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 ∈ ∂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 ϕ :Ωs → deformation field R ϕ(Ω ) 2×2 ϕ s W : R → [0, ∞] elastic energy density Ωs

2×2 Assumptions: F = ∇ϕ ∈ R

I W (F) = ∞ if det F ≤ 0 + coercivity.

I polyconvexity: W (F) = g(F, det F), g convex and lower semicontinuous. 2×2 > I multiplicative stress control: ∀F ∈ R+ : F DW (F) ≤ c1(W (F) + 1)

2 2 Example: W (F) = c1 |F| + c2(det F) − c3 log(det F).

1,p Admissible deformations V (Ωs) = { ϕ ∈ W (Ωs); ϕ = ϕD } ΓD Elastic energy E(t, ϕ, s) = R W (∇ϕ) dx − R h(t) · ϕ da Ωs ΓN

Reduced energy I(t, s) = infϕ∈V (Ωs ) E(t, s, ϕ)

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) = . ∞ else κ > 0 fracture toughness

Evolution criterion: ) local stability: κ ≥ −∂sI(t, s(t)),
⇔ 0 ∈ ∂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 minϕ∈V (Ωs ) E(t, s, ϕ)

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 minϕ∈V (Ωs ) E(t, s, ϕ)

Dissipated energy: ( κ(snew − sold) if snew ≥ sold R(snew − sold) = . ∞ else κ > 0 fracture toughness

Evolution criterion: ) local stability: κ ≥ −∂sI(t, s(t)),
⇔ 0 ∈ ∂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 ∈ C1([0, T ] × (0, L)).

Energy release rate in the polyconvex case

Griffith formula with Eshelby tensor: Z >
G(s, ϕ) := ∇ϕ DW (∇ϕ) − W (∇ϕ)I :(e1 ⊗ ∇θs) dx θ = 1 Ωs θ = 0

Theorem (K./Mielke/Zanini) I(·, ·) ∈ Clip([0, T ] × (0, L)) and

+ ∂s I(t, s) = min{ −G(s, ϕ); ϕ minimizes E(t, s, ·) } is lower semicontinuous, − ∂s I(t, s) = max{ −G(s, ϕ); ϕ minimizes E(t, s, ·) } 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, ϕ) := ∇ϕ DW (∇ϕ) − W (∇ϕ)I :(e1 ⊗ ∇θs) dx θ = 1 Ωs θ = 0

Theorem (K./Mielke/Zanini) I(·, ·) ∈ Clip([0, T ] × (0, L)) and

+ ∂s I(t, s) = min{ −G(s, ϕ); ϕ minimizes E(t, s, ·) } is lower semicontinuous, − ∂s I(t, s) = max{ −G(s, ϕ); ϕ minimizes E(t, s, ·) } 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 ∈ C1([0, T ] × (0, L)).

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 7 (29) ↑ abstract convergence principle:

ψn * ψ, E(s, ψn) → E(s, ψ), then ∂sE(s, ψn) → ∂sE(s, ψ) + ⇒ ∂s I(σ) ≥ min{ −G(s, ϕs); ϕs minimizes E(t, s, ·) }.

h→0 R → − (∇ϕs): ∇(e1 ⊗ ∇θs) dx = −G(s, ϕs) Ωs E Since ϕs was an arbitrary minimizer we may take the infimum: + min{ −G(s, ϕs); ϕs minimizes E(t, s, ·) } ≥ ∂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, ϕs ◦ Th ) − E(t, s, ϕs)

1
≥ h I(t, s + h) − I(t, s)

1
≥ h E(t, s + h, ϕs+h) − E(t, s, ϕs+h ◦ Th)

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 8 (29) h→0 R → − (∇ϕs): ∇(e1 ⊗ ∇θs) dx = −G(s, ϕs) Ωs E Since ϕs was an arbitrary minimizer we may take the infimum: + min{ −G(s, ϕs); ϕs minimizes E(t, s, ·) } ≥ ∂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, ϕs ◦ Th ) − E(t, s, ϕs)

1
≥ h I(t, s + h) − I(t, s)

1
≥ h E(t, s + h, ϕs+h) − E(t, s, ϕs+h ◦ Th) ↑ abstract convergence principle:

ψn * ψ, E(s, ψn) → E(s, ψ), then ∂sE(s, ψn) → ∂sE(s, ψ) + ⇒ ∂s I(σ) ≥ min{ −G(s, ϕs); ϕs minimizes E(t, s, ·) }.

