Isoperimetric inequalities and cavity interactions

Transcription

1 Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie - Paris 6, CNRS May 17, 011

2 Motivation [Gent & Lindley 59] [Lazzeri & Bucknall 95 Dijkstra & Gaymans 93] [Petrinic et al. 06] Internal rupture of rubber under hydrostatic tension Gent & Lindley 59 Oberth & Bruenner 65 Gent & Park 84 Dorfmann 03 Bayraktar et al. 08 Cristiano et al. 10 Rubber toughening of plastics (polystyrene, ABS, PMMA) Lazzeri & Bucknall 95 Cheng et al. 95 Steenbrink & Van der Giessen 99 Liang & Li 00 Ductile fracture by void growth and coalescence Goods & Brown 79 Tvergaard 90 Petrinic et al. 06

3 Mathematical analysis (Gent & Lindley 59, Ball 8, Stuart 85, Horgan & Abeyaratne 86, Sivaloganathan 86, Ertan 88, Meynard 9, Horgan 9,... ) Stored-energy density W : M n n 1 R Incompressible elastic ball [Ball 8] T(x) = DW ( Du ) Du T p(x)1 T R (x) = DW ( Du) p(x) cof Du(x), det Du(x) = 1 for x B(0, 1) \ {0} Radial symmetry: u(x) = u( x )x/ x Elastostatics, traction-free cavity surface Div T R (x) = 0, x B(0, 1) \ {0} T R (x) ν(x) = Pν(x), x B(0, 1) T(εθ) θ ε 0 0, θ = 1, θ R n extensions allowing for compressibility, Blatz-Ko and Varga materials, anisotropic loading, material anisotropy, elastodynamics, plasticity, elastic membrane theory, material inhomogeneity,...

4 Variational approach to cavitation (Ball 8, Ball & Murat 84, Müller & Spector 95, Sivaloganathan & Spector 00,... ) minimize Ω W (Du) dx among W 1,p deformations; conditions of invertibility, orientation preservation, incompressibility, loading at the boundary number of cavities, shapes, sizes, location of singularities; interaction between cavities; dependence on loading conditions, domain geometry, material parameters; void coalescence, alignment of cavities, crack formation [Petrinic et al. 06] [Xu & H. 11] [Lian & Li, preprint] lack of lower semicontinuity and quasiconvexity; det Du not weakly continuous; weak limit does not preserve incompressibility and invertibility

5 Connection with Ginzburg-Landau superconductivity Cavitation Ginzburg-Landau u(r, θ) = A + r e iθ min Du p ; det Du 1 u W 1,p Ω Du p A p x=0 r p u(r, θ) = e idθ min Du + 1 u H 1 Ω ε (1 u ) Du d x=0 r

6 Connection with Ginzburg-Landau Distributional determinant: Det Du = 1 n Div ( (adj Du)u ), appearing in nonlinear elasticity, geometric measure theory, liquid crystals, superconductivity,... Cavitation Deformation u : Ω R n, det Du = 1 Ginzburg-Landau Order parameter u : Ω R S 1 (when κ ), det Du = 0, M Det Du = 1 L n + v i δ ai, v i > 0 i=1 Det Du = M d i δ ai, i=1 d i Z (Ball 77, Müller & Spector 95, Sivaloganathan & Spector 00) (Béthuel, Brezis & Hélein 94, Sandier & Serfaty 07)

7 Basic estimate Ginzburg-Landau: u(x) = f (x)e iϕ(x), Du = Df + f Dϕ. For f 1, Du Dϕ, R ( R ) Du τ ϕ dr ds dr τ ϕ Ω ε B(a,r) ε B(a,r) πr = πd log R ε Hence Du dx Ω M i=1 πdi log R i ε i Ability to predict number of vortices, their vorticities and locations; energy estimates; repulsion and confinement effects

8 Radial symmetry, isoperimetric inequality Incompressible neo-hookean material; Ω ε := Ω \ M i=1 Bε(a i) R : Du τ u ds ds B(a,r) B(a,r) ( 1 ( ) Per E(a, r) τ u ds) = 4πr B(a,r) 4πr Isoperimetric inequality: (Per E) 4π E, implies Du 1 dx (cavity volume) log R ε. ε< x <R

