Fracture mechanics. code_aster, salome_meca course material GNU FDL licence (

Size: px

Start display at page:

Download "Fracture mechanics. code_aster, salome_meca course material GNU FDL licence ("

Tamsyn Crawford
5 years ago
Views:

1 Fracture mechanics code_aster, salome_meca course material GNU FDL licence (

$Why fracture$ 2 - code_aster and

2 Why fracture mechanics? 2 - code_aster and salome_meca course material GNU FDL Licence

3 Fracture mechanics: objectives and generalities Sane structure Initiation Propagation Failure Where? What shape? When? Propagate or not? Speed? Path? Does it fail? 3 - code_aster and salome_meca course material GNU FDL Licence

4 Fracture mechanics: objectives and generalities Damage mechanics Fracture mechanics Sane structure Initiation Propagation Failure Where? What shape? When? Propagate or not? Speed? Path? Does it fail? 4 - code_aster and salome_meca course material GNU FDL Licence

5 Applications of fracture mechanics Brittle fracture Ductile fracture Fatigue propagation Design Operation Lifetime assessment Maintenance Justification Repair 5 - code_aster and salome_meca course material GNU FDL Licence

6 Applications of fracture mechanics Cracking of a dam Cracks in UK AGRs Cracks in EDF turbines 6 - code_aster and salome_meca course material GNU FDL Licence

7 Outline Main criteria in fracture mechanics Linear fracture mechanics in code_aster Non linear fracture mechanics References 7 - code_aster and salome_meca course material GNU FDL Licence

8 Outline Main criteria in fracture mechanics Linear fracture mechanics in code_aster Non linear fracture mechanics References 8 - code_aster and salome_meca course material GNU FDL Licence

9 Basis on LEFM: Vocabulary Crack : mater discontinuity FR EN code_aster 2D 3D Front Fond Front - Tip FOND_FISS Point Edge Lèvres Lips LEVRE _SUP _INF Edge Face 9 - code_aster and salome_meca course material GNU FDL Licence

10 Basis on LEFM: Cracking modes y x z u x = 0 u y 0 u z = 0 u x 0 u y = 0 u z = 0 u x = 0 u y = 0 u z code_aster and salome_meca course material GNU FDL Licence

11 Basis on LEFM: local axis y M z x n r M θ a t Global axis Crack local axis σ ~ r 0 K i σ, a r Singular stress f θ u ~ r 0 K i σ, a r g θ 11 - code_aster and salome_meca course material GNU FDL Licence

Stress intensity factor K K depends on: crack geometry structure geometry loading conditions General cases: 2 examples: K I = σ πa cos 2 α 2a K II = σ πa cos α sin

12 Stress intensity factor K K depends on: crack geometry structure geometry loading conditions General cases: 2 examples: K I = σ πa cos 2 α 2a K II = σ πa cos α sin α Based on analytical solution, approximated solution or FEM calculations 2a b K I = σ πa cos πa b code_aster and salome_meca course material GNU FDL Licence

Polynomial fit K I = πa σ 0 i 0 + σ 1 i 1 a t + σ 2 i 2 a t 2 + σ3 i 3 a t

13 Stress intensity factor K Codified approaches Influence coefficients a σ σ x = σ 0 + σ 1 x t + σ 2 x t 2 + σ 3 x t 3 + σ 4 x t 4 t R 0 a x Calcul Polynomial fit K I = πa σ 0 i 0 + σ 1 i 1 a t + σ 2 i 2 a t 2 + σ3 i 3 a t 3 + σ4 i 4 a t code_aster and salome_meca course material GNU FDL Licence

14 Stress intensity factors K Mode ~ r 0 f σ ij ~ r 0 f u i K I σ θθ r, 0 2π r E 2π 8 1 ν 2 u 2 r K II σ rθ r, 0 2π r E 2π 8 1 ν 2 u 1 r K III σ θz r, 0 2π r E 2π ν u 3 r 14 - code_aster and salome_meca course material GNU FDL Licence

15 Contour integral: Rice Characterization of stress singularity Induced from energy conservation Independent of the considered contour For a plane cracked solid subjected to a mixed-mode load (modes I et II): u i J = w e n 1 σ ij n j x i C 1 ds x 2 C 1 n ds x 1 With w e = σ: ε the elastic energy density code_aster and salome_meca course material GNU FDL Licence

16 Energy release rate: G (Griffith) Griffith s hypothesis Cracking energy is proportional to separated surface (material properties ) Total energy = Potential energy + Cracking energy Minimum total energy principle 2D example : E tot l = W l + 2γl E tot l + l = W l + l + 2γ l + l Minimum total energy principle: l E tot l + l < E tot l l + Δl W l + l W l < 2γ l 16 - code_aster and salome_meca course material GNU FDL Licence

