Morphological Models of Random Structures [1st ed. 2021] 3030754510, 9783030754518
This book covers methods of Mathematical Morphology to model and simulate random sets and functions (scalar and multivar
288 33 29MB
English Pages 939 Year 2021
Preface
Acknowledgements
Symbols and Notations
Contents
1 Introduction
1.1 Use of a probabilistic approach
1.2 Aims of probabilistic models
1.2.1 Descriptive aspect
1.2.2 Predictive aspect
1.3 Types of models
1.4 Model construction
1.4.1 Characterization of a random model
1.4.2 Choice of basic assumptions
1.4.3 Computation of the functional T (K)
1.5 Some general properties of the models
1.6 Organization of the book
Part I Tools for Random Structures
2 Introduction to Random Closed Sets and to Semi-Continuous Random Functions
2.1 Introduction
2.2 Introduction to random closed sets (RACS)
2.2.1 Recall of basic definitions in probability theory
2.2.2 Definition of the random closed sets
2.2.3 The Choquet capacity
2.2.4 Measurability of a random closed set
2.2.5 Operations on random closed sets
2.2.6 Independent random sets
2.2.7 Spatial law
2.2.8 Geometrical interpretation of the Choquet capacity in the Euclidean case
2.3 Introduction to Random Functions
2.3.1 Semi continuous functions
2.3.2 Subgraph of a function
2.3.3 Choquet topology on Φ_f
2.3.4 The Borel σ algebra on Φ_f and Φ_g , and the Choquet capacity for RF
2.3.5 Multivariate Random Functions
2.3.6 Erosion and Dilation of functions
3 Quantitative Analysis of Random Structures
3.1 Introduction
3.2 Basic morphological transformations
3.2.1 Types of structures
3.2.2 Notion of structuring element
3.2.3 Dilations and Erosions
3.3 Basic morphological measurements
3.3.1 Definition of Minkowski functionals
3.3.2 Stereological aspects
3.3.3 Steiner formula
3.3.4 Specific measurements
3.3.5 Counting procedure and the Euler relations
3.3.6 Minkowski tensors
3.4 Size distributions
3.4.1 Axioms of size distributions; openings and closings
3.4.2 Linear size distributions
3.4.3 Two and three dimensional size distributions
3.4.4 Size distributions for functions
3.4.5 Stereological reconstruction of the size distribution of spheres
3.5 Morphological analysis of the spatial distribution
3.5.1 Random sets
3.5.2 Multi component random sets
3.5.3 Random functions
3.5.4 Covariance of orientations of vector fields
3.5.5 Definition of a Statistical Representative Volume Element (RVE) and problems of estimation
3.5.6 Distance functions
3.5.7 Random graphs
3.6 Geodesic criteria and connectivity
3.7 Robustness of morphological measurements to noise
3.8 Shape and texture classification and recognition
3.8.1 Automatic recognition of non metallic inclusions
3.8.2 3D Shape classification of particles with complex shapes
3.8.3 Texture analysis and classification
3.9 Conclusion
3.10 Exercises
3.10.1 Minkowski tensor of a cylinder
3.10.2 Variogram γ_1(h)
3.10.3 Correlation functions of two phase random composites
3.10.4 Specific measurements and derivatives of morphological moments
3.10.5 Covariance of a noisy binary image
3.10.6 Moments of inertia in R^3
Part II Models of Random Structures
4 Excursion Sets of Gaussian RF
4.1 Introduction
4.2 Random sets and truncated RF: the Gaussian case.
4.2.1 Construction of truncated Gaussian random sets
4.2.2 Second order central correlation function
4.2.3 Third order central correlation function
4.2.4 Order m central correlation function
4.3 Application to a food microstructure
4.4 Three Phase random model
4.4.1 Introduction
4.4.2 Three components models based on independent random sets
4.4.3 Three component model based on the Boolean model
4.4.4 Three component model based on truncated Gaussian random functions
4.5 Exercise
4.5.1 Combination of independent random sets
5 Stochastic Point Processes and Random Trees
5.1 Introduction
5.2 Tools to characterize point processes
5.3 The Poisson point process
5.3.1 Definition and properties of the Poisson point process
5.3.2 Distribution of points of a homogeneous Poisson point process
5.3.3 Simulation of a homogeneous Poisson point process
5.3.4 Distribution of points of a non stationary Poisson point process
5.4 Hard-core point processes
5.5 The Cox point process
5.5.1 Definition and main properties
5.5.2 Cox process with a random variable θ
5.5.3 Large scale behavior of the Cox process
5.6 Point processes on iteration of Boolean varieties
5.6.1 Two steps Boolean varieties
5.6.2 Three steps Boolean varieties in R^3
5.7 Gibbs point processes
5.8 Determinantal point processes
5.9 Introduction to models of random trees by branching processes
5.9.1 Discrete branching process
5.9.2 Continuous branching process
5.9.3 Discrete random trees generated by random functions
5.9.4 Biological applications
5.10 Exercises
5.10.1 Laplace transforms for sections of discs and of spheres
5.10.2 A model of random tree embedded in the Euclidean space
6 Boolean Random Sets
6.1 Introduction
6.2 Construction of the Boolean model
6.3 Choquet capacity
6.4 Some stereological properties
6.5 Connectivity numbers and counting
6.6 Connectivity and percolation of the Boolean model
6.7 IDRACS and semi Markovian RACS
6.8 Testing the Boolean model
6.8.1 Convexity of the primary grain A0
6.8.2 Dilation by convex sets
6.8.3 Infinite divisibility of the Boolean model
6.9 Identification of the Boolean model
6.10 Some examples of application
6.10.1 Boolean model of spheres
6.10.2 The Poisson Boolean model
6.10.3 Further applications of the Boolean model to materials
6.10.4 Multiphase textures
6.11 The Boolean model and the Poisson process
6.12 Cox Boolean model and multiscale models of RACS
6.12.1 Intersection of independent random sets
6.12.2 Introduction to the Cox Boolean model
6.12.3 Percolation of the Cox Boolean model
6.13 Boolean model and the Poisson varieties
6.13.1 Construction and properties of the linear Poisson varieties
6.13.2 Stable RACS
6.13.3 Boolean model on the linear Poisson varieties
6.13.4 Power laws variance scaling of the Boolean random varieties
6.13.5 Two steps iterated Poisson varieties
6.14 Multi component Boolean models
6.14.1 Construction of Boolean models with m components
6.14.2 Choquet capacity
6.14.3 Spatial law and covariances
6.14.4 Stability for the reunion
6.14.5 Multi component Boolean models on Poisson varieties
6.15 Exercises
6.15.1 Counting with the Euler relation
6.15.2 Hierarchical Boolean model on a mosaic
6.15.3 Fractal Boolean random set
6.15.4 A noise random set model
6.15.5 Simulation of percolating aggregates
6.15.6 Transverse and longitudinal covariances of a Boolean model of cylinders
6.15.7 Boolean varieties in R^2 and R^3 as limit case of Boolean models
7 Random Tessellations
7.1 Introduction
7.2 Reminder on random tessellations
7.3 Probability distributions of the classes
7.4 The Voronoi, Johnson-Mehl, and Laguerre tessellation
7.4.1 Voronoi tessellation
7.4.2 Johnson-Mehl and Laguerre random tessellations
7.5 Random tessellation generated by a geodesic distance
7.6 The Poisson tessellation
7.6.1 Definition and Choquet capacity of the Poisson hyperplanes
7.6.2 Stereological aspects of the Poisson hyperplanes
7.6.3 Characterization of the Poisson polyhedron containing the origin
7.6.4 Conditional invariance by erosion
7.6.5 Opening size distribution
7.7 Example of application to model concrete by multiscale Poisson polyhedra
7.8 The Cauwe tessellation
7.8.1 Construction of the Cauwe tessellation
7.8.2 Stereological properties
7.8.3 Choquet capacity on convex sets
7.8.4 Mixture of Poisson and Cauwe tessellations
7.9 Iteration of tessellations and the STIT model
7.10 Exercise
7.10.1 Superposition of random tessellations
8 The Mosaic Model
8.1 Introduction
8.2 Construction
8.3 Choquet capacity
8.4 Mosaic model in R^n
8.4.1 Calculation of P {N(K) = 1}
8.4.2 First order statistics
8.4.3 Second order statistics
8.4.4 Third order statistics
8.4.5 Higher order statistics
8.5 The Poisson mosaic
8.6 The STIT mosaic
8.7 The Mosaic random set
8.8 The multivariate mosaic model
8.9 Exercises
8.9.1 Multi component mosaic
8.9.2 Cross covariances of the multicomponent mosaic
8.9.3 Hierarchical mosaic model
9 Boolean Random Functions
9.1 Introduction
9.2 Construction of the Boolean random functions BRF
9.3 Choquet capacity of the BRF
9.4 Supremum stability and infinite divisibility
9.5 Characteristics of the primary functions
9.5.1 Transformation by anamorphosis
9.5.2 Moments of Z0∨(K) and mathematical expectation of theanamorphosed of Z0∨(K)
9.5.3 Geometrical covariogram of the primary function
9.6 Some stereological aspects of the BRF
9.7 BRF and counting
9.8 Identification of a BRF model
9.9 Test of the BRF
9.9.1 Convexity of AZ0t(z)
9.9.2 Change of support on convex sets
9.9.3 Supremum infinite divisibility
9.10 Examples of application to rough surfaces
9.10.1 Simulation of the evolution of surfaces and of stresses during shot peening
9.10.2 Simulation of the roughness transfer on steel sheets
9.11 Modeling three phase random media with some contact constraints
9.11.1 Microstructure of a current collector
9.11.2 Use of a Boolean random function
9.11.3 Use of truncated Gaussian random functions
9.12 Multiscale Boolean random functions
9.13 The Boolean varieties RF
9.14 The multivariate BRF and Varieties
9.14.1 Construction
9.14.2 Choquet capacity
9.14.3 Spatial law
9.14.4 Supremum stability
9.14.5 Distribution of maxima
9.14.6 Multivariate Boolean varieties RF
9.15 Exercises
9.15.1 BRF with cylinder primary random functions
9.15.2 A hierarchical BRF model
9.15.3 BRF with cones anamorphosis
10 Random Tessellations and Boolean Random Functions
10.1 Introduction
10.2 Extensions of the Voronoi tessellation
10.2.1 Random tessellations defined from local metrics
10.2.2 Calculation of the probability P (K)
10.3 Extension to Johnson-Mehl and to Laguerre random tessellations
10.4 Random tessellations and Boolean random functions
10.4.1 Connection between metric based random tessellations and some BRF
10.4.2 General random tessellations built from BRF
10.4.3 Random tessellations with thick boundaries
10.5 Some indications on model identification
10.6 Conclusion
11 Dead Leaves Models: from Space Tessellations to Random Functions
11.1 Introduction
11.2 Sequential random tessellations
11.2.1 Infinitesimal Boolean random set
11.2.2 Construction of the sequential random tessellations (Dead Leaves tessellation)
11.2.3 Probabilistic properties
11.2.4 Linear size distributions
11.2.5 Specific parameters
11.2.6 Higher order probabilistic properties
11.2.7 Statistics on intact grains and bias correction
11.2.8 Example of application to polycristalline salt
11.2.9 Iteration of Dead Leaves tessellations
11.2.10 Multi scale Cox DLRT
11.3 Random packings generated by the Dead Leaves model
11.3.1 Intact grains of the Dead Leaves model for the time homogeneous case
11.3.2 Intact grains of the Dead Leaves model for the general case
11.3.3 Examples of models of dense packings
11.3.4 Size distributions for dense packings
11.3.5 Pair correlation function of intact grains centers
11.3.6 Conclusion
11.4 Color Dead Leaves
11.4.1 Construction of the Color Dead Leaves
11.4.2 First order statistics
11.4.3 Second order statistics
11.4.4 Third order statistics
11.4.5 Correlation functions
11.4.6 An example of two phase medium
11.4.7 Specific connectivity numbers of the Color Dead Leaves
11.4.8 Some potential applications of the Color Dead Leaves
11.5 The Dead Leaves Random Functions (DLRF)
11.5.1 The infinitesimal Boolean Random Functions (IBRF)
11.5.2 Construction of sequential RF from IBRF
11.5.3 Probabilistic properties of the DLRF
11.5.4 Moment P (g, t)
11.5.5 Summits of the DLRF
11.5.6 Intact primary functions
11.5.7 Estimation of the parameters of the DLRF and tests
11.6 Multivariate DLRF and varieties
11.6.1 Construction of the Multivariate DLRF
11.6.2 Multivariate distribution in point x
11.6.3 Bivariate distributions
11.6.4 Cross covariances
11.6.5 Multivariate DLRF built on Poisson varieties
11.7 Dead Leaves of rank m
11.8 Transparent Dead Leaves
11.9 Application to the morphology of powders
11.9.1 Mixtures of Boolean models
11.9.2 Use of The DLRT
11.9.3 Use of The DLRF for grey level images
11.10 Sequential RF with Markovian jumps (MJF)
11.10.1 Construction of MJF
11.10.2 Properties of the domains A0(t) = ∪u≤tA00(u)
11.10.3 Probabilistic properties of the MJF
11.11 Elements of simulation
11.12 Conclusions
11.13 Exercises
11.13.1 Exponential distribution of intercepts (DLRT)
11.13.2 Correlation between contiguous intercepts (DLRT)
11.13.3 Linear size distributions (DLRT)
11.13.4 Construction of a random set from a DLRT
11.13.5 Composite grains (Color Dead Leaves)
11.13.6 Exponential covariance of a non Markov random mosaic (DLRF)
11.13.7 Statistics of intact grains
11.13.8 A two scale model of platelets
12 Sequential Cox Boolean and Conditional Dead Leaves Models
12.1 Introduction
12.2 Sequential Cox Boolean RACS
12.2.1 Notations and definition of the two components sequential Cox Boolean model
12.2.2 Probabilistic properties
12.2.3 Simplified expressions for the Choquet capacity
12.2.4 First variant of the sequential Cox Boolean model
12.2.5 Second variant of the sequential Cox Boolean model
12.2.6 Multi components version of the Cox sequential Boolean model
12.3 Sequential Cox Boolean RF
12.3.1 Notations and definition of the two components Sequential Cox BRF
12.3.2 Probabilistic properties
12.3.3 Simplified expressions for the Choquet capacity
12.3.4 Multi components Sequential Cox mosaic BRF
12.3.5 Extension of the Sequential Cox mosaic BRF to the
continuous case
12.4 Conditional Dead Leaves tessellation
12.4.1 Construction of the model
12.4.2 Probabilistic properties
12.4.3 Simplified expressions of probabilistic properties
12.5 Conditional color Dead Leaves
12.5.1 Notations and definition of the model
12.5.2 Probabilistic properties
12.5.3 Simplified expressions of probabilistic properties
12.5.4 Simplified expression for the Choquet capacity of the set B0(t)
12.5.5 Probabilities Pi(t) of the conditional color Dead Leaves
12.5.6 Covariances of the conditional color Dead Leaves
12.5.7 Expressions for Pi(t) and for the covariances in some specific cases
12.6 Conclusion
13 Sequential Alternate Random Functions
13.1 Introduction
13.2 Sequential alternate RF
13.2.1 Construction of SARF
13.2.2 Construction of coupled SARF
13.2.3 Univariate distribution
13.2.4 Bivariate distribution
13.2.5 Apparent maxima and minima of the primary RF
13.3 Multiscale SARF
13.4 Example of application to modeling of the Electro Discharge Textures (EDT)
13.5 Multivariate SARF
13.6 SARF varieties
13.7 Elements of simulation
13.8 Exercise
14 Primary Grains and Primary Functions
14.1 Introduction
14.2 Random primary grains
14.2.1 Population of spheres and of ellipsoids in R^3
14.2.2 Random parallelepipeds
14.2.3 The Poisson polyhedra
14.2.4 Grains generated by the DLRT
14.2.5 Random aggregates
14.2.6 Cylinder primary RF
14.2.7 Restriction of a stationary RF to a random compact set
14.2.8 Some RF with spherical thresholds
14.2.9 Boolean RF with a compact support
14.2.10 Dead Leaves RF with a compact support
14.3 Exercises
14.3.1 Boolean aggregates of rectangles
15 Dilution Random Functions
15.1 Introduction
15.2 Construction of the multivariate Dilution random functions DRF
15.3 Convolution of a DRF
15.4 Characteristic functional of a multivariate DRF
15.5 Moments of a multivariate DRF
15.6 Example of application to TEM micrographs
15.6.1 A model for thick slices images
15.6.2 Covariance of the TEM images
15.6.3 Numerical method to estimate the transitive covariogram
15.7 Random tokens
15.8 Positive DRF
15.9 DRF of the number of grains covering x
15.9.1 Univariate distribution
15.9.2 Bivariate distribution
15.9.3 Trivariate distribution
15.9.4 Random set A_m
15.9.5 Random tessellation generated by the A_m
15.9.6 Excursion random sets of N(x)
15.10 Convergence of the DRF towards a Gaussian RF
15.11 Multivariate Dilution varieties RF
15.12 Exercises
15.12.1 The Dilution RF and the Cox process
15.12.2 Cox Dilution RF
15.12.3 Hybrid model of spheres and cylinders
15.12.4 Model of indentation
16 Reaction-Diffusion and Lattice Gas Models
16.1 Introduction
16.2 Reaction-Diffusion models
16.2.1 Reaction-Diffusion equations
16.2.2 Discrete Reaction-Diffusion models
16.3 Random Functions and the linearReaction-Diffusion model
16.4 Examples of simulations of non- linear
Reaction-Diffusion Random Functions
16.4.1 Schlögl model
16.4.2 Turing structures
16.4.3 The complex Ginzburg-Landau model
16.4.4 Modifications of the Ginzburg-Landau model
16.5 Lattice gas models
16.5.1 Basic rules
16.5.2 Some indications on the evolution equations
16.5.3 Boundary conditions
16.5.4 Application to flow in porous media
16.5.5 Application to simulations of random media
16.5.6 Multi species lattice gas models
16.6 Conclusion
17 Texture Segmentation by Morphological Probabilistic Hierarchies
17.1 Introduction
17.2 Morphological texture descriptors
17.3 Texture classification
17.4 Probabilistic texture segmentation
17.4.1 Watershed texture segmentation
17.4.2 Probabilistic hierarchical segmentation
17.4.3 Higher order probabilistic segmentation
17.4.4 Probabilistic distances between sets
17.4.5 Use of random markers
17.4.6 Random markers and higher order fusion of regions
17.5 Conclusion
Part III Random Structures and Change of Scale
18 Change of Scale in Physics of Random Media
18.1 Introduction
18.2 From microscopic to macroscopic
18.3 Homogeneous medium and heterogeneous medium
18.4 Practical interest of change of scale methods
18.5 Principle of calculation of effective properties
18.6 Exact results
18.6.1 Geometrical average estimate
18.6.2 Self-consistent and effective medium estimates
18.6.3 Composite spheres assemblage
18.7 Perturbation expansion in electrostatics
18.8 Formal expansion of the effective dielectric permittivity of random media
18.9 Perturbation approach in elasticity and calculation of the effective elastic tensor
18.10 Bounds of effective properties
18.10.1 Variational principles
18.10.2 Bounds of order 2N + 1 derived from the classical variational principle
18.11 Third order bounds of the dielectric permittivity
18.11.1 Reminder on third order bounds for RF
18.11.2 Third order bounds for the mosaic model
18.11.3 Third order bounds of the Dilution model
18.11.4 Transformation of basic models
18.11.5 Combination of the basic random functions models
18.11.6 A hierarchical model
18.12 Third order bounds of the real dielectric permittivity of random sets
18.13 Third order bounds of the complex dielectric permittivity and spectral measure of random sets
18.13.1 Bounds in the complex plane
18.13.2 Spectral measure of a random set
18.14 Third order bounds for elastic moduli of random sets
18.14.1 Third order bounds for two-phase composites
18.14.2 Bounds of the bulk modulus K
18.14.3 Bounds of the shear modulus G
18.14.4 Bounds of the Young’s modulus E
18.14.5 Bounds of the Poisson coefficient ν
18.14.6 Functions ζ_1(p) and η_1(p)
18.15 Third order bounds of some models of random sets
18.15.1 Bounds for the Boolean model
18.15.2 Bounds for the hard spheres model
18.15.3 Bounds for the mosaic model
18.15.4 Bounds for the Dead Leaves model
18.15.5 Bounds for excursion sets of Gaussian RF
18.15.6 Combination of the basic random sets models
18.16 Case of porous media
18.17 Optimal conductivity of two-components porous media
18.17.1 Optimization of two conductivities
18.17.2 Upper bounds of the optimal properties
18.17.3 Conclusion
18.18 Fields fluctuations
18.18.1 Second order statistics of fields
18.18.2 Distribution function of fields
18.19 Conclusion
18.20 Exercises
18.20.1 Properties of the operator Γ
18.20.2 Calculation of the second order perturbation term in the scalar isotropic case
18.20.3 Calculation of a third order term
18.20.4 Field averages
18.20.5 Field averages in components
18.20.6 Hill-Mandel condition
18.20.7 Geometrical average effective property in R^2
19 Digital Materials
19.1 Introduction to Digital Materials and to numerical approaches
19.2 Homogenization of random media by numerical simulations
19.3 Fluctuations of apparent properties, and statistical RVE
19.3.1 The integral range
19.3.2 Practical determination of the size of the RVE
19.4 A case study: the Voronoi mosaic
19.5 FE Computation on 3D confocal microscope micrographs
19.5.1 Experimental effective physical properties
19.5.2 Numerical estimation of apparent physical properties by FE
19.5.3 Determination of the integral ranges
19.5.4 Size of the RVE
19.5.5 Phase connectivity and effective properties
19.6 Improving elastic properties from Boolean models
19.7 Gigantic RVE: 3D Poisson fibres
19.8 Stochastic Finite Elements
19.9 Numerical solutions of the Lippmann-Schwinger equation by iterations of FFT
19.10 Fields fluctuations in dielectric random media
19.10.1 Fields in dielectric autodual RS
19.10.2 Fields in a 3D Boolean model of spheres
19.10.3 Optical properties of paints
19.11 Elastic and thermal response of heterogeneous media from 3D microtomography
19.11.1 Hotspots in a granular material
19.11.2 Stress localization in a mortar microstructure
19.11.3 Elastic and thermal properties of lightweight concretes
19.12 Elastic and viscoelastic properties of multiscale random media
19.12.1 Bulk modulus of the Boolean model of spheres
19.12.2 Linear elastic properties and conductivity of multiscale Cox Boolean models
19.12.3 Nonlinear elastic and conductivity of multiscale Cox Boolean models
19.12.4 Elastic and viscoelastic properties of rubber with carbon black filler
19.13 Elastic properties of fibrous materials
19.14 Diffusion and fluid flows in porous media
19.14.1 Diffusion in random porous media
19.14.2 Estimation of the fluid permeability of porous media
19.15 RVE of acoustic properties of fibrous media
19.15.1 Thermoacoustic equations and homogenization of acoustic properties of porous media
19.15.2 Homogenization of acoustic properties of a periodic fibrous network
19.15.3 Acoustic fields simulated on random unit cells
19.15.4 Statistical RVE and integral ranges
19.16 Further examples of application
19.17 Conclusion
20 Probabilistic Models for Fracture Statistics
20.1 Introduction
20.2 Choice of a fracture criterion
20.2.1 Local fracture criteria
20.2.2 Global fracture criteria
20.3 Brittle fracture and weakest link
20.3.1 Introduction
20.3.2 Examples of stress fields used for the weakest link model
20.3.3 The Boolean random varieties and the weakest link model
20.3.4 Randomization of Boolean varieties
20.3.5 Weakest link model and iterated Boolean Varieties
20.3.6 The Dead Leaves varieties and the weakest link model
20.3.7 Competition between fracture mechanisms
20.3.8 Multicriteria and multiscale weakest link models
20.4 Fracture statistics models with a damage threshold
20.5 Fracture statistics model with a crack arrest criterion
20.5.1 Crack propagation and the Griffith’s criterion fort wo-dimensional random media
20.5.2 Types of probability distributions obtained from the models
20.5.3 Probability of fracture and scale effects for the Poisson Mosaic
20.5.4 Probability of fracture and scale effects for the Boolean Mosaic
20.5.5 Conclusions
20.6 Models of random damage
20.6.1 Introduction
20.6.2 Basic assumptions
20.6.3 Random damage under a homogeneous load
20.6.4 Random damage under a non homogeneous load
20.6.5 Conclusion
20.7 Elements of practical use of fracture statistics models in numerical simulations
20.8 Conclusion
20.9 Exercises
20.9.1 Crack propagation in a random polycrystal with grain boundary fracture energy
20.9.2 3D crack propagation in a random medium
21 Crack Paths in Random Media
21.1 Introduction
21.2 Probabilistic fracture of 3D random tessellations
21.2.1 Basic assumptions
21.2.2 Fracture of a random tessellation and percolation
21.3 Phase field model for crack initiation and propagation in random media with anisotropic fracture energy
21.3.1 Reminder on phase field models for fracture of homogeneous isotropic media
21.3.2 A phase field model for heterogeneous anisotropic fracture energy
21.3.3 Introduction to full fields estimation by FFT
21.3.4 Estimation of an effective toughness
21.4 Conclusion
References
Index
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Interdisciplinary Applied Mathematics 53
Dominique Jeulin
Morphological Models of Random Structures
Interdisciplinary Applied Mathematics Volume 53
Editors Anthony Bloch, University of Michigan, Ann Arbor, MI, USA Charles L. Epstein, University of Pennsylvania, Philadelphia, PA, USA Alain Goriely, University of Oxford, Oxford, UK Leslie Greengard, New York University, New York, USA Advisors L. Glass, McGill University, Montreal, QC, Canada R. Kohn, New York University, New York, NY, USA P. S. Krishnaprasad, University of Maryland, College Park, MD, USA Andrew Fowler, University of Oxford, Oxford, UK C. Peskin, New York University, New York, NY, USA S. S. Sastry, University of California, Berkeley, CA, USA J. Sneyd, University of Auckland, Auckland, New Zealand Rick Durrett, Duke University, Durham, NC, USA
Problems in engineering, computational science, and the physical and biological sciences are using increasingly sophisticated mathematical techniques. Thus, the bridge between the mathematical sciences and other disciplines is heavily traveled. The correspondingly increased dialog between the disciplines has led to the establishment of the series: Interdisciplinary Applied Mathematics. The purpose of this series is to meet the current and future needs for the interaction between various science and technology areas on the one hand and mathematics on the other. This is done, firstly, by encouraging the ways that mathematics may be applied in traditional areas, as well as point towards new and innovative areas of applications; and secondly, by encouraging other scientific disciplines to engage in a dialog with mathematicians outlining their problems to both access new methods as well as to suggest innovative developments within mathematics itself. The series will consist of monographs and high-level texts from researchers working on the interplay between mathematics and other fields of science and technology.
More information about this series at http://www.springer.com/series/1390
Dominique Jeulin
Morphological Models of Random Structures
123
Dominique Jeulin Centre de Morphologie Mathématique MINES ParisTech Fontainebleau, France
ISSN 0939-6047 ISSN 2196-9973 (electronic) Interdisciplinary Applied Mathematics ISBN 978-3-030-75451-8 ISBN 978-3-030-75452-5 (eBook) https://doi.org/10.1007/978-3-030-75452-5 Mathematics Subject Classification: 60-XX, 00A69, 35Q60, 35Q74, 35R60, 52A22, 60H15, 60D05, 60G60, 60G70, 60J75, 60K35, 60K40, 74A40, 74A45, 74E30, 74E35, 74Q20, 74R05, 74R10, 74S60, 76M28, 76M50, 76S05, 78A48, 78M40, 82B43 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Dedicated to my wife Anne-Marie, to my children François, Guillaume, Clotilde and Bénédicte In memory of my parents
Preface
This book covers methods of Mathematical Morphology to model and simulate random sets and functions (scalar and multivariate). These models concern many physical situations in heterogeneous media, where a probabilistic approach is required: fracture statistics of materials, scaling up of permeability in porous media, electron microscopy images (including multispectral images), rough surfaces, multicomponent composites, biological tissues, and also textures for image coding and synthesis. . . The common feature of these random structures is their domain of definition in n dimensions (with n ≥ 3), requiring more general models than standard Stochastic Processes. The present book is based on my own research developments and applications, most of them being available as papers in journals and proceedings. It develops various models of random structures available for applications, and details their probabilistic properties. A unified approach is followed in random structure modeling and simulations. The book covers all steps of modeling. Some topics detailed here are missing in previous books limited to random sets: models of scalar and multivariate random functions, multiscale models, use of random models to predict the physical behavior of microstructure (like effective properties, or fracture statistics). Concerning applications given to illustrate the theory, they are based on quantitative image analysis made on representative samples. The main topics of the present book cover an introduction to the theory of random sets, random space tesselations, Boolean random sets and functions, space-time random sets and functions (Dead Leaves, Alternate Sequential models, Reaction-Diffusion), prediction of effective properties vii
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Preface
of random media, and probabilistic fracture theories. The book details the construction of models, their main probabilistic properties, and their practical use from experimental data by means of examples of application. This book collects results of near 50 years of research in the area of random media, which are widely dispersed in scientific publications in various areas, and in lecture notes. In addition, some unpublished new results are provided. It is intended to make available to the scientific community tools of research in the area probabilistic modeling. It will be of interest for researchers and research engineers in the areas of applied mathematics, image analysis, and applications of models of random structures. It will be greatly profitable to theoreticians and practitioners of the simulation and prediction of physical properties of heterogeneous microstructures, as encountered in heterogeneous natural or man-made materials or in life sciences. It will be a source of inspiration for further research in these fields. Graduate students and teachers in applied probability will learn developments on the theory and applications of random structures. Fontainebleau, Dominique Jeulin August 30th 2020
Acknowledgements
A large part of the work presented in this book is the result of interactions with many persons. There is indeed a crucial impact of scientific meetings and of cooperation with colleagues which I want to acknowledge. My first steps in applied probability, Mathematical Morphology and Geostatistics started at the beginning of 70s, thanks to the teaching of P. Formery, G. Matheron and J. Serra in Paris School of Mines. They transmitted to me the basic knowledge and the foundations for my work in research. Furthermore, they communicated their enthusiasm at the very beginning of these new disciplines. Later on, I had the chance to cooperate closely for years with G. Matheron and J. Serra in Centre de Morphologie Mathématique. My first position was in Moscow State University by E.N. Kolomenski, to initiate a morphological analysis of microstructures and multivariate statistical analysis of geological data. Student in Mathematical Statistics with J. P. Benzécri, I learned multivariate data analysis. Then, I made a ten years research for steel industry in IRSID, starting with a work on iron ore mineralogical textures and their physical behavior in the blast furnace. The topic was wide and complex, but I enjoyed absolute trust from my management, mainly M. Schneider and M. Olette, to carry out a large European project. I was introduced to the field of Physics of random media by E. Kröner and J. Willis, when they visited me in IRSID in 1980. I was invited to CMDS (Continuum Models and Discrete Systems) meetings and to many other Conferences, where I met and got acquainted with scientists to whom I am indebted for their strong influence, like M. Beran, D. Bergman, M. ix
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Kachanov, S. Kanaun, K. Markov, G. Milton, M. Ostoja-Starzewski, P. Ponte Castaneda, P. Suquet, among others. A. Zaoui and T. Bretheau introduced me to the French community in Micromechanics. A probabilistic lighting was developed over years in this area through the Mecamat association and other groups, in which I cooperated with many colleagues, like M. Bornert, A. Fanget, D. François, F. Hild, F. Montheillet, S. Roux, H. Trumel, and in the Alea working group with K. Sab C. Soize. In the area of Image Analysis and Sterology, I was an active member of ISS (International Society for Stereology), where I worked with J.L. Chermant and M. Coster, since 1976. I had many exchanges with J. Bertram, B.V. Vedel Jensen, M. Jourlin and R.E. Miles. I would like to thank my German colleagues, for a long-standing cooperation: D. Stoyan (Freiberg), K. Schladitz and C. Redenbach (Kaiserslautern), J. Ohser (Darmstadt). Concerning interactions between Mathematics and Industry, I would like to thank L. Bonilla , V. Capasso and A. Micheletti for numerous exchanges within ECMI. I was welcomed in Paris School of Mines in 1986 by F. Mudry, with the mission of building bridges between Mathematical Morphology and Materials Science, which is the main topic of this book. This was made possible by a close cooperation and interaction, mainly with my colleagues of three research centers in Paris School of Mines, which I want to warmly thank: J. Angulo, S. Beucher, M. Bilodeau, E. Decencière, J.C. Klein, F. Meyer, F. Willot (Mathematical Morphology Center), C. Lantuéjoul (Geostatistics), Y. Bienvenu, G. Cailletaud, S. Forest, F. Grillon, M. Jeandin, F. N’Guyen, A. Pineau, J. Renard, J.L. Strudel, A. Thorel, J. P. Trottier (Centre des Matériaux P.M. Fourt). In addition to research, I had the pleasure to be involved in teaching courses on random media in Paris School of Mines and in other places, based on a large part of the content of this book. In this context, I am grateful to my former students, my fifty PhD students, and the numerous Postdocs, with whom luckily I was able to work over years on many topics with a strong support from industry. These topics are a fertile source of exciting problems in applied research. Physics of random media is a rich, active, and still promising domain of research and applications, which brought and still brings me great satisfaction. I had a long and fruitful cooperation with many partners, from both industrial and academic circles, to support students and to start new research projects. I am very grateful to them for their continuous support. I am pleased to thank Springer International Publishing, and particularly L. Kunz and M. Peters, for the preparation of this edition. Finally, I would never have been able to work on the material contained in this book without the relentless help of my wife Anne-Marie over so many years.
Symbols and Notations
Morphological operations Minkowski addition of sets A and K: A ⊕ K = ∪x∈A,y∈K {x + y} = ∪y∈K Ay = ∪x∈A Kx Minkowski substraction of sets A and K: A ª K = ∩y∈K Ay = (Ac ⊕ K)c ˇ = {x, Kx ∩ A 6= ∅} Dilation by a compact set K : A → A ⊕K ˇ Erosion by a compact set K : A → A ªK = {x, Kx ⊂ A} ˇ ⊕K Opening by a compact set K : AK = A ªK K ˇ Closing by a compact set K : A = A ⊕K ª K Indicator function of set A: 1A (x) = 1 if x ∈ A usc (upper semi continuous) transformation lsc (lower semi-continuous) transformation Convex hull C(A)
Random sets A, B: random closed sets (RACS) Ac : complementary set of A Aˇ = {−x, x ∈ A}: transposed set of A Ai : component of a multi-component random set B(r): closed ball with radius r K: compact set E: topological space F, G and K : closed, open and compact sets of E F K = {F ∈ F; F ∩ K = ∅, K ∈ K} xi
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Symbols and Notations
FG = {F ∈ F; F ∩ G 6= ∅, G ∈ G} Probability P GK (s): generating function of the random variable N (K) Choquet’s capacity T (K) = P {K ∩ A 6= ∅} = P {FK } = 1 − P {K ⊂ Ac } = 1 − Q(K) p = P {x ∈ A} q = P {x ∈ Ac } Covariance C(h) = P {x ∈ A, x + h ∈ A} Covariance Q(h) = P {x ∈ Ac , x + h ∈ Ac } Covariance Cij (h) = P {x ∈ Ai , x + h ∈ Aj } for a multi-component random set Three points Probability Q(h1 , h2 ) = P {x ∈ Ac , x+h1 ∈ Ac , x+h2 ∈ Ac } Segment l: P (l) = P {l ⊂ A}; Q(l) = P {l ⊂ Ac } Hexagon H(r): P (H(r)) = P {H(r) ⊂ A}; Q(H(r) = P {H(r) ⊂ Ac } R(x, A): distance between the point x and the set A Autodual random sets: A and Ac have the same Choquet capacity
Measurements μ(A): measure of A μn Lebesgue measure in Rn Volume of A: V (A) Integral of mean curvature of A: M (A) Surface area of A: S(A) Perimeter of A (in R2 ): L(A) Specific connectivity number in R2 : NA (A) Specific connectivity number in R3 : NV (A) − GV (A) Minkowski functionals of A: Wi (A) Minkowski tensors of A: Wνr,s (A) Size distribution in number: F (λ), in measure: G(λ) Geodesic distance between x ∈ A and y ∈ A: dA (x, y) Morphological tortuosity τ A (x, y) = dA (x, y)/ kx − yk
Random Functions Random Function Z(x) test function g(x) Φ(E): set of functions from E → R upper semi continuous random functions (usc RF) Φf ⊂ Φ: set of usc functions from E → R Choquet’s capacity T (g) = P {x ∈ DZ (g)}; DZ (g)c = {x, Z(x + y) < g(y), ∀y ∈ K} lower semi continuous random functions (lsc RF) Φg ⊂ Φ : set of lsc functions from E → R functional P (g) = P {x ∈ HZ (g)}; HZ (g) = {x, Z(x + y) ≥ g(y), ∀y ∈ K}
Symbols and Notations
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subgraph Γ ϕ of the function ϕ: Γ ϕ = {x, z}, x ∈ E, z ∈ R, with z ≤ ϕ(x) overgraph Γϕ : Γϕ = {x, z}, x ∈ E, z ∈ R, with z ≥ ϕ(x) excursion set above level z: AZ (z) = {x, Z(x) ≥ z} ∨: supremum; Z∨ (K) = ∨x∈K {Z(x)} ∧: infimum; Z∧ (K) = ∧x∈K {Z(x)} dilation of Z by a function g (with gˇ(x) = g(−x)): Z ⊕ gˇ(x) = ∨y∈Rn {Z(y) + g(y − x)} erosion of Z by a function g (with gˇ(x) = g(−x)): Z ª gˇ(x) = ∧y∈Rn {Z(y) − g(y − x)} opening of Z by g: Ψg (Z) = (Z ª gˇ) ⊕ g closing of Z by g: Ψ g (Z) = (Z ⊕ gˇ) ª g thresholding: AZ (z) = {x, Z(x) ≥ z} spatial law: F (x, z) = P {Z(x1 ) < z1 , ..., Z(xm ) < zm } with x ∈ E m and m z∈R spatial law: T (x, z) = P {Z(x1 ) ≥ z1 , ..., Z(xm ) ≥ zm } with x ∈ E m and m z∈R F (z), G(z) distribution functions (with density, or pdf f (z) and g(z)) S: coefficient of variation of a distribution D2 [Z]: variance of the random variable Z Hermite polynomials Hn (z) Bivariate distribution Fij (h, z1 , z2 ) = P {Zi (x) < z1 , Zj (x + h) < z2 } Bivariate distribution Tij (h, z1 , z2 ) = P {Zi (x) ≥ z1 , Zj (x + h) ≥ z2 } Bivariate distribution T2 (h, z1 , z2 ) = P {Z(x) ≥ z1 , Z(x + h) ≥ z2 } Covariance C(x, x + h) and second order central correlation function W 2 (x, x + h) Integral range An RVE: Representative Volume Element γ 1 (h), γ 2 (h): variograms of order 1 and 2 g(h): transitive covariogram Central correlation function of order m W m (x), with x ∈ E m Z(V ): average of Z(x) in volume V Mathematical expectation E{.} Φ(u), Φ(u1 , u2 ), Φ(u1 , u2 , ..., um ): characteristic functions of F (z), F (z1 , z2 ), F (z1 , z2 , ..., zm )
Models Random sets BRS: Boolean random set model: RACS A with primary grain A0 Intensity θ Primary grain A0 : geometrical covariogram K(h) = μn (A0 ∩ A0−h ) normalized covariogram r(h) = K(h)/K(0) s(h1 , h2 ) = μn (A0 ∩ A0−h1 ∩ A0−h2 )/K(0) IBRS: infinitesimal Boolean random set; time t; Intensity θ(t)
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Symbols and Notations
DLRT: Dead Leaves tesselation: Ni (t): specific number of intact grains; ϕA0 , ϕA0i : pdf of grains and of intact grains STIT: random tessellation stable under iterations Poisson varieties Vk (ω), with intensity θ(ω) Probabilistic Texture segmentation: probabilistic distance between regions Ai and Aj : P (Ai , Aj , d)
Random functions (RF) BRF: Boolean random function Z(x) Gaussian RF DLRF: Dead Leaves random function TDLRF: Transparent Dead Leaves random function IBRF: infinitesimal Boolean random function; time t; Intensity θ(t) MJF: sequential RF with Markovian jumps SARF: Sequential alternate random function 0 Zt0 (x): Primary random function, with subgraph Γ Zt = A0 (t) and sections AZt0 (z) ϕ(Z): transformation of the RF Z by the anamorphosis ϕ DRF: Dilution RF: Φ(U, X) and φt (U, X): multivariate characteristic functions of the RF Z(x) and Zt0 (x) Z ∗ pˇ(x): convolution of the RF Z(x) by a weight function p(x) Reaction-Diffusion RF; coefficients of diffusion Di
Change of scale in random media Elasticity: stress σ, strain e, elasticity tensor C; isotropic elasticity: bulk modulus K, shear modulus G, Young’s modulus E, Poisson coefficient ν Fluid flow: pressure gradient ∂i p, fluid velocity u, kinematical viscosity μ; macroscopic flow rate Q, macroscopic pressure gradient ∂i P , permeability K Electrostatics: dielectric displacement D, electric field E, potential φ, permittivity , energy U FE: Finite Elements KUBC: Kinematic Uniform Boundary Conditions SUBC: Static uniform Boundary Conditions PBC: Periodic Boundary Conditions FFT: Fast Fourier Transform RVE: Representative Volume Element Green’s function in electrostatics G(x, y) 2 Operator Γ (x, y) in electrostatics, with components Γij (x, y) = ∂x∂j ∂yi G(x, y) Hashin-Shrikman bounds H-S; upper H-S+ , lower H-S− ς 1 (p) and η 1 (p): Milton functions for Beran-Molyneux-McCoy third order bounds Pn (u): Legendre polynomials Fracture statistics: σR , fracture stress; Φ(σ), intensity of defects with critical stress σ c < σ
Symbols and Notations
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Random fracture energy Γ ; energy release rate G; fracture toughness Gc ; SVE: Statistical Volume Element in fracture Φ(σ) phase field in fracture; Γef f : effective toughness
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Use of a probabilistic approach . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Aims of probabilistic models . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Descriptive aspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Predictive aspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Types of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Model construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Characterization of a random model . . . . . . . . . . . . . . 1.4.2 Choice of basic assumptions . . . . . . . . . . . . . . . . . . . . . 1.4.3 Computation of the functional T (K) . . . . . . . . . . . . . . 1.5 Some general properties of the models . . . . . . . . . . . . . . . . . . . 1.6 Organization of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 2 2 3 3 4 5 7 7 8 8 9
Part I Tools for Random Structures 2
Random Closed Sets and Semi-Continuous Random Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Introduction to random closed sets (RACS) . . . . . . . . . . . . . . 2.2.1 Recall of basic definitions in probability theory . . . . . 2.2.2 Definition of the random closed sets . . . . . . . . . . . . . . 2.2.3 The Choquet capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Measurability of a random closed set . . . . . . . . . . . . . . 2.2.5 Operations on random closed sets . . . . . . . . . . . . . . . .
17 17 19 20 20 22 23 23 xvii
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2.2.6 Independent random sets . . . . . . . . . . . . . . . . . . . . . . . . 2.2.7 Spatial law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.8 Geometrical interpretation of the Choquet capacity in the Euclidean case . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Introduction to Random Functions . . . . . . . . . . . . . . . . . . . . . 2.3.1 Semi continuous functions . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Subgraph of a function . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Choquet topology on Φf . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 The Borel σ algebra on Φf and Φg , and the Choquet capacity for RF . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Multivariate Random Functions . . . . . . . . . . . . . . . . . . 2.3.6 Erosion and Dilation of functions . . . . . . . . . . . . . . . . . 3
25 25 25 27 28 29 30 31 36 36
Quantitative Analysis of Random Structures . . . . . . . . . . . . 41 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Basic morphological transformations . . . . . . . . . . . . . . . . . . . . 42 3.2.1 Types of structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.2 Notion of structuring element . . . . . . . . . . . . . . . . . . . . 43 3.2.3 Dilations and Erosions . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Basic morphological measurements . . . . . . . . . . . . . . . . . . . . . 45 3.3.1 Definition of Minkowski functionals . . . . . . . . . . . . . . 45 3.3.2 Stereological aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3.3 Steiner formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3.4 Specific measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.5 Counting procedure and the Euler relations . . . . . . . 53 3.3.6 Minkowski tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4 Size distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4.1 Axioms of size distributions; openings and closings . 57 3.4.2 Linear size distributions . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.4.3 Two and three dimensional size distributions . . . . . . . 61 3.4.4 Size distributions for functions . . . . . . . . . . . . . . . . . . . 62 3.4.5 Stereological reconstruction of the size distribution of spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.5 Morphological analysis of the spatial distribution . . . . . . . . . 65 3.5.1 Random sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.5.2 Multi component random sets . . . . . . . . . . . . . . . . . . . . 74 3.5.3 Random functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.5.4 Covariance of orientations of vector fields . . . . . . . . . . 81 3.5.5 Definition of a Statistical Representative Volume Element (RVE) and problems of estimation . . . . . . . 83 3.5.6 Distance functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.5.7 Random graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.6 Geodesic criteria and connectivity . . . . . . . . . . . . . . . . . . . . . . 89 3.7 Robustness of morphological measurements to noise . . . . . . 94 3.8 Shape and texture classification and recognition . . . . . . . . . . 96
Contents
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3.8.1 Automatic recognition of non metallic inclusions . . . 97 3.8.2 3D Shape classification of particles with complex shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.8.3 Texture analysis and classification . . . . . . . . . . . . . . . 102 3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.10.1 Minkowski tensor of a cylinder . . . . . . . . . . . . . . . . . . . 106 3.10.2 Variogram γ 1 (h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.10.3 Correlation functions of two phase random composites107 3.10.4 Specific measurements and derivatives of morphological moments . . . . . . . . . . . . . . . . . . . . . . . . 108 3.10.5 Covariance of a noisy binary image . . . . . . . . . . . . . . 109 3.10.6 Moments of inertia in R3 . . . . . . . . . . . . . . . . . . . . . . . 109 Part II Models of Random Structures 4
Excursion Sets of Gaussian RF . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.2 Random sets and truncated RF: the Gaussian case. . . . . . . . 113 4.2.1 Construction of truncated Gaussian random sets . . . 114 4.2.2 Second order central correlation function . . . . . . . . . 117 4.2.3 Third order central correlation function . . . . . . . . . . . 119 4.2.4 Order m central correlation function . . . . . . . . . . . . . . 120 4.3 Application to a food microstructure . . . . . . . . . . . . . . . . . . . . 121 4.4 Three Phase random model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.4.2 Three components models based on independent random sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.4.3 Three component model based on the Boolean model 127 4.4.4 Three component model based on truncated Gaussian random functions . . . . . . . . . . . . . . . . . . . . . . 128 4.5 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.5.1 Combination of independent random sets . . . . . . . . . 129
5
Stochastic Point Processes and Random Trees . . . . . . . . . . 131 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.2 Tools to characterize point processes . . . . . . . . . . . . . . . . . . . . 132 5.3 The Poisson point process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.3.1 Definition and properties of the Poisson point process133 5.3.2 Distribution of points of a homogeneous Poisson point process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3.3 Simulation of a homogeneous Poisson point process . 136
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5.4 5.5
5.6
5.7 5.8 5.9
5.10
6
5.3.4 Distribution of points of a non stationary Poisson point process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Hard-core point processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 The Cox point process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.5.1 Definition and main properties . . . . . . . . . . . . . . . . . . . 138 5.5.2 Cox process with a random variable θ . . . . . . . . . . . . . 140 5.5.3 Large scale behavior of the Cox process . . . . . . . . . . . 142 Point processes on iteration of Boolean varieties . . . . . . . . . . 143 5.6.1 Two steps Boolean varieties . . . . . . . . . . . . . . . . . . . . . . 144 5.6.2 Three steps Boolean varieties in R3 . . . . . . . . . . . . . . . 147 Gibbs point processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Determinantal point processes . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Introduction to models of random trees by branching processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.9.1 Discrete branching process . . . . . . . . . . . . . . . . . . . . . . 150 5.9.2 Continuous branching process . . . . . . . . . . . . . . . . . . . . 154 5.9.3 Discrete random trees generated by random functions158 5.9.4 Biological applications . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.10.1 Laplace transforms for sections of discs and of spheres160 5.10.2 A model of random tree embedded in the Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Boolean Random Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Construction of the Boolean model . . . . . . . . . . . . . . . . . . . . . 6.3 Choquet capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Some stereological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Connectivity numbers and counting . . . . . . . . . . . . . . . . . . . . . 6.6 Connectivity and percolation of the Boolean model . . . . . . . 6.7 IDRACS and semi Markovian RACS . . . . . . . . . . . . . . . . . . . . 6.8 Testing the Boolean model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Convexity of the primary grain A0 . . . . . . . . . . . . . . . . 6.8.2 Dilation by convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.3 Infinite divisibility of the Boolean model . . . . . . . . . . 6.9 Identification of the Boolean model . . . . . . . . . . . . . . . . . . . . . 6.10 Some examples of application . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10.1 Boolean model of spheres . . . . . . . . . . . . . . . . . . . . . . . . 6.10.2 The Poisson Boolean model . . . . . . . . . . . . . . . . . . . . . . 6.10.3 Further applications of the Boolean model to materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10.4 Multiphase textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 The Boolean model and the Poisson process . . . . . . . . . . . . . 6.12 Cox Boolean model and multiscale models of RACS . . . . . . 6.12.1 Intersection of independent random sets . . . . . . . . . . .
165 165 166 166 170 171 173 175 178 178 178 179 180 181 181 183 186 187 188 189 189
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6.12.2 Introduction to the Cox Boolean model . . . . . . . . . . . 191 6.12.3 Percolation of the Cox Boolean model . . . . . . . . . . . . . 197 6.13 Boolean model and the Poisson varieties . . . . . . . . . . . . . . . . 197 6.13.1 Construction and properties of the linear Poisson varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 6.13.2 Stable RACS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.13.3 Boolean model on the linear Poisson varieties . . . . . . 201 6.13.4 Power laws variance scaling of the Boolean random varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 6.13.5 Two steps iterated Poisson varieties . . . . . . . . . . . . . . . 208 6.14 Multi component Boolean models . . . . . . . . . . . . . . . . . . . . . . . 211 6.14.1 Construction of Boolean models with m components 211 6.14.2 Choquet capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.14.3 Spatial law and covariances . . . . . . . . . . . . . . . . . . . . . . 212 6.14.4 Stability for the reunion . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.14.5 Multi component Boolean models on Poisson varieties213 6.15 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 6.15.1 Counting with the Euler relation . . . . . . . . . . . . . . . . . 214 6.15.2 Hierarchical Boolean model on a mosaic . . . . . . . . . . . 215 6.15.3 Fractal Boolean random set . . . . . . . . . . . . . . . . . . . . . 218 6.15.4 A noise random set model . . . . . . . . . . . . . . . . . . . . . . 220 6.15.5 Simulation of percolating aggregates . . . . . . . . . . . . . 220 6.15.6 Transverse and longitudinal covariances of a Boolean model of cylinders . . . . . . . . . . . . . . . . . . . . . . 223 6.15.7 Boolean varieties in R2 and R3 as limit case of Boolean models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 7
Random Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Reminder on random tessellations . . . . . . . . . . . . . . . . . . . . . . 7.3 Probability distributions of the classes . . . . . . . . . . . . . . . . . . 7.4 The Voronoi, Johnson-Mehl, and Laguerre tessellation . . . . . 7.4.1 Voronoi tessellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Johnson-Mehl and Laguerre random tessellations . . . 7.5 Random tessellation generated by a geodesic distance . . . . 7.6 The Poisson tessellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Definition and Choquet capacity of the Poisson hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Stereological aspects of the Poisson hyperplanes . . . . 7.6.3 Characterization of the Poisson polyhedron containing the origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.4 Conditional invariance by erosion . . . . . . . . . . . . . . . . . 7.6.5 Opening size distribution . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Example of application to model concrete by multiscale Poisson polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
233 233 234 235 236 236 239 241 243 243 245 245 246 247 248
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7.8 The Cauwe tessellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.1 Construction of the Cauwe tessellation . . . . . . . . . . . . 7.8.2 Stereological properties . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.3 Choquet capacity on convex sets . . . . . . . . . . . . . . . . . 7.8.4 Mixture of Poisson and Cauwe tessellations . . . . . . . . 7.9 Iteration of tessellations and the STIT model . . . . . . . . . . . . 7.10 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10.1 Superposition of random tessellations . . . . . . . . . . . . .
249 250 250 251 253 253 255 255
The Mosaic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Choquet capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Mosaic model in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Calculation of P {N (K) = 1} . . . . . . . . . . . . . . . . . . . . 8.4.2 First order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Second order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Third order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.5 Higher order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 The Poisson mosaic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 The STIT mosaic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 The Mosaic random set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
257 257 257 259 260 260 261 261 263 264 265 267 268
8.8 The multivariate mosaic model . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.1 Multi component mosaic . . . . . . . . . . . . . . . . . . . . . . . . 8.9.2 Cross covariances of the multicomponent mosaic . . . . 8.9.3 Hierarchical mosaic model . . . . . . . . . . . . . . . . . . . . . . .
269 270 270 271 272
Boolean Random Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Construction of the Boolean random functions BRF . . . . . . 9.3 Choquet capacity of the BRF . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Supremum stability and infinite divisibility . . . . . . . . . . . . . . 9.5 Characteristics of the primary functions . . . . . . . . . . . . . . . . . 9.5.1 Transformation by anamorphosis . . . . . . . . . . . . . . . . . 9.5.2 Moments of Z∨0 (K) and mathematical expectation of the anamorphosed of Z∨0 (K) . . . . . . . . . . . . . . . . . . . 9.5.3 Geometrical covariogram of the primary function . . . 9.6 Some stereological aspects of the BRF . . . . . . . . . . . . . . . . . . 9.7 BRF and counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Identification of a BRF model . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Test of the BRF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.1 Convexity of AZt0 (z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.2 Change of support on convex sets . . . . . . . . . . . . . . . .
275 275 276 278 280 282 282 282 284 284 284 286 287 287 288
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9.9.3 Supremum infinite divisibility . . . . . . . . . . . . . . . . . . . . 9.10 Examples of application to rough surfaces . . . . . . . . . . . . . . . 9.10.1 Simulation of the evolution of surfaces and of stresses during shot peening . . . . . . . . . . . . . . . . . . . . . 9.10.2 Simulation of the roughness transfer on steel sheets . 9.11 Modeling three phase random media with some contact constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11.1 Microstructure of a current collector . . . . . . . . . . . . . . 9.11.2 Use of a Boolean random function . . . . . . . . . . . . . . . . 9.11.3 Use of truncated Gaussian random functions . . . . . . . 9.12 Multiscale Boolean random functions . . . . . . . . . . . . . . . . . . . 9.13 The Boolean varieties RF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.14 The multivariate BRF and Varieties . . . . . . . . . . . . . . . . . . . . 9.14.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.14.2 Choquet capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.14.3 Spatial law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.14.4 Supremum stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.14.5 Distribution of maxima . . . . . . . . . . . . . . . . . . . . . . . . . 9.14.6 Multivariate Boolean varieties RF . . . . . . . . . . . . . . . . 9.15 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.15.1 BRF with cylinder primary random functions . . . . . . 9.15.2 A hierarchical BRF model . . . . . . . . . . . . . . . . . . . . . . . 9.15.3 BRF with cones anamorphosis . . . . . . . . . . . . . . . . . . . 10 Random Tessellations and Boolean Random Functions . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Extensions of the Voronoi tessellation . . . . . . . . . . . . . . . . . . . 10.2.1 Random tessellations defined from local metrics . . . . 10.2.2 Calculation of the probability P (K) . . . . . . . . . . . . . . 10.3 Extension to Johnson-Mehl and to Laguerre random tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Random tessellations and Boolean random functions . . . . . . 10.4.1 Connection between metric based random tessellations and some BRF . . . . . . . . . . . . . . . . . . . . . . 10.4.2 General random tessellations built from BRF . . . . . .
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289 290 290 291 293 293 293 296 297 299 300 301 301 302 303 303 304 305 305 306 307 311 311 312 312 313 314 316 316 318
10.4.3 Random tessellations with thick boundaries . . . . . . . . 326 10.5 Some indications on model identification . . . . . . . . . . . . . . . . 326 10.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 11 Dead Leaves Models: from Space Tessellations to Random Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 11.2 Sequential random tessellations . . . . . . . . . . . . . . . . . . . . . . . . . 330
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11.4
11.5
11.6
11.7
11.2.1 Infinitesimal Boolean random set . . . . . . . . . . . . . . . . . 331 11.2.2 Construction of the sequential random tessellations (Dead Leaves tessellation) . . . . . . . . . . . . . . . . . . . . . . . 332 11.2.3 Probabilistic properties . . . . . . . . . . . . . . . . . . . . . . . . . 332 11.2.4 Linear size distributions . . . . . . . . . . . . . . . . . . . . . . . . . 335 11.2.5 Specific parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 11.2.6 Higher order probabilistic properties . . . . . . . . . . . . . . 337 11.2.7 Statistics on intact grains and bias correction . . . . . . 338 11.2.8 Example of application to polycristalline salt . . . . . . . 338 11.2.9 Iteration of Dead Leaves tessellations . . . . . . . . . . . . . 340 11.2.10Multi scale Cox DLRT . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Random packings generated by the Dead Leaves model . . . . 342 11.3.1 Intact grains of the Dead Leaves model for the time homogeneous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 11.3.2 Intact grains of the Dead Leaves model for the general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 11.3.3 Examples of models of dense packings . . . . . . . . . . . . . 345 11.3.4 Size distributions for dense packings . . . . . . . . . . . . . . 349 11.3.5 Pair correlation function of intact grains centers . . . . 349 11.3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Color Dead Leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 11.4.1 Construction of the Color Dead Leaves . . . . . . . . . . . . 353 11.4.2 First order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 11.4.3 Second order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 356 11.4.4 Third order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 11.4.5 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 11.4.6 An example of two phase medium . . . . . . . . . . . . . . . . 363 11.4.7 Specific connectivity numbers of the Color Dead Leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 11.4.8 Some potential applications of the Color Dead Leaves366 The Dead Leaves Random Functions (DLRF) . . . . . . . . . . . . 367 11.5.1 The infinitesimal Boolean Random Functions (IBRF) 367 11.5.2 Construction of sequential RF from IBRF . . . . . . . . . 369 11.5.3 Probabilistic properties of the DLRF . . . . . . . . . . . . . . 371 11.5.4 Moment P (g, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 11.5.5 Summits of the DLRF . . . . . . . . . . . . . . . . . . . . . . . . . . 381 11.5.6 Intact primary functions . . . . . . . . . . . . . . . . . . . . . . . . 382 11.5.7 Estimation of the parameters of the DLRF and tests 385 Multivariate DLRF and varieties . . . . . . . . . . . . . . . . . . . . . . . 387 11.6.1 Construction of the Multivariate DLRF . . . . . . . . . . . 387 11.6.2 Multivariate distribution in point x . . . . . . . . . . . . . . . 388 11.6.3 Bivariate distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 389 11.6.4 Cross covariances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 11.6.5 Multivariate DLRF built on Poisson varieties . . . . . . 391 Dead Leaves of rank m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
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11.8 Transparent Dead Leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 Application to the morphology of powders . . . . . . . . . . . . . . . 11.9.1 Mixtures of Boolean models . . . . . . . . . . . . . . . . . . . . . 11.9.2 Use of The DLRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9.3 Use of The DLRF for grey level images . . . . . . . . . . . . 11.10Sequential RF with Markovian jumps (MJF) . . . . . . . . . . . . 11.10.1Construction of MJF . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.10.2Properties of the domains A0 (t) = ∪u≤t A00 (u) . . . . . . 11.10.3Probabilistic properties of the MJF . . . . . . . . . . . . . . . 11.11Elements of simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.12Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.13Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.13.1Exponential distribution of intercepts (DLRT) . . . . . 11.13.2 Correlation between contiguous intercepts (DLRT) 11.13.3 Linear size distributions (DLRT) . . . . . . . . . . . . . . . . 11.13.4 Construction of a random set from a DLRT . . . . . . . 11.13.5Composite grains (Color Dead Leaves) . . . . . . . . . . . 11.13.6Exponential covariance of a non Markov random mosaic (DLRF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.13.7 Statistics of intact grains . . . . . . . . . . . . . . . . . . . . . . . 11.13.8A two scale model of platelets . . . . . . . . . . . . . . . . . . . 12 Sequential Cox Boolean and Conditional Dead Leaves Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Sequential Cox Boolean RACS . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Notations and definition of the two components sequential Cox Boolean model . . . . . . . . . . . . . . . . . . . 12.2.2 Probabilistic properties . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Simplified expressions for the Choquet capacity . . . . 12.2.4 First variant of the sequential Cox Boolean model . . 12.2.5 Second variant of the sequential Cox Boolean model 12.2.6 Multi components version of the Cox sequential Boolean model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Sequential Cox Boolean RF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Notations and definition of the two components Sequential Cox BRF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Probabilistic properties . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 Simplified expressions for the Choquet capacity . . . . 12.3.4 Multi components Sequential Cox mosaic BRF . . . . . 12.3.5 Extension of the Sequential Cox mosaic BRF to the continuous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Conditional Dead Leaves tessellation . . . . . . . . . . . . . . . . . . . . 12.4.1 Construction of the model . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Probabilistic properties . . . . . . . . . . . . . . . . . . . . . . . . .
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398 398 399 402 405 406 406 407 408 409 410 410 410 410 411 412 413 414 415 416 419 419 420 420 420 421 432 434 436 441 441 442 443 447 451 452 452 453
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12.4.3 Simplified expressions of probabilistic properties . . . 12.5 Conditional color Dead Leaves . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.1 Notations and definition of the model . . . . . . . . . . . . . 12.5.2 Probabilistic properties . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.3 Simplified expressions of probabilistic properties . . . . 12.5.4 Simplified expression for the Choquet capacity of the set B0 (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.5 Probabilities Pi (t) of the conditional color Dead Leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.6 Covariances of the conditional color Dead Leaves . . . 12.5.7 Expressions for Pi (t) and for the covariances in some specific cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
455 459 459 459 464
13 Sequential Alternate Random Functions . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Sequential alternate RF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Construction of SARF . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Construction of coupled SARF . . . . . . . . . . . . . . . . . . . 13.2.3 Univariate distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.4 Bivariate distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.5 Apparent maxima and minima of the primary RF . . 13.3 Multiscale SARF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Example of application to modeling of the Electro Discharge Textures (EDT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Multivariate SARF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 SARF varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7 Elements of simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.8 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
487 487 488 488 492 492 493 496 498
14 Primary Grains and Primary Functions . . . . . . . . . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Random primary grains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Population of spheres and of ellipsoids in R3 . . . . . . . 14.2.2 Random parallelepipeds . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.3 The Poisson polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.4 Grains generated by the DLRT . . . . . . . . . . . . . . . . . . . 14.2.5 Random aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.6 Cylinder primary RF . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.7 Restriction of a stationary RF to a random compact set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.8 Some RF with spherical thresholds . . . . . . . . . . . . . . . 14.2.9 Boolean RF with a compact support . . . . . . . . . . . . . . 14.2.10Dead Leaves RF with a compact support . . . . . . . . . . 14.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
509 509 510 510 514 515 516 517 523
464 465 470 471 485
499 502 506 506 507
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14.3.1 Boolean aggregates of rectangles . . . . . . . . . . . . . . . . . 529 15 Dilution Random Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Construction of the multivariate Dilution random functions DRF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Convolution of a DRF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Characteristic functional of a multivariate DRF . . . . . . . . . . 15.5 Moments of a multivariate DRF . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Example of application to TEM micrographs . . . . . . . . . . . . . 15.6.1 A model for thick slices images . . . . . . . . . . . . . . . . . . . 15.6.2 Covariance of the TEM images . . . . . . . . . . . . . . . . . . . 15.6.3 Numerical method to estimate the transitive covariogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Random tokens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8 Positive DRF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9 DRF of the number of grains covering x . . . . . . . . . . . . . . . . . 15.9.1 Univariate distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9.2 Bivariate distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9.3 Trivariate distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9.4 Random set Am . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9.5 Random tessellation generated by the Am . . . . . . . . . 15.9.6 Excursion random sets of N (x) . . . . . . . . . . . . . . . . . . . 15.10Convergence of the DRF towards a Gaussian RF . . . . . . . . . 15.11Multivariate Dilution varieties RF . . . . . . . . . . . . . . . . . . . . . . 15.12Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.12.1The Dilution RF and the Cox process . . . . . . . . . . . . . 15.12.2Cox Dilution RF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.12.3Hybrid model of spheres and cylinders . . . . . . . . . . . . 15.12.4Model of indentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Reaction-Diffusion and Lattice Gas Models . . . . . . . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Reaction-Diffusion models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Reaction-Diffusion equations . . . . . . . . . . . . . . . . . . . . . 16.2.2 Discrete Reaction-Diffusion models . . . . . . . . . . . . . . . 16.3 Random Functions and the linear Reaction-Diffusion model 16.4 Examples of simulations of non- linear Reaction-Diffusion Random Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 Schlögl model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.2 Turing structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.3 The complex Ginzburg-Landau model . . . . . . . . . . . . . 16.4.4 Modifications of the Ginzburg-Landau model . . . . . . 16.5 Lattice gas models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.1 Basic rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
531 531 532 532 533 535 537 537 538 539 539 542 542 543 543 543 544 546 547 549 549 550 550 552 553 554 557 557 558 558 562 563 567 568 569 572 574 576 576
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16.5.2 Some indications on the evolution equations . . . . . . . 16.5.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.4 Application to flow in porous media . . . . . . . . . . . . . . 16.5.5 Application to simulations of random media . . . . . . . 16.5.6 Multi species lattice gas models . . . . . . . . . . . . . . . . . . 16.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
576 578 578 581 584 587
Texture Segmentation by Morphological Probabilistic Hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Morphological texture descriptors . . . . . . . . . . . . . . . . . . . . . . 17.3 Texture classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Probabilistic texture segmentation . . . . . . . . . . . . . . . . . . . . . . 17.4.1 Watershed texture segmentation . . . . . . . . . . . . . . . . . . 17.4.2 Probabilistic hierarchical segmentation . . . . . . . . . . . . 17.4.3 Higher order probabilistic segmentation . . . . . . . . . . . 17.4.4 Probabilistic distances between sets . . . . . . . . . . . . . . 17.4.5 Use of random markers . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.6 Random markers and higher order fusion of regions . 17.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
589 589 590 591 592 592 592 599 601 603 608 611
Part III Random Structures and Change of Scale 18 Change of Scale in Physics of Random Media . . . . . . . . . . 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 From microscopic to macroscopic . . . . . . . . . . . . . . . . . . . . . . . 18.3 Homogeneous medium and heterogeneous medium . . . . . . . . 18.4 Practical interest of change of scale methods . . . . . . . . . . . . . 18.5 Principle of calculation of effective properties . . . . . . . . . . . . 18.6 Exact results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6.1 Geometrical average estimate . . . . . . . . . . . . . . . . . . . . 18.6.2 Self-consistent and effective medium estimates . . . . . 18.6.3 Composite spheres assemblage . . . . . . . . . . . . . . . . . . . 18.7 Perturbation expansion in electrostatics . . . . . . . . . . . . . . . . . 18.8 Formal expansion of the effective dielectric permittivity of random media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.9 Perturbation approach in elasticity and calculation of the effective elastic tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.10Bounds of effective properties . . . . . . . . . . . . . . . . . . . . . . . . . . 18.10.1Variational principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.10.2Bounds of order 2N + 1 derived from the classical variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.11Third order bounds of the dielectric permittivity . . . . . . . . . 18.11.1Reminder on third order bounds for RF . . . . . . . . . . . 18.11.2Third order bounds for the mosaic model . . . . . . . . . .
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18.11.3Third order bounds of the Dilution model . . . . . . . . . 18.11.4Transformation of basic models . . . . . . . . . . . . . . . . . . 18.11.5Combination of the basic random functions models . 18.11.6A hierarchical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.12Third order bounds of the real dielectric permittivity of random sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.13Third order bounds of the complex dielectric permittivity and spectral measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.13.1Bounds in the complex plane . . . . . . . . . . . . . . . . . . . . 18.13.2Spectral measure of a random set . . . . . . . . . . . . . . . . . 18.14Third order bounds for elastic moduli of random sets . . . . . 18.14.1Third order bounds for two-phase composites . . . . . . 18.14.2Bounds of the bulk modulus K . . . . . . . . . . . . . . . . . . . 18.14.3Bounds of the shear modulus G . . . . . . . . . . . . . . . . . . 18.14.4Bounds of the Young’s modulus E . . . . . . . . . . . . . . . . 18.14.5Bounds of the Poisson coefficient ν . . . . . . . . . . . . . . . 18.14.6 Functions ζ 1 (p) and η 1 (p) . . . . . . . . . . . . . . . . . . . . . . . 18.15Third order bounds of some models of random sets . . . . . . . 18.15.1Bounds for the Boolean model . . . . . . . . . . . . . . . . . . . 18.15.2Bounds for the hard spheres model . . . . . . . . . . . . . . . 18.15.3Bounds for the mosaic model . . . . . . . . . . . . . . . . . . . . 18.15.4Bounds for the Dead Leaves model . . . . . . . . . . . . . . . 18.15.5Bounds for excursion sets of Gaussian RF . . . . . . . . . 18.15.6Combination of the basic random sets models . . . . . . 18.16Case of porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.17Optimal conductivity of two-components porous media . . . . 18.17.1Optimization of two conductivities . . . . . . . . . . . . . . . . 18.17.2Upper bounds of the optimal properties . . . . . . . . . . . 18.17.3Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.18Fields fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.18.1Second order statistics of fields . . . . . . . . . . . . . . . . . . . 18.18.2Distribution function of fields . . . . . . . . . . . . . . . . . . . . 18.19Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.20Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.20.1Properties of the operator Γ . . . . . . . . . . . . . . . . . . . . . 18.20.2Calculation of the second order perturbation term in the scalar isotropic case . . . . . . . . . . . . . . . . . . . . . . . 18.20.3Calculation of a third order term . . . . . . . . . . . . . . . . . 18.20.4Field averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.20.5Field averages in components . . . . . . . . . . . . . . . . . . . . 18.20.6Hill-Mandel condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.20.7Geometrical average effective property in R2 . . . . . . .
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19 Digital Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701
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19.1 Introduction to Digital Materials and to numerical approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 19.2 Homogenization of random media by numerical simulations 702 19.3 Fluctuations of apparent properties, and statistical RVE . . . 19.3.1 The integral range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.2 Practical determination of the size of the RVE . . . . . 19.4 A case study: the Voronoi mosaic . . . . . . . . . . . . . . . . . . . . . . .
703 704 706 708
19.5 FE Computation on 3D confocal microscope micrographs . . 19.5.1 Experimental effective physical properties . . . . . . . . . 19.5.2 Numerical estimation of apparent physical properties by FE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5.3 Determination of the integral ranges . . . . . . . . . . . . . . 19.5.4 Size of the RVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5.5 Phase connectivity and effective properties . . . . . . . . 19.6 Improving elastic properties from Boolean models . . . . . . . . 19.7 Gigantic RVE: 3D Poisson fibres . . . . . . . . . . . . . . . . . . . . . . . . 19.8 Stochastic Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.9 Numerical solutions of the Lippmann-Schwinger equation by iterations of FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.10Fields fluctuations in dielectric random media . . . . . . . . . . . . 19.10.1Fields in dielectric autodual RS . . . . . . . . . . . . . . . . . . 19.10.2Fields in a 3D Boolean model of spheres . . . . . . . . . . . 19.10.3Optical properties of paints . . . . . . . . . . . . . . . . . . . . . .
712 713
19.11Elastic and thermal response of heterogeneous media from 3D microtomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.11.1Hotspots in a granular material . . . . . . . . . . . . . . . . . . 19.11.2Stress localization in a mortar microstructure . . . . . . 19.11.3Elastic and thermal properties of lightweight concretes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.12Elastic and viscoelastic properties of multiscale random media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.12.1Bulk modulus of the Boolean model of spheres . . . . . 19.12.2Linear elastic properties and conductivity of multiscale Cox Boolean models . . . . . . . . . . . . . . . . . . . 19.12.3Nonlinear elastic and conductivity of multiscale Cox Boolean models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.12.4Elastic and viscoelastic properties of rubber with carbon black filler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.13Elastic properties of fibrous materials . . . . . . . . . . . . . . . . . . . 19.14Diffusion and fluid flows in porous media . . . . . . . . . . . . . . . . 19.14.1Diffusion in random porous media . . . . . . . . . . . . . . . . 19.14.2Estimation of the fluid permeability of porous media
713 715 717 717 718 719 721 723 726 727 731 731
733 733 735 741 742 742 745 749 750 753 755 756 758
Contents
19.15RVE of acoustic properties of fibrous media . . . . . . . . . . . . . . 19.15.1Thermoacoustic equations and homogenization of acoustic properties of porous media . . . . . . . . . . . . . . . 19.15.2Homogenization of acoustic properties of a periodic fibrous network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.15.3Acoustic fields simulated on random unit cells . . . . . . 19.15.4Statistical RVE and integral ranges . . . . . . . . . . . . . . . 19.16Further examples of application . . . . . . . . . . . . . . . . . . . . . . . . 19.17Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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764 765 767 768 769 769 770
20 Probabilistic Models for Fracture Statistics . . . . . . . . . . . . . 773 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 20.2 Choice of a fracture criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 775 20.2.1 Local fracture criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 775 20.2.2 Global fracture criteria . . . . . . . . . . . . . . . . . . . . . . . . . 775 20.3 Brittle fracture and weakest link . . . . . . . . . . . . . . . . . . . . . . 776 20.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776 20.3.2 Examples of stress fields used for the weakest link model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 20.3.3 The Boolean random varieties and the weakest link model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 780 20.3.4 Randomization of Boolean varieties . . . . . . . . . . . . . . . 791 20.3.5 Weakest link model and iterated Boolean Varieties . . 792 20.3.6 The Dead Leaves varieties and the weakest link model799 20.3.7 Competition between fracture mechanisms . . . . . . . . . 801 20.3.8 Multicriteria and multiscale weakest link models . . . 805 20.4 Fracture statistics models with a damage threshold . . . . . . . 811 20.4.1 Fracture statistics models with a critical volume fraction of defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811 20.4.2 Critical density of defects . . . . . . . . . . . . . . . . . . . . . . 814 20.5 Fracture statistics model with a crack arrest criterion . . . . 816 20.5.1 Crack propagation and the Griffith’s criterion for two-dimensional random media . . . . . . . . . . . . . . . . . . . 816 20.5.2 Types of probability distributions obtained from the models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819 20.5.3 Probability of fracture and scale effects for the Poisson Mosaic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821 20.5.4 Probability of fracture and scale effects for the Boolean Mosaic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826 20.5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832 20.6 Models of random damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832 20.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832 20.6.2 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833 20.6.3 Random damage under a homogeneous load . . . . . . . 834 20.6.4 Random damage under a non homogeneous load . . . 837
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20.6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.7 Elements of practical use of fracture statistics models in numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.9.1 Crack propagation in a random polycrystal with grain boundary fracture energy . . . . . . . . . . . . . . . . . . 20.9.2 3D crack propagation in a random medium . . . . . . . 21
Crack Paths in Random Media . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Probabilistic fracture of 3D random tessellations . . . . . . . . . 21.2.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Fracture of a random tessellation and percolation . . . 21.3 Phase field for cracks propagating in random media . . . . . . . 21.3.1 Reminder on phase field models for fracture of homogeneous isotropic media . . . . . . . . . . . . . . . . . . . . 21.3.2 A phase field model for heterogeneous anisotropic fracture energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.3 Introduction to full fields estimation by FFT . . . . . . . 21.3.4 Estimation of an effective toughness . . . . . . . . . . . . . . 21.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
840 841 842 843 843 844 847 847 848 848 849 851 851 855 857 862 863
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915
1 Introduction
The purpose of this book is the development of probabilistic models giving a simplified representation of heterogeneous structures (Part I and Part II) and enabling us to predict some change of scale effects (micro-macro scaling) for the physical behavior of heterogeneous media (Part III). Our approach is directed towards probabilistic models. Other books were published on these subjects over years. To mention some monographs, and excluding proceedings, books [113], [591] review basic models of random sets, as introduced in Stochastic Geometry. Books [497], [498], [499] are concentrated on a basic type of model (namely the Boolean model), mainly from a theoretical point of view. Books [532] and [515] concern mainly image analysis, and are mostly algorithmic. The present book is a large extension of earlier works [257], [275], of a preliminary unpublished version, and of lecture notes on courses on random media [287], [297], [308]. We will make use of the probabilistic modeling paradigm, developing a synthetic approach, in contrast with data analysis. The main advantages of this approach are to make possible a representation and simulation of microstructures (based on the notion of « virtual materials » in materials science), after identification of models by image analysis. Once a proper model is available, it can be implemented for the prediction of probabilistic properties of heterogeneous media, and in the next step for the optimization of microstructures with respect to their physical behavior.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Jeulin, Morphological Models of Random Structures, Interdisciplinary Applied Mathematics 53, https://doi.org/10.1007/978-3-030-75452-5_1
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1. Introduction
1.1 Use of a probabilistic approach Why to use models of Random Structures? In the present case, there is no assumption about any specific role of Randomness, as opposed to Determinism [451], [462]. We will not use neologisms like disorder or chaos. In fact, the probabilistic approach is a choice of a methodology and of models [451], [462] useful to bring a practical and efficient solution to some problems such as: to estimate spoiled or missing data, structural, morphological, or physical properties of heterogeneous media. At least two practical reasons justify the probabilistic approach. • The first one is connected to a more or less important heterogeneity of microstructures at different scales, that seems to be opposed to structured aspects. The combination of these two aspects is often referred to as Textures, when dealing with images. How to account for these two points? How to replace fluctuating data at a small scale by global characteristics when going from Microscopic to Macroscopic? • The second one is the consequence of the mode of access to regionalized data are known from a sampling procedure: it can be for instance the result of a selection of objects in a population (grains of a powder, biological cells, trees, plants, individuals). In addition, there is often a partial knowledge of objects: ore deposits are explored by probes; the microstructure of a material is accessed from 2D or 1D slices; a polished section is examined with the microscope in a finite number of zones. From the sampling process of heterogeneous media, it is important to ask oneself about the representativeness of data: how to extrapolate partial available information to a full single object or to a full population of objects? To answer this question, two typical extreme models can be used: — a periodic medium, completely known from a single period; — a probabilistic model; The first case is satisfactory for ordered media like perfect crystals. The second case covers most real practical situations for heterogeneous media.
1.2 Aims of probabilistic models The models that will be introduced have two main aspects: a descriptive aim, and a predictive aim for problem solving.
1.2 Aims of probabilistic models
3
1.2.1 Descriptive aspect The descriptive part of models is oriented towards a simplified representation of a complex structure, to get: • a summary of structural data in few parameters (1, 2,...) for relation with the physical properties of a medium, as well as for classification of structures, as used in texture machine learning. • genetic models, namely simplified procedures of construction to simulate the physical processes driving the formation of the structure. Of course it is wise not to forget the more or less extreme simplifications, and therefore to remain modest about interpretations in terms of physical mechanisms! Nevertheless, the use of physical processes leads to models giving more or less realistic simulations, well suited to the representation of some phenomena (the reader can judge from the illustrations given in various chapters of this book).
1.2.2 Predictive aspect The predictive aspects of probabilistic models are illustrated from typical examples of problems to which satisfactory answers are given. • What can be obtained from partial data? This is the object of Geostatistics for regionalized data [437], [451], [462]. — To give the precision of a global estimation, such as the integral Z I= f (x)dx R
estimated from sampling points by I∗ = a
p=+∞ X
f (x0 + pa)
p=−∞
— To interpolate data at points without information (orebody deposit exploration, missing data such as non available slices for a 3D image analysis of a microstructure,..) by kriging. — To restore noisy data by filtering (e.g. by kriging) [139], [140], [121]. — To solve optimization problems by simulations. This is a common practice in game theory, operation research, orebody deposits exploitation, oil reservoir production (both involving conditional simulations that respect data at points with information), reliability computation, and in some cases materials conception and optimization [45],...
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1. Introduction
FIGURE 1.1. Nodular graphite in pig iron
• Further practical problems get solutions from a probabilistic approach. — Stereology: how to reach 3D morphological properties from 1D or 2D morphological properties (obtained on slices or on projections)? — Counting and measuring: how to count the number of imbricated objects in a texture, without segmenting the image? How to estimate a powder size distribution from images of overlapping objects? — Change of support: how the statistical properties of a medium change with the size and shape of the support of data? — Change of scale: how to predict the macroscopic physical behavior of a medium from its microscopic behavior? For instance, in Fracture Statistics of brittle materials, how to predict the fracture probability of parts (or planes, buildings,...) from data on small scale samples? This book illustrates through probabilistic models answers to the above mentioned problems. As suggested in [451], [462], it is also possible to construct algorithms based on theoretical probabilistic models, to be used more generally in a heuristic way (filtering, counting, change of support, change of scale,...)
1.3 Types of models The principal types of regionalized data and corresponding types of models can be commonly classified as follows: • i) Dispersions of small particles in a matrix (like non metallic inclusions in steel or nodular graphite in Fig. 1.1), modelled by realizations of stochastic point processes. • ii) Granular structures (polycrystals as in Fig. 1.2), assimilated to random tessellations of space (each class corresponding to a grain). • iii) Two phase (porous media as in Fig. 1.3) or multiphase structures (composite materials with several components as in Fig. 1.4) may be
1.4 Model construction
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FIGURE 1.2. Polycristal in steel
•
• •
• •
simulated by random sets (binary), or multicomponent random sets. iv) Rough surfaces (steel plate on Fig. 1.5, fracture surface,...), chemical concentration mappings (X ray images obtained with an electron microprobe), and more generally grey level images (video images, secondary electron images in a scanning electron microscope,...) can be represented by random functions. v) Multivariate data (multi species chemical mappings, components of a vector or of a tensor in every point x of space), modelled by multivariate random function models. vi) Sequential images (change of a microstructure during its formation, its deformations under solicitations, its degradation; successive grounds in a perspective view) are associated to sequential random sets or random functions. vii) Hierarchical data and hierarchical structures can be modelled by random trees. viii) Data on a network connecting vertices (roads, porous medium, cracks,...) and their properties depending on connectivity, can be modelled by random graphs. In Part II of this book, mostly models for points iii)-vi) are developed.
1.4 Model construction To build random structures, it is required to define: • a set of events and their combination (σ algebra), e.g. open sets on a topological space (the Borel σ algebra); • a probability. The proposed models are derived from the theory of Random Sets by G. Matheron [438],[441],[448] . We will see in chapter 2 that a random structure is characterized by its Choquet capacity, as summarized now.
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1. Introduction
FIGURE 1.3. A porous medium: metallurgical coke
FIGURE 1.4. Mineral textures in iron ore sinters
FIGURE 1.5. Roughness of a steel rolling mill
1.4 Model construction
7
1.4.1 Characterization of a random model A random closed set A (RACS) is known from the Choquet capacity T (K) defined on the compact sets K T (K) = P {K ∩ A 6= ∅} = 1 − P {K ⊂ Ac } = 1 − Q(K) An upper semi continuous random function (usc RF), is characterized by the Choquet capacity T (g) defined on the lower semi continuous functions (lsc) g with compact support K T (g) = P {x ∈ DZ (g)}; with DZ (g)c = {x, Z(y) < g(y − x), ∀y ∈ K} For the function g(x) = z for x ∈ K, T (g) = 1 − P {Z∨ (K) < z}; with Z∨ (K) = ∨x∈K {Z(x)} Similarly for a lower continuous random function (lsc RF) can be defined the functional P (g) with P (g) = P {x ∈ HZ (g)}; with HZ (g) = {x, Z(y) ≥ g(y − x), ∀y ∈ K} For the function g(x) = z for x ∈ K, P (g) = P {Z∧ (K) ≥ z}; with Z∧ (K) = ∧x∈K {Z(x)} Emphasis should be placed on the following points: • The Choquet capacity of a RACS is equivalent to the distribution function for a random variable. • Two models (RACS, usc RF) with the same functional T (K), T (g), or P (g) cannot be distinguished (theoretically as well as experimentally). • The functional T (K), (or T (g), or P (g)), connects theory and experiments; it is used to estimate the parameters of a model and to test its validity. To construct a model of random structure, the following approach will be used, as illustrated in Part II of this book.
1.4.2 Choice of basic assumptions Various types of basic assumptions can be used, some being mentioned for illustration. • A construction can be proposed from elementary processes and geometrical modes of interaction. This corresponds to the so called class of genetic models.
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1. Introduction
• One can define a RF as a solution of stochastic partial differential equations (e.g. Reaction-Diffusion models in chapter 16). • Prior general properties can be required for a given class of models, among which: stationarity, independent increments, infinite divisibility, Markov property (lack of memory), stable distribution after transformation,... • A model can be obtained as an asymptotic limit of families of models: the Gaussian RF (connected to the ”Central limit” theorem); the Boolean RF obtained by the supremum ∨ of independent RF (chapter 9).
1.4.3 Computation of the functional T (K) The functional T (K) (or T (g)) must be known as a function of the assumptions of the model, of its parameters, and of the compact set K or the function g. For a given model, the functional T is obtained: • by theoretical calculation • by estimation on simulations of the random structure, or on samples of the real structures. In the last case, it is possible to estimate the parameters from the ”experimental” T , and to test the validity of assumptions. The estimation of the functional T (K) is easily performed from the implementation of the basic operations of mathematical morphology, namely erosions or dilations. It is important to point out here the fact that the functions T (K) (K being variable) are coherent (which is not the case of any prior analytical model, as often used in the literature). After specification and validation of the model from available data, it is possible to use a predictive implementation of its properties (such as T (K) for compacts K not used during the identification step). Practical examples are the following: 3D properties deduced from 2D observations (stereology); change of support by ∨ or ∧ in the case of a change of scale in fracture statistics (chapter 20).
1.5 Some general properties of the models Most random structure models that we present are defined in the Euclidean space Rn : • they are more general than stochastic processes limited to the 1D space R, where the order relation is used; • they differ from discrete models defined on a grid, even if the discrete counterpart of the Euclidean models is easily defined.
1.6 Organization of the book
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The models depend on only few parameters, not to ask too much from the available data, and for a realistic experimental identification and test. They have mostly a stationary and ergodic version, to allow a statistical inference from a single realization. They own good stereological properties, for access from slices or projections. Finally there are facilities for their simulations.
1.6 Organization of the book This book is organized as follows (most chapters being completed by exercises with elements of solutions, which provide additional probabilistic properties useful for applications): A first part gives an introduction to random sets and random functions and to their main probabilistic properties (chapter 2). A practical point of view on the morphological analysis of random structures is given in chapter 3. A second part is dedicated to random structure models. Several families of models are reviewed, first excursion random sets obtained by truncating Gaussian random functions (chapter 4), stochastic point processes and random trees (chapter 5), the Boolean random set model and the Poisson varieties (chapter 6). The Boolean model and its generalized versions (chapters 6, 9) are suitable for connected structures (two phase or porous media, numerical functions). They privilege the set union for the binary version, the operators ∨ and ∧ operators for the function version. These models are built in two steps: implantation of random germs (according to a Poisson point process, or more generally to Poisson random varieties); translation to these germs of independent realizations of primary grains (compact random sets for the Boolean model, and semi continuous random functions for the Boolean RF). The Choquet capacity of these models, that owns interesting properties such as the infinite divisibility for ∪ or for ∨ are worked out theoretically, so that tests of validity are proposed. Basic random tessellations models (chapter 7) are used to generate granular media like metallic grains in steel. The mosaic model (chapter 8), obtained by affectation of independent realizations of a random function (scalar or multivariate) to classes of a random tessellation (chapter 7). Semi continuous versions of this model give a good description of granular microstructures, such as polycrystals, to simulate the variation in space of their crystallographic orientation or of their cleavage strength. Second and third order statistics are given. For the Poisson mosaic is obtained the law of the change of support by the operators ∨ and ∧ over a segment.
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The Boolean RF model (chapter 9) is a generalization to functions of the Boolean model for sets. It is also a first step in the generation of random tessellation models (chapter 10) generalizing basic conventional models. The sequential RF (namely various versions of the Dead Leaves model) of chapter 11 enable us to simulate sequences of images corresponding to the following situations: evolution of a microstructure during its formation, (for instance crystallizations of different components out of a melt), grounds in a perspective view, stacked grains of a powder examined in a scanning electron microscope. These models are built in the same way as the Boolean models, after replacing the operator ∨ by the first or the last occurrence of the values observed on every point of the space during the sequence. For the various Dead Leaves models (tessellation, multi component texture, or RF), this is similar to the covering process observed for falling leaves. A version of the model (Markov jumps RF) considers the occurrence of a family of primary RF inside a stationary random mask. Statistics of order 1, 2 and 3 are given as a function of the historical account (from parameter t) and of the equivalent properties of the primary functions. In some conditions, the properties of Zt∧ (K) are known for a connected compact set K. As a particular application, this makes possible to estimate an unbiased size distribution of a powder studied in the scanning electron microscope(SEM) in the case of mixes of powders with convex or non convex grains. Sequential Cox Boolean models and conditional Dead Leaves (chapter 12) introduce a local conditioning for the location of germs, to generate various spatial arrangements with aggregation or repulsion effects. The sequential alternate RF (chapter 13) are obtained from a combination of two families of primary random functions by means of the operators ∨ and ∧, in a similar way as for the Boolean RF. Their building process by a sequence of abrasions and of depositions, make them attractive to simulate rough surfaces (with possible applications in tribology or in geology). They restore a symmetry between summits and valleys, which is lost in the case of Boolean surfaces which favor one type of extrema. First and second order statistics are available for this model. The chapter 14 presents random primary grains that can be used as an input to the previous models. In fact are developed here models with a compact support. Several families are introduced: populations of spheres and of ellipsoids in R3 , Poisson polyhedra, random aggregates; several processes to build primary RF are proposed: cylinder RF, restriction of a stationary RF to a compact support, spherical primary RF, distance function to a random set, Boolean and Dead Leaves RF with a compact support. The Dilution RF model (chapter 15) is similar to the Boolean RF model, after replacing the operator ∨ by the more conventional operation +.
1.6 Organization of the book
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First Step of the Structure Point Process (Poisson) Random Variety Tessellation Zones of influence of Poisson polyhedra germs (Voronoï, Johnson-Mehl) Cauwe polyhedra Dead Leaves Tessellation TABLE 1.1. Construction of Random Tessellations First Step of the Structure Point Process Poisson Random Variety Tessellation Union of Primary grains Dilation of varieties by Primary grains Affectation Boolean Model (2 phases) of every class to Ai Sequential Construction with probability Pi Color Dead Leaves Color Dead Leaves Variety Multiphase Texture TABLE 1.2. Main steps of construction of some Multicomponent Random sets
In chapter 16 a family of spatial temporal models, Reaction Diffusion RF, is introduced. These models are solutions of stochastic non linear parabolic partial differential equations in the general case. They simulate the evolution of a structure made of various chemical species interacting by chemical reactions and propagating in space by diffusion. These models can be introduced at a microscopic scale in a discrete version combining Markov jump processes (for the chemical reaction) and random walks (for the diffusion); they can also be simulated by the lattice gas model reproducing a hydrodynamic behavior. The theoretical study of the general models is difficult because of the non linearity. The linear reaction-diffusion model is presented when using as initial conditions a stationary RF and a random source. This model including a particular case of dilution RF gives access to very smooth RF, sometimes indefinitely differentiable, contrary to the models given in the previous chapters, which only are mean square continuous. In each case, a multivariate version of the model is given. The introduced models are illustrated by some simulations. The articulation of most models and their common points are summarized and illustrated in Tables 1.1, 1.2, 1.3, 1.4, 1.5: two steps (or sequential) construction, and use of a primitive (primary grains or primary functions, random tessellation in the mosaic case). We concentrate on the random functions (scalar or multivariate), random tessellations or random sets being particular cases of the preceding constructions: Boolean model, Dead Leaves,...
In chapter 17 are introduced tools for a probabilistic hierarchical segmentation of various types of textures (scalar or multivariate), by progressively merging regions of a fine partition. It combines appropriate morphological operations and texture classification for supervised or unsu-
12
1. Introduction First Step Point Process Poisson Sup of Primary functions Inf of Primary functions Boolean RF of Primary functions Dilution RF (cf. Reaction-Diffusion)
of the Structure Random Variety Tessellation Dilation of varieties Random affectation by Primary functions of every class of a RV Boolean RF varieties with distribution G(z) Mosaic RF Dilution RF varieties
TABLE 1.3. Main steps of construction of some Random Functions First Step of the Structure Point Process Poisson Random Variety Sequential Random Functions Dead Leaves RF Dead Leaves RF varieties Markov jumps RF Sequential alternate RF Sequential alternate varieties (Sup and Inf of Primary functions) TABLE 1.4. Main steps of construction of some sequential Random Functions First Step of the Structure Point Process Poisson Random Variety Tessellation Multicomponent Random Grain multi c. Boolean Random Set random affectation Multivariate Boolean RF Boolean RF variety to each class Multivariate Dilution RF Dilution RF variety G(z1 , .., zn ) Multivariate Dead Leaves RF Dead Leaves RF variety Multivariate Multivar. Sequential alternate Sequential alternate RF var. Mosaic RF TABLE 1.5. Main steps of construction of some Multivariate Random sets and functions
pervised texture segmentations, a probabilistic distance between regions, and the use of various kinds of random markers carrying statistical information on textures. In each situation, required probabilities are computed in a closed form by simple algebra. This probabilistic approach of segmentation is made of some random sampling process in images, and makes use of random sets presented in this book. The third part presents changes of scale problems in heterogeneous media, occurring in the Physics of Random Media. This is concerned by the prediction of the macroscopic behavior of a physical system from its microscopic behavior. A first topic (chapter 18) concerns the estimation of the effective properties (namely the overall properties of an equivalent homogeneous medium) or homogenization of random heterogeneous media from their microstructure. The approach, using variational principles, provides bounds of the effective properties for linear constitutive equations. It is illustrated by third order bounds (from third order central correlation functions) results derived for some of the models of the first part, and for microstructures with widely separate scales. Most recent developments involve the numer-
1.6 Organization of the book
13
ical generation of simulated microstructures at different scales, also called Digital Materials, combined to a numerical resolution of partial differential equations PDE involved in the studied physical problem (chapter 19). Due to the limited size of simulations, the observed apparent effective properties can fluctuate between realizations, so that a rigorous statistical approach has to be followed to generate a representative volume element RVE and to provide size dependent intervals of confidence of the estimated physical properties. A second class of topics concerns fracture statistics models (chapter 20) based on random structure models studied in the first part of the book. In this case, they simulate at a point scale the variations of a fracture criterion (critical stress σ c , or critical stress intensity factor KIc for brittle materials; local fracture energy γ). The models proposed in this chapter are based on different macroscopic fracture criteria: the weakest link assumption is suitable for the sudden fracture of brittle materials; models with a damage threshold allow several sites with a crack initiation before fracture; models based on the fracture energy according to the Griffith’s criterion account for the propagation and arrest of cracks in a random medium. For every family of models the probability of fracture is worked out as a function of the loading conditions and of the parameters of the selected random structure models. Some useful aspects for applications, such as the prediction of expected scale effects, are derived. Models of random damage based on statistical populations of defects predict the average macroscopic stress-strain curves accounting for the development of damage under loading conditions and show a full range of possible behaviors, from a brittle to a ductile constitutive law. The proposed fracture models can be tested at different scales (including the microscopic scale, by use of image analysis). The diversity of the obtained theoretical distributions for fracture statistics offers new possibilities for the microstructure based interpretation and modelling of mechanical data obtained on materials. Finally, novel results concern the simulation of crack paths in random media (chapter 21), which can give access to a numerical estimation of the effective fracture energy (or toughness) of complex heterogeneous microstructures.
Part I Tools for Random Structures
2 Introduction to Random Closed Sets and to Semi-Continuous Random Functions
Abstract: This chapter recalls the construction and the main properties of random closed sets: i) The Choquet topology, for the closed sets of a space E, enables us to build a probability space of closed random sets, characterized by their Choquet capacity. ii) From the same approach extended to upper or lower semi continuous numerical functions, it is possible to build random functions for which the supremum or the infimum of its values inside a compact set K is a random variable.
2.1 Introduction Models developed in this book are derived from the theory and concept of random sets elaborated by G. Matheron [438], [441], [448]. To ensure a correct and rigorous mathematical construction of random structures, an appropriate probability space must be defined, as summarized in this technical chapter. To conceptually build random structures, two steps are required: i) a set of events and their combination (through the notion of σ algebra, like for instance open sets on topological space generating a Borel σ algebra); ii) a probability. The aim of this chapter is to introduce some classes of objects on which we will work later: random closed sets and semi continuous random functions (RF). Our purpose summarizes the construction and the properties of these general objects, with a recall of some important results. The inter© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Jeulin, Morphological Models of Random Structures, Interdisciplinary Applied Mathematics 53, https://doi.org/10.1007/978-3-030-75452-5_2
17
18
2. Random Closed Sets and Semi-Continuous Random Functions
ested reader is invited to consult [438], [441], [448], [463], [598], [600] for a complete study. Technical and may be seemingly abstract aspects for the practitioner should not hide the physical background of this mathematical construction. The connection between introduced concepts and the experimental aspects is illustrated in details in chapter 3 by the choice of some structuring elements and by their morphological interpretation. But before, it is interesting to give in this introduction some physical justifications that motivated the construction of a theory of random sets and of semi continuous RF [438], [441], [448], [598]. To characterize a set A in a space E (for instance the grains of a porous medium seen as a subset of the Euclidean space R3 ), it is possible to consider a reference set K ⊂ E, and to examine the mutual locations of K and of A, by means of the answer to the two following exclusive questions: i) K disjoint from A? (K ∩ A = ∅)? (2.1) ii) K hits A (K ∩ A 6= ∅)? For instance if E = R3 and if K is the point x, the previous test enables us to test whether x ∈ / A (i) or x ∈ A (ii), meaning to which component of the two-phase medium (A, Ac ) the point x belongs. It is natural to repeat the same process for K made of the pair of points {x, x + h}, or more generally of a countable set of points {x1 , x2 , ..., xn , ...}. By increasing the number of points, the number of positive answers to questions i) decreases and ii) increases, and correlatively a richer information is obtained about the structure of A. Let us assume now that x ∈ / A and that is required to estimate the distance separating the point x from the set A: R(x, A) = ∧y∈A {d(x, y)}
(2.2)
Let Bx (r) the closed ball with radius r and with center x: y ∈ Bx (r) ⇔ d(x, y) ≤ r. R(x, A) = ∨{r; Bx (r) ⊂ Ac } (2.3) Equation (2.3) involves sets Bx (r) containing a non countable set of points. To build random sets A, a coherent set of events (building a σ algebra) and a probability law P are needed. The theory of Random Sets by G. Matheron [438], [441], [448], and the parallel development of D. G. Kendall [378], is based on events of the type (2.1) for sets K having any number of points (finite, or non countable infinite). Thus it is possible to speak about the probability for a point x to belong to the set A, or for a n-uple {x1 , x2 , ..., xn } to belong to the set Ac (complementary set of A). This gives the notion of spatial law, common in the theory of stochastic processes. In addition, the event {Bx (r) ⊂ Ac } will be meaningful, and its probability can be defined.
2.2 Introduction to random closed sets (RACS)
19
When considering numerical functions (for instance a chemical concentration c(x) in every point x of space), it is often made use of continuous functions models. This class of function is too much restrictive for most phenomena encountered in practice: to come back to the notion of set, the indicator function of set A, 1A (x) (with 1A (x) = 1 if x ∈ A, else 0) is itself discontinuous at each transition between the two sets A and Ac . Another example is given by a function which remains constant inside every class of a space tessellation, but changes from one class to its neighbors (a physical example is a polycrystal, where the variable of interest might be a function of the orientation of every crystal. As a function of space, it is discontinuous for disorientation between grains). In addition, one may be interested in the building of the following functions, starting from the function f (x) and from the set B: i) f∨ (B) = ∨x∈B {f (x)} (2.4) ii) f∧ (B) = ∧x∈B {f (x)} where ∨ and ∧ represent the supremum and the infimum of the values taken by the function f (x) for x belonging to the set B. The two functions f∨ (B) and f∧ (B) defined by Eqs (2.4) give a change of support with the operators ∨ and ∧. Operator ∧ is used in chapter 20, when modelling the fracture of materials obeying to the weakest link criterion. The construction of random function models should allow us to consider the case of non continuous functions, and also to justify the use of random variables defined from a random function by means of one of the change of support given in Eqs (2.4). Random closed sets are characterized by events corresponding to Eq. (2.1) for a compact set K (2.1 i)) or an open set (2.1 ii)). For random functions (RF), are considered upper semi continuous (usc) and lower semi continuous (lsc) functions. When B is a compact set K, Eqs (2.4) define random variables for usc or for lsc RF respectively. Other types of random structures, that are not introduced here, can receive a correct mathematical construction: random tessellations [441] as detailed on chapter 8), random measures [362], random distributions [437], random graphs [76], random trees (see chapter 5).
2.2 Introduction to random closed sets (RACS) The notion of random set requires the construction of events (and of their combination) and of a probability. In a first step, we have to recall some usual definitions in probability theory [177].
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2. Random Closed Sets and Semi-Continuous Random Functions
2.2.1 Recall of basic definitions in probability theory Definition 2.1. A probability space is made of: i) a set Ω; ii) a σ algebra A; iii) a positive measure, with measure 1 over A, called a probability P . Definition 2.2. A σ algebra A is a family of sets A ∈ A, satisfying the following axioms: i) Ω ∈ A; ∅ ∈ A; (∅ is the empty set); ii) ∀A ∈ A, Ac ∈ A; iii) for any finite or denumerable family Ai (i ∈ I), Ai ∈ A ⇒ ∪i∈I Ai ∈ A. Therefore, ∩i∈I Ai ∈ A. The probability P satisfies: P {Ω} = 1 if Xi ∈ A andPXi ∩ Xj = ∅ (i 6= j) then P {∪i∈I Xi } = i∈I P {Xi } for I countable
As examples of σ algebra, consider:
• the set ℘(Ω) of the parts of the set Ω. • the set generated by the open sets of a topological space, or Borel σ algebra. Definition 2.3. A topological space is made of a set E and of a set G of subsets of E, called open sets, satisfying the following conditions: i) ∅ ∈ G; E ∈ G; ii) every finite or infinite union of open sets is an open set; iii) every finite intersection of open sets is an open set. A part G 0 ⊂ G builds a base of the topology ⇔ every open set G of E is the reunion of sets of G 0 .
2.2.2 Definition of the random closed sets Let E be a locally compact separable topological space (namely with a topology admitting a countable base). F, G and K define the sets of closed, open, and compact sets in E. Call σ f the smallest σ algebra defined on the closed sets F(E), containing the open sets G(F(E)) of the hit or miss (Choquet) topology, as defined later; σ f is the Borel σ algebra of this topology. Definition 2.4. A Random Closet Set (RACS) is defined by the triple (F(E), σ f , P ). This is a probability space defined over the closed sets of the space E. The elementary events composing the σ algebra σf are defined by: i) the RACS A does not hit the compact K; ii) the RACS A hits the open sets Gi . Definition 2.5. The Choquet topology (also named the ”hit or miss topology”) is built on the space of closed sets F(E) from the following open sets:
2.2 Introduction to random closed sets (RACS)
21
FIGURE 2.1. F and F 0 belong to the neighborhood of the Choquet topology defined by the compacts K1 and K2 and by the open sets G1 and G2 : K1 ∪K2 FG = F K1 ∩ F K2 ∩ FG1 ∩ FG2 1 G2
F K = {F ∈ F; F ∩ K = ∅, K ∈ K}
(2.5)
FG = {F ∈ F; F ∩ G 6= ∅, G ∈ G}
(2.6)
K or, for k finite, FG = 1 G2 ...Gk
{F ∈ F; F ∩ K = ∅, K ∈ K; F ∩ Gi 6= ∅, Gi ∈ G, i = 1, 2..., k}
(2.7)
= F K ∩ FG1 ∩ FG2 ∩ ... ∩ FGk The neighborhoods defined by Eq. (2.7) are made of all the closed sets F disjoint from the compact set K and hitting the opens sets G1 , G2 ,..., Gk . The open sets FG defined by Eq. (2.6) are based on the intersection logic. Similarly, Eq. (2.5) can be written K ⊂ F c , expressing the inclusion logic for the open set F c . An example of neighborhood is represented in Fig. 2.1. The topology on closed sets is used to study their convergence. This important topic is not discussed here. The space F, equipped with the Choquet topology, is compact, and admits a countable base. It is therefore possible to build a probability P over σ f , and to define a Random Closed Set. In the same way, and by duality, it is possible to define the notion of open random set, from a σ algebra σ g built on the open sets GK (K ∈ K) and G G (G ∈ G) from: GK = {G ∈ G; G ⊃ K, K ∈ K}
(2.8)
G G = {G0 ∈ G; G0 6⊃ G, G ∈ G}
(2.9)
A random open set is defined by a probability P over σ g . More generally, an equivalence relationship can be introduced for the sets A ∈ ℘(E), by
22
2. Random Closed Sets and Semi-Continuous Random Functions
˚ and the same adherence A. identifying the sets with the same interior A On these equivalence classes, one can build a σ algebra σ P generated by the following events: G0 ˚ ∈ GG , A ∈ F G0 ; G ∈ G, G0 ∈ G} MG = {A: A
(2.10)
As previously, Eq. (2.10) implies the simultaneous use of the intersection and reunion logics.
