Numerical Simulation of Delamination in Composites via Incremental Energy Minimization

Transcription

1 Numerical Simulation of Delamination in Composites via Incremental Energy Minimization Pavel Gruber 1 Martin Kružík 1,2 Jan Zeman 1 1 Fakulta stavební České vysoké učení technické v Praze 2 Ústav teorie informace a automatizace Akademie věd České republiky 19. konference studentů v matematice na školách VŠTEZ Tetřeví boudy června 2011 Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 1 / 27

2 Delamination phenomena: Modelling assumptions Inelastic phenomena concentrated at interfaces Decohesion displacement jumps at interfaces Rate-independent (quasi-static) approximation Frictionless contact conditions Small-strain approximation Energetic setting Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 2 / 27

3 Basic concepts of interfacial damage mechanics Constitutive characterization Mode I (opening) Mode II (shearing) full debonding decohesion perfect bonding full debonding decohesion perfect bonding Initial stiffnesses: k n, k t (Nm 3 ) Activation energies: G o n, G o t (Jm 2 ) Dissipated energies: G c n, G c t (Jm 2 ) Mode interaction parameters Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 3 / 27

4 Basic concepts of interfacial damage mechanics State variables T n G o n k n 1 T t G c n G o t G c t T k t t t 0 u n 0 u t 1 T n u t n u n Displacement jump decomposition u n = u n 0 u t = u u n n (1) Intensity of adhesion T n = ζk n u n T t = ζk t u t 0 ζ 1 Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 4 / 27

5 Basic concepts of interfacial damage mechanics Interaction criteria (VALOROSO & CHAMPANEY, 2006) Mode mixity parameter ψ( u ) = kt u t k n u n (mode I =) 0 ψ < + (= mode II) Mode interaction criteria ( G o (ψ) ) a1 ( ψ 2 G o (ψ) + ) a2 = 1 (1 + ψ 2 )G o n (1 + ψ 2 )G o t ( G c ) (ψ) b1 ( ψ 2 G c ) b2 (ψ) + = 1 (1 + ψ 2 )G c n (1 + ψ 2 )G c t parameters a 1, a 2, b 1, b 2 fitted from experiments or set equal to 2 Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 5 / 27

6 Motivation Challenges in modelling delamination phenomena Theoretical aspects 1 Incremental variational framework for mixed-mode delamination 2 Establish a posteriori energetic estimates Computational aspects 1 Oscillations of interfacial tractions for almost perfect interfaces (k ) 2 Unstable response for brittle interfaces, leading to oscillation of quantity of interests for coarse meshes 3 Efficient and reliable resolution of frictionless contact Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 6 / 27

7 Outline 1 Incremental energy minimization Incremental variational problems A-priori estimates Application to delamination problem 2 Numerical treatment Incremental optimization problems Alternate minimization Backtracking strategy 3 Examples Flexure tests of two-layer beams Fiber-reinforced composites Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 7 / 27

8 Incremental energy minimization Notation (MIELKE, LEVITAS & THEIL, 2002; MIELKE & ROSSI, 2007) Domain Ω R d, time interval I = [0; T ] State variables: q = (u, z) Q = U Z Displacements u : Ω R d Internal variables z : Ω R m Constitutive description Stored energy E(t, q) : I Q R Dissipation rate D (u, ż) : U Z [0; + ] State-dependent, positively 1-homogeneous in ż Results in (MR 2007) based on Uniform convexity of E(t, ) Finite dissipation Specific form of D(q, q) Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 8 / 27

9 Incremental energy minimization Incremental variational problems (MIELKE & ROSSI, 2007) Implicit time discretization 0 = t 0 τ < t 1 τ = t 0 τ + τ <... < t M τ = T Incremental problems Given q 0 τ = q(0) Q, solve for k = 1,..., M q k τ arg min E(t k, q) + D(u k 1 τ, z z k 1 τ ) (2) q Q Under standard coercivity and lower-semicontinuity conditions, an incremental solution exists and the discrete stability condition holds E(t k τ, q k τ ) E(t k τ, q) + D(u k 1 τ, z z k τ ) for all q Q Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 9 / 27

