Application of Continuum Damage Theory or f … · Application of Continuum Damage Theory or f...

Application of Continuum Damage Theory for Flexural

Fatigue Tests

Eng.°Luiz Guilherme Rodrigues de Mello

National Department of Infrastructure on Transportation

NUMBERS from BRAZIL`s HIGHWAYSFederal Highways (118.000km) Federal Highways Condition

Evaluation

79

84

92

93

94

95

96

97

98

99

00

01

02

03

04

05

06

07

08

Year

100%

80%

60%

40%

20%

0%

DNIT has approximately $4 billions for 2009/2010

INTRODUCTION

21.1

K

f KN ⎟⎠⎞

⎜⎝⎛=ε

nKAdNda

Δ= .

m

m

R

m DWD

α

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−=•

INTRODUCTIONWorks published on the subject:

H. J. Lee (1996). Uniaxial Constitutive Modeling of Asphalt Concrete Using Viscoelasticityand Continuum Damage Theory;

Park et al. (1996). A Viscoelastic Continuum Damage Model and its Application to UniaxialBehavior of Asphalt Concrete;

Y. R. Kim et al. (2002). Fatigue Performance Evaluation of West Track Asphalt MixturesUsingViscoelastic Continuum Damage Approach;

Daniel (2001). Development of a Simplified Fatigue Test and Analysis Procedure Using aViscoelastic Continuum Damage Model and its Implementation to WestTrack Mixtures;

Lundstrom & Isacsson (2003). Asphalt Fatigue Modeling Using Viscoelastic ContinuumDamage Theory;

Lundstrom & Isacsson (2004). An Investigation of the Applicability of Schapery’s WorkPotential Model for Characterization of Asphalt Fatigue Behavior;

Christensen & Bonaquist (2005). Practical Application of Continuum Damage Theory toFatigue Phenomena in Asphalt Concrete Mixtures

Baek et al. (2008). Viscoelastic Continuum Damage Model Based Finite Element Analysis ofFatigue Cracking;

Kutay et al. (2008). Use of Pseudostress and Pseudostrain Concepts for Characterization ofAsphalt Fatigue Tests;

CONTINUUM DAMAGE MECHANICS

“(…) The growth of short crack is them described in global way, using D instead ofthe crack length measure. The fact that short crack propagation does not follow thelong crack behavior is a good justification to use a more global parametrization, inthe framework of CD instead of Fracture Mechanics.” Chaboche & Lesne (1988)

“(…) Description of damage evolution in materials using micromechanical analysis isin principle very difficult as the character of the microstructure and interaction amongdefects is difficult to characterize based on details of the micro-cracks. Due to thisdifficulty, CD mechanics models often ignore the physical details of the defects andinstead simply focus on macroscopic response (e.g. stiffness) .” Lundstrom et al. (2007)

Why Continuum Damage Mechanics?

“(…) The evidence relating to the influence of microcracks on the mechanicalresponse of solids over a wide spectrum of circumstances is too obvious to beneglected.” Krajcinovic (1989)


A “proper” mathematical representation forthe damage variable;

Particular objective form for the strainenergy density (Green`s hyperelastic constitutivemodel);

Appropriate form of the damage evolutionlaws;

Three aspects of the CDM theory are associated with theselection of:

A A-A’=AD

AAAD D−

=damage

CONTINUUM DAMAGE MECHANICSClassical Continuum Damage

Scalar damage parameter: will never reaches the value D = 1.In this case we need a damage law (limit at D = 0,5 forexample).

But sometimes damage and microcracks distribution might appear to be different!

CONTINUUM DAMAGE MECHANICSMicrocracks in planesperpendicular to the tensileaxis:

sample will behave asdamaged

sample will behave asundamaged

If the sign of stress is reversed

Krajcinovic, 1989


( )εσ

εσ

.1

~

DE

−==

EED~

1−=Effectivestress

concept

Usually, damage parameter are related to mechanicalproperties

The concept of scalar damage has some advantages as theconstitutive equations for the damaged material areequivalent to those for the undamaged material, using theeffective stress instead of the nominal stress. (Krajcinovic 1989)


Correspondence principle

Work Potential Theory – WPT;

Damage evolution law;

Schapery (1984, 1990); Park et al. (1996)

CDM applied to viscoelastic materials


( )∫ ∂∂

−=t

R

R dtEE

t0

...1)( ττετε )(.1)( t

Et

R

R σε =

∫ ∂∂

−=t

dtEt0

.).()( ττετσ

Constitutive model for viscoelastic material

(convolution integral)

Schapery`s extended correspondence principle suggest that elasticand viscoelastic constitutive equations are identical, but stress andstrains in the viscoelastic body not necessarily are physical quantitiesbut pseudo-elastic variables.