2.2.3 The Choquet capacity Theorem 2.1. (G. Matheron & D.G. Kendall). The RACS A is characterized by a probability P defined on σ f , and completely determined by the functional T (K) (its Choquet capacity) defined on the compact sets K : T (K) = P {FK } = P {A ∩ K 6= ∅} = 1 − P {K ⊂ Ac } = 1 − Q(K) (2.11) or equivalently by: Q(K) = P {F K } = P {A ∩ K = ∅} = 1 − T (K)
(2.12)
The functional T (K) is increasing and lower semi continuous on K: Kn ↓ K ∈ K ⇒T (Kn ) ↓ T (K) Similarly, Q(K) is decreasing and lower semi continuous on K. The Choquet capacity must respect some consistency relationships, that can be compared to the case of multivariate distributions in probability theory. As a particular case, T (K) enables us to calculate the following probabilities Sk (equivalent to the Poincaré relations): K0 } Sk (K0 , K1 , ..., Kk ) = P {FK 1 K2 ...Kk
= Q(K) − where:
X
i
Q(K ∪ Ki ) +
X
i1 0) ˇ , from the product space — Dilation by a compact K : A → A ⊕K F × K onto F with ˇ = {x, Kx ∩ A 6= ∅} = ∪x∈A, y∈K {x − y} A⊕K
(2.16)
Kx = {x + y; y ∈ K} being the translated of K at the point x — Convex hull of a compact A: C(A)
Upper semi continuous transformations • Intersection ∩ : A, A0 → A ∩ A0 from the product space F × F into F. • In Rn : ˇ from the — erosion by a non empty compact K : A → A ªK 0 product space F × K into F with ˇ = {x, Kx ⊂ A} = ∩y∈K A−y AªK
(2.17)
For instance, if B(r) is the closed ball with radius r, when r → 0, lim(AªB(r)) ⊂ A but lim(A⊕B(r)) = A (the dilation is continuous). ˇ ⊕K — Opening by a compact K : AK = A ªK K ˇ — Closing by a compact K : A = A ⊕K ª K — For a family of usc transformations Ψi (i ∈ I), ∪i∈I Ψi (A) is usc if I is finite ∩i∈I Ψi (A) is usc Lower semi continuous transformations • • • •
Closure of the convex hull: C(A) Closure of Ac :Ac Boundary set: ∂A = A ∩ Ac if the set E is connected. For a family of lsc transformations Ψi (i ∈ I), ∪i∈I Ψi (A) is lsc ∩i∈I Ψi (A) is lsc if I is finite
2.2 Introduction to random closed sets (RACS)
25
2.2.6 Independent random sets The two random sets A1 and A2 are independent if: P {A1 ∈ F K1 and A2 ∈ F K2 } = P {A1 ∈ F K1 }P { A2 ∈ F K2 } = Q1 (K1 )Q2 (K2 )
(2.18)
2.2.7 Spatial law When considering finite parts I, I 0 of E, with I = {x1 , x2 , ..., xn ; xi ∈ E} and I 0 = {x01 , x02 , ..., x0n ; x0i ∈ E}, it is possible to define in ℘(E) the parts MII0 : MII0 = {A; A ⊃ I and A ∩ I 0 = ∅} (2.19) = {A; x1 ∈ A,..., xn ∈ A; x01 ∈ / A,..., x0n ∈ / A} For any random set A can be used the probability P {MII0 }. It is generated by the probability T (I) = P {(M I )c } = P {A ∩ I 6= ∅}, or spatial law of A: T (K) = 1 − P {x1 ∈ Ac , x2 ∈ Ac , ..., xn ∈ Ac }
(2.20)
Contrary to the Choquet capacity (which can use compact sets K with a non countable set of points), the spatial law cannot completely characterize the RACS A: T (I) = 0 for a stationary point process or for a stationary random set with measure 0. Similarly, starting from a RACS A, it is not possible to distinguish by means of T (I) the set A from the set A∪X, where X is a point process on E independent on A, since TA∪X (I) = TA (I), ∀I. An example is provided in chapter 3 about the Poisson point process.
2.2.8 Geometrical interpretation of the Choquet capacity in the Euclidean case • Consider now the Euclidean space Rn , with translations by vector x. The Choquet capacity of A for K at the origin of coordinates O can be expressed by: ˇ T (K) = P {K ∩ A 6= ∅} = P {O ∈ A ⊕ K} For K translated at x : ˇ T (Kx ) = P {Kx ∩ A 6= ∅} = P {x ∈ A ⊕ K}
(2.21)
The functional Q(K) is connected to the erosion operation by : ˇ Q(Kx ) = P {Kx ⊂ Ac } = P {x ∈ Ac ª K}
(2.22)
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2. Random Closed Sets and Semi-Continuous Random Functions
As mentioned in chapter 1, the functional T (K) connects theory and experimentation; it is used to estimate the parameters of a model and to test its validity. Experimental estimation of T (K) can be obtained by image analysis, using realizations of A, and dilation operation. In the general case, several realizations of A are required, and T (Kx ) is estimated from a frequency for every point x. For a stationary random set, T (Kx ) = T (K) (T (Kx ) is invariant by translations of K). If in addition the RACS A is ergodic, T (K) can be estimated from a single realization of A with: ˇ ∗ = VV (A ⊕ K) ˇ ∗ T (K)∗ = P {x ∈ A ⊕ K}
(2.23)
where VV is the volume fraction (for a RACS in R3 ). Eq. (2.23) provides a practical process to estimate T (K) from image analysis on samples of a microstructure. Every compact set K brings its own information on the set A, considered as a realization of a random set. For instance, for the following types of compact sets (the associated morphological content is detailed in chapter 3): • K = {x}
T (x) = P {x ∈ A}
(2.24)
Q(x, x + h) = P {x ∈ Ac ∩ Ac−h }
(2.25)
For a stationary RACS in R3 , T (x) = p = VV (A) • K = {x, x + h} T (x, x + h) = P {x ∈ A ∪ A−h }
Q(x, x + h) is the covariance of Ac . It depends only on h for a stationary random set. • K = B(r) (closed ball with radius r) T (Bx (r)) = P {x ∈ A ⊕ B(r)}
(2.26)
T (Bx (r)) provides a way to us to give the distribution of the random variable R(x, A) defined in Eqs (2.2, 2.3): Fx (r) = P {R(x, A) < r | x ∈ Ac } =
T (Bx (r)) − T (x) 1 − T (x)
(2.27)
The distribution function Fx (r) does not depend on x for a stationary RACS. The covariance and the distribution F (r) are useful for studying the spatial distribution of A. Generalizations of Eq. (2.27) to random sets with several components are introduced in chapter 3 to characterize mutual associations between components. A random set A is said to be autodual if the two sets A and Ac have the same probabilistic properties (if A is a RACS, we have to compare A
2.3 Introduction to Random Functions
27
and the closure of its complementary set, Ac ). This implies that for any compact set K, P (K) = P {K ⊂ A} = Q(K) = P {K ⊂ Ac } As a particular case, an autodual stationary random set must satisfy P {x ∈ A} = P {x ∈ Ac } = 1/2.
2.3 Introduction to Random Functions In the general case, a random function (RF) Z(x) (E → R) is characterized by its spatial law, defined for every m ∈ N : F (x, z) = P {Z(x1 ) < z1 , ..., Z(xm ) < zm } n where x ∈ E m and z ∈ R
(2.28)
When x = x1 and x = {x1 , x2 }, are obtained the univariate and bivariate distribution functions. As Eq. (2.20) for a random set, Eq. (2.28) is restricted to countable sets of points {x1 , x2 , ..., xm } in the space E. Noting C(x, x + h) the covariance and W 2 (x, x + h) the central second order correlation function, E{} being the mathematical expectation: C(x, x + h) = E{Z(x)Z(x + h)} W 2 (x, x + h) = E{Z(x)Z(x + h)} − E{Z(x)}E{Z(x + h)}
(2.29)
More generally, the central correlation function of order m, W m (x), where x ∈ E m is defined by: W m (x) = E{(Z(x1 ) − E{Z(x1 )})...(Z(xm ) − E{Z(xm )})}
(2.30)
For any RF, it is generally not possible to define probabilities involving events with a non countable set of points; for instance, the following changes of support do not define in general random variables : Z I = μ(dx) Z(x) (2.31) (stochastic integral for a measure μ) Z∨ (K) = ∨x∈K {Z(x); K ∈ K}
(2.32)
Z∧ (K) = ∧x∈K {Z(x); K ∈ K}
(2.33)
Whenever one of these operators is used for physical reasons, it is necessary to use appropriate RF models for which they are well defined. For instance, the stochastic integral in Eq. (2.31) can be defined for mean quadratic
28
2. Random Closed Sets and Semi-Continuous Random Functions
FIGURE 2.2. Upper semi continuous function (a); lower semi continuous function (b)
continuous RF, such as the variance D2 (I) of the integral I in Eq. (2.31) remains finite: Z Z 2 D (I) = μ(dx) W 2 (x, y) μ(dy)
In the same way, the expressions in Eqs (2.32,2.33) can be defined for usc or lsc RF.
2.3.1 Semi continuous functions As in [441], we use the following notations: • • • •
Φ(E) is the set of functions from E → R Φf ⊂ Φ is the set of usc functions from E → R Φg ⊂ Φ is the set of lsc functions from E → R Φh is the set of pairs (ϕ, ϕ) with ϕ< ϕ, ϕ∈ Φg , ϕ ∈ Φf
Definition 2.8. A function f ∈ Φ(E) is usc at point x, if ∀λ ∈ R, with λ > f (x) ⇒ ∃V (x), with ∀y ∈ V (x), f (y) < λ, V (x) being a neighborhood of x. A function f ∈ Φ(E) is lsc at point x, if ∀λ ∈ R, with λ < f (x) ⇒ ∃V (x), with ∀y ∈ V (x), λ < f (y). An usc function in x takes the higher value in a discontinuity point, while a lsc function takes the lower value in a discontinuity point. We will consider functions which are usc or lsc in every point x of E. If f is usc, −f is lsc. An usc function is schematized in Fig. 2.2 a) and a lsc in Fig. 2.2 b). For an usc function ϕ defined on a compact K, there is at least one point x of the compact K for which ϕ(x) = ∨y∈K {ϕ(y)} [114]. Similarly, for a lsc ϕ, ∃x ∈ K with ϕ(x) = ∧y∈K {ϕ(y)}: the upper bound of an usc function, and the lower bound of a lsc function are reached in at least one point of the compact K. For such events, it is therefore suitable to use usc or lsc functions (or even continuous, when this is not too restrictive).
2.3 Introduction to Random Functions
29
FIGURE 2.3. Upper (above the smooth curve) and Lower (under the smooth curve) regularized functions
The sets of functions Φf and Φg are stable for ∨ and ∧ operations in a finite number. With the order relationship ≤ (ϕ1 ≤ ϕ2 if ϕ1 (x) ≤ ϕ2 (x),∀x ∈ E ), the sets Φf and Φg have a lattice algebraic structure (like F(E) and G(E) for the inclusion ⊂). For a systematic study of compact lattices, the reader should report to [463]. Φf is stable under ∧ (and Φg under ∨) for any family of functions. Let ϕ be any function in Φ(E). It is possible to approximate ϕ from above and from below by two functions ϕ and ϕ (with ϕ≤ ϕ), where ϕ is the upper regularized function of ϕ and ϕ is the lower regularized function of ϕ; ϕ is the lower usc function majoring ϕ and ϕ is the upper lsc function minoring ϕ (Fig. 2.3) ϕ(x) = ϕ(x) =
inf
{sup ϕ(y)}
sup
{ inf ϕ(y)}
x∈G, G∈G y∈G
x∈G, G∈G y∈G
˚ and with the same adFor random sets, sets A with the same interior A herence A can be considered as identical. This is the same situation for functions ϕ with the same regularized versions (ϕ, ϕ).
2.3.2 Subgraph of a function There is a connection between Random Functions (RF) and Random Sets by subgraphs (Fig. 2.4) and overgraphs (Fig. 2.5). Definition 2.9. The subgraph Γ ϕ of the function ϕ is made of the pairs {x, z}, x ∈ E, z ∈ R, with z ≤ ϕ(x). Similarly, the overgraph Γϕ is made of the pairs {x, z}, x ∈ E, z ∈ R, with z ≥ ϕ(x) (Figs (2.4), (2.5)).
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2. Random Closed Sets and Semi-Continuous Random Functions
FIGURE 2.4. Subgraph Γ Z of function Z
FIGURE 2.5. Overgraph ΓZ of function Z
Proposition 2.1. The function ϕ is lsc ⇔ its overgraph Γϕ is a closed set in E × R; ϕ is usc ⇔ its subgraph Γ ϕ is a closed set in E × R [114]. Considering the closed subgraph of an usc function, or the closed overgraph of a lsc, links the theory of closed random sets to the theory of random functions. The following properties are satisfied: • Let any function ϕ ∈ Φ(E). Its upper regularized version ϕ admits for subgraphΓ ϕ and its lower regularized version ϕhas for overgraph Γϕ . • Let a family of functions ϕi ∈ Φf ; Γ ∧i∈I ϕi = ∩i∈I Γ ϕi = Γ ϕ . Since Γ ϕ is closed, ϕ ∈ Φf . Let Γ ∨i∈I ϕi = ∪i∈I Γ ϕi = Γ ϕ . In general, ϕ is not usc. An usc function is obtained from Γ ϕ . • By duality, a similar result is obtained for a family of lsc functions ϕ0i ∈ Φg : ∨i∈I ϕ0i = ϕ (lsc function with Γϕ = ∩i∈I Γϕi ). In general ∧i∈I ϕ0i = ϕ is not lsc. A lsc function is obtained from the overgraph ∪i∈I Γϕi . • Starting from a family of continuous functions ϕi , ∨i∈I ϕi is lsc, and ∧i∈I ϕi is usc.
2.3.3 Choquet topology on Φf If ϕ ∈ Φf , we have Γ ϕ ∈ F(E × R); therefore it is possible to work out a topology on Φf , starting from the Choquet topology built on F(E×R). This
2.3 Introduction to Random Functions
31
0
topology is generated by the open sets F K and FG0 , where K 0 ∈ K(E × R) and G0 ∈ G(E × R). For instance: K 0 = K × [a, +∞], K ∈ K(E), a ∈ R G0 = G×]b, +∞], G ∈ G(E), b ∈ R The open sets of the Choquet topology are generated by parts of Φf of the type: {ϕ∨ (G) > b} and {ϕ∨ (K) < a} With this topology, Φf is compact with a countable base; it is the same case for Φg with the topology induced by the open sets in G(E×R). it is therefore possible to build the corresponding Borel σ algebra and a probability.
2.3.4 The Borel σ algebra on Φf and Φg , and the Choquet capacity for RF The random usc and lsc functions are defined from the Borel σ algebra given by the Choquet topology of Φf and Φg . On Φf , this σ algebra is generated by the random variables {ϕ∨ (G); G ∈ G} and contains the events {ϕ∨ (K) < a; K ∈ K}. On Φg , the σ algebra is generated by the random variables {ϕ∧ (G); G ∈ G} and contains the events {ϕ∧ (K) < a; K ∈ K}. The random function Z(x) ∈ Φf is characterized by its Choquet capacity T (K 0 ) with 1 − T (K 0 ) = Q(K 0 ) = P {K 0 ⊂ (Γ Z )c } = P {∨x∈Ki Z(x) < ai }
(2.34)
obtained for the compact sets K 0 = ∪i=1,...,k Ki ×[ai , +∞] (Ki ∈ K, ai ∈ R, k ∈ N ). In fact it is enough to consider for the compacts Ki balls with rational radii located at points xi with rational coordinates. It is equivalent to characterize the usc RF Z(x) from lower semi continuous test functions g with compact support K. Consider the following event [274], [275]: A = {Z, Z(y) < g(y), ∀y ∈ K}
(2.35)
The Choquet capacity T (g) is given by: T (g) = 1 − Q(g) = 1 − P {A}
(2.36)
32
2. Random Closed Sets and Semi-Continuous Random Functions
Eq. (2.36) is equivalent to Eq. (2.34) when g(y) = ai for y ∈ Ki . For instance, if g(x) = z for x ∈ K, T (g) = 1 − Q(g) = 1 − P {Z∨ (K) < z}
(2.37)
and the Choquet capacity gives in this case the distribution function of the RF Z(x) after a change of support by ∨ on the compact set K. Another example derived from Eq. (2.36) is obtained with the lsc function g(xi ) = zi for points xi (i = 1, 2, ..., n) and else g(x) = +∞. In these conditions, T (g) = 1 − P {Z(x1 ) < z1 , ..., Z(xn ) < zn } and from 1 − T (g) is recovered the spatial law (2.28), which is therefore linked to the Choquet capacity (but which can be used for any type of RF). In the same way, a lsc RF can be characterized from upper semi continuous test functions g with compact support K. Consider the following event: B = {Z, Z(y) ≥ g(y), ∀y ∈ K} (2.38) and the probability P (g) = P {B} For instance, if g(x) = z for x ∈ K (and g(x) = −∞ for x ∈ / K), P (g) = P {Z(y) ≥ g(y), ∀y ∈ K} = P {Z∧ (K) ≥ z} For usc random functions with support in Rn , it is possible to translate the test function g to each point of Rn , and to call DZ (g)c the set of point x, where Eq. (2.35) is satisfied, as illustrated in Fig. 2.6. DZ (g) is a RACS. In that case: Theorem 2.4. A function Z(x) defined in Rn , upper semi continuous (usc), is characterized by its CHOQUET capacity T (g) defined over lower semi continuous functions (lsc) g with a compact support K T (g) = P {x ∈ DZ (g)} = 1 − Q(g) with DZ (g)c = {x, Z(x + y) < g(y), ∀y ∈ K}
(2.39)
For g(x) = z, else g(x) = +∞: 1 − T (g) = Q(g) = P {Z(x) < z} With AZ (z) = {x, Z(x) ≥ z} and DZ (g) = AZ (z). For further calculation, it is useful to give a general expression of DZ (g) from the closed excursion sets AZ (z) [274], [275]. Define
2.3 Introduction to Random Functions
33
FIGURE 2.6. Definition of the event A given by Eq. (2.36)
the closed set Bg (z)obtained from: Bg (z) = {x, g(x) ≤ z} The points x satisfying Eq. (2.35) verify: ∀z ∈ R, Bg (z) must be disjoint from AZ (z). Therefore ˇg (z))c } DZ (g)c = {x, ∀z ∈ R, x ∈ (AZ (z) ⊕ B c ˇg (z)) = ∩z∈R (AZ (z) ⊕ B
(2.40)
ˇg (z)) DZ (g) = ∪z∈R (AZ (z) ⊕ B
(2.41)
so that Let now Z1 (x) and Z2 (x) be two usc random functions. Consider Z(x) = Z1 (x) ∨ Z2 (x). Since (DZ1 ∨Z2 (g))c = DZ (g)c = {x, Z1 (x + y) ∨ Z2 (x + y) < g(y), ∀y ∈ K} = {x, Z1 (x + y) < g(y) and Z2 (x + y) < g(y) , ∀y ∈ K} = DZ1 (g)c ∩ DZ2 (g)c Therefore DZ1 ∨Z2 (g) = DZ1 (g) ∪ DZ2 (g)
(2.42)
and the Choquet capacity of Z1 (x) ∨ Z2 (x) is given by: TZ1 ∨Z2 (g) = P {x ∈ DZ (g)} = 1 − Q(g) = P {x ∈ DZ1 (g) ∪ DZ2 (g)} and Q(g) = P {x ∈ DZ1 (g)c ∩ DZ2 (g)c }
(2.43) (2.44)
34
2. Random Closed Sets and Semi-Continuous Random Functions
If the two RF Z1 (x) and Z2 (x) are independent, the two random sets DZ1 (g) and DZ2 (g) are independent, and from Eq. (2.18), Eq. (2.44) becomes: Q(g) = P {x ∈ DZ (g)c } = P {x ∈ DZ1 (g)c ∩ DZ2 (g)c } = P {x ∈ DZ1 (g)c }P {x ∈ DZ2 (g)c } = Q1 (g)Q2 (g) More generally, if Z(x) = ∨i=m i=1 Yi (x) and if the Yi (x) are independent realizations of the same RF Y (x) with P {x ∈ DY (g)c } = QY (g), c m QZ (g) = P {x ∈ DZ (g)c } = P {x ∈ ∩i=m i=1 DY i (g) } = QY (g)
(2.45)
Concerning the excursion set at level z, AcZ (z) = {x, Z(x) < z} = {x, Z1 (x) ∨ Z2 (x) < z} = {x, Z1 (x) < z and Z2 (x) < z} = AcZ1 (z) ∩ AcZ2 (z) so that AZ1 ∨Z2 (z) = AZ1 (z) ∪ AZ2 (z) For the function g(x) = z for x ∈ K, and g(x) = +∞: if x ∈ / K, £ ¤c DZ (g)c = {x, Z(x + y) < z, ∀y ∈ K} = AZ∨ (K) (z) AZ∨ (K) (z) = DZ∨ (K) (g)
What is the connection between AZ (z) and AZ∨ (K) (z)? We have: ˇ Z∨ (Kx ) < z ⇔ Kx ⊂ (AZ (z))c ⇔ x ∈ (AZ (z))c ª K Therefore ¡ ¢c c ˇ and AZ (K) (z) = DZ (K) (g) = AZ (z) ⊕ K ˇ AZ∨ (K) (z) = (AZ (z)) ª K ∨ ∨ (2.46) and T (g) = P {x ∈ DZ (g)} = 1 − Q(g) = 1 − P {Z∨ (K) < z}
(2.47)
The Choquet capacity gives the probability distribution of the RF Z(x) after a change of support by ∨ over the compact set K. For lsc random functions in Rn , the functional P (g) is defined on the usc functions g with compact support K and we have: Theorem 2.5. A RF Z(x) defined in Rn , lower semi continuous (lsc) is characterized by the functional P (g) defined over the upper semi continuous functions (usc) g with a compact support K
2.3 Introduction to Random Functions
35
FIGURE 2.7. Definition of the event B given by Eq. (2.38)
P (g) = P {x ∈ HZ (g)}; with HZ (g) = {x, Z(y) ≥ g(y − x), ∀y ∈ K} (2.48) HZ (g) is the set of points where Eq. (2.38) is satisfied, illustrated by Fig. 2.7. For example if g(x) = z for x ∈ K and g(x) = −∞ if x ∈ /K P (g) = P {Z(x + y) ≥ g(y), ∀y ∈ K} = P {Z∧ (K) ≥ z} and is obtained the probability distribution of the RF Z(x) after a change of support by ∧ on the compact set K. When g(x) = z and g(y) = −∞ for y 6= x HZ (g) = {x, Z(x) ≥ z} = AZ (z) As for the set DZ (g)c , it may be useful to express HZ (g) from the closed excursion sets AZ (z). Exchanging the roles of Z(x) and g(x) in Eq. (2.40), HZ (g) = ∩z∈R (AˇZ (z)c ⊕ Ag (z))c = ∩z∈R (AˇZ (z) ª Ag (z))
(2.49)
Let Z1 (x) and Z2 (x) be two lsc random functions. Consider Z(x) = Z1 (x)∧ Z2 (x): HZ (g) = {x, Z1 (x + y) ∧ Z2 (x + y) ≥ g(y), ∀y ∈ K} = {x, Z1 (x + y) ≥ g(y) and Z2 (x + y) ≥ g(y), ∀y ∈ K} = HZ1 (g) ∩ HZ2 (g) When g(x) = z for x ∈ K, HZ (g) = AZ∧ (K) (z) The connection between AZ (z) and AZ∧(K) (z) is as follows: ˇ Z∧ (Kx ) ≥ z ⇔ Kx ⊂ AZ (z) ⇔ x ∈ AZ (z) ª K
(2.50)
36
2. Random Closed Sets and Semi-Continuous Random Functions
and then ˇ HZ (g) = AZ∧ (K) (z) = AZ (z) ª K
(2.51)
so that ˇ (2.52) P (g) = P {x ∈ HZ (g)} = P {x ∈ AZ∧ (K) (z)} = P {x ∈ AZ (z) ª K} and the functional P (g) gives the probability distribution of the RF Z(x) after a change of support by ∧ over the compact set K. To conclude this section, recall that two models (RACS, usc RF, lsc RF) with the same functional T (K), T (g), or P (g) cannot be distinguished, theoretically as well as experimentally . In the next chapters, the functionals are obtained by theoretical calculation as a function of the assumptions, of the parameters of the model and of the compact K or the function g. This approach is easily extended to the multivariate case.
2.3.5 Multivariate Random Functions For a multivariate random function Z = {Z1 , Z2 , ..., Zm }, denote: g = {g1 , g2 , ..., gm }, each gi being a lsc function with compact support Ki ; K = m {K1 , K2 , ..., Km }; z = {z1 , z2 , ..., zm } (z ∈ R ); x = {x1 , x2 , ..., xm } (x ∈ n×m R ); yi = (xi , zi ). From the functions gi is defined DZ (g) = ∪i=m i=1 DZi (gi )
(2.53)
The multivariate Choquet capacity of a multivariate usc RF is given by 1 − T (g) = Q(g) = P {x ∈ / DZ (g)}
(2.54)
A particular case is obtained with gi constant (zi ) on the compact Ki translated at point xi : 1 − T (g) = Q(g) = P {Z1∨ (K1x1 ) < z1 , ..., Zm∨ (Kmxm ) < zm }
(2.55)
For a multivariate lsc RF, define: P (g) = P {x ∈ HZ (g)}; with HZ (g) = ∩i=m i=1 HZi (gi )
(2.56)
2.3.6 Erosion and Dilation of functions In Mathematical Morphology, it is common to use the dilation and erosion operations for functions. This is illustrated on a SEM image of a powder given in Fig. 2.8. They are defined as follows, for functions with a compact support K in Rn , with the usual notations [598]:
2.3 Introduction to Random Functions
37
FIGURE 2.8. Scanning Electron Microscope image (512×512) of a UO 2 powder (5.12μm wide) [336]
FIGURE 2.9. Erosion of image of Fig. 2.8 by an hexagon of size 10
• erosion of f by a function g (with gˇ(x) = g(−x)) (Fig. 2.9): f ª gˇ(x) = ∧y∈Rn {f (y) − g(y − x)}
(2.57)
with the convention +∞ − ∞ = +∞. The erosion has the following algebraic properties: — When the origin O belongs to K and for g(x) ≥ 0, the erosion is anti extensive: f ª gˇ(x) ≤ f (x); — The erosion is increasing for f : if f1 (x) ≥ f2 (x), ∀x ∈ Rn , f1 ª gˇ(x) ≤ f2 ª gˇ(x); — The erosion is decreasing for g; — It follows: f ª (ˇ g ∨ gˇ0 ) = (f ª gˇ) ∧ (f ª gˇ0 ) • dilation of f by a function g (Fig. 2.10): f ⊕ gˇ(x) = ∨y∈Rn {f (y) + g(y − x)}
(2.58)
38
2. Random Closed Sets and Semi-Continuous Random Functions
FIGURE 2.10. Dilation of image of Fig. 2.8 by an hexagon of size 10
with the convention +∞ − ∞ = +∞.The dilation has the following algebraic properties: — When the origin O belongs to K and for g(x) ≥ 0, the dilation is extensive: f (x) ≤ f ⊕ gˇ(x); — The erosion is increasing for f and g; — It comes: f ⊕ (ˇ g ∨ gˇ0 ) = (f ⊕ gˇ) ∨ (f ⊕ gˇ0 ) (f ⊕ gˇ) ⊕ gˇ0 = f ⊕ (ˇ g ⊕ gˇ0 ) 0 (f ª gˇ) ª gˇ = f ª (ˇ g ⊕ gˇ0 ) The operations defined by Eqs (2.57, 2.58) give after thresholding in z = 0 the sets defined by Eqs (2.38, 2.35). The erosion and dilation are dual operations active on functions: Definition 2.10. Let a transformation Ψ acting on functions. The transformation Ψ ∗ is said to be dual of Ψ if for any function f , Ψ ∗ (f ) = −Ψ (−f ) With this definition, it is easy to check that the dual transformation of ∧ is ∧∗ = ∨: if Ψ (f ) = ∨y∈K {f (y)} then Ψ ∗ (f ) = − ∨y∈K {−f (y)} = ∧y∈K {f (y)} Similarly, the dilation is the dual operation of the erosion. Two other dual operations acting on functions are defined: • the opening of f by g (Fig. 2.11): Ψg (f ) = (f ª gˇ) ⊕ g • the closing of f by g (Fig. 2.12):
2.3 Introduction to Random Functions
39
FIGURE 2.11. Opening of image of Fig. 2.8 by an hexagon of size 10
FIGURE 2.12. Closing of image of Fig. 2.8 by an hexagon of size 10
Ψ g (f ) = (f ⊕ gˇ) ª g with (Ψg (f ))∗ = −(−f ª gˇ) ⊕ g = −(−(f ⊕ gˇ) ⊕ g) = (f ⊕ gˇ) ª g) = Ψ g (f ) The opening and closing transformations are increasing for f and g and are idempotent (Ψ (Ψ (f )) = Ψ (f )). They are basic morphological filters used in image processing and can also be used for sizing, as a generalization of the set transformations presented in chapter 3.
3 Quantitative Analysis of Random Structures
Abstract: The measurements used to analyze the morphology of random media, experimentally available from image analysis, are introduced: measurements are obtained in a two-steps approach: transformation and basic measurement; their prototypes are the basic operations of mathematical morphology (erosion and dilation) by appropriate structuring elements, coupled with the Minkowski functionals. They are used to define various criteria for characterizing the sizes, the spatial distribution and the connectivity of objects like sets, functions or graphs. Tools introduced in this chapter can be used for pattern and texture recognition by means of machine learning algorithms, as illustrated by some examples.
3.1 Introduction Among others, the quantitative analysis of structures is helpful to summarize with numbers, which can be obtained by appropriate experiments, the content of more or less complex microstructures, or to relate the microstructure to the physical properties of materials. These measurements give a precise meaning to vague and common concepts, such as the size, shape, or distribution. In some cases, these data are useful indications on the genesis of the examined structures, possibly through the use of stochastic models such as those which are proposed in this book. This approach requires the definition of basic measurements and transformations. Since we are interested in random structure, we will use here information that © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Jeulin, Morphological Models of Random Structures, Interdisciplinary Applied Mathematics 53, https://doi.org/10.1007/978-3-030-75452-5_3
41
42
3. Quantitative Analysis of Random Structures
is compatible with the notion of random sets and random functions introduced in chapter 2. In the present chapter, we intend to present the basis of a quantitative analysis of microstructures that are useful for practical applications. These basis give a morphological interpretation of the abstract concepts introduced in chapter 2 around the Choquet capacity. To make the presentation as concise as possible, the following points are presented: after a recall of the basic transformations used in Mathematical Morphology, the basic measurements that can be obtained on sets are introduced; then, the concepts of size and distribution in space are studied by appropriate operations and measurements. Finally measurements on the connectivity are introduced.
3.2 Basic morphological transformations Most useful quantitative information is obtained in two steps [438],[598]: • A transformation Φ is performed on the studied structure. • A measurement is made on the transformed object. The choice of the transformation is primordial for the morphological content of the quantitative information that is looked for. A very common example is given by the determination of a size distribution of grains taken from a powder. This can be performed from a sieving of the particles (transformation), followed by a weighing (measurement). If now it is required to determine the proportion of different types of particles (according to their shapes), it is necessary to classify every object (transformation) before counting them (measurement). As explained in [598], some experimental constraints restrict the type of allowed morphological transformations and measurements: • the result should not depend on the arbitrary choice of the origin of coordinates (invariance by translations); • since the studied objects are known in an approximate way (due to the finite resolution of instruments, and also to the digitization on a grid of points), the operations and measurements should own some continuity property to be robust with the approximation and to converge to the desired measurement when refining the resolution; • most often specimens have to be sampled from a population of objects and a statistical analysis must be performed, involving, among others, the estimation of averages. This can be performed with a physical meaning for additive measurements only; • finally in quantitative microscopy the objects are known from sections in lower dimensional spaces (such as two dimensional sections made in three dimensional objects). This brings some stereological constraints on measurements if a 3D meaning is looked for.
3.2 Basic morphological transformations
43
The basic operations and measurements that are recalled below satisfy these constraints, for a correct experimental implementation.
3.2.1 Types of structures In this chapter the following types of structures are considered: • point processes; • binary media (modelled by random closed sets); • multi component media like several populations of inclusions in a matrix, mineralogical phases in an ore (modelled by random closed sets);we denote by Ai (i = 1, 2, ..., m) each of the m disjoint components (Ai ∩ Aj = ∅ for i 6= j); • functions, like grey level images obtained with a TV camera, denoted by Zi (x) (i = 1, 2, ..., m) for possible series of signals; • graphs.
3.2.2 Notion of structuring element A natural approach of the description of the morphology of objects is made through the use of comparisons with model shapes: e. g. a pore is spherical or elongated; a grain is polyhedral. This process is transposed in Mathematical Morphology by means of the notion of structuring element introduced in [438],[598]: consider the binary case and a set A; an object K (a compact set called a structuring element) with a well defined shape and size (point, sphere, segment,...) is translated on every point x of the Euclidean space Rn . For every location x, the answer to a question concerning the mutual location of K and A is recorded. In the case of a binary answer, positive answers build the indicator function of a set Φ(A) transforming the initial set A. The choice of the type of question and of the type of structuring element K leads to various morphological criteria. Since we operate on sets, we can work as well on isolated objects or on interconnected media such as on a porous medium, a mineralogical phase, or even the union of mineralogical phases.... The basic morphological operations are illustrated below on the binary image of Fe (Fig. 3.2) obtained by thresholding the grey level image of a Fe-Ag alloy (Fig. 3.1) [91].
3.2.3 Dilations and Erosions Consider the two following questions: i) Kx (K located at x) encounters A? ii) Kx is included in A? These questions enable us to define the dilation (Fig. 3.3) and erosion (Fig. 3.4) of A by the compact set K. We recall now the two equations (2.16,2.17) given in chapter 2.
44
3. Quantitative Analysis of Random Structures
FIGURE 3.1. Image of a two phases alloy made of Fe (black) and Ag (grey): width of the field 250μm [91]
FIGURE 3.2. Binary image of Fe (in white) obtained by thresholding of (Fig. 3.1); Ag appears in black
ˇ = {x, Kx ∩ A 6= ∅} = ∪x,y∈K {(x + y) ∈ A} = ∪y∈K A−y (3.1) A⊕K = ∪x∈A,y∈K {x − y} ˇ = {x, Kx ⊂ A} = ∩y∈K A−y = (Ac ⊕ K) ˇ c AªK
(3.2)
In Eqs (3.1,3.2), Kx = {x + y, y ∈ K} is the translated of K at point x; ˇ is obtained by transposition of K: K ˇ = {−x, x ∈ K}; the symbols ⊕ K and ª represent the addition and substraction of Minkowski. From Eqs (3.1,3.2), it is clear that the results of dilation and erosion operations are not independent: dilating the set A by K is equivalent to eroding Ac by K and taking the complement: the two operations dilations and erosions are said to be dual for the complementation. Among the properties of these operations, we can mention: ˇ 1) ª K ˇ 2 = A ª (K ˇ1 ⊕ K ˇ 2) (A ª K ˇ ˇ ˇ (A ∩ B) ª K = (A ª K) ∩ (B ª K) ˇ ˇ ˇ ˇ 2) A ⊕ (K1 ∪ K2 ) = (A ⊕ K1 ) ∪ (A ⊕ K
(3.3)
3.3 Basic morphological measurements
45
FIGURE 3.3. Dilation of Fig. 3.2 by an hexagon of size 2 pixels
FIGURE 3.4. Erosion of Fig. 3.2 by an hexagon of size 2 pixels
ˇ is convex. If A and K are convex sets, A ⊕ K ˇ If A is a convex set, A ª K is convex.
3.3 Basic morphological measurements 3.3.1 Definition of Minkowski functionals According to [598], the experimental constraints that restrict the choice of any potential morphological parameter μ are the following: • i) Invariance by translation: the result of a measurement should not depend on the location of the object A. For the analysis of shapes, it is usual to add a constraint of invariance by rotations. • ii) Continuity: we must have for sequences An → A, μ(An ) → μ(A). • iii) Additivity: we assume that we can add measurements made on disjoint objects. This warrants a physical meaning to averaging of measurements taken over samples. • iv) Local knowledge of objects is sufficient for an estimation of basic measurements inside a mask M : this is required for quantitative mi-
46
3. Quantitative Analysis of Random Structures
croscopy, where in general only parts of objects are known from fields of view in a microscope on sections. This point is illustrated at the end of subsection 3.3.5.
It can be shown in Integral Geometry [496],[226] that in the n dimensional space n + 1 measures satisfy the above mentioned constraints. These are the Minkowski functionals defined on the convex ring (finite union of convex sets). They have the following properties:
• μ(A) is bounded if the set A is bounded. • μ(λA) = λn−i μ(A) : the functionals are homogeneous and of degree n−i (0 ≤ i ≤ n): for two similar objects (with the same shape), the results of measurements differ from a scale factor. The functional of degree n− i is denoted Wi . The functional of degree n can be identified with the measure of Lebesgue μn . The functional of degree 1, Wn−1 , is related to the norm A of convex sets A by the expression A(A) = nWn−1 (A) ([448] p. 79). For a convex set A, the breadth in a given direction is given by the distance between two supporting hyperplanes En−1 with dimension n − 1 othogonal to this direction. The norm is proportional to average of the breadth over rotations. The norm satisfies A(A ⊕ B) = A(A) + A(B)
(3.4)
• μ(A ∪ B)+ μ(A ∩ B) = μ(A) + μ(B) (additivity). In the spaces R, R2 , and R3 , which are useful for applications, the Minkowski functionals Wi are the following:
• In the one dimensional space R W0 (A) = l(A) (length of the set A) W1 (A) = 2
(3.5)
• In the two dimensional space R2 W0 (A2 ) = A(A2 ) (area of the set A2 ) 2W1 (A2 ) = L(A2 ) (perimeter of the set A2 ) 2W2 (A2 ) = 2πN (A2 ) (connectivity number in R2 )
(3.6)
N (A) is the difference between the number of connected components of A and the number of holes that it contains. • In the three dimensional space R3
3.3 Basic morphological measurements
47
W0 (A) = V (A) (volume of the set A) 3W1 (A) = S(A) (surface area of the set A) 1 3W2 (A) = M (A) = 2
Z
∂A
µ
1 1 + R1 R2
¶
dS (integral of mean curvature)
¶ 1 dS R1 R2 ∂A (integral of total curvature or connectivity number in R3 ) 3W3 (A) = 4π(N − G) = N (A) =
Z
µ
(3.7) R1 and R2 are the principal radii of curvature at any point of the boundary ∂A of the set A. We denote kmin = R11 and kmax = R12 the two principal curvatures. (N − G) is the difference between the number of connected components N of A and its genus G. The genus of A is the maximal number of closed curves that can be drawn on its boundary ∂A without disconnecting it into separate parts: the genus of a sphere is equal to 0 and the genus of a torus is equal to 1. The connectivity numbers in R2 and in R3 are topological properties describing the connectivity of an object (it is used it in chapter 6 to explain the rheological behavior of Fe-Ag composites). It can be shown that in R2 , N (A) is obtained as the difference between two convexity numbers C(A) and C(Ac ) : ¶ µZ Z 1 N (A) = C(A) − C(Ac ) = dα − dα (3.8) 2π R>0 R0 R1 R2 0), for a convex compact set K, denoted by Kλ : ˇ λ ) ⊕ Kλ Φλ (A) = (A ª K This can be performed on sets of any shape, such as isolated particles or an interconnected medium, as illustrated by Fig. 3.5. It can be shown that Φλ (A) = AKλ = {x ∈ A; ∃y ∈ A with Kλ,y ⊂ A}
(3.25)
AKλ is the set of points of A covered by the Kλ when it is translated in space and constrained to remain included inside A. To come back to the analogy with the sieve analysis, it can be said that A behaves like a sieve with respect to Kλ , and that the set AKλ records all possible crossings of Kλ through the sieve A. The closing operation is the dual operation of the opening: ˇ λ ) ª Kλ = (Φλ (Ac ))c AKλ = (A ⊕ K
(3.26)
From performing closings of A by convex compact sets K, it is possible to perform a granulometry of the set Ac , as shown in Fig. 3.6. The size
3.4 Size distributions
59
FIGURE 3.6. Closing of Fig. 3.2 by an hexagon of size 6
distribution according to a given granulometry is obtained from measurements made on the results of the sizing operation: counting the number of objects, or measuring their volume in the n dimensional space.
3.4.2 Linear size distributions The morphological analysis by means of segments is relatively easy, since size distributions can be estimated from erosions [438],[225],[448],[598]. The extreme anisotropy of this structuring element is useful for studying the orientations of a structure from linear measurements in various directions. For the simplification of notation, we omit the direction of the segment. Consider a stationary RACS A and let Q(l) = P {l ⊂ Ac }. We give below the size distribution of linear intercepts of Ac ; the same results are obtained for A, after replacing Q(l) by P (l). The functional Q(l) is convex, and therefore admits a derivative Q0 (l) almost everywhere [448]. This is a consequence of the fact that P {F}ll1 , l2 > 0. In the case of three adjacent segments {l1 }, {l}, {l2 } , P {F}ll1 , l2 = Q(l) − Q(l + l1 ) − Q(l + l2 ) + Q(l + l1 + l2 ) ≥ 0 It can be shown that P {x ∈ Acl } = Q(l) − lQ0 (l)
(3.27)
This is easily understood when Q0 (l) remains finite, Q0 (l) being the specific number of intercepts in Ac longer than l: Eq. (3.27) can be derived from Q(l), using the formula of Steiner applied to every chord longer than l. In fact, this is also true if in the neighborhood of l = 0, Q(0) − Q(l) ' lβ with 0 < β < 1; this corresponds to a set A with a boundary having a non integer Hausdorff dimension d = β − 1 (or with a fractal boundary [434]), for which −Q0 (0) = +∞ (there is an infinity of chords with an infinitesimal length). From Eq. (3.27) can be obtained the measure distribution of intercepts G(l) (and its density g(l) when it exists), every chord having
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3. Quantitative Analysis of Random Structures
a weight proportional to its length since we use the probability for x to belong to chords longer than l : Q(l) − lQ0 (l) Q(0)
1 − G(l) =
(3.28) lQ00 (l) g(l) = Q(0) The average intercept of this distribution, L∗ is given by Z +∞ 2 ∗ L = Q(l) dl Q(0) 0
(3.29)
Other properties derived from the function Q(l): in any point x, the star is defined as the domain scanned by a beam of segments containing x and contained in Ac . The average value of the measure of this domain (length in R, area in R2 and volume in R3 ) for every location of x is the star in one (L∗ ), two (S ∗ ), or three dimensions (V ∗ ). For a non isotropic random set, is considered for Q(l) the average value over the orientations of the segment l. L∗ is given in Eq. (3.29). We have: 2π S = Q(0) ∗
V∗ =
4π Q(0)
Z
+∞
lQ(l) dl
0
Z
(3.30) +∞
l2 Q(l) dl
0
The stars can be defined as average sizes for media of any shape (particles, interconnected porous medium, matrix in a composite,...). For convex bodies, the stars give ”measure” averages of the length, area, or volume, according to the dimension of the space of interest. They have a stereological content, since the estimation is obtained from linear (1D) information. When Q0 (0) is finite, a number granulometry of intercepts is defined, every chord having the same weight (this is obtained by counting). The cumulative distribution function of intercepts F (l) and its distribution f (l) when it exists are given by: 1 − F (l) =
number of intercepts with length ≥ l Q0 (l) = 0 total number of intercepts Q (0) (3.31) −Q00 (l) f (l) = Q0 (0)
The corresponding ”number” average intercept L is given by
3.4 Size distributions
L=
−Q(0) Q0 (0)
61
(3.32)
The two types of linear granulometries are not independent. In fact, g(l) is proportional to f (l) and to l, as seen from Eqs (3.28,3.31). It is instructive to compare the ”measure” and the ”number” average intercept for the following example: f (l) = a exp(−al) (exponential distribution). In that case, 1 and g(l) = a2 l exp(−al) (gamma distribution). For this example, L = a 2 L∗ = , so that there is a factor two between the two averages. In applia cations, it is therefore wise to consider carefully which type of distribution is appropriate for the considered process, since completely different results can be derived...
3.4.3 Two and three dimensional size distributions Practitioners of image analysis use to perform two dimensional size distributions, most often by means of image transformations with openings by squares or hexagons. The calculation of the Euclidean distance on binary images [648] gives access to sizing from opening by discs. These developments were easily extended to the three dimensional space [476],[121],[215], when 3D images are available like in microtomography. In the two dimensional space, a ”measure” size distribution is estimated from openings by convex structuring elements Kλ : G(λ) =
P {x ∈ A} − P {x ∈ AKλ } P {x ∈ A}
(3.33)
An estimation of G(λ) is obtained from area fraction (AA ) or volume fraction (VV ) measurements after openings. To eliminate edge effects, this ˇ 2λ , measurement must be performed inside a mask contained into X ª K X being the field of measurement [598]. From Eq. (3.33) are calculated the moments of the area S of the largest K containing x and included in A (namely of the volume V in R3 ). For a disc with radius r E{S} = 2π
Z
+∞
0
E{S n } = 2nπ For a sphere of radius r
Z
0
(1 − G(r))r dr (3.34)
+∞
(1 − G(r))r2n−1 dr
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3. Quantitative Analysis of Random Structures
E{V } = 4π n
Z
+∞
(1 − G(r))r2 dr
0
E{V } = 4nπ
Z
0
(3.35)
+∞
(1 − G(r))r3n−1 dr
Similarly a ”measure” size distribution from closings by convex structuring elements Kλ is estimated from: F (λ) =
P {x ∈ AKλ } − P {x ∈ A} 1 − P {x ∈ A}
(3.36)
Another convenient way to define a size distribution for isolated objects is to perform a measurement satisfying the increasing condition with respect to the inclusion on each object (area in R2 , diameter,...), and to use the distribution function of this parameter. It can be shown that a classification of objects by this process defines an opening operation, which is therefore consistent with the axioms of a granulometry. A correct estimation is obtained after elimination of objects hitting the edges of the field of measurements, and application of the Miles-Lantuejoul correction [404] to the bias (larger objects have a higher probability to hit the edges). This type of sizing, based on the connectivity of objects, cannot be used for interconnected media. We illustrate 3D size distributions obtained from microtomography in the case of a wood fibrous network used for acoustic attenuation [548] (Fig. 3.7). In this network it is impossible to get a segmentation of individual fibres (Fig. 3.8), due to the interlocking of fiber and to the limited resolution of images (9.36μm per voxel). Using openings by rhombicuboctaedra on the fibres provides a size distribution of the radii of fibres weighted by their volume (Fig. 3.9), whatever their length and orientation. A unimodal distribution with a 40μm mode for the radius. The 3D size distribution of the porous network between fibres shows a more uniform size distribution between 10μm and 100μm (Fig. 3.10).
3.4.4 Size distributions for functions It is difficult, or even impossible, to define a size distribution of connected parts on a function, as it was possible for sets. On the other hand, the opening and closing size distributions are easily extended to functions Z(x) with support in Rn [598], using families of structuring elements. These operations can be viewed as standard granulometries applied to the subgraph or to the overgraph of the function. When using a "flat" structuring element ˇ λ ) ⊕ Kλ is obtained like a convex set Kλ ⊂ Rn each section of (Z ª K by opening the section AZ (z) by Kλ (ibidem for the closing operation), so
3.4 Size distributions
63
FIGURE 3.7. Microtomography of a fibrous network
FIGURE 3.8. Binary image of fibres
FIGURE 3.9. Size distribution of the fibrous network by morphological openings with rhombicuboctaedra
64
3. Quantitative Analysis of Random Structures
FIGURE 3.10. Size distribution of the porous network by morphological openings with rhombicuboctaedra
that a granulometry depending on the threshold z can be defined: G(λ, z) =
P {x ∈ AZ (z)} − P {x ∈ (AZ (z))Kλ } P {x ∈ AZ (z)}
A natural extension of the distribution functions (3.33, 3.36) is to replace the area fraction or the volume fraction by the integral of Z(x) over the R mask X, or equivalently by the average Z = μ 1(X) X Z(x)dx. Then are n obtained the distribution functions G(λ) = F (λ) =
Z − Z Kλ Z − Z K∞ Z Z
Kλ
K∞
−Z
−Z
The distribution functions G(λ) and F (λ) are good descriptors of textures [122, 123]. It was used to describe the topography of rough surfaces [327], using the granulometry of peaks by opening and of valleys by closing .
3.4.5 Stereological reconstruction of the size distribution of spheres The problem of the estimation of the diameter distribution of spheres in R3 from information on sections, known as "unfolding problem", is a basic stereological problem that was solved by many authors [664],[377],[225],[438]. We remind here the relationships linking the characteristics in R3 (average (3) number of spheres NV and distribution of diameters in number F3 (D))
3.5 Morphological analysis of the spatial distribution
65
to the induced characteristics of the same type in R2 (average number of (2) discs NV and distribution of diameters in number F2 (D)) and in R (aver(1) age number of chords NV and distribution of lengths in number F1 (D)) [598]. A first type of relation connects measurement in R 3 and in R : (3)
NV =
2 (1) 00 N F (0) π V 1
1 F10 (D) 1 − F3 (D) = D F100 (0)
(3.37)
A second relation is used for going from dimension i to i + 1 (i = 1, 2) (i+1)
NV
=
(i) Z +∞
NV π
0
Fi0 (h) dh h
(i) Z +∞
N (i+1) (1 − Fi+1 (D)) = V NV π
D
(3.38) F 0 (h) √ i dh h2 − D2
Eqs (3.37,3.38) involve Abel integral equations that underlie an ill-posed problem, that requires specific techniques for their resolution, such as reguralization [632]. Practical algorithms for their implementation on experimental data are provided in [641],[125], [659], to mention a few references. It may be tempting to use the solution of this unfolding problem to estimate 3D grain size distribution form 2D measurements on polished sections of polycrystals. To avoid to obtain inconsistent results, it should be better to avoid this approach which is strongly dependent on the spherical shape assumption.
3.5 Morphological analysis of the spatial distribution The spatial distribution of a microstructure is of major importance to characterize the heterogeneity. It has incidences on the behavior of materials (since for instance the heterogeneity of defects controls the dispersion of the fracture strength, as seen in chapter 20, as well as on the sampling process to apply when estimating structural properties. The aim of a morphological analysis of the spatial distribution is to provide quantitative criteria connected to the following notions: • Measurement of a scale, and of the homogeneity of a structure. • Measurement of preferential associations between the components of a microstructure.
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3. Quantitative Analysis of Random Structures
The presentation in this section will cover successively the following points: random sets, multicomponent random sets, random functions, and random graphs.
3.5.1 Random sets A random set A is obtained by means of thresholds applied to numerical data, or by a sequence of binary operations. The spatial distribution of A can be studied from the following information: the covariance, distance functions, closings by convex sets. The covariogram, the covariance and the correlation function Definition 3.2. Using the operation of erosion by the structuring element K = {x, x+h} with the orientation α enables us to define the geometrical covariogram K(h, α) of any bounded set A in Rn [437], [444] by Z ˇ K(h, α) = μn (A ª h) = μn (A ∩ A−h ) = 1A (x)1A (x + h) dx (3.39) Rn
Equation (3.39) can be rewritten as a convolution product ∗: K(h) = 1A ∗ ˇ1A
(3.40)
ˇ = A ª ˇl where the segment l links the points For a convex set A, A ª h x and x + h. The main properties of the geometrical covariogram are: • K(0, α) = μn (A) • K(h, α) = 0 for h ≥ bα where bα is the width of A in the direction α. Z K(h, α) dh = (μn (A))2 . • n R Z 1 4π S(A) − K 0 (0, α) dα = • In R3 , , provided that −K 0 (0, α) remains 4π 0 4 finite (this is a particular case of the results given in Table 3.1 and of the theorem of Crofton). When K(0) − K(h, α) ' hβ for h → 0, with 0 < β < 1, the boundary of the set A has a non integer Hausdorff dimension d = 3 − β. The anisotropy of the set A can be studied from the behavior of K(h, α) as a function of the angle α. For a random compact set, consider the average of K(h, α) over the realizations of A, and use the notation K(h, α) = E{μn (A ∩ A−h )} = μn (A ∩ A−h )
(3.41)
so that the expressions above are replaced by expectations. As particular cases K(h, α) provides the two first moments (and consequently the variance) of the Lebesgue measure of the bounded random set A :
3.5 Morphological analysis of the spatial distribution
Z
Rn
67
K(0, α) = E{μn (A)} K(h, α) dh = E{μn (A)2 }
(3.42)
Definition 3.3. The covariance C(h) of a random set A was introduced in chapter 2 as a particular case of the Choquet capacity, for the compact set K = {x, x + h}. It is equal to the probability that the two points x and x + h belong to the set A: C(x, x + h) = P {x ∈ A, x + h ∈ A}
(3.43)
For a stationary and ergodic random set A, the covariance does not depend on the location x. For a set in R3 , it can be expressed from the volume fraction VV by: ˇ C(h) = VV (A ∩ A−h ) = VV (A ª h)
(3.44)
The covariance can be estimated from images (for instance in planar sections) obtained inside a mask X by means of the geometrical covariograms of the set A ∩ X (KA∩X (h)) and of X (KX (h)): C ∗ (h) =
A((A ∩ X) ∩ (A ∩ X)−h ) KA∩X (h) = A(X ∩ X−h ) KX (h)
(3.45)
Using the convolution product given in Eq. (3.40), Eq. (3.45) becomes: C ∗ (h) =
1A ∗ ˇ1A KX (h)
(3.46)
and an alternative estimation is obtained by the Fourier transform F [385], since it factorizes the convolution product. We get, noting F −1 the inverse Fourier transform, and F (1A ) the complex conjugate of the Fourier transform: ¢¢ ¡ ¡ F −1 F 1A )F (1A ) ∗ C (h) = (3.47) KX (h) providing an estimation of the covariance for all orientations of he vector h. We will go back to this estimation in subsection 3.5.3. The result of the erosion by {x, x + h} depends on the vector h (by its modulus | h | and its orientation α) are characteristic of the size and distributions of connected objects building the set A. This is reflected by the variations of C(h). We can as well use the covariance Q(h) defined for the set Ac (with Q(0) = q = 1 − p). In fact, it contains no additional information, since Q(h) = P {x ∈ Ac , x + h ∈ Ac } = 1 − 2C(0) + C(h)
(3.48)
68
3. Quantitative Analysis of Random Structures
Eq. (3.48) shows that the covariance simultaneously characterizes the two sets (A, Ac ), while the two granulometries of A and Ac provide two separate pieces of information, which cannot be deduced from each other in general. Similarly, the second order statistics of a random set can be described by the order 1 variogram γ 1 (h) of the indicator function 1A (x), or by the order 2 variogram γ 2 (h). These are equal, and equivalent to the covariance for a stationary random set, since γ 1 (h) = 12 E{| 1A (x + h) − 1A (x) |} = 12 E{(1A (x + h) − 1A (x))2 } = γ 2 (h) = C(0) − C(h) = Q(0) − Q(h) = P {x ∈ A, x + h ∈ Ac } (3.49) The main properties of the covariance are the following, that we recall for random sets in R3 : • C(0) = P {x ∈ A} = p • The covariance of a stationary RACS must be continuous for h → 0 (it is quadratic mean continuous). A counter-example is given in chapter 6, inZsectionµ6.15.4. ¶ 1 4π ∂C(h, α) • − dα = SV (A) when the partial derivative reπ 0 ∂h h=0 mains finite. If C(0) − C(h) ' hβ for h → 0, with 0 < β < 1, the boundary of the random set A has a non integer Hausdorff dimension d = 3 − β. It is common to say in that case that A is a fractal set. • C(∞) = p2 (the covariance of a stationary and ergodic random set reaches a sill) • For the orientation α, C(h) reaches its sill at the distance aα , or range: C(aα ) = C(∞) = VV (A)2 = p2 . It does not depend on the orientation for an isotropic random set A. The range measures a characteristic length scale in the structure (for h ≥ a the events {x ∈ A} and {x + h ∈ A} are uncorrelated. Various scales can be disclosed from intermediary sills or inflections observed on the experimental covariance. This is the case for nested structures such as clusters, clusters of clusters, ... Periodicity in images appears as periodicity on the covariance. Pseudo periodicity and damped oscillations can be observed in the case of random packing of particles of the same size. For an anisotropic structure, the covariance C(h) must be studied as a function of the orientation α of the vector h. A convenient representation can be obtained through roses of directions [598],[125]. Examples of experimental covariances are given below. For the binary image of Fe of Fig. 3.2, the covariance (Fig. 3.11) shows a finite range (' 20 pixels) corresponding here to some average diameter of Fe particles. The image of Fig. 3.12 is generated by randomly located according to a sequential absorption model (chapter 5). Its experimental covariance given in (Fig. 3.13) presents oscillations around its asymptotic sill, resulting from the repulsion effect of the disc centers induced by the non overlapping con-
3.5 Morphological analysis of the spatial distribution
69
Composite Fe Ag Covariance of Fe 6500 ’Fe’ Sill 6000
5500
5000
4500
4000
3500 0
20
40
60
80
100
120
140
FIGURE 3.11. Covariance of Fe (Fig. 3.2) estimated by translations in the horizontal direction
FIGURE 3.12. Simulated image of non overlapping random discs
dition. The covariances of the fibrous medium shown in (Fig. 3.8), with transverse and axial section in Fig. 3.14 and in Fig. 3.15 present a transverse isotropy of the network (in Fig. 3.16 covariances in the Ox and Oy directions overlap with a range close to 500μm, while the covariance in the Oz direction has a shorter range close to 150μm). This is consistent with the fabrication of the fibrous network, obtained by deposition of isotropic layers of fibres parallel to the plane xOy. For physical applications is often made use of the correlation functions. The second order central correlation function W 2 (h) is deduced from C(h) for a two phase composite having the property Z = Z1 when x ∈ A and Z = Z2 when x ∈ Ac (chapter 2): W 2 (h) = E{(Z(x + h) − E(Z))(Z(x) − E(Z))} = (Z1 − Z2 )2 (C(h) − p2 ) = (Z1 − Z2 )2 (Q(h) − q 2 )
(3.50)
70
3. Quantitative Analysis of Random Structures Covariance non-overlapping discs (repulsion) 3000 C(h) Sill 2500
2000
1500
1000
500
0 0
50
100
150
200
FIGURE 3.13. Covariance of Fig. 3.12 estimated by translations in the horizontal direction
FIGURE 3.14. Transverse section xOy of the fibrous medium of (Fig. 3.8)
FIGURE 3.15. Axial section xOz of the fibrous medium of (Fig. 3.8)
3.5 Morphological analysis of the spatial distribution
71
FIGURE 3.16. Covariances in the Ox, Oy, and Oz directions of the 3D image (Fig. 3.8)
Higher order moments Generalizations of the covariance make use of m points to generate order m moments, namely the spatial law (chapter 2): P {x ∈ A, x + h1 ∈ A, x + h2 ∈ A, ..., x + hm−1 ∈ A} At the end of the present chapter in section 3.10.4, use is made of the moments of orders 3 and 4 to estimate specific connectivity numbers. In chapter 18 correlation functions of order m enter into bounds of effective properties of random media. These correlation functions are derived for two-phase composites in section 3.10.3.
Distance functions Definition 3.4. As introduced at the beginning of chapter 2, it is possible to define for every point x its distance R(x, A) to the set A :
R(x, A) = ∧y∈A {d(x, y)} = ∨{r; Bx (r) ⊂ Ac }
(3.51)
The sets of points for which R(x, A) < r is obtained by dilation of A by the ball B(r) : {x, R(x, A) < r} = {x, x ∈ A ⊕ B(r)}
(3.52)
The distribution function of the distance R(x, A) for points x ∈ / A (called the distribution of the first point of contact in [438], is given by (cf chapter 2):
72
3. Quantitative Analysis of Random Structures
Fx (r) = P {R(x, A) < r | x ∈ Ac } = P {x ∈ A ⊕ B(r) | x ∈ Ac } =
T (Bx (r)} − T (x) P {x ∈ A ⊕ B(r)} − P {x ∈ A} = c P {x ∈ A } 1 − T (x)
(3.53)
Or equivalently, for a stationary random set T (r) = 1 − F (r) =
1 − P {x ∈ A ⊕ B(r)} 1−p
(3.54)
The information contained in Eqs (3.53,3.54) is obtained either from a sequence of binary dilations of A, or from the distribution function of the digital version of a distance function. For practical applications are used: • dilations by squares or octagons on the square grid; • dilations by hexagons or dodecagons on the hexagonal grid; • direct calculation of distance functions [144], or even of Euclidean distance functions in R2 [648] or in R3 [215]. From the experimental distribution F (r) are derived various practical dF (r) data: the pdf f (r) = , the maximal distance a, when it is bounded dr (lowest value of a for which F (a) = 1), the median distance dM (F (dM ) = 1 ), moments of the distance and of the area of the largest ball Bx (r) 2 centered in x and included in Ac (using the relation (3.34) after replacing G(r) by F (r)). The distribution F (r) can be used for testing some models of random sets as the Boolean model (see chapter 6). For a fractal random set A (with a non integer dimension d), F (r) behaves like rβ for r → 0 with d = n − β. However a non integer experimental β cannot warrant an underlying fractal set, as shown in chapter 6 with a counter-example based on the Cox point process, so that care must be taken in the interpretation of experimental or simulated data. For illustration, dilation and erosion of the Fe image in Fig. 3.2 are shown in Fig. 3.3 and Fig. 3.4. The curves P {H(r) ⊂ Ag} and P {H(r) ⊂ F e} in Fig. 3.17 give access to the distribution of distances T (r) in Eq. (3.54).
Closing by convex sets As mentioned earlier, the closing operations by homothetic convex sets Kλ provide a size distribution of Ac . In the same time, it gives indications on the spatial distribution of the set A. As for the dilation by balls, it is convenient to interpret the closing by balls in terms of distance. Definition 3.5. Let Rf (x, A) be the radius of the largest ball By (r) with center in y, containing x and remaining included in Ac (compare to Eq. (3.51)):
3.5 Morphological analysis of the spatial distribution
73
Composite Fe Ag Hexagonal erosions and Distance Function 7000 ’Fe’ ’Ag’ 6000
5000
4000
3000
2000
Ag
Fe
1000
0 0
5
10
15
20
FIGURE 3.17. Curves P {H(r) ⊂ Ag} and P {H(r) ⊂ F e}
Rf (x, A) = ∨{r; By (r) ⊂ Ac ; x ∈ By (r)}
(3.55)
From Eqs (3.25,3.26), the set of points x for which Rf (x, A) < r is the intersection of Ac and of the closing of A by B(r), and therefore F1 (r) = P {Rf (x, A) < r | x ∈ Ac } P {x ∈ (A ⊕ B(r)) ª B(r)} − P {x ∈ A} = P {x ∈ Ac }
(3.56)
or equivalently for a stationary random set T1 (r) = 1 − F1 (r) =
1 − P {x ∈ (A ⊕ B(r)) ª B(r)} 1−p
(3.57)
The two distribution functions F (r) (Eq.(3.53)) and F1 (r) (Eq.(3.56)) give different information: if for instance in the plane Ac is made of a population of discs, R(x, A) varies continuously from 0 to 1 for r belonging to a disc of radius r, while Rf (x, A) keeps the value r for all points x inside the same disc. The choice of the operation (dilation or closing) depends on the type of information that is looked for. What is the connection between these two distributions of distances and the covariance? In fact, each of these three criteria bring its own information on the spatial distribution of a set A: • For the covariance, the two sets A and Ac play the same part; in addition, from an experimental point of view, the covariance is not too much sensitive to noise. • For the distance functions, separate data are obtained on A and Ac , and complementary pieces of information are obtained. They are sensitive to the presence of objects with zero Lebesgue measure (as seen later), such as points, lines, boundaries,... and also to the presence of noise.
74
3. Quantitative Analysis of Random Structures
In addition to the distance functions, further information is obtained from recording the variation with r of the connectivity number NA (A ⊕ B(r)) in R2 or (NV − GV )(A ⊕ B(r)). This type of function is sensitive to the more or less random distribution of objects and to the presence of clusters at different scales, as shown in applications [237],[238].
3.5.2 Multi component random sets The previous criteria can be extended to the case of multi component random sets. Each component can be studied separately as previously, but we are now interested by morphological properties involving at least the combination of two components (by means of appropriate composite structuring elements). Definition 3.6. In the case of m components Ai (i = 1, 2, ..., m), we can consider the closure of each component, and from the compacts sets Ki (i = 1, 2, ..., m) introduce a multi component version of the Choquet capacity depending on K = (K1 , K2 , ..., Km ): T (K) = 1 − Q(K) = 1 − P {K1 ⊂ A1 , K2 ⊂ A2 , ..., Km ⊂ Am }
(3.58)
In the next subsections, we consider the case of properties involving the associations of pairs of components: the cross covariances and distance functions. The cross covariograms and the cross covariances Definition 3.7. For the two sets Ai and Aj it is possible to define the cross geometrical covariogram as a generalization of Eq. (3.39): Z Kij (h, α) = μn (Ai ∩ Aj−h ) = 1Ai (x)1Aj (x + h) dx (3.59) Rn
The various cross geometrical covariograms are not independent, since they have to satisfy j=m X
Kij (h, α) = μn (Ai ) = Kii (0, α)
(3.60)
j=1
When considering the geometrical covariogram KXi ∪Xj (j, α) of the union Xi ∪ Xj , we have KXi ∪Xj (j, α) = Kii (h, α) + Kjj (h, α) + Kij (h, α) + Kji (h, α)
(3.61)
For sets with μn (Ai ∩ Aj ) = 0 (as for disconnected sets Ai ), Kij (0, α) = 0. Z 1 4π 0 Sij 0 3 When Kij (0, α) remains finite, in R , Kij (0, α) dα = , where 4π 0 4
3.5 Morphological analysis of the spatial distribution
75
Sij is the area of contact between Ai and Aj . For bounded random sets, the expectations of these relationships provides the average cross geometrical covariograms and the average areas of contact. Definition 3.8. The cross covariances Cij (h) of a multi component random set generalize the case of one component. Cij (h) is equal to the probability that the point x belongs to Ai and that x + h belongs to the set Aj : Cij (x, x + h) = P {x ∈ Ai , x + h ∈ Aj } (3.62) For a stationary and ergodic multi component random set, the covariances do not depend on the location x. In R3 , they can be expressed from the volume fraction VV by: Cij (h) = VV (Ai ∩ Aj−h )
(3.63)
The cross covariances can be estimated from images by means of the geometrical covariograms of the set Ai ∩ X (KA∩X (h)) and of X (KX (h)): ∗ (h) = Cij
KAi ∩X, Aj ∩X (h) A((Ai ∩ X) ∩ (Aj ∩ X)−h ) = A(X ∩ X−h ) KX (h)
(3.64)
Every Cij (h) brings information on the mutual association of the pair (Ai , Aj ). The morphological properties of the cross covariances are similar to those of the covariances (here is considered the case where the Ai build a tessellation of the space, so that P {x ∈ Aj ∩ Aj } = 0): • Cij (0) = 0 Xj=m • Cij (h, α) = Cii (0, α) = pi j=1
• CAi ∪Aj (j, α) = Cii (h, α) + Cjj (h, α) + Cij (h, α) + Cji (h, α) µ ¶ Z 1 4π ∂Cij (h, α) • dα = SVij (specific area of contact between the π 0 ∂h h=0 components Ai and Aj ), when the partial derivative remains finite. • Cij (∞) = pi pj (the cross covariances of a stationary and ergodic multi component random set reaches a sill) • For the orientation α, Cij (h) reaches its sill at the distance aijα , or range: Cij (aijα ) = Cij (∞) = pi pj . For h ≥ a the events {x ∈ Ai } and {x + h ∈ Aj } are uncorrelated.
A very simple but effective way to characterize multi component structures is to consider the behavior in h = 0 of the cross covariances Cij (h) by means of the specific areas of contact SVij [256], [257], [259]: • The distribution of contacts of the component Ai with the other components Aj (j 6= i), or transition probability from Ai to Aj for a random point on ∂Ai is given by:
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3. Quantitative Analysis of Random Structures
pij =
SVij SVi
(3.65)
In general, pij6= pji , since SVi 6= SVj . • The indices of coordination ic (ij) (generalization of the index introduced in [223]) are symmetric in i and j, and are obtained from ic (ij) = with SV =
Xi=m i=1
SVij SV SVi SVj
(3.66)
SVi (total area of contact of the medium).
To appreciate the preferential associations from these indices, reference values are needed. They can be given heuristically (for instance, a uniform SV 1 1 SV ' distribution of contact is expected to produce ij ' and SVi m−1 SVj m m so that a coordination index ic (ij) > reflects a particular attraction m−1 of components i and j, while the opposite situation should indicate some repulsion. Applications to complex mineralogical textures iron ore sinters are developed in [256], [257], [259]. Other reference values for the indices can be derived from models of textures. An example is presented in exercise (8.9.2). The main difficulty for applications concerns situations where the boundary of one component is very rough or prevalent. It is not the case SVij SVi when the ratios and are not too close to 1. SV SV Mutual associations and distribution of distances When considering degenerate components with dimension d < n, like cracks, grain boundaries, points, it is not any more possible to study their spatial distribution from the cross covariances Cij (h), which are equal to zero for any h. Therefore the approach had to be generalized [262] to the analysis of neighborhoods between various domains of an image, to study for instance cracking processes acting in a material: mutual location of cracks with respect to the microstructure, sites of crack initiation or of crack arrest,... These distribution criteria are based on the comparison between the microstructure located inside neighborhoods of a given component with increasing sizes, and the overall microstructure. This information is obtained from the basic specific measurements after isotropic dilations with balls of increasing radii. A first class of criteria is obtained from neighborhoods of Ai with increasing sizes, as introduced in [619]. Definition 3.9. Let Φj (k, r) be the function Φj (k, r) = P {x ∈ (Ak ⊕ B(r)) ∩ Aj }
(3.67)
3.5 Morphological analysis of the spatial distribution Distance to from a random point in component Ai
crack Ak
crack Ak
component Ai
crack ends Ek
component Ai
(in
R2 )
component Ai
crack ends Ek
77
Measurement SV ((Ai ⊕ B(r)) ∩ Ak ) SV (Ak ) VV ((Ak ⊕ B(r)) ∩ Ai ) VV (Ai ) AA ((Ek ⊕ B(r)) ∩ Ai ) AA (Ai ) NA ((Ai ⊕ B(r)) ∩ Ek ) NA (Ek )
TABLE 3.2. Association between components, at the scale r
and ρj (k, r) the ratio ρj (k, r) =
Φj (k, r) 1 − pk P {x ∈ (Ak ⊕ B(r))} − pk pj
(3.68)
When we observe ρj (k, r) > 1, a preferential association between components Ak and Aj , at the scale r, is expected. For ρj (k, r) < 1, a repulsion effect is expected between the two components at the same scale. The functions Φj (k, r) and ρj (k, r) are related to the following distance distributions: • From a random point x in Ack to the boundary of Ak (as in Eq. (3.54)): P {x ∈ (Ak ⊕ B(r))} − pk 1 − pk (3.69) • From a random point x in Aj to the boundary of Ak F (k, r) = P {d(x, Ak ) < r | x ∈ Ack } =
Fj (k, r) = P {d(x, Ak ) < r | x ∈ Aj } = We have ρj (k, r) =
Fj (k, r) F (k, r)
Φj (k, r) pj
(3.70)
(3.71)
For structures in R3 , the probability P appearing in Eqs (3.67-3.70) is estimated from the volume fraction VV . Other information is obtained by replacing the measurement VV with other specific properties as the specific surface area SV , or the connectivity number NA , as indicated in Table (3.2).
A second class of criteria involves dilations of the pair Ai , Aj , resulting into a symmetrical approach of mutual associations. Definition 3.10. Let Ψij (r) be the function
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3. Quantitative Analysis of Random Structures
Ψij (r) = P {x ∈ (Ai ⊕ B(r)) ∩ (Aj ⊕ B(r))}
(3.72)
The function Ψij (r) enables us to estimate the distribution Tij (r) of the maximum of the distances from a random point x in (Ai ∪ Aj )c to the boundaries of Ai and Aj : Tij (r) =
Ψij (r) − P {x ∈ (Ai ∪ Aj )} 1 − P {x ∈ (Ai ∪ Aj )}
(3.73)
This approach was followed to show preferential crack paths in porous hematite during the reduction of iron ore sinters [262]. The criteria introduced in this subsection are connected to the specific area of contact, when they are finite, by the following expression, obtained from averaging over the orientation of sections: Ã Ã ! ! ∂Φj (k, r) ∂Ψij (r) 4 =4 = SVij ∂r ∂r r=0
r=0
3.5.3 Random functions We consider now the case of multivariate random functions Zi (x), to which the notions introduced for sets can be extended. Bivariate distribution and second order statistics For every pair of images Zi (x) and Zj (x) (with possibly i = j), we can use the bivariate distribution: Tij (h, z1 , z2 ) = P {Zi (x) ≥ z1 , Zj (x + h) ≥ z2 } = P {x ∈ AZi (z1 ), x + h ∈ AZj (z2 )}
(3.74)
For non stationary RF, Tij (z1 , z2 , h) depends on the two points x and x+h. We consider here the stationary case. Eq. (3.74) should be compared to Eq. (3.62), showing that the bivariate distribution is made of a full set of cross covariances, parametrized by the thresholds (z1 , z2 ). The full experimental bivariate distribution builds a huge set of data, requiring rather long calculations. Practically, some classes of values for z1 and z2 can be used. We can limit the investigations to simpler functions summarizing the second order statistics, such as the covariances. Definition 3.11. The covariances Cij (h) are defined by [437] Z Z Cij (h) = E{Zi (x)Zj (x + h)} = Tij (h, z1 , z2 ) dz1 dz2 R R
(3.75)
3.5 Morphological analysis of the spatial distribution
79
Many properties of the covariances for random functions (in particular the sensitivity to the anisotropy) are similar to the random set case. They can be summarized as follows: • The behavior of Cij (h) for h → 0 is a picture of the spatial regularity of the studied images: a discontinuity in h = 0 of the covariances Cii (h) (called a nugget effect) is the result of microstructures at a scale lower than the scale of observation (minimal distance between two pixels in images), or of the presence of noise without spatial correlation. For noisy images, the cross covariance between two acquisitions (as obtained in a microscope) gives an estimation of the underlying covariance of the underlying signal without noise. A continuous behavior of Cii (h) in h = 0 is the property of mean square continuous RF, and twice differentiable covariances are obtained for mean square differentiable RF. Finally if Cii (0) − Cii (h) ' hβ when h → 0, with 0 < β ≤ 2, Zi (h) may own β a non integer Hausdorff dimension d = n + 1 − . This is true for a 2 Gaussian RF, but it is not always the case: the Poisson mosaic model studied in chapter 8 has an exponential correlation function with a linear behavior for h → 0, and therefore β = 1; however, this RF has an integer Hausdorff dimension. • For h → ∞, the covariances Cij (h) of stationary ergodic RF reach a sill E{Zi }E{Zj }. The sill is reached for the separation h = aij called the range. It is common to use the central second order correlation functions W ij2 (h) (chapter 2) with W ij2 (h) = Cij (h) − E{Zi }E{Zj }. For every RF Zi (x), the central correlation function is a positive type function; it means that for every set of points x1 , x2 , ..., xm and for the coefficients λ1 , λ2 , ..., λm , the following expression remains positive (D2 meaning the variance): D
2
Ãk=m X k=1
λk Zi (xk )
!
=
k=m,l=m X k=1 l=1
W ii2 (xl − xk ) ≥ 0
(3.76)
Theorem 3.3. (Bochner) Continuous central correlation functions W ii2 (h) are the Fourier transform of a positive integrable measure fi (ν) (the spectral density of the RF): Z W ii2 (h) = exp(2iπνh)fi (ν) dν Z (3.77) and fi (ν) = exp(−2iπνh)W ii2 (h) dh In Eq. (3.77), i is the imaginary complex number wit i2 = −1. The density fi (ν) is a distribution of the total variance of the RF Zi (x) according
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3. Quantitative Analysis of Random Structures
to the spatial frequencies ν. For instance, in the case of a stationary process Z(t) in R with a discrete and finite spectrum Z(t) =
k=n X
Φk exp(iν k t)
(3.78)
k=−n
where i is the complex number such that i2 = −1 and the Φk are uncorrelated random variables with mean 0. If to the frequency ν k corresponds the variance Fk , the variance of Z(t) is expressed as 2
2
D [Z(t)] = E{Z(t) } =
k=n X
k=−n
E{Φ2k }
=
k=n X
Fk
(3.79)
k=−n
Definition 3.12. The moments of the central second order correlation function of the RF Z(x) enable us to define the integral range An from [444],[451],[462],[598],[687] Z 1 An = 2 W 2 (h) dh (3.80) D [Z] Rn For an isotropic RF, Eq. (3.80) becomes An =
ωn D2 [Z]
Z
+∞
hn−1 W 2 (h) dh
(3.81)
0
f (0) . Eqs (3.80-3.81) can be applied D2 [Z] to the covariance of a random set A (in that case, D2 [1A (x)] = p(1 − p) where p = E{1A (x)}). The integral range is a good measure of the notion of scale: for instance, in the plane, an area S can be considered A2 as n = surface elements inside which the average values of Z(x) are S uncorrelated random variables. An interesting study of RF with zero or infinite integral range is given in [406]. Some random sets and random functions with an infinite integral range are derived from Poisson varieties, and are studied in chapter 6. Periodic structures have by construction a zero integral range. As a consequence, the algorithm (3.47) used in [385] to estimate the covariance from an image by means of a Fast Fourier transform provides an experimental covariance with a zero integral range, generating an artifact which does not correspond to the real situation. From Eq. (3.77), it results that An =
Definition 3.13. For random functions with stationary increments (and therefore more general than the stationnary RF), can be defined the variograms γ ij (h) as half the covariances of the increments Zi (x + h) − Zi (x) [437].
3.5 Morphological analysis of the spatial distribution
81
In the stationary case, the variograms are deduced from the covariances. For instance, γ 2 (h) =
1 E{(Z(x + h) − Z(x))2 } = C(0) − C(h) 2
(3.82)
In addition to the structural properties of the covariances and of the variograms, these functions are the basic tools for solving practical problems, such as some effects of a change of support, the calculation of a variance of estimation, and the interpolation by kriging. We give here a very brief summary of these topics, that are developed in [437], [444]. • When changing of support by a convolution operation (using a weighting function p(x)), a RF Y (x) becomes Z Z(x) = p(h)Y (x + h) dh = Y ∗ pˇ (3.83) Rn
The covariances and the variograms involving the RF Z(x) and Y (x) satisfy CZ (h) = CY ∗ Pˇ with P = p ∗ pˇ (3.84) CY Z (h) = CY ∗ pˇ Z γ(u)P (u) du γ Z (h) = γ Y ∗ Pˇ − I(P ) with P = p ∗ pˇ and I(P ) = γ Y Z (h) = γ Y ∗ pˇ − I(p)
Rn
(3.85) In particular, the change of the variance D2 (Y (x)) by convolution can be predicted, provided that the covariance (or the variogram) is known. However, the distribution function of Z(x) depends in general from the full spatial law of Y (x). • The first order variogram γ 1 (h) of the RF Z(x) is defined as follows: γ 1 (h) =
1 E{|Zi (x + h) − Zi (x)|} 2
It is more robust to outliers than the variogram γ 2 (h). When the bivariate distribution T2 (h, z1 , z2 ) = P {Z(x) ≥ z1 , Z(x+ h) ≥ z2 } is available, as will be the case for many models studied in this book, we have (see exercise 3.10.2): Z Z E {|Z(x + h) − Z(x)|} = T (z)dz − T2 (h, z, z)dz (3.86)
3.5.4 Covariance of orientations of vector fields One of the features of vector fields is the local orientation. It is common to consider as a statistical descriptor the distribution of orientations. For
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3. Quantitative Analysis of Random Structures
instance a uniform distribution of orientations over the sphere S n reflects the isotropy of the vector field. Another point of interest concerns the correlation of orientations at different scales. The tool provides a quantitative description to estimate the scales of orientations present in a vector field [12], [307]. Consider a random field of vectors U (x) in Rn . The components of U (x) in an orthonormal basis of Rn are U1 (x), U2 (x), ..., Un (x). To compare the orientations of the two vectors U (x) and U (x + h), it is convenient to consider their angle α. Using the scalar product (.) and the norm kk, cos2 α(x, x + h) =
(U (x).U (x + h))2 2
2
kU (x)k kU (x + h)k
Consider now a random unit vector field, where kU (x)k = 1. The covariance of orientations is defined by [12] Cova (x, x + h) = E{cos2 α(x, x + h)} In the case of a stationary random vector field, the covariance does not depend on x but depends on the vector h. We write it Cova (h). The covariance of orientations is estimated from the various covariances of the components of U . From the scalar product, 2
cos α(x, x + h) =
Ãi=n X
Ui (x)Ui (x + h)
i=1
=
i=n X
!2
Ui2 (x)Ui2 (x + h)
i=1
+2
i=n,j=n X
Ui (x)Uj (x)Ui (x + h)Uj (x + h)
i=1,j=1,i6=j
and therefore the covariance Cova (h) is the sum of the covariances of the scalars Ui2 and Ui Uj : Cova (h) =
i=n X i=1
CUi2 (h) + 2
X
CUi Uj (h)
i6=j
Each term of the covariance Cova (h) can be estimated using the Fourier Transform of the corresponding product Ui Uj . For a two-dimensional random vector fields, it involves the calculation of three covariances. As an example of illustration, consider a two scale orientation mosaic model (cf. chapter 8) defined as follows: start with a random tessellation of space in classes Ai . To each class Ai of the tesselation is allocated a unit
3.5 Morphological analysis of the spatial distribution
83
vector U on the sphere S n according to a given distribution of orientations ϕ. The orientations of two classes are independent. For this model we have, noting K(h) = E{μn (A ∩ A−h )} the geometrical covariogram of A and and r(h) = K(h) K(0) the reduced geometrical covariogram, Cova (h) = r(h) + (1 − r(h))
Z
cos2 αϕ(a)dα Sn
The range of the covariance Cova (h) is given by the range of the geometrical covariogram K(h) and therefore the scale of the orientations is controlled by the scale of the underlying random R tessellation. For a uniform distribution of orientations in R3 we have S 3 cos2 αϕ(a)dα = 13 [12]. In R2 , Z 2π 1 1 cos2 αdα = 2π 0 2
On grey level images, boundaries are detected from the gradient of the image. More precisely, on boundaries the vector gradient is orthogonal to the boundary, where it shows a high modulus [351]. When parallel boundaries appear on a large scale in an image, as for a microstructure showing aligned elongated objects, correlations of orientations are present. We can therefore study correlations of orientations in a scalar image Z(x) by estimation of the covariance of the orientations of the gradient. The components of the gradient are given by ∂Z(x) ∂xi . On an image, they are estimated from increments Z(x + δxi ) − Z(x). The components of the gradient must be divided by its norm, to get a unit vector for orientations characterization. In order to eliminate the effect of noise on the gradient, it is wise first to smooth the image by convolution by a Gaussian filter with a small standard deviation (typically 2 pixels). For large separations h, and for a uniform distribution of orientations, the covariance Cova (h) reaches its asymptotic value ( 12 in R2 and 13 in R3 ) when h > a, a being the range of the covariance, giving the range of the orientation. When the range a is larger than the typical size of objects present in an image, alignments of objects are expected, as in the example of the orientation mosaic model.
3.5.5 Definition of a Statistical Representative Volume Element (RVE) and problems of estimation The variogram, and the covariance of a stationary RF are the key geostatistical tools to compute variances of estimation and to provide estimators to restore spoiled data.
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3. Quantitative Analysis of Random Structures
Variance of estimation When estimating the average of a RF Z(x) over a volume V (briefly written Z(V )) from an average over a volume V 0 (Z(V 0 )), we obtain a result with a random error of estimation Z(V ) − Z(V 0 ). The variance of this error σ 2E (V, V 0 ) is given as a function of the variogram by [437], [444]: Z Z Z Z 2 1 γ(x − y) dxdy − γ(x − y) dxdy σ 2E (V, V 0 ) = V 02 V V0 V V0 V0 V0 (3.87) Z Z 1 − 2 γ(x − y) dxdy V V V A similar expression using the covariance is obtained for stationary RF. In Eq. (3.87), appear a structural property (through the variogram), and the geometry of implantation of the fields of measurements (through the domains of integration). In practical applications, simplified expressions can be used (for large domains V and V 0 , they contain the integral ranges An ), as illustrated for solving sampling problems in image analysis applied to steels and to iron ore sinters in [236]. This enables us to define the size of a statistically Representative Volume Element (RVE). For a stationary RF 2 2 Z(x), with expectation E{Z}, and point variance DRZ the variance DZ (V ) 1 ¯ of the average of Z(x) in the volume V (Z(V ) = V V Z(x)dx) is given by: Z Z 1 2 DZ (V ) = 2 W 2 (x − y) dxdy, (3.88) V V V For V À A3 (for finite A3 ), to first order in 1/V , A3 being the Integral Range we obtain an asymptotic value of the variance: 2 2 DZ (V ) = DZ
A3 V
(3.89)
V subvolumes in which A3 the average values of the RF Z(x) are uncorrelated random variables. 2 For infinite integral range the variance DZ (V ) decreases more slowly −1 than V . From empirical observations, a decrease in V −α with α < 1 was proposed [406]. In [317] the coefficient α is theoretically computed in the case of Boolean random varieties (see chapter 6). When A3 = 0, the 2 decrease of DZ (V ) is much faster than V −1 , which induces a fast convergence of Z(V ) towards its expected value. Media with a zero integral range satisfied the so-called "Hyperuniformity" condition [635]. Eq. (3.89) gives access to an experimental estimation of the integral range by plotting the 2 curve DZ (V ) obtained for different sizes of samples [451], [462], [406]. Its equivalent form when using a V −α scaling gives access to α and to a test of the validity of the scaling law [367]. From Eq. (3.89) the volume V is equivalent to k =
3.5 Morphological analysis of the spatial distribution
85
q 0.05 0.1 0.2 0.3 0.5 2σE (V ) 0.0054 0.00865 0.0134 0.0167 0.02 2σE (S) 0.021 0.032 0.049 0.0603 0.071 TABLE 3.3. Intervals of confidence for the Boolean model of spheres
The absolute error abs and relative error rela of the average value obtained for n independent realizations of volume V are deduced from the confidence interval (to 95%) abs
=
2DZ (V ) √ ; n
rela
=
abs
E{Z}
=
2DZ (V ) √ E{Z} n
(3.90)
The statistical RVE can be defined as the volume V (obtained for instance for n = 1 realization in the ergodic case) for which E{Z} is estimated with a given relative precision (for instance rela = 1%). It can be applied to the volume fraction (with Z(x) = 1A (x), E{Z} = p and D2 [Z] = p(1 − p)), to a stress field σ(x) or to a strain field ε(x) to define the RVE of effective properties 19, as initiated in [367]. For example we computed the numerical values of 2σ E (V ) and 2σ E (S) of a sample made of a cube with volume V = L3 and of a square of area S = L2 , to provide the interval of variation q ± 2σ of q = 1 − p expected for observations in the case of a Boolean model of spheres with diameter a with a/L = 0.1. This is given in Table 3.3, where it is clear that the expected dispersion of sections is much larger than the dispersion of volumes.
Estimation of spoiled or missing data by kriging Images can be spoiled by the presence of noise. The observed histogram can be very far from the underlying histogram, for instance when a proportional noise is observed as a consequence of a poissonisation [140], [141]. When data are missing at some points, or when we obtained noisy data, a best linear unbiased estimator Ym∗ of the unknown Ym (x) can be built from observations Zj (xα ): X α Ym∗ (x) = λj Zj (xα ) (3.91) j,α
It makes use of the cokriging algorithm, that extends the Wiener filter [667] to multivariate Intrinsic RF of order k [447]. Requiring a minimization of the variance of estimation and a non bias condition, optimal weights based on the second order statistics of the images are solutions of the cokriging system (written here with the covariances with the notation Cov): P α j,α λj Cov{Zj (xα ), Zk (xβ )} + μk = Cov{Zk (xβ ), Ym (x)} (3.92) P α m λ = δ j α j
86
3. Quantitative Analysis of Random Structures
FIGURE 3.18. Microprobe image of microsegregation of Al during solidification of a superalloy [139]
In practice, the Zk (xβ ) may be Ym (x) (interpolation), (multivariate) noisy (Fig. 3.18) or convoluted images. Appropriate models of Covariances (or of Variograms) are fitted to data and used for the estimations of weights λα j . Examples are given for the filtering of multivariate noisy images in [139], [140], [141], and for the filtering and deconvolution of noisy images by using models satisfying Eqs (3.84, 3.85) [121], [332]. More generally, for a stationary RF Y (x), a non linear estimation of any function Φ(Y (x)) (for instance the probability for Y (x) to exceed a given threshold inside a domain V , or a local distribution function) can be obtained by Disjunctive Kriging [449], [450] with X Φ(Y (x))∗ = fjα (Zj (xα )) (3.93) j,α
The functions fjα are solutions of the system E[Φ(Y (x)) | Zk (xβ )] =
X j,α
E[fjα (Zj (xα )) | Zk (xβ )]
(3.94)
involving conditional expectations and bivariate distributions (and therefore much more than the sole covariances). In practice a transformation of initial data is made to use specific bivariate distributions (e.g. Gaussian,...) admitting an isofactorial expansion over orthogonal functions (e.g. Hermite polynomials). The coefficients of the expansion of the functions fjα are solutions of linear systems. Applications of Disjunctive Kriging to the prediction of local probability for the measurements of micro segregations in steels are presented in [141], [142].
3.5 Morphological analysis of the spatial distribution
87
3.5.6 Distance functions The notions introduced for a random set can be extended to RF. For instance the distance given in Eq. (3.51) becomes for a point {x, z} in Rn ×R : R((x, z), Γ Z ) = ∧y∈Γ Z {d(x, y)} = ∨{r; Bx,z (r) ⊂ (Γ Z )c }
(3.95)
where Bx,z (r) is the ball in Rn+1 with radius r and center {x, z} in Rn × R. If Q(r, z) = P {Bx,z (r) ⊂ (Γ Z )c }, a family of distribution functions F (r, z) depending on the parameter z is defined from: 1 − F (r, z) =
Q(r, z) = P {R((x, z), Γ Z ) ≥ r} Q(0, z)
(3.96)
For balls in Rn (namely discs of Rn+1 with support parallel to Rn ), F (r, z) is the distribution function of the RF Z(x) after dilation by B(r), Z∨ (B(r)). Specific distances enable us to study some associations of random stationary components: • association between a RACS A and a RF Z(x). A can be a point process, a population of cracks,... and Z(x) a RF modelling the concentration of a chemical specie... Use can be made of G(r, z) = P {Z(x) < z and x ∈ A ⊕ B(r)} and ρ(r, z) =
G(r, z) F (z)P {x ∈ A ⊕ B(r)}
(3.97)
(3.98)
with ρ(r, z) → 1 for r → ∞ and ρ(r, z) = 1 if Z(x) and A are independent; ρ(r, z) < 1 when Z(x) takes lower values in the vicinity of A. • association between two RF Zi (x) and Zj (x). Use can be made of Φj (i, r, z) = P {(Zi∨ (B(r)) ∧ Zj (x)) ≥ z} and of ρj (i, r, z) =
Φj (i, r, z) (1 − Fj (z))(1 − Fi (r, z))
(3.99)
(3.100)
where Fi (r, z) is given by Eq. (3.96) for the component Zi (x) and Fj (z) = P {Zj (x) < z}. The index ρj (i, r, z) = 1 for independent RF Zi (x) and Zj (x); ρj (i, r, z) > 1 when values of Zj (x) higher than z are preferentially in the neighborhood of values of Zj (x) higher than z. This methodology is applied to elastic fields developed in the multiscale microstructure of concrete, on 3D real [171] or simulated images [173]. Stress fields are computed by FFT [479], and locally compared to the microstructure (chapter 19). A stress concentration is observed on a part of the
88
3. Quantitative Analysis of Random Structures
3D skeleton by zones of influence of the aggregates contained in concrete. These zones can be places were damage could initiate under compression of the material.
3.5.7 Random graphs A systematic study of random graphs is out of the scope of this book (in particular a good reference for the extension of mathematical morphology to graphs is [648]). We consider here graphs generated from structures. For example a grid of points (square or hexagonal in two dimensions) can generate various graphs; in a different context, a point process {xi } (for instance obtained with low size objects) can build a graph with vertices {xi } and edges made of segments linking any point to its nearest neighbors; similarly, a graph can be obtained from particles and their zones of influence, or from classes of a tessellation such as grains in a polycrystal. For the last case it may be interesting to examine neighborhood relationships such as: • correlations between crystallographic orientations of neighbor grains; • correlations between the area of a grain and of one of its neighbors,... Valued graphs are also of interest for physical applications involving propagations in a heterogeneous medium (light, sound, fluid, fracture, as studied in [275]). In fact we deal with planar valued graphs. Definition 3.14. A planar valued graph {V, E} is a set of summits V = {v1 , v2 , ...} and of edges E = {Eij } connecting pairs of nearest neighbors vi and vj . each edge Eij carries the value d(vi , vj ) ≥ 0 which is a local metrics. A path P (vs , vd ) between the source vs and the destination vd is a set of connected edges: P (vs , vd ) = {Es1 , E12 , ..., End } The length of the path L(P ) is obtained by addition of the d(vi , vj ) along the path: L(P (vs , vd )) = d(vs , v1 ) + d(v1 , v2 ) + ... + d(vn , vd ) and the distance between the two vertices vs and vd , d(vs , vd ) is the minimal value of L(P ) over all possible paths P (vs , vd ) on the graph {V, E}. Definition 3.15. From the distance d(vs , vd ), can be defined distance functions such as the distance from a vertex vs to a set of vertices Wd : d(vs , Wd ) = ∧vd ∈Wd {d(vs , vd )}
(3.101)
and distance between two sets of vertices Ws and Wd : d(Ws , Wd ) = ∧vs ∈Ws {d(vs , Wd )} = ∧vs ∈ Ws,
vd ∈Wd {d(vs , vd )}
(3.102)
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For a point process or for a tessellation, we can use the discrete distance generated by the adjacency graph of the classes Ci with d(Ci , Cj ) = 1 ⇐⇒ Ci and Cj are nearest neighbors. Bivariate distribution and second order statistics A function Z(v) may be defined on the vertices of the graph. For a random function (and possibly a random graph), a probabilistic approach may be used. Usually some assumption of homogeneity is made, so that the same random variable Z concerns all the vertices (this is equivalent to stationary and isotropic RF with support in Rn ). For this type of random structure can be studied first order (F (z)) and second order statistics, like a bivariate distribution defined for pairs of vertices separated by the distance h on the graph {V, E}. Similarly, correlation functions or variograms can be defined. There are nevertheless difficulties in building theoretical models, since information on orientation is lost and since {V, E} is not an Euclidean space. Specific models were developed for graphs defined on regular grids, where a spectral approach of covariances is available [224]. Distance functions From the non Euclidean distance on {V, E} a non Euclidean morphology can be built [600], [648] with its dilations. For instance, the binary dilation of size r applied to the set of vertices W , Γr (W ), is defined by Γr (W ) = {vk ∈ {V, E}, d(vk , W ) ≤ r}
(3.103)
Similarly the numerical dilation with size r of the function Z(v) is obtained as: Γr (Z) = Z∨ (B(r)) = ∨d(vk ,W )≤r {Z(vk )} (3.104) From the binary dilation Γr it is easy to estimate the distribution function of the first point of contact, generalizing the notions introduced in Eqs (3.54), or a measure of the association between several populations of vertices (generalization of criteria (3.67-3.73) of the multi component case). Here the volume fraction are replaced by frequencies (every vertex having a unit weight). In a same way, the criteria given in Eqs (3.96-3.100) are accessed from the dilation operation of Eq. (3.104).
3.6 Geodesic criteria and connectivity Many applications in Physics are based on some propagation phenomena, with different propagation velocities for heterogeneous media: propagation of light in optics (according to the principle of Fermat, the light propagates in a heterogenous medium according to the shortest time paths), of
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sound in acoustics, of a fluid in a porous medium, diffusion of a constituent, advance of a crack front... For all these domains where transport mechanisms are involved, the notion of connectivity is of prior importance, since without paths across a specimen (or without percolation), the overall macroscopic ”conductance” is zero. Indeed connectivity is an important morphological aspect for the prediction of physical properties of composites with components presenting a high contrast (microcracks, voids). This can be approached from the measurement of connectivity number N in R3 , which are topological measurements describing the overall connectivity of a given medium, as described in section 3.3.5. But more than giving only a binary description of percolation from the existence of connected paths across a medium, information like lengths of paths and its distribution can be useful. This is illustrated by the notions introduced in this section. When the structure can be modelled by a valued planar graph, it is possible to compute, by means of tools which generalize the Dijkstra’a algorithm for shortest paths [162], distances to a source on the valued graph, usually called the geodesic distance (namely the length of shortest paths). This is made after defining a valuation of the edges of the graph according to the physical problem of interest (inversely proportional to the light or sound velocity or to the coefficient of diffusion; proportional to the fracture energy for fracture processes) and after defining a source Ws and a destination Wd on images of the structure (for instance two opposite edges of a two dimensional field, or two opposite faces of a three dimensional field). Considering two points x and y belonging to a set A, the shortest paths connecting them, while constrained to remain in A, are called the geodesic paths in the geodesic mask A. When A is a connected component of an image, the geodesic distance dA (x, y) between two points x and y in A is the minimum length L of all the paths P = (p1 , p2 , p3 , ..., pn ) linking x and y included in A (Fig. 3.19) [609]. According to this definition, in the present image dA (x, y 0 ) = +∞, since x, y 0 belong to two different connected components of the set A. From a morphological point of view, this can be processed with geodesic dilations from x to y into A [598]. We have for the elementary geodesic dilation of the set (or seed) A0 ⊂ A into A, B(1) being the ball of radius 1: DA (A0 )(1) = (A0 ⊕ B(1)) ∩ A and by n iterations we obtain the front of propagation at step n ³ ´ DA (A0 )(n) = DA (A0 )(n−1) ⊕ B(1) ∩ A
(3.105)
(3.106)
In Fig. 3.20, the propagating fronts are circular, which is intuitive to estimate a Euclidean distance map in a continuum framework. However, when
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FIGURE 3.19. Geodesic distance
FIGURE 3.20. Propagating front to estimate the geodesic distance
working on digitized images on a grid of points, the type of graph (C6 , C4 ,or C8 in 2D, C6 or C26 , in 3D) strongly affects the obtained geodesic distance map [549]. The geodesic distance between Ws and any vertex x is then calculated. Various fast algorithms were developed for this calculation and are not presented here [648],[608]. In [549], the propagation by means of spherical structuring elements is approximated by fast marching [603] consisting of iteratively solving the eikonal equation (Eq. (3.107)) into a narrow band around the seed. The arrival time T (x) at a point x of the propagating front, starting from the seed A0 (T (x0 ) = 0 for x0 ∈ A0 ) is solution of Eq. 3.107, where f is the cost function, inverse of the speed of propagation: k∇T k = f
(3.107)
When the cost function is homogeneous and equal to 1, the solution of Eq. 3.107 leads to fronts propagating at the speed of one pixel (in 2D)
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and one voxel (in 3D) per iteration. The fronts travel outwards and do not return to the source of the propagation. For a directional propagation in a specific direction, half spheres are used to estimate the geodesic distance to a source. In [528] is given a fast algorithm to compute geodesic distances between all couples of points in a 2D images. From the geodesic distance function, useful information can be extracted. • Normalizing the geodesic distance dA (x, y) by the Euclidean distance kx − yk provided the morphological tortuosity τ A (x, y) [151]: τ A (x, y) =
dA (x, y) kx − yk
We have τ A (x, y) > 1 when f = 1, but we may have 0 ≤ τ A (x, y) < 1 in the case of a porous medium where there is no cost to propagate in pores, for which f = 0. For f = 0, we can speak about a geodesic deviation, because τ A (x, y) is not anymore a distance. This assumption was used to simulate crack propagation in porous media [328], [279]. • The geodesic distance between Ws and Wd (or length of shortest paths linking the source and the destination); with simulations of random media, the statistics of this parameter can be investigated, as shown in [279]. • The distribution of the geodesic distances in the image. • The shortest paths connecting Ws and Wd . They are obtained as follows. From the definition of distances (Eq. (3.103)), the source and the destination can be interchanged, as long as there is no orientation of the edges. Otherwise, the orientation of the edges must be reversed for a backward propagation. This situation is very common in optics, where use is made of direct and inverse propagations. This analogy is used to extract geodesic (or shortest paths) between two sets of vertices Ws and Wd . The vertices x belonging to the shortest paths between the source Ws and the destination Wd satisfy the relationship: dA (x, Ws ) + dA (x, Wd ) = dA (Ws , Wd )
(3.108)
Eq. (3.108) requires in general the use of two propagations, exchanging the roles of Ws and Wd . For each vertex x of the image, the sum LA (x) = dA (x, Ws ) + dA (x, Wd ) if dA (x, Ws ) ∧ dA (x, Wd ) > 0 else LA (x) = 0 gives the length of the shortest paths connecting Ws and Wd and going through x (we use the convention LA (x) = 0 for points x that do not belong to shortest paths between the source Ws and the destination Wd ). Similarly a morphological tortuosity can be defined for any point x from the Euclidean distance d(Ws , Wd ) by
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τ A (x) =
93
LA (x) d(Ws , Wd )
• By application of thresholds to the image of geodesic distances, the sequence of propagation of the front is displayed. The distribution of LA (x) or of τ A (x) characterizes connected paths in a medium. Shortest paths with a given tortuosity range can be extracted by source backtracking [549]. • Points that cannot be accessed during the propagation are located at the distance +∞ from the source. This enables us to detect closed pores in a porous specimen. In some situations like on images of particles obtained by nanotomography with the electron microscope [108], it is not possible to use propagations between well defined source and destination. In that case, local measurements of the morphological tortuosity are made on shortest paths connecting random pairs of points (x, y). Various applications of this type of measurements were developed in the following fields: 2D examples concern the fracture of polycrystalline graphite [328], diffusion in polymers [331],[89], and the fracture of simulated random media at different scales (porous media, polycrystals) [279]. In the two first types of studies, an excellent agreement is found between geodesic parameters and physical properties of interest measured on a macroscopic scale. In [151] a porous medium is studied in 3D (Fig. 3.21), geodesic paths being extracted for a range of tortuosity in Fig. 3.22. For this example, the propagation is restricted to percolating pores connected to two opposite faces of the field. The corresponding distributions of tortuosity in the x and y directions are shown in Fig. 3.23 and 3.24. In [549] the anisotropy of fibrous networks is characterized in 3D by propagations in fibres and in pores, using various orientations for the source and destination. This information completes with directional connectivity the information given by the covariance in section 3.5.1. The tortuosity of the glassy phase and of Zirconia in refractory materials is studied with this geodesic tools from 3D microtomography [431]. Finally geodesic paths in elastic-plastic fibrous composite materials are related to the location of shear bands in the matrix obtained by finite elements micromechanical computation [353]. Few theroretical results are available concerning geodesics in random media. In [674] upper bounds of geodesic distances are given in 2D and 3D porous Boolean and multiscale Boolean models for vanishing pore volume fractions. To account for transport properties and bottleneck effects in porous media, the evolution of morphological tortuosity is studied as a function of the size of spherical openings [618] or spherical erosions [109]. Grey level textures were characterized by morphological tortuosity to provide an automatic classification of rough surfaces, as detailed in section 3.8.3 [187], [188], [189].
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FIGURE 3.21. Connected pores
FIGURE 3.22. Shortest paths (with tortuosity range: 1.6 − 2.2) in a porous medium 3
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3.7 Robustness of morphological measurements to noise Specific measurements estimated by counting configurations of the neighborhood of each grid points are related to derivatives of second, third and fourth order moments at the origin (section 3.10.4). As a consequence, their sensitivity to noise differs. For instance it is possible to estimate properly the volume fraction from area measurement on a noisy images. It is not the
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FIGURE 3.24. Distribution of the tortuosity of paths (cuboctahedron) in the and y direction Area Perimeter Connectivity N Tortuosity τ noisy image 21129 23565 1156 1. 617 filtered image 21784 2356 13 1. 574 TABLE 3.4. Effect of noise on basic morphological measurements
case anymore for numbers of intercepts (and therefore the perimeter in 2D and the surface area in 3D), and connectivity numbers, as illustrated in the noisy image (256×256) of Fig. 3.25, to be compared to the filtered image of Fig. 3.26, obtained by a binary morphological filter: closing the image by an hexagon with size 1, followed by an opening by an hexagon with size 3. In Table 3.4 are collected rough data (Area in pixels, perimeter, connectivity number, and morphological tortuosity obtained from horizontal propagations in Ac on the hexagonal grid to connect the left and right vertical boundaries of the image). It appears that the area and the tortuosity are rather robust to noise (there is presently a compensation of missing white pixels in A by white noisy pixels in Ac ). The tortuosity of the noisy image is slightly larger (2. 7% increase), due to additional obstacles generated in Ac . On the other hand the perimeter and the connectivity number are surestimated by one and two orders of magnitude, as the result of first and second order derivations of the noisy image.
