14 Fracture Mechanics

March 29, 2018 | Author: Alexander Narváez | Category: Fracture Mechanics, Fracture, Plasticity (Physics), Fatigue (Material), Elasticity (Physics)


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Code_Aster, Salome-Meca course materialGNU FDL licence (http://www.gnu.org/copyleft/fdl.html) Fracture mechanics 2 - Code_Aster and Salome-Meca course material GNU FDL Licence Outline Introduction to fracture mechanics Objectives Crack vocabulary Main criteria in fracture mechanics Linear fracture mechanics in Code_Aster Computation of K by Displacement Jump Extrapolation method Computation of G by G-theta method Accounting for plasticity : limit of classical methods References 3 - Code_Aster and Salome-Meca course material GNU FDL Licence Fracture mechanics: objectives and generalities Objectives of fracture mechanics: determine the speed of propagation of an existing crack and its shape change. Under given loading conditions and boundary conditions, is the crack able to propagate? if yes, at what propagation rate? The prediction of the crack initiation is not a focus of fracture mechanics (for this application, damage mechanics may be more appropriate) Applications: Design (computation of fatigue life) Safety (for existing defaults) 4 - Code_Aster and Salome-Meca course material GNU FDL Licence Cracking in aeronautical industry Ductile failure in fatigue of fuselage shells Contribution of fracture mechanics: better estimation of service life of structures 5 - Code_Aster and Salome-Meca course material GNU FDL Licence Cracking in civil engineering Cracking on a surface of a dam Contribution of fracture mechanics: assessment of crack and repairing 6 - Code_Aster and Salome-Meca course material GNU FDL Licence Vocabulary for a mechanical problem with crack Crack : macroscopic geometrical discontinuities of matter Crack front : zone where matter sees its continuity (dimension N-2) Crack faces : parts of crack on which discontinuity occurs (dimension N-1) In 2 dimensions In 3 dimensions Superior crack face Inferior crack face Crack front (or crack tip) Cube with a circular crack Crack face Crack front 7 - Code_Aster and Salome-Meca course material GNU FDL Licence Crack in elastic solids 3 cracking modes [ ] [ ] [ ] [ ] [ ] [ ] ¦ ¹ ¦ ´ ¦ = ≠ = 0 0 0 z y x u u u [ ] [ ] [ ] [ ] [ ] [ ] ¦ ¹ ¦ ´ ¦ = = ≠ 0 0 0 z y x u u u [ ] [ ] [ ] [ ] [ ] [ ] ¦ ¹ ¦ ´ ¦ ≠ = = 0 0 0 z y x u u u x y z Singular stress (crack = geometrical singularity) ( ) ( ) [ ] ( ) ( ) [ ] ¦ ¦ ¹ ¦ ¦ ´ ¦ ∞ → ∞ → θ θ g r a K f r a K i r i r , , ~ ~ 0 0 σ u σ σ with polar coordinates With respect to crack front, and the crack size. Von-Mises stress at the crack front r x y q ( ) θ , r a 8 - Code_Aster and Salome-Meca course material GNU FDL Licence Expressions of displacement and stress fields 2D-plane 2D- antiplane Displacement field: Stress field: Displacement field: ( ) 4(1 ) sin 2 2 z III r u K E ν θ π + = sin 2 2 cos 2 2 III xz III yz K r K r θ σ π θ σ π ¦ | | = − ¦ | ¦ \ ¹ ´ | | ¦ = | ¦ \ ¹ ¹ Stress field: 9 - Code_Aster and Salome-Meca course material GNU FDL Licence Criteria for crack propagation in LEFM Stress intensity factors (Westergaard): local criterion Contour integral (Rice): global criterion Energy release rate (Griffith): global criterion Relation between parameters Aside: Fatigue (Paris’s law) 10 - Code_Aster and Salome-Meca course material GNU FDL Licence Stress intensity factors K