Dynamic Failure of Advanced Ceramics using Molecular

Dynamic Failure of Advanced Ceramics using Molecular

Dynamics Simulations

Author: Victor Juarros Supervisor: James Hogan

Centre for Design of Advanced Materials

University of Alberta, Canada

In collaboration with ETSEIB-UPC, Spain

Fall 2020

Mechanical Engineering Faculty of Engineering 10-203 Donadeo Innovation Center for Engineering www.engineering.ualberta.ca/mece Tel: 780.492.3598 9211 – 116 St NW, Edmonton, Alberta, Canada T6G 1H9 Fax: 780.492.2200

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ABSTRACT

This project has been mainly focused on the validation of the model of Al2O3 that will be used to

study the mechanical behavior of alumina under shock loading conditions through the application

of Molecular Dynamics simulations using LAMMPS software. The simulations were used to

estimate the parameters that are commonly used to validate the potential fields and the Molecular

Dynamics models before running complex simulations to study the behavior of materials (e.g.,

elastic constants, cohesive energy, lattice constants, and the radial distribution function). In

addition, a methodology for shock loading simulations with alumina was proposed. In this

methodology, the equilibration part, the boundary conditions, and the shock loading conditions

were described.

The project also contains a study of some B4C specimens that were analyzed by image processing

algorithms using MATLAB to understand the usual behavior of ceramics under ballistic impacts

and provides a basis on the topic. Altogether, this study presents a thorough understanding of the

dynamic failure of Al2O3 under ballistic impacts with regard to mechanical properties and failure

mechanisms, and provides insights for simulating with this type of material and with advanced

ceramics in general.


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CONTENTS 1. INTRODUCTION ............................................................................................................... 5

1.1 Motivation ........................................................................................................................... 5

1.2 Objectives ............................................................................................................................ 7

1.3 Goals ................................................................................................................................... 7

2. DYNAMIC FAILURE OF ADVANCED CERAMICS .................................................................. 8

2.1 Background ......................................................................................................................... 8

2.2 Crack propagation during dynamic failure in boron carbide ................................................. 9

3. MOLECULAR DYNAMICS ................................................................................................ 15

3.1 Introduction to Molecular Dynamics and LAMMPS ............................................................ 15

3.2 MD simulations applied to shock loading on materials ....................................................... 16

4. MOLECULAR DYNAMICS SIMULATIONS OF Al2O3 ........................................................... 18

4.1 Potential field .................................................................................................................... 18

4.2 Validation of the model and the potential .......................................................................... 20 4.2.1 Al2O3 polytypes .................................................................................................................................. 20 4.2.2 Elastic Constant Validation .............................................................................................................. 21 4.2.3 Cohesive Energy ................................................................................................................................ 24 4.2.4 Lattice Constants ............................................................................................................................... 25 4.2.5 Radial Distribution Function of density of surrounding matter as a function of the distance from a point ..................................................................................................................................................... 26

4.3 Equilibration and boundary conditions of the system ......................................................... 28

4.4 Methodology ..................................................................................................................... 31

5. CONCLUSIONS ............................................................................................................... 33

ACKNOWLEDGEMENTS ......................................................................................................... 35

REFERENCES ......................................................................................................................... 36


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1. INTRODUCTION

1.1 Motivation

Advanced ceramics are important in ballistic protection applications due to their high hardness,

high strength and low density. Some of the most used advanced ceramics are alumina (Al2O3),

boron carbide (B4C) and silicon carbide (SiC). Among these ceramic materials, alumina provides

the best cost-benefit ratio [1]. With a high modulus of elasticity and high hardness, it has a

relatively lower cost which stands out from the other materials by delivering equal or better

solutions for low-cost production and manufacturing. Moreover, Al2O3 with its adequately low

density (~ 3.9 g/cm3) and moderate strength in the family of advanced ceramics (e.g., stiffness of

370 GPa and compressive strength of 2,600 MPa for alumina AD995 reported by Coorstek. Inc

[2]), is still one of the preferred candidates for some protection applications in defense [3].

The study of the dynamic failure of ceramics is essential for a big range of applications, including:

developing new lightweight materials for shielding systems for military use (e.g., body armor and

breastplates, vehicle and aircraft protection; and cutting and polishing tools) [1]; and designing of

coatings to mitigate wear failure of machines in the oil and gas industry [1,4]. Researching the

fragmentation of materials is also important for improving blasting and milling operations in

mining through controlled fragmentation [5]. Moreover, it is also relevant in planetary and space

science and in other multiple industries, and therefore it is the main topic of this project [5].

The impact of a projectile involves failure mechanisms originating defects, including plasticity,

phase transformations, and fracture [6,7]. Most of these processes can form in periods as short as

a few microseconds [8]. Mukhopadhyay et al. [9] observed that due to the shock wave caused by

the impact, an increase of defect density took place. As the shock wave propagation continued

inside alumina, there may be shear as well as twist-induced micro-fracture initiation at a single

plane or at multiple planes in an oriented grain [10]. Micro-wing crack (Figure 1a) occurs due to

the tensile stress created on two neighboring suitably oriented grain boundaries (Figure 1b). Such


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high shear stress shows dislocations hampered at grain boundaries (Figure 1c) and dislocation

entanglement within single grains (Figure 1d).

Figure 1. SEM (a,b) and TEM (c,d) images of shock experiments at shock pressures levels three to six times as high as the Hugoniot Elastic Limit (HEL). Reprinted from Shock Deformation of Coarse Grain Alumina above

HEL. Copyright 2012 IOP Science

To better understand material behavior under shock conditions, Molecular Dynamics (MD)

simulations are a powerful technique to study and understand the behavior of the materials at an

atomic scale [11]. This method allows to explore the dynamic properties (e.g., elastic constants,

stress-strain and shock wave profiles) [12,13] directly from the interactions between atoms [14].

