Plate-like and shell-like inclusions with high rigidity

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Submitted on 3 Jul 2018

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Plate-like and shell-like inclusions with high rigidity

Anne-Laure Bessoud, Françoise Krasucki, Michele Serpilli

To cite this version:

Anne-Laure Bessoud, Françoise Krasucki, Michele Serpilli. Plate-like and shell-like inclusions with

high rigidity. Comptes Rendus. Mathématique, Centre Mersenne (2020-..) ; Elsevier Masson

(2002-2019), 2008, 346 (11-12), pp.697 - 702. �10.1016/j.crma.2008.03.002�. �hal-01829179�

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a,b a b a b 1 εp p = 1 p = 3 ε

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1 ε ω×] − ε, ε[ ω ω 1 εp p = 1 p = 3 p = 3 p = 1 p = 1 p = 3 E3 _{(O; e} 1, e2, e3) Ω+ _Ω− _∂Ω+ _∂Ω− _{ω =}_{∂Ω+_{∩ ∂Ω}−_}◦ R2 _{y = (y} α) ω Ω+ _Ω− _e₃ _−e₃ ε > 0 Ω±,ε _{:= {x}ε _:= x± εe3; x ∈ Ω±} Ωm,ε := ω×] − ε, ε[ Ωε := Ω−,ε∪ Ω+,ε∪ Ωm,ε Γ0⊂ (∂Ωε\ Γm,ε) Γm,ε:= ∂ω×] − ε, ε[ Ωε Vε _{= {(V , ) ∈ H}1_(Ωε_{; R}3_{) × H}1_(Ωm,ε_{; R}3_{); V} |Ωm,ε= ; V_|Γ₀ = 0}. Ωε ! (Uε_, ε_{) ∈ V}ε _{(V , ) ∈ V}ε A−,ε(Uε, Vε) + A+,ε_(Uε_{, V}ε_{) + A}m,ε₍ ε_, ε_{) = L(V ),} A±,ε(Uε, Vε) = " Ω±,ε (λ±,ε_eε pp(Uε)eεqq(Vε) + 2µ±,εeεij(Uε)eεij(Vε))dxε, Am,ε₍ ε_, ε_{) =} " Ωm,ε (λm,ε_eε pp( ε)eεqq( ε) + 2µm,εeεij( ε)eεij( ε))dxε. L(·) λ±,ε= λ±, µ±,ε= µ±, λm,ε₌ 1 εpλ m_{, µ}m,ε₌ 1 εpµ m_, p∈ {1, 3} λ± λm _µ± _µm

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ε ε Ω := Ω±,1_{∪ Ω}m _Ωm _{:= ω×] − 1, 1[} Ωm,ε _{V =} _{{(V , ) ∈} H1_{(Ω; R}3_{) × H}1_(Ωm_{; R}3_{); V} |Ωm = ; V_|Γ₀ = 0} ⎧ ⎨ ⎩ (U (ε), (ε)) ∈ V (V , ) ∈ V A−(U (ε), V ) + A+(U(ε), V ) + 1 εp−1A m αβ( (ε), ) + 1 εpA m α3( (ε), ) + 1 εp+1A m 33( (ε), ) = L(V ), A±_{(·, ·)} _Ω± _Am ij(·, ·) Amαβ( , ) = " Ωm & λmeσσ( )eτ τ( ) + 2µmeαβ( )eαβ( ) +µ m 2 ∂αu3∂αv3 ' dx, Am α3( , ) = " Ωm &µm 2 (∂αu3∂3vα+ ∂3uα∂αv3) + λm(eσσ( )e33( ) + e33( )eσσ( )) ' dx, Am33( , ) = " Ωm & (λm_{+ 2µ}m_)e 33( )e33( ) +µ m 2 ∂3uα∂3vα ' dx. (U(ε), (ε)) U (ε) = U0+ εU1+ ε2U2+ . . . , (ε) = 0+ ε 1+ ε2 2+ . . . . (U0 , 0₎ p = 1 _! (U0 , 0) ∈ VM (V , ) ∈ VM A−(U0, V ) + A+_(U0_{, V ) + A}m M( 0, ) = L(V ), VM = {(V , ) ∈ H1(Ω; R3) × H1(ω; R3); V|ω= , V|Γ0 = 0}, AmM( , ) = " ω & 4λm_µm λm+ 2µmeσσ( )eτ τ( ) + 4µ m_e αβ( )eαβ( ) ' dy p = 3 _! (U0_, 0_{) ∈ V} F (V , ) ∈ VF A−(U0, V ) + A+(U0, V ) + AmF( 0, ) = L(V ), VF = {(V , ) ∈ H1(Ω; R3) × H2(ω; R3); V|ω= , V|Γ0 = 0, eαβ( ) = 0}, Am F( , ) = 1 3 " ω & 4λm_µm λm_{+ 2µ}m∆τu3∆τv3+ 4µ m_∂ αβu3∂αβv3 ' dy ∆τ ω Ωm,ε p = 1

