Härtelt, M.

Riesch-Oppermann, H.

Scherrer-Rudiy, S.

English

KITopen

6.5 6.6 6.7 6.8 6.9ln@stressêMPaD-4-3-2-101nl@-nl@1-FDD700 800 900 1000stressêMPa0.020.050.10.20.50.70.90.99f_PKIT – University of the State of Baden-Wuerttemberg and National Research Center of the Helmholtz AssociationKarlsruhe Institute of Technology (KIT), Institute for Applied Materials (IAM)www.kit.eduSTAUUncertainties in Mechanics and Materials Research –Application in the field of Materials ResearchH. Riesch-Oppermann, M. Härtelt, S. Scherrer-RudiyHigh performancesliding and friction systemsbased on advanced ceramicsKarlsruhe Institute of Technology (KIT), IAM2 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011OutlineScope; historical accountAbout ceramics…The Weakest Link approach and Fracture MechanicsSTAU – the interface to Finite ElementsUncertainty analysis: Bootstrap & Bayes‘ methodsReducing uncertainties by pooling( uncertainty: predictions/boundary conditions )SummaryKarlsruhe Institute of Technology (KIT), IAM3 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Scope Iceramicsglasssteel (ferrite)tungstenhigh strength, low wear, heat resistance, chemical inertnessReliability of brittle materialsKarlsruhe Institute of Technology (KIT), IAM4 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Scope IImicrostructure (silicon nitride Si3N4)failure behaviourmechanical propertiesfields of interestKarlsruhe Institute of Technology (KIT), IAM5 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Scope IIIprobability theorymethodsfracture mechanicsengineeringElement 1Element 2Element 3Karlsruhe Institute of Technology (KIT), IAM6 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011From then to now (history)Griffith (~1920)Irwin (~1960)Freudenthal (~1950)the importance of material flawsmaterial flaws as stress raisersfracture of materials (mechanics) fracture of materials (statistics)Weibull (~1930)fracture mechanics based weakestlink approachHutchinson, Rice (~1970)Batdorf, Evans, Matsuo (~1980-90)Karlsruhe Institute of Technology (KIT), IAM7 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011From then to now IIFinite elementsstress analysis for complex geometriesspontaneous fracturestress analysis under complex loadsthe importance of microstructurecomputers (engineering) FE postprocessing (CARES, STAU)link between stress analysis and reliability analysiscyclic loading(microstructure)delayed (time-dependant) fractureKarlsruhe Institute of Technology (KIT), IAM8 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011The importance of microstructuresintering: pores, inclusions, grain boundariesmachining: surface flawscritical crack size; fracture toughnessphases, fibres, ...the importance of microstructure (high strength ceramics)FE postprocessing & microstructurecrack propagationcontact loading; thermal shock(surface treatment; hardening)crack resistance10 µmKarlsruhe Institute of Technology (KIT), IAM9 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Aspects of uncertaintystrength: characteristic strength, scatterlifetime: strength & crack propagation parametersdata from different sourcescrack propagation: power law parametersmaterial behaviour (inherent scatter)material characterization (data uncertainties)limited amount of datavarious models for strength, lifetime, crack propagation(surface treatment; hardening)unknown boundary conditionsmodelling; analysisKarlsruhe Institute of Technology (KIT), IAM10 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Reliability uncertaintyobtained by numerical analysis of mechanical stress fielduncertainty in input parameters transforms to resultsdata from different sourcesincluding routines for crack propagationFailure probabilitylifetime predictionditosize effectmechanisms of fracture (microcrack failure)Data base uncertainties prediction uncertaintiesKarlsruhe Institute of Technology (KIT), IAM11 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Strategies for uncertainty assessmentstrength: use appropriate pdflifetime: use stress/lifetime approach via suitable pdfuse data from different sourcescrack propagation: assess parameter uncertaintymaterial behaviour (inherent scatter)material characterization (data uncertainties)pooling: transferring of data to reference conditionsboth, probabilistic and non-probabilistic approaches !! parametric studies...modelling; analysisKarlsruhe Institute of Technology (KIT), IAM12 