Fracture initiation in confined incompressible materials · Fracture initiation in con ned...

Fracture initiationin confined incompressible materials

Duvan Henao Manrique

Facultad de Matemáticas

Pontificia Universidad Católica de Chile

Institute for Mathematics and its Applications27 Feb 2018

Duvan Henao Pontificia Universidad Católica de Chile

I IMA

I FONDECYT 1150038

I Millennium Nucleus CAPDE NC130017

I Xianmin Xu, Chinese Academy of Sciences

I Carlos Mora-Corral, Universidad Autónoma de Madrid

I Marco Barchiesi, Napoli Federico II

I Sylvia Serfaty, Paris VI, NYU

I V́ıctor Cañulef, PUC, UAM


I IMA

I FONDECYT 1150038





I Sylvia Serfaty, Paris VI

, NYU



I IMA

I FONDECYT 1150038








I IMA

I FONDECYT 1150038






I V́ıctor Cañulef, PUC

, UAM


I IMA

I FONDECYT 1150038








N. Petrinic, J. L. Curiel Sosa, C. R. Siviour, B. C. F. Elliot:Improved Predictive Modelling of Strain Localisation and Ductile

Fracture in a Ti-64Al-4V Alloy Subjected to Impact Loading.

J. Phys. IV France 134 (2006), 147–155.


O. van der Sluis et al.: From fibrils to toughness: multi-scale mechanics of

fibrillating interfaces in stretchable electronics. Materials 11 (2018), 231.


8 The European Physical Journal E

has also been observed for other materials, such as vis-cous liquids [17–19], rubbers [20] and viscoelastic liquids(uncrosslinked polymer melt) [21] when their volume isforcedly expanded. These results are reviewed by Shulland Creton [22].

Theoretical studies for the debonding process have alsobeen done. Gay and Leibler [23] discussed semiquantita-tively the effect of the surface roughness and air suctionin soft elastic materials on the stress-strain curves. Cre-ton and Lakrout [14], and Crosby et al. [24] developed atheoretical analysis for the cavitation condition based onfracture mechanics. Chikina and Gay [25] gave a scalingargument to predict the number of bubbles appearing inthe viscoelastic film as a function of defect density andseparation speed.

Extensive works have been done for the dynamics ofsingle bubbles placed in a viscous medium [26], elasticmedium [27–30], and viscoelastic medium [31,32]. Theanalysis of these results has been applied to interpret theobserved growth kinetics of cavities [33,34]. Finite elementcalculation has been done [35] for a single bubble placedin a viscoelastic filament being stretched. Cavitation andfibrillation have also been observed in molecular-dynamicssimulation [36–38], where the polymer chains sandwichedbetween substrates or grafted on the substrate are sep-arated. Though the scale is quite different, the observedphenomena are quite similar to each other.

In spite of these studies, it is still not clear how vis-coelasticity and multiple cavitation dynamics affect themechanical properties. Here we present a simple modelto calculate the stress-strain curve of a viscoelastic liquidin the debonding process. Though the model presentedhere is crude, it includes various effects which have notbeen included altogether in a single model but have beendiscussed individually in previous papers, such as the vis-coelasticity of the film, the cavity expansion, and the slip-page between the film and the substrate. We also believethat the descriptions in our model hold, as the first ap-proximation, for actual PSA which are viscoelastic solidsand show very complex deformations.

2 The block model

2.1 Outline of the model

We consider a thin adhesive layer sandwiched between tworigid substrates, and discuss the process when the sub-strates are pulled apart with constant speed (see Fig. 1).Let H0 and L0 be the initial thickness and the length ofthe film. We assume that the film is thin, i.e., L0 ! H0.

The debonding process involves a fairly complicateddynamics, i.e., defomation of the adhesives, expansion ofthe cavities, and slip of the adhesive at the interface [39].Our model is intended to describe all these processes. It isbased on the lubrication analysis of Newtonian fluids [40,41], but takes into account the various effects discussedabove in an approximate way.

In this paper, we explain the model for the two-dimensional case assuming that the flow takes place only

Fig. 1. Schematics of actual debonding process (left) andcorresponding model representations (right). (a) Initial state,(b) uniform deformation, (c) cavity expansion, and (d) fibrilla-tion are described. Final detachment of fibrils from the probeis omitted.

in the xz plane shown in Figure 1. The extension to athree-dimensional system will be discussed in a separatepaper.

In our model we divide the initial adhesive layer into Nrectangular blocks of equal size, and consider the transla-tion (slippage) and deformation of each block. We assumethat the deformation of each block is expressed by a super-position of the stretching in z-direction and Poiseulle-like(parabolic) deformation in x-direction. Therefore, the mo-tion of each block is characterized by two parameters: oneparameter, Xi, denotes the x coordinate of the center ofmass of the block i (i = −N/2, . . . , N/2), and the other pa-rameter, Ci, characterizes the magnitude of the parabolicdeformation. We shall call this model the block model.

In the block model, the shape and the location of cav-ities are ignored. The effect of cavities are included onlythrough the relation between the volume change and pres-sure.

