure
Stress
triaxiality
Damage threshold
Damage enhanced by the stress level
and the stress triaxiality
"H=
1
3tr" " eqhydrostatic stress von Mises stress
Y=1
2"e:E :"
e=
#eq2R
$
2E
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R! = 23 (1+!) + 3(1"2!) #
H#eq
$
%&&
'
())
2
Lemaitre's damage lawLemaitre's damage law
Elastic strain energy
Triaxiality function
D = Dc
amorage dune fissure
"H=
1
3tr" " eqhydrostatic stress von Mises stress
Y=1
2"e:E :"
e=
#eq2R
$
2E
Damage exponent
Damage strength
Critical damage
Accumulated plastic strain
Damage threshold
D =Y
S
"
#$
%
&'
s
p if p > pD
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R! = 23 (1+!) + 3(1"2!) #
H#eq
$
%&&
'
())
2
Lemaitre's damage lawLemaitre's damage law
Elastic strain energy
Triaxiality function
D = Dc
amorage dune fissure
"H=
1
3tr" " eqhydrostatic stress von Mises stress
Y=1
2"e:E :"
e=
#eq2R
$
2E
Damage strength
Critical damage
Stored energy
damage thresholdDamage exponent
D =Y
S
"
#$
%
&'
s
p if ws > wD
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D = a ! "eqMax
"u
#
$%
&
'(
2s
R)sp
Y
S!
"eqMax2 R
#
2ES=a
1/ s$
"eqMax2
"u2 R#
D =Y
S
"
#$
%
&'s
p
a1/s
=
!u
2
2ES
Maximum von Mises stress
(symmetric loading)
Ultimate stress
Lemaitre's damage law in fatigueLemaitre's damage law in fatigue
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1D symmetric fatigue loadings - no damage threshold
D = a ! "eqMax
"u
#
$%
&
'(
2s
R)sp
Y
S!
"eqMax2 R
#
2ES=a
1/ s$
"eqMax2
"u2 R#
D =Y
S
"
#$
%
&'s
p
a1/s
=
!u
2
2ES
Maximum von Mises stress
(symmetric loading)
Ultimate stress
+ cyclic plasticity law!" = !"(!#p )
so that
Lemaitre's damage law in fatigueLemaitre's damage law in fatigue
NR =(8ES)
s
KcycqDc
2("#)2s+q
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CalculatedCalculatedWhler curveWhler curve
+ cyclic plasticity law !" = !"(!#p )
1 10 100 1 103
1 104
1 105
1 106
1 107 cycles
100
1000
!Max
(MPa)
NR
200
! f = 220MPa
xper men s
500
! f"
=180MPa
!
Max
#$p
$0
!
M1=450
#$p1=0.027[ M2=340
#$p2=0.0035[
!
!
o e as en e
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R! = 23 (1+!) + 3(1"2!) #
H#eq
$
%&&
'
())
2
Lemaitre's damage lawLemaitre's damage law
Elastic strain energy
Triaxiality function
amorage dune fissure
"H=
1
3tr" " eqhydrostatic stress von Mises stress
Y=1
2"e:E :"
e=
#eq2R
$
2E
Stored energy
damage threshold
D =Y
S
"
#$
%
&'
s
p if ws > wD
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Damage threshold in terms of stored energyDamage threshold in terms of stored energy
Monotonic loading D=0 as long as !p
< !
pD
Damage threshold
In tension!pD "0.1...0.3 for metals
Fatigue loading D=0 as long as N
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Classical thermodynamics : variables R and p
p
ws
Classical
thermodynamics
Experiments or correction:
variables Q and q
ws =
R(p)dp =
0
p
! "eq # "y( )dp0p
!
ws = R(p)z(p)dp0
p
! = Q(q)dq0p
!
Correction : variables Q and q
z(p) =A
mp1!m
m
!y
stored energy ws!
"
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The stored energy depends on the choice on
thermodynamics variables
Unchanged hardening law: Q(q)=R(p) dq=z(p)dp
ws
variables R and r
0 0.5 1 1.5 20
0
p
variables Q and q
A=0.05, m=4.4
!
"
ws
0
!
