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A Vanishing Viscosity Approach in Fracture Mechanics Weierstrass Institute for Applied Analysis and Stochastics A vanishing viscosity approach in fracture mechanics Dorothee Knees jointly with A. Mielke, A. Schröder, C. Zanini Nonlocal Models and Peridynamics, Nov. 5–7, 2012, TU Berlin Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de Introduction Goal: Derive a rate independent model for crack propagation based on the Griffith criterion. Du E(u(t); s(t)) = `(t), 0 2 @R(s_(t)) + DsE(u(t); s(t)). Questions: The energy E is not convex in s ) The evolution might be dicontinuous. Suitable jump criteria? Hyperelstic material with polyconvex energy density: DsE well defined? Convergence of fully discretized models? Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 2 (29) Contents 1 Fracture model and vanishing viscosity solutions (finite strains) 2 FE-approximation of vanishing viscosity solutions (small strains) 3 Numerical example 4 Summary Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 3 (29) Energies and notation (2D) 2 ' :Ωs ! deformation field R '(Ω ) 2×2 ' s W : R ! [0; 1] elastic energy density Ωs 2×2 Assumptions: F = r' 2 R I W (F) = 1 if det F ≤ 0 + coercivity. I polyconvexity: W (F) = g(F; det F), g convex and lower semicontinuous. 2×2 > I multiplicative stress control: 8F 2 R+ : F DW (F) ≤ c1(W (F) + 1) 2 2 Example: W (F) = c1 jFj + c2(det F) − c3 log(det F): 1;p Admissible deformations V (Ωs) = f ' 2 W (Ωs); ' = 'D g ΓD Elastic energy E(t; '; s) = R W (r') dx − R h(t) · ' da Ωs ΓN Reduced energy I(t; s) = inf'2V (Ωs ) E(t; s;') Ball’77: Minimizers exist (not necessarily unique!) Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 4 (29) Dissipated energy: ( κ(snew − sold) if snew ≥ sold R(snew − sold) = : 1 else κ > 0 fracture toughness Evolution criterion: ) local stability: κ ≥ −@sI(t; s(t)); , 0 2 @R(s_(t)) + @sI(t; s(t)) complementarity: s_(t) κ + @sI(t; s(t)) = 0 Problems: (a) Discontinuous solutions may occur. (b) @sI well defined? The Griffith fracture criterion Griffith criterion (1921) The crack is stationary, if the (locally) released elastic energy is less than the energy dissipated to create the new crack surface. Energy release rate: G(t; s) = −@sI(t; s) = −@s min'2V (Ωs ) E(t; s;') Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 5 (29) The Griffith fracture criterion Griffith criterion (1921) The crack is stationary, if the (locally) released elastic energy is less than the energy dissipated to create the new crack surface. Energy release rate: G(t; s) = −@sI(t; s) = −@s min'2V (Ωs ) E(t; s;') Dissipated energy: ( κ(snew − sold) if snew ≥ sold R(snew − sold) = : 1 else κ > 0 fracture toughness Evolution criterion: ) local stability: κ ≥ −@sI(t; s(t)); , 0 2 @R(s_(t)) + @sI(t; s(t)) complementarity: s_(t) κ + @sI(t; s(t)) = 0 Problems: (a) Discontinuous solutions may occur. (b) @sI well defined? Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 5 (29) Why discontinuous solutions? Evolution law: Local stability: κ ≥ −@sI(t; s(t)), Complementarity: s_(t) κ + @sI(t; s(t)) = 0. s κ + @sI(t; s) > 0 κ + @sI(t; s) < 0 ? t Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 6 (29) Why discontinuous solutions? Evolution law: Local stability: κ ≥ −@sI(t; s(t)), Complementarity: s_(t) κ + @sI(t; s(t)) = 0. Formulation allowing for discontinuous solutions? I Model based on global minimization of the total energy I + R ! Global energetic formulation for rate independent processes (Mielke/Theil’99) ! Francfort/Marigo-Model (arbitrary cracks), shape memory alloys, finite strain elastoplasticity, damage and delamination models,... I Model based on viscous approximations General theory: Mielke/Efendiev ’06, Rossi/Mielke/Savaré ’08-12, Mielke/Zelik’10 Cracks and damage: Lazzaroni/Toader 11, K./Mielke/Zanini 08-10, K./Schröder 10-12, K./Rossi/Zanini 12, Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 6 (29) Remarks: + − Cl [@s I(t; s);@s I(t; s)] = Clarke-differential @s I of the mapping s 7! I(t; s). Cl @s I(t; ·):(0; L) !P(R) is upper semicontinuous as a set-valued mapping. Conclusion: If for all t; s the minimizers