17 Energy release rate: G (Griffith) F l F W l + l W l l W l + l W l l+δl G l+δl G U U 17 - code_aster and salome_meca course material GNU FDL Licence

18 Energy release rate: G (Griffith) Definition of G : variation of potential energy per virtual crack advance l l + dl G = dw dl 2D G = dw da 3D Potential energy Cracking energy 18 - code_aster and salome_meca course material GNU FDL Licence

19 G-theta method G = dw da Derivative difficult to compute directly G-θ method θ F = h t Γ θ : x x + h θ x F F Solution of variational equation: Γ 0 Gθ t ds = G θ = dw A 19 - code_aster and salome_meca course material GNU FDL Licence

20 G-theta method: implementation In 2D: θ 0 θ 0 R inf R sup θ 0 0 R inf R sup r In 3D: Discretisation of θ and G along front: s = 0 γ 0 s Γ 0 s = l s = 0 γ 1 s Γ 0 s = l s = 0 γ 2 s Γ 0 s = l φ 1 s s = 0 φ 2 s φ 3 s Γ 0 s = l 20 - code_aster and salome_meca course material GNU FDL Licence

21 Relation between parameters (Irwin) Linear elasticity G = 1 E K I 2 + K II ν E K III 2 Plane stress G = 1 ν E K I 2 + K II ν E K III 2 Plane strain, 3D G = J Plane elasticity (plane strain + plane stress) 21 - code_aster and salome_meca course material GNU FDL Licence

22 Material criterion Some material criterion Propagation if K I K Ic G G c or J Ic Concrete Hardened Steel Aluminium alloy Titanium Alloy Polymer K Ic 1MPa m K Ic 3MPa m K Ic 30MPa m K Ic 100MPa m K Ic 120MPa m! Be careful with FEM units! K s dimension: MPa m Glass, ceramics Aluminium Steel Pure metals G c 2J. m 2 G c 10kJ. m 2 G c 100kJ. m 2 G c 1MJ. m 2 G s dimension: J/m² or N/m 22 - code_aster and salome_meca course material GNU FDL Licence

Aside: fatigue s law (Paris) Principle of fatigue: Crack propagation by repetition of a weak load Paris fatigue propagation law da dn = c Km (c, m material parameters) Stage A : DK weak, slow or

23 Aside: fatigue s law (Paris) Principle of fatigue: Crack propagation by repetition of a weak load Paris fatigue propagation law da dn = c Km (c, m material parameters) Stage A : DK weak, slow or non propagation Stage B : DK moderate, propagation with a constant velocity Stage C : DK high, sudden failure See POST_RUPTURE operator 23 - code_aster and salome_meca course material GNU FDL Licence

24 Outline Main criteria in fracture mechanics Linear fracture mechanics in code_aster Non linear fracture mechanics References 24 - code_aster and salome_meca course material GNU FDL Licence

25 Fracture mechanics problem in code_aster Step 1 Meshing cracked structures Type of calculation: Thermo-Elastic (linear or non linear) Thermo-elastoplastic : See the end of the presentation Residual stresses (linear or non linear elasticity) Step 2 Thermo-mechanical computation Step 3 Crack definition Step 4 Computation of fracture mechanics parameters 25 - code_aster and salome_meca course material GNU FDL Licence

26 Crack definition in code_aster 26 - code_aster and salome_meca course material GNU FDL Licence

27 Crack definition in code_aster OUVERT In 2D In 3D FERME 27 - code_aster and salome_meca course material GNU FDL Licence

28 Crack definition in code_aster CONFIG_INIT NORMALE crack crack COLLEE DECOLLE (α<5 ) SYME No Yes LEVRE_SUP/INF LEVRE_SUP LEVRE_INF 28 - code_aster and salome_meca course material GNU FDL Licence

29 Displacement jump Displacement Jump Extrapolation Method (1) Operator POST_K1_K2_K3 N Analytical model (r 0): u 2 n ABSC_CURV_MAXI K 1 = E 2π 8 1 ν 2 u 2 r Extraction of node displacements along the crack front (normal direction) 5,0E-07 4,5E-07 4,0E-07 3,5E-07 3,0E-07 2,5E-07 2,0E-07 1,5E-07 1,0E-07 5,0E-08 0,0E+00 Computed displacement jump function K.sqrt(r) 0E+00 1E-05 2E-05 3E-05 4E-05 5E-05 6E-05 Curvilinear co-ordinate u 2 ~ r ABSC_CURV_MAXI 29 - code_aster and salome_meca course material GNU FDL Licence