"
ws
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Monotonic loading
Fatigue loading
ws =A(!u " !y )#p1/ m
ws =A(!eqMax " !y )p1/ m
Damage threshold in stored energy
ws =wD =A(!u " !y )#pD1/ m
$p =pD
monotonic pD ="pDcreep pD ="pD
fatigue pD ="pD#u$ #y
#eqMax $ #y
%
&''
(
)**
m
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Modeling a loading dependent damage threshold
ALERT School 2006 Rodrigue DESMORAT
More acurate case with kinematic hardening
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NumberNumber ofofcyclescyclestotorupturerupture ininfatiguefatigue
Sensitivity analysis
ND =!pD
2"p
#u$ #y
#eqMax$ #y
%
&
''
(
)
**
m
NR =ND +Dc
2!p
2ES
"eqMax2
R#
$
%
&&
'
(
))
s
!NR
NR=S
"pNR !"p
"p+STX
NR !TX
TX
eqMax
eqMax
+S#yNR
!# y
#y
+SENR !E
E+SS
NR !S
SSsNR
! s
s
+S #uNR !# u
#u
+S$NR
!$
$+Sm
NR !m
m+S%pD
NR!%pD
%pD
+SDcNR !Dc
Dc
2.9
eqMax
!#
#+S#
NR
1
3. .
+
2.2
1.942
0.7 0.5 0.5
8
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0
0.4
0.6
0.8
1
1.2
0.2 0.4 0.6 0.8 1 1.20
t
!
"!1
"!2
n1 n2
n2
NR2
n1
NR1
"!1
t
!
"!2
n1 n2
"!1=0.01
"!2=0.016
"!1=0.016
"!2=0.01
ND
NR
NR
10 103
104
105
0
0.2
0.4
.102
0.6
0.8
1
!"=0.01
!"=0.016
Two level fatigueTwo level fatigueloadingloading
(computations performed with ZeBuLon Finite Element code)
ALERT School 2006 Rodrigue DESMORAT
Loading dependency of the ratio ND/NR Nonlinear damage accumulation
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NonNonlinear creep-fatiguelinear creep-fatigue interactioninteraction
Computations without damage threshold
Computations with damage threshold
and kinematic hardening
NR
NR
F +
tR
tR
c =1
NR
NR
F +
tR
tR
c < 1
t R / tRc
0
1
c
NR / N
R
FLinear
interaction law
1
10
!M= 200MPa
!M= 180MPa
!M= 220MPa
!
t
"t
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OutlineOutline
I- Elasto-plastic ity / Continuum Damage Mechanics
1. Plasticity in thermodynamics framework
2. Damage and effective stress concept
3. Elasticity coupled with damage
4. Von Mises plasticity coupled with damage
II- Ampli tude damage laws
III- Damage evolution laws for fatigue
1. Lemaitre's damage law2. Quasi-brittle materials
3. Rocks or soils
4. Micro-defects closure effect - Mean stress effect
5. Damage post-processing
6. Jump-in-cycles procedure
IV- Toward an unified approach for damage and fatigue ?1. Basis of a thermodynamics framework
2. Application to metals, concrete, elastomers and rocks
V- High Cycle Fatigue 57
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Paas approach for fatiguePaas approach for fatigue
Elastic ity coupled with damage following one of the laws
Paas damage law
Peerlings damage law
From the time integration over one cycle
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Generalized damage lawGeneralized damage law
D =Y
S
!"#
$%&s
'
The previous laws can be rewritten in this form as
Damage governed by the main disspative mechanism through theintroduction of a cumulative measure of the irreversibilties %
ALERT School 2006 Rodrigue DESMORAT
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OutlineOutline
I- Elasto-plastic ity / Continuum Damage Mechanics
1. Plasticity in thermodynamics framework
2. Damage and effective stress concept
3. Elasticity coupled with damage
4. Von Mises plasticity coupled with damage
II- Ampli tude damage laws
III- Damage evolution laws for fatigue
1. Lemaitre's damage law2. Quasi-brittle materials
3. Rocks or soils
4. Micro-defects closure effect - Mean stress effect
5. Damage post-processing
6. Jump-in-cycles procedure
IV- Toward an unified approach for damage and fatigue ?1. Basis of a thermodynamics framework
2. Application to metals, concrete, elastomers and rocks
V- High Cycle Fatigue 60
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Rocks or soils - Non incompressible plasticityRocks or soils - Non incompressible plasticity
Deviatoric irreversible strain rate
Equivalent (von Mises) irreversible shear strain
Hydrostatic irreversible strain
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Damage evolution law ?????