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, ϕs ◦ Th ) − E(t, s, ϕs)

1
≥ h I(t, s + h) − I(t, s)

1
≥ h E(t, s + h, ϕs+h) − E(t, s, ϕs+h ◦ Th) ↑ abstract convergence principle:

ψn * ψ, E(s, ψn) → E(s, ψ), then ∂sE(s, ψn) → ∂sE(s, ψ) + ⇒ ∂s I(σ) ≥ min{ −G(s, ϕs); ϕs minimizes E(t, s, ·) }. h→0 R → − (∇ϕs): ∇(e1 ⊗ ∇θs) dx = −G(s, ϕs) Ωs E Since ϕs was an arbitrary minimizer we may take the infimum: + min{ −G(s, ϕs); ϕs minimizes E(t, s, ·) } ≥ ∂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 ∈ ∂s I(ti , sτ ) + ∂R τ + ν τ .

√ 0 1 2 Estimates: supτ ν kˆsτ kL2(0,T ) < ∞, ksτ − ˆsτ kL∞(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 ∈ 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 ∈ 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.  i i−1  i i−1 + i sτ −sτ sτ −sτ or equivalently 0 ∈ ∂s I(ti , sτ ) + ∂R τ + ν τ .

√ 0 1 2 Estimates: supτ ν kˆsτ kL2(0,T ) < ∞, ksτ − ˆsτ kL∞(0,T ) ≤ c(τ/ν) .

+ + − Problem: ∂sI is lower semicontinuous, only. (Recall: ∂sI ≤ ∂sI ).

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 9 (29) Vanishing viscosity model with polyconvex energies

Theorem (K./Mielke/Zanini) τ Let ν → 0. There exists s ∈ BV ([0, T ], R) and a subsequence τ & 0 with ∗ ˆsτ * s in BV ([0, T ], R) and ˆsτ (t) → s(t) for every t ∈ [0, T ].

Moreover, (a) s is non-decreasing,

− − (b ) κ + ∂sI (t, s(t)) ≥ 0 for every t ∈ [0, T ]\J(s),

+ + (c ) if κ + ∂sI (t, s(t)) > 0, then t ∈ D(s) and s˙(t) = 0,

+ + (d ) ∀ t ∈ J(s), ∀s∗ ∈ [s(t−), s(t+)] we have κ + ∂sI (t, s∗) ≤ 0.

J(s) jump set of s; D(s) set of differentiable points of s.

Cl Proof: A-priori estimates, Helley selection principle, continuity property of ∂s I, change of variables (s ↔ t) to obtain (d+).

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 10 (29) Vanishing viscosity model with polyconvex energies

Theorem (K./Mielke/Zanini) τ Let ν → 0. There exists s ∈ BV ([0, T ], R) and a subsequence τ & 0 with ∗ ˆsτ * s in BV ([0, T ], R) and ˆsτ (t) → s(t) for every t ∈ [0, T ].

Moreover, (a) s is non-decreasing,

− − (b ) κ + ∂sI (t, s(t)) ≥ 0 for every t ∈ [0, T ]\J(s),

+ + (c ) if κ + ∂sI (t, s(t)) > 0, then t ∈ D(s) and s˙(t) = 0,

+ + (d ) ∀ t ∈ J(s), ∀s∗ ∈ [s(t−), s(t+)] we have κ + ∂sI (t, s∗) ≤ 0.

J(s) jump set of s; D(s) set of differentiable points of s.

Complementarity condition: + − For every t ∈ D(s) exists g(t) ∈ [∂s I(t, s(t)), ∂s I(t, s(t))] with κ + g(t)
s˙(t) = 0, 0 ∈ ∂R(s˙(t)) + g(t).

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 10 (29) Proof of the jump condition (d)

Let t∗ ∈ J(s), s0, s1 ∈ (s(t∗−), s(t∗+)) arbitrary

0 1 0,1 τ→0 0,1 Continuity of ˆsτ : ∃ tτ < tτ with tτ → t∗ and ˆsτ (tτ ) = s0,1.