K ’s dimension: MPa√ √√ √m Characterisation of stress intensity factors ( ) ( ) ( ) [ ] [ ] | | ¹ | \ | − = = → → 2 2 0 0 2 1 8 lim 2 0 , lim u r E r r K r yy r I π ν π σ ( ) ( ) ( ) [ ] [ ] | | ¹ | \ | − = = → → 1 2 0 0 2 1 8 lim 2 0 , lim u r E r r K r xy r II π ν π σ ( ) ( ) ( ) [ ] [ ] | | ¹ | \ | + = = → → 3 0 0 2 1 8 lim 2 0 , lim u r E r r K r yz r III π ν π σ 11 - Code_Aster and Salome-Meca course material GNU FDL Licence R t Stress intensity factor K: some analytical solutions K depends on: - crack geometry - structure geometry - loading conditions Codified approaches: RCC-M and RSE-M for pipes Example: semi-elliptical crack with a/b ratio = 0.3 in a pipe with R/t ratio = 10 with 2 classical examples 2a α ∞ σ α α π σ α π σ sin cos cos 2 a K a K II I ∞ ∞ = = 2a b ∞ σ 2 1 cos − ∞ | ¹ | \ | = b a a K I π π σ | | ¹ | \ | | ¹ | \ | + | ¹ | \ | + | ¹ | \ | + | ¹ | \ | + = 4 4 4 3 3 3 2 2 2 1 1 0 0 t a i t a i t a i t a i i a K I σ σ σ σ σ π ( ) 4 4 3 3 2 2 1 0 | ¹ | \ | + | ¹ | \ | + | ¹ | \ | + | ¹ | \ | + = t x t x t x t x x σ σ σ σ σ σ 12 - Code_Aster and Salome-Meca course material GNU FDL Licence 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): With the elastic energy density. 1 x 2 x n ds 1 C ∫ | | ¹ | \ | ∂ ∂ − = 1 ds 1 1 C i j ij e x u n n w J σ ε σ : = e w 13 - Code_Aster and Salome-Meca course material GNU FDL Licence 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 l l + dl ? ( ) ( ) ( ) ( ) ( ) dl l dl l P dl l E l l P l E tot tot + + + = + + = γ γ 2 , 2 ( ) ( ) ( ) ( ) γ 2 < − + ⇔ < + dl l P dl l P l E dl l E tot tot γ 2 > ∂ ∂ − = l P G Definition of G : variation of potential energy per (virtual) crack advance Minimum total energy principle: 14 - Code_Aster and Salome-Meca course material GNU FDL Licence Generalization and illustration: G (Griffith) Generalisation: Illustration: A P G ∂ ∂ − = Potential energy Cracking energy F U Prescribed load G F U Prescribed displacement G G’s dimension: J/m² or N/m 15 - Code_Aster and Salome-Meca course material GNU FDL Licence Relation between parameters (Irwin) Linear elasticity In Code_Aster Computation of K Computation of G Usual values (limit criterion) ( ) ( ) 2 2 2 2 2 2 2 1 1 1 1 III II I III II I K E K K E G K E K K E G ν ν ν + + + − = + + + = Plane strain, 3D Plane stress J G = Plane elasticity (plane strain + plane stress) Propagation if ¹ ´ ¦ ≥ ≥ γ 2 G K K Ic i Aluminium alloy Titanium Alloy Hardened Steel Polymer Wood Concrete m MPa K Ic 30 ≈ m MPa K Ic 100 ≈ m MPa K Ic 3 ≈ m MPa K Ic 120 ≈ m MPa K Ic 2 ≈ m MPa K Ic 1 ≈ 16 - 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 (c, m material parameters) – Stage A : ∆K weak, slow or non propagation – Stage B : ∆K moderate, propagation with a constant velocity – Stage C : ∆K high, sudden failure m K c dN da ∆ = . 17 - Code_Aster and Salome-Meca course material GNU FDL Licence Cube containing circular crack Regular mesh near crack front Crack face Crack front Study of a fracture mechanics problem in Code_Aster Mandatory steps to perform crack analysis: Step 1: Meshing cracked structures (except for X-FEM method) Step 2: Thermo-mechanical computation Step 3: Post-processing : computation of fracture mechanics parameters Possible Thermo-mechanical computations Thermo-Elastic (linear or non linear) Residual stresses (linear or non linear elasticity) Thermo-elastoplastic : need to use specific tools of crack analysis 18 - Code_Aster and Salome-Meca course