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
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:
Nanoscale:
Microscale:
Hydration Products:
random inclusions at
nm scale
Entrained Air Voids:
random inclusions at
mm scale
Mesoscale:
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
http://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 OOF
http://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
Microstructural
Image
Cohesive
Interface
Local
damage &
degradation
Interface
law
Moving-Window
GMC Model
Represents local
behavior of
microstructure
Local
Properties
Strain-Softening
FE model
Determines global
deformation & failure
behavior
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)
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:
 ij  Cijklij
  22  c22

 

 33   c32
   c
 23   42
c23
c33
c43
c24   22 
 
c24   33 
c44   23 
• 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:
  22 ( x, y )  c22 ( x, y ) c23 ( x, y ) c24 ( x, y )  22 ( x, y ) 

 


  33 ( x, y )   c32 ( x, y ) c33 ( x, y ) c34 ( x, y )   33 ( x, y ) 
  ( x , y )  c ( x, y ) c ( x, y ) c ( x, y )    ( x , y ) 
43
44
 23
  42
 23

• 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
Microstructural
Image
Cohesive
Interface
Local
damage &
degradation
Interface
law
Moving-Window
GMC Model
Represents local
behavior of
microstructure
Local
Properties
Strain-Softening
FE model
Determines global
deformation & failure
behavior
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
Microstructural
Image
Cohesive
Interface
Local
damage &
degradation
Interface
law
Moving-Window
GMC Model
Represents local
behavior of
microstructure
Local
Properties
Strain-Softening
FE model
Determines global
deformation & failure
behavior
Model Development
Moving Window GMC: interfacial damage
• Cohesive interfacial debonding is used to model interfacial damage
• Objective: incorporate ITZ into model
mortar pixel

t

w
Area under curve = Gf
interface
w
aggregate pixel
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
w  R

With:
t



0
e  t

2
 

1
R    t 
    1   e   t

 
Gf

w
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 x3 strain
3.5
10
3
9
33- 33
8
2.5
6
 (MPa)
x3 direction
7
5
4
2
1.5
3
1
2
22- 22
0.5
1
0
0
1
2
3
4
5
x2 direction
6
7
8
9
10
0
0
Apply x2 strain
0.5
1

1.5
2
-4
x 10
Model Development
Multi-scale Microstructure Model: schematic
Microstructural
Image
Cohesive
Interface
Local
damage &
degradation
Interface
law
Moving-Window
GMC Model
Represents local
behavior of
microstructure
Local
Properties
Strain-Softening
FE model
Determines global
deformation & 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:
4
3.5
4.5
4
3
3.5
Secant Modulus (MPa)
33- 33
2.5
 (MPa)
x 10
2
1.5
1
22- 22
x3
3
2.5
2
1.5
x2
1
0.5
0.5
0
0
0.5
1

1.5
2
-4
x 10
0
0
0.5
1

1.5
2
-4
x 10
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
cieff  ci2 sin 2 (q )  ci3 cos2 (q )
q
x2 axis
2
ci-eff = effective property in
principle direction
ci-2 = GMC property, x2 dir
ci-3 = GMC property, x3 dir
Outline
Introduction
Motivation
Multi-Scale Model Development
Model Results & Discussion
Conclusions & Future Work
Model Results & Discussion
Direct Tension Experiments:
Determination of bond tensile strength
Model Results & Discussion
Sample GMC-FE Analysis: Direct Tension Experiments
Symmetric Digitized Microstructure
75
mm
HCP
w/c = 0.35
Granite
25
mm
75
mm
38 mm
37 x 37 pixels
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
Model Results & Discussion
Sample GMC-FE Analysis:
4 node, plane strain
finite elements
Softening Parameters
from GMC
Stochastic Interface
Properties in GMC:
i = (1 + ni) i
FE Model Parameters:
• 37x37 element mesh
• 1000 mm square elements
• Displacement increment
Model Results & Discussion
Sample GMC-FE Analysis: Results
Comparison: Deterministic interface properties & experiments
1.5
bulk  (MPa)
1
0.5
0
0
1
bulk 
2
-4
x 10
Model Results & Discussion
GMC-FE Analysis: Secant Modulus degradation
4
x 10
6
35
5.5
y-position (pixels)
30
5
4.5
25
4
20
3.5
3
15
2.5
10
2
5
1.5
5
10
15
20
25
x-position (pixels)
30
35
1
MPa
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
Mortar - aggregate
Mortar - mortar
Aggregate - Aggregate
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
Model Results & Discussion
Stochastic Analysis: Interface Fracture Energy Histogram
60
50
Frequency
40
30
20
10
0
0.5
1
1.5
2
2.5
3
Fracture Energy (N/m)
3.5
4
Model Results & Discussion
Sample GMC-FE Analysis: Stochastic Results
1.8
Mean
Maximum
Minimum
1.6
Peak Stress:
1.4
Experiments (11):
m = 1.72 MPa
 = 0.36 MPa
Bulk  (MPa)
1.2
1
Simulations (50):
m = 1.61 MPa
 = 0.04 MPa
0.8
0.6
0.4
0.2
0
0
0.5
1
Bulk m

1.5
2
-4
x 10
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