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Modellazione delle interfacce

Elio SaccoUniversità di Cassino e del Lazio MeridionaleDipartimento di Ingegneria Civile e Meccanica

sacco@unicas.it

Dottorato di Ricerca in Ingegneria Civile, Meccanica e Biomeccanica

19 giugno 201511:00 – 13:00; 15:00 – 17:00

aula 1S3

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An interface model coupling damage friction and unilateral effect

Elio SaccoUniversità di Cassino e del Lazio MeridionaleDipartimento di Ingegneria Civile e Meccanica

sacco@unicas.it

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In collaboration with:

Nelly Point, Sonia Marfia, Giulio Alfano, Jessica Toti, Frédéric Lebon, Raffaella Rizzoni, Serge Dumont, Francesco Freddi,

Roberto Serpieri

• Delamination• Coupling damage and friction in the interface• Dilatancy and interlocking• Coupling of interface damage with body damage• Interface model accounting for in-plane effects• Higher order asymptotic theory

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Outline of the presentation• Motivations and objectives• Multiscale interface model

– Main idea– Equilibrium and compatibility– Combining interface damage and friction– Damage evolution

• Micromechanical approach (m.s.)• Numerical examples (first part)• Thermodynamic considerations• Interlocking and dilatancy• Numerical examples (second part)• Conclusive remarks

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Motivations and objectives

• Material nonlinear effects localized in thin zones defined by narrow layers, where high strain gradients occur.

• Small thickness of these layers, it can be neglected in a mathematical model.

• Layer replaced with an interface where displacement discontinuities can take place.

• Interface models characterized by suitable constitutive relationships between the stresses acting on the interface and the displacement discontinuities.

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• Interface widely adopted in many engineering problems:– adhesion of joined bodies– interaction of heterogeneities in composite materials– opening of cracks for the evolution of potential fracture lines– formation of shear bands.

• Used at different scales:– geological scale, to reproduce tectonic movements– structural scale, to predict the construction response– scale of the material, to evaluate the overall response of

composite materials subjected to damage– nano-scale, to study the crack growth in layered nano-materials.

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σ < 0τinterface

• Friction plays a basic role in many problems involving crack growth

• Several proposed models couple interface decohesion and friction using softening plasticity (Maier and Cocchetti (2002); Bolzon and Cocchetti (2003), Giambanco et al. (2001), Gambarotta et al. (1997)).

• In other approaches material softening and friction are directly linked in the equations (Chaboche et al (1997), Raous et al (1999), Lin et al. (2001))

σ

τ

2D schematization

Initialfailure-locus (cohesive phase)

Final failure-locus (complete decohesion)

Objective: derive an interface model based on a micromechanicalapproach able to couple damage and friction

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T

N body 1

body 2

undamagedinterface

processzone

realcrack

A B C

Multiscale interface model

MacroscaleInterface

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T

N body 1

body 2

MacroscaleInterface

undamagedinterface

processzone

realcrack

A B C

Multiscale interface model

Undamaged material

A

Au=A Ad=0

MesoscaleRVE

Representative Volume Element

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T

N body 1

body 2

MacroscaleInterface

undamagedinterface

processzone

realcrack

A B C

Multiscale interface model

Partially damaged material

B

Au=part of A Ad=A-Au

MesoscaleRVE

Representative Volume Element

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T

N body 1

body 2

MacroscaleInterface

undamagedinterface

processzone

realcrack

A B C

Multiscale interface model

Fully damaged material

C

Au=0 Ad=A

MesoscaleRVE

Representative Volume Element

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Multiscale interface model

AA

BB

CC

partial decohesion

total decohesion

A B C

no decohesion

TN

partial decohesion

total decohesion

A B C

no decohesion

N TUndamaged part

Au=A Ad=0

Undamagedpart

Damage part

Au Ad

Damaged part

Au=0 Ad=A

Au=A Ad=0

Au Ad

Au=0 Ad=A

TN

TN

T

N

T

N

TN

TN

RVERepresentative Volume Element

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• on the undamaged part : suitable constitutive law