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Rocks or soils - Non incompressible plasticityRocks or soils - Non incompressible plasticity
Deviatoric irreversible strain rate
Equivalent (von Mises) irreversible shear strain
Hydrostatic irreversible strain
(a) D =Y
S"
#
$%
&
'(
s
"p
(b) D =Y
S"
#
$%
&
'(
s
)"p
2 possible extensions of Lemaitre's law
ALERT School 2006 Rodrigue DESMORAT
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OutlineOutline
I- Elasto-plastic ity / Continuum Damage Mechanics
1. Plasticity in thermodynamics framework
2. Damage and effective stress concept
3. Elasticity coupled with damage
4. Von Mises plasticity coupled with damage
II- Ampli tude damage laws
III- Damage evolution laws for fatigue
1. Lemaitre's damage law2. Quasi-brittle materials
3. Rocks or soils
4. Micro-defects closure effect - Mean stress effect
5. Damage post-processing
6. Jump-in-cycles procedure
IV- Toward an unified approach for damage and fatigue ?1. Basis of a thermodynamics framework
2. Application to metals, concrete, elastomers and rocks
V- High Cycle Fatigue 63
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Quasi-unilateral conditions
h=1
"= "
1#
hD
h=0.2
Physical mecanism
Mechanical behavior
EEt
Ec
"
!
microcracks and microcavitiespartially closed in compression
QUASI-UNILATERAL CONDITIONS
Elasticity different in tension and in
compression
Evolution of damage slower incompression than in tension
ONEstate of microcracking
=
ONEdamage variable
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Postive part in terms of principal stresses
Energy equivalence no possibility
Strain equivalence
!
" = " # #"
State potential introducing the micro-defects closure parameter
!" =1+#
2E
$ 2
1 %D+
%$ 2
1%hD
&
'(
)
*+%#
E
3$H
2
1%D+
%3$H
2
1% hD
&
'(
)
*+
Isotropic damage (Ladevze & Lemaitre, 1984)
Key : Gibbs potential can be continuously differentiated
h: micro-defects closure parameter
ALERT School 2006 Rodrigue DESMORAT
f
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State laws
Elasticity law and damage thermodynamics forceElasticity law and damage thermodynamics force
ALERT School 2006 Rodrigue DESMORAT
M ff
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Uniaxial case
Mean stress effectMean stress effect
ALERT School 2006 Rodrigue DESMORAT
M ffM t ff t
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Uniaxial case
Mean stress effectMean stress effect
For a non symmetric loading with as load ratio
R"=
"min
"Max
For a given stress amplitude, a larger load ratio(more time spent in
tension) gives a lower number of cycles to rupture(feature usually
represented as straight lines in Goodman and Haigh diagrams)
ALERT School 2006 Rodrigue DESMORAT
O liO tli
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OutlineOutline
I- Elasto-plastic ity / Continuum Damage Mechanics
1. Plasticity in thermodynamics framework
2. Damage and effective stress concept
3. Elasticity coupled with damage
4. Von Mises plasticity coupled with damage
II- Ampli tude damage laws
III- Damage evolution laws for fatigue
1. Lemaitre's damage law
2. Quasi-brittle materials
3. Rocks or soils
4. Micro-defects closure effect - Mean stress effect
5. Damage post-processing
6. Jump-in-cycles procedure
IV- Toward an unified approach for damage and fatigue ?1. Basis of a thermodynamics framework
2. Application to metals, concrete, elastomers and rocks
V- High Cycle Fatigue 69
DD ff t i f Fi it El t ltt i f Fi it El t lt
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R! = 23 (1+!) + 3(1"2!)