As a result of the difficult experimental conditions of the ballistic loading of materials, MD can

overcome these obstacles in the experimental processes and study the advanced ceramics under

different loading conditions without leading to high expense and investment in experimental

methods. Moreover, inherent to the MD simulations approach at sub-ns time scales, events that

appear at the very first instants after the impact on the sample can be carefully studied without

performing complex experiments.

Due to the high potential of MD simulations in Materials Science and Engineering, multiple studies

have used this tool in the field of research in advanced ceramics. For example, Awasthi and

Subhash [15] revealed that temperature plays a dominant role in the deformation behavior and

amorphization of boron carbide, and P. S. Branicio [16] performed large scale planar shock-wave


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MD simulations on AlN ceramics. These research studies with boron carbide used the ReaxFF MD

potential developed by An et al. [17]. In the case of pure alumina, very few studies have been

conducted. In relation to the potential, one of the most relevant studies to date is the one carried

out by Sinnott et al. [18] where the group developed the Charge optimized many-body (COMB)

potential for Al2O3. The COMB3 (3rd generation of the COMB) potential, validated for mechanical

loading, was applied to the study of Al-Al2O3 interfaces and the deformation of Al, Al2O3, and

Al2O3-coated Al nanowires subjected to tensile stress. Moreover, the COMB3 potential is also

stable for studies of a wide range of simulating conditions, including shock loadings [18,19,20]

and can be used to model other different heterogeneous materials [18].

1.2 Objectives The overall objectives of the present thesis are based on the understanding of the mechanical

behavior of advanced ceramics under shock loading conditions using Molecular Dynamics

simulations. These objectives include: learning about the software to perform the simulations, and

studying the dynamic failure of Al2O3 and its mechanical properties to find applications for

materials design. For this research, the simulations will be performed using LAMMPS software

[21].

1.3 Goals The goals of the project are:

• Develop the model validation of the Al2O3 structure that will be used to carry out the MD

simulations and choose the right potential that fits with the material and the experimental

conditions.

• Set the boundary conditions for the simulation and implement them in LAMMPS.

• Explore the state of the art of the research in advanced ceramics and the usefulness of

studying materials under shock loading conditions.


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2. DYNAMIC FAILURE OF ADVANCED CERAMICS

This project focuses on understanding the deformation and failure mechanisms that govern the

dynamic behavior of brittle materials (e.g., alumina, silicon carbide, boron carbide). During

experiments with this type of materials, damage accumulation, fracture and fragmentation

phenomena are studied. Molecular Dynamics simulations require a broad knowledge and

understanding of the subject, so it is important to acquire a good theoretical base before starting to

work with this type of simulations.

2.1 Background Dynamic Failure Mechanics is the study of failure phenomena in the presence of high local strain

rates. While static failure is characterized by the activation of a small number of large pre-existing

defects that eventually lead to failure, dynamic failure is governed by crack growth from a large

number of multi-sized defects. Coalescence of this large number of cracks lead to finer

fragmentation of the material [22].

At room and ambient temperatures, fracture in ceramics is typically brittle. It occurs without any

significant plastic deformation and the failure strain is very small. In ceramics, brittle fracture is

controlled by the extension of small flaws which are dispersed in a material or component's surface

and which behave like cracks [23]. Flaws can arise from the production process, but also from

handling and service. Brittle solids subjected to large amplitude shock waves can support

substantial shear stress (of order 2-10 GPa) without failing due to the slip systems in this type of

materials [23,24].

The development of shocks is due to the increase in sound speed as the pressure rises. These

phenomena are loads of very high amplitude and short duration. When failure occurs under intense

shock loading, the effect is usually observed as a wave splitting in the compressive shock front

[23,24]. At very low pressures, sound waves are propagated at a fixed velocity, but when the

pressure is sufficient, the sound speed also increases. It is this effect which causes a compression


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wave with a positive pressure gradient (pressure increases with time) to steepen up, because the

higher-pressure region of the pulse travels faster than the lower pressure region ahead of it [25].

The following is an analysis that was carried out as a background for this project to understand the

failure of advanced ceramics.

2.2 Crack propagation during dynamic failure in boron carbide

This part of the study consisted of the analysis of 7 dynamic compression experiments using the

Kolsky bar device. The analysis was performed with two different types of samples; some from

Saint Gobain, the supplier, and the other from a Chinese supplier (unknown). The data was

analyzed with image processing tools from the MATLAB software.

The aim of this analysis is to determine the lengths and the propagation speed of the first cracks

originated when the stress wave is created due to the impact of the incident bar and the transmitted

bar of the instrument. Additionally, different behaviors have been observed, in terms of the

propagation of the cracks and the size of the fragments, depending on the type of specimen.

Data analysis was done with image processing implemented algorithms from MATLAB software.

Through a previous calibration of the “ruler tool” with the length of the specimen (3.5 mm), it is

possible to know the length of the cracks once the crack tip had been observed. Performing this

process on a set of images from the same sample, the crack growth can be determined. Dividing

the growing length over the interframe time, the propagation speed of the crack is obtained. The

timestep of every picture is represented in the lower right corner (Figure 2).


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Figure 2. Pictures of one of the samples before and after the impact

Plotting the length and the propagation speed allowed to compare the results and to make

qualitative comments on the behavior of the samples under investigation. An example of a plot of

the length of the cracks obtained from one of the samples throughout time is shown below (Figure

3):

Figure 3. Length of the first cracks vs time

y = 0.0017x - 116.54

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

69,000 69,500 70,000 70,500 71,000 71,500 72,000

Leng

th [m

m]

Time [ns]

Cracks length

Crack_1

Crack_2

Crack_3

Linear (Crack_2)


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Looking at Figure 3, it can be observed that the growth of the very first cracks experience a linear

behavior throughout the first microseconds of time, while this behavior is no longer fulfilled in the

following cracks because the behavior of the material is different at the beginning due to its

breakage into smaller fragments. It can also be observed that the slope of the lines is more

pronounced in the cracks that appear after the very first ones.