The physical meaning of the pseudostrain (εR) corresponds to the linear viscoelastic stress for a given strain history


RC εσ .=

The pseudostrain concepts allow definition of the concept of pseudostiffness (C),which is the loss off stiffness solely due to loss of material integrity caused bydamage and defined as follows:

Park et al (1996)

How determine pseudostrain (sinusoidal loading)?

( )[ ]ϕθωεε ++= tEER

R sin...1 *0 ( )[ ]ϕθωεε ++= tS

ER

R sin...1 *0

Uniaxial condition (Lee 1996) Flexural condition


RC εσ .=

ijij

Wε

σ∂∂

= Rij

R

ijWε

σ∂∂

=

( )210 ..21. RRR CCW εε +=

),( DfW RR ε=

Work Potential Theory – WPT

Lee (1996) presented the pseudo-strain energy density function as thefollow:

Green’s hyperelastic model


Damage Evolution law

m

m

R

m DWD

α

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−=•

• α is a material constant;

• Initially, it could be defined as viscoelastic property: relaxation orcreep test;

( ) ( )( )

( ) ( )ααα

ε +−

+

=− −⎥⎦

⎤⎢⎣⎡ −=∑ 1/1

1

1/

11

2...

2 ii

N

iii

Ri ttCCID

Numericalintegration


Characteristic Curve C x D

2.10CDCCC −=

).exp( 21

CDCC −=

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 8.0E+04 1.6E+05 2.4E+05Nor

naliz

ed p

seud

o st

iffne

ss (

C)

Damage parameter (D)

Relationship that define the damage evolution

for a particular material/test/Specimen

(!)


Daniel & Kim 2002

Lundström & Isacsson 2003

Damage parameter - D

Pse

udo-

stiff

ness

-C

0

0.2

0.4

0.6

0.8

1

0 50000 100000 150000 200000 250000 300000

Pse

udo-

stiff

ness

Damage parameter

50/6070/100

160/220

MATERIALS AND METHODS

Layer % binder % Voids thickness(cm)

Conventional 5,5% 7% variable

Gap graded 7% 9% 5,0

Open graded 9% 18% 1,3

1,30 cm – open graded

Variableconventional(Dense)

38,0 cm - Base

Subgrade

5,0 cm – gap graded

18


19

Beams for felxural fatigue tests

Mixtures were tankenfrom construction sites;

molds were carefully filledwith AC (hetereogenity)

Compaction processusing haversine loading (2Hz and 2,8 MPa): close to fieldobservation

Approximately 250 fatigue tests


20


Height variation

Stif

fnes

sva

riatio

n

MATERIALS AND METHODSProjects

Loading modeTest temperature

TestsIdentification Mixture type 4°C 21°C 37°C

BC7 Conventional

Strain Control(haversine)

8 8 8 24

BC4 Open graded 8 8 - 16

KR7 Conventional 8 8 8 24

KRTR7 Conventional 8 8 8 24

SS7 Conventional 8 8 8 24

SS4 Open graded 8 8 - 16

BB3 Gap graded 8 8 8 24

BB4 Open graded 8 8 - 16

JR7 Conventional 8 8 8 24

JR3 Gap graded 8 8 8 24

JR4 Open graded 8 8 - 16

TG7 Conventional 8 8 8 24

TG3 Gap graded 8 8 8 24

TG4 Open graded 8 8 - 16

Total 320

23


Number of cycles

Stra

in

Wholer curves – 50% So

Number of cycles

Stra

in

Wholer curves – Wn/wn

Pronk & Erkens (2001)

MATERIALS AND METHODSIn summary the following conditions were used:

Air voids: 7% for the conventional specimens, 9% for thegap graded specimens and 18% for the open gradedspecimens.

Load condition: Constant strain level, 6 – 10 levels of therange (300-1900 microstrain)

Load frequency: 10 Hz (2 and 5 Hz for validation)

Test temperature: 4.4, 21, and 37.8oC for conventional andgap graded mixtures; 21.1, and 37.8oC for open gradedmixtures.