2 Complementarity condition: ∀ψ ∈ L ([s0, s1]), ψ ≥ 0 we have

1 Z tτ  +  0 0 ≥ κ + ∂sI (tτ (t), sτ (t)) ψ(ˆsτ (t)) ˆsτ (t) dt. 0 tτ

0 1 Change of variables: σ = ˆsτ (t), ˜tτ (σ) = min{ t ∈ [tτ , tτ ]; ˆsτ (t) = σ }

Z s1  +  0 ≥ κ + ∂sI (tτ (˜tτ (σ)), sτ (˜tτ (σ))) ψ(σ) dσ. s0

p Note: tτ (˜tτ (σ)) → t∗, sτ (˜tτ (σ)) − σ = sτ (˜tτ (σ)) − ˆsτ (˜tτ (σ)) ≤ c τ/ν → 0.

+ By lsc. of ∂s I we conclude that ∀σ ∈ [s0, s1]: 0 ≥ κ + ∂sI(t∗, σ).

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 11 (29) Special solutions

Theorem (K./Mielke/Zanini)

There exists a nondecreasing s ∈ BV ([0, T ], R) satisfying + + (b ) κ + ∂s I(t, s(t)) ≥ 0 for every t ∈ [0, T ]\J(s),

+ + (c ) if κ + ∂s I(t, s(t)) > 0, then t ∈ D(s) and s˙(t) = 0,

+ + (d ) ∀ t ∈ J(s), ∀s∗ ∈ [s(t−), s(t+)] we have κ + ∂s I(t, s∗) ≤ 0.

Moreover, there exists a measurable map ϕ :[0, T ] → V (ΩL) such that for all t + ϕ(t) minimizes E(t, ·, s(t)) and ∂s I(t, s(t)) = −G(s(t), ϕ(t)).

+ + Proof: smax(t) = max{ s(t); s local energetic sol. } satisfies (a),(b )–(d ).

Conclusion: For every t ∈ D(s) the complementarity condition is satisfied:

+
+ κ + ∂s I(t, s(t)) s˙(t) = 0 and 0 ∈ ∂R(s˙(t)) + ∂s I(t, s(t)).

Open question: viscous approximation ⇒ vanishing viscosity solutions with (b+)?

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 12 (29) Energy balance

Theorem (K./Mielke/Zanini)

+ Every special solution (i.e. with (b )) satisfies for every t0 < t1, ti ∈/ J(s),

ˆ Z s(t1) Z s(t+) X + ˆ I(t1, s(t1)) + κ(σ)dσ + −(κ(σ) + ∂s I + (t, σ))dσ s(t0) s(ˆt−) ˆt∈J(s)∩(t0,t1)

Z t1 − = I(t0, s(t0)) + ∂t I(t, s(t))dτ. t0

Proof: Switch to a parameterized formulation of the evolution problem, use a chain rule and (a), (b+)–(d+).

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 13 (29) Global energetic solutions s (S) Global stability κ + ∂s I(t, s) > 0 ˜ R ˜s ˜ I(t, s(t)) ≤ I(t, s) + s(t) κdσ ∀s ≥ s(t),

(E) Energy equality κ + ∂s I(t, s) < 0 R s(t) d R t d I(t, s(t)) + s(0) κ σ = I(0, s(0)) + 0 ∂t I(τ, s(τ)) τ. t

R s+
Jump condition: ∂sI(t, σ) + κ dσ = 0 s−

Example for viscosity solutions

Vanishing viscosity solutions (a) s nondecreasing, s κ + ∂s I(t, s) > 0 (b) κ + ∂sI(t, s(t)) ≥ 0 for all t ∈ [0, T ]\J(s),

(c) if κ + ∂sI(t, s(t)) > 0, then t ∈ D(s) and s˙(t) = 0,

(d) ∀ t ∈ J(s), ∀s∗ ∈ [s(t−), s(t+)] it holds κ + ∂s I(t, s) < 0 κ + ∂sI(t, s∗) ≤ 0. t

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 14 (29) Example for viscosity solutions