The covariance of the noisy image (Fig. 3.27) shows a sharp discontinuity when h → 0, as demonstrated in section 6.15.4 in chapter 6. For large h the covariance of the noisy image is similar to the covariance of the filtered image, as proved in section 3.10.5 for a particular model of binary noise. It is not anymore the case of the distribution function of the distance F (r), given by Eq. 3.54 or of openings and closings size distribution. To conclude this section, some morphological parameters like the covariance are robust to noise, and can be estimated on noisy images. Other parameters, like surface area, connectivity numbers or size distributions require to filter noise in images before measurement. However in some situations it will be possible to make an identification of a model from the
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FIGURE 3.25. Noisy binary image
FIGURE 3.26. Filtered image
covariance (Part II), and therefore to get access to most morphological parameters by means of the covariance measured on noisy images. This may be useful for real time measurements for process control with low quality images.
3.8 Shape and texture classification and recognition The morphological parameters and operations introduced in this chapter can be combined as multivariate descriptors for texture and shape classification, using multivariate statistical analysis, or more recent machine learning algorithms. A basic approach considers each field of measurements or each object described by a vector of parameters as a point with a weight in a multidimensional space. The data obtained on fields or on objects build a cloud in a space of high dimension. For a simplified representation of the cloud, a reduction of dimension is obtained after extraction of the axis of inertia. Data are projected on the most significant axis, based on their contribution to the total inertia. This is illustrated by examples applied to shape classification and recognition, then to textures. In some
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Covariance of a noisy image 3500 ’COVFNOI.DAT’ ’COVNOIS.DAT’ 0.3323**2*10000. 3000
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FIGURE 3.27. Covariances of the noisy image (dots) and of its filtered version
examples is combined a texture analysis by mathematical morphology and an implementation of models of random structures.
3.8.1 Automatic recognition of non metallic inclusions One of our early studies in this area [323], [324] was made on automatic recognition of non metallic inclusions, starting from the Diegarten chart made of 90 images (Fig. 3.28). A supervised learning was obtained by combination of morphological measurements on each image of the chart (12 parameters made of basic morphological measurements on the initial image, after dilation and after closing to detect clusters of inclusions) , followed by a Correspondence Analysis, and a classification made from the 4 first factors. The studied ended with an automatic procedure based on image analysis, to evaluate the steel cleanliness, as done before by hand.
3.8.2 3D Shape classification of particles with complex shapes The complex morphology of intermetallic particles in aluminium alloys was studied from 3D images obtained by microtomography at ESRF [538], [539], [540], [541], [542]. These images have a 0, 7µm3 resolution and contain thousands particles with a volume in the range 9 -24000µm3 . The particles are not convex (Fig. 3.29, 3.30). Standard morphological measurements (volume V , surface area S, sphericity index, granulometry by openings by spheres) are far from being exhaustive to describe their shape. They are completed by three other types of measurements: geodesic parameters, inertia parameters, and bivariate distribution of curvature.
1. Geodesic parameters are obtained from geodesic distances between voxels located inside a given particle, as a 3D generalization [538] the
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FIGURE 3.28. The Diegarten chart of non metallic inclusions
FIGURE 3.29. Intermetallic particle in an aluminium alloy
FIGURE 3.30. Intermetallic particle in an aluminium alloy
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2D approach [405]. For every point x ∈ A, define the geodesic propagation function P (x) from the geodesic distances between points x and y, dA (x, y): P (x) = max {dA (x, y)} y∈A
From function P (x), obtained by a 3D implementation, [540], one estimates a geodesic length Lg (or diameter), a geodesic radius Rg , and a geodesic elongation index Ig by comparison to the spherical case: Lg = max {P (x)} , Rg = min {P (x)} , Ig = x∈A
x∈A
π L3g 6 V
2. Inertia features, related to the Minkowski tensor W02,0 (subsection 3.3.6), are derived from principal components of inertia of the object A (I1 , I2 , I3 ), the eigen values of the inertia matrix of A, and directly estimated from the coordinates of voxels of A considered as points with a unit weight. After normalization of the principal moments of inertia, three reduced principal moments are obtained (λ1 > λ2 > λ3 ), satisfying the following relations (section 3.10.6), valid for any set A (even non connected) [540], [542]: λ1 + λ2 + λ3 = 1 λi 6 0.5 (i = 1, 2, 3) λ2 > 0.5(1 − λ1 ) Using these relations, any object is represented by a point in the plane λ1 , λ2 located inside a triangle typical of the shape (Fig. 3.31). For convex 3D objects, the summits of the triangle represent the three most distant types of mass distribution: spherical, flat, and thread type. Between these summits there is a continuous change of shape. On the edges of the triangle, objects show typical shapes: prolate ellipsoid, oblate ellipsoid , and planar ellipse. The orientation of the main axis of inertia (parallel to the eigen vector corresponding to the eigen value λ1 ) allows us to study the in situ distribution of orientations of particles, and their evolution during during rolling [538], [540]. 3. Bivariate distribution of curvatures concerns properties of the surface ∂A of the object: local radii of curvature introduced in section 3.3.1, as done to study the evolution of dendritic interfaces during solidification [9]. On each point x of a smooth surface ∂A is considered the → → normal vector − n . Each plane Πi containing − n intersects ∂A according to a curve Ci with radius of curvature Ri (x), and with curvature ki (x) = Ri1(x) . The minimum Rmin and maximum Rmax radii of curvature are obtained by rotation of Πi for two orthogonal planar sections. They give the principal curvatures in x, kmin (x) and kmax (x), character-
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FIGURE 3.31. Domain of 3D shapes in the plane (λ1 , λ2 )
izing the local shape of the interface. Points x where kmin (x) = kmax (x) correspond to a locally spherical interface; if kmin (x) = kmax (x) = 0, the interface is flat; if one of the curvatures is zero, the interface is cylindrical; if the two curvatures have opposite signs, point x belongs to a saddle point. The portions of the surface with positive curvatures (resp. negative) are concave (resp. convex). One can represent the cloud of points x of ∂A in the plane diagram (kmin (x), kmax (x)) [9]. In the present case, object A (or its surface ∂A) is described by the bivariate distribution of (kmin (x), kmax (x)), after discretization of each curvature in classes. This approach requires a surface meshing of particles and an implementation of the "marching cubes" algorithm [426], as detailed in [540], [541]. A classification of complex particles is looked for, based on previous measured parameters. Using a principal components analysis (PCA) to classify particles according to their volume, geodesic parameters, and moments of inertia, a good representation of data is obtained in a 3D factor space to provide an unsupervised classification in five major types of objects, as illustrated in Figs 3.32, 3.33, 3.34, 3.35, 3.36, showing representative elements of the complex particles classified from volumic morphological criteria [540], [542]: i) quasi spherical convex particles (b); ii) elongated (Ig ' 55), with a large and λ1 > 0.4 (a, e): iii) thread like (c); iv) prolate ellipsoids; v) flat (d). Each class contains 10% -30% of particles in the alloy after a weak deformation. The evolution of particles during hot rolling was studied by projecting on the initial factor space particles after hot rolling deformation [540], [542]. Complex shapes particles (ii, iii and v)) progressively disappear, remaining particles showing simpler shapes (i and iv). This evolution is explained and confirmed by a micromechanical study of the sensitivity of particle shape to their fragmentation capability [505], [506].
3.8 Shape and texture classification and recognition
FIGURE 3.32. (a)
FIGURE 3.33. (b)
FIGURE 3.34. (c)
FIGURE 3.35. (d)
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FIGURE 3.36. (e)
A second classification is deduced from bivariate distributions of the two curvatures in each point x of ∂A, studied by correspondence analysis and hierarchical automatic classification [541]. Five sub-populations emerge, with some overlaps with the previous classification: i) branched; ii), flat; iii) flat and concave; iv) slender; v) spherical interface. In a last step a probabilistic model of complex particles was proposed and tested [539]. The solid phase S of alloys is represented by a Boolean model of spheres (chapter 6) with a distribution of radii estimated from the experimental covariance C(h) as explained in section 6.10.1. With this model, the in situ particles are the connected components of S c , which remain bounded due to the very low volume fraction of S c (typically 0.0055). A good agreement between the real and the simulated microstructure is obtained. However, the variability of shapes of simulated particles is lower than real ones. Furthermore, the observed variability of curvatures in the diagram (kmin (x), kmax (x)) cannot be reproduced, since for the major part of points x ∈ ∂A in simulations, by construction kmin (x) = kmax (x).
3.8.3 Texture analysis and classification A non periodic texture can be represented as a realization of a random structure, characterized by morphological parameters as introduced in this chapter. A lot of applications of texture classification exist in the literature, with a particular renewal, due to the explosion of machine learning tools. We focus here on early examples, with industrial purposes like quality control (even on-line), mainly in the area of rough surfaces. 1. Texture classification can be made from global measurements when working at a small scale, so that a single texture is contained in each window X. This approach was developed for mineralogical textures in iron ore sinters [325], using unsupervised learning from data: on 400 fields of measurements with dimensions 350 × 290μm2 were measured basic morphological parameters (surface area, perimeters, and number of intercepts in the three main directions of the hexagonal grid) for the five phases of a sinter material, building a vector of description of dimension 25 for each field. By means of correspondence analysis a classification of 7 mineralogical facies was obtained from the factor plane 2-3 (the first
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FIGURE 3.37. Map of the analyzed zone, each field of measurement being represented by its mineralogical texture
factor carrying mainly the information on pore volume fraction). A map of the facies located on the polished section is obtained (Fig. 3.37), where large scale correlations between fields appear (for instance macropores), that can be subject to a further morphological study at a mesoscale. 2. More information is used, like morphological transformations (erosion, dilation, opening, closing curves for structuring elements with increasing sizes) applied to binary or grey level images: realizations of Markov random fields are used for texture classification in [606]. In [28] the same information is used to classify by means of correspondence analysis rough steel plates from images of their topography obtained by mechanical probing. It is also applied to simulations of models of random surfaces (truncated Gaussian to account for abrasion of peaks (chapter 4), Dead Leaves RF (chapter 11), sequential alternate RF (chapter 13)), in order to detect the morphological proximity between real and simulated surfaces and to enlighten the choice of the most suitable model. A similar approach is followed for the classification of the quality of the visual aspect of samples of plastering mortars [29] by means of a supervised learning from morphological measurements (speed of erosion and dilation, size distribution, watershed sizing for peaks and valleys). A satisfactory classification was obtained in the first factor plane provided by a correspondence analysis. The visual aspect of rough surfaces such as steel surfaces becomes of great importance for assessing the quality of the final product they are dedicated to, like for instance the visual aspect of pre-painted products [187], [188], [189]. The industrial motivation of such studies is to be able to provide
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an evaluation of quality as close as possible as the product output, both to be able to control the process and to avoid to deliver unacceptable products to the customer. To replace the manual evaluation of a quality index given by experts of visual inspection in the case of orange peel surfaces, a full automatic classification methodology was developed. In a first step, topographical images obtained by interferometric microscopy were introduced in a supervised learning procedure on a training image data base, involving measurement of previous morphological information, completed by the estimated distribution of tortuosity of grey level images (using a distance based on the absolute difference of elevations between adjacent pixels, and a normalization) as described in subsection 3.6. Multivariate factor analysis is implemented from training data, while the classification of unknown surfaces, including a test data set, is made by a Bayesian classification in the generated low dimension space. In a second step, topographic information was replaced by high definition camera images at the laboratory scale, and was then successfully implemented on-line on a rolling mill in Montataire plant. 3. A classification of each pixel of an image is required to obtain an automatic segmentation of textures contained in an image [122], or for instance to extract defects that might even fill a very small part of the analyzed area [123]. As opposite to the previous approach, based on global measurements in images, local information is now required for each pixel. A first step is the construction of multispectral images from a family of transformed images, so that a vector of description is available for each pixel. The local approach of texture can be made in different ways [122]: a. Transform the image by dilatations, erosions, openings, closings by Ki , gi ( "pixel" approach), and generation of a multispectral image from the collection of Ki or gi . It corresponds to filters, like granulometries. Here are stored the increments of morphological transformations between two size steps b. Consider a neighborhood B(x) of each pixel, and use a local estimate of T (K), T (g) inside B(x). From the estimates, generate a multispectral image from the collection of Ki or gi . Various sizes (up to 60 × 60 windows) were tested. The optimal sizes giving the lowest error rate when testing the learning procedure is retained. c. Use measures μi with a compact support Ki and estimate μi (A) or μi (Z), generating a multispectral image from the collection of Ki . As particular cases are recovered various types of linear filters, like multi-scale convolution by Gaussian kernels, wavelets, curvelets, but also local measurements of the Minkowski functionals for a random set A. i. This method was tested on patchworks made of realizations of various random sets (Boolean, Dead Leaves, Poisson mosaic) and ran-
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dom function (Boolean, Dead Leaves, Sequential Alternate) models. A supervised learning procedure based on linear discriminant analysis is applied on a set of 10000 randomly chosen pixels (instead of images in previous examples based on global measurements) to generate a learning data set. The same number of pixels is used for test. The best results were obtained on the generated multispectral gray level images, and on T (K) estimated in 35 × 35 windows for binary images. A correct classification of test pixels in the range 82−90% is obtained, misclassified pixels being mainly located close to the boundaries between different textures. ii. A second application concerns the automatic extraction of textures and surface defects on rough metallic surfaces [123]. Here very thin defects are drowned in strong textures, which makes them very difficult to extract, even by and advised observer. For this application, a database containing 800 images of scraping steel surface factory is used. These images (240 × 296 pixels, with a 1mm scale resolution) contain different defects on a non uniform background showing fluctuations of texture. Each image is characterized at the scale of pixels by 56 texture descriptors (30 morphological granulometries and 26 curvelets linear filtering [104]). Among the 800 images, 50 of them were randomly picked for the training of the system. The remaining images are used to validate the results. On these 50 images, 100 000 pixels belonging to the defect class and 100 000 pixels outside, corresponding to the background were randomly selected. The statistical supervised machine learning approach used in this study is decomposed in three steps: α) a dimensional reduction of the set of independent variables by principal component analysis; β) a supervised learning using linear discriminant analysis: γ) a variable selection based on forward selection. It appears in the present application that curvelet linear filters are the best descriptors for the classification task. Indeed, the curvelets (with both low and high frequency filters) are the 3 first selected variables and they represent 13 of the 20 selected variables. However the curvelets are not sufficient by themselves (keeping only theses descriptors leads to an error of 11.4 %). It turns out that the morphological granulometry with various structuring elements cooperates well with the curvelets. Concerning the results of pixels classification, a sensitivity of 99.5 %, a negative predictive value of 97 % and an accuracy of 98.5 % were reached. It shows that defects are well located by the approach, whatever their shape and structure. In a next step, the approach was optimized to make feasible a real-time on-line morphological image processing for an automatic extraction of learned defects.
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3.9 Conclusion Morphological characterization of a microstructure involves a large number of parameters, according to various criteria in order to get pertinent and significant information with regards to the application in mind. Every type of measurement covers a specific aspect of the structure. With respect to the large available amount of information, various strategies can be developed: • A possible reduction of data can be obtained by means of multivariate data analysis, as was seen in examples about shape classification or texture learning. • A synthesis of morphological information can be looked for from probabilistic models of random media, as discussed in Part II. • Use can be made of these morphological measurements to predict the macroscopic behavior of materials through models of change of scale models, as detailed in Part III.
3.10 Exercises 3.10.1 Minkowski tensor of a cylinder Using cylindrical coordinates (with unit vector ez along Oz), compute the Minkowski tensor W10,2 of a cylinder with radius R and height L [170] (subsection 3.3.6). Answer: Z 1 W10,2 (A) = n ⊗ ndS 3 ∂A ¶ µ 2 2 L 0,2 W1 (A) = πR ez ⊗ ez + (Id − ez ⊗ ez ) 3 2R
3.10.2 Variogram γ 1 (h) Give the expression of the first order variogram as a function of the bivariate distribution T (h, z1 , z2 ) = P {Z(x) ≥ z1 , Z(x + h) ≥ z2 }. Answer: We have |Z(x + h) − Z(x)| = Z(x+h)∨Z(x)−Z(x+h)∧Z(x) and P {Z(x + h) ≥ z, Z(x) ≥ z} = T2 (h, z, z). On the other hand, P {Z(x + h) < z, Z(x) < z} = P {x + h ∈ AZ (z)c , x ∈ AZ (z)c } = 1 − 2P {x ∈ AZ (z)} + P {x ∈ AZ (z), x + h ∈ AZ (z)} Therefore
3.10 Exercises
E {|Z(x + h) − Z(x)|} = 2
Z
T (z)dz − 2
Z
107
T2 (h, z, z)dz
3.10.3 Correlation functions of two phase random composites Let A be a stationary random set with a physical property Z: Z = Z1 if x ∈ A and Z = Z2 if x ∈ Ac . Express the central correlation functions of order m W m of the random function Z(x) as a function of the corresponding moments for the random set A. Answer: We have Z(x) = k(x)Z1 +(1−k(x))Z2 with k(x) = 1 for x ∈ A, else k(x) = 0. Therefore, E{Z(x)} = pZ1 + (1 − p)Z2 with p = E{k(x)}. It comes Z(x) − E{Z(x)} = (Z1 − Z2 )(k(x) − p) and W m (h1 , h2 , ..., hm−1 ) = (Z1 − Z2 )m E{(k(x) − p)(k(x + h1 ) − p)...(k(x + hm−1 ) − p)} = (Z2 − Z1 )m E{(k 0 (x) − q)(k 0 (x + h1 ) − q)...(k 0 (x + hm−1 ) − q)} (3.109) with the indicator function of the set Ac , k0 (x) = 1 − k(x) and q = 1 − p. It is clear from Eq. (3.109) that interchanging the properties Z1 and Z2 leave the even order moments unchanged, and changes the sign of the odd order moments. Consequently an autodual random set A owns zero central moments of odd order. For instance, the second order central correlation function W 2 (h) is given by W 2 (h) = (Z1 − Z2 )2 (Q(h) − q 2 ) = (Z1 − Z2 )2 (C(h) − p2 )
(3.110)
with C(h) = P {x ∈ A ∩ A−h }. W 2 (0) = (Z1 − Z2 )2 p(1 − p) and therefore W 2 (0) > 0 for a non trivial random set. The third order central correlation function is W 3 (h1 , h2 ) = (Z1 − Z2 )3 (Q(h1 , h2 ) − q(Q(h1 ) + Q(h2 ) + Q(h3 )) + 2q 3 ) = (Z1 − Z2 )3 (P (h1 , h2 ) − p(C(h1 ) + C(h2 ) + C(h3 )) + 2p3 ) (3.111) with Q(h1 , h2 ) = P {x ∈ Ac ∩ Ac−h1 ∩ Ac−h2 } and P (h1 , h2 ) = P {x ∈ A ∩ A−h1 ∩ A−h2 }. For h1 = h2 = 0, we have W 3 (0) = (Z1 − Z2 )3 p(1 − 1 p)(1 − 2p) , so that W 3 (0) = 0 for p = . As a consequence, we cannot 2 1 have W 3 (h1 , h2 ) = 0 for all couples (h1 , h2 ) when p 6= . 2
3.10.4 Specific measurements and derivatives of morphological
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moments Show that estimations of specific measurements obtained by counting IA , NA , NV − GV are related to derivative of morphological moments. Answer: consider IA obtained from the measurement N (01). We have NA (01) = C(a)−C(0), a being the distance separating two adjacent pixels. Therefore from a first order Taylor expansion we have NA (01) ' −aC 0 (0) and specific number of intercepts are related to the slope of the covariance at the origin. In the presence of noise, NA (01) → ∞ (section 6.15.4 in chapter 6). An estimation of the connectivity number of an isotropic random set in 2D, using Euler relation in R2 for the graph C4 , is given by a2 NA (A) = P (0, 0) − P (0, a) − P (a, 0) + P (a, a), where P (h1 , h2 ) = P {x ∈ A, x + h1 ∈ A, x + h2 ∈ A} is the three-points probability function, or third order moment (section 3.3.5). Using a Taylor expansion ∂ ∂ P (0, 0) + a P (a, 0) ∂y ∂y 2 ∂ P (0, 0) ' a2 ∂y∂x
a2 NA (A) ' −a
2
∂ so that NA (A) ' ∂y∂x P (0, 0). The estimator of the connectivity number makes use of the second order derivative of the third moment . As a consequence in the present of noise NA (A) → ∞ (section 3.7). Similarly for the connectivity number in R3 , using the Euler relation and the graph C6 ,
a3 (NV − GV ) = P (0, 0, 0) − P (a, 0, 0) − P (0, a, 0) − P (0, 0, a) +P (a, a, 0) + P (a, 0, a) + P (0, a, a) − P (a, a, a) where is introduced the fourth order moment P ((h1 , h2 , h3 ) = P {x ∈ A, x + h1 ∈ A, x + h2 ∈ A, x + h3 ∈ A} By means of a Taylor expansion, and after some calculation we get a3 (NV − GV ) ' a3
∂3 P (0, 0, 0) ∂x∂y∂z
so that the estimated connectivity number is expressed as the third order derivative of the fourth order moment.
3.10 Exercises
109
3.10.5 Covariance of a noisy binary image Starting from a random set A, we modify A by means of two binary noisy images N1 and N2 . We define AN a noisy version image of A by AN = (A ∪ N1 ) ∩ N2c where the sets A, N1 and N2 are assumed to be independent. Give the covariance CN (h) of AN as a function of the covariances of A, C(h), and of N1 and N2 , C1 (h) and C2 (h). Answer: We have CN (h) = CA∪N1 (h)Q2 (h), denoting by Q2 (h) the covariance of N2c = N2c . QA∪N1 (h) = Q(h)Q1 (h) and CA∪N1 (h) = 1 − 2qq1 + Q(h)Q1 (h) with Q(0) = q = 1 − p and Q1 (0) = q1 . Therefore CN (h) = Q2 (h)(1 − 2qq1 ) + Q(h)Q1 (h)Q2 (h) For h = 0 we get CN (0) = q2 (1 − 2qq1 ) + qq1 q2 = q2 (1 − qq1 ) For h 6= 0, Q2 (h) = q22 and Q1 (h) = q12 (see section 6.15.4 in chapter 6) and therefore, CN (h) = q22 (1 − 2qq1 ) + q12 q22 (1 − 2p + C(h)) The observed covariance CN (h) is an affine function of the underlying covariance C(h). This is illustrated by Fig. in section 3.7.
3.10.6 Moments of inertia in R3 Define I1 , I2 and I3 the 3 principal moments of inertia of a solid body in R3 . Using as coordinates axis the principal directions of the inertia tensor of a solid body with unit mass density, Z Z Z I1 = (x22 + x23 )dx1 dx2 dx3 Z Z Z I2 = (x21 + x23 )dx1 dx2 dx3 Z Z Z I3 = (x21 + x22 )dx1 dx2 dx3
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3. Quantitative Analysis of Random Structures
Give inequalities followed by principal moments of inertia. Answer: From the definition of I1 , I2 , I3 , Z Z Z I1 + I2 = (x22 + x23 + x21 + x23 )dx1 dx2 dx3 Z Z Z ≥ (x22 + x21 )dx1 dx2 dx3 = I3 The normalized moments λi = Ii /(I1 + I2 + I3 ) satisfy λ1 + λ2 ≥ λ3 Then λ1 + λ2 ≥ 1 − (λ1 + λ2 ) and λ1 + λ2 ≥ or 1 − λ3 ≥
1 2
1 1 =⇒ λ3 ≤ 2 2
After permutation of the indexes, we obtain λi ≤ 12 , and this inequality is followed by the largest normalized principal component. Using λ1 > λ2 > λ3 , we have 1 = λ1 + λ2 + λ3 ≤ λ1 + 2λ2 and then λ2 > 0.5(1 − λ1 ).
Part II Models of Random Structures
4 Excursion Sets of Gaussian RF
Abstract: Random sets models derived from truncated RF, with an application to Gaussian random functions, are introduced. Covariance and order m central correlation functions are given. The model is illustrated by examples of application.
4.1 Introduction Probably the most simple way to generate a model of random set is to start from some model of random function, and to apply some threshold to its realizations. A good and parsimonious candidate is the Gaussian RF, which is completely known from its second order correlation function. It has been used for a long time, mainly in geological applications. Indeed one of the strong points of this class of models is to open the way to a simple implementation of conditional simulations, in order to generate realizations of a random set which respect the information available on data points [461], [408].
4.2 Random sets and truncated RF: the Gaussian case. It is possible to generate random sets by truncating a random function, which generates a so-called "excursion set". For instance, AZ (z) is a ran© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Jeulin, Morphological Models of Random Structures, Interdisciplinary Applied Mathematics 53, https://doi.org/10.1007/978-3-030-75452-5_4
113
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4. Excursion Sets of Gaussian RF
dom closed set if the RF Z(x) is upper semi continuous. The probabilistic properties of AZ (z) are deduced from the properties of Z(x) : for any compact set K we have T (K) = 1 − P {Z∨ (K) < z}
(4.1)
and for a compact set K made of a finite number of points K = {x1 , ...., xm }, Eq. (4.1) gives the spatial law of AZ (z) : P {x1 ∈ AZ (z), ..., xm ∈ AZ (z)} = P (x1 , ..., xm , z) = P {Z(x1 ) ≥ z, ..., Z(xm ) ≥ z} P {x1 ∈ (AZ (z))c , ..., xm ∈ (AZ (z))c } = Q(x1 , ..., xm , z) = P {Z(x1 ) < z, ..., Z(xm ) < z}
(4.2)
For illustration, consider now the case of random sets generated by truncation of a Gaussian RF Z(x), as studied by G. Matheron [455],[461], [408], which we follow in this presentation. These are widely used for simulations of random sets [4], [407], with facilities concerning conditional simulations, as developed in the case of oil reservoir simulations [467]. Many publications concern the geometry of excursion sets derived from Gaussian RF (among others [5], [31]), mostly concentrated on extremal values, and on some integral geometry properties like the Euler characteristics, or connectivity number (Chapter 3). Applications to the simulation of foam microstructures are given in [572], [573]. Since for most Gaussian RF the distribution Z∨ (K) is not known in the general case of any compact set K, the Choquet capacity of excursion random sets derived from Gaussian RF is usually not available from a theoretical point of view, but can be estimated on real or simulated images. However, some probabilistic properties of these random sets, like their order m central correlation functions can be calculated as expansions in series of Hermite polynomials, as presented below.
4.2.1 Construction of truncated Gaussian random sets Starting from a stationary standard Gaussian RF Z(x) (with null expectation, unit variance, and covariance (or correlation function) ρ(h)), various stationary random sets can be defined by means of thresholds zi : • From a single threshold z, define the set AZ (z) with indicator function k(x) = 1Z(x)≥z (1Z(x)≥z = 1 when Z(x) ≥ 1) and its complementary set (AZ (z))c with the indicator function 1Z(x) 1, else dF = 2 log α
Continuous scaling A continuous version, generating a sequential Boolean mode, is obtained using θu = θ0 u−μ with u1 ≤ u ≤ 1 and ru = uν r0 i) Give the total intensity for u1 ≤ u ≤ 1. ii) Give the average surface area and the average volume of the grains. iii) What is the fractal dimension of this model, estimated as for the discrete scales, imposing ε = uν1 r0 ? Answer: i) Z 1 u1−μ − 1 θ= θu du = θ0 1 1−μ u1 ii) Grains A0i have for surface area Su = u2ν S0 and for volume Vu = u3ν V0 . The average surface area and volumes of grains are given by: E{S} = E{V } =
1 − μ u2ν−μ+1 −1 1 S0 1−μ u1 − 1 μ − 2ν − 1 1 − μ u3ν−μ+1 −1 1 V0 1−μ u1 − 1 μ − 3ν − 1
220
6. Boolean Random Sets
iii) It is required that limu1 →0 SV = ∞ with 0 < VV < 1 so that 2ν + 1 ≤ μ < 3ν + 1 Imposing ε = uν1 r0 μ−1 dF = ν
6.15.4 A noise random set model A sequence of random sets An in R3 is obtained by Boolean models of spheres A0n with diameter an and intensity θn satisfying π6 θn a3n = θ. We use a sequence An with limn→∞ (θn ) → ∞ and limn→∞ (an ) → 0. Give its covariance, its moments Q(r), Q(l) and its surface area. What is the limit of An when n → ∞? Answer: The Choquet capacity of An is given by Qn (K) = exp (−θn V (A0n ⊕ K)) and therefore qn = exp (−θn V (A0n )) = exp −θ = q Qn (h) = q 2−rn (h) , with rn (h) = Kn (h)/Kn (0) 3 1 r Q (r) = q ( 2 + an ) n
(6.65)
0 1+lrn (0)
Qn (l) = q SV n = qθn S (A0n ) = qθn a2n The limits of results (6.65) are given by q = exp −θ = qn r(h) = 0 for h 6= 0, else 1; r0 (0) = −∞ Q(h) = q 2 for h 6= 0, else q Q(r) = 0 for r 6= 0, else q Q(l) = 0 for l 6= 0, else q SV = +∞
(6.66)
The functions Q(h), Q(r), and Q(l) are discontinuous at the origin, which cannot be the case for a RACS. In fact the limit case A∞ is a binary noise, which is not closed, but with dense realizations (their topological closure is the full space R3 ) almost surely.
6.15.5 Simulation of percolating aggregates The method described here [346], [347], [349], [350] generates low cost and accurate simulations of multi scale aggregates of grains with a finite diam-
6.15 Exercises
221
eter without representation on a voxel grid. We restrict here to spherical grains. Two properties of the Poisson point process are used (for a three-scale model, three processes are used, with respective intensities θ1 , θ2 , θ3 ): • The numbers of points N1 and N2 contained in disjoint volumes V1 and V2 are independent variables • The points falling in V are uniformly distributed in V . For the simulation, information is stored concerning each sphere: coordinates (x, y, z) of the centre M , radius of the sphere (constant in the present simulation), and a label corresponding to the current aggregates number. Three types of spheres are used for for multi-scale simulations: individual spheres (the initial grains of the material) S1 with a constant radius R1 and with the centre M1 (x1 , y1 , z1 ); inclusion spheres S2 with a constant radius R2 and with the centre M2 (x2 , y2 , z2 ), and finally exclusion spheres S3 with a constant radius R3 and with centre M3 (x3 , y3 , z3 ). Zones of permitted location for M1 are inclusion spheres. Zones non permitted for M1 are the exclusion spheres. The one scale model is the standard Boolean model of spheres. The full volume V is subdivided into cubes V1i with edge D1 = 2 × R1 . They are scanned in a given order, for instance along axis Ox, Oy, and then axis Oz. In each volume V1i is generated the random number N1i of centres of spheres S1 located according to a Poisson point process with intensity θ1 . For each new centre of sphere S1i , uniformly distributed in a V1i , is performed: • Determination of the 13 adjacent volumes V1j of V1i • For all spheres S1i contained in these volumes, test of intersection given by Eq. (6.67) • If the test is negative, a new label is created for S1i , to start a new aggregate. Else, S1i belongs to the aggregate Agj of S1j and receives the label of Agj . If S1i already belongs to an existing aggregate, these two aggregates are merged and the concerned labels are updated. When all domains V1j were explored, the simulation is over. It is then easy to extract individual 3D aggregates from their labels, for a further analysis. For a two-scale model is just stored the centre M1 of a S1 if it is included in a sphere S2 . The simulation is built in two steps: • V is subdivided into cubes V2i with edge D2 = 2R2 , explored in a given order. In each V2i is generated a realization of a Poisson point process with intensity θ2 . • For the scale S1 , the previous one-scale procedure is applied and for each new sphere S1i , with a uniformly distributed centre in V1i , a new condition is added to Eq. (6.67). First the volume V2 to which S1i belongs, and its 26 adjacent cubic volumes, is determined; then for all S1j is performed the inclusion test given below by Eq. (6.68).
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6. Boolean Random Sets
A three-scale model is obtained by keeping only spheres with radius R1 and with a centre located outside of a Boolean model with large spheres S3 j with radius R3 (we have R3 > R2 > R1 ). The simulation is now obtained in three steps: • V is subdivided into cubes V3i with edge D3 = 2R3 explored in a given order. In each V3i is generated a realization of a Poisson point process with the intensity θ3 . • V is subdivided into cubes V2i with edge D2 = 2R2 explored in a given order. In each V2i is generated a realization of a Poisson point process with the intensity θ2 . • For S1 , the two scale simulation is performed, and for each new sphere S1i , uniformly distributed in a V1i , a new condition is added to Eqs (6.67) and (6.68). The volume V3 in which S1i is located, and its 26 adjacent volumes, is defined ; then for all S3 j is performed the exclusion test given in Eq. (6.69). If the result of the test is negative, S1i is deleted and the next S1i is examined. The tests of intersection (of exclusion or of inclusion) make use of the distance between grains and their geometrical properties. For instance, for two spheres S1 , S2 , with radii R1 , R2 , and centers with coordinates (x1 , y1 , z1 ) and (x2 , y2 , z2 ) we define: • Intersection test: the two spheres are in contact if the following inequality is satisfied: (x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2 6 (R1 + R2 )2 •
Inclusion test : the center of S1 is inside S2 when: (x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2 6 (R2 )2
•
(6.67)
(6.68)
Exclusion test: the center of S1 is outside S2 if: (x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2 > (R2 )2
(6.69)
Similar inequalities, but slightly more complex, exist when working on sphero-cylinders [347]. From the simulations is provided an estimation of the percolation threshold. A simulation percolates when there is at least on aggregate of objects connecting two opposite faces in a cubic domain. To know if a simulation percolates according to the z axis for objects of the smallest scale, we have to know the labels of objects belonging to cubes of the first level along axis z. These have to be compared to the labels of objects belonging to cubes of the last level along axis z. If they differ, the simulation does not percolate. Making this test to each change of level along axis z allows us to quickly stop a simulation that will not percolate.
6.15 Exercises
223
From multiple realizations of 3D simulations, the percolation threshold is by convention obtained when at least 50% of realizations percolate for a given volume fraction. A dichotomic search on volume fraction is implemented.
6.15.6 Transverse and longitudinal covariances of a Boolean model of cylinders Consider a Boolean model of random cylinders with a transverse anisotropy, to represent overlapping fibres [547]. The population of cylinders has a random radius R and a random length L. R and L are independent RV. Fibres are orthogonal to the Oz axis, with a uniform distribution of orientations in the xOy plane. Knowing the distributions f1 (r) and f2 (l) for R and for L, compute the theoretical covariance in the transverse direction (along Oz), or in any horizontal direction in the xOy planes (longitudinal covariance). Answer: The covariance of the pores is given as a function of the geometrical covariogram of cylinders, and depends on the direction of vector h: Q(h) = q 2 eθ(K(h)) = q 2−r(h) (6.70) Transverse covariance of pores, QZ (h). Consider h parallel to axis Oz. From Eq. (6.70), it is expressed as a function of the transverse reduced covariogram of the cylindrical fibre, rZCylinder (h), as a function of the average length Lf ibres and of the average geometrical covariogram of a population of disks with a random radius R following the distribution f1 (r) by KZCylinder (h) = Lf ibres .KZDisc (h) Therefore, rZCylinder (h) =
KZDisc (h) KZDisc (0)
(6.71)
The normalized transverse geometrical covariogram of the fibres does not depend on their lengths, as a result of the independence between R and L. In Eq. (6.71), KZDisc (h) is deduced from the geometrical covariogram of a disc KZDisc (h, r) and from f1 (r). We have, if 0 ≤ h ≤ 2.r (if h > 2.r, KZDisc (h, r) = 0): ⎛
and
KZDisc (h, r) = 2r2 ⎝arccos
µ
h 2r
¶
⎞ s µ ¶2 h ⎠ h − . 1− 2r 2r
(6.72)
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6. Boolean Random Sets
KZDisc (h) =
Z
+∞ h 2
KZDisc (h, r)f1 (r)dr
(6.73)
Longitudinal covariance of pores QxOy (h). Given h in the horizontal xOy plane (for instance parallel to the Ox axis), the average longitudinal geometrical covariogram of the fibre KxOyCyl (h) is given by Eq. (6.75). The geometrical covariogram of a cylinder with length l and radius r is given by Eq. (6.74), for hX ≤ l (for hX > l, KxOyCylinder (h, θ) = 0). KxOyCylinder (hX , hY , l) = (l − hX )KZDisc (hY )
(6.74)
Considering hX = h cos θ, hY = h sin θ and the uniform distribution of the orientations θ of the fibres, KxOyCyl (h) (6.75) Z +∞ Z π Z +∞ 1 = (l − h cos θ)f2 (l)dldθ KZDisc (h, r)f1 (r)dr h.sinθ π 0 h.cosθ 2 and rxOyCylinder (h) =
KxOyCyl (h) KxOyCyl (0)
(6.76)
6.15.7 Boolean varieties in R2 and R3 as limit case of Boolean models Consider Boolean varieties in R2 and R3 as limit case of Boolean models [313]. Show that with an appropriate choice of the intensity of Boolean models with segment primary grains in R2 and with cylinder primary grains in R3 , Poisson varieties are recovered. From this approach, give the connectivity numbers of Boolean varieties in R3 and the covariance of Boolean varieties in R2 and R3 . Answer: Convergence of Boolean models towards Poisson varieties In R2 , start from a Boolean model of segments with length l, a uniform distribution of orientations, and an intensity θ. We study the limit where θ → 0, l → ∞ with θl → θ1 finite. For any compact set K, Q(K) = exp(−θA(l ⊕ K)) For K = C(r), (disk with radius r)
6.15 Exercises
Q(r) = exp(−θ(lr + πr2 ) = exp(−θl(r +
225
πr2 )) l
and Q(r) →l→∞ exp(−θ1 r) For any convex set K, Q(K) = exp(−θA(K) + and
l L(K)) 2π
¶ µ L(K) Q(K) → exp −θ1 2π
θ1 The same result as for Poisson lines is obtained, using 2π = λ (see chapter 7). To compute the covariance of segments after dilation by K, it is necessary to compute the reduced geometrical covariogram of the projection of K in each direction of lines. In R3 , start from a Boolean model of cylinders Cy (r, l) (with radius r and height l) with uniform orientations in R3 , with intensity θ. Then,
q = exp(−θπr2 l) For any compact set K Q(K) = exp(−θV (Cy (r, l) ⊕ K)) and for any convex set K ¡ ¢ M (K) M Q(K) = exp −θ(πr2 l + (2πr2 + 2πrl +S(K) +V (K))) (6.77) 4π 4π
Fix now l and let r → ∞ with θr2 → θ1 . The model converges towards Poisson planes with thickness l. We get q = exp(−θ1 πl) (and therefore θ = 12 θ1 ), and for convex sets K, Q(K) = q exp(−θ1 M (K)/2)
If r is fixed and if l → ∞ with θl → θ1 we get isotropic Poisson fibres with radius r, with q = exp(−θ1 πr2 ) (and therefore θ = 2θπ1 ). Starting from a segment with length l dilated by the compact convex set K, V (K ⊕ l) = V (K) + l
S(K) 4
and lim (θlV (K ⊕ l)/l) = θ1
l→∞
S(K) 4
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6. Boolean Random Sets
so that Q(K) = q exp(−θ1 S(K)/4) Results given here for the Choquet capacity are limited to convex sets K. For any other set K, we have to consider µ ¶ V (Cy (r, l) ⊕ K) lim r→∞ r2 and lim
l→∞
µ
V (Cy (r, l) ⊕ K) l
¶
Connectivity number of Boolean varieties in R3 Start from the Miles formula valid for isotropic Boolean models with convex grains µ ¶ π 3 S 3 2 MS NV − GV = q θ − θ + θ ( ) 4π 6 4 For a cylinder Cy (r, l), V = πr2 l, S = 2πr(l + r), M = π(l + πr) Considering θ → 0, l → ∞ and θl → θ1 for Poisson fibres, the connectivity number of the complementary set is given by: µ ¶ π 2πr 3 l2 NV − GV ' q −θ2 2π 2 r + θ3 l3 ( ) 4π 6 4 µ 2 ¶ θ1 2 π 3 π 3 r3 NV − GV → q − 2π r + θ1 4π 6 8 µ ¶ µ ¶ 4 1 π π π3 = qr −θ21 + θ31 r2 = qπrθ21 − + θ1 r2 2 48 2 48 We have NV −GV = 0 for θ1 = 0, giving q = 1 and p = 0 (fibres percolate according to the heuristic on the percolation threshold), and also for θ1
24 π3 2 1 r = , or θ1 = 3 2 48 2 π r
giving 24 ) = 8. 788 7 × 10−2 π2 According to this result, the percolation threshold of the complementary set of fibres: is given by 8. 788 7 × 10−2 , which is higher than for the complementary set of of a Boolean model of spheres. q = exp(−θ1 πr2 ) = exp(−
6.15 Exercises
227
For the dilated Poisson planes, use θ → 0, r → ∞ and θr2 → θ2 . Then µ ¶ π 3 2πr2 3 2 1 2 2 NV − GV ' q −θ 2πr π r + θ ( ) 4π 6 4 µ ¶ 2 π 3 π3 π π3 2π + θ2 = q θ32 NV − GV → q −θ2 2r 6 8 6 8 As a result NV −GV = 0 for θ2 = 0, or q = 1 (the thick planes percolate) and for q = 0 (meaning that the complementary set of thick planes does not percolate, which is expected). For these two examples (Poisson fibres and thick Poisson planes), results on percolation deduced from zeros of the connectivity number are faithful to what is expected, which validates this heuristic approach in the present case. Covariance of Poisson fibres in the plane Consider Poisson lines dilated by the disk with radius r. Then, q = exp(−θK(0)) = exp(−2πθr) As a consequence, the average intercept of the primary grain induced on 1D sections is given by K(0) = 2πr. For a line orthogonal to direction α, if h ≤ 2r →→ − K(α, h .− u (α)) = 2r − h cos α After a uniform rotation of lines, K(h) = 2πr − 2h
Z
0
π 2
cos αdα = 2πr − 2h
and
h πr For Poisson fibres A with radius r dilated by the segment l, r(h) = 1 −
Q(l) = exp(−θL(A0 ⊕ l) = exp(−θ(2l + 2πr)) Then Q0 (l) = −2θQ(l) and Q0 (0) = −2θq
The average intercept of Ac is given by
−Q0 (0) 1 = q 2θ For checking, the average intercept can be deduced from the covariance, given by
228
6. Boolean Random Sets
Q(h) = q 2 exp(θK(h)) and Q0 (0) = θK 0 (0)q q 1 1 = = −Q0 (0) −θK 0 (0) 2θ Then for h ≤ 2r the covariance of Poisson fibres A with radius r dilated by the segment l is given by h
Q(h) = exp(−(2πθr + 2h) = q (1+ πr ) 2
Q(2r) = q 1+ π . For h ≥ 2r, K(α, h) = 2r − h cos α if 2r ≥ h cos α, else = 0 We have K(α, h) 6= 0 for cos α ≤
2r h,
so that arccos( 2r h)≤α≤
π 2.
Then
¶ Z π2 2r π cos αdα − arccos( ) − 2h 2 h arccos( 2r h ) ⎞ ⎛ s µ µ ¶2 ¶ 2 2r 2r ⎠ = 2πr 1 − arccos( ) − 2h ⎝1 − 1 − π h h
K(h) = 4r
µ
with K(2r) = 2πr − 4r. For h ≥ 2r
⎞ ⎛ s µ ¶2 2 2r ⎠ 2r h ⎝ r(h) = 1 − arccos( ) − 1− 1− π h πr h
When h → ∞, r(h) → 0 and admits an expansion in 2r h 2 π r(h) ' 1 − ( − arcsin( )) − π 2 h πr 2r h 2r2 2 2r ' − ' 2 π h πr h πh
1 h:
¶ µ 2r2 1 − (1 − 2 ) h
Covariance of Poisson fibres in R3 For Poisson fibres with radius r in R3 , select a vector h parallel to axis Oz, and consider the angle α between h and the normal to the plane orthogonal to Poisson lines. When α = 0, it is a horizontal. plane, and the projection of vector h on this plane is the null vector (cos ϕ = 0 and so ϕ = π2 ). For α = π2 , it is a vertical plane on which h is projected with modulus h (cos ϕ = 0 and so ϕ = 0). It comes:
6.15 Exercises
229
³ π ´ q = exp −θ 4πr2 2
and for a convex set K,
³ π ´ Q(K) = exp −θ S(A0 ⊕ K) 2
For a normal to the plane with angle π2 − ϕ with axis Oz, we have, Kr (h) being the geometric covariogram of the disk with radius r, K(h, ϕ) = Kr (h cos ϕ) At a small scale, for h ≤ 2r, K(h) = 2π
Z
π 2
Kr (h cos ϕ) cos ϕdϕ
0
When h ≥ 2r, we have to satisfy h cos ϕ ≤ 2r, so that arccos 2r h ≤ ϕ ≤ and K(h) = 2π with
Z
π 2
arccos
⎛
Kr (h) = 2r2 ⎝arccos
2r h
µ
Kr (h cos ϕ) cos ϕdϕ
h 2r
¶
−
h 2r
s
1−
µ
h 2r
¶2
Using this expression, when h ≤ 2r, K(h) Z 2 = 4πr
π 2
0
⎛
⎝arccos
µ
s
¶
−
h cos ϕ 2r
h cos ϕ 2r
¶
h cos ϕ − 2r
h cos ϕ 2r
1−
µ
h cos ϕ 2r
⎞ ⎠
¶2
and when h ≥ 2r K(h) Z 2 = 4πr
π 2
arccos
2r h
⎛
⎝arccos
µ
π 2,
s
1−
µ
h cos ϕ 2r
⎞
⎠ cos ϕdϕ
¶2
⎞
⎠ cos ϕdϕ
Explicit results are formulated from elliptic integrals [681], [676]:
230
6. Boolean Random Sets
K(2r) Z = 4πr2
π 2
0
= 4πr
2
Z
0
π 2
µ
¶ q 2 ϕ − cos ϕ 1 − (cos ϕ) cos ϕdϕ
(ϕ − cos ϕ sin ϕ) cos ϕdϕ = 4πr
and K(2r)/K(0) = 4
µ
1 4 π− 2 3
¶
2
µ
1 4 π− 2 3
¶
= 0.949 85
Covariance of dilated Poisson planes in R3 For dilated Poisson planes A, q = exp(−4πθr) = exp(−θK(0)) and then the average intercept of the grain induced by section on lines is given by K(0) = 4πr We have Q(l) = exp(−θM (A0 ⊕ K)) = exp(−θM (B(r) ⊕ l)) with M (B(r) ⊕ l) = 4πr + 2πrl
1 = 4πr + πl 2r
Then, Q(l) = q exp(−θπl) Q0 (l) = −θπQ(l)
and the average intercept of Ac is given by −q 1 = 0 Q (0) θπ
On a small scale (for h ≤ 2r), the same expression is obtained for the covariance and for linear erosions, since dilation of segments by l or by the doublet {x, x + h} give the same result. h
3
Q(h) = q exp(−θπh) = q 1+ 4r and Q(2r) = q exp(−2θπr) = q 2 When h ≤ 2r, choice can be made ¡ ¢of a vector h parallel to axis Oz. For 2r ≥ h sin ϕ, namely ϕ ≤ arcsin 2r h K(h, ϕ) = 2r − h sin ϕ
6.15 Exercises
231
and K(h) = 2π
Z
0
Ã
arcsin( 2r h )
(2r − h sin ϕ) cos ϕdϕ
! Z arcsin( 2r h ) 2r = 2π 2r( ) − h sin ϕ cos ϕdϕ h 0 Ã µ 2 µ ¶2 ! ¶ 4r2 4r 2r r2 4πr2 = 2π /2 = 2π −h −2 = h h h h h As a result, K(2r) = 2πr, so that K(2r)/K(0) =
1 2
3
and Q(2r) = q 2 .
7 Random Tessellations
Abstract: The probabilistic characterization of random tessellations is introduced. The construction and the main properties of some random tessellations (Voronoi, Johnson-Mehl, Laguerre, Poisson, Cauwe, and iterated tessellation models) and of their extensions are given.
7.1 Introduction Random tessellations of the Euclidean space Rn are typical models widely used to simulate polycrystals in random media. Other areas of applications cover satellite images, meshing of images for computer graphics [399] or for finite elements computation, and more generally results of a segmentation of an image in subdomains or classes based on some random process, like for instance the stochastic watershed [21]. The random tessellation models introduced in this chapter are relatives of the Boolean model, since they are also derived from the Poisson point process. They are useful by themselves, to model a medium like a population of grains, and can be used for the generation of primary grains for the Boolean model, or as a first step of a mosaic model (chapter 8). Some usual models are defined from distances to the points xk of a point process, usually the Poisson point process P. The Voronoi tessellation mostly studied by E. N. Gilbert [206], is built from the zones of influence of points xk . Its generalizations like the Johnson-Mehl and the Laguerre tessellations introduce a time sequence of points, or weights allocated to each point. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Jeulin, Morphological Models of Random Structures, Interdisciplinary Applied Mathematics 53, https://doi.org/10.1007/978-3-030-75452-5_7
233
234
7. Random Tessellations
Extensions of these models in the context of Boolean RF are presented in chapter 10. We introduce the Poisson tessellation studied mainly by R.E. Miles [483], [484] and by G. Matheron [443], [445], [448], and finally the Cauwe tessellation [107], [443]. Morphological properties of some of these random tessellations are presented in more details in [598], [113], [591].
7.2 Reminder on random tessellations Random tessellations were formalized by G. Matheron ([441], pp. 35-39), in the framework of the theory of random sets [448]. Definition 7.1. Consider a locally compact separable space E and subsets Ci of E, belonging to ℘(E). A tessellation Θ is a collection of classes Ci ∈ ℘(E) with ∪i Ci = E and Ci ∩ Cj = ∅ for i 6= j However, with this general definition the status of class boundaries is ambiguous ([441], pp. 35-36, [408] pp. 133-134), so that it is useful to replace the general set Π(E) by the set Πg (E) which partitions E into open classes Ci and points of their boundary F = (∪Ci )c . Starting from a tessellation Θ ∈ Π(E) a tessellation in Πg (E) is obtained from the interior of classes ˚i and from points of F . This is the preliminary step to define a proper σ C algebra and random tessellations. Definition 7.2. Noting Π(E) (namely Πg (E)) the set of tessellations of E (namely of tessellations with open (or point) classes), RA is the subset of Πg (E), such that for any A ⊂ E, A is contained in one class C. When G are open parts of E, subsets RG generate a σ algebra on Πg (E), σ(RG ), on which a probability can be constructed. A random tessellation Θ in Πg (E) is characterized by P (RG ) = P {G ⊂ Ci }. With this σ algebra, events (and their probability) like ”x belongs to a single class”, ”x1 and x2 belong to a single class, ”x1 , x2 , ... xm belong to k classes", or more generally "the compact set K is included in a single class" are well defined. The classes of a random tessellation in Rn are not necessarily simply connected components, as in the case of the Dead Leaves tessellation [275] in chapter 11 and other examples in chapter 10. In publications on random tessellations, the term "cell" is often used instead of "class". In this book we will use both terms. In what follows, random tessellations of the Euclidean space Rn are introduced. In the case of open classes C ⊂ G, and a closed boundary set F , with zero measure and smooth, use of the random closed sets approach can be made, as in many publications. Then it is possible characterize F by its Choquet capacity TF (K), or alternatively by
7.3 Probability distributions of the classes
235
QF (K) = P {K ⊂ F c }. This is not equivalent to the content of definition 7.2, since for non connected classes (as for the Dead Leaves model) and non connected compact sets K, we can have K ⊂ F c for K in a single class or in several classes. Therefore QF (K) does not give access to P {K ⊂ Ci }. However, when K is connected (like a segment l or a ball B(r)), QF (K) = P { K is included in a single class of the tessellation}. In particular, this approach using QF (K) works for random tessellations with convex classes (in that case their boundaries are portions of hyperplanes) often used in publications, but this type of tessellations is a strong limitation with respect to many real situations in materials, and to the general case studied in this book. This chapter and chapter 10 study some models of random tessellations in the Euclidean space Rn . Furthermore, locally finite random tessellations are considered, where the random number of classes hit by any bounded domain K is almost surely (i.e. with probability 1) a finite random number N (K). Then the Choquet capacity is deduced from T (K) = P {N (K) = 0}.
7.3 Probability distributions of the classes Consider a random class of a stationary and ergodic random tessellation, for which spatial averages converge towards mathematical expectations, and a random characteristic Z of this class (it might be multivariate, like its volume V , its surface area S, its diameter in R3 ) . Depending on the mode of selection of the class, two different kinds of probability have to be used: • Giving to every class Ci the same weight, Z is characterized by a ”number” probability density function F (dz, dV ) • Considering a class C0 containing a given point (for instance the origin), Z is characterized by a ”measure” probability density function F0 (dz, dV ): it is weighted proportionally to its volume in Rn . The two distributions are not equivalent, classes with a higher volume having a higher probability to contain a given point x. Exactly as for the linear size distributions in chapter 3, the first situation is coined a number weighting, while the second one is called a measure weighting. Considering a class C with a volume V and a random property Z, the ”measure” distribution of X, F0 (dz, dV ) and the ”number” distribution of Z, F (dz, dV ) are related as follows, as was proved by R.E. Miles from ergodic properties of the Poisson tessellation [483], [484]:
236
7. Random Tessellations
F0 (dx, dV ) =
V F (dx, dV ) E(V ) (7.1)
V −1 F0 (dx, dV ) F (dx, dV ) = E0 (V −1 ) where E(V ) and E0 (V −1 ) are the mathematical expectations of V and V −1 for the distributions F and F0 . In fact Eq. (7.1) is satisfied for random tessellations of Rn with good ergodic properties. An illustration of the difference between the two distributions is given in chapter 3 with the exponential and the gamma distribution corresponding to the number and to the measure distributions involved by a Poisson point process in R inducing polyhedra made of segments.
7.4 The Voronoi, Johnson-Mehl, and Laguerre tessellation 7.4.1 Voronoi tessellation The Voronoi tessellation is of wide use in many areas, such as the simulation of crystal growth in metallurgy, or in computational geometry, for instance in order to produce meshes for finite elements calculation. The usual model is based on the Poisson point process Definition 7.3. The Poisson Voronoi tessellation [206] is built from a Poisson point process P(θ) in Rn : every point x of the space is attributed to the class Ci containing the point xi ∈ P(θ) ⇔ x is closer to xi than to any other point of P(θ). The resulting tessellation is made of convex polyhedra in Rn , as the result of the intersection of half spaces. An example of simulation in R3 is given in Fig. 7.1. For this simulation, periodic boundary conditions are imposed on the distance function, in order to generate a periodic medium for further use in finite element calculations with periodic boundary conditions [367]. A physical interpretation of this model is the following: consider the points of P(θ) as seeds of crystals growing from a melt. The growth of every wall of the cells is stopped when the walls of neighboring cells enter in contact. At the end of the process, the cells build the zones of influence of the initial points of P(θ). Two-phase or multi-phase mosaic models of materials can be generated from 3D Voronoi tessellations, as was studied in [367]. Each class Ck is defined from Ck = {x ∈ Rn , d(x, xk ) < d(x, xl ), xk ∈ P, xl ∈ P, l 6= k}
(7.2)
7.4 The Voronoi, Johnson-Mehl, and Laguerre tessellation
237
FIGURE 7.1. Simulation of a 3D Voronoï tessellation
Every class of the tessellation is an open set, its closure being made of planar faces (planes in R3 and segments in R2 ) orthogonal to segments connecting neighbor points of P. Starting from the definition of the Euclidean distance, i=n X d (x, xk ) = (xi − xki )2 2
i=1
xi being the coordinates of point x in Rn , the boundary separating classes Ck and Cl is obtained from d2 (x, xk ) = d2 (x, xl )
(7.3)
Equation (7.3) is a linear equation with respects to coordinates xi , giving the equation of a hyperplane. In the definition the Euclidean metric may be replaced by the Lp metric [306], [309] with the integer p ≥ 1: dp (x, xk ) =
i=n X i=1
|xi − xki |p
(7.4)
The equation of the boundaries between classes is obtained by setting dp (x, xk ) = dp (x, xl ). For p = {1, 2} the separations between classes are planar. For p = 3, we get portions of quadrics. Increasing the value of p gives higher order polynomial surfaces, with degree p − 1 for p > 1. The obtained tessellations not Pi=n are p isotropic in the Euclidean space, since the balls defined by i=1 |xi | = rp are not isotropic, except for p = 2, giving spheres. For p = 1, the balls are hypercubes with edges orthogonal to directions given by (±1, ±1, ..., ±1). When p → ∞ we get the L∞ metric, where balls are hypercubes with
238
7. Random Tessellations
edges parallel to the coordinates system. When p = 1 and p = ∞, the separations are parallel to the faces of the corresponding hypercubes. It is possible to replace in the definition of the Voronoi tessellation the stationary Poisson point process by the non homogeneous one, with intensity θ(x). The probability P (K) is obtained from [306], [309]. Consider first the Voronoi models defined from a Poisson point process with intensity θ(x), and a homogeneous metric Lp . The ball B(x, r) with center x and radius r is defined from the metric Lp by B(x, r) = {y, dp (x, y) ≤ rp } The theorem below is a generalization of the results of Gilbert [206]. Theorem 7.1. Consider a Voronoi tessellation of space defined from the Poisson point process with intensity θ(x), and the metric Lp . The probability P (K) = P (K ⊂ Ck ) is given by Z P (K) = θ(dy) exp −θ(F (K, y)) (7.5) Rn
R
where θ(F (K, y)) = Rn θ(dx)1F (K,y) (x) is the measure of the Voronoi flower F (K, y) = ∪x∈K B(x, d(x, y)). For a constant intensity θ, the random tessellation is stationary, and equation (11.2) becomes Z P (K) = θ exp −θμn (F (K, y))dy, (7.6) Rn
μn being the Lebesgue measure in Rn . Proof. We observe the event {K ⊂ Ck } ⇐⇒ ∀x ∈ K, ∀l 6= k, d(x, xk ) ≤ d(x, xl ) ⇔ the ball with center x ∈ K and radius d(x, xk ) contains no point of the Poisson point process. Defining the Voronoï flower of K with center y by F (K, y) = ∪x∈K B(x, d(x, y)) [103], we have {K ⊂ Ck } ⇔ F (K, xk ) contains no point of the process, with probability exp −θ(F (K, xk )). Equation (11.2) is derived by randomizing the point xk , θ(dy) being the probability that the element of volume dy contains a point of the process, given in Eq. (11.2). When K is a connected compact set, P (K) gives the probability for K to be included in a single connected component of the class. The deterministic intensity θ(x) can be replaced by a realization of a positive random function Θ(x), the Poisson point process becoming a Cox process (chapter 5). This gives Cox based random tessellations, as illustrated by Fig. 7.2, obtained from the Cox point process of Fig. 5.3 in chapter 5. Their corresponding probability P (K) is deduced from Eq. (11.2) by taking its expectation with respect to the random intensity Θ.
7.4 The Voronoi, Johnson-Mehl, and Laguerre tessellation
239
FIGURE 7.2. Two scale Cox Voronoï tessellation generated from Poisson points located inside a BRS of discs with radius R = 32
The size distribution of Voronoi polyhedra enclosed in Voronoi tessellations is not known. However, from numerical simulations the number distribution of the volumes of 3D polyhedra was fit to a Γ distribution with a parameter b = 5.56 when the average volume is equal to 1 [435].
7.4.2 Johnson-Mehl and Laguerre random tessellations A sequential version of the Voronoi model was proposed by W. A. Johnson and R.F. Mehl for applications in metallography [359]. A detailed presentation is given in [500]. Definition 7.4. Starting from a sequence of Poisson seeds with intensity θ(t), P(θ(t)), crystals grow isotropically with the same rate. The growth of crystal boundaries is stopped when they enter in contact, while seeds falling at time t in the grains are removed. The resulting tessellation obtained at t = ∞ for θ(t) = θ and for a constant rate is made of non convex cells, with boundaries made of branches of hyperboloids in R3 . A 2D simulation is given in Fig. 7.3. The Johnson-Mehl tessellation [359], [387] results from a combination of germination (with a sequential intensity θ(t)) and growth (with a growth rate α(t)). The conventional model is based on a constant (with respect to time) germination and growth rate. During the process, germs falling inside growing crystals are deleted. Each Poisson germ {xk , tk }, generated at time tk with the growth rate α(tk ) generate the class Ck defined by: Ck = {x ∈ Rn , d(x, (xk , tk ))/α(tk ) + tk < d(x, (xl , tl ))/α(tl ) + tl , xk ∈ P, xl ∈ P, l 6= k}
(7.7)
240
7. Random Tessellations
FIGURE 7.3. 2D simulation of a Johnson-Mehl random tessellation
Immediate extensions of this model are obtained by means of a Lp metric [306], [309]. The Laguerre tessellation [412], [413] generalizes the Voronoi tessellation, by attaching to each Poisson point xk a random radius Rk . The power P (x, xk ) is defined by P (x, xk ) = d2 (x, xk ) − Rk2
(7.8)
The class Ck is defined from: Ck = {x ∈ Rn , P (x, xk ) < P (x, xl ), xk ∈ P, xl ∈ P, l 6= k}
(7.9)
Some germs xl generate empty classes, depending on the values of Rk2 and on the distance to other germs. Classes are again bounded by portions of hyperplanes. New random tessellations are derived from the Lp metric [306], [309], replacing in equation (10.10) P (x, xk ) by dp (x, xk ) − Rkp , where in general non planar boundaries are generated. Introducing a random radius Rk brings some flexibility, as compared to the Voronoi model with a single parameter, the intensity θ, and opens a wide field of applications, like the modeling of foams [413]. Sections of the Voronoi and the Johnson-Mehl tessellations in Rn by subspaces Rk (k < n) are not models of the same type (induced by a Poisson point process in Rk ): they do not own the stereological invariance of the Boolean model. Sections of Voronoi tessellations are some specific Laguerre tessellations, which remain induced models of the same type. In [422], 3D conditional simulations of Laguerre tesselations are generated by simulated annealing and having the same 2D section as a given polished section. It makes possible to produce 3D simulated images compatible with some available 2D image, which is a wide interest for applications, when 3D images are experimentally inaccessible.
7.5 Random tessellation generated by a geodesic distance
241
Few probabilistic properties of these models are known (average geometrical properties of the cells, such as the numbers of edges, vertices, faces,...). A detailed report is given in [206], [598], [620], [500], [598], [113]. It is interesting to relate these models to the Boolean RF models by means of the distance function, and to propose more general models [306], [309], as presented in chapter 10.
7.5 Random tessellation generated by a geodesic distance An alternative approach of previous models of random tessellations makes use of Poisson points xk and of a geodesic distance dG generated by some function f (x) [309]. For instance when using the topographic interpretation of f on a grid of points and the flooding from markers, we have dG (x, y) = |f (x) − f (y)|, x and y being neighbors on the grid. For any pair of points (x, y), their geodesic distance is given by the length of shortest paths connecting x to y. An extension of the Voronoi tessellation given by Eq. 7.2 is immediately obtained. Zones of influence of random points generated on a Riemannian manifold equipped with a Riemannian metric, namely a geodesic distance on the sphere, were already studied by R. Miles [485]. Definition 7.5. A geodesic random tessellation in the Euclidean space Rn is defined from the zones of influence of Poisson points based on a geodesic distance dG . The class CGk of the tessellation containing point xk of the Poisson point process P is defined by CGk = {x ∈ Rn , dG (x, xk ) < dG (x, xl ), xk ∈ P, xl ∈ P, l 6= k}
(7.10)
The zones of influence of the points with respect to the geodesic distance dG are obtained as the attraction basins of the points during the construction of the watershed of f (x) constrained by the markers xk [69]. They generate a random tessellation from the geodesic distance. This approach is followed for the segmentation of images by means of the so-called stochastic watershed [21], where the pertinence of boundaries of the segmentation is measured by their probability of occurrence estimated from realizations of random markers. This generation of random tessellations is illustrated in Fig. 7.5, where a realization of a geodesic Voronoi tessellation from the image f (x) in Fig. 7.4 is shown. The probability P (K) obtained for the Voronoi tessellation in Eqs (11.2) and (7.6) are easily extended to the geodesic case, where the Euclidean balls B(x, r) have to be replaced by geodesic balls
242
7. Random Tessellations
FIGURE 7.4. SEM image f (x) of a powder to generate a geodesic distance dG (x, y) = |f (x) − f (y)|
FIGURE 7.5. Voronoï tessellation generated from a realization of a Poisson point process and the geodesic distance dG induced by the gradient of image in Fig. 7.4
BG (x, r) = {y, dG (x, y) ≤ r} Theorem 7.2. Consider a random tessellation of space defined from the zones of influence CGk generated by a Poisson point process with intensity θ(x) and the geodesic distance dG . For any compact set K, the probability P (K) = P (K ⊂ CGk ) is given by Z P (K) = θ(dy) exp −θ(FG (K, y)) (7.11) Rn
R
where θ(FG (K, y)) = Rn θ(dx)1FG (K,y) (x) is the measure of the geodesic flower FG (K, y) = ∪x∈K BG (x, dG (x, y)). In the stationary case, for a constant intensity θ, equation (11.2) becomes Z P (K) = θ exp −θμn (FG (K, y))dy, (7.12) Rn
7.6 The Poisson tessellation
243
μn being the Lebesgue measure in Rn . Proof. We have K ⊂ CGk ⇐⇒ ∀x ∈ K, ∀l 6= k, dG (x, xk ) ≤ dG (x, xl ) ⇔ the geodesic ball with center x ∈ K and radius dG (x, xk ) contains no point of the Poisson point process. Defining the geodesic flower of K with center y by FG (K, y) = ∪x∈K BG (x, dG (x, y)), we get K ⊂ CGk ⇔ FG (K, xk ) contains no point of the process, with probability exp −θ(FG (K, xk )). Equation (7.11) is obtained by randomization of the point xk , θ(dy) being the probability that the element of volume dy contains a point of the process. The obtained random tessellation is not stationary. Using for f (x) a realization of a random function Y (x), FG (K, y) is a random set, and the probability is obtained by taking the expectation of Eqs (7.11, 7.12) with respect to the random variable θ(FG (K, y)). For a constant θ and a stationary random function Y (x), we get a stationary random tessellation. Similarly, the Johnson-Mehl and the Laguerre tessellations are extended to the geodesic case, replacing the distance d by a geodesic distance dG in the definition of the class Ck from the modified distance (7.7) and from the power (10.9).
7.6 The Poisson tessellation 7.6.1 Definition and Choquet capacity of the Poisson hyperplanes In the Euclidean space Rn , the Poisson linear variety of dimension n − 1, are the Poisson hyperplanes (cf. chapter 6). They enclose cells of a space tessellation, the Poisson polyhedra. Definition 7.6. An hyperplane H(u, r) in Rn is defined by a unit vector u orthogonal to H(u, r) (u ∈ 12 Sn ), and by a scalar r ∈ R, such that Pi=n 1 n i=1 ui xi = r; it can be viewed as an application H from 2 Sn × R in R . 1 Consider a Poisson point process in 2 Sn × R with intensity λn (dω)dx, λn being a positive Radon measure. The Poisson hyperplanes network A is a RACS, image by the application H of the point process [445],[448]. One should notice the present notation λn instead of θn−1 for the general case of the Poisson varieties, to keep the notation given in the literature. Theorem 7.3. The number of hyperplanes hit by a set K is a Poisson variable with intensity Z λ(K) = λn (dω)l(K(ω)) (7.13) 1 2 Sn
and in the isotropic case for a convex set K
244
7. Random Tessellations
FIGURE 7.6. Simulation of a planar Poisson tessellation
λ(K) = λn A(K) The Choquet capacity of the Poisson hyperplanes A is given by ÃZ ! T (K) = 1 − exp −
1 2 Sn
λn (dω)l(K(ω))
(7.14)
(7.15)
and in the isotropic case for a convex set K T (K) = 1 − exp −λn A(K)
(7.16)
Proof. These results are particular cases of Eqs (6.40-6.44). By construction, the hyperplanes induce by intersection on every orthogonal line with direction ω a Poisson point process with dimension with intensity λn (dω). The contribution of the direction ω to N (K), is the Poisson variable N (K, ω) with intensity λn (dω)l(K(ω)). The contributions of the various directions are independent. If the compact set is not convex, but is connected, Eqs (7.14) and (7.16) can be applied to the convex hull of K, C(K). An example of simulation of a 2D isotropic Poisson tessellation is shown in Fig. 7.6. Note that Poisson lines can be considered as a special Poisson point process in the plane: with a parametric representation (or using a Hough transform), each line D can be considered as a point with coordinates given by its distance d to the origin, and its angle α with Ox axis. A Poisson point process in the plane (d,α) is applied back to Poisson lines. Alternatively, Poisson lines can be obtained as a limit case of a Boolean model of segments (cf. section 6.15.7 in chapter 6)
7.6 The Poisson tessellation
245
7.6.2 Stereological aspects of the Poisson hyperplanes Let En−k be a plane of dimension n − k in the space Rn . The intersection En−k ∩A generates a Poisson hyperplanes network in Rn−k , with the induced intensity λn−k . For convex sets K, λn A(K) = λn−k An−k (K)
(7.17)
where An−k is the norm in Rn−k . From integral geometry, bn−k−1 A(K) = bn−1 An−k (K) and therefore λn
bn−1 = λn−k bn−k−1
(7.18)
2 For illustration, if λ2 = λ, we get λ1 = 2λ and λ3 = λ; for λ3 = λ, π π λ2 = λ and λ1 = πλ. 2 It is interesting and useful for applications (see sections 18.94 and 6.10.2) to give the particular form of Eq. (7.16) in R3 for λ3 = λ. • When K = B(r) • When K = l
T (B(r)) = 1 − exp − (4πλr)
(7.19)
T (l) = 1 − exp − (πλl)
(7.20)
• When K is a compact convex set in R2 T (K) = 1 − exp − and for a disc C(r) with radius r
³π 2
λL(K)
´
¢ ¡ T (C(r)) = 1 − exp − π2 λr
(7.21)
(7.22)
7.6.3 Characterization of the Poisson polyhedron containing the origin The complementary set Ac of the Poisson hyperplanes is made of separate polyhedra. Therefore the hyperplanes tessellate the space into Poisson polyhedra. In a first step, it is interesting to characterize the polyhedron containing the origin O, Π0 . For any compact set K, K ⊂ Π0 ⇔ C{{O} ∪ K} ⊂ Ac and therefore
246
7. Random Tessellations
P {K ⊂ Π0 } = 1 − T (C{{O} ∪ K}) = Q(C{{O} ∪ K})
(7.23)
For instance, the probability for x at the distance l from O to belong to Π0 is equal to Q(l) given in Eq. (7.20); the probability for x and y (with Ox = h1 , Oy = h2 and xy = h2 − h1 ) to belong to Π0 is equal to ³π ´ Q(T ) = exp − λ(|h1 | + |h2 | + |h2 − h1 |) 2
7.6.4 Conditional invariance by erosion The Poisson polyhedra satisfy a property of invariance by erosion. Consider the polyhedron Π0 and two compact sets K and K 0 containing the origin O. We have ˇ ⇔ K ⊕ K 0 ⊂ Π0 K 0 ⊂ Π0 ª K (7.24) and Q(K ⊕ K 0 ) = exp −λn A(C(K ⊕ K 0 )) = Q(K)Q(K 0 )
(7.25)
and for α ≥ 0, Q(αK) = Q(K)α It comes out that Eq. (7.25) is a characteristic property of the Poisson polyhedra [445]. Theorem 7.4. An open random set Π0 containing almost surely the origin O is a Poisson polyhedron ⇔ Q(K ⊕ K 0 ) = Q(K)Q(K 0 ) for any compact sets K and K 0 containing the origin O. Proposition 7.1. Eq. (7.25) is a semi-group equation which reveals a Markovian behaviour of the Poisson polyhedra: it can be reformulated ˇ | K ⊂ Π0 } = P {K 0 ⊂ Π0 } P {K 0 ⊂ Π0 ª K
(7.26)
From a measure point of view, conditionally to the fact that the origin ˇ the eroded set Π0 ª K ˇ is a Poisson polyhedron with O belongs to Π0 ª K, the same probability law as the initial polyhedron Π0 . This property is called a conditional invariance by erosions. It is a generalization of the lack of memory of the exponential distribution, law of the intervals between Poisson points on the line, which are the Poisson polyhedra in R. Using Eq. (7.1) connecting number and measure distributions, the conditional invariance by erosions can be expressed as follows, from the number point ˇ is of view: conditionally to the fact that for any polyhedron Π, Π ª K ˇ not empty, Π ª K as the same characteristics as Π. In particular, for any measurable property X such that X(∅) = 0, ˇ = E{X(Π)}P {Π0 ª K ˇ 6= ∅} = E{X(Π)}Q(K) E{X(Π ª K)}
(7.27)
7.6 The Poisson tessellation
or Q(K) =
ˇ E{X(Π ª K)} E{X(Π)}
247
(7.28)
For applications to polyhedra in R3 if X is the volume V , Eq. (7.28) becomes E{V (Π ª B(r))} = exp (−4πλr) (7.29) E{V (Π)} E{V (Π ª l)} K(l) = = exp (−πλl) E{V (Π)} K(0)
(7.30)
It can be shown [445] that: 6
E{V (Π)} = K(0) =
π 4 λ3
E{S(Π)} = −4K 0 (0) = E{M (Π)} =
24 π 3 λ2
(7.31)
3 λ
and more generally the Minkowski functionals in Rn Wk have the following mathematical expectation, where λ = λ2 E{Wk (Π)} =
bk bn−k
λk−n
(7.32)
7.6.5 Opening size distribution Consider a convex set K (for instance a ball B(r)) and a size distribution of the polyhedra by opening by the set μK (chapter 3) ˇ ⊕ μK)} = 1 − GK (μ) = P {O ∈ (Π0 ª μK
E{μn (ΠμK )} E{μn (Π)}
(7.33)
ˇ has the same In Eq. (7.33), use was made of Eq. (7.24). Since Π ª μK probabilistic properties as Π, conditionally to the fact that it is not empty, Eq. (7.33) becomes 1 − GK (μ) =
ˇ E{μn (Π ⊕ μK)} exp (−μλn A(K)) E{μn (Π)}
(7.34)
For instance, for K = B(r),µ by use of the Steiner formula (3.12-3.17), Xk=n n¶ E{Wk (Π)}μk , the following size distribE{μn (Π ⊕ B(μ))} = k=0 k utions are obtained:
248
7. Random Tessellations
in R : 1 − G1 (l) = λ1 l exp (−λ1 l) (gamma distribution) in R2 : 1 − G2 (r) = (1 + 2πλ2 r + π 2 λ22 r2 ) exp (−2πλ2 r) 4 in R3 : 1 − G3 (r) = (1 + 4πλ3 r + π6 λ23 r2 + 29 π5 λ33 r3 ) exp (−4πλ3 r) (7.35)
7.7 Example of application to model concrete by multiscale Poisson polyhedra A multiscale model of random polyhedra was developed to simulate concrete microstructures, starting from a wide granulometric range of gravels (from some millimeters up to 20mm) [172]. A simplified formulation of concrete was prepared for research purpose, starting from a mixture of 3 granulometric classes of gravels: sand [0 − 4mm], fine gravels [4 − 12.5mm] and gravels [10 − 20mm], with relative proportions 0.41, 0.17 and 0.42, respectively. The material was observed by microtomography. In a first step a model of random polyhedra has to be selected. The concrete formulation granulometry is modeled by a combination of three truncated granulometries representing, respectively sand, fine gravels and gravels. Three populations of Voronoi polyhedra and of Poisson polyhedra are tested as candidates to simulate each class of objects, each population depending on a size parameter ξ i . The determination of the two truncation parameters Λ1 and Λ2 , and of the three size parameters ξ 1 , ξ 2 and ξ 3 is achieved using a least squares minimization, using the nonlinear optimization algorithm of Nelder-Mead [516]. In Fig. 7.7 is shown the best fit obtained for three populations of Poisson polyhedra, using Eq. 7.35, with intensities λ1 = 0.518mm−1 , λ2 = 0.0695mm−1 , λ3 = 0.0418mm−1 , and Λ1 = 2.41mm, Λ2 = 6.15mm. A similar optimization approach was followed for Voronoi polyhedra, using the measure distribution of polyhedra volumes, deduced from the empirical approximation of the number distribution by a Γ distribution [435]. No equivalent quality was obtained for the fit, so that it was decided to use Poisson polyhedra to simulate the microstructure. In a second step, simulations of Poisson polyhedra are obtained with a vectorial implementation, each polyhedron being defined analytically by equations of the planes building its facets. The main advantages of such algorithms, as compared to generation on grids of points, is that they allow a faster generation, low memory cost, and facilities to binarize polyhedra with any arbitrary resolution for further use. A library of polyhedra was made available. Details on the implementation are given in [172]. In a third step, Poisson polyhedra are selected from the library, and sorted by volume. They are implanted in a domain D by a sequential absorption algorithm according to their decreasing volume, in order to maximize the filling ratio. During iterations, a new polyhedron is initially
7.8 The Cauwe tessellation
G(r) 1
Λ1
249
Λ2
0,8 0,6
Experimental granulometry
0,4
Fit with Poisson polyhedra truncated granulometries
0,2 0
2.41
5 6.15
10
r (mm) FIGURE 7.7. Comparison between the granulometry of the concrete formulation and the fitted size distribution of three populations of Poisson polyhedra [172]
uniformly implanted in D. The packing is carried out using an attractionrepulsion mechanism through a translation of the polyhedron in a uniform direction. From a temporary binarization, the intersection between this polyhedron and previously implanted polyhedra is tested. If there is no intersection, iterative small size translations are made until intersection, while the reverse is applied in case of intersection. Periodic simulations, to avoid edge effect and for further use in the computation of elastic fields [173], are made on cubes with 16003 volumes. A typical simulation (a 2D section is shown in Fig. 7.8) contains 106 aggregates, which are disconnected by application of a 3D watershed algorithm [71] to the inverse of the distance function inside the polyhedra phase. This disconnection is performed in order to avoid the percolation of aggregates obtained in the simulation. The volume fraction of aggregates after disconnection is around 64.3%, to be compared to the formulation (68.7%). The simulation is validated by the granulometry of the simulations, close to the input granulometry of polyhedra given in Fig. 7.7.
7.8 The Cauwe tessellation In his studies on the comminution of ores [107], Ph. Cauwe first tried to simulate the fragmentation process by iterations of Poisson planes splitting already existing rock grains, generating Poisson polyhedra. Therefore he tried to characterize the particles of ores from their size distribution, as estimated by image analysis and compared to the relations (7.35). After a disagreement between the experimental results and the theoretical prediction of the size distribution, showing an underestimation of small size intercepts by the model, he was lead to propose a new model of random tessellation of space in convex polyhedra [107] based on doublets of parallel
250
7. Random Tessellations
FIGURE 7.8. 2D section of a simulation of concrete microstructure in a 16003 cube (14.43 cm3 ) [172]
hyperplanes. The theoretical characterization of this model was made by G. Matheron [443]. A detailed presentation is given in [598].
7.8.1 Construction of the Cauwe tessellation Definition 7.7. The Cauwe tessellation is built as the Poisson tessellation in Rn , replacing every hyperplane by a doublet of parallel hyperplanes separated by a random distance H with the probability distribution Fn (h).
7.8.2 Stereological properties Sections of the Cauwe tessellation in subspace Rn−k are models of the same type with induced intensity λn−k and induced distance distribution Fn−k (h). As for Poisson polyhedra, Eq. (7.18) is satisfied. We have Fn−k (h) =
kbk bn−k−1 bn−1
Z
0
1
¡ ¢ k −1 un−k 1 − u2 2 Fn (hu)du
(7.36)
In practice it will be useful to recover higher dimensional distributions from lower dimensional observations, like for instance to recover F3 (h) from F1 (h) or F2 (h). When k = 2, Eq. (7.36) becomes: n−1 Fn−2 (h) = n−1 h
Z
h
xn−2 Fn (x)dx 0
from which an inversion formula is obtained by derivation:
7.8 The Cauwe tessellation
Fn (h) =
¤ 1 d £ n−1 F (h) h n−2 (n − 1)hn−2 dh
251
(7.37)
When k = 1, Eq. (7.36) becomes: Fn−1 (h) =
2bn−2 1 bn−1 hn−1
Z
0
h
xn−1 √ Fn (x)dx h2 − x2
(7.38)
To invert Eq. (7.38), first deduce Fn+1 from Fn−1 using Eq. (7.37), and then apply Eq. (7.38) to Fn+1 . It comes: 1 ω n−1 1 Fn (h) = π ω n−2 hn
Z
0
h
x d √ (xn Fn−1 (x)) 2 2 h − x dx
From previous results we have the following useful inversion equations: Z
h
¢ x d ¡ 3 √ x F2 (x) 2 2 h − x dx 0 Z h ¢ 2 x d ¡ 2 √ F2 (h) = x F1 (x) 2 2 πh2 0 dx h −x ¢ 1 d ¡ 2 F3 (h) = h F1 (h) 2h dh F3 (h) =
1 2h3
(7.39)
7.8.3 Choquet capacity on convex sets The Cauwe tessellation generates Cauwe polyhedra. The boundary of polyhedra is a RACS. To compute the Choquet capacity of the tessellation , we can consider the representation of hyperplanes doublets by random Poisson point doublets {x, x + h} in the space R × 12 Sn , where 12 Sn is the half hypersphere in Rn . A convex set K in Rn is projected in every direction ω of the hypersphere as a segment d(ω). Every direction ω contributes independently to Q(K). the elementary contribution of the angular sector [ω, ω + dω] is given by dQ(K) = exp (−λ(ω)dωμ(d(ω) ⊕ h)) and d (log Q(K)) = −λ(ω)dωμ(d(ω) ⊕ h) We have μ(d(ω) ⊕ h) = d(ω) + h, if d(ω) > h = 2d(ω) if h ≥ d(ω) and noting fn (h) the pdf of Fn (h)
252
7. Random Tessellations
μ(d(ω) ⊕ h) =
Z
d(ω)
(d(ω) + h) fn (h)dh + 2d(ω)
0
= d(ω) + d(ω)(1 − Fn (d(ω))) +
Z
Z
∞
fn (h)dh
d(ω)
d(ω)
hfn (h)dh
0
= d(ω)(2 − Fn (d(ω))) Z d(ω) (1 − Fn (h))dh − d(ω)(1 − Fn (d(ω))) + 0
= d(ω) +
Z
d(ω)
0
(1 − Fn (h))dh =
Z
d(ω)
0
(2 − Fn (h))dh
It comes log Q(K) = −λn (ω)
Z
dω 1 2 Sn
Z
d(ω)
(2 − Fn (x)) dx
0
Theorem 7.5. The Choquet capacity of the Cauwe random tessellation for convex sets K is given by à ! Z Z d(ω) T (K) = 1 − Q(K) = exp −λn (ω) dω (2 − Fn (x)) dx (7.40) 1 2 Sn
0
For a segment l and for a disc of radius r, we get from Eq. (7.40): ÃZ ! l
Q(l) = exp −λ1
Q(r) = exp −πλ2
0
(2 − F1 (x)) dx
µZ
0
2r
¶ (2 − F2 (x)) dx
and the number distribution of intercepts F (l) is deduced by ¶ µ Q0 (l) 1 1 − F (l) = 0 = 1 − F1 (l) Q(l) Q (0) 2 In practice, from linear and circular erosions, it is possible to estimate separately F1 (h) and of F2 (h), which have to satisfy the reciprocity relationship given by Eq. (7.38) with n = 2; This gives access to testing the validity of the model. The size distribution by openings by disks is derived in [443]. The cumulative size distribution is given by G(r) = [1 + πλ2 r (2 − F2 (2r))]2 Q(r)
7.9 Iteration of tessellations and the STIT model
253
This model was successfully used to describe the size distributions of various rocks after milling in different conditions [107]. The distribution function of the inter-plane distance F3 (h) was uniform in the range [0 − d], which overall gives a parsimonious model depending on only two parameters (λ3 and d).
7.8.4 Mixture of Poisson and Cauwe tessellations In the construction of the Cauwe tessellation it is possible to mix randomly point doublets (with probability p), and singletons (with probability 1 − p). If the two processes are independent, the resulting functional Q(K) is the product of corresponding functionals.
7.9 Iteration of tessellations and the STIT model The comminution process is mimicked by iterations of Poisson hyperplanes 7.6 to generate Poisson polyhedra, or of doublets of Poisson hyperplanes 7.8 to generate Cauwe polyhedra [443]. The same idea is used by W. Nagel and V. Weiss to generate "tessellations stable under iterations" (STIT), where the iterations of Poisson hyperplanes are made in situ. This process generates nested tessellations or equivalently a hierarchical network, useful to simulate hierarchical cracks [512]. This model is iteratively defined starting from a bounded convex window K (it is usually a parallelepiped located in Rn , but K can be a hypersphere with center in O). A sequential construction of Poisson hyperplanes and its use in iteration of tessellations is introduced as explained below. Definition 7.8. Consider a Poisson point process in 12 Sn × R ×R+ with intensity λn (dω)dxdt, λn being a positive Radon measure. The Poisson hyperplanes Ht (u, r) generated between time 0 and time t is a RACS, image by the application H of the point process, with intensity λn (dω)tdx. Theorem 7.6. The number of hyperplanes hit by a convex set K in the time interval [0, t] is a Poisson variable with intensity Z λt (K) = λn (dω)l(K(ω))t (7.41) 1 2 Sn
As a consequence, the random time interval T separating the occurrence of two hyperplanes hitting K follows an exponential distribution with time density λ(K): Z λ(K) =
1 2 Sn
λn (dω)l(K(ω))
(7.42)
254
7. Random Tessellations
The current tessellation Θ(τ ) at time τ < t is the union of random classes Ck (τ ). At time t = 0, K is included in a single class of the tessellation. Between τ and τ +dτ , a given class Ck (τ ) may be divided in two daughters by intersection with some hyperplane Hτ (u, r) which generates a new boundary. The sequential construction of Θ(t) requires a priority rule concerning the candidates Ck (τ ) for division. This can be done in different ways: i)
to a selected candidate class for division Ck (τ ) is given a sequence of exponential time Tki with time density λ(Ck (τ )) (given by Eq. (7.42), after replacement of K by Ck (τ ) ) and of hyperplanes Hτ +Tki (u, r) [512], [513], [514]. A test of intersection of the hyperplanes with Ck (τ ) isP performed for increasing i, as long as there is no intersection and τ + i Tki < t. The first hyperplane Hτ +Tki (u, r) with non empty intersection provides a new boundary Hτ +Tki (u, r) ∩ Ck (τ ) and two new classes Ck1 (τ + Tki ) and Ck2 (τ + Tki ). The process starts again with the updated tessellation Θ(τ + Tki ). If no intersection is found before t, Ck (τ ) is rejected and a new class is selected for the same sequence. The process is stopped after scanning unsuccessfully all classes in the window K. ii) for each class Ck (β(k)) is stored its birth time β(k) (with β(1) = 0 for t = 0), an exponential life time Tk with time density λ(Ck (τ )) and a Poisson hyperplane Hβ(k)+Tk (u, r) [421]. The classes are sorted with increasing death times β(k) + Tk . The class with minimal death time β(k) + Tk < t is intersected by Hβ(k)+Tk (u, r) to generate two new classes Ck1 (β(k) + Tk ) and Ck2 (β(k) + Tk ), to which are attributed two independent exponential life times with time density λ(Ck1 (β(k) + Tk ) and λ(Ck2 (β(k) + Tk ). The process is repeated on the updated tessellation Θ(β(k)+Tk ), as long as the date t is not reached. This second construction is more efficient, as it avoids the generation of rejected events.
By construction, the obtained classes are convex, as intersections of half spaces delimited by hyperplanes. Furthermore, Theorem 7.7. Each class of the STIT being generated by a sequence of independent Poisson hyperplanes is a Poisson polyhedron. However the polyhedra are arranged differently from the Poisson tessellation. If λn (dω) for the generation of hyperplanes is isotropic, the STIT is isotropic, and is made of isotropic Poisson polyhedra. The intersection of any line in Rn with the planar boundaries of classes of the STIT is a Poisson point process, as a result of the intersection of Poisson hyperplanes by a line. In [472] the construction of the STIT is directly constructed in Rn , as a particular model (using Poisson hyperplanes) of nested iteration of random tessellations. The STIT is the only case where nested tessellations are stable by iterations, meaning that up to a scaling it produces the same tessellation. Typical simulations illustrate available publications [512], [513], [514], [421].
7.10 Exercise
255
The morphological properties of classes of a STIT are identical to the properties of Poisson polyhedra given in section 7.6. Further properties, linked with the spatial arrangement of classes like specific number of nodes, among others, are given in R2 [514] and in R3 [513], [113]. In [421] a modification of the construction of the STIT is proposed to get more flexibility in the shape of simulated crack networks: the uniform location on l(Ck (ω) of an intersecting hyperplane Hτ (u, r) is replaced by a Gaussian distribution centered on the middle of segment l(Ck (ω), providing more regular shapes; optionally, the exponential life time Tk is parametrized by the Lebesgue measure of Ck instead of the average measure of its projections (surface area in R3 and perimeter in R2 ). However the resulting tessellations are not any more STIT. Remark 7.1. The STIT is not stable by intersection of independent STIT realizations, generating tessellations with Poisson polyhedra where the hierarchy of the nesting is lost. Therefore this intersection provide new models of random tessellations. Remark 7.2. The initial state for t = 0 can be any random tessellation, even with non convex or non connected classes such as a DLRT (chapter 11). At time τ > t, the division of each class Ck in twonclasses is made by o P intersection of Ck with the two half spaces Hτ− (u, r) = x, i=n u x < r i i i=1 n P o i=n + and Hτ (u, r) = x, i=1 ui xi > r . For large t, the iterated tessellation Θ(t) is a mixture of Poisson polyhedra and of fragmented classes of the initial tessellation. As noticed in [512], the hierarchy of the nested tessellation generated from K is described by a binary tree. This tree is the result of a continuous Markov branching process as in section 5.9.2 of chapter 5. Using the bottom-up construction process of the iteration of random tessellations in a compact domain K, it is possible to generate random trees, starting from the root in K (it can be its center). Each new branch, with random length given by the exponential life time Tk with time density λ(Ck (τ )), can start from some random point in Ck (τ ) until division of Ck (τ ). This is the possible source of simulation of interesting miscellaneous Markov random trees driven by iterative space tessellations. A detailed study of the morphological properties of such random trees remains to be done.
7.10 Exercise 7.10.1 Superposition of random tessellations Starting from independent realizations of random tessellations Θk , a new tesselation is defined, with classes given by intersection of classes Ck ∈ Θk . Give the functional P (RG ) of the resulting random tessellation.
256
7. Random Tessellations
Answer: For the class C = ∩Ck P (RG ) = P {G ⊂ C} = Πk P {G ⊂ Ck } When the individual random tessellations Θk are Poisson tessellations or STIT, or any combination of these two models, the class C = ∩Ck is also a Poisson polyhedron, which is stable by intersection. This is the consequence of the stability of Poisson hyperplanes by union (section 6.13.2 in chapter 6). This stability by intersection is not satisfied for the Cauwe RT.
8 The Mosaic Model
Abstract: This chapter introduces the mosaic model, where subdomains of space have a constant value (univariate or multivariate). It is built on a random tessellation of space. Its Choquet capacity is given and illustrated by order two and order three probability distributions. Special cases are the Poisson and the STIT mosaic models.
8.1 Introduction Some microstructures like polycrystals present constant properties inside each grain. A common probabilistic representation is to use a mosaic model, also called a cell model in the physical literature [487]. After a description of the construction of the mosaic model, we provide its Choquet capacity, which is detailed for the case of the mosaic model with support in the Euclidean space Rn . Previous results are applied to the Poisson mosaic, and extended to the multivariate case.
8.2 Construction The mosaic RF model is a model of random function [455],[264] which gives interesting simulation of polycrystals (with a different crystallographic in each grain, as in [16]) or a multi component medium. In chapters 6, 9, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Jeulin, Morphological Models of Random Structures, Interdisciplinary Applied Mathematics 53, https://doi.org/10.1007/978-3-030-75452-5_8
257
258
8. The Mosaic Model
FIGURE 8.1. Values on boundaries of classes
15, and in Exercise 8.9.3, it is used as a first step of the construction of hierarchical structures. The mosaic model is built in two steps:
Definition 8.1. Starting from a locally finite random tessellation Π [441] in a space E (for instance Rn ), and from a random variable Z with the cumulative distribution function G(z) = P {Z < z} and T (z) = 1 − G(z) to the elements x of every class Ci of the random tessellation is attributed a realization of the random variable Z. The realizations concerning two separate classes Ci and Cj are independent. This construction does not account for boundaries ∂C of the class C. In what follows, we consider open classes C ⊂ G and a boundary set F which is a closed set with null measure and with some regularity. Consider the part of boundary Fij separating the open classes Ci and Cj (with random values Zi and Zj for Z) (Fig. 8.1). For x ∈ Fij , several functions can be constructed, such as: Z1 (x) = Zi ∨ Zj
(8.1)
Z2 (x) = Zi ∧ Zj
(8.2)
Zi ∧ Zj < Z(x) < Zi ∨ Zj
(8.3)
or Z1 (x) is usc and Z2 (x) is lsc. The function Z(x) defined by the inequalities (8.3) admits for upper regularized version Z1 (x) and for lower regularized version Z2 (x). In what follows, mosaic models satisfying conditions (8.1,8.2 or 8.3) are studied, and similar inequalities for points x belonging to the boundary of more than two classes (the affectation of triple points in R2 , triple lines or quadruple points in Rn can be managed in a similar way), when considering probabilities of events involving a non countable set of points. Otherwise, the values taken at boundary points with a null measure will not affect the result.
8.3 Choquet capacity
259
Particular cases of random mosaics are the following: starting with a discrete distribution function G(z) with a finite number of values for z, multicomponent random sets models are obtained by affectation of classes to the phase Ai with the probability pi . When P {Z = 1} = p and P {Z = 0} = 1 − p, Z(x) obtained by application of Eq. (8.1) is the indicator function of a RACS A.
8.3 Choquet capacity Consider in E × R a compact set K 0 = K × [z, +∞]. Let N (K) be the random number of classes C of the tessellation Π hit by the compact set K in E. N (K) is a discrete random variable with generating function GK (s): GK (s) =
n=∞ X n=0
pn (K) sn with pn (K) = P {N (K) = n}
(8.4)
Proposition 8.1. The Choquet capacity of the Mosaic RF model satisfying the inequality (8.3), for a test function g(x) = z if x ∈ K and g(x) = +∞ if x ∈ / K, is given by: 1 − T (K 0 ) = Q(K 0 ) = 1 − T (g) = Q(g) = P {Z∨ (K) < z} = GK (G(z)) (8.5) We have also, for a test function g(x) = z if x ∈ K and g(x) = −∞ if x∈ / K, P (K, z) = P (g) = P {Z∧ (K) ≥ z} = GK (T (z)) (8.6)
n Proof. i) Conditionnally to N (K) = n, Q(K 0 ) = P {∨k=n k=1 Zk < z} = G(z) . After deconditioning with respect to the random variable N (K),
Q(g) =
n=∞ X
pn (K) G(z)n = GK (G(z))
n=0
n ii) For N (K) = n, P {∧k=n k=1 Zk ≥ z} = T (z) . After deconditioning, P (g) = GK (T (z)).
For Eqs (8.5,8.6) to be exact, we shall admit that for the random tessellations that we use in practice (particularly for stationary tessellations), the event {K hits some points of the boundary of one class but remains disjoint from the other classes of the tessellation owning the same boundary} has a null probability. This assumption is not satisfied in general. As a counter example, let us consider a periodic tessellation in Rn and take for K the closure of a class of the tessellation; the above mentioned event is realized to each node of the periodic network generating the tessellation. From Eqs (8.5,8.6), it is clear that for every value of z, there is for this model a separation between the variables in E (through the functional
260
8. The Mosaic Model
GK ) and the variable Z. In addition, for every value z of Z (or for any component Ai in the multicomponent case), the statistical properties are identical (up to the value G(z) or pi ). This symmetry involves that in the case of two components with p1 = p2 = 12 , the RACS A has the same statistical properties as Ac ; this is the autodual case.
8.4 Mosaic model in Rn We examine now the case of a mosaic RF with support in the Euclidean space Rn , built on a stationary random tessellation, and give the results obtained for specific compact sets K.
8.4.1 Calculation of P {N(K) = 1}
To compute some part of the Choquet capacity, we have to know the generating function GK (s), according to Eq. 8.5. Unfortunately this information is not available in general, but P {N (K) = 1} can be calculated. Proposition 8.2. Consider an ergodic stationary mosaic RF with support in Rn . When C is a random class of the underlying tessellation: P {N (K) = 1} = P {K ⊂ C} =
ˇ μn (C Ä K) μn (C)
(8.7)
Proof. For an ergodic stationary random tessellation, P {N (K) = 1} = P {K ⊂ C} = P {(K ⊂ C1 ) ∪ (K ⊂ C2 )... ∪ (K ⊂ Cn )...} ˇ = P {Kx ⊂ C} = P {UCi {x ∈ Ci Ä K}} ˇ = P {x ∈ Ui {Ci Ä K}} ˇ and the ergodicity, Using the incompatibility between events {x ∈ Ci Ä K} P ˇ Ci ⊂D μn (Ci Ä K) P {K ⊂ C} = lim μn (D) μn (D)→∞ P ˇ Ci ⊂D μn (Ci Ä K) P = lim μn (D)→∞ Ci ⊂D μn (Ci ) P ˇ N (D) Ci ⊂D μn (Ci Ä K) P = lim N (D) μn (D)→∞ Ci ⊂D μn (Ci ) ˇ μ (C Ä K) = n μn (C)
8.4 Mosaic model in Rn
261
8.4.2 First order statistics When K = {x}, since the tessellation is stationary and the boundary set F is with zero measure, P {N (x) = 1} = 1 and Gx (s) = s. Therefore the mosaic model and the random variable Z have the same distribution function: F (z) = P {Z(x) < z} = G(z)
8.4.3 Second order statistics When K = {x, x + h}, and using Eq. (8.7), we have N (K) = 1 with the K(h) probability r(h) = and N (K) = 2 with the probability 1 − r(h) K(0) where K(h) = μn (A ∩ A−h ) is the average geometric covariogram of a class A of the random tessellation (μn is the Lebesgue measure (or the volume) in Rn ). Therefore: Theorem 8.1. For the random mosaic, the distribution functions of Z(x)∨ Z(x + h) and of Z(x) ∧ Z(x + h) are given by: Q(h, z) = P {Z(x) ∨ Z(x + h) < z} = r(h)G(z) + (1 − r(h))G(z)2 (8.8) P (h, z) = P {Z(x) ∧ Z(x + h) < z} = r(h)T (z) + (1 − r(h))T (z)2
(8.9)
From Eqs (8.8,8.9) is deduced:
Q(h, z) − G(z)2 P (h, z) − T (z)2 = = r(h), ∀z T (z)Q(z) T (z)Q(z)
(8.10)
Theorem 8.2. Using K 0 = {(x, z1 ), (x + h, z2 )} is obtained the bivariate distribution: T (h, z1 , z2 ) = P {Z(x) ≥ z1 , Z(x + h) ≥ z2 } = r(h)T (z1 ∨ z2 ) + (1 − r(h))T (z1 )T (z2 )
(8.11)
F (h, z1 , z2 ) = P {Z(x) < z1 , Z(x + h) < z2 } = r(h)F (z1 ∧ z2 ) + (1 − r(h))F (z1 )F (z2 )
(8.12)
For every (z1 , z2 ) we have: T (h, z1 , z2 ) − T (z1 )T (z2 ) F (h, z1 , z2 ) − F (z1 )F (z2 ) = = r(h) T (z1 ∨ z2 ) − T (z1 )T (z2 ) F (z1 ∧ z2 ) − F (z1 )F (z2 )
(8.13)
Equations (8.10) and (8.13) can be used to test the validity of the model from experimental bivariate distributions. Other second order statistics of the RF Z(x) are the covariance and the variograms.
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8. The Mosaic Model
Theorem 8.3. The central second order correlation function W 2 (h) is obtained from W 2 (h) = E{(Z(x + h) − E(Z))(Z(x) − E(Z))} = D2 [Z] r(h)
(8.14)
where D2 [Z] is the variance of the random variable Z. Eq. (8.14) is easily derived from the fact that the only contribution to W 2 (h) is obtained for the two points {x, x + h} belonging to the same class of the tessellation, which occurs with the probability r(h). Theorem 8.4. The order one variogram γ 1 (h) is given by 1 E{| Z(x + h) − Z(x) |} 2 1 = (1 − r(h))E{| Zi − Zj |} = (1 − r(h))S 2
γ 1 (h) =
(8.15)
where Zi and Zj are two independent realizations of Z: | Zi − Zj |= Zi ∨ Zj − Zi ∧ Zj Therefore E{| Zi −Zj |} =
Z
(1−G(z)2 )dz−
Z
T (z)2 dz = 2
Z
G(z)(1−G(z))dz = 2S
where S is the coefficient of variation of the distribution G(z) [455]. Theorem 8.5. The order two variogram γ 2 (h) is 1 1 E{[Z(x + h) − Z(x)]2 } = (1 − r(h))E{[Zi − Zj ]2 }(8.16) 2 2 2 = (1 − r(h))D [Z]
γ 2 (h) =
The first and second order variograms satisfy: γ 1 (h) = γ 2 (h)
S D2 [Z]
(8.17)
The present relationship between the order one and the order two variograms differs from p the usual one in the case of diffusion Gaussian RF, where γ 1 (h) ' γ 2 (h). The assumption ”mosaic model” can be tested to the second order by one of the equations (8.10, 8.13, 8.17). Remark 8.1. Using for G(z) a Gaussian distribution, the point distribution of the mosaic model is also Gaussian, but its bivariate distribution cannot be bi-Gaussian. For instance, the bivariate distribution of the mosaic for z1 = z2 is given by, where f (z) is a Gaussian pdf:
8.4 Mosaic model in Rn
263
f (h, z, z) = r(h)f (z) + (1 − r(h))(f (z))2 and f (h, z, z) is not consistent with the bivariate Gaussian pdf g(h, z, z) given by µ ¶ z2 1 exp − g(h, z, z) = p 1 + r(h) 2π 1 − r2 (h)
except when r(h) = 0. It is therefore hazardous to use a heuristic approach based on the use of the Gaussian RF assumption, after transforming the data by anamorphosous of the univariate distribution (see section 19.8 in chapter 19).
8.4.4 Third order statistics Using K 0 = {(x, z1 ), (x + h1 , z2 ), (x + h2 , z3 )} is obtained the trivariate distribution. First is needed the probability pn (K). There are only three possibilities: • N (K) = 1 with the probability s(h1 , h2 ) =
μn (A ∩ A−h1 ∩ A−h2 ) (Fig. μn (A)
8.2) • N (K) = 2 with the probability P2 (h1 ) + P2 (h2 ) + P2 (h3 ) = P2 (h1 , h2 ) (Fig. 8.3) with: s(h1 , h2 ) + P2 (h1 ) = r(h1 ) s(h1 , h2 ) + P2 (h2 ) = r(h2 ) s(h1 , h2 ) + P2 (h3 ) = r(h3 ) h3 = h2 − h1 • N (K) = 3 (Fig. 8.4) with the probability 1 − s(h1 , h2 ) − P2 (h1 , h2 ). Theorem 8.6. It comes for the 3-points distribution function:
T (z1 , z2 , z3 ) = P {Z(x) ≥ z1 , Z(x + h1 ) ≥ z2 , Z(x + h2 ) ≥ z3 } = s(h1 , h2 ) [T (z1 ∨ z2 ∨ z3 ) + 2T (z1 )T (z2 )T (z3 ) − T (z1 )T (z2 ∨ z3 ) −T (z2 )T (z1 ∨ z3 ) − T (z3 )T (z1 ∨ z2 )] +r(h1 ) [T (z3 )T (z1 ∨ z2 ) − T (z1 )T (z2 )T (z3 )] +r(h2 ) [T (z2 )T (z1 ∨ z3 ) − T (z1 )T (z2 )T (z3 )] +r(h3 ) [T (z1 )T (z2 ∨ z3 ) − T (z1 )T (z2 )T (z3 )] (8.18) The central third order correlation function can be deduced from Eq. (8.18), or more simply obtained directly, since the only contribution comes from the occurrence of the three points in a single class of the tessellation: W 3 (h1 , h2 ) = E{(Z(x) − E(Z))(Z(x + h1 ) − E(Z))(Z(x + h2 ) − E(Z))} = s(h1 , h2 )E[(Z − E(Z))3 ] (8.19)
264
8. The Mosaic Model
FIGURE 8.2. Configurations of 3 points in class C
FIGURE 8.3. Configuration with 3 points in two classes
FIGURE 8.4. Configuration with 3 points in 3 classes
For this model, W 3 (h1 , h2 ) = 0 for any distribution G(z) symmetrical with respect to its expectation E(Z). For instance, this is the case of a 1 random set obtained when Z = 1 with the probability p = and Z = 0 2 with the probability 1 − p = 1/2, generating an autodual random set.