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011About ceramicsHigh performancesliding and friction systemsbased on advanced ceramicsKarlsruhe Institute of Technology (KIT), IAM13 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011About ceramics IIfrom powder to product: typical steps in ceramic processingHigh performancesliding and friction systemsbased on advanced ceramicsKarlsruhe Institute of Technology (KIT), IAM14 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011About ceramics IIIafter sintering: grinding to final geometry surface flawsmaterial characterization: four-point bend test (inert strength, lifetime under static & cyclic loads)typical Weibull diagram for4PB fracture strength of Si3N4 ceramic; F=i/(n+1): 6.5 6.6 6.7 6.8 6.9ln@stressêMPaD-4-3-2-101nl@-nl@1-FDD700 800 900 1000stressêMPa0.020.050.10.20.50.70.90.99f_PKarlsruhe Institute of Technology (KIT), IAM15 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011About ceramics IVhigh strength ceramics: strengthening mechanismscrack bridging5µmcrack bridging stresses affects critical crack size:  „R-curve effect“Karlsruhe Institute of Technology (KIT), IAM16 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011About ceramics Vcrack bridging stresses affects critical crack size:  „R-curve effect“Karlsruhe Institute of Technology (KIT), IAM17 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011About ceramics VIcritical crack size with respect to local stress fieldinert strength; K-concept:inert strength; strongly varying local stresses (contact; thermal shock):aYK IeqeqI σ=, aYKIeqIeq,=σijσijσ-400-2000200400600800100012000 10 20 30 40 50r [μm]σx [MPa]01234560 0.005 0.01 0.015 0.02 0.025a [mm]KI[MPa√m]constant stressstress gradientsKarlsruhe Institute of Technology (KIT), IAM18 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011About ceramics VIIinert strength; R-curve behaviour:)()()( aKaKaK brappltip −= ∫∫ −=abraappl dxxaxhdxxaxh00)(),()(),( σσiterative procedure for critical crack size necessary:Icctip KaK =)(Karlsruhe Institute of Technology (KIT), IAM19 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Basic ideas of the weakest link approachBasic idea (components & system): system failure, if weakest link fails• fracture mechanics: flaws as planar cracks• most unfavourable combination of stress, flaw size and orientationdetermines failure of a component• isotropy; i.e. flaws are uniformly distributed• size and orientation are random• flaws are independent (no interaction)probabilistic modelfailure modelKarlsruhe Institute of Technology (KIT), IAM20 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011The weakest link approach in brief( )nn QQ 111 −=− Mnn enMp −=![ ]1exp1 QMPf −−=flaw population: pnfailure of individual flaws: Q1Karlsruhe Institute of Technology (KIT), IAM21 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011The weakest link approach in detail (enter fracture mechanics)1st step (individual flaw) - k=1failure, if since a random variate with distribution F(a)( )ϕθ ,,xaa cr≥i.e. with probability ( )( ) ( )( )ϕθϕθ ,,1)(,, xaFdaafxaaP caccrr−==≥ ∫∞e.g. flaw at location (x, y, z) with given orientationbut random size aϕθIf location and orientation are also random: ( )∫ ∫ΩΩ−=Vc dVdaFVQ )(14111 πQ1 is the failure probability for a component with exactly one flaw of random size and orientation at a random location of  the component !!Karlsruhe Institute of Technology (KIT), IAM22 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011The weakest link approach in detail (enter fracture mechanics)2nd step (fixed number of flaws/dislocations/extrusions/…) - k=nflaws are independent (no interaction) 1 component with n flaws == n components, each with 1 flawconsider n components each with 1 flaw: each components fails with probability Q1survival probability 1-Qn for n components: ( )nn QQ 111 −=−number of flaws follows a Poisson distribution: M - average number of flaws per componentnn Qpprobability that a component has exactly n flaws and fails: Mnn enMp −=!Karlsruhe Institute of Technology (KIT), IAM23 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011The weakest link approach in detail (enter fracture mechanics)3rd step (summing it all up …)component with random number of flaws has failure probability: ( ) ( )1)1(010exp111!1 1 MQeeQenMQpP QMMnnMnnnnf −−=−=−−==−−∞=−∞=∑∑MQ1 is just the average number of critical flaws in the componentNow we need some information about the