We conducted molecular dynamics simulations to investigate the atomistic edge crack vacancy interactions in graphene. We demonstrate that the crack tip stress field of an existing crack in graphene can be effectively tailored (reduced by over 50% or increased by over 70%) by the strategic placement of atomic vacancies of varied shapes, locations, and orientations near its tip. The crack vacancy interactions result in a remarkable improvement (over 65%) in the fracture strength of graphene. Moreover, at reduced stiffness of graphene, due to a distribution of atomic vacancies, a drastic difference (~60%) was observed between the fracture strengths of two principal crack configurations (i.e. armchair and zigzag). Our numerical simulations provide a remarkable insight into the applicability of the well-established continuum models of crack microdefect interactions for the corresponding atomic scale problems. Furthermore, we demonstrate that the presence of atomic vacancies in close proximity to the crack tip leads to a multiple stage crack growth and, more interestingly, the propagating cracks can be completely healed even under a significantly high applied tensile stress level (~5 GPa). Our numerical experiments offer a substantial contribution to the existing literature on the fracture behavior of two dimensional nanomaterials.
Implementation of a tension-stiffening model for the cracking nonlinear analy...Luis Claudio Pérez Tato
The development of a smeared-crack model offers a general crack-modeling method that is independent of the structural configuration. It treats cracking as a constitutive material behaviour rather than a geometric discontinuity and lends itself well to implementation in large finite element codes. This paper deals with the implementation in XC of a constitutive model for reinforced concrete elements that takes into account the increase in stiffness of a cracked member due to the development of tensile stresses in the concrete between the cracks, effect known as tension-stiffening. The nonlinear analysis in XC of fiber-like sections with this constitutive model allows for a more general, direct and intuitive evaluation of the crack amplitude than applying the mostly specific formulae developed in the standards. The numerical results obtained by the program compare extremely well with existing designing results issued by other applied methods.
RESULTS OF FINITE ELEMENT ANALYSIS FOR INTERLAMINAR FRACTURE REINFORCED THERM...MSEJjournal1
The double cantilever beam (DCB) is widely used for fracture toughness testing and it has become popular
for opening-mode (mode I) delamination testing of laminated composites. Delamination is a crack that
forms between the adjacent plies of a composite laminate at the brittle polymer resin. This study was
conducted to emphasize the need for a better understanding of the DCB specimen of different fabric
reinforced systems (carbon fibers) with a thermoplastic matrix (EP, PEI), by using the extended finite
element method (X-FEM). It is well known that in fabric reinforced composites fracture mechanisms
include microcracking in front of the crack tip, fiber bridging and multiple cracking, and both contribute
considerably to the high interlaminar fracture toughness measured. That means, the interlaminar fracture
toughness of a composite is not controlled by a single material parameter, but is a result of a complex
interaction of resin, fiber and interface properties.
Implementation of 1D Brinson model in 3D SMA specimens Influenced by stress c...irjes
The application of small-size SMA components has grown rapidly in various industries over the last
few years. Due to their complex nature, local constitutive models for those components have proven somehow
impractical; whereas global models have been unable to take the stress concentration into consideration and
hence cannot predict the nucleation and stress patterns accurately, especially in small samples. Moreover, FE
analyses are impossible to conduct in cases where a large number of evaluations are required to design a suitable
SMA sample. This paper proposes a new iterative method to accurately calculate the stress and martensitic
volume fraction in samples with stress concentration using 1D Brinson model. The method is implemented in
the form of a computer code which is then refined iteratively using FEM model which is validated based on
experimental data. Final assessments show that the proposed method could predict the stress values accurately
at negligible computational costs.
Implementation of a tension-stiffening model for the cracking nonlinear analy...Luis Claudio Pérez Tato
The development of a smeared-crack model offers a general crack-modeling method that is independent of the structural configuration. It treats cracking as a constitutive material behaviour rather than a geometric discontinuity and lends itself well to implementation in large finite element codes. This paper deals with the implementation in XC of a constitutive model for reinforced concrete elements that takes into account the increase in stiffness of a cracked member due to the development of tensile stresses in the concrete between the cracks, effect known as tension-stiffening. The nonlinear analysis in XC of fiber-like sections with this constitutive model allows for a more general, direct and intuitive evaluation of the crack amplitude than applying the mostly specific formulae developed in the standards. The numerical results obtained by the program compare extremely well with existing designing results issued by other applied methods.
RESULTS OF FINITE ELEMENT ANALYSIS FOR INTERLAMINAR FRACTURE REINFORCED THERM...MSEJjournal1
The double cantilever beam (DCB) is widely used for fracture toughness testing and it has become popular
for opening-mode (mode I) delamination testing of laminated composites. Delamination is a crack that
forms between the adjacent plies of a composite laminate at the brittle polymer resin. This study was
conducted to emphasize the need for a better understanding of the DCB specimen of different fabric
reinforced systems (carbon fibers) with a thermoplastic matrix (EP, PEI), by using the extended finite
element method (X-FEM). It is well known that in fabric reinforced composites fracture mechanisms
include microcracking in front of the crack tip, fiber bridging and multiple cracking, and both contribute
considerably to the high interlaminar fracture toughness measured. That means, the interlaminar fracture
toughness of a composite is not controlled by a single material parameter, but is a result of a complex
interaction of resin, fiber and interface properties.
Implementation of 1D Brinson model in 3D SMA specimens Influenced by stress c...irjes
The application of small-size SMA components has grown rapidly in various industries over the last
few years. Due to their complex nature, local constitutive models for those components have proven somehow
impractical; whereas global models have been unable to take the stress concentration into consideration and
hence cannot predict the nucleation and stress patterns accurately, especially in small samples. Moreover, FE
analyses are impossible to conduct in cases where a large number of evaluations are required to design a suitable
SMA sample. This paper proposes a new iterative method to accurately calculate the stress and martensitic
volume fraction in samples with stress concentration using 1D Brinson model. The method is implemented in
the form of a computer code which is then refined iteratively using FEM model which is validated based on
experimental data. Final assessments show that the proposed method could predict the stress values accurately
at negligible computational costs.
Examples of Homogenization Techniques in the Material Modeling Under the Effe...Sardasht S. Weli
Explosions become a very attractive research area in the last decades.
