desmorat r damage and fatigue

7/25/2019 Desmorat R Damage and Fatigue

1/95

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

[email protected]

ALERT School 2006

Revue Europenne de Gnie Civil, Vol. 10, n6/7, pp. 849-877, 2006


2/95

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

ALERT School 2006 Rodrigue DESMORAT


3/95

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



4/95

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


5/95

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


6/95

I- ELASTO-PLASTICITY /CONTINUUM DAMAGE MECHANICS

!

"E E

"p "e Damage = scalar variable D

D = 1!E

E

!

"-



7/95

OutlineOutline








3. Rocks or soils








8/95

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


9/95

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

(

)*

+*



10/95

Thermodynamics frameworkThermodynamics framework



!" =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

(

)*

+*


Stored (blocked)

energy density ws


11/95

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



12/95

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 :



13/95

OutlineOutline








3. Rocks or soils








14/95

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



15/95

OutlineOutline








3. Rocks or soils








16/95

Elasticity coupled with damage

D=D(Y) ou D=D( )!+

!

"

E

E(1-D)

f=0

f


17/95


State laws


!" =1

2(1#D)$ : E : $

! ="#$

#%=E(1& D) : %

!Y ="#$

#D ! Y =

1

2

": E : "



18/95

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:



19/95

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



20/95

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 )[ ]



21/95

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



22/95

Anisotropic damage modeling

Local FE Non local FE

D22field D22field

classical mesh dependency

Nooru-Mohamed test (1992)

Desmorat, Gatuingt, Ragueneau (2004-2006)


23/95

OutlineOutline








3. Rocks or soils








24/95

Plasticity coupled with damage

!

"E E(1-D)

"p "e

E

f=0plasticit et endomagement

f


25/95



!" =

1

2(# $ #

p

):E(1$D):(# $ #

p

)+

G(r)

Strain partition

! = !e+ !

p


Stored (blocked)

energy density ws


26/95

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



27/95


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



28/95

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



29/95

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"



30/95

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"



31/95

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



32/95


!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



33/95

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



34/95

II- AMPLITUDE DAMAGE LAWS

Numbers of cycles N

ALERT School 2006 Rodrigue DESMORAT34

"

!

$!p $"


35/95

II- AMPLITUDE DAMAGE LAWS

or "Numbers of cycles N


$"=2"Min case of symmetric loading


36/95

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

ALERT School 2006 R. DESMORAT


37/95

Amplitude damage law in terms of s tress

"D

"N= g(D)G# ($#,R# )

R"=

"min

"Max

load ratio

R"=

#"M

"M

=1symmetric

loading



38/95

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



39/95

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 ?



40/95

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



41/95

OutlineOutline








3. Rocks or soils








42/95

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



43/95

R! = 23 (1+!) + 3(1"2!) #

H#eq

$

%&&

'

())

2




D = Dc


"H=

1


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



44/95

R! = 23 (1+!) + 3(1"2!) #

H#eq

$

%&&

'

())

2




D = Dc


"H=

1


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



45/95

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



46/95

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



47/95

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



48/95

R! = 23 (1+!) + 3(1"2!) #

H#eq

$

%&&

'

())

2





"H=

1


Y=1

2"e:E :"

e=

#eq2R

$

2E

Stored energy

damage threshold

D =Y

S

"

#$

%

&'

s

p if ws > wD



49/95

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


50/95

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!

"



51/95

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



52/95

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



53/95

Modeling a loading dependent damage threshold


More acurate case with kinematic hardening


54/95

NumberNumber ofofcyclescyclestotorupturerupture ininfatiguefatigue


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



55/95

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)


Loading dependency of the ratio ND/NR Nonlinear damage accumulation


56/95

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



57/95

OutlineOutline



2. Damage and effective stress concept

3. Elasticity coupled with damage





3. Rocks or soils








58/95

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



59/95

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 %



60/95

OutlineOutline









3. Rocks or soils








61/95

Rocks or soils - Non incompressible plasticityRocks or soils - Non incompressible plasticity

Deviatoric irreversible strain rate

Equivalent (von Mises) irreversible shear strain

Hydrostatic irreversible strain


Damage evolution law ?????


62/95

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



63/95

OutlineOutline









3. Rocks or soils








64/95

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



65/95

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


f


66/95

State laws

Elasticity law and damage thermodynamics forceElasticity law and damage thermodynamics force


M ff


67/95

Uniaxial case

Mean stress effectMean stress effect


M ffM t ff t


68/95

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)


O liO tli


69/95

OutlineOutline








1. Lemaitre's damage law

2. Quasi-brittle materials

3. Rocks or soils







DD ff t i f Fi it El t ltt i f Fi it El t lt


70/95

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


Damage gouverned by plasticity

D = Dc

Mesocrack initiation

Stress

triaxiality


D b ti i t ti f th l ti lD b ti i t ti f th l ti l


71/95

!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


O tliO tli


72/95

OutlineOutline










3. Rocks or soils







J i l f i di b bl k l diJ i l f i di b bl k l di


73/95

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)



74/95

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 ?


OutlineOutline


76/95

OutlineOutline




3. Elasticity coupled with damage4. Von Mises plasticity coupled with damage





3. Rocks or soils









77/95



!" =(1# D) w1($) + w2 ($ # $% ) + ws (q,a)

Criterion fonctiun f =!"

1#D# x #Q # !s





78/95



!" =(1# D) w1($) + w2 ($ # $% ) + ws (q,a)

Criterion fonctiun f =!"

1#D# x #Q # !s



Stored (blocked)

energy density


79/95


(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


OutlineOutline


80/95

OutlineOutline




3. Elasticity coupled with damage4. Von Mises plasticity coupled with damage





3. Rocks or soils







Damage andDamage and fatiguefatigue of concreteof concrete


81/95

Damage andDamage andfatiguefatigue of concreteof concrete

Calculated fatigue curve

Aas-Jackobsen formula

Hysteretic response in

compression

from time integration of the

generalized damage law


Damage andDamage and fatiguefatigue ofof elastomerselastomers


82/95

E : Green Lagrange strain tensorS : 2nd Piola-Kirchhoff stress tensor

Damage andDamage and fatiguefatigueofofelastomerselastomers

D =Y

S!

"# $

%&

s

'

' = E'

( dt


Drucker-Prager plasticity coupled with damageDrucker-Prager plasticity coupled with damage


83/95

Drucker-Prager plasticity coupled with damageDrucker Prager plasticity coupled with damage

recovers laws (a) and (b) with the relationship



84/95

IV- HIGH CYCLE FATIGUE

Mesoscopic RVE behavior remains elastic

Damage by post-processing elastic FE computations

DAMAGE_2005 post-processor


Two scale damage model


85/95

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)


Localization law for thermomechanical loading (basis)


86/95

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


87/95

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


Constitutive equations at microscale


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Constitutive equations at microscale

12

Thermo-elasto-plasticity coupled with damage

with linear kinematic hardening


Material parameters identification


89/95

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


Example of identification for 2 temperatures


90/95

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)


Characteristic effects reproduced


91/95

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)


Out of phase 3D random thermomechanical fatigue


92/95

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


FATHER structure

FATHER results over a cycle and in terms of crack initiation


93/95

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


94/95

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!


ReferencesReferences


95/95

"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

desmorat r damage and fatigue

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