In the case of propagation speed, the same linear pattern can be seen. For the analysis, the

comparison has been done according to the order of appearance, since the speed is lower in the

very first tips and faster in the last cracks. As in the case of the lengths, it has been differentiated

according to the different types of samples to obtain the desired qualitative results.

In the following plot, the average speed of every crack has been used to determine if there is a

similarity between cracks (Figure 4):

Figure 4. Propagation speed of every crack detected in the Saint Gobain samples

-

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

Crack_ Crack_1 Crack_2 Crack_3 Crack_4 Crack_5 Crack_6

Spee

d [m

/s]

Propagation speed

B1 [m/s]

B3 [m/s]

B4 [m/s]

B5 [m/s]


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Tracing a curve and uniting all the points, the similarity can be observed as the majority of the

cracks have an average propagation speed between 1,000 and 3,000 m/s (Figure 4). Velocities

above this range may not be relevant because the material enters a state of non-linearity between

stress and strain as fragments grow.

In the case of the samples of the Chinese supplier, the results are similar to the ones obtained with

the Saint Gobain tested specimens (Figure 5).

Figure 5. Propagation speed of every crack detected in the Chinese samples

It can be seen from Figure 5 that the propagation speed of the first cracks detected in the Chinese

samples is around 1,000 m/s and 2,000 m/s. The range of velocities of both types of specimens

indicates that the propagation speed in B4C samples using the Kolsky bar device is around 1,000

m/s and 3,000 m/s. Previous studies performed by Hogan et al. [26] measured the range of the

velocity of the axial cracks. This range was of approximately 1,800 m/s to 2,400 m/s with an

average of 2,300 m/s.

-

500

1,000

1,500

2,000

2,500

Crack_ Crack_1 Crack_2 Crack_3 Crack_4

Spee

d [m

/s]

Propagation Speed

L1 [m/s]

L4 [m/s]

L5 [m/s]

L6 [m/s]


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The evolution of the appearance of the number of fragments was also studied in this analysis.

Through an image processing algorithm, the fracture of the samples was accounted to observe the

fracture behavior of the material (Figure 6).

Figure 6. Evolution of the number of fragments in some of the analyzed specimens

Differences in the evolution of the fragments are clearly seen. The material of the S# samples had

a higher strain-rate than the L# and LT# samples (Figure 6), so it agrees with the material

embrittlement that occurs at a high strain-rate in ceramics [27,28,29].

Another approach to study the fragmentation of the B4C specimens was to plot the distribution by

sizes of the fragments obtained in the ballistic tests of both types of samples (Figure 7). It can be

seen in Figure 7a that the SG samples finished having smaller fragments than the ones obtained

with the SG specimens. In all cases of the SG samples, most of the fragments (over 50%) had a

0

10

20

30

40

50

60

0 2000 4000 6000 8000 10000 12000

Frag

men

ts

Time [ns]

Number of fragments vs Time

L1

L4

LT1

LT2

S2

S6


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surface smaller than 0.05 mm2, while bigger fragments distributed in different ranges of sizes.

However, the plot in Figure 7b showed that the CHN samples ended having bigger fragments.

Figure 7. Distribution by sizes of the fragments observed in the tests

This previous analysis has allowed us to find a relationship of the growth of the first cracks

generated in the samples studied from two different suppliers (CHN and SG). The first part of the

study has consisted in determining the length of the first cracks observed in the samples (Figure

2). It has demonstrated linearity in the results obtained. It also proposes some regression lines to

determine with accuracy the behavior of the cracks. The second part of the study has consisted in

calculating the propagation speed of the cracks studied previously. Comparing the velocities of

each crack, it has been possible to observe that most of the very first cracks experience the same

propagation speed, while the cracks originated later have different speeds, especially on the

0

10

20

30

40

50

60

70

L2 L3 LT2 LT3 S1 S3 S4 S6

a) Num. of fragments by size SG

(0 - 0.05) (0.05 - 0.1) (0.1 - 0.15) (0.15 - 0.2) ( 0.2 - )

0

10

20

30

40

50

60

70

L1 L4 LT1 LT2 S2 S6

b) Num. of fragments by size CHN

(0 - 0.05) (0.05 - 0.1) (0.1 - 0.15) (0.15 - 0.2) ( 0.2 - )


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samples from the Chinese supplier. Finally, the last part of the analysis concluded that materials

with a higher strain-rate present a higher and faster embrittlement than materials with lower strain-

rates.

3. MOLECULAR DYNAMICS

Molecular Dynamics simulations allow researchers to represent crack initiation, propagation and

interaction, as well as providing a detailed description of fragment sizes and speeds [30]. These

simulations give access to observations that cannot be easily monitored during experimental tests

as a result of the short time scales and small length scales that they occur over. Moreover, they

facilitate the understanding of the fragmentation process. This section of the project aims to explain

how MD simulations work to better understand shock loading in advanced ceramics, specifically

alumina.

The software that will be used to conduct this project is the LAMMPS software (Large-scale

Atomic/Molecular Massively Parallel Simulator), a classical molecular dynamics code with a

focus on materials modeling, developed and distributed by Sandia National Laboratories, a US

Department of Energy laboratory [21].

To develop a simulation program, it is necessary to have an overview of the general methodology,

which should include the assignment of the initial configuration and velocities, the treatment of

boundary conditions, and techniques to reduce computation time [31].

3.1 Introduction to Molecular Dynamics and LAMMPS Classical molecular dynamics is a commonly used computational tool for simulating the properties

of solids, liquids and molecules. Each of the N atoms or molecules in the simulation is treated as

a point mass and Newton’s equations are integrated to compute their motion. From the motion of


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the ensemble of atoms, a variety of microscopic and macroscopic information can be extracted.

