Graphical Abstract Figure
This molecular statics methodology is laid out to study the crack growth of any crystalline material under tensile loading to obtain mode-I SIF. Different crack positions relative to the loading boundary are studied for single-crystal silicon structures to obtain the relationship between the crack depth and SIF (the LEFM fracture criteria). The simulation results show a region of “bulk” where SIF and boundary load required for crack propagation are invariant concerning the crack depth. Away from the “bulk” region, the SIF and the boundary load required for crack propagation decreases linearly.
Graphical Abstract Figure
This molecular statics methodology is laid out to study the crack growth of any crystalline material under tensile loading to obtain mode-I SIF. Different crack positions relative to the loading boundary are studied for single-crystal silicon structures to obtain the relationship between the crack depth and SIF (the LEFM fracture criteria). The simulation results show a region of “bulk” where SIF and boundary load required for crack propagation are invariant concerning the crack depth. Away from the “bulk” region, the SIF and the boundary load required for crack propagation decreases linearly.
Abstract
In the present work, an atomistic scale investigation is done on crystalline silicon to understand the effect of crack depth from the loading (pulling) boundary on the critical near-tip state of stress. For various depths of embedded cracks, the near-tip stress field has been calculated at the critical state just before the crack propagation initiation. This atomistically calculated stress field is found to be quite close to those found using continuum linear elasticity. Thereafter, the critical stress intensity factor (SIF) is calculated for all cases by fitting the atomistically calculated normal stress over inverse square-rooted distance from the crack tip. It has been found that the closer the crack is located to the loading boundary (i.e., lesser depth), the lower is the (locally calculated) critical SIF. This implies that it is easier to initiate crack propagation when the crack is located closer to the loading boundary. The claim is also strengthened by a similar observation of (globally calculated) boundary stresses at the critical state just before crack propagation initiation.
References
1.
Nguyen
, B.
, Celler
, G.
, and Mazuré
, C.
, 2009
, “A Review of SOI Technology and its Applications
,” J. Integr. Circuit. Syst.
, 4
(2
), pp. 51
–54
. 2.
Aspar
, B.
, Moriceau
, H.
, Jalaguier
, E.
, Lagahe
, C.
, Soubie
, A.
, Biasse
, B.
, Papon
, A. M.
, et al., 2001
, “The Generic Nature of the Smart-Cut® Process for Thin Film Transfer
,” J. Electron. Mater.
, 30
(7
), pp. 834
–840
. 3.
Du
, J.
, Ko
, W. H.
, and Young
, D. J.
, 2004
, “Single Crystal Silicon MEMS Fabrication Based on Smart-Cut Technique
,” Sens. Actuat. A
, 112
(1
), pp. 116
–121
. 4.
Bitzek
, E.
, Kermode
, J. R.
, and Gumbsch
, P.
, 2015
, “Atomistic Aspects of Fracture
,” Int. J. Fract.
, 191
(1–2
), pp. 13
–30
. 5.
Swadener
, J. G.
, Baskes
, M. I.
, and Nastasi
, M.
, 2002
, “Molecular Dynamics Simulation of Brittle Fracture in Silicon
,” Phys. Rev. Lett.
, 89
(8
), pp. 19
–22
. 6.
Cramer
, T.
, Wanner
, A.
, and Gumbsch
, P.
, 2000
, “Energy Dissipation and Path Instabilities in Dynamic Fracture of Silicon Single Crystals
,” Phys. Rev. Lett.
, 85
(4
), pp. 788
–791
. 7.
Dontsova
, E.
, and Ballarini
, R.
, 2017
, “Atomistic Modeling of the Fracture Toughness of Silicon and Silicon-Silicon Interfaces
,” Int. J. Fract.
, 207
(1
), pp. 99
–122
. 8.
Lee
, G. H.
, Na
, S.
, Chung
, Y.
, and Beom
, H.
, 2021
, “Atomistic Aspects of the Temperature Effect on Fracture Toughness of a Silicon Single Crystal
,” Comput. Mater. Sci.