material GNU FDL Licence FOND = DEFI_FOND_FISS ( MAILLAGE = MA, FOND_FISS =_F( GROUP_NO = … / GROUP_MA =…) , / CONFIG_INIT =/’COLLEE’ /’DECOLLEE’ / LEVRE_SUP = _F(GROUP_MA = …), LEVRE_INF = _F(GROUP_MA = …), / NORMALE = (x, y, z), ); Crack definition in Code_Aster General case : Crack front definition Crack surface definition Superior face Inferior face Crack front n r Warning: - 3D front (orientation) - notch n n p p n Γ p r r r ∧ = 2D/3D 3D If CONFIG_INIT = ‘DECOLLEE’ 19 - Code_Aster and Salome-Meca course material GNU FDL Licence Computation of K and G in Code_Aster Two possibilities in Code_Aster: K or G (elasticity) Computation of stress intensity factors K: operator POST_K1_K2_K3 ☺ : easy to use, relatively precise : quite sensitive to mesh quality near the front, only quasi-planar cracks Computation of energy release rate: operator CALC_G ☺ : theoretically more precise and less mesh sensitive : regularity of results along crack front in 3D 20 - Code_Aster and Salome-Meca course material GNU FDL Licence Usage of the POST_K1_K2_K3 operator TABL_K = POST_K1_K2_K3 ( MODELISATION = "3D","AXIS","D_PLAN" or "C_PLAN", /FOND_FISS = FOND, /FISSURE =FISS MATER = …, RESULTAT ( or TABL_DEPL_SUP / TABL_DEPL_INF) ABSC_CURV_MAXI= …, ) Crack front (pre-defined with DEFI_FOND_FISS) Model definition Results of mechanics computation Maximal distance from the crack front for extrapolation X-FEM crack (pre-defined with DEFI_FISS_XFEM) 21 - Code_Aster and Salome-Meca course material GNU FDL Licence Displacement Jump Extrapolation Method (1) 0,0E+00 5,0E-08 1,0E-07 1,5E-07 2,0E-07 2,5E-07 3,0E-07 3,5E-07 4,0E-07 4,5E-07 5,0E-07 0E+00 1E-05 2E-05 3E-05 4E-05 5E-05 6E-05 Curvilinear co-ordinate D i s p l a c e m e n t j u m p Computed displacement jump function K.sqrt(r) Extraction of node displacements along the crack front (normal direction) ABSC_CURV_MAXI Analytical model: with: Operator POST_K1_K2_K3 ABSC_CURV_MAXI ( ) [ ] [ ] | | ¹ | \ | − = → 2 2 0 2 1 8 lim u r E K r I π ν [ ] [ ] n u 2 [ ] [ ] ( ) N . LEVRE_INF LEVRE_SUP 2 U U − = u N 22 - Code_Aster and Salome-Meca course material GNU FDL Licence Displacement Jump Extrapolation Method (2) 3 methods to extrapolate the displacement: Method 1 Prolongation until r = 0 for the right segments One value of K for each consecutive node couple [ ] [ ] r u 2 With quarter-node elements Without quarter- node elements 23 - Code_Aster and Salome-Meca course material GNU FDL Licence Displacement Jump Extrapolation Method (3) Slope of the line One value of K of each node of crack front Method 2: [ ] [ ] 2 u 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 Method 3 ( ) ∫ − = dmax 0 2 )] ( [ 2 1 ) ( dr r k r U k J Minimisation by least square error of J(k): One value of K Without quarter-node elements With quarter-node elements 24 - Code_Aster and Salome-Meca course material GNU FDL Licence Some advices about the usage of POST_K1_K2_K3 Remarks and advices: Operator limited to plane or quasi-planar cracks (possibility to define only one normal) Choice of ABSC_CURV_MAXI: in general so that the extrapolation is made on 3 to 5 elements Interesting verification: the relative error should be small enough Precision of computation: error < 10 % for validation tests; precision is tremendously increased by using quadratic ¼node elements (Barsoum elements, operator MODI_MAILLAGE). Mesh type: free or structured ? If possible, use structured (one extrapolation