To set up equations, we introduce the additional vari-able Pi which stands for the pressure in the region betweenthe block i−1 and i. The pressure Pi appears in two places,one is in the force balance equation for the block, and theother is in the equation controlling the size of cavities.

We shall now set up equations for the block modelbased on this picture.

2.2 Deformation of the block

Let us focus our attention on a certain block i. Initially theblock has width W0 = L0/N and height H0. When the ad-hesive layer is stretched by a factor λ in z-direction, theheight and the width of the block change to H = λH0and W = λ−1W0, respectively. We denote each mate-rial point by the dimensionless coordinate (ξ, ζ) definedin Figure 2. If the block keeps the rectangular shape,the material point (ξ, ζ) will occupy the position (Xi +λ−1W0ξ,λH0(ζ + 1/2)). In reality, the block is deformed.We assume that the deformation is parabolic and that the

Yamaguchi, Morita & Doi: Euro Phys. J. E 20 (2006) 7–17

Yamaguchi, Morita & Doi: Euro Phys. J. E 21 (2006) 331–339


Hydroxyl-terminated polybutadiene (HTPB)

Courtesy of: Robert Nevière, SNPE Matériaux Energétiques,Centre de Recherches du Bouchet, France


Sivaloganathan & Spector (JE ’00a, JE ’00b, JE ’02,PRSE ’02, SJAM ’03, SJAM ’06 - with Tilakraj - , . . . )

u(x)

u : Ω" ! R2

x

min

∫Ωε

|Du|2 dx subject to det Du ≡ 1


u(x)

u : Ω" ! R2

x

min

∫Ωε

|Du|2 dx subject to det Du ≡ 1


λ = 1.076

x u(x)


λ = 1.33

xu(x)


λ = 2.7

x u(x)


Xu & H., M3AS (2011)


Double cavitation 3

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−1.5

−1

−0.5

0

0.5

1

1.5

2

Body with pre-existing microcavities at (0, 0) and (0.3, 0).

min

{∫Ω

W (∇u) dx :u ∈W 1,p(Ω;Rn)u(x) = λx on ∂DΩ

δ = 0.01, λ = 2

}


Variational fracture

Burke, Ortner & Süli ’10 Bourdin, Francfort & Marigo ’09


Regularized model for cavitation and fracture

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1

Pre-existing microcavities at (0, 0) and (0.3, 0)

λ = 1.4, δ = 0.01, ε = 0.005, η = 10−7

min

∫Ω

(v 2 + η)W (∇u) + ε|∇v |2 + (1− v)2

4εdx



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1


λ = 1.5, δ = 0.01, ε = 0.005, η = 10−7

min

∫Ω

(v 2 + η)W (∇u) + ε|∇v |2 + (1− v)2

4εdx



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−0.8

−0.6

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0

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1


λ = 1.6, δ = 0.01, ε = 0.005, η = 10−7

min

∫Ω

(v 2 + η)W (∇u) + ε|∇v |2 + (1− v)2

4εdx



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−0.2

−0.1

0

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1

Ωδ = (−0.5, 0.5)× (0, 0.4) \3⋃

i=1

B((0, i10

), δ)

u(x1, x2) = (λx1, x2), λ = 3, δ = 0.01, ε = 0.005, η = 10−7

min

∫Ω

(v 2 + η)W (∇u) + ε|∇v |2 + (1− v)2

4εdx



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−0.2

−0.1

0

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Ωδ = (−0.5, 0.5)× (0, 0.4) \3⋃

i=1

B((0, i10

), δ)

u(x1, x2) = (λx1, x2), λ = 4, δ = 0.01, ε = 0.005, η = 10−7

min

∫Ω

(v 2 + η)W (∇u) + ε|∇v |2 + (1− v)2

4εdx


Fracture nucleation

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Gent & Lindley ’59

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Gent & Lindley ’59

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Rigid inclusion

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Two rigid inclusions

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λ = 1.33

xu(x)


λ = 1.753

x

u(x)


λ = 2.7

x u(x)


u(x)

Fraenkel asymmetry

D(E ) := minx∈R2

|E4B(x, rE )||E |

with πr 2E = |E |.


Fusco, Maggi & Pratelli (AM ’08):The sharp quantitative isoperimetric inequality

Per(E ) ≥√

4π|E |(1 + C · D(E ))


H. & Serfaty (CPAM ’13):Cavity interaction for a critical-exponent model

R = dist({a1, . . . , am}, ∂Ω)

⇒∫

Ωε

|Du|2 − 12

dx ≥m∑1

vi · logR

2ε

+m∑1

viD2i log

min{di ,√

viD2i }ε

− C

Bounded renormalized energy. Only three possible scenarii.



R = dist({a1, . . . , am}, ∂Ω)

⇒∫

Ωε

|Du|2 − 12

dx ≥m∑1

vi · logR

2ε

+m∑1

viD2i log

min{di ,√

viD2i }ε

− C

Bounded renormalized energy.

Only three possible scenarii.



R = dist({a1, . . . , am}, ∂Ω)

⇒∫

Ωε

|Du|2 − 12

dx ≥m∑1

vi · logR

2ε

+m∑1

viD2i log

min{di ,√

viD2i }ε

− C

Bounded renormalized energy. Only three possible scenarii.


