Modeling and Simulation of Functionalized Materials for Additive Manufacturing and 3D Printing: Continuous and Discrete Media: Continuum and Discrete Element Methods

308 Pages·2018·8.281 MB·English

by  Tarek I. Zohdi (auth.)

Lecture Notes in Applied and Computational Mechanics 60
Tarek I. Zohdi
Modeling and Simulation
of Functionalized
Materials for Additive
Manufacturing and 3D
Printing: Continuous and
Discrete Media
Continuum and
Discrete Element Methods
Lecture Notes in Applied and Computational
Mechanics
Volume 60
Series editors
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Peter Eberhard, University of Stuttgart, Stuttgart, Germany
e-mail: [email protected]
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More information about this series at http://www.springer.com/series/4623
Tarek I. Zohdi
Modeling and Simulation
of Functionalized Materials
for Additive Manufacturing
and 3D Printing: Continuous
and Discrete Media
Continuum and Discrete Element Methods
123
Tarek I.Zohdi
University of California
Berkeley, CA
USA
ISSN 1613-7736 ISSN 1860-0816 (electronic)
Lecture Notesin AppliedandComputational Mechanics
ISBN978-3-319-70077-9 ISBN978-3-319-70079-3 (eBook)
https://doi.org/10.1007/978-3-319-70079-3
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To my patient and loving wife, Britta
Schöenfelder-Zohdi and to the memory of my
close friend, colleague and mentor, David
Dornfeld.
Preface
Withinthelastdecade,severalindustrializedcountrieshavestressedtheimportance
of advanced manufacturing to their economies. Many of these plans have high-
lightedthedevelopmentofadditivemanufacturingtechniques,suchas3Dprinting
which, as of 2018, are still in their infancy. The objective is to develop superior
products,producedatloweroveralloperationalcosts.Forthesegoalstoberealized,
adeepunderstandingoftheessentialingredientscomprisingthematerialsinvolved
in additive manufacturing is needed. The combination of rigorous material mod-
eling theories coupled with the dramatic increase of computational power can
potentially play a significant role in the analysis, control, and design of many
emerging additive manufacturing processes. Specialized materials and the precise
design of their properties are key factors in these processes. Specifically,
particle-functionalized materials play a central role in this field, in three main
regimes:
(cid:129) (1) To enhance overall filament-based material properties, by embedding par-
ticles within a binder, which is then passed through a heating element and
deposited onto a surface,
(cid:129) (2) To “functionalize” inks by adding particles to freely flowing solvents
forming a mixture, which is then deposited onto a surface, and
(cid:129) (3) Todirectlydepositparticles,asdrypowders, onto surfacesandthen toheat
them with a laser, e-beam, or other external sources, in order to fuse them into
place.
The goal of these processes is primarily to build surface structures which are
extremely difficult to construct using classical manufacturing methods. The
objective of this monograph is to introduce the readers to basic techniques which
canallowthemtorapidlydevelopandanalyzeparticulate-basedmaterialsneededin
such additive manufacturing processes.
This monograph is broken into two main parts: “Continuum Method”
(CM) approaches and “Discrete Element Method” (DEM) approaches. The mate-
rials associated with methods (1) and (2) are closely related types of continua
(particles embedded in a continuous binder) and are treated using continuum
vii
viii Preface
approaches. The materials in method (3), which are of a discrete particulate char-
acter, are analyzed using discrete element methods. I am certain that, despite
painstaking efforts, there remain errors of one sort or another in this monograph.
Therefore, I would be grateful if readers who find such flaws could contact me at
[email protected].
This document is under copyright. No part can be copied, electronically
stored, transmitted, reproduced, or translated into another language without
written permission from Tarek I. Zohdi.
Berkeley, USA Tarek I. Zohdi
September 2017
Contents
1 Introduction: Additive/3D Printing Materials—Filaments,
Functionalized Inks, and Powders. . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Continuum Methods (CM): Basic Continuum Mechanics . . . . . . . . 9
2.1 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Kinematics of Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Deformation of Line Elements . . . . . . . . . . . . . . . . . . 11
2.3 Equilibrium/Kinetics of Continua. . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Postulates on Volume and Surface Quantities. . . . . . . . 12
2.3.2 Balance Law Formulations . . . . . . . . . . . . . . . . . . . . . 14
2.4 The First Law of Thermodynamics/An Energy Balance. . . . . . . 14
2.5 Linearly Elastic Constitutive Equations . . . . . . . . . . . . . . . . . . 16
2.5.1 The Infinitesimal Strain Case . . . . . . . . . . . . . . . . . . . 16
2.5.2 Material Response . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.3 Material Component Interpretation . . . . . . . . . . . . . . . 18
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 CM Approaches: Characterization of Particle-Functionalized
Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Basic Micro–Macro Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 Testing Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.2 The Average Strain Theorem . . . . . . . . . . . . . . . . . . . 24
3.2.3 The Average Stress Theorem . . . . . . . . . . . . . . . . . . . 25
3.2.4 Satisfaction of Hill’s Energy Condition . . . . . . . . . . . . 25
3.2.5 The Hill–Reuss–Voigt Bounds . . . . . . . . . . . . . . . . . . 26
3.2.6 Improved Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
ix
x Contents
4 CM Approaches: Estimation and Optimization of the Effective
Properties of Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1 Combining Bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Local Fields: Stresses and Strains . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Optimization: Formulation of a Cost Function . . . . . . . . . . . . . 34
4.4 Suboptimal Properties Due to Defects—Effects of
Pores/voids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5 CM Approaches: Numerical Thermo-Mechanical Formulations . . . 43
5.1 Transient Thermo-Mechanical Coupled Fields . . . . . . . . . . . . . 44
5.2 Iterative Staggering Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3 Temporal Discretization of Fields . . . . . . . . . . . . . . . . . . . . . . 50
5.4 The Overall Solution Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.5 Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.6 Summary and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.7 Chapter Appendix 1: Summary of Spatial Finite Difference
Stencils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.8 Chapter Appendix 2: Second-Order Temporal Discretization . . . 63
5.9 Chapter Appendix 3: Temporally Adaptive Iterative
Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.10 Chapter Appendix 4: Laser Processing. . . . . . . . . . . . . . . . . . . 67
5.10.1 Formulations for Particulate-Laden Continua . . . . . . . . 68
5.10.2 A Specific Numerical Example—Controlled
Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.10.3 Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.10.4 Extensions: Advanced Models for Conduction
Utilizing Thermal Relaxation . . . . . . . . . . . . . . . . . . . 75
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6 PART II—Discrete Element Method (DEM) Approaches:
Dynamic Powder Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.1 Direct Particle Representation/Calculations. . . . . . . . . . . . . . . . 86
6.1.1 Comments on Rolling. . . . . . . . . . . . . . . . . . . . . . . . . 86
6.1.2 Particle-to-particle Contact Forces . . . . . . . . . . . . . . . . 87
6.1.3 Particle-Wall Contact . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.1.4 Contact Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.1.5 Regularized Contact Friction Models. . . . . . . . . . . . . . 89
6.1.6 Particle-to-particle Bonding Relation . . . . . . . . . . . . . . 90
6.1.7 Electromagnetic Forces. . . . . . . . . . . . . . . . . . . . . . . . 90
6.1.8 Inter-particle Near-Field Interaction . . . . . . . . . . . . . . . 91
6.1.9 Magnetic Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.1.10 Interstitial Damping . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.2 Time-Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93