Euler characteristic

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Table of contents

Introduction
1 Morphological modeling: a generalized method based on excursion sets
1 Introduction
2 Review on correlated Random Fields
2.1 Basic definitions
2.2 Gaussian and Gaussian related distribution
2.3 Covariance functions
2.4 Numerical implementation
3 Excursion set theory
3.1 General principle
3.2 Measures of excursion set
3.3 Expectation formula
4 Application to cementitious materials at different scales
4.1 Meso-scale modeling of concrete
4.2 Micro-scale modeling of cement paste
5 Analytical model for size effect of brittle material
5.1 Correlation lengths as scale parameters
5.2 One-dimensional case
5.3 Validation, results and comments
6 Continuous percolation on finite size domains
6.1 Accounting for side effects
6.2 Representative Volume Element for percolation
7 Concluding remarks
2 Unified numerical implementation of quasi-brittle behavior for heterogeneous material through the Embedded Finite Element Method
1 Introduction
2 Kinematics of strain and displacement discontinuity
2.1 Jump in the strain field
2.2 Jump in the displacement field
2.3 Strong discontinuity analysis
3 Variational formulation
3.1 Three-field variational formulation
3.2 Assumed strain and double enhancement
3.3 Finite Element interpolation
4 Discrete model at the discontinuity
4.1 Localization
4.2 Traction-separation law
5 Resolution methodology
5.1 Integration and linearization with constant strain elements .
5.2 Solving the system
5.3 Application to spatial truss and spatial frame
5.4 Application to volume Finite Elements
6 Concluding remarks
3 Applications to cementitious materials modeling
1 Introduction
2 One-dimensional macroscopic loading paths
2.1 Analysis of the asymmetricmacroscopic response for traction and compression loading paths
2.2 Transversal strain
2.3 Dissipated energy
2.4 Induced anisotropy
2.5 Uniaxial cyclic compression loading
3 Representative Volume Element for elastic and failure properties
3.1 Experimental protocol
3.2 Young moduli analysis
3.3 Tensile and compressive strength
3.4 Comments
4 Application to the Delayed Ettringite Formation
4.1 Numerical simulation
4.2 Homogeneous mortar expansion
4.3 Residual Young modulus
5 Concluding remarks
Conclusions and perspectives
A Gaussian Minkowski Functionals
1 Volume of the unit ball
2 Probabilist Hermite polynomials
3 Gaussian volumes of spherical set in Rk
B Correlated Random Fields
1 Orthogonal decomposition of correlated random fields
2 Finite Element discretization of the Fredholm problem
3 The turning band method
Bibliography