A Multi-Scale Mechanics Method for Analysis of Random Concrete Microstructure

1. A Multi-Scale Mechanics Method for Analysis of Random Concrete Microstructure David Corr Nathan Tregger Lori Graham-Brady Surendra Shah Collaborative Research: Northwestern University Center for Advanced Cement-Based Materials Johns Hopkins University National Science Foundation Grant # CMS-0332356

2. Outline Introduction: Concrete Heterogeneity Motivation Multi-Scale Model Development Model Results & Discussion Conclusions & Future Work

3. Introduction Structural Analysis: • Typically uses homogeneous properties • Sufficient for average structural behavior However: • In extreme events, local maxima in stress and strain are of interest • Strongly dependent on heterogeneous microstructure and mechanical properties

4. Introduction Concrete Material Heterogeneity: Microscale: Nanoscale: Mesoscale: Hydration Products: random inclusions at nm scale Entrained Air Voids: random inclusions at mm scale Aggregate: random inclusions at mm scale

5. Outline Introduction Motivation: how we analyze heterogeneity 1. Simulated microstructures 2. Microstructural images Multi-Scale Model Development Model Results & Discussion Conclusions & Future Work

6. Motivation: Simulated Materials Simulated Materials: numerical representations of real materials At many length scales: 1. Angstrom/nanoscale: Molecular Dynamics 2. Microscale: hydration models: NIST model, HYMOSTRUC (Delft) 3. Mesoscale: particle distributions in a volume Advantages: 1. Computer-based “virtual experiments” 2. Inexpensive computational power Disadvantages: Assumptions must be made: 1. Size and shape of components 2. Particle placements 3. Dissolution & hydration rates, extents NIST Monograph http://ciks.cbt.nist.gov/~garbocz/monograph

7. Motivation: Microstructural Image Analysis Microstructure Image Analysis: using “images” of material structure to examine heterogeneity For mechanical properties, images can digitized and used as FE meshes: 1. Pixel methods: each pixel is a finite element 2. Object Oriented FEM (OOF): NIST software package 3. Voronoi cells method: hybrid finite element method Advantages: 1. FE method is well-established and robust 2. No assumptions about particle geometry 3. Applicable on any “image-able” length scale Disadvantages: 1. Computationally intensive 2. Subject to limitations of image 3. Singularities at pixel corners 4. Local properties are not unique: - dependent on boundary and loading conditions NIST OOF http://www.ctcms.nist.gov/oof/

8. Outline Introduction Motivation Multi-Scale Model Development Model Results & Discussion Conclusions & Future Work

9. Cohesive Interface Local damage & degradation Moving-Window GMC Model Represents local behavior of microstructure Interface law Local Properties Strain-Softening FE model Determines global deformation & failure behavior Model Development Multi-scale Microstructure Model: schematic Microstructural Image

10. Moving-Window Models Moving-Window Models image-based methods that address limitations of other methods to examine material heterogeneity Theory: for any location within a microstructure, use a finite portion (window) of the surrounding microstructure to estimate local properties Procedure: 1. Digitize microstructural image & define a moving window size 2. Scan window across microstructure, moving window 1 pixel at a time 3. For each window stop, use analysis tool to define local properties. 4. Map the local properties to an “equivalent microstructure” for subsequent analysis.

11. Moving-Window Models • Advantages: • Image-based, so no assumptions about components are necessary • Results in smooth material properties, suitable for simulation and FEM • Computationally efficient

12. Moving-Window Models Analysis of Windows: Generalized Method of Cells (GMC) GMC approximates the mechanical properties of a repeating composite microstructure • “Subcells” (pixels, single material) are grouped into “Unit Cells” (windows, predefined pixel size) • Results: approximation of constitutive properties: • FEM vs. GMC (inter-element boundary conditions): • FEM: requires exact displacement boundary continuity, no traction continuity • GMC: requires continuity on average for both traction and displacement

13. Moving-Window Models Moving Window GMC: • Equivalent microstructure gives mechanical properties at a location: • Equivalent microstructure features: • Includes local anisotropy and heterogeneity from original microstructure • Results can be used two ways: • Direct analysis with FEM • Input to stochastic simulation of mechanical properties • Using GMC on heterogeneous, non-periodic microstructure is an approximation: • Recent studies show errors in GMC approximation less than 1%

14. Cohesive Interface Local damage & degradation Moving-Window GMC Model Represents local behavior of microstructure Interface law Local Properties Strain-Softening FE model Determines global deformation & failure behavior Model Development Multi-scale Microstructure Model: schematic Microstructural Image

