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

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

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

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

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

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

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

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

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

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.

Moving-Window Models Advantages: 1.Image-based, so no assumptions about components are necessary 2.Results in smooth material properties, suitable for simulation and FEM 3.Computationally efficient

Moving-Window Models Analysis of Windows: Generalized Method of Cells (GMC) “Subcells” (pixels, single material) are grouped into “Unit Cells” (windows, predefined pixel size) Results: approximation of constitutive properties: GMC approximates the mechanical properties of a repeating composite microstructure 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

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%

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

Moving-Window Models Moving Window GMC: Sample Results digitize Moving-Window GMC: Contour plot of Elastic modulus in x 2 direction

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

Model Development Moving Window GMC: interfacial damage Cohesive interfacial debonding is used to model interfacial damage Objective: incorporate ITZ into model  tt w Area under curve = G f  interface mortar pixel aggregate pixel w

 tt GfGf w 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       With: w is additional displacement at subcell interfaces in GMC

Model Development Moving Window GMC: window boundary conditions Unidirectional strain conditions are used to examine window behavior Example: window behavior with increasing  22 and  33 Apply x 3 strain Apply x 2 strain

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

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

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:  11 x 2 axis 22 c i-eff = effective property in principle direction c i-2 = GMC property, x 2 dir c i-3 = GMC property, x 3 dir

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

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

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

Model Results & Discussion Moving-Window GMC Model: 3x3 pixel windows 1000  m / pixel E mortar = 25 GPa mortar = 0.2 E granite = 60 GPa granite = 0.25

Model Results & Discussion Sample GMC-FE Analysis: FE Model Parameters: 37x37 element mesh 1000  m square elements Displacement increment 4 node, plane strain finite elements Softening Parameters from GMC Stochastic Interface Properties in GMC:  i = (1 + n i )  i

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

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

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

Model Results & Discussion Stochastic Analysis: Interface Fracture Energy Histogram

Model Results & Discussion Sample GMC-FE Analysis: Stochastic Results  Peak Stress: Experiments (11):  = 1.72 MPa  = 0.36 MPa Simulations (50):  = 1.61 MPa  = 0.04 MPa

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

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

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

Acknowledgements National Science Foundation Grant # CMS Center for Advanced Cement-Based Materials