This is due to the increase of accidental and intentional explosions.
Historical structure were not designed and built against the extreme loading events.
Homogenization Techniques were developed to assess the masonry response.
Properties of the cake layer in the ultrafiltration of polydisperse colloidal...eSAT Journals
Abstract The properties of the cake layer formed during the ultrafiltration of polydisperse silica colloid using the Karman Cozeny equation have been studied and are compared to the experimental results obtained from SEM and AFM analysis. AFM analysis is used to give a direct comparison to SEM analysis in order to give a better understanding of the cake properties formed at different compositions and different pH’s. The trend of the filtration rate of binary mixtures of silica particles (W30 and X30) were analysed and correlated to the results of the specific resistance obtained from SEM analysis. The study involves analysing the fouled membrane from the top view and from the cross sectional view. This present work has found that results from the SEM and AFM analysis corresponded to each other in terms of particle arrangement and the average cake thickness. Keywords: polydisperse; cake resistance, colloidal -interaction
Vibrational Analysis Of Cracked Rod Having Circumferential Crack IDES Editor
The frequency ratio of torsional vibration of a rod without crack and of rod with crack subjected to torque at the free end for various crack depth and varying crack location is investigated. It is found that even a cracked of small depth is
dangerous at the fixed end, also as the crack depth is increases more than 50% of diameter of rod there is a considerable drop in natural frequency of the rod .
Numerical modeling of the welding defect influence on fatigue life of the wel...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
RESULTS OF FINITE ELEMENT ANALYSIS FOR INTERLAMINAR FRACTURE REINFORCED THERM...msejjournal
The double cantilever beam (DCB) is widely used for fracture toughness testing and it has become popular
for opening-mode (mode I) delamination testing of laminated composites. Delamination is a crack that
forms between the adjacent plies of a composite laminate at the brittle polymer resin. This study was
conducted to emphasize the need for a better understanding of the DCB specimen of different fabric
reinforced systems (carbon fibers) with a thermoplastic matrix (EP, PEI), by using the extended finite
element method (X-FEM). It is well known that in fabric reinforced composites fracture mechanisms
include microcracking in front of the crack tip, fiber bridging and multiple cracking, and both contribute
considerably to the high interlaminar fracture toughness measured. That means, the interlaminar fracture
toughness of a composite is not controlled by a single material parameter, but is a result of a complex
interaction of resin, fiber and interface properties.
RESULTS OF FINITE ELEMENT ANALYSIS FOR INTERLAMINAR FRACTURE REINFORCED THERM...msejjournal
The double cantilever beam (DCB) is widely used for fracture toughness testing and it has become popular
for opening-mode (mode I) delamination testing of laminated composites. Delamination is a crack that
forms between the adjacent plies of a composite laminate at the brittle polymer resin. This study was
conducted to emphasize the need for a better understanding of the DCB specimen of different fabric
reinforced systems (carbon fibers) with a thermoplastic matrix (EP, PEI), by using the extended finite
element method (X-FEM). It is well known that in fabric reinforced composites fracture mechanisms
include microcracking in front of the crack tip, fiber bridging and multiple cracking, and both contribute
considerably to the high interlaminar fracture toughness measured. That means, the interlaminar fracture
toughness of a composite is not controlled by a single material parameter, but is a result of a complex
interaction of resin, fiber and interface properties.
RESULTS OF FINITE ELEMENT ANALYSIS FOR INTERLAMINAR FRACTURE REINFORCED THERM...msejjournal
The double cantilever beam (DCB) is widely used for fracture toughness testing and it has become popular
for opening-mode (mode I) delamination testing of laminated composites. Delamination is a crack that
forms between the adjacent plies of a composite laminate at the brittle polymer resin. This study was
conducted to emphasize the need for a better understanding of the DCB specimen of different fabric
reinforced systems (carbon fibers) with a thermoplastic matrix (EP, PEI), by using the extended finite
element method (X-FEM). It is well known that in fabric reinforced composites fracture mechanisms
include microcracking in front of the crack tip, fiber bridging and multiple cracking, and both contribute
considerably to the high interlaminar fracture toughness measured. That means, the interlaminar fracture
toughness of a composite is not controlled by a single material parameter, but is a result of a complex
interaction of resin, fiber and interface properties.
Examples of Homogenization Techniques in the Material Modeling Under the Effe...Sardasht S. Weli
Explosions become a very attractive research area in the last decades.
This is due to the increase of accidental and intentional explosions.
Historical structure were not designed and built against the extreme loading events.
Homogenization Techniques were developed to assess the masonry response.
Properties of the cake layer in the ultrafiltration of polydisperse colloidal...eSAT Journals
Abstract The properties of the cake layer formed during the ultrafiltration of polydisperse silica colloid using the Karman Cozeny equation have been studied and are compared to the experimental results obtained from SEM and AFM analysis. AFM analysis is used to give a direct comparison to SEM analysis in order to give a better understanding of the cake properties formed at different compositions and different pH’s. The trend of the filtration rate of binary mixtures of silica particles (W30 and X30) were analysed and correlated to the results of the specific resistance obtained from SEM analysis. The study involves analysing the fouled membrane from the top view and from the cross sectional view. This present work has found that results from the SEM and AFM analysis corresponded to each other in terms of particle arrangement and the average cake thickness. Keywords: polydisperse; cake resistance, colloidal -interaction
Vibrational Analysis Of Cracked Rod Having Circumferential Crack IDES Editor
The frequency ratio of torsional vibration of a rod without crack and of rod with crack subjected to torque at the free end for various crack depth and varying crack location is investigated. It is found that even a cracked of small depth is
dangerous at the fixed end, also as the crack depth is increases more than 50% of diameter of rod there is a considerable drop in natural frequency of the rod .
