Delamination and fracture of thin films

Transcription

1 J.-F. Babadjian 1, B. Bourdin 2, D. Henao 3, A. León Baldelli 4, C. Maurini 4 1 Lab. Jacques-Louis Lions Univ. Pierre et Marie Curie 2 Department of Mathematics Lousiana State University 3 Facultad de Matemáticas Pont. Univ. Católica de Chile 4 Inst. J. L. Rond d Alembert Univ. Pierre et Marie Curie Oxford, 13 September 2012

2 ermal shock cracks: experiments Brittle fracture in thin films Thermal shock cracks on a glass slab (Geyer and Nemat-Nasser IJSS 92) Geyer & Nemat-Nasser (IJSS 1982) Periodic crack pattern with period doubling

3 Numerical results: comparisons with Bahr et al. Brittle fracture Width = 5, Thickness = 1, Element size =.01 Bourdin & ` = Maurini.02, 0 = (in 54 preparation) E l (u, α) = E(u, K) = Ω\K 1 2 A(e(u) θi) (e(u) θi) + G ch n 1 (K) C.Maurini (UPMC) Variational fracture mechanics: thermal cracks March 14, / 51 Ω approximated by (Bourdin, Maurini, in preparation) 1 2 (1 α)2 A(e(u) θi) (e(u) θi) + Ω ( ) 9α 64l + l α 2

4 Thermal shock on a cylinder Bourdin & Maurini (in preparation) elements, 101 time steps. Approx. 6h walltime on 256 cpus (Ranger, TACC) C.Maurini (UPMC) Variational fracture mechanics: thermal cracks March 14, / / 51

5 Thermal shock on a cylinder Bourdin & Maurini (in preparation) Numerical results: an overview of 3D results Cylinder - Cross sections elements, 101 time steps. Approx. 6h walltime on 256 cpus (Ranger, TACC) C.Maurini (UPMC) Variational fracture mechanics: thermal cracks March 14, / / 51

6 Thermal cracks 3D crack patterns Goehring, Mahadevan & Morris (PNAS 2009) Goehring, Mahadevan and Morris, PNA

7 Thermal cracks Goehring, Mahadevan & Morris (PNAS 2009) g, Mahadevan and Morris, PNAS 2009

8 Cracks in thin films Krämer et al. (MSE 2008)

9 Cracks in thin films Wu et al. (ASS 2011)

10 Cracks in thin films Tsui, McKerrow & Vlassak (JMR 2005)

11 Delamination Reis (PNAS 2009) Airwolf Aerospace

12 Delamination & Fracture León Baldelli et al. (in preparation)

13 Delamination & Fracture León Baldelli et al. (in preparation)

14 Delamination & Fracture León Baldelli et al. (in preparation)

15 Delamination & Fracture A 2D model for a thin film bonded on a substrate León Baldelli, Bourdin, Marigo & Maurini (CMT 2012) 1 mín u SBD(ω;R 2 ) 2 ω with Thin film on an elastic substrate with transverse fractures ( ) and debonded regio Ω (Film) Γ (Transverse fracture) Substrate Δ (Debonding) Kinematics Loadings u : film membrana "(u): film membra " 0: inelastic deform u 0: substrate displ Strain energy of the thin film (see e.g. Xia and Hutchinson JMPS 2000) A (e(u ) Z θi ) (e(u ) θi 2 2 ) Z P(u,, )= \ 2 A("(u) " 1 ω 0) ("(u) " 0 ) d + {z } \ 2 K(u u 0) { Film strain energy Film/Substrate Crack energy of + transverse H 1 (Jcracks u ) + and delaminated µ u 2 surfaces + κh 2 ( ) 2 ω\ S(, )=G 2D length( ) + D 2D surface( ) {z } {z } Transverse cracks Delamination A e = 2λµ λ + 2µ e ααe ββ + 2µe αβ e αβ Total energy functional: E(u,, )=P(u,, )+S(, )