#H#eq
$
%&&
'
())
2
DamageDamage fromfrompost-processing of Finite Element resultspost-processing of Finite Element results
After an elastic computation (Neuber correction)
After an elasto-(visco-)plastic computation
Uncoupled approach
D =Y
S
!"#
$%&
s
p si p > pDDamage
evolution law(Lemaitre)
Y =1
2
!e : E : !e =
"eq2R
#
2E
Elastic energy
Triaxiality function
Damage gouverned by plasticity
D = Dc
Mesocrack initiation
Stress
triaxiality
ALERT School 2006 Rodrigue DESMORAT
D b ti i t ti f th l ti lD b ti i t ti f th l ti l
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!eq(t), !H (t) = 13tr!(t)
p(t)
Y(t) =!eq
2 (t)R"(t )
2E
Time to damage initiation: p(tD ) =pD ! tD
which are computed
in elasto(-visco-)plasticity
Damage calculation:
D(t) = DdttD
t
! =Y(t)
S
"#$
%&'s
p(t)dttD
t
!
Time to rupture :
D(tR ) =D c! tR
Damage by time integration of the evolution lawDamage by time integration of the evolution law
ALERT School 2006 Rodrigue DESMORAT
O tliO tli
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OutlineOutline
I- Elasto-plastic ity / Continuum Damage Mechanics
1. Plasticity in thermodynamics framework
2. Damage and effective stress concept
3. Elasticity coupled with damage
4. Von Mises plasticity coupled with damage
II- Ampli tude damage laws
III- Damage evolution laws for fatigue
1. Lemaitre's damage law
2. Quasi-brittle materials
3. Rocks or soils
4. Micro-defects closure effect - Mean stress effect
5. Damage post-processing
6. Jump-in-cycles procedure
IV- Toward an unified approach for damage and fatigue ?1. Basis of a thermodynamics framework
2. Application to metals, concrete, elastomers and rocks
V- High Cycle Fatigue 72
J i l f i di b bl k l diJ i l f i di b bl k l di
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Jump-in-cycles for periodic by blocks loadingsJump-in-cycles for periodic by blocks loadings
IDEA:
Before damage growth, run the computation until a stabilized cycle is reached
accumulated internal sliding
(plastic strain) over a cycle
Assume a linear variation of the damage (with respect to N)
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Once damage has started, calculate
Number of cycles to be jumped
Divide by
the computation time
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IV- TOWARD AN UNIFIED APPROACH
FOR DAMAGE AND FATIGUE ?
ALERT School 2006 Rodrigue DESMORAT
OutlineOutline
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OutlineOutline
I- Elasto-plastic ity / Continuum Damage Mechanics
1. Plasticity in thermodynamics framework
2. Damage and effective stress concept
3. Elasticity coupled with damage4. Von Mises plasticity coupled with damage
II- Ampli tude damage laws
III- Damage evolution laws for fatigue
1. Lemaitre's damage law
2. Quasi-brittle materials
3. Rocks or soils
4. Micro-defects closure effect - Mean stress effect
5. Damage post-processing
6. Jump-in-cycles procedure
IV- Toward an unified approach for damage and fatigue ?1. Basis of a thermodynamics framework
2. Application to metals, concrete, elastomers and rocks
V- High Cycle Fatigue 76
Generalized damage lawGeneralized damage law
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Thermodynamics potential
Thermodynamics variables
!" =(1# D) w1($) + w2 ($ # $% ) + ws (q,a)
Criterion fonctiun f =!"
1#D# x #Q # !s
Generalized damage lawGeneralized damage law
ALERT School 2006 Rodrigue DESMORAT
Generalized damage lawGeneralized damage law
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Thermodynamics potential
Thermodynamics variables
!" =(1# D) w1($) + w2 ($ # $% ) + ws (q,a)
Criterion fonctiun f =!"
1#D# x #Q # !s
Generalized damage lawGeneralized damage law
ALERT School 2006 Rodrigue DESMORAT
Stored (blocked)
energy density
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Dissipation potential
(non associated model)
F = f+ FD
FD =S
(s +1)(1!D)
Y
S
"#$
%&'
s+1
Evolution laws (normality rule)
Generalized damage
evolution law
Cumulative measure ofthe internal sliding
D =Y
S
!
