NUMERICAL IMPLEMENTATION OF THE ARBITRARY CRACK FRONT FOR THREE DIMENSIONAL PROBLEMS

Transcription

1 NUMERICAL IMPLEMENTATION OF THE ARBITRARY CRACK FRONT FOR THREE DIMENSIONAL PROBLEMS EL KABIR Soliman 1, MOUTOU PITTI Rostand 2,3, DUBOIS Frederic 1, LAPUSTA Yuri 4, RECHO Naman 2,5, 1 Université de Limoges, Centre du génie Civil, GC2D, Egletons, France 2 Université Clermont Auvergne, CNRS, Institut Pascal, F Clermont-Ferrand, France 3 CENAREST, IRT, BP 14070, Libreville, Gabon 4 SIGMA, Institut Pascal, Aubière, France 5 EPF, Engineering school, 2 rue Fernand Sastre, Troyes, France 24/11/2017 Club Cast3M 2017 Paris 24 novembre 2017

2 Scientific context q Predicting the behavior of structures under mixed mode loading q Developing specific tools for three-dimensional configurations q Consider thickness effect under variable environments q Need for applications related to the inspection and diagnosis of structures Climatic loading Moisture variation Creep loading Opening mode In-plane Shear mode Out-of-plane shear mode Mode I Mode II Modes of fracture Mode III 2

3 Plan q Scientific context q Part I : three dimensional contour integral generalizations Analytical formulation (J "# and G "# % -integral) Numerical implementation q Part II : mixed mode three dimensional contour integral Analytical formulation (M "# % -integral) Mixed mode Numerical implementation q Conclusions and outlooks 3

4 Rice s integral J (# = W. n. σ 12. n 2. u 1,.. dγ 8 Integration domain size for 2D 3D Closed volume and surfaces integration domains The main advance of this form is the presence of an arbitrary crack front line enclosed by a 3D surface 4

5 The J-integral formulation is based on the Noether's theorem application (Noether 1918) : δl = ; ; δw. dv. dt? > = 0 A Gauss-Ostrogradski transformation allows us writing the Lagrangian's invariance in the form: ; W. n A σ 12. n 2. u 1,A. ds + ; σ 12. ε 12 W,A,A. dv C? = 0 the J "# -integral is defined as below: Stationary crack Crack lips pressure Crack propagation

6 1 2 3 θ = 0 θ = 0 S ST> θ = 0 S 1U c M θ = c c. dw dw θ = 0 on S 8IJK, θ = θ = 0 Definition of θ around a close crown c c. dw on C S 8 IJK and θ = 0 on S 8QR The average energy release rate can be calculated with an integration of along the crack front line divided by the crack width 6

7 The finite element implementation is based on a Double Cantilever Beam loaded in an open mode. 15mm Crack lips Fracture surface 80mm Crack front line uz = Uimp z r y r x r Specimen DCB Material is characterized by : - Elastic isotropic - Thickness =20 mm - Linear crack front line - E = 210MPa, ν = 0,3 Cylindrical integration domain Crack front line R c 7

8 Entrée : Opérateurs Option générale OPTI, DIME, ELEM, ECHO,.. Maillage 3D Points, DROI, SURF, VOLU.. Construction du champ θ MODE, Thermique.. modèle mécanique MODE, MATE, RIGI,.. Chargement + CL BLOQ, DEPI, FORC, Modèle mathématique GRAD, WORK, INTG,.. Sortie : G % "# Post-traitement 8

9 Rc = 1mm R = 22mm c We can shows the variations of the energy release rate versus R^: Average energy release rate (J/ m²) 3,10E+04 3,00E+04 2,90E+04 2,80E+04 2,70E+04 2,60E+04 2,50E Rc (mm) Numerical results validate the non-dependence of the integration domain with an average value of 30.3kJ/m² 9

10 Surface integration domains for the Bui s integral : J _` = W. n. σ 12. n 2. u 1,.. dγ 8 a _(8) ab c σ 1". u 1,.. da(γf J (# ( G) n r A( G) dl x 3 = 0 Integration domains Rc = 22mm Integration domain size for 2D model 10

11 Surface integration domains for the Bui s integral : J _` = W. n. σ 12. n 2. u 1,.. dγ 8 a _(8) ab c σ 1". u 1,.. da(γf J (# 5,0E+ 04 J 3D J 2D Amestoy's corr ection J A Energy release rate (J/ m²) 4,5E+ 04 4,0E+ 04 3,5E+ 04 3,0E+ 04 2,5E+ 04 2,0E+ 04 1,5E+ 04 1,0E Rc (mm) Comparison between J 2D and J 3D approaches 11

