Basic concepts and models in continuum damage mechanics Djimedo Kondo, Hélène Welemane, Fabrice Cormery To cite this version: Djimedo Kondo, Hélène Welemane, Fabrice Cormery. Basic concepts and models in continuum damage mechanics. Revue europã enne de gã nie civil, 2007, vol. 11, pp. 927-943. <10.3166/regc.11.927-943>. <hal-00776729> HAL Id: hal-00776729 https://hal.archives-ouvertes.fr/hal-00776729 Submitted on 16 Jan 2013 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
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S S D S D ω ω = S D S 1
S D = 0 ω = 0 ω S S D σ = F S σ = F F = S S D S(1 S D /S) = σ 1 ω σ σ = Eε E σ σ = (1 ω)eε ω ω = g(ε) g ω ε κ t κ(t) = max ε(τ) τ t ω = g(κ) g f(ε,κ) = ε κ
f 0; κ 0; κf = 0 f 0 ε κ κ κ κ σ = E s ε = (1 ω)eε E s = (1 ω)e D = 1 Es E ω = g(κ) f f(ε,κ) = ε κ D D = (1 ω)d 0 D 0 ω
σ = (1 ω)d 0 : ε σ σ = (1 ω) σ f f f(ε,κ) = ε eq (ε) κ ε eq ε ε eq = ε : ε ε eq = ε : D 0 : ε ε ε eq ω t ω c ω t ω c g t g c ε eq = 3 < ε I > 2 I=1
<. > ε I < ε I > ε I ω = α t ω t + α c ω c α t α c α t = 3 [ ] < ε t β I >< ε I > ; α c = I=1 ε 2 eq 3 [ ] < ε c β I >< ε I > α t = 1; α c = 0 ω = ω t α t = 0; α c = 1 ω = ω c I=1 ε 2 eq { 0 κ κ0 ω t = g t (κ) = 1 (1 At)κ0 κ A t exp[ B t (κ κ 0 )] κ κ 0 { 0 κ κ0 ω c = g c (κ) = 1 (1 Ac)κ0 κ A c exp[ B c (κ κ 0 )] κ κ 0 κ 0 = ft E f t E κ 0 A t B t A c B c β β
ω ω ω = Na 3 N a
ω = N i ηs 3/2 i S i η n Ò a 2c ǫ n D n n D = i ω i n i n i ω i n i i ε ω W(ε,ω) = 1 2 ε : D(ω) : ε
D(ω) ω = 0 D(ω) σ(ε,ω) σ(ε,ω) = W ε (ε,ω), F ω (ε,ω) F ω (ε,ω) = W ω (ε,ω) = 1 2 ε : D (ω) : ε D = D ω F ω D D = σ : ǫ Ẇ = W ω ω F ω = W ω f f(f ω,ω) = F ω κ(ω) κ ω κ ω κ(ω) = κ 0 (1 + ηω) κ 0 > 0 η > 0.
f ω = Λ F ω (F ω,ω) = Λ Λ = 0 f(f ω,ω) 0 f(f ω,ω) < 0 Λ > 0 f(f ω,ω) = 0 f(f ω,ω) = 0 Λ σ = L : ε L D(ω) L = D(ω) F ω ε F ω ε h f(f ω,ω) 0 f(f ω,ω) < 0 f(f ω,ω) = 0 f(f ω,ω) = 0 D(ω) h = κ 0 η F ω d D(ω) D(ω) D(ω) = 3K(ω)J + 2µ(ω)K. (K,µ) J K J = 1 31 1 K = I J 1 I D(ω) D(ω)
) 48(1 ν K(ω) = K 0 (1 2 0)ω 27(1 2ν 0 ) + 16ω(1 + ν 0 ) 2 ) 480(1 ν 0 )(5 ν 0 )ω µ(ω) = µ 0 (1 675(2 ν 0 ) + 64ω(4 5ν 0 )(5 ν 0 ) ν 0 D W ε D (ε,d) tr(ε),tr(ε 2 ),tr(ε 3 ),tr(d),tr(d 2 ),tr(d 3 ) tr(ε.d),tr(ε 2.D),tr(ε.D 2 ),tr(ε 2.D 2 ) D = 0 D W ε 2
W D W W = 1 2 [λ(tr(ε))2 + 2µtr(ε 2 )]+ 1 2 [ηtr(d)(tr(ε))2 + γtr(d)tr(ε 2 )] + αtr(ε)tr(ε.d) + χtr(ε 2.D) α β η γ tr(d) σ W σ = W ε = D(D) : ε D(D) = D 0 + α(1 D + D 1) + χ(d 1 + 1 D) + η tr(d)(1 1) + γ tr(d)(1 1) D 0 = 3k 0 J + 2µ 0 K D F D = W D D D f(f D,D) = F D (a 0 + a 1 tr(d)). a 0 a 1 a 0 a 1 D Ḋ = { 0, f < 0 f = 0 f < 0 Λ f F D, f = 0 f = 0
Λ Λ = tr(f D Ḟ D ) a 1 tr(f D ) ω D o (ω) D cl (ω) Γ ε Γ(ε) > 0 Γ(ε) 0 D(ω) = { D o (ω) = 3K o (ω)j + 2µ o (ω)k Γ(ε) > 0 D cl (ω) = 3K cl (ω)j + 2µ cl (ω)k Γ(ε) 0 (K o,µ o ) (K cl,µ cl ) 3
(K o,µ o ) (K cl,µ cl ) W C 1 [D(ω)] =D o (ω) D cl (ω) = s(ω) Γ Γ (ε) (ε) ε Γ(ε) = 0 ε ε s ω D o D cl [D(ω)] = 3[K o (ω) K cl (ω)]j + 2[µ o (ω) µ cl (ω)]k [D(ω)] ω,µ o (ω) = µ cl (ω) W µ(ω) = µ o (ω) = µ cl (ω) K 0 K cl (ω) = K 0 K o (ω) = K(ω) Γ [D(ω)] [D(ω)] = 3[K o (ω) K cl (ω)]j = [K o (ω) K cl (ω)]1 1
Γ(ε) = tr(ε) = 0 W(ε,ω) = 1 { 3K(ω)J + 2µ(ω)K tr(ε) > 0 2 ε : D(ω) : ε D(ω) = 3K 0 J + 2µ(ω)K tr(ε) 0 K(ω) µ(ω) σ = L : ε e ε L L = 3k 1 J + 2k 2 K 2k 3 (1 e + e 1) 2k 4 e e k 2 = µ(ω) k 4 = 2(µ (ω)) 2 h { k1 = K(ω) H 1 k 3 = K(ω) H 3 tr(ε) > 0 k 1 = K 0 k 3 = 0 tr(ε) 0 H 1 = (K (ω)trε) 2 H 3 = µ (ω)k (ω) h h h h = H 0 η+k (ω)tr(ε) 1+µ (ω)e K (ω) µ (ω) K(ω) µ(ω) K (ω) µ (ω) K(ω) µ(ω)
K(ω) µ(ω)