26

RESULTS AND DISCUSSIONS

0.00E+00

5.00E+01

1.00E+02

1.50E+02

2.00E+02

2.50E+02

3.00E+02

3.50E+02

0.00E+00

1.00E+03

2.00E+03

3.00E+03

4.00E+03

5.00E+03

6.00E+03

7.00E+03

0 100000 200000 300000 400000 500000 600000 700000 800000 900000

Mic

ro s

train

(æî)

Flex

ural

Stif

fnes

s -S

-(M

Pa)

Number of Repetitions - (N)

Flex

ural

stiff

ness

Number of cycles

Classical results:

27


Stain (microstrain)

Stre

ss

-2000

-1000

0

1000

2000

-300 -200 -100 0 100 200 300

Stre

ss (k

Pa)

Strain (με)

10th

50th

500th

5000th

20000th

90000th

Dissipatedenergy

28


29


(Rauhut & Kennedy 1982)

T (°C)

k2 VARIATION WITH

TEMPERATURE FOR

CONVENTIONAL MIXTURES

30

k2 VARIATION WITH

TEMPERATURE FOR ASPHALT

RUBBER MIXTURES


T (°C)

Despite the variability founded and the amount of tests, k2 will beless temperature susceptible for asphalt rubber mixtures (my pointof view!)

31

Kohls Ranch 100°FNf = 4.41E-10ε-4.09

R2 = 0.93

Kohls Ranch 70°FNf = 1.06E-12ε-4.62

R2 = 0.91

Kohls Ranch 40°FNf = 5.12E-18ε−5.91

R2 = 0.771.E-04

1.E-03

1.E-02

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

Stra

in L

evel

Cycles to Failure


32


Failure criteria!!

(Lundstrom et al. 2004)

33

-2000

-1000

0

1000

2000

-400 -200 0 200 400Stre

ss (k

Pa)

Strain (10-6 m/m)

N = 10N = 20N = 30N = 40


-2000

-1000

0

1000

2000

-2000 -1000 0 1000 2000

Stre

ss (k

Pa)

Pseudo strain- εR

N = 10N = 20N = 30N = 40

34

-2000

-1000

0

1000

2000

-400 -200 0 200 400Stre

ss (k

Pa)

Strain (10−6 m/m)

10505005002000090000

-2000

-1000

0

1000

2000

-2000 -1000 0 1000 2000Stre

ss (k

Pa)

Pseudo strain εR

105050050002000090000


35

Damage calculation


( ) )1(1

11

11

2

1 )4

.().(.2

ααα

ε +−+

=−

−⎥⎦⎤

⎢⎣⎡ −≅∑ ii

N

iii

Rm

ttCCID

Daniel (2001) showed that in the case of cyclic loadingsdamage can only accumulate during the tensile loadingportion of each cycle. Therefore, damage was calculated usingonly ¼ of the entire loading time, which was the approximateperiod for tensile stresses under haversine loading (uniaxialtests):

36

RESULTS AND DISCUSSIONSFor flexural conditions, the time should be different:

( ) )1(1

11

11

2

1 )2

.().(.2

ααα

ε +−+

=−

−⎥⎦⎤

⎢⎣⎡ −≅ ∑ ii

N

iii

Rm

ttCCID

37

0.E+00

4.E+04

8.E+04

1.E+05

2.E+05

0.0E+00 2.0E+05 4.0E+05 6.0E+05

Dam

age

para

met

er

Number of cycles

350,0 300,0 250,0212,5 187,5 175,0

0.E+00

1.E+05

2.E+05

3.E+05

4.E+05

5.E+05

0.0E+00 6.0E+04 1.2E+05 1.8E+05 2.4E+05 3.0E+05

Dam

age

para

met

er

Number of cycles

37°C21°C5°C


0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 5.0E+04 1.0E+05 1.5E+05 2.0E+05

Pse

udo

stiff

ness

(C)


300,0

250,0

Model

38


0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 5.0E+04 1.0E+05 1.5E+05 2.0E+05

Pse

udo

stiff

ness

(C)


350,0

300,0

250,0

212,5

Model

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 5.0E+04 1.0E+05 1.5E+05 2.0E+05

Pse

udo

stiff

ness

(C)


350,0300,0250,0212,5187,5175,0Model

α = 2,36

T = 21°C

2.10CDCCC −=

39

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 2.0E+04 4.0E+04 6.0E+04 8.0E+04

pseu

do s

tiffn

ess

(C)


975,0950,0900,0750,0675,0Model


0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 1.5E+05 3.0E+05 4.5E+05 6.0E+05

pseu

do s

tiffn

ess

(C)


275,0237,5225,0175,0162,5Model

T = 37°C

T = 4°C

40

Temperature C0 C1 C2 α

5°C 0,99 1,11E-05 0,84 3,17

21°C 1,03 2,39E-03 0,49 2,36

37°C 1,03 1,02E-02 0,41 1,79


T (°C)

Also k2 is lowerfor higher

temperature!

