Advanced constitutive model for bituminous

materials: materials: a research challenge for g f

road engineering

Prof. Hervé Di Benedetto 

ICPT Sapporo 07/08

OutlineOutline

• Introduction: bituminous materials and sollicitations • Introduction: bituminous materials and sollicitations on road

• Types of behaviour for bituminous materialsTypes of behaviour for bituminous materials

• The DBN model (thermo-visco-elastoplastic)

• Focus Focus Linear domain: Viscoelasticity (LVE)

Time-temperature superposition principle

• Importance of the type of behaviour: examples of numerical simulations

• Conclusion

2

Paris

Lyon.

3

Bitumen, mastic, bituminous mixture

Complex thermo-viscoplastic behaviour

• Bitumen: from fluid to brittle solidbehaviour

M i h “ l ”• Mastic : the “glue”Bitumen + fines (< 100μm)

• Bituminous mixture : used on roadAggregates: 80% to 85% in volume

(92% to 96% in weight)

4

(92% to 96% in weight)Bitumen: 12% to 20% in volume

(4% to 8% in weight)

Stress path I f l ( )

0.6 q (Mpa) z=-5 cmA3

Principal stress

In surface layer(s)

0.3

q (Mpa)z=-15 cmz=-25 cm

3

A2

B1

Bituminous layer(s) 25

0p (Mpa)

A1B3 B2

Bituminous layer(s)cm

-0.3 0 0.3 0.6

Rotation of axes

30

60

90Angle (°) z=-5 cm

z=-15 cmz=-25 cm

Rotation of axesB1

B3B2

-60

-30

0-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

z 25 cm

Isotropic and Elastic (Alisé LCPC)

B32

5-90

60

Distance from the center of the wheel (m)Cycles &Rotation of axes

Important aspects for bituminous layersImportant aspects for bituminous layers

Stiffness and evolution with time & temperatureFatigue and damage law evolutionFatigue and damage law evolutionPermanent deformation and accumulation of this deformationCrack and crack propagation in particular atCrack and crack propagation, in particular at low temperature

different “types or domains” of behaviour for6

different types or domains of behaviour forbituminous mixtures

Domains of behaviour(Di B d 90)(Di Benedetto 90)

|ε|LogFailure

|ε|Log

Influence of

+ coupling T-M 5

-2 Nonlinear

Permanent deformationif stress tests from 0

temperature

3

-4

linearDeformability

32Always great influence of Temperature and loading rate4

Linear viscoelaticity

LVE FATIGUE

41

Log(N)1 2 3 4 5-6

6

LVE FATIGUE

7Bituminous mixtures

Fundamental design methodTime increment

Mechanical - Empirical

Geometry Traffic Climate

Mechanical Empirical

Material propertiesResponse calculation

Elastic Behaviour

Sress & strain: σ, ε Rheologyfuture

Need a powerful

Performance model

gy

Need a powerful constitutive modelDamage increment: ΔD

8Cumulative damage: Σ(ΔD)

Behaviour and associated phenomenap• Linear viscoelasticity• Non linearity 1, 2• Non linearity• Fatigue• Healing Failure

|ε|Log

Influence of

+ coupling T-M 5

,

4g• Thixotropy• Crack propagation

-4

-2 Nonlinear

Deformability

Permanent deformationif stress tests from 0

Influence oftemperature

32

4

3• Permanent deformation• Brittle failure• Vi l ti fl

Log(N)1 2 3 4 5-6

4

6

Linear viscoelaticity

LVE FATIGUE

41

3

2 3• Viscoplastic flow• Thermo-mechanical coupling

2, 3

3, 5

• 3 D formalism /one D1,2,3,5TSRS Test

9Calculation stress path in road

DBN (Di Benedetto – Neifar) Model: Thermo-visco plastic

• Introduces non linearity and irreversibility but gives a linear behaviour in the small strain domain (asympyotic behaviour)

• Respects the time – temperature superposition i i l ( i th li d i )principle (even in the non linear domain)

10

One-dimensional formalism of the DBN Model

11

Generalisation of generalised KV body

EinfE1 E2 Ei En

η0

Generalisation

η1 η2 ηi ηn

For any material

) E P1

V

EP0 Vn+1

E Pna)