8.4.5 Higher order statistics The derivation of higher order statistics is rather cumbersome. We just outline the case of 4th and 5th order central moments: • Order four central correlation function W 4 (h1 , h2 , h3 ): the only contributions come from the occurrence of the four points in a single
8.5 The Poisson mosaic
265
μn (A ∩ A−h1 ∩ A−h2 ∩ A−h3 ) ) and of two μn (A) points in the class C1 while the two other points are in the class C2 (with the probability P22 , that we do not give explicitly). class (with the probability
W 4 (h1 , h2 , h3 ) =
(8.20)
μn (A ∩ A−h1 ∩ A−h2 ∩ A−h3 ) E[(Z − E(Z))4 ] + P22 [D2 [Z]]2 μn (A)
• Order five central correlation function W 5 (h1 , h2 , h3 , h4 ): the only contributions come from the occurrence of the four points in a single μ (A ∩ A−h1 ∩ A−h2 ∩ A−h3 ∩ A−h4 ) class (with the probability n ) and μn (A) of two points in the class C1 while the three other points are in the class C2 (with the probability P23 , that we do not give explicitly). W 5 (h1 , h2 , h3 , h4 ) =
μn (A ∩ A−h1 ∩ A−h2 ∩ A−h3 ∩ A−h4 ) E[(Z − E(Z))5 ] μn (A)
(8.21)
+P23 E[(Z − E(Z))3 ]D2 [Z] For this model, W 5 (h1 , h2 , h3 , h4 ) = 0 for any distribution G(z) symmetrical with respect to its expectation E(Z). This is also the case of all the central correlation functions of odd order.
8.5 The Poisson mosaic Generally, for any compact K in Rn , the generating function GK (s) required for the calculation of the Choquet capacity is unknown, as said before. We introduce now a particular case of mosaic model, the Poisson mosaic [264, 293], built from a Poisson tessellation of the space Rn ([448], and chapter 7). The classes of this tessellation are bounded by a network of hyperplanes (with dimension n − 1) orthogonal to the directions ω of the space. The hyperplanes orthogonal to the line Dω with the direction ω cut Dω according to a Poisson point process with intensity λn (ω)dω. We consider here the isotropic case (λn (ω) = ω). For this tessellation in R3 and with λ3 = λ, r(h) = exp(−πλ |h|) s(h1 , h2 ) = exp(− π2 λ(|h1 | + |h2 | + |h3 |))
(8.22)
266
8. The Mosaic Model
FIGURE 8.5. Simulation of a Poisson mosaic
For the isotropic tessellation in R2 and λ2 = λ, r(h) = exp(−2λ |h|) s(h1 , h2 ) = exp(−λ(|h1 | + |h2 | + |h3 |))
(8.23)
An example of simulation in 2D is given in Fig. 8.5. For the Poisson mosaic, • The central second order correlation function W 2 (h) is given by: W 2 (h) = D2 [Z] exp(−πλ |h|)
(8.24)
• The central third order correlation function W 3 (h1 , h2 ) is given by: π W 3 (h1 , h2 ) = E[(Z − E(Z))3 ] exp(− λ(|h1 | + |h2 | + |h3 |)) 2
(8.25)
It is possible to build a new mosaic random function from the summation of n independent Poisson mosaics (with parameters λi (i = 1, 2, ..., m) and with random variables Zi ): Z(x) =
i=m X
Zi (x)
i=1
For this model, W 2 (h) =
i=m X i=1
W 3 (h1 , h2 ) =
i=m X i=1
D2 [Zi ] exp(−πλi |h|)
(8.26)
π E[(Zi − E(Zi ))3 ] exp(− λi (|h1 | + |h2 | + |h3 |)) (8.27) 2
8.6 The STIT mosaic
267
This construction does not correspond P to the definition 8.1, since in i=m the present case the random variables i=1 Zi (x) are no more independent when considering two classes of the underlying random tessellation Π = ∩i=m i=1 Πi . When the Zi are independent realizations of the same RF Z(x) with a finite variance, from the Central Limit Theorem Pi=m 1 i=1 Zi (x) converges towards a Gaussian RF with the exponential n covariance n1 D2 [Z] exp(−πλ |h|). Other Gaussian RF with correlation function r(h) can be obtained, starting with a mosaic RF based on a random tessellation having the reduced geometrical covariogram r(h). For the Poisson mosaic are known the distribution functions of Z∨ (l) and of Z∧ (l), deduced from the fact that the number of hyperplanes Nl hitting the segment of length l follows a Poisson distribution with parameter πλ. The number of polyhedra met by l is Nl + 1 and therefore: Gl (s) =
n=∞ X n=0
pl (s)sn = s exp(πλl(s − 1))
Theorem 8.7. From Eqs (8.5,8.6), for the mosaic model satisfying the inequality (8.3),
and
P {Z∨ (l) < z} = Q(l, z) = G(z) exp(−πλl(1 − G(z)))
(8.28)
P {Z∧ (l) < z} = P (l, z) = T (z) exp(−πλl(1 − T (z)))
(8.29)
P (l, z)Q(l, z) = exp(−πλl), ∀z G(z)T (z)
(8.30)
Finally, the profiles of a Poisson mosaic on every line D build a Markov jumps RF.
8.6 The STIT mosaic Starting from STIT random tessellations (section 7.9 in chapter 7) produces a mosaic model where each initial class is a Poisson polyhedron. As a consequence, the central second order and third order correlation functions W 2 (h) and W 3 (h1 , h2 ) are the same as for the Poisson mosaic and are given by Eqs (8.24) and (8.25). Since the intersection of any line and a STIT tessellation generates a Poisson point process, the distribution functions of Z∨ (l) and of Z∧ (l) are also the same as for the Poisson mosaic, and are given by Eqs (8.28) and (8.29). These results hold on for mosaic models built on intersections of any number of STIT. Higher order moments, like order four or order five central correlation functions differ from the Pois-
268
8. The Mosaic Model
FIGURE 8.6. Simulation of a Poisson mosaic random set
son mosaic, as a consequence of a different spatial agencement of Poisson polyhedra in the STIT.
8.7 The Mosaic random set The Mosaic Random Set is a special RF model, a particular example of the multi-component version introduced in Exercise 8.9.1. For the random set, Z = 1 with the probability p and Z = 0 with the probability q = 1 − p. As particular cases of previous results, using the functions given in Eq. (8.22) it comes:
P {x ∈ A} = p
C(h) = P {x ∈ A, x + h ∈ A} = pr(h) + p2 (1 − r(h)) P {x ∈ A, x + h1 ∈ A, x + h2 ∈ A} = s(h1 , h2 )(p + 2p3 − 3p2 ) +(r(h1 ) + r(h2 ) + r(h3 ))(p2 − p3 ) + p3 When p = 1/2, an autodual random set is obtained. For the Poisson binary mosaic in R3 (an example of simulations in is given in Fig. 8.6), P (l) = P {l ⊂ A} = p exp(−πλlq) Q(l) = P {l ⊂ Ac } = q exp(−πλlp) The same results follow for a Mosaic Random Set built on a STIT (section 7.9 in chapter 7), or on intersections of STIT.
8.8 The multivariate mosaic model
269
8.8 The multivariate mosaic model The previous developments can be extended to the case of a multivariate RF with m components Z1 , Z2 , ...., Zm . To each class of the tessellation is affected a multiple of m random variables according to the multivariate distribution G12...m (z1 , z2 , ..., zm ) = P {Z1 < z1 , Z2 < z2 , ...., Zm < zm } or T12...m (z1 , z2 , ..., zm ) = P {Z1 ≥ z1 , Z2 ≥ z2 , ...., Zm ≥ zm } (Fig. 8.7). Each component Zi is a mosaic RF to which the previous results apply. Consider the case where every component respects the inequality (8.3). By the same reasoning as for the scalar case, it follows: Theorem 8.8. The Choquet capacity of the multivariate mosaic RF is obtained as: if g(xi ) = zi for xi ∈ K (i = 1, 2, ..., n), and g(x) = +∞ for x ∈ / K: i) 1 − T (g) = Q(g) (8.31) = P {Z1∨ (K) < z1 , Z2∨ (K) < z2 , ..., Zm∨ (K) < zm } = GK [G(z1 , z2 , ..., zm )] If g(xi ) = zi for xi ∈ K (i = 1, 2, ..., n), and g(x) = −∞ if x ∈ / K: ii) P (g) = P {Z1∧ (K) ≥ z1 , Z2∧ (K) ≥ z2 , ..., Zm∧ (K) ≥ zm } = GK [T (z1 , z2 , ..., zm )]
(8.32)
Similar distributions for m compacts K1 , K2 , ..., Km are more difficult to calculate, and require multivariate generating functions of the random variables N (K1 ),..., N (Km ), N (Ki ∩ Kj ),..., N (K1 ∩ K2 ∩ ... ∩ Km ). Theorem 8.9. The bivariate distributions are given by: Tij (h, z1 , z2 ) = P {Zi (x) ≥ z1 , Zj (x + h) ≥ z2 } = r(h)Tij (z1 , z2 ) + (1 − r(h))Ti (z1 )Tj (z2 ) with Tij (z1 , z2 ) = P {Zi ≥ z1 , Zj ≥ z2 }
(8.33)
Fij (h, z1 , z2 ) = P {Zi (x) < z1 , Zj (x + h) < z2 } = r(h)Fij (z1 , z2 ) + (1 − r(h))Fi (z1 )Fj (z2 )
(8.34)
Theorem 8.10. The central second order correlation function are given by: W ij2 (h) = E{(Zi (x + h) − E(Zi ))(Zj (x) − E(Zj ))} = σ ij r(h) with σ ij = E{(Zi − E(Zi ))(Zj − E(Zj ))}
(8.35)
Theorem 8.11. The central third order correlation function given by: W ijk3 (h1 , h2 ) = s(h1 , h2 )σ ijk with σ ijk = E{(Zi − E(Zi ))(Zj − E(Zj ))(Zk − E(Zk ))}
are
(8.36)
270
8. The Mosaic Model
FIGURE 8.7. Three component mosaic model built on a Poisson random tessellation
From Eqs (8.33,8.34), is deduced the following expression, that can be used for tests: Tij (h, z1 , z2 ) − Ti (z1 )Tj (z2 ) Fij (h, z1 , z2 ) − Fi (z1 )Fj (z2 ) = = r(h) Tij (z1 , z2 ) − Ti (z1 )Tj (z2 ) Fij (z1 , z2 ) − Fi (z1 )Fj (z2 ) (8.37) To close this chapter, we can mention some applications of the scalar mosaic model, for the roughness of anisotropic surfaces [264], for the prediction of effective properties of random media (or change of scale) ([487], [46], [16], chapter 18), and in fracture statistics (chapter 20).
8.9 Exercises 8.9.1 Multi component mosaic As a particular case of mosaic model, consider a multi component mosaic to simulate a medium with m phases Ai (i = 1, ..., m). Starting from a random stationary tessellation Π in Rn , every class C is affected independently to Ai with the probability pi . Express the functionals Ti (K) = P {Ai ∩K 6= ∅} and Pi (K) = P {K ⊂ Ki }. As a particular case, calculate the covariances Cii (h) = P {x ∈ Ai , x + h ∈ Ai }. Give also the cross covariances Cij (h) = P {x ∈ Ai , x + h ∈ Aj }. Starting from a Poisson tessellation in R3 , give the covariances, cross covariances and the functions Ti (l) and Pi (l) for a segment of length l. Answer: For every component Ai , 1 − Ti (K) =
n=∞ X n=0
pn (K)(1 − pi )k = GK (1 − pi )
(8.38)
8.9 Exercises
271
and by symmetry Pi (K) =
n=∞ X
pn (K)pi k = GK (pi )
(8.39)
n=0
From Eqs (8.38,8.39) it is clear that all the random components Ai have similar probabilistic properties through the generating function GK (s)), 1 up to their volume fraction pi . If pi = , all the components are strictly m equivalent, and we obtain a symmetrical model. The covariances are obtained from Eq. (8.39) for K = {x, x + h} with p1 = r(h) and p2 = 1 − r(h) Cii (h) = pi r(h) + p2i (1 − r(h)) = p2i + pi (1 − pi )r(h)
(8.40)
The cross covariances are given by Cij (h) = pi pj (1 − r(h))
(8.41)
For a Poisson mosaic, r(h) = exp(−πλ | h |) and Cii (h) = p2i + pi (1 − pi ) exp(−πλ |h|) Cij (h) = pi pj (1 − exp(−πλ |h|))
(8.42)
Gl (s) = s exp(πλl(s − 1)) 1 − Ti (l) = (1 − pi ) exp(−πλlpi ) Pi (l) = pi exp(−πλl(1 − pi ))
(8.43)
8.9.2 Cross covariances of the multicomponent mosaic For the multicomponent mosaic in R3 (with m components Ai ) defined in exercise (8.9.1) in chapter 8, give the expression of the cross covariances Cij (h), of the areas of contact SVij , and of the vicinity indices defined by Eqs (3.65,3.66). Answer:Using the same notations as in (8.9.1), the cross covariances are given by: Cij (h) = pi pj (1 − r(h)) • The areas of contact are (for an isotropic structure, starting from a random tessellation with the random class A0 ) SVij = −4pi pj r0 (0) = pi pj and SVi = pi (1 − pi )
E{S(A0 )} E{V (A0 )}
E{S(A0 )} E{V (A0 )}
272
8. The Mosaic Model
• The transition probability from Ai to Aj for a random point on ∂Ai is given by: SV pj pij = ij = SVi 1 − pi • The indices of coordination ic (ij) are ic (ij) =
Xk=m k=1
pk (1 − pk )
(1 − pi )(1 − pj )
The indices of vicinity depend only on the volume fractions pi , whatever the underlying random tessellation. It may be of interest to compare experimental values to theoretical indices to detect preferential associations or to test this model.
8.9.3 Hierarchical mosaic model A random function model is built in two steps as the mosaic model, but using a random tessellation and a random function model Zθ (x) depending on a set of parameters θ: in the first step is generated a realization of the stationary random tessellation model; inside every class Ci of the tessellation is considered a realization of the parameters and a realization of Zθ (x); the realizations concerning two different classes are independent. Therefore a two-scale structure is obtained by this process. We denote by E/θ the mathematical expectations concerning a RF Zθ (x) (for a given set of parameters θ), and Eθ the mathematical expectations with respect to the random variables θ. Give the expressions of the central second and third order correlation functions W 2 (h) and W 3 (h1 , h2 ) of the RF Z(x) obtained from this construction. Give the same expressions for the following particular cases: Zθ (x) is a mosaic RF (with the cumulative distribution function Gθ (z)); Zθ (x) is a mosaic model with non random parameters (with the cumulative distribution function G(z)); consider in this last case a Poisson mosaic for Zθ (x), and a Poisson tessellation for the first step of the hierarchical model. What happens after n iterations of this process? Finally consider the case when Zθ (x) is the indicator function of a stationary random set Aθ depending on the random parameters θ, and express the results as a function of the covariance Cθ (h) = P {x ∈ Aθ , x ∈ Aθ } and of the third order moment Pθ (T ) = P {x ∈ Aθ , x+h1 ∈ Aθ , x+h2 ∈ Aθ }. Other examples are considered in exercises (6.15.2,9.15.2). Answer: We write Z θ = E/θ {Zθ (x)}, Z = Eθ {Z θ }; W 2θ (h), W 3θ (h) are the central second and third order correlation functions of Zθ (x); we use the subscript 1 for the statistical properties of the first step mosaic. For the calculation of W 2 (h) and W 3 (h1 , h2 ), only the events concerning the occurrence of points {x} and {x + h}, or {x}, {x + h1 }, {x + h2 } has
8.9 Exercises
273
a non zero contribution (with h3 = h2 − h1 ). We get: ªª © © W 2©(h) =©r1 (h)Eθ E/θ (Zθ (x) − Z)(Zθ (x + h) − Z) ªª = r1 (h)Eθ E/θ (Zθ (x)£− Z θ + Z θ − Z)(Zθ (x + h) −¤Z θ + Z θ − Z) = r1 (h) Eθ [W 2θ (h)] + Eθ [(Z θ − Z)2 ] (8.44) W 3 (h1 , h2 )/s1 (h1 , h2 ) ªª © © = Eθ E/θ (Zθ (x) − Z)(Zθ (x + h1 ) − Z)(Zθ (x + h2 ) − Z)
© = Eθ W 3θ (h1 , h2 ) + (Z θ − Z)(W 2θ (hª1 ) + W 2θ (h2 ) + W 2θ (h3 )) +(Z θ − Z)3
(8.45)
When the RF used in the second step is a mosaic model, where the variance of the random variable Zθ is given by Dθ2 (Zθ ), we have: © ª W 2θ (h) = r2 (h)E/θ (Zθ (x) − Z θ )2
and therefore:
£ ¤ W 2 (h) = r1 (h) r2 (h)Eθ [Dθ2 (Zθ )] + Eθ [(Z θ − Z)2 ]
(8.46)
Similarly, to the third order:
© ª W 3θ (h) = s2 (h1 , h2 )E/θ (Zθ (x) − Z θ )3
W 3 (h1 , h2ªª )/s1 (h1 , h2 ) © © © ª = s2 (h1 , h2 )Eθ E/θ (Zθ (x) − Z θ )3 + Eθ {(Z θ − Z)E/θ (Zθ (x) − Z θ )2 (r2 (h1 ) + r2 (h2 ) + r2 (h3 )) + (Z θ − Z)3 } (8.47) When Zθ (x) is a mosaic model with non random parameters, we have Z θ = Z and Eqs (8.46, 8.47) simplify into: W 2 (h) = r1 (h)r2 (h)D2 (Z) © ª W 3 (h1 , h2 ) = s1 (h1 , h2 )s2 (h1 , h2 )E (Z − Z)3
(8.48) (8.49)
This process of construction can be iterated to any order n, giving simple expressions for the second and third order central correlation functions: W 2 (h) = r1 (h)r2 (h)...rn (h)D2 (Z) © ª W 3 (h1 , h2 ) = s1 (h1 , h2 )s2 (h1 , h2 )...sn (h1 , h2 )E (Z − Z)3
(8.50) (8.51)
When using Poisson tesselations models for each step of the construction (with parameters λ1 , λ2 , ...λn ), we deduce from Eqs (8.48-8.51) that the
274
8. The Mosaic Model
same results as for the Poisson mosaic (with replacing the parameter λ by λ1 + λ2 + ... + λn ), are obtained for the second and third order central correlation functions, which are not sensitive to the superimposition of scales (except for the value of the parameter). This not true any more for higher order moments. Finally consider the case of a two scale random set. Using the results of exercise 3.10.3, 2
W 2 (h) = r(h)(Eθ {Cθ (h)} − Z ) W 3 (h1 , h2 )
(8.52) (8.53) 3
= s1 (h1 , h2 )(Eθ {Pθ (h1 , h2 )} − Z(Eθ {Cθ (h1 ) + Cθ (h2 ) + Cθ (h3 )}) + 2Z )
9 Boolean Random Functions
Abstract: The Boolean RF are a generalization of the Boolean RACS. Their construction based on the combination of a sequence of primary RF by the operation ∨ (supremum) or ∧ (infimum), and their main properties (among which the supremum or infimum infinite divisibility) are given in the following cases: scalar RF built on a Poisson point process and on Poisson varieties; multivariate case to simulate multispectral data.
9.1 Introduction This chapter reviews a family of RF with of wide use for applications, the Boolean RF. An abridged version was published in [311]. It owns the interesting property of supremum (or infimum, according to the chosen type of construction) infinite divisibility. These models are particularly interesting for applications in physics, such as illustrated in chapter 20 about fracture statistics. The basic idea of the Boolean RF (BRF) was born about the modelling of rough surfaces by D. Jeulin (1979), by a generalization of the Boolean model of G. Matheron. The first presentations and applications are given in [326], [598]. In [264] an anisotropic version is developed. Out of the field of materials, other examples of applications are given for biomedical images [559], [558], [561], for Scanning Electron Microscope images [601], and for solving problems of exploitation of oceanographic reserves [111]. The first theoretical studies of the BRF are given in [326], [598], [264], © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Jeulin, Morphological Models of Random Structures, Interdisciplinary Applied Mathematics 53, https://doi.org/10.1007/978-3-030-75452-5_9
275
276
9. Boolean Random Functions
[599]. In [600], [601], J. Serra introduces a general BRF model, connected to a non stationary Poisson point process in Rn+1 . In [560],[558], F. Preteux and M. Schmitt proved some characteristic properties of the BRF, useful for the identification of a model from images. Finally, a generalization of the BRF at two levels was proposed and developed by D. Jeulin [274], [275]: introduction of Boolean varieties RF (including the Poisson point process as a particular case), and of the multivariate case. In what follows, the main properties of the BRF built on the Poisson point process are reviewed; then are introduced the Boolean varieties RF, and the multivariate case.
9.2 Construction of the Boolean random functions BRF This section concerns Boolean RF with support in the Euclidean space Rn . n Denote by μn (dx) and θ(dt) R the Lebesgue measure in R and a σ finite measure on R (such that B θ(dt) remains finite for every bounded Borel set B in R). We consider: • i) a Poisson point process P, with the intensity measure μn (dx) ⊗ θ(dt) in Rn × R; • ii) a family of independent lower semi continuous primary RF Zt0 (x), 0 with a subgraph Γ Zt = A0 (t) having almost surely compact sections AZt0 (z). Definition 9.1. The Boolean random function (BRF) with the primary function Zt0 (x) and with the intensity μn (dx) ⊗ θ(dt) is the RF Z(x) obtained by Z(x) = ∨(tk ,xk )∈P {Zt0k (x − xk )} (9.1) This process of construction is illustrated in Fig. 9.1. A possible physical interpretation of this construction is a random punching process digging holes on an initially flat surface. An example of such a process is given in section 9.10. An example of simulation with cone primary functions is shown in Fig. 9.2. Concerning this definition of the model, we report the following remarks. • i) This definition, given in [275], is more general than the one proposed by J. Serra in [600], [601]; it covers the previous definitions: — the Boolean islands, for which the measure θ(dt) is the Dirac distribution concentrated in a point t of R: θ(dt) = θδ t [326], [598],[264];
9.2 Construction of the Boolean random functions BRF
277
FIGURE 9.1. Illustration of the construction of a 1D Boolean random function
FIGURE 9.2. Boolean random function with cone primary functions (image 800 × 800)
— the ”generalized” BRF, where Zt0 (x) = Yt0 (x)+t, Yt0 (x) being a family of primary RF. The addition of t comes from a definition of the BRF as a non stationary Boolean RACS in Rn+1 with Poisson germs in Rn+1 and with primary random sets A0 (t) defined at the origin (0, 0) of the coordinates of Rn+1 . To introduce BRF on more general lattices, where the addition is not necessarily defined, this construction process cannot be used. • ii) The parameter t, that can be assimilated to z in the definition [600], [601], can also be interpreted as a time, leading to the notion of sequential RF (as developed in chapters 11, 12, 13). In these conditions, for the time interval (t, t + dt) is defined an infinitesimal BRF. • iii) It is possible to parametrize the primary functions by t ∈ Rk , with a σ−finite measure θ(dt) on Rn−k . This enables us to introduce a primary function depending on several indexes. Instead of Rk , an abstract space E and a measure θ defined on E can be chosen. Similarly, the Lebesgue measure on Rn , μn (dx), can be replaced by a σ−finite measure θ(dx) on
278
9. Boolean Random Functions
Rn , dropping the stationarity in Rn . This process can be used to build primary RF, as illustrated in chapter 14. • iv) From the definition (9.1), the ”floor” value of Z(x) is −∞. This value can be bounded (z0 ) by use of primary functions such that AZt0 (z0 ) = Rn , or by taking Y (x) = z0 ∨ Z(x). • v) From lower semi-continuous primary functions Z 0 (t) (with overgraph ΓZt0 ), it is possible to build a ∧ BRF, by replacing in Eq. (9.1) the operation ∨ by ∧. It is equivalent to build a ∨ BRF Y from the primary RF Yt0 (x) = −Zt0 (x) and to consider as a ∧ BRF Z(x) = −Y (x). For this reason, we limit this presentation mainly to the ∨ BRF given by Eq. (9.1). • vi) From the point of view of subgraphs (closed in Rn+1 for lower semicontinuous functions), the relation (9.1) should be compared to Eq. (6.1) defining the Boolean RACS in Rn : Γ Z = ∪(tk ,xk )∈P A0 (tk )xk
(9.2)
9.3 Choquet capacity of the BRF As mentioned in chapter 2 (Eq. (2.36)), a BRF is characterized by means of the functional T (g) defined on lower semi-continuous functions g with a compact support K : T (g) = P {x ∈ DZ (g)}; DZ (g)c = {x, Z(y) < g(y − x), ∀y ∈ K} with (see Eq. 2.41 in chapter 2) ˇg (z)) DZ (g) = ∪z∈R (AZ (z) ⊕ B Since DZ1 ∨Z2 (g) = DZ1 (g) ∪ DZ2 (g), for a BRF Z(x) : DZ (g) = ∪(tk ,xk )∈P DZt0 (g)xk k
(9.3)
and DZ (g) is a Boolean RACS with the primary grain DZt0 (g). Since DZ (g) corresponds to the event Ac (Z) = {∃y ∈ Rn , Z(y) ≥ g(x − y)}, the two following theorems result. Theorem 9.1. Consider a BRF Z(x) and a lower semi continuous function g translated in x. The number of primary functions Zt0 for which the event Ac (Zt0 ) is satisfied, follows a Poisson distribution with parameter Z R
μn (DZt0 (g)) θ(dt).
Theorem 9.2. The Choquet capacity of the BRF Z(x) is given by:
9.3 Choquet capacity of the BRF
µ Z ¶ 1 − T (g) = Q(g) = exp − μn (DZt0 (g)) θ(dt)
279
(9.4)
R
As particular functions g, let us examine the following cases: • i) If g(xi ) = zi for points xi (i = 1, 2, ..., m), and else g(x) = +∞, the spatial law is obtained: 1− µ T Z(g) = P {Z(x1 ) < z1 , ..., Z(xm ) < zm }
= exp −
R
μn (A (z1 )x1 ∪ ... ∪ A (zm )xm ) θ(dt) Zt0
Zt0
¶
(9.5)
For a single point x, is obtained the cumulative distribution function F (z) µ Z ¶ F (z) = P {Z(x) < z} = exp − μn (AZt0 (z)) θ(dt) (9.6) R
For two points x and x + h, Eq. (9.5) gives the bivariate distribution F (h, z1 , z2 ) as a function of the cross geometrical covariogram K(h, z1 , z2 , t) between the two sets AZt0 (z1 ) and AZt0 (z2 ) : F (h, zµ 1 , zZ 2 ) = P {Z(x) < z1 , Z(x + h) < z2 } ¶
= exp −
μn (AZt0 (z1 ) ∪ AZt0 (z2 )−h ) θ(dt) µZ ¶ = F (z1 )F (z2 ) exp μn (AZt0 (z1 ) ∩ AZt0 (z2 )−h ) θ(dt) R µZ ¶ = F (z1 )F (z2 ) exp K(h, z1 , z2 , t) θ(dt) R
(9.7)
R
From Eq. (9.7), it is clear that for the BRF we always have F (h, z1 , z2 ) ≥ F (z1 )F (z2 ), so that no negative correlation can occur. More generally, for the spatial law, from Eq. (9.5): P {Z(x1 ) < z1 , z2 ..., Z(xm ) < zm } ≥ F (z1 )F (z2 )...F (zm ) • ii) If g(x) = z for x ∈ K and else g(x) = +∞, K being a compact set, from Eq. (9.4) is given the distribution of Z(x) after a change of support by the operator ∨ taken over the compact set K (Z∨ (x) = ∨x∈K {Z(x)}); ˇ and in that case DZt0 (g) = AZt0 (z) ⊕ K µ Z ¶ ˇ 0 P {Z∨ (K) < z} = exp − μn (AZt (z) ⊕ K) θ(dt)
(9.8)
R
From the definition (9.1) and from Eq. (9.3), the following result is obtained: Proposition 9.1. The RF Z∨ (x) is a BRF with the primary function Z∨0 (x).
280
9. Boolean Random Functions
All previous results can be specialized to the Boolean island version of the model, when θ(dt) = θδ 0 (t) and Z00 = Z 0 . Starting form the Choquet capacity 9.4, the spatial law 9.4 and Eq. ( 9.8), 1 − T (g) = Q(g) = exp (−θμn (DZ 0 (g))) P {Z(x1 ) < z1 , ..., Z(xn ) < zn } = exp (−θμn (AZ 0 (z1 )x1 ∪ ... ∪ AZ 0 (zn )xn ))
(9.9)
The bivariate distribution is given by F (h, z1 , z2 ) = F (z1 )F (z2 ) exp (θK(h, z1 , z2 ))
(9.10)
and the change of support by the operator ∨ follows ¡ ¢ ˇ P {Z∨ (K) < z} = exp −θμn (AZ 0 (z) ⊕ K)
(9.11a)
One should notice that with the definition given in [600], [601], the Choquet capacity is obtained after replacing in Eq. (9.4) DZt0 (g) by DZt0 (g − t) and in Eq. (9.5) AZt0 (zk )xk by AZt0 (zk − t)xk .
9.4 Supremum stability and infinite divisibility Let Z1 (x) and Z2 (x) be two independent BRF with the primary functions 0 0 Z1t and Z2t , and the intensities θ1 (t) and θ2 (t). From Eq. (9.2) is obtained: Γ Z = Γ Z1 ∪ Γ Z2 = ∪(tk ,xk )∈P1 A01 (tk )xk ∪(tk ,xk )∈P2 A02 (tk )xk and therefore Γ Z is a Boolean model in Rn+1 ; as a consequence, Z(x) is a BRF with intensity θ(t) = θ1 (t) + θ2 (t). Proposition 9.2. Every supremum P of a family of independent BRF Zi (x) is a BRF with intensity θ(t) = i θi (t). As a consequence of the infinite divisibility of the Boolean model for ∪,
Theorem 9.3. Every BRF Z(x) is infinite divisible for ∨ : ∀n, Z(x) ≡ ∨k=n k=1 Zk (x) where the Zk are independent BRF with the same law. This results immediately from the expression of the Choquet capacity of the BRF 9.4 and from the Choquet capacity of ∨k=m k=1 Zk (x) derived from Eq. (2.45) in chapter 2: for any integer m, µ Z ¶m θ(dt) 1 − T (g) = Q(g) = exp − μn (DZt0 (g)) = Q1/m (g)m m R
9.4 Supremum stability and infinite divisibility
281
The Boolean RF can also be obtained as a limit model from a supremum superimposition of random mosaics [600], generalizing the results of [156], [598] for the Boolean random set: Proposition 9.3. Consider the supremum superimposition Zn of n independent mosaic RF Zi in Rd , built from an ergodic stationary random tessellations in Rd with a random class A0 , by a random affectation of independent realizations of a RV Z 0 to any class with the probability pn . If the θ probability pn satisfies pn = for a finite value θ, the RF Zn = ∨i=n i=1 Zi n θ converges, for n → ∞, towards a Boolean RF with intensity and μn (A0 ) 0 0 0 with primary RF Z (x) = Z for x ∈ A . The converse proposition is not true (for instance a BRF with spherical primary RF cannot be obtained as a limit case from random tessellations...). Proof. From results concerning the mosaic model (Eq. (8.6) in chapter 8, we have for the random set An = AZn (z) Qn (K) = P {K ⊂ Acn } = GK (1 − pn (1 − G(z)))n When p → 0, GK (1 − pn (1 − G(z))) =
j=∞ X j=0
pj (K)(1 − pn (1 − G(z)))j
and GK (1 − pn (1 − G(z))) → n→∞ 1 −
j=∞ X j=0
pj (K) jpn (1 − G(z))
= 1 − pn (1 − G(z))E{N (K)} where N (K) is the average number of classes of the random tessellation hit by K. Therefore, µ ¶n θ lim Qn (K) = 1 − E{N (K)}(1 − G(z)) p→0 n and lim Qn (K) = exp(−θE{N (K)}(1 − G(z))
n→∞
is obtained by use of the ergodic behavior of the random tessellation: let N be the number of classes hit by a large domain D. We have
282
9. Boolean Random Functions
N (K) = N P {K hits A0 } = N As when μn (D) → ∞,
N 1 → we get μn (D) μn (A0 ) E{N (K)} =
and
μn (A0 ⊕ K) μn (D)
μn (A0 ⊕ K) μn (A0 )
µ ¶ μ (A0 ⊕ K) 1 − T (K) = Q(K) = exp −θ n (1 − G(z)) μn (A0 )
which is the Choquet capacity of a BRF with primary function Z 0 (x) = Z 0 for x ∈ A0 .
9.5 Characteristics of the primary functions Some characteristics of the pair (intensity, primary function) can be determined from information on the BRF Z(x). These characteristics are directly deduced from the Choquet capacity (9.4) and from the derived properties (9.5-9.7).
9.5.1 Transformation by anamorphosis Let ϕ be an anamorphosis transformation (namely a monotonous nondecreasing transformation applying R into R). Let Y = ϕ(Z). Proposition 9.4. Every anamorphosis of a BRF, ϕ(Z), is a BRF obtained with the same intensity θ(t) and with the primary function Y 0 = ϕ(Z 0 ). Proof. We have Aϕ(Z) (z) = {x; ϕ(Z(x)) ≥ z} = {x; Z(x) ≥ ϕ−1 (z)} = AZ (ϕ−1 (z)) = ∪(tk ,xk )∈P AZt0 (ϕ−1 (z))xk k = ∪(tk ,xk )∈P AYt0 (z)xk = AY (z) k
This result enables us to restrict our study to strictly positive BRF, since it is always possible to transform any function Z into a positive function Y = ϕ(Z) (consider for instance the anamophosis obtained by an exponential transformation).
9.5.2 Moments of Z∨0 (K) and mathematical expectation of the anamorphosed of Z∨0 (K) Consider now positive BRF.
9.5 Characteristics of the primary functions
Proposition 9.5. We have: Z M (i, K) = − z i−1 log (P {Z∨ (K) < z}) dz R ∙Z ¸ Z 1 0 E (Zt∨ (K)(x))i dx θ(dt) = i R+ Rn Let Φ(z) be a strictly positive function. We have: Z − ϕ(z) log (P {Z∨ (K) < z}) dz ¸ Z R ∙Z 0 = E Φ(Zt∨ (K)(x)) dx θ(dt) R+
with Φ(z) =
Z
283
(9.12)
(9.13)
Rn
z
ϕ(u) du. 0
Proof. Denote by 1Zt0 ≥z (x) the indicator function of the set AZt0 (z) at point x (1Zt0 ≥z (x) = 1 if Zt0 (x) ≥ z and else 1Zt0 ≥z (x) = 0). For a given realization of the primary function, Z μn (AZt0 (z)) = 1Zt0 ≥z (x) dx Rn
and
Z
z
i−1
R+
1
Zt0 ≥z
(x) dz =
n
Z
Zt0 (x)
z i−1 dz =
0
(Zt0 (x))i i
By integration in R , Z Z (Zt0 (x))i z i−1 μn (AZt0 (z)) dz dx = i Rn R+ and by taking the mathematical expectation ½Z ¾ Z (Zt0 (x))i E z i−1 μn (AZt0 (z)) dz dx = i Rn R+ The moment M (i) is deduced by integration of the last expression over the measure θ(dt), and similarly for the moment M (i, K), after replacing Z and Z 0 by Z∨ (K) and by Z∨0 (K). Then, Z
R+
ϕ(z)1Zt0 ≥z (x) dz =
Z
Zt0 (x)
ϕ(z) dz = Φ(Zt0 (x))
0
and by integration in Rn Z Z 0 ϕ(z)μn (AZt (z)) dz = R+
Rn
Φ(Zt0 (x)) dx
284
9. Boolean Random Functions
After taking the mathematical expectation and after integration over θ(dt) the expression (9.13) is immediate for Z and for Z∨ (K).
9.5.3 Geometrical covariogram of the primary function Starting from the bivariate distribution given in Eq. (9.7), for z = z1 = z2 we obtain for a positive RF Z Z ∞ ¡ ¢ log P {Z(x) < z, Z(x + h) < z}/(F (z))2 0Z Z ∞ Z (9.14) K(h, z, z, t)θ(dt) dz = K(h, t)θ(dt) = R 0
R
with the notation K(h, t) = μn+1 (A0+ (t) ∩ A0+ (t)−h ) for the geometrical covariogram in Rn+1 of the positive part of the subgraph of Zt0 , A0+ (t). The Eq. (9.14) may be useful for the identification of primary functions from K(h, t), often simpler for calculations than the bivariate distribution deduced from the cross geometrical covariogram K(h, z1 , z2 , t).
9.6 Some stereological aspects of the BRF As for the Boolean model, a BRF defined in Rn generates by section in Rk (k < n) BRF with induced intensity and primary functions. This is a property connected to the Poisson point process. For some families of primary functions (for instance when the positive part of the subgraph is made in Rn+1 of spheres, similar cylinders, or similar parallelepipeds,...), it is possible to estimate the properties of the primary functions (up to the intensity), from the sole bivariate distribution known on profiles, through the Z function
K(h, t)θ(dt). As far as these primary functions are well suited
R
to real data, it can be relatively easy to implement them in applications, as done in chapter 6 with the Boolean model of spheres.
9.7 BRF and counting In this section, consider digital images with support in R2 , modelled by Boolean island BRF. Z ∞ • i) As in [600], [601], assume that the integral V = μ2 (AZ 0 (z)) dz 0
is known from a preliminary study. We wish to estimate θ for images considered as realizations of BRF with intensity θ. We can use here the numerical version of the algorithm presented in chapter 3 for the
9.7 BRF and counting
285
Boolean model, derived from the expression − log q = θK(0); from the distribution function F (z), Z ∞ − log (F (z)) dz = θV (9.15) 0
This counting algorithm is very convenient, since it does not require any segmentation or any choice of a threshold. Well suited to Boolean textures, it is weakly sensitive to noise, but it is sensitive to illumination conditions (through V ), which should remain strictly constant between a standard experiment (to estimate V ) and an image acquisition for counting. • ii) If the primary function Z 0 (x) owns convex sections AZ 0 (z), and if the average value of its single maximum is known (E{Z 0 }), a similar approach as for the Boolean model can be followed: C(z) = θ(1 − G(z)) F (z)
(9.16)
where C(z) is the convexity number of AZ 0 (z) (estimated from measurements) and G(z) is the distribution function of the maximum of Z 0 (x). From Eq. (9.16), Z ∞ C(z) (9.17) dz = θE{Z 0 } 0 F (z)
• iii) When the primary function Z 0 (x) has a single maximum, with unknown pdf g(z), the following relationship results [264]: g1 (z) =
θ g(z) F (z) NS
(9.18)
where g1 (z) is the pdf of the observed maxima of the BRF (that can be estimated experimentally), and NS is the number of maxima per unit of area. From Eq. (9.18) can be deduced g1 (z) (9.19) F (z) Z ∞ After integration in z of Eq. (9.19) (with g(z) dz = 1) is deduced θg(z) = NS
θ, and then g(z).
0
Irrespective of their domain of validity (concerning the assumptions on the primary function), the methods ii) and iii), proposed in [272], must be applied to images without noise, any counting of a convexity number or of a number of maxima being sensitive to noise.
286
9. Boolean Random Functions
9.8 Identification of a BRF model To identify a BRF from data, both the family of primary function Zt0 (x) and the measure θ(dt) must be known. However, the Choquet capacity (9.4), experimentally estimated from realizations of BRF, depends on the product of two factors: the intensity and a measure on the primary function. It is therefore not possible to know these two terms separately from their product, so that we have to face an indetermination. To raise it, we rely on the following results proved by M. Schmitt and F. Preteux [559], [560], [588]. We denote by (Zt0 , θ) the BRF defined by a choice of the primary function Zt0 and of the intensity θ(dt). Proposition 9.6. Characterization of a BRF. Consider a BRF (Zt0 , θ); i) If θ(R) = θ < ∞, the BRF admits a unique representation as a Boolean island (Z 0 , θδ), where Z 0 is centered on the projection on the plane z = 0 of the center m of the sphere in Rn+1 circumscribing the maxima of the primary function; ii) If θ(R) = +∞, the BRF can be uniquely represented by (Zt0 , θ), where the Zt0 are centered in m and where Z(x) = ∨(tk ,xk )∈P {Zt0k (x − xk ) + tk }. From experimental data, there is always access to a bounded range of variation for Z(x). We can therefore mostly consider Boolean islands. It will be the same situation for simulations. However, at the level of a theoretical model, it is often interesting to consider the case ii) with θ(R) = +∞. For instance the following BRF can be used (chapter 20): • The Weibull model is obtained by implantation of primary functions Zt0 (x) with a point support (Zt0 (x) = tδ(x), δ(x) being the Dirac distribution in Rn ) and θ(dt) = θm(z0 − t)m−1 for t ≤ z0 ≤ 0. With this definition, the BRF differs from −∞ on points of a Poisson process. It cannot be characterized by its spatial law, which is equal to zero. Use must be made of the Choquet capacity (9.4) for functions g having a support with non zero measure in Rn . For instance, the distribution function of Z∨ (K) is derived from Eq. (9.8): Z z0 ˇ dt P {Z∨ (K) < z} = exp − θm(z0 − t)m−1 μn (AZt0 (z) ⊕ K) −∞
ˇ =K ˇ for t ≥ z, else = ∅. It comes: with AZt0 (z) ⊕ K P {Z∨ (K) < z} = exp −θ(z0 − z)m μn (K)
(9.20)
In fracture statistics, the variable of interest is Z > 0 (the fracture stress), and use is made of the BRF Y (x) = −Z(x), which can be directly obtained with the intensity θ(dt) = θm(t − z0 )m−1 dt (t ≥ z0 ), by means of the operator ∧ instead of ∨, and starting from the value +∞ outside of the Poisson point process in Rn . For z ≥ z0 ,
9.9 Test of the BRF
P {Z∧ (K) ≥ z} = exp −(θ(z − z0 )m μn (K))
287
(9.21)
• The Pareto model is obtained with the same construction as the Weibull −θdt model, with the intensity θ(dt) = for t ≤ z0 ≤ 0 and else θ(dt) = 0: t Z z0 ˇ dt P {Z∨ (K) < z} = exp θμn (AZt0 (z) ⊕ K) t Z −z−∞ (9.22) 0 dt ¡z0 ¢θμn (K) = exp θμn (K) = z t −z
Using the operator ∧ instead of ∨, and starting from the value +∞ outside of the Poisson point process in Rn . For z ≥ z0 µ ¶θμn (K) z0 P {Z∧ (K) ≥ z} = z
(9.23)
9.9 Test of the BRF Tests proposed for testing the BRF model are derived from tests proposed for the Boolean model in chapter 6. In a first step, it is possible to work on sets obtained by applying thresholdsZon Z(x) at different levels zi , which (1 − Gt (zi ))θ(dt), where Gt (z) is
are Boolean models with intensity R
the distribution function of the maximum of Zt0 (x). Other tests can be directly applied to the function Z(x). They involve the following criteria: convexity of the sections AZt0 (z), change of support on convex sets, and infinite divisibility for ∨.
9.9.1 Convexity of AZt0 (z) This is the most often used test used in applications until now. It is based on an additional assumption, the convexity of the sections of the primary function, AZt0 (z). This is not satisfied in the general case. The test makes use of the Steiner formula to the distribution (9.8) P {Z∨ (K) < z} when K is a compact convex set. In these conditions, log(P {Z∨ (λK) < z}) and Z
similarly k
R
log(P {Z∨ (λK) < z}) dz are polynomials of degree k in λ for
K⊂R . It is easy to implement these tests, since they only require the estimation of the distribution functions after change of support by the operator ∨ on convex sets with increasing sizes λK. The first test, based on a threshold z, is the same as for the Boolean random set model. The second test may be the source of numerical difficulties, since we may obtain P {Z∨ (λK)
1.45) [149], [152]
required to display the full sequence on videos rather than on fixed images of the present illustrations. The linear stability analysis of these models is studied in the references, and not reproduced here.
16.4.1 Schlögl model The Schlögl model [587] is a three-species chemical scheme, based on four elementary reactions. This system can be reduced to a single component Zt (x), where the chemical reaction term is a polynomial of degree three in the concentration of Zt (x), leading to an unstable steady state Z = a0 and to two stable states Z = a1 and Z = a2 , with a1 < a0 < a2 . Z(x, t) is solution of : ∂Zt (x) (16.20) = D 4 Z + k0 − k1 Z + k2 Z 2 − k3 Z 3 ∂t with k0 = 0.002, k1 = 0.0390, k2 = 0.07125, k3 = 0.030625 in the present simulations Such a nonlinear dynamics is at the origin of an auto-catalytic behavior. When the system is initially set in the unstable state a0 with the addition of random noise, a bifurcation occurs. On a macroscopic level, spatially organized parts of the domain correspond to one of the stable states, as shown in Fig. 16.2. These spherical type regions grow and progressively coalescence, until a single state fills the whole domain. In two dimensions, similar random textures have already been simulated using a cellular automaton [137], [138] or a lattice gas [150], as illustrated in section 16.5.
16.4 Examples of simulations of non- linear Reaction-Diffusion Random Functions
569
16.4.2 Turing structures This class of reaction-diffusion models are at the origin of Turing type structures [640]. At least a two-species chemical scheme is required to produce such complex structures. One of the species acts as an activator with diffusion coefficient D1 , whereas the other component acts as an inhibitor with diffusion coefficient D2 À D1 , and hindering the spreading of the slower activator by its chemical action. The resulting patterns show a very low evolution rate and are therefore considered to be quasi-stable. Depending on the model, but also on its parameters and the initial concentrations, Turing structures show various typical spatial heterogeneities, with a generation of discontinuities and almost binary structures. They are mostly pseudo-periodic, with tilings, parallel bands with dislocations, mazes or stripes [77]. These models are particularly interesting for the simulation of ”natural” textures by morphogenesis in physics like metal alloys, in biology like furs of mammals, fishes or birds dress, shells, butterfly wings, etc... [511]. In R3 they can model tortuous random porous media. As examples are shown below some results for the Walgraef, the Maginu and the Brusselator models. Walgraef model The symmetrical model studied by D. Walgraef [652] is based on the following equations for two species: ⎧ ∂Z 1 ⎪ ⎨ ∂t = D1 4 Z1 + f (Z1 , Z2 ) ∂Z2 (16.21) ∂t = D2 4 Z2 − f (Z1 , Z2 ) ⎪ ⎩ 2 f (Z1 , Z2 ) = β Z2 Z1 − α Z1
The scales of resulting structures are function of the ratio D1 /D2 . Fig. 16.3 shows a binary structure resulting from a 3D simulation. This type of structure is fully connected, with a single connected component for each phase. Anisotropic structures are obtained by introducing a linear drift in the diffusion process, as shown in Fig. 16.4. A similar extension of this model was proposed to simulate dislocations in metals [651].
The Maginu model The Maginu model [433] involves two concentrations, solution of the following equations: . ( ∂Z1 Z1 3 ∂t = D1 4 Z1 + Z1 − 3 − Z2 (16.22) ∂Z2 Z1 − k Z2 ∂t = D2 4 Z2 + c
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16. Reaction-Diffusion and Lattice Gas Models
FIGURE 16.3. 3D simulation of a Turing structure (Walgraef symmetrical model; 10000 iterations; domain 2003 ; Z(x, t) > 1.8) [149], [152]
FIGURE 16.4. Anisotropic Turing structure (Walgraef model with a drift in the Ox direction; 500 iterations; domain 2003 ; Z(x, t) > 1.8) [149], [152]
Depending on parameters k, c and D1 /D2 , it generates Turing structures or oscillating chaotic behavior. The condition of spatial instability is given by: r √ √ (1 − 1 − k) c D1 ≤ with 0 < c < k D2 k In two dimensions, maze-like patterns with a high tortuosity are generated. For a higher dimension, intricate structures are obtained, as shown in the realization presented in Fig. 16.5, which looks like a highly branched and connected three dimensional network. The Brusselator model The Brusselator model was proposed by I. Prigogine and R. Lefever in [562]. The reactive system is generated by the following chemical reactions, where X is an activator (species 1), and Y an inhibitor (species 2):
16.4 Examples of simulations of non- linear Reaction-Diffusion Random Functions
571
FIGURE 16.5. 3D simulation of the Maginu model (domain 2003 ; 10000 iterations; Z1 > 0.6; k = 0.9; c = 0.45; D2 /D1 = 6) [149], [152]
FIGURE 16.6. 3D simulation of the Brusselator model (50000 iterations; domain 2003 ; α = 2.3, β = 3.4, D2 /D1 = 8; Z1 > 3.7) [149], [152]
k1
A → X,
k2
B + X → Y + D,
k3
2X + Y → 3X,
The two species concentrations are solutions of: ( 2 ∂Z1 ∂t = D1 4 Z1 + α − (β + 1) Z1 + Z1 Z2 ∂Z2 ∂t
= D2 4 Z2 + β Z1 − Z1 2 Z2
k4
X→E
(16.23)
Pseudo-periodic Turing structures can be generated, like in 2D hexagonal tiling of discs, with defects. Fig. 16.6 gives an example of such structures in 3D, which can be compared with a random set based on the implantation of spherical primary grains with a repulsion distance (namely, a hard-core model).
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16. Reaction-Diffusion and Lattice Gas Models
FIGURE 16.7. 3D simulation of the complex Ginzburg-Landau model (spiral waves, 2003 ) [149], [152]
16.4.3 The complex Ginzburg-Landau model This model gives an example of reaction-diffusion structures with a very different morphology. Many numerical studies of this model were published in one or two dimensions [398]. The complex time-dependent GinzburgLandau equation describes the evolution, following a Hopf bifurcation, of the slowly varying modulations in an extended system. It arises from the analysis of numerous physical situations like hydrodynamic flows, nonlinear optics, or uncommon cases of chemical systems. It describes a reactive system with two species of concentrations Z1t (x) and Z2t (x), having a representation in the complex plane Z(Z1 , Z2 ), solution of: ∂Z ∂t
2
= D 4 Z + A Z − B |Z| Z ⎧ ⎨ Z = Z1 + i Z2 A=α + iγ with ⎩ B=β + iδ
(16.24)
This equation admits spiral wave solutions, with oscillating and rotating behaviors (see a population of coupled spirals structures on a 3D simulation 2 in Fig. 16.7 and 16.8). The image of the modulus |Z| (Fig 16.9) contains local minima corresponding to the axis of the spiral structures (Fig. 16.10). A high level threshold of |Z|2 provides the boundaries of a population of cells of a space tessellation (Fig. 16.11), each cell containing a unique spiral structure. The size of cells is controlled by the diffusion coefficient D. During their slow evolution, these cells interact, with attraction phenomena, and in some cases collapse, reducing progressively the number of cells.
16.4 Examples of simulations of non- linear Reaction-Diffusion Random Functions
FIGURE 16.8. Binary image of spiral waves obtained from Fig. 16.7 [149], [152]
FIGURE 16.9. Modulus of 3D simulation corresponding to Fig. 16.7 [149], [152]
FIGURE 16.10. Modulus of Fig. 16.7, with with |Z|2 < 0.18 [149], [152]
573
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16. Reaction-Diffusion and Lattice Gas Models
FIGURE 16.11. Modulus of Fig. 16.7, with |Z|2 > 0.93 [149], [152]
16.4.4 Modifications of the Ginzburg-Landau model New models of non-linear Reaction-Diffusions were proposed by addition of one or two species to the original two species model [149]. Their interest is their ability to generate new morphologies that depend on the choice of their parameters.
A three species propeller model A third species with concentration Z3t (x) is added to the two species of the Ginzburg-Landau model, according to the following equations: ⎧ ∂Z1 2 ⎪ ∂t = D1 4 Z1 + αZ1 − γ 1 Z2 + (δ 1 Z2 − Z1 )|Z| ⎪ ⎨ ∂Z 2 2 ∂t = D2 4 Z2 + αZ2 − γ 1 Z1 − (δ 1 Z1 + Z2 )|Z| 2 ∂Z3 ⎪ ⎪ ∂t = D3 4 Z3 + (δ 2 Z1 − γ 2 Z2 )|Z| ⎩ 2 2 2 2 with |Z| = Z1 + Z2 + Z3
(16.25)
where α, γ 1 , γ 2 , δ 1 and δ 2 are control parameters. The square of the modulus |Z|2 introduced in Eq. (16.24) was modified by the introduction of Z3 . The reactive terms F1 and F2 are left unchanged, while F3 depends on Z3 only through |Z|2 . An example of simulation is given in Fig. 16.12, showing rotating propellers (like rotating spirals in the Ginzburg Landau model) generated on the interface between two components.
A four species membrane model A four species model is obtained by similar reactive functions as for the Ginzburg Landau model, with a permutation of variables:
16.4 Examples of simulations of non- linear Reaction-Diffusion Random Functions
575
FIGURE 16.12. 2D simulation of a propeller model (component Z3 ; 15000 iterations; domain 600 × 600; α = 1.5, δ 1 = 1.4, δ2 = 1, γ 1 = γ 2 = 1, D = 0.5) [149]
FIGURE 16.13. 2D simulation of the membrane model (component Z2 ; 60000 iterations; domain 600 × 600; α1 = α2 = β 1 = γ 1 = 1, β 2 = 2.0, γ 2 = δ 2 = −1, , D = 0.5) [149]
⎧ 2 ∂Z1 ⎪ ⎪ ∂t = D1 4 Z1 + α1 Z2 − α2 Z4 + (β 2 Z2 − β 1 Z1 )|Z| ⎪ ⎪ ∂Z2 = D 4 Z + α Z + α Z + (β Z + β Z )|Z|2 ⎪ ⎨ ∂t 2 2 1 1 2 3 2 1 1 2 2 ∂Z3 = D 4 Z + γ Z − γ Z + (δ Z − δ 3 3 2 4 1 Z3 )|Z| 1 1 2 4 ∂t ⎪ 2 ⎪ ∂Z4 ⎪ ⎪ ∂t = D4 4 Z4 + γ 1 Z3 + γ 2 Z2 + (δ 2 Z3 + δ 1 Z4 )|Z| ⎪ ⎩ with |Z|2 = Z12 + Z22 + Z32 + Z42
(16.26)
In the simulation shown in Fig. 16.13 the domain is split in two regions with homogeneous concentrations, that can be considered as two phases. With the chosen set of parameters (with δ 1 = 1.1), initially thick rotating regions become thinner and thinner under some kind of stretching effect, despite the absence of any mechanical stresses in the system. The membrane aspect of the obtained structure is clearly apparent in Fig. 16.13. For a slightly lower value of δ 1 (0.9), spirals appear after 7000 iterations, as in the Ginzburg Landau model.
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16. Reaction-Diffusion and Lattice Gas Models
16.5 Lattice gas models Another way to digitize these models, even on a more microscopic level, is the use of the lattice gas models. Here particles with a unit velocity and mass move on the vertices of a graph. This idea goes back at least to J. E. Broadwell [92], and was considerably extended to solve hydrodynamics problems, mainly in the field of turbulence at the end of the eighties [229], [191], [192]. A synthetic presentation on lattice gas models is provided in [579]. In this section, we give some elementary properties of these models, and illustrate them with some applications to random media.
16.5.1 Basic rules In two dimensions, the FHP model proposed by U. Frisch, B. Hasslacher and Y. Pomeau [191] is built on a hexagonal lattice; a population of particles having seven possible velocities (unit velocity with one of the six possible directions on the lattice, or null velocity) moves on the lattice. At most one particle per velocity is allowed on every point of the lattice, which corresponds to an exclusion principle. Therefore the gas can be described by a multi-component random sets, or by seven binary images (one per velocity), setting the so-called "molecular dynamic of the poor", as suggested by U. Frisch. During each time step, every particle moves to its nearest neighbor in the direction of its velocity; this results in a translation T of the corresponding binary images in the appropriate directions. In addition to the translation of particles, rules of interaction between particles are required, namely collision rules through the operator C for the particles of a gas. These redistribute the velocities with the following constraints: preservation of the total mass and momentum (and consequently of the kinetics energy) at every vertex of the graph. The choice of specific rules enables us to change the viscosity of the fluid for simulations. The process is a sequence of cycles involving the propagation of particles and the redistribution of their velocities. It is shown [192] that on a macroscopic scale, the velocity map satisfies the Navier Stokes equations, which enables us to simulate hydrodynamics. The great advantage of this model is its simplicity for implementation, even for complex boundary conditions such as those occurring during the evolution of a microstructure; since only binary images are used, there is no round-off errors, and the process can be iterated indefinitely. Three dimensional simulations were made possible by the construction of a specific lattice [192].
16.5.2 Some indications on the evolution equations We summarize now the main steps and the main assumptions that are used for the derivation of the Navier Stokes equations from the lattice gas model. Complete derivations are available in [192], [200], [614]. Starting
16.5 Lattice gas models
577
from the basic rules, the evolution equations of each population of particles are Boltzmann equations accounting for the balance of particles involved by the translation and the collision rules. At point x and at the time step t + δt we have for the number of particles of type i with the velocity ui : → Ni (t + δt, x + − u i ) = Ni (t, x) + ∆i (N )
(16.27)
Eq. (16.27) can be considered as a finite difference version of Eq. (16.28) below, obtained for δt → 0 and for a → 0, a being the size of the grid: ∂Ni − +→ u i . grad Ni = ∆i (N ) ∂t
(16.28)
When only head-on collisions between two particles occur, cross products between the variables Ni and Nj occur. For triple collisions, Ni Nj Nk are P present, and so on. Together with the preservation of mass ( i=n i=1 Ni = N ) Pi=n − → u ), Eq. (16.28) leads to the continuity and of momentum ( i=1 Ni → u i = N− → and to the momentum equations, where − u is the average velocity: ∂N → + div(N − u)=0 ∂t
(16.29)
∂(N uα ) ∂(Παβ ) + =0 (16.30) ∂t ∂α P where Παβ = k Nk ukα ukβ , ukα being the α component of the velocity − →. The next steps in the calculation uses a Chapman-Enskog expansion of u k −−−−→ Ni (t, x) as a function of the macroscopic variable u(x, t) and of its gradient, up to the second order. It leads to the Navier Stokes equations for incompressible flows, with transport coefficients depending on the ∆i (N ) terms (which reflect the collision rules) and on the solutions of the Boltzmann equations (16.27). These involve a dependence on at least the bivariate distributions of Ni and Nj at point x and time t. Equations for these bivariate distributions involve higher order distributions. To avoid a common regression ”ad infinitum”, new assumptions must be introduced. Usually is made the Boltzmann’s approximation, considering that the variables Ni (x, t) are independent at the order required by the multiple collisions. From this assumption, the Ni (x, t) are solutions of non-linear partial differential equations deduced from Eq. (16.28). Exact general solutions are known only in particular cases [200], [92]. A particular solution in the spatial stationary case for mutually excluding particles, as the binary images used in lattice gas simulations, is given by the Fermi-Dirac distribution, when is added the independence between every Ni (x, t) and Nj (x0 , t0 ). This approximation is claimed to be valid for the limit case of a low particle density N (x, t). In that case it comes 1 Ni = (16.31) → → 1 + exp (h + − q .− u i)
578
16. Reaction-Diffusion and Lattice Gas Models
→ where the vector − q and the constant h are deduced from the mass and momentum conservation equations. When the macroscopic velocity u is close to zero, the Fermi-Dirac distribution degenerates into Ni = d, where d is the average density, which is the low velocity equilibrium solution. An expansion of the Fermi-Dirac solutions in powers of the macroscopic velocity, together with the Chapman-Enskog expansion, gives an estimation of the fluid transport coefficients. Higher order statistics of the random sets generated by the lattice gas populations (such as for instance their covariances) are unknown, despite some attempts to derive them in a similar but different context [75], where only upper bounds of covariances could be found. If the binary velocity field is replaced by digital fields and if no exclusion rule is acting, the Fermi-Dirac distribution is replaced by the Maxwell-Boltzmann distribution → → Ni = exp (−h − − q .− u i)
(16.32)
Instead of working with binary images, some authors developed a model based on the use of local probabilities (0 ≤ Ni ≤ 1) and on the Boltzmann independence assumption in every point x for the collision rule [428], [622]. This model named the LBG (Lattice Boltzmann Gas) has a considerable success, since it provides velocity maps without noise, even for small size systems, contrary to the binary model. However, due to the approximations made in the derivation of the model, there is no guarantee that probabilities are obtained in each step of the calculation.
16.5.3 Boundary conditions Boundary conditions imply the behavior of particles when reaching obstacles: if on a vertex located inside a solid obstacle the collision rule is replaced by a bounce-back condition, the simulated fluid respects the usual no-slip condition (the average velocity being equal to zero on the boundary). On the edges of the field, periodic conditions (particles leaving one side of the field are reintroduced with the same velocity on the opposite side) as well as non periodic conditions (with random injection of particles on the open edges of the field, as made in [276]) can be used.
16.5.4 Application to flow in porous media Starting from images of porous media made of solid grains that cannot be accessed by the fluid particles, it is possible to simulate flows in porous media and to estimate their transport properties [577], [276]. • Firstly, by appropriate boundary conditions, or by randomly imposing a drift in the velocity at points of the fluid (which is an input of impulsion to fight against the dissipation due to the zero velocity on the bound-
16.5 Lattice gas models
579
aries), it is possible to impose a macroscopic pressure gradient to a fluid moving through a porous medium. For a given geometry (as for instance for flows in porous media), since the velocity map solution of the boundary conditions is unknown, we start from independently and uniformly distributed velocities. After some iterations (typically twice the length of the field), the velocity map is stabilized. This approach considering the velocity map is the Euler point of view. When the average velocity is proportional to the pressure gradient, it follows the Darcy’s law relating the macroscopic flux and the pressure gradient, the proportionality factor being the tensor of permeability. Thus it becomes possible to estimate the permeability of a porous medium from lattice gas simulations, using a method that operates at a ”submicroscopic” scale, where basic physical conservation rules are applied. On that scale, no partial differential equation is acting, and the boundary conditions are easily handled. The simulation leaves a population of particles evolve like a dynamic system, until a possible statistical equilibrium is reached. This is a typical simulation of a statistical physics problem. This approach was applied to various Boolean models of porous media, for which were studied the following points [276]: variability of the permeability induced by the distribution of grains in space; influence of the pore area fraction, of the grain size and of the anisotropy of porous media on the permeability. • Secondly, the dispersion in porous media can be studied from the Lagrange point of view: instead of considering the velocity field obtained in the lattice gas simulation, it is possible to mark a given particle and to follow its trajectory with time, which builds a random walk (a diffusion process or a Brownian motion with an advective velocity field u). In the present case, the random walk is just the result of the interaction between the marked particles and the other particles in the fluid, respecting the boundary conditions. During the simulation, the velocity of the particle is chosen at random among the possible velocities after each collision. As in [454], when a macroscopic Fick’s law is observed, the coordinates Xi (t) (i = 1, 2) of the trajectory of the marked particle (starting from x (xi ) at time t = 0) in the random velocity field u(x) are diffusion stochastic processes with expectation and covariance given by E[Xi (t)] = xi + ui t E[(Xi (t) − xi − ui t)(Xj (t) − xj − uj t)] = 2Dij t
(16.33) (16.34)
where ui is the average of the i component of the velocity, while the coefficients Dij build the effective diffusion tensor of an equivalent homogeneous porous medium. The macroscopic coefficients ui and Dij obtained from averages of various particles trajectories are valid for an equivalent homogeneous medium when are fulfilled the conditions for a
580
16. Reaction-Diffusion and Lattice Gas Models
macroscopic Fick’s law to exist; these conditions are unknown in general. For some random media (for instance for self similar RS, and therefore non stationary porous networks), Eqs (16.33, 16.34) are not valid, and a tα (with α 6= 1) behavior is observed (this is called anomalous diffusion). In practice for Boolean porous media simulations, we checked the validity of Eqs (16.33, 16.34) from the following experimental variograms (with X1 (t) = X(t) and X2 (t) = Y (t)), from which a fit to a parabolic curve is looked for: 2γ x (∆t) = E[(X(t + ∆t) − X(t))2 ] = 2Dx ∆t + u2x (∆t)2 2γ(∆t) = 2γ x (∆t) + 2γ y (∆t) = 2(Dx + Dy )∆t + (u2x + u2y )(∆t)2 (16.35) Additional information is obtained from the empirical distribution of the sojourn time τ of the particle in the simulated field, Fτ a (t). Consider particles starting from O and leaving the field on abscissa a at time τ a , we expect in the case of a constant velocity field (ux , uy ) and in an infinite homogeneous medium: Fτ a (t) = P {τ a < t)
(16.36)
Where X(t) ≥ a ⇒ τ a ≤ t and therefore P {X(t) ≥ a | τ a < t} =
P {X(t) ≥ a} Fτ a (t)
(16.37)
1 If ux = 0, by symmetry P {X(t) ≥ a | τ a < t} = and in these 2 conditions (16.38) Fτ a (t) = 2P {X(t) ≥ a) When ux À 0 and a > 0, P {X(t) ≥ a | τ a < t} ' 1 and Fτ a (t) = P {X(t) ≥ a)
(16.39)
In Eqs (16.38, 16.39) the probability P {X(t) ≥ a) for a Brownian motion with the drift ux is obtained by µ ¶ Z +∞ 1 (x − ux t)2 √ P {X(t) ≥ a) = exp − dx (16.40) 4Dx t 4πDx t a Eqs (16.38, 16.39 ,16.40) were largely used together with Eq. (16.35) to estimate the diffusive properties of random porous media from simulations [7]. Extensions of this approach include three-dimensional flow simulations [612], and two-phase flows in porous media [578]. Among the potential domains of application can be mentioned complex flows as encountered in oil reservoirs.
16.5 Lattice gas models
581
FIGURE 16.14. Growth of an aggregate with rule A (see text) after 500, 1000, 1500 and 2000 iterations [87], [88]
16.5.5 Application to simulations of random media An immediate extension of the lattice gas models is the simulation of aggregation (even multiphase) processes [87], [88]. Random aggregates, different from those studied in chapter 14. A mixture of a fluid and suspensions is simulated by means of marks (F and S). The standard collision rules are applied on each vertex of the lattice, and the marks are randomly distributed after the collisions. The behavior of the two types of particles F and S differ during the aggregation process: operating in a field containing obstacles, suspensions are allowed to aggregate (with the probability p+ ), and to be bounced back (with the probability 1 − p+ ) when they become the nearest neighbor of an obstacle. Additional conditions can be introduced for the aggregation, such as: number of aggregated particles in the neighborhood of a candidate, or directional conditions: the particle may be allowed to aggregate in a neighborhood made of a cone of 0, 60 or 120 degrees. For instance simulations shown in illustrations were made with the following rules of aggregation: A (0 degree), B (120 degrees), C (120 degrees as for B, and at least 2 of the 3 neighbors must be already aggregated). These rules can mimic some surface tension effects, as common in a solidification process. In addition, particles in the aggregate can leave it for the fluid with the probability p− , in order to simulate a disintegration process. The growth of an aggregate from a seed at the middle of the field, with a zero average velocity and with the aggregation rule A is illustrated by Fig. 16.14. The simulation is made on a 200x200 periodic system. These aggregates are very similar to what is obtained in the so-called DLA (diffusion limited aggregation) model [686]. Addition of a disintegration results into more compact aggregates. Replacing rules A and B by rule C gives less ramified aggregates and quasi dendritic textures as occur in a solidification process are obtained. Interesting morphologies are obtained with non zero velocity on the boundaries of the field and applying shear conditions, by forcing the flow in two opposite directions on the upper and the lower boundary. The probabilistic properties of these random aggregates, such as T (K), are not known. Some of them are studied in [87] from measurements on simulations. It is easy to estimate a ”fractal dimension” α from the Eq.
582
16. Reaction-Diffusion and Lattice Gas Models
FIGURE 16.15. Nucleation and growth from Poisson seeds: comparison of rules A,B, C after 2000 iterations [87], [88]
(16.41): A(r) = Krα−1 dr
(16.41)
where A(r) is the area of the portion of aggregates inside a crown of radius r and of thickness dr, centered on the origin of the aggregate. The parameter α was estimated for each type of aggregation rule, after averaging A(r) over 10 realizations, and keeping r < 20. The following results were obtained: α = 1.736, 1.672, and 1.813 for rules A, B, C. For comparison, the parameter α obtained for simulations of DLA aggregates built with random walks of particles on a square lattice is equal to 1.715 [469], [470]. This is similar to the results of simulations with the rule A. Another approach of aggregation concerns the nucleation and growth of a population of aggregates: by replacing the single seed by Poisson points, structures similar to the dendritic solidification out of a melt are obtained; on Fig. 16.15, an average of 40 seeds per 200x200 image is chosen. There is a competition between the seeds to trap the particles in suspension. In addition, the aggregates are allowed to coalesce, as seen for rule B. The structure is periodic vertically and horizontally. With a continuous introduction of random seeds, the nucleation generates a dispersion in the sizes of aggregates, as seen on Fig. 16.16.
The deposition of particles may be produced during sedimentation processes, such as for the formation of geological structures. Simulations of the deposition of particles submitted to a vertical force, such as the gravity were produced on a field closed on its lower boundary, and open on its vertical boundaries, with periodic conditions. This is illustrated in Fig. 16.17 for the three previous rules of aggregation. The obtained structures are very similar to a dendritic solidification on a cold plate. Finally if a probability of disintegration is added, a packing of the structure by the gravity field is generated.
16.5 Lattice gas models
583
FIGURE 16.16. Continuous germinaion and growth of aggregates. Comparison of rules A, B, C after 2000 iterations [87], [88]
FIGURE 16.17. Deposition of particles on the right side, simulating dendritic solidification on a cold plate. Comparison of rules A, B, C after 5000 iterations [87], [88]
Multiphase aggregates can be simulated by introduction of various colors (Ai ) for the particles of the suspension [88]. The probability of aggregation p+ is now replaced by a probability matrix P with coefficients pij where pij is the probability for a particle of the type i to aggregate to a clustered particle Aj . Similarly can be introduced a probability matrix P 0 for the disintegration. Other interesting applications of lattice gas models are briefly mentioned now: flows of suspensions [400]; in [86], [90] the filtration of suspensions in liquid iron is simulated, in order to be able to design new filter geometrical properties to optimize the retention of impurities. A model, based on aggregation and disintegration processes, was developed to reproduce the filtration process inside a channel of a ceramic filter. It was calibrated from various experimental data: measurement of the flux of liquid iron during the filtration; examination of polished sections of clogged filters, where morphological measurements on the obtained aggregates could be compared to the structures generated by the simulations. Suspensions of a gas and a liquid are simulated in [22], [23], while evaporation and phase separation in porous media are studied with the same model in [556]. This is obtained in
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16. Reaction-Diffusion and Lattice Gas Models
a clever way by introducing a force attracting particles a certain distance apart.
16.5.6 Multi species lattice gas models Reaction-diffusion models can be simulated by lattice gas after an introduction of a random reaction operator [150]. This is illustrated by simulations of a creation-anihilation model, of a multi-species model and by the hydrodynamic simulation of surface coating by projection of molten particles in a plasma [118]. Building a general reaction operator Reactions can be introduced in simulations by means of marking particles according to their species and by the addition of a reaction operator, as studied in [137], [138], [374]. The most general case of a k-species lattice-gas model, is obtained by associating one of the k possible species with each particle. The local concentration nα (t, x) of a chemical specie α is given by the number of particles of type α at a given lattice node x and at the time t (thus for the FHP-III model, 0 ≤ nα (t, x) ≤ 7). By extension, the chemical species α = 0 represents void particles. To the standard lattice-gas evolution rule C ◦ T (translations T and collisions C), is added a reactive step R: the chemical species α in the cell i may react and randomly change to any other species β, according to the conditional probability Pαβ (nα (t, x)). This procedure offers the main advantage to avoid computing local concentrations for every possible (or present) species in each node. The counterpart is the discretization effect on the local concentration of species, and the non trivial task to generate the probabilities Pαβ to reproduce a specific reactive system. The local reaction operator may be iterated ν times to balance the diffusion strength in comparison with the reaction rate, so that the lattice-gas basic evolution step becomes finally Rν ◦ C ◦ T . The operator R alone induces a Markov process depending on the transition probability given by matrix P . In the present case, a birth and death process (addition or deletion of a single particle at each evolution step) satisfying the exclusion rule of the lattice-gas model, is used. In many situations, combinations of such a reactive step with diffusion effects generate complex system behaviors, including spatial self-organization in the form of coexisting areas with distinct concentrations, that are considered as random structures. Applications to a creation-annihilation model The Schlögl model [587], already introduced in subsection 16.4.1, describes a simple three-molecular autocatalytic chemical scheme, based on four elementary reactions. A specific cellular automaton model for this particular system was proposed by D. Dab and J. B. Boon [137], [138], [7], using a
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two-dimensional square lattice and diffusive collisions produced by particle velocity random rotations. It was adapted to the lattice gas model [150], the number n of particles at point x ranging between 0 and 7, by means of experimental fittings of the reaction operator parameters. A reactive separation towards two distinct stable states with uniform particle concentrations ρ1 and ρ2 is promoted. Starting with initial conditions (at t = 0) with an unstable steady state with concentration ρ0 such as ρ1 < ρ0 < ρ2 , particle annihilation is made with probability P10 , and particle creation with probability P01 . For this model, the random number of particles N at point x is characterized by the following transition probability matrix describing the evolution of the population by reaction: P (n, n − 1) = p1 P10 (n) with p1 = n/7, for n = 1, 2, ..., 7 P (n, n + 1) = p0 P01 (n) with p0 = 1 − p1 , for n = 0, 1, ..., 6 P (n, n) = p1 P11 (n) + p0 P00 (n) The model is completely specified by the 14 free parameters P10 (n) and P01 (n), which can be reduced to 7 if a symmetric behavior is simulated (P10 (n) = P01 (n) for n = 1, ..., 6; P10 (7) = P01 (0)). In the framework of the lattice gas models, there is a strong coupling between reaction and hydrodynamic processes, since the number of particles per node is proportional to the local pressure. Therefore concentration gradients generate pressure gradients, as would be the case for instance in combustion processes. Results of simulation for a given set of parameters are shown in Fig. 16.18, where are generated random structures corresponding to two stable values of the concentration, obtained from local fluctuations of the density of particles. For this simulation the cycle R3 ◦ C ◦ T operates. The initial density is ρ0 = 0.2 particle per node. The non zero reaction probabilities are given by P01 (3) = 0.24, P01 (4) = 0.08, P01 (5) = 0.04 P10 (1) = 0.15, P10 (2) = 0.1 It is easy to modify the kinetics of the microstructure evolution by introduction of random obstacles on which particles are bounced back. In fact, the presence of obstacles disturbs the standard diffusion process mainly on a macroscopic level. Using scatterers as special lattice nodes where particle velocities are reversed at each iteration, the effective coefficient of diffusion decreases with the number of scatterers. Applications to a multi-species reactive decomposition model The creation-annihilation reaction model produces regions with very different particle densities, which therefore cannot be advected in the same
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FIGURE 16.18. Simulation of a lattice gas Schlögl model. Domain 500 × 500, with periodic boundary conditions; 100 iterations; grey levels are proportional to the number of particles, from 0 (white) to 7 (black) [150]
way. As a result, the introduction of particle flows or local particle deflection fields (ie. conditions with non zero velocity field) may lead to aberrant, non physical results. The example of basic three-species reactive model presented now allows only particles to change of color during the reaction step R. In this case, there is a conservation of the total number of particles per node, and therefore there is no transport due to a pressure gradient generated by the reaction. The reaction probabilities are defined so that the species with a low local concentration may react and change to other possible species. Fig. 16.19 presents results for this model obtained with periodic boundary conditions, and a vortex deflection imposed by particle velocity tangential deviations. In Fig. 16.20 an extruded 3 components structure is generated. In these simulation the initial density is 0.25 particles per node and color, and the reaction probabilities are given by: Pαβ (nα = 1) = 0.1, Pαβ (nα = 2) = 0.05, for α 6= β With the chosen probabilities the three species generate equivalent random sets. Nodes with a dominant color (>50%) have 3 different grey levels corresponding to the 3 species, other nodes being coded in black.
Simulation of surface coating In surface coating by plasma, molten droplets are projected on a substrate, where they are flattened before solidification. Two dimensional simulations of this process were developed with a specific lattice gas model [150], [118], [251]. Stable droplets in a carrier gas are obtained by adaptation of the twophase immiscible lattice gas generating a surface tension [578]. Droplets solidification during the impact on the substrate is obtained by a random aggregation, with probability Pa for particles of the droplets pointing towards the immediate neighborhood of an obstacle, as already used in [87],
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FIGURE 16.19. Simulation of a 3 species reactive lattice gas with a vortex deflection, after 2000 iterations (500 × 500) [150]
FIGURE 16.20. Reactive decomposition of 3 species in a closed pipe after 2000 iterations; flow interacting with an obstacle generating an extrusion (deflection to the right and periodic lateral boundary conditions) [150]
[88]. Affecting different Pa to different particles enables us to account for possible variations of temperature (or of melting temperatures) in a deposition process, as illustrated in Fig. 16.21 with two values of Pa (0.01 for 80% of nodes and 0.8 for 20% of nodes). Periodic lateral conditions are applied. Droplets have a radius of 32 pixels. The density is d = 0.7 particles per node. A vertical deflection with probability Pf = 0.15 is imposed to generate a projection on a substrate. Various conditions were simulated and are detailed in the references, like the effect of the roughness of the substrate or the porosity induced by projection of droplets on a fibrous network.
16.6 Conclusion To conclude this chapter, it is instructive to compare random functions obtained for the linear Reaction-Diffusion model to models obtained with
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FIGURE 16.21. Simulation of droplets deposition (600×300) after 60000 iterations [150]
the same family of primary RF but with a different construction, such as Boolean or sequential (Dead Leaves, Alternate Sequential) RF. If the first ones are very smooth (and even sometimes infinitely differentiable), the other RF are only mean square continuous. Intermediary situations can be accessed by non linear Reaction-Diffusion models. Due to the diffusive component of these models, their spatial scale usually increases with time, and the solutions of Eq. (16.1) asymptotically tends towards a homogeneous field of concentrations in general, which is not the case of models presented in other chapters. When it is the case, the choice of the wanted scale can therefore be controlled by the stopping time t of the simulations. The strength of non-linear random Reaction-Diffusion systems resides in their simplicity of implementation and on their strong physical background to produce a broad variety of morphologies. Complex phenomena can be studied from simple rules and with few adjustable parameters. The main drawbacks with their use in practical applications is presently the lack of theoretical results concerning the statistical properties of the generated structures as well as guidelines for model identification, as opposed to more conventional random sets models. This is undoubtedly a source of work for theory and also for applications in the future. The outline of lattice gas models and of their extension intends to demonstrate the potential use of these discrete models for random media generation. Initially proposed to solve complex flow problems at a microscale, they have a much wider field of applications like complex chemical reactive phenomena coupled to hydrodynamics.
17 Texture Segmentation by Morphological Probabilistic Hierarchies
Abstract: A general methodology is introduced for probabilistic texture segmentation in binary, scalar, or multispectral images. Textural information is obtained from morphological operations on images. Starting from a fine partition of the image in regions, hierarchical segmentations are set up in a probabilistic framework by means of probabilistic distances conveying the textural or morphological information, and of random markers accounting for the morphological content of the regions and of their spatial arrangement. The probabilistic hierarchies are built from binary or multiple fusion of regions.
17.1 Introduction Previous chapter were concerned by the characterization of random structures and by the development of models. We will now completely change our viewpoint, working on images as data, and randomness will be introduced by sampling random points in the image, or by random markers. Furthermore, the starting point of the segmentation will be a fine tessellation of space resulting from an over-segmentation, or given by a mesh or a realization of some random tessellation model. In this approach models of random structures are used to interact with the image of interest, the response of this interaction providing efficient means of image segmentation. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Jeulin, Morphological Models of Random Structures, Interdisciplinary Applied Mathematics 53, https://doi.org/10.1007/978-3-030-75452-5_17
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In many cases, images contain regions with different textures, rather than objects on a background or regions with homogeneous grey level easily segmented by thresholding. Automatic texture extraction is required in different areas such as for instance industrial control [123] or remote sensing [525], [526]. In these two last cases, in every pixel of images multivariate information is available, like results of morphological transformations applied to grey level or to binary images [122], [123], or like the wavelength response of a sensor in multispectral images [525], [526]. In this context a typical approach of segmentation makes use of pixel classification by means of multivariate image analysis [122], [123], sometimes combined with a watershed segmentation based on some multivariate gradient [525], [526]. The standard deterministic watershed segmentation extracts regions of interest in an image by construction of zones of influence of markers [69], [71]. In order to avoid the common over-segmentation resulting from an excess of noisy markers, for years a efforts were dedicated to a paradigm, namely the selection of pertinent markers, which can make sometime the morphological segmentation very tedious. An alternative approach to deterministic segmentation is to forget about this paradigm and to implement a probabilistic approach [21], which was proved very efficient in many applications. In what follows, a hierarchical probabilistic segmentation of textures, based on multivariate morphological information available on every pixel, is introduced. Previous publications on probabilistic segmentation of textures are rather sparse, and are reviewed in [312], [319]. In contrast with some of these previous developments, the present approach does not make use of any probabilistic or statistical model of random field for the image. Randomness is introduced by the process of sampling random points in the image, or by means of random markers probing the image like an observer would randomly scan a landscape before pointing out regions of interest. Furthermore, we operate on regions of a partition resulting from an over segmentation process and build hierarchies based on a probabilistic content. After a short reminder on morphological texture descriptors and texture classification, a probabilistic approach of hierarchical segmentation of textures, based on a probabilistic distance and on the introduction of various random markers, is developed. Preliminary results, followed by a more extensive presentation of the content of this chapter, are given in [312], [319].
17.2 Morphological texture descriptors Consider images as domains D in the n dimensional space Rn . As already discussed in chapter 3, every pixel x is described by a set of morphological parameters or transformations building a vector with dimension p in the
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parameter space Rp . For instance in the case of 2D multispectral images, n = 2 and p is the number of channels of each spectrum. Many types of transformations can be used. From experience, some standard families of morphological transformations Ψ [438], [598], performed on an initial image, are efficient as texture descriptors [122], [123]: dilations, erosions, openings or closings by convex structuring elements with size ρ. These operations are as well defined for binary images as for scalar grey level images. Smart texture descriptors, from the point of view of pixel classification, are increments of transformations with respect to the size ρ. When Ψ is an opening or a closing operation, the increments provide a granulometric spectrum, as used in various domains: binary textures [605], [122], rough surfaces [28], satellite imagery [546], to mention a few. In some specific situations, the transformed images are averaged in a local window K(x) around x, to provide local granulometries [216], [122], [123] or the output of linear filters, like curvelet transform [122], [123]. These descriptors are easily extended in a marginal way to the components of multispectral images.
17.3 Texture classification The morphological descriptors generate a vector field on the domain D, from which a classification of pixels in the various textures present in the image can be looked for. For this, a partition in classes Cβ must be built in the high dimensional parameter space. A convenient methodology is based on multivariate factor analysis to reduce the dimension of the data and to remove noise: in [525], [526] use is made of Factor Correspondence analysis FCA, well suited to positive data, like multi spectral images or like probability distributions as encountered in granulometric spectra; for heterogeneous data, Principal Component Analysis PCA can be a good method to produce the dimensional reduction [123]. Each of these analysis makes use of some specific distance in the parameter space, denoted by kZ(x1 ) − Z(x2 )k for the descriptors of the two pixels x1 and x2 . Various distances can be chosen (for instance the chi-squared between distributions in the case of FCA). This choice is highly application dependent. A classification of pixels is then made in the parameter space or in its reduced version, after keeping the most prominent factors, as illustrated for various applications to textures in chapter 3. This classification can be unsupervised, using random germs in the K-means algorithm or a hierarchical classification, as described in [48]. When the textures are documented by a set of representative pixels, supervised statistical learning methods can be implemented for later classification (see an extensive presentation in [234]). In [122] a Linear Discriminant Analysis LDA is used. It follows a PCA in [123].
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17.4 Probabilistic texture segmentation In this section we assume that in every pixel x in the image embedded in Rn , multivariate information (like multispectral data, or transformed images as described in section 17.2) is stored in a vector Z(x) with components Zα (x). For any pair of pixels x1 and x2 , a multivariate distance kZ(x1 ) − Z(x2 )k is defined in the parameter space Rp .
17.4.1 Watershed texture segmentation Considering points y in the neighborhood B(x) of point x, a multivariate gradient can be defined as [525], [526]: grad (Z(x)) = ∨y∈B(x) kZ(x) − Z(y)k − ∧y∈B(x) kZ(x) − Z(y)k The gradient image can be the starting point of the segmentation of the domain D into homogeneous regions Ai . In fact, it is expected that a texture sensitive gradient will provide weak values in homogeneous regions, and high values on the boundary Aij between two regions Ai and Aj . A separation of the domain D in homogeneous connected regions Ai is obtained by the construction of the watershed of the gradient image from markers generated by the minima of the gradient, as initially defined for scalar images [70], [477] and later extensively used for multispectral images [525], [526]. The main drawback of the watershed segmentation is its sensitivity to noise, resulting in systematic over segmentation of the image. This is alleviated by means of a careful choice of markers, driven by some local content, like for instant chosen from a multivariate classification [527]. Another approach, the stochastic watershed [21], makes use of random markers replacing the usual markers, enabling us to estimate a local probability of boundaries at each point x ∈ Aij . The main idea is to evaluate the strength of contours by their probability, estimated from Monte Carlo simulations in a first step, as developed in the scalar [21] and in the multispectral cases [525], [526], [527]. A 3D application to microtomographic images of granular media is given in Fig. 17.1 [175]. Simulations can be replaced by a direct calculation of the probability of contour for each boundary Aij between adjacent regions Ai and Aj [300], [478]. This was successfully applied for 3D multiscale segmentation of granular media using point markers [209] or oriented Poisson lines markers [210].
17.4.2 Probabilistic hierarchical segmentation In what follows, a new probabilistic segmentation is obtained by a hierarchical merging of regions from a fine partition of a domain D in regions Ai .
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FIGURE 17.1. Example of 3D microtomographic image segmentation by the Stochastic Watershed [175]
This initial partition can be obtained in a first step from the watershed of a gradient image, or from some classification of pixels. Given two regions Ai and Aj , not necessarily connected or even adjacent, will be estimated for various criteria the probability pij : pij = P {Ai and Aj contain different textures}
(17.1)
The probability pij can play the same role as a gradient (or a distance) between regions Ai and Aj . In a hierarchical approach, a progressive aggregation of regions is performed, starting from lower values of pij and updating the probability after fusion of regions containing similar textures. This approach was proposed for the case of random markers [300] and implemented in an iterative segmentation based on the stochastic watershed [209]. Using a probabilistic framework makes easier the combination of different criteria for the segmentation, as illustrated later. In the context of texture classification, the probability 1 − pij = P {Ai and Aj contain the same texture} is also a similarity index between regions Ai and Aj . Probabilistic distance Consider two points x1 and x2 in a domain D, and the multivariate distance kZ(x1 ) − Z(x2 )k. The choice of a specific multivariate distance (not necessarily Euclidean), with appropriate scaling of variables, is quite standard in multivariate data analysis, and is not discussed here. When the two points are located randomly in D, kZ(x1 ) − Z(x2 )k becomes a random variable, characterized by its cumulative distribution function P {kZ(x1 ) − Z(x2 )k ≥ d} = T (x1 , x2 , d). We have the following property: Proposition 17.1. For any d > 0, the distribution function T (x1 , x2 , d) is a distance in D. Proof. We have T (x1 , x1 , d) = 0 and T (x1 , x2 , d) = T (x2 , x1 , d). T satisfies the triangle inequality: for any triple (x1 , x2 , x3 ),
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kZ(x1 ) − Z(x2 )k ≤ kZ(x1 ) − Z(x3 )k + kZ(x2 ) − Z(x3 )k . Therefore kZ(x1 ) − Z(x2 )k ≥ d =⇒ kZ(x1 ) − Z(x3 )k + kZ(x2 ) − Z(x3 )k ≥ d and T (x1 , x2 , d) ≤ P {kZ(x1 ) − Z(x3 )k + kZ(x2 ) − Z(x3 )k ≥ d} ≤ T (x1 , x3 , d) + T (x2 , x3 , d). Definition 17.1. Consider two regions Ai and Aj in D, and two independent random points xi ∈ Ai , xj ∈ Aj . For any d > 0, the probability P (Ai , Aj , d) = P {kZ(xi ) − Z(xj )k ≥ d}
(17.2)
defines a probabilistic distance between Ai and Aj . By construction, P (Ai , Aj , d) is a pseudo-distance, since is not satisfied for every d, P (Ai , Ai , d) = 0. For any triple (Ai ⊂ D, Aj ⊂ D, Ak ⊂ D) and (xi ∈ Ai , xj ∈ Aj , xk ∈ Ak ), P (Ai , Aj , d) ≤ P (Ai , Ak , d) + P (Aj , Ak , d) as a result of the triangle inequality satisfied by T (xi , xj , d). The probabilistic distance can also be used between classes Cβ obtained for a partition in the parameter space as a result of a classification. In that case, define for two classes Cα and Cβ the probability pαβ = P {kZ(xα ) − Z(xβ )k ≥ d} estimated for independent random points xα ∈ Cα and xβ ∈ Cβ . For a classification of textures in homogeneous classes, the diagonal of the matrix P , with elements pαβ , is expected to be close to 0. This can drive the choice of the threshold d, based on the data used for the classification. Remark 17.1. For any pair of regions Ai and Aj in D, where the proportions of pixels belonging to class Cα are piα and pjα respectively, and for independent uniform random points xi ∈ Ai , xj ∈ Aj we have P P {xi ∈ Cα , xj ∈ Cβ } = piα pjβ . In that case we get P (Ai , Aj , d) = α,β piα pjβ pαβ . The calculation of the probabilistic distance between Ai and Aj is made faster after a preliminary storage of the probability matrix P . Probabilistic distance and hierarchical segmentation As mentioned before, use can be made of the probabilistic distance P (Ai , Aj , d) to build a hierarchical segmentation, starting from the lowest probability. Let P (Ai , Aj , d) < P (Ai , Ak , d) and P (Ai , Aj , d) < P (Aj , Ak , d), ∀k 6=
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i, k 6= j. Merging regions Ai and Aj with measures |Ai | and |Aj | (for instance area μ2 in R2 and volume μ3 in R3 ), generates a new region Al = Ai ∪ Aj . For any k, P (Ak , Al , d) =
|Ai | |Aj | P (Ak , Ai , d) + P (Ak , Aj , d) |Al | |Al |
Therefore, P (Ak , Ai , d) ∨ P (Ak , Aj , d) ≥ P (Ak , Al , d) ≥ P (Ak , Ai , d) ∧ P (Ak , Aj , d) > P (Ai , Aj , d). The probabilistic distance increases when merging two classes, so that it can be used as an index in the hierarchy. All remaining values P (Ak , Al , d) are updated after fusion of two regions, and the process can be iterated. Indeed, ∂(Ai , Aj ) = inf{p, Ai and Aj are included in the same region Al } is equivalent to the diameter of the smallest region of the hierarchy containing Ai and Aj , which satisfies the ultrametric inequality required to generate a hierarchy [48]. Alternatively, the probabilistic distance involved in every level of merging is used to generate an ultrametric distance, used to build the hierarchy [166]. The segmentation involved with the probabilistic distance is unsupervised in the general case. Remark 17.2. In the context of segmentation, a partition of the domain D is obtained by considering all subdomains obtained when cutting the hierarchy at a given level (probability) p. The choice of the threshold p can be driven by the results of the preliminary classification of pixels, using the values of the elements of the matrix P , or by the number of subdomains, that should correspond to the number of textures present in D. An estimate of this number can be derived from the spectral analysis of the matrix of the graph Laplacian derived from the matrix with elements 1−P (Ai , Aj , d), which is an adjacency matrix [234]. Remark 17.3. The proposed hierarchy generates non necessarily connected subdomains, since no adjacency condition is imposed in the choice of regions to be merged. This condition can be required, as is made for the construction of watersheds, by restricting the use of the probabilistic distances to adjacent regions. In addition to connectedness of segmented regions, it reduces the cost of calculations by limiting the number of pairs, instead of considering the full cross-product Ai × Aj . Intermediate constructions can involve the probabilistic distances of iterated adjacent regions. This approach can be followed in the first steps of the segmentation, to reduce the number of regions, and released for the remaining steps of the process, in order to allow for the extraction of non connected regions containing the same type of texture.
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Remark 17.4. Integration of the probability P (Ai , Aj , d) with respect to the threshold d gives the average distance kZ(xi ) − Z(xj )k for independent random points xi ∈ Ai , xj ∈ Aj . This average can be used in a hierarchy where the aggregation is made according to the average distance criterion [48]. Combination of probabilistic segmentations It may be useful to enrich the probabilistic distance by other probability distributions concerning the comparison of the content of two regions, in order to combine them for the segmentation. The choice of probability distributions on the cross-product Ai × Aj will be restricted according to the following definition. Definition 17.2. A probability P (Ai , Aj ) is said to be increasing with respect to the fusion of regions, when it satisfies P (Ak , Al ) ≥ P (Ak , Ai ) ∧ P (Ak , Aj ) for any i, j, k, with Al = Ai ∪ Aj . As shown before, the probabilistic distance satisfies the property given in definition 17.2. Other probability distributions with the same property will be introduced later. The property given in definition 17.2 is satisfied when P (Ai , Aj ) is increasing with respect to ⊂, which means that P (Ak , Al ) ≥ P (Ak , Ai ) when Ai ⊂ Al . However, this is not a necessary condition. We start from two probabilistic segmentations, based on separate aggregation conditions, involving the probability of separation of regions Ai and Aj , P 1 (Ai , Aj ) and P 2 (Ai , Aj ). P 1 and P 2 are assumed to own the fusion property of definition 17.2. It is possible to combine these probabilities according to different rules. For instance: 1. 2. 3. 4.
probabilistic independence: P (Ai , Aj ) = P 1 (Ai , Aj )P 2 (Ai , Aj ) more reliable event: P (Ai , Aj ) = P 1 (Ai , Aj ) ∨ P 2 (Ai , Aj ) least reliable event: P (Ai , Aj ) = P 1 (Ai , Aj ) ∧ P 2 (Ai , Aj ) weighting between the two events (with probabilities λ1 and λ2 ): P (Ai , Aj ) = λ1 P 1 (Ai , Aj ) + λ2 P 2 (Ai , Aj )
5. any combination P (Ai , Aj ) = Φ(P1 , P2 )(Ai , Aj ), where P is a probability increasing with respect to the fusion of regions according to definition 17.2. These rules are easily extended to more than two conditions of aggregation. Proposition 17.2. The previous rules of combination of the probability of separation of regions satisfy the property given in definition 17.2.
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Proof. Start from P (Ai , Aj ) = Φ(P1 , P2 )(Ai , Aj ). As before, consider Al = Ai ∪ Aj and the condition: P (Ai , Aj ) < P (Ai , Ak ) and P (Ai , Aj ) < P (Aj , Ak ), ∀k 6= i, k 6= j. For any region Ak P (Ak , Al ) =
|Ai | |Aj | P (Ak , Ai ) + P (Ak , Aj ). |Al | |Al |
and P (Ak , Ai ) ∨ P (Ak , Aj ) ≥ P (Ak , Al ) ≥ P (Ak , Ai ) ∧ P (Ak , Aj ) > P (Ai , Aj ) Local probability distributions The regions of the fine partition (or obtained after some steps of aggregation) can be characterized by some local probability distributions. Probabilistic classification If pixels xi in region Ai are attributed to various classes of textures Cα by a probabilistic classification, the probability piα = P {xi ∈ Cα } can be used as a probabilistic descriptor of Ai . Considering now independent uniform random points xi ∈ Ai and xj ∈ Aj , the probability (17.1) is written P (Ai , Aj ) = 1 − P {xi and xj belong to the same texture} X = 1 − pij = 1 − piα pjα
(17.3) (17.4)
α
Denoting Iα (x) the indicator function of class Cα , and I(x) the vector with components Iα (x), P {kIα (xi ) − Iα (xj )k = 0} = P {xi ∈ Cα , xj ∈ Cα } = piα pjα and therefore P (Ai , Aj ) = P {kI(xi ) − I(xj )k > 0}, so that P (Ai , Aj ) defined by (17.3) is a probabilistic distance corresponding to definition 17.1. It satisfies the property given in definition 17.2. This can be checked as follows: using Al = Ai ∪ Aj , plα =
|Ai | i |Aj | j pα + p |Al | |Al | α
and 1 − P (Ak , Al ) =
X α
pkα plα =
|Ai | X i k |Aj | X j k p p + p p |Al | α α α |Al | α α α
|Ai | |Aj | = (1 − P (Ai , Ak )) + (1 − P (Aj , Ak )) |Al | |Al |
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Therefore, P (Ak , Al ) = and
|Ai | |Aj | P (Ai , Ak ) + P (Aj , Ak ) |Al | |Al |
P (Ak , Ai ) ∨ P (Ak , Aj ) ≥ P (Ak , Al ) ≥ P (Ak , Ai ) ∧ P (Ak , Aj ) Illustrative examples of this situation in the binary case of two textures (α = 1, 2) are given in [319]. Local granulometric spectrum As indicated in section 17.2, granulometric information can be provided after transformation of the image by morphological opening or closing operations. A local granulometric spectrum can be obtained in each region Ai by averaging the components Iα (xi ) or Zα (xi ) over Ai . After normalization, are obtained local granulometric spectra in classes of sizes Cα where for size α, piα = P {xi ∈ Cα }. This local classification with respect to size can be introduced in the probability (17.3) to build a hierarchy. Alternatively different granulometries like opening and closing may be combined, for instance by linear combination as proposed in section 17.4.2, to generate a composite hierarchy. Local orientation Some textures present local orientation [351] which it is convenient to study → − from a vector field V (x), like the gradient vector. To remove the effect of noise in the gradient, a local orientation is extracted as follows: in every window K(x) is considered the cloud of points Mγ generated by connecting → − the origin to vector V (xγ ). The principal axes of inertia of the points Mγ are then extracted [351]. A degree of confidence of the orientation is provided by the ratio of the largest eigenvalue of the inertia matrix to its trace. Working with a partition of the domain D, the same approach is → − followed with the cloud of points generated by V (xi ), xi ∈ Ai . in a second step, the disorientation between Ai and Aj are characterized from the scalar → − → − product between the two main eigenvectors °V i and V°j . If°αij is°the °angle ° → ° →° →° → − → − °− °− °− between V i and V j with Euclidean norms ° V i °and ° V j °, for ° V i ° 6= 0 °− ° °→ ° and ° V j ° 6= 0: ³− → ´2 → − V i. V j cos2 αij = ° °2 ° °2 → ° °− → ° °− ° V i° ° V j ° The value cos2 αij is a proximity index between orientations of regions Ai and Aj , while 1 − cos2 αij is an angular distance, to be used in a similar
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way as P (Ai , Aj ) to build an orientation based segmentation, starting from a partition. Note that cos2 αij is used in chapter 3 to define a covariance of → − → − orientations between V (xi ) and V (xj ). An alternative way to account for the disorientation between two regions is to use the probability distribution → − → − of the quantity (1−cos2 αij ) obtained for V (xi ) and V (xj ), when xi and xj are independent uniform points in Ai and Aj . An orientational probabilistic distance can be defined by © ª P (Ai , Aj , d) = P (1 − cos2 αij ) ≥ d → − In the presence of pixels with a weak gradient V it may be wise to exclude them from the calculation of the orientational probabilistic distance, since the computation of cos2 αij is ill-defined in that case. Alternatively, we can ³− → − → ´2 use the probability distribution of the random variable V i . V j .
17.4.3 Higher order probabilistic segmentation In the previous section, the fusion construction of a hierarchy was limited to the fusion of pairs of regions, restricting to second order probabilities P (Ai , Aj ) and generating binary trees. This can be easily extended to higher order probabilities, to produce more general trees and hierarchies. The resulting aggregation process can be accelerated, as compared to the binary case, at a marginal computational cost. Consider m regions Ai1 , Ai2 , ..., Aim . For every pair Aik , Ail independent uniform random points xik , xil and the independent random variables kZ(xik ) − Z(xil )k are generated. We decide to merge the m regions when (∨ik ,il kZ(xik ) − Z(xil )k) < d. The probability of this event is given by: Y P {∨ik ,il kZ(xik ) − Z(xil )k < d} = P {kZ(xik ) − Z(xjl )k < d} (17.5) ik ,il
=
Y
ik ,il
(1 − P (Aik , Ail , d))
An indexed hierarchy is obtained by sorting the probabilities (17.5) with decreasing order. Alternatively the m order version of the probabilistic distance, sorted with increasing order, can be introduced: P (Ai1 , Ai2 , ..., Aim , d) = P {∨ik ,il kZ(xik ) − Z(xil )k ≥ d} Y =1− (1 − P (Aik , Ail , d)) ik ,il
(17.6)
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For instance in the case of a ternary hierarchy the probabilities (17.5) and (17.6) become: P {kZ(xi ) − Z(xj )k ∨ kZ(xi ) − Z(xk )k ∨ kZ(xj ) − Z(xk )k < d} = (1 − P (Ai , Aj , d)) (1 − P (Ai , Ak , d)) (1 − P (Aj , Ak , d))
(17.7)
and P (Ai , Aj , Ak , d) (17.8) = 1 − (1 − P (Ai , Aj , d)) (1 − P (Ai , Ak , d)) (1 − P (Aj , Ak , d)) By construction, the m order probabilities (17.5) are always lower than the corresponding second order probabilities, and the P (Ai1 , Ai2 , ..., Aim , d) are larger than the P (Ai , Aj , d). Once the binary probabilities P (Ai , Aj , d) are available, it is easy to work out higher order probabilities (17.5, 17.6, 17.7, 17.8). When starting from a partition with r regions, computing the r! Crm = m!(r−m)! order m probabilities may be expensive. However this can be easily performed when restricting the fusion to adjacent regions. When merging the m regions with the lowest probability P (Ai1 , Ai2 , ..., Aim , d) a new region Al = Ai1 ∪Ai2 ∪...∪A is generated. In a next step, all possible fusions of m − 1 regions with Al have to be considered. We get P (Ak1 , Ak2 , ...., Akm−1 , Al , d) |Ai1 | P (Ak1 , Ak2 , ...., Akm−1 , Ai1 , d) + ... |Al | ¯ ¯ ¯ ¯Ai m−1 + P (Ak1 , Ak2 , ...., Akm−1 , Aim−1 , d) |Al | =
so that ∨i=i1 ,...im−1 P (Ak1 , Ak2 , ...., Akm−1 , Ai , d) ≥ P (Ak1 , Ak2 , ...., Akm−1 , Al , d) ≥ ∧i=i1 ,...im−1 P (Ak1 , Ak2 , ...., Akm−1 , Ai , d) > P (Ai1 , Ai2 , ..., Aim , d) The probability P (Ai1 , Ai2 , ..., Aim , d) increases when merging m classes, and therefore satisfies the property given in definition 17.2. The hierarchy can also be built by combining distance probabilities of various orders m. To illustrate this point, consider the combination of m = 2 and m = 3. At a given step of the hierarchy the three regions Ai , Aj , and Ak are merged into Al = Ai ∪ Aj ∪ Ak if for any p
17.4 Probabilistic texture segmentation
601
P (Ai , Aj , Ak , d) < P (Ai , Ap , d) P (Ai , Aj , Ak , d) < P (Aj , Ap , d) P (Ai , Aj , Ak , d) < P (Ak , Ap , d) it comes for p, P (Al , Ap , d) =
|Ai | |Aj | |Ak | P (Ai , Ap , d) + P (Aj , Ap , d) + P (Ak , Ap , d) |Al | |Al | |Al |
and therefore P (Al , Ap , d) > P (Ai , Ap , d) ∧ P (Aj , Ap , d) ∧ P (Ak , Ap , d) > P (Ai , Aj , Ak , d) so that the index of the hierarchy increases after the fusion of the regions Ai , Aj , and Ak . Remark 17.5. There is a connection between the m order probabilistic distance and some image transformations in the case of a digitized image. Consider a neighborhood B(x) with m pixels. The criterion (∨ik ,il kZ(xik ) − Z(xil )k) becomes Y (x) = ∨x1 ∈B(x),x2 ∈B(x) kZ(x1 ) − Z(x2 )k In the case of a scalar image Z(x), Y (x) = ∨y∈B Z(y) − ∧y∈B Z(y) = Z(x) ⊕ B − Z(x) ª B and Y (x) is a standard morphological gradient. For a probabilistic classification of textures, the result (17.3) is easily extended to higher orders: P (Ai1 , Ai2 , ..., Aim ) = 1 − P {xi1 , xi2 , ..., xim belong to the same texture} XY piα1 piα2 ...piαm =1−
(17.9)
α
17.4.4 Probabilistic distances between sets As a particular case of a probabilistic distance, it is interesting to consider the distribution of the Euclidean distance d(x, y) between two random points x and y.
