Multiscale Modeling Approaches for Composites

✍ Scribed by George Chatzigeorgiou, Fodil Meraghni, Nicolas Charalambakis

• Publisher: Elsevier
• Year: 2022
• Tongue: English
• Leaves: 342
• Edition: 1
• Category: Library

✦ Synopsis

Multiscale Modeling Approaches for Composites outlines the fundamentals of common multiscale modeling techniques and provides detailed guidance for putting them into practice. Various homogenization methods are presented in a simple, didactic manner, with an array of numerical examples.

The book starts by covering the theoretical underpinnings of tensors and continuum mechanics concepts, then passes to actual micromechanic techniques for composite media and laminate plates. In the last chapters the book covers advanced topics in homogenization, including Green’s tensor, Hashin-Shtrikman bounds, and special types of problems. All chapters feature comprehensive analytical and numerical examples (Python and ABAQUS scripts) to better illustrate the theory.

✦ Table of Contents

About the authors
Foreword
Preface
Acknowledgments
Contents
1 Tensors
1.1 Tensors in Cartesian coordinates
1.2 Cartesian systems and tensor rotation
1.3 Tensor calculus
1.4 Examples in tensor operations
1.5 Voigt notation: general aspects
1.6 Operations using the Voigt notation
1.7 Tensor rotation in Voigt notation
1.8 Examples in Voigt notation operations
References
2 Continuum Mechanics.pdf
2 Continuum mechanics
2.1 Strain
2.2 Stress
2.3 Elasticity
2.3.1 General aspects
2.3.2 Special symmetries
2.3.2.1 Monoclinic materials
2.3.2.2 Orthotropic materials
2.3.2.3 Transversely isotropic materials
2.3.2.4 Isotropic materials
2.4 Reduction to 2–D problems
2.4.1 Plane strain
2.4.2 Plane stress
2.5 Examples
References
3 General concepts of micromechanics
3.1 Heterogeneous media
3.2 Homogenization
3.3 Homogenization principles
3.3.1 Average theorems
3.3.1.1 Average stress theorem
3.3.1.2 Average strain theorem
3.3.2 Hill–Mandel principle
3.3.3 Concentration tensors
3.3.3.1 Properties of concentration tensors
3.4 Bounds in the overall response
3.4.1 Voigt and Reuss bounds
3.4.2 Hashin–Shtrikman bounds
3.5 Examples
References
4 Voigt and Reuss bounds
4.1 Theory
4.1.1 Voigt upper bound
4.1.2 Reuss lower bound
4.2 Simple methods for fiber composites
4.3 Composite beams
4.3.1 Essential elements of beam bending theory
4.3.2 Beam made of two materials
4.4 Examples
References
5 Eshelby solution–based mean–field methods
5.1 Inclusion problems
5.1.1 Eshelby's inclusion problem
5.1.2 Inhomogeneity problem
5.2 Eshelby–based homogenization approaches
5.2.1 Eshelby dilute
5.2.2 Mori–Tanaka
5.2.3 Self–consistent
5.3 Examples
References
6 Periodic homogenization
6.1 Preliminaries
6.2 Theoretical background
6.3 Computation of the overall elasticity tensor
6.4 Particular case: multilayered composite
6.5 Examples
References
7 Classical laminate theory
7.1 Introduction
7.2 Stress–strain relation for an orthotropic material
7.2.1 From tensor to contracted (Voigt) notation
7.2.2 Hooke's law for orthotropic material in Voigt notation
7.3 Hooke's law for an orthotropic lamina under the assumption of plane stress
7.4 Stress–strain relations for a lamina of arbitrary orientation: off–axis loading
7.4.1 Stress and strain in global axes (x-y)
7.4.2 Off–axis stress–strain relations
7.4.3 Off–axis strain–stress relations
7.4.4 Engineering constants and induced coefficients of shear–axial strain mutual influence in an angle lamina
7.4.5 Example
7.5 Macromechanical response of a laminate composite thin plate
7.5.1 Laminate code and convention
7.5.2 Laminated thin plates and Kirchhoff–Love hypothesis
7.5.3 Kinematics of thin laminated plates and strain–displacement relation
7.5.4 Stress variation in a laminate
7.5.5 Force and moment resultants related to midplane strains and curvatures
7.5.6 Physical meaning of some coupling components of the laminates stiffness matrices
7.5.7 Workflow and summary
7.5.8 Example
References
8 Composite sphere/cylinder assemblage
8.1 Composite sphere assemblage
8.2 Composite cylinder assemblage
8.3 Eshelby's energy principle
8.4 Universal relations for fiber composites
8.5 Examples
References
9 Green's tensor
9.1 Preliminaries
9.1.1 Fourier transform
9.1.2 Betti's reciprocal theorem
9.2 Definition and properties
9.3 Applications of Green's tensor
9.3.1 Infinite homogeneous body with varying eigenstresses
9.3.2 Eshelby's inclusion problem
9.4 Examples
References
10 Hashin–Shtrikman bounds
10.1 Preliminaries
10.1.1 Positive and negative definite matrices
10.1.2 Calculus of variations
10.2 Hashin–Shtrikman variational principle
10.3 Bounds in a bi–phase composite
10.4 Examples
References
11 Mathematical homogenization theory
11.1 Preliminaries
11.2 Variational formulation
11.2.1 Functional spaces
11.2.2 Homogeneous body
11.2.3 Heterogeneous body with a surface of discontinuity
11.2.4 Approximating functions
11.2.5 Finite element method
11.3 Convergence of the heterogeneous problem
11.3.1 Weak convergence
11.3.2 Mathematical homogenization
11.4 Asymptotic expansion approach
11.5 Examples
References
12 Nonlinear composites
12.1 Introduction
12.2 Inelastic mechanisms in periodic homogenization
12.3 Inelastic mechanisms in mean–field theories
12.3.1 Inhomogeneity problem with two eigenstrains
12.3.2 Mori–Tanaka/TFA method for composites with inelastic strains
12.4 Examples
References
A Fiber orientation in composites
A.1 Introduction
A.2 Reinforcement orientation in a plane
A.3 Reinforcement orientation in 3–D space
A.4 Examples
References
Index