Abilash Nair: Dissertation - acumen - The University of Alabama

MULTISCALE SIMULATION OF POLYMER NANO-COMPOSITES (PNC) USING
MOLECULAR DYNAMICS (MD) AND GENERALIZED INTERPOLATION
MATERIAL POINT METHOD (GIMP)
by
ABILASH R. NAIR
A DISSERTATION
Submitted in partial fulfillment of the requirements
for the degree of Doctor in Philosophy in the
Department of Aerospace Engineering and Mechanics
in the Graduate School of
The University of Alabama
TUSCALOOSA, ALABAMA
2010

Copyright Abilash R. Nair 2010
ALL RIGHTS RESERVED

ABSTRACT
Recent mechanical characterization experiments with pultruded E-Glass / polypropy-
lene (PP) and compression molded E-Glass/Nylon-6 composite samples with 3-4 weight%
nanoclay and baseline polymer (polymer without nanoclay) confirmed significant im-
provements in compressive strength (∼122%) and shear strength (∼60%) in the nan-
oclay modified nanocomposites, in comparison with baseline properties. Uniaxial tensile
tests showed a small increase in tensile strength (∼3.4%) with 3 wt % nanoclay load-
ing. While the synergistic reinforcing influence of nanoparticle reinforcement is obvious,
a simple rule-of-mixtures approach fails to quantify the dramatic increase in mechanical
properties. Consequently, there is an immediate need to investigate and understand the
mechanisms at the nanoscale that are responsible for such unprecedented strength en-
hancements. In this work, an innovative and effective method to model nano-structured
components in a thermoplastic polymer matrix is proposed. Effort will be directed to-
wards finding fundamental answers to the reasons for significant changes in mechanical
properties of nanoparticle-reinforced thermoplastic composites. This research ensues a
multiscale modeling approach in which (a) a concurrent simulations scheme is developed
to visualize atomistic behavior of polymer molecules as a function of continuum scale
loading conditions and (b) a novel nanoscale damage mechanics model is proposed to
capture the constitutive behavior of polymer nano composites (PNC). The proposed re-
search will contribute towards the understanding of advanced nanostructured composite
materials, which should subsequently benefit the composites manufacturing industry.
ii

DEDICATION
Dedicated to my teachers
and
my parents Ambika and Rajendran Nair.
iii

LIST OF ABBREVIATIONS AND SYMBOLS
ρ current density of material
ρ 0 undeformed (initial) density of material
J Jacobian (or determinant) of deformation gradient
F deformation gradient tensor
C right Cauchy-Green deformation tensor
S 2 nd Piola-Kirchoff stress tensor
σ Cauchy stress tensor
ɛ small strain tensor
E Green-Lagrange strain tensor
D damage tensor
T ambient temperature
A Helmholtz free energy
kB
Boltzmann constant
∆t time-step for continuum update
∆τ time-step for discrete (atomistic) update
1 fs 1 femto(10 −15 ) seconds
1 ps 1 pico(10 −12 ) seconds
1 ˚A 1 Angstrom (10 −10 ) meters
1 nm 1 nano-(10 −9 ) meter
RVE Representative Volume Element
MD Molecular Dynamics
GIMP Generalized Interpolation Material Point Method
FEA Finite Element Analysis
PNC Polymer Nano-composite
iv

ACKNOWLEDGEMENTS
I would like to use this space to acknowledge the help and encouragement without
which this dissertation would not have come to realization. While the work undertaken in
this research is challenging, the words of wisdom provided by my advisor Dr. Samit Roy
encouraged me to learn new methodologies and equip myself with the necessary tools
to pursue this research. Being a student of mechanics, I have had the opportunity to
learn many elements of physics and chemistry that is an integral part of this dissertation.
This work was supported by the NSF-EPSCoR funded GRSP program. Applying to
the GRSP, has opened my eyes to an important part of a career in research: writing
proposals. I thank the AEM department at UA, for providing me with the teaching
assistantship that allowed me to complete this research. I like to thank Mr. Rezwanur
Rehman for his help with the nano-clay modeling and Mr. Kamesh Narasimhan for his
discussions on nano-mechanics of polymer nano-composites. I would like to thank my
committee members for taking the time to read my dissertation and providing comments
to improve on the manuscript. I thank Mr. Cameron Purvis at O.I.T. for his help with
compiling the parallel version of LAMMPS.
I thank my family whose constant support and encouragement helped me through
the hardest times in my academic career. I thank them for their trust and confidence in
my decisions. I would like to close by recognizing the invaluable help and support of my
close friends who have been family to me for the past four years. While it may not seem
long, most lifetimes are lived within a breath and its only in the briefest of moments like
these that we appreciate the ones who were there when it really mattered. It seems only
fitting that this dissertation marks the end of an era when I had my friends graduate and
move on, and now my time has come with the proverbial “passing of the buck”.
The air, pregnant with excitement beckons a future still unknown, but full of hope. To
the pages that have been written and the many pages that are waiting to be done.
v

CONTENTS
ABSTRACT .................................................................................. ii
LIST OF ABBREVIATIONS AND SYMBOLS ............................................ iv
ACKNOWLEDGEMENTS .................................................................. v
LIST OF TABLES ........................................................................... ix
LIST OF FIGURES .......................................................................... x
1. Introduction ............................................................................... 1
1.1. Motivation............................................................................ 2
1.2. Literature Review.................................................................... 7
2. Generalized Interpolation Material Point Method (GIMP) ............................ 16
2.1. Equations of Motion of a Deformable body........................................ 18
2.1.1. Discretizing governing equation ............................................ 19
2.2. Explicit Time Integration in GIMP ................................................ 26
2.2.1. A Finite Deformation Hyper-Elastic Model for Polymeric Materials .... 29
2.2.2. The Verlet Integration in GIMP............................................ 31
2.3. Dynamic Fracture Analysis using GIMP ........................................... 33
3. Molecular Dynamics (MD) ............................................................... 37
3.1. Introduction .......................................................................... 38
3.1.1. Equations of Motion......................................................... 41
3.1.2. The Verlet Algorithm ....................................................... 42
3.2. Molecular Force Fields .............................................................. 44
3.2.1. Bond stretching .............................................................. 46
3.2.2. Angle bendings .............................................................. 47
3.2.3. Bond torsion ................................................................. 47
3.2.4. Non-bonded Interactions .................................................... 48
vi

3.3. Modeling Thermoplastic Polymer Nanocomposites in MD ....................... 50
3.3.1. Thermoplastic Polymers .................................................... 51
3.3.2. Nanoclay ..................................................................... 55
3.4. Applications of the Molecular Dynamics Algorithm............................... 57
3.4.1. Determination of Free-Energy due to deformation ........................ 58
3.4.2. Multi-scale modeling of Thermoplastic composites ....................... 62
4. Concurrent Multiscale Modeling of Thermoplastic PNC ............................... 65
4.1. Introduction .......................................................................... 65
4.2. Embedded Statistical Coupling in GIMP .......................................... 69
4.3. Setup of Coupled Simulation model ................................................ 73
5. Hierarchical Modeling of Damage in PNCs ............................................. 77
5.1. Damage Mechanics using the Internal State Variable Approach ................. 79
5.1.1. Definition of Damage Tensor ............................................... 79
5.1.2. Thermodynamics of Damage and Constitutive equations ................ 81
5.2. A Nano-scale Damage Model ....................................................... 84
5.2.1. A Damage Model for Interphase Debond and Void Nucleation in
Thermoplastic Polymer Nanocomposites .................................. 92
6. Results ..................................................................................... 98
6.1. Concurrent Coupling of Thermoplastic Polymers ................................. 98
6.1.1. Case I: Benchmark of Coupling algorithm using a Simple Tension Test. 103
6.1.2. Case II: Mode I Crack Growth in Thermoplastic Polymer ............... 105
6.1.3. Case III: Mode II crack growth in Thermoplastic Polymer ............... 107
6.2. Free Energy Calculations of Thermoplastic Polymers ............................. 111
6.2.1. Calculation of Persistence Length of a Polymer ........................... 112
6.2.2. Calculation of Free energy of Polymers undergoing Mechanical Deformation
..................................................................... 115
6.3. Multi-scale Damage model for Thermoplastic Nanocomposites .................. 122
6.3.1. Simulation results for Thermoplastic Polymer Nano-composite .......... 124
vii

6.3.2. Simulation results for pure Polypropylene without Nanoscale Reinforcement.....................................................................
138
6.3.3. Comparison of Stiffness and Strength between Polymer Nanocomposite
and Pure Polymer .................................................... 143
6.3.4. Summary ..................................................................... 145
7. Discussion ................................................................................. 147
7.1. Significance of Research ............................................................. 147
7.2. Future Work ......................................................................... 149
7.2.1. Coupled simulation of Fiber reinforced PNC .............................. 149
7.2.2. Multi-scale modeling of Polymer Nanocomposite ......................... 151
7.2.3. A Generalized Three-Dimensional Damage Model ........................ 153
References ..................................................................................... 155
A. Derivation of Stress-Strain law based on Dilatational and Deviatoric response
of a material ............................................................................... 164
viii

LIST OF TABLES
3.1. Position of Nanoclay atoms in space group ........................................ 56
6.1. Values of scaling parameter λ to determine ∆A ................................... 119
6.2. Bond parameters for Nanoclay...................................................... 126
6.3. Angle parameters for Nanoclay ..................................................... 126
6.4. Torsion parameters for Nanoclay ................................................... 126
6.5. Non-bond parameters for Nanoclay-Polymer hybrid .............................. 126
6.6. Linear Regression and Analysis for PNC deformation Case 1 .................... 130
6.7. Linear Regression and Analysis for PNC deformation Case 2 .................... 133
6.8. Linear Regression and Analysis for PNC deformation Case 3 .................... 135
6.9. Linear Regression and Analysis for PNC deformation Case 4 .................... 137
6.10. Linear Regression and Analysis for PP deformation Case 1 ...................... 140
6.11. Linear Regression and Analysis for PP deformation Case 2 ...................... 140
6.12. Comparison of modulus of PP and PNC predicted by NDM ..................... 145
6.13. Comparison of strength of PP and PNC predicted by NDM ..................... 145
ix

LIST OF FIGURES
1.1. Change in compressive strength of E-glass/polypropylene composite with
nanoclay loading ..................................................................... 3
1.2. Change in compressive modulus of E-glass/polypropylene composite with
nanoclay loading ..................................................................... 4
1.3. Change in compressive modulus of pure polypropylene with nanoclay loading . 4
1.4. Length scales involved in multi-scale modeling .................................... 5
1.5. The multi-scale modeling paradigm ................................................ 6
2.1. The GIMP description .............................................................. 17
2.2. Schematic of continuum deformation............................................... 18
2.3. A two-dimensional GIMP particle .................................................. 20
2.4. The one-dimensional characteristic particle shape function ...................... 20
2.5. The one-dimensional grid interpolation function .................................. 23
2.6. The one-dimensional GIMP interpolation function................................ 25
2.7. The two-dimensional GIMP interpolation function ............................... 26
2.8. The GIMP crack ..................................................................... 34
2.9. Schematic for determining strain energy release rate in GIMP................... 35
3.1. Common molecular dynamics force fields for polymers ........................... 45
3.2. Schematic of bond stretching ....................................................... 46
3.3. Schematic of angle bending ......................................................... 47
3.4. Schematic of bond torsion........................................................... 48
3.5. Schematic of non-bonded interactions .............................................. 49
3.6. Schematic of van der Waals interaction energy .................................... 50
3.7. Chemical formula of polypropylene................................................. 53
3.8. Coarse grained polypropylene monomer ........................................... 53
3.9. Profile views of nanoclay in XY and XZ planes ................................... 55
3.10. Calculation of free energy change using thermodynamic integration............. 61
x

3.11. Schematic of the MD simulation model for polymer nanocomposite ............. 62
3.12. Volumetric and deviatoric deformation of simulation box ........................ 63
4.1. Schematic of Concurrent coupling .................................................. 66
4.2. Schematic of the Direct Coupling methodology ................................... 67
4.3. Schematic of the Statistical Coupling .............................................. 68
4.4. Layout of Coupled Model ........................................................... 69
4.5. Setup of atomistic model for Concurrent coupling ................................ 74
4.6. Setup of ESCM/GIMP model for Concurrent coupling ........................... 75
5.1. Development of nano-scale damage model (NDM) ................................ 78
5.2. Characterization of damage entities ................................................ 80
5.3. Representative volume element (RVE) and nano-clay orientation in damage
model ............................................................................ 85
5.4. Schematic of interface debond and void nucleation in damage model for
deformations along principle axis ................................................... 86
5.5. Schematic of interface debond in RVE under simple-shear deformation ......... 87
5.6. Schematic of nano-clay with dimensions ........................................... 93
5.7. Schematic of voids in pure polymers and polymer nano-composites ............. 94
5.8. Determination of nano-scale damage parameters .................................. 96
6.1. Layout of zones in coupled simulation ............................................. 99
6.2. Layout of a non-periodic MD simulation box ...................................... 101
6.3. Computation of displacement gradient ............................................. 101
6.4. Schematic of coupled model in a simple tension setup ............................ 103
6.5. Time progression of vertical displacements (v) for concurrent simulation
(Case 1) .............................................................................. 104
6.6. Time progression of displacement gradient (∂v/∂Y ) for concurrent simulation
(Case 1) ....................................................................... 105
6.7. Setup of coupled model for mode I crack growth .................................. 106
6.8. Time progression of displacement gradient (∂v/∂Y ) for concurrent simulation
(Case 2) ....................................................................... 107
xi

6.9. Time progression of vertical displacement (v) for concurrent simulation
(Case 2) .............................................................................. 108
6.10. Fracture surface of mode I crack from concurrent simulation .................... 108
6.11. Setup of coupled model for mode II crack growth ................................. 109
6.12. Time progression of horizontal displacement (u) for concurrent simulation
(Case 3) .............................................................................. 110
6.13. Time progression of displacement gradient ∂u/∂Y for concurrent simulation
(Case 3) ......................................................................... 111
6.14. Time progression of displacement gradient ∂v/∂X for concurrent simulation
(Case 3) ......................................................................... 112
6.15. Schematic of polymer model for determination of bending stiffness EI ......... 113
6.16. Dependence of loading rate on bending stiffness .................................. 114
6.17. Schematic of the dilatational and deviatoric deformation used to determine
free energy change associated with mechanical deformation ...................... 116
6.18. Free-energy versus volumetric strain ............................................... 120
6.19. Free-energy versus deviatoric strain ................................................ 121
6.20. Schematic of primary RVE deformations in MD .................................. 123
6.21. Structure of nano-clay with dimensions ............................................ 125
6.22. σ11 versus ɛ11 for PNC deformation (Case 1) ...................................... 130
6.23. σ22 versus ɛ11 for PNC deformation (Case 1) ...................................... 131
6.24. σ33 versus ɛ11 for PNC deformation (Case 1) ...................................... 131
6.25. Deformed and undeformed configurations of PNC deformation (Case 1) ........ 132
6.26. σ11 versus ɛ22 for PNC deformation (Case 2) ...................................... 133
6.27. σ22 versus ɛ22 for PNC deformation (Case 2) ...................................... 133
6.28. σ33 versus ɛ22 for PNC deformation (Case 2) ...................................... 134
6.29. Deformed and undeformed configurations of PNC deformation (Case 2) ........ 134
6.30. σ11 versus ɛ33 for PNC deformation (Case 3) ...................................... 135
6.31. σ22 versus ɛ33 for PNC deformation (Case 3) ...................................... 136
6.32. σ33 versus ɛ33 for PNC deformation (Case 3) ...................................... 136
xii

6.33. σ23 versus γ23 for PNC deformation (Case 4) ...................................... 137
6.34. σ13 versus γ13 for PNC deformation (Case 5) ...................................... 138
6.35. σ12 versus γ12 for PNC deformation (Case 6) ...................................... 139
6.36. σ11 versus ɛ22 for PP deformation (Case 1) ........................................ 141
6.37. σ22 versus ɛ22 for PP deformation (Case 1) ........................................ 141
6.38. σ33 versus ɛ22 for PP deformation (Case 1) ........................................ 142
6.39. Deformed and undeformed configurations of PP deformation (Case 1) .......... 142
6.40. Schematic of void formation in PP under simple shear deformation ............. 143
6.41. σ12 versus γ12 for PP deformation (Case 2) ........................................ 143
6.42. Deformed and undeformed configurations of PP deformation (Case 2) .......... 144
7.1. Morphology of a typical Polymer nano-composite................................. 152
7.2. Schematic of randomly oriented nano-clay ......................................... 153
xiii

Chapter 1
Introduction
The use of polymer matrix composites (PMC) in aerospace and civil engineering, as
well as in sports and leisure applications is rapidly increasing. PMC has found wide range
of applications in the structural components where the substitution of PMC for metals has
substantially improved performance and reliability. Among the most important features
in PMC, the following unique properties make it a very attractive engineering material:
high specific static and fatigue strength, as high as eight times that of steels, and high
corrosion resistance.
Carbon fiber-reinforced polymer matrix composites are now being used widely in
aerospace industry for structural components. The new Boeing 787 and the Airbus A380
are cases in point. In addition to their extensive use in aircraft, composites are becoming
a material of choice in the automotive, marine, civil infrastructure construction and
consumer products industries. Several major manufacturers are prototyping concept cars
almost totally with composite materials. Other current examples include race cars, racing
boats, top-of-the-line motorcycles, bicycles and snowmobiles just to name a few. Potential
new application areas include all-composite bridge I-beams, bridge deck, dry board and
other building components.
The high strength in PMC derives from the high strength in the fibers embedded in
the matrix. Fibers have high strength in tension, however, their compressive strength
is generally much lower due to the fact that under compression, the fibers tend to fail
through micro-buckling well before compressive fracture occurs in conventional compos-
ites. Matrix material has to contribute significantly to the overall compressive strength
of the PMC, as matrix strength is much lower than that of fibers. The overall com-
pressive strength of the PMC is, in general, only about 50% of the tensile strength or
1

lower. Similarly, significant improvement in the interlaminar shear strength of a compos-
ite can be achieved using nanoparticle reinforcement, since it is also a matrix dominated
property. As most composite structures are used under multiaxial stress states, typically
involving compression and/or shear in certain orientations, the lower compressive and/or
shear strength is a major obstacle for the full realization of light-weight yet high-strength
potential of composite applications.
1.1 Motivation
Preliminary results on nanoclay reinforced pultruded E-glass/polypropylene (PP) test
specimens published by Roy et al. (2007) indicates significant improvements in the com-
pressive strength as well as compressive modulus of nanoclay reinforced E-glass/PP com-
posite, as shown in fig.(1.1) and fig.(1.2). Significant enhancements in compressive mod-
ulus of neat polypropylene resin was also observed (fig.1.3). Interestingly, more than
100% improvement in mechanical properties (of both neat resin and composite) was ob-
served with only 10 wt% addition of nanoclay. Further, more than 60% improvement in
interlaminar shear strength was recorded for 3 wt% nanoclay loading (Roy et al., 2007).
While the synergistic reinforcing influence of nanoparticle reinforcement is obvious, a
simple rule-of-mixtures approach fails to quantify the dramatic increase in mechanical
properties. Consequently, there is an immediate need to investigate and understand
the mechanisms at the nanoscale that are responsible for such unprecedented strength
enhancements. It is envisioned that a better understanding of the mechanisms at the
nanoscale will lead to optimization of processing variables at the macroscale, which in
turn will lead to the manufacture of nanocomposites more efficiently and at lower cost.
Figure 1.4 shows the length scales involved in a typical nanoparticle reinforced fiber com-
posite. The nanoclay reinforcements are nanometer (10 −9 meter) thick particles that
interact intimately with the polymer matrix molecules to provide strengthening of the
2

Figure 1.1: Change in compressive strength of E-glass/polypropylene composite with
nanoclay loading
polymer matrix at the nano-scale. The size of the intercalating polymer molecules are
roughly on the order of the nanoclay thickness, depending on the degree of polymeriza-
tion. Moving up the scale we find that the carbon fiber diameter is usually on the order
of 5 to 10 microns (1 micron, or 1µm = 10 −6 meter), although its length may extend
up to several meters. The composite (matrix reinforced with fiber) is in the macro or
engineering-scale of analysis. The combination of fiber reinforcements in the polymer ma-
trix gives the macro-scale composite orthotropic properties (Jones, 1998). It is evident
from the discussions above that the nanoscale interaction between polymer molecules
and nanoclay-species is a crucial factor in determining the macro-scale strength of the
composite. Hence, a satisfactory theoretical model must appeal to the physics of the
problem at the nano-scale and at the same time be robust enough to predict perfomance
at a macroscale. During material fracture, the reinforcing forces ahead of the crack tip
3

Figure 1.2: Change in compressive modulus of E-glass/polypropylene composite with
nanoclay loading
Figure 1.3: Change in compressive modulus of pure polypropylene with nanoclay loading
4

Figure 1.4: Length scales involved in multi-scale modeling
is governed by the complex inter play between non-bonded interactions and the driving
forces that want to extend the crack at the nanoscale. While, the influence of atomistic
forces can be simulated to a satisfactory extent with robust molecular dynamics based
algorithms (Allen & Tildesley, 1987), the technique is usually intractable when it comes
to the study of larger systems under the influence of an external forcing environment. On
the other hand, in a purely continuum analysis, the micro-structural detail of the ma-
terial is overlooked. In the interest of lowering computational expense, an approximate
phenomenological model is usually used to describe the microscale processes. Most con-
tinuum scale models however, do not address the obvious length and time scale issues in
fracture processes. This deficiency could undermine predictions in material behavior. A
robust multiscale simulation technique will be cognizant of the discrete material behavior
at atomistic level and will correlate microstructural phenomena to macroscale loading or
environmental condition. However, the challenge is to seamlessly overcome disparities
in both time and length scales and transfer information efficiently. As can be seen in
fig.(1.5), in the modeling of atomistic systems there has been mainly two computational
methodologies of interest to researchers: quantum mechanics (QM) based calculations
5