16 Delamination & Fracture Leo n Baldelli, Bourdin, Maurini D.Henao & J.-F.Babadjian, B.Bourdin, A.Leo n, C.Maurini

17 Delamination & Fracture

18 Linear plate theory (e.g. Ciarlet vol. II) Ω ε = ω (0, ε) R 3 v ε = (v1 ε, v 2 ε, v 3 ε) H1 (Ω ε (0, ε)) 2 1 e 11 e 12 e 13 2µ 2 e 21 e 22 e 23 + λ(e αα + e 33 ) 2 Ω ε e 31 e 32 e 33 v3 ε(x 1, x 2, εx 3 ) = εu3 ε(x 1, x 2, x 3 ) v ε α(x 1, x 2, εx 3 ) = ε 2 u ε α(x 1, x 2, x 3 ) Γ ĺım = 1 2 Ω 2λµ λ + 2µ e αα(u)e ββ (u) + 2µe αβ (u)e αβ (u), e α3 (u) = e 33 (u) = 0.

19 Dimension reduction Le Dret & Raoult, JMPA 95: nonlinear membrane model Fonseca & Francfort, JRAM 98: 3D-2D in optimal design Bhattacharya & James, JMPS 99: martensitic thin films Braides, Fonseca & Francfort, IUMJ 00: inhomogeneous thin films Mora & Scardia, JDE 12: covergence of equilibria of physical plates...

20 Brittle thin films Braides & Fonseca (AMP 2001), Bouchitté, Fonseca, Leoni & Mascarenhas (ARMA 2002): Ω ε = ω (0, ε), u GSBV p (Ω; R 3 ), ( ) ( J ε = W α u 1 ε 3u + u + u, ν α (u), 1 ) ε ν 3(u) dh 2 Ω J u ϑ ϑ symmetric, positively 1-homogeneous, Lipschitz, linear at infinity F p W (F ) C(1 + F p ), continuous Γ-converges to ω Q W ( αu) + R ϑ(u + u, ν α (u)) dh 1 J u ω u SBV p (Ω; R 3 ) : 3 u = 0, ν 3 (u) = 0.

21 Brittle thin films Giacomini CVPDE 05: Ambrosio-Tortorelli for the quasistatic evolution Babadjian CVPDE 06: Convergence of the quasistatic evolutions

22 Variational delamination Bhattacharya, Fonseca & Francfort (ARMA 2002): Ω ε = ω ( εs, εh), u W 1,p (Ω + ; R 3 ) W 1,p (Ω ; R 3 ), ( ) ( ) α u 1 ε 3u + W α u 1 ε 3u + ε α 1 [u] γ Ω ω Ω + W + 1 C F p C W (F ) C(1 + F p ), continuous Γ-converges to ω hqw + + sqw (small debonding with small energy, independent oscillations in each layer; in the limit u is continuous and independent of x 3 ).

23 Cohesive interfacial energy ω [u] y Ansini AA 04: nonlinear Neumann sieve Ansini, Babadjian & Zeppieri M3AS 07: multiscale Neumann sieve Ansini-Braides JMPA 02, AAP 01; Attouch-Picard RSMUPT 87; Conca JMPA 1985, 1987; Damlamian RDMUPT 85; Del Vecchio AMPA 87; Murat 85; Sanchez-Palencia 80, 81, 85 Roubíček, Scardia, Zanini CMT 09: kz[u] 2 Γ c dh n 1 Γ c Freddi, Paroni, Roubíček & Zanini ZAMM 11: transverse fracture as a form of delamination

24 1D delamination and debonding León Baldelli, Bourdin, Author's Marigo personal & Maurini copy CMT 12: 1 2 (u (x) t) 2 1 dx + 2 u(x)2 dx + #(Γ) + γh 1 ( ) Fracture of thin film ω\γ Ω\Γ Total Energy t (a) D.Henao 1.0 & J.-F.Babadjian, B.Bourdin, A.León, C.Maurini

