Fissuration en milieux isotrope et orthotrope via les intégrales invariantes: prise en compte des effets environnementaux

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1 G E M H Fissuration en milieux isotrope et orthotrope via les intégrales invariantes: prise en compte des effets environnementaux R. Moutou Pitti 1,2, H. Riahi 1,2, F. Dubois 3, N. Angellier 3, A. Châteauneuf 1,2 1 Clermont Université, Université Blaise Pascal, Institut Pascal, CLERMONT-FERRAND, France 2 CNRS, Institut Pascal, AUBIERE, France 3 University of Limoges, Heterogeneous Material research Group, EGLETONS, France This work is sponsored by French National Research Council through the ANR JCJC Project CLIMBOIS N ANR-13-JS and Labelled by ViaMeca rostand.moutou_pitti@univ-bpclermont.fr Club Cast3M Novembre 2014, Paris, France 1

2 Outline Path independent integrals formulation 1 Path independent integrals formulation (J, M, A) Conclusions and perspectives 2

3 x 2 Path independent integrals formulation Integration path Intégrale J Intégrale M Intégrales T et A Strain energy deformation dl x 1 Cracked body Ω p j1 : Energy momentum tensor Noether s theorem x 2 (1) Virtual displacement (2) Boundary conditions on crack lips ds x 1 Integration on surface V enclosed by (2) Rice s integral =0 =0 3

4 Intégrale J Intégrale M Intégrales T et A x 2 Integration path C 0 x 2 C i dl x 1 Cracked body Ω C e ds x 1 - C 0 : continuously varying from (1,0) to (0,0) - C i : θ=(1,0) Integration on V enclosed by (1) - C e : θ=(0,0) (1) and Green-Ostrogradsky theorem s 4

5 Intégrale J Intégrale M Intégrales T et A Energy release rate Fracture modes separation plan σ plan ɛ M-integral Noether theorem s Bilinear expression of the strain energy density Real fields(fem) Virtual fields (auxiliary problem) M-integral formulation Relation between M-integral and SIF K I et K II 5

6 Intégrale J Intégrale M Intégrales T et A T and A integrales Temperature variation Noether theorem s Bilinear expression of the strain energy density Real fields(fem) Virtual fields (auxiliary problem) T-integral formulation A-integral formulation A 1 : Classical term A 2 : temperature variation effect 6

7 Intégrale J Intégrale M Intégrales T et A Improvement of the A-integral formulation Applied forces on the crack lips x 2 O x 1 B.C. on the crack lips - A 1 A 2 et B 2 B 1 : - A 1 A 2 : - B 2 B 1 : Energy momentum tensor Noether theorem s A-integral formulation A 1 : Classical term A 2 : temperature variation effect A 3 : effect of pressure applied on the crack lips 7

8 Intégrale J Intégrale M Intégrales T et A Improvement of the A-integral formulation Crack growth process fissure x 2 Δa crack growth no crack growth x 1 A-integral formulation A 1 : Classical term A 2 : temperature variation effect A 3 : effect of pressure applied on the crack lips A 4 : effect of crack growth 8

9 Mode I Path independent integrals formulation Mode I Mixed mode (I and II) Axisymmetric Pressure on crack lips Thermal load Material properties E = dan/mm 2 ν = 0,3 c c 1 2 c c 3 4 c c 5 6 c7 Applied load σ = 1 dan/mm 2 Geometry parameters 2L = 400 mm 2H = 1200 mm 2a = 200 mm Rectangular plate with central crack subjected to a far-field tensile stress: (a) Geometry and loads, (b) Finite elements mesh, (c) Deformed shape Results for plan strain condition (Rooke et al. 76; Wilson, IJF 79) Path independence verification of (a) the energy release rate GI, (b) the stress intensity factor KI 9

10 Mixed mode (I and II) Path independent integrals formulation Mode I Mixed mode (I and II) Axisymmetric Pressure on crack lips Thermal load Rectangular plate with central inclined crack subjected to a tensile stress (a) Geometry and loads, (b) Finite elements mesh, (c) Deformed shape Results for plan stress condition c c 1 2 c c 3 4 c c 5 6 c7 10

11 Axisymetric problem P Path independent integrals formulation Mode I Mixed mode (I and II) Axisymmetric Pressure on crack lips Thermal load R i t a a c c 1 2 c c 3 4 c c 5 6 c7 P Material properties E = Pa ν = 0,3 Applied load P = N Geometry parameters R i = 1 m t = 0,1 m a = 0,05 m Analytical FEM 11

12 Thermal load Temperature field Path independent integrals formulation Mode I Mixed mode (I and II) Axisymmetric Pressure on crack lips Thermal load c c 1 2 c c 3 4 c c 5 6 c7 Applied load T0 = 100 C Material properties E = dan/mm 2 ν = 0,3 α = C -1 Geometry parameters 2H = 800 mm 2W = 200 mm a = 100 mm Rectangular plate with edge crack subjected to temperature field (a) Geometry, (b) Finite elements mesh, (c) Thermal load, (d) Deformed shape Computing in plan strain Path independence verification of (a) the energy release rate GI, (b) the stress intensity factor K I 12

13 Pressure on the crack lips Mode I Mixed mode (I and II) Axisymmetric Pressure on crack lips Thermal load Applied load σ = 1 dan/mm 2 c c 1 2 c c 3 4 c c 5 6 c7 Material properties E = dan/mm 2 ν = 0,3 Rectangular plate with central crack subjected to a tensile stress (a) Geometry and loads, (b) Finite elements mesh, (c) Deformed shape Geometry parameters 2b = 400 mm 2h = 1200 mm 2a = 200 mm Results for plan strain condition Path independence verification of (a) the energy release rate G I, (b) the stress intensity factor K I 13