41


1.E+02

1.E+03

1.E+03 1.E+04 1.E+05 1.E+06

Stra

in (μ

m)

Number of cycles (Nf)

Power (37°C)

Power (21°C)

Power (5°C)

Comparison shouldbe made using a

pavement structuralanalysis!

42

Open graded mix at T = 21°C

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 1.0E+04 2.0E+04 3.0E+04

Pse

udo

stiff

ness

(C)


800700600600575Modelo


0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 1.0E+05 2.0E+05 3.0E+05

Pse

udo

stiff

ness

(C)


375300287,5212,5187,5150Modelo

Gap graded mix at T = 5°C

43

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 1.0E+05 2.0E+05 3.0E+05

pseu

do s

tiffn

ess

(C)


375,0300,0212,5187,5150,0Model275,0


0.E+00

2.E+03

4.E+03

6.E+03

8.E+03

Flex

ural

stif

fnes

s (M

Pa)

300,0 375,0 275,0 150,0 212,5 187,5

44

0.00

0.20

0.40

0.60

0.80

1.00

0.E+00 5.E+04 1.E+05 2.E+05

pseu

do s

tiffn

ess

(C)


JR7 - 21°C

JR3 - 21°C

JR4 - 21°C

RESULTS AND DISCUSSIONSComparison should be made using a pavement

structural analysis!

45

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 5.0E+04 1.0E+05 1.5E+05 2.0E+05

pseu

do s

tiffn

ess

(C)


250 - 5 Hz

250 - 5 Hz

187,5 - 5 Hz

300 - 5 Hz

225 - 2 Hz

Model - 10 Hz


0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 8.0E+04 1.6E+05 2.4E+05

pseu

do s

tiffn

ess

(C)


Model - 10 Hz

240 - 5 Hz

225 - 5 Hz

225 - 2 Hz

200 - 5 Hz

Characteristic curve should be independent of frequencyloading:

46


Daniel (2001) and Lundstrom et al. (2003) showedcharacteristic curve is also temperature independent. In myopinion, parameter α (also k2) will be temperature dependentand a unique C x D curve will be able for differenttemperatures.

Why not temperature comparison forcharacteristic curves?

47

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 5.0E+04 1.0E+05 1.5E+05

Pse

udo

stiff

ness

(C)


200,0150,0125,0112,5105,0Strain-controlled modelStress-controlled model

RESULTS AND DISCUSSIONSCharacteristic curve should be independent of loading mode:

0.0

0.2

0.4

0.6

0.8

1.0

0.0E+00 3.0E+04 6.0E+04 9.0E+04

pseu

do s

tiffn

ess

(C)


350,0300,0250,0225,0150,0Strain-controlled modelStress-controlled model

48

RESULTS AND DISCUSSIONSAs illustrated by different works (Lee et al. 2003; Kim et al.2006), parameter α is directly related with coefficient asfollows: α.22 =k

Conventional mix Asphalt rubber mix

49

RESULTS AND DISCUSSIONSPrediction of fatigue life:

50

RESULTS AND DISCUSSIONSAs we mention before, comparison among different mixturesshould be done by pavement structural analysis. This can bedone using numerical analysis. For example VECD+FEPsoftware (Dr. Richard Kim):

51

RESULTS AND DISCUSSIONSPreliminary results:

N = 5.0 x 107

WHAT DO WE HAVE RIGHT NOW

VEPCD: ViscoElasticPlastic Continuum Damage

Probably some interaction between ContinuumDamage and Fracture Mechanics

(Chehab 2002)

53

The evolution of internal damage in hot mix asphalt (HMA) can beproperly evaluated using the framework of the Continuum DamageTheory to determine its characteristic curve.

The characteristic curves proved to be unique for a wide range ofimposed strain amplitudes for both conventional and asphalt-rubbermixes.

Tests with different temperatures, however, were better fitted byadopting different values of parameter a in the damage evolutionrelation. This parameter reduces with the increase in temperature.

The results of this research, using bending fatigue tests, corroboratethe findings of other researchers about the uniqueness of thecharacteristic curve obtained from uniaxial fatigue tests subjected todirect tension.

CONCLUSIONS

54

Despite the fact that some mechanical properties are indirectlyobtained from bending tests, these are simpler to perform and areavailable and well known in several research centers.

The uniqueness of the characteristic curve is an auspicious factsince it provides a means to characterize a HMA with fewer laboratorytests than other approaches. It also implies that such curves can beimplemented in numerical codes to simulate the behavior of flexiblepavements subject to a wide range of in field load conditions.

CONCLUSIONS

55

ACKNOWLEDGEMENTS

Application of Continuum Damage Theory or f … · Application of Continuum Damage Theory or f...

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