The variational approach to fracture: Jean-Jacques Marigo

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1 The variational approach to fracture: main ingredients and some results Jean-Jacques Marigo Ecole Polytechnique, LMS

2 Part I : Griffith theory The classical formulation The extended formulation The issue of the crack path Numerics Part 2 : Barenblatt theory The issue of crack initiation Fatigue Part 3 : Perspectives Process zone and microcracking Dynamics and the balance of energy

3 Objective : Determine the evolution of cracks with the loading D Ω \ Γ t Γ t l(t) Main assumptions of the classical theory: Crack = surface of discontinuity of the displacement Quasi-static approach (Linear) elasticity Griffith surface energy Rate independent law based on energy balance Crack path = curve (in 2D) Smooth evolution in time F t Extra assumptions: 2D (plane strain or plane stress) Isotropy Path given time = loading parameter t l(t)?

4 Quasi-static Linear elasticity (t, l) u displacement field at equilibrium with the loading at time t 0=divσ + f t in Ω \ Γ l σ = Aε(u) in Ω \ Γ l u = U t on D σn = F t on Ω \D σn =0 on Γ l u = Argmin v C(t,l) Ω\Γ l 1 2 Aε(v) ε(v)dx f t(v) Potential energy P(t, l) = Ω\Γ l 1 2 Aε(u) ε(u)dx f t(u) Surface energy S(l) =G c l Total energy E(t, l) =P(t, l)+s(l) Potential energy release rate G(t, l) = P l (t, l)

5 Griffith evolution law (along the given path) 1. (Irreversibility) l(t) 0 2. (Yield criterion) G(t, l(t)) G c ( 3. (Energy balance) G(t, l(t)) G c ) l(t) =0 with an initial condition Remark: Only valid for smooth evolution. At least t l(t) continuous

6 Griffith evolution law (along the given path) 1. (Irreversibility) l(t) 0 2. (Yield criterion) G(t, l(t)) G c ( 3. (Energy balance) G(t, l(t)) G c ) l(t) =0 with an initial condition Remark: Only valid for smooth evolution. At least t l(t) continuous Theorem of existence for Griffith s law : In the case of proportional loading, Griffith s law admits a (continuous) solution t l(t) in the interval [t 0,t 1 ] where the crack length grows from l 0 to l 1 if and only if l E(1, l) is strictly convex in the interval [l 0, l 1 ]. Remark: For proportional loading, f t = tf 1,... by definition, u(t, l) =tu(1, l) and P(t, l) =t 2 P(1, l) by linearity. Remark : When l E(1, l) is not strictly convex, then the evolution of l(t) is necessarily discontinuous, but Griffith s law is not able to treat such a case.

7 E G(l) l Issue : Since non convexity is rather the rule, how to modify Griffith s law to find the evolution in such cases?

8 Change of the energy balance 3. New energy balance : E(t, l(t)) = E(0, l(0)) + t 0 E t (s, l(s))ds Equivalent when the evolution is smooth de E (t, l(t)) dt t E (t, l(t)) = (t, l(t)) l(t) ( l = G(t, l(t)) + G c ) l(t) = 0 Different when the evolution is discontinuous E(t, l + (t)) = E(t, l (t)) i.e. ) P(t, l (t)) P(t, l + (t)) = G c (l + (t) l (t)

9 Change of the yield criterion Other fundamental properties: When l E(1, l) is strictly convex in the interval [l 0, l 1 ], then the solution of Griffith s law is unique and for each t [t 0,t 1 ], l(t) minimizes E(t, l) over the interval [l 0, l 1 ].

10 Change of the yield criterion Other fundamental properties: When l E(1, l) is strictly convex in the interval [l 0, l 1 ], then the solution of Griffith s law is unique and for each t [t 0,t 1 ], l(t) minimizes E(t, l) over the interval [l 0, l 1 ]. 2. Stability Condition : E(t, l(t)) E(t, l), l [l(t), l(t) +h] The yield criterion is a first order stability criterion It is equivalent to the Stability Condition only in the convex case A stronger condition should be global minimization

11 Modification of Griffith law (along the given path) 1. (Irreversibility) t l(t) increasing 2. (Stability criterion) E(t, l(t)) E(t, l) for l [l(t), l(t)+h) 3. (Energy balance) E(t, l(t)) = E(0, l(0)) + t 0 E t (s, l(s))ds Remark : (i) In the strictly convex case, the formulations are equivalent (ii) In the non strictly convex case, the new formulation admits (non smooth) solutions.

12 The question of the path In the classical approach : additional criteria K i [[θ]] (Local Symmetry Principle) K 2 =0 (G-max criterion) [[θ]] maximizes G Remark : Which criterion for anisotropic materials? in antiplane setting? at in interface? at a boundary? In the variational approach : extend the stability condition to virtual cracks

13 The generalized evolution law (including the path criterion) D Ω \ Γ (t, Γ) u u = Argmin v C(t,Γ) Ω\Γ 1 2 Aε(v) ε(v)dx f t(v) Γ P(t, Γ) = Potential energy Ω\Γ 1 2 Aε(u) ε(u)dx f t(u) F t Γ = any set S(Γ) = Γ Surface energy G c (x, ν(x))dh n 1 (x) Total energy E(t, Γ) = P(t, Γ) + S(Γ)

14 The new three extended items Revisited Griffith theory 1. (Irreversibility) t Γ(t) increasing 2. (Stability criterion) E(t, Γ(t)) E(t, Γ), Γ Γ(t), Γ close to Γ(t) 3. (Energy balance) E(t, Γ(t)) = E(0, Γ(0)) + t 0 E t (s, Γ(s))ds Remark : (i) Valid in a very general context: 3D, heterogeneous, anisotropic bodies; (ii) Allows discontinuous solutions; (iii) Contains a path selection criterion. Francfort, Larsen, Dal Maso,...

15 Numerical implementation Griffith energy 1 E i (u) = Ω\S u 2 Aε(u) ε(u) dx f i(u)+g c H n 1 (S u ) Regularization by a non local damage law (Ambrosio-Tortorelli, Bourdin) ( ) Ei l 1 α 2 (u, α) = 2 (1 α)2 Aε(u) ε(u) dx f i (u)+g c + l α α dx 4l Ω Ω small numerical length Classical FEM Gamma-convergence result Alternate (quadratic) minimization Some issues: Irreversibility Non interpenetration (Royer, Lancioni, Maurini,...)

16 Numerical experiments

17 Numerical experiments

18 L brittle unbreakable brittle U ΗL

19 L brittle unbreakable brittle U ΗL

20 L bittle unbreakable brittle U ΗL

21 L bittle unbreakable brittle U ΗL

22 L bittle unbreakable brittle U ΗL

23 Barenblatt theory The issue of crack initiation Fatigue

Models for dynamic fracture based on Griffith s criterion

$Models for dynamic fracture based on Griffith s criterion$ Models for dynamic fracture based on Griffith s criterion Christopher J. Larsen Abstract There has been much recent progress in extending Griffith s criterion for crack growth into mathematical models