Micromechanical analysis and multi-scale modeling using the Voronoi cell finite element method

714 Pages·2011·16.189 MB·English

by  Somnath Ghosh

Mechanical Engineering
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s Micromechanical
Micromechanical Analysis and Multi-Scale Modeling i
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Using the Voronoi Cell Finite Element Method g r
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Somnath Ghosh th m
e e Analysis and

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n n
o i Multi-Scale
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Modeling
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As multi-phase metal/alloy systems and polymer, ceramic, or metal matrix composite materials are t y
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increasingly being used in industry, the science and technology for these heterogeneous materials have s
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advanced rapidly. By extending analytical and numerical models, engineers can analyze failure characteristics l s
e
of the materials before they are integrated into the design process. Micromechanical Analysis and Multi- m a Using the Voronoi Cell Finite
Scale Modeling Using the Voronoi Cell Finite Element Method addresses the key problem of multi-scale n
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failure and deformation of materials that have complex microstructures. The book presents a comprehensive n d
Element Method
computational mechanics and materials science–based framework for multi-scale analysis. t
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The focus is on micromechanical analysis using the Voronoi cell finite element method (VCFEM) developed e u
by the author and his research group for the efficient and accurate modeling of materials with non- t l
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uniform heterogeneous microstructures. While the topics covered in the book encompass the macroscopic i
o
-
scale of structural components and the microscopic scale of constituent heterogeneities such as inclusions d S
or voids, the general framework may be extended to other scales as well.
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The book presents the major components of the multi-scale analysis framework in three parts. Dealing
l
with multi-scale image analysis and characterization, the first part of the book covers 2D and 3D image- e