"#
$
%&
s
'
! = "!# dt
ALERT School 2006 Rodrigue DESMORAT
OutlineOutline
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OutlineOutline
I- Elasto-plastic ity / Continuum Damage Mechanics
1. Plasticity in thermodynamics framework
2. Damage and effective stress concept
3. Elasticity coupled with damage4. Von Mises plasticity coupled with damage
II- Ampli tude damage laws
III- Damage evolution laws for fatigue
1. Lemaitre's damage law
2. Quasi-brittle materials
3. Rocks or soils
4. Micro-defects closure effect - Mean stress effect
5. Damage post-processing
6. Jump-in-cycles procedure
IV- Toward an unified approach for damage and fatigue ?1. Basis of a thermodynamics framework
2. Application to metals, concrete, elastomers and rocks
V- High Cycle Fatigue 80
Damage andDamage and fatiguefatigue of concreteof concrete
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Damage andDamage andfatiguefatigue of concreteof concrete
Calculated fatigue curve
Aas-Jackobsen formula
Hysteretic response in
compression
from time integration of the
generalized damage law
ALERT School 2006 Rodrigue DESMORAT
Damage andDamage and fatiguefatigue ofof elastomerselastomers
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E : Green Lagrange strain tensorS : 2nd Piola-Kirchhoff stress tensor
Damage andDamage and fatiguefatigueofofelastomerselastomers
D =Y
S!
"# $
%&
s
'
' = E'
( dt
ALERT School 2006 Rodrigue DESMORAT
Drucker-Prager plasticity coupled with damageDrucker-Prager plasticity coupled with damage
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Drucker-Prager plasticity coupled with damageDrucker Prager plasticity coupled with damage
recovers laws (a) and (b) with the relationship
ALERT School 2006 Rodrigue DESMORAT
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IV- HIGH CYCLE FATIGUE
Mesoscopic RVE behavior remains elastic
Damage by post-processing elastic FE computations
DAMAGE_2005 post-processor
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Two scale damage model
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Two scale damage model
4
Initial 3D thermolastic computation: !ij(t) ou "ij(t), T(t)
Scale transition law: Eshelby-Krner law with thermal expansion
Plasticity and damage at microscale
D(t)
ALERT School 2006 Rodrigue DESMORAT
Localization law for thermomechanical loading (basis)
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Localization law for thermomechanical loading (basis)
LE
!"! +=
#
:~ 1
Real problem: with: )1(~
DEE !=
Initial Eshelby problem: *1:
LE
!"! +=
#
Deviatoric part: with:
Hydrostatic part:
LD
D
LD
GD
D
!
"
! +
#
=
21
* pLD !! =
L
H
D
HL
KD
D
!
"
! +
#
=
31
*with: T
L!=
"#
6 ALERT School 2006 Rodrigue DESMORAT
Localization law for thermomechanical loading
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Localization law for thermomechanical loading
"D
=
1
1# bD"D# b((1#D)"
p#"
p)( )
"H
=
1
1# aD"H# a (1# D)$
#$[ ]%T( )
Deviatoric part:
Hydrostatic part:
"
=1
1# bD"+
(a # b)D3(1#aD)
"kk1+ b((1#D)"p
#"p)$
%& '
()+ a (1#D)
*
#*[ ]1#aD
+T1
Thermal effect if:
D = 0 and &!&
'D !0 even if &= &
recovers the law proposed by Sauzay andDesmorat (2000) for isothermal cases
)1(3
1
!
!
"
+
=a
)1(15
)54(2
!
!
