dottorato di ricerca in ingegneria civile, meccanica e ... σ < 0 τ interface • friction plays...
Post on 22-Mar-2018
220 Views
Preview:
TRANSCRIPT
1
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
2
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
3
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
4
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
5
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.
6
• 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.
7
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
8
T
N body 1
body 2
undamagedinterface
processzone
realcrack
A B C
Multiscale interface model
MacroscaleInterface
9
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
10
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
11
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
12
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
13
• 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
14
• 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
15
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
16
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
17
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
18
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
19
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
20
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
21
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
22
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
23
(a) (b)
(a)
(b)
24
25
Numerical examples (first part)
Test to measure properties of the fiber/matrix interface within a composite
26
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
27
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.
28
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
29
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.
30
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.
31
Crack path
32
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
33
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
34
Animated deformation showing txy.Magnification factor: 5000.
35
Animated deformation showing the σy.Magnification factor: 5000.
36
• 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 =
37
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=
38
iso-damage curvessII
sI
∆0
∆sep
αI0
αII0
αIIf
αIf
𝐺𝐺𝑐𝑐𝑐𝑐 = 𝐺𝐺𝑐𝑐𝑐𝑐𝑐𝑐
Severe restriction:
same damage
same damage force
consequences
39
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
40
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
=
= ∑
( )( ) ( ) ( ) ( )
11
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 …
41
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
42
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
43
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
44
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
45
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.
46
• 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.
47
• 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)
48
• 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.
top related