Numerical modeling of the welding defect influence on fatigue life of the wel...inventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
RESULTS OF FINITE ELEMENT ANALYSIS FOR INTERLAMINAR FRACTURE REINFORCED THERM...msejjournal
The double cantilever beam (DCB) is widely used for fracture toughness testing and it has become popular
for opening-mode (mode I) delamination testing of laminated composites. Delamination is a crack that
forms between the adjacent plies of a composite laminate at the brittle polymer resin. This study was
conducted to emphasize the need for a better understanding of the DCB specimen of different fabric
reinforced systems (carbon fibers) with a thermoplastic matrix (EP, PEI), by using the extended finite
element method (X-FEM). It is well known that in fabric reinforced composites fracture mechanisms
include microcracking in front of the crack tip, fiber bridging and multiple cracking, and both contribute
considerably to the high interlaminar fracture toughness measured. That means, the interlaminar fracture
toughness of a composite is not controlled by a single material parameter, but is a result of a complex
interaction of resin, fiber and interface properties.
RESULTS OF FINITE ELEMENT ANALYSIS FOR INTERLAMINAR FRACTURE REINFORCED THERM...msejjournal
The double cantilever beam (DCB) is widely used for fracture toughness testing and it has become popular
for opening-mode (mode I) delamination testing of laminated composites. Delamination is a crack that
forms between the adjacent plies of a composite laminate at the brittle polymer resin. This study was
conducted to emphasize the need for a better understanding of the DCB specimen of different fabric
reinforced systems (carbon fibers) with a thermoplastic matrix (EP, PEI), by using the extended finite
element method (X-FEM). It is well known that in fabric reinforced composites fracture mechanisms
include microcracking in front of the crack tip, fiber bridging and multiple cracking, and both contribute
considerably to the high interlaminar fracture toughness measured. That means, the interlaminar fracture
toughness of a composite is not controlled by a single material parameter, but is a result of a complex
interaction of resin, fiber and interface properties.
RESULTS OF FINITE ELEMENT ANALYSIS FOR INTERLAMINAR FRACTURE REINFORCED THERM...msejjournal
The double cantilever beam (DCB) is widely used for fracture toughness testing and it has become popular
for opening-mode (mode I) delamination testing of laminated composites. Delamination is a crack that
forms between the adjacent plies of a composite laminate at the brittle polymer resin. This study was
conducted to emphasize the need for a better understanding of the DCB specimen of different fabric
reinforced systems (carbon fibers) with a thermoplastic matrix (EP, PEI), by using the extended finite
element method (X-FEM). It is well known that in fabric reinforced composites fracture mechanisms
include microcracking in front of the crack tip, fiber bridging and multiple cracking, and both contribute
considerably to the high interlaminar fracture toughness measured. That means, the interlaminar fracture
toughness of a composite is not controlled by a single material parameter, but is a result of a complex
interaction of resin, fiber and interface properties.
RESULTS OF FINITE ELEMENT ANALYSIS FOR INTERLAMINAR FRACTURE REINFORCED THERM...msejjournal
The double cantilever beam (DCB) is widely used for fracture toughness testing and it has become popular
for opening-mode (mode I) delamination testing of laminated composites. Delamination is a crack that
forms between the adjacent plies of a composite laminate at the brittle polymer resin. This study was
conducted to emphasize the need for a better understanding of the DCB specimen of different fabric
reinforced systems (carbon fibers) with a thermoplastic matrix (EP, PEI), by using the extended finite
element method (X-FEM). It is well known that in fabric reinforced composites fracture mechanisms
include microcracking in front of the crack tip, fiber bridging and multiple cracking, and both contribute
considerably to the high interlaminar fracture toughness measured. That means, the interlaminar fracture
toughness of a composite is not controlled by a single material parameter, but is a result of a complex
interaction of resin, fiber and interface properties.
Structural Changes in the Surface Layer of Deep Rolled Samples Due to Thermal...IJERA Editor
Deep rolling processes initiate plastic deformations in the surface layer. The local characteristics of deformation are dependent on the induced stress expressed by the local stress tensor. Equivalent stresses above yield strength cause plastic deformation. Additionally the intrinsic energy, e. g. the dislocation density, is enhanced and the residual stress state is changed. The effects to a deep rolled surface from an increase in temperature are mainly dependent on the material, the microstructure, the initial residual stress state, the inclusion density, the distribution of soluted alloying elements and the plastic deformation. In the described experiments the interactions between deformation and temperature of the steel grade AISI 4140 (42 CrMo 4) used for all further experiments in a transregional Collaborative Research Center (CRC) were to be examined. The most simple investigation methods were chosen deliberately to allow a better statistical support of correlations between introduced strains and material reactions for a wide variation of process parameters. Since the visual effects by light microscopy in AISI 4140 were very small, the experiments were repeated with german grade 18 CrNiMo 7-6 (comparable to AISI 4820). This paper focuses on the micro structural changes in defined deep rolled surface regions due to an increase in temperature. The work described is part of the Collaborative Research Center “Process Signatures”, collaboration between Bremen University, Technical University Aachen, Germany and Oklahoma State University Stillwater, USA.
Influence of contact friction conditions on thin profile simulationVan Canh Nguyen
The paper presents the development of the Finite Element model for simulation of thin
aluminium profile extrusion of both solid and hollow shapes. The analysis has shown that the material
flow in simulation is very dependent on the friction model. Experimental and theoretical studies show
that friction traction on the interface between the tool and the deformed material can be represented as
a combination of adhesive friction force and the force that is required to deform surface asperities. In
aluminium extrusion we can clearly distinguish two different areas with respect to friction conditions
such as sticking and sliding and transient zones between them. The lengths of these zones are also
dependent on variation of the choke angle and actual thickness of the profile. To get these values the
material flow problem is to be coupled with the simulation of the tools deformation. A series of
experiments with specially designed tools have been done to investigate how the bearing length and
choke angle may influence the extension of different friction zones and by these means vary the
material flow pattern. The friction models have also been tested with industrial profiles of complex
shapes and have shown good correspondence to reality.
When a ductile material with a crack is loaded in
tension, the deformation energy builds up around the crack tip
and it is understood that at a certain critical condition voids are
formed ahead of the crack tip. The crack extension occurs by
coalescence of voids with the crack tip. The “characteristic
distance” (Lc) defined as the distance b/w the crack tip & the void
responsible for eventual coalescence with the crack tip. Nucleation
of these voids is generally associated with the presence of second
phase particles or grain boundaries in the vicinity of the crack tip.