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Proposition 17.3. Consider two uniform random points x and y located in the set Ai ⊂ Rn . The distribution of distances d(x, y) is given by Z |Bx (d) ∩ Ai | dx (17.10) P {d(x, y) < d} = 2 |Ai | Ai where Bx (d) is the ball with center x and radius d. Proof. Given point x in Ai , we have d(x, y) < d ⇐⇒ y ∈ Bx (d) and y ∈ Ai . i| . After deconditioning The probability of this event is given by |Bx (d)∩A |Ai | with respect to the uniform location of x in Ai , the result (17.10) follows. Proposition 17.4. Consider two uniform random points x ∈ Ai and y ∈ Aj . The distribution of distances d(x, y) is given by Z Z |Bx (d) ∩ Aj | |By (d) ∩ Ai | P {d(x, y) < d} = dx = dy (17.11) |A | |A | |Ai | |Aj | i j Ai Aj where Bx (d) is the ball with center x and radius d. By construction, it satisfies the criterion of fusion given in definition 17.2. Proof. Given a random point x in Ai , we have d(x, y < d ⇐⇒ y ∈ Bx (d) |B (d)∩A | and y ∈ Aj . The probability of this event is given by x |Aj | j . After deconditioning with respect to the uniform location of x in Ai , the result (17.11) is obtained. Alternatively this probability is computed, given a random point y in Aj . The two expressions are equal since, noting 1Ai (x), 1Aj (y) and 1Bx (d) (y) the indicator functions of the sets Ai , Aj and Bx (d), we get: Z Z 1 |Bx (d) ∩ Aj | 1A (x)1Aj (y)1Bx (d) (y)dydx dx = |Ai | |Aj | |Ai | |Aj | Rn i Ai Z 1 1A (x)1Aj (y)1B0 (d) (y − x)dydx = |Ai | |Aj | Rn i Z 1 1A (x)1Aj (y)1By (d) (x)dydx = |Ai | |Aj | Rn i Z |By (d) ∩ Ai | dy |Ai | |Aj | Aj The probability distributions (17.10) and (17.11) involve average values of the measures |Bx (d) ∩ Ai |, |Bx (d) ∩ Aj | and |By (d) ∩ Ai |. In fact, |Bx (d) ∩ Ai | is the cross geometrical covariogram between Ai and the ball B(d) for the separation x. If Ai = B(R), we get, noting KB(R) (x) the geometrical covariogram of the ball B(R): |Bx (R) ∩ Ai | = KB(R) (x)
17.4 Probabilistic texture segmentation
and P {d(x, y) < 2R} =
Z
KB(R) (x)
Ai
2
|Ai |
603
dx = 1
as expected. In practice, the probability distributions (17.10) and (17.11) can be estimated from sampling the locations of the ball Bx (d). These probabilities of distances may be used for some morphological characterization of each set Ai and for each pair Ai , Aj . When considering m regions in a fine partition, they allow us the calculation of m order probabilities, after introduction in the probability given in (17.5) to calculate P {∨ik ,il d(xik , xil ) < d}. These probabilities of distance between sets carry some morphological content on the partition, to be introduced in a hierarchical segmentation based on the shape and spatial distribution of regions Ai . For a given pair of regions Ai and Aj , the probability of fusion decreases when their spatial separation increases, so that the process promotes the fusion of neighbor regions. This information can be combined with the previously defined probability distances carrying a textural content. Another way to introduce morphological data on regions of the partition is obtained by the introduction of random markers.
17.4.5 Use of random markers Following the approach proposed for the stochastic watershed [21] we can introduce random markers to randomly select regions for which the previous probabilistic segmentation will be performed. Doing so, some morphological content on the regions of the hierarchy and on their location in D is accounted for, in addition to the previous probabilistic textural information. The aim of this section is to calculate the probability PR (Ai , Aj ) of selection of two regions (Ai , Aj ) by random markers, as introduced in [300]. Reminder on random allocation of germs Use ng random points (or germs) xk with independent uniform distributions in the domain D containing r regions. The probability pi for a germ to fall in Ai is given by r X |Ai | pi = pi = 1. , with |D| i=1 By construction, the allocation of germs in the regions of a partition follows a multinomial distribution (Ni being the random number of germs in Ai ) with multivariate generating function: n o Nr 1 N2 G(s1 , s2 , ..., sr ) = E sN s ...s r 1 2
Starting with ng = 1 (use of a single germ):
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G1 (s1 , s2 , ..., sr ) = p1 s1 + p2 s2 + ... + pr sr For ng ≥ 1, the numbers Ni are the sum of n independent binary random variables, and Gng (s1 , s2 , ..., sr ) = G1 (s1 , s2 , ..., sr )ng = (p1 s1 + p2 s2 + ... + pr sr )ng so that ng ! pk1 pk2 ...pkr r k1 !k2 !...kr ! 1 2 with k1 + k2 + ... + kr = ng
P {N1 = k1 , N2 = k2 , ..., Nr = kr } =
(17.12)
An interesting case is asymptotically obtained when |D| → ∞ and ng → ∞, ng with |D| → θ. For these conditions, the multinomial distribution converges towards the multivariate Poisson distribution. We have: log Gng (s1 , s2 , ..., sr ) = ng log (p1 s1 + p2 s2 + ... + pr sr ) =ng log(1 + p1 (s1 − 1) + p2 (s2 − 1) + ... + pr (sr − 1)) and log Gng (s1 , s2 , ..., sr ) → θ |A1 | (s1 − 1) + θ |A2 | (s2 − 1) + ... + θ |Ar | (sr − 1) so that: lim Gng (s1 , s2 , ..., sr ) = π i=r i=1 exp(θ |Ai | (si − 1))
ng →∞
The random numbers N1 , N2 , ..., Nr are independent Poisson random variables with intensities θi = θ |Ai |: P {Ni = k} =
θki exp −(θi ) k!
(17.13)
Using Poisson points as markers, the number of germs for each realization follows a Poisson distribution with parameter θ |D|. Calculation of the probability PR (Ai , Aj ) for point markers Random markers are used to select regions of a partition by reconstruction. With this process, the reconstructed regions for any realization of the random germs are left intact, while regions without germs are merged. Considering many realizations of the germs, the probability PR (Ai , Aj ) for the two regions to remain separate is computed. Proposition 17.5. For ng independent uniformly distributed random germs, the probability PR (Ai , Aj ) for the two regions Ai , Aj to remain separate is given by: n PR (Ai , Aj ) = 1 − (1 − pi − pj ) g (17.14)
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605
Proof. The pair (Ai , Aj ) is merged ⇐⇒ {Ni = 0 and Nj = 0} ⇐⇒ Ni + Nj = 0 ⇐⇒ N (Ai ∪ Aj ) = 0. Working on images, the probabilities PR (Ai , Aj ) computed for all pairs (Ai , Aj ) are easily ranked in increasing order. A hierarchical fusion of regions is obtained by starting with the lowest probability PR (Ai , Aj ). After fusion of two regions with Al = Ai ∪ Aj the probabilities PR (Ak , Al ) are updated. The pair (Ak , Al ) is merged ⇐⇒ {Nk = 0 and Nl = 0} ⇐⇒ Nk + Nl = 0 ⇐⇒ N (Ak ∪ Al ) = 0 ⇐⇒ N (Ai ∪ Aj ∪ Ak ) = 0. Then: PR (Ak , Al ) = 1 − (1 − pi − pj − pk )
ng
> PR (Ai , Aj )
and the probability PR (Ak , Al ) is increasing with respect to the fusion of regions as in definition 17.2. In general no conditions of connectivity or of adjacency of regions are required for the fusion process. It is easy to force the connectivity by working on connected components of regions, or to limit the fusion to adjacent regions. The random germs can be generated by a Poisson point process. Proposition 17.6. For Poisson point germs with intensity θ, the probability PR (Ai , Aj ) for the two regions Ai , Aj to remain separate is given by: PR (Ai , Aj ) = 1 − exp [−θ (|Ai | + |Aj |)]
(17.15)
and PR (Ak , Al ) is increasing with respect to the fusion of regions as in definition 17.2. The morphological content in the probabilities (17.14, 17.15) only depends on the Lebesgue measure (area in R2 and volume in R3 ) of regions. It increases with the measure of regions, larger regions resisting more to fusion. For a pair of regions, PR (Ai , Aj ) is maximal when |Ai | = |Aj |, so that the random markers hierarchy tends to generate by fusion regions with homogeneous sizes, the regions with lower measure disappearing first. Calculation of the probability PR (Ai , Aj ) for Poisson lines and Poisson flats markers It can be interesting to obtain other weightings of regions with a probabilistic meaning, like the perimeter in R2 or the surface area in R3 . Restricting to the Poisson case, it is easy to make this extension, provided use is made of appropriate markers. For this purpose, consider now isotropic Poisson lines in R2 , isotropic Poisson planes and Poisson lines in R3 [448], [275] (chapter 6). Oriented Poisson lines in R3 were used as markers in the context of the stochastic watershed (and so with another type of probability), and applied to the segmentation of granular structures [210].
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Proposition 17.7. Consider stationary isotropic Poisson lines with intensity λ as random markers in R2 . The probability PR (Ai , Aj ) for the two regions Ai , Aj to remain separate is expressed as a function of the average projected length l of the projection of Ai ∪ Aj in directions ω: ∙ ¸ Z π l(Ai (ω) ∪ Aj (ω))dω (17.16) PR (Ai , Aj ) = 1 − exp −λ 0
when Ai ∪ Aj is a connected set, PR (Ai , Aj ) is given by: PR (Ai , Aj ) = 1 − exp [−λL (C(Ai ∪ Aj ))]
(17.17)
where L is the perimeter and C(Ai ∪ Aj ) is the convex hull of Ai ∪ Aj . Proposition 17.8. Consider stationary isotropic Poisson lines with intensity λ as random markers in R3 . The probability PR (Ai , Aj ) for the two regions Ai , Aj to remain separate is expressed as a function of the average projected area A of the projection of Ai ∪ Aj in directions ω: ∙ Z PR (Ai , Aj ) = 1 − exp −λ
2πster
A(Ai (ω) ∪ Aj (ω))dω
0
¸
when Ai ∪ Aj is a connected set, PR (Ai , Aj ) is given by: h π i PR (Ai , Aj ) = 1 − exp −λ S(C(Ai ∪ Aj )) 4
(17.18)
(17.19)
where S is the surface area and C(Ai ∪ Aj ) is the convex hull of Ai ∪ Aj . For random markers in R3 made of stationary isotropic Poisson planes with intensity λ, the probability PR (Ai , Aj ) for the two regions Ai , Aj to remain separate is expressed as a function of the average projected length l of the projection of Ai ∪ Aj in directions ω: ∙ Z PR (Ai , Aj ) = 1 − exp −λ
0
2πster
l(Ai (ω) ∪ Aj (ω))dω
¸
(17.20)
when Ai ∪ Aj is a connected set, the probability PR (Ai , Aj ) is given as a function of the integral of mean curvature A by: PR (Ai , Aj ) = 1 − exp [−λA(C(Ai ∪ Aj ))]
(17.21)
It is possible to combine various types of Poisson markers (points and lines in R2 , points, planes and lines in R3 ) with their own intensities. For instance, when Ai ∪ Aj is a connected set in R2 , PR (Ai , Aj ) = 1 − exp [− {θ (|Ai | + |Aj |) + λL (C(Ai ∪ Aj ))}]
(17.22)
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607
where a weighting by the area and the perimeter of the regions acts for the segmentation. Similarly in R3 is introduced a weighting of the volume, and the surface area and integral of mean curvature of C(Ai ∪ Aj ) in the process of segmentation. Calculation of the probability PR (Ai , Aj ) for compact markers Further morphological information on the regions is accounted for when introducing compact random markers (not necessarily connected). In the process of selection of regions of a partition by reconstruction, point mark0 ers are replaced by a compact grain A located on Poisson points, and generating a Boolean model A (chapter 6). Proposition 17.9. For compact markers A0 generating a Boolean model with intensity θ, the probability PR (Ai , Aj ) for the two regions Ai , Aj to remain separate is given by: ¯¤ £ ¯ PR (Ai , Aj ) = 1 − exp −θ ¯(Aˇi ⊕ A0 ) ∪ (Aˇj ⊕ A0 )¯ (17.23) ¯ ¯ ¯ ¯ ¯¢¤ £ ¡¯ 0¯ 0¯ 0 ˇ ˇ ˇ ˇ ¯ ¯ ¯ = 1 − exp −θ Ai ⊕ A + Aj ⊕ A − (Ai ⊕ A ) ∩ (Aj ⊕ A0 )¯
Proof. The pair (Ai , Aj ) is merged ⇐⇒ Ai ∪ Aj is outside the Boolean model with primary grain A0 . The expression (17.23) is the Choquet capacity T (K) of the Boolean model when K = Ai ∪ Aj .
The compact markers can be random sets (for instance spheres wit a random radius). In that case, the measures || are replaced by their mathematical expectations with respect to the random set A0 . Using for A0 a ball with radius ρ, PR (Ai , Aj ) increases until a constant value when the distance between Ai and Aj increases from 0 to 2ρ: the probability to merge two regions is higher when their distance is lower. Combination of textural and of morphological information Random markers, conveying morphological content on the partition and on its evolution in the hierarchy, are now combined to the previous textural content (probabilistic distance, or local probability information). For instance, we can decide to merge two regions when they are not reconstructed by markers (with a marker dependent probability 1−PR (Ai , Aj )) and when the textures they enclose are similar (with a probability 1 − P (Ai , Aj , d)). In this context the probability pij (17.1) becomes P (Ai , Aj ) = Φ(PR , Pd )(Ai , Aj ) (17.24) = PR (Ai , Aj ) + P (Ai , Aj , d) − PR (Ai , Aj )P (Ai , Aj , d) By construction, this composite probability is increasing with respect to the fusion of regions as in definition 17.2, and generates a hierarchy for the
608
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Texture Segmentation by Morphological Probabilistic Hierarchies
segmentation, by updating each terms of (17.24) according to the previous rules. Alternatively, the probabilities of distance between sets introduced in section 17.4.4 can replace PR (Ai , Aj ) in the formulation (17.24). Other textural information can be introduced in the probability (17.24), such as orientational information, as discussed in section 17.4.2, or extinction values: in the case of watershed segmentation, the flooding algorithm relies on the minimal values zij of the function to be flooded, like grad (Z(x)), on boundaries Aij between adjacent regions Ai and Aj . Let F (z) be the cumulative distribution function of the extinction values. It is equivalent to sort the values zij or F (zij ), since the distribution function is a monotonous transformation of data. Using F (zij ) instead of zij gives a probabilistic content to the flooding. In this context, 1 − F (zij ) is a probabilistic distance between Ai and Aj . Consider the regions Ai , Aj and Ak . Aj and Ak are merged by flooding, if zjk < zij and zjk < zik , and consequently zjk < zij ∧ zik . After fusion, we obtain Al = Aj ∪ Ak and zil = zij ∧ zik . Therefore, the probabilistic distance 1 − F (zil ) increases by fusion, as required in definition 17.2. Combining this probabilistic distance to the various probabilities PR (Ai , Aj ) generated by random markers provides a hierarchical segmentation, even outside of the field of textures. This hierarchical segmentation resulting from a progressive fusion of adjacent regions is close to the segmentation given by the stochastic watershed, but remains different.
17.4.6 Random markers and higher order fusion of regions Consider now random germs and decide that regions without germs are merged. For the fusion of m regions, the previous second order results are easily extended. Proposition 17.10. For ng independent uniformly distributed random germs, the probability PR (Ai1 , Ai2 , ..., Aim ) for the m regions Ai1 , Ai2 , ..., Aim to remain separate is given by: ng
PR (Ai1 , Ai2 , ..., Aim ) = 1 − (1 − pi1 − pi2 − .... − pim )
(17.25)
Proof. The regions Ai1 , Ai2 , ..., Aim are merged ⇐⇒ {Ni1 = 0 and Ni2 = 0 and .... Nim = 0} ⇐⇒ Ni1 + Ni2 + .... + Nim = 0 ⇐⇒ N (Ai1 ∪ Ai2 ∪ ... ∪ Aim ) = 0. A hierarchical fusion of regions is obtained by starting with the lowest probability PR (Ai1 , Ai2 , ..., Aim ). After fusion of m regions with Al = Ai1 ∪ Ai2 ∪...∪ Aim the probabilities PR (Ai1 , Ai2 , ..., Aim ) are updated. The regions (Ak1 , Ak2 , ...., Akm−1 , Al ) are merged
17.4 Probabilistic texture segmentation
609
© ª ⇐⇒ Nk1 = 0, Nk2 = 0,..., Nkm−1 = 0 and Nl = 0 ⇐⇒ Nk1 + Nk2 + ... + Nkm−1 + Nl = 0 ⇐⇒ N (Ak1 ∪ Ak2 , ∪.... ∪ Akm−1 ∪ Al ) = 0. Then, PR (Ak1 , Ak2 , ...., Akm−1 , Al ) ¢ng ¡ = 1 − 1 − pi1 − pi2 − .... − pim − pk1 − pk2 − ... − pkm−1 > PR (Ai1 , Ai2 , ..., Aim ) and the probability PR (Ai1 , Ai2 , ..., Aim ) is increasing with respect to the fusion of regions as in definition 17.2. The extension of previous results to various markers (Poisson points, compact markers, Poisson lines or Poisson planes) is straightforward. The following results are obtained. Proposition 17.11. For Poisson point germs with intensity θ, the probability PR (Ai1 , Ai2 , ..., Aim ) for the m regions Ai1 , Ai2 , ..., Aim to remain separate is given by: PR (Ai1 , Ai2 , ..., Aim ) = 1 − exp [−θ (|Ai1 | + |Ai2 | + ... + |Aim |)]
(17.26)
and PR (Ai1 , Ai2 , ..., Aim ) is increasing with respect to the fusion of regions as in definition 17.2. Proposition 17.12. For compact markers A0 generating a Boolean model with intensity θ, the probability PR (Ai1 , Ai2 , ..., Aim ) for the m regions Ai1 , Ai2 , ..., Aim to remain separate is given by: PR (Ai1 , Ai2 , ..., Aim ) (17.27) ¯¤ £ ¯ 0 0 0 = 1 − exp −θ ¯(Aˇi1 ⊕ A ) ∪ (Aˇi2 ⊕ A ) ∪ ... ∪ (Aˇim ⊕ A )¯
Proposition 17.13. Consider stationary isotropic Poisson lines with intensity λ as random markers in R2 . The probability PR (Ai1 , Ai2 , ..., Aim ) for the m regions Ai1 , Ai2 , ..., Aim to remain separate is expressed as a function of the average projected length l of the projection of Ai1 ∪ Ai2 ∪ ... ∪ Aim in directions ω:
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PR (Ai1 , Ai2 , ..., Aim ) ∙ ¸ Z π = 1 − exp −λ l(Ai1 (ω) ∪ Ai2 (ω) ∪ ... ∪ Aim (ω))dω
(17.28)
0
when Ai1 ∪ Ai2 ∪ ... ∪ Aim is a connected set, it is given by: PR (Ai1 , Ai2 , ..., Aim ) = 1 − exp [−λL (C(Ai1 ∪ Ai2 ∪ ... ∪ Aim ))] (17.29) where L is the perimeter and C(Ai1 ∪ Ai2 ∪ ... ∪ Aim ) is the convex hull of Ai1 ∪ Ai2 ∪ ... ∪ Aim . Proposition 17.14. Consider stationary isotropic Poisson lines with intensity λ as random markers in R3 . The probability PR (Ai1 , Ai2 , ..., Aim ) for the m regions Ai1 , Ai2 , ..., Aim to remain separate is expressed as a function of the average projected area A of the projection of Ai1 ∪ Ai2 ∪ ... ∪ Aim in directions ω: PR (Ai1 , Ai2 , ..., Aim ) ¸ ∙ Z 2πster A(Ai1 (ω) ∪ Ai2 (ω) ∪ ... ∪ Aim (ω))dω = 1 − exp −λ
(17.30)
0
when Ai1 ∪ Ai2 ∪ ... ∪ Aim is a connected set, it is given by: h π i PR (Ai1 , Ai2 , ..., Aim ) = 1 − exp −λ S(C(Ai1 ∪ Ai2 ∪ ... ∪ Aim )) (17.31) 4
For random markers in R3 made of stationary isotropic Poisson planes with intensity λ, the probability PR (Ai1 , Ai2 , ..., Aim )
for the m regions Ai1 , Ai2 , ..., Aim to remain separate is expressed as a function of the average projected length l of the projection of Ai1 ∪ Ai2 ∪ ... ∪ Aim in directions ω: PR (Ai1 , Ai2 , ..., Aim ) ¸ ∙ Z 2πster l(Ai1 (ω) ∪ Ai2 (ω) ∪ ... ∪ Aim (ω))dω = 1 − exp −λ
(17.32)
0
when Ai1 ∪ Ai2 ∪ ... ∪ Aim is a connected set, it is given by: PR (Ai1 , Ai2 , ..., Aim ) = 1 − exp [−λA(C(Ai1 ∪ Ai2 ∪ ... ∪ Aim ))] (17.33)
17.5 Conclusion
611
As previously (cf. the combination(17.24)), the morphological content carried by random markers can be combined to the textural content given by the probability (17.6) in the construction of a hierarchy based on m order probabilities. Alternatively, m order probabilities of distance between regions (section 17.4.4) can play the same role.
17.5 Conclusion The probabilistic hierarchical segmentation tools introduced in this chapter are flexible enough to handle various types of textures (scalar or multivariate) and their spatial distribution, by progressively merging regions of a fine partition. Combining appropriate morphological operations and texture classification, successfully implemented in previous studies on texture segmentation mentioned in the references, supervised or unsupervised texture segmentations are obtained. A probabilistic distance between regions, carrying statistical information on textures, is defined. From this distance, hierarchies involving progressive binary or multiple fusions of regions with similar textures are built. Additionally, morphological information on the regions of the fine partition and on merged regions of the hierarchy is accounted for in the process, through the use of a probabilistic distance between regions of the initial partition, or of various kinds of random markers. In each situation, the required probabilities are computed in a closed form by simple algebra, from the set of probability distributions (17.2) estimated on the initial partition, as a function of the content and of the location of the regions of the partition evolving during the construction of the hierarchy. When using random markers of different kinds generating some random sets (like Poisson points, Poisson lines, or even a Boolean model for compact markers), the classes of the partition of the image to be merged play the same role as the compact sets usually involved in the calculation of the Choquet capacity. This way to process images for segmentation, based on the same probabilistic framework, is the reverse approach of the standard characterization of random sets, as done in the previous chapters of this book. Finally, the top-down construction of the hierarchy can also be used to generate random trees, starting from their leaves, as introduced in chapter 5. It remains to study the morphological properties of such trees, at least from numerical simulations.
Part III Random Structures and Change of Scale
18 Change of Scale in Physics of Random Media
Abstract: Problems of change of scale in physics of random media handled in this chapter concern the prediction of the macroscopic behavior of a physical system from its microscopic behavior. Most physical properties of random media are concerned by this approach, like for instance the dielectric permittivity in electrostatics, the permeability K for flows in porous media, or elastic moduli of composites. This topic concerns the definition an calculation of effective (or macroscopic) properties of an equivalent homogeneous medium from lower scale information by homogenization tools. In a first step, models of RS or of RF are used to represent maps of properties on a microscopic level. Then, the homogenization problem is addressed by a perturbation expansion of the physical fields of interest and by the calculation of bounds of effective properties for models of random structures by means of correlation functions of local properties. This illustrated in the case of linear constitutive behavior by third order bounds and their application to models of RF and of random sets studied in previous chapters. Finally local fluctuations of fields on a point level are characterized by their second order moments derived from the effective properties.
18.1 Introduction An important area of application of models of random media concerns the prediction of the macroscopic behavior of a physical system from its microscopic behavior, going much further than the common engineering practice © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Jeulin, Morphological Models of Random Structures, Interdisciplinary Applied Mathematics 53, https://doi.org/10.1007/978-3-030-75452-5_18
615
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18. Change of Scale in Physics of Random Media
based on the construction of empirical correlations between macroscopic properties and microstructure. This so-called ”homogenization” process goes back to the early preoccupations of statistical physics when was developed the theory of gas kinetics from the mechanical interaction of particles with different velocities. Two types of approaches can be followed: we can first look for estimations of the effective properties (namely the overall properties of an equivalent homogeneous medium) of random heterogeneous media from their microstructure. Using variational principles, bounds of the effective properties for linear constitutive equations are established from a limited amount of statistical information, as will be illustrated from third order bounds, more accurate than the Hashin and Shtrikman (H-S) bounds solely based on the volume fractions in the case of isotropic media. Another technique of estimation of the effective behavior involves numerical solutions of partial differential equations on simulations of random media, as illustrated in chapter 19. These two approaches can be followed for all physical properties involving a conservation law, as for instance the dielectric permittivity [53], [50], [62], [334], [334], [487],[490], [492], the magnetic permeability of materials [232], the permeability of porous media [438], [440], the elastic moduli of composites [54], [492], [495], [64], [65], [233], [393], [468], [488], [670]. In what follows, the estimation of the overall properties of random composites from perturbation expansions, or from bounds based on a limited amount of statistical information are introduced. Tighter bounds are derived from additional statistical information, such as the infinite set of correlation functions of the microstructure. They are illustrated by examples of applications to some models of random sets. The results of this chapter were partly published in [292], [296], [298], [299]. The prediction of effective properties by numerical techniques on realizations of random media and the need to explore the representativeness of the results by means of a statistical representative volume element RVE is introduced and illustrated by examples of application in chapter 19.
18.2 From microscopic to macroscopic From a macroscopic point of view, the behavior of a physical system is considered as the responses of a medium to solicitations. For current engineering practice, useful physical variables are well defined (for instance stresses and strains in mechanics) and are related by constitutive equations expressing physical laws. The meaning of the used physical variables, and the validity of the constitutive equations always depend on the scale of observation of physical phenomena: for instance in fluid mechanics it is possible to work on a ”microscopic scale”, where the physical system is made of a population of moving particles in interaction. On this scale it is possible to speak about
18.3 Homogeneous medium and heterogeneous medium
617
velocity momentum, as for the lattice gas model mentioned in chapter 8, but not about pressure or temperature. These last two ”macroscopic” variables have a physical meaning for a volume of matter containing a high number of particles, where only the collective behavior of a population is kept. By a change of scale, a new physical model emerges, replacing a discrete system by a continuum, for which are defined new variables such as pressure, temperature, concentration... From a general point of view, we mean by change of scale the problem of the prevision of the macroscopic behavior of a physical system from its macroscopic behavior. This problem is very wide, and is of interest for all the physics. In this part, our purpose is limited to some examples of different types, for which a new approach is provided. What can be expected from a model of change of scale? In addition to the change of physical status and to the emergence of new variables or of macroscopic behavior laws, as mentioned about fluid mechanics, a change of scale model should give an answer to the following problems.
• Do a macroscopic behavior law exist for a given heterogeneous medium (problem of emergence)? • If it exists, what is its expression and what are its coefficients (or the effective properties of the medium)? • What is the variability of effective properties measured on specimens with a finite size, as a function of their microscopic variability (scale effect on the fluctuations of properties)?
18.3 Homogeneous medium and heterogeneous medium In this section, is considered a continuum embedded inside a domain (usually bounded) B within the Euclidean space Rn . In each point x of B, we consider a set of physical properties P (x) (for instance a stress or strain tensor, a velocity, a temperature; this is illustrated in Table 18.1). P (x) builds a field defined on the domain B. The field P (x) is solution of a problem built from the following conditions: • application of conservation principles (for instance momentum, energy), which from a local balance generally involves systems of partial differential equations PDE. Equilibrium condition imposes that for a given flux F (x), div F = 0 (18.1) • choice of boundary conditions given on the boundaries ∂B of the domain B;
618
18. Change of Scale in Physics of Random Media
• use of constitutive equations linking several variables. For instance, the following linear laws linking are commonly used: — Hooke’s law between the stress tensor σ and the strain tensor e in linear elasticity through the fourth rank tensor of elasticity C: σ = Ce
(18.2)
— Navier linearized law linking for a fluid the pressure gradient (with components ∂i p) to the Laplacian of the velocity u through the kinematical viscosity μ: ∂i p = μ4ui (18.3) — Darcy’s law between the macroscopic flow rate (with components Qi ) and the macroscopic pressure gradient (with components ∂i P ) for a flow in a porous medium, K being the second rank tensor of permeability: 1 Qi = − K ij ∂i P (18.4) μ — For electrostatics of a dielectric medium, proportionality between the electric displacement D and the electric field E, being the second rank dielectric permittivity tensor, and φ the potential: D = E = − grad φ
(18.5)
Laws (18.2-18.5) are defined for each point x, where only local information takes part to the relations between the variables. For a homogeneous medium, the variables L, μ, K, (which are positive definite tensors) in Eqs (18.2-18.5) remain constant in space. The solution to the problem is a field P (x) in B, compatible with the boundary conditions. For the various instances given above, Eq. (18.1) is formulated as: div σ = 0 div(grad p) = 0 div Q = 0 div D = 0
for for for for
continuum mechanics fluid flows macroscopic flows electrostatics
(18.6)
For the quoted linear constitutive equations, existence and uniqueness of the solution for a given geometry are proved. In addition, on a macroscopic scale the same linear constitutive laws apply, with so-called effective properties. The geometry can be involved in a complex way in the solution; consider for instance a porous medium, for which are studied elastic properties or flow properties, where usual boundary conditions prescribe a null velocity for the fluid on the grain boundaries.
18.3 Homogeneous medium and heterogeneous medium
619
"Flux" Field Property 1 Heat Flux Temp Gradient Thermal Conductivity 2 Dielectric Displacement Electric Field Dielectric Permittivity 3 Stress Strain Elastic Moduli 4 Velocity Pressure Grad Permeability TABLE 18.1. Problem: Heat Conduction (1), Electrostatics (2), Elasticity (3), Fluid Flow in porous media (4)
The medium is heterogeneous when the constitutive equations can change in space, as a result of: • different types of constitutive equations (for instance linear for some places and non linear for other places within the domain B); • variations in space of the coefficients of the equations. They can be modelled either by random functions [438], [582], or by periodic fields [582]. In the case of heterogeneous media, the change of scale problem can be formulated as follows: • Is it possible to replace an heterogenous medium (with support B) by an equivalent homogeneous medium from a macroscopic point of view? • If yes, what is the macroscopic constitutive equation and what are the values of its macroscopic coefficients (or effective properties)? It is generally not straightforward to answer to these questions. To be convinced of that point, it is enough to recall the case of the composition of permeability studied by G. Matheron [438], for which a macroscopic Darcy’s law and a macroscopic permeability exist in the case of uniform flows, while for radial flows between two concentric contours, the macroscopic permeability depends on the geometry and remains different from the case of uniform flows. These difficulties are usually forgotten from the abundant literature on effective properties. In the rest of this book we leave out technical details of the methods of computation of effective properties (or homogenization techniques) for periodic structures [582] or for random media [50], [51], [115]. The probabilistic study of the fluctuations of properties is far less advanced. The few works on this topic are limited to approximations of the point histogram (or of the point variance of the field P (x) or of an effective property) in an infinite medium [51], [52], [392], [74], as introduced in section 18.18. They do not deal with fluctuations of properties of bounded domains B, and with possible scale effects. Presently this problem can be only studied by numerical methods of simulations on realizations of random media (like finite elements [101], [424], [643], [367], [368], or like Fast Fourier Transform [678]) (chapter 19), or on a discrete medium with the lattice gas model for complex flows [22], [23], [276], [577], [578], and chapter
620
18. Change of Scale in Physics of Random Media
16). To correctly handle scale depending fluctuations of effective properties, the notion of statistical representative volume element (RVE) is introduced in chapters 3 (section 3.5.5) and 19 (section 19.3). Some recent theoretical work was made to consider the convergence of apparent properties towards effective properties when considering observations with increasing scales [25].
18.4 Practical interest of change of scale methods The interest of change of scale methods is straightforward for applications in material sciences. • They help to account for the microstructure to describe the overall behavior of heterogeneous media. As a result, they can contribute to the optimization of materials [45]. Another important aspect concerns the implementation of constitutive equations containing microscopic information for introduction into computation codes used for the design of parts or of structures. • They connect the microstructure to physical properties of materials useful for their applications. These last ones can combine several properties, possibly studied by the same theoretical approach from the single microgeometrical information. • They can contribute to the optimization of the elaboration process when they are applied to the chemical engineering of reactors (physical or chemical kinetics in heterogenous media). In the remaining sections of this chapter the principles of calculation of effective properties by a perturbation expansion of involved fields, or of bounds are presented. This is applied to third order bounds of some of the models of random structures studied in the first part of this book. Then in chapter 19, the calculation of fields by numerical resolution of PDE and their use to estimate effective properties are introduced. In chapter 20 is considered the case of fracture of random media, based on the previously introduced models of random structure: Boolean random varieties, Dead Leaves models, mosaic model. They give indications on fluctuations of properties at different scales. In that sense, they are more fruitful than previously mentioned approaches oriented towards the estimation of effective properties of infinite ergodic media.
18.5 Principle of calculation of effective properties For simplicity is considered first the case of a vector field, namely the electric field E in a dielectric medium with a real permittivity. The average
18.5 Principle of calculation of effective properties
of any field P (x) in B with the volume V is defined as: Z 1 hP i = P (x)dx V B
621
(18.7)
For instance the average of the electric field E and of the displacement field D are: Z 1 hEi = E(x)dx V B Z Z (18.8) 1 1 D(x)dx = (x)E(x)dx hDi = V B V B
The effective dielectric tensor ∗ of the equivalent homogeneous medium contained in B is defined in such a way that Eq. (18.5) is satisfied on a macroscopic scale: hDi = ∗ hEi (18.9)
In general, ∗ depends on the applied boundary conditions on ∂B, as mentioned earlier about radial flows [438], on the local permittivity of the components and on the geometry of the medium. The most current applied boundary conditions (BC) are as follows: • Dirichlet BC: a uniform electric field E0 is applied on ∂B, or equivalently, introducing the potential φ(x) so that E0 = − grad φ, φ(x) = −E0 .x, ∀x ∈ ∂B • Neumann BC: D.n = D0 for x ∈ ∂B with normal n. • Periodic BC: φ(x) = −E0 .x + φP (x), where φP (x) is a periodic fluctuation, with φP (x+ ) = φP (x− ), x+ and x− being to homologous points on ∂B. D must be antiperiodic: on two homologous boundaries, D+ .n+ + D− .n− = 0.
Furthermore conditions must be satisfied on the interface between a pair of components. For perfect interfaces, the potential is continuous on the interface, but its normal derivative makes a jump, so that the normal component of E makes a jump on the interface, but its tangential component is continuous. In the remaining parts of this chapter, the interfaces are assumed to be perfect. The imperfect interface conditions involve interface discontinuities of φ(x) (with a discontinuity proportional to the normal component of D) or of the normal component of D. This is usually handled by means of an interphase layer with a vanishing thickness. A weak interface leads to a decrease of the effective permittivity, as shown from bounds taking into account of a lower permittivity of the interface [639]. The case of interfaces with a high conductivity (or equivalently permittivity) is presented in [639] and detailed in [489] for ellipsoidal inclusions. The effective dielectric tensor can equivalently be defined from the energy U (x) = 12 E(x) (x)E(x) = 12 E(x)D(x) (the energy is a quadratic form in the case of a linear constitutive equation, as it is the case for linear electrostatics):
622
18. Change of Scale in Physics of Random Media
1 (18.10) hEi ∗ hEi 2 The relations given by Eqs (18.8-18.10) can be applied to any heterogeneous medium in a bounded region B, provided the field E(x) is expressed from the applied boundary conditions and from the partial differential equation expressing the Gauss’ law: hU i =
div D =
X ∂ (Di (x)) = 0 ∂xi i
(18.11)
This procedure may be used for a periodic medium from the knowledge of the cell building the period [355], [582]. When applying periodic BC on the period, or homogeneous BC on a finite domain B the averages of the fields are given by [355], [582] (see subsection 18.20.4): hEi = E0
(18.12)
hDi = D0
(18.13)
or Starting from the local energy U (x) = 12 E(x) (x)E(x) = 12 E(x)D(x) and averaging over the domain B, the following relation is satisfied, from which the effective dielectric tensor can equivalently be defined (in linear elasticity of heterogeneous media, this result is called the Hill-Mandel lemma for heterogeneous media [240], [393], [582]), as shown in subsection 18.20.6: 1 1 1 (18.14) hEDi = hEi hDi = hEi ∗ hEi 2 2 2 Eq. 18.14 shows that the random fields E and D are uncorrelated for the homogeneous or periodic applied boundary conditions. It holds for any pair (E, D) where E = − grad φ and div(D) = 0 satisfying these boundary conditions. This relation is independent on the microstructure contained in a domain B, and does not rely on the linear constitutive relation (18.5) connecting D and E. When a constant electric field E0 is applied on ∂B, from Eq. (18.14) is deduced hU i =
hDi =
∗
E0
(18.15)
When a constant displacement field D0 is applied on ∂B, Eq. (18.14) yields: ¡ ¢∗ hEi = −1 D0 (18.16)
Therefore, in the case of periodic or of homogeneous BC on ∂B, ¡ applied ¢∗ the effective dielectric permittivity ∗ , (or its inverse −1 ) is obtained from the average hDi (or hEi). In practice it can be implemented from a numerical solution of the electrostatic problem, knowing the geometry of
18.5 Principle of calculation of effective properties
623
the medium and the boundary conditions, as will be seen in section 3.5.5, in chapter 19. The procedure based on the average energy can be extended to non linear constitutive equations, for which the function U is not any more quadratic. This is done in [627], [554], [672], [38], [555], [673]. For the effective elastic properties of a random medium, a similar approach is made, where in accordance to Table 18.1 the potential is now replaced by the displacement filed u(x), the electric field by the deformation field e(x), and D by the stress field σ(x). Eq. (18.14) becomes: hσei = hσ >< ei
(18.17)
Eq. (18.17) holds for any pair (e, σ) where e derives from a displacement field u(x) ¶ µ 1 ∂ui ∂uj eij (x) = + (18.18) 2 ∂xj ∂xi and div(σ) = 0 satisfying boundary conditions given below. It does not rely on the linear constitutive relation (18.2) connecting σ and e. For micromechanics in a domain B, the common applied boundary conditions (BC) are as follows [367]:
• Dirichlet BC: a uniform deformation e is applied on ∂B, or equivalently u(x) = e.x, ∀x ∈ ∂B. It is also called Kinematic Uniform BC (KUBC). • Neumann BC: for the normal stress components on the boundary ∂B with normal n, σ.n = t: a uniform traction t is applied on ∂B. This corresponds to Static Uniform BC (SUBC). • Periodic BC: u(x) = e.x + uP (x), where uP (x) is a periodic fluctuation vector, with uP (x+ ) = uP (x− ), x+ and x− being to homologous points on ∂B. The traction t must be antiperiodic: on two homologous boundaries, σ + .n+ + σ − .n− = 0. As for the electrostatics case, homogenization tools presented in this chapter are valid for perfect interfaces between components, insuring the continuity of the displacement and of the normal stress components. For imperfect interface, discontinuities of displacements u(x) and of traction σ.n(x) are allowed on the interface [47], [100], giving different effective elastic moduli C ∗ . The effect of imperfect (or a contrario more than perfect) depends on the scale of phases, this effect being amplified by smaller size with a constant volume fraction, as a result of an increasing specific surface area of contact SV ij . This explains the reinforced effect of nanoparticles, as illustrated by numerical simulations for the conductivity of nanocomposites [688], while in the case of perfect interfaces there is no size or scale effect for the homogenization in linear electrostatics and elasticity.
624
18. Change of Scale in Physics of Random Media
18.6 Exact results Before turning to the general case, it is useful to remind "exact results" often used in applications, and to give the background of their validity.
18.6.1 Geometrical average estimate For two symmetric phases A1 (with permittivity 1 ) and A2 (with permittivity 2 ), the case of p = 0.5 produces an autodual random set, where the two phases have the same probabilistic properties. It is the case of a mosaic model, or of any truncated Gaussian random function. For this model in R2 , the effective permittivity ∗ is given by the geometrical average [373], [438], [168]: √ ∗ = (18.19) 1 2 Eq. (18.19) is also valid for a complex permittivity [168]. More generally, a log-symmetrical positive random function (for instance a permittivity field modelled by a lognormal random function) is such that log Z(x) and − log(Z(x) have the same spatial law. The effective permittivity in R2 is given by the geometrical average (see exercise 18.20.7) log
∗
= E{log (x)}
This result is not true in R3 ! These exact results are very useful to check the validity of any numerical simulation scheme in R2 . Similarly and in the same conditions, in 2D elasticity the planar effective shear modulus G∗ for two incompressible components satisfies [56]: p G∗ = G1 G2
18.6.2 Self-consistent and effective medium estimates Various simplified schemes were developed to produce estimates of the effective dielectric permittivity or elastic moduli of media made of particles immersed in a matrix. The most popular model is based on the so-called effective medium approximation of A.G. Bruggeman [96]. An example of practical application of this approximation to the estimation of the elastic properties and of the thermal conductivity of multicomponent materials is presented in [533]. Consider a multi-phase medium with m types of inclusions in Rd (with volume fractions pi , and dielectric permittivity i ). In the Self-Consistent Assumption, the interaction of inclusions outside of i is the same as the effect of a homogeneous medium with the effective permittivity ∗ . The average (over all populations of inclusions) of the perturbation of the field with respect to a uniform field is assumed to cancel, by a so-called
18.6 Exact results
625
self-consistency condition. For spherical inclusions, it comes: i=m X
− ∗ i + (d − 1) i
pi
i=1
∗
=0
(18.20)
so that ∗ is obtained by solving a quadratic equation, derived from the implicit approximation. For a two phase medium with spherical inclusions, p α + α2 + 4(d − 1) 1 2 ∗ (18.21) SCS = 2(d − 1) with α = 1 (dp1 − 1) + 2 (dp2 − 1) Eq. (18.21) is a symmetric expression, invariant by exchanging phases 1 and 2. ∗ given by Eq. (18.21) is within the Hashin- Shtrikman bounds presented in section 18.10, but not within third order bounds in general. It was shown by G. W. Milton that the self-consistent estimate of Eq. (18.21) is valid for a specific iterated hierarchical microstructure, obtained by iteration of embedded spheres containing smaller composite spheres containing the two components [493], [494]. This iterative scheme is also true for elasticity. Some limit cases of Eq. (18.21) are given by
•
2/ 1
= +∞: "Super-Permittivity" (or super-conduction): ∗ 1
=
1 1 − dp2
= +∞ for p2 = d1 . This critical proportion of phase 2 correspond to an effective percolation threshold, which is not a geometrical percolation threshold. • 2 / 1 = 0: "Low Permittivity" (or insulation): ∗
∗ 1
=1−
d p2 d−1
(18.22)
= 0 for p2 = d−1 d . Phase 2 percolates according to this model for a critical proportion different from the previous case and from the geometrical percolation threshold. The difference of these critical values are rather surprising with respect to the morphological properties, so that self-consistent estimates should be considered with care. ∗
Similar schemes were developed for elastic properties [97], [380], [493], [494], [495]. In that case, for two-phase composites estimates of the effective bulk modulus K ∗ and shear modulus G∗ = G∗ are solutions of the system:
626
18. Change of Scale in Physics of Random Media
p1
K1 − K ∗ K2 − K ∗ + p =0 2 K1 + 4G∗ /3 K2 + 4G∗ /3 G1 − G∗ G2 − G∗ + p2 =0 p1 ∗ G1 + F G2 + F ∗ G∗ (9K ∗ + 8G∗ ) with = F∗ 6(K ∗ + 2G∗ )
(18.23)
After calculation, K ∗ and G∗ are solution, obtained by iteration, of the following system, K∗ =
∗
G =
K1 K2 + (p1 K1 + p2 K2 ) 43 G∗ p2 K1 + p1 K2 + 43 G∗ G∗ ( 32 K ∗ + 43 G∗ ) K ∗ +2G∗ G∗ ( 32 K ∗ + 43 G∗ ) K ∗ +2G∗
G1 G2 + (p1 G1 + p2 G2 ) p2 G1 + p1 G2 +
(18.24)
(18.25)
As for the permittivity, one should be cautious with the use of Eqs. (18.24, 18.25), mainly in the case of very low or of very high elastic moduli, as a result of peculiar apparent percolation thresholds. In [363], [365] the self consistent scheme is extended by the effective field method, where each inclusion in a matrix is considered as isolated in some effective external field generated by the other inclusions, reducing the many-particles problem to a single particle problem, after simplification of interactions. This approach gives estimations of effective properties in static as well as in dynamic problems [366] of random heterogeneous media made of separate inclusions embedded in a matrix. In [364] is presented an application to the elastic properties of materials containing two different populations of inclusions (namely spheres and cylinders), called three phase hybrid composites, using various models of random sets: sequence of two independent Boolean models, a dilution model presented in section 15.12.3 in chapter 15, and a color Dead Leaves model. The effective moduli of three phase hybrid composites depend on the various set covariances involved in the model. In [343], it is applied to electromagnetic wave propagation in random media modelled by one scale and two-scale Boolean models of spheres.
18.6.3 Composite spheres assemblage In [231], [232] the effective permittivity ∗ of a two-phase medium in R3 made of composite concentric spheres with real scalar permittivity 1 (with volume fraction p1 ) and 2 (with volume fraction p2 = 1−p1 ), with 2 > 1 , was shown to be ∗
=
2
+
1/(
1−
p1 2 ) + p2 /(3 2 )
(18.26)
18.7 Perturbation expansion in electrostatics
when
2
is attributed to the shell of spheres and ∗
=
1
+
1/(
2−
1
627
to the core, and
p2 1 ) + p1 /(3 1 )
(18.27)
after exchange of the shell and core, p1 and p2 . It is easy to check that the effective permittivity of Eq. (18.26) is larger than ∗ given by Eq. (18.27). In fact these two values are HS bounds for two-phase isotropic media (see subsection 18.10.1). In R2 the same result is obtained by replacement of the factor 3 by 2 in the denominator of Eqs (18.26) and (18.27). A rigorous construction of such composite spheres assemblages as a random set is given by sequential intact grains of the Dead Leaves tessellation [321] (section 11.3 in chapter 11), for which theoretical size distributions of spheres are available. Other geometries made of coated laminates give the same effective properties [629], [427], as systematically studied by G. Milton [495].
18.7 Perturbation expansion in electrostatics From a perturbation expansion is derived an approximate solution of Eq. (18.11) for the vector field E [94], [438], [440], [50], [393], [241], [416]. This approximation can be used as input in Eq. (18.41) to estimate the effective ∗ . It can also enter into a variational principle as seen in section 18.10. The same results are obtained for the effective thermal conductivity and for transport properties like the permeability of the electrical conductivity, which are given by the same equations. Consider a large domain B containing a stationary random medium, submitted to a constant macroscopic field E{E} = E (with components E{Ei } = E i ) applied on ∂B. Define E 0 (x) = E(x) − E{E} and 0 (x) = (x) − E{ }. The field E 0 (x) is solution of the following equation, derived from Eq. (18.11): div(E{ }E 0 ) = − div( 0 E) (18.28) This comes from the fact that div(D) = 0 = div( 0 E) + E{ } div(E) = div( 0 E) + E{ } div(E 0 ) When the random medium satisfies E{ ij } = δ ij (for instance if the medium is stationary and isotropic, like a polycrystal with a uniform distribution of orientations, or if ij (x) = (x)δ ij with any macroscopic geometrical anisotropy), Eq. (18.28) can be interpreted as a Poisson equation, after introduction of a potential φ0 with E 0 = − grad φ0 ∆φ0 = div( 0 E)
(18.29)
628
18. Change of Scale in Physics of Random Media
A formal solution of Eq. (18.29) is given by means of the Green’s function G(x, y) (or harmonic potential) solution of: ∆x G(x, y) = −δ(x − y) for x ∈ B G(x, y) = 0 for x ∈ ∂B
(18.30)
In R2 for an infinite medium B = R2 , log |x − y| 2π
(18.31)
1 4π |x − y|
(18.32)
G(x, y) = − In R3 for B = R3 , G(x, y) =
More generally in Rd (d > 2), if Sd (B(1)) is the surface area of the ball with radius 1, 1 G(x, y) = (18.33) Sd (B(1))(d − 2) |x − y|d−2 The solution of Eq. (18.29) can be written Z 0 φ (x) = − G(x, y) div( 0 (y)E(y)) dy B
=−
X Z i
B
G(x, y)
∂ 0 ( (y)E(y))i dy ∂yi
P
0 where ( 0 (y)E(y))i = k ik (y)Ek (y) is the i component of the vector 0 ( (y)E(y)). After integration by parts, and accounting for the condition given in Eq. (18.30), the potential follows XZ ∂ φ0 (x) = G(x, y)( 0 (y)E(y))i dy ∂y i B i
The components of E 0 = − grad φ0 , and therefore the components of E are obtained by partial derivation of φ0 X Z ∂2 Ej (x) = E j − G(x, y) 0ik (y)Ek (y) dy i,k B ∂xj ∂yi (18.34) = E j − (Γ 0 E)j (x) where is introduced the operator Γ . Eq. (18.34) has the form of the Lippmann-Schwinger equation of the quantum mechanical scattering theory ([393], p. 111). For infinite media with homogeneous boundary conditions, or for periodic media, the operator Γ acts by a convolution, and
18.7 Perturbation expansion in electrostatics
629
therefore the solution of Eq. (18.34) can be obtained numerically by iterations (starting from an initial field Ek (y) in the integral), using Fourier transforms on images of the field (x) [174], [507]. The operator Γ of Eq. (18.34), operating for the j component Ej (x) is expressed as Γ (x, y) =
X i
∂2 G(x, y) ∂xj ∂yi
(18.35)
and the component Γij (x, y) is given by ∂2 G(x, y) ∂xj ∂yi
Γij (x, y) = From Eq. (18.34) is deduced ∙ I +Γ
0
¸
(18.36)
E(x) = E
I being the identity operator. The following formal Neumann expansion gives E(x): E(x) = E +
µ (−1) Γ
n=∞ X n=1
n
0
¶n
E=
n=∞ X
E (n) (x)
(18.37)
n=0
It can be shown that the expansion of Eq. (18.37) converges if the dielectric permittivity can be expressed as = 0 (I + γ) ° kγk < 1 [465] (with ° 0with ° ° ° ° 0 = , the present development converges if ° ° < 1). This happens if
and −1 remain bounded (and this excludes media where at some places (x) = 0 or (x) = ∞). The reason for this criterion of convergence comes from the fact that the operator Γ is a projector (with a norm less than one) on the subspace of gradients, ® when E(x) is defined in the Hilbert space with the scalar product E 1 , E 2 = E{E 1 E 2 }: as shown in subsection 18.20.1, Γ ∗Γ =Γ
The j component of the term of order n in Eq. (18.37) is given by: (0)
Ej (x) = E j (1) Ej (x)
=−
1X
i1 k1
Z
∂2 1 1 G(x, x ) B ∂xj ∂xi1
(18.38) 0 1 i1 k1 (x )E k1
1
dx
630
18. Change of Scale in Physics of Random Media (2) Ej (x)
1 X
=
2
i2 k2
Z
Z
∂2 2 2 G(x, x ) B ∂xj ∂xi2
∂2
2 1 B ∂xi2 ∂xi1
=
Z
1 X 2
G(x2 , x1 )
0 1 i1 k1 (x )E k1
(n)
(−1)n X
i1 k1 ...in kn
1X
in kn
i1 k1
dx1
2
dx1 dx2
Z
∂2 ∂2 G(x, xn )... G(x2 , x1 ) n 2 ∂xi2 ∂x1i1 B n ∂xj ∂xin
0 n 0 1 in kn (x )... i1 k1 (x )E k1
=−
X
∂ ∂ G(x, x2 ) G(x2 , x1 ) 2 2 ∂xi2 ∂x1i1 B 2 ∂xj ∂xi2
i1 k1 i2 k2
n
dx2
(18.39) 2
0 2 0 1 i2 k2 (x ) i1 k1 (x )E k1
Ej (x) =
0 2 i2 k2 (x )
Z
∂2 n n G(x, x ) ∂x ∂x j B in
dx1 ...dxn
(n−1) n 0 n (x ) in kn (x )Ekn
dxn (18.40)
18.8 Formal expansion of the effective dielectric permittivity of random media If the field (x) is modelled by an ergodic random multivariate function, for domains B converging to Rn the spatial averages converge towards mathematical expectations. In these conditions the random field E(x) (and then D(x) = (x)E(x)) given by Eq. 18.37 is stationary and ergodic: from (n) Eq. (18.37) it is easy to check that E{Ej (x)} = 0, since the central ª ©0 correlation function of order n, E i2 k2 (xn )... 0i1 k1 (x1 ) , does not depend on x and provides a null contribution after integration by parts. Therefore E{E(x)} = E. In addition, from ergodicity hEi = E. As a consequence, the averages h.i in equations (18.9, 18.14, 18.15, 18.16) become mathematical expectations E{.}, and: E{D} = E{U } = E
©1
∗
E{E}
ª 1 1 ∗ 2 E(x)D(x) = 2 E{E}E{D} = 2 E{E} E{E}
(18.41)
The two definitions of ª∗ given in Eq. (18.41) are equivalent. The factoriza©1 tion of E 2 E(x)D(x) , derived here from Eq. (18.14), can also be obtained from the stationarity of D and E [438], [440]: E 0 (x) is a stationary random field E 0 , with a zero average, deriving from a potential φ0 = −E00 .x + φ0S (x)
18.8 Formal expansion of the effective dielectric permittivity of random media
631
having stationary increments, with E 0 (x) = − grad φ0 . The electrostatic energy U (x) is given by: U (x) =
1 1 1 E(x)D(x) = ED(x) + E 0 (x)D(x) 2 2 2
By expectation, E{U } = E
½
¾ 1 1 1 E(x)D(x) = EE{D} + E {E 0 (x)D(x)} 2 2 2
The second term is equal to zero since E 0 (x)D(x) = − div(φ0 (x)D(x)) = − grad φ0 (x).D(x) = E00 .D(x) − grad φ0S (x).D(x) from application of Eqs (18.5 ,18.11). Then E {E 0 (x)D(x)} = −E{div(φ0 (x)D(x))} = − div(E{φ0 (x)D(x)}) = 0 from the sationarity of grad φ0 (x).D(x). For a random medium, the effective dielectric tensor defined in Eq. (18.41) is obtained from the expectation E{D} = E{ E} when a constant macroscopic field E{E} is applied on ∂B. If the random medium is statistically isotropic, all the orientations are equivalent and therefore ∗ is an isotropic second order tensor that can be summarized by a scalar. The expectation E{D} = E{ E} is generally unknown. Approximations are obtained from trial fields which carry some information on the random function (x). To estimate the effective properties of the random medium, it is necessary to calculate E{D(x)} (cf. Eq. (18.41)): E{D(x)} = E{ (x)E(x)} = E{( 0 (x) + )(E 0 (x) + E)} = E{ 0 (x)E 0 (x)} + E = E{Di (x)} =
X
∗ Ej j ij
∗
E(x)
X = E i + E{
0 (x)Ej0 (x)} j ij
X = E i + E{
(n) 0 (x)Ej (x)} jn ij
By introduction of the perturbation expansion of Eqs (18.38-18.40) for (n) Ej (x) and by identification and rearrangement,
632
18. Change of Scale in Physics of Random Media ∗ ik1
Z
= δ ik1 +
Xn=∞ n=1
(−1)n n
X
ji1 k1 ...in kn
∂2 ∂2 G(x, xn )... G(x2 , x1 ) n 2 ∂xi2 ∂x1i1 B n ∂xj ∂xin
E
©
0 0 1 0 n ij (x) i1 k1 (x )... i2 k2 (x )
ª
(18.42)
dx1 ...dxn
• The coefficient of order n of the expansion (18.42) for ∗ik1 involves central correlation functions of order n + 1 of the random field ij (x). It is expected to introduce more and more information on the random medium by increasing the order of the development. In [440] G. Matheron assumed that 0 = αγ with a small scalar α and a tensor γ. The coefficient of αn in the expansion is called the Schwydler tensor of order n, generalizing the work of this last author made at the order 2 [594], [595]. • The perturbation expansion gives useful results for heterogeneous media with a low contrast in properties. • When the random medium is statistically isotropic, and when B = Rd , it can be shown that the second term of the perturbation expansion (for n = 1) does not the details of the second order corre© depend on ª lation function E 0ij (x) 0i1 k1 (x1 ) (see exercise 18.20.2). In that case, it is necessary to introduce at least the term depending on third order correlation functions to get estimates of effective properties depending on the microstructure. This point motivated the work of G. Matheron [439], when he was criticizing the rule of geometrical averaging which was in that time more or less systematically applied by practitioners to solve the problem of composition of permeability by change of scale. • When the random medium is such that E{ (x)} is not a scalar, the perturbation expansion has to be changed: consider first a new system of coordinates for which E{ (x)} is diagonal (with the eigenvalues i ). Following [402] in the context of homogeneous anisotropic media, it is possible to recover the Poisson equation from the change of variable xi x0i = √ and to work out a new perturbation expansion, that will i
not be considered here. In [50] p. 225 is considered the case where the eigenvalues do not depend on x, as in a polycrystal. The anisotropic case can be handled formally as the scalar case: consider a reference medium with a constant dielectric permittivity tensor 0 (in the standard perturbation expansion is used 0 = E{ }, but this is not necessary). In replacement of Eq. (18.30), a Green’s function G0 (x, y) (depending on 0 ) can be introduced by div(
0
gradx G0 (x, y)) = −δ(x − y) for x ∈ B G0 (x, y) = 0 for x ∈ ∂B
(18.43)
18.9 Perturbation approach in elasticity and calculation of the effective elastic tensor
633
In the previous developments, using 0 (x) = (x) − 0 , Eq. (18.34) becomes: X Z ∂2 Ej (x) = E0j − G0 (x, y) 0ik (y)Ek (y) dy ik B ∂xj ∂yi (18.44) = E0j − (Γ0 0 E)j (x) After the introduction of the operator Γ0 (depending on Γ the operator ,
E(x) = E0 +
n=∞ X
E = E0 − Γ0 0 E n
(−1)n (Γ0 0 ) E0 =
n=1
0)
replacing (18.45)
n=∞ X
E (n) (x)
(18.46)
n=0
The other results remain, after replacing G(x, y)/ by G0 (x, y). • The expansion given by Eqs (18.37, 18.38-18.40) can be used to estimate the cross covariance between Ei (x) and of Ej (x) [53], [50]. For an expansion of E(x) limited to n terms, the covariance of the field is expressed as a function of all correlation functions of (x) up to order 2n. In [50] pp. 230-235, the expansion of E(x) is limited to n = 2 and the truncated covariance limited to terms involving correlation functions of (x) up to order 3. As a particular case, the variance of Ei (x) limited to the first term is given in [438]. In [52] bounds of the variance are derived. Similarly an estimation of the covariance between Di (x) and of Dj (x) can be obtained. • In [371], [372], the propagation of elastic or of electromagnetic waves in random media is studied in a similar way (with appropriate Green’s functions), from a second order perturbation expansion. Electromagnetic wave propagation is studied by related techniques in [190], [580].
18.9 Perturbation approach in elasticity and calculation of the effective elastic tensor Consider now a domain B containing a stationary and ergodic random elastic medium (with the average elastic tensor C), submitted to a macroscopic loading giving a random stationary strain with expectation e. The perturbation approach for elasticity is very similar to the case of electrostatics, starting from the equilibrium equation div σ= 0 but with tensors of higher ranks. Noting e0 (x) = e(x)− e and C0 (x) = C(x) − C, the field e0 (x) can be expressed as the integral equation:
634
18. Change of Scale in Physics of Random Media
e0 (x) = −
R
V
Γ(x − y) : C0 (y) : e(y) dy
(18.47)
0
= −(ΓC e)(x)
where is introduced the modified Green’s operator in deformation, Γ, defined for a homogeneous medium with the elastic tensor C. This last one plays here the same role as the reference medium with moduli C0 used to define the polarization field introduced in the Hashin and Shtrikman approach [232], [233]. A numerical resolution of the Lippmann-Schwinger equation (18.47) [393] by iterations of FFT on images can be performed in a very efficient way ([507] and [675], among others). From Eq. (18.47) is deduced: £ ¤ I + ΓC0 e(x) = e I being the identity operator; ε(x) is then given from the following formal expansion: e(x) = e +
n=∞ X n=1
n=∞ X ¡ ¢n (−1)n ΓC0 e = e + e(n) (x)
(18.48)
n=1
The convergence of this expansion requires that C and S = C−1 remain bounded (as a consequence are excluded media where locally C(x) = 0 (porous medium) or S(x) = 0 (rigid inclusions)). The perturbation method can give useful results only for heterogeneous media with a limited contrast between its components. Eq. (18.48) is for elasticity the equivalent formulation of Eqs (18.38-18.40) obtained for electrostatics. The term of order n in Eq. (18.48) is an integral operator containing the product C0 (x1 )...C0 (xn ) for n points x1 ,..., xn . It can be checked that each term e(n) (x) has a zero expectation: after replacement of the space integral by a mathematical expectation E, as a result of the ergodicity, there appears the moment of order n, E {C0 (x1 )...C0 (xn )}, which does not depend on x, and therefore brings a zero contribution after integration by parts. In addition, each term e(n) (x) is a kinematically admissible field for the studied loading problem. In a second step, similarly to electrostatics, effective elastic moduli can be derived from the perturbation expansion, and truncation at the order n. To estimate the effective properties of the random medium, it is required to compute the space average of σ(x). Noting C∗ the tensor of homogenized elastic moduli: ¡ ¢ ® hσi = C∗ e = hCei = C0 + C (e0 + e) 0 (18.49) = hC0 eP i + Ce ® 0 (n) = Ce + nC e
18.10 Bounds of effective properties
635
After introduction of the perturbation expansion (18.48) of e(n) (x), identification and reordering, and after replacing space averages by mathematical expectations as a result of ergodicity, P C∗ = C + n=∞ n=1 R (18.50) Γ(x, xn )... Γ(x2 , x1 )E {C0 (x)C0 (x1 )...C0 (xn )} dx1 ...dxn Vn
The coefficient of order n of the expansion of C∗ depends on central correlation functions of order n + 1 of the random field C(x). The morphological information on the random medium increases with the order of the expansion. When the medium is statistically isotropic, and when V = Rd , the second term (n = 1) does not depend on details of the second order correlation function, as in the electrostatic case. For a medium without any spatial correlations, where the RF C(x) can be modelled by a white noise, it was shown in many papers of E. Kröner that C∗ and ∗ in electrostatics are obtained as solutions of the self-consistent problem [394]. Therefore the self-consistent estimates (subsection 18.6.2) correspond to a specific type of microstructure, namely a "disordered" medium in the terminology of E. Kröner [393], [394]. However this result is matter of discussion, especially in M. Hori’s work, in [241] and in the six companion papers over years 1973-1977, where a considerable amount of energy is expended to try to prove that the self-consistent medium can be obtained as a limit case of the spherical mosaic model where cells have an infinitesimal size. However complications arise in this context, since the mosaic model differs from the white noise model with null correlation functions for a rank larger than 3.
18.10 Bounds of effective properties An approximate estimation of the overall properties of random composites can be obtained from bounds based on a limited amount of statistical information [50], [393]. The commonly used bounds for isotropic media are Hashin and Shtrikman (H-S) bounds based on the volume fractions [232]. However many different morphologies can be encountered for a given volume fraction. Tighter bounds can be derived from additional statistical information, such as the infinite set of their correlation functions [50],[393],[241]. They are obtained by a combination of variational principles and of trial fields. In practice it is difficult to obtain results beyond the third order correlation functions, which already provide interesting bounds that are useful for the comparison of microstructures according to their incidence on the behavior of composite media.
636
18. Change of Scale in Physics of Random Media
18.10.1 Variational principles Variational principles are derived from properties of operators involved by the partial differential equations expressing the conservation laws of the studied physical problem, like Eq. (18.11) for the electrostatic case. The next subsections are limited to the electrostatic case, for the sake of simplicity. Elements on classical variational principles are given in [543], §11-12. Self-adjoint operators The following result will be useful to build bounds of effective properties. Theorem 18.1. Let A be a self-adjoint operator, and f an element of R. For the solution u of the Equation Au = f
(18.51)
the expression F (u) = (Au, u)−2(u, f ) is stationary and equal to −(u, f ). If A is positive definite, −(u, f ) is a minimum of F (u), while if A is negative definite, −(u, f ) is a maximum of F (u). Proof. The variation δF corresponding to a variation δu of u is given by δF (u) = δ((Au, u) − 2(u, f )) = (Au, δu) + (Aδu, u) − 2(δu, f ) = (Au, δu) + (δu, Au) − 2(f, δu) = 2((Au, δu) − 2(f, δu)) = 2(Au − f, δu) and therefore δF (u) = 0 ⇐⇒ Au = f When Au = f we get F (u) = −(u, f ). The second variation δ(δF (u)) is given by δ(δF (u)) = 2(Aδu, δu) and in the vicinity of the extremum, F (u + δu) − F (u) ' 2(Aδu, δu) F (u + δu) − F (u) > 0 if A is definite positive and F (u + δu) − F (u) < 0 if A is definite negative. Classical variational principle Consider in the domain B functions φ1 (x) and φ2 (x), and the scalar product Z 1 (φ1 , φ2 ) = hφ1 φ2 i = φ (x)φ2 (x)dx V B 1
18.10 Bounds of effective properties
637
If B contains an heterogeneous dielectric medium (with a dielectric tensor (x)), the Gauss law (Eq. (18.41)) is written div(D) = div( E) = − div( grad φ) = 0
(18.52)
The operator Aφ = − div( grad φ) defines in every point x a linear operator A acting on φ. We have: Z 1 (Aφ, φ) = − φ(x) div( (x) grad φ(x))dx (18.53) V B and div(φ grad φ) = (grad φ) (grad φ) + φ div( grad φ) → therefore, noting − n the normal to the boundary ∂B (with the surface element dS): (Aφ, φ) =
= For
1 V
Z
1 V 1 V
B
Z
Z
(grad φ(x)) (x)(grad φ(x))dx −
B
1 V
(grad φ(x)) (x)(grad φ(x))dx −
∂B
Z
1 V
(div(φ(x)) (x)(grad φ(x))dx
B
Z
∂B
→ φ(x) (x)(grad φ(x)).− n dS
→ φ(x) (x)(grad φ(x)).− n dS = 0, 1 (Aφ, φ) = V
Z
(grad φ(x)) (x)(grad φ(x))dx > 0
B
The linear operator A is definite positive for any of the following null boundary conditions: φ(x) = 0, or E(x) = − grad φ(x) = 0, or D(x) = − (x) grad φ(x) = 0 when x ∈ ∂B in addition, for an infinite random medium, µ Z ¶ 1 → − lim φ(x) (x)(grad φ(x)). n dS = 0 V →∞ V ∂B when (x) remains bounded, since the modulus of the integrand remains S bounded and → 0 for similar domains B. As a result of Theorem 20.69, V it comes:
638
18. Change of Scale in Physics of Random Media
Theorem 18.2. The potential φ, solution of Eq. (18.52), minimizes the energy Z 1 2 hU i = (Aφ, φ) = grad φ(x) (x) grad φ(x)dx (18.54) V B This can be restated as follows: the electrical field E(x), Z solution of the 1 equation div( (x)E(x)) = 0, minimizes the energy E(x) (x)E(x)dx V B (direct variational principle) and equivalently, the displacement, solution of Z 1 D(x) −1 (x)D(x)dx (dual variadiv(D(x)) = 0 minimizes the energy V B tional principle). For an ergodic random medium, the integrals are replaced by mathematical expectations. Inversely, for any field E(x) = − grad φ(x) minimizing the expression (Aφ, φ) is such that div D(x) = − div ( (x) grad φ(x)) = 0 which is obtained from Euler-Lagrange equations derived from a variational principle. If now the unknown solution E(x) is replaced by any admissible approximation (namely a trial field) E ∗ (x) = − grad φ∗ (x), an estimation of the energy hU ∗ i is obtained with hU ∗ i ≥ hU i, where hU i is given by Eq. (18.10). A similar conclusion holds, if replacing the unknown D(x) by a trial field D∗ (x) with div(D∗ ) = 0. Wiener bounds A first application of the classical variational principle gives the generalized Wiener bounds [241]. If the trial field is E(x) = E{E} = E, 2U = E{E (x)E} ≥ 2U0 = E ∗ E and EE{ }E ≥ E ∗ E For a trial field D(x) = D, 2U = E{D
−1
(x)D} ≥ 2U0 = D( ∗ )−1 D
and DE{
−1
}D ≥ D( ∗ )−1 D
The obtained inequalities show that the tensors E{ } − ( ∗ )−1 are definite positive. This can be written: (E{
−1
})−1 ≤
∗
≤ E{ }
∗
and E{
−1
}−
(18.55)
18.10 Bounds of effective properties
639
For the scalar case (for instance in the case of a statistically isotropic random medium), Eq. (18.55) is applied to the scalars , −1 = 1/ , ∗ . For a locally isotropic medium with E{ (x)} = , the eigenvalues ∗i of ∗ satisfy the following inequalities, resulting from Eq. (18.55): (E{
−1
})−1 ≤
∗i
≤
(18.56)
This corresponds to the well known arithmetic and harmonic averages, corresponding to layered microstructures with series or parallel arrangements. Hashin-Shtrikman variational principle and H-S bounds Another variational principle can be established [232], starting from equation (18.45) relating E and 0 E E = E0 − Γ0 0 E Generalizations were given by J.R. Willis in [669], [670], [671], [673], [582], which is followed in this presentation. From Eq. (18.45), and noting the ”polarization” τ = 0 E = ( − 0 )E, ( −
−1 τ 0)
+ Γ0 τ = E0
(18.57)
where τ can be considered as an unknown. The operator ( − 0 )−1 + Γ0 acting on τ is a self adjoint operator; from application of theorem 20.69, the solution τ of Eq. (18.57) is given for an extremum of: H(τ ) = (τ , ( −
0)
−1
τ ) + (τ , Γ0 τ ) − 2(τ , E0 )
(18.58)
The extremum is given by (−τ , E0 ) = −( E, E0 ) + ( 0 E, E0 ) = −( E, E − E 0 ) + ( 0 (E0 + E 0 ), E0 ) 0 0 0 Rwith 0E = E − E0 = −Γ0 τ . We have ( 0 E , E0 ) = ( 0 E0 , E ) = 0 since E (x)dx = 0. Similarly, B Z Z 1 1 0 0 D(x)E (x)dx = − D(x) grad φ0 (x) dx ( E, E ) = V B V B Z 1 div(φ0 (x)D(x))dx =− V B Z 1 → φ0 (x)D(x).− n dS = 0, as φ0 (x) = 0 for x ∈ ∂B =− V ∂B
As a consequence, the extremum of H(τ ) is 2(U0 − U ) with 2U = ( E, E). It is easy to show that the operator ( − 0 )−1 +Γ0 is definite positive (and therefore the extremum of H(τ ) is a minimum) if 0 is chosen such that for every x ∈ B, ( − 0 ) remains positive definite. In that case, 2(U0 − U ) is a
640
18. Change of Scale in Physics of Random Media
minimum of H(τ ). This is due to the fact that the operator Γ0 is positive definite: (τ , Γ0 τ ) = −(τ , E 0 ) = −( 0 E, E 0 ) = (− E, E 0 ) + ( 0 E, E 0 ) = ( 0 E 0 , E 0 ) + ( 0 E0 , E 0 ) = ( 0 E 0 , E 0 ) ≥ 0 It can also be shown that the operator ( − 0 )−1 + Γ0 is definite negative if 0 is chosen such that for every x ∈ B, ( − 0 ) remains negative definite. In that case, 2(U0 − U ) is a maximum of H(τ ). Therefore, for any choice of τ ∗ (it has the advantage to have no particular constraint as in the case of a trial field used in the classical variational principle), H(τ ∗ ) ≥ 2(U0 − U ) when ( − H(τ ∗ ) ≤ 2(U0 − U ) when ( −
0)
is always positive definite (18.59) ) 0 is always negative definite
Applying to the comparison medium (with the constant permittivity 0 ) and to the random medium the electric field E = E0 , the H-S variational principle becomes H(τ ∗ ) ≥ (E, (
−
∗
is always positive definite (18.60) ∗ ∗ H(τ ) ≤ (E, ( 0 − )E) when ( − 0 ) is always negative definite 0
)E) when ( −
0)
The inequalities (18.60) remain after taking the mathematical expectation E{H(τ ∗ )}. Starting from any trial polarization τ , a field E 0 is defined by application of Eq. (18.45), Γ0 being a projector: E 0 = E − E0 = −Γ0 τ From the definition of the operator Γ0 (Eqs (18.43,18.44)), div( 0 E 0 ) = div(−τ ) Therefore the field 0 E 0 is not admissible as a displacement field; however, the following field D0 (x) admits a null divergence, and can be used as a trial field in the classical variational principle: D0 (x) =
0E
0
+ τ = − 0 Γ0 τ + τ
To illustrate the Hashin-Shtrikman (H-S) variational principle in the case of a N phases multi-component random medium (the component Ai having the indicator function ki (x) and the dielectric permittivity tensor i ), consider the following trial polarization τ ∗ , where the τ i are vectors chosen to get an extremal value of H(τ ∗ ):
18.10 Bounds of effective properties
∗
τ =
i=N X
τ i ki (x)
641
(18.61)
i=1
Introducing τ ∗ given by Eq. (18.61) into Eq. (18.58), it comes H(τ ∗ ) =(
i=N X
(18.62)
τ i ki (x),
i=1
−2(
i=N X i=1
i=N X
( i−
−1 τ i ki (x)) 0)
i=N X
+(
τ i ki (x), Γ0 (
i=1
i=N X
τ i ki (x)))
i=1
τ i ki (x), E)
i=1
Taking the mathematical expectation E{H(τ ∗ )}, noting pi = E{ki (x)}, Cij (x, y) = E{ki (x)kj (y)} and accounting for the definition of the operator Γ0 in Eqs (18.44,18.45), E{H(τ ∗ )} =
i=N X i=1
pi τ i ( i −
(18.63) −1 τi 0)
+
i=N,j=N X i=1 j=1
(τ i , Γ0 Cij (x, y)τ j ) − 2
i=N X
pi (τ i , E)
i=1
E{H(τ ∗ )} in Eq. (18.63) is extremal (it is a minimum when ( − 0 ) is definite positive, and a maximum when ( − 0 ) is definite negative) for a choice of τ i satisfying pi τ i ( i −
−1 0)
+
j=N X j=1
Γ0 Cij (x, y)τ j − pi E = 0
(18.64)
Pi=N This extremal value is given by − i=1 pi (τ i , E) = −(E{τ }, E). From the inequalities (18.60), the obtained extremum of E{H(τ ∗ )} gives the lowest upper bound for (E, ( 0 − ∗ )E) when ( − 0 ) is everywhere positive definite (and the highest lower bound when ( − 0 ) is everywhere negative definite) for the trial polarization τ ∗ given by Eq. (18.61). For an isotropic random medium, a reference medium with a scalar permittivity 0 can be introduced, and Cij (x, y) = Cij (|x − y|). In that case, Γ Γ0 = , with the operator Γ given in Eq. (18.34). The following result is 0
valid in the d dimensional space Rd for the isotropic case (exercise 18.20.2): (Γ0 Cij (x, y), τ j ) =
1 (Cij (0) − Cij (∞)) τ j d 0
(18.65)
642
18. Change of Scale in Physics of Random Media
and therefore the second order bounds obtained from the used trial polarization τ ∗ in the Hashin-Shtrikman variational principle do not depend on the second order statistics for an isotropic geometry. ⎛ ⎞ j=N j=N X X 1 1 ⎝ τ i pi − Γ0 Cij (x, y)τ j = pi pj τ j ⎠ = (τ i pi − pi E{τ }) d d 0 0 j=1 j=1
After report of this result, Eq. (18.64) becomes τ i( i −
−1 0)
+
1 (τ i − E{τ }) − E = 0 d 0
(18.66)
and τi = with Ci =
µ µ
E{τ } E+ d 0
¶
Ci−1
−1 + 0)
( i−
I d 0
(18.67) ¶
, I being the unit tensor
From Eq. (18.67), it comes E{τ } =
j=N X j=1
µ ¶ j=N X E{τ } pj τ j = E + pj Cj−1 K with K = d 0 j=1
(18.68)
Solving Eq. (18.68) gives µ ¶−1 K E{τ } = EK I − d 0 Ã
(18.69)
µ ¶−1 ! K and the extremum −(E{τ }, E) of E{H(τ )} is − EK I − ,E . d 0 By application of the H-S principle (18.60), the tensor ∗ can be bounded, since à ¶−1 ! µ K − EK I − , E ≥ (E, ( 0 I − ∗ )E) d0 ∗
and therefore
µ
K −K I − d0
¶−1
≥
0I
−
∗
when ( − 0 I) remains positive definite (the opposite inequality being satisfied when ( − 0 I) remains negative definite). In the first case a lower bound, and in the second case an upper bound, are derived:
18.10 Bounds of effective properties
643
• ( − 0 I) ≥ 0 if for every eigenvalue ji of the tensors i we have ji − 0 ≥ 0. This is satisfied when 0 ≤ m , where m = inf i,j ( ji ). In this case, choosing 0 = m , from Eq. (18.60), the lower bound is: ∗
≥
mI
+K
µ
K I− d0
¶−1
(18.70)
• ( − 0 I) ≤ 0 if for every eigenvalue ji of the tensors i we have ji − 0 ≤ 0. This is satisfied when 0 ≥ M , where M = supi,j ( ji ). In this case, choosing 0 = M , from Eq. (18.60), the upper bound is: ∗
≤
MI + K
µ ¶−1 K I− d 0
(18.71)
• When for every component i, i is a scalar, a particular case of the previous results is obtained [232], namely the H -S bounds, K=
j=N X j=1
(
j
−
0
pj −1 ) +
Choosing 0 = m = inf i ( i ) or alternatively (18.70, 18.71) become ∗
∗
≥
m
≤
M
+
(18.72)
1/( 0 d)
K
K d 0 K + K 1− d0
0
=
M
= supi ( i ), Eqs
(18.73)
1−
(18.74)
For illustration, the upper H-S+ and lower H-S− bounds of the dielectric permittivity of a two-phase isotropic composite are given by Eqs (18.26) and Eq.( 18.27). Any measured or calculated overall property of an isotropic composite should be enclosed in this interval. These bounds are reached on microstructures made of an assemblage of composite Hashin spheres (subsection 18.6.3). Other random microstructures like multi scale layered composites or multi scale dilated Poisson hyperplanes reach these bounds (subsection 18.15.6). These special examples can be called optimal microstructures. The (H-S) bounds are narrower than the Wiener bounds (they introduce additional information about the microstructure, namely the assumption of isotropy). Wiener and (H-S) bounds depend only on the volume fractions and on the values of the properties of the components, and therefore are the same for very different microstructures.
644
18. Change of Scale in Physics of Random Media
These first and second order bounds were derived in a very elegant way by G. Matheron [466], starting from the partial derivative of the effective permittivity ∗ with respect to i (giving the second order moment of the electric field E in component Ai ), without resorting to any variational principle ([495] p. 346). • For isotropic elastic media with an isotropic geometry, the elastic properties depend on the two elastic constants, the bulk modulus K and the shear modulus G. The H-S bounds of a "well ordered" [233], [495] twophase isotropic composite in R3 , where K1 ≤ K2 (with volume fraction p1 ), G1 ≤ G2 (to insure that the 4th rank tensor C2 −C1 is positive definite) are given by GHS−
(18.75) 2
= p1 G1 + p2 G2 −
p1 p2 (G1 − G2 ) p1 G2 + p2 G1 + G1 (9K1 + 8G1 )/(6(K1 + 2G1 ))
GHS+
(18.76) p1 p2 (G1 − G2 )2 = p1 G1 + p2 G2 − p1 G2 + p2 G1 + G2 (9K2 + 8G2 )/(6(K2 + 2G2 )) K HS− = p1 K1 + p2 K2 −
p1 p2 (K1 − K2 )2 p1 K2 + p2 K1 + 4G1 /3
(18.77)
K HS+ = p1 K1 + p2 K2 −
p1 p2 (K1 − K2 )2 p1 K2 + p2 K1 + 4G2 /3
(18.78)
The bounds K HS+ and K HS− are the effective bulk moduli of the composite Hashin spheres assemblage. The bounds GHS+ and GHS− are the effective shear moduli of infinite rank hierarchical laminates [495]. If K1 ≤ K2 and G1 ≥ G2 (non well ordered moduli), bounds of the shear modulus become the Walpole bounds [653], [495] obtained by exchanging K1 and K2 in Eqs (18.75, 18.76). • Bounds for the non scalar case when the eigenvalues of the permittivity tensor do not depend on x, as in a polycrystal, are considered in [50], p. 253. • The H-S bounds are rather inefficient for a high contrast of properties, since the difference H-S+ −H-S− can be extremely large. However they are very useful (and cheap) to get a range of expected properties, and also to check the results given by numerical simulations. In particular in case of a porous medium (with pore volume fraction 1 − p), the lower bound for ∗ , K ∗ or G∗ is zero, while for a composite where for one component = +∞, K = +∞ or G = +∞, the upper bound is +∞. For a porous medium upper H-S bounds are given by:
18.10 Bounds of effective properties HS+
K HS+ GHS+
2p 3−p 4pGK = 3(1 − p)K + 4G pG = 6(1−p)(K+2G)2 1 + 5G(4K+3G)+(3K+G)2 =
645
(18.79)
18.10.2 Bounds of order 2N + 1 derived from the classical variational principle In general, it is not possible to know exactly the effective (or macroscopic) permittivity ∗ , except for some specific geometries (some examples are given in section 18.6). In fact, the exact prediction of ∗ requires a very large amount of information (the set of all n point correlation functions, which naturally appear in the perturbation expansion (18.37) of the solution of equation (18.11) combined to equation (18.5) for infinite domains B containing realizations of ergodic random media [50],[393], [438], [439]. Using a limited amount of statistical information in electrostatics, bounds (upper bound + and lower bound − ) are derived from the perturbation expansion of the electric field and a variational principle on the stored electrostatic energy [53], [54], [50]. From this principle, which is equivalent to the conservation law (18.11), the replacement of the unknown solution E(x) by a suitable approximation (or trial field) E ∗ (x) provides an estimate hU ∗ i of the energy, with hU ∗ i ≥ hU i, where hU i is obtained for the solution (equation 18.11). The same inequality is obtained by replacing the unknown D(x) by a trial field D∗ (x) with div(D∗ ) = 0. By using the following trial field in the classical variational principle [53], [54], [50] n=N X (n) λn Ej (x) EjN (x) = E j + n=1
(n)
where Ej (x) is the term of order n of the perturbation expansion of the electric field (18.37), the λn minimizing the upper bound of the effective permittivity are obtained as the solution of a linear system with coefficients depending on correlation functions up to the order 2N + 1 (order 3 for N = 1). Similarly for a trial field DjN (x)
= Dj +
n=N X
(n)
λn Dj (x)
n=1
With for n > 1 [241]: (n)
(n)
(n−1)
Dj (x) = Ej (x) + 0 Ej
(n−1)
(x) − E{ 0 Ej
(x)}
646
18. Change of Scale in Physics of Random Media
the lower bound of order 2N + 1 are obtained. By increasing the order N of the expansion, narrower bounds are obtained, since they involve an increasing amount of information about the microstructure. In practice, the expansion is stopped to the first term, in order to get third order bounds detailed in section 18.11, correlation functions of higher order being usually unknown. Similarly in elasticity of random media, replacing in the classical variational principle the unknown solution for the deformation e(x) by a kinematically admissible approximation (or trial field) e∗ (x), the obtained approximation of the elastic potential energy hw e∗ i satisfies hw e∗ i ≥ hwi, e where hwi e corresponds to the solution of the boundary value problem. A similar result is obtained by replacing the unknown stress field σ(x) by a trial field σ ∗ (x) with div(σ∗ ) = 0 in the complementary energy he ui. A first example is obtained from the trial fields e∗ (x) = e and σ ∗ (x) = σ. They give the Voigt upper bound CV and the Reuss CR lower bound (in the sense of order 4 tensors), which depend only of the volume fractions of components in the case of a composite, and are equivalent to the Wiener bounds for the permittivity: CV = E {C(x)} n o ¡ R ¢−1 C = E (C(x))−1
By introduction in the classical variational principle an admissible trial field n=N X eN (x) = e + λn e(n) (x) (18.80) n=1
where e(n) (x) is the term of order n of the perturbation expansion of the strain field (18.48), the λn minimizing the upper bound of the homogenized elastic moduli are obtained as the solution of a linear system with coefficients depending on correlation functions up to the order 2N + 1 (order 3 when N = 1). Similarly, an admissible trial field [54], [50] σ N (x) = σ +
n=N X
λn σ (n) (x)
(18.81)
n=1
enables us to compute the lower bound of order 2N + 1. By increasing the order N of the expansion, tighter bounds are obtained. Stopping to the first order of the expansion, third order bounds are given, and will be detailed in section 18.14.
18.11 Third order bounds of the dielectric permittivity
647
18.11 Third order bounds of the dielectric permittivity The third order bounds (upper bound + and lower bound − ), derived from a perturbation expansion of the electric field and a variational principle on the stored electrostatic energy [50] as in section (18.10) (for the linear behavior considered in this chapter, this energy is quadratic), can be expressed in Rd . For materials with a continuous variation of properties (such as polycrystals or materials presenting gradient of properties like microstructures with segregations), that can be modeled by random functions instead of random sets, very few results are available in the literature. In this section are presented derivations of third order bounds for some types of random functions models corresponding to different micro geometrical arrangements [292]. For other models, third order probabilistic properties entering into the calculation of bounds are given. The bounds to be obtained in this general situation are of interest to compare the expected macroscopic response of various continuous textures.
18.11.1 Reminder on third order bounds for RF Consider random media with a scalar dielectric permittivity that can be modelled by a stationary and isotropic random function (RF) (x) > 0 that remains bounded. Using a perturbation expansion of the electric field, limited to the first order, as a trial field in a variational principle on the stored electrostatic energy, it is possible to derive third order bounds [49], [50] , calculated from third order moments of the RF (x). These are expressed by Eqs (18.82) and (18.83) below [49], [50],[241], [487], where the lower and the upper bounds are written − and + , and the expectation of a random variable Z is denoted by E{Z}: 1/
−
=E
½ ¾ 1
+
−
µ
4 E 3
½ 0 ¾¶2
1 2
4E { } E
©
1 ª /
02 02
2
3E { }
© ª 1 E 02 1 =E{ }− 3 E { } 1 + 3E { } /E {
(18.82) +J
02 } I
(18.83)
In Eqs (18.82) and (18.83), the centred RF is 0 (x) = (x)−E{ }. The values I and J are given by Eqs (18.86) and (18.87) [49], [50],[241], expressed as a function of the following third order correlation functions W 3 (h1 , h2 ) = W 3 (x, y, θ) = E{ 0 (x) 0 (x + h1 ) 0 (x + h2 )}
(18.84)
W (h1 , h2 ) = W (x, y, θ) = E{ 0 (x + h1 ) 0 (x + h2 )/ (x)}
(18.85)
648
18. Change of Scale in Physics of Random Media
with |h1 | = x, |h2 | = y, and u = cos θ, θ being the angle between the vectors h1 and h2 : © ª Z Z Z E 03 1 +∞ dx +∞ dy +1 2 2 (3u − 1)W 3 (x, y, θ)du in 3D + IE { } = x y ©9 03 ª 2 Z0 +∞ Z0 +∞ Z−1+π E 1 2 dy dx W 3 (x, y, θ) cos(2θ) dθ in 2D + IE { } = x y 4 π 0 0 0 (18.86) ©
ª
Z Z Z 1 +∞ dx +∞ dy +1 2 (3u − 1)W (x, y, θ)du in 3D JE { } = + x y © 902 ª 2 Z0 +∞ Z0 +∞ Z−1+π / E 1 2 dy dx W (x, y, θ) cos(2θ) dθ in 2D + JE { } = x y 4 π 0 0 0 (18.87) The evaluation of the integrals in Eqs (18.86) and (18.87) can be made in some cases analytically, but most often in practice, numerically. For some models, it may be easier to use the third order moment W3 instead of the central moment W 3 2
E
02
/
W3 (h1 , h2 ) = W3 (x, y, θ) = E{ (x) (x + h1 ) (x + h2 )}
(18.88)
Then Eq. (18.86) becomes © ª © ª 2 IE { }2 = E 03 /9 − E { } E 02 9 Z Z Z 1 +∞ dx +∞ dy +1 2 + (3u − 1)W3 (x, y, θ)du in 3D x y 2 0 0 −1 © ª © ª 1 2 IE { } = E 03 /4 − E { } E 02 4 Z Z Z 1 +∞ dx +∞ dy +π + W3 (x, y, θ) cos(2θ) dθ in 2D x y π 0 0 0
(18.89)
Similarly, it may be convenient to use the following moment W in Eq. (18.87) ½ µ ½ ¾¶¾ 1 1 W (x, y, θ) = E 0 (x + h1 ) 0 (x + h2 ) −E (18.90) (x) Then
18.11 Third order bounds of the dielectric permittivity
©
2
1 2
02
ª
JE { } = E / Z +∞ Z +∞ Z +1 dx x
dy y
2 /9 + E 9
½ ¾ 1
E
©
02
ª
(3u2 − 1)W (x, y, θ)du in 3D ½ ¾ © ª © ª 1 1 2 JE { } = E 02 / /4 + E E 02 4 Z Z Z 1 +∞ dx +∞ dy +π + W (x, y, θ) cos(2θ) dθ in 2D x y π 0 0 0
+
0
0
649
−1
(18.91)
Third order bounds have following properties: they increase with I and J for I > 0 and for J < 0 respectively. They are tighter than the H-S bounds [232], calculated as a function of the minimum and of the maximum of (x). When the RF (x) is symmetrical with respect to its expectation E { }, I = 0, and therefore the upper bound does not depend on the morphology as it becomes 1 © 02 ª /E { } (18.92) + =E{ }− E 3 When the RF (x) and 1/ (x) have the same probabilistic properties, in particular when E { } E {1/ } = 1, the lower bound becomes the Wiener bound, which does not depend on the morphology: 1/
−
= E {1/ 1}
(18.93)
In R2 for this last situation and an isotropic RF (x), the effective permittivity is exactly given by the geometrical average of (x) [438], [440], as seen in subsection 18.20.7.
18.11.2 Third order bounds for the mosaic model The mosaic model [440] or ”cell” model ,[241], [487] is built in two steps (chapter 8): starting with a random tessellation of space into cells, to every cell A0 is affected independently a realization of the random variable . The parameters of this model are the distribution function of and the probabilistic properties of the tessellation (there is here a separation of the space and of the physical variables). The moments W 3 and W are given by: © ª W 3 (x, y, θ) = s(x, y, θ)E 03 (18.94) © 02 ª W (x, y, θ) = s(x, y, θ)E / where s(|h1 | , |h2 | , θ) is the probability that the three points {x}, {x + h1 }, {x + h2 } belong to the same random cell A0 (noting μd the Lebesgue measure in Rd , μd its average over the realizations of A0 , and A0h obtained 0 by translation © 03 ª of A by the vector h), given in Eq. (18.125) When for the RV , E = 0, then I = 0. When and 1/ have the same distribution,
650
18. Change of Scale in Physics of Random Media
E { } E {1/ } = 1 and J does not depend on the morphology. For the mosaic model, Miller defined a parameter G (Eq. (18.126)) depending only on the random cell geometry by means of the© function s(|h1 | , |h2 | , θ), that ª can be expressed as a function of I when E 03 6= 0: G=I
E { }2 E { 03 }
The parameter G does not depend on the scale of the tessellation. In three dimensions, 19 ≤ G ≤ 13 (G = 19 for spheres and G = 13 for plates). In two dimensions, 14 ≤ G ≤ 12 (G = 14 for disks and G = 12 for needle shapes). However, it is difficult to produce a feasible construction of random tessellations corresponding to these extreme values. For the Poisson tessellation of space by Poisson random planes in R3 (with the intensity λ), and by Poisson lines in R2 (chapter 7), the cells are Poisson polyhedra and Poisson polygons, and s(|h1 | , |h2 | , θ) is given in Eq. (18.129). From Eqs (18.126), in R2 , G = 1 − ln 2 [440] and in R3 , G = 16 [416], [417]. The same results are obtained from a STIT mosaic (chapter 8). For the Dead Leaves model (chapter 11) when the grain A0 (t) and the random variable are independent, a mosaic model built on a Dead Leaves tessellation [439] is obtained. The shape of the resulting cell is non convex. The parameter G is obtained from s(x, y, θ) expressed in Eq. (18.131). For Dead Leaves built on Poisson polyhedra as primary grains [416]: G ' 0.170 in R3 and G ' 0.311 in R2 . Anisotropic cells (with a uniform orientation) are studied in [416], [417]: G increases with the cell anisotropy.
18.11.3 Third order bounds of the Dilution model The Dilution RF model (chapter 15) is obtained by implantation of random primary functions Z 0 (x), with a compact support A0 (with possible overlaps) on Poisson points xk with the intensity λ: X Z(x) = Z 0 (x − xk ) xk
In the present case, the primary function is taken as follows: starting from a RV Z 0 , Z 0 (x) = Z 0 for x ∈ A0 and Z 0 (x) = 0 for x ∈ A0c . The considered random permittivity is (x) = Z(x) + 0 ; the set covered by grains A = ∪xk A0xk builds a Boolean model (chapter 6). Here (x) = 0 for x ∈ Ac , and therefore the medium can be viewed as a matrix with a constant permittivity 0 (the set Ac ) embedding a set A with a random permittivity (x) > 0 . The average, second and third order central correlation functions of this model are given by, where K(0 = μd (A0 ):
18.11 Third order bounds of the dielectric permittivity
E{ } = 0 + λK(0)E{Z 0 } © 02 ª E = λK(0)E{Z 02 } © ª W 3 (h1 , h2 ) = λK(0)E 03 s(h1 , h2 )
651
(18.95)
Introducing Eq. (18.95) in Eqs (18.83) and (18.89), gives the upper bound + as a function of the parameters of the model, and namely as a function of the Miller parameter G corresponding to the random grain A0 . For any population of similar grains (like for instance a population of spheres with a random radius R), G remains constant, and therefore keeping constant λK(0) gives the upper bound + for a family of random functions © same ª (x). When E 03 > 0, + increases with G (plate reinforcement giving a higher upper bound than spheres). Unfortunately, the moments W (x, y, θ) and W (x, y, θ) needed to calculate the lower bound are usually unknown for this model.
18.11.4 Transformation of basic models A simple transformation of a primary RF Z(x) model by a function φ generates new RF (x) = φ(Z(x)). When applied to the mosaic model, a mosaic model with the same geometry and with a new random variable = φ(Z) is recovered. In the present case an exponential transformation is applied to the dilution RF and to the Gaussian RF. The correlation functions W2 (h), E { (x)/ (x + h)}, W3 (h1 , h2 ), and E { (x + h1 ) (x + h2 )/ (x)}, which are required for the calculation of third order bounds, are given below.
Exponential of a Dilution model If Z(x) is a dilution RF built with a primary grain A0 as before, and now a constant variable Z 0 for x ∈ A0 , the average, second and third order correlation functions of (x) = exp(Z(x)) are given by (after correction of some misprints in [292]): E{ } = exp (λK(0) (exp(Z 0 ) − 1)) E{1/ 1} = exp (λK(0) (exp(−Z 0 ) − 1)) W2 (h) = E{ (x) (x + h)} = exp (λK(0)r(h)(exp(2Z 0 ) − 2 exp(Z 0 ) + 1) +2(exp(Z 0 ) − 1) E { (x)/ (x + h)} = exp (λK(0)2(1 − r(h))(cosh(Z 0 ) − 1)) (18.96) where r(h) =
μd (A0 ∩ A0h ) . K(0)
652
18. Change of Scale in Physics of Random Media
h 3 log W3 (h1 , h2 ) = λK(0) (exp(Z 0 ) − 1) s(h1 , h2 ) (18.97) ³ 2 (exp(Z 0 ) − 1) (r(h1 ) + r(h2 ) + r(h2 − h1 )) +3 (exp(Z 0 ) − 1)]
log E { (x + h1 ) (x + h2 )/ (x)} = λK(0) ³³ ´ 2 2(cosh(Z 0 ) − 1) − (exp(Z 0 ) − 1) s(h1 , h2 )
(18.98)
−2 (r(h1 ) + r(h2 )) (cosh(Z 0 ) − 1)
´ 2 +r(h2 − h1 ) (exp(Z 0 ) − 1) + 2 exp(Z 0 ) + exp(−Z 0 ) − 3
Lognormal model If Z(x) is a multivariate reduced Gaussian RF, with second order central correlation function ρ(h), consider (x) = exp(Z(x)). The average, second and third order correlation functions of (x) and of 1/ (x) for this model are given by √ 1 E{ } = E{ } = e W2 (h) = E{ (x) (x + h)} = exp (1 + ρ(h)) E { (x)/ (x + h)} = exp (1 − ρ(h)) (18.99) µ ¶ 3 W 3 (h1 , h2 ) = exp + ρ(h1 ) + ρ(h2 ) + ρ(h2 − h1 ) 2 µ ¶ 3 E { (x + h1 ) (x + h2 )/ (x)} = exp − ρ(h1 ) − ρ(h2 ) + ρ(h2 − h1 ) 2
18.11.5 Combination of the basic random functions models Starting from basic models, it is interesting to consider some combinations, to describe more complex structures. For instance, the presence of multiple scales in a medium, or of fluctuations of the morphological properties can be observed in materials. The purpose of this subsection is to propose some models to account for these points. Addition of independent RF The simplest way to generate large families of RF is to consider the superimposition of independent primary RF by addition. For instance, starting from n independent positive RF Zi (x), define:
18.11 Third order bounds of the dielectric permittivity
(x) =
i=n X
653
Zi (x)
i=1
For this model, the average, second and third order correlation functions of (x) for this model are given by E{ } = W 3 (h1 , h2 ) =
i=n X i=1
i=n X
E{Zi }, W 2 (h) =
i=n X
W 2i (h)
(18.100)
i=1
W 3i (h1 , h2 )
i=1
The derivation of the other moments like W (h1 , h2 ) is difficult in the general case, so that only the upper third order bound can be expressed in this case. Starting from mosaic or dilution models (with parameter Gi for the scale P Zi ), for the RF (x) when W 3 (0) = i=n i=1 W 3i (0) 6= 0, an equivalent G is given by: i=n X G= W 3i (0)Gi /W 3 (0) i=1
For instance, if every scale Zi (x) admits the same G, the sum owns the same parameter.
Multiplication of independent RF In some cases, it may be useful to consider a multiplicative process to generate RF i=n Y (x) = Zi (x) i=1
This can be converted to the previous sum by using the logarithm. A direct calculation gives: E{ } = W3 (h1 , h2 ) =
i=n Y i=1
i=n Y
E{Zi }, W2 (h) =
i=n Y
W2i (h)
(18.101)
i=1
W3i (h1 , h2 )
i=1
From Eqs (18.101), the third order upper bound are derived after calculation of the term I (Eq. 18.89). The other terms, required for the calculation of the lower bound, are not available in general. However,Pthe lognormal model (which remains lognormal by product, with ρ(h) = ( i=n i=1 ρi (h))/n), or the exponential of a dilution RF can be used for this purpose.
654
18. Change of Scale in Physics of Random Media
18.11.6 A hierarchical model As in the case of random sets [334], a hierarchical model with two separate scales can be built in two steps: starting with a random tessellation of space, to every cell is affected a realization of a RF Z(x) (the realizations in distinct cells being independent). Any type of RF (mosaic, dilution model, lognormal, ...) can be used in the second step. For this model, using the subscript H for the hierarchical model: E{ } = E{Z}; E{1/ } = E{1/Z} W H2 (h) = r(h)W 2 (h) W H3 (h1 , h2 ) = s(h1 , h2 )W 3 (h1 , h2 ) W H (h1 , h2 )) = s(h1 , h2 )W (h1 , h2 )
(18.102)
For cells much larger than the scale of the secondary RF, it turns out that the two parameters I and J (Eqs (18.86) and (18.87)) remain the same, with a good approximation, for the initial and for the hierarchical model, which gives the same third order bounds.
18.12 Third order bounds of the real dielectric permittivity of random sets After a recall of the third order bounds in the special case of two phase media, we examine in section 18.15 their practical derivation for some generic random sets and their combination [334]. This can be used as guidelines to select the most appropriate microstructures and to design materials for given physical properties. Consider random composites made of two phases A1 (with fraction p) and A2 (with fraction q = 1 − p) having a scalar real dielectric permittivity 1 and 2 (with 2 > 1 ) (the same approach could be followed for other physical properties such as the thermal or electrical conductivity, the permeability of porous media,...). The composite is modelled by a stationary and isotropic random set A (with A = A2 and Ac = A1 ). In that case the central correlation function of order three of the random function (x) generated by the composite is (Eq. (3.111) in chapter 3): W 3 (h1 , h2 ) = ( 1 − 2 )3 (Q(h1 , h2 ) − q(Q(h1 ) + Q(h2 ) + Q(h3 )) + 2q 3 ) = ( 1 − 2 )3 (P (h1 , h2 ) − p(C(h1 ) + C(h2 ) + C(h3 )) + 2p3 ) (18.103) with Q(h1 , h2 ) = P {x ∈ Ac ∩ Ac−h1 ∩ Ac−h2 } and P (h1 , h2 ) = P {x ∈ A ∩ A−h1 ∩ A−h2 }
18.12 Third order bounds of the real dielectric permittivity of random sets
655
Third order bounds (upper bound + and lower bound − ), derived from the first order perturbation expansion of the electric field and the variational principle on the stored electrostatic energy [50] as in section (18.10) (for the linear behavior considered in this chapter, this energy is quadratic), can be expressed in Rd (d = 1, 2, 3) as follows, using notations of G. Milton [492] and of S. Torquato [633], [634]: −/ 1
(18.104)
1 + ((d − 1)(1 + q) + ζ 1 − 1) β 21 + (d − 1) (((d − 1)q + ζ 1 − 1)) β 221 = 1 − (q + 1 − ζ 1 − (d − 1)) β 21 + ((q − (d − 1)p) (1 − ζ 1 ) − (d − 1)q) β 221 +/ 2
=
(18.105)
1 + ((d − 1)(p + ζ 1 ) − 1) β 12 + (d − 1) (((d − 1)p − q) ζ 1 − p) β 212 1 − (1 + p − (d − 1)ζ 1 ) β 12 + (p − (d − 1)ζ 1 ) β 212
In Eqs (18.104,18.105), + − − is of the order of ( and β 12 and β 21 are given by: β ij =
− j i + (d − 1) i
1
−
4 2)
when
1
→
2,
(18.106) j
The function ζ 1 (p) [492] depends on the volume fraction p for a given model of random set, as illustrated later. It is deduced from the probability P (h1 , h2 ) that the three points {x}, {x+h1 }, {x+h2 } belong to A1 . Noting this probability P (|h1 | , |h2 | , θ), with u = cos θ, θ being the angle between the vectors h1 and h2 , Z +∞ Z +∞ Z +1 9 dy dx ζ 1 (p) = (3u2 − 1)P (x, y, θ)du in R3 x y 4pq 0 0 −1 (18.107) Z +∞ Z +∞ Z π 4 dy dx ζ 1 (p) = P (x, y, θ) cos(2θ) dθ in R2 x y πpq 0 0 0 The evaluation of the integrals in Eqs (18.107) can be made in some cases analytically, but most often numerically. For a better numerical convergence it is convenient to replace in the integrals P (x, y, θ) by P (x, y, θ) − P (x)P (y)/p , P (h) being the probability that the two points {x}, {x + h} belong to A1 . In [417], [416] the Gauss-Legendre quadrature is used, adapting the number of points (usually a few tens) to the convergence. For a function P (x, y, θ) which does not depend on θ, the integrals in Eqs (18.107) are identically equal to zero. For a function P (|h3 |) (with h3 = h2 − h1 2 2 2 and |h3 | = |x| + |y| − 2 |x| |y| cos θ), Eqs (18.107) becomes: (see exercise 18.20.3)
656
18. Change of Scale in Physics of Random Media
Z
+∞ dx x
0
Z
0
Z
+∞ dx x
+∞ dy y
0
Z
0
Z
+∞ dy y
+1
−1
Z
0
(3u2 − 1)P (|h3 |)du =
4 9
(P (0) − P (∞)) in R3
π
P (|h3 |) cos(2θ) dθ =
π 4
(P (0) − P (∞)) in R2
(18.108) Exchanging phases A1 and A2 enables us to define the function ζ 2 (q) with ζ 2 (q) = 1 − ζ 1 (p). The function ζ 1 (p) satisfies 0 ≤ ζ 1 (p) ≤ 1. For ζ 1 (p) = 1 or 0 (and only in these cases), the two bounds + and − coincide and are equal to the H-S upper (ζ 1 (p) = 1) or lower (ζ 1 (p) = 0) bound given in Eqs (18.73) (this is illustrated in subsection 18.15.6). For given p, 1 and 2 , the two bounds increase with the function ζ 1 (p) (while the H-S bounds remain fixed), so that higher values of the effective properties are expected. This point can be used to compare the expected properties of materials with different morphologies, as seen later. If the two phases A1 and A2 are symmetric, the case of p = 0.5 produces an autodual random set (the two phases having the same probabilistic properties), for which ζ 1 (0.5) = 0.5. This case corresponds to a third order central correlation function equal to zero. Therefore in two and three dimensions, the third order bounds of a symmetric medium present a fixed point at p = 0.5. In addition, it is known [438],[440] that for an autodual random set in two dimensions the effective permittivity is equal to the geometrical average of the two permitivities.