Figure 1.5: The multi-scale modeling paradigm
are used to simulate atomic (or molecular) systems, however as system sizes increase the
molecular dynamics (MD) method has proven to be far more effective and efficient com-
putationally although, not as accurate (Allen & Tildesley, 1987). In this research, the
molecular domain is simulated under free boundary conditions acting under the influence
of external forces from the macro-scale using molecular dynamics. Molecular dynamics
is a subset of computational chemistry codes which creates system trajectories consistent
with thermo-dynamic ensembles (Allen & Tildesley, 1987). The molecular dynamics pro-
gram LAMMPS (large-scale atomic/molecular massively parallel simulator) (Plimpton,
1995) is a scalable parallel computation algorithm with the added functionality to model
polymer systems. In general the results to an atomistic MD computation (in particular
polymer models) are system size dependent. However, with parallel MD algorithms such
as LAMMPS, large (over 10,000 atoms) molecular systems can be simulated and hence
6

makes it a more attractive choice over other approaches such as ab-initio calculations,
which in general are limited to the calculation of a few 100 atoms (Leach, 1996). At the
engineering scale of a coupled simulation most researchers have used the finite element
analysis (FEA) to model continuum response (see e.g. Liu et al. (2006)). In this research,
the continuum will be modeled using the generalized interpolation material point method
or GIMP (Bardenhagen & Kober, 2004). GIMP is a class of arbitrary Euler-Lagrange
(AEL) based meshless particle-in-cell method hence it automatically avoids mesh en-
tanglement and distortion problems that usually plague FEA during simulation of large
deformation. Furthermore, since GIMP discretizes the model into material points, it
forms a natural connection to the discrete molecular system to which it is coupled (Liu
et al., 2006). Using an implicit time-stepping strategy in GIMP one can also eliminate
the errors associated with explicit GIMP dynamics (Nair & Roy, 2009).
1.2 Literature Review
As described above the need for low cost, low weight and multifunctional materials
has propelled interest in studying advanced materials and in particular nano-composites
(Alivisatos et al., 1998). The vast size differences in a “multi-scale” system allows for the
improvement of material properties through reinforcement at the molecular level. In gen-
eral, polymers can either be thermoplastics or thermosets. Thermoplastic polymers are
tougher and has better impact resistance compared to thermosets. The harder, more rigid
and brittle behavior of thermosets are a manifestation of the strong covalent cross-links
between polymer chains which makes the macromolecule resistant to chain deformation
and rearrangement. The molecular morphology in a thermoset also affects its post cure
characteristics. In general cured thermoset polymers do not soften on re-heating, and
will only char and break down at high temperatures. Thermoplastics on the other hand,
owing to its weak inter molecular forces of attraction in between chains is amenable to re-
7

molding and reshaping on the application of heat. These properties renders thermoplastic
polymers recyclable and feasible to applications in green technology. This research will
focus on thermoplastic polymers and the effect of nanoclay platelets on the thermoplastic
polymer matrix.
Beginning in the early 90’s there has been an interest in the application of nanoclay
reinforced polymers as structural materials. Researchers at Toyota (Kojima et al., 1993)
were the first to use nanoclay reinforced thermoplastic (nylon-6) resin in manufacturing
car bumpers. They had observed a significant improvement in mechanical and impact
properties of the composite (polymer + nanoclay) with low weight additions of nanoclay.
The improvement in mechanical properties provided impetus for research on nanoclay
reinforced composites for multi-functional applications. It has been shown that nanoclay
reinforced polymers had improved barrier properties (Berta et al., 2006; Golebiewski
& Galeski, 2007), moisture diffusion (Sun et al., 2007) in addition to enhancement of
mechanical properties (Nair et al., 2002; Subramaniyan & Sun, 2006a,b; Roy et al., 2007;
Yalcin et al., 2008; Dong & Bhattacharyya, 2008). Application of nanoclay has also been
extended to various resin systems including epoxies (Park & Jana, 2003; Timmerman
et al., 2002), high density polyethylene (HDPE)/wood composite (Lei et al., 2007). In this
research we are focused on understanding the reasons behind significant improvement in
compressive and shear dominated properties of the matrix due to nanoclay reinforcement.
The strength and stiffness enhancements provided by fiber reinforced composites with
improved matrix dominated properties (such as compression and shear behavior) makes
it an attractive choice for engineering structural applications.
Montmorillonite (MMT) clay occurs in nature in the form of platelets with relatively
large aspect ratios (∼100:1). The nanoclay allows both for an intercalated and/or ex-
foliated morphology in the material system in which its dispersed (Mani et al., 2005).
The enhancement of mechanical properties of the nanocomposite has been found to be
8

strongly correlated to the quality of dispersion of nanoclay in the resin system (Chen
et al., 2004). In general most naturally occurring nanoclays are organophobic and hy-
drophyllic hence, nanoclays are surfactant modified to render them organophyllic. The
extent of nanoclay reinforcement is a function of its spatial distribution within the host
polymer and the compatability of surfactant species with the polymer molecules (Yang &
Tsai, 2006). In addition to the decreased processability, a high weight percent of nanoclay
also degrades the mechanical performance of the nanocomposite due to the formation of
agglomerates in the polymer matrix. On the other hand, in a well dispersed system the
polymer molecules tend to ingress into gallery spacings in between individual nanoclay
platelets. Due to topological constraints imposed by the nanoclay, the local dynamics
(i.e. chain relaxation, segmental mobility) of the polymer molecules is subdued (Krish-
namoorti et al., 1996). It has been realised that the intercalation of the polymer molecule
into the nanoclay galleries is entirely energetic (Krishnamoorti et al., 1996; Yang & Tsai,
2006) and a favorable interaction between polymer and nanoclay is required to overcome
the entropic loss due to confinement of polymer molecules.The small thickness of the
nanoclay implies a larger surface area that allows for enhanced load transfer through
non-bonded interactions at the polymer-particle interface.
Yang & Tsai (2006) obeserved that the decreased mobility of the polymer molecules
suggests that the viscoelastic response of the polymer could be used as an indicator of
dispersion of nanoclay in the polymer system. The hypothesis is verified by the enhanced
creep resistance of polymer nanocomposites (Yang et al., 2006). The introduction of nan-
oclay into the polymer also affects phase transformations and crystallinity of the polymer
system. Yuan & Misra (2006); Yuan et al. (2008) has observed that the introduction of
nanoclay act as nucleation sites for the potential growth of spherulites (spherical crystal-
lites) in the bulk material. Yuan & Misra (2006) observed that the growth of spherulites
is correlated to the change in the primary mechanism of failure (in dynamic, IZOD
9

tests) which is plastic deformation from crazing and vein-type in neat polypropylene to a
microvoid-coalescence-fibrillation process in the nanocomposite. In addition to the effect
of surfactant modification on nanoclay reinforcement the effects of nanoclay stacking and
orientation on the mechanical properties of thermoplastic polymers is well documented
(Galgalia et al., 2004; Looa & Gleason, 2004; Srinath & Gnanamoorthy, 2005).
In recent years, numerous modeling methodologies has been proposed to provide better
insight into the physics of nanoclay reinforcement. Zeng et al. (2008) provides an extensive
review of the work that has been done in this field. In studying the system behavior
of nanocomposites molecular dynamics (MD) has been the primary method of choice.
Kuppa et al. (2003) used MD simulation to study poly-ethylene oxide (PEO) /nanoclay
hybrid structures. They were able to correlate the results from the analysis with wide-
angle neutron diffraction (WAND) and differential scanning calorimetry (DSC) studies
of purely intercalated PEO in MMT. Sinsawat et al. (2003) used coarse-grained MD to
study the influence of polymer matrix architecture on nanoclay intercalation. Atomistic
simulations have also been able to show that the effective dispersion and intercalation is
strongly dependent on the functionalization of the constituent groups in the nanoclay,
polymer matrix (Smith et al., 2003; Minisini & Tsobnang, 2005; Katti et al., 2006).
Simpler theoretical models have also been suggested to explain the dynamics of melt
intercalation in polymer nanocomposites (Vaia & Giannelis, 1997). Tsai & Sun (2004)
used continuum shear lag to predict load transfer efficiency of nanoclay platelets. The
stress transfer between the (polymer) matrix and nanoclay inclusion was found to be
dependent on the degree of exfoliation of the nanoclay in the polymer matrix.
For the purpose of engineering applications, constitutive modeling of nanoclay rein-
forced polymers is of much interest to researchers. Drozdov et al. (2010) developed a
visco-elastic continuum model of polypropylene matrix with nanoclay fillers. The model
was able to capture the retardation of stress relaxation with increase in nanoclay content.
10

Sheng et al. (2004) determined the elastic modulii of MXD6-clay and nylon-6 compos-
ites using a micromechanical model introduced in Danielsson et al. (2002). The analysis
was able to incorporate geometry parameters of the hybrid structure such as the par-
ticle aspect ratio (L/t) and silicate layer gallery spacing (d001) in determining material
properties of the nanocomposite. The model was able to closely predict experimental
observations for moduli better than Halpin-Tsai (HT) or Mori-Tanaka (MT) based cal-
culation. Hbaieb et al. (2007) used MT to predict stiffness of randomly oriented nanoclay
polymer composites. Zhu & Narh (2004) calculated effective tensile modulus of nanoclay
polymers using a micromechanics representative volume element (RVE) model of aligned
nanoclay platelets in a polymer matrix. Their analysis included an “interface” region
between the nanoclay and the bulk polymer to simulate actual load transfer conditions
at the atomistic level. While the material models suggested above were successful in
predicting (at least qualitatively) the enhancements of modulus for nanocomposites, the
biggest drawback is the assumption of a well bonded continous interface at the nanoscale
between nanoclay and polymer material. In reality, the length scale issues are unavoid-
able, a realistic model of the material must explicitly account for the discrete material
behavior especially at interfaces. The objective of this research is to incorporate the
interactions at the interfaces and its associated properties by multiscale modeling.
In studying molecular systems, a multiscale and multi-physics simulation approach is
considered computationally more viable since relying on a single time or length scale can
take a huge amount of computational resources and can potentially lead to inaccurate
results. In recent years numerous efforts have been directed toward modeling nanocom-
posites in order better understand the reasons behind the enhancement in mechanical
properties, even with slight addition (a few weight percent) of nano-materials (Valavala
et al., 2007; Buryachenko et al., 2005; Riddick et al., 2006b,a; Abraham et al., 1999; Ogata
et al., 2001; Wagner & Liu, 2003; Xiao & Belytschko, 2004; Ma et al., 2006).There are
11

two unique choices in multiscale modeling, namely, the hierarchical and concurrent sim-
ulation schemes. In the hierarchical approach the physical system is studied in isolation
to far-field stimuli and the results are translated to a continuum response using curve fits
and/or statistical averaging. Valavala et al. (2007) used energy equivalence of continuum
and atomistic models of polymer systems to characterize the hyperelastic stress-strain
response of polymers (in particular, polycarbonate and polyimide). Buryachenko et al.
(2005) used the Eshelby and Mori-Tanaka methods to determine the effective proper-
ties of nanocomposite materials. Riddick et al. (2006a), employed molecular dynamics
(MD) to determine the force-displacement curves between nano inclusions (CNT) and the
polymer system. Riddick et al. (2006b) used equivalent modeling of carbon nanotubes
in polymers to study the fracture toughness of polymer matrix composites with carbon
nanotubes (CNTs) embedded in them.
In general, it was observed that the strengthening behavior observed in nanocomposite
materials due to nano sized inclusions can only be explained using theoretical formalisms
that appeal to fundamental physics of interactions at nanometer length-scales. True ma-
terial behavior cannot be accurately simulated through effective or “smeared” continuum
analysis since these techniques assume a continuous and “perfect” bonding behavior of
the nanocomposite system or in some cases “smears away” the interface altogether. By
definition the “smeared” analyses fail to capture the local details which are responsible
for the stiffness and strength enhancements at the nanoscale. It is evident that the stress-
strain response of the system is a function of localized behavior and that the strength
enhancement is primarily due to attenuation of internal damage due the complex interac-
tion of voids/stress concentrators with the nanosized inclusions in the material. It is for
this reason that regular continuum material models that are commonly used in engineer-
ing cannot accurately predict the material response of nanocomposite systems. From a
design standpoint, it is necessary that we are able to describe the material response with
12

no prior assumptions in order to reliably and accurately model responses of the material
under all loads and ambient conditions.
Advances in computational tools has allowed us to examine and understand small
scale phenomena using molecular dynamics and macro-scale responses using continuum
level simulations. However, due to limited computer resources and the inefficiency and
inaccuracy involved in using a single method to analyze large (or small) scale events has
led to the coupling of two (or more) methods of analysis. For example, the continuum
method (such as finite element analysis (FEA)), can be used to solve the governing
continuum field equations subject to given traction and displacement boundary conditions
and then apply the forces and tractions from its analysis as boundary conditions to the
region described by atomic interactions (e.g. molecular dynamics). By taking advantage
of this synergy between these vastly different methods we get the appropriate resolution
to examine extremely localized events such as crack initiation, dislocations etc.
However, the challenge is to simultaneously process all information available from
both length scales and efficiently pass this information continually across both time and
length scales. Concurrent simulations have the unique advantage that it can tie local
events at nanoscale to stimuli from a larger time and length scale during real time simu-
lation. One of the earliest forays into concurrent coupling was by Abraham et al. (1999)
in their scheme, MAAD. The scheme involved simultaneous coupling of tight-binding
(TB), molecular dynamics (MD) and finite element analysis (FEA). The scheme was
successfully applied to study fracture behavior of silicon systems. However, the method
had some shortcomings in it in that it was limited by the timestep of the tight-binding
step (on the order of 10 −15 seconds) which made the procedure computationally demand-
ing. Ogata et al. (2001) used FEA to concurrently couple with density function theory
(DFT) to study the surface oxidation of silicon system. Some of the more recent efforts in
multiscale concurrent simulations are the bridging-scale technique (Wagner & Liu, 2003;
13

Xiao & Belytschko, 2004) and the MD-MPM coupling technique by (Ma et al., 2006). In
Ma et al. (2006), the atoms within the handshake region of the MD domain are coupled
point-wise to MPM particles. The continuum is hence refined down to atomic sizes and
the boundary material points are assumed to overlap with boundary atoms of the MD
zone at all times. The outcome of this refinement is that the molecular dynamics domain
induces spurious high frequency oscillations into the continuum domain as the simula-
tion proceeds due to thermal vibrations of the atoms. Recently, Saether et al. (2009),
proposed a statistical averaging technique to overcome the high frequency vibrations, in
their concurrent coupling scheme to study grain boundary structures in aluminum.
Although there have been many attempts to model standard solid lattice structures
and couple them to various continuum methods, to our knowledge, large scale concur-
rent coupling of nanoparticle reinforced polymer systems has not been attempted before
partly due to the complexity involved in modeling polymers. Under ambient conditions
the polymer model is amorphous (depending on the degree of crystallinity) and lacks
specific ordered structure. Further, a polymer MD model takes longer time to equilibri-
ate as compared with an ordered lattice. In this dissertation we propose two approaches
to modeling nanocomposites. First, we propose to couple the generalized interpolation
material point method (continuum scale) and molecular dynamics (discrete scale) in sim-
ulating atomic interactions in polymer systems using the statistical averaging technique
proposed by Saether et al. (2009) in order to visualize the nanoscale failure mechanism
in correlation with macroscale mechanical loads. Next we propose a continuum dam-
age mechanics (CDM) based equivalent 3D RVE model of nanoclay-polymer that can be
incorporated in to continuum scale analysis to capture the strength of nanoclay compos-
ites as a function of interface degradation or micro-void nucleation. We envision that
the work in this paper will contribute towards the understanding of nanoscale interac-
tions in nanostructured composite materials. The dissertation is organized as follows: In
14

Chapter 2, the GIMP algorithm is discussed along with its application to modeling crack
propagation. In Chapter 3 the MD algorithm is introduced along with a technique to de-
termine free energy from molecular simulations of polymer (or polymer nanocomposite)
systems. The coupling algorithm for coupling MD and GIMP using the embedded statis-
tical scheme (ESCM) is discussed in Chapter 4. In Chapter 5, a hierarchical simulation
method is proposed to model stiffness and damage behavior of nanoclay composites. In
Chapter 6, results from the proposed analyses methods are presented, followed by a list
of conclusions and recommendations for future research, in Chapter 7.
15

Chapter 2
Generalized Interpolation Material Point Method (GIMP)
The material point method (MPM) was first suggested by Sulsky et al. (1995). It
belongs to a class of particle-in-cell (PIC) algorithms first developed at Los Alamos Na-
tional laboratories. MPM was initially proposed to model and solve complex problems
in fluid flows however its mixed Euler-Lagrange formulation has made the method vi-
able to applications in finite-deformation solid mechanics (Sulsky et al., 1994; Sulsky &
Schreyer, 1996). In particular, it provides as an alternative to FEA which is susceptible to
numerical instabilities like mesh entanglement in large deformation simulations. In recent
years, MPM has increasingly found application in a wide variety of engineering problems
such as fluid-structure interactions (York et al., 2000; Guilkey et al., 2003), simulating
granular materials (Zhang et al., 2009; Wieckowski, 2004), studying crack growth (Tan
& Nairn, 2002; Daphalapurkar et al., 2007), analysis of multi-phase flows (Gilmanov &
Acharya, 2008; Zhang et al., 2008) and multi-scale modeling (Ma et al., 2005; Lu et al.,
2006; Ayton et al., 2001).
The material point method (MPM) is a class of arbitrary Euler-Lagrange (AEL)
scheme. The solution method uses Lagrangian material points embedded in an Eulerian
mesh. The use of Lagrangian material points simultaneously avoids the need to advect
state variables (such as stress and strain) and the numerical diffusion problems associated
with a purely Eulerian mesh (see, fig.2.1). The Eulerian mesh on the other hand, ensures
that mesh entanglement is avoided. However, certain drawbacks such as cell-crossing
anomalies, that are primarily attributed to shortcomings in the original MPM interpo-
lating functions are a major concern in applying MPM to real world large deformation
problems. In the classic MPM algorithm the equations have to be re-ordered and written
in terms of momentum in order to account for problems with particles crossing cells of
16

Figure 2.1: The GIMP description
the grid in case of large deformations. This leads to unsymmetric equations that have to
be solved on the background grid. The generalized interpolation material point method
(GIMP) as proposed by Bardenhagen & Kober (2004), explicitly accounts for the shape
of the particle and the background grid structure in formulating the interpolation basis
functions. In this way GIMP naturally overcomes the cell crossing anomalies of MPM,
and provides a better time description of the solution.
The advantages of using a meshless particle method in couple atomistic-continuum
simulation are many as listed in Ma et al. (2005) and Lu et al. (2006). In particular,
the particle description of the material form a natural transition from a continuum to
the discrete atoms in the handshake region. In GIMP, the exchange of information
between the discrete atomistic system and the continuous domain is enhanced through
the use of the finite sized particles. In this Chapter the theoretical foundation for GIMP
17

is introduced and an numerical implementation of the explicit time-stepping algorithm
is discussed. The equations of motion and GIMP discretization is described in section
2.1. In section 2.2 an algorithmic implementation of the GIMP equations and the Verlet
explicit time-stepping strategy in GIMP is developed. Finally, section 2.3 discusses the
application of GIMP in studying crack propagation problems.
2.1 Equations of Motion of a Deformable body
Figure 2.2: A solid undergoing continuous deformation φ(X, t) which maps every point
X in the reference body to x on the deformed body
Consider a solid deformable body, undergoing continuous deformation φ(X, t) which
maps the equilibrium (initially undeformed; zero stress state) to a deformed volume Ω
at time ‘t’. Let x be the current coordinates of a point X on the undeformed body such
that x ≡ x(X, t). The deformation causes the body to develop a stress state σ(x, t)
18

where σ is the Cauchy stress tensor. The following equations for conservation of mass
and momentum must be satisfied by the deformed body,
dρ
dt
+ ρ∇ · v = 0 (2.1)
ρa = ∇ · σ + ρb (2.2)
Where ρ is the current mass density, a is the acceleration of the body, v is the velocity
and b is the body force. It should be noted that d/dt denotes the material derivative.
The weak form of the momentum equation can be found by multiplying eq.(2.2) by an
admissible velocity test function δv and integrating the resulting equation over the current
volume Ω of the body gives,
Ω
ρa · δvdΩ +
Ω
σ · ∇δvdΩ =
Ω
ρb · δvdΩ +
∂Ωτ
τ · δvdS (2.3)
∂Ωv is the part of the body where velocity is specified while ∂Ωτ is the part of the
surface where a traction force is specified. The current boundary ∂Ω of the volume Ω is
the union of the two disjoint set of surfaces, i.e., ∂Ω = ∂Ωv ∪ ∂Ωτ.
2.1.1 Discretizing governing equation
The following section is a review of the discretization procedure in GIMP. For further
details on the scheme, please refer to Bardenhagen & Kober (2004). Like MPM in GIMP,
the particle is characterized by the particle characteristic function χp. In MPM, this
function is simply the delta function (Sulsky et al., 1995). However, in GIMP the particle
function is chosen such that the particle is given a discrete shape and simultaneously
satisfies the partition of unity (Bardenhagen & Kober, 2004). Shown in fig.(2.3) is a
typical (rectangular) GIMP particle (in two-dimensions) with Cartesian coordinates of
the centroidal location given by xp = (x (1)
p , x (2)
p ) and (l (1)
p , l (2)
p ) are the half-lengths of the
19

particle in each direction.
X2
X1
xp
(x (1)
p , x (2)
p )
2l (1)
p
2l (2)
p
Figure 2.3: A two-dimensional GIMP particle
In GIMP, the simplest choice of particle characteristic function is:
χp(x) = H(x − (xp − lp)) + H(x − (xp + lp)) (2.4)
Where, H(x) is the unit step function, xp is the particle centroid position while lp is
Figure 2.4: The one-dimensional characteristic particle shape function
the half length of the particle. Figure (2.4) is the plot of the one-dimensional particle
shape function. We can extend the one-dimensional formula to further dimensions using
20

a multiplicative decomposition, i.e., for a space coordinate x = (x1, x2, x3), we have
χp(x) = χp(x1) · χp(x2) · χp(x3).
We can now define particle properties using the characteristic shape function eq.(2.4),
Vp =
Ω
χp(x)dΩ (2.5)
Where, Ω is the volume of the continua at current time ’t to be discretized. Using this
definition the particle mass and momenta can be defined as follows, if ρ is the current
density field of the body, then,
mp =
pp =
It follows from above that ρp = mp
Vp and vp = p p
mp
Ω
Ω
ρ(x)χp(x)dΩ (2.6)
ρ(x)v(x)χp(x)dΩ (2.7)
is the particle mass density and
velocity, respectively. The following discretization procedure ensures the conservation of
mass and momenta (Bardenhagen & Kober, 2004). In GIMP for any material point data
fp, the continuous representation of data f(x) is possible using the characteristic material
point function, i.e.,
f(x) =
fpχp(x) (2.8)
p
We can now re-write eq.(2.3) using the rate of change of particle momentum density
( ˙p p
Vp ), Cauchy stress σp of the particle and above mentioned material point representation
eq.(2.8).
21

p
Ω∩Ωp
˙p p
χp(x) · δvdΩ +
Vp
=
p
Ω∩Ωp
mp
Vp
p
Ω∩Ωp
χp(x)b · δvdΩ +
σpχp(x) : ∇δvdΩ
∂Ωτ
τ · δvdS (2.9)
Where, Ωp is the domain of the particle. Similar to the representation of particle data,
eq.(2.8), continuous data on the grid data can be represented using grid interpolation
functions. For example if g(x) is a continuous variable and gi is the grid nodal data,
then,
g(x) =
giSi(x) (2.10)
i
Where, Si(x) is the grid interpolation function, A bi-linear (in two-dimensions) shape
function is used in the formal MPM methodology Sulsky et al. (1995). Shown in Fig.(2.5)
is a one-dimensional tent function. As before, the shape function in 3-dimensions is
written as, Si(x) = Si(x1) · Si(x2) · Si(x3). The interpolation function a point ‘x’ to a
node ‘i’ of coordinate Xi in a regular grid of cell spacing ‘L’ is given by,
⎧
⎪⎨
Si(x) =
⎪⎩
0 x − Xi ≤ −L
1 + x−Xi
L
1 − x−Xi
L
−L < x − Xi ≤ 0
0 < x − Xi ≤ L
0 x − Xi > L
22
(2.11)

Figure 2.5: The one-dimensional grid interpolation function
Using eq.(2.10) we can now write the admissible velocity field as a summation of the
nodal velocities, δv =
i δvi. Substituting, this approximation into eq.(2.9) and taking
into account the arbitrariness of the nodal velocities, we arrive at the discrete form of
equations that needs to be solved at nodes of the background grid,
˙p i = f int
i + f ext
i
23
(2.12)

Where,
˙p i =
p
f int
i = −
Sip ˙p p
p
f ext
i = f b
i + f τ
i
f b
i =
f τ
i
=
p
∂Ωτ
σp · ∇SipVp
mpbpSip
τ Si(x)dS
(2.13)
In the above equation ˙p i, f int
i and f ext
i are nodal rate of momentum, internal and external
forces, respectively. The external force on the node can be due to a spatially varying
traction force on the particles τ (x) acting on surface ∂Ωτ or a body force b acting on
the volume enclosed by Ω. In the above equations, Sip and ∇Sip represent the volume
averaged GIMP interpolation function and derivative. They are defined as follows (in
one-dimension),
Sip(xp) = 1
=
2lp
⎧
⎪⎨
⎪⎩
xp+lp
xp−lp
Si(x)dx (2.14)
0 xp − Xi ≤ −L − lp
(L+lp+(xp−Xi)) 2
4Llp
1 + xp−Xi
L
1 − (xp−Xi) 2 +l 2 p
2Llp
1 − xp−Xi
L
(L+lp−(xp−Xi)) 2
4Llp
−L − lp < xp − Xi ≤ −L + lp
−L + lp < xp − Xi ≤ −lp
−lp < xp − Xi ≤ lp
lp < xp − Xi ≤ L − lp
L − lp < xp − Xi ≤ L + lp
0 L + lp < xp − Xi
24

∇Sip(xp) = 1
2lp
xp+lp
xp−lp
∇Si(x)dx (2.15)
In the above equations, ∇ ≡ ∂/∂xp. Shown in fig.(2.6) is the plot of the GIMP
interpolation function in one-dimension. The extension of the interpolation function to
three dimensions is simply given by, Sip = Sip(x (1)
p ) · Sip(x (2)
p ) · Sip(x (3)
p ). For example in
fig.(2.7) the two-dimensional interpolation function is shown for a particle located at the
origin of a uniformly spaced grid.
Figure 2.6: The one-dimensional GIMP interpolation function
25

Figure 2.7: The two-dimensional GIMP interpolation function
2.2 Explicit Time Integration in GIMP
In this section the algorithmic implementation of GIMP is discussed. The solution strat-
egy follows explicit time-stepping and hence numerical stability and accuracy of the al-
gorithm is time-step (∆t) dependent. For time-steps lower than the CFL (Courant et al.,
1928) condition for elastic wave propagation in the material good numerical accuracy can
be achieved (Reddy, 2006; Bathe, 1982).
At the beginning of timestep ‘t’ the mass mp, velocity v t p, stress σ t p external and
internal forces on particle ‘p’ is extrapolated to its nearest node neighbors,
26