The physics of the model is contained in a potential energy function for the system from which

individual force equations for each atom are derived [21].

The methodology involves defining the system configuration and boundary conditions, and the

environmental conditions to be maintained during simulations. The atoms are relaxed to move to

those positions where they can have minimum potential energy of the system. After equilibration

and defining of loading and boundary conditions, the simulation is performed. During the

simulation, as the atomic coordinates evolve with time, the positions are recorded in a trajectory

file. Mechanical properties (e.g., Young’s modulus, ultimate strength, yield stress) [32] can then

be analyzed.

3.2 MD simulations applied to shock loading on materials

The timescales involved in shock loadings are difficult to deal with. The risetime for shock waves

and for the rise of pressure and particle velocity are of the order of picoseconds to nanoseconds

(10-12 – 10-9 s). Such timescales are very hard to deal with experimentally and usually powerful

devices such as ultra-high-speed cameras and fast digital processing devices are required to

perform shock loading experiments. There are several methods commonly used in MD to simulate

the passage of shocks through material. The goal is to cause changes that are associated with a

large strain imposed upon the material in question [31].

The most common used method is the one known as piston driven shock, where a piston drives

into the material with a certain velocity. In the figure below, the piston’s velocity is Up, therefore

the material behind the shock travels at the same speed as the piston. The increase in density due

to the action of the piston causes a shock that travels at a velocity of Us (Figure 8) [33] Therefore,

the potential energy and the kinetic energy increase both due to the acceleration of the material

[31].

.


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Figure 8. Piston driven shock. Reprinted from A beginner’s guide to the modeling of shock/uniaxial/quasi-isentropic compression using the LAMMPS molecular dynamics simulator. Copyright 2013 Oscar Guerrero

The piston impacts into the specimen from the left side. As it can be seen in the following figure,

each row is a snapshot of the system (Figure 9) [33].

Figure 9. Variation of the system under the piston shock. Reprinted from A beginner’s guide to the modeling of shock/uniaxial/quasi-isentropic compression using the LAMMPS molecular dynamics simulator. Copyright 2013

Oscar Guerrero

In LAMMPS, the way to model this type of experiment is using the fix wall/piston command. This

command acts as a moving wall which reflects particles in the system and drives it with an effective

infinite-mass piston. As it has been explained above, this piston is capable of driving shock waves.

A momentum mirror technique is used to mimic the rebound of the shock wave in the material. If


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an atom (or the wall) moves such that an atom is outside the wall by a distance delta, then it is put

back inside the face by the same delta, and the velocity relative to the moving wall is flipped [21].

4. MOLECULAR DYNAMICS SIMULATIONS OF Al2O3

The approach to perform these kind of simulations with a model of Al2O3 requires some previous

steps. The very first procedure in this methodology is to choose the right potential field that will

be used to carry out the MD simulations. Once the potential field is chosen, the next step is to

validate the model of the material that will be used for these simulations. The following step is to

design an equilibration of the system that will perform the relaxation of the sample and that will

be the first part of the simulation that will have to include the script at the beginning of it. Once

completed these previous and essential steps, it is now possible to start writing the script that will

run the simulations and obtain the necessary results to conduct the study.

4.1 Potential field In LAMMPS, potentials are defined between pairs of atoms that are within a cutoff distance and

the set of active interactions typically changes over time. So as to say, interatomic potentials

describe the forces that govern interactions of atoms in the system and they are essential to perform

MD simulations [21].

There are a number of potentials that have been developed by different research groups since the

beginning of Molecular Dynamics simulations [34,35,36]. Nowadays, the most used potentials are

the following:

- EAM Potentials (Embedded Atom Model Potentials)

- COMB Potentials (Charge-Optimized Many-Body Potentials)

- ReaxFF Potentials (Reactive Force-Field Potentials)


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These three types of potentials have been used on Al2O3 studies by various research groups. For

example, Zhang et al. [37] developed a ReaxFF potential that was used to explore and predict the

structural, energetic, and adhesion properties of Al2O3 and Al-Al2O3 systems (e.g., the cohesive

energy, lattice parameter, bulk modulus and surface energies). Sinnott et al. [18] developed a third-

generation charge-optimized many-body potential (COMB3 Potential) for Al2O3, applied to the

deformation of Al2O3 nanowires subjected to tensile stress. The attractive features of the COMB3

potential include also dynamic charge [18]. The family of COMB potentials have allowed

researchers to perform MD simulations of heterogeneous material systems. This formulation can

be adapted to model metallic, covalent, ionic and van der Waals bonding in the same simulation

cell [38]. As the COMB3 potential is fully optimized for compounds of Al and O and due to the

fact that it is proper and stable for shock loading conditions, it has been considered as currently

the most suitable potential to be used in the simulations for this project. Its formalism is described

below [18,38].

The total energy ET of a system of atoms is given by (Eq. 1):

𝐸! =#$𝐸"#$%&(𝑞") + $#𝐸"'#()*+)𝑟"' , 𝑞" , 𝑞',

'

+𝐸"',)-%)𝑟"' , 𝑞" , 𝑞',. + 𝐸".)%/*)𝑞" , 𝑟"',

"

+ 𝐸"012)𝑟"', + 𝐸"3/**(𝑞") + 𝐸"4)**)𝑟"',.(𝐸𝑞. 1)

where Eiself is the self-energy of atom i (including atomic ionization energies and electron

affinities), Eishort is the bond-order potential between atoms i and j, EijCoul are the Coulomb

interactions, Eipolar is the polarization term for organic systems, EivdW is the van der Waals energy,

Eibarr is a charge barrier function, Eicorr are angular correction terms and rij is the distance of the

atoms i and j, and qi and qj are charges of the atoms [39,40]. The angular correction terms are

primarily used to modify energies due to specified angles outside the bond order terms, also

denoted as Legendre polynomial terms [18].