"
"
=b
7 ALERT School 2006 Rodrigue DESMORAT
Constitutive equations at microscale
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Constitutive equations at microscale
12
Thermo-elasto-plasticity coupled with damage
with linear kinematic hardening
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Material parameters identification
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Material parameters identification
E, (, &, Cy, "f S, s, h = 0.2, Dc=0.3
Parameters at RVE mesoscale
8
Parameters at microscale
Plastic
modulus
Asymptotic
fatigue limit
Damage
parameters
(Lemaitre's law)
one tension curve (with plasticity)
one Whler curve
2 exp. curves necessary per temperature
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Example of identification for 2 temperatures
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Example of identification for 2 temperatures
E, (, &, Cy, "f S, s, h = 0.2, Dc=0.3
Parameters at mesoscale Damage
Low T Higher T
100
1000
10 10 10 10 10 102 3 4 5 6 7
"Max "Max
NR NR
Model with difference tension/compression
(h=0.2, s=4)
Model with difference tension/compression
(h=0.2, s=3)
8 ALERT School 2006 Rodrigue DESMORAT
Characteristic effects reproduced
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Characteristic effects reproduced
Nonlinear damage accumulation
Mean stress effect
Effect in trension-compression, no effect in shear
Biaxial effects
Thermal and thermomechanical fatigue
Fatigue of structures (3D model)
Complex, non proportional or random loading
(rate form model)
3 ALERT School 2006 Rodrigue DESMORAT
Out of phase 3D random thermomechanical fatigue
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A thermo-hydraulic computation gives the temperatureand stresseshistory.
DAMAGE post-processor gives D(t)and the time to mesocrack initiation.Here around 200 h in accordance with the observations of micro-cracks
initiation
Loading
sequence made
of 1000 points
3D stresses
Temperature
Out of phase 3D random thermomechanical fatigue
ALERT School 2006 Rodrigue DESMORAT
FATHER structure
FATHER results over a cycle and in terms of crack initiation
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FATHER first cycle at the most loaded point
D
time
p
point NR (amorage) temps en h
C11_m50i 139330 387,0C11_m60i 108350 301,0C11_m70i 94942 263,7C11_m80i 73596 204,4
C11_m90i 76618 212,8C14_m50i > 1E6 > 2780C14_m60i 691274 1920,2C14_m70i > 1E6 > 2780C21_m60i > 1E6 > 2780C24_m70i > 1E6 > 2780
FATHER results over a cycle and in terms of crack initiation
Time to crack initiationcomputed with DAMAGE_2005
Initiation observedbewteen 200h and 300 h
timeRodrigue DESMORAT
CONCLUSIONCONCLUSION
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CONCLUSIONCONCLUSION
Continuum Damage Mechanics allows for the estimation of the crack
initiation conditions in fatigue
Post-processing approaches efficient
Rate form of damage laws allows to handle complex loadings
Anisothermal conditions naturally taken into account
Rate form damage laws will be also helfull for coupling with other
physics (THM, diffusion problems)
Coupling with non associated plasticity possible by use of the (damage)
effective stress concept
Many materials, many applications concerned
Still a lot to do!
ALERT School 2006 Rodrigue DESMORAT
ReferencesReferences
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"Mechanics of solid materials", J. Lemaitre and J.L. Chaboche, Oxford University Press,
1991 (in english), Dunod, 1985 (in french)
"Modlisation et estimation rapide de la plasticit et de lendommagement", R. Desmorat,
Habilitation Diriger des Recherches de l'Universit Pierre et Marie Curie, 2000.
"Two scale damage model for quasi-brittle and fatigue damage", R. Desmorat, J. Lemaitre,
Handbook of Materials Behavior Models, chapter Continuous Damage, section 6.15, p. 525-
535, 2001.
"Thermodynamics modelling of internal friction and hysteresis of elastomers.", S.
Cantournet & R. Desmorat, C. R. Mcanique, 331,p. 265-270, 2003.
"Phenomenological constitutive damage models", R. Desmorat, chapter VII of the book
Local Approach to Fracture , CNRS Summer School MEALOR 2004, Ed. J. Besson,
Presses de lEcole des Mines de Paris, 2004.
"Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures", J. Lemaitre
et R. Desmorat, Springer, 2005.
"Continuum Damage Mechanics for hysteresis and fatigue of quasi-brittle materials and
structures", R. Desmorat, F. Ragueneau, H. Pham, International Journal of Numerical andAnalytical Methods for Geomaterials, in press 2006.
"Damage and fatigue: Continuum Damage Mechanics modeling for fatigue of materials and
structures" R Desmorat Revue Europenne de Gnie Civil vol 10 p 849-877 2006