15. Moving-Window Models Moving Window GMC: Sample Results digitize Moving-Window GMC: Contour plot of Elastic modulus in x2 direction

16. Cohesive Interface Local damage & degradation Moving-Window GMC Model Represents local behavior of microstructure Interface law Local Properties Strain-Softening FE model Determines global deformation & failure behavior Model Development Multi-scale Microstructure Model: schematic Microstructural Image

17. Model Development Moving Window GMC: interfacial damage • Cohesive interfacial debonding is used to model interfacial damage • Objective: incorporate ITZ into model mortar pixel s st s w interface Area under curve = Gf w aggregate pixel

18. Model Development Moving Window GMC: interfacial damage • Cohesive interface present at every interface within window: • Cohesive properties vary depending on type of interface: • measured experimentally or estimated from literature s With: st    Gf  w is additional displacement at subcell interfaces in GMC w

19. Model Development Moving Window GMC: window boundary conditions • Unidirectional strain conditions are used to examine window behavior • Example: window behavior with increasing e22 and e33 Apply x3 strain Apply x2 strain

20. Cohesive Interface Local damage & degradation Moving-Window GMC Model Represents local behavior of microstructure Interface law Local Properties Strain-Softening FE model Determines global deformation & failure behavior Model Development Multi-scale Microstructure Model: schematic Microstructural Image

21. Model Development Moving Window GMC: local property database • FEM is supplied with local properties, as predicted from GMC • Complete s-e behavior not feasible because of storage restrictions • Solution: supply orthotropic secant moduli at regular intervals • FEM can interpolate to reconstruct approximate secant modulus:

22. e1 q x2 axis e2 Model Development Moving Window GMC: Strain-Softening FEM • Current SS-FEM model is for monotonic tensile loading • Softening on plane orthogonal to principle tensile strain • GMC properties incorporated with a strain angle approximation: ci-eff = effective property in principle direction ci-2 = GMC property, x2 dir ci-3 = GMC property, x3 dir

23. Outline Introduction Motivation Multi-Scale Model Development Model Results & Discussion Conclusions & Future Work

24. Model Results & Discussion Direct Tension Experiments: Determination of bond tensile strength

25. 75 mm HCP w/c = 0.35 Granite 75 mm 25 mm 38 mm Model Results & Discussion Sample GMC-FE Analysis: Direct Tension Experiments Symmetric Digitized Microstructure 37 x 37 pixels

26. Model Results & Discussion Moving-Window GMC Model: 3x3 pixel windows 1000 mm / pixel Emortar = 25 GPa nmortar = 0.2 Egranite = 60 GPa ngranite = 0.25

27. Model Results & Discussion Sample GMC-FE Analysis: 4 node, plane strain finite elements Softening Parameters from GMC Stochastic Interface Properties in GMC: si = (1 + ni) si • FE Model Parameters: • 37x37 element mesh • 1000 mm square elements • Displacement increment

28. Model Results & Discussion Sample GMC-FE Analysis: Results Comparison: Deterministic interface properties & experiments

29. Model Results & Discussion GMC-FE Analysis: Secant Modulus degradation

30. Model Results & Discussion Stochastic GMC-FE Analysis: Procedure • Parameters governing debonding are uncertain • Randomly generated, 10% c.o.v. for each parameter • Uncertainty defined before moving-window analysis • Look at effect of uncertainty in fracture properties on global specimen behavior

31. Model Results & Discussion Stochastic Analysis: Interface Fracture Energy Histogram

32. Model Results & Discussion Sample GMC-FE Analysis: Stochastic Results Peak Stress: Experiments (11): m = 1.72 MPa s = 0.36 MPa Simulations (50): m = 1.61 MPa s = 0.04 MPa e

33. Outline Introduction Motivation Multi-Scale Model Development Model Results & Discussion Conclusions & Future Work

34. Conclusions Moving-Window models address shortcomings of other heterogenous material models: • No assumptions about geometry of material components necessary • Unique properties • Computationally efficient Current multiscale model: • Cohesive debonding • Moving-Window GMC • Strain-softening FEM • Stochastic interface properties

35. Future Work • 3D microstructure models • Straightforward extension of MW-GMC and FEM • Data storage a problem • Compressive Behavior • Stochastic Simulation

36. Acknowledgements • National Science Foundation Grant # CMS-0332356 • Center for Advanced Cement-Based Materials