based microstructure generation and tessellation into Voronoi cells. The second part develops VCFEM M
for micromechanical stress and failure analysis, as well as thermal analysis, of extended microstructural
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regions. It examines a range of problems solved by VCFEM, from heat transfer and stress–strain analysis
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of elastic, elastic–plastic, and viscoplastic material microstructures to microstructural damage models
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including interfacial debonding and ductile failure. Establishing the multi-scale framework for heterogeneous
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materials with and without damage, the third part of the book discusses adaptive concurrent multi-scale i
n
analysis incorporating bottom-up and top-down modeling.
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Including numerical examples and a CD-ROM with VCFEM source codes and input/output files, this book
is a valuable reference for researchers, engineers, and professionals involved with predicting the
performance and failure of materials in structure–materials interactions.
Ghosh
94378
Somnath Ghosh
6000 Broken Sound Parkway, NW
Suite 300, Boca Raton, FL 33487
711 Third Avenue INCLUDES CD-ROM
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Abingdon, Oxon OX14 4RN, UK
Micromechanical
Analysis and
Multi-Scale
Modeling
Using the Voronoi Cell Finite
Element Method
CRC Series in
COMPUTATIONAL MECHANICS
and APPLIED ANALYSIS
Series Editor: J.N. Reddy
Texas A&M University
Published Titles
ADVANCED THERMODYNAMICS ENGINEERING, Second Edition
Kalyan Annamalai, Ishwar K. Puri, and Miland Jog
APPLIED FUNCTIONAL ANALYSIS
J. Tinsley Oden and Leszek F. Demkowicz
COMBUSTION SCIENCE AND ENGINEERING
Kalyan Annamalai and Ishwar K. Puri
CONTINUUM MECHANICS FOR ENGINEERS, Third Edition
Thomas Mase, ROnald Smelser, and George E. Mase
DYNAMICS IN ENGINEERING PRACTICE, Tenth Edition
Dara W. Childs
EXACT SOLUTIONS FOR BUCKLING OF STRUCTURAL MEMBERS
C.M. Wang, C.Y. Wang, and J.N. Reddy
THE FINITE ELEMENT METHOD IN HEAT TRANSFER AND FLUID DYNAMICS,
Third Edition
J.N. Reddy and D.K. Gartling
MECHANICS OF LAMINATED COMPOSITE PLATES AND SHELLS: THEORY
AND ANALYSIS, Second Edition
J.N. Reddy
MICROMECHANICAL ANALYSIS AND MULTI-SCALE MODELING USING
THE VORONOI CELL FINITE ELEMENT METHOD
Somnath Ghosh, Ohio State University, Columbus, USA
NUMERICAL AND ANALYTICAL METHODS WITH MATLAB®
William Bober, Chi-Tay Tsai, and Oren Masory
PRACTICAL ANALYSIS OF COMPOSITE LAMINATES
J.N. Reddy and Antonio Miravete
SOLVING ORDINARY AND PARTIAL BOUNDARY VALUE PROBLEMS
IN SCIENCE and ENGINEERING
Karel Rektorys
STRESSES IN BEAMS, PLATES, AND SHELLS, Third Edition
Ansel C. Ugural
Micromechanical
Analysis and
Multi-Scale
Modeling
Using the Voronoi Cell Finite
Element Method
Somnath Ghosh
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
CRC Press
Taylor & Francis Group
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© 2011 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20111012
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I dedicate this book to the Lotus Feet of Shri Sai Baba, the beacon of my life.
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Contents
Preface xxi
About the Author xxv
1 Introduction 1
2 Image Extraction and Virtual Microstructure Simulation 5
2.1 Multi-Scale Simulation of High-Resolution Microstructures . 6
2.1.1 Resolution Augmentation Problem . . . . . . . . . . . 9
2.1.2 Wavelet-Based Interpolation in the WIGE Algorithm. 10
2.1.2.1 A brief discussion of wavelet basis functions. 10
2.1.2.2 Wavelet interpolated indicator functions . . . 11
2.1.3 Gradient-Based Probabilistic Enhancement of
Interpolated Images in the WIGE Algorithm . . . . . 14
2.1.3.1 Accounting for relative locations of the
calibrating and simulated micrographs . . . . 19
2.1.3.2 A validation test for the WIGE algorithm . . 20
2.1.4 Binary Image Processing for Noise Filtering . . . . . . 22
2.2 Three-Dimensional Simulation of Microstructures with
Dispersed Particulates . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 Serial Sectioning and Computer Assembly for
Microstructure Simulation . . . . . . . . . . . . . . . . 27
2.2.2 Generating Equivalent Microstructures from Section
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.3 Stereological Methods in Size and Shape Distributions 33
2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 2D- and 3D-Mesh Generation by Voronoi Tessellation 39
3.1 Two-Dimensional Dirichlet Tessellations in Plane . . . . . . 41
3.2 Mesh Generator Algorithm . . . . . . . . . . . . . . . . . . . 42
3.2.1 Dirichlet Tessellations with Dispersed Points . . . . . 43
3.2.2 Non-Convex and Multiply Connected Domains . . . . 45
3.2.3 Effects of Size and Shape of Dispersed Inclusions . . . 47
3.2.3.1 Modified tessellation for irregular shapes . . 49
3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Voronoi Tessellation for Three-Dimensional Mesh Generation 52
3.4.1 Algorithm for Point-Based Tessellation . . . . . . . . . 53
vii
viii Micromechanical Analysis and Multi-Scale Modeling Using the VCFEM
