1 / 27

Variational Fracture and FEM with strong discontinuities

Evolution problems in damage plasticity and fracture Udine 19-21 2012. Variational Fracture and FEM with strong discontinuities. Maurizio Angelillo*, joint work with Enrico Babilio # , Antonio Fortunato*. * Università di Salerno # Università di Napoli Federico II.

deon
Télécharger la présentation

Variational Fracture and FEM with strong discontinuities

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Evolution problems in damage plasticity and fracture Udine 19-21 2012 Variational Fracture and FEM with strong discontinuities Maurizio Angelillo*, joint work with Enrico Babilio#, Antonio Fortunato* * Università di Salerno # Università di Napoli Federico II

  2. Objective: FEM code for Variational Fracture (small strains, 2d) Basic Ref’s on VF: De Giorgi & Ambrosio 88, Braides 92, Dal Maso 96, Del Piero 97, Francfort & Marigo 98, Dal Maso & Toader 02 What is Variational Fracture? … Energy to be minimized is the sum of bulk and interface terms … Energy depends not only on displacements but also on the “jump set” G: Introduction M. A., Enrico Babilio, Antonio Fortunato, Numerical solutions for crack growth based on the variational theory of fracture, Computational Mechanics, 2012 (on line)

  3. Introduction Restrictions: 2d linear elasticity & Griffith type interface energy bulk energy (linear elasticity & isotropy) interface energy (g = Gc : critical energy ….two types)

  4. Introduction • Numerical difficulties in using Variational Fracture • Non interpenetration: fracture has a definite sign … constraint. • Regularity of G … basic mathematical question … solved with the Direct Method of the Calculus of Variations -> Global Minima (Ambrosio 89) ……. G is regular if the interface energy has nice properties …..… sufficient ingredients: Concavity & Activation term • In reality evolution follows local not global minima … sensitivity to energy barriers … later • Permanence of fracture: Irreversibility  evolutionary global minimization (Francfort & Marigo 98): the set G(t+Dt)  G(t) for any time t and for any Dt>0 …..incremental formulation even if the evolution is quasi-static. • Main difference static vs dynamics: in the second case the evolution is ruled (at any time t) by the Euler equation of the functional +

  5. Numerical implementation of Variational Fracture: state of the art • Phase fields approaches to Variational Fracture • Standard numerical strategy (Bourdin et al 00, Del Piero et al 04): regularization. • Ambrosio-Tortorelli approach (kind of damage: fractures appear smeared; by reducing the damage parameter e get possible discontinuity in the limit). • Other approaches (similar): • phase fields (Borden 2011), eigenstrains (Ortiz, Pandolfi: e-neighborhood)….. • No strong discontinuities: Fractures are smeared over strips • Extremely fine meshes required to locate cracks Borden 2011…mesh size: h= 3 Micron

  6. Numerical implementation of LEFM: state of the art • Strong discontinuity approaches. Finite element methods are used extensively in Griffith’s type LEFM. • The most commonly used FE models: the virtual crack closure technique (Review: Krueger 04) and, more recently the Extended Finite Element Method (X-FEM, Moes, Dolbow, Belitschko 99). • Cracks represented as discrete strong discontinuities, either by remeshing and inserting discontinuity lines, or by enriching the shape functions inside the elements with discontinuous components. • The two methods follows the evolution of a given crack (no branching, no nucleation)…tracking the evolution of complex fracture lines (surfaces) with X-FEM has proven to be a tedious task. • Evolution is usually ruled through ad hoc criteria (MTS , MRR). • Our aim: introduce the VF concept into the strong discontinuity approach.

  7. Numerical implementation of Variational Fracture • Our code: • Modelling nucleation and propagation of cracks within VF through FE with gaps …. Strong discontinuities • Local Minima … Minimization strategy: Descent Minimization • The VF model requires the ability to locate and approximate the crack. fixed FE mesh non convergent … our mesh is variable (mesh nodes are taken as further unknowns : minimization over variable triangulations). • Main issue with numerical local minimization: how the material moves from an equilibrium state to another nearby state of lower energy …ability to surmount small energy barriers

  8. … sensitivity to energy barriers … • With a Griffith type interface energy fracture nucleation is always brutal: (descent directions for the energy do not exist in absence of singularities, such as a pre-existing crack) … sensitivity to energy barriers … • In classical FE approximations descent paths (leading to nearby local minima) do not ever exist since an infinite stress is required to overcome the energy barrier and displacements based on classical finite elements have gradients that are bounded in terms of the mesh size (inconvenient remedied in X-FEM). Relaxation of the interface energy

  9. Relaxation of the interface energy g =Gc … critical energy

  10. Relaxation of the interface energy With our approach we introduce a sort of FE length scale (mesh size h) The parameters: l, m, g=Gc, are material constants The parameters: k, s°, t° are artificial constants depending on h: k, artificial stiffness … penalty for non-interpenetration … gives some interface elasticity s°, t°, artificial stress thresholds: changing the ratio s°/ t°…

  11. Time discretization The given displacement u(on the constrained part ∂D) is time dependent. The time interval [0, τ] is discretized into n time steps δ: τi = i δ , i = {1, 2, . . . , n} At any loading step τi , minimal states of the energy are searched through an iterative descent procedure. Fracture opened when stress over the threshold. If a fracture opens up at step τi, it is maintained at any step τj , j >i (becomes a piece of the boundary with fracture energy threshold γ = 0, penalty for a < 0 is left).