Note that V t
p = mp
ρ t p
m t i =
p
p t i =
p
f int,t
i = −
f ext,t+∆t
i
=
Sipmp
Sipmpv t p
p
∂Ωτ
σ t p∇SipV t
p
(2.16)
(2.17)
(2.18)
τ t+∆t Si(x)dS (2.19)
is the volume of the particle at the beginning of the timestep with
ρ t p as its current mass density. The momenta on the grid is then updated using a forward
differencing scheme.
p t+∆t
i
= p t i + ∆tf tot,t+∆t
i
Where, the total force on the grid is given by f tot,t+∆t
i
= f int,t
i
the grid update the particle positions and velocities are updated next.
x t+∆t
p = x t p + ∆t
v t+∆t
p = v t p + ∆t
i
i
Sip
Sip
p t+∆t
i
m t i
f tot,t+∆t
i
m t i
(2.20)
ext,t+∆t
+ f . Following
i
(2.21)
(2.22)
Note that for large deformation problems, the particle lengths in each direction
(l (1)
p , l (2)
p ) is tracked by the GIMP algorithm as the material deforms. The advantages
of tracking lengths are two-fold: the particle volumes are tracked and it circumvents
particle-splitting (e.g., Ma et al. (2006)) in finite deformation simulations. The imple-
mentation of corner tracking as suggested by Ma et al. (2006) alleviates this issue to a
large extent. In order to update particle stresses, the gradient of the particle displace-
27

ments have to be determined. This gradient can be found using gradient of the GIMP
particle shape function and nodal velocity field of the background grid.
Where, ∂∆up
∂x t
∂∆up
= ∆t
∂xt i
∇Sip
p t+∆t
i
m t i
(2.23)
is a two-tensor and the gradient of the incremental displacement field
with respect to the particle coordinates (x t p) at the beginning of the time-step ‘t’ (Bathe,
1982). For finite deformation, the incremental deformation gradient can be determined
from,
dFp = I + ∂∆up
∂x
(2.24)
Where, I is the second order identity tensor. The total deformation gradient is given by
the multiplicative decomposition (which follows from the the chain rule of differentiation
(Malvern, 1969)),
F t+∆t
p
= F t
p · dFp
For small deformation, the displacement gradient can be approximated as,
∂ut+∆t p
∂X = ∂utp ∂∆up
+
∂X ∂xt The small strain tensor follows from above,
ɛ t+∆t
p
(2.25)
(2.26)
= 1
⎡
⎤
T
t+∆t
t+∆t
∂u
⎣ p ∂up +
⎦ (2.27)
2 ∂X ∂X
Where, T is the transpose of the deformation tensor. The Cauchy stress (Malvern,
1969) on the particle can be updated using the deformation gradient eq.(2.25) or eq.(2.26)
depending on the severity of the deformation coupled with the material constitutive law.
28

For a linearly elastic isotropic material with Lame constants λ and µ undergoing small
deformation, the cauchy stress on the particle can be updated using,
Where, e t+∆t
p
σ t+∆t
ij,p
= λet+∆t p δij + 2µɛ t+∆t
ij,p
(2.28)
= tr(ɛt+∆t p ) the trace of the small deformation strain tensor and δij is
the Kronecker delta function. After the particle state variables are updated, the grid
is reset while the particle positions are updated using eq.(2.21). While the Lagrangian
material points are “convected” in this fashion, the background grid is restored to its
initial (undeformed) state. The arbitrary Euler-Lagrange solution strategy naturally
overcomes issues involved with mesh entanglement in large deformation problems.
2.2.1 A Finite Deformation Hyper-Elastic Model for Polymeric
Materials
In this dissertation a hyperelastic material model is assumed to simulate large deformation
behavior of thermoplastic polymers. Consider the following strain energy potential “W”
(from Valavala et al. (2007)) for a compressible hyperelastic material.
W = c1 (I3 − 1) 2 + c2
I1
I 1/3 +
3
I3 2
I2 3
− 30
(2.29)
Where, I1, I2 and I3 are the invariants of the right Cauchy-Green deformation tensor
C (C = F T F, where F is the deformation gradient) in continuum mechanics (Malvern,
1969). They are defined as follows,
29

I1 = tr(C) (2.30)
I2 = 1 2 2
[tr(C)] − tr(C )
2
(2.31)
I3 = det(C) (2.32)
Where, tr(C) and det(C) is the trace and determinant of tensor C. The Lagrangian
strain tensor E is defined as follows,
E = 1
(C − I) (2.33)
2
The 2 nd Piola-Kirchoff stress S is energy conjugate to the Lagrangian strain tensor E.
Hence we have,
S = ∂W
∂E
= ∂W
∂C
· ∂C
∂E
= 2∂W
∂C
(2.34)
Substituting eq.(2.29) into eq.(2.34), and employing the chain rule of differenciation
we get,
S = 2
6c1I3(I3 − 1) − c2
3
+2c2
1
I 1/3
3
I1
I 1/3 + 6
3
I3 2
I2 3
+ 3I1I 2 2
I 2 3
C −1
I
I − 6c2
2 2
I2 C (2.35)
3
Where, the following identities were employed in the above derivation,
∂I1
∂C
= I
∂I2
∂C = I1I − C (2.36)
∂I3
∂C
=
−1
I3C
30

The Cauchy stress in the current configuration can be found using the Piola transfor-
mation:
σ = 1
J FSFT
σ = 1
J
2
3
6c1I3(I3 − 1) − c2
I1
I 1/3
+2c2
1
I 1/3
3
+ 3I1I 2 2
I 2 3
3
+ 6 I3 2
I 2 3
I
B − 6c2
2 2
I2 B
3
2
I
(2.37)
(2.38)
Where, J = det(F) is the Jacobian of the deformation gradient F, B is the left
Cauchy-Green tensor (B = FF T ). As can be seen, J = √ I3 = det(C) = det(B).
2.2.2 The Verlet Integration in GIMP
As observed in the earlier section the internal force on the nodes eq.(2.18) will always
“lag” behind the current external force hence the dynamic force equilibrium is satisfied
only in the limit of small time-step in an explicit time-stepping strategy. A better esti-
mate of the internal nodal force vector is possible through implicit iterations where the
error in solution per time-step is reduced by iterating to a pre-decided error norm (Nair
& Roy, 2009). However, in the interest of expediting calculations during a multi-scale
coupled simulation an alternate explicit time-stepping strategy is employed in this re-
search, namely the “Verlet” integration scheme wherein a mid-step internal force vector
is determined.
The algorithm begins with interpolating particle state variables to nodes of back-
ground grid using equations (2.16) - (2.19). A temporary velocity field is determined
using the following equation,
v ∗ i =
p t i + ∆t
2
f int,t
i
31
m t i
ext,t+∆t
+ f
i
(2.39)

The temporary nodal velocity field is used to update the positions and the deformation
gradients of the material points, using equations (2.21) and (2.23).
x t+∆t
p = x t p + ∆t
∂∆u ∗ p
∂x
= ∆t
i
i
∇Sipv ∗ i
Sipv ∗ i
(2.40)
(2.41)
The “mid” timestep deformation gradient is used to create the mid-interval deforma-
tion gradient,
F t∗
p = F t
p · I + ∂∆u∗
p
∂x
(2.42)
The updated Cauchy stress will follow from the updated deformation gradient eq.(2.42)
and the material constitutive law. For an isotropic material undergoing small deformation
the Cauchy stress is updated using eq.(2.28). However, in this dissertation a hyper-elastic
material model is assumed to simulate large deformation behavior of thermoplastic poly-
mers. The Cauchy stress is updated using equations (2.41), (2.25) and (2.38). The
updated Cauchy stress is used to update the internal nodal force vector.
f int,t∗
i
= −
p
σ t∗
p ∇SipV t
p
(2.43)
In the above equation t ∗ denotes the mid-step update of nodal and particle variables.
It should be further noted that since the equations of motion on the grid are solved using
an updated Lagrangian formulation (Bathe, 1982) the gradient of the GIMP interpolation
function ∇Sip is with respect to the coordinate of the particle and node at the beginning
of time-step ‘t’, i.e., x t p and Xi. The total force and nodal momentum is next updated,
32

f tot,t+∆t
i = f int,t∗
i + f ext,t+∆t
i
p t+∆t
i = m t iv ∗ i + ∆t
2
f tot,t+∆t
i
(2.44)
(2.45)
The updated momentum and force is used to update the particle velocities and stresses
using equations (2.22), (2.23), (2.24), (2.25) and (2.38). For a compressible hyper-elastic
material with initial density ρ 0 , the particle densities can updated using,
Where, J t+∆t
p
= det(F t+∆t
p ).
ρ t+∆t
p = ρ0
J t+∆t
p
2.3 Dynamic Fracture Analysis using GIMP
(2.46)
An important application of the GIMP algorithm is in the modeling of crack propagation
and failure analysis of structures. Contact and friction properties of crack surfaces can
be easily modeled by incorporating the necessary normal and tangential behavior of the
crack surface into the GIMP analysis. Since the GIMP methodology is mesh independent,
crack extension and propagation in GIMP is easier to simulate than conventional FEA
(Duflot et al., 2010). By construction, the Lagrangian particles track evolution of state
variables (such as stress, strain, temperature etc.) within the bulk of the continuum hence
in a crack propagation situation the particles could be also used to track the extension
of crack surface. This unique property allows easier definition of crack since the analysis
grid does not have to be remeshed in order to accommodate new crack geometry, instead
the material point field variables can be adjusted during simulation time to include the
new singularity (see, Nairn (2003)).
In this research the crack in material point (CRAMP) algorithm by Nairn (2003) is
33

Figure 2.8: The GIMP crack
employed to track and simulate propagation of crack surface. The CRAMP algorithm has
been used to calculate dynamic strain energy release rates (SERR) using virtual crack
closure technique (VCCT) (Krueger, 2004) and the dynamic (i.e., with inertial effects
included) version of the J-integral (Guo & Nairn, 2004). The CRAMP algorithm allows
for the definition of multiple nodal velocity fields for those material points that lie along
the crack surface, this approach implicitly allows the definition of discontinuous stress
and displacement field around the vicinity of the crack. In CRAMP the crack surface
is defined as line segments (in 2D) or planes (in 3D). At the beginning of the GIMP
update the position of the material point relative to the crack surface is determined using
a line crossing algorithm. This is illustrated as follows: consider for example, the crack
schematic in fig.(2.8). According to the CRAMP formulation an imaginary line is drawn
from material point mp1 to node 1, since the line intersects the crack surface from above
34

we accordingly assign that material point mp1 is on “top” of the crack surface similarly
we find that mp2 is at the “bottom” in its position relative to the crack surface. Next
in order to account for the discontinuity (crack) multiple velocity fields are introduced
at nodes 1 and 2 (and other nodes in the neighborhood of the crack). At these special
nodes the two equations of motion are solved; those for the GIMP particles above the
crack surface and for those below the crack surface. At the other nodes (such as node 3
in fig.2.8) the regular gimp update follows. Further details on the scheme can be found
in Nairn (2003).
Figure 2.9: Schematic for determining strain energy release rate in GIMP
In this work the SERR at the crack tip is calculated using the VCCT approach (Tan
& Nairn, 2002). Figure 2.9 is a schematic of mode I crack in GIMP. In the schematic
points 1 and 1’ are nodes behind the crack tip located on top and bottom of the crack
surface respectively. Note that the displacements are exaggerated to show the opening of
the crack. Further, it should be understood that according to the CRAMP scheme points
35

1 and 1’ are the same node but with multiple velocity fields there exists a discontinuity in
the velocity at this node (as indicated by the unequal displacements, uy(1) and uy(1 ′ ))a
unique velocity field. The crack tip is located at node 2 and node 3 is located ahead of
the crack tip. For a traction free crack, the crack closure integral in mode I is given by,
GI = lim
δa→0
1
2δa
δa
0
σyy(x)∆uy(x − δa)dx (2.47)
Similarly the mode II SERR (GII) can be determined using the following virtual crack
closure integral,
GII = lim
δa→0
1
2δa
δa
0
σxy(x)∆ux(x − δa)dx (2.48)
To determine the nodal stress values the interpolation scheme suggested in Tan &
Nairn (2002) can be used. Writing the stresses as a linear interpolation between nodal
values at nodes (2) and (3) and the displacements as linear interpolations between nodes
(1) or (1’) and (2), and substituting into the crack closure integral leads to,
GI = u(1) y − u (1′ )
y
(2σ (2)
yy + σ (3)
yy ) (2.49)
12
GII = u(1) x − u (1′ )
x
(2σ (2)
xy + σ (3)
xy ) (2.50)
12
36

Chapter 3
Molecular Dynamics (MD)
The classical dynamical equations of motion are valid for slow and heavy particles with
typical velocities v ≪ c where, c is the speed of light and masses m ≫ me, me being the
electron mass (Liu et al., 2006). Therefore a typical molecular dynamics (MD) simulation
can predict trajectories of a molecular system evolving under the influence of internal
forces which the constituent entities of the atomistic system exert on each other. Thus,
molecular dynamics algorithm finds application in many areas of atomistic simulation. It
has been used to calculate stiffness of novel engineering materials such as nanoclay (Katti
et al., 2005; Suter et al., 2007, 1995), carbon nanotube (CNT) (Frankland et al., 2002;
Liew et al., 2004), transport properties such as thermal conductivity (Maruyama, 2002;
Abramson et al., 2002; Berber et al., 2000; Volz & Chen, 1999), electrical conductivity
(Yuge et al., 2005; Rey-Castro & Vega, 2006; Tang et al., 2001) and viscosity (Erpenbeck,
1984; Hess, 2002). Other popular molecular simulation techniques includes ab initio based
approaches such as density functional theory (DFT), tight-binding (TB) calculations (see
e.g.Leach (1996)). However, first principle calculations come at a high computational
cost. Molecular dynamics based methods on the other hand can leverage state-of-the-
art computing to study larger molecular systems. Billion atom simulations with MD is
possible under parallel computing scenarios (Vashishta et al., 1996; Abraham et al., 2002;
Rountree et al., 2002).
The efficiency enhancements in MD are solely due to the empirical energy functions
that are used to describe the inter-atomic interaction. However, these enhancements come
at the price of accuracy of trajectory prediction. In general, MD is unable to accurately
predict long term responses of the atomistic system due to quantuum errors and curve
fitting approximations of force field parameters. Some authors have hence, proposed cou-
37

pling quantuum mechanics based methods to molecular dynamics e.g.(Abraham et al.,
1999; Khare et al., 2007; Qian et al., 2008) in the process zone to achieve a better pre-
diction of material behavior.
In studying the system behavior of nano-composites, molecular dynamics (MD) has
been the primary method of choice. Kuppa et al. (2003) used MD simulation to study
poly-ethylene oxide (PEO) /nanoclay hybrid structures. They were able to correlate the
results from the analysis with wide-angle neutron diffraction (WAND) and differential
scanning calorimetry (DSC) studies of purely intercalated PEO in MMT. Sinsawat et al.
(2003) used coarse-grained MD to study the influence of polymer matrix architecture
on nanoclay intercalation. In this chapter we focus on multi-scale modeling strategies
for thermoplastic polymer nano-composites using the MD algorithm. We begin with
description of statistical thermodynamics in the canonical (or NVT) ensemble and its
applications to this research. Later, we derive the thermostatted equations of motion
and the common force fields that are used in the modeling of polymers.
3.1 Introduction
Molecular dynamics simulations can be viewed as a bridge between the deterministic
behavior of the macroscopic material system and the statistical nature of atomistic in-
teractions. If we can imagine the position (r) and momenta (p) of a N atom MD system
represented in a 6N dimensional space (r N , p N ) called phase space, during simulation
time the MD system will occupy various regions of the phase space as the system evolves.
We can imagine the collection of possible configurations in phase space (also called mi-
crostates) of the atomistic system to lie inside a 6N dimensional hyper-sphere called
an ensemble. The concept of an ensemble plays a central role in the determination of
macroscopic properties of the atomistic system. Consider for example the evolution of an
observable F (r N , p N ) (F is any variable of interest; it may refer to the internal energy,
38

Helmholtz energy, enthalpy etc.) for the N-atom system. Then according the postu-
late by Gibbs and Boltzmann the macroscopic observable is an average over all possible
micro-states in phase-space of the atomistic model. Mathematically this is written as,
〈F 〉 =
dp N dr N F (r N , p N )ρ(r N , p N ) (3.1)
Where, dp N dr N is short-hand for the 6N space volume element,
dp N dr N = dp 1 xdp 1 ydp 1 z . . . dp N x dp N y dp N z dr 1 xdr 1 ydr 1 z . . . dr N x dr N y dr N z ,the double integral alludes
to the volume integral in 6N dimensional space and ρ(r N , p N ) is the probability associated
with a particular micro-state. In a statistical sense, the ensemble average of the observable
is the expectation value of the observable. In the molecular dynamics method the classical
equations of motion are time integrated to simulate the behavior of the atomistic system
in its phase space. Hence, to extract the macroscopic observable in MD, a time averaging
has to be performed.
F τ = 1
τ
τ
F (r
t=0
N (t), p N (t))dt (3.2)
In the limit of sufficiently long simulation time the following equality will hold,
lim
τ→∞ F τ = F = 〈F 〉 (3.3)
This is the statement of the ergodic hypothesis. It is the fundamental assumption
that bridges molecular dynamics and statistical mechanics. The probability distribution
ρ(rN, p N) is of particular importance in statistical mechanics. For a NVT (constant
number of particles N, constant volume V and temperature T) ensemble also called the
canonical ensemble, the probability distribution is assumed to follow the Boltzmann
distribution,
39

ρ(r N , p N ) = exp −H(r N , p N )/kBT
QNV T
(3.4)
Where, H(r N , p N ) is the total energy (Hamiltonian of the system) associated with a
particular configuration and QNV T is the partition function for the NVT ensemble, kB is
the Boltzmann constant and T is the temperature. The partition function for the NVT
ensemble is written as follows,
QNV T =
1
N!h 3N
dp N dr N exp −H(r N , p N )/kBT
(3.5)
Where, N! is a factor used to prevent ‘over-count’ of the particle states and h is
a quantuum-correction factor usually chosen to be the Plank’s constant . Once the
partition function has been established, all thermodynamic properties can be defined.
For example in a NVT ensemble, the internal energy (U) and Helmholtz free energy (A)
can be derived from the partition function from eq.(3.5),
U =
=
2 kBT ∂Q
Q ∂T
dp N dr N H(r N , p N ) exp −H(rN , pN )/kBT
Q
(3.6)
(3.7)
A = −kBT ln Q (3.8)
= kBT ln dp N dr N
N N H(r , p )
exp
ρ(r
kBT
N , p N
) (3.9)
While the statistical nature of thermodynamic quantities are well established, the
calculation of these variables from simulation is dependent on the efficiency and quality
of MD algorithm. In the following sections, the equations of motion in a thermostatted
atomistic system are derived and the application of the MD algorithm is discussed.
40

3.1.1 Equations of Motion
From Newton’s law, we know that the classical equations of motion for a point mass is,
mia τ i = −∇x τ i U + Fi, i=1,2,. . .,N (3.10)
Where, mi, x τ i and a τ i could be considered as the mass, current position and acceler-
ation vector of atom i at time τ, ∇x τ i is the spatial derivative in the three orthogonal
space directions, Fi is a non-conservative or external force on the i th particle, and U is the
energy of interaction of atom i with all other atoms in the molecular system. In section
3.2, the various forms of energy interactions are discussed. For a non conservative system,
vector force Fi could act as a dissipative force. For example, the force could represent a
velocity dependent damping force,
F S
i = −miγv τ i (3.11)
In the above equation, γ is the damping coefficient that adds a momentum dependent
retardation on the atom ‘i’ while v τ i is the current velocity of atom ‘i’. Here, F S
i is called
the viscous, or Stoke’s friction force. In this dissertation, temperature control is achieved
using the Langevin thermostat (Allen & Tildesley, 1987). The formulation is based on
the addition of a stochastic external force Ri that represents thermal collisions of the
atom ‘i’ with hypothetical solvent molecules. In the case of a thermally excited system
the stochastic force will simulate Brownian dynamics of the atoms in the solute. The
thermostatted equation of motion is now,
mia τ i = −∇x τ ,iU + Ri(τ), i=1,2,. . .,N (3.12)
It should be noted that the stochastic force Ri is chosen such that it satisfies the
41

following relationships,
3.1.2 The Verlet Algorithm
τ
1
lim Ri(t)dt = 0
τ→∞ τ 0
(3.13)
τ
1
lim Ri(t) · Rj(t0 + t)dt = aδ(t0)δij
τ→∞ τ 0
(3.14)
The typical molecular dynamics algorithm simulates evolution of the atomistic system in
its phase space through time integration of the equations of motion. The time integration
is performed for a given set of boundary (usually a periodic boundary condition, see Allen
& Tildesley (1987)) and prescribed temperature and pressure conditions (or alternatively,
prescribed temperature and volume). For a simulation algorithm to be successful, it needs
to satisfy the following conditions (Allen & Tildesley, 1987),
1. It should be fast, and require little memory.
2. It should permit the use of a large time step ∆t.
3. It should duplicate the classical trajectory as closely as possible.
4. It should satisfy the known conservation laws of energy and momentum, and be
time-reversible.
5. It should be simple in form and easy to program.
The Verlet algorithm (Verlet, 1967) is the most widely used method of integrating
equations of motion. The method is based on current positions x τ , accelerations a τ , and
the positions x τ−∆τ from the previous step. Using the Taylor series expansion of the
vector functions around the known system configuration at time ‘τ’,
x τ+∆τ = x τ + ∆τv τ + (1/2)∆τ 2 a τ + . . . (3.15)
42

x τ−∆τ = x τ − ∆τv τ + (1/2)∆τ 2 a τ − . . . (3.16)
Combining the above two equations we get the update for position as follows,
x τ+∆τ = 2x τ − x τ−∆τ + ∆τ 2 a τ
The velocities can be calculated as follows,
v τ = xτ+∆τ − x τ−∆τ
2∆τ
(3.17)
(3.18)
We see that the error in eqn.(3.17) is of the order ∆τ 4 while the error in velocity
(eq.3.18)is of the order ∆τ 2 . We can observe from eqn.(3.17) the position update is
time-reversible (since, x τ+∆τ and x τ−∆τ play symmetrical roles). The algorithm however
introduces some numerical imprecision since in update eq.(3.17), a small term O(∆τ 2 )
is added to the difference of large terms O(∆τ 0 ). To improve on these shortcomings a
half-step ‘leap-frog’ scheme (Hockney, 1970) was introduced. In this algorithm a mid-step
velocity is determined as follows,
x τ+∆τ = x τ ∆τ
τ+
+ ∆τv 2 (3.19)
∆τ
∆τ
τ+ τ−
v 2 = v 2 + ∆τa τ
(3.20)
The velocities for energy calculations are computed using a mid-time interval update,
v τ+∆τ =
∆τ
∆τ
τ−
vτ+ 2 + v 2
2
(3.21)
However, as evident from the above equation (eq.3.21) the leap-frog algorithm does
not handle round-off errors in predicting velocities . In order to overcome these errors
a velocity Verlet algorithm was proposed (Swope et al., 1982). The algorithm uses the
43

following update equations,
x τ+∆τ = x τ + ∆τv τ + 1
2 ∆t2 a τ
v τ+∆τ = v τ + 1
2 ∆τ a τ + a τ+∆τ
(3.22)
(3.23)
In general, when the velocity Verlet algorithm is implemented a mid-step velocity is
determined,
∆τ
τ+
v 2 = v τ + ∆τ
2 aτ
The position and velocity updates then follow in a single update step,
(3.24)
x τ+∆τ = x τ ∆τ
τ+
+ ∆τv 2 (3.25)
v τ+∆τ ∆τ
τ+
= v 2 + 1
2 ∆τaτ+∆τ
(3.26)
At this point, the kinetic energy is calculated and the potential energy is calculated in
the force loop. The velocity Verlet algorithm as implemented in the molecular dynamics
algorithm LAMMPS (Plimpton, 1995) is the atomistic trajectory predictor algorithm
used in this dissertation.
3.2 Molecular Force Fields
In general quantuum mechanics (QM) based methods provide a better time-description
of atoms in phase space. However since the energy calculated in a QM calculation is
a function of the motions and interactions of electrons within each atom the method is
computationally restricted to solving smaller atomistic systems. The Born-Oppenheimer
approximation allows the energy of the system to be written as a product of mutually
independent functions of nuclear and electron coordinates (Leach, 1996). In MD, the
44

position and configuration of electrons within the shell of atoms are ignored and focus
is entirely on the energy due to the position of nuclei with respect to each other. This
approximation allows MD to be robust and scalable to larger thermodynamic systems.
The molecular force fields employed in MD algorithm are in general empirical force fits
to quantuum mechanical calculations. In certain cases the molecular dynamics algorithm
is able to provide results as good as the highest quantuum mechanical calculations, for a
fraction of computer time (Leach, 1996). For a N-body system, the total energy in the
system can be written as the sum of contributions from individual potentials,
U(r) =
Vij(ri, rj) +
Vijk(ri, rj, rk) + . . . (3.27)
i
j
Figure 3.1: Common molecular dynamics force fields for polymers
i
Where, Vij is a two-body interaction potential and Vijk is the potential due to three-
body interaction. As discussed above molecular mechanics is based on simple interaction
of nuclei centers, hence the energy of interactions can be written purely as a function
of nuclei coordinates. In a polymer system, the energy contributions are due to bonded
interactions such as bond stretching (Ebond), the opening and closing of angles between
45
j
k

onds (Eang), bond torsions (Etors), and non-bonded interactions such as van der Waals
forces (Enb). The total energy (U) for these components can be written as a sum of these
individual contributions,
U(r) = Ebond + Eang + Etors + Enb
(3.28)
In the following sections some the popular molecular force field forms are discussed.
3.2.1 Bond stretching
Figure 3.2: Schematic of bond stretching
The energy due to bond stretch accounts for the contribution to the total configu-
rational energy of the system due to stretching (or compressing) of the intra-molecular
bonds during simulation. The simplest form of this energy force field is the harmonic
potential given below.
Ebond =
bonds
kl 2
(l − l0)
2
(3.29)
Where, kl is the stiffness of the bonds, l is the current length of the bond while l0
is the ‘equilibrium’ length that the bond adopts when all other terms in the force field
46

contribute (Leach, 1996). Note that in the summation above, the energy contribution
from all bonds in the system are accounted for.
3.2.2 Angle bendings
Figure 3.3: Schematic of angle bending
Angle bending interactions are usually included in describing molecules to include
the energetic contribution of angles deviating from their equilibrium configurations. It is
usually described by Hooke’s law,
Eangle =
angles
kθ
2
(θ − θ0)
2
(3.30)
Where, kθ is the stiffness of the bonds, θ is the current angle of the bond while θ0 is
the ‘equilibrium’ angle.
3.2.3 Bond torsion
The bond stretch and angle bending accounts for the stiffer response in the molecular
system to configurational change (Leach, 1996). In general, the conformational change
of the polymer is attributed to the complex interplay between torsional and non-bonded
47