Although approximate, Lc assumes a special significance since it
links the fracture toughness to the microscopic mechanism
considered responsible for ductile fracture. The knowledge of the
“characteristic distance” is also crucial for designing the size of
mesh in the finite element simulations of material crack growth
using damage mechanics principles. There is not much work
(experimental as well as numerical) available in the literature
related to the dependency of “characteristic distance” on the
fracture specimen geometry. The present research work is an
attempt to understand numerically, the geometry dependency of
“characteristic distance” using three-dimensional FEM analysis.
The variation of “characteristic distance” parameter due to the
change of temperature across the fracture specimen thickness was
also studied. The work also studied the variation of “characteristic
distance”, due to the change in fracture specimen thickness.
Finally, the ASTM requirement of fracture specimen thickness
criteria is evaluated for the “characteristic distance” fracture
parameter. “Characteristic distance” is found to vary across the
fracture specimen thickness. It is dependent on fracture specimen
thickness and it converges after a specified thickness of fracture
specimen. “Characteristic distance” value is also dependent on the
temperature of ductile material. In Armco iron material, it is
found to decrease with the increase in temperature.
The determination of the stress distribution around Pin-Loaded Holes in Mechanically Fastened Joints is presented to the Engineering Department, University of Technology - Jamaica
Fracture mechanics based estimation of fatigue life of weldsAvinash B
This paper presents a mechanism based approach for lifetime prediction of welded joints, subjected to a multiaxial nonproportional
loading. Assuming the existence of crack-like flaws after the welding process, the stage of a fatigue crack
initiation becomes insignificant and the threshold for the initial crack propagation can be taken as a criterion for very
high cycle fatigue (VHCF) whereas crack growth analysis can be used for low and high cycle fatigue (LCF, HCF). The
proposed deterministic method, which is based on the welding process simulation, thermophysical material modeling
and fracture mechanics, considers the most important aspects for fatigue of welds. Applying worst case assumptions,
fatigue limits derived by this method can be then used for the fatigue assessment of complex welded structures. The
capability of the approach is validated by S-N curves provided at the Recommendations of the IIW.
Pullout Behavior of Geotextiles: Numerical PredictionIJERA Editor
The mechanism of soil-geosynthetic interaction is usually complex and depends on the nature of the reinforcement, as well as on the characteristics of the surrounding soil. The strength parameters of the interface are the key for the design of reinforced soil slopes. Usually, these parameters are defined from laboratory pullout tests. The absence of test results implies on conservative assumptions and higher costs. The possibility of using computer programs for analyzing the load transfer mechanism arises as an attractive design tool. This paper presents the numerical simulation of pullout tests, conducted in large equipments. The numerical predictions of the load and displacement distribution along the geosynthetic length were compared to instrumented test results, available in the literature. The analyses revealed to be satisfactory and consistent with the experimental results. Thus, it becomes possible to reduce the uncertainties in the design of the anchorage length for the reinforcement by previously performing studies with computer programs that simulates stress x strain behavior of geotechnical problems.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Similar to Tailoring fracture strength of graphene (20)
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
TOP 10 B TECH COLLEGES IN JAIPUR 2024.pptxnikitacareer3
Looking for the best engineering colleges in Jaipur for 2024?
Check out our list of the top 10 B.Tech colleges to help you make the right choice for your future career!
1) MNIT
2) MANIPAL UNIV
3) LNMIIT
4) NIMS UNIV
5) JECRC
6) VIVEKANANDA GLOBAL UNIV
7) BIT JAIPUR
8) APEX UNIV
9) AMITY UNIV.
10) JNU
TO KNOW MORE ABOUT COLLEGES, FEES AND PLACEMENT, WATCH THE FULL VIDEO GIVEN BELOW ON "TOP 10 B TECH COLLEGES IN JAIPUR"
https://www.youtube.com/watch?v=vSNje0MBh7g
VISIT CAREER MANTRA PORTAL TO KNOW MORE ABOUT COLLEGES/UNIVERSITITES in Jaipur:
https://careermantra.net/colleges/3378/Jaipur/b-tech
Get all the information you need to plan your next steps in your medical career with Career Mantra!
https://careermantra.net/
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
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ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
1. Tailoring Fracture Strength of Graphene
M. A. N. Dewapriya and S. A. Meguid
Mechanics and Aerospace Design Laboratory, Department of Mechanical and Industrial
Engineering, University of Toronto, Toronto, Ontario M5S 3G8, Canada
Abstract
We conducted molecular dynamics simulations to investigate the atomistic edge crack-vacancy
interactions in graphene. We demonstrate that the crack-tip stress field of an existing crack in
graphene can be effectively tailored (reduced by over 50% or increased by over 70%) by the
strategic placement of atomic vacancies of varied shapes, locations, and orientations near its tip.
The crack-vacancy interactions result in a remarkable improvement (over 65%) in the fracture
strength of graphene. Moreover, at reduced stiffness of graphene, due to a distribution of atomic
vacancies, a drastic difference (~60%) was observed between the fracture strengths of two
principal crack configurations (i.e. armchair and zigzag). Our numerical simulations provide a
remarkable insight into the applicability of the well-established continuum models of
crack-microdefect interactions for the corresponding atomic scale problems. Furthermore, we
demonstrate that the presence of atomic vacancies in close proximity to the crack-tip leads to a
multiple-stage crack growth and, more interestingly, the propagating cracks can be completely
healed even under a significantly high applied tensile stress level (~5 GPa). Our numerical
experiments offer a substantial contribution to the existing literature on the fracture behavior of
two-dimensional nanomaterials.
Keywords: Crack-tip stress field, edge crack, fracture of graphene, crack healing, crack-vacancy
interaction, tailoring of fracture properties
2. 2
1. Introduction
The extraordinary stiffness, fracture strength, and flexibility of graphene [1], a two-dimensional
(2D) hexagonal lattice of sp2
bonded carbon, demonstrate great potentials in a rich variety of
applications such as flexible electronics [2–4], nanoelectromechanical systems [5,6], and
nanocomposites [7–9]. Even though an atomically perfect graphene sample has an extraordinary
fracture strength of 130 GPa [1], pristine graphene samples of large area are yet to be developed
for the proposed applications. Even with a pristine graphene sample, atomic flaws such as
vacancies and adatoms are difficult to avoid in real engineering fabrication of graphene-based
nanodevices [5,10–15]. A recent experiment revealed that the fracture toughness of graphene is
~4 MPa√m [16], which is significantly lower than conventional engineering materials [17]. This
relatively small fracture toughness poses serious limitations in the use of graphene for numerous
structural applications [18]. Therefore, it is important to study the effect of the presence of and
interactions between atomistic cracks and defects upon the fracture behavior of graphene. In
addition, investigating potential toughening mechanisms for graphene is highly desirable in its
commercialization for real life engineering applications.