V1 Vn

12→ Choice of EP and V for mixes

Model For Bituminous mixturesModel For Bituminous mixturesDBN model (Di Benedetto, Neifar)

EPEPEP1E0EPnEP1E0EPn

(Τ) (Τ)(Τ) (Τ)

Instantaneousstrain/stress

& brittle η1(Τ) ηn (Τ)η1(Τ) ηn (Τ)& brittle

Each EP body behaves as a non cohesive granularEach EP body behaves as a non cohesive granular material

Δ Δσ εf ( )13

Δ Δσ εj jf= h( )

Behaviour of EPi body: generalisation of the M i lMasing rule

EP EPEP EP

σnv

η (Τ)

EP1E0EPn

η (Τ)η (Τ)

EP1E0EPn

η (Τ)si+

Vi i l di f+

η1(Τ) ηn (Τ)η1(Τ) ηn (Τ)

Ei1

Virgin loading curve f

ssio

n

Each EPi :Ei

Ei 1

Ei

E

1

i

1

Ei

1

Ei1 Com

pres si

+

si- = -k si

+

1 1

EiEi ε

nsio

n

14 si-

Virgin unloading curve f-

Ten

Three-dimensional formalism of the DBN Model

15

EP1E0EPnEP1E0EPn

3 D formalismη1(Τ) ηn (Τ)η1(Τ) ηn (Τ)

3 D formalism

• Elastoplastic 3D model for EPi

• For viscous branch: only one scalar equation and a scalar equation and a mapping rule

S ti D ithSame equation 1D with:

andv v vp vpσ σ ε ε→ →& &

Mapping rule : direction of d(σf)

and v v vp vpσ σ ε ε→ →

16 Di Benedetto et al 07

Model calibrationCalibration for small strainCalibration for small strain

Linear viscoelastic (LVE) behaviourComplex modulus testsDetermination of moduli Ej, Gi and viscosities ηj

≠ fr & ≠ T j, i ηj

Calibration at failure (in the ductile domain).Viscoplastic behaviorFailure criterion (Di Benedetto)

≠ ε & ≠ T .

dDetermination of limits sj+ et sj

-

Calibration at failure (in the brittle domain)Not treated

Calibration at failure (in the brittle domain)Brittle failure at low temperature / high rateF il i i

17Failure criterionDetermination of brittle limits

≠ T

Linearity domain (Bituminous mix)Non linearity

1|E*|/|E*0|

LGM, fréquence=2.5 HzFrequency 2.5 Hz

TemperatureLinearity domain (90%)0.8

0.9qFrequency 2.5 Hz

Temperature( )

~ VEL behaviour

0.7

In the roadbehaviour

0.5

0.6

89%, 5°C 89%, 23°C 89%, 41°C

In the road

0 3

0.493%, 5°C96%, 5°C

93%, 23°C

96%, 23°C

93%, 41°C

96%, 41°CDoubbaneh 920.3

10 100 1000Strain amplitude ε0 (10+6 m/m)

HDB 18

Tension/compression cyclic tests at same frequency and different ε0

Linearity domain (Binders & mixes) 3 4

3.6

3.8

4

MPa

)

1E+0850 d 50 d

2.6

2.8

3

3.2

3.4

mpl

ex M

odul

us (M Go*

0.95 Go*

1E+06

1E+07

mic

rost

rain

) 50 pen - unaged 50 pen - agedMortar - 50 pen HRA - 50 penDBM - 50 pen

i li i

T

2

2.2

2.4

0.1 1 10Shear Strain (percent)

Com

1E+03

1E+04

1E+05

trai

n Li

mit

(m LVE strain limit

Mixtures(p )

1E+01

1E+02

1E+02 1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 1E+09 1E+10 1E+11

LVE

St

LVE strain limits for 50 pen bitumen

Airey et al: IJRMPD 3/2003

Binders

1E 02 1E 03 1E 04 1E 05 1E 06 1E 07 1E 08 1E 09 1E 10 1E 11

Complex Modulus (MPa)