Figure 3.4: Schematic of bond torsion
interactions. The existence of energy barriers to rotation about chemical bonds is of prime
importance in defining the structure property of the polymer molecule. The barrier to
rotation arises from the repulsive forces in the non-bonded interactions between the end
atoms of the torsional unit, hence in principle, it is possible to model structure changes
through non-bonded interactions. Most torsional interactions are almost modeled using
a cosine series expansion (Leach, 1996). One form of interaction is,
Etors =
N
tors n=0
Alternatively it could also be expressed as,
Etors =
Vn
[1 + cos(nω − γ)] (3.31)
2
N
tors n=0
3.2.4 Non-bonded Interactions
Cn cos(ω) n
(3.32)
Most physical properties of the molecular system can be explained by the non-bonded
interactions between atoms of the physical system. The non-bonded interactions can
be either due to the electro-static forces that occur between ionically charged atoms or
48

Figure 3.5: Schematic of non-bonded interactions
due to the quantuum-mechanical nuclear interactions between atoms. In this work, the
electrostatic (or Columbic) interactions are ignored and hence, the contribution to non-
bonded interactions are entirely attributed to the dispersive-repulsive forces called “van
der Waals” (Allen & Tildesley, 1987) that quantifies the quantuum-statistical interactions
between atoms of the molecular system. These are short ranged nuclear forces with an
influence zone in the vicinity of a few Angstroms. The van der Waal force is in general
the sum of instantaneous attractive and repulsive forces between atoms (Leach, 1996).
The dispersive or London force is an attractive force due to instantaneous induced dipoles
with in the atoms. The repulsive forces on the other hand is primarily attributed to the
forces arising from the Pauli exclusion principle. Figure (3.6) is the typical energy versus
inter-atomic separation profile for van der Waals interactions.
The Lennard-Jones potential is a common empirical potential used to model non-
bonded interactions and is given by,
Enb =
σ 4ɛ
r
i
j>i
12
−
σ
6
r
(3.33)
Where, r=|rij| is the distance between atoms i and j, ɛ is the depth of the potential
well and σ is the distance at which inter-particle energy is zero.
49

Figure 3.6: Profile of van der Waals interaction energy versus interatomic separation
3.3 Modeling Thermoplastic Polymer Nanocomposites in MD
In the context of MD simulation, modeling of thermoplastic or thermoset polymers rep-
resents significant computational challenges. Some of the more predominant challenges
are listed below:
1. A realistic description of material atomistic configuration at given ambient condition
is inherently difficult. Most thermoplastic polymers have an amorphous (i.e., lack
of specific long range order) configuration at room temperature. For example,
polypropylene is semi-crystalline at room temperature i.e., it has regions of ordered
lamellar crystal configuration embedded in a matrix of amorphous molecules. The
visco-elasticity of the polymer material is entirely attributed to the amorphous
matrix while the crystal structure gives the material stiffness.
50

2. The macro-scale behavior (transport properties, thermo-mechanical behavior etc.)
of the polymer (or polymer nanocomposite) atomistic model is dependent on the
force fields used in the simulation. Given in sec.(3.2) is a list of energy contribu-
tions that are commonly considered in modeling polymers. Because there are few
force fields designed specifically for in-organic polymers, there has been an issue of
transferability of force fields to study polymeric molecular systems.
3. The equilibration (or energy relaxation) of polymer molecules to its minimum en-
ergy takes longer compared with regular solid lattice structures. The difficulties in
equilibrating structures arise from configurational as well as energetic sources of the
polymer in the phase space that add to the free energy of the system. It is known
that the parameters for minimum energy configuration is dependent on a multitude
of parameters.
Additional difficulties arise when nanoparticles such as nanoclay, is introduced into
the polymer atomistic model. The challenge is to develop a computationally viable model
that is able to address some of the issues listed above. In this dissertation, in the interest of
saving computational time we use a coarse grained polymer system using the parameters
described in Wei et al. (2002). In the following sections methods are suggested to model
thermoplastic polymers (sec.3.3.1), a model for nanoclay (sec.3.3.2) and finally a working
model for thermoplastic polymer nanocomposite (sec.3.3.2).
3.3.1 Thermoplastic Polymers
In the MD modeling of polymer systems, coarse graining has found wide acceptance.
Coarse graining significantly reduces the computational time by eliminating degrees of
freedom from the system that would be considered in the full atomistic model. For
example, a full MD model of polypropylene will typically include the intra- and inter-
51

molecular bonded and non-bonded interactions of all the constituent atoms of the atom-
istic model. A coarse grained model simplifies these interactions by creating an “equiv-
alent” or “pseudo” atom that has bonded and non-bonded properties equivalent to the
monomer of the full atomistic system. In this manner, the computations per monomer is
effectively reduced. The reduced degrees of freedom translates to computational savings
for the MD model. Another advantage of coarse graining is that it overcomes the high
frequency vibration of the C-H bonds in the polymer which automatically allows for larger
timestep in the MD simulation. While coarse graining has obvious advantages, certain
details of the chemical structure of the molecular model are inherently overlooked such
as the chain rotation and its related effect of on chain configuration and relaxation. For
a full realization of polymer properties these structural details may be of significance and
needs to be considered for future studies. In this dissertation, the atomistic model for
polypropylene uses the coarse-grained bead-spring model with parameters suggested by
Wei et al. (2002). In Wei et al. (2002), the −CH2− molecule is modeled as a single atom
or “repeating unit” for their coarse grained model of polyethylene. In polypropylene, the
repeating unit (or monomer) is (−C3H6−)n where n is the number of repeated units in
the polymer. The structure of molecule is given in fig.3.7.
The coarse grained model collapses the entire structure to two backbone pseudo atoms
as shown in fig.3.8. The mass of the polypropylene monomer is 42g/mol. In this coarse
grained model the mass of each pseudo monomer is taken to be 21g/mol. An alternate
coarse grained model could possibly be if the molecules CH2,CH3 and CH are consid-
ered as separate entities, this would provide an improved description of the monomer
with marginal increases in computational demand. In order to generate the amorphous
polymer model we employ a random walk algorithm to place the coarse-grained beads
at randomly chosen sites inside the volume box. All polymer chains were assumed to
have length of 20 beads, to minimize entropic contributions to the Helmholtz free energy.
52

Figure 3.7: Chemical formula of polypropylene
Figure 3.8: Coarse grained polypropylene monomer
After the polymer is created, the following steps are chosen to equilibrate the model at
the required ambient temperature. Note that the simulations in this work are in the NVT
canonical ensemble.
53

1. As mentioned above (sec.3.1.1), the Langevin thermostat is used to regulate the
temperature of the thermodynamic system at a desired level. At first a soft non-
bond potential is employed to remove the over-lapping centers of the polymer beads
that might have occupied the same site (or in the immediate vicinity) during the
random walk operation. The NVE dynamics (with no thermostatting) automati-
cally acts to remove the overlap of the centers and hence reduce the energy of the
system.
2. At first we run the MD model of polymer at a temperature at least three (3) times
higher than the ambient temperature of the model. For example, for the polymer
model considered in this dissertation the system was thermostatted at 1000K using
explicit rescaling of atomic velocities (see, Plimpton (1995)) in order ensure that
the polymer molecules fill the simulation box and are not restricted to a single
region of the simulation box. In general a ‘high’ temperature run ensures that the
volume box is evenly filled by the polymer molecules. It should be noted here that
if chain entanglements are created during the random walk procedure it will be
hard to move chains away from each other. The self avoiding random walk (SAW)
could be used to create the polymer model, in which case chain entanglements are
automatically avoided.
3. Next the temperature is reduced to desired level (the temperature for this work is
300K) and run using NVE and velocity rescaling to get a uniform distribution of
temperatures (the kinetic energy of the system was monitored).
4. Finally, the Langevin thermostat is employed and dynamics is used to equilibrate
the model until total energy of the system is minimized. Alternatively the auto-
correlation function (ACF) for the atomistic positions of the beads could be mon-
itored to ensure system convergence to equilibrium. In principle, the equilibrium
54

configuration of the polymer should be independent of the initial condition for the
atomistic system. The decay of the auto-correlation function ensures that the fi-
nal configuration of the system will be truly un-correlated to its initial state. An
additional note must be made that running a purely NVT simulation can build
significant but spurious hydrostatic stresses inside the MD model that may be
detrimental to future simulations and could potentially lead to erroneous results.
A NPT (constant number of particles N, pressure P and temperature T ) MD at
zero pressure is suggested to relieve these spurious internal stresses.
3.3.2 Nanoclay
Figure 3.9: Profile views of nanoclay in XY and XZ planes
Montmorillonite (MMT) is a well-known clay mineral. Its structure essentially consists
of two silica tetrahedral sheets sandwiching an edge-shared octahedral sheet of either
aluminum or magnesium hydroxide, known as t-o-t sheets. Most naturally occuring
MMT has intercalated sodium (Na + ) or calcium (Ca +2 ) ions in the gallery space. The
55

crystal system of the nanoclay is monoclinic with lattice parameters, a = 5.20˚A,b =
9.20˚A,c = 10.13˚A and α = 90 ◦ ,β = 99 ◦ and γ = 90 ◦ . The nanoclay belongs to the C2/m
space group. From Toth et al. (2004), we have the fractional coordinates of the nanoclay
system as given in Table 3.1.
Table 3.1: Position of Nanoclay atoms in space group C2/m in terms of fractional coordinates
ξ, η, ζ
Atoms ξ η ζ
Al 0.000 0.333 0.000
O1 0.481 0.500 0.320
O2 0.172 0.728 0.335
O3 0.348 0.691 0.110
OH 0.419 0.000 0.105
H 0.320 0.000 0.170
Si 0.417 0.329 0.270
Figure 3.9 is the structure of the nanoclay species used in this simulation. The
alumina-silicate structure in this work does not consider the inclusion of Na + or K +
ions because a fully exfoliated nano-clay morphology is considered in this dissertation.
The following steps are chosen to create and equilibrate the polymer nano-composite
model at the required ambient temperature.
1. The nanoclay model is embedded in a coarse grained propylene model generated
using the randolm walk algorithm. As before, an energy minimization is run using
the soft potential approach. In this case however, the non-bonded potential applied
to both the polymer-nanoclay interaction and the polymer-polymer interaction.
The nanoclay does not participate in this NVE dynamics (i.e., the force on it is
zeroed at the end of every run).
2. Once the overlap energies are dissipated, NVE dynamics coupled with high temper-
ature thermostatting (with explicit velocity rescaling) is used to evenly distribute
the polymer beads around the simulation box. Once again the nanoclay does not
56

move during polymer dynamics.
3. Next the polymer molecules are held fixed while NVE is performed on the nanoclay
to equilibrate its configuration. In this manner the nanoclay will equilibrate under
the influence of forces from the surrounding polymer matrix in addition to its own
internal forces.
4. In the final step the Langevin thermostat is employed and atomistic dynamics is
used to equilibrate the polymer nanocomposite (coarse grained polymer + nanoclay)
model till desired energy equilibrium is achieved.
3.4 Applications of the Molecular Dynamics Algorithm
As shown above the molecular dynamics algorithm can be used to develop a realistic
model of atomistic interactions. As a numerical scheme it provides a viable alternative
to computationally expensive quantuum mechanical calculations. Some of the applica-
tions of the MD algorithm has been to study free-energy (Muller-Plathe, 1991; Kollman,
1993; Mavrantzas & Theodorou, 1998; Zhao & Aluru, 2008), transport properties such
as diffusivity (Pant & Boyd, 1993; Heffelfinger & van Swol, 1994; Charati & Stern, 1998),
thermal conductivity (Maruyama, 2002; Abramson et al., 2002; Berber et al., 2000; Volz
& Chen, 1999) etc., in addition to mechanical property characterization of novel mate-
rials where an experimental characterization of properties is difficult (Suter et al., 2007,
1995; Frankland et al., 2002; Liew et al., 2004). In this section two cases of interest are
considered: (a) the MD method is used to study evolution of free energy, specifically the
Helmholtz free energy of the polymer system as a function of mechanical strain in the
body and (b) a MD based scheme is developed to study constitutive relations of poly-
mer nano-composites using the virial stress definition coupled with continuum damage
mechanics (CDM).
57

3.4.1 Determination of Free-Energy due to deformation
The thermodynamic work that can be extracted from a system is dependent on the free-
energy in the system. In polymers, free-energy dominates the deformation process and
must be quantified for a complete description of constitutive behavior. This scenario is
especially true for long chain macromoecules where the configurational entropy of the
system is predominant and cannot be ignored. Through the evaluation of free energy we
can quantify both energetic and entropic contributions in the deformation of polymers.
The MD algorithm has been used to calculate free energy of reactions, solvation, phase
change and mechanical deformation.
However, the determination of free-energy of a thermodynamic system through en-
semble averaging methods such as MD or Monte-Carlo (MC) is difficult due to inherent
short comings in these algorithms which is explained as follows: For most macromolecules
there may exist multiple minimum energy configurations separated by low-energy bar-
riers. In most cases of the MD (or MC) simulation the sampling space is limited to a
small zone (or lower energy levels) of the phase space. On the other hand, inspection of
eq.(3.9) reveals that the higher energy levels makes important contributions to the Hel-
moholtz energy due to the exp (H(rN, p N)/kBT ) term in the ensemble integral. Hence
an adequate phase-sampling must be performed in order to accurately determine the free
energy. There are multiple methods of interest in the determination of the Helmholtz
energy, such as (Leach, 1996): thermodynamic perturbation, thermodynamic integra-
tion, slow-growth method etc. In this work we shall examine the use of thermodynamic
integration for the purpose of free-energy of solids undergoing deformation.
While the calculation of absolute free energies are difficult for materials with disor-
dered phases, the change in free energy of an amorphous system can be quantified. Since,
the free energy of a thermodynamic system is a state variable (i.e., it is path independent)
58

the thermodynamic integration (TI) method seeks to determine the change in free energy
between two energetically distinct systems that exist at locations ‘0’ and ‘1’ separated in
phase space. In evaluating the change in free energy TI prescribes a phase trajectory to
the system such that we can sufficiently sample all energy levels that exist between states
0 and 1. The mathematical realization of this procedure is conceived by re-writing the
Hamiltonian of the system as an explicit function of the variable λ. The Hamiltonian of
the system is simply the sum of kinetic energy K and potential energy U. The parameter
λ is introduced as a perturbation in the parameters of the interaction energy functions
that are in U (see section 3.2). Consider the total potential energy of the system as
follows,
U(r) =
bonds
+
i
kl
2 (l − l0) 2 +
j>i
4ɛij
angles
σij 12 r
kθ
2 (θ − θ0) 2 +
−
σij
6
r
N
tors n=0
Vn
[1 + cos(nω − γ)]
2
(3.34)
We then prescribe the values of the constants in the energy field (e.g., kl, kθ, etc.) in
the above equation (3.34), such that as λ is ramped from 0 to 1, it forces the system to
evolve from a state 0 (at λ = 0) to a final state 1 (at λ = 1). That is we have,
kl(λ) = k 0 l (1 − λ) + k 0 l λ (3.35)
kθ(λ) = k 0 θ(1 − λ) + k 1 θλ (3.36)
Vn(λ) = V 0
n (1 − λ) + V 1
n λ (3.37)
ɛij(λ) = ɛ 0 ij(1 − λ) + ɛ 1 ijλ (3.38)
σij(λ) = σ 0 ij(1 − λ) + σ 1 ijλ (3.39)
Where, the superscripts 0 and 1 pertains to the constants that characterize the various
interaction energies in each state. Hence, we have an explicit relation for the Hamiltonian
59

(and hence the free-energy) of the system as a function of the internal variable λ. We
can now determine the change in free energy of the system (∆A) at state 1 relative to a
state 0 as follows,
From eq.(3.8) it follows,
get,
∆A = A 1 − A 0
λ=1
∂A
∂λ
=
λ=0
1 ∂Q
= −kBT
Q ∂λ
(3.40)
∂A
dλ (3.41)
∂λ
(3.42)
Substituting for the partition function Q for the canonical ensemble from eq.(3.5) we
∂A
∂λ
1
=
Q
∂H
=
∂λ
dp N N ∂H
dr
λ
∂λ exp (−H/kBT ) (3.43)
(3.44)
Where, λ denotes the ensemble average for a given value of λ, and H is the
Hamiltonian. We can then evaluate the change in free-energy (∆A) by eq.(3.45), which
is the area under the curve in fig.(3.10).
∆A =
λ=1
λ=0
∂H
dλ (3.45)
∂λ λ
The Helmholtz energy can hence be evaluated by finding the ensemble average in
eq.(3.45) for various values of λ (between 0 and 1) and then using numerical quadrature
to determine the integral. In this dissertation a representative TI method has been
suggested to find the Helmholtz free energy due to deformation for the polymer model
described in section 3.3.1. For the sake illustration we consider only the interaction
60

Figure 3.10: Calculation of free energy change using thermodynamic integration
energies due to changes in bond-length and non-bonded interactions. The free energy
is determined by parameterzing the non-bonded interactions in the internal energy as a
function of λ. That is we write,
Enb =
σ(λ)
4ɛ(λ)
r
i
j>i
12
6
σ(λ)
−
r
(3.46)
Where, ɛ(λ) = ɛ 0 (1 − λ) + ɛ 1 λ and σ(λ) = σ 0 (1 − λ) + σ 1 λ. We can now write the
Hamiltonian for the system as,
H(p N , r N ; λ) =
N
i=1
p i · p i
Where, p i is the momentum of atom ‘i’ and U(λ) is given as,
bonds
i
j>i
mi
U(λ) = kl
2 (l − l0) 2 +
σ(λ)
4ɛ(λ)
r
+ U(λ) (3.47)
12
6
σ(λ)
−
r
(3.48)
Thus we can derive a closed form analytical expression for the derivative of the Hamil-
61

tonian with respect to λ,
∂H
∂λ
=
i
j>i
4(ɛ 1 − ɛ 0 )
+6(σ 1 − σ 0 ) ɛ
σ
2
σ 12
σ
6
−
r r
σ
12
σ
6
−
r r
(3.49)
During MD simulation, the time average of the quantity ∂H/∂λ is calculated for each
value of λ. A three-point Gaussian quadrature (Reddy, 2006) is used to evaluate the
integral, hence 3 separate MD simulations has to be run for each step of deformation
state. The results from this analysis is given in Chapter 6.
3.4.2 Multi-scale modeling of Thermoplastic composites
Figure 3.11: Schematic of the MD simulation model for polymer nanocomposite
One of the primary challenges in modeling novel materials is the characterization
of mechanical properties such as stiffness and strength. While the widely accepted en-
gineering solution is to a subscribe a phenomenological model, MD presents a case of
representing stress-strain response based on actual atomistic behavior. In this work we
62

consider the stress-strain (constitutive) response of polymer nanocomposite employing
the atomistic stress definition (Zhou, 2003). It is envisoned that a hierarchical atomistic
simulation based model will be computationally effective in characterizing the stiffness
and damage induced degradation of material stiffness. To our knowledge this approach
has not been attempted.
The procedure involves deforming the simulation box as per a prescribed deformation
gradient F. Note that although a homogeneous deformation field is assumed, local in-
homogeneities such as micro-void formation, void coalescence, interface debond (in the
nanocomposite) etc. are allowed to evolve in the MD system. In this dissertation we are
particularly interested in modeling a nanoclay platelet oriented along the local x1-axis of
the simulation box as shown in fig.(3.11).
Figure 3.12: Schematic of deformation of simulation box under a volumetric and deviatoric
(e.g. simple shear) deformation gradients
The specified orientation of nanocomposite gives the nanoscale RVE orthotropic prop-
erties. This assumption enables us to decompose deformation into its deviatoric and vol-
umetric parts and study each mode of deformation in isolation (as shown in fig.3.12). In
this research the stress-strain response characteristics of the system involves deforming
63

the volume box (in volumetric or deviatoric deformation) and determining the stress that
develops inside the specimen. Historically, researchers employing atomistic simulation
algorithms have used the virial stress to define stress in the discrete system,
¯Παβ = 1
Ω
i
−miv α i v β
i
+ 1
2
j>i
r α ijf β
ij
(3.50)
Where, α, β are indicial variables that could be either =1, 2 or 3, mi is mass of atom
‘i’, Ω is the volume of the atomistic system, r α ij is α component of the vector joining
atoms i and j, while f α ij is the interaction force between atoms i and j as derived from the
total potential energy of system (eq.3.27). In Zhou (2003), the definition of true-stress
(or Cauchy stress) as a measure of internal force between the particles (or atoms) of the
system is invoked to show that the commonly accepted format of virial stress (eq.3.50) is
incorrect. Further, it is shown that the virial stress in its definition in eq.(3.50) is only a
measure of momentum flux through an imaginary plane fixed in space. The Cauchy stress
is now defined as the ensemble average (or time-averaged through MD) of the following
quantity,
¯σαβ = 1
2Ω
i
j>i
r α ijf β
ij
(3.51)
The stress measure converges to the Cauchy tensor in the continuum sense, when a
sufficiently large time and volume average is performed. Later in Chapter 5 we develop
a phenomenological model for the failure of nano-composites using the applied strain
and measured stress (eq.3.51), and compare the results with the baseline case where the
polymer does not contain nano-particle reinforcement.
64

Chapter 4
Concurrent Multiscale Modeling of Thermoplastic PNC
Structural and configurational characteristics of a polymer are both time and tem-
perature dependent. In particular, polymer characteristics such as viscoelastic relaxation
is a manifestation of polymer chain rearrangements and cooperative motion of polymer
molecules as a function of time. The recent interest in nanoclay technology to enhance
polymer properties has demanded further scientific enquiry to better explain and un-
derstand the physics of reinforcement. In this work the focus is on modeling material
behavior using multi-physics based algorithms to capture both the nano-scale (discrete)
behavior and the continuum behavior at the macroscopic length scales.
The theoretical model of thermoplastic polymer nano-composites must capture the
molecular physics at the atomistic scale in order to provide a better description of ma-
terial behavior. It is envisioned that molecular physics based modeling approaches will
eventually help in better design and risk assessments of polymer based materials for
structural applications. The purpose of this chapter is to provide a physical rather than
phenomenological model of the molecular rearrangements and configurational changes in
polymer in relation to macroscale loading through a concurrent atomistic and continuum
coupled simulation.
4.1 Introduction
Over the past 10 years numerous analysis techniques has been proposed to tie nanoscale
events to macroscale stimuli during simulation time. These simulations fall into the
category of concurrent analysis. Typically in concurrent simulations, the continuum
response is simulated using the finite element method and the region of interest or process
65

zone (see, fig.4.1) is simulated using the MD algorithm. The information exchange occurs
through the handshake zone where a region of pad atoms occupy the same space as the
continuum finite elements.
Figure 4.1: Schematic of Concurrent coupling
Depending on the description of the overlap region, the concurrent coupling algo-
rithm can be broadly divided in to two kinds: (a) direct coupling (e.g. Wagner & Liu
(2003); Xiao & Belytschko (2004) etc.) and the recently proposed (b) statistical coupling
technique (Saether et al., 2009).
In direct coupling (DC), the finite element mesh is gradually refined from a coarse
mesh starting from the external boundaries of the model down to atomistic sizes at the
handshake region. At the interface zone the atoms from the atomistic domain is coupled
one-on-one to each interface finite element node, as the arrows in figure 4.2 indicate.
While DC based methods is a widely employed for coupling atomistic simulations to
continuum, there are some inherent disadvantages:
1. Tying individual atoms to finite element nodes at the interface means cumbersome
mesh algorithms to realize the model setup.
66