20

4.2 Validation of the model and the potential Once the potential has been chosen, the following step is to choose and validate the model that

will be used to perform the simulations. The validation process consists of several steps. To

properly validate the model, various parameters have to be calculated (e.g., cohesive energy, lattice

constants, free surface energies and elastic constants). Once these parameters are calculated, they

must be compared with other studies and experimental results.

4.2.1 Al2O3 polytypes The structures that will be studied in this report are: Al2O3 mp-1938 (orthorhombic, Pbcn) (Figure

10a) [41]; Al2O3 mp-776475 (cubic la-3) (Figure 10b) [42]; and Al2O3 mp-2254 (orthorhombic,

Pna2_1) [(Figure 10c) [43]. The box surrounding the atoms in each structure is defined by the

lattice constants (ao, bo and co) of each model. The values of these lattice constants define the type

of the unit cell structure (e.g., cubic, orthorhombic, rhombohedral)

The objective is to find that these structures are suitable for the chosen potential and that the

parameters that validate the model (e.g., elastic constants, cohesive energy, radial distribution

function) are close to the ones experimentally observed and the ones predicted by other potentials.

The reason why different crystal structures need to be studied is because the importance of

determining the validity of the potential chosen. It is important to see if the COMB3 potential

models well the known crystal structures within the Al-O system for going forward and begin to

perform the MD simulations.


21

a) b) c) Figure 10. Structures in the Aluminum (blue) and Oxygen (red) system that were considered in this work

4.2.2 Elastic Constant Validation Elastic constants are quantities that measure an object or substance's resistance to being deformed

elastically, so as to say, they describe the stiffness of a material [18,21,44]. These parameters

express the relation between the stress and the strain on the materials within the stress range that

the materials exhibit elastic behavior. Some of the elastic constants include: the bulk modulus, the

shear modulus, and the Poisson’s ratio.

The ELASTIC script from the LAMMPS library has been used for the calculation of the elastic

constants. It has been adapted to different Al2O3 structures using the COMB3 potential. This script

computes the elastic constants for a system by applying small strains and performing static energy

minimizations. It performs thirteen energy/force minimizations: one for relaxing the initial system

and twelve for relaxing systems in which a small strain of magnitude is applied [18, 21].

The system virial pressures Pi are recorded for each of the thirteen relaxed configurations. Two

estimates for the Cij matrix are obtained through the following equations (Eq. 2, Eq. 3):

(𝐸𝑞. 2)

𝐶"'5 = −𝑃")∆𝜀', − 𝑃"(0)

∆𝜀

𝐶"'6 = −𝑃"(0) − 𝑃")−∆𝜀',

∆𝜀 (𝐸𝑞. 3)


22

The negative out front comes from the fact that the system-wide stress state is si = - Pi. The value

of each component of the elastic tensor (matrix Cij) is obtained by averaging the positive and

negative strain, as well as the symmetric components of the tensor, using the following (Eq. 4):

Below are shown the results obtained in applying and adapting the ELASTIC script to the different

Al2O3 polytypes presented. In the case of the unit cell Al2O3 mp-1938 (orthorhombic, Pbcn), the

results obtained after the simulation (a) are similar to the ones presented by other research studies

like the ones performed by Sinnott et al. (b) [18] and by de Jong et al. (c) [45]. Sinnott et al. noted

that an accuracy within 30% compared to other results is reasonably good and that deviations in

the results are common when calculating the elastic constants. Other alumina structures presented

above (e.g., Al2O3 mp-776475 (cubic, la-3), Al2O3 mp-2254 (orthorhombic, Pna2_1)) were also

used to calculate the elastic constants and they all showed accurate results as well.

𝐶"' =𝐶"'5 + 𝐶"'6 + 𝐶'"5 + 𝐶'"6

4 (𝐸𝑞. 4)

𝐶"'(/)[𝐺𝑃𝑎] =

⎝

⎜⎜⎛

790 200 117 330 280 481

_0_ _0_ _0__0_ _0_ _0__0_ _0_ _0_

_18 _0_ _0_ 85 _0_ 254⎠

⎟⎟⎞

𝐶"'(3)[𝐺𝑃𝑎] =

⎝

⎜⎜⎛

714 129 176139 348 −71164 −94 456

_0_ _0_ _0__0_ _0_ _0__0_ 0_ _0_

_2_ _0_ 0_ 189 0_ 199⎠

⎟⎟⎞

; 𝐶"'(4)[𝐺𝑃𝑎] =

⎝

⎜⎜⎛

450 146 87 394 150 466

_0_ _0_ _0__0_ _0_ _0__0_ 0_ _0_

164 0 0 120 0 143⎠

⎟⎟⎞


23

In the case of the Al2O3 mp-2254 (orthorhombic, Pna2_1), the results obtained (d) compared to

the ones calculated by de Jong et al. (e) [45] are as follows:

When performing this kind of studies, it is also important to look at other elastic constants like the

bulk modulus, the shear modulus and the Poisson’s ratio. All these three parameters can be

calculated from the elements of the elastic tensor Cij. According to the Voigt and Reuss

approximation, the bulk modulus and shear modulus for orthorhombic crystals can be written as

[46, 47, 48] (Eq. 5, Eq. 6, Eq. 7, Eq. 8. Eq. 9):

𝐶"'(1)[𝐺𝑃𝑎] =

⎝

⎜⎜⎛

467 178 117 267 96 243

_0_ _0_ _0__0_ _0_ _0__0_ 0_ _0_

90 _0_ 0_ 130 0_ 171⎠

⎟⎟⎞

; 𝐶"'($)[𝐺𝑃𝑎] =

⎝

⎜⎜⎛

333 119 149 343 116 375

_0_ _0_ _0__0_ _0_ _0__0_ 0_ _0_

76 0 0 124 0 98⎠

⎟⎟⎞

𝐵: = ;<∙ (𝐶;; + 2 ∙ 𝐶;= + 2 ∙ 𝐶;> + 𝐶== + 2 ∙ 𝐶=> + 𝐶>>) (𝐸𝑞. 5)