Vanishing viscosity solutions (a) s nondecreasing, s κ + ∂s I(t, s) > 0 (b) κ + ∂sI(t, s(t)) ≥ 0 for all t ∈ [0, T ]\J(s),

(c) if κ + ∂sI(t, s(t)) > 0, then t ∈ D(s) and s˙(t) = 0,

(d) ∀ t ∈ J(s), ∀s∗ ∈ [s(t−), s(t+)] it holds κ + ∂s I(t, s) < 0 κ + ∂sI(t, s∗) ≤ 0. t

Global energetic solutions s (S) Global stability κ + ∂s I(t, s) > 0 ˜ R ˜s ˜ I(t, s(t)) ≤ I(t, s) + s(t) κdσ ∀s ≥ s(t),

(E) Energy equality κ + ∂s I(t, s) < 0 R s(t) d R t d I(t, s(t)) + s(0) κ σ = I(0, s(0)) + 0 ∂t I(τ, s(τ)) τ. t

R s+
Jump condition: ∂sI(t, σ) + κ dσ = 0 s−

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 14 (29) 2 FE-approximation of vanishing viscosity solutions (small strains)

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 15 (29) Deformation energy Z Z E(t, s, v) = W (ε(v)) dx − `(t) · v da for v ∈ K (Ωs). Ωs ΓN u(t, s) = argmin E(t, s, ·) unique minimizer. K (Ωs )

Reduced energy I(t, s) = infv∈K (Ωs ) E(t, s, v) = E(t, s, u(t, s)).

1 Lip Minimizers unique =⇒ I ∈ C ([0, T ] × (0, L)), ∂sI(t, ·) ∈ Cloc (0, L)

Notation in the small strain setting

Assumptions:

I Small strain elasticity: 2 Cs u :Ωs → R displacements, ε(u) linearized strains

I Elastic energy density: Ωs 1 W (ε(u)) = 2 Cε(u): ε(u), C elasticity tensor.

I Non-interpenetration conditions on the crack Cs: 1 K (Ωs) = { v ∈ H (Ωs); v = 0, [[v]] · ~n ≥ 0 on Cs } ΓD

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 16 (29) Notation in the small strain setting

Assumptions:

I Small strain elasticity: 2 Cs u :Ωs → R displacements, ε(u) linearized strains

I Elastic energy density: Ωs 1 W (ε(u)) = 2 Cε(u): ε(u), C elasticity tensor.

I Non-interpenetration conditions on the crack Cs: 1 K (Ωs) = { v ∈ H (Ωs); v = 0, [[v]] · ~n ≥ 0 on Cs } ΓD

Deformation energy Z Z E(t, s, v) = W (ε(v)) dx − `(t) · v da for v ∈ K (Ωs). Ωs ΓN u(t, s) = argmin E(t, s, ·) unique minimizer. K (Ωs )

Reduced energy I(t, s) = infv∈K (Ωs ) E(t, s, v) = E(t, s, u(t, s)).

1 Lip Minimizers unique =⇒ I ∈ C ([0, T ] × (0, L)), ∂sI(t, ·) ∈ Cloc (0, L)

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 16 (29) Fully discretized incremental minimization problem

N IN (t, s) = min{E(t, v, s); v ∈ Ks } k−1 k k ˜s − sN  N k−1 sN ∈ Argmin{IN (tN , ˜s) + τN RνN ; ˜s ∈ Z , ˜s ≥ sN }. τN

ν 2 Rν (η) = 2 η + κη.

Note: IN (t, ·) piecewise constant in s.

Fully discretized viscous minimization problem

Parameters:

τN time step size νN viscosity

ρN hN mesh size ρN crack increment

hN

N Spaces: Ks ⊂ K (Ωs) finite element space + contact conditions on Cs N 1 MN Z = {σN , . . . , σN } discrete crack lengths of distance ρN

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 17 (29) Fully discretized viscous minimization problem

Parameters:

τN time step size νN viscosity

ρN hN mesh size ρN crack increment

hN

N Spaces: Ks ⊂ K (Ωs) finite element space + contact conditions on Cs N 1 MN Z = {σN , . . . , σN } discrete crack lengths of distance ρN

Fully discretized incremental minimization problem

N IN (t, s) = min{E(t, v, s); v ∈ Ks } k−1 k k ˜s − sN  N k−1 sN ∈ Argmin{IN (tN , ˜s) + τN RνN ; ˜s ∈ Z , ˜s ≥ sN }. τN

ν 2 Rν (η) = 2 η + κη.

Note: IN (t, ·) piecewise constant in s.

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 17 (29) Local stability and complementarity conditions

Incremental minimization problem:

k−1 k k ˜s − sN  N k−1 sN ∈ Argmin{IN (tN , ˜s) + τN RνN ; ˜s ∈ Z , ˜s ≥ sN }. τN

k N The choice ˜s = sN ± ρN ∈ Z leads to (a) local stability:

k k−1 sN −sN 1 k k
νN ρN 0 ≤ κ + νN + IN (tN , sN + ρN ) − IN (tN , sN ) + . τN ρN τN

(b) complementarity condition:

 k k−1  k k−1 sN −sN 1 k k k k
νN ρN sN −sN κ + νN + IN (tN , sN ) − IN (tN , sN − ρN ) − ≤ 0. τN ρN 2τN ρN

√ 0 1/2 Estimates: supN νN kˆsN kL2(0,T ) < ∞, ksN − ˆsN kL∞(0,T ) ≤ c(τN /νN ) . sN piecewise constant, ˆsN piecewise linear interpolation.

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 18 (29) Convergence to a vanishing viscosity solution

Assumption N A1 ∀δ, µ > 0 ∃Nδ,µ ∈ N such that ∀N ≥ Nδ,µ, s ∈ Z ∩ [δ, L − δ] it holds

1
IN (t, s + ρN ) − IN (t, s) − ∂sI(t, s) ≤ µ. ρN

Theorem (K./Schröder 10)

ρν τ Assume A1 and that τ → 0, ν → 0. Then there exits a vanishing viscosity solution s ∈ BV ([0, T ]) and a subsequence ∗ with ˆsN * s in BV ([0, T ]) and pointwise for all t. 1 Moreover, uN (t) → u(t) strongly in H (ΩL), where u(t) = argmin E(t, ·, s(t)).

Note: All BV -weak∗-cluster points are vanishing viscosity solutions.

Questions: Sufficient conditions for A1?

Relation between mesh size hN and crack increment ρN ? −→ regularity of displacement fields

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 19 (29) Regularity of minimizers with contact condition

∈ (Ω ) = { ∈ 1 (Ω );[ ] · ≥ C } For v K s v HΓD s v n 0 on s :

E(v, s) = R 1 ε(v): ε(v) dx − R `(t) · v da Ωs 2 C ΓN

Theorem (K. 10)

Let be u ∈ K (Ωs) a minimizer of E with respect to K (Ωs). Then 3 3 −δ u ∈ B 2 (Q ∩ Ω ) ⊂ H 2 (Ω ∩ Q) for all δ > 0. Q 2,∞ s s

Q contact

Comparison: Neumann conditions on the crack ⇒ 1 3 2 2 u = |x − xs| v(ϕ) + smooth terms ∈ B2,∞(Ωs ∩ Q).

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 20 (29) Regularity of minimizers with contact condition

∈ (Ω ) = { ∈ 1 (Ω );[ ] · ≥ C } For v K s v HΓD s v n 0 on s :

E(v, s) = R 1 ε(v): ε(v) dx − R `(t) · v da Ωs 2 C ΓN

Theorem (K. 10)

Let be u ∈ K (Ωs) a minimizer of E with respect to K (Ωs). Then 3 3 −δ u ∈ B 2 (Q ∩ Ω ) ⊂ H 2 (Ω ∩ Q) for all δ > 0. Q 2,∞ s s

Proof: Derive estimates for finite differences of ∇u:

I tangential: uhe1 := u(· + hη(·)e1), h > 0, is an admissible test for the variational inequality Q + uniform convexity of the elastic energy E η = 1

− 1 2 =⇒ sup h kuh(· + he1) − uk 1 ≤ c η = 0 h>0 H (Ωs ∩Q)

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 20 (29) Regularity of minimizers with contact condition