1E+07

1E+08

rain

) Radial SBS PMB - unagedRadial SBS PMB - agedM t di l SBS PMB

for 50 pen bitumen and mixtures

T

1E+05

1E+06

mit

(mic

rost

r Mortar - radial SBS PMBHRA - radial SBS PMBDBM - radial SBS PMB

T

1E+02

1E+03

1E+04

VE S

trai

n Li

m

LVE strain limits for radial SBS PMB

d i t

MixturesBinders

HDB 19

1E+011E+02 1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 1E+09 1E+10 1E+11

Complex Modulus (MPa)

LV and mixturesAirey et al: IJRMPD 3/2003

Linear case (small strain domain)Linear case (small strain domain)

EP1E0EPnEP1E0EPn

η1(Τ) ηn (Τ)η1(Τ) ηn (Τ)

No plastic deformation l ti li t Me

20

elastic compliance tensor M i

Mapping rule (linear case)

σklσ(t)

σ σf(t).σklσ(t)

σ σf(t).σf(t).

kl

σv(t)

σ σ (t)

σ(t).kl

σv(t)

σ σ (t)σ (t)

σ(t).σ(t).

σf(t)

( )

σfσf(t)

( )

σf

σijσij

Isotach case(Bituminous materials)(Bituminous materials)

σv and dσf have the

21same direction

Callibration in the small strain Callibration in the small strain domain : linear viscoelasticity

• Sinusoidal loading interpretation in the g pfrequency domain : Complex parameters (E*, G*, ν*), )

22

EP 1 E 0 EP n

Calibration for the small strain domain for E

η 1(T) ηn(T)

(Linear Visco-Elasicity)

&

3 Dim caseCreep functionη1 (T) ηn(T)⇒

E1 En

E0-E00ν1 νn

ν00 & ν0( ) hF t at=

Creep function

* ( )( )

( 1)

hiE

h

wtw

-

=GE0

E00 k

hParabolic

νi( 1)a hG +

110 11

−

⎟⎞

⎜⎛

η1(T) η n(T)

Optimisation η

h

0

10

1 )T( E1

E1E* ⎟⎟

⎠

⎞⎜⎜⎝

⎛η+

+= ∑=j ωi jj

* 0 00E E−

Optimisation 2S2P1D(7 constants)

+ 2

Time-Temperature

* 0 0000 k h 1

E EE ( ) E

1 ( ) ( ) ( )i

i i iωτ

δ ωτ ωτ ωβτ− − −= +

+ + +

+ 2

23

Time Temperatureprinciple(W.L.F.)

ηj(T) = ηj(T0) aT τ(T) = τ0(T0) aT

2S2P1D in 3Dim model (Di Benedetto et al 2007)(Di Benedetto et al, 2007)

1hk0

0*

)()()(1EEE)(E −−−

∞

+++−

+=ωβτωτωτδ

ωτiii

i)()()( β

* 0 000 k h 1( )

1 ( ) ( ) ( )i

i i iν ν ν

υ υυ ωτ υ

δ ωτ ωτ ωβτ− − −−

= ++ + +

&

E00, E0 , ν00, ν0 δ, τ, η, h, k & time-temperature superposition principle (C & C )

( ) ( ) ( )ν ν νβν00 & ν0

E0-E00

♦modelling of binders mastics & mixes

superposition principle (C1 & C2) 11 constants

E00 k

h

♦modelling of binders, mastics & mixes♦allows the introduction of a prediction formula providing the mix complex

η

h formula providing the mix complex modulus and mix Poisson’s Ratio from binder ones

24

ηNo simple analytical expression in

the time domain

Two devices for bituminous materials: mixes, mastics & bitumenmaterials: mixes, mastics & bitumen

T/C test (2 types)Focus on small strain

T/C test (2 types)H=160mm, φext=80mm

Annular Shear RheometerRheometer (ASR)

• Local strain measurements from some 10-6 to some 10-2

• High stress and strain resolutions• High stress and strain resolutions• precise loading conditions• Temperature control

25

e pe atu e co t o• Sinusoidal loading up to 10Hz H=40mm, φext=105mm, th=5mm

Kind of tests and measurementsLVE Theory

Tension/compression( ) sin( )t ts s f= +A i l t

Complex Young’s modulus

1 01( ) sin( )t te e w=1 01( ) sin( )t ts s w f= +Axial stress

Axial strainE*=(σ01/ε01) ejφ

2 02( ) sin( )t t ne e w f= - +Radial strain Poisson’s ratio ν*=(ε01/ε02) ejφν( 01 02)