12 Semi cylinder surrounding the crack front line 1 d w Crack front line R c d w Localized integration domain c p C w 0 dw = 10mm dw = 8mm dw = 6mm dw = 4mm dw = 2mm dw = 1mm Average value 3,20E+04 3,15E+04 Energy release rate (J/ m²) 3,10E+04 3,05E+04 3,00E+04 2,95E+04 2,90E P osition on the crack front line (mm) 12

13 Crack front line Layer Average crack length (mm) J 3D J 2D JA

14 Using CAD software we can define our specimen as : (b) (a) (c) DCBVI specimen (a), two dimensional crack tip (b) and three-dimensional elliptical crack front (c) 14

15 Step I : Geometry Step II : Meshing Step III : FE analysis - DCB specimen - Crack front - Couronne champ teta - fichier.stp - GMSH - Identification lignes, surfaces, volumes Contrôle maillage Physical Group.UNV LIRE UNV CL + Chargement + Champ θ Traitement incohérences (ELIM,..) Modèle mathématique G % "# Post traitement : ü Calcul de G à partir du l intégrale G_teta_3D ü Vérification de l indépendance du chemin d intégration ü Tracer l évolution de K1 le long du cfl ü 15

16 DCB Mesh with theta field : Typical FE meshes of the ½ half DCBVI specimen 16

17 Maillage 30mm Theta fields definition (30mm) 17

18 Radial loading (700N) z y BLOQ x Boundary conditions La deformée(castem) 18

19 30 mm 25 mm 20 mm 15 mm 2 mm Vue filiaire des segments 19

20 Plan q Scientific context q Part I : three dimensional contour integral generalizations Analytical formulation (J "# and G "# % -integral) Numerical implementation q Part II : mixed mode three dimensional contour integral Analytical formulation (M "# % -integral) Mixed mode Numerical implementation q Conclusions and outlooks 20

21 The M-integral formulation is based on the Noether's theorem application : δl = ; ; δw. dv. dt = 0? > A Gauss-Ostrogradski transformation allows us writing the Lagrangian's invariance in the form: W. n A ij. n it 2. u C 1,A Q,k ij il Q,k. n 2. v 1,A. ds +? ij it Q,n. δu 1,A,o + ij il Q,n. δv 1,A,o W,A(u) W,A(vp. dv = 0 M "# -integral

22 Integration domain size for 2D M % "# -integral 3D Closed volume and surfaces integration domains M % "# = 1 2 ; P A2. θ A,2. dv 1? 1 2 ; σ 12 l. u 1,A + σ 12 T. v 1,A. n 2. θ A. ds 2 C qr ; σ 12 l. ε 12 T,A + σ 12 T. ε 12 l,a σ 12 l. ε 12 T,A + σ 12 T. ε 12 l,a. θ A. dv 3? s t 22

23 The finite element implementation is based on a Double Cantilever Beam loaded in an open mode. 15mm Crack lips Fracture surface Finite Element Mesh Crack front line 80mm Cylindrical integration domain Specimen DCB 23

24 Rc = 1mm R = 22mm c We can shows the variations of the energy release rate versus R^: 60 Opening mode energy release rate G K(G(0 )) KI(M(0 )) KI(M(90 )) KI(M(45 )) Titre de l'axe Numerical results validate the non-dependence of the integration domain with an average value. 24

25 Shear mode out-of-plan shear mode 14 0,12 energy release rate G KII(M(0 )) K(G) KII(M(90 )) KII(M(45 )) energy release rate G 0,10 0,08 0,06 0,04 0,02 KIII(M(0 )) KIII(M(90 )) KIII(M(45 )) Rext 0, Rext Numerical results validate the non-dependence of the integration domain with an average value. 25

26 Conclusions and outlook q Numerical development of the contour integral concept for 3D problems q The generalization toward its G "# % and M "# % implementation form q Several numerical applications are proposed q Toward implementation three dimensional mixed mode crack problem qelliptical crack front q Numerical development of the contour integral concept for 3D problems. q Generalization of the local mechanical fields q New integral taking into account climatic effect q 3D fractures coupling hygrothermal with mechanics effects q Coupled 3D fracture mechanic probabilistic methodology q Confrontation FE / Experimental results 26