Multi-scale modeling of imperfect interfaces

Multi-scale modeling of imperfect interfaces Serge Dumont, Frédéric Lebon, Raffaella Rizzoni

International Workshop MULTI-SCALE MODELING AND CHARACTERIZATION OF INNOVATIVE MATERIALS AND STRUCTURES Cetara, Amalfi Coast - May 1-5, 2013

Plan of the presentation • Motivation • Methodology • Stiff interfaces in Elasto-statics (with a focus on an energy approach) • An example of soft interface • A Justification of Coulomb’s friction ? • Conclusion Multi-scale modeling

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Motivation

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Tyre Cornering

Large speed Multi-scale modeling

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Pellet-Cladding interaction Pressurized Water Reactor

µm scale Multi-scale modeling

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Molecular Adhesion

Spatial Slicer Multi-scale modeling

nm scale

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Forming Process

Pressurized Water Reactor Multi-scale modeling

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Modelling of masonry structures

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Interfaces in pavements

Roughness

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Conclusions • Multi-scale problems (from nm to structure) • How to take into account scale changes ?

• Various problems • A (too ?) large number of models Web of Science® Topic=(interface + mechanics) Results: 7,661

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(At least) two families of models • Phenomenological 3

2,5

2

1,5

manip Aron E=25961,21brique pleine E=8300 Brique pleine E=8300 Brique sans picots E=25961,21Brique sans picots

1

0,5

0 -0,05

0

0,05

0,1

0,15

0,2

0,25

-0,5

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(At least) two families of models • Phenomenological 3

2,5

2

1,5

manip Aron E=25961,21brique pleine E=8300 Brique pleine E=8300 Brique sans picots E=25961,21Brique sans picots

1

0,5

0 -0,05

0

0,05

0,1

0,15

0,2

0,25

-0,5

Coulomb ? Multi-scale modeling

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(At least) two families of models • Deductive 1 2 3 4 5

Behavior, Fissure, roughness, etc. At local scale Multi-scale modeling

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Methodology

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Deductive models : Introduction Thin Layers (Soft ?)

Perfect Adhesion ?

At least one small parameter (multiscale) : thickness

Behavior ? Roughness ? Curvature ?

ε

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Design (Structure finite elements) Low thickness Low stiffness ? Non linear ?

Large number of DOF Ill-conditioned systems Amplification of negative effects, Time step ??

Fiability and cost of numerical computations ???

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Asymptotic studies

Interphase

ε

Interface Interface law (memory of characteristics ?) Multi-scale modeling

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Asymptotic studies

Interphase

lim →

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Interface

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Asymptotic studies

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Asymptotic studies

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Elasto-statics

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Plane Stiff Interface

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Stiff Interface

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Stiff Interface

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Stiff Interface

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Stiff Interface

Lebon, F., Rizzoni, R.: Asymptotic analysis of a thin interface: the case involving similar rigidity, International Journal of Engineering Sciences, Vol. 48, pp. 473-486, 2010. Multi-scale modeling

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Stiff Interface

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Mathematical results (order 0)

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Mathematical results (Order 1)

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Lebon, F., Rizzoni, R.: Asymptotic analysis of a thin interface: the case involving similar rigidity, International Journal of Engineering Sciences, Vol. 48, pp. 473-486, 2010. Multi-scale modeling

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Rescaled energy

Expansions

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Hypothesis

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Isotropic materials

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Curved Stiff Interfaces

Lebon, F., Rizzoni, R. : Imperfect interfaces as asymptotic models of thin curved elastic adhesive interphases, Mechanics Research Communications, accepted, 2013. Multi-scale modeling

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Numerical Results

Dumont, S., Lebon, F., Rizzoni, R. : An asymptotic approach to the adhesion of thin stiff films, Mechanics Research Communications, in preparation. Multi-scale modeling

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Numerical Results

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The problem is not stable, we add Stable for D=0

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Numerical Results Validation

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Real

Approximated

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An example of soft interface

Rekik, A., Lebon, F.: Identification of the representative crack length evolution for a multi-level interface model for quasi-brittle masonry, International Journal of Solids and Structures, Vol. 47, pp. 3011-3021, 2010. Rekik, A., Lebon, F.: Homogenization methods for interface modeling in damaged masonry, Advances in Engineering Software, Vol. 46, pp. 35-42, 2012. Multi-scale modeling

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A justification of Coulomb friction ?