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θ∈ C2_{(ω; R}3₎

α(y) := ∂αθ(y)

S := θ(ω) θ(y) α_(y)

α_{(y) ·}

β(y) = δβα 3(y) = 3(y) :=

1(y)∧ 2(y) | 1(y)∧ 2(y)| S aαβ aαβ bαβ bβα Γσ αβ S aαβ:= α· β, aαβ:= α· β, bαβ:= 3· ∂β α, bβα:= aβσbασ, Γσαβ:= σ· ∂β α. Ωm,ε _{:= ω×] − ε, ε[} _Γ±,ε _{:= ω × {±ε}} _xε Ωm,ε xε α= yα S = θ(ω) 2ε > 0 Θm,ε_(Ωm,ε_{) ⊂ R}3 _Ωm,ε Θm,ε_{: Ω}m,ε → R3 Θm,ε_(xε_{) := θ(y) + x}ε 3 3(y), xε= (y, xε3) = (y1, y2, xε3) ∈ Ω m,ε . Θε_{: Ω}ε_{→ R}3 Θε _:= ⎧ ⎨ ⎩ Θ±,ε _Ω±,ε Θm,ε _Ωm,ε , Θ ±,ε_(Γ±,ε_{) = Θ}m,ε_(Γ±,ε_), Θ±,ε_{: Ω}±,ε_{→ R}3 _Ω±,ε _Ω±,ε Ωε ⎧ ⎨ ⎩ (Uε_, ε_{) ∈ V}ε ₍ ε_, ε_{) ∈ V}ε A−,ε(Uε, Vε) + A+,ε(Uε, Vε) + Am,ε( ε, ε) = L(Vε), A±,ε(Uε, Vε) := " Ω±,εA ijkℓ,ε ± eεkℓ( ε)eεij( ε) ( g±,ε_dxε_, Am,ε₍ ε_, ε_{) :=} " Ωm,ε Aijkℓ,ε m eεkℓ( ε)eεij( ε)√gm,ε dxε. Aijkℓ,ε _{:= λ}ε_gij,ε_gkℓ,ε_{+ µ}ε_(gik,ε_gjℓ,ε_{+ g}iℓ,ε_gjk,ε₎

gε _:= _(gε ij) (gεij) := (∂iΘε· ∂jΘε) (gij,ε) := (gijε)−1 Θε Ωε _Ω (U0_, 0₎ p = 1 _! (U0_, 0_{) ∈ V} M (V , ) ∈ VM A−(U0, V ) + A+(U0, V ) + AmM( 0, ) = L(V ), VM = {(V , ) ∈ H1(Ω; R3) × H1(ω; R3); V|ω= , V|Γ0 = 0}, Am M( , ) = " ω aαβστ_γ στ( )γαβ( )√a dy

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aαβστ _:= 4λm_µm λm_+2µmaαβaστ+ 2µm_(aασ_aβτ_{+ a}ατ_aβσ₎ γαβ( ) := 1₂(∂αuβ+ ∂βuα) − Γσαβuσ− bαβu3 p = 3 ! (U0_, 0_{) ∈ V} F (V , ) ∈ VF A−(U0, V ) + A+(U0, V ) + AmF( 0, ) = L(V ), VF = {(V , ) ∈ H1(Ω; R3) × H2(ω; R3); V|ω= , V|Γ0 = 0, γαβ( ) = 0 ω}, Am F( , ) = 1 3 " ω aαβστ_ρ στ( )ραβ( )√a dy ραβ( ) := ∂αβu3− Γσ αβ∂σu3− bαβu3+ bσα(∂βuσ− Γτβσuτ) + bτβ(∂αuτ − Γσατuσ) + (∂αbτβ+ Γτασbσβ− Γσαβbτσ)uτ VM VF AmM