In continuum fracture mechanics, it has been well established that the interaction between
crack and a micro defect in close proximity plays an important role in the overall failure
mechanism of quasi-brittle materials [19–25]. It has also been demonstrated that the crack-tip
stress field in a linear elastic continuum can be controlled by strategically placing a microdefect
near the crack-tip [26,27]. Recent nanoindentation tests of graphene containing vacancies have
revealed that the catastrophic failure of graphene can be transformed into a local failure by
controlling its defect concentration [28]. This observation suggests that existing cracks in
graphene can be arrested by introducing a controlled distribution of topological defects, which
has been realized for some other nanomaterials [29,30].
Most of the recent efforts have been focused on enhancing the fracture strength of
graphene by introducing topological defects such as pentagon-heptagons [31–35] and grain
boundaries [36] without paying attention to the complex stress state surrounding crack-defect
interactions, which could unveil rich atomistic mechanisms of 2D materials. In fact, many
existing molecular dynamics (MD) studies on fracture properties of graphene have focused on
the central crack problem partly due to the convenience of using periodic boundary conditions in
3. 3
MD. In addition, advanced continuum-based design tools for characterizing crack-defect
interaction [19–27] have not been tested at the atomic scale, which is critically important
considering the limited applicability of continuum concepts at the nanoscale [37–43]. If
applicable, the continuum tools such as design envelopes to ascertain crack-tip stress shielding
and amplification zones due to the crack-defect interactions [26,27] can be very useful for
nanoscale design of 2D materials [44].
In this work, we focus our attention on exploring the influence of interactions between
existing edge nanocracks in graphene and arbitrarily located atomic vacancies at the crack-tip.
Our atomistic simulations demonstrate that vacancies of varied shapes, locations, and
orientations can be effectively used to shield the existing cracks from high stress levels leading
to crack arrest. Moreover, our numerical uniaxial tensile tests reveal that the crack-vacancy
interactions in graphene lead to a multiple-stage crack growth and crack healing under
remarkably high applied tensile stress levels (~5 GPa), providing a new route for strain
engineering of graphene-based devices.
2. Molecular dynamics simulations
We conducted numerical uniaxial tensile tests of graphene samples using LAMMPS MD
simulator [45] with AIREBO potential [46], where the cutoff distance was modified to be 2 Å
[32,34,47,48]. At the beginning of all MD simulations, energy of the graphene samples was
minimized using the conjugate gradient algorithm. Then, before applying strain, the samples
were allowed to relax over 25 ps and the time step was selected to be 0.5 fs. Canonical (NVT)
ensemble was used for the simulations, where temperature was maintained at 300 K using
Nośe-Hoover thermostat. An initial random displacement perturbation (maximum of 0.01 Å)
along x, y, and z directions was imposed on carbon atoms to facilitate reaching the equilibrium
configuration. In fact, inducing out-of-plane displacement perturbations is essential in these
simulations because the Nośe-Hoover thermostat induces artificial thermal expansion in the
absence of out-of-plane deformation [49]. After the graphene samples reached equilibrium, strain
was applied to the samples along the y-direction (εyy) at a rate of 0.001 ps-1
. The planar
dimensions of the simulated samples were selected to be 60 nm × 60 nm, and the length of edge
crack was 10 nm. The selected simulation domain ensures that its boundaries do not fall within
the K-dominant region which is at and near the crack-tip region, where the stress field decays
4. 4
quite rapidly. Virial stress [50] was used for fracture characterization. Visual Molecular
Dynamics package [51] was used to visualize deformations and fracture of graphene.
Figure 1 shows a typical MD simulation sample of graphene containing an atomic vacancy
interacting with an edge crack. The origins of the two rectangular coordinate systems xy and x0y0
are taken at the tip of the edge crack and at the center of the vacancy, respectively. The
orientation angle of the vacancy is ϕ, and its major and minor diameters are 2c and 2b,
respectively. The distance between the tip of the crack and the center of the vacancy is taken to
be r. The inclination angle between the x-axis and the line joining the tip of the crack and the
center of the vacancy is θ. The value of 2c is selected to be 3.6 nm (see figure 1).
Figure 1 A typical MD simulation sample of graphene containing an edge crack and an
interacting atomic vacancy.
3. Results and discussion
3.1 Crack tip stress distribution
The stress distribution of individual carbon atoms at the crack-tip can be very informative in
characterizing the fracture behavior of graphene. In order to obtain the time averaged stress of
atoms at the incipient crack propagation, we conducted two sequential MD simulations as
5. 5
outlined in one of our earlier publications [52]. Figures 2a and 2b show the stress distributions at
the armchair and zigzag crack-tips, respectively. These stress distributions resemble the ones
predicted by the continuum linear elastic fracture mechanics (LEFM) [40]. The peak stress at the
tip of the armchair crack at the incipient crack propagation is 17% higher than the corresponding
stress of the zigzag crack. This is due to the high far field (or applied) stress σ0 level of the
armchair crack configuration and, more importantly, the different bond arrangements at the
crack-tips. In contrast to the isolated bond perpendicular to the crack at the zigzag crack-tip (see
inset of figure 2a), the two inclined bonds at the armchair crack-tip (inset of figure 2b)
accommodate part of the applied tensile strain by adjusting the bond angles, which allows the
atom at the crack-tip to carry a higher strain prior crack growth leading to a higher atomic stress.
According to LEFM, the critical stress intensity factor (SIF) of a single edge-cracked sample
under mode-I loading (where the crack surfaces are displaced directly apart) KIC can be defined
as follows [17]:
aK πσfIC 12.1= (1)
where a is the initial crack length, and σf is the fracture stress, i.e. the critical far field stress at
the incipient crack propagation. The computed KIC for armchair and zigzag cracks are 4.04 and
3.97 MPa√m, respectively, which are in excellent agreement with the experimentally measured
value 4 MPa√m [16].
