TOLERANT APPROACH WITH

Transcription

1 A XFEM-BASED PROBABILISTIC DAMAGE TOLERANT APPROACH WITH MORFEO/CRACK FOR ABAQUS Frédéric Lani 1,3 Eric Wyart 2 Laurent D Alvise 3 1 Université Catholique de Louvain, Institute of Mechanics, Materials and Civil Engineering, Louvain-la-Neuve, Belgique 2 Centre Recherche en Aéronautique Cenaero ASBL, Gosselies Belgique 3 GeonX SA, Gosselies - Belgique

2 FRAMEWORK Damage tolerance: a defect is present in the structure The structure can fail in service (static or fatigue crack propagation & final failure) Example statement of work: design the part so that it can indeed carry the ultimate load after a given number of cycles above the expected part service life

3 DESIGN CRITERIA Stability under ultimate load: in LEFM No growth approach: or under high cycle fatigue (e.g. vibrations) Fatigue Life N f > N under service loads with and after N cycles. Definition of inspection intervals to check the crack length and the failure criteria

4 USUALLY: DETERMINISTIC DESIGN The initial crack is placed at the most critical place at the locus of the maximum principal stress orthogonal to the max principal stress direction or maximizing the mode I (opening fracture mode) The imposed crack length is the smallest detectable size or the maximum observed size A-values, B-values are used for the material properties

5 DETERMINISTIC DESIGN Is usually over-conservative the probability of presence of the defect is 1 the probability of having the most critical size, orientation, the highest load, the worst material characteristics is 1 But can be under-conservative eg: the so-called most critical crack is stable under the static ultimate load BUT is more dangerous from a fatigue standpoint

6 DETERMINISTIC DESIGN Consequences: use of safety factors mass excess compared with optimal design no estimation of risk (or supposedly no risk taken) small inspection intervals

7 PROBABILISTIC DESIGN The probability of defect occurrence is not 1 Input parameters obey a distribution «The sizing is made for instance so that the probability of failure is below a threshold for a given number of cycles» «The inspection interval is determined so that the probability of failure is below a threshold»

8 PROBABILISTIC FRACTURE MECHANICS ANALYSIS COMPUTATIONAL CHAIN Limit state function Distribution of input variables (probability density functions ) Accurate solver to compute the stress intesity factors and the number of cycles Method to compute the probability of failure = cumulative density function (CDF), e.g. Monte-Carlo (MC) First Order Reliability Method (FORM) Second Order Reliability Method (SORM)

9 COMPUTATIONAL CHAIN Matlab drives the chain, Morfeo/Crack drives the crack propagation simulation, Abaqus is the FE solver, Abaqus CAE is the geometric modeler Matlab (MC, FORM, RBN, KBF) pdfparameters.txt customdata.txt sifs.txt cycles.txt model-propagation.odb model.cae Abaqus CAE (nogui) with Morfeo Crack features (modelscript.py ) model.inp crack.stp Morfeo Crack & Abaqus std. (XFEM solution, SIF, LevelSet update)

10 LIMIT STATE FUNCTION In this example: compute the probability that the part fails before reaching a number of cycles N

11 COMPUTATIONAL CHAIN Matlab drives the chain, Morfeo/Crack drives the crack propagation simulation, Abaqus is the FE solver, Abaqus CAE is the geometric modeler Matlab (MC, FORM, RBN, KBF) pdfparameters.txt customdata.txt sifs.txt cycles.txt model-propagation.odb model.cae Abaqus CAE (nogui) with Morfeo Crack features (modelscript.py ) model.inp crack.stp Morfeo Crack & Abaqus std. (XFEM solution, SIF, LevelSet update)

12 DISTRIBUTIONS OF INPUT VARIABLES Parameters of the Paris law (C,n) lognormally distributed Loading is normally distributed

13 DISTRIBUTIONS OF INPUT VARIABLES PDF for initial crack length

14 DISTRIBUTION OF INPUT VARIABLES Initial crack orientation: PDF uniform between lower and upper bounds

15 COMPUTATIONAL CHAIN Matlab drives the chain, Morfeo/Crack drives the crack propagation simulation, Abaqus is the FE solver, Abaqus CAE is the geometric modeler Matlab (MC, FORM, RBN, KBF) pdfparameters.txt customdata.txt sifs.txt cycles.txt model-propagation.odb model.cae Abaqus CAE (nogui) with Morfeo Crack features (modelscript.py ) model.inp crack.stp Morfeo Crack & Abaqus std. (XFEM solution, SIF, LevelSet update)

16 SOLVER: FRAMEWORK Linear elastic fracture mechanics Stress intensity factors computation Fatigue crack growth & Life assessment Step-by-step modelling process Proportional & constant amplitude loading XFEM extended Finite Element Method Level Set Method, Fast Marching Method Crack propagation modelling without remeshing High accuracy on stress singularity & SIF

17 Displacement field SOLVER: XFEM PRINCIPLE First term: like FEM Two kinds of enrichment: Displacement discontinuity Stress singularity

