isoperimetric inequalities and cavity interaction · 2011-12-17 · [petrinic et al. ’06] min z...

Isoperimetric inequalities and cavity interaction

Duvan Henao and Sylvia Serfaty

Facultad de MatematicasPontificia Universidad Catolica de Chile

Laboratoire Jacques-Louis LionsUniversite Pierre et Marie Curie - Paris 6, CNRS

26 September 2011

Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction

Motivation

[Gent & Lindley ’59]

Internal rupture of rubberunder hydrostatic tension

Gent & Lindley ’59Oberth & Bruenner ’65

Gent & Park ’84Dorfmann ’03

Bayraktar et al. ’08Cristiano et al. ’10

[Lazzeri & Bucknall ’95

Dijkstra & Gaymans ’93]

Rubber toughening of plastics(polystyrene, ABS, PMMA)

Lazzeri & Bucknall ’95Cheng et al. ’95

Steenbrink & Van der Giessen ’99Liang & Li ’00

[Petrinic et al. ’06]

Ductile fracture by voidgrowth and coalescence

Goods & Brown ’79

Tvergaard ’90

Petrinic et al. ’06


Mathematical modelling (Gent & Lindley ’59, Ball ’82, Stuart ’85, Horgan &

Abeyaratne ’86, Sivaloganathan ’86, Ertan ’88, Meynard ’92, Horgan ’92, . . . )

Stored-energy density W : Mn×n1 → R

Incompressible elastic ball

T(x) = DW(Du)DuT − p(x)1

TR(x) = DW(Du)− p(x) cof Du(x),

detDu(x) = 1 for x ∈ B(0, 1) \ 0

Radial symmetry: u(x) = u(|x|)x/|x|

Elastostatics, traction-free cavity surface

Div TR(x) = 0, x ∈ B(0, 1) \ 0

TR(x)ν(x) = Pν(x), x ∈ ∂B(0, 1)

T(εθ)θε→0−→ 0, |θ| = 1, θ ∈ Rn

Ball ’82

extensions allowing for compressibility, Blatz-Ko and Varga

materials, anisotropic loading, material anisotropy,

elastodynamics, plasticity, elastic membrane theory,

material inhomogeneity, . . .


Mathematical problem

min

∫Ω|Du|p dx, n − 1 < p ≤ n

subject to

I u : Ω ⊂ Rn → Rn

I u ∈W 1,p(Ω,Rn)

I detDu = 1 a.e.

I u(x) = λx on ∂Ω

I (DetDu)s = v1δa1 + v2δa2 .

Goal: to determine

I optimal location of singularities

I for v1 + v2 given, optimal values of v1 and v2

I optimal shape of the cavities

I interaction between cavities; dependence on loading conditions, domaingeometry, material parameters; void coalescence, alignment of cavities, crackformation


Motivation

[Xu & H. ’11][Lian & Li, ’11]

[Petrinic et al. ’06]

min

∫B(0,1)

(|Du|

32 + (detF− 1)2 − c log detF

)dx

subject to u(x) = λx on ∂Ω(model of Sivaloganathan & Spector, 2000, 2006)


Motivation (Ball ’77, Ball & Murat ’84,

Muller & Spector ’95, Sivaloganathan & Spector ’00)

I Weak continuity of detDu

detDu =1

nDiv

((adjDu)u

), u ∈W 1,n

I lower semicontinuity of∫W (Du); quasiconvexity

I invertibility

I Ginzburg-Landau (Bethuel, Brezis & Helein ’94,

Sandier & Serfaty ’07):

I |Du| ∼ A/r compared to |Du| ∼ d/rI (DetDu)s =

∑viδai compared to (DetDu)s =

∑diδai .


