a multi-scale mechanics method for analysis of random concrete microstructure david corr nathan...

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

David CorrNathan Tregger

Lori Graham-BradySurendra Shah

Collaborative Research:

Northwestern University Center for Advanced Cement-Based Materials

Johns Hopkins University

National Science Foundation Grant # CMS-0332356

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:


m scale

Aggregate:


mm scale

Outline

Introduction

Motivation: how we analyze heterogeneity 1. Simulated microstructures2. Microstructural images




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 Monographhttp://ciks.cbt.nist.gov/~garbocz/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 OOFhttp://www.ctcms.nist.gov/oof/

Outline

Introduction

Motivation




Model Development

Multi-scale Microstructure Model: schematic

Moving-Window GMC Model

Represents local behavior of

microstructure

LocalProperties

CohesiveInterface

Local damage &

degradation

Interface law

Strain-Softening FE model

Determines global deformation &

failure behavior

MicrostructuralImage

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 localproperties

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.


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


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

23

33

22

444342

243332

242322

23

33

22

ccc

ccc

ccc

ijijklij C

• 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 GMC:

• Equivalent microstructure gives mechanical properties at a location:

),(

),(

),(

),(),(),(

),(),(),(

),(),(),(

),(

),(

),(

23

33

22

444342

343332

242322

23

33

22

yx

yx

yx

yxcyxcyxc

yxcyxcyxc

yxcyxcyxc

yx

yx

yx

• 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




microstructure

LocalProperties

CohesiveInterface

Local damage &

degradation

Interface law



failure behavior



Moving Window GMC: Sample Results

digitize

Moving-Window GMC: Contour plot of Elastic modulus in x2 direction

Model Development




microstructure

LocalProperties

CohesiveInterface

Local damage &

degradation

Interface law



failure behavior


Model Development

Moving Window GMC: interfacial damage• Cohesive interfacial debonding is used to model interfacial damage

• Objective: incorporate ITZ into model

t

wArea under curve = Gf

interface

mortar pixel

aggregate pixel

w

t

Gf

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

Rw

te

te

tR

2

1

1

0

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

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

x2 direction

x 3 d

irect

ion

0 0.5 1 1.5 2

x 10-4

0

0.5

1

1.5

2

2.5

3

3.5

(

MP

a)

22

-22

33

-33

Apply x3 strain

Apply x2 strain

Model Development



microstructure

LocalProperties

CohesiveInterface

Local damage &

degradation

Interface law



failure behavior



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:

0 0.5 1 1.5 2

x 10-4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4

Sec

ant M

odul

us (

MP

a)

x2

x3

0 0.5 1 1.5 2

x 10-4

0

0.5

1

1.5

2

2.5

3

3.5

(

MP

a)

22

-22

33

-33

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

x2 axis

2

)(cos)(sin 23

22 iieffi ccc

ci-eff = effective property in principle direction

ci-2 = GMC property, x2 dirci-3 = GMC property, x3 dir

Outline

Introduction

Motivation




Direct Tension Experiments:

Determination of bond tensile strength



Sample GMC-FE Analysis: Direct Tension Experiments

Symmetric Digitized Microstructure

HCPw/c = 0.35

Granite

38 mm

25 mm

75 mm

75 mm

37 x 37 pixels


Moving-Window GMC Model:

3x3 pixel windows

1000 m / pixel

Emortar = 25 GPa mortar = 0.2

Egranite = 60 GPa granite = 0.25


Sample GMC-FE Analysis:

FE Model Parameters:

• 37x37 element mesh

• 1000 m square elements

• Displacement increment

4 node, plane strainfinite elements

Softening Parameters from GMC

Stochastic Interface Properties in GMC: i = (1 + ni) i


Sample GMC-FE Analysis: Results

0 1 2

x 10-4

0

0.5

1

1.5

bulk

bulk

(

MP

a)Comparison: Deterministic interface properties & experiments


1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6x 10

4

5 10 15 20 25 30 35

5

10

15

20

25

30

35

x-position (pixels)

y-po

sitio

n (p

ixel

s)MPa

GMC-FE Analysis: Secant Modulus degradation


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

Parameter Mean Std. Dev. Parameter Mean Std. Dev. Parameter Mean Std. Dev.

α 300.0 30.00 α 60.0 6.00 α 90.0 9.00

β 96.0 9.60 β 30.0 3.00 β 50.0 5.00

σt 0.8 0.08 σt 2.3 0.23 σt 10.0 1.00

Gf (model) 1.9 0.44 Gf (model) 78.4 16.9 Gf (model) 81.3 14.6

Gf (literature) 1.4 Gf (literature) 72.3 Gf (literature) 76.8

Mortar - aggregate Mortar - mortar Aggregate - Aggregate


Stochastic Analysis: Interface Fracture Energy Histogram

0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

40

50

60

Fracture Energy (N/m)

Fre

quen

cy

0 0.5 1 1.5 2

x 10-4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Bulk

Bul

k

(M

Pa)

MeanMaximumMinimum


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




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-0332356

• Center for Advanced Cement-Based Materials

a multi-scale mechanics method for analysis of random concrete microstructure david corr nathan...

Documents

nist model

window modelsadvantages

scan window

limitations of image

moving window size

finite portion window

local properties

heterogeneous microstructure