9 Radial symmetry, isoperimetric inequality Incompressible neo-hookean material; Ω ε := Ω \ M i=1 Bε(a i) R : Du τ u ds ds B(a,r) B(a,r) ( 1 ( ) Per E(a, r) τ u ds) = 4πr B(a,r) 4πr Isoperimetric inequality: (Per E) 4π E, implies Du 1 dx (cavity volume) log R ε. ε< x <R Equality is attained only for radially symmetric deformations

10 Radial symmetry, isoperimetric inequality Incompressible neo-hookean material; Ω ε := Ω \ M i=1 Bε(a i) R : Du τ u ds ds B(a,r) B(a,r) ( 1 ( ) Per E(a, r) τ u ds) = 4πr B(a,r) 4πr Isoperimetric inequality: (Per E) 4π E, implies Du 1 dx (cavity volume) log R ε. ε< x <R Equality is attained only for radially symmetric deformations round cavities

11 Radial symmetry, isoperimetric inequality Incompressible neo-hookean material; Ω ε := Ω \ M i=1 Bε(a i) R : Du τ u ds ds B(a,r) B(a,r) ( 1 ( ) Per E(a, r) τ u ds) = 4πr B(a,r) 4πr Isoperimetric inequality: (Per E) 4π E, implies Du 1 dx (cavity volume) log R ε. ε< x <R Equality is attained only for radially symmetric deformations round cavities (Sivaloganathan & Spector 010)

12 Lower bound Ω ε := Ω \ M i=1 Bε i (a i) u H 1 (Ω ε, R ), incompressible, Det Du = L + If B(a i, R) Ω then Du 1 Ω ɛ dx (v v M ) log M (v i + πε i )δ ai i=1 R (ε ε M )

13 Lower bound Ω ε := Ω \ M i=1 Bε i (a i) u H 1 (Ω ε, R ), incompressible, Det Du = L + If B(a i, R) Ω then Du 1 Ω ɛ dx (v v M ) log M (v i + πε i )δ ai i=1 R (ε ε M ) Ball construction (Jerrard, Sandier)

14 Lower bound Ω ε := Ω \ M i=1 Bε i (a i) u H 1 (Ω ε, R ), incompressible, Det Du = L + If B(a i, R) Ω then Du 1 Ω ɛ dx (v v M ) log M (v i + πε i )δ ai i=1 R (ε ε M ) Ball construction (Jerrard, Sandier) Does not consider individual cavity sizes or distances between cavities;

15 Lower bound Ω ε := Ω \ M i=1 Bε i (a i) u H 1 (Ω ε, R ), incompressible, Det Du = L + If B(a i, R) Ω then Du 1 Ω ɛ dx (v v M ) log M (v i + πε i )δ ai i=1 R (ε ε M ) Ball construction (Jerrard, Sandier) Does not consider individual cavity sizes or distances between cavities; insufficient to determine optimal locations, or study cavity interactions.

16 Quantitiative isoperimetric inequality (Per E) 4π E

17 Quantitiative isoperimetric inequality where D(E) := min { E B E (Per E) 4π E ( 1 + CD(E) ) } : B ball, B = E, (Fraenkel asymmetry of E) which depends on the shape of E only (not on its size). Bernstein 1905; Bonnesen 4; Fuglede 89; Hall, Hayman & Weitsman 91; Hall 9; Fusco, Maggi & Pratelli 08

18 Two cavities Du 1 dx (v 1 + v ) log R Ω ε B R ɛ d/ ( + C v 1 D(E(a 1, r)) + v D(E(a, r)) ) dr ε r R + C(v 1 + v ) D(E(a, r)) dr d r d Ω

19 Two cavities Du 1 dx (v 1 + v ) log R Ω ε B R ɛ d/ ( + C v 1 D(E(a 1, r)) + v D(E(a, r)) ) dr ε r R + C(v 1 + v ) D(E(a, r)) dr d r d Ω Energy is minimized if circles go to circles

20 Two cavities Du 1 dx (v 1 + v ) log R Ω ε B R ɛ d/ ( + C v 1 D(E(a 1, r)) + v D(E(a, r)) ) dr ε r R + C(v 1 + v ) D(E(a, r)) dr d r d Ω Energy is minimized if circles go to circles This is not always possible: π(r 1 + R ) (πr1 + πr ) = ( v 1 + v ) v 1 v = v 1 v R 1 = v 1π v 1 v v π Possible only if πd > v 1 v

