Are damage gradient models applicable up to ultimate fracture ? Eric Lorentz V. Godard

1. Are damage gradient models applicable up to ultimate fracture ? Eric LorentzV. Godard

2. Motivation : safety related to electricity generation FOCUS • Crack propagationin concrete structures • (characteristic size : several meters)

3. Outline • Continuum damage and crack propagation • A constitutive law designed for robustness and efficiency • Numerical applications

4. Continuum damageand crack propagation

5. The damage law : assumptions and commitments • Modelling assumptions for sake of simplicity • No crack closure • Damage isotropy • No distinction between traction and compression • No irreversible strain • Complying with macroscopic quantities of interest • Elastic properties • Fracture energy • Critical stress • Adjustable parameters • « crack thickness »  compatible with the structure size • Softening part of the law  so as to gain desirable properties

6. A non local law is necessary because of the high gradients of macroscopic fields Localisation control through gradient damage laws • Local constitutive laws lead to spurious damage localisation • Introduction of the damage gradient into the constitutive law • Usual meaning of the stress field • Thermodynamical framework • Limited intrusion in a finite element code • Compatibility with usual solution algorithms • 2 additional unknowns / element vertex • Availability of a symmetrical tangent matrix Dependenceon mesh size Dependence onmesh orientation Benallal & Marigo, 2007 Bourdin et al., 2000 Dimitrijevic & Hackl, 2008 Fremond & Nedjar, 1996 Liebe et al., 2001 Lorentz & Andrieux, 1999 Pijaudier-Cabot & Burlion, 1996

7. State variables Parameters Strain field Hooke’s tensor Stiffness function Damage field Nonlocal coefficient Dissipated energy Thermodynamical potentials  Generalised Standard Material Elastic strain energy Helmholtz’ free energy Dissipation potential Boundary conditions Pointwise interpretation Stress Yield function Consistency Description of the gradient constitutive law

8. Questions • How far can I push the model ? • Is it able to describe a total loss of stiffness ? • Is it able to go up to ultimate fracture ? • Is it necessary to introduce a transition to a real crack ? • Related expected qualities • Robustness  Does the model always provide a result ? • Reliability  What confidence can I grant to the result ? • Efficiency  How long have I been staring at my computer ?

9. A constitutive law designedfor robustness and efficiency

10. Application in case of rectilinear and stable propagation • The damage model is consistent with coarser formulations : Griffith, cohesive zone model BUT • The computation exhibits a lot of snap-backs which slow down the convergence • The number of iterations increase dramatically as soon as points are broken force Extrernal work displacement Propagation length CZM = 5 mn CDM = 5 h

11. Possible explanations for the drawbacks • Spurious snap-backs because the strain remains bounded ? • Excessive iterations because of loading / unloading issues ? stress. displ. In order to avoid a snap-backthe strain should be at least a Dirach Only a single point reaches a = 1 Wherever else, the strain is zero damage Without enforcing damage increase Would the band width reduce ? x

12. Tuning the constitutive law to remedy the drawbacks • Assumption • The shape of the local softening response is not significant • But the peak stress and the fracture energy are prescribed • Exploration among many types of constitutive laws • Design the stiffness function A(a) and the dissipated energy w(a) • Submitted to monotonicity and convexity constraints • Validation • Closed-form solution on a 1D problem • The constitutive law is retained if : (1) Increase of the band width (2) No snap-back at the scale of the localisation band

13. (1) Increase of the band width • Some constitutive laws used in the litterature • Liebe, Steinmann, Benallal (2001) • Bourdin, Francfort, Marigo (2008), • Lorentz, Benallal (2005) • Power-law constitutive relations None OK OK for someparameters

14. (2) No snap-back at the scale of the localisation band • Observations • Snap-back for m < 2 • Finite opening for m = 2 • Asymptotic fracture for m > 2 Average strain

15. The selected constitutive law • Quadratic constitutive laws • Closed-form expression for the identification process 2 elasticity parameters and 3 damage parameters Peak stress Fracture energy Final band width

16. Numerical applications

17. Perforated plate

18. L-shaped panel 400 000 dof 25 h CPU

19. Conclusion • Continuum damage is adapted to predict crack propagation • Nonlocal formulation : damage gradient • Numerical demonstration on concrete structures • Robustness, reliability and reasonable efficiency are achieved • Appropriate choice of the constitutive law • Complying with a critical stress and a fracture energy • Up to ultimate fracture • Consistent with coarser models (Griffith, cohesive zone models) • A future transition to discontinuous models what for ? • Easier access to crack opening (leakage) • Modelling of large relative motions between crack lips

20. Preview : mesh adaptation prescribed v rigid inclusion ( u = v = 0 ) Final damaged zone Final mesh Salomé_Méca (Code_Aster, Homard)

21. Thank you

22. Mesh dependency, Feyel (2005) Jirasek & Zimmermann (2005) Why using continuum damage for crack path prediction ? • Following the element edges ? • Initial stiffness  CPU cost • Approximating a curve with fixed segments • Ensuring crack path continuity • In order to compute the correct dissipation • Difficult to ensure step by step 3D continuity • Crack orientation criterion • Possibly inacurate in 2D (fixed crack) • Theoretically questioned in 3D

23. Newton Closed-form Solution algorithm Energetic formulation non linear non convex constraints Decomposition – coordination

24. Trapezium specimen