18.13 Third order bounds of the complex dielectric permittivity and spectral measure of random sets 18.13.1 Bounds in the complex plane An extension of bounds to the complex dielectric permittivity was developed by D. Bergman [62] and by G. Milton [490], [495], from analytic expansions. As before, consider a random composite made of two phases A1 (with volume fraction p) and A2 (with volume fraction p2 = 1 − p1 ). In the case of absorbing or conductive media, the dielectric permittivity de00 pends on the frequency ω of the electric field and is complex : = 0 − i ( 0 is the dielectric constant, and 00 the dielectric loss factor). The bounds derived by M. Beran for real permittivity [49] are not available anymore. The derivation of the bounds on the effective permittivity ∗ of a two component composite with complex permittivity 1 (phase A1 ) and 2 (phase A2 ) [490] is based on the Bergman’s method [62], which is valid in the quasi static case (when the wave length is much larger than the characteristic length of the microstructure). For this, it is convenient to introduce
18.13 Third order bounds of the complex dielectric permittivity and spectral measure
the complex variable τ , τ =(
1
+
2 )/( 1
−
2)
∗
is expressed as a function of 2 and τ , which can be represented in the following closed rational form [490], e
=
2
m Y
i=0
(τ − τ 0i )/(τ − τ i ),
(18.109)
where the constants τ i and τ 0i are real and must satisfy (18.110), 1 > τ 0 > τ 00 > τ 1 > τ 01 ...τ 0m > −1,
(18.110)
In two dimensions, the poles and zeroes in [0,1] define (by interchange) those in [-1,0]. Using (18.109) and the method described in [490], the third order Be0 00 ran’s bounds can be generalized to the complex plane ( , ). The bounds for complex 1 and 2 restrict ∗ to a region Ω n of the complex plane which depends on the available information on the microstructure of the composite material. They depend on some constants Rn , calculated from the perturbation development of ∗ [49] and summarizing the microstructural information on the composite: ¯ ∂ n e ( 1 , 2 ) ¯¯ Rn = (18.111) ¯ ∂ n1 1 = 2 =1 These constants Rn are:
• R1 = p for any composite material. 2 • R2 = − p(1 − p) if the material is statistically isotropic in Rd . d d+1 1−p • R3 = p(1 − p)(ζ 1 + ). d d−1 2 • R4 = 3p(1 − p) (2 − p) − 2R3 (3 − 2p) for 2 dimensional materials. For higher values of n, Rn can be calculated in terms of higher-order correlation functions. When the constants τ i and τ 0i are allowed to vary subject to the appropriate restrictions (18.110) and (18.111), while keeping τ , 2 , and Rn (n = 1, 2, ..., J) fixed [62], [490], [491], [492], [495], ∗ , given by Eq. (18.109), covers the domains Ω n of the complex plane. This region, which provides rigorous bounds on ∗ , is more restricted when the used morphological information is larger. Regions Ω, Ω 1 , Ω 2 , Ω 3 , and Ω 4 are enclosed by circular arcs (the pair of arcs An B n An−1 and An B n B n−1 bound Ω n ). Given 1 , 2 , p, d and ζ 1 , the points An and B n (for n = 1, 2, 3 and 4) are first plotted.
657
658
18. Change of Scale in Physics of Random Media
• Without any information on the geometry, ∗ is confined to the region 0 00 Ω( 1 , 2 ) of the complex plane ( , ) bounded by the arc O 1 2 and the straight line 1 2 . • Knowing the volume fractions of the components p and 1 − p , ∗ is confined to a smaller region Ω 1 ( 1 , 2 ; p; d), which is bounded by the two circular arcs A1 B 1 1 and A1 B 1 2 [490]. ½ 1 A = p 1 + (1 − p) 2 B 1 = (p/ 1 + (1 − p)/ 2 )−1 • For a statistically isotropic structure, ∗ is confined to a smaller region Ω 2 ( 1 , 2 ; p; d) of the complex plane (d = 2 or 3), bounded by the two circular arcs A2 B 2 A1 and A2 B 2 B 1 [490]. ⎧ d(1 − p) 1 ( 2 − 1 ) ⎪ ⎪ A2 = 1 + ⎪ ⎪ ⎨ d 1 + p( 2 − 1 ) ⎪ ⎪ ⎪ ⎪ ⎩ B2 =
dp 2 ( 1 − 2 ) d 2 + (1 − p)( 1 − 2 ) • Knowing the Milton’s function ζ 1 , the region Ω 3 ( 1 , 2 ; p; d; ζ 1 ) is bounded by the two circular arcs A3 B 3 A2 and A3 B 3 B 2 [492]. ⎧ 3 A /1 ⎪ ⎪ ⎪ ⎪ = (1 + ((d − 1)(2 − p) + ζ 1 − 1)β 21 ¢ ⎪ ⎪ ⎪ ⎪ +(d − 1)((d − 1)(1 − p) + ζ 1 − 1)β 221 ⎪ ⎪ ⎪ ⎪ / (1 − (2 − p − ζ 1 − (d − 1))β 21 ⎪ ⎪ ¢ ⎨ +((1 − p − (d − 1)p)(1 − ζ 1 ) − (d − 1)(1 − p))β 221 ⎪ ⎪ ⎪ ⎪ B3/ 2 ⎪ ⎪ ⎪ ⎪ = (1 + ((d − 1)(p + ζ 1 ) − 1)β 12 ⎪ ⎪ ¢ ⎪ 2 ⎪ ⎪ +(d − 1)(((d − 1)p − (1 − p))ζ − p)β 12 1 ⎪ ¢ ⎩ ¡ / 1 − (1 + p − (d − 1)ζ 1 )β 12 + (p − (d − 1)ζ 1 )β 212 i− j with β ij = i + (d − 1) j 2
+
• In the case of a two dimensional material, fourth order bounds can be defined. The region Ω 4 ( 1 , 2 ; p; d = 2; ζ 1 ) is bounded by the two arcs A4 B 4 A3 and A4 B 4 B 3 [492]. ⎧ 1 + (1 − p)β 21 − p(1 − ζ 1 )β 221 ⎪ 4 ⎪ A = ⎪ 1 ⎪ ⎨ d=2 1 − (1 − p)β 21 − p(1 − ζ 1 )β 221 ⎪ ⎪ ⎪ ⎪ 4 ⎩ Bd=2 =
2
1 − pβ 21 − (1 − p)ζ 1 β 221 1 + pβ 21 − (1 − p)ζ 1 β 221
When 1 and 2 are both real and positive the bounds reduce to Wiener, Hashin-Shtrikman, Beran and Milton bounds respectively, points Ai and B i giving lower and upper bounds: all the regions Ω n become segments on the real axis.
18.13 Third order bounds of the complex dielectric permittivity and spectral measure 3D material - Boolean Model - eps1={-2,3} eps2={1,1} - 3rd order
Volume fraction p1 1 (spheres)c
spheres
0.5 3 -20 -1.5
2.5 -1 -0.5 Real Epsilon 0
1.5 0.5
1
2 Imag Epsilon
1
FIGURE 18.1. Third order bounds of the complex permittivity for the Boolean model of spheres and (spheres)c with 1 = (−2 + 3i) and 2 = (1 + i) [341], [342]
For the case of a complex permittivity, the third and fourth order regions move from the point A2 to the point B 2 when ζ 1 changes from 0 to 1. Given 1 and 2 , and possibly p, a random texture can be looked for, from the desired value of ∗ , and consequently of ζ 1 . This approach can be successful when the segments (real case) or the domains (complex case) corresponding to different textures do not overlap too much, as illustrated in Fig. 18.1. This happens for a limited contrast of real permittivity, or for some relative positions of the three starting points of the geometric construction of the regions Ω n in the complex plane, which imply 01 02 + 001 002 > 0, providing a limitation of the method. A better separation of the domains would require higher order morphological information. It was applied to various types of random textures, like multi-scale models of random sets [341], [342], showing that third order bounds could generate separate domains in the complex plane for different random sets models. An example is given in Fig. 18.1 for the Boolean model of spheres for all volume fraction (using the function ζ 1 (p) given in Eq. 18.122), each section by p constant giving previously defined region region Ω 3 . For this example, the exchange of the morphology of the two phases gives separate domains. This approach on third order bounds was extended to the complex elastic moduli K and G for isotropic two phase viscoelastic composites [204], [495].
18.13.2 Spectral measure of a random set The analytic expansion in the complex plane of the complex effective permittivity of random sets gives access to a spectral measure which formally separates the geometry and the dielectric permittivity of components [62], [490], [212], [495]. It was extended to multicomponent random media by K. Golden and G. Papanicolaou by means of analytic expansion with re-
659
660
18. Change of Scale in Physics of Random Media
spect to several complex variables [213], [495]. For a random set with scalar permittivity, the effective permittivity ∗ can be expressed as a function of the complex variable s: 1 s= 1 − 1/ 2 The spectral measure μ is defined from ∗
F (s) = 1 −
=
2
Z
1 0
dμ(z) s−z
(18.112)
In Eq. (18.112) F (s) is an analytical function when s remains outside of the interval [0, 1]. The measure μ(z) is the spectral measure of the self adjoint operator ∇(−∆)(∇)1A1 , where ∇ is the divergence, −∆ the Laplacian, and 1A1 (x) the indicator function of the set A1 with permittivity 1 [692]. When the spectral measure has a continuous density, the spectral density function m(z) with dμ(z) = m(z)dz is obtained by the Stieljes inversion formula [692]: m(x) = −
1 lim (Im F (x + iy)) π y→0+
The moments μ(n−1) of the measure dμ(z) are defined by [212]: μ(n−1) =
Z
1
z n−1 dμ(z)
0
and F (s) is expressed as a function of the moments by F (s) =
n=∞ X n=1
μ(n−1) sn
In general, for a random medium the spectrum of the operator ∇(−∆)(∇)1A1 is continuous. For a discrete spectrum, F (s) in Eq. (18.112) owns a partial fraction representation [692], where the poles sn are the eigenvalues of the operator ∇(−∆)(∇)1A1 , α(s) and β(s) are two polynomials: F (s) =
X An α(s) ' s − sn β(s) n
In [692], [370] the function F (s) is given for the Hashin coated spheres (subsection 18.6.3) and for the self-consistent model (subsection 18.6.2). For the estimation (18.26), FHS + (s) =
p1 s − p2 /3
18.13 Third order bounds of the complex dielectric permittivity and spectral measure
For the estimation (18.21), FSCS (s) ¶ µ q 1 2 2 = 3s + 3p1 − 1 − 9s − 6(1 + p1 )s + (1 − 3p1 4s The function F (s) may be reconstructed from a limited range of measurements of ∗ (ω k ) for various frequencies ω k [692], after solving an ill posed problem by reguralization [632]. A range of 25 frequencies for FHS + (s) and of 40 frequencies for FSCS (s) gave a satisfactory reconstruction of poles and residues (p1 for FHS + (s)). For FSCS (s) a correct reconstruction of the spectral measure m(z) on reconstructed poles is obtained after a limitation of the degrees of the two polynomials α(s) and β(s) to degrees 15 and 9. A correct estimation of p1 from the sole knowledge of ( ∗ , 1 , 2 ) is obtained, even in the presence of noise for the measurement of ∗ (ω k ). √ In the case of a 2D autodual random set, since ∗ = 1 2 , the corresponding function FAD (s) is given by r r √ 1 1 2 1 FAD (s) = 1 − =1− =1− 1− s 2 2 1.3.5...(2n − 3) 1 11 1 1 + ... + + ... = + 2 s 2.4 s2 2.4.6...2n sn and the corresponding spectral moments are μ(n−1) =
1.3.5...(2n − 3) 2.4.6...2n
The strength of the spectral measure is to concentrate some morphological information, which can be estimated by inverse homogenization, from the measurement of macroscopic effective properties. Its weakness is that the underlying morphological information must be of a limited extent, since for a given set of values ( ∗ , 1 , 2 ) there is no unicity of the microstruc√ ture: for instance in 2D, autodual random sets for which ∗ = 1 2 and ∗ in 3D given by Eq. (18.26) (valid for Hashin coated spheres and also for a superposition of multiscale dilated Poisson planes as shown in subsection 18.15.6) can be built with very different morphologies and therefore with a different Choquet capacity. Therefore, the will to reconstruct full information on a random set from its spectral measure residing behind effective properties is illusory and out of reach.
661
662
18. Change of Scale in Physics of Random Media
18.14 Third order bounds for elastic moduli of random sets In the case of random materials with elastic isotropic components, and with an isotropic geometry, the third order bounds were given by M. Beran and J. Molyneux for the bulk modulus K [54] and by J.J. McCoy for the shear modulus G [468]. These bounds are valid for multiphase media, and more generally for elastic moduli modelled by random functions [50], [438], [439], [393], [263].
18.14.1 Third order bounds for two-phase composites Consider now two-phase random composites, modelled by a stationary and isotropic random set A (A = A1 and Ac = A2 ). The same approach can be followed for the bounds of transverse moduli, in the case of materials showing an isotropic transverse morphology [492]. Third order bounds depend on the moduli of the two phases, the volume fraction p, and on two morphological functions ζ 1 (p) (Eq. 18.107) and η1 (p) defined later in Eq. (18.120) (0 ≤ ζ 1 (p) ≤ 1 and 0 ≤ η 1 (p) ≤ 1). Using the notations of G.J Berryman and G. Milton [64], [65], [492], for any composite with properties P1 (with the probability p) and P2 (with the complementary probability q = 1 − p), define the following averages (with ζ 2 (q) = 1−ζ 1 (p) and η2 (q) = 1−η 1 (p)): hP i = pP1 + qP2 hP iζ = ζ 1 (p)P1 + ζ 2 (q)P2 hP iη = η1 (p)P1 + η 2 (q)P2
18.14.2 Bounds of the bulk modulus K The upper K+ and lower bounds K− are expressed by means of the auxiliary function Λ(x): Λ(x) =
¿
1 K + 43 x
À−1
4 − x 3
K+ = Λ(hGiζ ) ÿ À ! −1 1 K− = Λ G ζ
(18.113) (18.114)
The third order bounds of K dependent only on the function ζ 1 (p). If ζ 1 (p) = 1 or 0, K+ = K− (= K HS− (ζ 1 (p) = 1) or K HS+ (ζ 1 (p) = 0)). The Hashin and Shtrikman bounds are therefore a limiting case of third order bounds of K.
18.14 Third order bounds for elastic moduli of random sets
663
18.14.3 Bounds of the shear modulus G The upper G+ and lower bounds G− are expressed by means of the auxiliary function Γ (y), and of the following expressions Θ and Ξ: Γ (y) =
¿
1 G+y
À−1
−y
h i Θ = 10 hGi2 hKiζ + 5 hGi h3G + 2Ki hGiζ + h3K + Gi2 hGiη / hK + 2Gi
2
h i Ξ = 10 hKi2 h1/Kiζ + 5 hGi h3G + 2Ki h1/Giζ + h3K + Gi2 h1/Giη / h9K + 8Gi2
G+ = Γ G− = Γ
µ
µ
Θ 6
¶
1 6Ξ
¶
(18.115) (18.116)
When η1 (p) = ζ 1 (p) = 1 (0) and when K1 = K2 = ∞, G+ = G− = GHS+ (GHS− )
18.14.4 Bounds of the Young’s modulus E Starting from the expression of the elastic modulus E as a function of K and G, 1 1 1 = + E 9K 3G bounds of E deduced from bounds of K and G, since E increases with these variables (however these third order bounds are not optimal): 1 1 1 = + E+ 9K+ 3G+
(18.117)
1 1 1 = + E− 9K− 3G−
(18.118)
18.14.5 Bounds of the Poisson coefficient ν The Poisson coefficient is expressed as a function of K and G by:
664
18. Change of Scale in Physics of Random Media
ν=
3K − 2G 1 3 G = − 2(3K + G) 2 2 3K + G
ν increases with K and decreases with G, so that the following non optimal bounds ν + (G− , K+ ) and ν − (G+ , K− ) are obtained: ν+ =
1 3 1 3 G− G+ ; ν− = − − 2 2 3K+ + G− 2 2 3K− + G+
(18.119)
K+ − K− and G+ − G− are of the order of (K1 − K2 )4 and (G1 − G2 )4 when K1 → K2 and G1 → G2 .
18.14.6 Functions ζ 1 (p) and η 1 (p) The function ζ 1 (p) [492] , obtained from the probability P (h1 , h2 ) for the three points {x}, {x + h1 }, {x + h2 } to belong to A1 , are given in Eq. (18.107). The function η1 (p) is defined by: 5ζ (p) 150 η 1 (p) = 1 + 21 7pq in R3
Z
+∞ 0
dx x
Z
+∞ 0
dy y
Z
+1
P4 (u)P (x, y, θ)du
−1
(18.120)
where P4 (u) is the Legendre polynomial of order 4 with P4 (u) =
1 (35u4 − 30u2 + 3) 8
After exchanging the phases A1 and A2 , define the function η2 (q) with η2 (q) = 1 − η 1 (p). The function η 1 (p) satisfies 0 ≤ η 1 (p) ≤ 1. The integrals in Eqs (18.107, 18.120) are calculated analytically in some cases, but more generally numerically. For a better numerical convergence, it may be convenient to replace in the integrals P (x, y, θ) by P (x, y, θ) − P (x)P (y)/p, P (h) being the probability that the two points {x}, {x + h} belong to A1 . For two symmetrical phases A1 and A2 , the case p = 0.5 produces an auto-dual random set (the two phases having the same probabilistic properties), for which ζ 1 (0.5) = η 1 (0.5) = 0.5. This case corresponds to a third order central correlation function equal to zero. As a consequence, the third order bounds of a symmetrical medium have a fixed point in p = 0.5. Developments on non linear composites, with a non quadratic energy, provide new third order bounds using the same functions ζ 1 (p) and η 1 (p), using a comparison medium with a linear behavior [38],[555]. Previous general results on third order bounds are used in section 18.15 for some models of random sets, useful for applications.
18.15 Third order bounds of some models of random sets
665
18.15 Third order bounds of some models of random sets 18.15.1 Bounds for the Boolean model The Boolean model ([438],[448], chapter 6) is obtained by implantation of of random primary grains A0 (with possible overlaps) on Poisson points xk with the intensity given by λ in this subsection: A = ∪xk A0xk . For this model,
¢ ¡ ˇ = 1−q T (K) = 1−Q(K) = 1−exp −λμd (A0 ⊕ K)
ˇ μd (A0 ⊕ K) 0 μd (A ) (18.121)
ˇ = ∪−x∈K A0x is the result of the dilation of A0 by K. As where A0 ⊕ K already studied in chapter 6, particular cases of Eq. (18.121) give the covariance Q(h) = P {x ∈ Ac , x + h ∈ Ac } and the three point probability P (x, y, θ) entering into Eqs (18.107). Contrary to the mosaic model, the Boolean model is not symmetric (and not autodual for p = 1/2). Therefore, different sets of bounds are obtained when exchanging the properties of A and Ac . The known functions ζ 1 (p) for the Boolean model were obtained by numerical integration of these equations. With a good approximation, linear functions of p are obtained: ζ 1 (p) ' αp + β. The value ζ 1 (0) (β for the approximation) is deduced for p → 0 (or λ → 0 in Eq. (18.121)): ζ 1 (0) = a with a given by Eq. (18.127) for the random grain A0 (and consequently for any size distribution of this grain, since G is invariant by similarity of A0 ). Therefore, in R3 ζ 1 (0) = η 1 (0) = 0 for spheres, and ζ 1 (0) = 14 for Poisson polyhedra. In R2 , ζ 1 (0) = 0 for disks and ζ 1 (0) = 3 − 4 ln 2 ' 0.2274 for Poisson polygons. We have also η1 (0) = [4(5E − 1) − 5(9G − 1)] /6. The functions ζ 1 (p) and η 1 (p) were estimated in [637],[638] for spheres, and ζ 1 (p) in [361],[416],[417] for discs. A direct estimation of bounds for discs is presented in [33]. It is given in [416],[417] for polygons in R2 , including Poisson polygons. Here random rectangles are Poisson polygons generated by anisotropic Poisson lines from two orthogonal directions. These results (and ζ 2 (p) = 1 − ζ 1 (1 − p) obtained by exchange of the two phases) are summarized by: ζ 1 (p) ' 0.5615p for spheres [64], [637],[638], [633] c ζ 2 (p) ' 0.5615p + 0.4385 for (spheres) 2 2 ζ 1 (p) ' 3 p for discs in R [361],[416],[417] η 1 (p) ' 0.711p [64], [637],[638], [633] η 2 (p) ' 0.711p + 0.289 (sphere)c ζ 2 (p) ' 23 p + 13 for (discs)c in R2
(18.122)
666
18. Change of Scale in Physics of Random Media
FIGURE 18.2. AlN textures (AlN in black; Y rich phase in white), where AlN can be represented by the complementary set of a Boolean model of spheres [544]
ζ 1 (p) ' 0.503p + 0.2274 with p < 0.5 for Poisson polygons in R2 [416],[417] ζ 2 (p) ' 0.5057p + 0.2669 for (Poisson polygons in)c R2 ζ 1 (p) ' 0.495p + 0.236 with p < 0.5 for random rectangles in R2 [416],[417] ζ 1 (p) ' 0.601p + 0.079 with p < 0.5 for squares in R2 [416],[417] (18.123) For the models presented in Eqs (18.122), (18.123) ζ 2 (p) > ζ 1 (p), so that the third order bounds increase when (x) = 2 > 1 for x ∈ Ac . This is due to the fact that it is easier for the ”matrix” phase Ac to percolate than for the overlapping inclusions building A. For the sphere and disks models, ζ 2 (p) is larger than the ζ 1 (p) of the corresponding mosaic models. This is illustrated in the case of the thermal conductivity λ of ceramic materials [544], modelled by two Boolean models of spheres with a constant radius (in Figs. 18.2 and 18.3, the morphology of the low conducting bright phase is exchanged for two AlN textures): the third order bounds increase when λ(x) = λ1 > λ2 for x ∈ Ac (Fig. 18.4). Therefore when the more insulating phase (with λ2 = 10) has a globular shape modelled by a Boolean model of spheres, it is less efficient to stop the heat flux than when it can be modelled by the complementary set of a Boolean model of spheres with the same volume fraction but with a morphology acting as a barrier to the heat flux. With this low contrast and for this low volume fraction of the Y rich phase, the third order bounds separate the two microstructures with respect to the effective conductivity, which can be useful from an engineering point of view. It is clear here that third order bounds are sensitive to the morphology in a more accurate way than the H-S bounds which do not distinguish the two textures with respect to the effective conductivity. Third order bounds of elastic moduli of Boolean models of spheres are shown in Figs. 18.5 and 18.6: for the studied contrast and for a volume fraction of phase 1 lower than 0.3, the macroscopic elastic moduli are higher if this phase is affected to the complementary set of a Boolean model of spheres, this last one percolating while the union of spheres does not percolate (the percolation threshold of a boolean model of spheres is given by 0.2895 ± 0.0005 [571]). This result is again an illustration of the fact that
18.15 Third order bounds of some models of random sets
667
FIGURE 18.3. AlN textures (AlN in black; Y rich phase in white), where AlN grains can be represented by a Boolean model of spheres [544] Thermal conductivity 100 AlN=spheres AlN=spheres AlN=complement AlN=complement
95
90
85
80
75
70
65 Volume fraction of AlN phase 60 0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
FIGURE 18.4. Third order bounds of the thermal conductivity of AlN materials (λ1 = 100 W/(m.K) for AlN λ2 = 10 W/(m.K) for the second phase) [544]
third order bounds are more sensitive to percolation effects than Hashin and Shtrikman bounds.
18.15.2 Bounds for the hard spheres model A hard spheres model is obtained by a population of spheres in equilibrium under a potential preventing from any overlap [638], [633]. By construction, the fraction of spheres p is limited to 0.64 for this process. The functions η1 (p) and ζ 1 (p) were computed numerically, and fit to the following expressions [638], [633]: η1 (p) ' 0.48274p for p ≤ 0.64 ζ 1 (p) ' 0.21068p − 0.04693p2 + 0.00247p3
(18.124)
668
18. Change of Scale in Physics of Random Media Bornes de k (k1=100,mu1=100;E1=225,Nu1=0.125 k2=15,mu2=5; E2=13.5,Nu2=0.35) Booleen 110 100
Booleen+ BooleenBooleenc+ BooleencHS+ HS-
90 80
k
70 60 50 40 30 20 10 0
0.2
0.4 0.6 Fraction Materiau 1
0.8
1
FIGURE 18.5. Third order bounds of K for the Boolean model of spheres (Phase 1: K1 = 100, G1 = 100, E1 = 225, ν 1 = 0.125; Phase 2: K2 = 15, G2 = 5, E2 = 13.5, ν 2 = 0.35). The spheres are attributed to phase 1 (Boolean) or to phase 2 (Booleanc ) Bounds of G (K1=100,G1=100;E1=225,Nu1=0.125 K2=15,G2=5; E2=13.5,Nu2=0.35) Boolean 110 100
Boolean+ BooleanBooleanc+ BooleancHS+ HS-
90 80 70
G
60 50 40 30 20 10 0 0
0.2
0.4 0.6 Fraction Material 1
0.8
1
FIGURE 18.6. Third order bounds of G for the Boolean model of spheres (Phase 1: K1 = 100, G1 = 100, E1 = 225, ν 1 = 0.125; Phase 2: K2 = 15, G2 = 5, E2 = 13.5, ν 2 = 0.35). The spheres are attributed to phase 1 (Boolean) or to phase 2 (Booleanc )
18.15 Third order bounds of some models of random sets
669
In its range of validity (p < 0.64), 0 ≤ ζ 1 (p) ≤ 0.116 and therefore remains close to zero, at least for low values of p. As a consequence for this model when 2 (with 2 > 1 ) is affected to spheres, ∗ is close to the upper HS bound given by Eq. (18.26). On the contrary, when 1 is affected to spheres, ∗ is close to the lower HS bound given by Eq. (18.27). This fact can be qualitatively understood since in a hard sphere model, the microstructure is not too far from the Hashin assemblage of spheres, despite the presence of a single scale. From numerical simulations on microscope images of a fibre composite [536] where the thermal conductivity and the elastic moduli of fibres are much higher than for the matrix, this point was confirmed on data for the effective transverse conductivity and the effective transverse elastic moduli estimated from finite elements calculations of fields at a mesoscale (around 1mm2 for fibres with a 16μm average diameter). Results of numerical simulations are very close to the lower H-S bound. Therefore fluctuations of effective properties of this kind of composite material can be easily predicted from fluctuations of local area fraction combined to the locally estimated lower bounds.
18.15.3 Bounds for the mosaic model Remind here that the mosaic model [440], chapter 8, or ”cell” model [487], [488] is built in two steps: starting with a random tessellation of space into cells, every cell A0 is affected independently to the random set A with the probability p and to Ac with the probability q. The medium is symmetric in A and Ac , which may be exchanged (changing p into q). From a direct calculation of third order bounds based on the third order central correlation function, M. N. Miller defined a parameter G depending only on the random cell geometry by means of the function s(|h1 | , |h2 | , θ) (where μd is the Lebesgue measure in Rd , μd its average over the realizations of A0 , and A0h is obtained by translation of A0 by the vector h): s(|h1 | , |h2 | , θ) = G=
1 1 + 9 2
Z
1 1 G= + 4 π 1 9
+∞ dx x
0
Z
Z
+∞ dx x
0 1 3
+∞ dy y
0
Z
0
Z
+∞ dy y
μd (A0 ∩ A0h1 ∩ A0h2 ) μd (A0 )
(18.125)
+1 −1
Z
(3u2 − 1)s(x, y, θ)du in three dimensions +π
s(x, y, θ) cos(2θ) dθ in two dimensions
0 1 4
where ≤ G ≤ in three dimensions and ≤ G ≤ For the mosaic model Eq. (18.107) becomes [492]:
1 2
(18.126) in two dimensions.
670
18. Change of Scale in Physics of Random Media grain spheres needles plates E 1/5 3/8 1 G 1/9 1/6 1/3 TABLE 18.2. Parameters E and G for some mosaic models
ζ 1 (p) = p + a(q − p) with a = ζ 2 (p) = ζ 1 (p)
d2 G − 1 d−1
(18.127)
and therefore ζ 1 (p) varies linearly with the proportion p, with a slope 1−2a (0 ≤ a ≤ 1). The two extreme cases are the following: a = 0 for G = 19 in R3 and for G = 14 in R2 (this corresponds to spheres and disk cells respectively); a = 1 for G = 13 in R3 and for G = 12 in R2 (this corresponds to spheroidal cells of plate and needle shapes respectively). However, no correct construction of random tessellations corresponding to these extreme values seems to exist. Similarly, for the mosaic model, η1 (p) = p + (q − p) [4(5E − 1) − 5(9G − 1)] /6
(18.128)
E being another parameter depending on the random tessellation [604], [492]. These coefficients were obtained for several types of tessellation with spheroidal grains, and are given in Table 18.2, where needles are cylinders with infinite length, and plates are domains between two parallel planes (it is rather difficult to imagine an isotropic random tessellation of space from these two types of grains). The spheres mosaic can be rigorously constructed by means of as the sequential intact grains of the Dead Leaves tessellation [321] (section 11.3 in chapter 11). The parameters E and G are not sensitive to the scale of the grain A0 , and are therefore invariant for a population of similar grains. Since ζ 1 (0.5) = η1 (0.5) = 0, in two and three dimensions, the third order bounds present a fixed point for all mosaic models at p = 0.5. This is illustrated in Figs. 18.8 and 18.9.
A particular random mosaic can be obtained from a Poisson tessellation of space by Poisson random planes in R3 (with the intensity λ, which is a scale parameter that does not appear in the calculation of G) and by Poisson lines in R2 [448], [598]. The cells of these tessellations are Poisson polyhedra and Poisson polygons (chapters 7, and 8). For the Poisson mosaic model built from this tessellation (and also from the STIT mosaic presented in chapters 7, and 8), s(|h1 | , |h2 | , θ) (18.129) µ µ ¶¶ q 2 2 = exp −λ |h1 | + |h2 | + |h1 | + |h2 | − 2 |h1 | |h2 | cos θ
18.15 Third order bounds of some models of random sets
671
From Eqs (18.126), G is calculated analytically: in R2 , G = 1 − ln 2 [440] and in R3 , G = 16 [416], as for the so-called needles mosaic. Therefore, a = 3 − 4 ln 2 in R2 and a = 14 in R3 . For random rectangles, G = 0.309, close to isotropic Poisson polygons [417]. Anisotropic cells (but distributed with a uniform orientation) are studied in [416], [417]: G (and consequently ζ 1 (p) for a ≤ 12 ) increases with the cell anisotropy, as shown in Table 18.3 where L is the aspect ratio. Third order bounds of the dielectric permittivity of spheres and plate mosaics are compared to the case of Boolean models of spheres in Fig. 18.7. For the chosen contrast (100),these models show well separate bounds for high and low volume fraction. Starting from the partial derivatives of ∗ with respect to p (area fraction of A2 with permittivity 2 ), G. Matheron derived further bounds for 2D mosaic models with two components [466]. Noting m = p 2 + q 1 and m0 = p 1 + q 2 , and with ∗SCS being the self-consistent estimate given in Eq. (18.21), these bounds are: p p 1 2 + qm 1 (2m − 1 ) 1 ∗ ∗ p , for p ≤ ≤ SCS ≤ 2 pm0 + q 1 (2m − 1 ) p qm + p 1 (2m0 − 1 ) 1 p ≤ ∗ ≤ ∗SCS , for p ≥ (18.130) 1 2 2 q 1 2 + pm0 1 (2m0 − 1 ) An illustration of third order bounds for elastic properties of the mosaic models mentioned in Table 18.2 is given in Figs 18.8 and 18.9 (each bound has a fixed point for various models when p = 0.5).
18.15.4 Bounds for the Dead Leaves model The Dead Leaves model (chapter 11) is obtained sequentially by implantatation of random primary grains A0 (t) on a Poisson point process: in every point x is kept the last occurring value (x, t) during the sequence. In this way, non symmetric random sets are obtained if two different families of primary grains are used for A and for Ac . When using the same family of primary grains for the two sets, a mosaic model built on a Dead Leaves random tessellation is obtained. It is known that the shape of the resulting cell is non convex (and even non connected!), due to the overlaps occurring during the construction of the model. For the symmetric case, the calculation of G can be made from the knowledge of the function s(x, y, θ) :
s(|h1 | , |h2 | , θ) =
μd (A0 ∩ A0h1 ∩ A0h2 ) μd (A0 ∪ A0h1 ∪ A0h2 )
(18.131)
Calculations were made for Poisson polyhedra as primary grains [416]: G ' 0.170 in R3 and G ' 0.311 in R2 , which is slightly larger than for the
672
18. Change of Scale in Physics of Random Media 3D Mosaic and the Boolean sphere models 100 Spheres Spheres Plates Plates Boolean Boolean Boolean c Boolean c
10
1 0
0.2
0.4
0.6
0.8
1
FIGURE 18.7. Third order bounds of the dielectric permittivity of mosaic models (spheres, plates), Boolean model of spheres; spheres are attributed to phase 1 (Boolean) or to phase 2 (Booleanc ); 2 = 100, 1 = 1 Bounds of K (K1=100,G1=100;E1=225,Nu1=0.125 K2=15,G2=5; E2=13.5,Nu2=0.35) Mosaic models 110 100
Spheres+ SpheresPlates+ Plates Needles+ NeedlesHS+ HS-
90 80
K
70 60 50 40 30 20 10 0
0.2
0.4 0.6 Fraction Material 1
0.8
1
FIGURE 18.8. Third order bounds of K for mosaic models (spheres, needles, plates) and H-S bounds. Phase 1: K1 = 100, G1 = 100, E1 = 225, ν 1 = 0.125; Phase 2: K2 = 15, G2 = 5, E2 = 13.5, ν 2 = 0.35
18.15 Third order bounds of some models of random sets
673
Bounds of G (K1=100,G1=100;E1=225,Nu1=0.125 K2=15,G2=5; E2=13.5,Nu2=0.35) Mosaic models 110 100
Spheres+ SpheresPlates+ Plates Needles+ NeedlesHS+ HS-
90 80 70
G
60 50 40 30 20 10 0 0
0.2
0.4 0.6 Fraction Material 1
0.8
1
FIGURE 18.9. Third order bounds of G for mosaic models (spheres, needles, plates) and H-S bounds. Phase 1: K1 = 100, G1 = 100, E1 = 225, ν 1 = 0.125; Phase 2: K2 = 15, G2 = 5, E2 = 13.5, ν 2 = 0.35 L G Rectangle Mosaic 1 0.270 2 0.291 3 0.317 4 0.339 6 0.370 8 0.390 12 0.415 16 0.430 32 0.458 64 0.476 200 0.491
G Dead Leaves 0.293 0.302 0.313 0.323 0.336 0.345 0.355 0.362 0.373 0.382 0.387
TABLE 18.3. G versus L (1 0: the two bounds are improved by iteration of the intersection of random sets with different scales. A limiting case is obtained by iterations for n → ∞, where 1+p ζ (1) 2 1 1+p (n) ηH1 (p) → n→∞ η 1 (1) 2 (n)
ζ H1 (p) → n→∞
If the basic structure is a mosaic model with a = 0, or more realistically the complementary set of a Boolean model of spheres in R3 or of 1+p 1+p (n) (n) discs in R2 , ζ H1 (p) → and η H1 (p) → . For the corresponding 2 2 Boolean model of spheres with an infinite range of widely separate sizes, 1−p (n) (∞) . For this model, ζ H1 (1) = 0. One may be tempted to use ζ H1 (p) → 2
678
18. Change of Scale in Physics of Random Media
this construction as an input into the iterative construction of the intersection of random sets; however this cannot be done since the assumption of well separate scales would be violated. Starting from a mosaic model with a = 1 (plates in R3 or to needle cells in R2 , for which it seems difficult to provide a geometrical construction), or more realistically starting from a Boolean variety of dilated flats [274], (∞) (n) [275] (chapter 6), ζ 1 (1) = η1 (1) = 0 and therefore ζ H1 (p) = η H1 (p) = 0. The complementary set is a union of mosaics (or a union of dilated Poisson (∞) (∞) planes), for which ζ H1 (p) = ηH1 (p) = 1. In this last case, if 2 > 1 is attributed to the set obtained by union, the limiting structure admits two equal third order bounds (and therefore an effective property) equal to the upper H-S bound (Fig. 18.10). In the opposite, if 2 is attributed to the intersection of mosaics or to the complementary set of multiscale dilated Poisson planes, the limiting structure admits an effective property equal to the lower H-S bound. The obtained limit structure admits also two equal third order bounds for the bulk modulus K. If K2 > K1 is attributed to the multiscale union of multiscale dilated Poisson planes, K ef f = K− = K+ = K HS+ If K2 > K1 is attributed to the complementary set of multiscale dilated Poisson planes intersection of dilated Poisson planes, K ef f = K− = K+ = K HS− This construction provides a new optimal structure corresponding to HS bounds, based on the mosaic model, or on the Boolean model. On the other hand, by this process are obtained two separate bounds for the shear modulus G. However they are very close if the materials are well ordered, that is if (K2 − K1 )(G2 − G1 ) > 0. Structures with optimal properties are generated, based on multiscale models. It is practically the same for all examples known from the literature, like Hashin sphere assemblage are like multiscale random layered media introduced in subsection 18.6.3.
18.16 Case of porous media For a porous medium, the lower bound of properties is zero: − = K− = 1 1 G− = E− = 0, ν − = − , ν + = . The upper third order bounds can be 2 2 used as an estimator of effective permittivity and elastic moduli. They are expressed as a function of the volume fraction of the solid phase p (with A1 = pores and A2 = solid), , K, G, and of the 3-points morphological properties of the random set A1 , ζ 1 (p) and η 1 (p):
18.16 Case of porous media
679
Effective permittivity - Union of 3D mosaics with G=1/3 (Plates) 100 one scale one scale 100 scales 100 scales infinity infinity
10
1 0
0.2
0.4
0.6
0.8
1
FIGURE 18.10. Upper and lower third order bounds of the dielectric permittivity for multiscale union of plates mosaic as a function of q = 1 − p with 1 = 1 and 2 = 100
pζ 1 (p) ζ 1 (p) + 1−p 2
(18.139)
4pGKζ 1 (p) 3(1 − p)K + 4Gζ 1 (p)
(18.140)
+
K+ = G+ =
=
pG 6(1 − p)(K + 2G)2 1+ 5G(4K + 3G)ζ 1 (p) + (3K + G)2 η1 (p)
When η 1 (p) = ζ 1 (p) = 1, in Eq. (18.79).
+
=
HS+
(18.141)
, K+ = K HS+ and G+ = GHS+ given
The upper bound + is plotted as a function of the pore fraction q = 1 − p for various models of random set in Fig. 18.11. For the Hard Spheres model, only the part of the curve corresponding to a porosity lower than 0.64 should be considered. As expected from the corresponding function ζ 1 (p), + is very close to the H-S+ upper bound. On the opposite, when considering Hard Spheres for the solid phase, only the part of the curve corresponding to a porosity larger than 0.36 is correct for an upper bound. It is the lowest upper bound of the plots, but in reality ∗ = 0 for this model corresponding to non overlapping solid grains floating in a porous medium... Concerning the Boolean model of porous spheres, + (q) is close to the H-S+ upper bound. For q > 0.95, the solid phase does not percolate any more (see section 6.6 in chapter 6), so that ∗ = 0 for this range of pore volume fraction. For the complementary set of the Boolean model of spheres, the solid phase is made of a union of spheres, which explains the
680
18. Change of Scale in Physics of Random Media
Hashin-Shtrickman Boolean sphere Boolean sphere c Hard Core sphere Hard Core sphere c Mosaic sphere Mosaic plates Mosaic needles self consistent
100
80
60
40
20
0 0
0.2
0.4
0.6
0.8
1
FIGURE 18.11. Upper bound + for various porous media as a function of pore volume fraction q = 1−p. For the mosaic of plates a finite contrast is used ( 1 = 100 and 2 = 1) to avoid on the plot the discontinuity of + for q → 0
fact that the corresponding curve + (q) is one of the lowest. For q > 0.7, the solid phase does not percolate, so that ∗ = 0 and + (q) overestimates ∗ in this range of pore fractions. Concerning the mosaic models, their upper bounds reach a common points for q = p = 0.5, as expected. From Eq. (18.139) the curve + (q) for the plates mosaic (where ζ 1 (p) = q = 1−p) presents a discontinuity point for q → 0, since in that case, +
and
+
2(1 − q) for q > 0 (p < 1) 3 = for q = 0 (p = 1) =
In Fig. 18.11 this discontinuity is smoothed by computing + for a two phase medium with 1 = 100 and 2 = 1 (instead of 2 = 0), as in Fig. 18.7. This rather unexpected behavior of the plates mosaic may be explained by a seemingly very low percolation threshold of plates but, as already explained, such a morphology is not physically achievable, so that this extreme case should be avoided in practice. The self-consistent approximation decreases linearly with pore fraction, as expected from Eq. (18.22) and ∗ = 0 for q = 2/3. The upper bounds of K and G are compared for various models of random structures in Figs 18.12 and 18.13 (excluding here the plates mosaic showing again a discontinuity for q → 0), using the functions given in section 18.15. For spherical grains (corresponding for instance to the simulation of the beginning of the sinter process), the lower bounds of moduli are obtained (the effective moduli remaining close to zero for the hard spheres
18.17 Optimal conductivity of two-components porous media
681
Bounds of K (K=250,G=160) 300 Hashin-Shtrikman Boolean spheres Booleanc spheres Hard Spheres Hard Spheres c mosaic spheres mosaic needles
250
K
200
150
100
50
0 0
0.2
0.4
0.6
0.8
1
Pore Fraction
FIGURE 18.12. Upper third order bound of K for various porous random structures (solid phase: K = 250, G = 160). The pores are spherical for the Boolean model or the hard sphere model (only the part of the curve corresponding to a porosity lower than 0.6 should be considered); the solid grains are spherical for the complementary set of these models (Booleanc and Hard Spheresc (only the part of the curve corresponding to a porosity larger than 0.4 is correct))
model). For spherical pores (hard spheres, boolean model of spheres, spherical mosaic), higher upper bounds are obtained, and they are closer to the upper Hashin-Shtrikman bound. Third order upper bounds of the Boolean model of spheres are used to predict elastic moduli of gypsum [345]. The dual problem concerns media where for component Ai , i = +∞ (infinite dielectric constant), or Ki = Gi = +∞ (rigid elastic medium). The upper bound goes to infinity, but the lower bound can be used as an estimate of effective properties.
18.17 Optimal conductivity of two-components porous media For some material properties, it may be of interest to consider the optimization of the microstructure with respect to various physical characteristics. For instance in the case of SOFC applications, anionic and cationic conductions are considered. They are obtained by two separate components with their own conductivities. The purpose of this section is to show which choice of composition can be made, in order to simultaneously optimize the two effective conductivities [1], [316].
682
18. Change of Scale in Physics of Random Media Bounds of G (K=250,G=160) 160 Hashin-Shtrikman Boolean spheres Booleanc spheres Hard Spheres Hard Spheres c mosaic spheres mosaic needles
140
120
G
100
80
60
40
20
0 0
0.2
0.4
0.6
0.8
1
Pore Fraction
FIGURE 18.13. Upper third order bound of G for various porous random structures (solid phase: K = 250, G = 160). The pores are spherical for the Boolean model or the hard sphere model (only the part of the curve corresponding to a porosity lower than 0.6 should be considered); the solid grains are spherical for the complementary set of these models (Booleanc and Hard Spheresc (only the part of the curve corresponding to a porosity larger than 0.4 is correct))
18.17.1 Optimization of two conductivities Consider a porous material, where the solid phase is made of two random sets A1 and A2 with conductivities with type 1 and 2 σ 1 and σ 2 , such that σ 2 = ασ1 , with α < 1. Write P (x ∈ A1 ) = p1 P (x ∈ A2 ) = p2 P (x ∈ (A1 ∪ A2 )c = p = 1 − p1 − p2 Concerning the conductivity of type 1, it is equal to zero for the set A2 , so that the medium behaves like a porous medium with pore volume fraction p + p2 . Similarly, the conductivity of type 2 is equal to zero for the set A1 , so that the medium behaves like a porous medium with pore volume fraction p + p1 . For a given pore volume fraction p, the volume fractions p1 and p2 can be freely changed between 0 and 1 − p. The medium owns two f effective conductivities σ ef 1 (σ 1 , p1 ) and f ef f σ ef 2 (σ 2 , p2 ) = σ 2 (σ 2 , 1 − p − p1 ) f f and σ ef monotonously increase with p1 The effective conductivities σ ef 1 2 ef f and p2 respectively, and therefore σ 1 is a continuous monotonous in-
18.17 Optimal conductivity of two-components porous media
683
f creasing function of p1 , while σ ef is a continuous monotonous decreasing 2 function of p1 , for any given porosity p. The two effective conductivities satisfy f ef f σ ef 1 (σ 1 , 0) = 0; σ 1 (σ 1 , 1 − p) < σ 1 f σ ef 2 (σ 2 , 0)
= 0;
f σ ef 2 (σ 2 , 1
(18.142) − p) < σ 2 < σ 1
The following property holds: Proposition 18.1. As a result of the condition given in equation (18.142), f f and σ ef and due to the monotonicity of the functions σef 1 2 , there is a single f ef f volume fraction p1 maximizing the function inf(σ ef 1 (σ 1 , p1 ), σ 2 (σ 2 , 1−p− f p1 )). This "optimal" volume fraction is solution of the equation σ ef 1 (σ 1 , p1 ) = f σ ef 2 (σ 2 , 1 − p − p1 ). f be the unique solution of the equation σ ef Proof. Let popt 1 1 (σ 1 , p1 ) = ef f opt σ 2 (σ 2 , 1 − p − p1 ). When p1 ≤ p1 , f ef f ef f inf(σ ef 1 (σ 1 , p1 ), σ 2 (σ 2 , 1 − p − p1 )) = σ 1 (σ 1 , p1 )
which increases with p1 . When p1 ≥ popt 1 , f ef f ef f inf(σ ef 1 (σ 1 , p1 ), σ 2 (σ 2 , 1 − p − p1 )) = σ 2 (σ 2 , 1 − p − p1 )
which decreases with p1 . f f Therefore the optimal values of σ ef and σ ef can be estimated by di1 2 chotomy, starting from two initial values for p1 . This considerably reduces the number of simulations required for the estimation. It is possible to separate the optimization for the fluid flow permeability and the conductivities, looking first for the pore volume fraction p giving the desired permeability K ef f , and in a second step operating on the parameter p1 .
18.17.2 Upper bounds of the optimal properties Dealing with porous media, use can be made of upper bounds of the effective properties, the lower bound being equal to zero. For instance third order bounds will depend on the solid volume fraction, the solid property and the Milton parameter ζ 1 (p) depending on the three points probabilities of the random set. For illustration, consider now the Hashin-Shtrikman upper bound, valid for any isotropic microstructure. It is less performing than the third order bound, but will give more simple expressions for the calculation of a bound of the optimized property. In the three-dimensional space, 2σ 1 p1 (HS + )1 = 3 − p1
684
18. Change of Scale in Physics of Random Media
and (HS + )2 =
2σ 2 p2 2ασ 1 (1 − p − p1 ) = 3 − p2 3 − (1 − p − p1 )
A Hashin-Shtrikman upper bound of the optimal property is obtained by taking the supremum of inf 0 pc given in [679] and in section 19.12 (fit for a contrast C = 104 ) gives ∗ = 144. 35 when C = 103 , which is quite an excellent estimation. The RVE VRV E to estimate ∗ with a relative precision better than 5% is given by VRV E = 323 , 643 , 1283 for a contrast C = 10, 100, 1000 and p = 0.3. Point variance and integral ranges are estimated from 2563 simulations. Results are summarized in Table 19.14. The microstructure integral range A3 , theoretically calculated from the covariance, is underestimated by 1015% in simulations. Except for a low contrast and a low volume fraction, the integral range of Dx is larger than A3 and strongly increases with contrast, like the point variance. As previously in 2D, for a low contrast the covariances of Dy and Dz show a hole effect in directions Ox and Oy (Dy ), Ox and Oz (Dz ) (Fig. 19.16). Histograms of Dx , Dy and Dy are estimated from full field and shown on Fig. 19.17 for each component, for p = 0.3 and C = 100. They deviate strongly from a Gaussian distribution for Dx , showing tails for high values of Dx . Histograms are peaked for Dy and Dz . The three components of D have a very wide dispersion in phase 2 made of the union of spheres, while in the matrix the field is more homogeneous. VV /C 0.2 0.3 0.5
10 1.64 2.14 3.56
100 2.52 4.86 18.24
1000 3.64 13.89 145
TABLE 19.13. Effective permittivity ∗ of Boolean models of spheres with volume fractions p = 0.2, 0.3, 0.5 as a function of contrast C for a matrix with 1 = 1 [502],
19.10.3 Optical properties of paints In [32] the prediction of optical properties of deposit models for paints is investigated. The coating is a mixture of various components with different permittivity. The optical properties (and their change with wavelength) of
732
19. Digital Materials p C Variance of Dx IRDx (pixel3 ) A3 0.2 10 3.3 436 100 95 377 1000 1876 795 0.3 10 5.6 558 100 275 710 1000 10152 1683 0.5 10 8.5 873 100 548 2238 1000 14734 8935
(pixel3 ) A3 (theoretical) 445 484
424
462
351
408
TABLE 19.14. Point variance of Dx , integral ranges of Dx and of the volume fraction of Boolean models of spheres with volume fractions p = 0.2, 0.3, 0.5 [502], [352]
FIGURE 19.17. Histograms of the three components of D obtained for a 3D Boolean model of spheres: global (G); in phase 1 (P1); in phase 2 (P100); contrast C = 100 [502]
19.11 Elastic and thermal response of heterogeneous media from 3D microtomography
the layer are governed by the effective permittivity of the mixture, since the optical index is given by n = ( ∗ )0.5 . Paint coatings are simulated by a random deposition process of nanoparticles with different shapes: spheres, or rhombus-shaped inclusions. The typical size of the nanoparticles being much smaller than the light wavelengths, electrostatics can be used to estimate ∗ . This is made from electrostatic fields computed by FFTon simulated paint layers (in 12003 voxels volumes), using the dielectric function of the TiO2 pigment and of the matrix glass. Since the deposit shows transverse isotropy, this property appears in the permittivity tensor with the following diagonal components: ∗xx = ∗yy and ∗zz . Hot spots (similar to the thermal case in subsection 19.11.1) are generated on the field maps, around corners of very close rhombus nanoparticles, or between two nearly-touching spherical inclusions, when the zone of contact is orthogonal to the applied macroscopic field E. For these simulations, this effect is stronger for the imaginary part of the displacement field, which generates color on the macroscopic scale. The size of the RVE of D(x) is much larger than the pigment size, in particular for its imaginary part as a result of hot spots. The relative precision of the estimation of the average real part of D is 0.82%. It is not so good, but acceptable for the average of its imaginary part (8.84%).
19.11 Elastic and thermal response of heterogeneous media from 3D microtomography The three examples introduced in this section are based on 3D FFT computation of fields in granular materials from images obtained by microtomography: the first one concerns a propellant material made of spheroidal grains embedded in a polymer matrix [680]. The second material is a mortar [171]. The third example concerns lightweight concretes [644]. After a morphological segmentation of images, FFT computations provide thermal and elastic fields. In addition to the estimation of the apparent properties and of their statistical RVE, a morphological analysis of the fields provide instructive information on local field concentrations.
19.11.1 Hotspots in a granular material A cylinder specimen of a propellant, with diameter 7.4 mm and height 2.6 mm, is analyzed by microtomography, from which a 926×926×463 image is extracted [680]. A first segmentation is given by automatic thresholding using Otsu’s algorithm on the grey level histogram, maximizing the interclass variance [535]. In this segmentation, some overlap appears between grains in Fig. 19.18 (b). Such false contacts enhance grains percolation that might introduce bias in the estimation of apparent properties for a high contrast
733
734
19. Digital Materials
FIGURE 19.18. 2D section of segmentation by Ostu’s algorithm (b) and by the stochastic watershed (a) Grains are slightly connected in 3D in image where p = 0.613 (b), and unconnected in (a) where p = 0.537 [680]
between components. Therefore a second segmentation is produced by a stochastic watershed [21] with a number of germs n deduced from the covariance [175]. In the resulting segmentation shown on Fig. 19.18 (a), there is a separation between all propellant grains, sometimes with a 1 voxel width.
The thermal response of the image is obtained by the FFT method, with the following conductivities, obtained from experimental measurements: λ1 = 0.15J/mKs for the matrix and λ2 = 0.609J/mKs for grains. A macroscopic temperature gradient with averages h∂i T i = ∆T0 δ ij is applied, with periodic BC. The elastic response is computed for isotropic elastic moduli of components with two sets of parameters: for the matrix, i) G = 1/2, K = 1/3; ii) G = 1/2, K = 2/3; elastic properties of grains are the same up to a factor 2. Two macroscopic conditions are imposed: hydrostatic strain loading with heij i = e0 δ ij ; shear strain loading with heij i = e0 (δ ik δ jl + δ il δ jk , heij i = e0 (δ ik δ jk − δ il δ jl . Full field calculation is made on the two segmentations. With this very low contrast, about 12 iterations give a converged solution for the fields (measured by the maximal absolute value of the divergence). The apparent conductivity is isotropic with, for the connected microstructure, λapp ' 0.387, close to the lower HS bound HS− (0.368), as expected for a granular medium (see chapter 18). The integral range of the heat flux is A3 = 133 and its point variance Dq2 = 0.23. For the unconnected image, λapp ' 0.3306. The slight difference is a result of the lower grain volume fraction in case (a). The statistical relative precision is very good, due to the low contrast: rela = 0.4%. The apparent elastic properties in case i) are Gapp = 0.6534, and K app = 4455, very close to HS − . With this very low contrast a recourse to numerical
19.11 Elastic and thermal response of heterogeneous media from 3D microtomography
homogenization seems superfluous. However, the examination of full fields with a relatively high space resolution gives an interesting insight on the local response of the medium. Concerning the local response, local hot spots of the heat flux appear in the matrix at the interface between close grains, on parts of grain boundaries orthogonal to the macroscopic temperature gradient, and with a higher intensity for the connected image. Similar local maxima are seen for the component of the temperature gradient parallel to the macroscopic gradient, as illustrated in Fig. 19.20. Histograms (inside grains and in the matrix) of the component ∂1 T parallel to the macroscopic field are given for the two segmented images in Fig. 19.19. They are similar in the two cases. The tail of the pdf in the matrix reveals the presence of local high values of ∂1 T . Hot spots are extracted by thresholding the local flux to a value equal to twice the average flux, and are shown in Fig. 19.21. These hotspots might alter the sustainability of this material if submitted to a thermal shock during its storage. The elastic local response shows similar hotspots on mean and shear strain components (but with a lower intensity than for thermal fields) in the matrix, for a macroscopic hydrostatic loading, but are not orientation dependent and are therefore more numerous. The shear loading gives rise to hotspots more dependent on the applied field and grain boundaries orientation. Some additional analytical results, obtained by field expansions at a low contrast, are given in [680]. The main interest of this numerical study is the impact of local morphological singularities on fields amplification (as would also do sharp corners), in spite of a low contrast of properties making trivial the homogenization problem in the present case. Similar hotspots are observed by numerical simulations on stress during the viscoplastic deformation of polycrystals [576].
19.11.2 Stress localization in a mortar microstructure A 3D greyscale image (1000×1200×1000 voxels, 25μm per voxel) of a mortar sample is obtained by microtomography [171]. Three components are present: voids, fine aggregates, and the surrounding cement paste named matrix. The rough image cannot be used as it is for micromechanical calculations: in a first step a segmentation has to be performed. First, noise is removed by convolution by a Gaussian filter. In a second step, two thresholds are applied from entropy maximization of the histogram. Then a drift in illumination is suppressed by substraction a 2D moving average on squares with edge d = 250 pixels. To remove false connections between aggregates inducing a large stiff connected component, a multiscale separation is obtained by watersheds of the inverse distance function. Details are given in [171]. In the final segmented image, a 3-phase microstructure is obtained (Fig. 19.22): aggregates (VV = 0.334), cement matrix (VV = 0.64) and
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P∂ T(t) 1
grains (conn.) grains (unconn.)
3
matrix (conn.)
2
matrix (unconn.)
1 00
1
2
3
t4
FIGURE 19.19. Histograms of ∂1 T parallel to the applied field in grains and in the matrix for the two segmented images [680]
FIGURE 19.20. Component ∂1 T parallel to the applied field. Detail of the connected image [680]
voids (VV = 0.0235). Volume fractions of aggregates and voids are lower than in the real mortar material, due to the image resolution and to the loss of larger aggregates (with diameter larger than 3.15mm) in this sample. Elastic fields are computed by iterations of FFT for two types of loading conditions and with periodic BC, as before: hydrostatic strain loading, and shear strain loading, with hexy i = heyx i = e0 . To deal with the presence of voids, the augmented Lagrangian algorithm [479] is used on images with 5003 or 7353 voxels. For elastic fields calculation, the Poisson modulus is the same for all solid components (ν = 0.2). The contrast is monitored by the ratio of Young’s moduli of aggregates and of the matrix, χ = E(a)/E(m) = 10−8 , 3, 100, 1000, 10000 with a fixed value for E(m). The contrast 3
19.11 Elastic and thermal response of heterogeneous media from 3D microtomography
FIGURE 19.21. 3D picture of hot spots obtained by thresholding the local heat flux (q1 (x) ≥ 2 hq1 i) [680]
corresponds to the real medium. Other ratios enhance the contrast and allow us to anticipate the situations of damage (10−8 ) or of creep (χ ≥ 100). When E(m) = 20M P a andE(a) = 60M P a, the apparent Young’s modulus is E app = 27.7GP a, consistent with the experimental value (26.4GP a), which is not surprising for a low contrast (3). Concerning the RVE for elastic fields, rela = 5% is reached at low contrasts (χ ≤ 100) for volumes of the order of V0 = 5003 voxels. In the rigid case, the precision of the results remains between 5.4% and 11.4% for the used volumes. The morphological RVE for rela = 5% is much larger than the specimen size for pore volume fraction (16763 ), as an effect of its low value. An example of normalized mean stress component σ m /[E(m)hem i] is displayed on a 2D section in Fig. 19.23. For a high contrast a strain concentration is observed between aggregates for an hydrostatic loading. Localization patterns are even stronger for a shear loading. Histograms of σ m and σ xy , for hydrostatic and shear strain loading are given in Figs 19.24 and 19.25. For the lowest contrast (χ = 3), the field histogram is strongly peaked and almost Gaussian. For higher contrasts histograms become asymmetric, with power low tails eα in the area of large strains: for χ = 1000, α = −7.5 (hydrostatic loading) and α = −5.2 (shear loading). Finally, the local stress field response is analyzed by means of morphological tools based on distances, introduced in section 3.5.6 of chapter 3. To highlight the stress enhancement by aggregates or by voids in the matrix Am , the mean mj (r) of the stress field in the matrix is estimated as a function of the distance r to a reference set Aj , by averaging the field in region Xr : Xr = (Aj ⊕ B(r)) ∩ (Aj ⊕ B(r − 1))c ∩ Am
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FIGURE 19.22. 3D microstructure of mortar (1.25 × 1.25 × 1.25cm3 ): fine aggregates (blue), voids (red), matrix (transparent) [171]
FIGURE 19.23. 2D sections of the normalized mean stress component σ m /[E(m)hem i] at contrasts χ = E(a)/E(m) = 100 for hydrostatic strain loading; 3D image size 1.25 × 1.25 × 1.25cm3 with 25μm/voxel [680]
The set Aj is made of aggregates, or of the boundary of their zones of influence (SKIZ) obtained by the watershed of the distance to aggregates. From the experimental curves mj (r) [171], at high contrasts the parallel stress field highest values are located at large distance r of the aggregates, or alternatively close to the SKIZ. Such regions are candidate for developing damage zones during creep. A second tool compares the distribution of distances to a component Ak of a random point x ∈ Ack , F (r), to the same distribution of distances for x ∈ Aj , Fj (r). It is estimated on the 3D images as shown in Fig. 19.26. The dimensionless ratio ρj (r) given below shows at scale r a preferential association (resp. a repulsion effect) between sets Ak and Aj when ρj (r) > 1 (resp. ρj (r) < 1):
19.11 Elastic and thermal response of heterogeneous media from 3D microtomography
Pσ (t) m
4
-8
0 χ=10 χ=3
10 3 2
χ=10
χ=3 3
χ=10
2
χ=10
4
1
10
0
10
χ=10
0
χ=10
1
χ=10
3
χ=10
-8
1
0
4
2
3
2
t=σm
FIGURE 19.24. Full field histogram of the mean stress σm under hydrostatic strain loading; tails in log-log scale [171]
Pσ (t) xy
χ=3 2 χ=10
0
1,5
10
χ=10
χ=10
3
χ=3
4
χ=10
1 0
-8
0,5
χ=10
χ=10
10
χ=10
χ=10
-1
4
3
-8
2
0
1
2
3 t=σ xy
FIGURE 19.25. Full field histogram of the shear stress component σxy under shear strain loading; tails in log-log scale [171]
ρj (r) =
Fj (r) F (r)
It is applied with Ak generated by the higher stressed domains (5% of the overall volume) and Aj made of aggregates for hydrostatic strain loading (Fig. 19.27) and of voids for shear strain loading (Fig. 19.28), very similar behaviors being obtained by exchange of the loading conditions. It appears that regions of high stress are not located close to aggregates, and at high contrasts, high stress regions are not located around voids. At low contrast (χ = 3) there is a preferential short range (low r) association between high stresses and voids, as seen on Fig. 19.28: for a low aggregates/matrix contrast, the effect of aggregates can be neglected and the elastic response
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FIGURE 19.26. 2D section of the set used to estimate ρ(5); Aggregate (Aj ) in dark; areas with large values of the stress component (Ak ) in light gray; Aj ∩ (Ak ⊕ B(r)) in white [171]
ρa(r) 1
hydrostatic
0,8 0,6
χ=10 χ=3 χ=10
0,4
χ=10 χ=10
0,2
10
20
30
40
50
60
-8
2 3 4
70
r (voxels)
FIGURE 19.27. Correlation ρa (r) between aggregates and high stress region for hydrostatic strain loading [171]
is analogous to that of a homogeneous matrix with isolated voids generating stress concentration. In a complementary study [173], the same approach is made on virtual concrete materials generated by multiscale simulation of three families of Poisson polyhedra, as detailed in chapter 7. A contrast χ = 100 is used, and voids are not considered. In a first step, results are obtained for non overlapping spheres and non overlapping Poisson polyhedra. From the examination of average stresses as a function of the distance to aggregates, stress concentration is observed on the aggregate-matrix interface, due to the presence of sharp corners on polyhedra, and close to the SKIZ of aggregates, as for the real medium. In a second step, elastic fields are computed
19.11 Elastic and thermal response of heterogeneous media from 3D microtomography
ρv(r) 1,2
χ=3
1 0,8
χ=10 χ=10
0,6
χ=10
0,4
χ=10
0,2 0
10
20
-8
2
4
3
shear
30
40
50
r (voxels)
FIGURE 19.28. Correlation ρv (r) between voids and high stress region for shear strain loading [171]
by FFT on 3D simulations (16003 with 0.09mm/voxel) with three scales of polyhedra. For comparison a separation of scales is assumed, to introduce in a smaller size computation (in terms of voxels: 10243 with 0.24mm/voxel), a two components medium containing large gravels and a composite matrix made of cement and of smaller gravels with elastic properties estimated from a former homogenization. This simplified approach lowers the computational cost. The predicted moduli are very close: K app /Km = 9.0 (3 scales), 10.5 (scale separation); Gapp /Gm = 10.3 (3 scales), 11.5 (scale separation). The relative precision of estimates is very high, since the RVE for 3 3 rela = 5% are in the range 266 -308 , depending on the moduli and on the type of calculation.
19.11.3 Elastic and thermal properties of lightweight concretes Lightweight concretes are studied in [644], from micromotomographic images (10243 voxels with 3.23μm per voxel). The two samples are highly porous, with 0.5 and 0.33 pore volume fractions. After image segmentation, pore size distributions are estimated from morphological closings. The difference between FFT predictions of effective elastic moduli and measurements for the 0.33 pore volume fraction sample, where they are overestimated, may be the result of microcracks in the cement paste, which cannot be resolved in the microtomographic images. Similarly, computations of thermal fields with a varying thermal conductivity of the cement paste makes possible its identification. Again, the discrepancy between the thermal conductivity of the concrete with 0.33 pore volume fraction, identified from calculations, and the thermal conductivity of the other concrete, requires the introduction of micropores in its cement paste, suggesting further investigations at a higher magnification. This example of study illustrates
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the facilities for physical properties identification from comparison between measured and homogenized effective properties
19.12 Elastic and viscoelastic properties of multiscale random media Multiscale models of random structures give a wide flexibility to reproduce various morphological situations, and to design new materials with new effective properties. In this section, the bulk modulus of a Boolean model of spheres is estimated from simulations by FFT [678]. Linear elastic properties and conductivity are estimated for multi-scale Cox Boolean models. Then the approach is extended to nonlinear mechanical and electric behavior. Finally, elastic [250] and viscoelastic [179] properties of rubber reinforced by carbon black filler are estimated from Cox Boolean models accounting for the heterogeneous distribution of filler.
19.12.1 Bulk modulus of the Boolean model of spheres The effective bulk modulus of the Boolean model of spheres (with diameter 18 voxels) is estimated from 10 realizations with size 2563 covering the full range of volume fractions [678]. Two extreme situations are explored: the porous case (for spheres with null moduli) and the rigid case (where in fact K1 = G1 = 103 in spheres and K2 = G2 = 1 in the matrix). Elastic fields computation is made by means of the augmented Lagrangian version of the FFT algorithm, well adapted to infinite contrast [479], applying a macroscopic deformation equivalent to a hydrostatic loading hem = 1i. The normalized effective bulk moduli K ∗ /K2 (porous medium) and K ∗ /K1 (rigid spheres) are given in Figs 19.29 and 19.30 as a function of the sphere volume fraction f . In these figures are shown for comparison the self-consistent estimate SC, Hashin-Shtrikman bounds HS, and Beran third order bounds TOB. In the present case, the self-consistent estimate and bounds are not pertinent, owing to the infinite contrast of properties between components. The effect of percolation is clear in these figures: K ∗ vanishes for a 0.95 pore volume fraction (the solid phase being then disconnected) and starts to increase in the rigid case for a sphere volume fraction close to 0.29. 3 Close to the solid phase percolation threshold f 0 , K ∗ /K2 ∼ 6.5 |f 0 − f | 0 for |f − f | < 0.1. For the rigid case, a power law perfectly fits the data for f ≥ 0.29: K ∗ /K1 ' 2.39(f − 0.29)2.35 for f > 0.29 Second order correlations functions of the mean component σm of the stress field, averaged over the 3 main directions of the grid, are given in [678]. Contrary to the electrostatic case with an oriented applied field in
19.12 Elastic and viscoelastic properties of multiscale random media
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section 19.10, present correlation functions are isotropic and no hole effect appears (this may be the result of infinite contrast). The range of correlation functions of porous media is close the diameter of the spheres for the lower pore volume fraction (for f < 0.25) and then larger than the range of the microstructure. In the rigid case, the correlation length is significantly higher than the range of microstructure, except for volume fractions lower than the percolation threshold. Integral ranges are estimated from the statistics of field averages at different scales, and are plotted in Fig. 19.31 as a function of the sphere volume fraction, after normalization for a sphere of diameter 1. The integral range of the microstructure is obtained by direct calculation from the theoretical expression of the covariance of the Boolean model of spheres. It is lower than for the mean stress component, except for very high pore volume fraction or for sphere volume fraction lower than the percolation threshold for the rigid case. κ0/κ
1
κ0/κ 0.8
-2
SC HS
0.6
10 -4 10
10
-2
10
0.4
-1
f’-f
TOB
0.2 0
0
0.2
0.4
0.8
0.6
f
1
FIGURE 19.29. Normalized effective bulk modulus K ∗ /K2 as a function of the pore volume fraction f , Boolean model of spheres [678] κ0/κ 3.10
3
κ0/κ 2
2.10
3
10
SC
1
0
0.4
0.8 f TOB
3
10
HS
HS TOB 1
0
0.2
0.4
FFT
0.6
0.8
f
1
FIGURE 19.30. Normalized effective bulk modulus K ∗ /K1 as a function of the rigid spheres volume fraction f , Boolean model of spheres [678]
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FIGURE 19.31. Integral range of the mean component of the stress field σm , for porous and rigidly Boolean model of spheres as a function of sphere volume fraction f . A3 is normalized for a sphere of diameter 1 [678]
Pσ (t) m
porous f=0.2
12
f=0.9 8
f=0.4 f=0.65
4 0
0
0.5
1
σm
FIGURE 19.32. Histograms of the mean component σ m in the matrix (resulting from a hydrostatic macroscopic deformation hem = 1i), in the porous Boolean model [678]
Field histograms given in Fig. 19.32 are not Gaussian, and non-symmetric in the porous case, where the field histogram has a mode around the average. For large f , the histogram is bimodal with large values of the distribution for σ m = 0. When f ≥ 0.7, a single peak remains for σ m = 0 and there is a long tail for positive σm . The change of the shape of the distribution for f = 0.7 corresponds to the appearance of dead zones in the matrix where the strain and stress fields are null. In the rigid case in Fig. 19.33 histograms always show a single mode and is symmetric as long as f ≥ 0.3, when rigid spheres percolate.
19.12 Elastic and viscoelastic properties of multiscale random media
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Pσ (t) m
1
rigid
0.8
f=0.1 f=0.3
0.6
f=0.5 f=0.7 f=0.9
0.4 0.2 0
-5
0
5
σm
FIGURE 19.33. Histograms of the mean component σ m in the matrix (resulting from a hydrostatic macroscopic deformation hem = 1i), in the rigid Boolean model [678]
19.12.2 Linear elastic properties and conductivity of multiscale Cox Boolean models Two-scale and three scale random media generated by Cox Boolean model can show a lowering of the percolation threshold as compared to the Boolean model (chapter 6). The impact of this change of morphology on the effective properties of composites is studied by means of numerical simulations for a high contrast of properties in for various Cox Boolean model [679]. In this subsection are reported results obtained for 2 an 3 iterations of Cox Boolean model. The two-scale model is a three parameters model, obtained by keeping Poisson germs falling in a first large spheres (with radius R1 ) Boolean model, the two scales having the same volume fraction f 0.5 . Small spheres of the second scale and final model have a radius R2 ¿ R1 . The three scale model is similar, with an additional intermediary scale and a final radius R3 ¿ R2 ¿ R1 , each scale having the volume fraction f 1/3 . Iterations can go on up to infinity, and third order bounds are given for a large separation of scales and for any number of iterations in subsection 18.15.6 in chapter 18. To estimate the elastic fields by FFT, hydrostatic (heij i = δ ij ) and shear strain (heij i = 1 − δ ij ) loading conditions are applied. For the matrix, K = G = 1.For the porous medium σ = 0 in pores and for the rigid spherical inclusions K = G = 104 . In addition, the conductivity is estimated from the electric field obtained by application of a homogeneous electric field E0x in the Ox direction. FFT calculations are based on the augmented Lagrangian algorithm [479]. For each type of microstructure, four realizations are generated for domains with volumes 5123 of 7503 voxels. The elastic tensor of the reference medium is chosen on low resolution
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rigid, hydrostatic loading
4
10
∞
3
HS TOB
TOB
(B3)
10
∞
2
TOB
(IB)
10
TOB (B)
1
10
HS 1
0
0.2
0.4
0.6
0.8
f
1
FIGURE 19.34. FFT estimates and bounds of the normalized bulk modulus K ∗ /K2 (log scale), as a function of the volume fraction of rigid inclusions f ; Boolean model (B), iterated two-scale (IB) and three-scale (B3) Cox Boolean models; Hashin and Shtrikman bounds (HS); Beran third-order bounds for the Boolean model (TOB) and for infinite iterations (TOB ∞ ) [679]
subgrids (323 voxels) to get the faster convergence. Depending on the microstructure, it varies from K = G = 0.1 for porous media to 103 for rigid spheres, the convergence being obtained between 40 and 500 iterations, depending on the microstructure. A statistical relative precision of nearly 1—5% is obtained for nearly all microstructures. In a first step a test is made to define the scale ratio for which a separation of scales is acting, enabling to replace a two-scale medium by a one scale medium where spheres have elastic moduli obtained by a first homogenization. It is tested by means of increasing scale ratio; a scale ratio equal to 0.1, i. e. an order of magnitude, ensures correct results for the present infinite contrast. It is expected that lower scale ratio can be considered for a separation of scales in the case of lower contrasts of properties. Results for elastic properties of two and three iterations of the same Boolean model (using scale separation for three iterations) are illustrated in Figs 19.34 and 19.35 for the rigid case, and in Figs 19.36, 19.37 for the porous case. For the rigid case, at large sphere volume fractions (f ≥ 0.6) the threescale model (B3) has bulk and shear moduli close to Beran bounds. The reinforcement effect from two scales to three scales is very significant for all volume fractions. For instance when f = 0.3, for the bulk modulus a factor close to 8 is obtained between two and three iterations, and a factor 3 when comparing one scale and two iterations. Concerning strain fields maps, em and exy are close to 0 in large regions located between aggregates of quasi rigid spheres. This effect takes place for f = 0.2 and is slightly more important for f = 0.3. For porous media, results are given for one-scale model (B), two-scale iterated Boolean model (IB), and three non iterated two-scale models
19.12 Elastic and viscoelastic properties of multiscale random media μ0/μ
rigid, shear loading
4
10
3
747
HS TOB ∞ TOB
10
(B3)
2
(IB)
10
∞
TOB
(B)
1
10
1
0
0.2
0.4
0.6
TOB HS 0.8
f
1
FIGURE 19.35. FFT estimates and bounds of the normalized shear moduliusG∗ /G2 (log scale) as a function of the volume fraction of rigid inclusions f ; Boolean model (B), iterated two-scale (IB) and three-scale (B3) Cox Boolean models; Hashin and Shtrikman bounds (HS); Beran third-order bounds for the Boolean model (TOB) and for infinite iterations (TOB ∞ ) [679]
(MB20), (MB30), and (MB50), with the following volume fractions of large spheres f1 = 0.2, 0.3, 0.5. Beran third order bound provides a good estimate of the elastic properties for f ≤ 0.2. Close to the percolation threshold, the deformation is made easier by large clusters of voids and stress concentrations in a small subset of the medium, this point being stronger for the iterated model (IB), and corresponds to higher levels of field heterogeneities. The elastic properties of two scale models (MB-20), (MB-30), (MB-50) and (IB) are significantly lower than that of the one-scale model (B), a non uniform distribution of voids weakening the material. From a practical point of view, it means that the counterpart of a reinforcement of a material by a multiscale distribution of rigid inclusions could result in a larger degradation of the elastic moduli by damage (from the break up of the interface or of inclusions), in comparison to the one scale situation. About the electric behavior, the effective conductivity, estimated for a 104 contrast C, is much more sensitive to percolation effects than the elastic properties. When spheres are highly conducting, the non uniform dispersion of inclusions, as in models IB or MB, leads to higher overall conductivity, whereas when inclusions are perfectly insulating, the macroscopic conductivity is lower in multiscale microstructures than for the one-scale Boolean model (B). Volume fraction variation of the conductivity near the percolation threshold, scales with different power law exponents for the standard Boolean model (λ∗ ' 1.5C(p − pc )1.5 just above the percolation threshold) and for two scale models (3 just above the percolation threshold), the factor 2 being predicted from the separation of scales [679]. Concerning the RVE size, results given in [678] indicate that the largest RVE sizes for the one-scale model correspond to rigidly-reinforced media
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1
porous hydrostatic loading
0.8
(TOB)
0.6
(MB-20)
(B)
0.4
(IB) (MB-30)
0.2 0
0
(MB-50) 0.2 0.4
HS
0.6
0.8
f
1
FIGURE 19.36. FFT estimates and bounds of the normalized bulk modulus K ∗ /K2 as a function of the volume fraction of rigid inclusions f ; Boolean model (B), iterated two-scale (IB) and one-scale model (B), two-scale iterated Boolean model (IB), and three non iterated two-scale Cox Boolean models (MB20), (MB30), and (MB50), Hashin and Shtrikman bounds (HS); Beran third-order bounds for the Boolean model (TOB) [679] μ0/μ
1
porous shear loading
0.8 0.6 0.4 0.2 0
0
(MB-20)
(B) (IB)
(MB-30) (MB-50) 0.2 0.4
TOB HS
0.6
0.8
f
1
FIGURE 19.37. FFT estimates and bounds of the normalized shear moduli G∗ /G2 as a function of the volume fraction of rigid inclusions f ; Boolean model (B), iterated two-scale (IB) and one-scale model (B), two-scale iterated Boolean model (IB), and three non iterated two-scale Cox Boolean models (MB20), (MB30), and (MB50), Hashin and Shtrikman bounds (HS); Beran third-order bounds for the Boolean model (TOB) [679] 2 2 with Vv ≈ 0.7 (cf. Fig. 19.31). Accordingly, the variances DZ (V ) and DZ , with Z = σ m (mean stress or 1/3 of the trace of the stress tensor), are computed for the two-scales Cox Boolean model when Vv1 = Vv2 = 0.7 for a contrast higher than 10000 of the bulk and shear moduli, and a scale ratio equal to 10 . It is found that the integral range of the two-scales medium is increased by a factor close to 5.8 times, whereas the point variance increases by a factor 32. A 10% precision (i.e. rela = 0.1) is achieved when V ≈ 5503 , which corresponds to an increase of the RVE by a factor 5.8×32 = 5.73 . The drastic increase of the point variance in the case of the two-scale random medium underlies very large local stresses, that could induce more damage
19.12 Elastic and viscoelastic properties of multiscale random media
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in the matrix if it contains defect, as compared to the one scale model for the same volume fraction. Larger RVE, combined to the difficulty of observation of multiscale microstructures in single images, make the experimental study of such materials much more challenging than their theoretical approach. However, in the case of a large separation of scales (for typically an order of magnitude as illustrated from simulations detailed in [679]), investigations made at different resolutions are of common practice, and can be combined with iterative models, where the homogenization starts from the smaller scales.
19.12.3 Nonlinear elastic and conductivity of multiscale Cox Boolean models In [682], this work is extended to the nonlinear response of Cox Boolean models for elasticity and conductivity. The nonlinearity is managed by a power law constitutive equation for deviatoric stress in the matrix, with exponent n (0 ≤ n ≤ 1), n = 1 corresponding to linear elasticity, and n = 0 to a rigid perfectly plastic behavior. With this law, σ eq = ym eneq Simulations are made on 5123 images on Boolean models of spheres with diameter 20 voxels. Spheres are either rigid or porous. In these conditions the macroscopic behavior is assumed to follow the same power law. In the rigid case, the effective bulk modulus K0 , defined by K0 = hσ m i /(3 hem in ) increases with n, with a higher intensity for sphere volume fraction beyond the percolation threshold. An opposite trend is observed for the effective yield stress y0 defined by hσieq = y0 heineq more sensitive to volume fraction for VV < 0.4. From limit analysis, y0 = ym for large domains, when a plane is included in the matrix, which cannot be the case for a Boolean model, except when VV = 0. For porous spheres, the effective yield stress y0 is weakly sensitive to n, while K0 strongly decreases with the non linearity (when n → 0). Concerning the non linear conductivity, it is ruled by the following law between current J(x) and electric field E(x): Ji (x) = λm |E(x)|n−1 Ei (x) and the effective conductivity λ0 is obtained from n
|hJi| = λ0 |hEi|
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For a given insulating sphere volume fraction, λ0 is maximal when n = 0.7. Two scale microstructures are generated by a Cox Boolean model where the first scale is a Boolean model of spheres with diameter 50 voxels, and the second scale is made of spheres with diameter 5 voxels, the two scales having the same volume fraction. When n = 0 and for rigid spheres, a reinforcement is observed in two-scale medium for VV < 0.3, while a weakening effect appears for large sphere volume fractions. As for the linear case, a two scales distribution of pores weakens the overall behavior when VV ≤ 0.3.
19.12.4 Elastic and viscoelastic properties of rubber with carbon black filler In this subsection is considered the prediction of properties of rubber reinforced by carbon black fillers (CB), from simulation of the microstructure. In [249] a three steps Cox Boolean model is identified from TEM micrographs, as summarized in chapter 6. In [179] a variant of this model replaces the exclusion zones made of a Boolean model of spheres by a Johnson Mehl mosaic, and carbon black aggregates are located in another Boolean model of spheres. Elastic properties The prediction of elastic properties of carbon black rubber composites is made by FE and by FFT in [250]. First, an approximate estimate of the effective properties is obtained for a two-scale material containing non overlapping aggregates. Composite spherical aggregates are made of percolating CB spheres (a Boolean model with volume fraction p1 ) and of the elastomeric component (with volume fraction 1 − p1 ). The elastic properties of the aggregates are estimated by the third order upper bound of a Boolean model of spheres with volume fraction p1 . In a second step, these spherical aggregates (with a radius much larger than the radius of CB spheres, involving a separation of scales) are implanted in the same elastomeric matrix according to a hard-core process forbidding overlaps between aggregates, with volume fraction p2 , giving an overall volume fraction p = p1 p2 . The overall moduli are now estimated by the H-S lower bound computed for this composite, and using the same matrix properties as before. As shown in [301] and on Fig. 19.38 , the estimates of the shear modulus G, obtained for different values of p1 are consistent with the experimental measurements obtained for various mixes, a more uniform distribution in space giving a lower G. For a given CB volume fraction p, G increases when p1 decreases, making easier the percolation in the two-scale model (the extreme value p1 = 1 gives the standard hard-core two-components composite with moduli close to the lower H-S bound). A more accurate estimate is obtained by computer intensive numerical simulations introduced now [250], but the analytical model easily provides a good estimate of the elastic moduli.
19.12 Elastic and viscoelastic properties of multiscale random media
751
8
7
HS0.5 0.4 0.35
Shear modulus G (MPa)
6
5
4
3
2
1 0
0.05
0.1
0.15 Vv carbon black
0.2
0.25
FIGURE 19.38. Prediction of the effective shear modulus of carbon black CB elastomeric composite materials with a two scale hard-core process; curves from bottom: lower HS bound (p1 = 1, p1 = 0.5, p1 = 0.4, p1 = 0.35. Matrix: K = 3000MP a, G = 1MP a. CB: K = 66667MP a, G = 30769Mpa. Symbols show the experimental results obtained on various mixes with the same matrix and CB
For FE calculation, priority is been given to generate a FE mesh that follows the microstructure interfaces, according to the conforming meshing technique. From 3D images, the isosurface describing the interface between carbon black and the matrix is built with polygons, using the marching cubes algorithm. A triangulation of the surface mesh is made and finally a free linear tetrahedral mesh is built on the basis of the conforming optimized surface triangulation, the meshing being performed by a Delaunay triangulation. This first step before field computation is very tedious and requires some manual intervention. For calculations, the following elastic properties obtained from experimental measurements are used: for rubber, EM = 3M P a and ν M = 0.49983. The rubber is quasi-incompressible, with a Poisson ratio very close to 0.5. For the filler, ECB = 80000M P a and ν CB = 0.3, generating a high contrast with the matrix. The volume fraction of the filler is 14%. A domain decomposition is used and detailed in [250]. Specific difficulties are encountered, since the behavior of the elastomeric matrix is quasi-incompressible. This can lock the displacements and therefore can increase the apparent elastic moduli of the material. After several calculations, a Poisson ratio equal to 0.49 was retained to estimate the apparent shear modulus from FE calculations. In addition, FFT calculations are made with the augmented Lagrangian algorithm, on 3D images with different volumes: 2003 , 3503 , and 5003 for a voxel size of 3.2nm. It is applied to two versions of simulations, images used as input for FE, which are not periodic, and a slightly modified version to
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get periodic images. Contrary to FE, there is no difficulty to use the true Poisson ratio of the elastomer in FFT calculations. The shear modulus estimated by FE from SUBC and KUBC converges for largest volumes (from (800nm)3 . The elastic response of the discontinuous model, as computed by FFT for larger volumes, is in good agreement with FE computations. The obtained shear modulus is Gapp = 2.4M P a, close to the lower Hashin Shtrikman bound, while the measured one is Gex = 1.81M P a. This overestimation may be due to the presence of a layer of elastomer with weaker properties around carbon black particles, or to the morphological parameters identification from TEM images. Integral ranges of the field estimated by application of Eq. 19.2 are BC BC ASU = 3 × 107 nm3 and AP = 1.06 × 107 nm3 , insuring a correct 3 3 app statistical estimation of G for the largest simulated volumes (cube with edge equal to 1500nm and volume V = 337.5 × 107 nm3 ). An analysis of local fields in the microstructure, submitted to shear loading, is performed to characterize the interaction between aggregates and the elastomeric matrix. The local and global strain amplification are studied in the matrix from the trace of the strain tensor. The observed maximum local amplification value is close to 3, for a global amplification value equal to 1.14. The matrix located between two aggregates seems to be subjected to an increasing strain amplification when the distance between aggregates decreases. Histograms of the trace of the strain and stress tensors in the matrix are close to Gaussian distributions.
Prediction of viscoelastic properties by FFT To predict the viscoelastic properties from the microstructure of carbon black reinforced rubber studied from TEM images [179], simulations of Cox Boolean models are generated by a vectorial representation avoiding to operate on a grid of points [176]. This representation makes easy the study of percolation of carbon black aggregates, depending on the choice of morphological parameters. For this purpose, a two-scales Cox Boolean model implants random spheres in a Johnson-Mehl mosaic. In a second step, viscoelastic moduli are estimated from FFT computations on digitized images. Two components are considered: the filler with an elastic behavior, and the matrix with a viscoelastic behavior obtained from complex elastic moduli, viscous effect being accounted for by their imaginary part. For this purpose, a specific version of the Morph-Hom FFT software was developed by F. Willot. The same approach as for elasticity is followed, using complex properties and fields. Here the discrete scheme of [677] is used to compute the fields by iterations of FFT. A reference medium with real elastic moduli K0 = 0.51(Re K1 + Re K2 ) and G0 = 0.51(Re G1 + Re G2 ) is used for numerically solving the LippmannSchwinger equation. The method is validated by comparison to analytical
19.13 Elastic properties of fibrous materials
753
Re(μ) [MPa]
3
Boolean Johnson-Mehl
2
Hashin coating
polymer
1 -2
10
-1
10
0
10
1
10
2
10
3
10
4
10 ω [rad/s]
FIGURE 19.39. Frequency dependence of the effective real part of the complex shear modulus Gapp [179]
estimates for a periodic array of spheres with cubic symmetry given in [119]. Viscoelastic computations are performed on the Boolean and JohnsonMehl two-scales models. The local properties are set to 78 400 and 30 000 MPa for the bulk and shear moduli of the filler. For the polymer, a viscoelastic law with Prony series given in [401] is used for 23 frequencies. Domains with volume (0.8μm)3 digitized on 4003 voxels are simulated. Hydrostatic strain loading and shear strain loading are applied. The relative precision of the estimated moduli is calculated from the variance of the fields at different scales. At a frequency ω = 117Hz, the relative precision for the real part and the imaginary part of K app is equal to 0.2% and to 0.3%. For the the real part and the imaginary parts of Gapp , it is equal to 17% and 3.8% respectively. The frequency dependence of the real and imaginary parts of the bulk modulus is close to the prediction of the Hashin sphere assemblage, due to the low contrast between the two components. For Gapp , the contrast is of the order of 3×104 leading to very different results: the real part of Gapp is higher for the Johnson Mehl exclusion zones than for the Boolean model of spheres, and above the Hashin coated spheres assemblage, as shown on Fig. 19.39. The imaginary part of the complex shear modulus Gapp is similar for the different models, and close to the imaginary part of the complex shear modulus of the polymer (Fig. 19.40).
19.13 Elastic properties of fibrous materials A composite with a PA6 matrix and long parallel glass fibres is studied in [536]. The study by image analysis and FE element calculations of thermal and elastic fields are made on transverse sections, where fibres are displayed as non-overlapping discs with heterogeneous locations, mainly due to the
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10
10
Boolean Johnson-Mehl
-1
Polymer
-2
Hashin coating 10
-3
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10 ω [rad/s]
FIGURE 19.40. Frequency dependence of the effective imaginary part of the complex shear modulus Gapp [179]
presence of plies in the manufacturing process. The average volume fraction is 0.4, and the average fibre diameter 16μm. For homogenization, the following properties are used: thermal conductivity λf ibre = 1W mK −1 , λmatrix = 0.2 W mK −1 , Ef ibre = 72000M P a, Ematrix = 2002M P a, ν f ibre = 0.22, ν matrix = 0.39. The RVE is estimated for all properties of the composite. The most sensitive property to fluctuations of the microstructure is the transverse shear modulus, for which the RVE size is the largest: for a square with 854μm edge, containing about 1310 fibres, its relative precision is 18.7% for a single image, and therefore 1.87% for 100 images. In some composites, a microstructure is made of non overlapping objects (for instance spheres), described by a hard-core process. For the sequential absorption version of this model generated with spheres of a single radius, numerical simulations show that the obtained effective properties are close to the lower H-S bound for a “soft” matrix, and close to the higher H-S bound for “soft” spheres (like pores). This is also observed on a microscopic scale in the present case , the effective conductivity, bulk and shear moduli of images estimated by FE computations being quite well predicted by the lower 2D H-S bound on that scale [536], so that fluctuations of effective properties between images can immediately be deduced from fluctuations of fibre area fraction at the scale of images of the microstructure. In [14] microtomographic images and 3D simulations of various random networks of finite fibers having a length of the order of the size of the samples, various distributions of orientations and radius distributions, are made with the volume fraction VV ' 0.15 to predict the elastic properties and the conductivity by FFT. A glass-fiber reinforced polymer is examined by microtomography with a voxel sampling of 3.5μm. After application of a segmentation procedure to separate fibres, length-weighted radius and orientation distributions are estimated from 3D images. Curvature parameters estimated in [12] are introduced as input in a random-walk-based model of three-dimensional
19.14 Diffusion and fluid flows in porous media
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fiber systems [11] to generate virtual fibrous media with controlled bending of fibres. Various types of generated microstructures range from isotropic to transversely isotropic, and to orthotropic media, representing plausible virtual fibrous materials for fibres with average length 1.19mm (or equivalently 340 voxels). The anisotropy of the macroscopic responses and the size of the corresponding representative volume element (RVE) are examined numerically. Field calculations are made by FFT on simulations with typical size 6003 voxels. The elastic and thermal properties of the system studied in [536] and given above are used. It is found that the variance of the properties on a volume V scales as a power law ∼ 1/V a with α < 1 and in the range 0.71 − 0.87, in conformity to Eq. (19.5). This is a consequence of the long-range correlations in the microstructure, in comparison to the size of simulations, resulting in a slower decrease of the variance of local averages as usual, but faster than for the Poisson fibres where α = 2/3, as seen in section 19.7. Concerning the RVE size, a relative precision in the range 0.29 − 1.86% is reached with 10 realizations in cubic domains with 4003 voxels for the full set of parameters of interest: fibre volume fraction VV , elastic moduli and thermal conductivity for all considered fibrous systems. Lower size volumes can be used from flat specimens accounting for the main orientations of the fibrous system. The overall properties of the fiber composites are computed for varying fiber curvature and orientation distributions, and are compared to available analytical bounds. The fiber arrangement strongly influences the elastic and thermal responses, while the incidence of fiber curvature can be neglected. The effective Young’s modulus of an orthotropic model is close to the real one. The shear modulus is close to the lower Hashin—Shtrikman bound for fibres with Oz preferred orientation, as found in [536]. The bulk modulus is not very sensitive to the different orientation distributions, but bending tends to decrease it. The bulk modulus is relatively close to the lower bound for all types of simulations. The effective conductivity is high along the fiber directions as the very long fibers conduct heat through the material. As observed in [536], the thermal conductivity is close to the lower Hashin— Shtrikman bound in the directions orthogonal to fibres.
19.14 Diffusion and fluid flows in porous media Transport properties of porous media have many important consequences, like oil recovery in petroleum reservoirs, or chemical reactions in heterogeneous media. It is therefore interesting to consider diffusion of species in porous media, and fluid permeability. These topics are well documented in
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chapter 6 of [50] and in [634], among others. Specific difficulties are inherent to random porous media, namely the complexity of pore boundaries, and also the infinite contrast between pores and the solid phase.
19.14.1 Diffusion in random porous media In this subsection is considered the diffusion of species in pores, being filled with a gas or with a fluid, while penetration into the solid phase is forbidden. To estimate the effective coefficient of diffusion of the random medium, use can be made of random walks simulations, or of the numerical solution of the Fick’s equation. It is illustrated by two examples. Diffusion and random walks A first approach, close to the physical origin of diffusion, is to generate random walks in pores from random origins. These random walks are reflected on the solid walls. As already presented about lattice gas in chapter 16, the variogram of coordinates of trajectories, γ x (∆t), and the distribution of sojourn times τ a (t), Fτ a (t) = P {X(t) ≥ a), provide an estimation of the effective diffusion coefficient by means of Eqs 16.34, 16.35 and 16.40 [454]. This approach is followed to characterize the porous medium introduced in section 3.6 about morphological tortuosity in chapter 3 [151]. Random walks are generated by Brownian motion of 105 particles starting at time t0 from a random point with coordinates xi in open pores (with volume fraction 0.2), located in an ellipsoid with axis rx = 300 = 300, ry = 100, rz = 3, to account for the strong anisotropy of this porous medium. At every time step the current particle can move on a cubic grid from location with coordinates (Xi (t), Xj (t), Xk (t)) by one space step to (Xi0 , Yj0 , Yk0 ), with a set of probabilities pi , pj and pk such that a drift similar to a main flow is introduced. From the random walk, P {Xi0 = Xi (t) + 1} = pi , P {Xi0 = Xi (t)} = p0i , P {Xi0 = Xi (t) − 1} = 1 − pi − p0i When a macroscopic Fick’s law emerges, the coordinates Xi (t) generate a stochastic diffusion process with expectation and covariance given by E[Xi (t)] = xi + ui t E[(Xi (t) − xi − ui t)(Xj (t) − xj − uj t)] = 2Dij t
(19.16) (19.17)
where ui is the average of the component ui of the velocity (this notation should not be confused with the Fourier coordinates used in other chapters of this book), and the coefficients Dij build the tensor of diffusion of the porous medium, provided the Fick’s law emerges on a macroscopic scale. It may happen that instead of Dij t in Eq. 19.17 a time power law Atα
19.14 Diffusion and fluid flows in porous media
757
(with α 6= 1) is observed, reflecting an anomalous diffusion. It can be interpreted as a change of the coefficient of diffusion with time, with an effective coefficient of diffusion D ∼ Atα−1 , and consequently a dependence on the scale of observation. The use of covariances given in Eq. 19.17, and of the time variograms of trajectories for i = j are useful for short time and small space scales, while the distribution of sojourn times, obtained after closing boundaries of the domain of simulations B in one or two directions provides information on the large scales behavior of the porous medium, closer to estimates obtained by homogenization of Fick’s law introduced later. In the present case, random walks are implemented with p0i = p0j = p0k = 0.2, with biased and unbiased walks (where u = 0). For the estimation of Fτ a (t), a bias is introduced in the wanted direction, for instance pi = 0.4 − 0.5 From obtained experimental variograms γ x (t) ' 0.19t0.95 and γ y (t) ' 0.38t0.6 at a small scale (2 − 4mm), the diffusion behavior is underdiffusive with respect to Fick’s law, particles being trapped and slowed in elongated pores, mainly in the plane orthogonal to y direction. The coefficients of diffusion in different directions, given in Table 19.15, increase with the scale of observation, the long range connectivity of this porous medium appearing for long times and long distances. The anisotropy of diffusion reflects the anisotropy of the morphological tortuosity, with which D decreases. Dx (4mm) Dx (70mm) Dy (2mm) Dy (30mm) Dz (20mm) Dz (70mm) 0.13 0.23 0.025 0.11 0.09 0.14 TABLE 19.15. Random walks estimations of coefficients of diffusion (in voxels2 t−1 ) of an anisotropic porous medium (scale of measurement in mm) [151]
Homogenization of the diffusion coefficient An alternative way to estimate the effective diffusion coefficient D∗ is to consider time stationary solutions of the Fick’s law, when for concentration t (x) Zt (x), ∂Z∂t = 0. In this situation, a similar problem as the effective dielectric permittivity or the electrical conductivity has to be solved, where the concentration Z(x) acts as an electric potential φ(x), grad Z plays the role of the electric field E, and the flux of matter replaces the dielectric displacement D. Therefore estimating D∗ can be done by iterations of FFT, exactly as for the effective permittivity ∗ or conductivity, the solid phase behaving like an insulating component. Estimations of Fick diffusion by FFT in multiscale random porous media reproducing the nanosotructure of catalyst supports made of γ-alumina are given in [655]. First, the coefficient of diffusion of Toluene is measured by NMR experiments in 3 alumina samples containing nanoparticles with different shapes and with pore volume fraction close to 0.7. The three sam-
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ples are modelled by Boolean models with different primary grains: cubes C ((4nm)3 ), platelets P (7.1 × 3 × 3nm3 ), and rods R (16 × 2 × 2nm3 ) with uniform orientations. Simulations are made on cubic samples with volume (300nm)3 and (9093 voxels. For comparison, a Boolean model of spheres S with radius 2.5nm is also simulated. For the four models, the relative precision is better than 1%. Estimated D∗ are very close for each model, differing at most from 10% (this is expected from results on the effective conductivity of porous Boolean models reported in [679], showing weak differences for various Cox Boolean models in the case of insulat∗ ∗ ing inclusions), ranking as follows: DS∗ > DC > DP∗ > DR . In a second step, two scale models are introduced in simulations: aligned platelets AP , aligned and aggregated platelets AAP ; additionally, a two-scale model Cox Boolean model of platelets identified from TEM images of a real material RM [654], outlined in subsection 11.13.8 of chapter 11, is used to study the impact of the spatial distribution of alumina platelets on the effective coefficient of diffusion. Estimated D∗ , with a relative precision better than ∗ ∗ ∗ 1%, ranks as follows: DS∗ > DP∗ > DAP > DAAP > DRM . The lowest coefficient of diffusion, corresponding to the real material, is the result of the highest tortuosity factor in the porous medium as a consequence of partial alignment of platelets. To conclude, the various virtual materials introduced in simulations show a better coefficient of diffusion than the real material in this range of pore volume fractions, and could be used if a better diffusion is looked for.
19.14.2 Estimation of the fluid permeability of porous media The estimation of permeability of porous media from simulations can be made by lattice gas as introduced in chapter 16. In [586] the Lattice Boltzmann model is used to estimate the permeability of non woven fibrous networks simulated by anisotropic 3D Poisson fibres. Other efficient numerical techniques are based on finite difference schemes, and more recently involve FFT calculations, in a context different from solving the LippmannSchwinger equation [2], [681]. The emergence of Darcy’s law on a macroscopic level is a direct outcome of the linear character of the Stokes equation obtained from Navier-Stokes equations by linearization while neglecting inertial effects, for a fluid with viscosity μ [438], [453]: ∂i p = μ4ui (19.18) combined to the continuity equation for incompressible fluids ∂i ui = 0
(19.19)
and with the following boundary condition (no slip) applying on pore boundary ∂A:
19.14 Diffusion and fluid flows in porous media
759
ui (x) = 0 for x ∈ ∂A On a macroscopic scale, the fluid flow satisfies Darcy’s law relating the fluid flux Qi to the pressure gradient ∂i P : 1 hui i = Qi = − K ij ∂i P μ The permeability has the dimension of a surface (length2 ), and is usually expressed in cm2 . Bounds of permeability Some upper bounds of the effective permeability of random porous media are available: S. Prager [557], cited in [50], derived a bound involving the 3 points probability function of the random set by introduction in a variational principle of a particular trial function for the local stress in the fluid. G. Matheron derived a bound based on the statistics of linear intercepts (described a random variable L) in the pores [453]. In Rn for isotropic porous media and for a pore volume fraction p, k∗ ≤
© ª p E L2 12n
This upper bound can be explicitly calculated for many models of random sets, like Boolean models with convex grains, L following an exponential distribution for the complementary set of grains, or like for the Poisson mosaic. In [165] an upper bound is based on the covariance C(h) = C(d(x, y)) = C(x, y) of the random set A (with specific surface area SV ) representing pores, and on its first and second order derivatives: ¶ Z µ 2p ∂ 2 ∞ p2 ∂ 2 ∗ C(x, y) − C(x, y) + C(h) dh (19.20) k ≤ 3 0 Sv2 ∂x∂y SV ∂x In Eq. (19.20) the first and second terms are the covariance for two points ∂2 on ∂A ( ∂x∂y C(x, y)), and for one point on ∂A and one point in pores A ∂ ( ∂x C(x, y)). The objective of remaining parts of this section is to estimate the effective permeability tensor K ∗ (reduced to a scalar k ∗ in the isotropic case) from images of a porous medium, by numerical resolution of Eqs (19.18) and (19.19). Permeability and diffusion A first approach can make use of random walks [50], as is the case for diffusion in porous media. Combining Eqs (19.18) and (19.19),
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∆p = 0 A formal solution of Eq. (19.18) is expressed by means of the random Green’s function G(x, y), solution of ∆G(x, y) = δ(x − y) with the boundary conditions G(x, yS ) = 0 for yS ∈ ∂A. In the present case, G(x, y) is a random distribution depending on the random set A. The solution in domain B containing the porous medium provides the velocity field u(x). Its component ui (x) is given by: Z ∂p(y) 1 G(x, y)dy ui (x)1A (x) = 1A (x)1A (y) ∂xi B μ By averaging this equation, ¾ ½ Z ∂p(y) 1 pui = 1A (x)1A (y)G(x, y) dy E ∂xi B μ If the correlation between pui =
Z
B
∂p(y) ∂xi
and 1A (x)1A (y)G(x, y) can be neglected,
E {1A (x)1A (y)G(x, y)} dy
1 ∂p μ ∂xi
and the effective permeability k ∗ is given by Z 1 ∗ k = −p E {1A (x)1A (y)G(x, y)} dy B p
(19.21)
As noticed by M. Beran, it is possible to interpret the integral in Eq. (19.21) as the conditional probability for a random particle starting from x ∈ A to reach y ∈ A, before being absorbed by ∂A. This probability can be estimated in images of a porous medium by random walks starting in A, with absorption on ∂A, while for diffusion ∂A is a reflecting barrier. However, the conditions of validity of the assumption to neglect correlations between ∂p(y) ∂xi and 1A (x)1A (y)G(x, y) are not known. FFT solution of Stokes equations The Darcy permeability of random porous media with a full range of pore volume fractions is studied by solving the Stokes equations by FFT on periodic simulations of Boolean models of spheres [2] and of cylinders [681]. A numerical method of resolution was developed by A. Wiegmann [666]. It relies on the resolution of four Poisson equations (one for pressure p and one for each component of the velocity ui ), the inversion of the Laplacian being obtained by FFT. The no-slip condition is constrained by
19.14 Diffusion and fluid flows in porous media R
761
RVE for = 5%
120
System size computed RVE length A κ /R, L = 256
110 100
RVE length B κ /R, L = 256
90
RVE length B κ /R, L = 512
80
fcA = 3.17%
70
RVE length A f /R
60
RVE length B f /R
RVE length A κ /R, L = 512
fcB = 28.95%
50 40 30 20 10 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
f1
FIGURE 19.41. Normalized RVE of the permeability (l/R for a 5% relative precision) as a function of the pore volume fraction f , for cases A and B. The RVE for volume fraction is shown by thick lines; percolation thresholds figure as vertical lines [2]
forces introduced along ∂A. In [2] FFT solutions are obtained for a Boolean model of spheres with diameter d = 2R = 15 voxels generating two porous media: solid phase is the union of spheres (case A, with permeability kA ), or alternatively pores are the union of spheres (case B, with permeability kB ). Simulations in cubic volumes 1003 , 2563 and 5123 are generated to cover the full range of pore volume fractions. For a large range of pore volume fractions, the size of the statistical RVE (Fig. 19.41), calculated from the integral range and the point variance of the velocity field is much larger than the microstructural RVE (for estimation of the volume fraction) or than for elastic moduli or thermal conductivity. It strongly increases when reaching the percolation threshold. It is interesting to compare the permeability of the Boolean model to the prediction given by the popular Carman-Kozeny formula, originally derived for parallel cylinder capillaries [389], [106]: k CK =
p3 cSV2
where the constant c depends on the shape of capillaries and is empirically estimated. It is clear that k CK → ∞ when SV → 0, or alternatively when the diameter of capillaries grows to infinity. The flow permeability is sensitive to scale, a homothety of the microstructure by a factor λ changing the permeability k in λk. This is the effect of the surface of pore-solid interface, which does not occur for other properties like elastic or electric properties of random media in the presence of perfect interfaces. For beds of spherical particles c = 5, value that can be used for a Boolean model of spheres. The two curves for A and for Ac predict the same value kCK for p = 1−p = 0.5,
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which is not the case for the effective permeability of Boolean models plotted in Fig. 19.42, together with the Carman-Kozeny estimate and the Doi upper bound. For a Boolean model with convex random grains A0 , where q = P {x ∈ Ac }, SV is given by SV = −q log q
S(A0 ) V (A0 )
For a Boolean model of spheres with radius r, and for cylinders with radius r and height l 3 SV sphere = − q log q rµ ¶ 1 1 + q log q SV cylinder = −2 r l For Boolean Poisson fibres with radius r and for dilated Poisson flats with thickness l 2 SV f ibres = − q log q r 2 SV planes = − q log q l At a given pore fraction, the permeability of case B (with a lower percolation threshold) is larger than for case A. The Carman-Kozeny estimate is correct for case A in the range 0.3 < p < 0.8, and for case B when 0.7 < p < 0.9. It can be said that this formula cannot be applied to the Boolean model, missing the behavior of k ∗ with p for a large domain beyond the percolation threshold. Local information on the velocity is given by velocity maps [2] and by histograms. At lowest pore volume fraction p = 0.11 in case A, the velocity field is localized on some bright spots, the density of these spots increasing with p. Some histograms of the longitudinal (parallel to the main flow) and transversal (orthogonal to the main flow) components of the velocity field are shown in Figs 19.43, 19.44. Histograms are non symmetric, and show a peak for u1 = 0 and for u2 = 0, due to zones without fluid circulation. Streamlines are paths tangent to the velocity vector at each point. They are extracted and their tortuosity is measured. It is slightly lower for case A, except for pore volume fraction close to the percolation threshold. Lower tortuosity implies higher permeability, tortuosity decreasing with pore volume fraction. For a given p, the morphological tortuosity, deduced from geodesic paths is much smaller than the tortuosity deduced from fluid flow hu2 i characteristics (streamlines and the static viscous tortuosity τ 0 = hui2
19.14 Diffusion and fluid flows in porous media
763
2
κ (cm ) UB
κCK-B κCK-A κ κA (L=256) κA (L=512) κB (L=256) κB (L=512) fc=3.2%, 29.0%
7
10 6 10 5 10 4 10 3 10 2 10 1 10 1 0
0.2
0.4
1
0.8
0.6
f
FIGURE 19.42. Effective permeability (in logarithmic scale) of the Boolean models (case A and B for spheres with a conventional diameter equal to 2m), as a function of the pore volume fraction f ; for comparison, Carman-Kozeny estimate kCK and Doi and upper-bound kU B [2] Pu1 (t) 0
10
f B = 0.3 f B = 0.43 f B = 0.95
−1
10
−2
10
−3
10
−4
10
−5
10
−6
10
0
0.5
1
1.5 t
FIGURE 19.43. Histogram of the longitudinal component of the velocity field u1 in case B, for three pore volume fraction [2] Pu2 (t) 0
10
f B = 0.3 f B = 0.43 f B = 0.95
−1
10
−2
10
−3
10
−4
10
−5
10
−6
10
−0.5
−0.25
0
0.25
0.5
t
FIGURE 19.44. Histogram of the transverse component u2 in case B, for three pore volume fraction [2]
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[356]). As a consequence the overall permeability cannot be predicted directly from the morphological tortuosity alone, which should be completed with information on the width of paths, as done in [109]. The same approach is made for Boolean models of cylinders [681] with different aspect ratios l/r. In case B, where pores are union of cylinders, the Carman-Kozeny prediction is in agreement with FFT calculations for a wider range of pore volume fractions (0.3 < p < 0.9). For flow inside flat cylinders, kCK is very close to k ∗ for the full range of p, as expected since this geometry is faithful to the set of cylindrical capillaries. Interesting limit case concern Poisson fibres (for l → ∞) and Poisson thick planes (r → ∞). In [73] a different scheme is introduced to compute velocity fields on images by FFT. It makes use of a variational principle and trial force fields to produce an upper bound of the permeability. Therefore results obtained by this method should be compared to other calculations. It is used to estimate the permeability of the Voronoi mosaic for the full range of pore volume fractions on simulated images with size 5123 . The integral range of the velocity is much larger than the integral range of volume fraction. The RVE size shows similar variations with p as obtained for the Boolean model of spheres in [2], and is the largest close to the percolation threshold (0.1453 in the present case for site percolation). The Carman-Kozeny estimate seems to fit the obtained permeability in a very short domain of pore volume fractions. It would be interesting to estimate the Darcy permeability of multi-scale models such as Cox Boolean models. However this task may be insurmountable, since gigantic RVE are expected (as for Poisson fibres in section 19.7).