21 Circular cavities (πd v 1 v )

22 Distorted cavities Xu & H. 11 Lian & Li 11 H. & Serfaty 11

23 Vanishing volume ratio (πd > v 1 v ) v 1 =.5v v 1 = 10v v 1 = 100v

24 Scale invariance in elasticity (Ball & Murat, 1984): The condition πd > v 1v is to be compared with: Q λq λq λq Q Q Q Q λq λq Q λq Related works: Ortiz & Reina (010), Lopez-Pamies, Idiart & Nakamura (011)

25 Estimate on the distortions Du 1 dx (v 1 + v ) log R Ω ε B R ɛ d/ C ε C min ε<r<d/ d<r <R ( E a1,r D(E a1,r ) + E a,r D(E a,r ) ) dr R + C E a,r D(E a,r ) dr r d r ( E a1,r D(E a1,r ) + E a,r D(E a,r ) + E a,r D(E a,r ) ) min Proposition: E 1 E E, E 1 E =, E 1 E. Then { log d ε, log R d ( ) ( E D(E) + E 1 D(E 1) + E D(E ) E E \ (E1 E) C 1 E + E 1 E E 1 + E E 1 E } ) 3

26 Lower bound Theorem: Ω ε := Ω \ (B ε (a 1 ) B ε (a )), u H 1 (Ω ε, R ) incompressible Du 1 (v 1 + v ) log R Ω ε ε ( min{v + C(v 1 + v ) 1, v } (v 1 + v ) πd ) ( { }) 4 v1 + v log min v 1 + v 4πd, R d, d ε provided B( a1+a, R) Ω

27 Upper bound Theorem: a 1, a R, v 1 v. For all δ [0, 1] there exists a [a 1, a ] and a piecewise smooth homeomorphism u : R \ {a 1, a } R such that Det Du = 1 L + v 1 δ a1 + v δ a and for all R > 0 B(a,R)\(B ε(a 1 ) B ε(a )) Du C(v 1 + v + πr ) + (v 1 + v ) log R ε ( ( + C(v 1 + v ) (1 δ) log R ) v + δ d 4 log d ) + v 1 + v ε

28 Upper bound Theorem: a 1, a R, v 1 v. For all δ [0, 1] there exists a [a 1, a ] and a piecewise smooth homeomorphism u : R \ {a 1, a } R such that Det Du = 1 L + v 1 δ a1 + v δ a and for all R > 0 B(a,R)\(B ε(a 1 ) B ε(a )) Du C(v 1 + v + πr ) + (v 1 + v ) log R ε ( ( + C(v 1 + v ) (1 δ) log R ) v + δ d 4 log d ) + v 1 + v ε ( ) min{v1 Terms in lower bound: C(v 1 + v ),v ( { }) } (v 1 +v ) πd 4 log min v1 +v v 1 +v 4πd, R d, d ε

29 Geometric construction d 1 d Ω 1 Ω dδ d 1 a d 1 1 d a d d d

30 Geometric construction d 1 d Ω 1 Ω dδ d 1 a d 1 1 d a d d d Ratio Ω 1 Ω = v 1 v

31 Geometric construction d 1 d Ω 1 Ω dδ d 1 a d 1 1 d a d d d Ratio Ω 1 Ω = v 1 v ; u(x) λx on Ω 1 Ω, λ 1 := v 1 + v Ω 1 Ω = v 1 Ω 1 = v Ω

32 Cavity shapes δ = 0.1 δ = 0.4 δ = 0.9

33 Angle-preserving maps Ω 1 Ω Ω 1 Ω d 1 a 1 d 1 d a d a u(x) = λa + f (x) x a x a, λn 1 := v 1 + v Ω 1 Ω = v 1 Ω 1 = v Ω. det Du(x) = f n 1 (x) f r (x) ( ) x a r n 1 1 f n (x) = x a n n + A x a

34 Dirichlet conditions Reference configuration Deformed configuration

35 Dirichlet conditions Reference configuration Deformed configuration Necessary condition: π(r R 1 ) = 3π 18 (1 δ)(v 1 + v ).