Main result

I Ωε = Ω \ (Ba1,ε,ε) ∪ Ba2,ε,ε))

I (uε, a1,ε, a2,ε) is a minimizer of∫

Ωε|Du|n subject to

I Ω contains a ball B(

a1,ε+a2,ε

2 ,R)

, v1,ε + v2,ε <1

4n+1nωnRn

I DetDu = Ln + v1,εδa1,ε + v2,εδa2,ε (with v1,ε, v2,ε fixed)

I u(x) = λx on ∂Ω, λn|Ω| = |Ω|+ v1,ε + v2,ε

I condition INV


Main result

Then, as ε→ 0, one (and only one) of the following occurs

i) the cavities tend to be spherical: D(imT(uε,Bai,ε))→ 0 and the cavity pointsare well separated:

π|a2 − a1|2 ≥ C1(v1 + v2) exp

−C2(1 + |Ω|v1+v2

)(v2

v1+v2

) nn−1

ii) one of the cavities ‘closes up’ and the other becomes spherical

iii) |a2,ε − a1,ε| = O(ε) and the cavities are distorted:

lim infε→0

v1D(E(a1,ε, ε))n

n−1 + v2D(E(a2,ε, ε))n

n−1

v1 + v2> C

(v2

v1 + v2

) nn−1


Main result


i) the cavities tend to be spherical: D(imT(uε,Bai,ε))→ 0

and the cavity pointsare well separated:

π|a2 − a1|2 ≥ C1(v1 + v2) exp

−C2(1 + |Ω|v1+v2

)(v2

v1+v2

) nn−1



lim infε→0

v1D(E(a1,ε, ε))n

n−1 + v2D(E(a2,ε, ε))n

n−1

v1 + v2> C

(v2

v1 + v2

) nn−1


Main result



π|a2 − a1|2 ≥ C1(v1 + v2) exp

−C2(1 + |Ω|v1+v2

)(v2

v1+v2

) nn−1



lim infε→0

v1D(E(a1,ε, ε))n

n−1 + v2D(E(a2,ε, ε))n

n−1

v1 + v2> C

(v2

v1 + v2

) nn−1


Main result



π|a2 − a1|2 ≥ C1(v1 + v2) exp

−C2(1 + |Ω|v1+v2

)(v2

v1+v2

) nn−1



lim infε→0

v1D(E(a1,ε, ε))n

n−1 + v2D(E(a2,ε, ε))n

n−1

v1 + v2> C

(v2

v1 + v2

) nn−1


Infinite energy

I Ωε := Ω \⋃M

i=1 Bεi (ai )

I u ∈ H1(Ωε,R2), incompressible, DetDu = L2 +M∑i=1

(vi + πε2i )δai

If B(ai ,R) ⊂ Ω then∫Ωε

|Du|2 − 1

2dx ≥ (v1 + · · ·+ vM) log

R

2(ε1 + · · ·+ εM)

Ball construction (Jerrard, Sandier); classical isoperimetric inequality

Case p < n: cavities feel each other much more rapidly (in terms of πd2

v1+v2,

cf. Ball & Murat ’84)


Radial symmetry, isoperimetric inequality

Incompressible neo-Hookean material; Ωε := Ω \⋃M

i=1 Bε(ai ) ⊂ R2:∫∂B(a,r)

|Du|2

2ds ≥

∫∂B(a,r)

|∂τu|2

2ds

≥ 1

4πr

(∫∂B(a,r)

|∂τu| ds

)2

=

(Per E(a, r)

)2

4πr

Isoperimetric inequality: (PerE)2 ≥ 4π|E |, implies∫ε<|x|<R

|Du|2 − 1

2dx ≥ (cavity volume) · log

R

ε.

Equality is attained only for radially symmetric deformations

⇒ round cavities (Sivaloganathan & Spector 2010)




i=1 Bε(ai ) ⊂ R2:∫∂B(a,r)

|Du|2

2ds ≥

∫∂B(a,r)

|∂τu|2

2ds

≥ 1

4πr

(∫∂B(a,r)

|∂τu| ds

)2

=

(Per E(a, r)

)2

4πr


|Du|2 − 1


R

ε.


⇒ round cavities

(Sivaloganathan & Spector 2010)




i=1 Bε(ai ) ⊂ R2:∫∂B(a,r)

|Du|2

2ds ≥

∫∂B(a,r)

|∂τu|2

2ds

≥ 1

4πr

(∫∂B(a,r)

|∂τu| ds

)2

=

(Per E(a, r)

)2

4πr


|Du|2 − 1


R

ε.


⇒ round cavities (Sivaloganathan & Spector 2010)


Quantitiative isoperimetric inequality

(Per E )2 ≥ 4π|E |

(1 + CD(E )2

)where

D(E) := min

|E4B||E | : B ball, |B| = |E |

, (Fraenkel asymmetry of E)

which depends on the shape of E only (not on its size).

Bernstein 1905; Bonnesen ’24; Fuglede ’89; Hall, Hayman & Weitsman ’91;

Hall ’92; Fusco, Maggi & Pratelli ’08


Quantitiative isoperimetric inequality

(Per E )2 ≥ 4π|E |(1 + CD(E )2

)where

D(E) := min

|E4B||E | : B ball, |B| = |E |

, (Fraenkel asymmetry of E)

which depends on the shape of E only (not on its size).

Bernstein 1905; Bonnesen ’24; Fuglede ’89; Hall, Hayman & Weitsman ’91;

Hall ’92; Fusco, Maggi & Pratelli ’08


Two cavities

∫Ωε∩BR

|Du|2 − 1

2dx ≥ (v1 + v2) log

R

2ε

+ C

∫ d/2

ε

(v1D(E(a1, r))2 + v2D(E(a2, r))2

) dr

r

+ C(v1 + v2)

∫ R

dD(E(a, r))2 dr

r

d

Ω

Energy is minimized if ‘circles go to circles’

This is not always possible:

π(R1 + R2)2 − (πR21 + πR2

2 )

= (√v1 +

√v2)2 − v1 − v2

= 2√v1v2

Possible only if πππd2 > 2√v1v2

2√v1v2

R1 =√

v1π √

v2π


Circular cavities (πd2 ≥ 2√v1v2)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.0 -0.5 0.0 0.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-3 -2 -1 0 1

-3

-2

-1

0

1

2

3

-5 -4 -3 -2 -1 0 1 2 3


Vanishing volume ratio (πd2 > 2√v1v2)

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-3 -2 -1 0 1 2

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-3 -2 -1 0 1

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5

v1 = 2.5v2 v1 = 10v2 v1 = 100v2


Distorted cavities

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Xu & H. ’11 Lian & Li ’11 H. & Serfaty ’11


Estimate on the distortions

∫Ωε∩BR

|Du|2 − 1

2dx− (v1 + v2) log

R

2ε

≥ C

∫ d/2

ε

(|Ea1,r |D(Ea1,r )2 + |Ea2,r |D(Ea2,r )2

) dr

r+ C

∫ R

d|Ea,r |D(Ea,r )2 dr

r

≥ C minε<r<d/2

d<r′<R

(|Ea1,r |D(Ea1,r )2 + |Ea2,r |D(Ea2,r )2 + |Ea,r′ |D(Ea,r′ )

2)

min

log

d

2ε, log

R

d

Proposition: E1 ∪ E2 ⊂ E , E1 ∩ E2 = ∅, |E1| ≥ |E2|. Then

|E |D(E)2 + |E1|D(E1)2 + |E2|D(E2)2

|E |+ |E1 ∪ E2|≥ C

(|E2|

|E1|+ |E2|

)2(

1− |E \ (E1 ∪ E2)|2√|E1||E2|

)3


Lower bound

Theorem: Ωε := Ω \ (Bε(a1) ∪ Bε(a2)), u ∈ H1(Ωε,R2) incompressible∫Ωε

|Du|2 − 1

2≥ (v1 + v2) log

R

2ε

+ C (v1 + v2)