25 Total Energ 10 1D delamination and debonding León Baldelli, Bourdin, Marigo & Maurini CMT 12: t (a) x (b) x (c) Fig. 17 Coupled experiment #2: transverse fracture and debonding. Three transverse fractures and debonding. a Energy plot shows an evolution consisting in three subsequent transverse fractures and debonding. Boundary layer effects appear during the very first phase of debonding, as the linear growth of the debonding energy in the very first debonding phase indicates, b plot of the displacement and damage fields, the film exhibits three transverse cracks at the end of the loading phase, c plot of the displacement and debonded domain (shaded in light gray). The latter is not symmetric for each part of the bar since the total energy is insensitive to the arrangement of the debonded domain

26 1D delamination and debonding Author's personal copy A. A. León Baldelli et al. León Baldelli, Bourdin, Marigo & Maurini CMT 12 (a) 10 8 (b) E t n 4 2 E t n t Fig. 7 Energy curves Ê t (n) with possible debonding and transverse fracture for γ = 2.2, L = 6. Each curve is for a specific number n 1oftransversefractures,correspondingtothevalueattheintersectionwiththeaxist = 0. The solid continuous lines indicate the energy is obtained for a state without debonding, the dashed line for a state with debonding t t* (a) (b) u x 0 0 u x Delamination 1 0and fracture 1 of2thin films 3 D.Henao 3 2& J.-F.Babadjian, 1 0 B.Bourdin, 1 2A.León, 3C.Maurini

27 Multi-layer asymptotic analysis Ω = ω (0, 1), Ω = ω ( 1, 0); ω R 2 Vanishing Young s modulus in the bonding layer { (λ, µ) in Ω (λ ε, µ ε ) = ε 2 (λ, µ ) in Ω Rescaled energies J ε(u, Ω) = 1 e 11 e 12 ε 1 2 e 13 2µ 2 e 21 e 22 ε 1 e 13 + λ(e αα + ε 1 e 33) 2 Ω ε 1 e 31 ε 1 e 32 ε 2 e 33 2 J ε(u, Ω ) = 1 εe 11 εe 12 e 13 2µ 2 εe 21 εe 22 e 13 + λ(e αα + ε 1 e 33) 2 Ω e 31 e 32 ε 1 e 33 u H 1 (Ω Ω ), u(, 1) = 0

28 Multi-layer asymptotic analysis Energies: J ε (u, Ω) = 1 e 11 e 12 ε 1 2 e 13 2µ 2 e 21 e 22 ε 1 e 13 + λ(e αα + ε 1 e 33 ) 2 Ω ε 1 e 31 ε 1 e 32 ε 2 e 33 J ε (u, Ω ) = 1 2µ εe 2 11 εe 12 e 13 2 εe 21 εe 22 e 13 + λ(e αα + ε 1 e 33 ) 2 Ω e 31 e 32 ε 1 e 33 u H 1 (Ω Ω ), u(, 1) = 0, then

29 Multi-layer asymptotic analysis Energies: J ε (u, Ω) = 1 e 11 e 12 ε 1 2 e 13 2µ 2 e 21 e 22 ε 1 e 13 + λ(e αα + ε 1 e 33 ) 2 Ω ε 1 e 31 ε 1 e 32 ε 2 e 33 J ε (u, Ω ) = 1 2µ εe 2 11 εe 12 e 13 2 εe 21 εe 22 e 13 + λ(e αα + ε 1 e 33 ) 2 Ω e 31 e 32 ε 1 e 33 u H 1 (Ω Ω ), u(, 1) = 0, then Theorem J ε (u, Ω) + J ε (u, Ω ) Γ-converges to 1 2 ω [ 2λµ λ + 2µ e ααe ββ + 2µe αβ e αβ ] + µ 2 ω u α u α

30 Delamination and fracture For u SBV (Ω Ω Ω ; R 3 ), u = w a.e. in Ω, define ( F ε (u) = u A ε 2 3 u 2) ( dx + ε 2 u u 2) dx Ω Ω ( + (ν u ), 1 ) ε (ν u) 3 dh 2 + (ε(ν u ), (ν u ) 3 ) dh 2. Ω J u Ω J u Theorem F ε converges to ω u A 0 2 dx + u w 2 dx { u w 1} for u SBV (ω, R 3 ) and máx α u α L M. + H 1 (J u ) + H 2 ({ u w > 1}),