• independent constitutive laws on damaged and undamaged parts

Main idea

• on the damaged part : contact with friction

• damage evolution depends only on the response of the undamaged part

A

Damaged partUndamaged part

(1-α) A α A

Ideally ‘perfectly flat’ surface

thickness = 0

α ∈ [ 0 , 1 ] : damage parameterRepresentative Volume Element

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• Equilibrium: additive decomposition of the interface stress:

( )1 u dα α= − +σ σ σuσ dσ

σ

• Kinematic compatibility: u d= =s s sus ds

s

Compatibility and equilibrium

Damaged partUndamaged part

(1-α) A α A

localization

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ds

us

desdis

uu= =σ K s K s

s

0

0

00n

t

KK

=

K

dσ Coulomb friction law

( )( ) 01 0

0

dn n nd

didt tt

h s K sσs sKτ

− = = −

σ ( )1

1

1

1 0

0 0

if sh s

if s

>= <

damaged part Coulomb friction law (no dilatancy)

( )φ d d dσµ τ= +σ

0λ ≥ ( )φ 0d ≤σ ( )φ 0dλ =σ

undamaged part Linear elastic law

Combining interface damage and friction

0di

d

λ ϕτ

= ∂ ∂

s

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Damage evolution (2D)

• damage activated in both mode I and mode II

• linear softening in mode I and mode II

• different values of fracture energy in mode I and mode II

• mixed mode accounted for1 2

1 21 22 2

o o o o

c c

s sG Gσ τη η= =

first cracking relative displacement

peak value of the stressfracture energy

Mode 1

Area = Gc1

so1 sc1

σo

σ

τ

Mode 2Area = Gc2

- sc2sc2so2

- so1I

τo

sT

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Damage evolution (2D)

• damage activated in both mode I and mode II

• linear softening in mode I and mode II

• different values of fracture energy in mode I and mode II

• mixed mode accounted for1 2

1 21 22 2

o o o o

c c

s sG Gσ τη η= =

2 2

21 221 2 1 21 with

s s s sη η η = − + = +

ss s

equivalent relative displacement ratio

2 21 2

1 2

1o o

s ss s

β

= + − 1max min 1

1history

βαη β

= , +

Damage evolution

Mode 1

Area = Gc1

so1 sc1

σo

σ

τ

Mode 2Area = Gc2

- sc2sc2so2

- so1I

τo

sT

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Micromechanical approach (m.s.)

undeformedRVE

deformed RVE

h

h

sB’

sB’’

s

sB’

sB’’

s

B’

B’’

T

N

T

N

T

N

relative displacement

{ }TT Ns s=s

average strain { } /TNT NE E h= =E s

average shear and normal stress 1 1,NT NT N NV VdV dV

V Vσ σΣ = Σ =∫ ∫

local normal stress in the direction of potential fracture neglected

Homogenization problem

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Material =σ Cε

Unilateral contact and friction

0

0, 00,

σ τσ τ τ

= =< <limit shear stress associated to the normal stress

0τ

0 , 0 , 0N Nd dσ σ≥ ≤ =

Damage evolution governed by • the overall relative displacement acting on the RVE • classical Linear Fracture Mechanics (LFM)

Constitutive laws

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Solution procedure: definition of the subproblems1. Problem (p1) considers the RVE subjected to eE (i.e. s).

The relative displacement at the crack is denoted as ed .

2. Problem (p2), the relative displacement c e= −d d is prescribed between the crack mouths, while the overall relative displacement is enforced to be zero.

3. Problem (p3), the RVE is subjected to a relative displacement { }0 TTp=p

at the crack mouths, corresponding to the frictional sliding, leaving the overall relative displacement equal to zero.

s

dcde df

p1 p2 p3

s

dcde df

p1 p2 p3

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Solution of three linear elastic problems

Solution s1 s2 s3 Average strain, E eE 0 0 Average stress, Σ eΣ cΣ fΣ Stresses at the crack, τ σ 0 c cτ σ f fτ σ Relative displacement at crack, d ed c e= −d d { }0 T

Tp=p

s

dcde df

p1 p2 p3

s

dcde df

p1 p2 p3

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Possible mechanical situations

Open crack solution s1:

Average strain Average stress Stress at the crack Relative displacement at crack

e=E E e=Σ Σ 00

τσ

==

e=d d

Closed crack with no-sliding solution s1+s2:

Average strain Average stress Stress at the crack Relative displacement at crack

e=E E e c= +Σ Σ Σ

c

c

τ τ

σ σ

=

= =d 0

Closed crack with sliding solution s1+s2+s3:

Average strain Average stress Stress at the crack Relative displacement at crack

e=E E e c f= + +Σ Σ Σ Σ

c f

c f

τ τ τ

σ σ σ

= +

= + { }0 T

Tp= =d p

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(a) (b)

(a)

(b)

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Numerical examples (first part)

Test to measure properties of the fiber/matrix interface within a composite

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Load applied in two steps:

1. constant field of inelastic strains to simulate the experimentally measured chemical and thermal matrix shrinkage, kept constant;

2. vertical load applied by prescribing the vertical displacement of the top side of the punch and increasing it incrementally.

FEM scheme

Shrinkage effectPunching effect

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Load-displacement curve obtained for the polyester / epoxy composite (Ef/Em = 0.625), comparison with the experimental and other numerical results.(a)-(b) linear elastic behavior of the matrix/fiber interface; (b)-(c) process zone develops on the top of the interface; (c) process zone completely developedand crack begins topropagate from the top towards the bottom; (c)-(d) stable propagation; (d) unstable propagation.

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Contour plots of the shear stress

just after the application of matrix shrinkage

just before the peak load is reached

after complete damage has occurred, that is during the final sliding phase

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Masonry wall loaded in compression and shear studied by Raijmakers and Vermeltfoort (1992)

Loading:

initial compression obtained by prescribing the vertical displacement of the top (vertical reactions 30KN);

vertical displacements kept constant during the analysis;

horizontal displacement of the top-right corner incremented left-ward.

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Each half brick discretized with 2 × 2 4-noded,plane stress elements with enhanced strains.

Interface elements placed on the brick/mortar and on the brick/brick interfaces, to simulate the possible failure of a brick.

Numerically computed horizontal reaction Fplotted vs prescribed horizontal displacement; comparison with the experimental data.

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Crack path

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ICOLD benchmark test

Concrete

E = 24 GPa ν=0.15 γ = 24 KN m-3

Concrete/soil interface

Gc1 (J m-2) Gc2 (J m-2) η σo (MPa) τo (MPa)

90 350 0.9 0.3 0.7

Material properties

Water pressure effect

ICOLD - 5th International Benchmark workshop on numerical analysis of Dams. Theme A2: Imminent failureflood for a concete gravity dam, Denver, 1999.

( ) ( )MPa.q

qyqyq

o

ow

01=+= α

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Crack-mouth-Opening-Displacement (COD) vs. overload multiplier αfor different values of the water pressure decay parameter ρ

0

0,125

0,25

0,375

0,5

-0,002 0,002 0,006 0,01

ρ = 0

ρ = 50

ρ = 100

ρ = 500

ρ = 1000

α

COD (m)COD

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Animated deformation showing txy.Magnification factor: 5000.

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Animated deformation showing the σy.Magnification factor: 5000.

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• H1) A natural state exists

Thermodynamic considerationsHelmholtz free energy

( ) ( ) ( ), , ,u dψ α ψ α ψ α= +s s s

Xψ ψα

∂ ∂= = −

∂ ∂σ

s

Thermodynamic driving forces

sII

sI

∆0

∆sep

sI0 sIf

sII0

sIIf

naturalstate

complete separation

• H2) Initial elastic behavior in the neighborhood of the origin

• H3) A region of complete decohesion exists

• H4) Uncoupled elastic behavior in pure modes I and II2

0n ts sψ∂

=∂ ∂

• H5) Equivalent relative displacement

• H6) Linear elastic behavior for α=0 ( ) ( ) ( )2 21 1,2 2n n t tK s K sα αψ α α α= +s

0ntKα =

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Damage evolution recast as a complementary normality law