∈ (Ω ) = { ∈ 1 (Ω );[ ] · ≥ C } For v K s v HΓD s v n 0 on s :

E(v, s) = R 1 ε(v): ε(v) dx − R `(t) · v da Ωs 2 C ΓN

Theorem (K. 10)

Let be u ∈ K (Ωs) a minimizer of E with respect to K (Ωs). Then 3 3 −δ u ∈ B 2 (Q ∩ Ω ) ⊂ H 2 (Ω ∩ Q) for all δ > 0. Q 2,∞ s s

Proof: Derive estimates for finite differences of ∇u:

I normal: “solve” the variational inequality for the missing derivative ∂2∇u Q (variant of an argument by η = 1 Frehse/Ebmeyer/Kassmann’02, Kassmann/Madych ’07)

− 1 η = 0 =⇒ sup h 2 kuhe − uk 1 ≤ c h>0,i∈{1,2} i H (Ωs ∩Q)

3 2 =⇒ u ∈ B2,∞(Ωs ∩ Q).

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 20 (29) A technical lemma for the interchange of derivatives

Steklov regularization:

Z h i 1 Mh(w)(x) := w(x + rei ) dr, x ∈ QR/2. h 0 i 2 2 Mh : L (QR ) → L (QR/2) linear, continuous with uniform bound, i i hMh∂i w = 4hw = w(x + hei ) − w(x).

Z Z Z 1 2 2 1 2 2s 2 2 2s 4hw dx + 4hMhw dx ≤ c |h| =⇒ 4hw dx ≤ c |h| Q3R/2 QR/2 QR/2

Remark. Original version by Ebmeyer/Frehse/Kassmann, Kassmann/Madych based on Fourier trafo. By direct estimation of the integrals extendable to p 6= 2.

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 21 (29) Condition A1 in case of contact conditions

Geometry: Th regular, uniform triangulation of Ω0 into triangles, h = hN mesh size;

compatible with crack CL (Ω ) = { ∈ (Ω ); , τ ∈ T } Spaces: Kh L v K L v τ affine h , Kh(Ωs) = K (Ωs) ∩ Kh(ΩL), N Z ⊂ { nodes of Th lying on the crack CL}

crack increment ρN ≥ hN = h

Theorem (K./Schröder ’10) 1 −δ N 2 For all sN ∈ Z ∩ (, L − ) it holds ku(sN ) − uN (sN )k 1 ≤ c,δh , H (ΩL) N

1 −δ 1
2 −1
IN (t, sN + ρN ) − IN (t, sN ) − ∂sI(t, sN ) ≤ c,δ ρN + hN ρN . ρN

1 1 Possible choice for the convergence theorem: ρ = τ ≈ h 4 , ν = h 8

Observe: Much coarser discretization for the crack than for the displacement field! Proof: Falk approximation theorem for variational inequalities

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 22 (29) Condition A1 in case of contact conditions

Geometry: Th regular, uniform triangulation of Ω0 into triangles, h = hN mesh size;

compatible with crack CL (Ω ) = { ∈ (Ω ); , τ ∈ T } Spaces: Kh L v K L v τ affine h , Kh(Ωs) = K (Ωs) ∩ Kh(ΩL), N Z ⊂ { nodes of Th lying on the crack CL}

crack increment ρN ≥ hN = h

Theorem (K./Schröder ’10) 1 −δ N 2 For all sN ∈ Z ∩ (, L − ) it holds ku(sN ) − uN (sN )k 1 ≤ c,δh , H (ΩL) N

1 −δ 1
2 −1
IN (t, sN + ρN ) − IN (t, sN ) − ∂sI(t, sN ) ≤ c,δ ρN + hN ρN . ρN

3 −δ Proof: Falk approximation theorem for variational inequalities + H 2 -regularity of u:

 1 1 
2 2 ku − uN k 1 ≤ c inf N ku − vk 1 + kAs(u)k + k`(t)k 2 ku − vk ∗ , H (Ωs ) v∈Ks H (Ωs ) W L (ΓN ) W

1 +δ ∗ 1 ∗ 2 ∗ where W ⊂ V = (H (Ωs)) is a dense subspace. Here: W = (H (Ωs)) . ΓD ΓD