  12. Space discretization: FE approximation Energy: E=E(x(j), u(j)) Each node j has four degrees of freedom x1( j ), x2( j ), u1( j ), u2( j ) ϕ(ε), written, explicitly, as a function of the displacement u as well of the reference coordinates x, defined over the triangles

  13. q([[u]])=q((x(j), u(j)), written, explicitly, as a function of the displacement u as well of the reference coordinates x, defined over the interface triangles Remark. Triangles

  14. FE approximations and radical meshes Rate of convergence. Our method: constant strain triangular elements …. with uniform meshes an rα singularity with 0 < a < 1 yields no better than O(ha) convergence for the usual H1 variational formulation. The number of unknowns we can manage, in terms of computer time, with our approach is still rather low considering the O(h1/2) convergence, but there is a point we must consider. Approximation of large displacement gradients at the crack tip requires an extreme refinement of the mesh to achieve an acceptable convergence rate. For H1 methods it is known that graded mesh refinement can be used to recover the optimal convergence rate (Radical Mesh). Polynomials of degree p with radical meshes yield full O(hp) convergence with H1 schemes.

  15. Mesh modification in mixed mode

  16. FE approximations and radical meshes Rate of convergence. Our method: constant strain triangular elements …. with uniform meshes an rα singularity with 0 < a < 1 yields no better than O(ha) convergence for the usual H1 variational formulation. The number of unknowns we can manage, in terms of computer time, with our approach is still rather low considering the O(h1/2) convergence, but there is a point we must consider. Approximation of large displacement gradients at the crack tip requires an extreme refinement of the mesh to achieve an acceptable convergence rate. For H1 methods it is known that graded mesh refinement can be used to recover the optimal convergence rate (Radical Mesh). Polynomials of degree p with radical meshes yield full O(hp) convergence with H1 schemes.

  17. Previous work: Fracture Nucleation in 1d bar

  18. Previous work: Validation with Classical Linear Fracture Mechanics a: e=10-3 cm, k = 103 N/cm3, s° 10 N/cm3, t° = 104 N/cm3 b: e=10-3 cm, k = 104 N/cm3, s° 100 N/cm3, t° = 104 N/cm3 Mesh too coarse to capture the gradients of the displacement field near the crack … when trying to reduce apparent compliance, overall response becomes too stiff and too much energy tends to be dissipated.

  19. Previous work: Fracture Propagation in 2d, mixed mode: num convergence

  20. Example 1: Propagation in 2d. EFFECT OF s° Material very strong in shear: interface energy q’ Final given displacement: SH= 0.001 L, SV=0.001 L

  21. Example 2: NUCLEATION IN 2d. RUPTURE OF A STRETCHED AND SHEARED STRIP. Material very strong in shear: interface energy q’ Mesh topology Final given displacement: SH= 0.0007 B, SV=0.0007 B

  22. Example 3: NUCLEATION IN 2d. RUPTURE OF A STRETCHED AND SHEARED STRIP Material very weak in shear: interface energy q” Given displacement (pure shearing): SH= 0.001 B, SV=0

  23. Example 4: NUCLEATION IN 2d. RUPTURE OF A STRETCHED AND SHEARED STRIP Material very weak in shear: interface energy q” Given displacement (mixed mode at 22.5°): SH= 0.000924 B, SV=0.000383 B

  24. Example 5: NUCLEATION IN 2d. RUPTURE OF A STRETCHED AND SHEARED STRIP Material very weak in shear: interface energy q” Given displacement (mixed mode at 45°): SH= 0.000707 B, SV=0.000707 B

  25. Example 6: NUCLEATION IN 2d. RUPTURE OF A STRETCHED AND SHEARED STRIP Material very weak in shear: interface energy q” Given displacement (pure shearing): SH= 0.001 B, SV=0 Mesh adaptation blocked in elasic phase

  26. CONCLUDING REMARKS • A FE approximation of Griffith type fracture in 2d VF has been presented. • The way we propose to approximate cracks in variational fracture is different from regularization methods : the displ. jumps occur on the skeleton of the mesh … mesh is variable. • Local minima of the energy with respect both to displacements and jump sets are searched through descent methods. • To overcome small energy barriers (either physical or artifacts of the FE approximation) the interface energy is relaxed. • To follow the evolution of cracks the real quasi-static trajectory is approximated with a sequence of states (step by step approx.) • The closeness of the approximate trajectory to the real one depends on a sensible choise of the parameters (s°, t°, ..)

  27. CONCLUDING REMARKS Pro: The results we show are encouraging (self adaptation works) • We could reproduce the results of CL Fracture Mechanics • We got good indications of convergence as the mesh is refined • The mesh is self adapting to singularities and curved crack paths • Though the approach is necessarily restricted to triangular elements, mesh minimality produces efficient geometries (radical meshes) • Variable mesh seems to have the potentiality to describe branching with a coarse mesh  dynamics Contro: • Parameters are “mesh dependent” and “step dependent” • Apart from very rough bounds on the parameters, we were not able to identify any simple rule to set s°, t° , … case by case … look for more efficient ways to get out of energy wells. • Reproducing the when in mixed mode requires a great effort in tuning the parameters. • The descent method is still numerically too inefficient to be pushed at the refinement levels required by the singularities (optimize the code, reduce unknowns)

More Related