( ) ( ) ( )0, , 00 0 0

f X Xf f

α α α

α α

= − ≤

≥ ≤ ≤

s s

Consistency condition

0f =

system of two differential equations to be satisfied by ˆ Xα

( )ˆ eqsα α=

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iso-damage curvessII

sI

∆0

∆sep

αI0

αII0

αIIf

αIf

𝐺𝐺𝑐𝑐𝑐𝑐 = 𝐺𝐺𝑐𝑐𝑐𝑐𝑐𝑐

Severe restriction:

same damage

same damage force

consequences

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tangential displacement

norm

al d

ispl

acem

ent

Pure Mode I

st

τ 0

τ

sn

Pure Mode II

𝐺𝐺𝑐𝑐𝑐𝑐

𝐺𝐺𝑐𝑐𝑐𝑐𝑐𝑐

• This restriction is too severe and makes experimental calibration unfeasible as:

– pure mode II and mode II laws are generally different

– typically 𝐺𝐺𝑐𝑐𝑐𝑐𝑐𝑐 > 𝐺𝐺𝑐𝑐𝑐𝑐

σ 0 σ

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Modelling interlocking and dilatancy via a multi-scale approach

Smooth surface at the large-scale

Assumption: interface micro-geometry is periodic and is composed of a finite number of inclined flat planes

RVE

• sum of free energies on each microplane

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ), 1k k k k ku dψ α α ψ α ψ= − +s s s

( )

1

mNk

kψ ψ

=

= ∑

( )( ) ( ) ( ) ( )

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mNk k k k

u dk

D D=

= − +∑σ σ σ

1

2

ss

=

s

• each plane is ideally flat in that:

𝐺𝐺𝑐𝑐𝑐𝑐 = 𝐺𝐺𝑐𝑐𝑐𝑐𝑐𝑐, 𝐾𝐾𝑛𝑛𝑛 = 𝐾𝐾𝑡𝑡𝑛, 𝑠𝑠𝑐𝑐𝑛 = 𝑠𝑠𝑐𝑐𝑐𝑐𝑛 …

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s2 (mm)

τ(M

Pa)

p

q

s

q’

s

r

s

s

s

s

s

t

s

s

u

s

s

σs2

ABC

Typical mode II response to monotonic loadingAssessment of single point behavior

hh/3h/3h/3

θint

θint

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0 3.0MPaσ =

0 3.0MPaτ =2

1 0.3kJ/mcG =2

2 0.3kJ/mcG =0.9η =0.5µ =

Fixed parameterson each plane

Prescribed slip s2, and s = -2 MPa

Interface micro-geometry

hh/3h/3h/3

αd

ad

sII (mm)

τ(M

Pa)

Varying angle qint

Increase of mode II fracture energyAssessment of single point behavior

Transition: bilinear shape polynomial/exponential-like shape

Increase of ‘apparent’ mode II fracture energy

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Structural assessment: DCB-UBM numerical-experimental comparison(Exp. data from Sorensen and Jacobsen, 2009)

Experiment:

• Nonlinear FE simulations of mixed-mode DCB-UBM test on composite laminate specimens (glass/polyester)

• M1/M2 controlled via a wire and roller arrangement

• Response in terms of J integral

• Mode-mixity range is spanned all through

• Experimental results plotted in J-d curves

• d is the norm of the crack tip displacement

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Structural assessment: DCB-UBM numerical-experimental comparison

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 1 2 3 4 5 6

Frac

ture

Res

ista

nce

J R[N

/mm

2 ]

Relative displacement norm at the initial crack tip [mm]

Exp -M1/M2 = -1

Num M1/M2 = -1

Exp -M1/M2 = -0.52

Num M1/M2 = -0.52

Exp -M1/M2 = 0.25

Num M1/M2 = 0.25

Exp -M1/M2 = 0.5

Num M1/M2 = 0.5

Exp -M1/M2 = 0.87

Num M1/M2 = 0.87

• fracture energy 𝐺𝐺𝑐𝑐𝑐𝑐 = 𝐺𝐺𝑐𝑐𝑐𝑐𝑐𝑐 set on mode I

• (unique) values for 𝜇𝜇 and αd set by curve-fitting other mixed-mode responses

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Conclusive remarks

• Cohesive-zone model coupling damage and friction.