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 22 (29) Example: Mode I crack

Observed convergence rate: Error: 1 ∂sI(s) − h (Ih(s + h) − Ih(s)) ≤ ch1 h

Condition A1, improved estimates neglecting contact on the crack

Theorem (K./Schröder ’10)

No contact conditions on Cs, mesh locally translation invariant parallel to the crack. Then A1 holds with

1
2 −1 IN (t, sN + ρN ) − IN (t, sN ) − ∂sI(t, sN ) ≤ c(ρN + hN + hN ρN ) ρN

√ Possible choice for the convergence theorem: ρN = τN = hN , νN = hN

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 23 (29) Condition A1, improved estimates neglecting contact on the crack

Theorem (K./Schröder ’10)

No contact conditions on Cs, mesh locally translation invariant parallel to the crack. Then A1 holds with

1
2 −1 IN (t, sN + ρN ) − IN (t, sN ) − ∂sI(t, sN ) ≤ c(ρN + hN + hN ρN ) ρN

√ Possible choice for the convergence theorem: ρN = τN = hN , νN = hN

Example: Mode I crack

Observed convergence rate: Error: 1 ∂sI(s) − h (Ih(s + h) − Ih(s)) ≤ ch1 h

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 23 (29) I Local character of the Griffith formula: Z ∂sI(t, s) = − E(∇u): e1 ⊗ ∇θs dx A∩Ωs

I Assumption on the mesh =⇒ N ∃ projection PsN : V (ΩsN ) → V (ΩsN ), diffeomorphism TsN ,ρN :ΩsN → ΩsN +ρN N such that ∀v ∈ V (ΩsN +ρN )

supp (PsN (v ◦ TsN ,ρN ) − v ◦ TsN ,ρN ) ⊂ A.

Comments on the proof

Proof. ∈ 3/2 (Ω ), ∈ 2(Ω ∩ A˜) I Regularity: u B2,∞ s u A˜ H s Local error estimates for FEM (Nitsche/Schatz’74): I Cs

θs = 1 θs = 0 ku − uhk 1 ≤ ch kuk 2 ˜ + ku − uhk 2 H (A) H (Ωs ∩A) L (Ωs ) A   ≤ ch kuk 2 + kuk 3/2 H (Ωs ∩A˜) B2,∞(ΩS )

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 24 (29) I Assumption on the mesh =⇒ N ∃ projection PsN : V (ΩsN ) → V (ΩsN ), diffeomorphism TsN ,ρN :ΩsN → ΩsN +ρN N such that ∀v ∈ V (ΩsN +ρN )

supp (PsN (v ◦ TsN ,ρN ) − v ◦ TsN ,ρN ) ⊂ A.

Comments on the proof

Proof. ∈ 3/2 (Ω ), ∈ 2(Ω ∩ A˜) I Regularity: u B2,∞ s u A˜ H s Local error estimates for FEM (Nitsche/Schatz’74): I Cs

θs = 1 θs = 0 ku − uhk 1 ≤ ch kuk 2 ˜ + ku − uhk 2 H (A) H (Ωs ∩A) L (Ωs ) A   ≤ ch kuk 2 + kuk 3/2 H (Ωs ∩A˜) B2,∞(ΩS )

I Local character of the Griffith formula: Z ∂sI(t, s) = − E(∇u): e1 ⊗ ∇θs dx A∩Ωs

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 24 (29) Comments on the proof

Proof. ∈ 3/2 (Ω ), ∈ 2(Ω ∩ A˜) I Regularity: u B2,∞ s u A˜ H s Local error estimates for FEM (Nitsche/Schatz’74): I Cs

θs = 1 θs = 0 ku − uhk 1 ≤ ch kuk 2 ˜ + ku − uhk 2 H (A) H (Ωs ∩A) L (Ωs ) A   ≤ ch kuk 2 + kuk 3/2 H (Ωs ∩A˜) B2,∞(ΩS )

I Local character of the Griffith formula: Z ∂sI(t, s) = − E(∇u): e1 ⊗ ∇θs dx A∩Ωs

I Assumption on the mesh =⇒ N ∃ projection PsN : V (ΩsN ) → V (ΩsN ), diffeomorphism TsN ,ρN :ΩsN → ΩsN +ρN N such that ∀v ∈ V (ΩsN +ρN )

supp (PsN (v ◦ TsN ,ρN ) − v ◦ TsN ,ρN ) ⊂ A.