3D approachAnnular Shear RheometerAnnular Shear Rheometer

0( ) sin( )t t tt t w f= +Shear stress( ) i ( )t tSh t i

Shear modulus G* ( / ) jφ

26

0( ) sin( )t tg g w=Shear strain G*=(τ0/γ0) ejφτ

1D approach

Validation of the time-Temperature superposition principle superposition principle

(linear domain)

27

104

105

Tref = 10°C aTE

50

60

70 -18.5°C -8.4°C 0.1°C

E*

103

10

-18.5°C-8 4°C(M

Pa)

30

40

50 10.1°C 20.3°C 30.8°Cmaster curve(°

)

102

8.4 C 0.1°C 10.1°C 20.3°C30 8°C

|E*|

(

10

20

φ E

10-4 10-3 10-2 10-1 100 101 102 103 104 105 106101

30.8 C master curve

aT * frequency (Hz)10-4 10-3 10-2 10-1 100 101 102 103 104 105 106

0 Tref = 10°C

aT * frequency (Hz)

aTEBBSG - M1B9

T q y ( ) T q y ( )

0 45

0.50

0.55 -18.5°C -8.4°C 0.1°C10 1°C

0

2 -18.5°C 20.3°C -8.4°C 30.8°C 0.1°C master curve10 1°C a

ν*Bituminous mixture: BBSG

0 35

0.40

0.45 10.1°C 20.3°C 30.8°C master curve

|ν*|

-2

φ ν (°)

10.1 C aTνBituminous mixture: BBSG

0 25

0.30

0.35

-6

-4

28 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106

0.25Tref = 10°C

aT * frequency (Hz)

aTν

10-4 10-3 10-2 10-1 100 101 102 103 104 105 106

-6Tref = 10°C

aT * frequency (Hz)

Bitumen (B 50/70) : master curves

29

Shift factor : aT

Close shift factor for E* and ν∗; fixed by the binder

Binder E*Binder ν*

Mastic E*Mastic ν*

1( )log RT

C T Ta − −=

Binder νMix E*

Mastic ν

2log

( )TR

aT T C− +

30

Prediction of the mix VEL behaviour from binderfrom binder

31

Prediction of the mix VEL behaviour from binder : complex modulusbinder : complex modulus

0_mix* * EE ( ) E E (10 )T Tαω ω= +

E

mix 00_mix binder0_binder

E ( , ) E E (10 , )E

T Tω ω= +

3 constantsE2 3 constants

translationΕ*mix(10−αω,T)

Ε*binder(ω,T)E1

E EE32

E∞_binder E∞_mixE0_mix

Prediction of the mix VEL behaviour from bi d P i ’ tibinder : Poisson’s ratio

**00 _00 ( ) E( ) mix mixE ii ωτν ωτ ν −−

=0 00 0 _ 00 _Emix mixEν ν− −

2 constants

• 5 constants to obtain the 3D mix behaviour

2 constants

5 3from the binder one

• Verified by 2S2P1D (and DBN) if 6 parameters Verified by 2S2P1D (and DBN) if 6 parameters are the same for binder and mix:

δ η h k C & C

33

δ, η, h, k, C1 & C2

Examples of simulations : 2S2P1D & l k b b d dDBN & link between binder and mix

34

Modelling E and ν : BBSG (M1A2)

N li d t f E d1,20

Normalised master curves for E and νratio

0 80

1,00*

0( ) EE iωτ −

0,60

0,8000 0EE −

*0( )iν ωτ ν−

0,40

0

00 0

( )ν ν−

Model

0 00

0,20Model

0,001,E-06 1,E-04 1,E-02 1,E+00 1,E+02 1,E+04 1,E+06 1,E+08 1,E+10

aT fr

35E0 (MPa) Einf (MPa) k h Delta Tau Beta ν0 ν∞50 40000 0,21 0,58 1,9 8,00E-01 20 0,42 0,19

ConstantsT

PMB3open-graded mix

mix (50/70 BBSG)

