Cellular and Porous Materials in Structures and Processes

Title Page

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PREFACE

Table of Contents

Fracture Mechanics of Foams

1 Fundamentals of Fracture Mechanics

1.1 Introduction

1.2 Linear Elastic Fracture Mechanics

1.3 Crack tip stress and displacement fields in anisotropic materials

2 Experimental Determination of Fracture Toughness of Foam Materials

2.1 Tear Test for Flexible Cellular Materials

2.2 Standard Test Methods for Plane-Strain FractureToughness and Strain Energy Release Rate of Plastic Materials

2.3 Fracture Toughness Experimental Results

2.4 Impact Fracture Toughness

3 Micromechanical Models for Foams Fracture

4 Concluding Remarks

Bibliography

Finite Element Modeling of Foams

1 Introduction

2 Homogenization and the Unit Cell Method

3 Micro-Mechanical Finite Element Models of Cellular Materials

3.1 Introduction

3.2 Open-Cell Foams

3.3 Closed-Cell Foams

3.4 Open-Cell Foams with Hollow Struts

4 Micro-Mechanical Models - Methods and Results

4.1 Elastic Properties

4.2 Yielding

4.3 Buckling

4.4 Densification

4.5 Fracture

5 Optimization of Foam Density Distribution

6 Summary

Bibliography

Plasticity of Three-dimensional Foams

1 Fundamentals of Continuum Mechanics

1.1 Stress Tensor and Decomposition

1.2 Invariants

1.3 Constitutive Equations

1.4 Linear Elastic Behaviour: Generalised Hooke'sLaw for Isotropic Materials

2 Constitutive Relationships for Pressure Sensitive Materials: Systematic Overview

3 Simple Cubic Cell Models based on Beams and Shells for Open and Closed Cell Materials

3.1 Relative Density

3.2 Geometrical Moment of Inertia

3.3 Young's Modulus

3.4 Shear Modulus and Poisson's Ratio

3.5 Yield Stress

4 Procedures to Determine the Influence of the Hydrostatic Stress on the Yield Behaviour

5 Implementation of New Constitutive Equations into Commercial Finite Element Codes

5.1 One-Dimensional Drucker-Prager Yield Condition

5.2 Integration of the Constitutive Equations

5.3 Mathematical Derivation of the Fully Implicit Backward Euler Algorithm

5.4 Example Problem: Return Mapping for Ideal Plasticity and Linear Hardening

Bibliography

Thin-walled Structures Made of Foams

1 Introduction

1.1 Plates as Structural Elements

1.2 Foams as a Material for Structural Elements

2 Direct Two-dimensional Plate Theory

2.1 Classical Approaches in the Plate Theory

2.2 Governing Equations

2.3 Material-independent Equations

2.4 Two-dimensional Constitutive Equations

2.5 Basic Equations in Cartesian Coordinates

3 Stiffness Identification

3.1 Orthotropic Material Behavior

3.2 Classical Stiffness Values

3.3 Non-classical Stiffness Values

3.4 Special Case - Isotropic Behavior

4 Examples of Effective Stiffness Properties Estimates

4.1 Homogeneous Plate

4.2 Classical Sandwich Plate in Reissner's Sense

4.3 Functionally Graded Materials and Foams

4.4 On the Plates Made of Nanofoams

5 Symmetric Orthotropic Plate - Static Case

5.1 Bending Problem - One-dimensional Case

5.2 Bending Problem - Two-dimensional Case

5.3 Bending of an Isotropic Plate

5.4 Bending of an Elastic Plate Made of FGM (SymmetricCase)

6 Dynamics of Plates Made of an Elastic Foam

6.1 Equations of Motion for a Symmetric Isotropic Plate

6.2 Free Oscillations and Dispersion curves of a Rectangular Plate

7 Plate Made of a Linear Viscoelastic Material

7.1 Constitutive Equations

7.2 Effective Properties

7.3 Bounds for the Eigen-values

7.4 Quasi-static Behavior of a Symmetric Orthotropic Plate

7.5 Examples of Effective Stiffness Relaxation Functions

7.6 Bending of a Viscoelastic Plate

8 Plate Theory Deduced from the Cosserat Continuu

8.1 Two-dimensional Governing Equations

8.2 Reduction of theThree-dimensional Micropolar Equations

9 Summary

Bibliography

Plasticity Theory of Porous and Powder Metals

1 Introduction

2 Fundamentals of the Theory of Plasticity

2.1 Rigid Perfectly/Plastic Solids

2.2 Rigid Plastic Hardening Solids

2.3 Rigid Viscoplastic Solids

2.4 Maximum Friction Law and Singular VelocityFields (Rigid Perfectly/Plast ic Material)

2.5 Maximum Friction Law and Other Models of Pressure-independent Plasticity

3 Plasticity Theory for Porous and Powder Metals Based on the Associated Flow Rule

3.1 Preliminaries

3.2 Yield Criteria and the Associated Flow Rule for Porous and Powder Materials

3.3 Additional Remarks on the Yield Criteria

3.4 Simple Analytic Example

4 Plasticity Theory for Porous and Powder Metals Based on Non-associated Flow Rules

4.1 Stress Equations

4.2 Kinematic Theories

4.3 The Coaxial Model

4.4 The Double-shearing Model

4.5 The Double-slip and Rotation Model

5 Qualitative Behavior of Plastic Solutions for Porous and Powder Metals in the Vicinity of Frictional Interfaces

5.1 Preliminaries

5.2 Statement of the Problem

5.3 Solution for Stresses

5.4 Solutions for Velocities

5.5 Frictional Boundary Condition

5.6 Solution for Pressure-independent Plasticity

5.7 Singularity in Velocity Fields

6 Applications

Bibliography

Impact of Cellular Materials

1 Introduction

2 Wave Propagation in a Cellular Rod

3 Rigid Object Strikes on a Cellular Rod of Fixed End

3.1 Basic Assumptions

3.2 Shock Wave Analysis

4 Rigid Object Strikes on a Free Cellular Rod

5 Concluding Remarks

Bibliography