Earlier, Gong and Meguid studied the interaction between a semi-infinite crack and an
elliptical vacancy located near its tip (see figure 1) under mode I loading [26]. In the absence of
the vacancy, the singular stress field near the crack-tip can be described by using the
corresponding SIF aK πσ0I 12.1= . However, the presence of the elliptical vacancy in close
proximity to the crack tip influences the crack-tip stress field and leads to a modified SIF; we
will call v)-(c
IK . When a collinear elliptical vacancy is located ahead of the crack, i.e. θ = 0 and
ϕ = 0, the solution for the normalized SIF under mode I loading I
v)-(c
I KK can be explicitly
expressed up to the order (c/r)4
as follows [26]:
6. 6
( ) ( ) ...49124623
128
1
1
4
1
1
4
432
22
2
2
I
v)-(c
I
+⎟
⎠
⎞
⎜
⎝
⎛
−++⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
+=
r
c
r
c
K
K
ββββ (2)
where β is b/c. For the case of a circular vacancy (i.e. β = 1), equation (2) reduces to
...
4
1
2
1
1
42
I
v)-(c
I
+⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+=
r
c
r
c
K
K
(3)
Considering the leading order solution up to order (c/r)2
, a general solution for any combination
of r, θ, and ϕ can be given as,
( ) ( )C
r
c
G
r
c
K
K 2
2
2
2
I
v)-(c
I
1
2
1
8
1 ββ +⎟
⎠
⎞
⎜
⎝
⎛
+−⎟
⎠
⎞
⎜
⎝
⎛
+= (4)
where C and G are explicitly expressed as follows:
(5a)
(5b)
In order to characterize the crack-vacancy interaction, we use normalized crack-tip stress
c
tip
v)-(c
tip σσ , where v)-(c
tipσ is the crack-tip stress along the y-direction in the presence of an
interacting vacancy and c
tipσ is the crack-tip stress in the absence of any interacting vacancy. The
normalized crack-tip stress c
tip
v)-(c
tip σσ was computed at an applied tensile strain εyy level of 1% for
various arrangements of the interacting vacancies. The values of c
tipσ for the armchair and zigzag
cracks are 63.9 and 56 GPa, respectively. Figures 2c-2g demonstrate that the presence of
vacancies greatly influences the stress field of zigzag crack compared to that of the armchair
crack, as a result of their underlying crystal structures at the crack-tips. It can be seen in Figures
2c and 2f that the collinear (i.e. θ = ϕ = 0) nanocracks and circular vacancies result in an increase
in the crack-tip stress field (known as stress amplification effect, i.e. c
tip
v)-(c
tip σσ > 1). The oriented
vacancies (see figures 2d and 2g) with the orientation angle θ > 60°, result in a decrease in the
crack-tip stress field (known as stress shielding effect, i.e. c
tip
v)-(c
tip σσ < 1). The stress shielding
C = cos
3θ
2
⎛
⎝
⎜
⎞
⎠
⎟cos
θ
2
⎛
⎝
⎜
⎞
⎠
⎟
G = 2cos 2φ + θ( ) + 4cos 2φ − θ( ) + 8cos 2φ − 2θ( ) − 6cos 2φ − 3θ( )
− 8cos 2φ − 4θ( ) − 3cos 3θ( ) + 3cos θ( )
7. 7
and stress amplification effects are due to the fact that the presence of vacancies leads to a
change in the density of the stress trajectories at the crack-tip. Increased and decreased densities
of the stress trajectories correspond to the amplification and shielding effects, respectively. The
maximum shielding effect that can be achieved by means of the studied arrangements of
vacancies is 55%, which is in the case of the zigzag crack interacting with a circular vacancy
located at r = 2.8 nm and θ = 118°. Figure 2e shows that the amplification effect due to a
nanocrack can be successfully tailored (~20%) by regulating the obliquity of the nanocrack.
More remarkably, Figures 2c-2g show that the continuum-based analytical solutions given in
equation (3) and (4) are able to accurately capture the trends of the crack-tip stress fields
obtained from the atomistic simulations. Here, we compare I
v)-(c
I KK given by the analytical
solutions with c
tip
v)-(c
tip σσ obtained from the atomistic simulations. The normalized crack-tip
stress c
tip
v)-(c
tip σσ is a comparable quantity to the corresponding normalized stress intensity factors
[17]. It should be noted that the higher order terms of equation (3) and (4) have been neglected,
which limits their accuracy. As demonstrated in the Supporting Information (figure S1), the
accuracy improves when (c/r) becomes smaller. However, the continuum expressions are unable
to predict the influence of the underlying crystal structures (i.e. armchair and zigzag) on the
crack-tip stress field, which sets a limit on developing a unified continuum fracture mechanics
framework at the atomic scale. However, due to the remarkable accuracy and the extremely high
computational efficiency, the analytical solutions can be employed to create detailed design
envelopes to ascertain the crack-tip shielding and amplification zones as a result of the presence
of atomic vacancies ahead of the crack-tip in graphene (see figure S2 in the Supporting
Information).
8. 8
Figure 2 The effect of atomic vacancies on the crack-tip stress field. Figures (a) and (b) show
averaged stress σyy distributions at the tips of armchair and zigzag cracks at the incipient crack
propagation, respectively. Figures (c-g) show variation of the normalized crack-tip stress
c
tip
v)-(c
tip σσ with r, θ, and ϕ. Figures (c), (d), and (e) are for the collinear (θ = ϕ = 0), oriented (r is
fixed and ϕ = 0), and oblique (r and θ are fixed) nanocracks, respectively. Figures (f) and (g) are
for the collinear and oriented circular vacancies, respectively. Insets in Figures (c-g) show the
stress σyy distribution at an armchair crack-tip due to an applied tensile strain εyy of 1% in the
presence of an atomic vacancy at the specified location.