19.15 RVE of acoustic properties of fibrous media For engineering purpose, predicting acoustic properties of random media from their microstructure is a first step to design new materials with improved properties, like a better sound attenuation. One of the challenging problems to solve concerns the fluctuations of the microstructure and of the local physical fields involved in the homogenization of the acoustic behavior [550]. The results outlined in this section concern the estimation of acoustic properties of random fibres by numerical homogenization, and the study of corresponding RVE. The work was made in the context of the Silent Wall project, oriented towards the design of an insulating system using fibrous media with optimized acoustic absorption properties. On the microscopic scale, viscous and thermal dissipations of energy occur in the air saturating voids of a porous medium. To model the acoustic behavior of random porous media, use is made of a thermoacoustic formalism on realizations. Properties like harmonic acoustic velocity and temperature are homogenized by averaging.
19.15 RVE of acoustic properties of fibrous media
765
19.15.1 Thermoacoustic equations and homogenization of acoustic properties of porous media The viscous and thermal dissipation of the acoustic energy is handled from the simulation of acoustic fields [199], [545], [418], [645]. The variations →
of three harmonic variables, the flow velocity U (Eq. 19.22), the pressure P (Eq. 19.23) and the fluid temperature T (Eq. 19.24), allow to define → acoustic velocity u , pressure p and temperature τ by: →
→
→
U = U0 + u ei ω t
(19.22)
iωt
(19.23)
iωt
(19.24)
P = P0 + p e T = T0 + τ e
The harmonic variables are solution of four fully coupled equations, the linearized Navier Stokes equation for a compressible fluid (Eq. (19.25)), the equation of continuity (Eq. (19.26)), the heat equation (Eq. (19.27)), and the equation of the perfect gases (Eq. (19.28)) inside voids saturated by air. The air is considered as a visco-thermal fluid and as a perfect gas at the microscopic scale. The ambient pressure and temperature are respectively defined as P0 and T0 . →
→
i ω ρ0 u = −∇p +
³η
3
+ζ
´
µ ¶ → → → ∇ ∇. u + η ∆ u →
→ → ρ + ∇. u = 0 ρ0 i ω ρ0 Cp τ = κ ∆τ + i ω p µ ¶ p τ ρ (p, τ )|P0 ,T0 = ρ0 − P0 T0
iω
(19.25) (19.26) (19.27) (19.28)
In Eq. (19.25), p is the pressure, η is the dynamic viscosity, and ζ the second viscosity. In Eq. (19.26) (19.27), ρ is the volumic mass (with value ρ0 at equilibrium). In Eq. (19.27) Cp is the constant pressure specific heat, κ the thermal conductivity. Considering a porous medium as an equivalent fluid with a rigid solid phase (here the fibrous network), the interface Γ between fibres and pores → is considered as an adiabatic rigid boundary, implying acoustic velocity u (Eq. 19.29) and temperature τ (Eq. 19.30) equal to zero on Γ . → uΓ
τΓ
→
= 0 =0
(19.29) (19.30)
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19. Digital Materials
Working on a periodic domain B, periodic BC are applied for the three → complex acoustic variables u , p and τ for periodic homogenization [30]. The acoustic absorption of the medium is estimated from the effective tensor of dynamic viscous permeability Kv and from the effective scalar thermal permeability K 0 of the porous medium contained in B, assuming that viscous and thermal phenomena can be handled separately. Applying a macroscopic pressure gradient gradx p, the effective values of Kv (ω) and K 0 (ω), depending on the frequency ω of the acoustic wave, are respectively estimated from Eq. 19.31 (expressing the Darcy’s law) and from Eq. 19.32: Kv (ω) gradx p η K 0 (ω) hτ i = iωp κ
hui = −
(19.31) (19.32)
The thermoacoustic equations are numerically solved with the FE software Comsol Multiphysics, for frequencies in the range 0 Hz - 4032 Hz. The microstructure is meshed, and Eqs 19.29 and 19.30 are considered in the fibrous phase. The mesh at the interface Γ is refined in order to properly solve the equations in the boundary layer. The effective density tensor ρef f of the macroscopically equivalent medium is estimated from Eq. 19.33: η φ −1 (19.33) K iω The effective scalar compressibility modulus χef f is given by Eq. 19.34, with P r the Prandtl number of the air, and γ = 1.4, the perfect gas constant: µ ∙ ¸¶ 1 ρ0 P r i ω K 0 χef f (ω) = γ − (γ − 1) (19.34) γ P0 η φ ρef f =
→
→
→
From the wave vector Q = Q ξ defined from the unit vector ξ and parameters ρef f and χef f , are defined the effective characteristic velocity cef f (Eq. 19.35) and the effective characteristic acoustic impedance Zcef f (Eq. 19.36) of the homogenized porous medium:
cef f
Zcef f
v u→ u ξ T ρ−1 → t ω ef f ξ = = χef f Q v u→ → u T 1 t ξ ρef f ξ = φ χef f
(19.35)
(19.36)
19.15 RVE of acoustic properties of fibrous media
767
Finally, parameters ρef f and χef f can be used to calculate the acoustic impedance Zd (Eq. 19.37) of a macroscopic medium having a thickness equal to d: ⎤ ⎡ −2 i ω d cef f 1+e ⎦ Zd = Zcef f ⎣ (19.37) −2 i ω d cef f 1−e
The coefficient of reflection R(ω) (Eq. 19.38) is a complex number estimated from Zd and from Z0 = ρ0 c0 , the impedance of the air saturating the pores, and leads to the coefficient of absorption α(ω) (Eq. 19.39) of the corresponding d-thick macroscopic material. Zd − Z0 Zd + Z0 α (ω) = 1 − |R (ω)|2
R (ω) =
(19.38) (19.39)
19.15.2 Homogenization of acoustic properties of a periodic fibrous network A first example of calculation applies to a square cell containing a single fibre, to represent the transverse section of infinite parallel cylinders with the same radius. The macroscopic coefficient of absorption α(ω) strongly depends on the fibre radius, for a given pore volume fraction. In acoustics, the dissipation of energy occurs at the interface pore-solid, in the so-called boundary layer, with a thickness δ BL directly linked to the frequency f of the propagating sound wave (ρ0 and η being the density and the viscosity of the air filling the pores): r η (19.40) δ BL = f π ρ0 As a consequence, it is expected that reducing the fibre radius, implying an increasing surface area SV for a given pore volume fraction, could increase acoustic absorption in some frequency range. As shown in [550], this is true only for a range of sizes. In any case, the behavior of acoustic waves strongly depends on the scale of the microstructure, which is not the case of static effective properties. Interestingly, the prediction obtained on this very simplified geometry luckily coincides with experimental coefficients of absorption of the material when using a fibre radius equal to the average radius (RF = 42μm) measured by morphological openings on microtomography (as shown in chapter 3). This perfect agreement between calculation and measurements, obtained without any parameter fit is partially due to the a relatively homogeneous size distribution and to a high pore volume
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FIGURE 19.45. Harmonic fields of acoustic temperature modulus |τ | (K) for three frequencies f and two fibre radii (RF = 42 μm and 100 μm). Acoustic stimulation in the vertical Oy direction, sources at the top of the geometries. Fibres arrangement is kept constant [550]
fraction (0.64). It would be difficult to find such a simple representation for more complex porous media like heterogeneous fibrous networks, or foams.
19.15.3 Acoustic fields simulated on random unit cells Twenty realizations are generated for a model of random fibrous media of infinite parallel fibres, with cross sections made of three Boolean models of discs with radius RF = 42μm, 100μm and 300μm, with average pore volume fraction 0.64. Square fields have the following edge lengths: 2.52, 6, 18mm. An example of harmonic fields (acoustic temperature modulus |τ | (K)) for three frequencies f and two fibre radii (RF = 42 μm and 100 μm) is shown in Fig. 19.45. The source of the acoustic wave, propagating in the vertical Oy direction, is on the top of the field. The boundary layer (blue areas with |τ | ≈ 0 at the Γ air-fibre interface) gets thinner at high frequency, and an overlap of neighboring boundary layers is seen at low frequencies for → small fibres. Acoustic fields u and τ are not homogeneous and local fluctuations appear, resulting in variations of the acoustic properties between realizations. For each radius, the average absorption coefficient is estimated from permeabilities E{Kv } and E{K 0 } averaged over the n = 20 independent periodic microstructures. Concerning the radius 42μm, the prediction of α(ω) fits the experimental results, as for the simplified cell. For larger fibres (RF = 100 μm and 300 μm) the frequency evolutions are globally similar with almost identical amplitudes, but a so-called “frequency shift” is observed between regular and random microstructures.
19.16 Further examples of application
769
19.15.4 Statistical RVE and integral ranges The size of the statistical RVE is estimated for properties from results on the dispersion. Concerning the area fraction, the integral range of sections is directly calculated from the covariance of the Boolean model of discs (0.0049mm2 , 0.0277mm2 , 0.2495mm2 ), in agreement with estimations from simulations. The relative precision for the pore volume fraction on a single realization is rela = 3.7%. → Variables u and τ being complex, the estimation of the RVE operates on the following four fields: Re [uy ], Im [uy ], Re [τ ] and Im [τ ]. Detailed results are available in [550]. Integral ranges A2 , rela and the RVE size of the four average scalar fields decrease with the frequency for RF = 42 μm and RF = 100 μm. This can be explained by the frequency dependence of the thickness of the boundary layer δ BL given in Eq. ( 19.40) involving a decrease of correlation lengths at high frequency. The relative precision of the four scalar fields is much better than 10 %. The RVE size does not change with frequency for RF = 300 μm. Integral ranges of fields are much larger than the morphological integral ranges. Concerning the acoustic absorption α(ω), its relative precision for 20 realizations is better than 5% for RF = 42 μm and RF = 100 μm. Finally, in the present case thermal properties fluctuate more and develop longer correlation lengths than velocity fields. The thermal properties are therefore the limiting factor to define a representative volume element.
19.16 Further examples of application Many more examples of numerical homogenization of Digital Materials are available. To name a few recent results, in [415] FFT calculations of micromechanical fields are made on polycrystals with an elasto-viscoplastic behavior. In [55] is implemented a FFT scheme for a non-local polycrystal plasticity theory using a mesoscale version of the field dislocation mechanics theory. In [164] it is applied to two-phase laminate composites with plastic channels and elastic second phase. In [196], the thermal expansion and elastic properties of strongly anisotropic granular media, simulated by locally anisotropic random tessellations [309], are estimated by FFT calculations. In [198], it is extended to the thermoelastic properties of microcracked polycrystals, where the role of transgranular and of intergranular cracks is investigated. In [683] self-consistent estimate for cracked polycrystals with hexagonal symmetry are compared to FFT numerical computations carried out on a Voronoi tesselation for the polycrystal and a Boolean model of cracks. In contrast with self consistent estimates, FFT results do not indicate a percolation threshold in highlyanisotropic crystals at low-crack density, but rather a strong weakening of the elastic moduli.
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19. Digital Materials
19.17 Conclusion The domain of Digital Materials, and more generally Digital Media, is rapidly growing and will become more and more important in the search of practical solutions for engineering purpose, for the conception of new materials and microstructures with optimized functionality. The examples presented in this chapter are of limited extent concerning the kinds of results that can be obtained by application of this paradigm, and are intended to illustrate the richness of information that can be provided from a numerical approach of fields in microstructures. From available data on simulations summarized in this chapter, effective properties (linear and nonlinear elastic moduli, linear and nonlinear conductivity, flow permeability) of the Boolean model of spheres and of its multiscale versions are now well documented. In addition to efficient solutions of homogenization problems, they give us the possibility to get local information on fields in real or in simulated media, and to explore very specific localization phenomena like stress concentration of the formation of hot spots, giving access to some kinds of digital experiments. Such a morphological information is usually not available from more conventional experiments. Application of Digital techniques to microstructures can be useful and efficient only when operating with reliable data on a microscale. In particular, in addition to high quality 3D images, there is some needs to obtain data on the local physical properties of various components in materials, and on the quality of their interfaces. This requires the implementation of experiments, beside the development of models, even sometimes down to a nanoscale, like the dielectric function measured in a high resolution TEM by Electron Energy Loss Spectroscopy (EELS) experiments [537]. Furthermore, models cannot be useful without reliable tools of identification of morphological and of physical properties, that can involve the resolution of ill-posed mathematical problems. In micromechanics, a combination of full field calculations with field analysis of displacements from digital correlations is a useful tool for the identification of local mechanical properties [239]. Finally it is worth to underline some methodological issues when implementing numerical simulations on random media. Traditional approaches to solve physical or micromechanical problems on heterogeneous microstructures make use of analytical or of numerical calculations on some heterogeneous cell, assumed to be sufficiently representative in order to correctly predict its macroscopic behavior. This type of simplification, producing some idealized microstructure and successfully applied in the example of acoustic properties in section 19.15, can hardly work properly in many real situations where microstructure have a complex morphology: starting from 3D micrographs as illustrated in this chapter, it may become very hard to decide which component can be assimilated to a matrix, or inversely to some reinforcing inclusions. Furthermore, connectivity properties are often
19.17 Conclusion
771
difficult to capture correctly a very schematic simplified description. It is therefore more efficient to operate directly on Digital Materials, whatever real or virtual. When using simulations of a microstructure, it may be tempting to get rid of fluctuations by fixing some parameters of realizations, like for instance their volumetric composition, as seen in many publications. This cheap but naive approach may be the source of uncontrolled biases, the effective behavior depending nonlinearly on such parameters, and also on other key-points like the percolation of some given component. Hoping to be able to make a correct macroscopic prediction with a single small representative realization satisfying some constraints is a methodology very close to the above mentioned traditional approaches. Unless ergodicity conditions are reached by means of a single large enough simulation (which can be checked through the estimation of integral ranges), there is no warranty on the quality of the prediction, due to some uncontrolled features (like for instance the occurrence or not of some percolation on a small scale), or due to the generation of bias by the effect of some BC, as illustrated by some of the studies presented in this chapter. Computing facilities for physical fields estimation are more and more of easy access, so that data can be obtained with a decreasing cost. It does not prevent to check the reliability of estimations, avoiding the biases induced from edge effects of BC, by control of the evolution of apparent properties with the size of domains of calculation, by comparison of results obtained with different BC, and finally by following precepts induced by the statistical RVE which is strongly dependent on physical properties and is mostly much larger than the RVE required for estimating volume fractions.
20 Probabilistic Models for Fracture Statistics
Abstract: As a consequence of microstructural heterogeneities, fluctuations in the mechanical properties of materials are observed in experiments. This requires a probabilistic approach to relate the microstructure to the overall materials properties, and to predict scale effects in the fluctuations of properties. In this chapter, the problem of the strength of materials is addressed, and various models of random structures developed for fracture statistics are introduced. As any fracture criterion is sensitive to microstructural heterogeneities, such as flaws with low strength, or defects inducing a local stress concentration, large effects of small scale heterogeneities are observed for fracture phenomena. The approach combines the selection of appropriate fracture criteria and random structure models. It enables us to predict the probability of fracture of heterogeneous media under various loading conditions. Key words: Fracture Statistics, Change of Scale, Random Structures, Weakest Link, Crack Arrest, Critical Damage, Brittle Fracture.
20.1 Introduction To predict the overall behavior of heterogeneous media from their structure is of importance in many fields of applied and engineering sciences: in materials science, this is the way to design and produce materials with customized microstructures, as far as the final use properties are concerned. When considering the physical properties of heterogeneous media and their © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Jeulin, Morphological Models of Random Structures, Interdisciplinary Applied Mathematics 53, https://doi.org/10.1007/978-3-030-75452-5_20
773
774
20. Probabilistic Models for Fracture Statistics Aim of a probabilistic approach: ⎧ ⎪ Applied Field ⎪ ⎪ ⎪ σ(x) ⎨ P{fracture} = P Microstructure ⎪ ⎪ ⎪ σ c (x) ⎪ ⎩ Shape
evaluation of Loading Local Fracture ⎪ ⎪ ⎪ stress ⎪ ⎭ Sample ⇓ Change of scale ⎫ ⎪ ⎪ ⎪ ⎪ ⎬
TABLE 20.1. Principle of Fracture Statistics calculations
fluctuations at various scales, it is necessary to introduce appropriate techniques and models. This problem is encountered in many situations, such as flows in porous media, elastic properties, strength of composite materials... It is solved by different methods (closed-form calculations, simulations), using in any case models of random media. • The macroscopic (or effective) properties (like overall elasticity moduli of elastic composites) can be estimated from partial knowledge on the microstructure by variational methods. In that case, approximations of average properties of infinite random media are obtained, using homogenization techniques ([438],[582], chapter 18). • Another property of interest in materials science is the strength, much more sensitive on local heterogeneities than the effective properties. For practical applications, it is useful to calculate the probability of fracture as a function of the loading conditions and of the microstructure as shown in Table 20.1. It has many implications on the reliability analysis of parts and components at various scales, in many industrial fields (aeronautics, nuclear plants, civil engineering, etc.). The statistical distribution of the properties of finite domains can be obtained theoretically, as illustrated later. Scale effects, like the change of mean properties and of their variance with the size of specimens, can be predicted and compared to experimental data. For this purpose, there is need of a specific approach: in fracture, there is no smoothing effect by scaling, and there is a great sensitivity to local defects, with a large effect of tails of the distributions on the macroscopic fracture behavior, as seen below.
In this chapter, some probabilistic continuum models of brittle fracture are reviewed. They enable us to predict fracture probabilities of materials under various loadings and on different scales. The approach covers the following steps: • Choice of local (i.e. punctual) and of macroscopic fracture criteria; the first type accounts for crack initiation, and the second for the fracture of a specimen.
20.2 Choice of a fracture criterion
775
• Construction of random structure models, defined on a point scale, for which the calculation of the fracture probability and of scaling laws is possible; this usually involves simplifications, such as the use of the stress field seen by an equivalent homogeneous medium, using the socalled local approach, as introduced by A. Pineau [552]. This results in appropriate change of supports, as introduced below. Probabilistic models for fracture statistics can be implemented for a volume element, for a part, or for a structure, in order to predict their lifetime, reliability, and to manage their maintenance. In applications, the loading conditions may be deterministic or random, and use can be made of analytical techniques or simulations. After a presentation of the used fracture criteria, each of them and proposed random function models with a point support are separately examined to cover the following cases [284], [286]: fracture statistics of brittle materials; models with a damage threshold; crack arrest in random media, random damage.
20.2 Choice of a fracture criterion The first step required to develop fracture statistics models is the choice of local and of global fracture criteria. A local criterion is sensitive to the fracture initiation, while a global or macroscopic criterion accounts for the fracture of a specimen.
20.2.1 Local fracture criteria Various local criteria can be used: the fracture is initiated at points in the structure where some intrinsic mechanical property of the material is exceeded, as the result of the applied load. Usually, this property is the critical stress σc (x), or more generally the critical stress intensity factor KIc (x) for the tensile fracture in linear elastic brittle materials [273],[280],[281]. When there is competition between several fracture mechanisms, as for cleavage and intergranular fracture in rocks and in metals, multivariate criteria and multivariate random function models can be used [281],[275]. The local fracture energy γ(x) corresponding to the creation of a fracture surface is used for crack propagations [278],[283].
20.2.2 Global fracture criteria The following macroscopic fracture criteria, involving different fracture assumptions, were proposed by D. Jeulin [273],[281],[275].
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20. Probabilistic Models for Fracture Statistics
• The weakest link model is well suited for the brittle fracture of materials, as in the cleavage of steel at low temperature [552]; it corresponds to a sudden propagation of a crack after its initiation. • Models with a damage threshold generalize the previous one; they are valid for a fracture with several potential sites for crack initiation. • Models with a Griffith crack arrest criterion compare, for each step of a crack path, the local fracture energy to the stored energy G(x) due to the deformation of the material. Formally, when considering the overall condition of fracture of a sample, the first type of criterion uses a change of support of the information by the operator ∧ (infimum) over the domain of interest; the second family is connected to a change of support by convolution; finally, the last criterion involves a change of support by the operator ∨ (supremum) over the crack front. In the present approach, an equivalent homogeneous medium with a random critical stress is used. This simplification, which separates the applied field and the critical field, enables us to obtain closed form results without any simulation. This is justified for media with a single component, like polycrystals in metals or in rocks. However, this approach cannot account for small scale stress fluctuations induced by the microstructure when the components have different mechanical behaviors, resulting in a strong coupling between the applied stress field and the local fracture criterion. This is introduced in chapter 21 by means of numerical simulations based on a phase field approach [322]. The three above-mentioned macroscopic fracture criteria and the associated models are detailed in the following parts of the chapter. Theoretical probabilistic models are available for the following assumptions: weakest link assumption, coming back to Weibull (1939) [660], [661]: critical volume (or a critical density of defects); Griffith crack arrest for media with a random fracture energy; growth of random damage.
20.3 Brittle fracture and weakest link 20.3.1 Introduction Based on the weakest link assumption, these models assume the fracture of a part, as soon as for a single point x0 , we have σ(x0 ) > σ c (x0 ) (Fig. 20.1). It corresponds to the immediate propagation of a crack after its nucleation. To estimate the probability of fracture of a specimen B with this assumption and for proportional loading conditions, it is necessary to know the probability distribution of the minimum of the values σ(x)−σ c (x) over the loaded domain B. With the weakest link assumption, fracture statistics are governed by the most critical defects present in a
20.3 Brittle fracture and weakest link
777
microstructure, and therefore the pure statistical expected scale effect is the decrease of strength with the size of specimens, to a constant value for the largest samples. The probability of fracture can be expressed as: P {no fracture of By } = P {∧x∈By (σ c (x) − σ(x − y)) ≥ 0}
(20.1)
The stress field σ(x) depends on the local variations of the constitutive law of the material (namely on the local elastic moduli for an elastic material); it is therefore somewhat correlated to the local critical field σ c (x). The calculation of the probability P in Eq. (20.1) requires the following steps: • Estimation of the local stress field, possibly using the formal expression based on the use of Green’s functions [50],[393],[670] as in chapter 18. • By thresholding the difference σ c (x)−σ(x) to the value zero, one obtains a random set A. One then has to estimate the probability for the set B to miss the random set A. This is a part of the general characterization of the random sets [438],[448]. To our knowledge, this program has not yet been fulfilled. It is a tremendous task. This can be accessed from numerical simulations on realizations of random media as was shown for unidirectional fibre composites [40]. Otherwise, simplifications must be introduced. This is done for some random structure models when using the deterministic field σ(x) seen by an equivalent homogeneous medium as in the "local approach of fracture" [552]. This simplification is used in the remaining part of this chapter. For a stationary random function σ c (x), the probability of non fracture of a specimen B under the deterministic stress field σ(x) is given by: P {no fracture of B} = P (σ) = P {x ∈ Hσc (σ)}
(20.2)
In Eq. (20.2), Hσc (σ) is the set of implantations of the specimen B for which the minimum of the values σ c (x) − σ(x) remains positive. In general, the probability law (20.2) is not available. However it can be calculated in a closed form for specific models of microstructures like Boolean RF (chapter 9), more general than the Weibull model, as developed by D. Jeulin [273],[275],[281],[280],[285].
20.3.2 Examples of stress fields used for the weakest link model To illustrate the potential applications of the models, two kinds of stress fields are used in Eqs (20.1,20.2): • A uniform stress field applied on a cubic sample C0 with side L: σ(x) = σ for x ∈ C0 .
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20. Probabilistic Models for Fracture Statistics
FIGURE 20.1. Illustration of the weakest link assumption; fracture initiated at point x0
A non-uniform stress field in a cracked metallic specimen (Fig. 20.2), in an elastic-plastic material (with yield stress σ y ) under planestress conditions (a uniform stress field σ orthogonal to the crack plane is applied at the infinity). The crack tip is surrounded by a plastic region with radius 2ρ(θ). Outside the plastic region, the stress field σ(r) (r = 0 at the point O located at the distance r(0) from the crack tip (Fig. (20.2) is the elastic stress field for r ≥ ρ : at the point P (OP = r and −−→ −→ (OP , Ox) = θ), we have: σ ij (r) =
KI gij (θ) √ 2πr
with the stress intensity factor KI for a large size plate with a central crack with length 2a orthogonal to the tensile stress: √ KI = σ πa For a given crack size a, and the macroscopic stress σ, the region of the specimen (e.g. a cube of side L) under plastic yielding is a cylinder with axis L and with section: B (σ/σy )2 a (B = B(1) being obtained for (σ/σ y )2 a = 1; we will neglect the changes of ρ with the angle θ). From now on, we make the two following assumptions: i) fracture by cleavage can only be initiated into the plastic region B, when a local critical cleavage stress σ c is reached. ii) inside the plastic region B, the variable stress field (Fig. 20.2) is replaced by a homogeneous stress field equal to the maximum stress λσ y (Fig. 20.3). With this simplification, the probability of fracture for a given applied stress is overestimated. But the general line of the results will not be altered.
20.3 Brittle fracture and weakest link
779
FIGURE 20.2. Stress field at the crack tip of an elastic-plastic material; plane stress conditions
FIGURE 20.3. Stress field at the crack tip of an elastic-plastic material; simplified stress field (constant in the plastic zone) for the calculations
It is instructive to compare the behavior of a sample under a uniform stress field and of a cracked specimen: in both cases for our approach, the domain where fracture occurs is under a homogeneous stress field. In the first case the part of the material where σ > σ c is the sample itself. In the second case, the region of interest is the plastic region submitted to the homogeneous field λσ y (independent on the applied stress σ, and constant for a material with a constant yield stress σ y ). The size of the region increases with the applied stress σ and with the crack length a, 4 and its volume is proportional to (σ/σy ) a2 . So the macroscopic stress σ induces a volume effect proportional to σ 4 . Therefore, it is expected that each kind of applied stress field will bring its own piece of information on the random critical stress field σ c (x) in the material. This is illustrated in the next sections for different models σ c (x). In addition, any change of the yield stress σy due to specific conditions (e.g. a decrease in temperature) can be accounted for in our results, and will modify fracture statistics of cracked specimens.
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20. Probabilistic Models for Fracture Statistics
20.3.3 The Boolean random varieties and the weakest link model This subsection presents a wide class of models for which fracture statistics corresponding to the weakest link model are available. Their field of application is not restricted to micromechanical problems, as seen in the other chapters of this book. Reminder on the origin of weakest link statistics The standard weakest link model [660], [661] is based on the assumption that fracture in a brittle material is initiated on the most critical defect, that controls the full fracture process. For this model, it means that when there is at least one point x in a specimen where the applied principal stress component σ(x) is larger than the local critical stress σ c (x), the specimen is broken. Usually it is assumed that the occurrence or absence of critical defects (generating fracture) of any volume elements generate a set of independent events. After a decomposition of the volume V into links vi and assuming that there is a fracture of the volume V when a single link vi is broken, a classical computation for independent events gives: Y P {Non fracture} = P {Non fracture of vi } i
For vi → 0, P {fracture} ' Φ((σ(x))dx, with Φ increasing with the loading and P {Non fracture of dx} ' 1 − Φ((σ(x))dx. Therefore with these assumptions, Z P {Non fracture of V } = exp(− Φ(σ(x))dx) = exp(−V Φ(σ eq )) V
where the equivalent stress is defined from Z 1 Φ(σ eq ) = Φ((σ(x))dx V V Due to the probabilistic independence of elementary volume elements dx and dy, whatever their size and location, this assumption of the standard weakest link model is in fact equivalent to a distribution of point defects in a matrix with σ c = ∞, according to a Poisson point process in space (chapter 5), with intensity Φ(σ), where Φ(σ) is the average number per unit volume of defects with a critical stress σ c lower than σ. To resort to a Poisson point process to locate critical defects appears in a simplified way in [183]. Any increasing function Φ(σ) can be used for this purpose. For a homogeneous applied stress field σ(x) = σ, it comes: P {Non fracture of V } = exp(−V Φ(σ))
20.3 Brittle fracture and weakest link
781
For the Weibull model, the function Φ(σ) is power a law in σ: Φ(σ) = θ(σ − σ0 )m and P {Non fracture of V } follows a Weibull distribution, as reminded below. In that case σ eq is named the Weibull stress. There are thousands of papers using this distribution for strength of materials. In most cases, users are not aware of the implicit existence of an underlying Poisson point process for defects in this context. The Weibull distribution belongs to the family of Gumbel distributions involved in the statistics of extreme values [222], [423], obtained as an asymptotic limit by imposing a stability condition on the distribution of the normalized supremum X (or the infimum) of n independent RV with the same distribution F in the following sense : (F (an x + bn ))n →n→∞ G(x) with a normalization (an x + bn ), depending on the distribution G, chosen for the convergence to be satisfied. This normalization has no specific physical meaning. Starting from any distribution F , the distribution of (an x + bn ) converges towards asymptotic distributions, namely extreme value or max-stable distributions, under certain conditions. For these asymptotic distributions, the distribution of the normalized maximum of n realizations is of the same type as the distribution of a single realization of the RV. Generally speaking, any distribution F (x) can be considered as the distribution of the supremum (resp. infimum) of independent RV with distribution F (x)1/n (resp. 1 − (1 − F (x))1/n ). There is therefore no prior physical reason for the macroscopic fracture stress σR to follow a Weibull distribution nor any asymptotic distribution of extremes and anyway it is not essential to resort to extreme value theory to design probabilistic models of fracture. The extension of max-stable distributions to random processes generates so-called max-stable processes [154], among which emerge particular cases of Boolean RF. In [95] is given an example of Boolean stochastic process (BRF in 1D), as a limit case of infinite supremum of normalized Brownian motions. Construction of the Boolean random varieties The Boolean random varieties describe structures with different geometrical defects: points or grains, fibres, strata (chapter 6). They are generalizations of the well-known Boolean model proposed by G. Matheron [438], [448] as a model of random set, initially used for simulating porous media. In the present application, they are used as random models to represent the critical stress field σ c (x). In fact, as for other models used later on, it is possible to use them (or their multivariate version) to simulate any physical properties of composite media (chapter 18), such as for instance their moduli of elasticity. For these models, it is possible to derive theoretical models of change of scale including size and shape effects (concerning the specimen B), and also microstructure effects.
782
20. Probabilistic Models for Fracture Statistics
We will recall here the general results and illustrate them for the uniform stress field and for the stress field developed at the tip of a macroscopic crack in elastic-plastic materials. The construction of the Boolean random varieties is given in chapters 6 and 9. As already mentioned, a geometrical interpretation of the varieties is the following: • Consider a Poisson point process Pk (ω) with intensity θk (ω)μn−k (dx) on the linear varieties of dimension (n − k), with the orientation ω, and containing the origin. In each point xi (ω) of the process, a variety of dimension k, Vk (ω)xi , orthogonal to the direction ω, is located. A realization of the variety of dimension k is obtained as: Vk = ∪xi (ω) Vk (ω)xi For fracture statistics, we operate in R3 , so that are considered the following Poisson varieties: points (k = 0), lines (k = 1), and flats (k = 2). In addition, we will limit ourselves to isotropic models, but anisotropic structures (with appropriate θk (ω)) can be used when necessary. • In a second step, consider independent primary random functions Z 0 (x), lower semi-continuous, with a subgraph admitting compact sections almost surely and with Z 0 (x) ≤ σ m . Starting with a homogeneous medium with strength Z(x) = σ m , the ∧ Boolean random Variety of dimension k is built as follows: Z(x) = ∧{Z 0 (x − yi ); yi ∈ Vkxi ; xi ∈ Pk }
(20.3)
Simulations of Boolean RF in the plane are shown in chapter 9. With this construction, various types of defects (with a lower σ c ) in a homogeneous matrix are obtained: random grain defects, fibre defects, and strata defects. For these models, the probability of fracture of Eq. (20.2) is expressed as: P {no fracture of B} = exp(−θ¯ μ(HZc 0 (σ − σm )))
(20.4)
In Eq. (20.4), μ is equal to volume V (for grains), π/4S (S being the surface of the area) for fibres, or to the integral of mean curvature M for strata (chapter 6). It is assumed in Eq. (20.4) that HZc 0 (σ − σ m ) is a convex set for the case of fibre and strata defects. In Eq. (20.4), the average measure μ is taken over the realizations of Z 0 . A slightly more general formulation of Eq. (20.4) is obtained (chapter 9), starting from a family of primary random functions Zt0 , implanted on random Poisson varieties Vk (t) with intensity θ(t) (t being equivalent to a time parameter): Z P {no fracture of B} = exp(− μ ¯ (HZc t0 (σ − σ m ))θ(dt)) (20.5) R
20.3 Brittle fracture and weakest link
783
The general properties of these models for fracture statistics applications are the following : • By construction, the fracture stress allows correlations at a microscale (depending on the size of the support of the defects Zt0 (x)). • When considering the fracture strength at a macroscale (namely the scale of the specimen B), the following effects are expected to be observed : size effect (through the set of points HZc 0 (σ − σ m )), depending t both on B and on Zt0 ; shape effect for the specimen B ; microstructure effect (grains, fibres or strata give different weights to the probability of failure, through V , S or M ). Some examples of Boolean Varieties We develop now some specific examples of models, useful for applications, based on two separate constructions, and with σ m = +∞. Theoretical expressions for a homogeneous stress field are given. 1. Consider first defects Zt0 with a random closed set support A00 . The critical stress of the defect σ c is a random variable Z 0 , independent of A00 , implanted in space with the intensity θ(σ). The resulting field σ c (x) is a mosaic made of domains where σ c is constant. It is a good simulation of a multiphase material containing grains with different sizes and strengths. Since we are interested in brittle fracture in traction, we restrict ourselves to the scalar case with σ > 0 (σ being here the local maximal principal stress). In the case of a uniform stress field over the specimen B, the distribution of the fracture strength σ R of B is deduced from Eq. (20.5) as: ˇ P {σ R ≥ σ} = P {no fracture of B} = exp(−¯ μ(A00 ⊕ B)Φ(σ))
(20.6)
In Eq. (20.6), μ ¯ is the average over the realizations of the random closed set A00 and Z σ
θ(t)dt
Φ(σ) =
0
In practice, any increasing non-negative function Φ(σ) can be used in Eq. (20.6). This is illustrated below by well-known examples. For a non uniform stress field σ(x) and point defects, Eq. (20.5) becomes, as seen above: Z P {non fracture of B} = exp(− Φ(σ(x))dx) = exp(−Φ(σ eq )) B
(20.7) with V Φ(σ eq ) = B Φ(σ(x))dx. This formulation in terms of equivalent stress σeq enables us to put together the results of fracture tests obtained under different stress fields, when estimating the parameters of a model. R
784
20. Probabilistic Models for Fracture Statistics
This type of approach is followed in [60] for the Weibull statistics, σ eq being the Weibull stress. a. For defects with intensity θ(σ) = mθ(σ −σ0 )m−1 with m > 1 and σ > σ0 , we get Φ(σ) = θ(σ − σ 0 )m , and σ R obeys a Weibull distribution, with σ m u = 1/θ: µ ¶m σ − σ0 0 ˇ P {σ R ≥ σ} = exp(−¯ μ(A0 ⊕ B) ) (20.8) σu This distribution is very popular amongst practitioners of fracture statistics, as can be seen for instance from the numerous references given in [230],[382]. The classical version of the distribution is limited to point defects A00 , where the more severe defects (with a low σ c ) are much sparser than the less critical defects. This model is expected to be suited to high grade materials with a thorough control of defects. From Eq. (20.8), it is easy to calculate the expectation E[σ R ] and the ˇ variance of σ R , as a function of V = μ ¯ (A00 ⊕ B): E{σ R } = σ 0 + V −1/m σ u Γ (1 + 1/m)
(20.9)
D2 [σ R ] = σ 2u V −2/m (Γ (1 + 2/m) − Γ 2 (1 + 1/m)) (20.10) Z ∞ tx−1 exp(−t)dt. where Γ (x) is the Eulerian function Γ (x) = 0
When σ0 = 0, the coefficient of variation of σR , D[σ R ]/E{σ R } does not depend on V , which can be easily checked from experimental data: D[σ R ]/E{σ R } =
(Γ (1 + 2/m) − Γ 2 (1 + 1/m))1/2 Γ (1 + 1/m)
(20.11)
Eqs. (20.9,20.10) express a scale effect for the expectation and the variance of the strength, which always decrease with the size of the specimen, but differently according to the kind of Boolean Variety. This scale effect is also observed on the median σM of the fracture strength: σ M = σ0 + KV −1/m (20.12) The variance of the Weibull distribution decreases with the parameter m. When m → ∞, this distribution converges towards a Dirac distribution δ(σ − σ 0 ), reflecting a single and deterministic fracture stress σ 0 . Typical values of m depend on the material: m ∼ 5 for ceramics (which are very brittle) and m ∼ 20 for steels at low temperature (a comparison is shown on Fig. 20.4). In [61] a value of 6.25 for the modulus m of ceramic mullite-alumina fibres was estimated, using the following procedure: to fit m from experimental data like
20.3 Brittle fracture and weakest link
y
785
1.0 0.8 0.6 0.4 0.2 0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
x
FIGURE 20.4. Weibull distribution: m = 5 , θV = 1 (red); θV = 8 (green); m = 20, θV = 1 (black); θV = 8 (blue)
the fracture stress of specimens, use can be made of a Weibull plot: ordering k data with increasing values, an estimate of the cumulative distribution is given by affecting the probability r/(k + 1) to the value of rank r. Plotting log(log((k + 1 − r)/(k + 1))) as a function of σ(r) provide a straight line with slope m. From thousands simulations with m = 4, it was shown that a very good estimation of m was obtained from samples made of 30 fibres. Sometimes two lines appear with two slopes in the Weibull plot, reflecting the presence of a bimodal distribution with two parameters m1 and m2 , in accordance with Eq. (20.13). When using data from specimen loaded by a non homogeneous stress field, like for instance in bending tests, one should use the empirical statistics of Φ(σ eq ) defined by Eq. (20.7) (it should follow an exponential distribution), and estimate m by iterations. By means of Φ(σ eq ), data obtained for different testing conditions (stress field, volume of specimens) can be used together for the fit. b. Similarly, we can use a bimodal Weibull distribution, where there is a superposition of defects from two Weibull populations (with parameters (θ1 , σ 01 , m1 ), and (θ2 , σ 02 , m2 )) (a comparison between two standard Weibull distributions and a bimodal distribution is made on Fig. 20.5); in that case, Φ(σ) = θ1 (σ − σ 01 )m1 + θ2 (σ − σ 02 )m2
(20.13)
c. For defects with intensity θ(σ) = θ/σ u , the distribution of strength is exponential, and can be considered as a particular Weibull distribution, with m = 1. P {σ R ≥ σ} = exp(−θV σ/σ u )
(20.14)
786
20. Probabilistic Models for Fracture Statistics
y
1.0 0.8 0.6 0.4 0.2 0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
x
FIGURE 20.5. Weibull distribution, m = 5 (green); m = 20 (blue); bimodal Weibull distribution (red); θV = 1
with expectation σ u /(θV ), variance (σ u /(θV ))2 and coefficient of variation 1. θ for σ ≥ σ0 , Φ(σ) = d. For defects with intensity θ(σ) = (m − 1)σ m 1 1 − m−1 when m > 1. σ σm−1 0 e. For defects with intensity θ(σ) = θ/σ for σ ≥ σ 0 , P {σ R ≥ σ} = (σ 0 /σ)θV
(20.15)
ˇ as previously. Contrary to the Weibull model, the for V = μ ¯ (A00 ⊕ B), last two populations of defects are dominated by defects with very low critical stress σ c , that might reflect a poor quality of a material submitted to these statistics. The distribution given in Eq. (20.15) is well known as the distribution of Pareto [358]. Its expectation and variance are infinite for θV ≤ 1 and θV ≤ 2 respectively, namely for specimens B with a low size. When θV > 1, we have: σ 0 θV θV − 1
(20.16)
σ 20 θV (θV − 2)(θV − 1)
(20.17)
E{σ R } = For θV > 2 , D2 [σR ] = and
D[σ R ]/E{σ R } =
s
θV − 1 θV (θV − 2)
(20.18)
The last properties, given by Eqs. (20.16-20.18), decrease with the size of the specimen B. The size effect is higher for the Pareto model than for the Weibull distribution. A comparison between two Pareto
20.3 Brittle fracture and weakest link
y
787
1.0 0.8 0.6 0.4 0.2 0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
x
FIGURE 20.6. Weibull distribution: m = 5 (green); m = 20 (blue); θV = 4; Pareto distribution: θV = 5, σ0 = 0.61 (black); θV = 20, σ0 = 0.88 (black)
and two Weibull distributions with the same medians is given in Fig. 20.6. f. When Φ(σ) is a sigmoidal function, a saturation effect is observed on the cumulative distribution of defects : ∙ µ µ ¶m ¶¸ σ − σ0 Φ(σ) = θ 1 − exp − (20.19) σu for σ ≥ σ 0 . Such a distribution was observed for the fracture of carbon fibres [39]: using a multifragmentation test on a single carbon fibre embedded in an epoxy matrix, it is possible by means of acoustic emission to estimate the cumulative distribution of the critical stress of defects on fibres. As opposed to conventional determinations of defects statistics by means of single fracture tests on fibres (for which the weakest link assumption can be applied!), a much larger sample of defects is accessed; for larger critical stresses, it departs from the power law function involved in the Weibull model, which is recovered for the lower stresses by a Taylor’s expansion of Eq. (20.19), since when σ ' σ 0 , µ ¶m σ − σ0 Φ(σ) ' σu The obtained results were introduced in Finite Elements (FE) simulations to study the damage of fibre-matrix composite layers [40]. 2. A second family of Boolean Varieties models uses defects made of random closed sets with a constant critical stress σ c as previously, but the value of σ c depends on the realization of the primary grain A00 , as follows. Consider random grains A0 (d) (homothetic with ratio d of random grains A0 = A0 (1)). For each grain, σ c = √ψ(d). In fracture mechanics, it is common to assume that ψ(d) = K/ d, each grain A0 (d) behaving like a crack with length d. For the model, with A0 (u) = uA0 , in the case of a
788
20. Probabilistic Models for Fracture Statistics
homogeneous stress field, P {σ R ≥ σ} = exp(−
Z
∞ K 2 /σ 2
ˇ θ(u)¯ μ(A0 (u) ⊕ B)du)
(20.20)
In Eq. (20.20), we can use the Steiner formula [438], [448], [598], Eqs (3.12-3.17) in chapter 3) when the sets B and A0 (u) are convex : for Boolean grains ˇ = V (B) + S(B) uM + M (B) u2 S + V u3 V (A0 (u) ⊕ B) 4π 4π
(20.21)
for Boolean fibres ˇ = S(B) + M (B) uM + u2 S S(A0 (u) ⊕ B) 4π
(20.22)
for Boolean strata ˇ = M (B) + uM M (A0 (u) ⊕ B)
(20.23)
In Eqs (20.21-20.23), V , S, and M are the average volume, surface area, and integral of mean curvature, over the realizations of the random set A0 . Depending on the choice of θ(u) (i.e. of the grain size distribution of the model), various distributions can be obtained. When μ(B) → ∞, we recover asymptotically the Weibull distribution for θ(u) = θu−m (with m > 4 for grains, m > 3 for fibres, and m > 2 for strata), and the Pareto distribution for θ(u) = θ/u. This type of model is used for the fracture of a silicon nitride ceramic [57],[58],[59]. In this material it is shown from fractographic analysis that the brittle fracture occurs on the most severe defects (inclusions and porosities with a size ranging between 20μm and 70μm). The size distribution of defects is estimated from image analysis on polished sections, using an optical confocal microscope. Using a spherical shape assumption, it is possible to estimate the distribution of diameters by means of a standard stereological reconstruction procedure [641]. The experimental distributions are fitted to the following three-parameters model, with a power law tail for the most severe defects: µ ¶b+1 b c f (d) = (20.24) c d+a The assumption of a Poisson distribution of defects is validated from the study of the statistics of the number of defects per field of measurement [59]. A procedure of identification of the model was proposed. It combines image analysis data and the strength data obtained for various mechanical tests (4 points bending, biaxial bending, and tensile tests), using the stress distribution induced from the defects size distribution
20.3 Brittle fracture and weakest link
789
given by Eq. (20.24). For simplification, the size of the defects is neglected, as compared to the size of specimens, so that they are replaced by points. A good fit could be obtained, when accounting for the differences of microstructure observed between the tensile test specimens and the others. Fracture statistics of a cracked elastic-plastic specimen To get more simple notations in this part, consider Boolean varieties for i) grains, ii) strata and iii) fibres, using a primary random function Z 0 . As usual, AZ 0 (z) is the set of points x where Z 0 (x) ≥ z. We assume that the sets AZ 0 (z) are convex. We can apply Eq. (20.4), with the stress σ = λσ y < σ m , and with the domain B restricted to the cylinder generated by the plastic region defined above. For a cylinder with section B and height L, V = LA(B) S = L(B)L + 2A(B) M = π(L + 12 L(B))
(20.25)
L(B) and A(B) being the perimeter and the area of the section of the plastic region. From Eq. (20.3), we get (stating b = 1 − G(σ m − λσ y ) and σ m − λσ y = σ 0 ) P {σ R ≥ σ} = ! " õ ¶ µ ¶2 4 1 σ σ a2 bA(B) + aL(B)M (AZ 0 (σ0 )) i) exp −θ L σy 4π σ y µ ¶4 µ ¶2 1 σ σ 1 a2 A(B)M (AZ 0 (σ 0 )) + aL(B)S(AZ 0 (σ 0 )) 2π σ y 8 ¤¤σ y +V (AZ 0 (σ 0 )) " " ## µ ¶2 1 σ ii) exp −θ bπ(L + aL(B)) + M (AZ 0 (σ 0 )) 2 σy " " õ ¶ ! µ ¶4 2 σ σ M (AZ 0 (σ0 )) π iii) exp −θ abL(B) + a2 bA(B) L +2 2 σy 2 σy ## µ ¶2 M (AZ 0 (σ0 )) σ + aL(B) + S(AZ 0 (σ 0 )) σy 4 (20.26) From Eq. (20.26), it is clear that a cracked specimen of an elastic-plastic material nearly follows a Weibull distribution with the power 4, at least for large specimens. This limits the influence of the distribution of the defects critical stress on the size effect, as seen now. "
Size effects The study of size effects is very important from a practical point of view. In the present context, it describes the change of statistical properties of the strength when increasing the size of samples. This has many consequences
790
20. Probabilistic Models for Fracture Statistics
in relating data measured on the laboratory scale to the large scale behavior of parts or even of buildings... From an experimental point of view, it is not difficult to examine size effects by considering the change of the mean or of the median strength with the size of specimens. For the weakest link models, the size effect is the decrease of the median strength with the volume of the specimen, since the probability to observe a critical defect increases with size. Its analytical shape depends generally on the choice of the model and on the statistical properties of the defects (size, shape, critical stress). Now some results derived in the case of the Boolean random varieties are given [273], [285]. We will now compare the effect of a uniform stress field, or of a nonuniform stress field developed in a cracked specimen on the size effects for the Boolean varieties. To give explicit results, we will consider particular primary functions, where Z 0 (x) admits a set of minima which is a random variable Z 0 < σ m with probability law G(z): G(z) = P {min{Z 0 (x); x ∈ A00 } < z}
(20.27)
Using the same Z 0 (x) for the three kinds of Boolean varieties, with the distribution function G satisfying: µ ¶α σ (20.28) G(σ) = σ < σm α > 0 σm the median value σM of the macroscopic convex samples under a uniform tension (or any quantile of this distribution) changes with their volume V in the following way : i) σ M = KV −1/α ii) σ M = KV −1/3α iii) σ M = KV −2/3α
(20.29)
The median strength always decreases with the size of the sample; the size effect increases with the type of structure in the following order: Boolean strata, Boolean fibres, Boolean functions. A stratified medium should be less sensitive to size effects in fracture, when the stress field is homogeneous (and whatever the distribution G(z) in the model). The size effect of the cracked specimen can be studied from the basic Eq. (20.26) as a function of the length L. It is clear that the probability of fracture at the level σ will decrease with L (as the probability to encounter a weak zone will increase). For large specimens, we have the following asymptotic results on the size effect (→ ∞): i) if the size of the sets AZ 0 (σ0 ) can be neglected as compared to the plastic region B, from Eq. (20.26 i)), 1
1
σM = KL− 4 = KV − 12
(20.30)
20.3 Brittle fracture and weakest link
791
for a cube of side L. ii) For Boolean strata, it is easier to express the size effect as a change of the average fracture stress σ with L. For any microstructure, σ = K exp −cL = K exp −cV
1 3
(20.31)
for a cube of side L with c = θπ(1 − G(σ m − λσ y )). iii) with the same conditions as in case i) (we can neglect M (AZ 0 (σ 0 )) as compared to L(B)), the following size effect for the Boolean fibres is expected: 1
1
σ M = KL− 2 = KV − 6
(20.32)
for a cube of size L. Approximations involved in Eqs (20.30- 20.32) are valid for Boolean structures with zero range. They show that the asymptotic size effects do not depend on the details of the distribution function G(σ m −σ) (as was the case for a uniform stress field in the specimen). In particular the size effect of case i) recovers the results of Beremin [60]: large cracked elastic plastic materials follow a Weibull distribution with exponent m = 4. The sensitivity to size effects (and namely to the length of the crack front) depends on the type of microstructure used to describe the distribution of defects; it increases in the following order: Boolean functions, Boolean fibres, Boolean strata. We therefore get an opposite conclusion to the homogeneous stress field case: a stratified medium should be more sensitive to the size effect in fracture for a cracked specimen. This results from a higher probability for the crack front to hit a low strength domain. It is possible to draw the following practical conclusions from the comparison between the uniform stress field and the cracked specimen, in order to minimize the size effect on the average strength of the material: • for parts without cracks, under a homogeneous stress field, a stratified medium should be less sensitive to the size of the part; • when cracks may be present in a specimen, a Boolean microstructure (i.e. a structure with finite range, where the domains with lowest strength have a finite size) should be less sensitive to the size of the part; • to minimize the size effect for the two situations (cracked or uncracked sample), a good compromise would be to use a material with a fibrous distribution of lower strength domains.
20.3.4 Randomization of Boolean varieties In some cases it may be interesting to consider a random intensity θ changing from one specimen to the other, to account for a large heterogeneity of the defects distribution. As in chapters 5 and 6), this can reproduce large scale variations of a regionalized intensity θ.
792
20. Probabilistic Models for Fracture Statistics
Eq. (20.7) can be written P {non fracture of B} = exp(−
Z
Φ(σ(x))dx) Z Φ1 (σ(x))dx) = exp(−θμ(B)
(20.33)
B
B
with Φ1 (σ(x)) = Φ(σ(x))/(θμ(B)) If θ is a random variable (changing from one specimen to the other), with the Laplace transform ϕ(λ), as proposed in [657] for fibre composites with random diameters. By mathematical expectation of Eq. (20.33), we obtain: Z P {non fracture of B} = ϕ(μ(B) Φ1 (σ(x))dx) (20.34) B
For a stable distribution, with index α < 1 [177] ϕ(λ) = exp(−aλα ) and µ µ ¶α ¶ Z P {non fracture of B} = exp −a μ(B) Φ1 (σ(x))dx B
For example, in the case of a Weibull model with a homogeneous stress field: Φ1 (σ) = θ(σ − σ 0 )m α
P {non fracture of B} = exp (−a (μ(B)(σ − σ 0 )m ) )
• for a fixed size, we observe a Weibull distribution with the parameter m0 = mα < m, resulting in a larger dispersion; • a weaker size effect (in V −α/m ) than for the conventional model is expected.
20.3.5 Weakest link model and iterated Boolean Varieties In what follows the weakest link model is applied to the case of point processes generated by iteration of Boolean Varieties [318], as introduced in chapter 5. These special cases of Cox processes allow for clustering of defects on Poisson varieties. A comparison is made with the standard Poisson-based model and between the various models, when using the same function Φ(σ) for point defects.
20.3 Brittle fracture and weakest link
793
Fracture statistics for Poisson point defects on Poisson lines in R2 As seen in chapter 5, a two steps point process can be used to locate random defects: 1. Poisson lines in R2 (isotropic case), with intensity θ1 2. On each Poisson plane, a 1D Poisson point process of point defects acting in fracture statistics, with intensity θ replaced by Φ(σ) in Eq. (5.41) (chapter 5). Using the Laplace transform ϕL (λ, K) of the random intercept length L(K), P {σ R ≥ σ}L = exp (−θ1 L(K)(1 − ϕL (Φ(σ), K)))
(20.35)
When K is the disc with radius r, P {σ R ≥ σ}L = Q(r, σ) = exp [2πrθ1 (ϕL (Φ(σ), r) − 1)]
(20.36)
Comparison of Fracture statistics for Poisson points and for points on lines in R2 In the plane, the average number of Poisson points contained in the disc of radius r is E{NP (r)} = πr2 θ2 The average number of Poisson lines hit by the disc is 2πrθ1 . Therefore, the average number of points of the two-step process on lines is 2πrθ1 θE{L}, E{L} being the average chord of the disc. We have −πK 0 (0) = 2πr and r then −K 0 (0) = 2r, so that E{L} = πr2 2r = π 2 . The average number of points on lines is given by : E{NP (r)} = 2πrθ1 θπ
r = π 2 r2 θ1 θ 2
To compare the two fracture statistics, consider the same average number of defects in the disc, so that θ2 = πθ1 θ We have log (P {σ R ≥ σ}P ) − log (P {σ R ≥ σ}L ) µ ¶ πθ1 θ 2 = 2πrθ1 1 − ϕL (θ, r) − πr 2πrθ1 ³ ´ π = 2πrθ1 1 − ϕL (θ, r) − θ r 2
Using the parameter α = 2θr,
794
20. Probabilistic Models for Fracture Statistics
³ π π ´ 2πrθ1 1 − (StruveL(−1, α) − BesselI(1, α)) − α 2 4
From numerical calculation, it turns out that this expression remains negative for any α and then (P {σ R ≥ σ}P ) < P {σR ≥ σ}L This inequality is satisfied for any intensity Φ(σ). In 2D, it is easier to break a specimen with Poisson point defects than with point defects on Poisson lines for a given average number of critical defects. Fracture statistics for Poisson point defects on Poisson planes in R3 As earlier, locate point defects according to a two steps point process: 1. Poisson planes in R3 (isotropic case), with intensity θ2 2. On each Poisson plane,a 2D Poisson point process of point defects, with intensity θ replaced by Φ(σ) in Eq. (5.43). Considering the Poisson tessellation generated by Poisson planes, this model figures out point defects located on grain boundaries, generating intergranular fracture. We get P {σ R ≥ σ}π = exp [−θ2 M (K)(1 − ψ A (Φ(σ), K ∩ π))]
(20.37)
with: • M (K): integral of mean curvature of K • A(K ∩ π): area of sections of K by a random plane π, with Laplace transform ψ A (λ, K ∩ π) In the case of a spherical specimen with radius r, P {σ R ≥ σ}π = exp [−4πrθ2 (1 − ψ(πΦ(σ), r))] with ψ(λ, r) =
exp(−λr2 )
R r √λ 0
√ r λ
(20.38)
exp(y 2 )dy
Fracture statistics for Poisson point defects on Poisson lines in R3 A model of long fibres network with point defects is obtained from Poisson lines, where we replace θ by Φ(σ) in Eq. (5.45): h π i P {σ R ≥ σ}D = exp − θ1 S(K)(1 − ϕL (Φ(σ), K)) (20.39) 4 In the case of a spherical specimen with radius r,
20.3 Brittle fracture and weakest link
P {σ R ≥ σ}D = ∙ exp −π2 θ1 r2 (1 −
2 (2rΦ(σ))2
795
(20.40) ¸ [1 − (1 + 2rΦ(σ)) exp(−2rΦ(σ))]
Comparison of Fracture statistics for Poisson points and for points on planes Consider the fracture of a sphere of sphere of radius r containing a random number of points NP (r) with a given average. For the standard Poisson point process, E{NP (r)} =
4 3 πr θ3 3
For Poisson points on Poisson planes, E{NP (r)} =
8 2 3 π r θ2 θ 3
For a fixed average number of defects in the sphere of radius r, θ3 = 2πθ2 θ Using the same intensity Φ(σ) = θ for the two processes, log(P {σ R ≥ σ}P ) − log(P {σ R ≥ σ}π ) µ ¶ θr2 = 4πrθ2 1 − ψ(θπ, r) − 2 and P {σ R ≥ σ}π < P {σ R ≥ σ}P for r2 Φ(σ) < 1.8 Given the same statistics of defects Φ(σ), for low applied stresses, or at a small scale, the "intergranular" fracture probability is higher than the standard probability of fracture. For high applied stresses the reverse is true, and the material is less sensitive to "intergranular" fracture. The two probability curves cross for r2 Φ(σ) ' 1.8. Comparison of Fracture statistics for Poisson points and for points on lines Consider again the fracture of a sphere of sphere of radius r containing a random number of points NP (r) with a given average. For the standard Poisson point process, 4 E{NP (r)} = πr3 θ3 3
796
20. Probabilistic Models for Fracture Statistics
For Poisson points on Poisson lines, E{NP (r)} =
4 2 3 π r θ1 θ 3
Given the average number of defects in the sphere of radius r, θ3 = πθ1 θ Using the same intensity Φ(σ) = θ for the two processes, and the auxiliary variable α = 2rθ, log(P {σ R ≥ σ}D ) − log(P {σ R ≥ σ}P ) µ ¶ 2 2 2 2 = π θ1 r (1 − (1 + α) exp(−α)) − 1 + α α2 3 and P {σ R ≥ σ}D < P {σ R ≥ σ}P Therefore, given the same statistics of defects Φ(σ), the "fibre" fracture probability is higher than the standard probability of fracture. The material is more sensitive to point defects on fibres. Comparison of Fracture statistics for Poisson points on planes and for points on lines For a given average number of defects in the sphere of radius r, θ2 θπ =
1 θ1 θD 2
Taking 2rθD = πr2 θπ = α, we get π 2 θ1 r2 = 4πrθ2 . Using the same intensity Φ(σ) = θ = θπ = θD for the two processes, log(P {σR ≥ σ}π ) − log(P {σ R ≥ σ}D ) ¶ µ 2 2 2 = π θ1 r ψ(θπ, r) − 2 (1 − (1 + α) exp(−α)) > 0 α and therefore P {σ R ≥ σ}π > P {σ R ≥ σ}D for any distribution Φ(σ) of defects and it is easier to break a specimen with defects on fibres than with defects on planes. Fracture statistics for defects obtained in the three steps iteration Consider now a model of a long fibres in random planes, with point defects located on the fibres, where θ is replaced by Φ(σ) in Eq. (5.47).
20.3 Brittle fracture and weakest link
797
log(P {σ R ≥ σ}3 iterations ) = θ2 M (K) (Eπ {exp [θ1 L(K ∩ π) (ϕL (Φ(σ), K ∩ π) − 1)]} − 1) For fracture statistics of the sphere with radius r, log(P {σ R ≥ σ}3 iterations ) = µZ r ¶ 4πθ2 r exp [2πθ1 u (ϕL (Φ(σ), u) − 1)] f (u, r)du − 1 0
Comparison of Fracture statistics for Poisson points and for the three steps iteration Study now the fracture statistics of a sphere of radius r containing a random number of points NP (r) with a given average. For the standard Poisson point process, 4 E{NP (r)} = πr3 θ3 3 For Poisson points on Poisson lines on Poisson planes, E{NP (r)} =
4 3 πr (θ2 θ1 θ2π 2 ) 3
Given an average number of defects in the sphere of radius r, θ3 = 2π2 θ2 θ1 θ To compare fracture statistics of Poisson points and of the three iterations case, we use the ratio 4 θ3 r3 4 2π 2 θ2 θ1 θ 2 2 2 r = π θ1 θr2 = 3 4θ2 r 3 4θ2 3 With auxiliary variables 2θr = α and θ1 r = β, we have to compare 1 2 1 2 3 π θ 1 αr = 3 π αβ to Z r 1− exp [−2πθ1 u (1 − ϕL (ϕL (θ, u))] f (u, r)dr 0
Using
u r
= y and du = rdy, 1− = 1−
Z
1
0
Z
0
1
y exp [−2πθ1 ry (1 − ϕL (θ, ry))] p dy 1 − y2 y exp [−2πβy (1 − ϕL (θ, ry))] p dy 1 − y2
798
20. Probabilistic Models for Fracture Statistics
A comparison is made by numerical calculation of the integral over α, for a given β. For β = 0.01, 0.1, 1 and 10, P {σ R ≥ σ}3 iterations > P {σR ≥ σ}P Comparison of Fracture statistics for Poisson points on Poisson planes and for the three steps iteration Fixing the average number of points in the sphere with radius r, for Poisson points on Poisson planes, E{Nπ (r)} =
8 2 3 π r θ2π θπ 3
and for 3 iterations, E{NP (r)}3 iterations =
4 3 πr (θ2 θ1 θ2π2 ) 3
To keep the same average values, fix 2θ2π θπ = 2πθ2 θ1 θ Taking θπ = θ to get the same statistics over points, and θ2 identical for the two models, in order to keep the same scale for the Poisson polyhedra, we get πθ1 = 1 and θ1 = 1/π. With auxiliary variables 2θr = α and θ1 r = β, we have to compare 1 − ψ(θπ, r) and 1−
Z
1 0
y exp [−2πβy (1 − ϕL (θ, ry))] p dy 1 − y2
For β = 0.01, 0.1, 0.5, 0.75, numerical calculations give
P {σR ≥ σ}3 iterations > P {σ R ≥ σ}π For β = 1, P {σ R ≥ σ}3 iterations < P {σR ≥ σ}π when α < 1.99 P {σ R ≥ σ}3 iterations > P {σR ≥ σ}π when α > 1.99 For β = 2, 10, P {σR ≥ σ}3 iterations < P {σ R ≥ σ}π Conclusion The models point processes studied in this section were designed to simulate some specific clustering of points, namely on random lines in R2 and R3 and on random planes in R3 . A possible application is to model point
20.3 Brittle fracture and weakest link
799
defects in materials with some degree of alignment. The derived general theoretical results are useful to compare geometrical effects on the sensitivity of materials to fracture.
20.3.6 The Dead Leaves varieties and the weakest link model To model the random function σ c (x) we will use homogeneous DLRF models in R3 (chapter 11), where the primary random functions Z 0 (x) satisfy the conditions of Theorem (11.12): AZ 0 (σ) ∩ A00 = ∅ almost surely, and ∧x Z 0 (x) = σm . This excludes mosaic microstructures, but simulates media where the critical stress reaches a minimal value σm on some grain boundaries. With this assumption, and for a model with t → ∞, the probability of fracture (20.1) becomes when σ > σ m : P {no fracture of B} = V (HZ 0 (σ − σ m )) i) Dead Leaves functions ˇ V (A00 ⊕ B) Z
μ1 (HZ 0 (ω) (σ(ω)−σ m ))dω
ii) Dead Leaves strata
iii) Dead Leaves fibres
Z
Z
ˇ μ1 (A00 (ω)⊕B(ω))dω
(20.41)
μ2 (HZ 0 (ω) (σ(ω)−σ m ))dω
Z
ˇ μ2 (A00 (ω)⊕B(ω))dω
In Eqs (20.41), μ1 (Z(ω)) (resp. μ2 (Z(ω))) represents the length (resp. the area) of the random function Z(x) after orthogonal projection on the line D (resp. the plane π) orthogonal to the direction ω. When applying a uniform stress field σ > σ m to a convex specimen B, and when A00 is convex, Eqs (20.41) become: P {no fracture of B} = i) Dead Leaves functions ii) Dead Leaves strata
iii) Dead Leaves fibres
Z
Z
P {σ R ≥ σ} = ˇ V (AZ 0 (σ − σm ) ª B) 0 ˇ V (A ⊕ B) 0
ˇ μ1 (AZ 0 (ω) (σ − σ m ) ª B(ω))dω M (A00 )
(20.42)
+ M (B)
ˇ μ2 (AZ 0 (ω) (σ − σ m ) ª B(ω))dω ˇ S(A00 ⊕ B)
With this class of models, the following size effect are obtained: when ˇ = ∅), the fracture A00 is bounded, for large specimens (such as A00 ª B
800
20. Probabilistic Models for Fracture Statistics
strength is the constant σ m , so that fracture becomes deterministic. Therefore, size effects are restricted to small samples. As for the Boolean varieties, there is a shape effect: for small size specimens, the size effect on the average fracture stress decreases in the order: Dead Leaves, Dead Leaves fibres, Dead Leaves strata. But here, the underlying microstructure is a strengthening of a matrix with a constant strength σ m , while it was a population of defects in the previous case. An interesting example is built with Poisson polyhedra (with intensity θ) for A00 : we consider a primary function Z 0 (x) satisfying: AZ 0 (σ) = A00 ª Ba (σ/Z) where Ba (σ/Z) is the sphere with radius (σ/Z) and Z is a realization of a random variable, independent of A00 and with finite moments E[1/Z] and σ = E[Z]. When Z is given, we have: ˇ = V (A00 ª Ba ((σ − σ m )/Z) ª B) ˇ V (AZ 0 (σ − σ m ) ª B) 0 = V (A0 ) exp(−θM (B)) exp(−θ4π(σ − σ m )/Z) For Poisson polyhedra, we have ([448], and chapter 7): V (A00 ) =
6 π 4 θ3
S(A00 ) =
24 3 0 2 M (A0 ) = θ 3 π θ
In the case of a uniform stress field σ applied to the convex specimen B, Eq. (20.42) becomes, if ϕ(λ) is the Laplace transform of the random variable 1/Z : P {σ R ≥ σ} =
V (A00 ) exp(−θM (B))ϕ(4πθ(σ − σ m )) ˇ V (A00 ⊕ B)
(20.43)
When B is a cubic sample with side L, Eq. (20.43) becomes: P {σ R ≥ σ} =
exp(−3πθL)ϕ(4πθ(σ − σ m )) π 4 θ3 3 3π3 θ2 2 L + L + 3πθL + 1 6 4
(20.44)
From Eq. (20.44) can be deduced the average fracture stress σR : σR − σm =
(σ m + σ) exp(−3πθL) π θ 3 3π 3 θ2 2 L + L + 3πθL + 1 6 4 4 3
(20.45)
For large sizes L, the size effect is: σR − σm = K
1 exp(−3πθL) = KV −1 exp(−3πθV 3 ) 3 L
(20.46)
20.3 Brittle fracture and weakest link
801
This size effect is stronger than for the Boolean strata in a cracked specimen, as seen from Eq. (20.31). For a stress field induced at the crack tip of an elastic-plastic material, we can use Eq. (20.43) with σ = λσ y and B a cylinder of axis L generated by the plastic zone. We have: Ã ! µ ¶2 1 σ M (B) = π L + L(B)a 2 σy µ ¶2 µ ¶4 σ σ S(B) = L L(B)a + 2 a2 A(B) σy σy µ ¶4 σ a2 A(B) V (B) = L σy Therefore, Eq. (20.43) becomes: P {σ R ≥ σ}
¶2
(20.47)
σ L(B)a)ϕ(4πθ(σ − σ m )) σy 1 V (B) + 4π (S(B)M (A00 ) + M (B)S(A00 )) + V (A00 )
V (A00 ) exp(−θπL) exp(−θL =
µ
For large L, the following size effect results: σR − σm = K
1 1 exp(−πθL) = KV − 3 exp(−πθV 3 ) L
(20.48)
The size effect of the cracked specimen is lower than the size effect obtained for the uniform stress field (Eq. 20.46). However it is still more important for the Dead Leaves structure than for the Boolean strata.
20.3.7 Competition between fracture mechanisms In practice, the fracture process of materials is often very complex, and may result from a competition or a cooperation between various mechanisms. It is therefore of importance to work out models based on a multi-criterion approach, as illustrated now. This generalization offers no theoretical difficulty. It only implies an extension of the number of parameters involved in the model, which requires more data for their estimation than in the case of more simple models. Generalization of the weakest link model The fracture of a material may involve a competition between various mechanisms (for instance between cleavage and intergranular fractures in steels [553]. This competition may result from different stress criteria (such as
802
20. Probabilistic Models for Fracture Statistics
tensile, versus shear), or from different populations of defects. In the first instance, we must consider multivariate fields σ i (x) and σ ic (x) (i = 1, 2, ..., n for n mechanisms). In that case, the probability law given in Eq. (20.1) becomes [285]: P {no fracture of By } ª © = P ∧x∈By (σ 1c (x) − σ 1 (x − y)) ≥ 0, ..., ∧x∈By (σ nc (x) − σ n (x − y)) ≥ 0 (20.49) Note that in general the random functions involved in Eq. (20.49) are not independent. Appropriate multivariate random field models are available for such situations [274], as introduced in previous chapters. When different populations of defects are present in the material (corresponding either to the same fracture stress criterion, or to different fracture stress criteria as below), we may encounter the following situations: (i) the spatial distribution of defects is so heterogeneous that a single population is present in each specimen (defect k with probability pk ), (ii) the spatial distribution of defects is very homogeneous, two separate populations being either correlated (see subsection 20.3.7) or independent (the defect k occurring with probability pk ), (iii) the specimen V is made of disconnected subdomains Vi ; each Vi contains a single class of defects i. This is developed in subsection 20.3.8. For small volumes V , (iii) is equivalent to (i). Considering for instance the weakest link assumption in a homogeneous stress field σ, where defects are distributed according to a Poisson point process (which is a particular Boolean random function), we have: Tk (σ) = P {no failure of By by defects k} = exp (−V (B)Φk (σ)) (20.50) For situation (i), we get: T (σ) = P {no failure of By } =
k=n X
pk Tk (σ) =
k=1
k=n X
pk exp (−V (B)Φk (σ))
k=1
Xk=n
(20.51)
For a small volume V , T (σ) ' 1 − V (B) pk Φk (σ). In case (ii) and k=1 for independent defects, Ã ! k=n k=n Y X T (σ) = Tk (σ) = exp −V (B) pk Φk (σ) (20.52) k=1
k=1
For small values (V → 0), situations (i), (ii), (iii) become equivalent. In case (iii),
20.3 Brittle fracture and weakest link
Ã
T (σ) = exp −V (B)
k=n X k=1
Vk Φk (σ) V (B)
!
803
(20.53)
For large volumes V , Eq. (20.53) is equivalent to Eq. (20.52) (situation (ii)), with pk = Vk /V . Specific models of this type can be developed on the same basis for applications, as seen now for polycrystals.
Example of application to a polycrystal modelled by a Voronoi random mosaic As an illustration, we now consider an example of a model introduced by D. Jeulin [275] for the case of polycrystal materials, such as steels or ceramics. Results obtained for a single fracture criterion, and for a competition between cleavage and intergranular fracture are presented now. Approximate results, valid for large specimens, are given. - A mosaic model is obtained from a random tessellation of space. Each class of the tessellation may represent grains of a polycrystal, while its boundaries can model grain boundaries. For cleavage fracture, one may assume that the fracture of any grain is obtained for a random critical stress σ c following a probability law F (σ) = P {σ c < σ}. For two grains, independent realizations of σ c are considered. In this model, the weakest link assumption is recovered for each crystal (after initiation in a grain, a crack propagates and stops on the grain boundaries). Call GB (s) and G(s) the generating functions of the random number of grains contained in the sample B, N (B), and of the number of grains in B, N (σ), where σ c < σ (for simplification, we assume a homogeneous stress field σ(x) = σ in a stationary random tessellation); we have: GB (s) =
n=∞ X n=0
P {N (B) = n}sn
G(s) is obtained in two steps : when N (B) = n, N (σ) is a binomial random variable with parameter p = F (σ) and with generating function (ps + 1 − p)n . Therefore G(s) = GB (ps + 1 − p), from which pn can be calculated. - The probability p0 , required for the application of the weakest link model is obtained by p0 = GB (1 − p). As an example, let us consider the Voronoi mosaic built from the zones of influence of a Poisson point process in R3 (with intensity θ) (chapter 7). Approximately N (B) = N +1, where N follows a Poisson distribution with parameter θ0 = θV (B). For this model: GB (s) = s exp θ0 (s − 1) Therefore G(s) = (ps + 1 − p) exp θ0 p(s − 1)
804
20. Probabilistic Models for Fracture Statistics
We have p0 = (1 − p) exp −θ0 p, and for the weakest link assumption, P {σ R ≥ σ} = (1 − F (σ)) exp(−θμn (B)F (σ))
(20.54)
The scaling laws in Eq. (20.54) depend on the distribution F (σ). For a macroscopic crack in an elastic plastic material (σ ' λσ y in the plastic zone), with m0 = F (λσ y ), ³ ´ 4 (20.55) P {σ R ≥ σ} = (1 − m0 ) exp −θm0 La2 A(B) (σ/σy ) )
For a specimen containing a crack with length 2a the random strength follows a Weibull distribution with parameter m = 4, independently on the distribution F (σ), as already obtained in the case of the Boolean varieties. - A competition between two fracture mechanisms, namely cleavage and intergranular fracture (as can occur in steels [425], [553]), can be easily introduced from the mosaic model. A further approach, based on crack fronts and on percolation effects, is given in section 21.2, in chapter 21 [322]. Use a distribution Fg (σ) for the fracture of grains by cleavage. In addition, a random fracture stress with law Fi (σ) is independently affected to each interface. The number of grains Ng and of interfaces Ni in the sample B are correlated, since they satisfy the Euler equation: in R3 , 1 = Ng − Ni + Ne − Ns , where e and s are edges and summits of the tessellation. It is known that for a tessellation in equilibrium in R3 (i.e. minimizing the interfacial energy) N i /N g ' 7 [429]. For calculation, the bivariate generating function GB (s1 , s2 ) of Ni and Ng is needed, to estimate the distribution of the number of critical microstructural elements (grains + interfaces) where σc < σ, with generating function G(s)) . In the case of a homogeneous stress field, with pg = Fg (σ) and pi = Fi (σ), for Ng = ng and Ni = ni fixed, the numbers of critical grains and interfaces follow binomial distributions with parameters pg and pi , and with generating functions (pg s + (1 − pg ))ng and (pi s + (1 − pi ))ni . The sum of these two variables admits as a generating function the product of the two, and therefore G(s) = GB (pg s + 1 − pg , pi s + 1 − pi ). This can be simplified if Ni = αNg (α = 7 on average in R3 ): α
G(s) = GB ((pi s + 1 − pi ) (pg s + 1 − pg ))
(20.56)
In Eq. (20.56), GB (s) is the generating function of the number of grains hit by the sample B. For a Voronoi mosaic, approximately: G(s) (20.57) α α 0 = (pi s + 1 − pi ) (pg s + 1 − pg ) exp θ ((pi s + 1 − pi ) (pg s + 1 − pg ) − 1) The weakest link model is obtained from p0 = G(0):
20.3 Brittle fracture and weakest link
P {σ R ≥ σ}/(1 − Fi (σ))α (1 − Fg (σ)) = exp(−θμn (B)((1 − Fi (σ))α (1 − Fg (σ)) − 1))
805
(20.58)
In the case of a macroscopic crack in an elastic-plastic material, we use mi0 = Fi (λσ y ) and mg0 = Fg (λσ y ) in Eq. (20.58). From the volume of the plastic zone, we get again a Weibull distribution with parameter m = 4, as in the case of Eq. (20.54). For applications of these models to real materials, the following data are required: GB (s) or GB (s1 , s2 ) (mixed fracture), Fi (σ), Fg (σ), α. In simplified versions (the Voronoi model, or a constant number of grains and interfaces hit by B, that can be estimated by image analysis and stereological techniques), the histograms Fi and Fg may be estimated from experiments involving a single fracture mechanism, applying Eq. (20.54). Then the assumptions of the model can be tested from a comparison between experimental fracture statistics and the theoretical predictions deduced from Eq. (20.58).
20.3.8 Multicriteria and multiscale weakest link models As seen before, in a medium with heterogeneities at different scales, several populations of defects may exist in various sub-volumes Vi inside a domain with volume V . For this type of situation theoretical models based on multi-criteria and multi-scale weakest link models are presented below [304]. They are the source of new fracture statistics behavior in multi-scale distributions of defects, that would be helpful for the explanation of the observed behavior of very heterogeneous materials as studied in [20]. Consider r populations of defects and a decomposition Pi=rof the specimen i=r V in various sub-volumes Vi : V = ∪i=1 Vi , with V = i=1 Vi . Every subvolume Vi contains a sub-class of Poisson defects with intensity Φi (σ). In what follows, no specific assumption on the type of Φi (σ) is made (it should increase monotonously with the loading, and therefore possible mixing of various kinds of point defects are considered, like Weibull models for power laws, with different moduli. The sub-volumes generate a family of random variables summing to V , with a multivariate Laplace transform φ(λ1 , λ2 , ...., λr ): φ(λ1 , λ2 , ...., λr ) = E{exp(−
i=r X
λi Vi )}
i=1
.
Applying the weakest link assumption to each sub-volume, the fracture probability of volume V is given by: • For fixed volumes Vi , in the case of a homogeneous applied stress:
806
20. Probabilistic Models for Fracture Statistics
1 − P {fracture} = exp(−
i=r X
Vi Φi (σ))
(20.59)
i=1
• Taking the mathematical expectation of Eq. (20.59) with respect to Vi , the fracture probability is given by: 1 − P {fracture} = φ(Φ1 (σ), Φ2 (σ), ...., Φr (σ))
(20.60)
For a non homogeneous applied stress field σ(x), Eq. (20.59) becomes, when the Vi are given: 1 − P {fracture} = exp(− noting Vi Φi (σ ieq (Vi )) =
Z
i=r X
Vi Φi (σ ieq (Vi )))
(20.61)
i=1
Φi (σ(x))dx
(20.62)
Vi
Taking the mathematical expectation with respect to the random variables Vi Φi (σ ieq (Vi )), the fracture probability is expressed by means of the Laplace transform of these variables, defined by: φH (λ1 , λ2 , ...., λr ) = E{exp(−
i=r X
λi Vi Φi (σ ieq (Vi )))}
i=1
1 − P {fracture} = φH (1, 1, ...., 1)
(20.63)
For very large samples, compared to the microstructure, modelled by a stationary multi-component random set, with components Ai , every Φi (σieq (Vi )) is replaced by: Z 1 Φi (σ ieq ) = Φi (σ(x))dx, V V after substitution of the average in a random Vi by the overall average in the total volume V . With this approximation, the fracture probability is given by Eq. (20.60), replacing each Φi (σ) by Φi (σ ieq ): 1 − P {fracture} = φ(Φ1 (σ 1eq ), Φ2 (σ 2eq ), ...., Φr (σ req ))
(20.64)
Discrete mosaic model Start with a space tesselation into n sub-volumes Vi with volume v = Vn (this could model a polycrystal with isovolume grains, containing various types of defects). We have r populations of defects with intensities Φk (σ)
20.3 Brittle fracture and weakest link
807
(i = 1, 2, ...r) and probability of occurrence pk . For a nonhomogeneous applied stress field, in what follows σ must be replaced by σieq for each Vi . A discrete mosaic model obtained as follows: to each Vi is independently attributed the population of defects with type k , with the probability pk . By construction, the numbers of cells Nk with type k generate a system of multinomial random variables, with generating function n o n Nr 1 N2 Gr (s1 , s2 , ..., sr ) = E sN s ...s = (p1 s1 + p2 s2 + ... + pr sr ) r 1 2 (20.65) Replacing the si by exp(−λi v) in (20.65), we obtain the multivariate Laplace transform of the random volumes Vi , and therefore: 1 − P {fracture} =
à i=r X
pi exp(−vΦi (σ))
i=1
! Vv
(20.66)
With the approximation (20.64), expression (20.66) becomes:
1 − P {fracture} =
à i=r X
pi exp(−vΦi ((σ ieq )))
i=1
A limiting case is obtained when v → 0: Ã
1 − P {fracture} ' exp −V
i=r X i=1
! Vv
pi Φi (σ)
(20.67)
!
which gives back the result of Eq. (20.52). Consider for example a Voronoi tessellation, built from the points of a Poisson process with intensity θ. The number of cells N in a domain with volume V is a Poisson random variable with parameter θV . The multivariate distribution of cells volumes Vi is not known. Neglecting the fluctuations V of volumes of cells, and noting v = θV = θ1 , ⎧Ã !N ⎫ i=r ⎬ ⎨ X 1 pi exp(− Φi (σ)) 1 − P {fracture} = E ⎭ ⎩ θ i=1
By expectation with respect to N ,
1 − P {fracture} = exp θV
à i=r X
! 1 pi exp(− Φi (σ)) − 1 θ i=1
808
20. Probabilistic Models for Fracture Statistics
Continuous model with two modes of fracture Consider a random set A (and its complementary set Ac ), with p = P {x ∈ A}. Assume that the random set A contains Poisson point defects with intensity Φ1 (σ), and Ac contains Poisson point defects with intensity Φ2 (σ). The probability of fracture can be deduced from the bivariate distribution of the random volumes V1 and V2 (with V1 + V2 = V ) of A and Ac in the domain with volume V . This bivariate distribution is not known in general, but can be deduced from the Beta distribution. Let X be the random variable giving the proportion of A in a sample. The pdf f (x) of the random variable X (0 ≤ X ≤ 1) is assumed to follow a Beta distribution, with parameters α and β, which is exact in the case of a Poisson sequence generated on a line in a 1D process: f (x) =
Γ (α + β) α−1 (1 − x)β−1 x Γ (α)Γ (β)
(20.68)
The average and the variance of this distribution are given by: E(X) = V ar(X) =
α α+β αβ (α + β)2 (α + β + 1)
For a random set A with volume fraction p and with centred covariance W 2 (h), E(X) = p Z Z 1 V ar(X) = 2 W 2 (x − y) dxdy V V V
(20.69)
For large volumes, when V À An (An being the integral range deduced from the integral of the covariance, here assumed to be finite), the variance becomes: An V ar(X) = p(1 − p) (20.70) V Parameters α and β are expressed as a function of p and of V ar(X) or of An for large specimens, that are microstructural properties: α =p α+β αβ (α + and
β)2 (α
+ β + 1)
=
β An α α+β α+β V
20.3 Brittle fracture and weakest link
V − 1) An V β = (1 − p)( − 1) An
α = p(
809
(20.71)
The Laplace transform of the bivariate distribution of variables XV and (1 − X)V is given by: φ(λ1 , λ2 ) = E {exp (−λ1 V X − λ2 V (1 − X))} = exp(−λ2 V )E {exp (−V X(λ1 − λ2 ))} = φ((λ1 − λ2 )V ) exp(−λ2 V )
(20.72)
where φ(λ) is the Laplace transform of the Beta distribution given by Eq. (20.68): k=∞ Γ (α + β) X (−1)k λk Γ (α + k) φ(λ) = (20.73) Γ (α) k! Γ (α + β + k) k=0
and
Γ (α + β) Γ (α + k) (α + k − 1)...(α + 1)a = Γ (α) Γ (α + β + k) (α + β + k − 1)....(α + β) =
(20.74)
(p( AVn − 1) + k − 1)...p( AVn − 1) ( AVn − 1 + k − 1)...( AVn − 1)
The probability of fracture is obtained by replacing expression (20.74) in Eqs (20.72, 20.73). Replacing the term V by V (Aˇ0 ⊕ K) in Eq. 20.74, an estimation of the Choquet capacity of a Cox Boolean random set is obtained from Eq. (6.32), when the random variable V ((Aˇ0 ⊕K)∩A) follows a Beta distribution (see chapter 6). This can occur for sets Aˇ0 ⊕ K that cannot be considered as negligible with respect to A. For very large samples (V → ∞), the term in Eq. (20.74) becomes pk and φ(λ) = exp(−pλ). Then, φ(λ1 , λ2 ) = exp(−λ2 V ) exp −pV (λ1 − λ2 ) = φ(λ1 , λ2 ) = exp −V (pλ1 + (1 − p)λ2 ) and the fracture probability is given by: 1 − P {rupture} = exp −V (pΦ1 (σ) + (1 − p)Φ2 (σ))
(20.75)
For instance if the two functions Φ1 (σ) and Φ2 (σ) are power laws with exponents m1 and m2 , Eq. (20.75) gives back for the limit case (V → ∞) a bimodal Weibull distribution as Eq. (20.13), while at a lower scale a different distribution is obtained.
810
20. Probabilistic Models for Fracture Statistics
Multi-scale weakest link model Previous results can be specialized to a single population of defects Φ(σ) = Φ1 (σ), using Φ2 (σ) = 0 in the previous developments. This corresponds to a multi-scale model, where the defects are located in a random set A. Using the Laplace transform of the Beta distribution φ(λ) and Eq. (20.72), 1 − P {fracture} = φ(Φ(σ)V ) For very large samples (V → ∞), the term in Eq. (20.74) becomes pk and φ(λ) = exp(−pλ). Then 1 − P {fracture} = exp −V (pΦ(σ)). Therefore, very large scale effects of this model are similar to the standard model, but different scale effects are obtained at a very small scale and at intermediary scales. For the Discrete Model of Mosaic, V
1 − P {fracture} = (p exp(−vΦ(σ)) + 1 − p) v For a vanishing p, 1 − P {fracture} ≈ exp(−
V (1 − p exp −vΦ(σ)) v
For the Voronoi mosaic, neglecting fluctuations of volumes of cells, 1 − P {fracture} = exp(−pθV (1 − exp −
Φ(σ) ) θ
As in [275] and in subsection 20.4.2, this approach can be generalized to fracture controlled by a critical density of defects. The fracture probability is then worked out from the probability P {N defects = k} available for the multi-scale models, from its generator function GV (s) = E{sN }: GV (s) = φ((s − 1)Φ(σ)V )
(20.76)
The probability to find k critical defects in a domain with volume V is deduced from Eq. (20.76) by use of the appropriate Laplace transform φ (for instance the Laplace transform of the Beta distribution). Continuous model with r modes of fracture The previous model can be generalized to the case of r modes of fracture, the Beta distribution being replaced by a multivariate Dirichlet distribution with parameters α1 , α2 ,...., αr (it is exact for the multi-component 1D Poisson mosaic) f (x1 , x2 , ..., xr ) =
Γ (α0 ) i=r αi −1 i=r Γ (α ) Πi=1 xi Πi=1 i
20.4 Fracture statistics models with a damage threshold
811
Pi=r Pi=r−1 with α0 = xi . Each random variable Xi i=1 αi and xr = 1 − i=1 follows a Beta distribution with parameters αi and α0 − αi , with average αi (α0 −αi ) E( Xi ) = αα0i and variance V ar(Xi ) = (α . For illustration consider 2 0 ) (α0 +1) a random mosaic built on a random tesselation of space, by affecting independently to each class (or cell) the mode of fracture i with the probability pi . On a large scale for a random mosaic, V ar(Xi ) = pi (1 − pi ) AVn (the Dirichlet distribution implies that the integral range An does not depend on the mode i). For this model, α0 = 1 − AVn and αi = pi (1 − AVn ). The expansion of the multivariate Laplace transform φ(λ1 , λ2 , ...., λr ) depends on the moments μk1 ....kr : μk1 ....kr
n o Γ (Pi=r α ) Γ (αi + ki ) i j=r kj = E Πj=1 Xj = i=ri=1 Π i=r P Πi=1 Γ (αi ) i=1 Γ ( i=r i=1 (αi + ki ))
The Laplace transform of the Dirichlet distribution is obtained from: ! Ã n=∞ X (−1)n X k1 kr λ1 ...λr μk1 ....kr (20.77) φ(λ1 , λ2 , ...., λr ) = n! n=0 k1 +...+kr =n
The fracture probability is given by replacing in Eq. (20.77) the λi by V Φi (σ). In practice, the expansion can be stopped after a finite rank n.
20.4 Fracture statistics models with a damage threshold It is possible to generalize the weakest link criterion in various ways, as shown below [275], [280], [284]. When relaxing the weakest link assumption, critical damage models, based on a critical volume fraction, or on a critical intensity (depending on the nature of the defects), can be proposed.
20.4.1 Fracture statistics models with a critical volume fraction of defects Consider in a bounded specimen B a stress field σ(x) applied to a material with the critical stress field σ c (x). The indicator function 1σc (x) of the domain D is defined as: 1σc (x) = 1 if σ c (x) < σ(x), else 0 We have μn (D) =
Z
B
1σc (x)dx. Assume that B will not fail, as long as
μn (D) < pc , where pc is a parameter of the model. This parameter μn (B)
812
20. Probabilistic Models for Fracture Statistics
indirectly accounts for percolation effects, since when pc increases, the domain D (where the breakage is expected to occur) progressively invades the specimen B. This assumption can be used for non brittle damaging materials (neglecting the redistribution of the stresses during the process, or the growth of microcracks initiated on the sites with low critical stress) or for ductile fracture with cavities growing from inclusions. In the probabilistic approach, σ c (x) is assumed to be a realization of a random function, and therefore 1σc (x) is the indicator function of the random set D. Then, ½ ¾ μn (D) P {no fracture of B} = P (20.78) < pc μn (B) Solving Eq. (20.78) as a function of the probabilistic properties of σ c (x) and of the geometry of B (and of σ(x)) requires finding the solution of a change of support by convolution and to estimate the probabilistic properties of the RV μn (D). Since no general solution of Eq. (20.78) is known, appropriate models or simplifications must be used, as is proposed below. It is easy to calculate the average and the variance of the random variable μn (D), depending on σ c (x). We have Z E[μn (D)] = F (σ(x), x)dx (20.79) B
where F (σ(x), x) = P {σ c (x) < σ(x)} is the univariate distribution of the critical stress σ c (x). Even when σ is constant, the distribution F (σ(x), x) may depend on x in B for a non stationary random function σc (x). The geometrical covariogram of D, Kσ (h), is given by:
Kσ (h) =
Z
E [1σc (x)1σc (x + h)] dx =
B∩Bh
Z
F2 (σ(x), σ(x + h)) dx
B∩Bh
where F2 (σ(x), σ(x + h)) = P {σ c (x) < σ(x), σ c (x + h) < σ(x + h)} is the bivariate distribution of σ c . The variance D2 [μn (D)] is obtained as: D2 [μn (D)] =
Z
Rn
Kσ (h)dh −
∙Z
B
¸2 F (σ(x))dx
(20.80)
From now on, consider large specimens B and random functions σ c (x) for which the Central Limit Theorem can be applied. Asymptotically, μn (D) is assumed to be a Gaussian RV with average and variance given by Eqs. (20.79,20.80). Therefore,
20.4 Fracture statistics models with a damage threshold
1 P {no fracture of B} = √ π
Z
pc −m √ 2D
−∞
¡ ¢ exp −y 2 dy
813
(20.81)
where m = E[μn (D)]/μn (B) and D2 = D2 [μn (D)]/μn (B)2 . In the case of a stationary RF σ c (x), and a uniform stress field σ over B, m = F (σ) and D2 = m(1 − m)A(σ)/μn (B) (20.82) Eq. (20.82) can be applied when A(σ), the integral range of the random set deduced from σ c (x) < σ is different from 0 and remains finite. We have Z F2 (σ, σ, h) − F (σ)2 dh (20.83) A(σ) = F (σ)(1 − F (σ)) Rn with F2 (σ, σ, h) = P {σ c (x) < σ, σ c (x + h) < σ} The following empirical expressions [406] can also be used for the variance D2 , with α < 1 for A(σ) = 0 and α > 1 for A(σ) = ∞:
D2 = m(1 − m)B(σ)/μn (B)α For the general mosaic model introduced above (not necessarily based on a Voronoi tessellation), in Eq. (20.83), A(σ) = A3 , and does not depend on σ. Then, Z Z A3 =
∞
r(h)dh = 4π
R3
h2 r(h)dh
0
for an isotropic medium, with r(h) = K(h)/K(0), K(h) being the geometrical covariogram of a random class A0 of the tessellation (this can be measured by image analysis on polished sections of the material). From the law given by Eq. (20.81), general properties of the fracture probabilities can be easily deduced: • For a given stress field σ(x), P {fracture} decreases with the threshold pc , and P {fracture} = 1/2 for m = pc . • For a uniform stress field σ, the median macroscopic strength σM is obtained from F (σ M ) = pc . There is therefore no size effect for σ M in this model, which can be checked experimentally. The median σ M increases with pc , while the variance D2 [σR ] decreases with V (B). When V (B) → ∞, the fracture strength σ R converges to the constant σR = F −1 (pc ): a deterministic fracture behavior results on a large scale, towards a fracture strength depending both on F (σ) and on pc . For the stress field generated at the crack tip of an elastic-plastic material, and with the simplifications introduced above, the volume V where fracture can occur by cleavage is limited to the plastic zone. We have:
814
20. Probabilistic Models for Fracture Statistics
V = La2 α (σ/σ y )
4
where L is the length of the crack front, a the crack length, and α the 2 area of the section of the plastic zone for a (σ/σ y ) = 1. The convergence towards the normal distribution is insured for a very large plastic zone, which involves A0 ¿ V , where A0 is the integral range (assumed finite) obtained for the threshold σ 0 . This corresponds to large stresses fulfilling: (σ/σ y )4 À
A0 αLa2
When these conditions are satisfied, the part of the probability of fracture corresponding to the larger stresses is obtained as: Z t ¡ ¢ 1 P {σ R ≥ σ} = √ exp −y 2 dy (20.84) π −∞ where
√ (pc − m)a Lα σ t= p 2A0 m0 (1 − m0 ) σ y
and m0 = F (λσ y ). For pc ≥ m0 , Eq. (20.84) is not valid any more. In that case, no fracture of the material can occur for any bounded stress σ. For pc < m0 , the material breaks almost surely when σ tends to infinity.
20.4.2 Critical density of defects The previous models cannot be applied to a material containing defects with a zero measure (points, thin microcracks, etc.). An alternative criterion is a critical number density, θc . For a given applied stress field σ(x), N (σ) is the random number of defects where σ c (x) < σ(x). Assume that the fracture occurs when N (σ) > k, with k = θc μn (B) (k increases with the critical density θc , and with the volume of B). This illustrated by Fig. 20.7. For a Boolean RF, the overlay of defects is allowed, and N (σ) follows a Poisson distribution with parameter θ0 =
Z
T 0
¡ ¢ μn HZt0 (σ − σ m )c θ(dt) = kθ1
• When k < 1 (for specimens B with a small size), the weakest link criterion is recovered, and P {no fracture of B} = exp −θ0 = p0 • When k ≥ 1,
20.4 Fracture statistics models with a damage threshold
815
FIGURE 20.7. Comparison between the weakest link assumption and the critical density of defects: volume v with a single defect breaks, as opposite to volume V where N(σ) < θc μ(B)
P {no fracture of B} = p0 + p1 + ... + pi + ... + pk = 1 − Fk (θ1 ) (20.85) In the case of point defects and a uniform stress field over B, θ0 = θV (B)ϕ(σ) (where ϕ(σ) is an increasing positive function of σ, as earlier) θ and θ1 = ϕ(σ). In that case, (and approximately for large B), θ1 does θc not depend on the geometry of the specimen B. In Eq. (20.85), pi = P {N (σ) = i} =
(θ1 k)i exp −kθ1 i!
When ϕ(σ) is derivable, the distribution Fk (θ1 ) admits a density fk (θ1 ) ³ n ´k 1 for σ such that θ/θc ϕ(σ) = 1 (with e = exp 1). and a mode e (n − 1)! This mode is unique when ϕ(σ) is strictly monotonous. When increasing the size of B such that μn (B) → ∞, the pdf fk (σ) converges toward a Dirac distribution: the fracture strength of the material becomes a constant value θ σ R , such that ϕ(σ R ) = 1. This behavior is very similar to the previous θc model, based on a critical volume function. It is also shown [275] that, unlike the weakest link model, the large specimens are less sensitive to the most severe defects (with low σ c ). This type of approach is followed in the case of two dimensional woven composites [40]: the critical density model is used for the identification of defects on yarns made of a SiC/SiC composite, from mechanical tests, and for the simulation of damage by FE calculations. It appears that the tested specimens (with the two lengths 50mm and 180mm) present a damage threshold during tensile tests. This damage threshold is practically independent on the size, so that a critical damage density criterion was applied to estimate the density of defects from these tests. The density was chosen according to a Weibull model with σ 0 = 0 and m = 2.45 in Eq. (20.8).
816
20. Probabilistic Models for Fracture Statistics
20.5 Fracture statistics model with a crack arrest criterion In this section, microstructural information along the crack path is used to estimate the probability of fracture of random media, as a function of microgeometrical characteristics, namely the spatial distribution of the specific fracture energy γ(x). The approach is limited to crack propagation in two dimensional media. Some specific random media are considered (the Poisson mosaic, and the Boolean mosaic RF). Various loading conditions producing stable or unstable crack propagations, and the case of crack initiation from defects, followed by crack propagation, are studied. The models predict the probability laws of mechanical properties like strength, toughness (for crack initiation and arrest) and microcrack length. Indications on scale effects are provided. These results differ from the weakest link and the critical damage approaches developed above and even an increase of strength with the size of specimens is observed for very slowly decreasing tails of γ(x).
20.5.1 Crack propagation and the Griffith’s criterion for two-dimensional random media The aim of this part is to calculate for some two dimensional random media the probability of fracture by application of the Griffith’s criterion concerning the crack arrest during its propagation. Brittle random media, with a homogeneous constitutive law (namely elastic with Youngs’ modulus E), but with a random fracture energy Γ (x) are considered. As in previous works [278], [283], [284], [116], [117], [397]), it is assumed that a possible fracture path P (s, d) connecting the source s to a destination d must satisfy, for each point x belong the path: 2Γ (x) ≤ G(x)
(20.86)
The energy release rate G(x), which is the elastic energy stored in the loaded specimen, depends on the location of the crack front x, on the loading conditions and also on the overall crack path from s to x. Given Eq.(20.86) a candidate crack path P must satisfy: ∨{2Γ (x) − G(x); x ∈ P } ≤ 0
(20.87)
where ∨ is the supremum value of the expression in brackets. For a given RF Γ (x), the probability of fracture must be calculated from Eq. (20.87). This is a very difficult task in general, since on a microscopic scale, the crack path has a complex shape, depending on the microstructure. As a result, the function G(x) in Eq. (20.87) is a realization of a RF correlated to Γ (x). Simplifications are introduced, and furthermore specific models
20.5 Fracture statistics model with a crack arrest criterion
817
are used to give closed form results, as shown below. The potential crack paths P can be assumed to be realizations of a diffusion stochastic process independent on Γ [116], [117], [397]. This involves crack paths which do not depend locally on the underlying microstructure. This last one is indirectly reflected by two parameters: a crack diffusion coefficient D and a coefficient of smoothing. Following an approach developed elsewhere [266], [278], [279], using a minimal fracture energy assumption, the crack paths P would be the shortest paths (or geodesic paths) on the field Γ (x), that can be determined by image analysis. Crack fronts can be obtained by numerical simulations from a variational approach of the Griffith’s criterion, as shown in chapter 21. Considering G(x) obtained for a straight propagation of the crack, it is easy to calculate the criterion given by Eq. (20.87) on simulations and to estimate a fracture probability. However, in this section, we propose to calculate in a closed form the probability of extension of linear cracks along a segment, according to a crack advance in mode I in an elastic medium with homogeneous elastic properties, namely here a constant Young’s modulus E. The following crack configurations and loading conditions are examined, as shown in Figs 20.8, 20.9, 20.10, 20.11 [283]. Surface crack A surface crack with initial length a0 in a finite medium can propagate along a ligament with length b under two loading conditions (Fig. 20.8 and Fig. 20.9): (i) a uniform stress orthogonal to the crack is applied at infinity; in that case, the energy release rate G(x) increases along the crack path, resulting in an unstable crack propagation for a homogeneous medium. For short cracks (a0 +x < b/2), it can be approximated by the semi-infinite medium expression (up to a factor 1.25): G(a0 + x) = πσ 2 (a0 + x)/E
(20.88)
(ii) a concentrated load (also written σ for convenience) is applied at the mouth of the crack; for this configuration, G(x) decreases while the crack propagates, resulting in a stable crack propagation which may be stopped; with the same approximation as for Eq. (20.88), we have (with k ' 2.15): G(a0 + x) = kσ 2 /((a0 + x)E)
(20.89)
Internal crack An internal crack with length 2a inside an infinite medium is uniformly loaded at infinity by a stress σ; it can undergo an unstable propagation from its two ends (Fig. 20.10), with:
818
20. Probabilistic Models for Fracture Statistics
FIGURE 20.8. Surface crack (initial length a0 ) in a specimen with width (a + b), and variation of the energy release rate G(a), as a function of the crack advance, for unstable crack propagation
FIGURE 20.9. Surface crack (initial length a0 ) in a specimen with width (a + b), and variation of the energy release rate G(a), as a function of the crack advance, for stable crack propagation
FIGURE 20.10. Internal crack (with initial length a0 ) and variation of the energy release rate G(a), as a function of the crack advance. The other branch of the curve is obtained by symmetry [283]
G(a0 + x) = πσ 2 (a0 + x)/E
(20.90)
Ring test and crack arrest In order to study the dynamic fracture of materials and the crack arrest properties, a ring test was developed [160],[248]. For this type of specimen
20.5 Fracture statistics model with a crack arrest criterion
819
FIGURE 20.11. Ring test configuration [160], [248] with an initial crack length a0 : unstable propagation, followed by a stable crack propagation and arrest [283]
(Fig. 20.11), submitted to a diametral load σ, the stress intensity factor follows a parabolic variation along the crack path. In the present approach, the dynamic fracture properties are not accounted for and calculations are made in a static case for a parabolic shape of G(a0 + x) as a function of the crack advance (with initial length a0 ): it starts with an unstable propagation, followed by a stable crack propagation and arrest according to Eq. (20.91), where ∆R is the difference between the two radii of the ring: G(a0 + x) = σ 2 (a0 + x)(k − (a0 + x))/(E(∆R)3/2 ) = h(a0 + x)(k − (a0 + x))
(20.91)
with h = σ 2 /(E(∆R)3/2 . The maximal value of G along the crack path is given by Gmax = hk 2 /4. The factor k is approximately equal to 3/4.