Bitumen mastic 48%

open graded mix

BitumenMastic 32%Mastic 48%Open- graded mixMi (BBSG) bitumen

mastic 32%s c 8%

Mix (BBSG)2S2P1D

bitumen

PMB3BitumenMastic 32%Mastic 48%Open- graded mixMix (BBSG)Mix (BBSG)2S2P1D

E l ti ith Evolution with granular concentration

36

concentration

prediction from bitumen dataPrediction

(from f i )

mix data

transformation)

37

G*(ω) = G0 + G∞ − G0

1 + δ(iωτ)-k + (iωτ)-h + (iωβτ)-1 2S2P1D Parameters ( ) ( ) ( β )

4 constant parameters:k h δ β

Master curve, Tr = 10° C

1.E+10

1.E+11

k, h, δ, βOnly 3 parameters function of the aggregate 1.E+06

1.E+07

1.E+08

1.E+09G

*| (

Pa)

B5070M5070CDMX30 gg g

structure (filler concentration,.):G0 G τ0

1.E+02

1.E+03

1.E+04

1.E+05

|G M5070CDMX40M5070CDMX50M5070CDMX55Mix 50/702S2P1D Model

Material G0 (Pa) G∞ (Pa) k h δ τ0 = τ (10°C) β

B5070 0 9 50E+08 0 21 0 55 2 3 9 18E 05 450

G0, G∞, τ01.E-08 1.E-06 1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06 1.E+08aT.frequency (Hz)

B5070 0 9.50E+08 0.21 0.55 2.3 9.18E-05 450

M5070U100µ30 150 1.85E+09 0.21 0.55 2.3 1.40E-04 450

M5070U100µ40 250 4.10E+09 0.21 0.55 2.3 1.73E-04 450

HD

M5070U100µ50 350 6.30E+09 0.21 0.55 2.3 1.83E-04 450

M5070U100µ55 2000 8.80E+09 0.21 0.55 2.3 2.18E-04 450

Mix 50/70 6.00E+07 1.40E+10 0.21 0.55 2.3 7.00E-02 450HDB 38

Mix 50/70 6.00E+07 1.40E+10 0.21 0.55 2.3 7.00E 02 450

Generalisation of the Time-Temperature f psuperposition principle

• Linear domainDifferent experimental validations already shown: p yvalidity in 3 dim & same aT for E* and ν∗

• Non linear domain Cyclic compression & cyclic tension test

• High frequenciesHigh frequenciesBack analysis of wave propagation (linear domain)

39

Waves propagation

• Back analysis

*

Impact Wave propagation

*(1 )1(1 )(1 2 )( )

p

EC

ν

φ ν ν ρ

−=

+ −

10

0 20 40 60 80 100 120 140

10

( )( )cos( )2φ ρ

4

6

8

10

4

6

8

10

Capteur 1 Capteur 2

Wave arrival

piezoelectric sensors4

-2

0

2

4

-2

0

2

piezoelectric sensors

-10

-8

-6

-4

-10

-8

-6

-4

Wave departure

400 20 40 60 80 100 120 140

Temps (μs)

Waves propagation WaveWaves propagation

Master curve |E0,0TREF °C ENTPE3-2

Wavef=30kHz

|1,E+05

0,8- 19°C

26°C

1,E+04

Pa

)

51,4

40,9

30,2

42°C

°1,E+03

|E*|

(M

P 30,2

20,8

11,6

T (°C)

1,E+02 -9,9

-20,1

1,E+01

1,E-09 1,E-07 1,E-05 1,E-03 1,E-01 1,E+01 1,E+03 1,E+05 1,E+

41

a T . frequency (Hz)

Complex modulus test

Wave propagation: master curveWave propagation: master curve

Wave propagation : ~ 30 kHz• Master curve (shift aT)

1,E+04

1,E+05p p g

Classical tests (< 10 Hz) - 19°C26°C

42°C

1,E+03

| (M

Pa)

Modeling (DBN)

42°C

1,E+01

1,E+02

|E*

2S2P1DENTPE3-2test 26°Ctest -19°C

1,E+001,E-11 1,E-08 1,E-05 1,E-02 1,E+01 1,E+04 1,E+07 1,E+10

test 42°CTref = 0°C

aT . frequency (Hz)

Moduli & Validation of the t-T at high frequency

42

g q y

Importance to chose an appropriate Importance to chose an appropriate model for simulation