9. 9
3.2. Crack arrest
We further characterized the atomistic crack-vacancy interaction by computing the critical SIF
for the crack-vacancy arrangements considered in figure 2. We defined the normalized critical
SIF IC
v)-(c
C KK , where v)-(c
CK is the critical mode I SIF of a sample containing an edge crack
interacting with a vacancy, which is defined as
aK πσ v)-(c
f
v)-(c
C 12.1= (6)
where v)-(c
fσ is the fracture stress of the sample. It should be noted that equation (6) is analogous
to equation (1), and v)-(c
fσ obtained from the MD simulations takes into account the influence of
atomic vacancies.
In spite of the fact that vacancies have a greater influence on the crack-tip stress field of the
zigzag crack than the armchair crack (see Figures 2c-2g), Figures 3a-3e show that the vacancies
have a comparatively higher influence on v)-(c
CK of armchair crack compared to the zigzag crack.
This observation is due to the fact that the vacancies are mostly located along the propagation
path of the armchair crack, which results in a crack-vacancy coalescence leading to possible
crack arrest. Conversely, the vacancies are not aligned along the propagation path of the zigzag
crack except for the collinear vacancies. Figures 3d and 3e show that the presence of circular
vacancies at both the armchair and zigzag cracks leading to a remarkable crack arrest, i.e.
IC
v)-(c
C KK > 1, for almost all of the studied vacancy arrangements. Specifically, our results reveal
that the presence of strategically located holes in close proximity to the crack tip can enhance the
fracture strength of armchair and zigzag cracks by 56% and 67%, respectively. Moreover, a
two-stage crack propagation is observed in both the zigzag and armchair cracks in the presence
of atomic vacancies as shown in Figures 3f and 3g, respectively.
10. 10
Figure 3 Fracture characterization of the crack-vacancy systems considered. Figures (a-e) depict
the variation of the normalized critical SIF with r, θ, and ϕ. Figures (a), (b), and (c) are for the
collinear, oriented, and oblique nanocracks, respectively. Figures (f) and (g) are for the collinear
and oriented circular vacancies, respectively. Insets in Figures (a-e) depict the relative position of
the corresponding atomic vacancy with respect to the crack-tip. Figures (f) and (g) show the
stress σyy distribution and the crack propagation of graphene containing armchair and zigzag
cracks, respectively.
11. 11
3.3. Crack healing
Self-healing of cracks and other defects in graphene [53–55] provide a potential route for
developing higher strength and greater resistance to fracture. Our numerical experiments reveal
that the partially propagated cracks (see Figures 3f and 3g) can heal completely though it is
under applied tensile stresses that can be as high as 5 GPa. Figure 4a shows the variation of the
spatially averaged stress of the process zone (an area of 2 nm × 2 nm ahead of the crack-tip
containing 170 atoms) during the loading and unloading of a graphene sample containing an
armchair crack and an interacting circular vacancy. The loading and unloading were carried out a
strain rate of 0.001 ps-1
. The sudden decrease in the stiffness at the end of region I is due to a
partial crack propagation, and it continues to propagate in the region II (see Figures 4c and 4d).
The unloading was initiated at time T = 33 ps. The stiffness of the sample in region III is reduced
by 23% compared to the stiffness in region I due to the partial crack propagation. Crack healing
initiates at the beginning of region IV, where the number of broken bonds decreases from six to
five (see Figures 4e and 4f), which slightly increases the effective stiffness of the process zone. It
should be noted that the initial crack healing occurs under σ0 of 10.5 GPa, and the partially
propagated crack has completely healed when σ0 is reduced to 5 GPa during the subsequent
unloading (see figure 4g). As shown in figure S3 of Supporting Information, crack healing was
observed even under continuous unloading.
12. 12
Figure 4 Healing of a partially propagating armchair crack in the presence of an interacting
circular vacancy located at r = 2.9 nm and θ = 88°. (a) Variation of the spatially averaged stress
σyy of the process zone during the loading and unloading of graphene. The unloading was paused
at time T = 54 ps and then the sample was equilibrated for 15 ps (region IV). The far field stress
σ0 during the equilibration was 10.5 GPa. Subsequent unloading was carried out for 6 ps
(region V) and the sample was then equilibrated for another 15 ps (region VI). Figures (b-d)
depict snapshots of the crack-tip during the loading stage. The arrows demonstrate the broken
bonds at the crack surface due to the partial crack propagation. Figures (e-g) depict the stress σyy
distributions at the corresponding points during the unloading stage.
13. 13
3.4. The effects of an extended distribution of vacancies
Finally, we evaluated the influence of an array of vacancies on the crack-tip stress field and the
failure mechanism of graphene. We considered four distributions of atomic vacancies arranged
in a 2D array of 10 nm × 10 nm with various spacing among the individual vacancies along x
and y directions (dx, dy). Figures 5a and 5b show two vacancy distributions corresponding to
armchair crack-tip. In order to obtain the stiffness of the area in the vicinity of the crack-tip
containing multiple atomic vacancies (i.e. the damaged zone), we conducted numerical uniaxial
tensile test of graphene samples with dimensions of 10 nm × 10 nm containing the corresponding
periodic distributions of vacancies (see figure S4 in Supporting Information for details). Table 1
gives the stiffness of graphene containing different distributions of vacancies representing the
damaged zone. Figure 5c shows linear variations of the normalized crack-tip stress c
tip
v)-(c
tip σσ
with the stiffness of the damaged zone as predicted by the continuum damage mechanics models
[23–25]. However, the normalized critical SIF IC
v)-(c
C KK in figure 5d shows completely different
relationships with the stiffness due to the unique atomistic crack propagation pattern involving
crack deflection, crack branching, and crack bridging as shown in figure 5e. More interestingly, a
dramatic difference (~60%) exists between the critical SIFs of armchair and zigzag cracks at
lower stiffness values (~0.7 TPa) and, in fact, the stiffness-SIF relationships of the armchair and
zigzag cracks are completely different. Figure 5e demonstrates that the catastrophic failure of
graphene [16,40,56] can be transformed into a number of sequential local failures as a result of
varied atomistic vacancy distributions, which agrees with a recent experiment [28]. This result
further demonstrates the rich potential of employing the combination of atomic vacancies and the
underlying crystal structure of graphene in controlling its fracture strength and crack propagation
pattern.