Lebon F., Ronel-Idrissi, S. : Asymptotic studies of Mohr-Coulomb and Drucker-Prager soft thin layers, International Journal of Steel and Composite Structures, Vol. 4, pp. 133-148, 2004 Multi-scale modeling

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A justification of Coulomb friction ? The NonAssociated Plasticity Case Plane problems, small deformations, perfect adhesion, adherents : elastic Adhesive x 1

ε/2

•p

-ε/2 Γ1 x2

S

bp (σ , e ) = c em + (tgθ − tgϕ)[sm − c] e

•p

+ χKe (e ) + χKσ (σ )

Drucker-Prager

•p   • p • p  K e = e , e m ≥ tgθ e d    ) K σ = {σ , tσ ≤ c − tgϕ nσ } Mohr-Coulomb

Γ0

•p

•p

K σ = {σ , s ≤ c − tgϕ s m }

Bε

•p

•p

•p

bp (σ,e ) = cem − tgϕ[nσ −c] e

•p

+ χK)e (e ) + χK)σ (σ) Multi-scale modeling

) • p • p  K e = e , e m ≥ 0   89

A justification of Coulomb friction ? Formal asymptotic expansions

σ 12 = µ ([u ] − [u ]) e 1

Tangential

LOCAL PROBLEM

p 1

σ 22 = (λ + 2µ )([u2e ] − [u2p ])

Normal ∂σ i 2  =0  ∂y2  e = ee + e p   σ ij = λ ( ekk ) e δ ij + 2 µ ( eij ) e  s ≤ c − tgϕ sm  •p   If s < c − tgϕ sm then e = 0 •p   If s = c − tgϕ sm then e = −k s

Multi-scale modeling

DP

MC

• p u  = − k s.n , k > 0  

• p u  = − k t , k > 0   Flow direction ? Threshold ? 90

A justification of Coulomb friction ? Numerical Quantization

• 3 examples

87

65

43 89 ε/2

Thickness (mm) 1 Substrata Young modulus (Gpa) 200 Substrata Poisson ratio 0.3 Thin layer Young modulus (Gpa) 30 Thin layer Poisson ratio 0.3 Cohesion (Mpa) 1 Friction angle (°) 30 Dilatance angle (°) 0 -2 F1 (N/mm) a) (3.6E )*step b) (3.6E-2)*step F2 (N/mm) a) 0 -2 b) (1.8E )*step Finite element 8-node quadrangle

-ε/2 148

60

40

S

100

2

1

1

100

1

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Thickness (mm) 0.5 Substrata Young modulus (Gpa) 240 Substrata Poisson ratio 0.38 Thin layer Young modulus (Gpa) 30 Thin layer Poisson ratio 0.3 Cohesion (Mpa) 1 Friction angle (°) 30 Dilatance angle (°) 0 F1 (N/mm) (0.8E-2)*step 91 Finite element 6-node triangle

A justification of Coulomb friction ? Numerical results 1 Elasticity Step 13 - Ratio tangential stress (Mpa)/tangential displacement (mm) along the thin layer

3

1,20E+04 1,10E+04

1

1,00E+04 9,00E+03 8,00E+03

Step 30 -Ratio tangential stress (Mpa)/tangential displacement (mm) along the thin layer

7,00E+03 6,00E+03

3,50E+04

5,00E+03 4,00E+03 400

450

500

550

600

650

700

750

3,00E+04

800

∠12/u1

Nodes

Step 15 - Ratio tangential stress (Mpa)/tangential displacement (mm) along the thin layer

2,50E+04 2,00E+04 1,50E+04

1,20E+04

1,00E+04

1,10E+04

4

1,00E+04

104

154

204

254

304

354

404

Nodes

9,00E+03

∠12/u1

2

54

8,00E+03 7,00E+03 6,00E+03 5,00E+03 4,00E+03 400

450

500

550

600

650

700

750

800

Nodes

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A justification of Coulomb friction ? Numerical results 2 Plasticity Step 22 - Ratio normal stress (Mpa) /normal displacement (mm) along the thin layer