𝐵? = 𝜒 ∙ [𝐶;; ∙ (𝐶== + 𝐶>> − 2 ∙ 𝐶=>) + 𝐶== ∙ (𝐶>> − 2 ∙ 𝐶;>) − 2 ∙ 𝐶>> ∙ 𝐶;= + 𝐶;=∙ (2 ∙ 𝐶=> − 𝐶;=) + 𝐶;> ∙ (2 ∙ 𝐶;= − 𝐶;>) + 𝐶=> ∙ (2 ∙ 𝐶;> − 𝐶=>)]6;(𝐸𝑞.6)

𝐺: = 115 ∙

(𝐶;; − 𝐶;= − 𝐶;> + 𝐶== − 𝐶=> + 𝐶>> + 3 ∙ 𝐶@@ + 3 ∙ 𝐶AA + 3 ∙ 𝐶BB)(𝐸𝑞. 7)

𝐺? = 15 ∙ N4𝜒 ⋅

[𝐶;; ∙ (𝐶== + 𝐶>> + 𝐶=>) + 𝐶== ⋅ (𝐶>> + 𝐶;> + 𝐶>> ⋅ 𝐶;= − 𝐶;= ∙ (𝐶=> + 𝐶;=) ⋅ 𝐶;>

⋅ (𝐶;= + 𝐶;>) − 𝐶=> ∙ (𝐶;> + 𝐶=>)] + 3 ∙ P1𝐶@@

+1𝐶AA

+1𝐶BB

QR 6;(𝐸𝑞. 8)

𝜒 = 𝐶;> ∙ (𝐶;= ∙ 𝐶=> − 𝐶;> ∙ 𝐶==) + 𝐶=> ∙ (𝐶;= ∙ 𝐶;> − 𝐶=> ∙ 𝐶;;) + 𝐶>> ∙ (𝐶;; ∙ 𝐶== − 𝐶;== ) (𝐸𝑞. 9)


24

Using the Hill model, the shear modulus (G) and the bulk modulus (B) are the arithmetic averages

of Voigt and Reuss bounds [49]. They are calculated as shown in the following equations (Eq. 10,

Eq. 11):

In the case of Al2O3 mp-2254, the results obtained using the ELASTIC script from LAMMPS were

compared with the ones calculated by de Jong et al. [45] for the same Al2O3 structure. These

results, using all the equations above, are displayed in the table below (Table 1):

Table 1. Values of elastic constants obtained for the Al2O3 mp-2254 (orthorhombic Pna2_1) model

This project de Jong et al. [45]

Bulk modulus 197 GPa 202 GPa

Shear modulus 115 GPa 102 GPa

Poisson’s ratio 0.28 0.24

As it can be seen in Table 1, the bulk modulus obtained is almost equal to the one obtained by

Jong et al. [45] (< 2.5% deviation) and the values for the shear modulus and the Poisson’s ratio

differ by less than 16%. Changing the minimization parameters in the ELASTIC script and

applying the conditions to a pre-relaxed specimens could led to smaller deviations in the results.

4.2.3 Cohesive Energy

The cohesive energy of a solid refers to the energy required to separate constituent atoms apart

from each other and to bring them to an assembly of neutral free atoms [50], so as to say, the

𝐵 = 𝐵: + 𝐵?

2 (𝐸𝑞. 10); 𝐺 = 𝐺: + 𝐺?

2 (𝐸𝑞. 11)


25

energy required to break all the bonds between atoms. The cohesive energy can be obtained

calculating the total energy of the system and dividing it by the total number of atoms.

In order to calculate the cohesive energy for the model Al2O3 mp-2254, a minimization of the

system has been performed by relaxing all directions of the system to 0.0 Pa of pressure for all

atoms. The algorithm to perform the simulation is the one known as conjugate gradient (or steepest

descent), and this equilibrates the system and then calculates the minimum value of the cohesive

energy. The value obtained was -5.13 eV/atom, very close to the one calculated by Sinnott et al.

[18] that obtained a value of -5.29 eV/atom. These two results only differ by ~3%, fact that the

value of the cohesive energy obtained can be considered as validated.

4.2.4 Lattice Constants

Values for lattice constants have also been determined and compared with other values from

literature. In contrast to elastic constants, lattice constants describe better accuracy when

comparing them with other research studies. In the following table (Table 2) are presented the

values for the Al2O3 orthorhombic Pna2_1 unit cell obtained experimentally by Halvarsson et al.

[51] and Liu and Skogsmo [52] and with MD simulations in this project and by Sinnott et al. using

the COMB3 potential [18].

Table 2. Comparison of the lattice constants for the Al2O3 model studied in this project

This project Sinnott et al. [18] Halvarsson et al. [51] Liu and Skogsmo [52]

ao (Å) 4.89 4.78 4.84 4.84

bo (Å) 8.40 8.07 8.31 8.33

co (Å) 9.02 8.83 8.94 8.96


26

As it can be seen in Table 2, the values of the lattice constants obtained with MD simulations and

using the COMB3 potential are in a narrow range, very close to those observed and calculated by

the research studies mentioned above [18, 51, 52].

4.2.5 Radial Distribution Function of density of surrounding matter as a function of the distance from a point

Finally, to accomplish the validation of the model that will be used to carry out the simulations,

the total radial distribution function (RDF) has been calculated (Figure 11). The RDF has the useful

property that the quantity Gab(r) gives the number of atoms in a ring of thickness dr at a

distance r from another atom [53]. The RDF gab(r) between types of particles a and b is [54,55]

(Eq. 11):

which is normalized so that the RDF becomes 1 for large separations in a homogenous system.