Figure 4.2: Schematic of the Direct Coupling methodology
2. Time-step for dynamic problems are severely effected due to refined mesh at the
interface.
3. For finite temperature simulations of the MD domain, the atoms are under constant
thermal motion, for example in solid lattices the atoms undergo thermal vibration
about its equilibrium position. These vibrations emanate out of the MD domain
into the continuum region in the form of high frequency noise, and can potentially
destroy the numerical solution.
An alternative to direct coupling, is the recently developed statistical coupling tech-
nique (Saether et al., 2009). In this work, the statistical coupling method also called the
embedded statistical coupling method (ESCM) is employed to study polymer systems.
Some of the salient features of the coupling technique are listed here as follows:
1. The handshake zone is divided into volume cells through Voronoi tessellations (as
shown in fig.4.3) such that the interface node ‘communicates’ information with
the atomistic domain with a group of atoms rather than individual atoms. This
construction has a two-fold advantage (1) the finite element mesh does not have to
67

e refined down to atomistic size, and (2) the volume cell construction allows for
proper statistical sampling of atomistic trajectories and can hence provide a better
description of the material behavior
2. Through the statistical coupling methodology the acoustic vibrations (phonons)
emitted from the thermal environment of the atomistic zone can be captured and
diffused at the handshake region hence allowing for larger time-steps in the contin-
uum update (see, Saether et al. (2009)).
Figure 4.3: Schematic of the Statistical Coupling methodology (from Saether et al. (2009))
The primary advantage of using concurrent coupling is the ability to avoid finite
boundary (or free edge) effect. Concurrent simulation strategies also imply computa-
tional advantages by cutting down on the computational overhead involved in atomistic
simulations by restricting the calculations to a narrow process zone. In the following
section ESCM is extended to couple GIMP (see, Chapter 2) to MD simulations.
68

4.2 Embedded Statistical Coupling in GIMP
Figure 4.4: Layout of Coupled Model
In this section the statistical scheme developed in (Saether et al., 2009) is extended to
couple the GIMP algorithm to molecular dynamics (MD). GIMP is a meshless particle
scheme, originally developed by (Bardenhagen & Kober, 2004) as an extension to the
material point method (see e.g., Sulsky et al. (1995)) to improve the numerical stability
of the method. As reviewed in Chapter 2, GIMP particles are constructed as finite
domain entities that deform in a manner consistent with the deformation of the material
body. In ESCM,volume cells (VC) are constructed at the overlap domain of MD/GIMP
(also called handshake zone, see fig.4.4) to communicate information between the internal
69

MD environment and the external continuum (GIMP) region. The volume cells serves
the three-fold purpose of (a) determining averaged displacements of the atomic entities
within it (b) transferring forces from the continuum region down to the atoms on the MD
boundary (c) preventing the MD surface from degradation when forces are applied on it
(d) prevent the artificial “heating” of the MD atoms due to the loading wave ingressing
from the continuum environment or reflection of phonons emanating inside the atomistic
region at the MD/GIMP interface. In GIMP, it is easy to setup the VCs since the
deformation of the particle domain is tracked during real-time simulation. Hence, in a
continuous deformation field the contiguous particle domains remain contiguous even after
deformation. The construction of volume cells also enables easier spatial discretization
of the atomistic/continuum overlap domain.
In ESCM, the MD domain is divided into three (3) sub domains as shown in the
fig.(4.4). The surface volume cell (SVC), internal volume cell (IVC) and the interior MD
region. Since the material point (mp) domain serve as the volume cells, the IVC is also
referred to as IVC mp, similarly SVC is also called SVC mp. The IVC calculates time-
averaged displacements of the atoms (referred to here on as ∆u IVC
p
) within each IVC and
applies this as incremental displacement of the material point of each IVC at time ‘t’,
Where, x t−∆t
p
x t p = x t−∆t
p
+ ∆u IVC
p
(4.1)
is the position of the material point from the previous timestep. The
material point IVC uses the time-averaged displacements and calculates the internal force
on the background nodes of GIMP due to this change in position. This force is applied
as an equally distributed external force on the atoms within the IVC. The purpose of the
SVC is to prevent the degradation of the MD surface atoms due to the internal forces
from the continuum.
The continuum is updated using the explicit Verlet integration algorithm introduced
70

in Chapter 2. In the interest of brevity, the equations are not repeated here. Given
below is the explicit time-integration of the atomistic equations of motion that follows
the continuum update.
In the atomistic update, the IVC atoms that are a part of the IVC material points is
under the influence of an external force referred to as, f IVC
a , where the subscript a refers
to atom within the IVC ‘p’. This external force is calculated as,
f IVC
a
=
tot,t+∆t
i
Sipfi N IVC
a
(4.2)
Where, Sip is the GIMP interpolation function between IVC material point ‘p’ and
node ‘i’ and N IVC
a
is the number of atoms in this IVC. As can be seen the interpolation
function for each atom in the IVC is not calculated, instead all atoms of an IVC are
assumed to have the same interpolation function. Further, we determine the velocities
and accelerations of the SVC atoms as follows,
v ∗,SVC
a =
a SVC
a =
i
i
Sipv ∗ i
Sip
f int,t
i
ext,t+∆t
+ f
i
m t i
(4.3)
(4.4)
The following algorithm details the update of the atoms within the interior (abbrevi-
ated as INT), IVC and SVC parts of the MD domain using a Verlet integration scheme.
In the description to follow the interaction energy between atoms in the MD zone is
designated as the scalar function ‘φ’. For example, in a simple Leonard-Jones system,
σ 12
σ 6
φ = 4ɛ − where, r=|rij| is the distance between atoms i and j. Also defined
r r
is the discrete time step of molecular dynamics algorithm as, ∆τ = ∆t
N
where, ∆t is
the coarse time-step and N is the number of MD steps. Begin time integration of MD
71

equations of motion.
1. Update velocities of atoms in the interior (INT) and IVC atoms of the MD zone
at the beginning of time-step τ of MD. As mentioned in Chapter 3, the Langevin
thermostat (see e.g., Allen & Tildesley (1987)) is used to maintain the temperature
of the atomic zones. As per its formulation a stochastic damping force f T a is ap-
plied on atom ‘a’ such that the overall temperature of the system is maintained at
‘T’ (temperature units). In general, the Langevin equations are used to simulate
brownian dynamics in a thermally excited system. For an atom ‘a’ the mid-time
step velocities (denoted as v ∗ ) in the INT and IVC zones are,
Where, ∇x τ p
v ∗,INT
a = v τ,INT
a + ∆τ
−∇x
2
τ p
v ∗,IVC
a = v τ,IVC
a + ∆τ
2
−∇x τ p
T=Γ φ + fa mINT a
T=0 φ + fa mIVC a
+ f IVC
a
(4.5)
(4.6)
is the spatial derivatives along the three orthogonal spatial directions.
This indicates that the spatial derivatives are with respect to the current coordinates
of the atom. Please note that ‘Γ’ is the temperature of the atomic zone.
2. Update the position of all atoms in all zones of the MD region,
x τ+∆τ
a
= x τ a + ∆τv ∗,I
I ∈ (IVC,SVC,INT) (4.7)
3. Update velocities atoms in the interior (INT), IVC and SVC atoms of the MD zone
at the beginning of time-step ‘∆τ’ of the MD. For an atom ‘a’ the updated velocities
72

are,
Where, ∇ x τ+∆τ
p
v τ+∆τ,INT
a = v ∗,INT
a + ∆τ
2
v τ+∆τ,IVC
a = v ∗,IVC
a + ∆τ
2
v τ+∆τ,SVC
a = v ∗,SVC
a + ∆τ
2 aSVC a
−∇ x τ+∆τ
p
−∇ x τ+∆τ
p
m INT
a
φ + f T=Γ
a
φ + f T=0
a
m IVC
a
+ f IVC
a
(4.8)
(4.9)
(4.10)
is the spatial derivative along the three orthogonal space direc-
tions. This indicates that the spatial derivatives are with respect to the updated
coordinates of the atom. Note that at the end of every MD update the following
displacement average is recorded,
∆u IVC
p
= 1
t+N∆τ
N
τ=t
N IVC
a
a=1
τ,IVC xa − xt,IVC
a
N IVC
a
(4.11)
At the end of the MD update the control switches back to the GIMP algorithm and
the continuum update follows using the IVC displacements calculated during simulation
time (eq.4.1). The proposed algorithm was implemented to study fracture initiation and
evolution in the thermoplastic polymer system (introduced in Chapter 3) in relation to
macro-scale loading conditions. The results from the simulations are presented in Chapter
6.
4.3 Setup of Coupled Simulation model
The algorithmic implementation of the scheme described above is realized using Visual
C++ programming. However, the need to scale to larger systems combined with the fact
that both the continuum and atomistic methods are particle based methods necessitates
the use of parallel implementation with distributed computing. This dissertation focusses
73

on the effectiveness of the new method to model polymers and leaves room for further
improvements including computational efficiency. In this section, the details of setting
up a coupled simulation in a continuum/atomistic solution paradigm is explained. Figure
4.5 is a schematic of the atomistic model setup.
Figure 4.5: Setup of atomistic model for Concurrent coupling
74

Figure 4.6: Setup of ESCM/GIMP model for Concurrent coupling
Given below are the steps followed in the typical setup of the coupled thermoplastic
polymer model.
1. The thermoplastic polymer amorphous structure is generated using the random
walk operation.
2. A soft non-bond potential is employed to remove the over-lapping centers of the
polymer beads that might have occupied the same site (or in the immediate vicinity)
during the random walk operation. The NVE dynamics (with no thermostatting)
automatically acts to remove the overlap of the centers and hence reduce the energy
of the system.
3. The energy minimized MD model is then embedded in a volume box surrounded
by a regular ordered lattice. The atoms of the ordered lattice will eventually couple
to the continuum material points through the IVC and SVC.
75

4. The system is equilibrated at high temperature using eplicit velocity rescaling in
order to evenly distribute the polymer molecules inside the volume box
5. Finally, the system temperature is reduced and MD is performed until the desired
energy equilibration is achieved.
After the atomistic coupled setup domain is complete, the particles cells are embedded
around the atomistic domain and the IVC and SVC atom lists are constructed using a
linear search algorithm depending on the positions of the MD atoms relative to the
particle domain (see, fig.4.6).
76

Chapter 5
Hierarchical Modeling of Damage in PNCs
While the weight savings and property enhancements that can be realized using nano
composites are obvious, the availability of physical models to capture these enhancements
in relation to macro-structural load history is limited. This is due to the complexities
involved in the modeling of polymeric materials and the obvious length and time scales
involved in the structural material. On the other hand the significant enhancements in
the nanoclay nanocomposites are attributed to the richness of polymer-nanoclay interac-
tions at the interface. These interactions occur on a length scale which is comparable to
the interatomic separation. Further, the heterogeneity of the interacting species at the
nanoscale adds to the level of complexity. In this chapter, a novel RVE based model of
the nanoclay/polymer interface is proposed. The model attempts to capture the stiffness
enhancement due to nanoscale reinforcement through tracking the change in energy over
baseline as measured through the framework of volume-averaged atomistic stresses. Fur-
ther, the model attempts to capture the stiffness degradation using an internal state vari-
able (ISV) approach. The proposed analysis correlates the dissipation of internal energy
of the system to deterioration of interface and bulk properties of the nanoclay polymer
composite. The model employs a smeared analysis approach by assuming orthotropic
symmetry to material properties on the local representative volume element (RVE) coor-
dinate frame (we assume that the nanoclay is oriented along the local x1 axis). Figure 5.1
is a schematic to explain the development of the damage model. As shown in the figure,
the model constructs an equivalent model with a smeared representation of the nanoclay,
polymer system. As the “smeared” model deforms a critical value of strain occurs at
which damage appears in the nano-scale RVE (see, fig.5.1(c))(the damage initiation and
evolution model is discussed in detail later). In this research, the damage in the equivalent
77

model actually represents the debond at the polymer/nanoclay interface and micro-void
coalescence in the bulk polymer (regions of polymer away from the nanoclay/polymer
interface). Later, through atomistic simulation we will show that the loss in stiffness and
material decohesion in the form of interface debond and micro-void coalescence are well
correlated, and hence can be considered to be the primary modes of damage initiation
and evolution. It is envisioned that the proposed method will help in the constitutive
modeling of nano-composites by incorporating several elements of nano-scale physics into
a simple phenomenological model.
Figure 5.1: Development of the equivalent damage model (a) the nanoscale polymer
nanocomposite (b) the nanoscale RVE with equivalent elastic orthotropic properties (c)
interface debond and void coalescence in the actual polymer nanocomposite (d) damage
development in the smeared model
As mentioned above, we have the continuum treatment of damage evolution. In this
effort, we review the thermodynamics of a deformable solid, followed by specializing the
Clausius-Duhem equation to develop the constitutive relations with embedded dissipation
78

due to damage evolution. In the following sections a nanoscale damage model (NDM) is
introduced and internal state variable to define damage initiation and damage evolution
is defined.
5.1 Damage Mechanics using the Internal State Variable Ap-
proach
In this work the internal state variable (ISV) approach proposed by Talreja (1990) to
model damage in composites will be modified and extended to model damage at the
nano-scale. The proposed RVE is assumed to have a linear elastic response until failure
initiation after which the damage evolution is described by the model in section 5.2.1. We
begin with the review of the preliminary definitions and assumptions of the ISV model.
5.1.1 Definition of Damage Tensor
In Talreja (1990), the damage behavior is entirely described by a tensorial damage entity.
In general, the damage entity is any micro-structural change in the material brought
about by an internal dissipative mechanism. For example, in the macro-scale composite
the damage entity may refer to the cracks in the matrix, fiber-matrix debond, inter-ply
delamination, etc. Realistically, it is possible that multiple damage entities may coexist in
side the material specimen and evolve at mutually independent rates. Hence, the damage
entity is a concept to unify the multiple damage multiple modes as a single quantity, the
combined effect of which initiates and describes the progression of material degradation
in relation to macro-scale loading history, within the framework of thermodynamics.
Let V be a finite volume surrounding a point P within the bulk material with dis-
tributed damage at multiple damage modes (fig.5.2). Consider a damage entity inside
the volume V. Let S be its surface area and n be the unit outward normal to the surface.
79

Figure 5.2: Characterization of damage entities
An influence vector function a is then defined on the surface. The vector a essentially
represents a localized disturbance due to the damage entity in an otherwise uniform field.
A definition for a with respect to nano-scale damage is provided later in this chapter. A
second order tensor dij is defined as follows,
dij =
S
ainjdS (5.1)
Where, ai and nj are the components of the vectors a and n in three dimensional
Cartesian coordinates. In the general formulation we assume that there are ‘N’ distinct
damage modes in the material. We can define a damage tensor for each damage mode α
(where, α = 1, 2, . . . , N) as follows,
D (α)
ij
= 1
V
80
kα
(dij)kα
(5.2)

Where, kα is the number of damage entities in each mode. If m is the tangential
unit vector defined on surface S such that nimi = 0, nini = 1 and mimi = 1 (indicial
summation on i implied) we can then write
and
ai = a(ni + mi) (5.3)
Substituting eq.(5.3) into eq.(5.2) and dropping α for convenience we get,
Where,
Dij = D 1 ij + D 2 ij (5.4)
= 1
V
d 1 ij =
d 2 ij =
k
S
S
1
(dij)k + (d 2
ij)k
aninjdS (5.5)
aminjdS (5.6)
In this research we retain only the normal part of a in order to retain symmetry of
the damage tensor. Further, we assume only two distinct modes of damage hence,
Dij = 1
V
2
k=1
S
aninjdS
k
5.1.2 Thermodynamics of Damage and Constitutive equations
(5.7)
To derive the thermo-mechanical constitutive equations of damage we assume Truesdell’s
principle of equipresence (Truesdell & Noll, 1992). The principle simply states that an
independent variable assumed to present in one constitutive equation of the material
must exist in all other constitutive relations as well, unless its presence contradicts the
81

fundamental principles (see e.g. Malvern (1969), sec.6.7, page 379) for any constitutive
relation. In Talreja (1990) the independent variables chosen are, the small strain tensor
ɛ, temperature T , temperature gradient g = ∇T and damage tensor D (we have dropped
the superscript α for convenience) . Hence we can define,
σ = σ(ɛ, T, g, D) (5.8)
ψ = ψ(ɛ, T, g, D) (5.9)
η = η(ɛ, T, g, D) (5.10)
q = q(ɛ, T, g, D) (5.11)
˙D = ˙ D(ɛ, T, g, D) (5.12)
Where, σ is the Cauchy stress tensor, ψ is the specific Helmholtz free energy, η is
the specific entropy, q is the heat flux vector and ˙ D rate of damage accumulation. The
choice of stress, energy is assumed to satisfy the governing equations for linear momentum,
angular momentum and energy balance (Talreja, 1990). The Clausius-Duhem in-equality
(Malvern, 1969) is an alternate expression of the second law of thermo-dynamics. Given
by,
σ : ˙ɛ − ρ ˙ ψ − ρη ˙
q · g
T −
T
≥ 0 (5.13)
For a finite-deformation damage model formulation we can re-write the above equation
in the material frame as follows,
S : ˙ E − ρ0 ˙ Ψ − ρ0N ˙ Γ −
Q · G
Γ
≥ 0 (5.14)
Where, S is the second Piola-Kirchoff stress, E is the Green-Lagrange strain tensor, ρ0
is density of the reference configuration. If F is the deformation gradient with X and x as
the material and spatial coordinates respectively, then Q(X, t) = JF −1 q(x, t), N(X, t) =
82

η(x, t), Γ(X, t) = T (x, t), G = ∇Γ. We also have the thermodynamic relation:ψ = e−T η
(or, Ψ = U − ΓN). However for the material frame formulation eq.(5.8) to eq.(5.12) have
to be re-written in terms of independent variables in the material frame.
Assuming small strains and differentiating eq.(5.9) with respect to time gives,
˙ψ = ∂ψ
∂ɛ
∂ψ
: ˙ɛ +
∂T ˙ T + ∂ψ ∂ψ
· ˙g +
∂g ∂D : D ˙ (5.15)
Substituting eq.(5.15) in to eq.(5.13) we get the internal dissipation inequality,
σ − ρ ∂ψ
: ˙ɛ − ρ η +
∂ɛ
∂ψ
T˙ − ρ
∂T
∂ψ ∂ψ
· ˙g − ρ
∂g ∂D : D˙ q · g
−
T
≥ 0 (5.16)
Since eq.(5.16) must hold for any arbitrary rates ˙ɛ, ˙
T and ˙g, we get the relations,
σ = ρ ∂ψ
∂ɛ
η = − ∂ψ
∂ψ
∂g
∂T
(5.17)
(5.18)
= 0 (5.19)
Combining equations (5.17) through (5.19) and (5.8) through (5.11) gives,
The dissipative inequality 5.16 reduces to,
σ = σ(ɛ, T, D) (5.20)
ψ = ψ(ɛ, T, D) (5.21)
η = η(ɛ, T, D) (5.22)
− ρ ∂ψ
∂D : D˙ q · g
−
T
83
≥ 0 (5.23)

Further assuming that the formulation is isothermal (i.e. g = ∇T = 0) and for purely
mechanical response we arrive at,
With the internal dissipation inequality,
σ = σ(ɛ, D) (5.24)
ψ = ψ(ɛ, D) (5.25)
˙D = ˙ D(ɛ, D) (5.26)
∂ψ
∂D : ˙ D ≤ 0 (5.27)
Hence, for a purely mechanical response, formulation of the damage tensor D and free
energy ψ defines the constitutive behavior of the deformable body. The rate equation for
stress, σ is now from eq.(5.25) and eq.(5.17),
σij ˙ = ∂2ψ ∂ɛij∂ɛkl
ɛkl ˙ + ∂2ψ Dkl
∂ɛij∂Dkl
˙
(5.28)
The implications of the above relation in the context of hyperelastic materials is that
the constitutive relation remains polyconvex (Schroder & Neff, 2003) until initiation of
damage. The onset of damage and damage accumulation (as quantified by the rate
equation above), the material deviates from polyconvex constitutive behavior due to
damage induced softening.
5.2 A Nano-scale Damage Model
The internal state variable introduced above is now specialized for the case of damage in
a nano-scale representative volume element (RVE). In this formulation the nanoclay is
assumed to be oriented along the direction of the local 1 axis of the RVE (as shown in fig.
84

Figure 5.3: Representative volume element (RVE) and nano-clay orientation in damage
model
5.1), hence even though the orientation of the nanoclay is accounted for through material
properties, exact description of interfaces and features of the inclusion are avoided. Some
of the key features of the damage model and primary assumptions are as follows,
1. The material exhibits orthotropic behavior and preserves orthotropic symmetry
even after damage initiation (hence, Dij = 0 if i = j).
2. The damage is small hence Dij ≪ 1 and higher-order terms of damage tensor
components can be ignored.
3. Primary assumption We define nanoclay damage as that which occurs due to inter-
face debond growth (mode of damage, α = 1) along the local 2-axis of the material
(see, fig.5.4) and the micro-void coalescence in the x1 and x3 directions (α = 2,
respectively) in the bulk polymer.
The primary assumption can be explained further as follows: We assume that the
interface debond occurs normal to the local 1-3 plane of the nanoclay/polymer interface
and the D (1)
22 (or D2) term of the damage tensor dominates the damage evolution for
the case of interface debond. For void nucleation (and coalescence) we assume that the
85

Figure 5.4: Schematic of interface debond and void nucleation in damage model for
deformations along principle axis
terms D (2)
11 (D1) and D (2)
33 (D3) terms dominate for voids that oriented along the local x1
and x3 axis, respectively. However, for the case of small deformation simple shear the
formulation needs to be refined further. Consider the case of simple shear in the x1 − x2
plane, as shown in fig.(5.5). We assume that principal stretch in the body (which is
normal to the plane oriented 45 ◦ to the x1 direction) accounts for the interface debond in
the material. In Chapter 6, we shall revisit these assumptions and examine them more
closely.
For the remaining part of this chapter, for notational convenience we switch to the
Voigt notation in order to condense the final form of the equations. We use σ1 = σ11,σ2 =
σ22,σ3 = σ33,σ4 = σ23,σ5 = σ13 and σ6 = σ12 a similar number convention follows for the
tensorial components of ɛ and D. In order to derive the necessary constitutive relations
we now consider the free-energy of the system ψ. The free energy can be partitioned into
86

Figure 5.5: Schematic of interface debond in RVE undergoing simple shear deformation
in x1 − x2 plane with the principal stretch axis x ′ 2 oriented 45 ◦ to the local x2 axis
its undamaged (superscript ‘0’ ) and damaged (superscript ‘1’ ) parts.
ψ = ψ 0 + ψ 1
(5.29)
We first consider the undamaged part of the energy response. Since we do impose or-
thotropic symmetry on the material the energy can be further partitioned to a pure
volumetric and deviatoric (isochoric) deformation state, that is we have,
ψ 0 = ψ 0 vol + ψ 0 dev
(5.30)
A quadratic energy function is chosen such that the stress-strain relation prior to damage
initiation remain linear elastic; Hooke’s law behavior. Hence, we have
ψ 0 vol = c1ɛ 2 1 + c2ɛ 2 2 + c3ɛ 2 3 + c4ɛ1ɛ2 + c5ɛ1ɛ3 + c6ɛ2ɛ3 (5.31)
ψ 0 dev = c7ɛ 2 4 + c8ɛ 2 5 + c9ɛ 2 6 (5.32)
87