14 Mode I Mixed mode (I and II) Axisymmetric Pressure on crack lips Thermal load Effect of thermal load and pressure on the crack lips (a) Temperature field distribution, (b) (b) Path independence verification of the stress intensity factor K I T 0 = 0 C à T 1 = 30 C 14

15 Orthotropic fields Virtual fields CTS specimen Independence domain x 2 Plan stress condition Temperature variation E 2 E 1 x 1 Plan strain condition Hyp1 : γ = f(e 1,v 12,α 1 ) A 1 : Classical term A 2 : temperature variation effect Hyp2 : β = g(e 1,v 12,α 1 ) A 3 : effect of pressure applied on the crack lips A 4 : effect of crack growth 15

16 Orthotropic fields Virtual fields CTS specimen Independence domain Virtual fields computation Anisotropic material Orthotropic material c 16 = c 26 = 0 A B x 2 Compliance matrix x 2 Crack φ x 1 [C] = Crack φ x 1 R crack R orthotropy R crack R orthotropy Case I Case II Case III Case VI 16

17 Orthotropic fields Virtual fields CTS specimen Independence domain Virtual fields computation Virtual displacement field Opening mode Shear mode 17

18 Orthotropic fields Virtual fields CTS specimen Independence domain Virtual fields computation Virtual stress field Opening mode Shear mode 18

19 Orthotropic fields Virtual fields CTS specimen Independence domain Validation on CTS (Compact Tension Shear) specimen Arcan Wood CTS specimen geometry Mesh of the CTS specimen E 1 = 600MPa Parameters E 2 =15000MPa G 12 = 700MPa 19

20 Orthotropic fields Virtual fields CTS specimen Independence domain Numerical results for stress intensity factors without thermal load Opening mode Shear mode 20

21 Orthotropic fields Virtual fields CTS specimen Independence domain Numerical results for stress intensity factors with thermal load Path independence verification of stress intensity factor for ΔT = 10 C : (a) Opening mode K I, (b) (b) Shear mode K II Path independence verification of stress intensity factor for ΔT = -10 C : (a) Opening mode K I, (b) Shear mode K II 21

22 Analytical Formulation Fracture parameters Incremental formulation Viscoelastic SIF factors Expérience de fluage/formulation intégrale σ(t) σ 0 t 0 t ε(t) ε 0 t 0 Effet de fluage t Tenseur de fluage Intégrale de BOLTZMANN Arcan Modèle rhéologique de Kelvin-Voigt Wood Aq ( p) = ò - 1 V 2 Formulation de l intégrale A pour le comportement viscoélastique é ë ( p) s v ( ij,k u p) i - ( p) s u ( ij v p) ( i,k -gdt, j v p) k - Y k ( ( ) -gdt ( v p) i,k - Y k, j ) ù û q k, jdv Terme classique A 1 Terme A 2 : chargement thermique 22

23 Analytical Formulation Fracture parameters Incremental formulation Viscoelastic SIF factors Paramètres de rupture dans le cas viscoélastique Facteur d inténsité de contraintes Mode I Mode II u K I ( p) = u K II Aq ( p) v ( K p) I =1; v ( ( K p) II = 2) ( p) C 1 ( ( p) =1) ( p) = Aq ( p) v K I ( p) = 0; v K II C 2 ( p) and Complaisances viscoélastiques Taux de restitution d énergie viscoélastique 1 1 u ( p) 2 u ( p) K K ( p) 2 ( p) ( p) I G v G v C1 C 8 1 ( p) 2 2 ( Gv G v and Gv G v p p ( p) 2 p) p II with 8 0,1,... N 2 23

24 Analytical Formulation Fracture parameters Incremental formulation Viscoelastic SIF factors Formulation incrémentale en fluage Décomposition du tenseur de déformation Histoire du chargement Matrice des matérieux Equation d équilibre 24

25 Analytical Formulation Fracture parameters Incremental formulation Viscoelastic SIF factors Numerical results for stress intensity factors Réponse différée Réponse instantanée Opening mode Shear mode 25

26 1. Improve the analytical formulation of T and A integrales a. Temperature variation effect b. Pressure on crack lips c. Crack growth process 2. Generalization for orthotropic material 3. Generalization for viscoelastic material 3. Implementation in FE software a. Accurate results b. Integration domain independency A. Moisture variation and mechanosorptive law B. Viscoelastic crack growth using mixed mode process zone C. Reliability assessment (uncertainties) 26

27 G E M H Fissuration en milieux isotrope et orthotrope via les intégrales invariantes: prise en compte des effets environnementaux R. Moutou Pitti 1,2, H. Riahi 1,2, F. Dubois 3, N. Angelier 3, A. Châteauneuf 1,2 1 Clermont Université, Université Blaise Pascal, Institut Pascal, CLERMONT-FERRAND, France 2 CNRS, Institut Pascal, AUBIERE, France 3 University of Limoges, Heterogeneous Material research Group, EGLETONS, France This work is sponsored by French National Research Council through the ANR JCJC Project CLIMBOIS N ANR-13-JS and Labelled by ViaMeca rostand.moutou_pitti@univ-bpclermont.fr Club Cast3M Novembre 2014, Paris, France 27