flaw size distribution F(a): ⎥⎥⎦⎤⎢⎢⎣⎡Ω⎟⎟⎠⎞⎜⎜⎝⎛−−= ∫ ∫ΩVmeqf dVdVP00 411exp1σσπand finally we obtain a size effect relation: ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛Ω⎟⎟⎠⎞⎜⎜⎝⎛−−= ∫ ∫ΩmmeffmVmeqf bVVdVdVP*0*00**0exp1exp1411exp1 σσσσσσσπreqraaF 2)( σ∝∝ −fracture mechanicsKarlsruhe Institute of Technology (KIT), IAM24 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011The size effect in the weakest link approacha look at geometry…what we learn from the final result: ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛Ω⎟⎟⎠⎞⎜⎜⎝⎛−−= ∫ ∫ΩmmeffmVmeqf bVVdVdVP*0*00**0exp1exp1411exp1 σσσσσσσπthe component strength distribution is a Weibull distribution with parameters m and bthe component size leads to a geometry dependence of the distribution parameter bit is possible to define a geometry-independent distribution parameter σ0the size effect is described by the volume integral Veff using a reference stress value σ*Karlsruhe Institute of Technology (KIT), IAM25 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011The weakest link model – ready for FE use( )[ ]⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡Ω⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎪⎪⎩⎪⎪⎨⎧⎪⎪⎭⎪⎪⎬⎫⎥⎥⎦⎤⎢⎢⎣⎡Ψ+⎟⎟⎠⎞⎜⎜⎝⎛−−= ∫ ∫Ω−−∈Vmnneqtf dVdVtP2120,00max411exp1σσπ τFracture at time t is governed by themaximum load in [0, t]! Numerical integration of stress field: Calculation of Pf(t) by FE-Postprocessing (self-developed STAU postprocessor) - non-linear problem! spontaneous failure( )''0 00 dttBneq∫ ⎟⎟⎠⎞⎜⎜⎝⎛=ΨτσσσSub-critical crack growthKarlsruhe Institute of Technology (KIT), IAM26 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011The local risk of rupture – a design toolConsider partial volume Vtusing the following events:F: given flaw is located in Vt P(F)=Vt/V and K: given flaw is critical P(K)=Q1We need the probability P(F|K):P(F|K) = P(FK)/P(K) (Bayes‘ theorem)We obtain the probability P(FK) from:( ) ( )( )( )( )KPFKPVPVPxtt VtV 00limin  flaw criticalin  flaw criticallim→→==rπand thus:Karlsruhe Institute of Technology (KIT), IAM27 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011The basic idea of STAU (FE interface)stress analysisfailure descriptionreliability / failure probability / local risk of fractureFinite Elementsfracture mechanicsWeibull-TheoryKarlsruhe Institute of Technology (KIT), IAM28 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011The interface to Finite Elements⎥⎥⎦⎤⎢⎢⎣⎡Ω⎟⎟⎠⎞⎜⎜⎝⎛−−= ∫ ∫ΩVmeqf dVdzyxVP00),,,,(411exp1σβασπwhat we have to calculate (simple case):from Finite Element stress analysis we obtain stress tensor at node n: ),,( nnnij zyxσwe use the interpolation functions to generate additional integration points0.2 0.4 0.6 0.8 10.20.40.60.81large difference (red) between linear interpolation and m=5...10 curves!integration of σm would be very inaccurate!Karlsruhe Institute of Technology (KIT), IAM29 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011The options for Finite Element modelsgeometry: 2D model (plane stress, plane strain)3D modelaxisymmetric modelanalysis: surface flawsvolume flawsinterfacesoptions: spontaneous fracturesub-critical crack propagationcyclic crack propagationstress gradientsR-curve behaviour (not all to combine!)Karlsruhe Institute of Technology (KIT), IAM30 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Results: Rolling contact fatigue (RCF) testSchmierstoff-TauchbadStahlscheibeKeramikrolle Si3N4ω1ω2FF lubri antSi3N4-diskSteel-diskDamage after 105 rotationslubricant: friction coefficient μ=0.085F=1700Nrelative slip: ~22%max. principal stress: ~1100 MPaStress distributionRCF tests: Iyas Khader, Fraunhofer Institute IWM, FreiburgKarlsruhe Institute of Technology (KIT), IAM31 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Results: STAU analysisContact damage:  initiation of macroscopic flawsPredicting of a certain flaw density on the surfaceSi3N4rollsubarea of surfaceSTAU: probability of the initiation of one flawin the considered subareaSize of subarea ↔ crack densityResults refer to crack density of 1 crack per 250 μm along the circumferenceKarlsruhe Institute of Technology (KIT), IAM32 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Results: Failure probability100 101 102 103 104 105 106 107 108 1090.99940.63210.12660.01810.00250.0003-10-8-6-4-20240 5 10 15 20 25ln(Z) Zln(ln(1/(1-Pf(Z))))Pf(Z)CCG waterCCG airSCG airf01 expmZPN⎡ ⎤⎛ ⎞⎢ ⎥= − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦Weibull CDFm – slope of the curveN0 - characteristic lifetime(63%-quantile)Z – no. of rotationsProbability to initiate one macroscopic crack every 250 μmHighest failure probability obtained for fatigue parameters in water. Karlsruhe Institute of Technology (KIT), IAM33 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Results: Failure probability100 101 102 103 104 105 106 107 108 1090.99940.63210.12660.01810.00250.0003-10-8-6-4-20240 5 10 15 20 25ln(Z) Zln(ln(1/(1-Pf(Z))))Pf(Z)CCG waterCCG airSCG airRelation with experimental crack density after 10 hInitiation probability low for parameters in (air) Karlsruhe Institute of Technology (KIT), IAM34 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Bootstrapping: assessing the uncertainty by resamplingUse original sample to get confidence intervals for failure probabilityH. Riesch-Oppermann et al., Engng. Fract. Mech.original samplebootstrap-populationbootstrap-samplesestimator (m, σu) or P_f bootstrap-replications for estimatorKarlsruhe Institute of Technology (KIT), IAM35 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Assessment of material uncertainties– statistical evaluation01002003004005000 200 400 600 800 1000Tσ(T)inert strength dataalumina F99,7 bootstrapresamplingBayesian posteriori analysis(generalized Maximum-Likelihood method)Karlsruhe Institute of Technology (KIT), IAM36 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Pooling: reduction in uncertainty by combining samplesScaling of strength data to ambient temperature01002003004005000 200 400 600 800 1000Tσ(T)scaled dataoriginal dataSample size enlarged by poolingKarlsruhe Institute of Technology (KIT), IAM37 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Reduction of scatter by intelligent pooling– ceramic component01002003004005000 200 400 600 800 1000Tσ(T)scaled dataoriginal datapooling; collect all resultsin one sample of sizen=5x15! original data at RT;sample of sizen=15 for inert strengthKarlsruhe Institute of Technology (KIT), IAM38 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Uncertainty assessment and reduction by pooling & resamplingKarlsruhe Institute of Technology (KIT), IAM39 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Prediction of scatter in Pf – bootstrap resultsPf=0.01Pf=0.0395% quantile5% quantilenote: large impact of crack propagation and 2-3 mag’s uncertainty in results!Karlsruhe Institute of Technology (KIT), IAM40 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Pooling: reduction in uncertainty by combining sampleslg Δσ++++ + ++ + ++ +lg N(1)0N(3)0N(2)0N(1)σΔ(2)σΔ(3)σΔ+ + ++ +++++ +++Increase number of data available for one load levelConversion usingWeibull-distribution:( )( ) (1) 0(1)0poolpooli iNN NN⎛ ⎞= ⎜ ⎟⎝ ⎠Relation between Δσ and N0 follows from S-N-curve fitNi(1)Ni(pool)Pooled sampleKarlsruhe Institute of Technology (KIT), IAM41 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011Pooled crack growth curves10-0710-0810-0910-1010-1110-1210-1310-14-0.5 -0.4 -0.3 -0.2 -0.1 0( )R=0.5R=0.1n=20n=31da/dN[m/Cycle]log(ΔK/KIc)Scatter (uncertainty in n) is decreased by poolingvariation of n with R remains10-0710-0810-0910-1010-1110-1210-1310-14-0.5 -0.4 -0.3 -0.2 -0.1 0da/dN[m/Cycle]n=23n=33log(ΔK/KIc)R=0.5 R=0.1unpooled pooledM. Härtelt et al., J.Am.Ceram.Soc., (2011), in pressKarlsruhe Institute of Technology (KIT), IAM42 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011SummaryCeramicsWLT statisticsfracture mechanicsthe role of microstructurepooling: using data efficientlygetting an idea about possible inhomogeneitiesuncertainties: (bootstrap) and Bayes(modelling uncertainties)Karlsruhe Institute of Technology (KIT), IAM43 25.08.2011 H. Riesch-Oppermann  KIT-IFMA Summerschool 2011AcknowledgementsFinancial support by the “Deutsche Forschungsgemeinschaft” (DFG) isgratefully acknowledged. The work was performed within the framework of the Collaborative Research Centre 483 “High performance sliding and friction systems based on advanced ceramics” at the University of Karlsruhe (now: Karlsruhe Institute of Technology KIT).High performancesliding and friction systemsbased on advanced ceramics

Uncertainties in mechanics and materials research. Application in the field of materials research

https://publikationen.bibliothek.kit.edu/230083883/3815855

Uncertainties in mechanics and materials research. Application in the field of materials research

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