In order to construct the damaged response of the system and we into account the in-
variance of the tensorial quantities with respect to tensor transformations and we employ
the irreducible integrity basis (Spencer, 1971). Using the integrity basis Adkins (1959)
we have the following set of invariants for damage in modes, α = 1 and 2.
D1,D2,D3,D 2 4,D 2 5,D 2 6, D4D5D6, ɛ4D4,ɛ5D5, ɛ6D6, ɛ4ɛ5D6, ɛ5ɛ6D4, ɛ4ɛ6D5, ɛ4D5D6,
ɛ5D4D6, ɛ4D5D6.
Assuming small damage the second order terms vanish that is the coefficients for
terms containing D 2 4,D 2 5,D 2 6, D4D5D6 is zero. Since orthotropic symmetry is preserved at
all times in the true principal coordinates, the shear-extension coupling coefficients must
be absent hence, ɛ6D6, ɛ4ɛ5D6, ɛ5ɛ6D4, ɛ4ɛ6D5, ɛ4D5D6, ɛ5D4D6, ɛ4D5D6 are neglected.
As a consequence the energy partitioning is still feasible,
ψ 1 = ψ 1 vol + ψ 1 dev
Admissible free energy function that satisfies the prescribed criterion is,
ψ 1 vol = A1ɛ 2 1D1 + A2ɛ 2 2D1 + A3ɛ 2 3D1 + A4ɛ 2 1D2 + A5ɛ 2 2D2
+ A6ɛ 2 3D2 + A7ɛ 2 1D3 + A8ɛ 2 2D3 + A9ɛ 2 3D3
+ A10ɛ1ɛ2D1 + A11ɛ1ɛ2D2 + A12ɛ1ɛ2D3 + A13ɛ1ɛ3D1
+ A14ɛ1ɛ3D2 + A15ɛ1ɛ3D3 + A16ɛ2ɛ3D1 + A17ɛ2ɛ3D2
(5.33)
+ A18ɛ2ɛ3D3 (5.34)
ψ 1 dev = A19ɛ 2 4D1 + A20ɛ 2 5D1 + A21ɛ 2 6D1 + A22ɛ 2 4D2 + A23ɛ 2 5D2
+ A24ɛ 2 6D2 + A25ɛ 2 4D3 + A26ɛ 2 5D3 + A27ɛ 2 6D3 (5.35)
The Cauchy stress in the system can now be derived using equation (5.17). Hence, the
constitutive law for polymer nanocomposite can be written in short-hand (Voigt notation)
88

as,
Where, {σ} is the 6 × 1 vector {σ1, σ2, σ3, σ4, σ5, σ6}, while
{σ} = [C 0 ] + [C 1 ] {ɛ} (5.36)
{ɛ} is the vector of tensorial strain components {ɛ1, ɛ2, ɛ3, ɛ4, ɛ5, ɛ6}
[C 0 ] is the 6 × 6 undamaged orthotropic stiffness matrix,
⎡
[C 0 ⎢
] = ⎢
⎣
2c1 c4 c5 0 0 0
c4 2c2 c6 0 0 0
c5 c6 2c3 0 0 0
0 0 0 2c7 0 0
0 0 0 0 2c8 0
0 0 0 0 0 2c9
⎤
⎥
⎦
(5.37)
Where, the constants ci(i = 1, 2, 3 . . . 9) are related to the undamaged nanocomposite
material stiffnesses as follows,
c1 = C0 11
2 , c2 = C0 22
2 , c3 = C0 33
2 , c4 = C0 12
2 , c5 = C0 13
2 , c6 = C0 23
2
c7 = C 0 44, c8 = C 0 55, c9 = C 0 66 (5.38)
Where, the undamaged material stiffness components C 0 ij can be written in terms of
undamaged material constants E 0 1, E 0 2, E 0 3 (Young’s modulus along x1,x2 and x3 axis
directions, respectively), ν 0 23, ν 0 13, ν 0 12 (Poisson’s ratio effect in the 2-3, 1-3 and 1-2 planes),
89

µ 0 23, µ 0 13 and µ 0 12 (shear modulus in the 2-3, 1-3 and 1-2 planes repectively).
Where,
Υ =
ν 0 32 = ν 0 E
23
0 2
E0 3
ν 0 31 = ν 0 E
13
0 3
E0 1
ν 0 21 = ν 0 E
12
0 2
E0 1
C 0 11 = E 0 1(1 − ν 0 23ν 0 32)Υ (5.39)
C 0 22 = E 0 2(1 − ν 0 13ν 0 31)Υ (5.40)
C 0 33 = E 0 3(1 − ν 0 12ν 0 21)Υ (5.41)
C 0 23 = E 0 2(ν 0 32 + ν 0 12ν 0 31)Υ (5.42)
C 0 13 = E 0 3(ν 0 13 + ν 0 12ν 0 23)Υ (5.43)
C 0 12 = E 0 1(ν 0 31 + ν 0 21ν 0 32)Υ (5.44)
C 0 44 = µ 0 23 (5.45)
C 0 55 = µ 0 13 (5.46)
C 0 66 = µ 0 12 (5.47)
1
1 − ν 0 12ν 0 21 − ν 0 23ν 0 32 − ν 0 13ν 0 31 − 2ν 0 21ν 0 32ν 0 13
90

The damaged stiffness [C 1 ] is given as,
Where,
⎡
[C 1 ⎢
] = ⎢
⎣
C 1 11 C 1 12 C 1 13 0 0 0
C 1 12 C 1 22 C 1 23 0 0 0
C 1 13 C 1 23 C 1 33 0 0 0
0 0 0 C 1 44 0 0
0 0 0 0 C 1 55 0
0 0 0 0 0 C 1 66
⎤
⎥
⎦
(5.48)
C 1 11 = 2 (A1D1 + A4D2 + A7D3) (5.49)
C 1 22 = 2 (A2D1 + A5D2 + A8D3) (5.50)
C 1 33 = 2 (A3D1 + A6D2 + A9D3) (5.51)
C 1 12 = A10D1 + A11D2 + A12D3 (5.52)
C 1 13 = A13D1 + A14D2 + A15D3 (5.53)
C 1 23 = A16D1 + A17D2 + A18D3 (5.54)
C 1 44 = 2 (A19D1 + A22D2 + A25D3) (5.55)
C 1 55 = 2 (A20D1 + A23D2 + A26D3) (5.56)
C 1 66 = 2 (A21D1 + A24D2 + A27D3) (5.57)
The number constants to be determined can be reduced by using the primary assump-
tions that were mentioned earlier in the section. Since we assume that the primary
damage modes in each principal direction of the nanocomposite does not interact, we
have A4=A7=A2=A8=A3=A6=0. The argument is extended to characterize the effect
of damage on Poisson’s ratio while setting, A12=A14=A16=0. From, our discussions on
91

simple shear in the x1 − x2 or the x2 − x3 plane, we assume that D2 term dominates the
damage evolution in these planes, such that D1, D3 ≪ D2. Hence, A19=A25=A21=A27=0.
Finally we assume that the damage entity for simple shear in the x1 − x3 (i.e., void co-
alescence) is mainly dominated by the D3 term with the damage opening given by the
maximum principle stretch. The damaged stiffness response reduces to,
C 1 11 = 2A1D1 (5.58)
C 1 22 = 2A5D2 (5.59)
C 1 33 = 2A9D3 (5.60)
C 1 12 = A10D1 + A11D2 (5.61)
C 1 13 = A13D1 + A15D3 (5.62)
C 1 23 = A17D2 + A18D3 (5.63)
C 1 44 = 2A22D2 (5.64)
C 1 55 = 2A26D3 (5.65)
C 1 66 = 2A24D2 (5.66)
The constants Ai are characterized through atomistic MD simulations. In Chapter 6, the
results for the damage model are presented. The above model can describe the evolution
of damage using the damage entity. In the next section we provide an analytical model
for the damage entity based on the inter-atomic interaction potentials.
5.2.1 A Damage Model for Interphase Debond and Void Nu-
cleation in Thermoplastic Polymer Nanocomposites
In the proposed damaged model the two primary and competing modes of damage in
a polymer nanocomposite is the interphase debond behavior and void nucleation in the
92

Figure 5.6: Schematic of nano-clay with dimensions
process zone due to the atoms moving apart in the direction of maximum stretch. We
arbitrarily assign interphase debond as damage mode ‘1’ and void formation in the bulk
as mode ‘2’. For a general state of stress, all damage modes are assumed to coexist in
the material with no interaction between the constituent damage entities. Consider a
nano-particle oriented along the local 1-direction with the normal to the damage surface
given by n = {0, 1, 0} T as shown in fig.(5.6), then we assume the damage entity due
interphase debond is defined as,
D2 = 1
b 3 2a2Lnwn if λ > λC where λ = r/σ (5.67)
Where, Ln is the length of the nanoclay, wn is the width of nanoclay and b is the length
of the cubic volume box in each dimension and σ is the zero-point of the Leonard-
Jones potential. The variable ‘a2’ is the damage influence function for interphase debond
(magnitude of influence function) as introduced in eq.(5.3). For damage mode 2, consider
ellipsoidal voids in the nanoscale RVE oriented along principal stretch directions x1 (n =
{1, 0, 0} T ) and x3 (n = {0, 0, 1} T ) (see, fig.5.7). Then we assume the damage entity due
93

Figure 5.7: Schematic of undeformed spherical void in the undeformed polymer (or polymer
nanocomposite) and deformed ellipsoidal voids oriented along stretch directions x1
and x3 axis, with b1, b2 and b3 as the major axes of the ellipsoid.
to voids in the respective principal stretch directions to be given as,
D1 = 1
b 3 πa1b 2 2 if λ > λC
D3 = 1
b 3 πa3b 2 1 if λ > λC
(5.68)
(5.69)
Where, b1, b2 and b3 are lengths of the major axes of the ellipsoidal void as shown in
fig.(5.7), and a1 and a3 are the damage influence functions for void growth in the local 1
and 3 directions. In order to construct the influence function for the degenerate damage
modes, we consider the non-bonded interaction in the bulk polymer and at the local
nano-particle and polymer molecule interface. The force due to non-bonded interaction
94

has the following functional formula,
F (r) = − 24
r ɛ
2
σ
12 −
r
σ
6
r
(5.70)
Where, ɛ is the interaction energy, σ is the zero-point of the energy, while r is the distance
between the atoms at the interface of the polymer nano-clay. In MD simulations, the
extent of interphase debond (or void coalescence) and the related damage behavior is a
function of the inter-atomic potentials. Through multi-scale simulation we can embed
the characteristic atomistic debond behavior into the nano-scale continuum RVE model
to capture the evolution of damage in the system. In this manner, the multi-scale model
avoids the need for a full atomistic description of the nano-scale damage and enables
material scientists and engineers to easily incorporate the model to any design analysis
software to study nano-composites. In this effort, we define the influence function for the
three cases of damage to be,
a1(λ) = b3
πb2
1 −
2
24ɛ
1
2
rFmax λ
a2(λ) =
a3(λ) = b3
πb 2 1
12
6
1
−
λ
b3
1 −
2Lnwn
24ɛ
12
6
1 1
2 −
rFmax λ λ
1 − 24ɛ
12
6
1 1
2 −
rFmax λ λ
(5.71)
(5.72)
(5.73)
Where, λ is the maximum (principle) stretch of the continuum RVE (to be discussed in
detail in Chapter 6), while Fmax is the maximum value of the function F (r). Physically,
the damage influence function represents the evolution of opening displacement of the
damage entity following damage initiation (i.e., when λ > λC). To determine Fmax we
take the first derivative of eq.(5.70) w.r.t. ‘r’ and equate it to zero to obtain the maximum
95

Figure 5.8: Determination of Fmax with the nano-scale damage parameters σ and rmax
indicated.
as shown in fig.(5.8). This gives,
rmax = 6
26
σ (5.74)
7
Where, rmax is the value of ‘r’ for which F (r) is at a maximum. Substituting, eq.(5.74)
in to eq.(5.70), gives
Fmax = −2.396 ɛ
σ
Substituting for Fmax in to equations 5.71 to 5.73 we get,
a1(λ) = b3
πb2
1
1 + 10.015 2
2
λ
b
a2(λ) =
3
1
1 + 10.015 2
2Lnwn
a3(λ) = b3
πb 2 1
1 + 10.015
2
13
7
1
−
λ
13
7
1
−
λ λ
13
7
1 1
−
λ λ
(5.75)
(5.76)
(5.77)
(5.78)
It is evident from the above equations that the minimum value for Di (i=1, 2 or 3) is 0
96

(no damage, no separation) while 1 indicates complete damage and complete separation.
The critical value of stretch for damage initiation is when r = rmax as shown in fig.(5.8),
giving,
λc = rmax
σ
6 26
=
7
(5.79)
The components of the damage tensor can now be found by substituting equations (5.76)
through (5.78) in to eq.(5.67), eq.(5.68) and eq.(5.69), respectively.
Di(λ) = 1 + 10.015
2
13 1
−
λ
7
1
λ
where, i = 1, 2 or 3 (5.80)
We observe that all components of the damage tensor reduce to the same damage function,
due to the fact that although the geometry of the damages are unique, the non-bond
interactions that govern damage evolution is the same for all damage entities. In general,
due to the heterogeneities of the interacting species at the nano-scale, the critical value of
stretch λC at which damage (debond) initiates needs to be scaled to a value ascertained
from atomistic simulations. In particular, if ‘α’ is a correction factor determined from
simulations then the corrected value of critical stretch λ ′ C
λ ′ C = αλC
is given by,
(5.81)
In Chapter 7, a discussion is presented on generalizing the current damage formulation
for a generalized 3-dimensional RVE model.
97

Chapter 6
Results
In this chapter the results to the multi-scale modeling techniques proposed in this
dissertation are tabulated. Beginning in section 6.1 we look at an innovative scheme to
concurrently couple atomistic polymer simulations to macro-scale continuum during sim-
ulation time. In section 6.2, a technique to model the Helmholtz free-energy is discussed
followed by section 6.3 where the results to the damage model are discussed.
6.1 Concurrent Coupling of Thermoplastic Polymers
In this section the results to 2-D coupling of MD simulation of atomistic thermoplastic
polymer to continuum simulation of polymer using GIMP (see, Chapter 2) with the
coupling algorithm proposed in Chapter 4 is discussed. The coupling method is used
to solve three specific cases of polymer deformation: I. A coupled polymer model under
simple tension (with no embedded crack), II. coupled polymer with an evolving mode I
(crack face opening mode) crack and III. polymer model with an evolving mode II (crack
tip under shear deformation) crack. In each of these cases, the displacement fields (note:
u is the displacement along X axis and v is the displacement along Y ) and displacement
gradients (i.e., ∂u ∂u ∂v , , ∂X ∂Y ∂X
∂v and ) for the coupled simulation particles (material points
∂Y
and atomistic entities) are determined as a function of time.
For the larger continuum, the response of the system is assumed to be in the form
of the hyperelastic material model introduced in section 2.2.1 with material properties
c1 = 34.3MPa and c2 = 9.59MPa that corresponds to a Young’s modulus of E=1.5GPa
and Poissons ratio of 0.33 for small deformation. In all simulations the far-field (or macro-
scale) loading is simulated using an applied stress σapp = 2.5kPa. In order to mitigate
98

the effect of dynamic stress waves from unloading the crack tip during simulation time,
the load in each case is linearly ramped after t = 25 ps (1 pico = 10 −12 ). Figure 6.1
Figure 6.1: Layout of the interior (INT), internal volume cell (IVC) and surface volume
cell (SVC) zones in MD simulation, with dimensions indicated in ˚A
shows the layout of the atomistic zone that was used in all simulations. The interior MD
zone (i.e., the process zone) is a cube of dimension 40˚A × 40˚A × 40˚A. The IVC and
SVC regions are a 5 ˚A wide zones as indicated in fig.(6.1). The force field parameters for
the thermoplastic polymer atomistic model are chosen from Wei et al. (2002). The total
energy of the system is assumed to be a sum of the following contributions,
U(r) = Ebond + Eangle + Etorsion + Enon-bond
99
(6.1)

Where,
Ebond =
bonds
Eangle =
angles
Etorsion =
torsions n=0
kl 2
(l − l0)
2
kθ
2
(cos θ − cos θ0)
2
3
Enon-bond =
σ 4ɛ
r
i
j>i
Cn cos n (ω)
12
−
σ
6
r
(6.2)
The constants are as follows: kl = 346kJ/mol/˚A 2 , l0 = 1.53˚A, kθ = 520J/mol,θ0 =
112.813 ◦ , C0 = 8, 832J/mol, C1 = 18, 087J/mol, C2 = 4, 880J/mol, C3 = −31, 800J/mol
and ɛ = 0.498J/mol,σ = 2.95˚A. It should be noted that the cut-off radius σ was adjusted
to 2.95˚A from its recommended value of 3.95˚A in Wei et al. (2002). This adjustment
in value was required to prevent void formation in the MD model during equilibration.
The typical MD simulation algorithm requires the MD domain to adhere to one of the
classical ensembles, e.g. NVT, NPT, NPH etc.(see Allen & Tildesley (1987)), these
ensembles necessitates the use of periodic boundary conditions. However, the simula-
tion box is typically non-periodic in a coupled simulation. In order to circumvent this
problem, the MD box is mapped to a bigger box (with much greater volume) without
rescaling the internal coordinates of the MD atoms. In this manner we can ‘trick’ the
atomistic simulation into believing that the simulation domain is indeed periodic (see
fig.6.2). The primary objective of the coupled simulation is to ensure that the continuity
of field variables are preserved during simulation time. Hence, in this section the contour
plots of displacements and displacement gradients of particles are included in order to
illustrate the effectiveness of proposed coupled simulation in maintaining continuity of
field variables. The timestep for the continuum simulations was 0.5fs (1 femto = 10 −15 ).
For the purpose of temporal scaling (see Chapter 4) we use a factor of 50, such that
100

Figure 6.2: Layout of a non-periodic MD simulation box
the scaled atomistic time-step (∆τ) is 0.01fs. Zhou (2003) argues that in order to accu-
rately determine the value of continuum stress for a discrete system long time-averages
and a sufficiently large atomistic (discrete) system size is required. Since the temporal
space over which the system is averaged (N = 50 timesteps) is quite small, we choose
to plot the gradients of the displacement field rather than the instantaneous values of
Cauchy stress. Given below is a description of the method employed in this work to
determine the displacement gradients by post-processing the coupled simulation data. In
Figure 6.3: Layout of grid and particle in post-processing of concurrent coupled simulation
to compute displacement gradient
101

this description we designate all particles (i.e., continuum material points and atomistic
entities) as p. Let x k p and x k+1
p
be the current and updated of the particle at step ‘k’ (k
assumes integral values). Further, let Xp be the undeformed coordinate of the particles
such that x k=1
p
= Xp. We superpose the displacement solution at step ‘k’ on a regular
and structured finite element grid (see fig.6.3). From the displacement solution we have
the incremental nodal displacement vector ∆u k computed as, ∆u k = x k+1 − x k . Then,
assuming that the incremental displacements are small such that the particle does not
cross its current element boundaries, we can compute the incremental displacements on
node I of the background grid using the finite element shape functions NpI (the shape
function between node ‘I’ and particle ‘p’). That is we have,
∆U k
I =
M I p
p=1
NpI∆u k p
(6.3)
Where, M I p is the number of particles in the neighborhood of node I. This number is
simply the number of atoms that have a non-zero contribution to node I, as dictated by
the atom to node shape function NpI. An average nodal incremental displacement vector
is computed as follows,
∆U k
I = ∆UkI
M I p
(6.4)
For a particle ‘p’ that lies within the boundaries of element ‘e’ (see, fig.6.3), the gradient
of the incremental displacement field ‘∂∆u k p/∂x k ’ can be computed at each particle using
the following equation,
∂∆uk N
p
=
∂xk (e)
∂NpI
∆Uk
∂xk I
I=1
(6.5)
Where, N (e) is the number of nodes in element ‘e’. It should be noted that the derivative
of the shape function ∂NpI/∂x k is with respect to current coordinates x k p of the particle.
102

The deformation gradient for particle p, F k
p can then be updated using,
F k+1
p
= F k
p ·
I + ∂∆uk p
∂x k
The displacement gradient (Malvern, 1969) can then be found using,
∂u k+1
p
∂X
(6.6)
= Fk+1
p − I (6.7)
It should be noted that the a coarse mesh will act to “smear” the atomistic details of
deformation in computing the displacement gradients. The solution for displacement
gradients are hence inherently mesh size dependent.
6.1.1 Case I: Benchmark of Coupling algorithm using a Simple
Tension Test
Figure 6.4: Schematic of coupled model in a simple tension setup
Given in fig.(6.4) is the schematic of the thermoplastic polymer tension specimen with
103

the appropriate boundary conditions. The system evolves under the influence of the stress
applied on the edge located at Y=400˚A. Figure (6.5) and (6.6) is the time progression
Figure 6.5: Time progression of vertical displacements (v) of the coupled simulation
particles at (a) t = 1fs (b) t= 20ps (c) t = 40ps and (d) t=60ps. In this view the
particle displacements are plotted along the vertical axis and undeformed coordinates of
the particles are on the X-Y plane. The specimen is loaded at the edge on the right which
corresponds to the line Y=400˚A
of the displacements and gradients, respectively. The figures are viewed in the Y-Z with
the undeformed coordinates of the 2-D simulation plotted in the X-Y plane and the field
variable (displacement/displacement gradient) plotted along Z. As can be seen there is a
large variation in displacements and gradients of the particles inside the atomistic zone.
This is attributed to the thermal vibrations due to non-zero, finite temperature simulation
inside the MD zone. However, we see that the average behavior of the atoms satisfies the
necessary continuity requirements.
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Figure 6.6: Time progression of displacement gradient (∂v/∂Y ) of the coupled simulation
particles at (a) t = 0.5 fs (b) t= 10 ps (c) t = 20 ps and (d) t=30 ps. In this view the
displacement gradients are plotted along the vertical axis and undeformed coordinates
of the particles are on the X-Y plane. The specimen is loaded at the edge on the right
which corresponds to the line Y=400˚A
6.1.2 Case II: Mode I Crack Growth in Thermoplastic Polymer
Next we consider a more complex application of the coupling algorithm. Shown in fig.(6.7)
is the layout of the coupled model to study mode I crack growth in a thermoplastic
polymer system. The pre-existing continuum crack shown in the figure is extended by
applying an external applied stress as indicated. The strain energy release rate (SERR)
at the crack tip in the continuum is calculated using the virtual crack closure integral
discussed in sec.(2.3). The critical value of SERR for mode I crack growth is chosen to
be GIc = 10 −7 J/m 2 , with an initial flaw size (a) of 20 ˚A (fig.(6.7)). Since, the primary
focus of the algorithm is crack growth within the process zone (i.e. atomistic polymer
zone), the chosen value for critical SERR was assumed small (and in general physically
105

Figure 6.7: Schematic of coupled model for mode I crack growth in thermoplastic polymer
unrealistic) in order to extend the crack surface into the MD simulation zone within
a reasonable number of coupled computational steps. Figure 6.8 shows the evolution
of gradient of the vertical (v) displacement field in the Y-direction (∂v/∂Y ) at various
instants of simulation. As can be seen the simulation shows that as the crack extends into
the MD domain, significant displacement gradients occur. The spread of high gradients
inside the amorphous polymer MD domain (contour level red in fig.6.8(d)) indicate that a
diffused damage occurs inside the MD domain and is not a singular crack tip as one would
expect in a simulation of crack propagation in an ordered solid lattice. Figure 6.9 shows
the vertical displacement (v) of the particles as the mode I crack enters the MD domain.
The contours indicate good continuity of displacements in the coupled simulation of a
mode I propagating crack. The simulations methodology also allows the user to examine
fracture surfaces in detail with embedded atomistic information (see, fig.6.10).
106

Figure 6.8: ∂v/∂Y gradient profiles for a propagating 2D mode I crack in coupled simulation
of a thermoplastic polymer system at times t = (a) 0.5 fs (b) 10 ps (c) 25 ps, and
(d) 80 ps
6.1.3 Case III: Mode II crack growth in Thermoplastic Polymer
In the final coupled simulation benchmark example we consider mode II shear crack prop-
agation in a thermoplastic polymer. In short beam shear tests of E-glass/polypropylene
nanoclay and E-glass/nylon-6 nanoclay composites significant improvements in mechani-
cal properties were observed. It is envisioned that the coupled simulation scheme proposed
here can be utilized to study nano-composite specimens in shear. The coupling simulation
metholodology becomes all the more relevant in the context of nano-composite simula-
tion since nanoclay platelets have unusually high aspect ratios making them difficult to
model in regular sized MD domains while keeping the computational costs involved to
a minimum. However, in this example the MD model is of a purely amorphous poly-
mer system with no embedded nanoclay (fig.6.11). As in mode I simulation, the initial
107

Figure 6.9: Vertical displacement (v) contours for a propagating 2D mode I crack in
coupled simulation of thermoplastic polymer system at times t = (a) 0.5 fs (b) 10 ps
(c) 25 ps, and (d) 80 ps, with displacement (v) plotted along Z-axis and undeformed
coordinates of the body in the X-Y plane
Figure 6.10: Fracture surface of mode I crack with the atoms within the SVC (red), IVC
(yellow) and interior (green) zones indicated
108