18 SOLVER: ABAQUS ENHANCED BY THE MORFEO/CRACK PLUG-IN Morfeo/Crack for Abaqus is a proprietary plug-in by GeonX and Cenaero compatible with Abaqus CAE (menu Plug-In) Dedicated to automated Fatigue Crack Propagation using the joint XFEM capabilities of Abaqus and Morfeo accurate SIF computation methods within Morfeo the Level Set propagation technique implemented in Morfeo Morfeo/Crack for Abaqus features can also be used in Python scripts (morfeo module)

19 SOLVER: MORFEO/CRACK FOR ABAQUS PRINCIPLE

20 SOLVER: MORFEO/CRACK FOR ABAQUS

21 SOLVER: MORFEO/CRACK FOR ABAQUS

22 COMPUTATIONAL CHAIN Matlab drives the chain, Morfeo/Crack drives the crack propagation simulation, Abaqus is the FE solver, Abaqus CAE is the geometric modeler Matlab (MC, FORM, RBN, KBF) pdfparameters.txt customdata.txt sifs.txt cycles.txt model-propagation.odb model.cae Abaqus CAE (nogui) with Morfeo Crack features (modelscript.py ) model.inp crack.stp Morfeo Crack & Abaqus std. (XFEM solution, SIF, LevelSet update)

23 MORFEO/CRACK FOR ABAQUS A parameterized model with parameters crack length crack orientation material properties load accessible via system call abaqus cae nogui=modelscript.py

24 MAIN INGREDIENTS OF MODELSCRIPT.PY Besides the geometrical data, loads, parameters which can be changed using built-in mdb.models members. The following commands must be included import customkernel import morfeo2 platemdb = openmdb( model.cae') customdata.txt contains a set of variable input data f = open("customdata.txt","r") mdb.customdata.paris_exponent = Paris_n mdb.customdata.paris_coefficient = Paris_C mdb.customdata.load_ratio = load_ratio mdb.customdata.max_increment = crack_increment mdb.customdata.number_of_steps = number_steps mdb.customdata.force_planar = 0 morfeo2.writeinput( modeljob','')

25 COMPUTATIONAL CHAIN Matlab drives the chain, Morfeo/Crack drives the crack propagation simulation, Abaqus is the FE solver, Abaqus CAE is the geometric modeler Matlab (MC, FORM, RBN, KBF) pdfparameters.txt customdata.txt sifs.txt cycles.txt model-propagation.odb model.cae Abaqus CAE (nogui) with Morfeo Crack features (modelscript.py ) model.inp crack.stp Morfeo Crack & Abaqus std. (XFEM solution, SIF, LevelSet update)

26 PROBABILITY ESTIMATION Monte-Carlo type

27 PROBABILITY ESTIMATION First Order Reliability Method Concept: find the locally equivalent normal distribution transform to standard normal parameters using the Rosenblatt transformaiton find the most probable point using the Rackwitz Fiessler algorithm approximated the limit state function by an hyperplane in the space of normal variables and compute the probability of failure & reliability index

28 SIMPLE EXAMPLE SENT specimen 2 independent random input variables a with a th = 10mm and a max = 15mm normally distributed with mean = 50MPa and stdv = 0,03*50MPa W = 60mm h = 320mm C = 1, m/cycle/(mpa m) n n = 3,63 R = 0 a h W

29 PROBABILITY ESTIMATION NUMBER OF CALLS TO Morfeo/Crack for Abaqus 2 random var., target 10-6 Method Computation of g # Calls to the FE model Monte-Carlo Direct FE Minimum 10 6 n/a Monte-Carlo Monte-Carlo Konwledge Based Fit of K Radial Basis Functions Network for N # Calls to the meta-model 5 Minimum Minimum 10 6 FORM Direct FE Minimum 18 n/a FORM KBF 5 Min. 18 FORM RBF Network 625 Min. 18

30 ESTIMATED PROBABILITY OF FAILURE Validated against Harkness et al. 1992

31 SECOND EXAMPLE SENT specimen 5 independent random input variables a with a th = 10mm and a max = 15mm normally distributed with mean = 50MPa and stdv = 0,03*50MPa uniform between 0 and 45 C lognormally distributed with mean C = 1, m/cycle/(mpa m) n and stdv C = 0,01 C n lognormally distributed with mean n = 3,63 and stdv n = 0,05 n W = 60mm h = 320mm R = 0 a h W

32 XFE SIMULATION

33 SECOND EXAMPLE Selected approach: MC and FORM on KBF

34 SECOND EXAMPLE: KBF Only 26 Crack propagations simulations were performed using Morfeo/Crack for Abaqus to estimated the KBF Estimate Nf at =0 Dependence in

35 RESULTS

36 CONCLUSION A simple computational chain has been developed to compute a probability of failure after fatigue crack propagation based on Morfeo/Crack for Abaqus The FORM method with knowledge based metamodels is computationally efficient MC analysis, even with smart sampling is hardly feasible unless massive distributed computing is used RBF network is feasible, with adaptive sampling when no Knowledge based metamodel is exploitable

37 REMARKS Accuracy & convergence Validation Complex crack propagation Stochastic FE