36 Dirichlet conditions Theorem: Suppose πr 1, π(r R 1 ) C(v 1 + v )(1 δ); R 1 d. Then B(a,R)\(B ε(a 1 ) B ε(a )) + C(v 1 + v ) Du C(v 1 + v + πr ) + (v 1 + v ) log R ε ( ( (1 δ) log with u B(a,R ) radially symmetric. ) (v1 + v)(1 δ) v + δ πd 4 log d ), + v 1 + v ε

37 Dirichlet conditions Theorem: Suppose πr 1, π(r R 1 ) C(v 1 + v )(1 δ); R 1 d. Then B(a,R)\(B ε(a 1 ) B ε(a )) + C(v 1 + v ) Du C(v 1 + v + πr ) + (v 1 + v ) log R ε ( ( (1 δ) log ) (v1 + v)(1 δ) v + δ πd 4 log d ), + v 1 + v ε with u B(a,R ) radially symmetric. ( ) ) Previous upper bound: C(v 1 + v ) (1 δ) (log πr πd + δ 4 v log d + v 1 +v ε

38 Dirichlet conditions Corollary: Ω = B R, R d. For every v 1 v there exist a 1, a Ω with a a 1 = d and a Lipschitz homeomorphism u : Ω \ {a 1, a } R such that Det Du = L + v 1 δ a1 + v δ a, u Ω λid, and B(a,R)\(B ε(a 1 ) B ε(a )) + C(v 1 + v ) min δ [δ 0,1] Du C(v 1 + v + πr ) + (v 1 + v ) log R ε ( ( (1 δ) log with δ 0 := max{0, 1 Ω 4πd Cπd }. ) (v1 + v)(1 δ) v + δ πd 4 log d ), + v 1 + v ε

39 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)).

40 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω.

41 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV,

42 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV, Det Du ε = L in Ω ε,

43 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV, Det Du ε = L in Ω ε, sup u ε L <

44 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV, Det Du ε = L in Ω ε, sup u ε L < and Ωε Du (v 1,ε + v,ε) log diam Ω ε + C( Ω + v 1,ε + v,ε).

45 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV, Det Du ε = L in Ω ε, sup u ε L < and Ωε Du (v 1,ε + v,ε) log diam Ω ε + C( Ω + v 1,ε + v,ε). Then there exists u 1 p< W 1,p (Ω) H 1 loc(ω) and convergent subsequences.

46 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV, Det Du ε = L in Ω ε, sup u ε L < and Ωε Du (v 1,ε + v,ε) log diam Ω ε + C( Ω + v 1,ε + v,ε). Then there exists u 1 p< W 1,p (Ω) H 1 loc(ω) and convergent subsequences. i) if min{v 1, v } = 0, the only cavity opened is circular

47 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV, Det Du ε = L in Ω ε, sup u ε L < and Ωε Du (v 1,ε + v,ε) log diam Ω ε + C( Ω + v 1,ε + v,ε). Then there exists u 1 p< W 1,p (Ω) H 1 loc(ω) and convergent subsequences. i) if min{v 1, v } = 0, the only cavity opened is circular ii) Suppose v 1 v > 0.

48 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV, Det Du ε = L in Ω ε, sup u ε L < and Ωε Du (v 1,ε + v,ε) log diam Ω ε + C( Ω + v 1,ε + v,ε). Then there exists u 1 p< W 1,p (Ω) H 1 loc(ω) and convergent subsequences. i) if min{v 1, v } = 0, the only cavity opened is circular ii) Suppose v 1 v > 0. If a 1 a

49 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV, Det Du ε = L in Ω ε, sup u ε L < and Ωε Du (v 1,ε + v,ε) log diam Ω ε + C( Ω + v 1,ε + v,ε). Then there exists u 1 p< W 1,p (Ω) H 1 loc(ω) and convergent subsequences. i) if min{v 1, v } = 0, the only cavity opened is circular ii) Suppose v 1 v > 0. If a 1 a then the cavities are circular;

50 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV, Det Du ε = L in Ω ε, sup u ε L < and Ωε Du (v 1,ε + v,ε) log diam Ω ε + C( Ω + v 1,ε + v,ε). Then there exists u 1 p< W 1,p (Ω) H 1 loc(ω) and convergent subsequences. i) if min{v 1, v } = 0, the only cavity opened is circular ii) Suppose v 1 v > 0. If a 1 a then the cavities are circular; if moreover v 1 + v < 4π dist(a, Ω),

51 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV, Det Du ε = L in Ω ε, sup u ε L < and Ωε Du (v 1,ε + v,ε) log diam Ω ε + C( Ω + v 1,ε + v,ε). Then there exists u 1 p< W 1,p (Ω) H 1 loc(ω) and convergent subsequences. i) if min{v 1, v } = 0, the only cavity opened is circular ii) Suppose v 1 v > 0. If a 1 a then the cavities are circular; if moreover v 1 + v < 4π dist(a, Ω), then they are well separated:

52 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV, Det Du ε = L in Ω ε, sup u ε L < and Ωε Du (v 1,ε + v,ε) log diam Ω ε + C( Ω + v 1,ε + v,ε). Then there exists u 1 p< W 1,p (Ω) H 1 loc(ω) and convergent subsequences. i) if min{v 1, v } = 0, the only cavity opened is circular ii) Suppose v 1 v > 0. If a 1 a then the cavities are circular; if moreover v 1 + v < 4π dist(a, Ω), then they are well separated: π a a 1 C(v 1 + v ) exp 4(1 + Ω π(diam Ω) v 1 +v )(C + log v 1 +v ) C v (v 1 +v )

53 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV, Det Du ε = L in Ω ε, sup u ε L < and Ωε Du (v 1,ε + v,ε) log diam Ω ε + C( Ω + v 1,ε + v,ε). Then there exists u 1 p< W 1,p (Ω) H 1 loc(ω) and convergent subsequences. i) if min{v 1, v } = 0, the only cavity opened is circular ii) Suppose v 1 v > 0. If a 1 a then the cavities are circular; if moreover v 1 + v < 4π dist(a, Ω), then they are well separated: π a a 1 C(v 1 + v ) exp 4(1 + Ω π(diam Ω) v 1 +v )(C + log v 1 +v ) C v (v 1 +v ) iii) if a 1 = a

54 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV, Det Du ε = L in Ω ε, sup u ε L < and Ωε Du (v 1,ε + v,ε) log diam Ω ε + C( Ω + v 1,ε + v,ε). Then there exists u 1 p< W 1,p (Ω) H 1 loc(ω) and convergent subsequences. i) if min{v 1, v } = 0, the only cavity opened is circular ii) Suppose v 1 v > 0. If a 1 a then the cavities are circular; if moreover v 1 + v < 4π dist(a, Ω), then they are well separated: π a a 1 C(v 1 + v ) exp 4(1 + Ω π(diam Ω) v 1 +v )(C + log v 1 +v ) iii) if a 1 = a then a,ε a 1,ɛ = O(ε) C v (v 1 +v )

55 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV, Det Du ε = L in Ω ε, sup u ε L < and Ωε Du (v 1,ε + v,ε) log diam Ω ε + C( Ω + v 1,ε + v,ε). Then there exists u 1 p< W 1,p (Ω) H 1 loc(ω) and convergent subsequences. i) if min{v 1, v } = 0, the only cavity opened is circular ii) Suppose v 1 v > 0. If a 1 a then the cavities are circular; if moreover v 1 + v < 4π dist(a, Ω), then they are well separated: π a a 1 C(v 1 + v ) exp 4(1 + Ω π(diam Ω) v 1 +v )(C + log v 1 +v ) C v (v 1 +v ) iii) if a 1 = a then a,ε a 1,ɛ = O(ε) and the cavities are distorted:

56 Compactness (ε 0) Theorem: Ω ε = Ω \ (B ε(a 1,ε) B ε(a,ε)). Assume that {a i,ε } is compactly contained in Ω. Suppose that u ε H 1 (Ω ε, R ) satisfy condition INV, Det Du ε = L in Ω ε, sup u ε L < and Ωε Du (v 1,ε + v,ε) log diam Ω ε + C( Ω + v 1,ε + v,ε). Then there exists u 1 p< W 1,p (Ω) H 1 loc(ω) and convergent subsequences. i) if min{v 1, v } = 0, the only cavity opened is circular ii) Suppose v 1 v > 0. If a 1 a then the cavities are circular; if moreover v 1 + v < 4π dist(a, Ω), then they are well separated: π a a 1 C(v 1 + v ) exp 4(1 + Ω π(diam Ω) v 1 +v )(C + log v 1 +v ) C v (v 1 +v ) iii) if a 1 = a then a,ε a 1,ɛ = O(ε) and the cavities are distorted: lim inf ε 0 v 1D(E(a 1,ε, ε)) + v D(E(a,ε, ε)) v 1 + v v > C (v 1 + v. )

57 Summarizing Connection between cavitation and Ginzburg-Landau theory Role of isoperimetric inequalities in elasticity (c.f. Müller 90) Relation between quantities in the reference and deformed configuration (c.f. Ball & Murat 84; surface energy) Repulsion effect, role of incompressibility Void coalescence Explicit test maps (angle-preserving) Dirichlet conditions (Dacorogna-Moser flow; Rivière-Ye) Compactness (Struwe 94)