Figure 6.11: Schematic of coupled model for mode II crack growth in thermoplastic
polymer
crack length is chosen to be 20 ˚A. The critical SERR for extension of crack surface
GIIc, is chosen to be 10 −7 J/m 2 . The reasons for choosing this un-realistically low value
is the same as before. Figure 6.12, shows the contours of horizontal displacement (u)
as a function of time as the crack is propagated in the continuum. We see at the final
time-step the crack face has entered the MD domain, as indicated by the discontinuity in
displacements (fig.6.12(d)). Figure 6.13 and 6.14 are contours of displacement gradients
∂u
∂Y
and ∂v
∂X
at various instances. The figures indicate a discontinuity in the gradients to
the displacements especially as the crack impinges on the MD domain. The discontinuity
could be an indication of difference in material properties between the two zones. The
redistribution and localization of stresses inside the MD domain could also add to the
discontinuity. It is suggested that future simulations should try to incorporate better
material models in the continuum domain such that the continuity of gradients can be
achieved and thereby ensuring a reliable and uniform energy density around the coupling
domain. It is also observed that the use of non-bonded interactions to transfer forces
109

Figure 6.12: Horizontal displacement (u) contours for a propagating 2D mode II crack
in coupled simulation of thermoplastic polymer system at times t = (a) 0.5 fs (b) 25 ps
(c) 30 ps, and (d) 32.5 ps, with displacement (v) plotted along Z-axis and undeformed
coordinates of the body in the X-Y plane
from the continuum down to atomistic domain contributes to the dissipation of energy
at the MD/continuum interface. The coarse grained amorphous polymer system creates
voids inside the MD system during the equilibration process. The formation of voids
effectively reduces the interaction of the IVC atoms with the atoms inside the interior
zone. This concept is not unlike the mean-free path in gases. The lower the mean-free
path the better is the momentum transfer between gas particles. Needless to say, the
formation of gaps (or alternatively increase in mean-free path) affects the ability of the
handshake region to transfer internal forces from the continuum domain to the atomistic
system. It is envisioned that switching to a full atom (non-coarse grained) MD system
will help in mitigating some of the issues encountered with the current coupling scheme
by ensuring a better packing density of polymer atoms with a reduced amount of voids
110

and other related anomalies.
Figure 6.13: Contours of displacement gradient ∂u/∂Y for a propagating 2D mode II
crack in coupled simulation of thermoplastic polymer system at times t = (a) 0.5 fs
(b) 25 ps (c) 30 ps, and (d) 32.5 ps, with ∂u/∂Y plotted along Z-axis and undeformed
coordinates of the body in the X-Y plane
6.2 Free Energy Calculations of Thermoplastic Polymers
The atomistic deformation and the related thermodynamics of a material are a function
of the energetic as well as entropic contributions of the material. In disordered polymers,
configurational entropy plays a significant role in its elastic properties (Buehler & Wong,
2007). Hence, for a complete description of the polymer deformation the model must
include the relevant thermodynamics related to the deformation in order to develop a
complete description of the deformation process. In this work, we propose a scheme to
calculate the Helmholtz free energy of a thermoplastic polymer undergoing mechanical
deformation. Before we proceed to calculation of the Helmholtz free energy, we also look
111

Figure 6.14: Contours gradient ∂v/∂X of vertical displacement (v) for a propagating 2D
mode II crack in coupled simulation of thermoplastic polymer system at times t = (a) 0.5
fs (b) 25 ps (c) 30 ps, and (d) 32.5 ps, with ∂v/∂X plotted along Z-axis and undeformed
coordinates of the body in the X-Y plane
at the entropic contribution of individual polymer chains of the thermoplastic polymer
model used in this research.
6.2.1 Calculation of Persistence Length of a Polymer
The entropic contribution of individual polymer chains is quantified using the persistence
length (Binder, 1995). In general it indicates the transition of the molecules from energetic
dominated process to an entropic deformation for individual polymer chains. The central
idea of the following computations is to capture the configurational entropy of the polymer
chain. For a single polymer chain the number of configurations depend on the flexibility of
the polymer chain. In other words, if the chain is pliable the configurations the molecule
can exist in increases and by definition it attributes to the increases in entropy of the
112

system. A mechanics based approach to determine the flexibility of the polymer chain is
to calculate the bending stiffness EI using simulation. The persistence length ξp is then
related to the bending stiffness as,(Buehler & Wong, 2007)
ξp = EI
kBT
(6.8)
Where, kB is Boltzmann’s constant and T is the temperature. To determine the bending
stiffness of the molecule we fix both ends of the polymer chain by restricting all trans-
lational movement of the beads at either end of the polymer model. A force is applied
at the center of mass of the polymer chain using steered molecular dynamics (SMD, see
Plimpton (1995)) as illustrated in fig.(6.15). The spring constant for SMD was chosen to
be 10 kcal/mol/˚A 2 . As indicated in the figure we use the coarse-grained bead model of
the thermoplastic polymer used in this research with 4 beads. The interaction properties
of bond, angle and dihedral interactions switched on. The parameters for these force
fields are given in section 6.1. Steered molecular dynamics were performed for 5 rates of
Figure 6.15: Schematic of polymer model for determination of bending stiffness EI
spring deformation ( ˙r), ˙r = 0.01, 0.025, 0.05, 0.075 and 0.1 ˚A/fs and the bending stiffness
113

was determined for each rate of loading using the following beam bending equation,
EI =
FappL 3
48d
(6.9)
Where, L is the equilibrated length of the polymer chain and d is the vertical displacement
of the center of center of mass. The bending stiffness were determined for each loading
rate and plotted as shown in fig.(6.16)). A linear curve fit (indicated as EI(L-S) in fig.6.16)
was used to extrapolate the result to ˙r = 0 in order to determine the bending stiffness EI
at zero loading rate. In this manner we can eliminate all inertial effects in the bending
stiffness that can arise from using the MD algorithm.
Figure 6.16: Bending stiffness EI of coarse-grained polymer model as a function of loading
rate ˙r
The equation to curve-fit was EI(L-S)=7.027×10 −28 ˙r+2.433×10 −29 Nm 2 was obtained
from the linear curve-fir shown in fig.(6.16). Substituting for EI = 2.433 × 10 −29 Nm 2 in
eq.(6.8) and setting T=300K, we get ξp = 58.74˚A. This roughly translates to the length
of 40 beads (assuming bond length of 1.53˚A) in the coarse grained system. Hence, if
114

the unstretched length of coarse grained polymer chain is below 58.74˚A we can expect
the energetic contributions of the polymer to dominate its elastic strain energy; for chain
lengths above 58.74˚A the entropic contributions cannot be ignored.
6.2.2 Calculation of Free energy of Polymers undergoing Me-
chanical Deformation
In the previous section, the conditions governing entropic contributions to elasticity from
single chain polymer bead model were derived. However, a similar derivation does not
exist for the MD model of the bulk polymer system. In general, entropy driven forces in
MD models are harder to quantify. The Helmholtz free energy however, is a state variable
that combines the energetic and entropic contributions of the polymer system 1 . In this
section we use thermodynamic integration method that was introduced in sec.(3.4.1)
to determine the free energy change associated with mechanical deformation. In this
work the Helmholtz energy is determined for a volumetric and simple shear deformation
process in a pure thermoplastic polymer system as shown in the schematic in fig.(6.17).
The deformed states are realized through the deformation of the periodic MD simulation
box. The update equations for the box coordinates in dilatational (i.e., pure volumetric)
deformation at step ‘k’ is given by,
x k 1 = (1 + α k )X1
x k 2 = (1 + α k )X2
x k 3 = (1 + α k )X3 (6.10)
Where, x k i is the deformed coordinates of the box and Xi is the undeformed coordinate,
α k is the scaling factor to dilate the simulation box. The update equations for deviatoric
1 Helmholtz energy, A = U − T S, U is the potential energy, T the temperature and S is entropy
115

Figure 6.17: Schematic of the dilatational and deviatoric deformation used to determine
free energy change associated with mechanical deformation
deformation are given by,
x k 1 = X1 + β k (X2 + X3)
x k 2 = X2 + β k (X1 + X3)
x k 3 = X3 + β k (X1 + X2) (6.11)
Again, β k is the scaling factor by which the box is deformed. The volumetric and devia-
toric deformation gradients F vol and F dev respectively, can be derived from eq.(6.10) and
116

(6.11), as
F vol,k =
F dev,k =
⎡
⎢ 1 + α
⎢
⎣
k 0
0
1 + α
0
k 0
0 0 1 + αk ⎤
⎥
⎦
⎡
⎤
⎢
⎣
1 β k β k
β k 1 β k
β k β k 1
⎥
⎦
(6.12)
(6.13)
The desired strain measure can then be calculated from the deformation gradient. For
small strains, i.e., α k , β k ≪ 1 the small strain tensor for the two cases will simply be
given as,
ɛ vol,k =
ɛ dev,k =
⎡
⎢ α
⎢
⎣
k 0
0
α
0
k 0
0 0 αk ⎤
⎥
⎦
⎡
⎤
⎢
⎣
0 β k β k
β k 0 β k
β k β k 0
⎥
⎦
(6.14)
(6.15)
Five states of deformation were used in this computation with values of α (1) = 0.0 (un-
deformed configuration), α (2) = 1.25 × 10 −3 , α (3) = 2.5 × 10 −3 , α (4) = 3.75 × 10 −3 and
α (5) = 5.00 × 10 −3 , similarly, β (1) = 0.0 (undeformed configuration), β (2) = 1.25 × 10 −3 ,
β (3) = 2.5 × 10 −3 , β (4) = 3.75 × 10 −3 and β (5) = 5.00 × 10 −3 The Helmholtz energy is
determined using the thermodynamic integration method discussed in section 3.4.1. As
described in sec.3.4.1, the non-bonded parameters of the thermoplastic polymer model is
ramped from λ = 0 to λ = 1 during simulation time, and the change in the Hamiltonian
117

of the system with respect to internal scaling parameter λ is determined. For the example
considered here the Hamiltonian for the system (H) is given as,
H =
N
i=1
p i · p i
mi
+ U(λ) (6.16)
Where, U(λ) denotes a change in potential energy of the system as a function of the scaling
parameter λ. In this example we assume that the functional form for the potential energy,
including bond stretching and non-bonded interactions, is given by,
U(λ) =
bonds
kl
2 (l − l0) 2 +
σ(λ)
4ɛ(λ)
r
i
j>i
12
6
σ(λ)
−
r
(6.17)
With,ɛ(λ) = ɛ 0 (1 − λ) + ɛ 1 λ and σ(λ) = σ 0 (1 − λ) + σ 1 λ. Substituting the eq.(6.17) in to
eq.(6.16) and taking derivative of the resultant equation with respect to λ gives,
∂H
∂λ
= 4(ɛ
i j>i
1 − ɛ 0 σ 12
σ
6
) −
r r
+6(σ 1 − σ 0 ) ɛ
σ
12
σ
6
2 −
σ r r
(6.18)
The values for constants are as follows: kl = 346kJ/mol/˚A 2 , l0 = 1.53˚A, ɛ 0 = 0.0498J/mol,ɛ 1 =
0.498J/mol, σ 0 = 0.95˚A and σ 1 = 2.95˚A. The change in Helmholtz energy is then simply
calculated as (see, sec.3.4.1),
∆A =
λ=1
λ=0
∂H
dλ (6.19)
∂λ λ
Three point Gaussian quadrature is used to determine the integral in eq.(6.19). The
gaussian quadratue is a numerical tool commonly used to determine the integral of a
118

function f(x) with in the domain -1 to 1.
I =
1
Where, NG is the number of Gauss points and w i G
Gauss point x i G (xi G
and three-point integration rule,
−1
NG
f(x)dx ≈ w i Gf(x i G) (6.20)
i
is the weight associated with the ith
∈ [−1, 1]). For a change in limits of integration from [-1,1] to [0,1]
1
0
f(x)dx ≈ 1
2
3
w i i xG + 1
Gf
2
Hence, eq.(6.19) can be evaluated numerically using the equation,
∆A = 1
2
3
i=1
i=1
w i G
∂H
∂λ λi
The values of λi for which the ensemble average
∂H
∂λ λi
(6.21)
(6.22)
is evaluated are listed in Table
6.1. In the determination of the change in free energy for each deformation case, three
Table 6.1: Values of scaling parameter λ to determine ∆A
i w i G x i G λi
1 0.556 -0.775 0.113
2 0.889 0 0.500
3 0.556 0.775 0.887
sets of simulation must be carried out. The Langevin thermostat was used to maintain
temperature of the MD system at 300K. The time step for MD simulations was set at 0.5
fs. The algorithm to determine the change in free energy was implemented using the MD
simulation code LAMMPS. The steps involved in determining the free energy change are
listed below:
1. For step k: Update deformation state of the MD simulation box using the appro-
119

priate coordinate update equations (eq.6.10 or 6.11).
2. Set i = 1 and repeat the following steps from i = 1 to i = 3.
(a) Choose λ = λi (from Table 6.1)
(b) Run MD for 40 ps and determine the time average of
∂H
∂λ λi
eq.6.18) for the last 10 ps.
(computed using
3. Evaluate change in free energy ∆A using eq.(6.22) and proceed to step k + 1.
Figures 6.18 and 6.19 are the results to the Helmholtz energy computation for volumetric
and deviatoric deformations respectively.
Figure 6.18: ∆A versus volumetric strain trace(ɛ vol ) = ɛ vol
kk
The results to the Helmholtz computations show a “-” due to the fact that the ther-
modynamic system corresponding to λ = 0 has a larger free energy relative to the system
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Figure 6.19: ∆A versus deviatoric strain ɛdev 12 = ɛdev 13 = ɛdev 23
at λ = 1 (the original thermoplastic polymer). It has been shown using the thermody-
namic integration approach that we are able to clearly quantify the increase in Helmholtz
energy with mechanical deformation. However, as discussed in Chapter 3, absolute val-
ues of this increase is computationally difficult to obtain for the amorphous system we
have chosen to simulate. In the view of this author, the ability to quantify the change
in Helmholtz energy with respect to its mechanical behavior holds promise for future
research into determining a material model which is able to capture to both the ener-
getic and entropic contributions of the polymer to its free energy. Given in appendix A
is the theoretical formalisms to extract a stress-strain law based on the volumetric and
deviatoric response of the system interms of the hydrostatic stress and deviatoric stress
tensor. It is recommended that for future simulations that all parameters of the energy
121

functional eq.3.34 (Chapter 3) are varied as a function of λ in order to sample all possible
occupation volumes of the phase space.
6.3 Multi-scale Damage model for Thermoplastic Nanocompos-
ites
While section 6.1 attempts to couple atomistic simulation to continuum solution during
simulation time in this section, we will look at constitutive modeling of the nanoclay
polymer composite introduced in Chapter 3 using a purely atomistic (MD) simulations
approach. The primary objective of this section will be to characterize and correlate the
stress-strain behavior of the MD nano-composite model to the damage model introduced
in Chapter 5. In determining material properties of the nanocomposite (and pure poly-
mer) system the simulation scheme employs kinematic deformation (i.e., strain control)
of the simulation box followed by proper ensemble averaging (Zhou, 2003) to determine
the mechanical stress that develops in the bulk. In this work the volume averaged Cauchy
stress is the measure of internal force that develops in the atomistic model in response to
the applied deformation. It is determined using a time average of the following equation,
σαβ = 1
2Ω
i
j
r α ijf β
ij
(6.23)
Where, α, β are indicial variables that could be either = 1, 2 or 3, r α ij is α component of
the vector joining atoms i and j, while f α ij is the interaction force between atoms i and
j that is derived from the total potential energy of system. Figure 6.20 is a schematic
of the two primary deformations used to study stress response of the system. While
the name “molecular dynamics” implies a dynamic stress-strain response of the system
at extremely fine time-scales, the algorithm can be employed to determine quasi-static
material properties from sufficiently long simulations and time-samplings. Given below
122

Figure 6.20: Schematic of the two primary deformations (extension and simple shear)
used to determine material properties and the associated deformation gradient for each
mode of deformation. Shown here are extension in x1 direction and shear in the x1 − x2
plane
are the steps that were followed in this research to determine the stress-strain response
of the discrete atomistic system,
1. Equilibrate atomistic model: The typical steps involved in equilibrating a MD ther-
moplastic polymer model and a MD polymer/nano composite model are discussed
in section 3.3 of Chapter 5.
2. For step k, we have the deformation gradient F k . Apply F k to deform MD simula-
tion box. As the simulation box is deformed, remap coordinates of atomistic entities
into the deformed box. While orthogonal deformation of the box is straightforward,
simple shear deformations require a triclinic box to deform the box to required state.
The MD algorithm LAMMPS (Plimpton, 1995) has the ability to run simulation
in a orthogonal and non-orthogonal (triclinic) box settings.
3. The time step used in these calculations were 0.5 fs. Run MD simulations for 5 ps,
the volume averaged stress components (6 components) were written to file every 5
123

fs. The simulations are run with the Langevin thermostat set at 300K. For all cases
of deformation the linear momentum of the center of mass (COM) is set to zero
in order to prevent rigid translation of the atomistic entities. The stress averaging
can then be performed as a post-processing of the MD data. However, this would
stretch memory requirements.
4. At the end of the MD run, set k = k+1 and return to step 2. If we have reached
maximum strain proceed to next step.
5. After the simulations are completed the results are averaged (as discussed above)
to calculate the stress for each strain update. A 13 point moving average of the
data is performed to further smooth the stress-strain response.
It should be mentioned that although the kinematics of deformation has been defined in
terms of the deformation gradient F, this is not a finite deformation model. The strain
to failure is assumed to be small enough such that the non-linear terms of the finite
deformation strain measures can be ignored in the interest of solution tractability.
6.3.1 Simulation results for Thermoplastic Polymer Nano-composite
A polymer nano-composite specimen was created and equilibrated using the steps de-
scribed in sec.3.3.2. The structure of the nanoclay used in this is shown in fig.(6.21),
with an aspect ratio of 4. The paramaters for bonded interactions for the nanoclay were
obtained from Katti et al. (2005), while the non-bonded parameters were obtained from
the CVFF force field. The total potential energy for the nanoclay is given by the following
energy equation,
U(r) = Ebond + Eangle + Etorsion + Enon-bond
124
(6.24)

Figure 6.21: Structure of nano-clay with dimensions (unequilibrated) indicated with the
constituent atoms.
Where,
Ebond =
kl(l − l0) 2
bonds
Eangle =
angles
Etorsion =
torsions
kθ(cos θ − cos θ0) 2
Enon-bond =
σ 4ɛ
r
i
j>i
V [1 + cos(nϕ − δ)]
12
−
σ
6
r
(6.25)
In order to tabulate the force field parameters we use O1 to indicate the oxygen in the
octahedral sheet and O2 as surface oxygen. Further, C represents the coarse grained
polymer atom. Listed in Tables 6.2 through 6.4 are the bonded interaction parameters.
The non-bonded interactions between polypropylene coarse grained atoms and nanoclay
and the internal nanoclay non-bonded interactions are listed in Table 6.5. The parame-
125

Table 6.2: Bond parameters for Nanoclay
Bond Atom i Bond Atom j l0(˚A) kl(kcal/mol/˚A 2 )
Al O1 1.955 382.617
O1 H 0.988 827.209
Si O2 1.626 492.387
Si O1 1.62 558.593
Table 6.3: Angle parameters for Nanoclay
Bond Atom i Bond Atom j Bond Atom k θ0( ◦ ) kθ(kcal/mol/rad 2 )
Al O1 H 118 34.933
Si O1 Al 124.5 155.754
O2 Si O1 110.6 97.493
Al O1 Al 109.5 198.27
O1 Al O1 90/180 80.0
Table 6.4: Torsion parameters for Nanoclay
Bond Atom i Bond Atom j Bond Atom k Bond Atom l nϕ V (kcal/mol)
O2 Si O1 Al 0 0.722
Table 6.5: Non-bond parameters for Nanoclay-Polymer hybrid
Bond Atom i Bond Atom j σij(˚A) ɛij(kcal/mol)
O O 2.86 0.228
Si Si 4.05 0.04
Al Al 4.05 0.04
C O 3.33 0.094
C Al 3.96 0.0395
C Si 3.96 0.0395
O Si 3.404 0.0955
O Al 3.404 0.0955
Si Al 4.053 0.04
ters for the various energetic interactions inside the thermoplastic polymer system is still
the same as mentioned in sec.(6.1). The volume percentage of nano-clay is 4.72% which
is the ratio of nanoclay volume to the volume of the simulation box. For a 1500 coarse
grained polymer system, the weight fraction of the polymer nano-composite modeled in
this work is 6.8%, calculated as WNC/(WNC +WPP) where WNC is the weight if nano-clay
and WP P is weight of the polymer. The aspect ratio of the nano-composite that is, the
126

length along x direction divided by the width (z dimension) is 4.19, which is significantly
lower than conventional polymer nano-clay composites. For the purposes of model de-
velopment this model of nano-clay could serve as the foundation for future work. The
polymer nano composite has the nano-clay oriented along the long x1 axis of the box (see
fig.6.20). This particular orientation of nano-clay gives the RVE orthotropic properties.
Consequently, with a set of six strain controlled simulations the entire stiffness matrix can
be numerically characterized. The following six simulations were employed to completely
characterize mechanical behavior of the polymer nano-composite:
1. Case 1: Extension along x1 axis with the dimensions in x2 and x3 constrained
2. Case 2: Extension along x2 axis with the dimensions in x1 and x3 constrained
3. Case 3: Extension along x3 axis with the dimensions in x1 and x2 constrained
4. Case 4: Simple shear in x2 − x3 plane
5. Case 5: Simple shear in x1 − x3 plane
6. Case 6: Simple shear in x1 − x2 plane
For the sake of completeness we revisit some of the pertinent equations from Chapter
5. We know that the stiffness matrix [C] is a linear combination of the damaged and
undamaged stiffness.
[C] = [C 0 ] + [C 1 ] (6.26)
127

Where, [C 0 ] is the 6 × 6 undamaged orthotropic stiffness matrix,
⎡
[C 0 ⎢
] = ⎢
⎣
2c1 c4 c5 0 0 0
c4 2c2 c6 0 0 0
c5 c6 2c3 0 0 0
0 0 0 2c7 0 0
0 0 0 0 2c8 0
0 0 0 0 0 2c9
⎤
⎥
⎦
(6.27)
Where, the elastic constants ci(i = 1, 2, 3 . . . 9) are to be determined from MD simulations.
The damaged stiffness [C 1 ] is given as,
⎡
[C 1 ⎢
] = ⎢
⎣
C 1 11 C 1 12 C 1 13 0 0 0
C 1 12 C 1 22 C 1 23 0 0 0
C 1 13 C 1 23 C 1 33 0 0 0
0 0 0 C 1 44 0 0
0 0 0 0 C 1 55 0
0 0 0 0 0 C 1 66
128
⎤
⎥
⎦
(6.28)

Where,
C 1 11 = 2A1D1 (6.29)
C 1 22 = 2A5D2 (6.30)
C 1 33 = 2A9D3 (6.31)
C 1 12 = A10D1 + A11D2 (6.32)
C 1 13 = A13D1 + A15D3 (6.33)
C 1 23 = A17D2 + A18D3 (6.34)
C 1 44 = 2A22D2 (6.35)
C 1 55 = 2A26D3 (6.36)
C 1 66 = 2A24D2 (6.37)
The inelastic (damage) constants Ai are characterized through atomistic MD simulations
in the following sections. The scaling factor (α) to scale the stretch (eq.5.81) to account
for the heterogeneities in the MD that are not accounted for in the continuum RVE model
is α = 0.861.
Case 1: Extension along x1 axis with deformations in x2 and x3 constrained
As the polymer model is strained along x1 axis, a fully three dimensional stress state
develops within the material model due to the constraint on the Poisson’s contraction.
Figures 6.22 to 6.24 are the stress-strain curves obtained from MD simulation (simula-
tion results shown as data points) and the nanoscale damage model (NDM). Since, the
dilatational strain in the local x1 direction is the only non-zero component of strain and
promotes void nucleation and coalescence, we can characterize constants A1, A10 and A13
using the constitutive equations listed above and the damage definition of D1. Given in
Table 6.6 are the results to the analysis. Also, listed is the correlation coefficient of the
129

linear fit. Shown in fig.(6.25) is the deformation of the atoms in the PNC in response to
case 1 deformation. As can be seen from the deformed configuration of the composite we
see micro-voids that have coalesced into larger damage entities.
Table 6.6: Linear Regression and Analysis for polymer nanoclay composite (PNC) defor-
mation Case 1
Property Value (GPa) R 2
C 0 11 1.155 0.989
C 0 12 0.763 0.993
C 0 13 0.585 0.976
2A1 -0.99 0.999
A10 -0.60 0.997
A13 -0.47 0.999
Figure 6.22: σ11 versus ɛ11 for polymer nanoclay composite (PNC) deformation (Case 1)
130

Figure 6.23: σ22 versus ɛ11 for polymer nanoclay composite (PNC) deformation (Case 1)
Figure 6.24: σ33 versus ɛ11 for polymer nanoclay composite (PNC) deformation (Case 1)
131