14. 14
Figure 5 The effect of an array of single vacancies at the crack-tip on the fracture characteristics
of graphene. Figures (a) and (b) show stress σyy distribution at an armchair crack-tip due to an
applied tensile strain εyy of 1%. Figures (c) and (d) show the variations of the normalized crack-
tip stress and the critical SIF with the stiffness of the process zone, respectively. Figure (e)
depicts the propagation of an armchair crack in the presence of an array of single vacancies.
15. 15
Table 1 Stiffness of graphene samples containing various distributions of single atomic
vacancies.
armchair zigzag
dx
(nm)
dy
(nm)
S
(TPa)
dx
(nm)
dy
(nm)
S
(TPa)
0.8 0.7 0.69 0.7 0.8 0.71
1.3 1.0 0.76 1.0 0.8 0.78
1.3 1.3 0.78 1.2 1.3 0.85
1.7 1.5 0.82 1.5 1.7 0.90
4. Conclusions
In summary, our atomistic simulations, complemented by a continuum-based analytical model,
shed light on previously overlooked aspect concerning the tailoring of the fracture properties of
graphene. Our results demonstrate that crack-vacancy interaction can be effectively used to
transform the crack-tip stress field, fracture strength, crack propagation pattern, and even crack
healing of graphene; thus, providing another dimension in the design space of graphene. In this
regards, the atomistic crack-vacancy interactions supersede the other recently studied methods of
improving crack arrest in graphene. For example, the atomic vacancies can lead to a reduction in
the crack-tip stress field by as much as 55% and ultimately increase the fracture strength by 67%.
In addition, our numerical experiments provide a remarkable insight into the conditions leading
to crack arrest and the potential toughening mechanisms of graphene due to the crack-vacancy
interactions. In addition, our numerical experiments elucidate the pronounced effects of the
underlying crystal structure on the atomistic fracture characteristics of 2D materials.
Supporting Information
Information on the accuracy of the analytical model, design envelops for the crack-vacancy
interactions, crack healing under continuous unloading, and stress-strain relations of the
damaged zones are provided.
16. 16
Acknowledgements
The authors wish to thank NSERC and the Discovery Accelerator Supplement for their kind
support of this research. Computing resources were provided by WestGrid and Compute/Calcul
Canada.
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19. 19
Supporting Information for
Tailoring Fracture Strength of Graphene
M. A. N. Dewapriya and S. A. Meguid
Mechanics and Aerospace Design Laboratory, Department of Mechanical and Industrial
Engineering, University of Toronto, Toronto, Ontario M5S 3G8, Canada
1. The effects of higher order terms on the analytical solution given by eq. 4
We computed the percentage error of the normalized stress intensity factor I
v)-(c
I KK (see
equation 4) when it is compared with the normalized crack-tip stress (c)
tip
v)-(c
tip σσ obtained from
MD simulations as follows:
100errorPercentage
)c(
tip
)vc(
tip
I
)vc(
I
)c(
tip
)vc(
tip
×
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= −
−−
σ
σ
σ
σ
K
K
(s1)
As shown in figure S1a, the agreement between the analytical solution and the MD simulations is
excellent for smaller holes (i.e. for smaller c/r) demonstrating the influence of the higher order
terms on the accuracy of the analytical solution.
In order to improve the accuracy of the existing leading order solution in equation 4, a
correction factor (κ) can be introduced into the solution using a few representative values of
(c)
tip
v)-(c
tip σσ obtained from the MD simulations. As evidenced by figure S1a, the correction factor
depends on the orientation θ and the diameter of the hole d. For example, when d = 1.4 nm, the
correction factor κ can be computed using the four MD simulation results given in Figure S1a as
follows:
20. 20
05.101.122 2
I
)vc(
I
)c(
tip
)vc(
tip
++−=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= −
−
αα
σ
σ
κ
K
K
(s2) where α = θ×10-3
, and the corrected normalized stress intensity factor can be expressed as
( )I
v)-(c
IcorrectedI
v)-(c
I KKKK κ= .
Figure S1 The effects of the hole diameter d on the accuracy of the leading order solutions given
in eq. 4. (a) Variation of the percentage error for three hole diameters, where the inset depicts the
relative position of the hole with respect to the crack-tip and r is kept fixed at ~3.2 nm. Figures
(b-d) show the stress σyy distribution at the crack-tip due to an applied tensile strain εyy of 1% for
the holes located at θ = 31°.
21. 21
2. Design envelops developed using the analytical solution given by equation 4
Figure S2 Design envelops depicting the influence of the atomic vacancies on the crack-tip
stress field of an armchair crack: for the (a) oriented holes and (b) oriented and oblique
nanocracks.
22. 22
3. Crack healing under continuous unloading
Figure S3a shows the variation of the averaged stress of the process zone (2 nm × 2 nm
containing 170 atoms) during the loading and the continuous unloading of a graphene sample
with an armchair edge crack and an interacting circular vacancy. Partial crack propagation occurs
at the end of the region I, and it continues to propagate as shown in Figures S3c and 3d. The
unloading was started at T = 33 ps. During the unloading, crack healing first occurs at T = 54 ps
and it continues through the region IV, however there is no significant increase in the effective
stiffness of the process zone due to the crack healing. Half of the initially propagated crack has
been healed when the applied stress is completely removed at T = 66 ps, and the crack is entirely
healed during the subsequent relaxation of the graphene sample as shown in figure S3g.
23. 23
Figure S3 Healing of the initially propagated armchair crack in the presence of a circular
vacancy of diameter 3.6 nm located at r = 2.9 nm and θ = 88°. (a) Variation of the spatially
averaged stress σyy of the process zone during the loading and unloading of the graphene sample.
Figures (b-g) show snapshots of the crack-tip at various growth stages. The arrows in Figures
(c-e) indicate the broken bonds at the crack surface due to the partial crack propagation. Figures
(f) and (g) show considerable crack healing with time.
4. Stress-strain relations of graphene samples containing an edge crack and a distribution
of single vacancies
Figure S4 Stress distribution of graphene samples. Figures (a) and (b) show the stress-strain
relations for three graphene samples with various vacancy configurations as demonstrated in
Figures (c-e). In addition, Figures (c-e) show stress σyy distribution of the graphene samples due
24. 24
to an applied tensile strain εyy of 1%. Size of the sample-1 is 10 nm × 10 nm and it is
60 nm × 60 nm for the sample-2 and -3.