H. & Serfaty ’13:

B

(√v2a1 +

√v1a2√

v1 +√

v2,√

4|a1 − a2|2 + 32(v1 + v2))⊂ Ω


Cañulef & H. (in preparation):

For v1 = v2 = · · · = vm, set

σ =

mπmin

{mini

(R0 − |ai |)2,mini 6=j

( |ai − aj |2

)2}πR20

.

I This is the packing density of the largest disjoint collection ofthe form {B(ai , ρ) : i ∈ {1, . . . , n}} contained in B.

I

λ2 <1

1− σ .


I When m = 11 (Melissen GD ’94) the maximum packing densityis

11(1 + 1sin π

9

)2 ≈ 0.7145;it yields

λ <

√(1 + sin π9 )

2

1 + 2 sin π9 − 10 sin2 π9≈ 1.8714


I

σ = min

mini(

1− |ai |R0)2

vi∑vk

,mini 6=j

|ai − aj |2

R20

(√vi∑vk

+√

vj∑vk

)2 .

I Both when σ ≥ 1 and in the case when σ < 1 and λ2 < 11−σ .


I(

B(ai , di ))ni=1

disjoint

I

vi <1

1−∑πd2k

πR20

· πd2i .


Let m ∈ N, R0 > 0, and Ω := B(0,R0) ⊂ R2. We say that((ai )

ni=1, (vi )

ni=1

)is an attainable configuration if ai ∈ Ω and vi > 0

for all i ∈ {1, . . . ,m}, and there exist evolutionsI zi ∈ C 1([1, λ],R2) of the cavity centres, andI Li : [1, λ]→ [0,∞) of the cavity radii,

such that

m∑i=1

πL2i (t) = (t2 − 1)πR20 ∀ t ∈ [1, λ]

and for each i ∈ {1, . . . ,m}i) L2i belongs to C

1([1, λ], [0,∞));ii) zi (1) = ai and Li (1) = 0;

iii) πL2i (λ) = vi ; and

iv) for all t ∈ [1, λ] the disks B(zi (t), Li (t)) are disjoint andcontained in B(0, tR0).


Suppose(

(ai )ni=1, (vi )

ni=1

)is attainable. Assume that for every ε the

map uε minimizes∫

Ωε|Du|2 dx among u satisfying

I (INV)

I u(x) = λx for x ∈ ∂Ω;I det Du ≡ 1;I and | imT(u,Bε(ai ))| = vi + O(ε2).

Then there exists a C = C(n,R0, (ai )

ni=1, (vi )

ni=1

)such that∫

Ωε

|Duε|22

dx ≤ C +(

m∑i=1

vi

)| log ε|.

Moreover, there exists a subsequence (not relabelled) andu ∈ ⋂1≤p

{div v = 0 in E ,

v(x) = g(x)ν(x) on ∂E ,(1)

where

E = B(z0, r0) \n⋃

k=1

B(zk , rk) ⊂ R2, (2)

g ∈ C 1,α(

n⋃k=0

∂B(zk , rk)

)and

∫∂B0

g =n∑

k=1

∫∂B(zk ,rk )

g . (3)


∀i ≥ 1 ri ≥ d ,∀i ≥ 1 B(zi , ri + d) ⊂ B(z0, r0), and

mini ,j≥1i 6=j

dist(B(zi , ri ),B(zj , rj)) ≥ 2d ,(4)

for some generic length d .


TheoremLet n ∈ N and 0 < δ < 1. There exists a universal constant C3(δ)such that whenever z1, ...zn ∈ R2 and d , r0, ..., rn > 0 satisfy dr0 ≥ δand (4), we have that for every g verifying (3) it is possible toconstruct a solution to (1) for which

‖v‖∞ ≤ C3

((( r0d

)1+α+ B

( r0d2

)3+ B2

( r0d3

)3)‖g‖∞+

(r 2α+10dα+1

+ Br 2+α0d5

)[g ]0,α

),

‖Dv‖∞ ≤ C3(C1‖g‖∞ + C2[g ]0,α +

rα0dα‖g ′‖∞ +

r 2α0dα

[g ′]0,α

),

where

B = B(E ) := |E | 12 CP(E )(

d−12 CP(E ) + d

12

)n

12 r

12

0 , (5)

C1 = r1+α0 d

−α−2 + Br 30 d−7 + B2r 30 d

−10, andC2 = r

2α+10 d

−2−α + Br 2+α0 d−6.


∆φ = 0 in E ,∂φ∂ν

= g(x) on ∂E ,(6)

with∫E φ = 0 and then choosing v = Dφ+ D

⊥ψ whereD⊥ψ := (∂z2ψ,−∂z1ψ) is a divergence-free covector field that cancelsout the tangential parts of Dφ on ∂Bi , ∀i . Concretelyψ(z) = ϕ(z)− ζ

(2dist(z,∂E)

d

)ϕ(q(z)) where ϕ is the solution to

∆ϕ = 0 in E ,∂ϕ∂ν

=∂φ

∂τon ∂E ,

(7)


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