(minv2

1 , v22

(v1 + v2)2− πd2

v1 + v2

)log

(min

4

√v1 + v2

4πd2,R

d,d

ε

)

provided B( a1+a2

2 ,R) ⊂ Ω


Upper bound

Theorem: a1, a2 ∈ R2, v1 ≥ v2. For all δ ∈ [0, 1] there exists a∗ ∈ [a1, a2]and a piecewise smooth homeomorphism u : R2 \ a1, a2 → R2 such thatDetDu = 1 · L2 + v1δa1 + v2δa2 and for all R > 0∫

B(a∗,R)\(Bε(a1)∪Bε(a2))

|Du|2

2≤ C(v1 + v2 + πR2) + (v1 + v2) log

R

ε

+ C(v1 + v2)

((1− δ)

(log

R

d

)+

+ δ 4

√v2

v1 + v2log

d

ε

)

Terms in lower bound: C(v1 + v2)

(minv2

1 ,v22

(v1+v2)2 − πd2

v1+v2

)log(

min

4√

v1+v24πd2 ,

Rd, dε

)


Other difficulties

I Explicit test maps (angle-preserving)

I Dirichlet conditions (Dacorogna-Moser flow; Riviere-Ye)

I Compactness (Struwe ’94)


Summarizing

I Connection between cavitation and Ginzburg-Landau theory

I Role of isoperimetric inequalities in elasticity (c.f. Muller ’90)

I Relation between quantities in the reference and deformedconfiguration (c.f. Ball & Murat ’84; surface energy)

I Repulsion effect, role of incompressibility

I Void coalescence


Geometric construction

2d

2d1 2d2

2dδ

2d

a2a1

Ω1 Ω2

d1 d1 d2 d2

Ratio|Ω1||Ω2|

=v1

v2; u(x) ≡ λx on ∂Ω1 ∪ ∂Ω2,

λ2 − 1 :=v1 + v2

|Ω1 ∪ Ω2|=

v1

|Ω1|=

v2

|Ω2|



2d

2d1 2d2

2dδ

2d

a2a1

Ω1 Ω2

d1 d1 d2 d2

Ratio|Ω1||Ω2|

=v1

v2

; u(x) ≡ λx on ∂Ω1 ∪ ∂Ω2,

λ2 − 1 :=v1 + v2

|Ω1 ∪ Ω2|=

v1

|Ω1|=

v2

|Ω2|



2d

2d1 2d2

2dδ

2d

a2a1

Ω1 Ω2

d1 d1 d2 d2

Ratio|Ω1||Ω2|

=v1

v2; u(x) ≡ λx on ∂Ω1 ∪ ∂Ω2,

λ2 − 1 :=v1 + v2

|Ω1 ∪ Ω2|=

v1

|Ω1|=

v2

|Ω2|


Cavity shapes

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-3 -2 -1 0 1

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

δ = 0.1 δ = 0.4 δ = 0.9


Angle-preserving maps

a2a1

Ω1 Ω2

d1 d1 d2 d2

Ω2Ω1

a∗

u(x) = λa∗ + f (x)x− a∗

|x− a∗|, λn − 1 :=

v1 + v2

|Ω1 ∪ Ω2|=

v1

|Ω1|=

v2

|Ω2|.

detDu(x) =f n−1(x)∂f∂r (x)

rn−1≡ 1 ⇔ f n(x) = |x− a∗|n + A

(x− a∗

|x− a∗|

)n


Dirichlet conditionsδ = 0.1

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

π(R22−R2

1 ) = 3.06(v1+v2)(1−δ)

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

δ = 0.4

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

π(R22−R2

1 ) = 3.12(v1+v2)(1−δ)

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

δ = 0.4

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

π(R22−R2

1 ) = 2.46(v1+v2)(1−δ)

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

Necessary condition: π(R22 − R2

1 ) ≥ 2π−318π

(1− δ)(v1 + v2).

