Multi-scale methods in material mechanics

Characterization of Heterogeneous MaterialsbyMulti-Scale SimulationsP. Wriggers,I. Temizer, M. Hain & S. LöhnertInstitute of Continuum MechanicsLeibniz Universität HannoverLjubljana - 24.3.2009Heterogeneous Material: Concrete[cm][m][µm][nm]

Micro-Structure of MaterialsCement pasteGranular material (e.g. sand)Aluminium-Boron-Alloy (x250) Woven fabrics Contact surfacesOutline and Goals• Micro-scale modelling leads to better understanding offailure mechanism and hence leads to better materialdescriptions• Material behaviour is different on different length scales=> Multi-Scale Analysis• Difficulty: Not all data are known for micro-scalecomputation (e.g. constitutive data)• Homogenization is often necessary for practicalapplications (possible way out: model adaptivity)• Statistical analysis is necessary

Modelling of Microstructure/Homogenizationvolumetric:Torquato (2002)Nemat-Nasser (2004)Zohdi/Wriggers (2005)contact:Orlik (2004)Stupkiewicz (2007)Temizer/Wriggers (2008)thin structures:Geers et al. (2007)Hirschberger et al. (2008)Grytz/Meschke (2008)Löhnert/Wriggers (2008)• Bulk Inhomogeneities: high oscillatory stress fields• Volumetric homogenization:➡ reduces the complexity of the macrostructural analysis➡ construction of an effective material micromechanical testingHomogenization Procedure for a MaterialValidationbyExperimentsMicrostructureMacroscopicMaterial EquationNumerical Modeland AnalyisUpscaling

Requirements for Homogenization• Statistical representativenessL1 >> L2 >> L3• 3-D Computations for quantitativeresults• Converging finite element model• Sufficient number of particles• Monte Carlo simulations forstatistical evaluation

• volume fraction2 Vol% unhydrated clinker84 Vol% hydration products14 Vol% pores• elastic propertiesAcker [2001]; Constantinides, Ulm [2004]• finite-element representation• principal stress64 µm64 µm64³ hexahedron elements820 000 DOFssolution time 5minHomogenizationnumerical SimulationHomogenizationGoal: Material parameters of effective material12

Elastic Behaviour„Natural“ b.c. (window-method)- RVE is embedded in a matrix of averaged stiffness- Discretization 3dHomogenization- correlation between mean values- volume average RVE- bounds for displacement and stress b.c.can also be obtained for f.e. simulations [Zohdi & Wriggers (2001)]Numerical homogenization- 3D isotropic material

• 3D isotropic material• 2D isotropic material- width of window- size of RVE• isotropy of RVE• 3D isotropic material:- probability density of 9 200 RVEs (each 820 000 DOFs)• comparison withexperimental dataexperiments from IBAC

• meso-scopic elastic constituents fluctuate statistically• inhomogeneous meso-scopic behavior160 mmone representation ofYoung‘s modulusone representation ofprincipal stressHomogenization of a Neo-Hooke MaterialRVE with particles73 particles915.864 dofsdeformation due todisplacement b.c.

Upscaling: Micro- and homogenised structurePlate with particles of different sizeDeformation and StressesLöhnert & Wriggers 2008Upscaling: Micro- and homogenised structurePlate with particles of different size216 particles1802490 dofsDeformed state and stressesLöhnert & Wriggers 2008

outlineSimulation of water permeabilityKoster, Brameshuber (2007)network of connected tubes- pore-size distribution

Simulation of water permeabilityDegreee of water fillingKoster, Brameshuber (2007)256 µmwh=0.70wh=0.80Simulation of damage due to freezingConstitutive model for water filled poresKinematic splitLinear elastic constitutive equationσ = C ɛ uConstitutive data of iceJ. Pohe 1993G. W. Scherer, 1999No visco-plastic effects dince not known formicro pores

Constitutive model for water filled poresthermal strainfreezing temperature depending on radiiStockhausen [1981]Damage at micro-structural levelPrincipal stressesDamageDefinition of homogenized damage

Statistical tests of 64 RVEs (wh=0.80)64 µmdamage within RVEPolygons for median value and standard deviationHain & Wriggers 2008Homogenized damage due to frostfluctuates statistically

HomogenizationLimits• RVE concept does not work for- material instabilities and localization- local buckling- occurance of cracks- dynamics• Way out- true multi-scale method- model adaptivityMulti-Scale Analysis

Multi-Scale AnalysisMulti-Scale Analysis

Multi-Scale AnalysisMulti-Scale Analysis

Material Map ApproachInterpolationMaterial Map ApproachVirtual experimentsat RVEstorage in a DatabaseComponent of the2. Piola-Kirchhoff stressesEffective strain energy

Material Map ApproachComparision map to direct RVE approachInelastic behavior of hydration productscontinuum mechanical approachfor a virgin materialfor a non-virgin materialVerfication through multi-scale model

Multi-scale analysis★ perform micro-structural analysis at eachintegration point★ ALGO:• Obtain strains and microstructures atGauss integration point• Apply• Nonlinear finite element solution ofmicrostructural model• Obtain effective values• Compute on Macro-levelMulti-scale analysissecant moduli is obtained directly from homogenizationNumerical Simulation(on 8 processor SMP)Model 1:128 multi-scale elements16³ RVE per integration point1 week CPU time (speed-up 7.4)820 MB history data64 µmModel 2:16 multi-scale elements64³ RVE per integration point6 weeks CPU time (speed-up 7.2)2.1 GB history datadamage within specimenon micro-scale

Multi-scale analysismaterial parametersNumerical effort for 2 scalesEmbedded multi-scale: FE 2• Micro-scale model 100x100x100 FE• Same resolution for macro-scale• Each macro-scale finite elementcontains complete micro-scaleCoupled multi-scale computation• Mikro-Skale:• Macro-Scale:FEFEDevelopment of adaptive methodsfor efficient FE 2 multi-scale analysis

Adaptive multi-scale analysisAdaptive multi-scale analysis

Adaptive multi-scale analysisdoes not result in a gradient-sensitivity for homogeneous micro-structuresGradient sensitivityAdaptive multi-scale analysis

Adaptive multi-scale analysisAdaptive multi-scale analysis

Adaptive multi-scale analysisAdaptive multi-scale analysis

Adaptive multi-scale analysisAdaptive multi-scale analysisLöhnert & Belytschko 2007Löhnert et al. 2008

Conclusion• Homogenization schemes:– Constitutive parameters– Random distribution of constitutive data• FE 2 is expensive– Understanding of material behaviour at microscale– Pre-computed maps of constitutive data– Adaptive schemesReferences• M. Hain and P. Wriggers: Computational Homogenization of micro- structuralDamage due to Frost in Hardened Cement Paste, Finite Elements in Analysis andDesign, 44, 233-244, 2008• M. Hain and P. Wriggers: On the numerical homogenization of hardened cementpaste, Computational Mechanics, 42, 197-212, 2008• I. Temizer and P. Wriggers: An adaptive method for homogenization in orthotropicnonliner elasticity, CMAME, 196, 3409-3423, 2007• S. Löhnert and P. Wriggers:” Effective behaviour of elastic heterogeneous thinstructures at finite deformations” Computational Mechanics, 41, 595-606, 2008• I. Temizer and P. Wriggers: “On a mass conservation criterion in micro-to-macrotransitions”, Journal of Applied Mechanics - Transaction of the ASME, 75, No.054503, 2008