Figure 6.25: Deformed and undeformed configurations of polymer nanoclay composite
(PNC) deformation (Case 1)
Case 2: Extension along x2 axis with deformations in x1 and x3 constrained
Figures 6.26 to 6.28 are the stress-strain curves obtained from MD simulation (simulation
results shown as data points) and the nanoscale damage model (NDM). In this case, the
stretch in local 2-axis direction (also direction of applied maximum stretch) debonds the
interface of the polymer nanocomposite. The constants A11, 2A5 and A17 are found using
the constitutive equations listed above and the damage definition of D2. Given in Table
6.7 are the results to the analysis. Shown in fig.(6.29) is the deformation of the atoms in
the PNC in response to case 2 deformation with the interface debond clearly indicated.
132

Table 6.7: Linear Regression and Analysis for polymer nanoclay composite (PNC) deformation
Case 2
Property Value (GPa) R 2
C 0 12 0.708 0.979
C 0 22 1.153 0.994
C 0 23 0.563 0.960
A11 -0.548 0.996
2A5 -0.979 0.998
A17 -0.435 0.998
Figure 6.26: σ11 versus ɛ22 for polymer nanoclay composite (PNC) deformation (Case 2)
Figure 6.27: σ22 versus ɛ22 for polymer nanoclay composite (PNC) deformation (Case 2)
133

Figure 6.28: σ33 versus ɛ22 for polymer nanoclay composite (PNC) deformation (Case 2)
Figure 6.29: Deformed and undeformed configurations of polymer nanoclay composite
(PNC) deformation (Case 2)
Case 3: Extension along x3 axis with deformation in x1 and x2 constrained
Figures 6.30 to 6.32 are the stress-strain curves obtained from MD simulation and the
nanoscale damage model (NDM). The stretch in local 3-axis direction (also direction of
134

maximum stretch) causes voids to appear in the polymer nanocomposite. The constants
A15, A18 and A9 are found using the constitutive equations listed above and the damage
definition of D1. Given in Table 6.8 are the coefficients obtained from results to the
analysis, using linear regression.
Table 6.8: Linear Regression and Analysis for polymer nanoclay composite (PNC) deformation
Case 3
Property Value (GPa) R 2
C 0 13 0.733 0.973
C 0 23 0.633 0.973
C 0 33 0.648 0.993
A15 -0.588 0.997
A18 -0.504 0.995
2A9 -0.528 0.994
Figure 6.30: σ11 versus ɛ33 for polymer nanoclay composite (PNC) deformation (Case 3)
Case 4: Simple shear in x2 − x3 plane
In this section we characterize the stiffness and damage coefficients for simple shear in the
x2 −x3 plane. As mentioned in Chapter 5, for the case of pure shear in small deformation
the volumetric strain is zero. However for simple shear deformations in the local x1−x2 or
135

Figure 6.31: σ22 versus ɛ33 for polymer nanoclay composite (PNC) deformation (Case 3)
Figure 6.32: σ33 versus ɛ33 for polymer nanoclay composite (PNC) deformation (Case 3)
x2−x3 planes, the RVE is susceptible to interface debond due to the non-zero stretch that
occurs as shown in fig.(5.5). Since the influence function is a uniform radial atomistic
interaction field as set by the Leonard Jones potential, the damage in shear can be
characterized by simply tracking the maximum principle stretch in the RVE. Figure 6.33
shows the result for MD and the NDM curve fit model through linear regression. The
values for the linear regression model are listed in Table 6.9.
136

Table 6.9: Linear Regression and Analysis for polymer nanoclay composite (PNC) deformation
Case 4
Property Value (GPa) R 2
C 0 44 0.189 0.941
A22 -0.337 0.959
Figure 6.33: σ23 versus γ23 for polymer nanoclay composite (PNC) deformation (Case 4)
Case 5: Simple shear in x1 − x3 plane
While the debond behavior seems to be well correlated to the simple shear deformation
in case 4, the dilatational behavior is seen to have no affect on the stress response of the
PNC as shown by the stress-strain curves in fig.(6.34). Hence, the damage coefficients,
A20=A26=0. The undamaged stiffness coefficient C 0 55 is determined to be 38.851MPa.
Case 6: Simple shear in x1 − x2 plane
The arguments presented earlier for the debond growth in simple shear deformation in
case 4 applies here. However, it is seen that a neat debond separation behavior at the
interface is not observed; Instead, a distinct stick-slip stress-strain response with very
little to no discernible linear elastic region is seen to occur. This observation seems
to indicate that the polymer nano-composite is undergoing shear induced elasto-plastic
137

Figure 6.34: σ13 versus γ13 for polymer nanoclay composite (PNC) deformation (Case 5)
deformation at the macro-scale. It is seen that a similar behavior is observed for the
stress-strain response for case 4 deformation, however the “stick” and “slip” of the stress
versus increasing strain is not as pronounced as it is here. This could be attributed to
the lower interaction length in the direction of applied shear strain. Further discussion
on this observation is provided in Chapter 7. Figure 6.35 is the result for MD. Linear
regression was not not performed for these results since a linear elastic region could not
be discerned for the computation of the necessary stiffness and damage variables.
6.3.2 Simulation results for pure Polypropylene without Nanoscale
Reinforcement
In an effort to quantify the improvements in mechanical properties of the polymer nanocom-
posite and to clearly delineate the constituent damage modes for the pure polymer and
polymer nanocomposite system, the pure thermoplastic polymer model was created using
the steps described in section 3.3.1. We assume that the failure in the material model is
due to the dilatational strains in the material that causes voids to nucleate and coalesce
within the model when subject to deformation beyond a critical value. In the constitutive
138

Figure 6.35: σ12 versus γ12 for polymer nanoclay composite (PNC) deformation (Case 6)
modeling of pure polymer systems two cases of deformation needed to be studied using
MD simulation. Following along the lines of the derivation for elastic-damage constitutive
law for polymer nano composites, we propose a similar set of equations for the isotropic
stiffness of the polypropylene MD model,
Where,
C12 = C 0 12 + C 1 12 (6.38)
C22 = C 0 22 + C 1 22 (6.39)
C23 = C 0 23 + C 1 23 (6.40)
C66 = C 0 66 + C 1 66 (6.41)
C 1 22 = 2B1D2 (6.42)
C 1 12 = B2D2 (6.43)
C 1 23 = B3D2 (6.44)
C 1 66 = 2B4D2 (6.45)
139

The damage coefficients Bi for the pure polymer are characterized in the next section.
Note that in the following calculations the principle stretch determined from the defor-
mation gradient is scaled by a factor of α = 0.941.
Case 1: Extension along x2 axis with the deformations in x1 and x3 constrained
Figures 6.36 to 6.38 are the stress-strain curves obtained from MD simulation (simulation
results shown as data points) and the nanoscale damage model (NDM). In this case, the
stretch in local 2-axis direction (also direction of maximum stretch) creates void(s) to
nucleate with in the polymer system. The constants B1, B2 and B3 are found using the
constitutive equations listed above and the damage definition of D2. Given in Table 6.10
are the results to the analysis. Shown in fig.(6.39) is the deformation of polypropylene
(PP) in response to case 1 deformation with the void formation clearly indicated.
Table 6.10: Linear Regression and Analysis for pure polypropylene (PP) deformation
Case 1
Property Value (GPa) R 2
C 0 12 0.447 0.981
C 0 22 0.559 0.962
C 0 23 0.498 0.92
B1 -0.304 0.994
2B2 -0.405 0.994
B3 -0.355 0.997
Case 2: Simple shear in x1 − x2 plane
Table 6.11: Linear Regression and Analysis for pure polypropylene (PP) deformation
Case 2
Property Value (GPa) R 2
C 0 66 0.109 0.941
B4 -0.757 0.960
140

Figure 6.36: σ11 versus ɛ22 for pure polypropylene (PP) deformation (Case 1)
Figure 6.37: σ22 versus ɛ22 for pure polypropylene (PP) deformation (Case 1)
For the case of simple shear in the x1 − x2 plane, we assume that the principal stretch
causes the void to nucleate and grow along a direction inclined 45 ◦ to x2 (see, fig.6.40).
Figure 6.41 shows the result for MD and the NDM curve fit model using linear regression.
The values for the linear regression model are listed in Table 6.11. Also, shown in figure
6.42 is the atomistic deformation in simple shear of the polymer MD model. It should
be noted that the deformation map did not conclusively show a void opening on surface,
141

Figure 6.38: σ33 versus ɛ22 for pure polypropylene (PP) deformation (Case 1)
Figure 6.39: Deformed and undeformed configurations of pure polypropylene (PP) deformation
(Case 1)
indicating perhaps that the void nucleated inside the bulk polymer.
142

Figure 6.40: Schematic of void formation in pure polypropylene RVE undergoing simple
shear deformation in x1 − x2 plane with the principal stretch axis x ′ 2 oriented 45 ◦ to the
local x2 axis
Figure 6.41: σ12 versus γ12 for pure polypropylene (PP) deformation (Case 2)
6.3.3 Comparison of Stiffness and Strength between Polymer
Nanocomposite and Pure Polymer
The Young’s moduli and failure strengths for the cases of extensional and shear deforma-
tion can determined for the polymer nano-composite (PNC) and the pure polymer using
the characterization data from atomistic MD simulations. Assuming orthotropic symme-
143

Figure 6.42: Deformed and undeformed configurations of pure polypropylene (PP) deformation
(Case 2)
try in material properties for the PNC, we can use the following equation to determine
the Young’s modulus in the local 2 direction of the nano-composite, E 0,PNC
2 .
E 0,PNC
2
= C0 11C 0 22C 0 33 + 2C0 23C 0 12C 0 13 − C0 0 2
11C23 − C0 0 2
22C13 − C0 0 2
33C12 C0 22C 0 0 2
33 − C23 (6.46)
Where, C 0 11 = 1.155GPa, C 0 12 = 0.708GPa, C 0 13 = 0.585GPa, C 0 22 = 1.153GPa, C 0 23 =
0.563GPa and C 0 33 = 0.648GPa. Similarly, assuming that the pure polymer has an
isotropic response we can determine the Young’s modulus of the pure polymer E 0,PP
is given as,
E 0,PP =
2
C0 22 + C0 12C 0 0 2
11 − 2C12 C0 22 + C0 12
(6.47)
Where, C 0 22 = 0.559GPa and C 0 12 = 0.447GPa. Table 6.12 is a summary of the Young’s
modulus and stiffness enhancements of the polymer nano-composite relative to the pure
144

polymer system as predicted by the MD algorithm.
Table 6.12: Comparison of modulus of Polypropylene (PP) and Polymer Nano-composite
(PNC) systems as predicted by Nano-scale Damage Model (NDM)
C 0,PP
22 (GPa) C 0,PNC
22 (GPa) % increase E 0,PP (GPa) E 0,PNC
2 (GPa) % increase
0.559 1.153 106.26 0.162 0.567 250
In order to compare the strengths of the polypropylene (σ PP ) and polymer nano-
composite system (σPNC 2 ) in uni-axial extension along the local x2 axis we find the max-
imum value of stress that can be obtained from the nano-scale damage model (NDM).
From the results presented in the sections above we arrive at the results tabulated in
Table 6.13.
Table 6.13: Comparison of strength of Polypropylene (PP) and Polymer Nano-composite
(PNC) systems as predicted by Nano-scale Damage Model (NDM)
σ PP (MPa) σ PNC
2 (MPa) % increase
149.04 189.37 27.06
The values for shear modulus enhancements in the x1−x2 plane for the polymer nano-
composite could not be tabulated since a linear stress-strain response was not observed in
the MD simulations. It is important to note that while the MD results of pure polymer
indicates a clear drop in stress after damage is initiated (akin to brittle failure), the PNC
response in shear (in both x1 − x2 and x2 − x3 plane) exhibits an elastic-plastic response
indicating a prolonged yield type behavior.
6.3.4 Summary
In this section, the parameters for the proposed damage model from Chapter 5 were char-
acterized using molecular dynamics simulations. It is observed from the linear regression
curve fits to the MD simulations results that while the behavior of the polymer models
145

can be well correlated to the damage model for the case of pure volumetric deformation,
the shear behavior needs review and further research. However, the current model for
shear failure is consistent with the assumptions that were made for the damage model
(see Chapter 5). Interestingly, it is also observed that while the polypropylene had failed
(clear drop in stress) at about 37% shear strain, the polymer nanocomposite exhibited a
stick slip behavior indicating an elastic-plastic response in stress. This observation could
possibly explain the strength enhancements that are seen in polymer nano-composites.
146

Chapter 7
Discussion
In this dissertation two novel methods of modeling polymers and polymer nanocom-
posites were proposed. While the complexities involved in modeling these materials are
numerous, the proposed methods are able to determine an applicable multi-scale model
of the material behavior by appealing to fundamental physics and the statistical nature
of atomistic interactions at the nanoscale. In this chapter the significant contributions of
this research with respect to modeling of polymer nanocomposites is discussed, followed
by a discussion on avenues for further research and development.
7.1 Significance of Research
Listed below are some of the salient features and possible applications of the modeling
work in this dissertation.
1. Coupling Meshless methods to Atomistic Simulation: In this work, we have ad-
dressed and shown the application of meshless methods in coupling continuum to
MD simulations. While traditional concurrent coupling methods have used finite
element (FEA) based analysis of the continuum, in this work we have clearly shown
the advantages of using a meshless methods. They are,
(a) The particles act as a natural transition from the continuum domain to atom-
istic particles. As shown in Chapter 4, we have taken advantage of the particle
description in GIMP to create the spatial and temporal connection to atomistic
behavior in the coupling method.
(b) The meshless method allows for finite deformation analysis of the MD system
147

since it is not as susceptible to mesh entanglement issues as conventional finite
element.
(c) Atomistic interactions with crack growth in the continuum is now easier to
study in coupled simulation due to the ease with which cracks can be modeled
in GIMP, a method inherently independent of mesh.
2. Concurrent Coupling of Polymers While there exists numerous techniques to con-
currently couple atomistic models of regular lattice structures of solids to continuum
analysis, a similar analysis methods for polymers are limited. The primary appli-
cation of this analysis scheme is the ability to scale the MD simulation box to
larger system sizes. Concurrent coupling, enables the user to model larger mate-
rial systems while subscribing to macro-scale loading conditions without tagging on
additional computational expenses. Further, since the modeling technique treats
finite boundary effects more effectively than an equivalent MD simulation, it allows
for the focussed study of the polymer in relation to macroscale loading conditions.
3. Damage Modeling of Polymer Nanocomposites A simple small-strains phenomeno-
logical model has been proposed to simulate damage behavior in nanocomposites.
The damage model uses the exact non-bonded interactions at the atomistic scale to
determine the thermodynamics of the polymer nanocomposite as a function of its
mechanical behavior. Since the model is able to simulate the atomistic interaction
in the nano composite without resorting to a full MD simulation, the model can
be implemented in a continuum analysis to simulate deformational response in a
nano-composite all the way to failure.
4. Modeling Fiber reinforced Polymer/Nanocomposite matrix The primary objective
of this research was to explain the strength enhancements in compression and shear
properties of nano composites with very little addition of nanoclay to the polymer
148

matrix. The modeling methodologies proposed in this work can be applied in a
multi-scale analysis of the fiber reinforced composite with nano clay inclusions in
the polymer matrix. The methodology provides engineers a design scale analysis
and material evaluation theory for structural applications in which weight and cost
cutting measures are of primary importance.
7.2 Future Work
While the proposed model is a step towards understanding and accurate simulation of
nanocomposites, there remains scope for further improvement and extended research
opportunities. In this section specific algorithmic enhancements to the coupled simulation
method and the multi-scale damage model are discussed. It is envisioned these topics
would provide as a road map for future research and development.
7.2.1 Coupled simulation of Fiber reinforced PNC
From Chapter 6, it is evident that the coupled simulation technique is a viable alternative
to a million atom MD simulation. As mentioned before, concurrent coupling can mitigate
the finite edge effects that are normally unaccounted for in a periodic cell MD simulation.
We have seen through coupled simulation that that the macroscale loading conditions
creates a multi-axial stress field around the MD domain which in general is hard to
simulate in a regular periodic purely atomistic simulation without artificially constraining
the atomistic domain. Secondly, the use of coupled simulation also enables the user to
scale the size of the system without the need for additional atoms. With the concurrent
coupling scheme, the continuum domain can be considered as an extension of the atomistic
region with the process zone (i.e., location in model where damage develops) restricted
around the MD domain. However, certain challenges remain, some of which are listed
below,
149

1. Correct transfer of energy through the handshake zone is required to derive mean-
ingful results out of coupled simulation. Dissipation of strain-energy at the interface
due to incompatible material properties is the primary source of inaccuracy for all
concurrent coupling algorithms. In the atomistic model of polymers these dissipa-
tive mechanisms are enhanced due to the disordered material structure of polymer
molecules and the random walk motion of the molecules at finite temperature. A
one to one coupling of these systems are inherently difficult to implement. The
statistical coupling technique proposed in this work can help mitigate some of these
issues. However, in the interest of improvement it is recommended that a purely pe-
riodic MD simulation is first carried out to determine material properties and then
using these material properties in the simulation of the bulk (continuum system).
In this manner the material definition is consistent in both the bulk (continuum)
and the local (MD domain) sense. It further ensures that the deformations and
the associated stress field in the continuum zone will be compatible to those from
atomistic simulation.
2. An important application of the analysis scheme is to study behaviour of thermo-
plastic composites in relation to macroscale loading. In this work we have seen the
use of MD simulation to study the characteristic response of polymer (and poly-
mer/nanoclay) atomistic model to various deformation states. In nanocomposites
it is evident that the aspect ratio of the nanoclay platelets plays a central role in
reinforcing the polymer system at the nano scale. Most nanoclay platelets have
an aspect ratios in the range of about 50:1 to 100:1. In addition to the geometry
distribution of the particles, weight fractions of nanocomposites are equally impor-
tant in determining the mechanical behaviour of the composite. Hence, a realistic
atomistic model will require significantly large number of simulation atoms and a
relatively large sized atomistic domain. However it is envisioned that with coupled
150

simulation the computational costs can be considerably reduced by modeling parts
of composite with a smeared continuum model that is reliant on the deformation
mechanics of the bulk rather than the details at the local scale.
3. It is recommended that for a better description of the continuum response the im-
plicit GIMP algorithm be used such that the time-integration errors per continuum
update associated with an explicit time-stepping GIMP are avoided. An implicit
analysis will also allow for higher time-step in the continuum. However, this will
imply that the MD simulations will have to be run for longer periods (assuming
that the MD time-step remains the same). Hence a parallel implementation of the
GIMP/MD coupling is recommended to reduce the computational time per coupled
simulation update.
7.2.2 Multi-scale modeling of Polymer Nanocomposite
For a given specimen of polymer nanocomposite, the nanoclay can exist in a exfoliated or
intercalated morphology as shown in fig.7.1. In general, the configuration of the polymers
around the nanoclay and the spatial distribution within the bulk polymer influences the
property enhancements of the polymer nanocomposite. Hence, it will be beneficial to
include real nano-scale system parameters such as gallery spacing, nanoclay aspect ratio,
weight fraction of the embedded particle etc. in defining the damage entity. Further in
Chapter 6, the nano-scale damage model (NDM) was used to make strength predictions
for the composite and the pure polymer system. While the stiffness and strength enhance-
ments in the PNC with respect to pure polymer is readily observed, the absolute values of
strength and stiffness predictions for the two systems (pure polymer and polymer nano-
composite) do not compare well with experimental results 1 . These anomalies could be
attributed to system size of the MD and the particular choice of coarse grained atomistic
1 modulus of polypropylene is between 0.8-2 GPa, while strength of polypropylene ˜40MPa
151

Figure 7.1: Schematic of Nano-clay fully exfoliated and intercalated morphology in Polymer
nano-composite
model. In general, the purely amorphous polymer system will predict lower stiffness and
higher strength since the crystalline nature of polypropylene is entirely omitted from the
MD model. Further, while the use of coarse-graining expedites MD computations, the
predictions of mechanical properties are in general not good, as can be observed from the
results. Stiffness and strength are properties extensive to the MD system hence, simu-
lations are required to determine an optimal MD system size which will be viable both
computationally and physically as a representation of the actual polymer system. This
exercise will also enable in the design of a reliable NDM for the PNC and pure polymer.
152

Figure 7.2: Schematic of randomly oriented Nano-clay in global coordinates X,Y,Z and
local coordinates x1, x2, x3
7.2.3 A Generalized Three-Dimensional Damage Model
In general, with most commonly employed manufacturing techniques it is difficult to
fabricate well-aligned nano-clay nano composites. Hence, the final product will invariably
have some nanoclay platelets that are not aligned to global XYZ directions of macro-scale
loading (see, fig.7.2). The nanocomposite will hence be under the influence of a multi-
axial state of stress. The RVE model can be extended for a general state of deformation
in which the local axis of the nanoclay do not coincide with the loading directions of the
macro-scale composite.
Consider a point within the macro-composite that is in a general state of deformation,
represented by the deformation gradient F ∗ . Let, R be an orthogonal transformation that
153

converts the coordinates (X,Y,Z) in the global frame to local coordinates (x1, x2, x3) of
the RVE, or mathematically: R : R · X ↦−→ x.
In general, R will have the following properties,
R −1 = R T and R T R = I (7.1)
Where the superscript T denotes transpose and I is the identity tensor. We can then
define the deformation gradient F in the local coordinate frame as follows,
F = RF ∗ R T
or, in indicial notation, Fij = rimrjnF ∗ mn
(7.2)
We then employ the following procedure to determine the effective stretch λ along the
direction of vector n={n1, n2, n3} (n is the normal vector to the damage surface, see
Chapter 5). If N is the principle direction vector, λ ∗ the principle stretch values of the
deformation gradient F and C is the right deformation tensor (Malvern, 1969), then we
obtain the complete set of principle values (λ ∗ 1,λ ∗ 2,λ ∗ 3) solving the following eigen value
problem,
(C − λ ∗2 I) · N = 0 (7.3)
Let Ni be the principle direction vector that corresponds to the eigen value λ ∗ i . Further,
let the components of Ni in the local coordinate be given as, Ni =
N (1)
i , N (2)
i , N (3)
i .
Then the stretch λ can be determined for a general state of deformation as (indicial
summation implied),
λ = λiNi · n = λ ∗ 1N1 · n + λ ∗ 2N2 · n + λ ∗ 3N3 · n (7.4)
154

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163

5),
Appendix A
Derivation of Stress-Strain law based on Dilatational and
Deviatoric response of a material
Following the Claussius-Duhem inequality (Malvern, 1969) we can know (see, Chapter
σij = ρ ∂ψ
ɛij
(A.1)
Where, ρ is the current density of the system, ψ is the specific Helmholtz free energy.
For the small strain tensor we can define the volumetric (dilatational) (ɛii) and deviatoric
(¯ɛij) strains as follows,
ɛii = tr(ɛ) = ɛ11 + ɛ22 + ɛ33 (A.2)
¯ɛij = ɛij − 1
3 δijɛkk
(A.3)
Where, i, j, k are indicial variables and repeated indices imply summation (eq.A.2),
δij is the Kronecker delta function. Assuming that the energy can be partitioned in to a
pure volumetric and deviatoric parts we have,
∂ψ ∂ɛkk
σij = ρ
∂ɛkk ∂ɛij
+ ∂ψ
∂¯ɛlm
Differenciating eq.(A.3) with respect to ɛij we get,
∂¯ɛlm
∂ɛij
= ɛlm
ɛij
− 1
3 δlm
∂ɛkk
∂ɛij
∂¯ɛlm
∂ɛij
= δilδjm − 1
3 δijδlm
Where the identity ∂ɛkk
∂ɛij = δij has been used. Substituting eq.(A.5) in eq.(A.4),
164
(A.4)
(A.5)

∂ψ
σij = ρ δij +
∂ɛkk
∂ψ
∂¯ɛlm
∂ψ
= ρ δij +
∂ɛkk
∂ψ
∂¯ɛij
δilδjm − 1
3 δijδlm
− 1
3 δij
∂ψ
∂¯ɛll
(A.6)
(A.7)
We see that the last term in the above equation is zero since, trace(¯ɛ) = 0. Hence the
constitutive law for the material can be derived as,
∂ψ
σij = ρ δij +
∂ɛkk
∂ψ
∂¯ɛij
(A.8)
The eq.(A.8) is not applicable for fully anisotropic material behavior. The volumetric
(σii) and deviatoric (¯σij) stress tensor can now be derived using equation A.8,
σii =
∂ψ
ρ δii +
∂ɛkk
∂ψ
∂¯ɛii
= 3ρ ∂ψ
∂ɛkk
¯σij = σij − 1
3 δijσkk
= ρ ∂ψ
∂¯ɛij
Hence, in this manner the material stress-strain response can be deduced.
165
(A.9)
(A.10)
(A.11)