This is why it is close to 1 for larger values of r. The RDF effectively counts the average number

of b neighbors in a shell at distance r around an a particle and represents it as a density.

The radial cumulative distribution function is (Eq. 12): and the average number of b particles within radius r is (Eq. 13):

𝑔/3(𝑟) = (𝑁/ ∙ 𝑁3)6;#C!

"D;

#U𝛿)W𝑟" − 𝑟'W − 𝑟,XC"

'D;

(𝐸𝑞. 11)

𝐺/3(𝑟) = Y 4 ∙ 𝜋 ∙ 𝑟= ∙ 𝑔/3(𝑟)*

E𝑑𝑟(𝐸𝑞. 12)

𝑁/3(𝑟) = 𝜌 ∙ 𝐺/3(𝑟)(𝐸𝑞. 13)


27

Figure 11. Total radial distribution function obtained with MD simulations for the Al2O3 orthorhombic

Pna2_1 unit cell studied in this project. The peaks of the curve indicate the presence of the Al-O, Al-Al and

O-O bonds that are present in the structure

The RDF has the other useful property that peaks in the function coming from well-defined atom–

atom pairs reflect the precise shape of the pair-probability distribution. These peaks will have

a Gaussian shape [53]. This plot is a way to show the probability to find the bonds between atoms

in the model and it proofs that it is well validated if it has the corresponding peaks at the distances

(values of r) that are proper for the bonds.

Therefore, looking at the average number of aluminum and oxygen nearest neighbors can be

determined. Davis and Gutierrez [56] found that the first peak of the RDF function (Al-O bonding)

was around 1.8 Å (Figure 12) and the value obtained in this project is 2.0 Å. From Figure 11 it

can be seen that the other peaks related to the O-O bonding and the Al-Al bonding are located at

2.4 Å and 2.8 Å, respectively. Looking at other RDFs obtained with other MD simulations and

experiments from literature, the peak values of the RDF obtained in this project are consistent. For

example, Lizárraga et al. [57], who studied the structural characterization of amorphous alumina,

found the Al-O peak at 2.0 Å, the O-O peak at 2.2 Å, and the Al-Al peak at 2.4 Å. Other examples

of studies performed with alumina and that obtained very similar results are the ones carried out

by Poe et al. [58], and Lamparter and Kniep [59]. As an example, Figure 12 shows the total radial


28

distribution function obtained by Davis and Gutierrez [56], where the different peaks of the RDF

compare well with those obtained in this project for the Al2O3 model.

With the calculation of the elastic constants (e.g., elastic tensor, bulk modulus, shear modulus and

Poisson’s ratio), the cohesive energy and the total radial distribution function and the validation of

all these parameters with other results from literature, the model Al2O3 mp-2254 (orthorhombic,

Pna2_1) and the COMB3 potential have been validated and can be now considered suitable to

perform the MD simulations that will study the dynamic failure of Al2O3 under dynamic loading.

4.3 Equilibration and boundary conditions of the system

In Molecular Dynamics, systems can be fully described by the number of particles, their positions,

velocities, orientations, and angular momenta. From such a configuration, macroscopic quantities

such as temperature or pressure can be computed. The set of configurations forms a so-called

ensemble [60]. There are different possible types of ensembles that can be implemented in

LAMMPS’ scripts. The different statistical ensembles are [21]:

Figure 12. Total distribution function for Al2O3. Reprinted from Structural, elastic, vibrational and electronic properties of amorphous Al2O3 from ab initio calculations. Copyright 2011 IOP Science.


29

- Microcanonical (NVE)1

- Canonical (NVT)

- Isobaric-isothermal (NPT)

- Isoenthalpic-isobaric (NPH)

In order to characterize an ensemble, it is sufficient to determine three thermodynamic variables,

(e.g., number of particles N, volume V, and total energy E) [60].

Before performing the shock loading or any other type of simulation, the initial structure must be

fully relaxed. This allows the system to achieve the desired temperature and to have the total linear

momentum equal to zero. A way to control these parameters is to assign the initial velocity of the

particles using the Maxwell-Boltzmann distribution [61].

During this process to bring the simulation cell to the desired temperature, each of the 3 axes (x,

y, z) need to each have their own boundary conditions. In this case, all boundary conditions are set

as periodic (PBC). The implementation of PBC can be understood as surrounding the system with

replicates of the simulation cell (Figure 13). In a simulation, periodic conditions would result in

atoms near a boundary interacting with atoms on the other side of the system as if they were nearby,

and atoms leaving the simulation cell would reappear on the opposite side [62]. In Figure 13, the

central cell (filled with grey) represents the simulation box.


30

Figure 13. Two-dimensional representation of the periodic boundary condition.

Once the temperature riches the right value, the next step is to perform the main simulation that is,

in this case, the shock loading. As it is a completely different process, the boundary conditions for

the simulation cell need to be changed. For the alumina model that will be studied in this project,

the boundary conditions that will be used are the following: PBC for axes x and y and non-periodic

boundary conditions for the z axis (shrink-wrapped s on the side where the piston impacts the

sample and fixed f on the opposite side of the piston impact) [15, 21, 63, 64].

The styles f and s mean the box is non-periodic, so that particles do not interact across the boundary

and do not move from one side of the box to the other. For style f, the position of the face is fixed.

If an atom moves outside the face it may be lost. For style s, the position of the face is set so as to

encompass the atoms in that dimension (shrink-wrapping), no matter how far they move [21].


31

4.4 Methodology The following methodology is a proposal for performing the shock loading simulations using the

Al2O3 mp-2254 (orthorhombic, Pna2_1) model that has been validated in this project. All

simulations are performed using LAMMPS [21] and OVITO [66] is used for the visualization of

the results.

