Post on 16-Jan-2016
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:
random inclusions at
m scale
Aggregate:
random inclusions at
mm scale
Outline
Introduction
Motivation: how we analyze heterogeneity 1. Simulated microstructures2. 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 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
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
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.
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
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 Models
Moving Window GMC:
• Equivalent microstructure gives mechanical properties at a location:
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23
33
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• 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
LocalProperties
CohesiveInterface
Local damage &
degradation
Interface law
Strain-Softening FE model
Determines global deformation &
failure behavior
MicrostructuralImage
Moving-Window Models
Moving Window GMC: Sample Results
digitize
Moving-Window GMC: Contour plot of Elastic modulus in x2 direction
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
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
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
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:
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
Multi-Scale Model Development
Model Results & Discussion
Conclusions & Future Work
Direct Tension Experiments:
Determination of bond tensile strength
Model Results & Discussion
Model Results & Discussion
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
Model Results & Discussion
Moving-Window GMC Model:
3x3 pixel windows
1000 m / pixel
Emortar = 25 GPa mortar = 0.2
Egranite = 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 strainfinite elements
Softening Parameters from GMC
Stochastic Interface Properties in GMC: i = (1 + ni) i
Model Results & Discussion
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
Model Results & Discussion
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
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
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
Model Results & Discussion
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
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-0332356
• Center for Advanced Cement-Based Materials