20.5.2 Types of probability distributions obtained from the models For a given geometry and random medium, the probability P (a, b, σ) of straight propagation of a crack from length a to length (a + b) is calculated as a function of a, b, and σ. It can be interpreted in various ways, according to the variable of interest, like the effective toughness Gc , the critical length of defects ac , or the same properties at crack arrest, Ga and la (i) For a finite size specimen of width (a + b), P (a, b, σ) is the probability of fracture after initiation from a, under the considered loading condition. A rigorous approach would require introducing functions G(x) obtained for finite size specimens, as available in the fracture mechanics literature. For simplification, we use here the functions valid for infinite or semi-infinite media given in the previous subsection. Let σ R and Gc be the overall fracture stress and toughness of the cracked specimen, as measured by mechanical tests of fracture from pre-cracked samples, while ac is the critical length of a microcrack. Their probability laws are related
820
20. Probabilistic Models for Fracture Statistics
to P (a, b, σ) by: P (a, b, σ) = P {fracture} = P {σ R < σ} = P {Gc < G(a)} = P {ac < a} (20.92) (ii) For a finite size specimen of width larger than (a + b), 1 − P (a, b, σ) gives the probability of arrest of the crack before reaching the length (a + b). Let Ga and la be the arrest toughness of the medium and the length of the crack at arrest. When G increases with the size of the crack, their probability laws are given by: P (a, b, σ) = P {fracture} = P {Ga ≥ G(a + b)} = P {la ≥ a + b} (20.93) In Eq. (20.93) the symbol ≥ for Ga is replaced by < when G decreases with the crack length. The various probability distributions obtained from Eqs (20.92, 20.93) for a given random function model Γ (x) are related in a coherent way. In practice, they are very useful for testing the model from experimental data obtained at different scales: mechanical data (σR , Gc , σ a , Ga ), as well as crack size distribution observed for given loading conditions (ac , la ). (iii) When changing the size of the specimens, size effects are predicted on the probability distribution of the mechanical properties and on their moments, such as their average value. For large size specimens, asymptotic results can provide different behaviors, as illustrated in the next subsections for specific random function models. (iv) A combination of crack initiation and propagation (or arrest) is modelled in a simple way as follows: consider in the random medium critical defects located according to a random point process with intensity ϕ(σ) (the average number of defects per unit area, with critical stress σ c < σ). For the defects where σc < σ, a microcrack with length a0 is initiated, a0 being a random variable, or more simply a constant material property. For a low intensity of defects, their crack paths during propagation are uncorrelated (with respect to Γ (x)), and the complex calculation of G resulting from a network of microcracks can be avoided; therefore microcrack interactions are neglected. In addition, for large specimens it can be assumed that the probability of propagation of a0 to the length (a0 +b) is independent on the location x of the defect. Let Gσ (s) be the probability generating function of the discrete probability distribution pn for the random number of critical defects N (σ) contained in a rectangular specimen with sides b and d. Given N (σ) = n, it becomes: P {non fracture | N = n} = (1 − P (a0 , b, σ))
n
(20.94)
and therefore: P {non fracture } = Gσ (1 − P (a0 , b, σ))
(20.95)
20.5 Fracture statistics model with a crack arrest criterion
821
When accounting for the probability distribution of the independent locations x of the defects, P (a0 , b, σ) in Eq. (20.95) must be replaced by the expectation E{P (a0 , b, σ)}, obtained as an average of P (a0 , b, σ) over these locations. An important particular case is given by a Poisson point process (for which when N (σ) = n, the defects are independently and uniformly distributed in the material) with intensity ϕ(σ), as used in weakest link models, with generating function: Gσ (s) = exp{bdϕ(σ)(s − 1)} For this model Eq. (20.95) becomes: P {non fracture } = exp{−bdϕ(σ)P (a0 , b, σ)}
(20.96)
Similar expressions are obtained for surface cracks, when restricting the nucleation on defects located on stripes along the edges of the specimen. Illustrations are given in the next two subsections in the case of two models of RF for Γ (x): the Poisson mosaic, and the Boolean random functions.
20.5.3 Probability of fracture and scale effects for the Poisson Mosaic The Poisson mosaic (chapter 8) is a particular model, which might simulate a random polycrystal with local changes of fracture energy. We recall here that it is built in two steps: • a Poisson tessellation with parameter λ delimits the grain boundaries; they are made of Poisson lines in the plane for a two-dimensional medium; • for grains of the tessellation, the fracture energy are independent realizations of a random variable Γ , with the cumulative distribution function F (γ) = P {Γ < γ} ; this is also the distribution function of the random function Γ (x) built by the random mosaic. The probability of fracture of a specimen loaded according to Eqs. (20.8820.91) is derived from the fact that, for the Poisson mosaic, the number of grains hit by a segment of length l follows a Poisson distribution with parameter λl. In the next subsections is handled the isotropic Poisson mosaic. For the anisotropic case, the fracture statistics depend on the orientation of the crack with respect to the microstructure. An extreme situation is the layered Poisson mosaic (with parallel Poisson lines in a single direction), where the same results are obtained by using the parameter λ sin α instead of λ in all formulas, α 6= 0 being the angle between the crack and the orientation of layers. For a crack parallel to the layers, on each realization the observed fracture toughness is the toughness of a homogeneous material with a constant fracture energy given by a realization of the RV Γ , reflect-
822
20. Probabilistic Models for Fracture Statistics
ing a non ergodic situation. The probability distribution of toughness in this degenerate situation is the distribution of homogeneous samples with random fracture energy Γ . Replacing the Poisson mosaic by a STIT mosaic or by a mosaic built on the intersection of independent STIT tessellations (chapter 8) gives exactly the same results as for the Poisson mosaic. An extension of the present model for the case of grain boundary reinforcement is given in subsection 20.9.1.
Surface crack In this subsection, it is assumed that a surface crack with initial length 2a propagates in the random medium until the length 2(a + b). i) Unstable crack propagation For the unstable crack propagation given in Eq. (20.88) (Fig. 20.8), or for any loading where G increases with x, the probability of fracture P (a + b) is given by: " # Z 2(a+b)
P (a + b) = F (G(a)/2) exp −λ
2a
(1 − F (G(u)/2))du
(20.97)
Eq. (20.97) is obtained as follows: consider the evolution of the probability of fracture P (l) for an infinitesimal growth dl; from l and l + dl, the crack stays inside the same Poisson polygon (with probability 1 − λdl) with fracture energy γ, or it enters a new polygon (with probability λdl) with random fracture energy γ satisfying G(l + dl) ≥ 2γ. As a result, P (l)dl) = P (l) [1 − λdl + λdlF (G(l + dl)/2)] To first order in dl, dP (l) = −λP (l)(1 − F (G(l)/2)) and the probability of Eq. (20.97) is given after integration between 2a and 2(a + b) with the initial condition for the end of a crack with length 2a located inside a polygon: P (a + b) = F (G(a)/2) The probability of fracture of the specimen increases (and converges toward 1), when the crack length 2a or the applied stress increases. It also increases for lower λ corresponding to a coarser microstructure: with this model, small grains improve the ability to resist the crack growth, as a result of a higher probability to meet grains with a large fracture energy along the same crack path. For the same reason, the probability of fracture decreases with the length b of the crack path.
20.5 Fracture statistics model with a crack arrest criterion
823
Considering now samples with a similar geometry (with a constant ratio a/b), scale effects depending on the distribution F (γ) are observed. First, if the range of the distribution is finite (F (γ) = 1 for γ > γ c ), the probability of fracture becomes equal to 1 beyond the critical fracture length 2ac given by 2ac = 4γ c E/(πσ2 ). If the distribution F (γ) possesses a tail such as 1 − F (γ) ' γ −α when γ becomes infinite, the scale effects strongly depend on the positive coefficient α: a)
if α = 1, for large specimens the asymptotic probability of fracture becomes independent on their size (there is no scale effect) and P (a + b) = y λc with y = a/(a + b) and c = E/(2πσ 2 )
b)
(20.98)
if α 6= 1, the large scale behavior of the probability of fracture becomes: £ ¤ P (a + b) = exp −λ(2c)α a1−α (1 − y α−1 )/(α − 1) (20.99)
When α < 1, the growth of F (γ) towards 1 is so slow that the crack is stopped with a probability 1 by grains with fracture energy as large as required for the crack arrest condition (P (a+b) converges to 0 for increasing sizes). When α > 1, P (a + b) converges to 1 for increasing sizes, as for a distribution with a finite range. As previously mentioned, from the probability P (a + b), other statistics are derived from the knowledge of F (γ); to illustrate this, we just consider here asymptotic results for a power law tail of the distribution: • Replacing c by its expression, the fracture stress distribution is obtained. Scale effects concerning the change of the median strength σ M with the size of the specimen are the following: √ — if α = 1, σ M ' − log y, and there is no scale effect; ¯1 1−α ¯ — if α 6= 1, σ M ' L 2α ¯1 − y α−1 ¯ 2α , where L is the size of the specimen; therefore σ M increases with L for α < 1 and decreases with L for α > 1. • The statistics for the toughness Gc is obtained from Eqs (20.92,20.98,20.99), and from Gc = a/c: — if α = 1: P {Gc < G} = y (λa/G) The median toughness is proportional to λa. — if α 6= 1: ∙ ¸ λ2α a(1 − y α−1 ) P {Gc < G} = exp − (α − 1)Gα The median toughness is proportional to (λa)1/α .
(20.100)
(20.101)
824
20. Probabilistic Models for Fracture Statistics
• The probability distribution for the critical length ac (for 2a < L) is: — if α = 1: P {ac < a} = (2a/L)λc
(20.102)
— if α 6= 1:
£ ¤ P {ac < a} = exp −λ(2c)α (L/2)1−α (1 − (2a/L)1−α )/(α − 1) (20.103) The median critical length is proportional to the specimen size L.
• The probability distribution for the arrest length la (with la ≥ a) is: — if α = 1: P {la ≥ l} = (a/l)λc
(20.104)
la follows a Pareto distribution, and its median value is proportional to a; — if α 6= 1: £ ¤ P {la ≥ l} = exp −λ(2c)α a1−α (1 − (a/l)α−1 )/(α − 1) (20.105) The median length at arrest is proportional to a for α ≤ 1 and to aα for α > 1.
• The probability distribution for the toughness at arrest Ga is obtained from the expression Ga = (a + b)/c, and from G0 = a/c (with G ≥ G0 ): — if α = 1 P {Ga ≥ G} = (G0 /G)λc
(20.106)
Ga follows a Pareto distribution, with a median value proportional to a; — if α 6= 1 £ ¤ P {Ga ≥ G} = exp −λ(2c)α a1−α (1 − (G0 /G)α−1 )/(α − 1) (20.107) The median value of Ga is proportional to a. ii) Stable crack propagation For the stable crack propagation given in Eq. (20.89), or for any loading where G decreases with x, P (a + b) becomes (start from Eq. (20.97), and revert the crack trajectory: " # Z 2(a+b)
P (a + b) = F (G(a + b)/2) exp −λ
2a
(1 − F (G(u)/2))du
(20.108)
P (a + b) converges towards 0, whatever the distribution F , contrary to the other loading condition. The crack is stopped almost surely for speci-
20.5 Fracture statistics model with a crack arrest criterion
825
mens with increasing sizes, the probability for the crack to hit a grain with fracture energy Γ ≥ G(x) converging to 1. Internal crack For the loading given by Eq. (20.89) the probability of fracture P (a + b) corresponding to the configuration of Fig. (20.10) is given by: " # Z 2(a+b)
P (a + b) = P0 (a) exp −2λ
(1 − F (G(u)/2))du
2a
(20.109)
In Eq. (20.109), P0 (a) is the probability for the initial crack (2a) to start from its both ends. It is given by: P0 (a) = r(2a)F (G(2a)/2) + (1 − r(2a))F (G(2a)/2)2 r(2a) = exp(−2λa)
(20.110)
The first term in Eq. (20.110) expresses that the two ends of the crack are in the same cell of the Poisson tessellation and the second term that they belong to two different cells. Up to P0 (a), the fracture probability given by Eq. (20.109) is very similar to Eq. (20.97). Therefore the above derived statistics and the corresponding scale effects, given for large specimens, still apply for the internal crack problem, provided the factor λ is replaced by 2λ. Parabolic loading For the loading corresponding to Eq. (20.91) and Fig. (20.11), the probability of fracture P (a + b) is given by (Gmin being the smallest value of G along the crack path): " # Z a+b
P (a + b) = F (Gmin ) exp −λ
a
(1 − F (G(u)/2))du
(20.111)
As a particular case, it is interesting to consider as before a distribution F (γ) with a power law tail. Explicit results are easily derived when α = 1, for which Eq. (20.111) becomes, h being given in Eq. (20.91): P (a + b) =
∙
a k − (a + b) a+b k−a
¸2λ/(kh)
(20.112)
For this model, there is no scale effect, as for the previous cases. Using Eq. (20.91) the probability laws of la and of Ga are given by: P (la ≥ l) =
∙
a k−l k−a l
¸2λ/(kh)
(20.113)
826
20. Probabilistic Models for Fracture Statistics
with a ≤ l ≤ k
∙
a k−l P (Ga ≥ G) = k−ak+l
¸2λ/(kh)
(20.114)
with l2 = k 2 − 4G/h and G < Gmax . Crack initiation and propagation
Starting from Eq. (20.96) and using the results given above, the probability of fracture may be derived for various situations. It is interesting to examine surface and internal crack initiation. It turns out that, with the simplifications made to obtain Eq. (20.96), the observed scale effects are the same in the two cases so that the presentation will be limited to surface crack initiation by means of Eqs (20.97-20.99) for a distribution F (γ) with a power law tail. • if α = 1 :
"
∙
a0 P {non fracture} = exp −a0 dϕ(σ) a0 + b
¸λE/(πσ2 ) #
(20.115)
Scale effects are obtained for similar specimens, when b and d are replaced by kb and by kd. — When λE/(πσ 2 ) = 1, there is no scale effect. — When λE/(πσ2 ) > 1, P {non fracture} converges to 1 for large scales. This corresponds to low stresses. — When λE/(πσ2 ) < 1, P {non fracture} converges to 0 for large scales. This corresponds to large stresses. • if α 6= 1: P {non fracture} = exp [−a0 dϕ(σ)P (a0 , b; σ)]
(20.116)
with "
" ∙ ∙ ¸α ¸α−1 ## λ a0 E 1−α P (a0 , b; σ) = exp − a0 1− α − 1 σ2π a0 + b — When α < 1, P {non fracture} converges to 1 for large scales. — When α > 1, P {non fracture} converges to 0 for large scales.
20.5.4 Probability of fracture and scale effects for the Boolean Mosaic The Boolean RF were used earlier in this chapter when applying the weakest link criterion. A different construction is used in the present section.
20.5 Fracture statistics model with a crack arrest criterion
827
A particular class is given by the Boolean mosaic, which is obtained as follows: • a material with a constant fracture energy γ 0 is considered; for simplification in the notations, assume that γ 0 = 0; it is equivalent to examining the case of cracks with lengths larger than 2a0 , that should be propagating in the homogeneous material according to the criteria given by Eqs. (20.86, 20.87). • on every point of a Poisson point process (with intensity θ(u)), is implanted a random function Γ 0 (u). The value Γ (x) of the fracture energy at point x is given by the supremum over all the ”primary” random functions Γ 0 covering x. For the general model, any dependence of Γ 0 on u is allowed. In the present case, a constant fracture energy (γ = u ≥ γ 0 ) inside realizations of a random grain A0 is considered. The measure θ is R such that θ(u)du remains finite.
The main morphological difference with the Poisson mosaic is that, if the previous model is well suited for simulating a polycrystal, the Boolean random function with convex grains A0 is a good simulation of a matrix with a constant fracture energy γ 0 containing reinforcing inclusions with a larger fracture energy, as opposed to defects for the weakest link model. The probability of fracture deduced from the criterion (20.87) is obtained for a given crack path by: P {non fracture} = exp(−μ0 )
(20.117)
In Eq. (20.117), the coefficient depends on the loading conditions through G(x), on the measure θ(γ), and on the random set A0 , that is assumed to be a convex set. This equation is obtained as a particular case of the properties of the Boolean RF. It is developed below for the different loading conditions. As for the Poisson mosaic, it is possible to work out from Eq. (20.117) the probability distributions of various mechanical properties (toughness, length of crack at arrest, and so on). This is not presented here. An extension of this model to the case of a 3D crack front is given in subsection 20.9.2. Surface crack i) Unstable crack propagation In the case of the unstable crack propagation corresponding to Eq. (20.88) and to Fig. 20.9 the following expressions are obtained for the mentioned family of primary random functions Γ 0 : μ0 = 0 if 2γ < G(a) μ0 = A(A0 ⊕ 2b) = A(A0 ) + L(A0 )2b/π if 2γ ≥ G(a + b) μ0 = A(A0 ⊕ 2l(γ)) = A(A0 ) + L(A0 )2l(γ)/π with l(γ) = (2γ − G(a))E/(πσ 2 ) if G(a) ≤ 2γ ≤ G(a + b)
(20.118)
828
20. Probabilistic Models for Fracture Statistics
In Eq. (20.118), A(A0 ) and L(A0 ) are the average area and perimeter of the random convex set A0 . We have: Z +∞ Z G(a+b)/2 0 μ0 = A(A ⊕2b) θ(u)du + A(A0 ⊕2l(u))θ(u)du (20.119) G(a+b)/2
G(a)/2
For a given measure θ, the probability of fracture is given exactly by a combination of Eqs. (20.117, 20.118, 20.119). The probability of fracture converges towards 1 when increasing separately the crack length 2a or the applied stress σ. The effect of the size of the inclusions is the following: the probability of fracture decreases with the size of the reinforcing inclusions A0 . This is due to the fact that, with this size, the overall area fraction of higher strength material increases. If θ(γ) = 0 when γ ≥ γ c , the probability of fracture reaches 1 for large specimens where the crack length 2a is beyond the critical fracture length 2ac given by 2ac = 4γ c E/(πσ 2 ). For other measures, the asymptotic behavior of large size specimens depends on the tail of the measure θ for large σ. The case of θ(γ) = θγ −α with α > 1 is presented for illustration. With this intensity, the cumulative distribution function of the random function Γ is: ∙ ¸ θ F (γ) = P {Γ (x) < γ} = exp − (20.120) γ 1−α α−1 For large values of γ, a power law tail of the distribution is obtained, since: 1 − F (γ) '
θ γ 1−α α−1
This is equivalent to the Poisson mosaic model with similar distributions, leading to Eqs. (20.98, 20.99).
• If α = 2, μ0 = 4θc[A(A0 )/a − 2l(A0 )(log y)/π]. It results: 0
P (a + b) = y 8θcL(A )/π exp[−4θcA(A0 )/a]
(20.121)
with y and c given in Eq. (20.98) to which Eq. (20.121) can be compared; it corresponds to a similar situation for a different model: the fracture probability slightly increases with the size of similar specimens, and converges towards a constant depending only on the geometry (through y) and on the microstructure (through θ and L(A0 )). For this specific intensity , there is no size effect on the fracture statistics. • If α 6= 2 : ¸ h c iα−1 1 ∙ y α−2 − 1 μ0 = 4θ A(A0 ) + 2L(A0 )a/π (α − 1) (20.122) a α−1 2−α
20.5 Fracture statistics model with a crack arrest criterion
829
When α < 2, μ0 diverges for increasing sizes of the specimen, so that the probability of fracture converges to 0. When α > 2, μ0 converges to 0, and the probability of fracture of large specimens converges to 1. This is consistent with the results obtained for a Poisson mosaic and a similar distribution function (after replacement of α in the previous case by α − 1 in the present case). The same scale effects for the median strength as before are observed. ii) Stable crack propagation For the stable crack propagation, Eq. (20.119) becomes, with G(a) given by Eq. (20.89): μ0 = 0 if 2γ < G(a + b) μ0 = A(A0 ⊕ 2b) = A(A0 ) + L(A0 )2b/π if 2γ ≥ G(a) μ0 = A(A0 ⊕ 2(b − l(γ))) = A(A0 ) + L(A0 )2(b − l(γ))/π with l(γ) = kσ 2 /(2Eγ) − a if G(a + b) ≤ 2γ ≤ G(a)
(20.123)
In the present case, Eq. (20.119) becomes, with G(a) given by Eq. (20.89): μ0 = A(A0 ⊕ 2b)
Z
+∞
θ(u)du +
G(a)/2
Z
G(a)/2
G(a+b)/2
A(A0 ⊕ 2(b − l(u)))θ(u)du
(20.124) It is easy to deduce from Eq. (20.124) that the coefficient μ0 diverges with the size of the specimen, so that the probability of fracture becomes 0 for large samples, as for the Poisson mosaic model. Internal crack For the loading given by Eq. (20.90) corresponding to the configuration of Fig. 8 (20.10) , the following expressions of μ0 are obtained, when considering the approximation of cracks much larger than the size of the inclusions A0 : μ0 = 0 if 2γ < G(a) μ0 = 2A(A0 ⊕ b) = 2A(A0 ) + L(A0 )2b/π if 2γ ≥ G(a + b) μ0 = 2A(A0 ⊕ l(γ)) = 2A(A0 ) + L(A0 )2l(γ)/π with l(γ) = (2γ − G(a))E/(πσ 2 ) if G(a) ≤ 2γ ≤ G(a + b)
(20.125)
Therefore μ0 is given by : μ0 = 2A(A0 ⊕ 2b)
Z
+∞
G(a+b)/2
Z θ(u)du + 2
G(a+b)/2
G(a)/2
A(A0 ⊕ l(u))θ(u)du
(20.126) Since Eq. (20.126) is very similar to Eq. (20.119) obtained for the unstable propagation of a surface crack, the previously obtained results concerning scale effects are observed again for this internal crack configuration.
830
20. Probabilistic Models for Fracture Statistics
Parabolic loading For the loading corresponding to Eq. (20.91) and Fig. (20.11), μ0 is expressed as follows, with Gmax given in Eq. (20.91) and Gmin = min(G(a), G(a + b)) The approximation for b much larger than the size of the inclusions A0 is made. μ0 = 0 if 2γ < Gmin μ0 = A(A0 ⊕ b) = A(A0 ) + L(A0 )b/π if 2γ ≥ Gmax μ0 = 2A(A0 ) +rL(A0 )2l(γ)/π Gmax − 2γ if G2 with 2l(γ) = b − 2 h = max(G(a), G(a + b)) ≤ 2γ ≤ Gmax
(20.127)
• For b < k − 2a, G(a + b) > G(a), and for G(a) < 2γ < G(a + b): μ0 = A(A0 ) + L(A0 )l(γ)/π r Gmax − 2γ with l(γ) = k/2 − a − h • For b > k − 2a, G(a + b) < G(a), and for G(a + b) < 2γ < G(a): μ0 = A(A0 ) + L(A0 )l(γ)/π r Gmax − 2γ with l(γ) = a + b − k/2 − h Therefore μ0 is given by: μ0 = A(A0 ⊕ b)
Z
+∞
θ(u)du +
Gmax Z G2 /2
+
G min
Z
Gmax /2 G2 /2
A(A0 ⊕ l(u))θ(u)du
(20.128)
0
A(A ⊕ l(u))θ(u)du
Specific results are deduced from Eq. (20.128) for particular measures θ(γ). Crack initiation and propagation Combining Eq. (20.96) to the results given in the previous subsections, the probability of fracture may be derived for surface and internal crack initiation. The case with θ(γ) = θ−γ is studied below for large scale effects. i) Surface microcracks • If α = 2, using Eq. (20.121) enables us to draw the following conclusions:
20.5 Fracture statistics model with a crack arrest criterion
831
— When 8θcL(A0 )/π = 1, there is no scale effect on the probability of fracture. — When 8θcL(A0 )/π > 1, P {Non fracture} converges to 1 for large scales. — When 8θcL(A0 )/π < 1, P {Non fracture} converges to 0 for large scales. • If α 6= 2, using Eq. (20.122) we get — When α < 2, μ0 diverges for large specimens, and P {Non fracture} converges to 1 for large scales. — When α > 2, μ0 converges to 0 and P {Non fracture} converges to 0 for large scales. The asymptotic behavior of the probability of fracture is very similar to the Poisson mosaic case with the same fracture energy distribution. ii) Internal microcracks The internal microcracks with length 2a0 , are assumed to be much smaller than the reinforcing inclusions A0 . Therefore a different approximation than in the case leading to Eq. (20.126) is made, which gives: μ0 = 0 if 2γ < G(a0 ) μ0 = A(A ⊕ 2b) = A(A0 ) + L(A0 )2b/π if 2γ ≥ G(a0 + b) μ0 = A(A0 ⊕ 2l(γ)) = A(A0 ) + L(A0 )2l(γ)/π with l(γ) = 2γE/(πσ 2 ) if G(a0 ) ≤ 2γ ≤ G(a0 + b) 0
• It comes for α = 2 (see Eq. (20.121)): ∙ ∙ ¸¸ 2b 0 0 μ0 = 4θc A(A )/a0 + L(A )/π − 2 log y0 a0 + b
(20.129)
(20.130)
a0 . For large sizes b, diverges so that P (a0 , b, σ) expoa0 + b nentially converges to 0 and P {Non fracture} converges to 1 for large scales. This is different from the macroscopic crack a and from the surface nucleation. • If α 6= 2 with y0 =
∙
∙ ∙ ¸¸ 1 y α−2 − 1 0 0 α−1 a0 A(A ) + 2L(A )/π by (α − 1) α−1 2−α (20.131) When α < 2, μ0 diverges for large scales, in such a way that P (a0 , b, σ) exponentially converges to 0 and P {Non fracture} converges to 1. When α > 2, μ0 converges towards a finite value for large scales, so that P {Non fracture} converges to 0. Again similar results as for the Poisson mosaic are obtained. 4c μ0 = 4θ a0
¸α−1
832
20. Probabilistic Models for Fracture Statistics
20.5.5 Conclusions It was possible to derive a theoretical probability of fracture of unstable or stable straight crack propagation, due to various loading configurations, in two types of random media, namely the Poisson mosaic (or alternatively the STIT mosaic) and Boolean random functions. Resulting size effects are similar for the two models, and are strongly dependent on the tail of the cumulative distribution function of the fracture energy γ modelled by a power law with exponent α: for instance, for a large surface or internal crack, in the case of a very slow growth of F (γ) (α < 1), the probability for large specimens to encounter a locally large fracture energy is so important that the probability of fracture decreases to 0. When α = 1, the probability of fracture of large specimens is a constant depending on the geometry and on the microstructure, but not on the size. When α > 1, in any large specimen, a long crack is subject to propagation until fracture. Similar results are obtained for crack initiation and propagation on critical defects. The observed size effects may differ from the results of the weakest link model where the average strength of a material always decreases with the specimen size. Additional probability laws concerning the fracture strength, the toughness (at crack initiation or at crack arrest), and the size distributions of critical microcracks and stable microcracks were derived. These results can be used to interpret experimental data obtained on materials.
20.6 Models of random damage 20.6.1 Introduction The occurrence of damage in a loaded material is followed by a progressive loss of integrity, ending by its ruin. This damage is the result of a local degradation of weaker parts in the material, that can be generated by the presence of defects. The description of damage and of its growth is usually made in a continuous and deterministic way, by means of standard models of Continuum Damage Mechanics [420]. On the other hand, the presence of defects and of their heterogeneity can be accounted for by a probabilistic approach. This is generally made in the case of brittle materials by the weakest link model and its generalizations, as presented in section 20.3. In what follows, a probabilistic models of damage based on the presence of random defects is introduced [294]. After a presentation of the main assumptions, the cases of random damage generated under homogeneous, then heterogeneous loading conditions are considered.
20.6 Models of random damage
833
20.6.2 Basic assumptions Consider a homogeneous elastic material (with the elasticity tensor C) containing point defects. Under the action of a stress field σ(x) (σ(x) = σ in the case of a uniform load), any defect located at x, with a critical stress σ c lower than σ(x), generates a damaged zone with volume v centered in x. The damaged zone can be replaced by a domain v with a constant elasticity tensor CD . Here is considered the extreme case where damage zones behave like pores (CD ≡ 0). This is equivalent to the notion of statistical volume element SVE (containing at least a critical defect) introduced in [41], [42] for the FE simulation of damage in composites. One model of this type is obtained as a tessellation of space into cells with a volume v, the cell covering x remaining unbroken with the probability P (σ(x)). Another model considers point defects distributed according to a Poisson point process, Φ(σ) being the average number of defects with a critical stress lower than σ, per unit volume of material. As reminded in the beginning of this chapter, the ”weakest link” model proposed by Weibull [660], [661], where it is assumed that a specimen is broken as soon as it contains at least one critical defect, is obtained when the intensity is given by a m power law: Φ(σ) = (σ/σ u ) . In the present approach, critical defects at Poisson points xk generate damaged zones A0k (namely pores in what follows) with centers in xk . Therefore the damaged part of a specimen builds a Boolean model with primary grain A0 (chapter 6). This primary grain can be random, and can be oriented in a given direction, in the case of anisotropic damage. In examples below, isotropic and uniform damaged zones are introduced, generating a Boolean model with spheres having a constant volume v. In [158], a probabilistic model of damage for dynamic fracture is based on a Boolean model of spheres with radii increasing with time, to account for the propagation of a shock wave. The next assumption in the model concerns the redistribution of stress in the non damaged zone. As in the case of the damage of bundles of fibres, a uniform load sharing is assumed [143], [390], [607], [623]. This means that the amplification of stresses generated in the material by the presence of damage is considered to be made uniformly in the non damaged parts. With this simplification, the damage behaves as if it was totally diffuse, no localization being permitted. This is a model which is opposite to the weakest link assumption, where the fracture process is located on the weakest defect. The progression of damage in the material processes as follows: defects are ranked with an increasing severity (σc − σ(x)); the most critical defect is converted into a damaged domain; the load is then redistributed over the matrix, and the process goes on for the remaining defects. The classes of materials to which these assumptions can be applied are quasi brittle materials presenting a diffuse damage during their degradation.
834
20. Probabilistic Models for Fracture Statistics
20.6.3 Random damage under a homogeneous load Consider first a discrete model involving a specimen with a finite volume V containing n cells with volume v, each one with a critical stress σ c . It can represent a random tessellation of space into cells, but it as originally introduced for bundles of fibres [143], [390], [607], [623]. The probability for every cell, submitted to the stress σ, to remain undamaged is P (σ). Cells are ordered according to their random critical stress σ1 ≤ σ 2 ≤ ... ≤ σ k ≤ ... ≤ σ n . After the progression of damage, k cells (k < n) are damaged. If the specimen is submitted to a uniform macroscopic stress σ, every non damaged cell is submitted to the stress n/(n − k)σ, owing to the uniform load sharing assumption (this simplification does not account for effects such as the connectivity of non damaged cells, or as the percolation of damage). Assuming that the ruin of the specimen is obtained when every cell is damaged, this occurs when for every k < n, the critical stresses remain lower than the applied stress. The probability of rupture under the macroscopic stress σ is given by [143], [390], [607], [623]: P {fracture} (20.132) ½ ¾ n n = P σ 1 < σ, σ 2 < σ, ...σ k < σ, ...σ n < nσ n−1 n−k+1 ½ ¾ n−1 n−k+1 1 = P σ 1 < σ, σ 2 < σ, ... σk < σ, ... σ n < σ n n n ½ ¾ n−1 n−k+1 1 = P max(σ1 , σ 2 , ... σ k , ... σ n ) < σ n n n n−1 n−k+1 1 The sequence (σ 1 , σ 2 , ... σ k , ... σ n ) represents the evolution n n n of the macroscopic stress during every step of the progression of the damage. For each realization of the critical stresses, this sequence owns a maximum (σk increasing with k, while the factor (n − k + 1)/n decreases), which is the ultimate stress of the specimen, σ ult . It is easy to generate simulations of the random variable σ ult from n independent realizations of σ according to the distribution P (σ). The probability of fracture (20.132) can be expressed as a function of n and of P (σ) [143], [623]. Asymptotic results concern the average and the variance of σ ult : for large n, σ ult is found to maximize σ ult P (σ ult ) [143], [623]; this is expected, since for large n every term of the sequence of macroscopic stresses corresponds to some σ i P (σ i ). The proportion of non damaged material P (σ ult ) follows a Gaussian distribution with the variance P (σ ult )(1 − P (σ ult ))/n (due to the independence of the n cells and to the stationarity of the process), and therefore σ ult follows a Gaussian distribution with the average σ ult P (σult ) and the variance σ 2ult P (σ ult )(1 − P (σ ult ))/n, as derived in [143], [623]. The rate of convergence with n, which is low, is studied [607].
20.6 Models of random damage
835
P (σult ) 0.05 0.1 0.2 0.3 0.5 2D(σult , V ) 0.0054 0.00865 0.0134 0.0167 0.02 TABLE 20.2. Variability of the proportion of undamaged material
Consider now a continuous model obtained as a limit case of large specimens. The average volume fraction of the specimen where the critical stress is larger than σ is given by P (σ). For the Poisson point defects, generating a Boolean model, P (σ) = exp (−Φ(σ)v) (20.133) When the undamaged zones of the material are loaded with the homogeneous stress field σ, their average volume fraction is equal to P (σ), and then the average macroscopic stress σ M acting on the material is: σ M = σP (σ)
(20.134)
From Eq. (20.134), the average macroscopic stress is expected to reach a maximum when σ = σult as in the discrete model of bundles of fibres, since P (σ) decreases with σ. For a specimen B with a finite volume V , fluctuations of P (σ) (and therefore of P (σ ult )) are expected. Its variance D2 (σ, V ) is given, as a function of the covariance Q(h) of the undamaged zone (with Q(0) = q = P (σ)), by (see chapter 3): Z Z 1 D2 (σ, V ) = 2 (Q(x − y) − q 2 ) dxdy (20.135) V V V For a large specimen (with V À A3 ), Eq. (20.135) is expressed as a function of the integral range in R3 , A3 , by q(1 − q)A3 V Z 1 with A3 = (Q(h) − q 2 ) dh q(1 − q) R3
D2 (σ, V ) =
(20.136) (20.137)
For a Boolean model of spheres, the value of two standard deviations (giving the interval of confidence P (σ ult ) ± 2D(σ, V ) for P (σ ult ) expected on realizations) is given in Table 20.2, when the specimen is a cube with the volume V = L3 and when the diameter a of spheres (damaged zones) is such that a/L = 0.1. The same statistical property follows for σ ult , since from Eq. (20.134), the variance of σ ult is obtained from the variance of P (σ ult ), after multiplication by σ 2ult .
As in section 20.3.8, assume that the volume fraction Pσ of the specimen B where the critical stress is larger than σ, is a RV Pσ with average P (σ)
836
20. Probabilistic Models for Fracture Statistics
given by Eq. (20.133) and with variance given by Eq. (20.135) follows a Beta distribution (Eq. (20.68)), with cumulative distribution function: Z Γ (α + β) x α−1 u (1 − u)β−1 du (20.138) Bi (x; α, β) = Γ (α)Γ (β) 0
with parameters α and β given as a function of the average and of the variance of P (σ) by Eqs (20.71). As long as σ M = σPσ ≤ σ ult P (σ ult ), it is assumed that the behavior of B is located before the peak illustrated in Fig. 20.12. When reaching the peak, fracture of B is assumed to be initiated. Then there is no fracture of B as long as σM = σPσ ≤ σ ult P (σult ), or equivalently as long as Pσ ≤ σ ult P (σ ult )/σ. With this condition, on a mescoscopic scale B, P {non fracture of B} = Bi (σ ult P (σ ult )/σ; α, β)
(20.139)
When increasing σ, P {non fracture of B} decreases, as expected, and P {non fracture of B} = 1 for σ ≤ σ ult P (σ ult ): therefore with these assumptions, there is a critical stress σ c = σ ult P (σ ult ) (corresponding to an average macroscopic stress σ c P (σ c )) below which no fracture of B can occur. From Eq. (20.139), it is possible to calculate the distribution function of the fracture stress σ R of the undamaged zone, and of the macroscopic fracture stress σ MR of B as a function of characteristics of the model (through Φ(σ)v): P {non fracture of B} = P {σ R < σ} = P {σ MR < σP (σ)}
(20.140)
Eq. (20.140) is valid for σMR in the domain where σP (σ) increases with σ, corresponding to σP (σ) < σ ult P (σult ). The macroscopic average relationship between the stress σ M and the deformation ε is known, provided the effective elastic tensor C(σ) is known as a function of the damage induced in the material at the level of stress σ acting on the matrix. This effective tensor depends on the geometrical arrangement of damage (and therefore on its configuration for a given volume V ). For simplification, assume that C(σ) is given by the effective tensor of an infinite porous medium with the solid volume fraction P (σ). It is therefore obtained by homogenization of a random medium. This can be made by FE calculations as in [41], [42], or by estimation from some knowledge on the probabilistic properties of the medium. For instance, we can use for a porous medium the upper bound involving the three points probability of the medium, known for the Boolean model of spheres as given in chapter 18. The average stress-strain relation is deduced in a parametric way (as a function of σ) from: σM = σP (σ), ε = (C(σ))−1 σP (σ)
(20.141)
20.6 Models of random damage m 1 2 3 σult /σu 1 0.707 0.693 P (σult ) 0.368 0.606 0.716 σult P (σ ult ) 0.368 0.429 0.497 c 1.311 0.805 0.629
4 0.707 0.779 0.551 0.533
5 0.725 0.819 0.593 0.470
10 0.794 0.905 0.719 0.324
25 0.879 0.961 0.845 0.202
837
50 0.925 0.980 0.906 0.142
TABLE 20.3. Effect of m on the ultimate strength and c (Weibull type defects)
For an isotropic elastic medium in traction, the relation between the elongation ε, the traction σ and the Young’s modulus E(σ) is given by ε = σP (σ)/E(σ). To illustrate the approach, consider specimens in tension, with defects on Poisson points. Therefore, P (σ) is given by Eq. (20.133). For defects obeying to the Weibull distribution, the ultimate stress σ ult and the fraction of non damaged material are given by µ
¶1/m 1 σ ult = σ u mv 1 P (σ ult ) = exp(− ) m
(20.142) (20.143)
In Table 20.3 is given the variation, as a function of m, of σ ult /σ u (taking q v = 1), of P (σ ult ) and of the reduced coefficient of variation 1−P (σ ult ) standard deviation of P (σ ult ) and of c = P (σ ult ) , from which the ratio average p σ ult is deduced by c A3 /V . The ultimate strength σ ult and the maximum of σ M , σ ult P (σ ult ), increase with m, while the proportion of damaged material 1 − P (σ ult ) and the fluctuations decrease. Increasing m results in a more deterministic behavior. Note that for m ≥ 3, the volume fraction of the damaged material is lower than the percolation threshold of the phase made of spheres for the Boolean model, (estimated to 0.2895 from simulations) which means that the physical behavior of the model is correct. Concerning the σ M − ε curve, high values of m generate a more brittle elastic material, while a low value of m results in a ductile type behavior, as illustrated in Fig. 20.12. The curves own a common point obtained for σ = σu . σ For defects corresponding to a Pareto distribution, Φ(σ) = m log and σu ³ σ ´mv u P (σ) = for σ ≥ σ u . When mv > 1, σ ult = σ u and P (σult ) = 1: σ large specimens of the material are brittle with a constant ultimate stress.
20.6.4 Random damage under a non homogeneous load Consider now a non homogeneous load σ(x) acting on a specimen B of volume V , where the changes of σ in space occur slowly as compared to the
838
20. Probabilistic Models for Fracture Statistics Stress-strain curves for Poisson-Weibull defects 1.2 m=1 m=5 m=10 m=25
1
stress
0.8
0.6
0.4
0.2
0 0
0.005
0.01
0.015
0.02
0.025 strain
0.03
0.035
0.04
0.045
0.05
FIGURE 20.12. Average macroscopic stress-strain (σ M /σu ) relation for Weibull populations of defects with various m and a constant median σu log 2 [294]
scale of the microstructure. It can be generated by the flexion of a specimen, by its shape, or by the presence of holes. It can also be generated by the presence of components with different elastic properties in a composite material. Any field σ(x) generates a random variable Σ by taking any point x at random. We use the cumulative probability distribution function of the load over the specimen, G(σ) = P {Σ < σ}, and its density of probability g(σ). Defects generate a non homogeneous damage, where the local volume fraction of non damaged material is given by P (σ(x)). As before, assume that the load is uniformly shared over the non damaged material. In addition, the geometry of the applied stress field σ(x) remains unchanged during the progression of damage, which can be accepted for fields with a long range variation as compared to the microstructure. This results in a model where the details of the microstructure act only on average. For convenience of notation, consider the traction component σ, which remains positive for the progression of damage to occur (nevertheless, the volume averages in Eqs (20.144, 20.145) below can be written for any tensor σ). For these assumptions, the average macroscopic traction stress σ M of the material, obtained by averaging the local stress over non damaged domains, is given by Z ∞ Z 1 σM = σ(x)P (σ(x))dx = σg(σ)P (σ)dσ (20.144) V B 0 The average proportion of non damaged material, PM (σM ), is: Z ∞ Z 1 PM (σ M ) = P (σ(x))dx = g(σ)P (σ)dσ V B 0
(20.145)
20.6 Models of random damage
839
Increasing the load results from Eq. (20.144) in a change of the distribution g(σ). For instance, a proportional loading is obtained when we change σ(x) to kσ(x); without damage, σ M is changed to kσ M . If damage occurs, σ M becomes σ Mk , and PM (σ M ) becomes PM (k) with: Z ∞ Z 1 σ Mk = kσ(x)P (kσ(x))dx = k σg(σ)P (kσ)dσ (20.146) V B 0 Z ∞ Z 1 P (kσ(x))dx = g(σ)P (kσ)dσ PM (k) = V B 0 As in Eq. (20.134), σ Mk is expected to reach aRmaximum σ ult , obtained for ∞ k = kult , when k increases, since the integral 0 σg(σ)P (kσ)dσ decreases with k. The proportion of non damaged specimen when the ultimate stress is reached is given by PM (kult ). An approximate expression for the variances of σ ult and of PM (kult ) is derived: starting from a discrete model made of n cells of volume v [294], and considering the limit case of a continuum, we get for D2 (σ ult , V ), when V À A3 : 2 Z A3 kult 2 σ(x)2 P (kult σ(x))(1 − P (kult σ(x)))dx D (σ ult , V ) = V V B 2 Z A3 kult = σ 2 g(σ)P (kult σ)(1 − P (kult σ))dx (20.147) V B A first example is obtained by considering a stress field σ(x) generating a uniform distribution of stress ranging from 0 to σ max . Assume that the population of defects is such that P (σ) follows a uniform distribution between 0 and σ 0 . For a progressive proportional loading, with σ max < σ 0 , the aver3σ 0 3 age σult = and the proportion of damaged material 1 − PM (σ ult ) = . 16 8 This results are lower than for a homogeneous loading σ (σ ult = 0.5σ 0 and PM (σult ) = 0.5). The variance of σ ult for a homogeneous load is equal to A3 σ 20 , and it is 5.7375 times higher for the uniform distribution of stress. V 16 A second example is provided by the presence of N zones (i = 1, ..., N ) in the material where for the zone i, with volume fraction pi , the applied stress is equal to kσ i . These zones can result from a non homogeneous loading, or from the presence of phases with different Young’s moduli in the material. The average macroscopic stress σM derived from Eq. ( 20.144) becomes: σM = k
i=N X
pi σi P (kσi )
(20.148)
i=1
The ultimate stress is given by the maximum, over the values of k, of σM . Depending on the values of pi and σi , various stress-strain behaviors are predicted. Starting from a Poisson-Weibull population of defects, each term
840
20. Probabilistic Models for Fracture Statistics Macro Stress - Micro Stress for Poisson-Weibull defects (m=5); proportion p with an overload (2.6) 0.6 p=0.5 p=0.0 p=0.2 p=0.7 p=0.3 p=0.23
0.5
macroscopic stress
0.4
0.3
0.2
0.1
0 0
0.2
0.4
0.6
0.8 1 1.2 stress in domain 1
1.4
1.6
1.8
2
FIGURE 20.13. Average Macroscopic stress (σM /σu ) - Stress in domain 1 for two zones (σ1 = 1, σ2 = 2.6, m = 5) for various values of p2 [294] p2 0. 0.2 0.3 0.5 σult 0.725 0.4747 0.4154 0.4419 P (σ ult ) 0.819 0.6547 0.573 0.864 P (σ ult )1 0.818 0.818 0.997 P (σ ult )2 0.001 0.000 0.731
0.7 0.5 0.85 0.998 0.787
TABLE 20.4. Two zones, one being overloaded by a factor 2.6
µ ¶1/m σu 1 , and the linear σi mv combination of Eq. ( 20.148) can present a broad variety of shapes. This is illustrated in Fig. 20.13 (m = 5) and in Table 20.4 for two zones, one with volume fraction p2 being overloaded by a factor 2.6. In some cases, the macroscopic stress presents two maxima when the loading parameter k increases. The ultimate stress is lower than for the homogeneous loading. As seen in Table 20.4, lower values of p2 , the overloaded region is fully damaged, and for higher values of p2 , the under-loaded regions remain without damage. The effective modulus of every zone can be calculated as a function of the damage, from which is deduced a relation between the average macroscopic stress and strain. of Eq. ( 20.148) shows a maximum for σ =
20.6.5 Conclusion The models of random damage introduced in this section are based on statistical population of defects, which can be estimated from appropriate tests as in [41], [42] for fibre composites. They allow to predict the average macroscopic stress-strain curves with a full range of possible behaviors from a brittle to a ductile constitutive law. Moreover, they give indications on the expected dispersion of the stress strain curves, and of the ultimate stress, as a function of loading conditions and of the size of specimens. The size
20.7 Elements of practical use of fracture statistics models in numerical simulations
841
effect predicted from these models differ radically from the weakest link model, since a deterministic non linear constitutive law is obtained instead of a brittle elastic behavior in the latter case.
20.7 Elements of practical use of fracture statistics models in numerical simulations The main interest of probabilistic approaches to fracture for applications is their ability to predict expected scale effects. For this purpose, models must be tested at different scales (including the microscopic scale). The diversity of theoretical distributions for fracture statistics is the fruit of a microstructure based interpretation and modeling of mechanical data obtained on materials. Nevertheless, their is an intrinsic limitation of purely probabilistic models, the difficulty to introduce rigorously diffuse and growing damage. An easier way to fulfill this task is to implement numerical techniques, combined with Monte Carlo simulations, at the price of a high numerical cost. Typical examples of damaging materials at different scales, for which many numerical simulations of fracture are available, are fibre composites or metals. For instance a methodology was developed over years to study the fracture statistics of unidirectional composites for fibre fracture, transverse fracture, and for the fracture of laminate composites [41], [42], [43]. The mains steps of this methodology are as follows: 1. Identification of the population of defects from appropriate experiments 2. Determination of a Statistical Volume Element SVE (namely the elementary volume element broken during the progression of damage, which should contain at least one critical defect) 3. Introduction of statistical information (local fracture stress σR of SVE) in meshes of a FE calculation 4. Study of the fracture behavior on a first scale (RVE): fracture of the SVE if σ > σ R or σ eq > σ R 5. If necessary, material considered as a set of RVE to study its behavior on the next scale The main difficulties are: • The determination of the SVE • The choice of the appropriate fracture statistics and of its change with scale, with a possible use of models detailed in this chapter, provided the correct corresponding fracture criterion (weakest link, critical density,...) is known Similar approaches were developed for the ductile fracture statistics of steels [148], [159], [68].
842
20. Probabilistic Models for Fracture Statistics
For instance in [148], the fracture of a C-Mn steel containing clusters of MnS inclusions, which are germs for cavities during the fracture process, was simulated by axisymmetric FE calculations, using a continuum damage mechanics approach based on the Gurson potential controlling the growth of cavities. At the beginning of the simulation, an initial random distribution of cavities on the SVE mesh (with size 250μm, defined by the average distance between clusters) is generated by means of its local volume fraction; a lognormal distribution of volume fractions is used after fitting the data on MnS volume fraction obtained by quantitative image analysis on fields with size 250μm. In [159], the statistics of the fracture toughness of duplex stainless steels (austeno-ferritic) is simulated in two dimensions to study the effect of the embrittlement of the ferrite by aging. In this case again, the Gurson plastic criterion is used to predict fracture. In these simulations the random variable of interest is the local damage rate for the growth of cavities. Its probability distribution was measured by quantitative metallography on specimens, and then introduced in the generation of random damage rates on the SVE corresponding to the mean size of austenitic grains which generate clusters of microcracks (1mm2 ). In these two last types of simulations, the decrease of ductility and of its scatter (size effect) is well predicted by the model. From these examples of simulations, general guidelines can be derived with the following steps: 1. Selection of a damage parameter (density of point defects, microcrack network parameter, cavity volume fraction, cavity growth rate,...), connected to the microstructure 2. Statistical properties of this parameter obtained by image analysis (or by micromechanical tests): probability distribution function over domains with a given size 3. Simulations of the damage parameter as initial conditions for the prediction of its evolution by means of FE In any type of numerical simulations, morphological models of random media, as well as change of scale models to generate correct simulations on different scales, can be useful in order to introduce pertinent and coherent data in simulations, as well as for a correct interpretation of the obtained results.
20.8 Conclusion The models introduced above present some simplifications coming from the basic assumptions recalled at the beginning of the chapter. Their main advantages are the following:
20.9 Exercises
843
• Various morphologies of microstructural defects (inclusions in a matrix, polycrystals, fibres and strata,...), inducing correlations on various ranges, can be described and simulated. • Exact theoretical results for the probabilistic properties of fracture, coherent at different scales, are available. • Depending on few parameters (2, 3 or more), they can be tested from data on various scales: on the macroscopic scale, by means of the experimental distributions obtained from mechanical tests on various specimens geometries; on a microscopic scale, by means of image analysis measurements after localization of the defects [57], [58], [59]. • Various scaling laws are obtained, according to the chosen fracture criteria, or to the appropriate random structure models, reflecting the situations occurring with experimental data: the overall strength of specimens may decrease with their size (weakest link), may be size invariant (damage threshold), or may even increase (crack arrest). However, one must remain aware of the fact that different combinations of fracture criteria and microscopic models can result in the same size effects, as underlined in [275]. Therefore it is unwise to draw definite conclusions solely on size effects, without any indication on the microstructure and on the micro-mechanisms of fracture. • The models can be easily introduced in post-processing calculations in a FE code [57], [58], [59]). Extensions combine the simulation of population of defects after their identification from the models and mechanical tests, and the simulation of the progression of damage in composites by FE calculation [40], [334], [41], [42]; this is a way to account for stress redistribution in microcracking processes, which is difficult to handle solely by analytical calculations. The use of these models is not restricted to the simulation of critical fracture criteria, as proposed in this chapter. In fact they are able to simulate other physical random media with a microstructure, including the distribution of multivariate or tensor properties, that are also used for other purposes such as homogenization calculations, as illustrated in chapter 18.
20.9 Exercises 20.9.1 Crack propagation in a random polycrystal with grain boundary fracture energy In the random crack arrest model of section 20.5, the case of a Poisson mosaic or of a STIT mosaic is revisited by the introduction of a random facture energy Γb with distribution function Fb (γ) for the edge of each Poisson polygon, in addition to the random fracture energy of the grains. In this context, a straight propagating crack can be stopped by a grain bound-
844
20. Probabilistic Models for Fracture Statistics
ary with a high fracture energy. Give the fracture probability replacing Eq. (20.97) in this case, for an unstable crack propagation. What happens for weak or strong grain boundaries? Answer: In the presence of a fracture energy of grain boundaries, Eq. (20.97) becomes: P (a + b)
"
= F (G(a)/2) exp −λ
Z
2(a+b)
2a
(20.149) #
(1 − F (G(u)/2)Fb (G(u)/2))du
To prove this result, one needs to consider transitions between adjacent grains; the crack advance is possible, provided G(l + dl) ≥ 2Γ ∨ Γb , where Γb is the energy of fracture of the boundary between the adjacent grains and Γ is the fracture energy of the second grain. Therefore, P (l)dl) = P (l) [1 − λdl + λdlF (G(l + dl)/2)Fb (G(l + dl)/2)] and to first order in dl, dP (l) = −λP (l)(1 − F (G(l)/2)Fb (G(l + dl)/2)) In Eq. (20.149), (1 − F (G(u)/2)Fb (G(u)/2)) ≥ (1 − F (G(u)/2)) As a consequence, P (a + b) given by Eq. (20.149) is always less or equal than P (a + b) given by Eq. (20.97), and there is always a reinforcement of the microstructure by the presence of a term γ b . For weak boundaries, on average Γb < Γ and Fb (γ) < F (γ). Assume that Fb (γ) = 1 for γ ≥ γ 0 , while F (γ) < 1. In these conditions, F (γ)Fb (γ) = F (γ) for γ ≥ γ 0 and P (a + b) is given by the previous result of Eq. (20.149) in section 20.5 for long crack propagation, so that weak boundaries do not exert any significant reinforcement. For strong boundaries, Fb (γ) > F (γ). Assume now that F (γ) = 1 for γ ≥ γ 0 , while Fb (γ) < 1. In these conditions, F (γ)Fb (γ) = Fb (γ) for γ ≥ γ 0 and there is a strong decrease of the probability of fracture P (a+b) given by Eq. (20.149), as compared to Eq. (20.97): boundaries with a high fracture energy can significantly reinforce the effective toughness.
20.9.2 3D crack propagation in a random medium Consider the random crack arrest model of section 20.5, for a planar crack front CF in 3D propagating in mode I, in the case of an unstable propagation of the crack front as shown in Fig. 20.8. The evolution G(a) is not known exactly in this case [181] and numerical simulations show a profile
20.9 Exercises
845
of G(a) decreasing close to the free boundaries of a parallelepiped specimen with thickness d. To make some indicative probabilistic calculation, the following simplifications can be used: •
G(a + x) is expressed by Eq. (20.88) showing a linear increase with crack advance, and keeps the same value along the crack front, neglecting boundary effects. • The straight crack front is blocked along its progression as soon as it meets zones x where 2Γ (x) > G(x), keeping its shape without any bending. It moves again forward when increasing the load, as soon as ∨{2Γ (x) − G(x); x ∈ CF } ≤ 0. • With these conditions, the crack front can reach the location 2(a + b), provided ∨{2Γ (x) − G(x); x ∈ R(2b, d)} ≤ 0, where R(2b, d) is the rectangle with length 2b and width d, and the probability of fracture is expressed by P (a, b, d, σ) = P {∨{2Γ (x) − G(x); x ∈ R(2b, d)} ≤ 0}
(20.150)
. Give the fracture probability derived from Eq. (20.150) for the case of a 3D Boolean mosaic RF to model Γ (x). Answer: The conditions to derive Eqs (20.118) and (20.119) are again satisfied. we have P (a, b, d, σ) = exp(−μ0 ) where now μ0 is the volume of the grain after dilation by rectangles. Using the Steiner formula for random convex grains A0 : μ0 = 0 if 2γ < G(a) μ0 = V (A0 ⊕ R(2b, d)) 0
) = V (A0 ) + 18 (4b + 2d)S(A0 ) + bd M(A if 2γ ≥ G(a + b) π 0
) μ0 = V (A0 ⊕ R(2l(γ), d)) = V (A0 ) + 18 (4l(γ)) + 2d)S(A0 ) + l(γ)d M (A π
with l(γ) = (2γ − G(a))E/(πσ2 ) if G(a) ≤ 2γ ≤ G(a + b)
(20.151) In Eq. (20.151), V (A0 ), S(A0 ) and M (A0 ) are the average volume, surface area and integral of mean curvature of the random convex set A0 . We have: Z +∞ θ(u)du (20.152) μ0 = V (A0 ⊕ R(2b, d)) Z +
G(a+b)/2
G(a+b)/2
G(a)/2
V (A0 ⊕ R(2l(u), d))θ(u)du
846
20. Probabilistic Models for Fracture Statistics
The probability of fracture converges towards 1 when increasing separately the crack length 2a or the applied stress σ. The effect of the size of the inclusions is the following: the probability of fracture decreases with the size of the reinforcing inclusions A0 . Parameters, like the shape of θ(γ), have the same effect on the probability of fracture as in the 2D case, with the addition of a size effect due to the width d, such that P (a, b, d, σ) decreases with d: increasing the width of the specimen increases the probability to arrest the crack front CF by increasing the probability for CF to hit inclusions with a high fracture energy.
21 Crack Paths in Random Media
Abstract: Simple probabilistic models of fracture of 3D polycrystals involving transgranular and intergranular cracks are based on undamaged paths and appropriate percolation thresholds. In a second part, theoretical extensions of phase field models for crack initiation and propagation to the case of locally heterogeneous and anisotropic fracture energy are presented. They are implemented by means of iterations of Fourier transforms, replacing the standard Finite Elements approach, to generate full field solutions. When the phase field problem is numerically solved on simulations of a random medium, its effective fracture energy can be estimated and its fluctuations characterized with the corresponding RVE.
21.1 Introduction In heterogeneous materials, local fracture energy can show fluctuations, represented by realizations of some random function: reinforced composites present different fracture energies in the matrix and in embedded particles; polycrystalline materials may have different transgranular or intergranular fracture energies; cleavage fracture energy in polycrystals may be strongly anisotropic in each crystal, due to the presence of weak planes. This chapter first introduces some probabilistic models of fracture of 3D random tessellations of space, modelling polycrystals. The second part presents a phase field model to simulate crack initiation and propagation © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Jeulin, Morphological Models of Random Structures, Interdisciplinary Applied Mathematics 53, https://doi.org/10.1007/978-3-030-75452-5_21
847
848
21.
Crack Paths in Random Media
in heterogeneous media with anisotropic fracture energy. The full content of this chapter is available in [322].
21.2 Probabilistic fracture of 3D random tessellations Granular media like polycrystals can be modelled by random tessellations (chapter 7) The crystallographic orientation of each class or cell can be chosen independently or with some prescribed correlation between orientations of neighboring cells. Usual probabilistic models of brittle fracture, like cleavage fracture, are based on the weakest model, like the Weibull model (chapter 20). Statistical cleavage fracture models make use of point random defects generated by a Poisson point process in the grains. Extensions of the weakest link model to intergranular fracture and to a competition between intergranular and transgranular fracture were proposed, with point defects located on the grain boundaries, leading to other size effects [275]. Crack paths were considered for 2D fracture, assuming that the fracture paths minimize the overall surface energy [279]. Simulations involving a competition between intergranular and transgranular fracture are introduced in this section. We give an introduction to the fracture statistics of random tessellations related to percolation [310]. In this simplified approach, a sudden brittle fracture of the medium is assumed: for a given applied homogeneous stress σ, all elements with fracture stress lower than σ are simultaneously broken, so that there is no progressive fracture and no stress redistribution in the material.
21.2.1 Basic assumptions Consider a locally finite random tessellation of R3 ma de of open classes Ci with closure C i The common boundary between Ci and Cj is Cij = C i ∩ Cj. For cleavage fracture, each class can be independently cracked (or broken) with probability Pg (σ), σ being the applied stress field (for instance a tensile stress σ). Alternatively, for intergranular fracture, each boundary Cij can be independently cracked (or broken) with probability Pgb (σ). It will be convenient to map the random tessellations onto a random graph, every class being represented by a node Ci (or a site) of the graph, and every boundary Cij being represented by an edge of the graph. Increasing Pg (σ) increases the proportion of broken nodes, while increasing Pgb (σ) increases the proportion of broken edges, so that fracture of tessellations can be studied on a random graph.
21.2 Probabilistic fracture of 3D random tessellations
849
21.2.2 Fracture of a random tessellation and percolation To break a specimen in at least two pieces, a connected fracture surface without holes must pass through the sample. Transgranular fracture To break a sample by transgranular fracture, a crack front must go across grains. As long as intact grains are surrounded by the crack front, the specimen remains unbroken. This can be restated as follows: the specimen remains unbroken, as long as at least one path connecting unbroken grains (including those surrounded by the crack front) can cross the specimen. When no connected path with unbroken grains can go across the specimen, a complete crack front without holes is subject to propagate. Therefore the possible fracture of the specimen is related to the site percolation of the graph generated by the random tessellation. We have the following result: Proposition 21.1. Transgranular fracture of a random tessellation cannot occur if 1−Pg (σ) > ps , ps being the site percolation threshold of the random graph. Fracture will occur if 1 − Pg (σ) < ps , and then Pg (σ) > 1 − ps . Knowing Pg (σ) and ps provides a fracture stress σR obtained by solving Pg (σR ) = 1 − ps . The percolation threshold ps is available for systems with infinite size, and therefore the fracture stress σ R is obtained, as expected. For finite systems and a given applied stress σ, we have to know the probability of percolation of realizations, where sites are left intact with the probability 1 − Pg (σ). Proposition 21.2. For finite size systems subject to transgranular fracture, P {σ R ≥ σ} = P {no fracture at σ} = P {the sites of the random graph percolate}
(21.1)
Since the percolation of the finite random graph depends on Pg (σ), the fracture probability given in Eq. (21.1) depends on the applied stress σ, and will generally be estimated from simulations of random graphs, from which size effects can be investigated. Intergranular fracture To break a sample by intergranular fracture, a crack front must pass across grain boundaries. We have now to consider paths along edges of the graph generated by the random tessellation: the possible fracture of the specimen is related to the edge (or bond) percolation of the graph generated by the random tessellation. We have the following result:
850
21.
Crack Paths in Random Media
Proposition 21.3. Intergranular fracture of a random tessellation cannot occur if 1 − Pgb (σ) > pb , pb being the bond percolation threshold of the random graph. Fracture will occur if 1 − Pgb (σ) < pb , and then Pgb (σ) > 1 − pb . Knowing Pgb (σ) and pb provides a fracture stress σ R obtained by solving Pgb (σ R ) = 1 − pb . For systems with infinite size, the percolation threshold pb is available, and therefore the fracture stress σ R is known, as expected. As before, for finite systems and a given applied stress σ, the probability of percolation has to be estimated on simulations, where edges are left intact with the probability 1 − Pgb (σ). Proposition 21.4. For finite size systems subject to intergranular fracture, P {σ R ≥ σ} = P {no fracture at σ} = P {the bonds of the random graph percolate}
(21.2)
Again, the fracture probability given in Eq. (21.2) depends on σ through Pgb (σ) and will generally be estimated from simulations of random graphs, from which size effects can be investigated. Competition between transgranular and intergranular fracture Assume now that the two fracture mechanisms (transgranular and intergranular) can occur in the same material. In that case, there is no fracture as long as there is no transgranular and no intergranular fracture of the material, so that must be satisfied 1 − Pg (σ) > ps and 1 − Pgb (σ) > pb These inequalities can be rewritten as Pg (σ) < 1 − ps and Pgb (σ) < 1 − pb Since Pg (σ) and Pgb (σ) are monotonous non decreasing functions of σ, there is no fracture as long as −1 σ < Pg−1 (1 − ps ) and σ < Pgb (1 − pb )
and therefore −1 σR = inf(Pg−1 (1 − ps ), Pgb (1 − pb ))
In the case of systems with finite size, the probability of no fracture (21.1) and (21.2) become: P {σR ≥ σ} = P {no fracture at σ} (21.3) = P {the sites and the bonds of the random graph percolate}
21.3 Phase field for cracks propagating in random media
851
The probability given in Eq. (21.3) now depends on σ through Pg (σ) and through Pgb (σ), and can be estimated from simulations as before. Case of the Poisson Voronoi tessellation Site and bond percolations of the Poisson Voronoi tessellation were studied in 3D by simulations [254]. From their results ps = 0.1453 ± 0.002 and pb = 0.0822 ± 0.005. With these data, transgranular fracture can happen for Pg (σ) > 0.854 7 while intergranular fracture requires Pgb (σ) > 0.917 8.
21.3 Phase field model for crack initiation and propagation in random media with anisotropic fracture energy Efficient numerical simulations of crack initiation and propagation were developed from phase field models implementing the minimization of the total energy (elastic energy and surface energy generated by extending cracks) stored in a medium. Most available schemes concern media with isotropic fracture energy. All presently available schemes are restricted to homogeneous media, and are implemented in Finite Elements codes. In what follows, a short reminder on the main phase field models for fracture, and on the extended versions to anisotropic fracture energy is given. A new phase field model for the case of heterogeneous media with a locally anisotropic fracture energy, and its possible numerical implementation by FFT is introduced, extending a preliminary work [320].
21.3.1 Reminder on phase field models for fracture of homogeneous isotropic media A breakthrough in the simulation of fracture of materials was initiated from a variational principle introduced in [180]. It can be formulated as follows: the total energy (potential energy and energy dissipated by fracture) is minimized. The total energy of a domain Ω containing a crack Γ (surface in R3 ) submitted to a deformation field ε(x) is given by Z Z E(ε, Γ ) = E(ε) + E(Γ ) = W (ε(x))dx + gc dS (21.4) Ω\Γ
Γ
where E(ε) is the stored elastic energy and E(Γ ) is the fracture surface energy, according to the Griffith criterion of fracture (as already used in chapter 20), and gc is the toughness (specific fracture surface energy). This approach was followed much earlier for the simulation of crack paths by minimization of the fracture energy, but neglecting the fluctuations of
852
21.
Crack Paths in Random Media
stored elastic energy [266], [279]. In 2D, this corresponds to the Fermat principle, and to the extract geodesics (or shortest paths) in a heterogeneous medium. It was successfully applied to the fracture of porous carbon electrodes [266], and to the simulation of fracture in porous or reinforced media, as well as intergranular and cleavage fracture of polycrystals [279]. By asymptotic homogenization of periodic media, it was shown that the geodesics are the solution of the minimization of fracture energy obtained by Γ convergence in the case of anti-plane shear [85]. In a slightly different context, geodesic paths coincide with plastic flow patterns in elastoplastic composites [353]. Extension to the 3D case requires the calculation of energy minimizing surfaces, as developed in [167]. 3D implementations make use of heuristic algorithms developed on graphs [83], [211], like graph cuts algorithms. These algorithms are closely related to the approach used on the fracture of random tessellations introduced in section 21.2. Based on the theoretical results of [85], a FFT based method was developed to extract minimal surfaces for homogenization of the fracture energy [589]. In the area of image segmentation, variational criteria were introduced by Mumford and Shah [510], where some energy involving integrals over the domain and over the unknown boundary of objects have to be minimized. Elegant approximate numerical solutions are obtained after introduction of an appropriate precursor of a phase field [17]. Using the same approach here, after introduction of an Ansatz, the phase field Φ(x) (0 ≤ Φ(x) ≤ 1) with Φ(x) = 1 on the crack and Φ(x) = 0 away from the Rcrack, and of a small scale parameter l, the volume integral is replaced by Ω (1 − Φ)2 W (ε(x))dx and the surface integral E(Γ ) is replaced by a volume integral of the crack surface density function, [80], [82]: Z 1 l gc γ(Φ) = gc ( Φ2 + ∇Φ.∇Φ)dx (21.5) 2l 2 Ω The energy E(ε, Φ) modified by the field Φ(x) is an approximation of the total energy (21.4). In the variational approach, the phase field Φ(x) and the stress field σ(x) minimize the total energy E(ε, Φ): Z Z 1 l (1 − Φ)2 W (ε(x))dx + gc ( Φ2 + ∇Φ.∇Φ)dx (21.6) E(ε, Φ) = 2l 2 Ω Ω The first term corresponds to the stored elastic energy for a damaged medium, (1 − Φ)2 being interpreted as a scalar damage parameter reducing the elasticity tensor. The main advantage of the phase field formulation is to replace surface integrals involved in the fracture integrals by volume integrals. In [583], other functions of the scalar Φ are proposed. The second term gc γ(Φ) appears when considering a specific solution of the profile of Φ orthogonally to a linear crack in 2D, d(x) being the distance to the crack, namely [17], [18], [80], [82], [481], [482]:
21.3 Phase field for cracks propagating in random media
853
Φ(x) = exp(−d(x)/l) It is easy to show that this profile minimizes the second integral in the 2D case. When the scale parameter l → 0, the approximated energy E(ε, Φ) converges to Eq. (21.4) in the sense of the Γ convergence, and the field Φ minimizing E(ε, Φ) Γ converges to the indicator function of the crack resulting from the minimization of Eq. (21.4) [17], [18], [84]. The length scale l is sometimes considered as a material parameter, which can be estimated from the experimental fracture stress [551], [78], [519]. In [693] its impact on the results of fracture simulations is studied. Other versions of the crack surface density function were introduced: • adding a higher order term in l4 (∆Φ)2 provides smoother profiles Φ(x) around the crack [78]. • an alternative crack surface density function replaces the Φ2 term by Φ [182], [551], [628]: Z 1 l gc γ(Φ) = gc ( Φ + ∇Φ.∇Φ)dx (21.7) 2 Ω 2l This change provides a finite non zero elastic limit (instead of zero elastic limit when using Φ2 ), and generates cusp shaped localization profiles around the crack in 1D [551]. The crack initiation and propagation problem under an increasing load is given by the two fields (ε(x) or σ(x), and Φ(x)) minimizing the total energy given by Eq. (21.6). Unfortunately, no algorithm is presently available to provide the solution, but approximate solutions can be obtained by heuristic approaches [80], [82]. An optimization procedure, based on alternate minimization (over the displacement u, and Φ), and on a Backtracking algorithm is proposed [80], [81], [82] and implemented in various examples presented by these authors. Other authors (see for instance [227], [481], [482]) implement a separate minimization of the two integrals given in Eq. (21.6). Of course, the obtained solution gives only an approximation of the overall minimization problem, since Φ is present in the two integrals. Applying the Euler-Lagrange equation separately to each integral generates appropriate coupled PDE for the fields σ(x), and Φ(x). From the first term, the equilibrium equation of elasticity is obtained (∇σ = 0). Far from the crack field is recovered an elasticity problem for a non damaged material. In [80] is presented an example of application to a thermo-elastic problem, the elastic energy containing an additional term with the coefficient of thermal expansion and the temperature. In that case, a numerical solution of the inhomogeneous thermo-elastic problem is needed, instead of the standard elastic problem.
854
21.
Crack Paths in Random Media
Following [481], [482], the Euler Lagrange equation derived from the second term of Eq. (21.6) is given by: Φ − l2 ∆Φ = 0 in Ω, and ∇Φ.n = 0 on ∂Ω, n being the outward normal of ∂Ω
(21.8)
To take into account the loading history parametrized by the variable t, the PDE (21.8) governing the phase field Φ is modified as follows [482], [518], [519]: 2(1 − Φ)H −
gc (Φ − l2 ∆Φ) = 0 in Ω (21.9) l with Φ = 1 for x ∈ Γ , and ∇Φ.n = 0 on ∂Ω
It is completed by the equilibrium equation for the stress field, with body force f ∇σ(Φ) = f in Ω (+ boundary conditions) (21.10) The coupled equations (21.9) and (21.10) are solved alternatively for increments of external loading. In [482], [518], [519], [520] they are solved by standard Finite Elements. Like in [80], [81], it is possible to plot the two terms of Eq. (21.6) (bulk energy and surface energy), as a function of the increasing load. This can be used to estimate an effective toughness of an heterogeneous medium, when working with an initial crack of length a. In Eq. (21.9) H = H(x, t) represents the history of the energy density during the loading. It was introduced in [481] to account for possible loading/unloading sequences, and to warrant the growth of damage during the loading. We have H(x, t) = max {Ψ + (x, s)} 0≤s≤t
+
where Ψ is the tensile part of the elastic strain density. For an isotropic elastic medium, Ψ + (ε) = with [481], [520]
n¡ ¢ o ¢2 λ¡ 2 hT r( )i+ + μT r ε+ 2
ε+ =
X i
i
< εi > ni ⊗ ni
ε being the eigenvalues of ε and ni its eigen vectors; hyi± = (y ± |y|) /2 and ε± are the compression and the tensile parts of strain. The constitutive law relating stress and strain, accounting for unilateral effects, is given by [481]: ¡ ¢© ª σ = (1 − Φ)2 + k λ hT r( )i+ 1 + 2με+ + λ hT r( )i− 1 + 2με− (21.11)
21.3 Phase field for cracks propagating in random media
855
k being a small parameter to control the stability of calculations (according to some authors it can be set equal to 0). An illustrative example of application of this phase field model to a real material is given in [520], where a comparison is made on predicted crack paths in 3D, and real crack paths observed by microtomography at different steps of loading of plaster and concrete specimens. Some extensions to homogeneous anisotropic fracture energy Very few extensions of the phase field model were proposed for the anisotropic fracture energy. They are implemented by an adaptation of the surface term in the phase field energy given by Eq. (21.6). These extensions are reviewed in [322]. They are based on higher order expansions of the surface energy, anisotropic crack surface density functions, or use of several phase fields Φi with their own length scales li , he Γ convergence for schemes involving several phase fields Φi and length scales li remaining presently unproven.
21.3.2 A phase field model for heterogeneous anisotropic fracture energy In this section we consider an heterogeneous medium (polycrystal, mutiphase medium,...) where the anisotropic fracture energy can change from → place to place. It will be represented by a vector field − γ (x) with modulus → − → − → − |γ| and unit vector u γ , so that γ (x) = |γ(x)| u γ (x). We denote by γ x , → γ y , and γ z the coordinates of − γ in an orthonormal basis. For further → − applications to random media, γ (x) will be considered as a realization of a random vector field. Consider in R3 a closed set A with surface boundary Σ. Define in every point x the signed distance d(x) by d(x) = inf{d(x, y), y ∈ Ac } if x ∈ A, d(x) = − inf{d(x, y), y ∈ A} if x ∈ Ac The signed distance function allows us to define level sets given by d(x) = α. The boundary of A, ∂A is obtained by: ∂A = {x, d(x) = 0}. In each point x, ∇d defines a unit vector (|∇d| = 1), normal Rto the level set containing → x. The surface energy of ∂A is given by γ ∂A = ∂A (− γ (x).∇d)dS. → As an example, consider a monocrystal with the fracture energy − γ. A → − planar surface with normal given by n (nx , ny , nz ) generates a specific fracture energy (per unit area) → → γ = γ x nx + γ y ny + γ z nz = |γ| − u γ .− n = |γ| cos θ → → θ being the angle between vectors − n and − γ . To look for the minimal fracture energy is equivalent to look for the minimum of (cos θ)2 (or of
856
21.
Crack Paths in Random Media
→ → → → (− u γ .− n )2 ). It is obtained for θ = 0 and therefore for − uγ ⊥ − n . When : 0 < γ x ¿ γ y , and 0 < γ x ¿ γ z , a minimum is given for a planar surface with a normal orthogonal to the plane yOz; this planar surface is orthogonal → to the − x direction. It is a "weak plane" with minimal fracture energy γ x . In the framework of the phase field approach, the distance function is replaced by the phase field Φ(x) (0 ≤ Φ(x) ≤ 1) with Φ(x) = 1 on the crack and Φ(x) = 0 away from the crack. In the case of anisotropic fracture energy, replace now in Eq. (21.5), to 1 2 minimize in the scalar and homogeneous case, the term gc ( 2l Φ + 2l ∇Φ.∇Φ) by: 1 l 2 → F (x, Φ, ∇Φ) = |γ(x)| Φ2 + |γ(x)| (∇Φ.− (21.12) u γ (x)) 2l 2 The field Φ corresponding to the minimum of fracture energy is solution of the PDE obtained by means of the Euler-Lagrange equation applied to Eq. (21.12). It comes: Φ l F∇Φ = {(F∇Φ )x , (F∇Φ )y , (F∇Φ )z } ∂Φ ∂Φ with components (F∇Φ )x = l |γ(x)| (uγ )x = lγ x , ∂x ∂x ∂Φ ∂Φ (F∇Φ )y = lγ y , and (F∇Φ )z = lγ z ∂y ∂z FΦ = |γ(x)|
and therefore F∇Φ is a vector ⎡with components lD∇Φ, where D is a diag⎤ γx 0 0 onal diffusion matrix D(x) = ⎣ 0 γ y 0 ⎦. The Euler-Lagrange equation is 0 0 γz given from: FΦ =
∂ ∂ ∂ (F∇Φ )x + (F∇Φ )y + (F∇Φ )z = div(F∇Φ ) = div(lD(x)∇Φ) ∂x ∂y ∂z
and Φ is solution of |γ(x)| Φ − l2 div(D(x)∇Φ) = 0
(21.13)
For a scalar and homogeneous surface fracture energy, for which D(x) is a diagonal matrix proportional to the identity, where γ x = γ y = γ z = γ, we recover from Eq. (21.13): γ(Φ − l2 ∆Φ) = 0 Introducing the history term [482] Eq. (21.9) becomes in the non homogeneous anisotropic case:
21.3 Phase field for cracks propagating in random media
857
1 2(1 − Φ)H − ( |γ(x)| Φ − l2 div(D(x)∇Φ)) = 0 in Ω (21.14) l with Φ = 1 for x ∈ Γ , and ∇Φ.n = 0 on ∂Ω Alternatively, in the case of anisotropic fracture energy the crack surface density function entering into Eq. (21.7) to minimize becomes: F (x, Φ, ∇Φ) =
1 l 2 → |γ(x)| Φ + |γ(x)| (∇Φ.− u γ (x)) 2l 2
(21.15)
Using the Euler-Lagrange formulation, Eq. (21.13) becomes: 1 |γ(x)| − l2 div(D(x)∇Φ) = 0 2
(21.16)
which gives in the scalar homogeneous case: 1 γ( − l2 ∆Φ) = 0 2 Starting from crack surface density function in Eq. (21.7), we have to solve: 1 1 2(1 − Φ)H − ( |γ(x)| − l2 div(D(x)∇Φ)) = 0 in Ω (21.17) l 2 with Φ = 1 for x ∈ Γ , and ∇Φ.n = 0 on ∂Ω
21.3.3 Introduction to full fields estimation by FFT The crack initiation and propagation problem requires the resolution of the elastic equilibrium Eq. (21.10) and of the phase field Eq. (21.9) in the homogeneous case. For a non-isotropic elastic medium, the constitutive relation between strain and stress (21.10) and the term Ψ + (ε) have to be rewritten from the corresponding elasticity tensor C. The PDE governing the two fields (Φ and the displacement u or the deformation ε) have to be solved alternatively for loading increments. They are usually solved by means of Finite Elements. Consider inhomogeneous media, where the elastic moduli and the fracture energy are not homogeneous in space. They are considered as realizations of random fields, so that Φ(x) and H(x) are also realizations of random fields. In that case, it is known that an efficient approach to solve the equilibrium equation is to use iterations of FFT, as initiated in [507], and further developed (see [675] among others). The implementation of iterations of FFT to compute stress fields in the case of anisotropic elastic polycrystals containing cracks was successfully tested and compared to Finite Elements calculations [197], its main advantage being to operate with fast algorithms on high resolution images (up to 19443 , out of reach
858
21.
Crack Paths in Random Media
of Finite Elements), without needs to mesh the microstructure. An application to the computation of thermo-elastic fields in polycrystalline TATB containing transgranular and intergranular cracks was made on 5123 simulated images [198]. It is therefore worth looking for a similar approach to solve the phase field Eqs (21.9), (21.14), or (21.17). FFT solution of the phase field PDE in homogeneous media Consider first the crack surface density function (21.5). For the scalar case, starting from Eq. (21.9), we have to solve: 2(1 − Φ)H =
γc (Φ − l2 ∆Φ) l
The coefficients of Φ on the right side are constant, while H is a field H(x) depending on the history of the elastic solution. The equation to solve is a Helmholtz equation with a second member depending on the solution Φ. It can be solved iteratively by means of the 3D Green’s function G(x − x0 ) [668]: µ ¶ 1 |x − x0 | 0 G(x − x ) = exp − 4π |x − x0 | l Start with Φ0 (x) = 0. After n iterations we have Z 2l Φn+1 (x) = G(x − x0 )(1 − Φn (x0 ))H(x0 )dx0 γc For cracks propagating in the half space z ≥ 0, the solution Φ with zero normal derivative on the plane z = 0 is obtained with the Green’s function [668]: G(x − x0 )
⎛ ¡ ¢1 ⎞ (x − x0 )2 + (y − y 0 )2 + z 2 2 ⎝− ⎠ = 1 exp l 2π ((x − x0 )2 + (y − y 0 )2 + z 2 ) 2 1
An alternative way to solve Eq. (21.9), given H, is to take the Fourier transform of the two members of the equation. Noting F(Φ) the Fourier transform of Φ, the following iterative relation comes: 2l 2 F ((1 − Φn )H) = F(Φn+1 ) + l2 |w| F(Φn+1 ) γc where |w|2 is the squared norm of vector w in the Fourier space. Therefore F(Φn+1 ) =
2l F ((1 − Φn )H) γ c 1 + l2 |w|2
(21.18)
21.3 Phase field for cracks propagating in random media
859
and Φn+1 is expressed as a function of Φn after inverse Fourier transformation. For the homogeneous anisotropic case, we have to solve (with |γ| = (γ 2x + 1 2 γ y + γ 2y ) 2 ) 1 2(1 − Φ)H = ( |γ| Φ − l2 div(D∇Φ)) l where ∂2Φ ∂2Φ ∂2Φ div(D∇Φ)) = γ x 2 + γ y 2 + γ z 2 ∂x ∂y ∂z Taking the Fourier transform of the two members of the equation, 2lF ((1 − Φ)H) = |γ| F(Φ) + l2 (γ x wx2 + γ y wy2 + γ z wz2 )F(Φ) wx , wy , wz being the three components in the Fourier space. As for the isotropic case, an iterative scheme based on the Fourier transform can be used to find the solution. Eq. (21.18) becomes: F(Φn+1 ) = 2l
F ((1 − Φn )H) |γ| + l2 (γ x wx2 + γ y wy2 + γ z wz2 )
(21.19)
Considering now the crack surface density function (21.7) for the scalar case, we have to solve 2(1 − Φ)H = or γ c l∆Φ = and ∆Φ =
γc 1 ( − l2 ∆Φ) l 2
1 γc − 2(1 − Φ)H 2 l
2 11 − (1 − Φ)H 2 2l γcl
which is a Poisson equation with a second member depending on the solution Φ. The corresponding Green’s function is the harmonic potential: G(x − x0 ) =
1 4π |x − x0 |
Φ can be obtained by iterations, starting with Φ0 (x) = 0. After n iterations, ¶ µ Z 1 1 1 2l 0 0 Φn+1 (x) = − 2 − (1 − Φn (x ))H(x ) dx0 l 4π |x − x0 | 2 γ c As before, the Poisson equation can be iteratively solved by Fourier transforms: µ ¶ 2 11 2 F (1 − Φn )H− 2 = |w| F(Φn+1 ) γcl 2l
860
21.
Crack Paths in Random Media 2
and for |w| 6= 0, F(Φn+1 ) =
F
³
2l γ c (1
− Φn )H− 12
l2 |w|2
´
For the homogeneous anisotropic case, Eq. (21.19) becomes for |w|2 6= 0, 1 with |γ| = (γ 2x + γ 2y + γ 2y ) 2 F(Φn+1 ) =
F
³
2l |γ|c (1
− Φn )H− 12
´
l2 (γ x wx2 + γ y wy2 + γ z wz2 )
FFT solution of the phase field PDE in heterogeneous media In the case of heterogeneous media the crack surface density function (21.5), it is useful to introduce a reference homogeneous medium, with fracture energy γ 0 . Using for the scalar case the perturbation γ(x) = γ 0 (x) + γ 0 , we have to solve 2(1 − Φ)H =
1 0 ((γ (x) + γ 0 )Φ − l2 div((γ 0 (x) + γ 0 )∇Φ)) l
It comes: γ0 (Φ − l2 ∆Φ) + l γ = 0 (Φ − l2 ∆Φ) + l
2(1 − Φ)H =
¢ 1¡ 0 γ (x)Φ − l2 div(γ 0 (x)∇Φ l 1 J(γ 0 (x), Φ) l
where J(γ 0 (x), Φ) = γ 0 (x)Φ − l2 div(γ 0 (x)∇Φ Using the same approach as for Eq. (21.18), the solution Φ is obtained by iterations of (noting i the imaginary number with i2 = −1): 2l F ((1 − Φn )H) 1 F(J(γ 0 (x), Φn )) − (21.20) 2 γ 0 1 + l2 |w| γ0 1 + l2 |w|2 1 ³ ´ (2lF ((1 − Φn )H) − F(γ 0 (x)Φn ) = γ 0 1 + l2 |w|2
F(Φn+1 ) =
∂ ∂ Φn ) + wy F(γ 0 (x) Φn ) ∂x ∂y ¶ ∂ 0 +wz F(γ (x) Φn )) ∂z −l2 i(wx F(γ 0 (x)
(21.21)
For each iteration, the gradient ∇Φn has to be calculated before taking the Fourier transform.
21.3 Phase field for cracks propagating in random media
861
Similarly, an iterative scheme based on a reference anisotropic medium → → → with fracture energy (γ 0x , γ 0y , γ 0z ) can be used. From − γ (x) = − γ 0 (x)+ − γ 0, we get 1 (( |γ 0 | + |γ| − |γ 0 |)Φ − l2 div((D0 + D0 )∇Φ)) l 1 1 = ( |γ 0 | Φ − l2 div(D0 ∇Φ)) + (( |γ| − |γ 0 |)Φ − l2 div(D0 ∇Φ)) l l
2(1 − Φ)H =
After rearrangement, 1 1 ( |γ 0 | Φ − l2 div(D0 ∇Φ)) = 2(1 − Φ)H− (( |γ| − |γ 0 |)Φ − l2 div(D0 ∇Φ)) l l Taking the Fourier transform of the two members of the previous equation, after multiplication by l, we get: |γ 0 | F(Φ) + l2 (γ 0x wx2 + γ 0y wy2 + γ 0z wz2 )F(Φ)
= 2lF(1 − Φ)H − F (( |γ| − |γ 0 |)Φ) + l2 F(div(D0 ∇Φ)) with F(div(D0 ∇Φ)) = −i(wx F(γ 0x
∂Φ ∂Φ ∂Φ ) + wy F(γ 0y ) + wz F(γ 0z )) ∂x ∂y ∂z
Eq. (21.19) becomes, to provide an iterative scheme: (21.22) F(Φn+1 ) 2 0 2lF ((1 − Φn )H) −F (( |γ| − |γ 0 |)Φn ) + l F(div(D ∇Φn )) = . (|γ 0 | + l2 (γ 0x wx2 + γ 0y wy2 + γ 0z wz2 ) Considering now the crack surface density function in Eq. (21.7) for the scalar case, and following [174] for an implementation by FFT, a reference homogeneous medium with fracture energy γ 0 is introduced. We have to solve: 11 0 ((γ (x) + γ 0 ) − l2 div((γ 0 (x) + γ 0 )∇Φ)) 2l ¶ µ 1 1 0 γ 1 γ (x) − l2 div((γ 0 (x) + γ 0 )∇Φ) = 0 ( − l2 ∆Φ) + l 2 l 2
2(1 − Φ)H =
so that γ 1 γ0 1 l ∆Φ 0 = + l 2 l l 2
µ
¶ 1 0 2 0 γ (x) − l div((γ (x) + γ 0 )∇Φ) − 2(1 − Φ)H 2
After Fourier transformation of the two members of this equation, and for |w|2 6= 0,
862
21.
Crack Paths in Random Media
l2 |w|2 F(Φn+1 ) =
¢ 2l 11 ¡ F ((1 − Φn )H)− F γ 0 (x) + γ 0 ) − l2 div(γ 0 (x)∇Φn γ0 2l
The solution Φ is obtained by iterations of µ 2l 1 11 F(Φn+1 ) = (21.23) F ((1 − Φn )H) − F(γ 0 (x) + γ 0 ) γ 0 l2 |w|2 2l ¶ ∂ ∂ ∂ 0 0 0 −li(wx F(γ (x) Φn ) + wy F(γ (x) Φn ) + wz F(γ (x) Φn )) ∂x ∂y ∂z Using now a reference anisotropic medium with fracture energy (γ 0x , γ 0y , → → → γ 0z ), from − γ (x) = − γ 0 (x) + − γ 0 , we get the following iterative scheme: 2(1 − Φ)H =
1 1 ( |γ| − l2 div((D0 + D0 )∇Φ)) l 2
Therefore 1 1 1 − div(D0 ∇Φ) = 2(1 − Φ)H− ( |γ| + l div(D0 ∇Φ)) l l 2 After Fourier transformation, we get the iterative scheme, for |w|2 6= 0: F(Φn+1 )
µ 1 1 = 2 2lF ((1 − Φn )H) − F( |γ|) l (γ 0x wx2 + γ 0y wy2 + γ 0z wz2 ) 2
(21.24)
∂ ∂ ∂ −l i(wx F(γ (x) Φn ) + wy F(γ 0 (x) Φn ) + wz F(γ 0 (x) Φn )) ∂x ∂y ∂z 2
0
¶
21.3.4 Estimation of an effective toughness It is interesting to estimate the effective toughness of fracture energy with the present approach. Once the phase field Φ(x) was computed from images of a material or from simulations of the microstructure, the fracture surface energy dissipated in domain Ω is obtained from the surface energy term of Eq. (21.6). It is derived from Eq. (21.12) or (21.15) (depending on the choice of the crack surface density function) as a function of the increasing load by: Z l 1 2 → E1 (Γ ) = |γ(x)| Φ(x)2 + |γ(x)| (∇Φ.− u γ (x)) dx 2l 2 Ω or Z 1 l 2 → E2 (Γ ) = |γ(x)| Φ(x) + |γ(x)| (∇Φ.− u γ (x)) dx 2l 2 Ω
Similarly an effective cracked surface area Aef f in the domain is estimated from
21.4 Conclusion
Aef f =
Z
863
Φ(x)dx Ω
The phase field Φ(x) and the energy E1 (Γ ) or E2 (Γ ), as well as the effective cracked surface area Aef f depend on the loading conditions and on their history. This can be used to estimate an effective toughness of an heterogeneous medium, when working with an initial crack of length a and applying an homogeneous tensile stress σ on the boundary of the domain, reproducing the conditions of a tensile test on a cracked specimen. Esti→ mations are obtained on realizations of − γ (x) and of the random elasticity tensor. Scale depending fluctuations of the integrals E1 (Γ ), E2 (Γ ), and Aef f can be studied by geostatistical tools as previously done in chapters 3 and 18, in order to define a statistical RVE. Then an effective fracture energy Γef f can be defined by Γ1ef f =
E1 (Γ ) E2 (Γ ) or Γ2ef f = Aef f Aef f
→ In the case of a local anisotropy (through vector − γ (x)) or of a global anisotropy of the microstructure, the effective fracture energy will depend on the direction of propagation of the initial main crack with respect to the orientation of the microstructure, as observed for correlations between toughness and geodesic distances in polycrystalline graphite containing anisotropically distributed elongated voids [266]. Examples of numerical homogenization of the toughness of layered media by a phase field approach [244] show that for a straight propagation in a medium with homogeneous elastic moduli the effective toughness corresponds to its maximal value along the crack path. It corresponds to the assumption of our probabilistic approach presented in section 20.5 of chapter 20. An alternative estimation of toughness is obtained by integration of the curve relating the macroscopic load to the displacement. This approach is followed in studies of optimization of the effective toughness of composite materials with respect to the morphology of reinforcement particles by means of phase field models [135], [136]. Made in the spirit of [45] developed for effective properties, it accounts for the interfacial toughness through the introduction of an additional phase field to describe the interface, and it is solved by Finite Elements numerical calculations.
21.4 Conclusion In many materials like polycrystals, the surface fracture energy cannot be described by a scalar, due to a strong crystallographic anisotropy. This requires extensions of conventional models of fracture. In the first section, probabilistic models of fracture accounting for transgranular and intergran-
864
21.
Crack Paths in Random Media
ular fracture are proposed, based on the existence of non damaged paths in 3D random polycrystals modelled by random tessellations. In the second section, some phase field models of crack initiation and propagation are extended to the case of non homogeneous and anisotropic fracture energy. This approach leads to phase field solutions of non homogeneous Helmholtz and Poisson equations, for which full fields solutions can be obtained by iterations of FFT. These theoretical extensions remain to be tested by means of numerical simulations on models of microstructures, a preliminary implementation in the case of a homogeneous and isotropic fracture energy being introduced in [570]. Once the phase field problem is numerically solved on simulations of a random medium, its effective fracture energy and the corresponding statistical RVE can be estimated as in the case of effective properties in chapter 19.
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Index
Acoustic properties, 764, 767 Anamorphosis, 115, 155, 282, 307, 370, 406, 491, 524—526, 722 Apparent properties, 702, 708, 736 Autodual random set, 26, 107, 116, 264, 268, 354, 366, 485, 624, 660, 686, 727 Bivariate distribution, 78, 88, 261, 280, 372, 389, 408, 493, 541, 543 Boolean RF, 275, 293, 316, 318, 500, 777, 781, 802, 814, 826, 845 compact support, 526 Cox, 297, 323 infinitesimal, 367, 488 sequential Cox, 441, 447 Boolean RS, 126, 165, 607, 664, 718, 730, 742, 749, 760, 767, 833, 834, 836 Cox, 189, 550, 744, 749, 750, 752, 809 Fractal, 218 infinitesimal, 331
Multi component, 211 sequential Cox, 420, 432, 434 multicomponent, 436 Boolean varieties, 197, 224, 299, 677, 688, 719, 779, 781, 783, 787, 788, 790—792 Bounds of effective properties, 635, 644 Change of Scale, 615 Change of support by convolution, 27, 81, 280, 532, 566 by infimum, 19, 27, 35, 267 by supremum, 19, 27, 34, 267 Characteristic function, 115, 533, 535 Choquet’s capacity, 6, 25, 166, 421, 443, 449 functions, 31, 259, 278, 301, 443, 448 sets, 22, 166, 420 Closing functions, 38 sets, 57, 73
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. Jeulin, Morphological Models of Random Structures, Interdisciplinary Applied Mathematics 53, https://doi.org/10.1007/978-3-030-75452-5
915
916
Index
Concrete, 248, 735, 740 Continuous transformation, 24 Correlation function of order m, 27, 107, 120, 133, 169, 631 Counting, 48, 52, 53, 59, 60, 94, 108, 132, 158, 171, 172, 214, 284 Covariance , 27, 721, 729 functions, 78, 374, 541, 566 cross, 390, 408, 535, 538, 566, 633 orientations, 81 sets, 26, 66, 109, 168, 223, 227, 228, 230, 356, 462, 470 cross, 74, 125, 271, 358, 462, 470 Covariogram cross, 74, 535 geometrical, 66, 168, 284, 510, 514, 526 geometrical, normalized, 168, 512, 515 transitive , 535, 538 Crack arrest, 776, 815, 816, 819, 842— 844, 863 path, 847, 848 Damage critical density of defects, 776, 814 random, 832, 833, 838 threshold, 776, 811 Darcy’s permeability, 578, 618, 755, 758, 760, 765 Dead Leaves color, 352, 490, 670, 727 conditional, 459 compact support, 527 intact grains, 137, 338, 342, 344, 382, 403, 415 multivariate RF, 387 rank m, 393 RF, 367, 405, 500
tessellation, 332, 401, 516 conditional, 452 Cox, 341 iteration, 340 transparent RF, 397 varieties RF, 391, 799 Dielectric permittivity, 618, 630, 642, 646 Diffusion, 557, 856 Diffusion in porous media, 579, 755, 759 Digital Material, 701, 718, 840 Dilation functions, 37 sets, 43 Dilution RF, 531, 650, 651, 727 Cox, 552 multivariate varieties, 549, 566 Dirichlet BC, 621, 694, 697, 713 Distance function, 18, 71, 76, 86, 89, 737, 739 Distribution function Beta, 194, 808—810, 835 fields, 690, 731, 734, 737, 743, 744, 762 Gamma, 141 Gaussian, 115, 722, 812, 834 Pareto, 287, 786, 788, 824, 837 Stable, 141 Weibull, 159, 286, 780, 783—785, 789, 791, 792, 804, 805, 809, 836 Effective properties, 616, 620, 630, 634 Elastic moduli, 618, 634, 643, 661, 708, 714, 718, 719, 736, 740, 742, 745, 750, 753, 754 Elasticity linear, 618, 633, 689, 724 nonlinear, 748 Electrostatics, 618, 627, 723, 726, 732 Erosion conditional invariance, 246
Index
functions, 37, 378 sets, 43 Excursion set, 34, 114, 127, 296, 377, 490, 547, 672, 727 FFT, 703, 723, 730, 732, 733, 736, 741, 742, 745, 750, 752, 754, 757, 758, 760, 769, 857, 861 Fibrous media, 201, 202, 401, 404, 753, 763, 764, 767 Field average, 622, 694, 696, 705 Finite Elements computation, 703, 707, 713, 718, 719, 721, 750, 753, 766, 815, 841 Stochastics, 721 Fluid flow, 577, 578, 618, 755, 758, 760 Food microstructure, 121, 712, 718 Fracture criteria global, 775 local, 775 Fracture Defects, 774, 776, 780— 789, 791—798, 800—802, 805— 807, 809—811, 814, 816, 819— 821, 832—834, 836—842 Fracture energy, 775, 851 anisotropic, 855 Fracture probability, 773, 774, 776— 778, 783, 790, 795, 799, 802, 804, 806, 807, 809—812, 814, 815, 819, 822—828, 834—836, 843, 844 Fracture toughness effective, 816, 819, 821, 823, 824, 827, 832, 841, 844, 851, 854, 862, 863 Gaussian RF, 114, 127, 262, 296, 549, 672, 721 Geodesic distance, 89—93, 98, 158, 241, 863 Geodesic paths, 90, 93, 762, 817, 852
917
Geometrical average, 624, 632, 648, 655, 686, 691, 698, 699 Green’s function, 564, 627, 692— 694, 724, 858 Griffith, 776, 815, 816, 844, 851 Hard-core, 137, 196, 330, 342, 502, 519, 571, 668, 750, 754 Hashin composite spheres, 344, 626, 643, 677, 687, 753 Hashin-Shtrikman bounds, 638, 677, 680, 708, 714, 718, 734, 742, 750, 751, 754, 755 Hausdorff dimension, 49, 59, 66, 68, 72, 79, 118, 119, 141, 169, 218, 581 Hermite polynomials, 86, 115, 117, 119, 120, 722 Hierarchy, 215, 272, 306, 416, 594, 653, 673 Homogenization, 615, 620, 630, 702 Image analysis, 53, 398, 590, 788 filtering, 85, 735 microtomography, 62, 93, 98, 174, 186, 248, 592, 702, 733, 735, 754, 767, 855 segmentation, 104, 589, 733, 735 SEM, 10, 36, 121, 124, 183, 186, 293, 315, 330, 369, 387, 398, 401, 405, 406, 702 simulation, 221, 409, 506, 580, 718, 719, 727, 730, 732, 742, 744, 748, 749, 754, 757, 760, 767, 769, 770 TEM, 189, 190, 195, 532, 537, 702, 750—752, 757, 770 Infinite divisibility addition, 534 supremum, 280 union, 175, 200 Integral range, 80, 704, 728, 731, 734, 743, 747, 751, 760, 768
918
Index
KUBC, 623, 695, 697, 709, 713, 718, 719 Lattice gas, 575 Lippmann-Schwinger equation, 628, 633, 723 Localization hotspots, 732, 734 Lognormal RF, 651, 699 lsc (lower semi-continuous) function, 28, 258 random function, 7 transformation, 24 Machine learning, 95, 102, 105, 591 Markov process, 154, 334, 371, 406, 407, 415, 494 Max-stable RF, 280, 781 Minkowski tensors, 55, 106 Minkowski’s functionals, 45—51, 55, 57, 104, 169, 177, 180, 208, 247, 383, 385, 386 MJF, 406 Moments of inertia, 56, 99, 109 Morphological tortuosity, 89, 756— 758, 762 Mosaic RF, 649 Mosaic RS, 668 Voronoi, 707 Multiscale random media, 138, 189, 215, 272, 297, 306, 653, 675, 741, 744, 748, 749 Multivariate mosaic, 269 Multivariate RF, 36, 300 Multivariate statistical analysis, 95, 97, 100, 591 Neumann BC, 621, 694, 697, 713 Neumann expansion, 629, 723 Norm, 46, 245 Numerical simulation, 702, 718, 719, 840 Opening
functions, 38 sets, 57 Optical properties , 732 Overgraph, 29 Pareto, 287 PBC, 621, 623, 694, 695, 697, 713, 733 Percolation, 173, 190, 197, 221, 226, 625, 626, 665, 667, 680, 733, 742—744, 747, 749, 750, 752, 760, 762, 764, 769, 771, 837, 849 Perturbation expansion, 627, 633 Phase field, 851, 852 Point process Cox, 137 Determinantal, 149 Gibbs, 149 Poisson, 133, 604 Poisson mosaic, 265, 649, 669, 810, 821, 843 Poisson polyhedra, 183, 245, 515, 541 Poisson varieties, 80, 175, 188, 197—199, 201, 204, 205, 243, 391, 506, 549, 782, 792 iteration, 143, 208, 605 Polycristalline salt, 338 Porous media, 578, 644, 678, 680, 719, 740, 742, 745, 755, 758, 764 Powder morphology, 398 Primary function, 282, 305, 523, 525 Primary grain, 510 Probabilistic distance, 593, 601 Probabilistic segmentation, 592, 599 Random aggregate, 517, 529, 580 Random closed sets RACS, 6, 19, 20 Random graph, 87 Random marker, 317, 603
Index
Random packing, 342 Random tessellation, 82, 83, 175, 215, 233—235, 238, 240, 241, 243, 254, 255, 281, 311, 312, 314, 316, 326, 546, 649, 653, 669, 673, 675, 833, 848, 849 Random tokens, 539 Random tree branching process, 150, 521, 522 hierarchy, 158, 611 Reaction-Diffusion RF, 558, 567 Ginzburg-Landau , 571, 574 lattice gas, 583 linear model, 563 Schlögl , 568 Turing structure, 568 Rough surface, 103, 290, 499, 554 Rubber, 749 RVE, 83, 703, 706, 707, 717, 719, 731, 733, 734, 737, 747, 751, 752, 754, 755, 757, 760, 768 Second order correlation function, 27, 69, 79, 117, 132, 139, 261, 266, 360, 374, 632, 693 Segment , 59, 170, 267, 335, 379, 410 Self-consistent, 624, 670, 688, 708 Semi Markovian RACS, 175 Sequential alternate RF, 487 Cox, 498 multivariate, 502 varieties, 506 Sequential model, 329, 419, 487 Shape analysis, 95 Simulation plasma coating, 586 Size distribution , 57, 349, 400 measure, 59 number, 60 spheres, 64, 511 Spatial law, 25, 27, 69, 280 Specific connectivity number 2D, 53, 73, 108, 171, 214, 364 3D, 52, 54, 73, 108, 171, 214, 364
919
Specific measurements, 50, 108, 336 Spectral measure, 658 Steiner’s formula, 49, 169, 383, 385, 787 STIT mosaic, 267, 649, 669, 821, 843 SUBC, 623, 695, 697, 709, 713 Subgraph, 29, 276, 489 SVE, 841 Tessellation Cauwe, 249 Cox, 238, 323 from local metrics, 312, 314 Johnson-Mehl, 239, 314 Laguerre, 240, 314 Poisson, 243, 256, 270, 272, 649, 670, 675, 794, 825 STIT, 253, 256 Voronoi, 236, 312, 803, 805, 807, 810, 851 Thermal conductivity, 708, 713, 718, 719, 733, 741, 753 Third order bounds, 646, 654, 656, 661, 664, 678, 725, 731, 742, 744, 747, 750 Third order correlation function, 119, 132, 139, 169, 263, 266, 361, 377, 541, 647, 654 Three components RS, 124, 293, 553 Three points Probability, 108, 169, 193, 359, 655, 664, 836 usc (upper semi continuous) function, 28, 258 random function, 7 transformation, 24 Variance estimation, 80, 83, 704 fields, 684, 686, 706, 727 point, 684, 705 scaling, 84, 204, 706, 720, 754
920
Index
Variational principle, 635, 851 Variogram of order 1, 81, 106, 179, 261, 262, 374, 375, 495, 496 Variogram of order 2, 80, 261, 262, 374, 375, 496, 501, 566, 579, 756 Viscoelasticity, 752 Watershed segmentation, 592 stochastic, 592, 733 tessellation, 241, 317, 323 Weakest link, 19, 775, 776, 780, 789, 792, 799, 801—805, 809, 841 Weibull, 286, 776, 777, 780, 784— 787, 792, 805, 815, 833