10 Energetic rate-independent systems A-posteriori estimates of incremental solution Two-sided energy estimates t k τ t k τ t k 1 τ t k 1 τ s E(s, q k τ ) ds E(t k τ, q k τ ) + D(u k 2 τ, z k τ z k 1 τ ) s E(s, q k 1 τ ) ds E(t k τ, q k τ ) + D(u k 1 τ E(t k τ, q k τ ) = E(t k τ, q k τ ) E(t k 1 τ, q k 1 τ ), z k τ z k 1 τ ) (3) Work in progress (with M. KRUŽÍK) Convergence of numerical approximations u k τ,h U h and z k τ,h Z h Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 10 / 27

11 Application to delamination problem Geometry (KOČVARA, MIELKE & ROUBÍČEK, 2006) Γ 12 Ω 1 Γ D Γ D Ω 2 Γ 23 Ω 3 Body Ω R d with the Lipschitz boundary Γ Time-dependent Dirichlet boundary conditions at Γ D Collection of disjoint bodies Ω α with Lipschitz boundaries Γ α, α = 1, 2,..., m (Γ D,α = Γ α Γ D ) Internal boundaries Γ αβ = Γ α Γ β (α > β) with normal vectors n αβ Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 11 / 27

12 Application to delamination problem State variables and data (KOČVARA, MIELKE & ROUBÍČEK, 2006) Domain displacements u α : Ω α R d Internal variable ζ αβ : Γ αβ R Function spaces U = Z = m α=1 m { u α W 1,2 (Ω α ; R d } ), u ΓDα α = 0 m α=1 β=α+1 {ζ αβ L (Γ αβ )} Hard-device loading w D C 1 (I; W 1/2,2 (Γ D ; R d )) Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 12 / 27

13 Application to delamination problem Stored energy (KOČVARA, MIELKE & ROUBÍČEK, 2006; KRUŽÍK, Z & GRUBER) Stored energy E(t, q) = Bulk energy m E α (t, ε(u α )) + α=1 E α (t, ε) = 1 2 m m α=1 β=α+1 E αβ (u α Γαβ u β Γαβ, ζ αβ ) Ω α C α (ε + ε(u D,α (t))) : (ε + ε(u D,α (t))) dω Interfacial energy from (1) ζ Γ E αβ ( u, ζ) = αβ 2 k αβ u u + f αβ (ψ( u ), ζ) dγ for u n 0 and 0 ζ 1 a.e. on Γ αβ + otherwise Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 13 / 27

14 Application to delamination problem Dissipation function (KOČVARA, MIELKE & ROUBÍČEK, 2006; KRUŽÍK, Z & GRUBER) Dissipation rate D(q, ż) = m m α=1 β=α+1 D αβ (u α u β, ζ αβ, ζ αβ ) Interfacial dissipation D αβ ( u, ζ) G c αβ = (ψ( u )) ζ dγ for ζ 0 Γ αβ + otherwise (KMR, 2006) results for time-incremental solutions Full FEM-based discretization P 1 elements for u P 0 elements for z Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 14 / 27

15 Numerical treatment Strategy and algebraic re-formulation Ω h α Γ h αβ Ω h β Discrete state variables Nodal displacements: u α Interfacial displacement jumps: u αβ Adhesion intensity variables: z αβ (u, u ) K u = {(v, v ) : B d v = 0, B e v = v, B i v 0} z K z = {y : 0 y 1} Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 15 / 27

16 Numerical treatment Discrete energy functions Energy stored in domains E Ω,h (t, u) = 1 2 (u + u D(t)) T K (u + u D (t)) K = K 1... K m Energy stored at interfaces E Γ,h ( u, z) = 1 2 u T k(z) u + f h (ψ( u ), z) f h (ψ, ) convex in z Interfacial dissipation: z 0 D h ( u, z) = g(ψ( u )) T z Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 16 / 27

17 Numerical treatment Incremental optimization problems Incremental energy (2) I k,h (z, u, u ) = 1 2 ut Ku + u T Ku D (t k ) u T k(z) u + f h (ψ( u ), z) z T g(ψ( u k 1 )) Incremental optimization problems Given (z 0, u 0, u 0 ) K z K u solve for k = 1,..., M (z k, u k, u k ) = arg min 0 z z k 1 min I k,h(z, u, u ) (u, u ) K u Large-scale, sparse, separately convex and constrained problem Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 17 / 27

