Isoperimetric inequalities and cavity interaction Duvan Henao and Sylvia Serfaty Facultad de Matem´ aticas Pontificia Universidad Cat´ olica de Chile Laboratoire Jacques-Louis Lions Universit´ e Pierre et Marie Curie - Paris 6, CNRS 26 September 2011 Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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Isoperimetric inequalities and cavity interaction · 2011-12-17 · [Petrinic et al. ’06] min Z B(0;1) jDuj32 + (detF 1)2 c log detF dx subject to u(x) = x on @ (model of Sivaloganathan
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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
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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
)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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
)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
Other difficulties
I Explicit test maps (angle-preserving)
I Dirichlet conditions (Dacorogna-Moser flow; Riviere-Ye)
I Compactness (Struwe ’94)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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|
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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|
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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|
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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 .)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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
)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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
)
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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 .
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
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.
Duvan Henao and Sylvia Serfaty Isoperimetric inequalities and cavity interaction
Scale invariance in elasticity (Ball & Murat, 1984):
The condition πd2 > 2√v1v2 is to be compared with: