Fracture Mechanics, Damage and Fatigue Non Linear Fracture Mechanics: Numerical Methods

Transcription

1 University of Liège Aerospace & Mechanical Engineering Fracture Mechanics, Damage and Fatigue Non Linear Fracture Mechanics: Numerical Methods Ludovic Noels Computational & Multiscale Mechanics of Materials CM3 Chemin des Chevreuils 1, B4000 Liège Fracture Mechanics Numerical Methods

2 Numerical Methods LEFM Crack propagation Cohesive models XFEM Ductile material Extension to the brittle methods Damage models Multiscale methods Atomistic models Fracture Mechanics Numerical Methods 2

3 LEFM Definition of elastic fracture Strictly speaking: During elastic fracture, the only changes to the material are atomic separations As it never happens, the pragmatic definition is Therefore The process zone, which is the region where the inelastic deformations Plastic flow, Micro-fractures, Void growth, happen, is a small region compared to the specimen size, and is at the crack tip Linear elastic stress analysis describes the fracture process with accuracy Mode I Mode II Mode III (opening) (sliding) (shearing) Fracture Mechanics Numerical Methods 3

4 LEFM SIF (K I, K II, K III ): asymptotic solution in linear elasticity Crack closure integral Energy required to close the crack by an infinitesimal da If an internal potential exists with AND if linear elasticity AND if straight ahead growth J integral Energy that flows toward crack tip If an internal potential exists Is path independent if the contour G embeds a straight crack tip BUT no assumption on subsequent growth direction If crack grows straight ahead: G=J If linear elasticity: Can be extended to plasticity if no unloading (power law) Fracture Mechanics Numerical Methods 4

5 Crack growth criteria Crack growth criterion is G G C LEFM Stability of the crack is reformulated (in 2D) Stable crack growth if Unstable crack growth if Crack growth direction s y Method of the maximum hoop stress Crack criterion with & 2a eqq b e rr q * x In case of mixed loading This corresponds to s Fracture Mechanics Numerical Methods 5

6 Crack propagation A simple method is a FE simulation where the crack is used as BCs The mesh is conforming with the crack lips Fracture Mechanics Numerical Methods 6

7 Crack propagation A simple method is a FE simulation where the crack is used as BCs (2) Mesh the structure in a conforming way with the crack Extract SIFs K i (see lecture on SIF) Use criterion on crack propagation Example: the maximal hoop stress criterion with crack propagation direction obtained by & If the crack propagates Move crack tip by Da in the q*-direction A new mesh is required as the crack has changed (since the mesh has to be conforming) Not used Involves a large number of remeshing operations (time consuming) Is not always fully automatic Requires fine meshes and Barsoum elements Fracture Mechanics Numerical Methods 7

8 Cohesive elements The cohesive method is based on Barenblatt model This model is an idealization of the brittle fracture mechanisms Separation of atoms at crack tips (cleavage) As long as the atoms are not separated by a distance d t, there are attractive forces (see overview lecture) Cohesive zone tip y Crack tip s y s y (d) r p 2a d t r p x G C d t d For elasticity (recall lecture on cohesive zone) So the area below the s-d curve corresponds to G C if crack grows straight ahead This model requires only 2 parameters Peak cohesive traction s max (spall strength) Fracture energy G C Shape of the curves has no importance as long as it is monotonically decreasing Fracture Mechanics Numerical Methods 8

9 Cohesive elements Insertion of cohesive elements Between 2 volume elements Computation of the opening (cohesive element) Normal to the interface in the N - t (+) deformed configuration N Normal opening d (+) t (+) Sliding Resulting opening with b c the ratio between the shear and normal critical tractions Definition of a potential Potential to match the s y (d) traction separation law (TSL) curve Traction (in the deformed configuration) derives from this potential G C d t d Fracture Mechanics Numerical Methods 9

10 Cohesive elements Computational framework How are the cohesive elements inserted? First method: intrinsic Law Cohesive elements inserted from the beginning So the elastic part prior to crack propagation is accounted for by the TSL Drawbacks: Requires a priori knowledge of the crack path to be efficient Mesh dependency [Xu & Needelman, 1994] Initial slope that modifies the effective elastic modulus» Alteration of a wave propagation This slope should tend to infinity [Klein et al. 2001]» Critical time step is reduced Second method: extrinsic law Cohesive elements inserted on the fly when failure criterion (s>s max ) is verified [Ortiz & Pandolfi 1999] Drawback: Complex implementation in 3D especially for parallelization Failure criterion incorporated within the cohesive law Failure criterion external to the cohesive law Fracture Mechanics Numerical Methods 10

11 Cohesive elements Examples Fracture Mechanics Numerical Methods 11

12 Cohesive elements Advantages of the method Can be mesh independent (non regular meshes) Can be used for large problem size Automatically accounts for time scale [Camacho & Ortiz, 1996] Fracture dynamics has not been studied in these classes Really useful when crack path is already known Debonding of fibers Delamination of composite plies No need for an initial crack The method can detect the initiation of a crack Drawbacks Still requires a conforming mesh Requires fine meshes So parallelization is mandatory Could be mesh dependant Fracture Mechanics Numerical Methods 12

13 extended Finite Element Method How to get rid of conformity requirements? Key principles For a FE discretization, the displacement field is approximated by Sum on nodes a in the set I (11 nodes here) u a are the nodal displacements N a are the shape functions x i are the reduced coordinates XFEM New degrees of freedom are introduced to account for the discontinuity It could be done by inserting new nodes ( ) near the crack tip, but this would be inefficient (remeshing) Instead, shape functions are modified Only shape functions that intersect the crack This implies adding new degrees of freedom to the related nodes ( ) Fracture Mechanics Numerical Methods 13

14 Key principles (2) extended Finite Element Method New degrees of freedom are introduced to account for the discontinuity J, subset of I, is the set of nodes whose shape-function support is entirely separated by the crack (5 here) u* a are the new degrees of freedom at node a Form of F a the shape functions related to u* a? Use of Heaviside s function, and we want +1 above and -1 below the crack In order to know if we are above or below the crack, signed-distance has to be computed lsn(x i, x i* ) Normal level set lsn(x i, x i *) is the signed distance between a point x i of the solid and its projection x i* on the crack with H(x) = ±1 if x >< Fracture Mechanics Numerical Methods 14

15 extended Finite Element Method Key principles (3) Example: removing of a brain tumor (L. Vigneron et al.) At this point A discontinuity can be introduced in the mesh Fracture mechanics is not introduced yet New enrichment with LEFM solution Zone J of Heaviside enrichment is reduced (3 nodes) A zone K of LEFM solution is added to the nodes ( ) of elements containing the crack tip LEFM solution is asymptotic only nodes close to crack tip can be enriched y b a is the new degree b at node a (more than one see next slide) Y b is the new shape function b (more than one see next slide) Fracture Mechanics Numerical Methods 15

16 Key principles (4) extended Finite Element Method New enrichment with LEFM solution (2) y' y x' Y b and y b a from LEFM solutions x Fracture Mechanics Numerical Methods 16

17 extended Finite Element Method Key principles (5) New enrichment with LEFM solution (3) But We still have We have determined 4 independent shape functions Y b Fracture Mechanics Numerical Methods 17

18 extended Finite Element Method Key principles (6) New enrichment with LEFM solution (4) Vectors of unknowns y b and shape functions Y b are now defined We have 12 new degrees of freedom on the LEFM-enriched nodes Remark: as Y 1 is discontinuous we do not need Heaviside functions for K-nodes Fracture Mechanics Numerical Methods 18

19 extended Finite Element Method Key principles (7) How are evaluated? New level sets Normal level set lsn(x i, x**) is the normal signed distance between a point x i of the solid and the crack tip x i** Tangent level set lst(x i, x**) is the tangential signed distance between a point x i of the lsn(x i, x ** ) lst(x i, x ** ) solid and the crack tip x i** Fracture Mechanics Numerical Methods 19

20 Crack propagation criterion Requires the values of the SIFs extended Finite Element Method Using y b a as y' y x' x was substituted by Fracture Mechanics Numerical Methods 20

21 extended Finite Element Method Crack propagation criterion Requires the values of the SIFs (2) A more accurate solution is to compute J But K I, K II & K III have to be extracted from» Define an adequate auxiliary field u aux» Compute J aux (u aux ) and J s (u+u aux )» On can show that the interaction integral (see lecture on SIFs)» If u aux is chosen such that only K i aux 0, K i is obtained directly Then the maximum hoop stress criterion can be used with & Fracture Mechanics Numerical Methods 21

22 Numerical example Crack propagation (E. Béchet) extended Finite Element Method Advantages: No need for a conforming mesh (but mesh has still to be fine near crack tip) Mesh independency Computationally efficient Drawbacks: Require radical changes to the FE code New degrees of freedom Gauss integration Time integration algorithm Fracture Mechanics Numerical Methods 22

23 Ductile materials True s Failure mechanism Plastic deformations prior to (macroscopic) failure of the specimen Dislocations motion void nucleation around inclusions micro cavity coalescence crack growth Griffith criterion should be replaced by s TS s p 0 True e Numerical models accounting for this failure mode? Fracture Mechanics Numerical Methods 23

24 Ductile materials Introduction to damage (1D) As there are voids in the material, only a reduced surface is balancing the traction Virgin section S Damage of the surface is defined as F F So the effective (or damaged) surface is actually And so the effective stress is Resulting deformation Hooke s law is still valid if it uses the effective stress So everything is happening as if Hooke s law was multiplied by (1-D) Isotropic 3D linear elasticity Failure criterion: D=D C, with 0 < D C <1 But how to evaluate D, and how does it evolve? Fracture Mechanics Numerical Methods 24

25 Ductile materials Evolution of damage D for isotropic elasticity Equations Stresses Example of damage criterion Y C is an energy related to a deformation threshold There is a time history Either damage is increased if f = 0 Or damage remains the same if f <0 Example for Y C such that damage appears for e = 0.1 s / (2EY C ) 1/2 e But for ductile materials plasticity is important as it induces the damage Fracture Mechanics Numerical Methods 25 e

26 Ductile materials Gurson s model, 1977 Assumptions Given a rigid-perfectly-plastic material with already existing spherical microvoids Extract a statistically representative sphere V embedding a spherical microvoid Porosity: fraction of voids in the total volume and thus in the representative volume: F V n V void F with the material part of the volume V Define Material rigid-perfectly plastic elastic deformations negligible Macroscopic strains and stresses: e & s Microscopic strains and stresses: e & s The Macroscopic strains are defined by Fracture Mechanics Numerical Methods 26

27 Ductile materials Gurson s model, 1977 (2) Macroscopic strains n V void V Gauss theorem V Stresses In V microscopic stresses s are related to the microscopic deformations e In terms of an energy rate As the energy rate has to be conserved Gurson solved these equations For a rigid-perfectly-plastic microscopic behavior Which leads to a new macroscopic yield function depending on the porosity Fracture Mechanics Numerical Methods 27

28 Ductile materials Gurson s model, 1977 (3) Gurson deduced the yield surface accounting for the voids with If f V = 0, it corresponds to J2 plasticity When voids are present, a hydrostatic stress state can induce plasticity If Or Eventually, Sign to be selected as for f V = 0, p cannot induce plasticity Fracture Mechanics Numerical Methods 28

29 Ductile materials Gurson s model, 1977 (4) Shape of the new yield surface s e /s 0 p Normal flow p /s 0 p What remains to be defined is the evolution of the porosity f V Fracture Mechanics Numerical Methods 29

30 Ductile materials Gurson s model, 1977 (5) Evolution of the porosity f V Volume of material is constant as Elastic deformations are neglected Plastic deformations are isochoric cst Void porosity and macroscopic volume variation are linked But volume variation can be expressed from the deformation tensor as elastic deformations are neglected Fracture Mechanics Numerical Methods 30

31 Ductile materials Gurson s model, 1977 (6) Eventually The porosity is actually not an independent internal variable Yield surface Normal flow with Assumptions were Rigid perfectly-plastic material Initial porosity (no void nucleation) No voids interaction No voids coalescence More evolved models have been developed to account for Hardening Voids nucleation, interactions and coalescences Fracture Mechanics Numerical Methods 31

32 Ductile materials Hardening Yield criterion remains valid but one has to account for the hardening of the matrix In this expression, the equivalent plastic strain of the matrix is used instead of the macroscopic one Values related to the matrix and the macroscopic volume are dependant as the dissipated energies have to match Voids nucleation Increase rate of porosity results from Matrix incompressibility Creation of new voids cst Represented by 1 void The nucleation rate can be modeled as strain controlled Fracture Mechanics Numerical Methods 32

33 Ductile materials Voids interaction 1981, Tvergaard In Gurson model a void is considered isolated The presence of neighboring voids decreases the maximal loading as the stress distribution changes With 1 < q < 2 depending on the hardening exponent Fracture Mechanics Numerical Methods 33

34 Ductile materials Voids coalescence 1984, Tvergaard & Needleman When two voids are close (f V ~ f C ), the material loses capacity of sustaining the loading If f V is still increased, the material is unable to sustain any loading with Fracture Mechanics Numerical Methods 34

35 Ductile materials Softening response Loss of solution uniqueness mesh dependency F, d F F F F, d F, d Dd Dd Dd F F F F, d Dd Dd Dd Fracture Mechanics Numerical Methods 35

36 Ductile materials Softening response (2) Requires non-local models Dr.-Ing. Frederik Reusch, University of Dortmund, Deparment of Mechanical Engineering Mechanics, Fracture Mechanics Numerical Methods 36

37 Why multiscale? Multiscale Methods Previous methods are models based on macroscopic results The idea is to simulate what is happening at small scale with correct physical models and to extract responses that can be used at macroscopic scale Gurson s model is actually a multiscale model Fracture Mechanics Numerical Methods 37

38 Principle 2 BVPs are solved concurrently Multiscale Methods The macro-scale problem The meso-scale problem (on a Material response Macro-scale Extraction of a mesoscale Volume Element meso-scale Volume Element) Requires two steps Downscaling: BC of the BVP mesoscale BVP from the macroscale deformationgradient field L macro >>L VE >>L micro Upscaling: The resolution of the mesoscale BVP yields an homogenized macroscale behavior Fracture Mechanics Numerical Methods 38

39 Multiscale Methods Example: Failure of composite laminates Heterogeneous materials: failure involves complex mechanisms Fiber rupture Debonding Pull out Delamination Matrix rupture Bridging Use Multi-scale model with non-local continuum damage within each ply Cohesive elements for delamination Fracture Mechanics Numerical Methods 39

40 Multiscale Methods Failure of composite [90 o / 45 o / -45 o / 90 o / 0 o ] S - open hole laminate 90 o -ply (out) 45 o -ply -45 o -ply 90 o -ply (in) 0 o -ply Fracture Mechanics Numerical Methods 40

41 Multiscale Methods Example: Failure of polycrystalline materials The mesoscale BVP can also be solved using atomistic simulations Polycrystalline structures can then be studied Finite element for the grains Cohesive elements between the grains Material behaviors and cohesive laws calibrated from the atomistic simulations Grain size: 6.56 nm Grain size: 3.28 nm Fracture Mechanics Numerical Methods 41

42 Multiscale Methods Atomistic models: molecular dynamics Newton equations of motion are integrated for classical particles Particles interact via different types of potentials For metals: Morse-, Lennard-Jones- or Embedded-Atom potentials For liquid crystals: anisotropic Gay-Berne potential The shapes of these potentials are obtained using ab-initio methods Resolution of Schrödinger for a few (<100) atoms Example: Crack propagation in a two dimensional binary model quasicrystal It consists of particles and it is stretched vertically Colors represent the kinetic energy of the atoms, that is, the temperature The sound waves, which one can hear during the fracture, can be seen clearly Prof. Hans-Rainer Trebin, Institut für Theoretische und Angewandte Physik Universität Stuttgart, Fracture Mechanics Numerical Methods 42

43 Lecture notes References Lecture Notes on Fracture Mechanics, Alan T. Zehnder, Cornell University, Ithaca, Other references «on-line» Book Fracture Mechanics, Piet Schreurs, TUe, Fracture Mechanics: Fundamentals and applications, D. T. Anderson. CRC press, Fatigue of Materials, S. Suresh, Cambridge press, Fracture Mechanics Numerical Methods 43