avoided, better precision a priori) ; Computation on an unstructured mesh Computation on a structured mesh 25 - Code_Aster and Salome-Meca course material GNU FDL Licence Computation of K and G in Code_Aster Two possibilities in Code_Aster: K or G (elasticity) Computation of stress intensity factors K: operator POST_K1_K2_K3 ☺ : easy to use, relatively precise : quite sensitive to mesh quality near the front, only quasi-planar crack Computation of energy release rate : operator CALC_G ☺ : theoretically more precise and less mesh sensitive : regularity of results along crack front in 3D 26 - Code_Aster and Salome-Meca course material GNU FDL Licence Linear fracture mechanics in Code_Aster: G-theta method Operator CALC_G G-theta method: Lagrangian derivation of the global energy of the system Properties: G, local energy release rate, is solution of the following variational equation Derivative difficult to compute directly ( ) : F M M M η ηθ → + Family of transformations from reference configuration Represent virtual crack propagation A P G ∂ ∂ − = ( ) ( ) ( ) 0 0 dW u G mds d η η θ θ η Γ = ⋅ = Γ = − ∫ crack front normal to front Ω ∂ ∀ to tangent θ 27 - Code_Aster and Salome-Meca course material GNU FDL Licence G-theta method: introduction of the theta field 0 θ θ 0 R inf n | || |θ | || | =0 | || |θ | || | =| || |θ 0 | || | R inf R sup R sup Remark: computation is made between R inf and R sup Geometrical definition of theta field Basics of test functions for theta field Spatial discretization of G { } , 1,..., i i P Θ = ∈Θ = % θ θθ θ In 2D: In 3D: 28 - Code_Aster and Salome-Meca course material GNU FDL Licence G-theta method: 3D case ( ) ( ) ( ) ( ) Θ ∈ ∀ Γ = ∫ Γ θ θ m θ , . 0 ds s s s G ( ) ( ) ( ) [ ] P i ds s s p G i i N j j j , 1 , . 0 1 ∈ ∀ Γ = | | ¹ | \ | ∫ ∑ Γ = θ m θ [ ] ( ) ( ) ( ) ¦ ¦ ¦ ¹ ¦ ¦ ¦ ´ ¦ Γ = = ∈ = ∫ ∑ Γ = i i i j ij i N j j ij b ds s m s p a P i b G a θ θ 0 . with , 1 1 Two families of smoothing: utilisation of LEGENDRE polynomials with degree from 0 to 7 utilisation of shape functions of elements of crack front: ‘LAGRANGE’ Linear Quadratic 29 - Code_Aster and Salome-Meca course material GNU FDL Licence TAB_G = CALC_G ( RESULTAT = resu, THETA = _F( FOND_FISS = FOND, (or FISSURE = ...), R_INF = ri, R_SUP = rs, ), SYME_CHAR = "SYME" or "SANS " EXCIT =_F( CHARGE = charmeca, FONC_MULT = ff,) LISSAGE = _F( LISSAGE_THETA = … LISSAGE_G = … DEGRE = 0 7 Operator CALC_G Loading may influence G. Advice: do not use this keyword, by default all loads are taken into account Results from mechanical computation Definition of theta field Smoothing options in 3D Non mandatory 30 - Code_Aster and Salome-Meca course material GNU FDL Licence Operator CALC_G: computational options OPTION = / ‘CALC_G’ / ‘CALC_G_GLOB’ / ‘CALC_K_G’ Computation of G in 2D & 3D (local) Computation of G in 3D (global) Computation of K in 2D & 3D (local) Usual options mandatory keyword: FOND_FISS / FISSURE 31 - Code_Aster and Salome-Meca course material GNU FDL Licence Operator CALC_G: local vs global values Global values: options CALC_G_GLOB and G_MAX_GLOB, … Γ o Local values: options CALC_G, CALC_K_G, G_MAX The value G(s) printed in the result table corresponds to local value in J/m² To induce a mean local value (J/m²), we need to divide by the crack length l : ( ) ( ) absc. curviligne de m 1 , o s s s ⋅ = ∀ Γ θ θθ θ The global G (J/m) printed in the result table corresponds to an uniform crack propagation. G Particular case: in 2D-axisymetric, the ‘local’ G (option CALC_G) corresponds to the energy by unit of radian. In order to obtain a local value of G, we need to divide by its radius R. ( ) ( ) ( ) ( ) ∫ Γ Θ ∈ ∀ = Γ 0 , . θ θ θ ds s m s s G ( ) ( ) θ Γ = = ∫ Γ l ds s G l G 1 1 0 ( ) θ Γ = R G 1 ( ) θ Γ 32 - Code_Aster and Salome-Meca course material GNU FDL Licence Operator CALC_G: advice Remarks and advice: Theoretically, results do not depend on the contour of theta. Advise to choose - R inf different from 0 (imprecise computational results at crack front) - R sup ‘not too large’ (for example 5 or 6 elements) - if possible, verify the independency for different contours - compare G and G_Irwin in result table Remark: in practice, we use mesh with a tore around the crack front … No obligation to use a tore in the mesh around the crack front If a tore is meshed around the crack front, results will be more regular if the radii of the theta field correspond to the radius of the tore 33 - Code_Aster and Salome-Meca course material GNU FDL Licence Operator CALC_G: advice for 3D Choice of smoothing in 3D : need to use different smoothing methods and compare the obtained results ! Energy release rate for an elliptical crack (relative quite coarse mesh) 0,0E+00 5,0E+03 1,0E+04 1,5E+04 2,0E+04 2,5E+04 3,0E+04 0 0,005 0,01 0,015 0,02 0,025 0,03 0,035 Curvilinear co-ordinate G LAGRANGE: no smoothing oscillations can occur LAGRANGE_REGU: decrease of oscillations if nodes along crack front are dense and regularly spaced LEGENDRE: smooth results, but results from the nodes at the extremities of the crack front should be used with care 34 - Code_Aster and Salome-Meca course material GNU FDL Licence G-theta method: computation of stress intensity factors Definition: g(u,v) symmetric bilinear form associated with G(u) ( ) 1 ( , ) ( ) ( ) 4 g u v G u v G u v = + − − ( , ) ( ) g u u G u = Properties: singular displacements are orthogonal two to two referring to the scalar product defined by g(u,v) ( ) ( ) ( ) 0 , , , = = = S III S II S III S I S II S I u u g u u g u u g ( ) ( ) ( ) 0 , , , = = = S III R S II R S I R u u g u u g u u g ( ) ( ) , , R S S S R S S S I I II II III III I I II II III III G g u u g u K u K u K u u K u K u K u = = + + + + + + 2 2 2 ( , ) ( , ) ( , ) ( , ) S S S S S S I I I II II II III III III G g u u K g u u K g u u K g u u = = + + ( ) ( ) 2 , 1 S I I E K g u u ν = − ( ) ( ) 2 , 1 S II II E K g u u ν = − ( ) 2 , S III III K g u u µ = (so ) Demonstration: Rice’s integral (=G in elasticity) and symmetric properties of singular functions R S S S I I II II III III u u K u K u K u = + + + 35 - Code_Aster and Salome-Meca course material GNU FDL Licence Additional physical zone: Between free surface (crack) and sound zone, interaction zone. Particular behaviour law: Assuming the existence of a surface energy depending on displacement jump Cohesive law = relation between and force vector of separation (stress) Parameters of cohesive law: critical surface energy and critical stress Particular finite elements: Classical joint elements, linear, with regularisation: PLAN_JOINT, AXIS_JOINT, 3D_JOINT ; an additional parameter PENA_ADHERENCE Discontinuous elements including, linear, without regularisation: PLAN_ELDI, AXIS_ELDI Interface elements, quadratic, mixed formulation: PLAN_INTERFACE, AXIS_INTERFACE, 3D_INTERFACE ; (an additional parameter PENA_LAGR) Cohesive zone model (CZM): generalities