30 Displacement Jump Extrapolation Method (2) 3 methods to extrapolate the displacement: Method 1 With quarter-node elements Without quarternode elements u 2 2 r = 64 1 ν2 2 2πE 2 K 1 2 One value of K for each consecutive node couple Maximal value 30 - code_aster and salome_meca course material GNU FDL Licence

31 Displacement Jump Extrapolation Method (3) Method 2: u 2 2 = r Without quarter-node elements With quarter-node elements 64 1 ν2 2 2πE 2 K 1 2 Method 3 Minimisation by least square error of J(k): J k = 1 2 max _absc U r k r 2 dr 0 One value of K Maximal slope Printed results: - in a table (resu file): only the max values of method 1, - in a table (resu file): an estimation of the relative difference between the 3 methods, - in the mess file (if INFO=2): computing details 31 - code_aster and salome_meca course material GNU FDL Licence

32 Usage of the POST_K1_K2_K3 operator FEM X-FEM From mechanical calculation Use different material from the one in Result! Type of mesh of the crack REGLE LIBRE Maximal distance of calculation 32 - code_aster and salome_meca course material GNU FDL Licence

33 POST_K1_K2_K3: advices Limited to plane or quasi-planar cracks (possibility to define only one normal) Choice of ABSC_CURV_MAXI: between 3 to 5 elements Precision of computation: error < 10 % for validation tests Precision is better if crack mesh is REGLE Verifications: Compare with different ABSC_CURV_MAXI Check errk1, errk2 and errk3 < 1% 33 - code_aster and salome_meca course material GNU FDL Licence

34 Operator CALC_G 34 - code_aster and salome_meca course material GNU FDL Licence

35 Operator CALC_G Example: NB_POINT_FOND=7 FEM X-FEM LAGRANGE only If crack has several fronts several CALC_G R_SUP R_INF 35 - code_aster and salome_meca course material GNU FDL Licence

36 Operator CALC_G Result from mechanical calculation 36 - code_aster and salome_meca course material GNU FDL Licence

37 Operator CALC_G Option = CALC_G In 3D/2D plane the local value G(s) is in J/m² in 2D-axisymetric, G is the energy by unit of radian. In order to obtain a local value of G, we need to divide by its radius R. G = 1 R Γ θ Option = CALC_K_G Also compute stress intensity factors Use of mathematical properties of G to separate contributions of K1, K2 and K3 Option = CALC_G_GLOB DO NOT USE! 37 - code_aster and salome_meca course material GNU FDL Licence

38 Operator CALC_G Smoothing options in 3D Default OR LISSAGE_THETA=LEGENDRE LISSAGE_G=LEGENDRE DEGRE=N LISSAGE_THETA=LAGRANGE LISSAGE_G=LAGRANGE THETA: [NB_POINT_FOND=N] 38 - code_aster and salome_meca course material GNU FDL Licence

39 Operator CALC_G Choice of smoothing in 3D : need to use different smoothing methods and compare the obtained results! LAGRANGE: no smoothing, oscillations can occur Reference (analytical solution) LAGRANGE with NB_POINT_FOND=20 (33): decrease of oscillations LEGENDRE: smooth results Results at the extremities of the crack front should be used with care Good if G is polynomial Energy release rate for an elliptical crack (test case sslv154a) Quadratic mesh with Barsoum elements 39 - code_aster and salome_meca course material GNU FDL Licence

40 Operator CALC_G To calculate only at given steps 40 - code_aster and salome_meca course material GNU FDL Licence

41 Operator CALC_G: general advice R_INF > 0 (imprecise computational results at crack front) ~ 2 elements R_SUP not too large (for example 5 or 6 elements) Use OPTION= CALC_K_G If LISSAGE LAGRANGE : If N front nodes > 25 use NB_POINT_FOND NB_POINT_FOND = N/5 or N/10 5 NB_POINT_FOND 50 Verifications: Compare with different R_INF and R_SUP Compare between LISSAGE LEGENDRE and LAGRANGE Compare different values of NB_POINT_FOND 41 - code_aster and salome_meca course material GNU FDL Licence

42 General advice Use a mesh with a tore around the crack front Not mandatory Results will be more regular if the R_SUP radius of the tore Use BlocFissure plugin in salome_meca to insert a crack in a mesh with a tore Computation on a structured mesh Computation on an unstructured mesh 42 - code_aster and salome_meca course material GNU FDL Licence

43 General advice General advices for meshed cracks: Element type : prefer quadratic elements with Barsoum << << crack crack crack Linear Quadratic Quadratic + Quarter nodes See MODI_MAILLAGE with OPTION= NŒUD_QUART POST_K1_K2_K3 CALC_G advice advice 3D if free or structured mesh 3D if structured mesh 43 - code_aster and salome_meca course material GNU FDL Licence