• Damage and friction treated separately: a different damage model can be chosen without modifying the friction model, and viceversa.

• Friction is introduced into the model, by a mesomechanic approach, through an additional stress vector acting on the damaged part of the representative interface area and using a simple Coulomb-friction law (no softening plasticity required).

• Damage-friction model recovered by (more sophisticated) mesomechanical analysis, developing a specific procedure.

• Cohesive-zone model successfully used for several applications.

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• General thermodynamic restrictions to model parameters were investigated:- for ideally planar surfaces: use of a single damage variable, in

combination with a normality hypothesis requires 𝑮𝑮𝒄𝒄𝒄𝒄 = 𝑮𝑮𝒄𝒄𝒄𝒄𝒄𝒄• Interlocking effect via a multiscale approach:

- for non planar surfaces: ‘apparent’ mode II fracture energy emerges as the joint effect of a nonzero friction angle and a nonzero interlocking angle.

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• Numerical-experimental comparisons of mixed-mode debonding of DCB-UBM tests on laminated composite specimens show that: - increase of apparent fracture energy under increasing mode

II/mode I ratio is predicted with good overall agreement- easier and physically clearer procedures are made available for

the separate evaluation, and interpretation of:decohesion, friction, interlocking.

Future work: - Extension to 3D problems- Incorporation of finite length of asperities (finite dilatancy)- Inclusion of asperity-related degradation (wear, crushing)

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• R. Serpieri, E. Sacco, G. Alfano. A thermodynamically consistent derivation of a frictional-damage cohesive-zone model with different mode I and mode II fracture energies. European Journal of Mechanics A/Solids 49, 13-25, 2015 (http://dx.doi.org/10.1016/j.euromechsol.2014.06.006).

• F. Freddi, E. Sacco. An interface damage model accounting for in-plane effects. International Journal of Solids and Structures 51, 4230 - 4244, 2014 (DOI 10.1016/j.ijsolstr.2014.08.010).

• R. Rizzoni, S. Dumont, F. Lebon, E. Sacco. Higher order model for soft and hard elastic interfaces. International Journal of Solids and Structures 51, 4137-4148, 2014 (DOI 10.1016/j.ijsolstr.2014.08.005).

• J. Toti, S. Marfia, E. Sacco. Coupled body-interface nonlocal damage model for FRP detachment. Comput. Methods Appl. Mech. Engrg. (http://dx.doi.org/10.1016/j.cma.2013.03.010) 260, 1–23, 2013.

• E. Sacco, F. Lebon. A damage–friction interface model derived from micromechanical approach. International Journal of Solids and Structures, doi: http://dx.doi.org/10.1016/j.ijsolstr.2012.07.028, 49: 3666–3680, 2012.

• S. Marfia, E. Sacco, J. Toti. A coupled interface-body nonlocal damage model for FRP strengthening detachment. Computational Mechanics, 50:335–351, DOI 10.1007/s00466-011-0592-7, 2012.

• E. Sacco, J. Toti Interface Elements for the Analysis of Masonry Structures. International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 11, pp. 354-373, 2010.

• G. Alfano, E. Sacco, Combining interface damage and friction in a cohesive-zone model. International Journal for Numerical Methods in Engineering, Vol. 68. pp. 542-582, 2006

• G. Alfano, S. Marfia, E. Sacco, A cohesive damage-friction interface model accounting for water pressure on crack propagation. Computer Methods in Applied Mechanics and Engineering, Vol. 196, pp.192-209, 2006

• N. Point, E. Sacco Mathematical properties of a delamination model. Math. Comput. Modeling, Elsevier Science Ldt, Great Britain, vol. 28, n. 4-8, pp 359-371, 1998.

• N. Point, E. Sacco Delamination of beams: an application to DCB specimen. Int. J. Fracture, Kluwer Academic Publishers, The Netherland, vol. 79, pp. 225-247, 1996.

• N. Point, E. Sacco A delamination model for laminated composites. Int. J. Solids Structures, Elsevier Science Ldt, Great Britain, vol. 33-4, pp. 483-509, 1996.