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 24 (29) 3 Numerical example

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 25 (29) s 10 global 9 local kappa+dsI=0 8 7 6 −∂ I(1, ·): Solutions: 5 s 4 crack length 3 2 1 0 s 0 20 40 60 80 100 t time

A numerical example

Computations: A. Schröder, HU Berlin 2 monotone loading: `1 ∈ L (ΓN ), `(t) = t `1.

R 1 R Deformation energy: E(t, s, v) = ε(v): ε(v) dx − t `1 · v da. Ωs 2 C ΓN

2 2 density quadratic ⇒ I(t, s) = t I(1, s), ∂sI(t, s) = t ∂sI(1, s).

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 26 (29) A numerical example

Computations: A. Schröder, HU Berlin 2 monotone loading: `1 ∈ L (ΓN ), `(t) = t `1.

R 1 R Deformation energy: E(t, s, v) = ε(v): ε(v) dx − t `1 · v da. Ωs 2 C ΓN

2 2 density quadratic ⇒ I(t, s) = t I(1, s), ∂sI(t, s) = t ∂sI(1, s).

s 10 global 9 local kappa+dsI=0 8 7 6 −∂ I(1, ·): Solutions: 5 s 4 crack length 3 2 1 0 s 0 20 40 60 80 100 t time

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 26 (29) A numerical example

I Finite element discretization with continuous, piecewise bilinear functions on quadrilaterals, CG-like contact solver (Braess et al. ‘04), SOFAR (Scientific Object Oriented Finite Element Library for Application and Research)

10 10 h=0.5 9 9 h=0.25 h=0.125 8 8 h=0.0625 h=0.03125 7 7 h=0.015625 6 6 5 5 4 4 crack length crack length 3 h=0.5 3 h=0.25 2 h=0.125 2 h=0.0625 1 h=0.03125 1 h=0.015625 0 0 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 time time

Globale energetische Lösung Viskositätslösung ρ = h, ν = 0, τ = 0.01 ρ = h, ν = 0.013h0.5, τ = 0.1h

k k νN k−1 2 k−1 N k−1 sN ∈ Argmin{IN (tN , ˜s) + (˜s − s ) + κ(˜s − s ); ˜s ∈ Z , ˜s ≥ s }. 2τN N N N

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 27 (29) Summary

I Vanishing viscosity approach + Griffith fracture criterion (finite strains)

I Convergence analysis for the approximation of vanishing viscosity solutions with FE-discretization in space (small strains) 1 1 I with contact conditions: ρ = τ ≈ h 4 , ν =√h 8 I without contact conditions: h = ρ = τ, ν = h.

I Similar analysis for the case with contact? Local error estimates (Nitsche/Schatz’74) for variational inequalities?

I Extension to the Lazzaroni/Toader model (single crack, free crack path)??

I Extension to damage models (collaboration with R. Rossi, C. Zanini)

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 28 (29) References

I Efendiev/Mielke, On the rate independent limit of systems with dry friction and small viscosity, J. Convex Analysis, 2006.

I Knees/Mielke, Energy release rate for cracks in finite–strain elasticity, M2AS, 31, 501–528, 2008.

I Knees/Mielke/Zanini, On the inviscid limit of a model for crack propagation, M3AS, 2008.

I Knees/Mielke/Zanini, Crack propagation in polyconvex materials, Physica D, 2010.

I Knees/Rossi/Zanini, A vanishing viscosity approach to a rate-independent damage model, M3AS, (accepted).

I Knees/Schröder, Computational aspects of quasi-static crack propagation, DCDS-S, 2013.

I Knees/Schröder, Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints, M2AS, accepted.

I Negri/Ortner, Quasi-static crack propagation by Griffith’s criterion, M3AS, 2009. I Toader/Zanini, An artificial viscosity approach to quasistatic crack growth, Boll. Unione Mat. Ital., 2009.

Vanishing viscosity approach in fracture mechanics D. Knees ·· Page 29 (29)