3

4,00E+04 3,50E+04

1

3,00E+04 2,50E+04

Step 50 - Ratio normal stress (Mpa) /normal displacement (mm) along the thin layer

2,00E+04

7,00E+04

1,50E+04 1,00E+04 400

6,00E+04 450

500

550

600

650

700

750

800

5,00E+04

∠2/u2

Nodes

4,00E+04 3,00E+04

Step 24 - Ratio normal stress (Mpa) /normal displacement (mm) along the thin layer

2,00E+04

4,00E+04

1,00E+04 4

3,50E+04

104

154

204

254

304

354

404

Nodes

3,00E+04

◊ 2/u2

2

54

2,50E+04 2,00E+04 1,50E+04 1,00E+04 400

450

500

550

600

650

700

750

800

Nodes

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A justification of Coulomb friction ? Numerical results 3 Plastic yield 3

Step 22- Plastic Yield along the thin layer

Real yield

Step 50- Plastic Yield along the thin layer

2,2 Simplified Drucker-Prager 2,4

s ≤ c − tgϕ s m

Simplified Drucker-Prager

1,8

Plastic Yield (Mpa)

1

Plastic Yield MPa

2

1,6 1,4

Real Drucker-Prager

1,2 1

2,2

2 Real Drucker-Prager

Simplified

1,8

1,6

400

450

500

550

600

650

700

750

800

4

54

104

154

204

254

304

354

Nodes Nodes

Step24- Plastic Yield along the thin layer

404

σ.n ⊗sn

Step 50- Relative difference between the plastic yields

2,6 Simplified Drucker-Prager

Relative difference

2

Plastic Yield (Mpa)

2,4 2,2 2 1,8 1,6 1,4

Real Drucker-Prager

1,2 1 400

0,03 0,02 0,01 0 -0,01

4

54

104

154

204

-0,02

500

550

600

650

700

750

304

354

404

deviatoric

-0,03

spheric

-0,04 -0,05

450

254

Nodes

800

Nodes

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Stress vector

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A justification of Coulomb friction ? Numerical results 4 Plastic strain Evolution of the plastic strain components

3

4,00E-05

Strain12 Node710

3,50E-05

Strain12 Node780

2,00E-05 1,50E-05

Strain22 Node780

Strain12 Node710

3,50E-05

Strain22 Node710

3,00E-05

Plastic Strain

Plastic Strain

1

Strain12 Node650

Real strain

4,00E-05

Strain22 Node710

3,00E-05 2,50E-05

Evolution of the plastic strain components

Strain22 Node650

1,00E-05

Strain12 Node650

2,50E-05

Strain12 Node780

2,00E-05 1,50E-05

Strain22 Node780

ep

Strain22 Node650

1,00E-05

Simplified

5,00E-06 5,00E-06

0,00E+00 12

14

16

18

20

0,00E+00

22

-5,00E-06

12

Step

14

16

18

20

22

-5,00E-06

Strain11

Step

Strain11

Step 50: Equivalent plastic strain along the thin layer

Evolution of the plastic strain components

ep.n ⊗sn

5,0E-04 4,00E-04

Strain22 Node16

3,00E-04

Strain22 Node50

2

Plastic Strain

2,50E-04

Strain12 Node80

2,00E-04

4,5E-04

Strain12 Node16

Strain12 Node50

Equivalent plastic strain

3,50E-04

Strain22 Node80

1,50E-04 1,00E-04

Real plastic strain

4,0E-04

Simplified plastic strain

3,5E-04 3,0E-04 2,5E-04 2,0E-04 1,5E-04 1,0E-04 5,0E-05 0,0E+00

5,00E-05

0

100

200

300

400

0,00E+00

25

30

35

40

45

Nodes

50

-5,00E-05

Step

Strain11

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A justification of Coulomb friction ? Numerical results 5 Conclusions • Local problem : quantization • Plastic yield : simplified version • Plastic strain : simplified version σ.n

• Contact law

[u]

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Conclusion • • • •

Elasto-statics plane/curved stiff interfaces Efficient solver Elasto-statics plane soft interfaces Plasticity vs Coulomb

Outlook • Elasto-statics curved soft interfaces • Comparison stiff/soft • Elasto-dynamics Multi-scale modeling

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Thank You for Your Attention

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