The MD simulation cell of alumina is modified from the original 63-atom unit cell. This modified

specimen had an overall size of about 19.55 Å × 33.59 Å × 270.75 Å consisting of 86,672 atoms

(Figure 14). Simulations are performed with the system initially at 5 K and 300 K to evaluate the

effect of temperature in the shock loading process.

A Nosé-Hoover thermostat [65] (NVT ensemble) is used to bring the system to a temperature of 5

K or 300 K depending on the case, and a pressure thermostat (NPT ensemble) is used to bring the

system to a pressure of 1.0 bar for 5 ps, while maintaining the temperature to 5K or 300 K. For the

equilibration, fully periodic boundary conditions are implemented. Shock simulations, however,

require non-periodic boundary conditions in the shock direction (z axis in the case of this study).

Therefore, after the equilibration, the periodic boundary conditions have to be only applied to the

x and y directions.

Figure 14. Modified specimen from the original 63-atom unit cell that is used in the simulations. The structure has the following sizes: 19.55 Å x 33.59 Å x 270.75 Å with a total number of 86,672 atoms.

The relaxed system length in the z direction is 271.85 Å. Shock loading conditions are

implemented using the “fix wall/piston” command of LAMMPS through which the z axis receives

the impact of a piston that acts as a shock loading, described by a momentum mirror, which reflects


32

back any arriving atom by inverting its velocity in the z direction. The velocity of the piston is set

from up = 10.0 to up = 25.0 Å/ps (1 km/s to 2.5 km/s). Pressure, temperature and kinetic energy are

monitored during the entire impact process as well as strains in all directions of the sample. The

shock loading simulation is run in a NVE ensemble for a total of 20,000 steps at 1.0 fs interval,

which is equivalent to 20 ps.


33

5. CONCLUSIONS

In this project, a first study to understand the deformation and failure mechanisms that govern the

dynamic behavior of brittle materials using image processing algorithms in MATLAB. This

analysis measured the evolution and the speed propagation of the cracks formed in different

samples of B4C under ballistic impacts. These results matched with other experimental studies and

the propagation velocities were limited to a range of 1,800 m/s to 2,000 m/s.

The first part of the research process with alumina consisted in validating the appropriate potential

for the case of study and, once the COMB3 potential was validated and chosen as the right potential

for the shock loading simulations on alumina, the next step was to validate the model. In this part

of the project, the model Al2O3 mp-2254 (orthorhombic, Pna2_1) was chosen as the suitable model

for the study. For its validation, the elastic constants (e.g., elastic tensor, bulk modulus, shear

modulus and Poisson’s ratio) were obtained and the results matched very closely (deviations ~2.5%

in most of the cases) to the experimental results obtained with the same model in other research

studies. The cohesive energy was also calculated and the value obtained was -5.13 eV/atom, almost

equal to the values obtained by other researchers who worked with this model of Al2O3.

Additionally, lattice constants were also calculated to validate both the model and the COMB3

potential. The values obtained agreed again with other research studies. The last part of the

validation consisted in obtaining the total radial distribution function that showed that the model

studied in this project was suitable as the peaks of the curve matched well with other rdfs obtained

in previous studies.

This project has been mainly focused on the validation of the model of Al2O3 that has been used

for the simulations and the study of the state of the art of the research in advanced ceramics (e.g,

alumina and boron carbide). Therefore, it has been a previous study that will be useful for other

projects that study alumina and, specifically, its behavior under shock loadings using MD

simulations since it has validated a potential and a model that are both suitable for this kind of


34

simulations. However, the development of the project does not have to end here. The following

ideas could be tested:

• Apply the methodology proposed in this project to study the behavior of alumina under

shock loading conditions. Besides simulating at temperatures of 5 K and 300 K, these

simulations could be performed at a bigger range of temperatures to explore the influence

of temperature in the deformation behavior and amorphization on ceramics revealed in

previous studies [15]. Changing the initial conditions of the simulation to observe different

behaviors of the model after ballistic impacts could also be explored (e.g., simulations in

different temperature or pressure conditions that can mimic the behavior of the material in

a bigger range of real situations).

• Calculate deviations of the validation parameters (e.g., elastic constants, lattice constants,

cohesive energy) by changing the input conditions of the scripts to assign an uncertainty to

the results of the validation of the model.

• Computing simulations with bigger/different Al2O3 previously validated unit cells to fully

validate the COMB3 potential and to explore difficulties that may occur for other types of

Al2O3 that have not been studied in this project. MD simulations allow researchers to study

the materials in their ideal conditions, but in physical experiments it is very difficult to

obtain materials that have a single type of structure, so it is very important to validate the

most types of unit cells possible.

• Considering other planes and directions where the piston could impact the samples to

explore if the material experiences different situations depending on the way it is loaded as

it has been revealed that different type of fragmentation occur when the shock wave

propagation travels through different oriented grains [10].

• Connecting Molecular Dynamics simulation and Phase-Field modelling (PF). The PF

method requires the input of some properties (e.g., bulk properties like density and heat

capacity, kinetic coefficients and interfacial free energies) [67, 68, 69]. MD simulations

can provide a link between an interaction potential and all the required properties [67].


35

ACKNOWLEDGEMENTS

This project could not have been possible without the guidance and assistance of Dr. James D.

Hogan and all the people in the group of the Centre for Design of Advanced Materials at the

University of Alberta.

I would like to thank Dr. Sinnott of the Pennsylvania State University for providing useful

information about the COMB3 potential that was used to perform the simulations of this project.

I also thank Matthew Guziewski, of the U. S. Army Research Laboratory, and Mauricio Ponga, of

the University of British Columbia, for their advice on MD simulations and for helping me solve

doubts and obstacles that I had during the realization of the project.

Finally, I gratefully acknowledge all the people that answered my questions in the LAMMPS mail

list and those who shared with me their knowledge and helped me improve the scripts for the

Molecular Dynamics simulations.


36

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