are unique, then I 2 C1([0; T ] × (0; L)). Energy release rate in the polyconvex case Griffith formula with Eshelby tensor: Z > G(s;') := r' DW (r') − W (r')I :(e1 ⊗ rθs) dx θ = 1 Ωs θ = 0 Theorem (K./Mielke/Zanini) I(·; ·) 2 Clip([0; T ] × (0; L)) and + @s I(t; s) = minf −G(s;'); ' minimizes E(t; s; ·) g is lower semicontinuous; − @s I(t; s) = maxf −G(s;'); ' minimizes E(t; s; ·) g is upper semicontinuous: Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 7 (29) Energy release rate in the polyconvex case Griffith formula with Eshelby tensor: Z > G(s;') := r' DW (r') − W (r')I :(e1 ⊗ rθs) dx θ = 1 Ωs θ = 0 Theorem (K./Mielke/Zanini) I(·; ·) 2 Clip([0; T ] × (0; L)) and + @s I(t; s) = minf −G(s;'); ' minimizes E(t; s; ·) g is lower semicontinuous; − @s I(t; s) = maxf −G(s;'); ' minimizes E(t; s; ·) g is upper semicontinuous: Remarks: + − Cl [@s I(t; s);@s I(t; s)] = Clarke-differential @s I of the mapping s 7! I(t; s). Cl @s I(t; ·):(0; L) !P(R) is upper semicontinuous as a set-valued mapping. Conclusion: If for all t; s the minimizers are unique, then I 2 C1([0; T ] × (0; L)). Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 7 (29) " abstract convergence principle: n * , E(s; n) ! E(s; ), then @sE(s; n) ! @sE(s; ) + ) @s I(σ) ≥ minf −G(s;'s); 's minimizes E(t; s; ·) g. h!0 R ! − (r's): r(e1 ⊗ rθs) dx = −G(s;'s) Ωs E Since 's was an arbitrary minimizer we may take the infimum: + minf −G(s;'s); 's minimizes E(t; s; ·) g ≥ @s I(σ). Comment on the proof For h > 0 define a family of inner variations via 1 h Th :Ωs ! Ωs+h; x 7! x + hθ(x)( ) 0 θ = 1 1 −1 θ = 0 h E(t; s + h;'s ◦ Th ) − E(t; s;'s) 1 ≥ h I(t; s + h) − I(t; s) 1 ≥ h E(t; s + h;'s+h) − E(t; s;'s+h ◦ Th) Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 8 (29) h!0 R ! − (r's): r(e1 ⊗ rθs) dx = −G(s;'s) Ωs E Since 's was an arbitrary minimizer we may take the infimum: + minf −G(s;'s); 's minimizes E(t; s; ·) g ≥ @s I(σ). Comment on the proof For h > 0 define a family of inner variations via 1 h Th :Ωs ! Ωs+h; x 7! x + hθ(x)( ) 0 θ = 1 1 −1 θ = 0 h E(t; s + h;'s ◦ Th ) − E(t; s;'s) 1 ≥ h I(t; s + h) − I(t; s) 1 ≥ h E(t; s + h;'s+h) − E(t; s;'s+h ◦ Th) " abstract convergence principle: n * , E(s; n) ! E(s; ), then @sE(s; n) ! @sE(s; ) + ) @s I(σ) ≥ minf −G(s;'s); 's minimizes E(t; s; ·) g. Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 8 (29) Comment on the proof For h > 0 define a family of inner variations via 1 h Th :Ωs ! Ωs+h; x 7! x + hθ(x)( ) 0 θ = 1 1 −1 θ = 0 h E(t; s + h;'s ◦ Th ) − E(t; s;'s) 1 ≥ h I(t; s + h) − I(t; s) 1 ≥ h E(t; s + h;'s+h) − E(t; s;'s+h ◦ Th) " abstract convergence principle: n * , E(s; n) ! E(s; ), then @sE(s; n) ! @sE(s; ) + ) @s I(σ) ≥ minf −G(s;'s); 's minimizes E(t; s; ·) g. h!0 R ! − (r's): r(e1 ⊗ rθs) dx = −G(s;'s) Ωs E Since 's was an arbitrary minimizer we may take the infimum: + minf −G(s;'s); 's minimizes E(t; s; ·) g ≥ @s I(σ). Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 8 (29) Minimizers satisfy + i ν i i−1 (a) local stability: κ + @sI (ti ; sτ ) + τ (sτ − sτ ) ≥ 0 i i−1 i i−1 + i sτ −sτ sτ −sτ (b) complementarity: κ + @sI (ti ; sτ ) + ν τ τ = 0. i i−1 i i−1 + i sτ −sτ sτ −sτ or equivalently 0 2 @s I(ti ; sτ ) + @R τ + ν τ : p 0 1 2 Estimates: supτ ν k^sτ kL2(0;T ) < 1, ksτ − ^sτ kL1(0;T ) ≤ c(τ/ν) . + + − Problem: @sI is lower semicontinuous, only. (Recall: @sI ≤ @sI ). Time incremental, viscous evolution model i τ time step size, sτ crack length at time ti = iτ, ν > 0 viscosity Time incremental minimization with viscosity term i i−1 Find sτ ≥ sτ such that i−1 i σ − sτ ν i−1 2 i−1 sτ 2 Argmin I(ti ; σ) + τR + σ − sτ ; σ ≥ sτ : τ 2τ Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 9 (29) Time incremental, viscous evolution model i τ time step size, sτ crack length at time ti = iτ, ν > 0 viscosity Time incremental minimization with viscosity term i i−1 Find sτ ≥ sτ such that i−1 i σ − sτ ν i−1 2 i−1 sτ 2 Argmin I(ti ; σ) + τR + σ − sτ ; σ ≥ sτ : τ 2τ Minimizers satisfy + i ν i i−1 (a) local stability: κ + @sI (ti ; sτ ) + τ (sτ − sτ ) ≥ 0 i i−1 i i−1 + i sτ −sτ sτ −sτ (b) complementarity: κ + @sI (ti ; sτ ) + ν τ τ = 0.
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