18 Numerical treatment Alternate minimization (BOURDIN, FRANCFORT & MARIGO, 2000; BOURDIN, 2007) Alternate minimization algorithm 1: Set j = 0, z (0) = z k 1, u (0) = u k 1, u (0) = u k 1 2: repeat 3: Set j = ( j + 1 4: Solve u (j), u (j)) = arg min I k,h(z (j 1), u, u ) (u, u ) K u 5: Solve z (j) = arg min 0 z z k 1 I k,h (z, u (j), u (j) ) 6: until z (j) z (j 1) δ 7: Set u k = u (j), u k = u (j), z k = z (j) Step 5 can be performed element-by-element at interface Step 4 well-suited for duality solvers based on FETI method (FARHAT & ROUX, 1991; KRUIS & BITTNAR, 2008; DOSTÁL; 2009) Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 18 / 27

19 Numerical treatment Backtracking (BOURDIN, 2007; MIELKE, ROUBÍČEK & Z, 2010; BENEŠOVÁ, 2009 ) Convergence to a critical point of the objective function Not good enough theory relies on the global minimization Backtracking algorithm 1 : Set k = 1, z 1 = 1, z 0 = 1, z (0) = 1 2 : repeat 3 : Determine z k using the alternate minimization algorithm : for time t k and initial value z (0). 4 : Set z (0) = z k 5 : if two-sided estimate (3) is satisfied with tolerance η 6 : Set k = k : else 8 : Set k = k 1 9 : end 10: until k M Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 19 / 27

20 Examples Flexure tests (VALOROSO & CHAMPANEY, 2006) modified ENF prescribed displacement ENF 2x3 mm MMF initial crack 18 mm 60 mm 120 mm ENF test (displacements are magnified 5 ) Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 20 / 27

21 Examples Material data Domains: E = 75 GPa, ν = 0.3 Interfaces Material parameter Ductile Brittle k n = k t (GNm 3 ) G o n = G o t (Jm 2 ) G c n = G c t (Jm 2 ) Interaction parameters a i, b i 2 2 f(ψ, ζ) = ζ 0 ( G o G o ) (ψ) 2 (ψ) 1 η + G o (ψ)/g c dη (ψ)η Algorithmic settings: δ = 10 5, η = 10 3 In-house experimental MATLAB implementation Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 21 / 27

22 Examples Energetics of ENF test, h 0, ductile interface, M = 40, overall time s Energy [Nmm] h=0.5 mm h=0.75 mm h=1 mm Total energy Dissipation 200 Stored bulk energy 100 Stored interfacial energy Time [-] (No back-tracking activated) Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 22 / 27

23 Examples ENF test, ductile interface, snapshots of delamination evolution t = 0.17 t = 0.25 t = 0.33 t = 0.50 t = 0.58 t = 0.62 t = 0.75 t = 0.83 t = 0.91 t = 1 Displacement are scaled 5 Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 23 / 27

24 Examples Energetics of modified ENF test, h = 1 mm, brittle interface 150 No backtracking Backtracking Total energy Energy [Nmm] Dissipation Stored bulk energy Stored interfacial energy Time [-] Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 24 / 27

25 Examples Energetics of MMF test, h 0, brittle interface Energy [Nmm] h=0.5 mm h=0.75 mm h=1 mm Dissipation Total energy Stored bulk energy Stored interfacial energy Time [-] (No back-tracking activated) Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 25 / 27

26 Examples Snapshots of MMF test t = 0.14 t = 0.22 t = 0.31 t = 0.48 t = 0.57 t = 0.66 t = 0.74 t = 0.83 t = 0.92 t = 1 Displacement are scaled 5 Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 26 / 27

27 Computational homogenization Debonding in fiber-reinforced composites t = 0.4 t = 0.6 t = 0.8 t=1 Gruber, Kružík, Zeman (CTU & CAS) Delamination via Incremental Energy VŠTEZ 27 / 27