Elasticity versus Viscoelasticity

43

Elasticity versus ViscoelasticityElasticity versus Viscoelasticity

• FEM calculation of the 5 point bending test (french standard NF P 98-286) use to study fatigue of mixtures surfacing on orthotropic steel briges

Millau Bridge

343 meters

2460 meters

44

Identified problem for ortotropic bridgesIdentified problem for ortotropic bridges

crackmaximum tensile strain

crackat the top of the bituminous surfacing

bituminous mix

supporting line deck plate

Trough web

45

5 point bending test (principle)Shoes with rubber

cyclic loading 4 Hz

web

steel plate weldsPP

Critical zone (crack)

46 tensionCompression

FEM CalculationS l d li h li l i i i• Steel and sealing sheet : linear elastic isotropic

• Mix surfacing (isotropic)El i ( d l fi d b d f )Elastic (modulus fixed by temperature and frequency)

Viscoelastic with ν (Posson’s ratio) constant

Viscoelastic with ν function of time (DBN model with 20 Viscoelastic with ν function of time (DBN model with 20 elements)

65 mm 65 mm130 mm 130 mm190 mm

i ( )i ( )P (Figure 6)

e = 60 mm

P (Figure 6)

bituminous

P sinus (4 Hz) P sinus (4 Hz)

e = 60 mm

e = 12 + 3 mm33 mm

mix

steelt

sealing sheet

10 mm12 mm

10 mm

support

47

20 mm20 mm 230 mm 230 mm2*34+12 mm

FEM Calculation 2.02.42.8

t=0s LVE1 (10°C) LVE2 (30°C)

FEM Calculation

Calculation at point « A » 0 40.81.21.6 T=10°C

t=10000scycle n°40000

MPa

)

T=30°Ct=10000scycle n°40000

10 °CCalculation at point « A »

-0 8-0.40.00.4

σ (M

10 C

30 °CBig difference

2 42.8

LE1 (10°C) -4000 -3000 -2000 -1000 0 1000-1.6-1.20.8

T=10°C; t=41sT=30°C; t=0.1s

Elastic

Viscoelastic

g

1.21.62.02.4 LE1 (10 C)

LE2 (30°C) ε (10-6)Elastic

10 °Cy A

0.00.40.8

σ (M

Pa)

30 °C

x

yσxx→ σεxx → ε

A

B

1 6-1.2-0.8-0.4 No compression in the

Linear Elastic (LE) case

B

48-4000 -3000 -2000 -1000 0 1000

-1.6

ε (10-6)

FEM Calculation (30°C)

P ~

49

- Powerful constitutive law for

bituminous materials &

Implementation for road calculation - Implementation for road calculation

and design

A research challengeA research challenge

With i t t ti l i li tiWith important practical implication

50

Thank Youどうも ありがとう

Thank You

Merciどうも ありがとう！

Merci

H é Di B d ttHervé Di BenedettoENTPE/CNRSSC indexed

journal

Some publications on the DBN Model

NEIFAR M., DI BENEDETTO H., :Thermo-viscoplastic law for bituminous mixesp fInt. Jl Road Materials and Pavement Design, Vol 2, pp. 71-95, N°1/2001.

OLARD F., DI BENEDETTO H.OLARD F., DI BENEDETTO H.The “DBN” Model : A Thermo-Visco-Elasto-Plastic Approach for Pavement Behavior Modeling. Application to Direct Tension Test and Thermal Stress Restrained Specimen Test Journal of the AAPT 33 p 2005Thermal Stress Restrained Specimen Test, Journal of the AAPT, 33 p., 2005

DI BENEDETTO H., NEIFAR M., SAUZEAT C. and OLARD F.Three dimensional thermo viscoplastic behaviour of bituminous materials:Three-dimensional thermo-viscoplastic behaviour of bituminous materials: the DBN model, Int. Jl Road Materials and Pavement Design, Vol. 8, N°2, pp. 285-316, 2007

DI BENEDETTO H., DELAPORTE B., SAUZEAT C,. Three-dimensional linear behavior of bituminous materials: experiments

52

and modeling, ASCE jl of Geomechanics, Volume 7, N°2, pp. 149-157, 2007