δ r δ σ r r ∂ Ψ ∂ = Ψ δ r c G c σ 36 - Code_Aster and Salome-Meca course material GNU FDL Licence Cohesive zone model (CZM): possible laws CZM_LIN_REG + δ r σ C σ C C G σ 2 + δ r σ C σ Keyword RUPT_FRAG of DEFI_MATERIAU: MAT = DEFI_MATERIAU ( RUPT_FRAG =_F( GC=… SIGM_C =… PENA_ADHERENCE =… PENA_LAGR =… ),); Principal parameters For joint elements For interface elements CZM_EXP_REG 0 κ κ 0 κ κ + δ r σ C σ C C G σ 2 C C G σ 2 κ n δ n σ C σ CZM_OUV_MIX CZM_TAC_MIX 37 - Code_Aster and Salome-Meca course material GNU FDL Licence Summary for brittle behaviour: Specificities for some laws: CZM_OUV_MIX pure mode I CZM_TAC_MIX both sides of crack must be meshed Orientation: Cohesive force direction: all element; MODI_MAILLAGE (ORIE_FISSURE =_F(GROUP_MA= )) Local basis of crack: interface elements; AFFE_CARA_ELEM(MASSIF=_F(ANGL_REP= )) STAT_NON_LINE=(CARA_ELEM= ) Cohesive zone model (CZM): some rules Element JOINT DISCONTINUITE INTERFACE Type Linear Linear Quadratic Thickness Null or non null Non null Null or non null Material Linear or non Linear Linear or non Possible laws CZM_LIN_REG CZM_EXP_REG CZM_EXP CZM_OUV_MIX CZM_TAC_MIX Regularisation PENA_ADHERENCE None PENA_LAGR (optional) x r z r y r X r Y r Z r 38 - Code_Aster and Salome-Meca course material GNU FDL Licence Accounting for confined plasticity by plastic correction we replace a crack of length a by a virtual crack of length a + r y , where r y is the plastic zone size (Irwin’s approach) we compute stress intensity factors by analytical formula (RCC-M approach) Simple accounting for plasticity If the loading is radial and monotone, we can compute G for a crack with non-linear elastic behaviour (ELAS_VMIS_TRAC or ELAS_VMIS_LINE under COMP_ELAS) Advanced models for complex situations (see doc Aster): Research at EDF R&D GTP approach for extended plasticity (addition of a plastic term in computation of G) Gp approach (extended energetic approach to account for plasticity for brittle fracture) Damage law ENDO_FRAGILE for brittle fracture and ROUSSELIER ductile fracture Specific cohesive law CZM_TRA_MIX for ductile fracture Non Linear fracture mechanics 2 1 6 I y s K r π σ | | = | \ ¹ y cp I a r K K a α + = α coefficient dependent on ratio crack length / pipe thickness 39 - Code_Aster and Salome-Meca course material GNU FDL Licence Code_Aster references General user documentation Application domains of operators in fracture mechanics of Code_Aster and advices for users [U2.05.01] Notice for utilisation of cohesive zone models [U2.05.07] Realisation for a computation of prediction for cleavage fracture [U2.05.08] Documentation of operators Operators DEFI_FOND_FISS [U4.82.01], DEFI_FISS_XFEM [U4.82.08], CALC_G [U4.82.03] et POST_K1_K2_K3 [U4.82.05] Reference documentation Computation of stress intensity factors by Displacement Jump Extrapolation Method [R7.02.08] Computation of coefficients of stress intensity in plane linear thermoelasticity [R7.02.05] Energy release rate in linear thermo-elasticity [R7.02.01] Energy release rate in non-linear thermo-elasticity [R7.02.03] Energy release rate in non-linear thermo-elasticity-plasticity: GTP approach [R7.02.07] Elastic energy release rate en thermo-elasticity-plasticity by Gp approach [R7.02.16] 40 - Code_Aster and Salome-Meca course material GNU FDL Licence End of presentation Is something missing or unclear in this document? 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