44 Outline Main criteria in fracture mechanics Linear fracture mechanics in code_aster Non linear fracture mechanics References 44 - code_aster and salome_meca course material GNU FDL Licence

45 Non Linear fracture mechanics Accounting for confined plasticity by plastic correction (RCC-M ZG5110 appendix) Replace crack length a by a virtual crack a + r y (Irwin s approach) with: r y = 1 6π K I σ s 2 σ s : yield stress r y : plastic zone size Compute corrected stress intensity factors K cp = αk I a + r y a With: α = 1 α = 1 + 0,15 r y 0,05 t a 0,035 t a α = 1,6 r y 0,05 t a 2 0,05 t a < r y 0,12 t a r y > 0,12 t a 45 - code_aster and salome_meca course material GNU FDL Licence

46 Non Linear fracture mechanics Accounting for confined plasticity by 3D approach: STAT_NON_LINE CALC_G RELATION ELAS_VMIS_LINE ELAS_VMIS_LINE ELAS_VMIS_PUIS ELAS_VMIS_PUIS ELAS_VMIS_TRAC ELAS_VMIS_TRAC VMIS_ISOT_LINE ELAS_VMIS_LINE VMIS_ISOT_PUIS ELAS_VMIS_PUIS VMIS_ISOT_TRAC ELAS_VMIS_TRAC!! Loading must be radial and monotonous See DERA_ELGA in CALC_CHAMP Compare VMIS_ISOT_ and ELAS_VMIS_ in STAT_NON_LINE Compare by activating or not CALCUL_CONTRAINTE= NON in CALC_G 46 - code_aster and salome_meca course material GNU FDL Licence

47 Outline Main criteria in fracture mechanics Linear fracture mechanics in code_aster Non linear fracture mechanics References 47 - code_aster and salome_meca course material GNU FDL Licence

48 References General user documentation Application domains of operators in fracture mechanics of code_aster and advices for users [U ] Notice for utilisation of cohesive zone models [U ] Realisation for a computation of prediction for cleavage fracture [U ] Documentation of operators Operators DEFI_FOND_FISS [U ], CALC_G [U ] et POST_K1_K2_K3 [U ] Reference documentation Computation of stress intensity factors by Displacement Jump Extrapolation Method [R ] Computation of coefficients of stress intensity in plane linear thermoelasticity [R ] Energy release rate in linear thermo-elasticity [R ] and non-linear thermo-elasticity [R ] Elastic energy release rate en thermo-elasticity-plasticity by Gp approach [R ] Other references : Plasticité et Rupture - Jean-Jacques Marigo course Formation ITECH Mécanique de la rupture 48 - code_aster and salome_meca course material GNU FDL Licence

49 End of presentation Is something missing or unclear in this document? Or feeling happy to have read such a clear tutorial? Please, we welcome any feedbacks about code_aster training materials. Do not hesitate to share with us your comments on the code_aster forum dedicated thread code_aster and salome_meca course material GNU FDL Licence

50 Appendix for TP forma05b Best results with quadratic and Barsoum CALC_G : Rinf 2 elements Rsup 5 elements POST_K1_K2_K3 : ABSC_CURV_MAXI 5 elements crack crack Quadratic Barsoum R_SUP ABSC_CURV_MAXI CALC_G or POST_K1_K2_K3 in 2D? R_INF Similar results Both very accurate K and G independent of parameters 50 - code_aster and salome_meca course material GNU FDL Licence

51 Appendix for TP forma07a DEFI_FOND_FISS in 3D Define crack front with GROUP_MA FOND_FISS=_F( GROUP_MA='LFF', GROUP_NO_ORIG='NFF1', GROUP_NO_EXTR='NFF2'), Define LEVRE_SUP and LEVRE_INF (optional with CALC_G) CALC_G or POST_K1_K2_K3? Some advice POST_K1_K2_K3 : 3D if structured mesh CALC_G : 3D if free mesh Use OPTION= CALC_K_G Accuracy : LISSAGE in CALC_G LAGRANGE+NB_POINT_FOND Better than LEGENDRE Better than LAGRANGE Computational time : LEGENDRE Faster than LAGRANGE+NB_POINT_FOND Faster than LAGRANGE Oscillations of G and K : LEGENDRE Better than LAGRANGE+NB_POINT_FOND Better than LAGRANGE 51 - code_aster and salome_meca course material GNU FDL Licence

Continuation methods for non-linear analysis

Continuation methods for non-linear analysis FR : Méthodes de pilotage du chargement Code_Aster, Salome-Meca course material GNU FDL licence (http://www.gnu.org/copyleft/fdl.html) Outline Definition of