3.4.2 Surface-Based Voronoi Tessellation for Ellipsoidal
Heterogeneities . . . . . . . . . . . . . . . . . . . . . . 57
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Microstructure Characterization and Morphology-Based
Domain Partitioning 63
4.1 Characterization of Computer-Generated Microstructures . . 66
4.1.1 Statistical Analysis . . . . . . . . . . . . . . . . . . . . 69
4.1.1.1 Mean and standard deviation of microstruc-
tural parameters . . . . . . . . . . . . . . . . 71
4.1.1.2 Cumulative distribution and probability
density functions . . . . . . . . . . . . . . . . 72
4.1.1.3 Second-order intensity function and pair
distribution function . . . . . . . . . . . . . . 75
4.2 Quantitative Characterization of Real 3D Microstructures . . 79
4.2.1 Particulate Geometry . . . . . . . . . . . . . . . . . . 79
4.2.1.1 Mean and standard deviations . . . . . . . . 79
4.2.1.2 Size effects . . . . . . . . . . . . . . . . . . . 80
4.2.1.3 Shape effects . . . . . . . . . . . . . . . . . . 81
4.2.2 Spatial Distribution Patterns . . . . . . . . . . . . . . 82
4.2.2.1 Second-order statistics. . . . . . . . . . . . . 82
4.2.2.2 Local area and volume fractions . . . . . . . 85
4.2.2.3 Neighbor distances . . . . . . . . . . . . . . . 86
4.2.3 Morphological Anisotropy . . . . . . . . . . . . . . . . 89
4.2.3.1 Nearest-neighbor orientation . . . . . . . . . 90
4.2.3.2 Mean intercept cell length. . . . . . . . . . . 91
4.2.3.3 Measures of particle orientation . . . . . . . 91
4.3 Domain Partitioning: A Pre-Processor for
Multi-Scale Modeling . . . . . . . . . . . . . . . . . . . . . . 92
4.3.1 Functions for Microstructure Characterization. . . . . 93
4.3.2 Size Descriptors . . . . . . . . . . . . . . . . . . . . . 93
4.3.3 Shape Descriptors . . . . . . . . . . . . . . . . . . . . 93
4.3.4 Spatial Distribution Descriptors. . . . . . . . . . . . . 94
4.3.4.1 Covariance function . . . . . . . . . . . . . . 95
4.3.4.2 Cluster index . . . . . . . . . . . . . . . . . . 97
4.3.4.3 Cluster contour . . . . . . . . . . . . . . . . 98
4.3.5 Characterization of the W319 Microstructure . . . . . 99
4.3.6 Identification of Effective Spatial Distribution
Descriptors . . . . . . . . . . . . . . . . . . . . . . . . 101
4.3.7 Domain Partitioning Using Characterization Functions 104
4.3.8 StatisticalHomogeneityandHomogeneousLengthScale
(L ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
H
4.3.9 Multi-Scale Domain Partitioning Criteria . . . . . . . 106
4.3.10 NumericalExecutionoftheMDPMethodontheW319
Alloy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Contents ix
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5 The Voronoi Cell Finite Element Method (VCFEM) for 2D
Elastic Problems 115
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.1.1 VCFEM Depiction and Motivation . . . . . . . . . . . 118
5.2 Energy Minimization Principles in VCFEM Formulation . . 119
5.2.1 Euler Equations and Weak Forms . . . . . . . . . . . 121
5.3 Element Interpolations and Assumptions . . . . . . . . . . . 124
5.3.1 Equilibrated Stress Representation . . . . . . . . . . . 124
5.3.2 Augmented Stress Functions Accounting for Interfaces 127
5.3.2.1 Example of stress function convergence . . . 131
5.3.3 Stress Functions for Irregular Heterogeneities Using
Numerical Conformal Mapping . . . . . . . . . . . . . 132
5.3.3.1 Multi-resolution wavelet functions for
irregular heterogeneities with sharp corners . 138
5.3.3.2 Implementation of multi-resolution wavelet
stress functions . . . . . . . . . . . . . . . . . 140
5.3.4 Compatible Displacement Interpolations on Element
Boundary and Interfaces . . . . . . . . . . . . . . . . . 141
5.4 Weak Forms in the VCFEM Variational Formulation . . . . 142
5.4.1 Kinematic Relations . . . . . . . . . . . . . . . . . . . 142
5.4.2 Traction Reciprocity Conditions . . . . . . . . . . . . 143
5.5 Solution Methodology and Numerical Aspects in VCFEM . . 144
5.5.1 Scaled Representation of Stress Functions ΦM and ΦI 145
5.5.2 Numerical Integration of Element Matrices . . . . . . 146
5.5.2.1 Integration scheme for heterogeneities of
irregular shape . . . . . . . . . . . . . . . . . 147
5.6 Stability and Convergence of VCFEM . . . . . . . . . . . . . 150
5.6.1 Numerical Implementation of Stability Conditions . . 154
5.6.2 Alternative Method of Constraining Rigid-Body Modes
at Internal Interface . . . . . . . . . . . . . . . . . . . 155
5.7 Error Analysis and Adaptivity in VCFEM . . . . . . . . . . 157
5.7.1 Characterization of Error in Traction Reciprocity . . . 160
5.7.2 Characterization of Error in Kinematic Relationships. 164
5.7.3 The Adaptation Process . . . . . . . . . . . . . . . . . 165
5.7.3.1 Displacement adaptation for traction
reciprocity . . . . . . . . . . . . . . . . . . . 165
5.7.3.2 Stress adaptation for improved kinematics . 168
5.8 Numerical Examples with 2D Adaptive VCFEM . . . . . . . 169
5.8.1 MicrostructureswithDifferentDistributionsofCircular
Heterogeneities . . . . . . . . . . . . . . . . . . . . . . 170
5.8.1.1 Composite material microstructure. . . . . . 172
5.8.1.2 Material microstructures with voids . . . . . 176
5.8.2 Effect of Heterogeneity Size on Adapted Solutions . . 178