(Parameters:√

(v1 + v2)/(πd2) = 1.5, v2/v1 = 0.3, d = 1, R1 ≈ d .)


Dirichlet conditions

Theorem: Suppose πR21 , π(R2

2 − R21 ) ≥ C (v1 + v2)(1− δ); R1 ≥ 2d . Then∫

B(a∗,R)\(Bε(a1)∪Bε(a2))

|Du|2

2≤ C(v1 + v2 + πR2) + (v1 + v2) log

R

ε

+ C(v1 + v2)

((1− δ)

(log

(v1 + v2)(1− δ)

πd2

)+

+ δ 4

√v2

v1 + v2log

d

ε

),

with u|∂B(a∗,R2) radially symmetric.

Previous upper bound: C(v1 + v2)

((1− δ)

(log πR2

πd2

)+

+ δ 4

√v2

v1+v2log d

ε

)


Dirichlet conditions

Corollary: Ω = BR , R ≥ 2d . For every v1 ≥ v2 there exist a1, a2 ∈ Ω with|a2 − a1| = d and a Lipschitz homeomorphism u : Ω \ a1, a2 → R2 suchthat DetDu = L2 + v1δa1 + v2δa2 , u|∂Ω ≡ λid, and∫

B(a∗,R)\(Bε(a1)∪Bε(a2))

|Du|2

2≤ C(v1 + v2 + πR2) + (v1 + v2) log

R

ε

+ C(v1 + v2) minδ∈[δ0,1]

((1− δ)

(log

(v1 + v2)(1− δ)

πd2

)+

+ δ 4

√v2

v1 + v2log

d

ε

),

with δ0 := max0, 1− |Ω|−4πd2

Cπd2 .


Compactness (ε→ 0)

Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)).

Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫

Ωε

|Du|2

2≤ (v1,ε + v2,ε) log

diam Ω

ε+ C(|Ω|+ v1,ε + v2,ε).

Then there exists u ∈ ∩1≤p<2W1,p(Ω) ∩ H1

loc(Ω) and convergent subsequences.



Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω.

Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫

Ωε

|Du|2

2≤ (v1,ε + v2,ε) log

diam Ω

ε+ C(|Ω|+ v1,ε + v2,ε).





Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,

DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫Ωε

|Du|2

2≤ (v1,ε + v2,ε) log

diam Ω

ε+ C(|Ω|+ v1,ε + v2,ε).





Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε,

sup ‖uε‖L∞ <∞ and∫Ωε

|Du|2

2≤ (v1,ε + v2,ε) log

diam Ω

ε+ C(|Ω|+ v1,ε + v2,ε).





Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞

and∫Ωε

|Du|2

2≤ (v1,ε + v2,ε) log

diam Ω

ε+ C(|Ω|+ v1,ε + v2,ε).





Theorem: Ωε = Ω \ (Bε(a1,ε) ∪ Bε(a2,ε)). Assume that ai,ε is compactlycontained in Ω. Suppose that uε ∈ H1(Ωε,R2) satisfy condition INV,DetDuε = L2 in Ωε, sup ‖uε‖L∞ <∞ and∫

Ωε

|Du|2

2≤ (v1,ε + v2,ε) log

diam Ω

ε+ C(|Ω|+ v1,ε + v2,ε).




Scale invariance in elasticity (Ball & Murat, 1984):

The condition πd2 > 2√v1v2 is to be compared with:

Qλ Q

λQ

Qλ

Q

Q

λ

λ

λ

Q

Q

Q

Q Q

Q

Related works: Ortiz & Reina (2010), Lopez-Pamies, Idiart & Nakamura (2011)


isoperimetric inequalities and cavity interaction · 2011-12-17 · [petrinic et al. ’06] min z...

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