, 195
, p. 110489
. 9.
Masolin
, A.
, Bouchard
, P. O.
, Martini
, R.
, and Bernacki
, M.
, 2013
, “Thermo-Mechanical and Fracture Properties in Single-Crystal Silicon
,” J. Mater. Sci.
, 48
(3
), pp. 979
–988
. 10.
Budarapu
, P. R.
, Javvaji
, B.
, Reinoso
, J.
, Paggi
, M.
, and Rabczuk
, T.
, 2018
, “A Three Dimensional Adaptive Multiscale Method for Crack Growth in Silicon
,” Theor. Appl. Fract. Mech.
, 96
, pp. 576
–603
. 11.
Zhuo
, X. R.
, and Beom
, H. G.
, 2015
, “Size-Dependent Fracture Properties of Cracked Silicon Nanofilms
,” Mater. Sci. Eng. A
, 636
, pp. 470
–475
. 12.
Bailey
, N. P.
, and Sethna
, J. P.
, 2003
, “Macroscopic Measure of the Cohesive Length Scale: Fracture of Notched Single-Crystal Silicon
,” Phys. Rev. B - Conden. Matter Mater. Phys.
, 68
(20
), pp. 2052041
–2052048
. 13.
Cheng
, S. H.
, and Sun
, C. T.
, 2014
, “Size-Dependent Fracture Toughness of Nanoscale Structures: Crack-Tip Stress Approach in Molecular Dynamics
,” J. Nanomech. Micromech.
, 4
(4
), pp. 1
–8
. 14.
Molaei
, F.
, 2022
, “Molecular Dynamics Simulation of Edge Crack Propagation in Single Crystalline Alpha Quartz
,” J. Mole. Graph. Model.
, 111
, p. 108085
. 15.
Budarapu
, P. R.
, Gracie
, R.
, Bordas
, S. P. A.
, and Rabczuk
, T.
, 2014
, “An Adaptive Multiscale Method for Quasi-Static Crack Growth
,” Comput. Mech.
, 53
(6
), pp. 1129
–1148
. 16.
Peng
, P.
, Liao
, G.
, Shi
, T.
, Tang
, Z.
, and Gao
, Y.
, 2010
, “Molecular Dynamic Simulations of Nanoindentation in Aluminum Thin Film on Silicon Substrate
,” Appl. Surf. Sci.
, 256
(21
), pp. 6284
–6290
. 17.
Adnan
, A.
, and Sun
, C. T.
, 2010
, “Evolution of Nanoscale Defects to Planar Cracks in a Brittle Solid
,” J. Mech. Phys. Solids
, 58
(7
), pp. 983
–1000
. 18.
Buehler
, M. J.
, van Duin
, A. C.
, and Goddard
, W. A.
, 2006
, “Multiparadigm Modeling of Dynamical Crack Propagation in Silicon Using a Reactive Force Field
,” Phys. Rev. Lett.
, 96
(9
), pp. 1
–4
. 19.
van Duin
, A. C.
, Strachan
, A.
, Stewman
, S.
, Zhang
, Q.
, Xu
, X.
, and Goddard
, W. A.
, 2003
, “ReaxFF SiO Reactive Force Field for Silicon and Silicon Oxide Systems
,” J. Phys. Chem. A
, 107
(19
), pp. 3803
–3811
. 20.
Munetoh
, S.
, Motooka
, T.
, Moriguchi
, K.
, and Shintani
, A.
, 2007
, “Interatomic Potential for Si-O Systems Using Tersoff Parameterization
,” Comput. Mater. Sci.
, 39
(2
), pp. 334
–339
. 21.
Zhang
, S.
, Zhu
, T.
, and Belytschko
, T.
, 2007
, “Atomistic and Multiscale Analyses of Brittle Fracture in Crystal Lattices
,” Phys. Rev. B - Conden. Matter Mater. Phys.
, 76
(9
), pp. 1
–10
. 22.
Griffith
, A. A.
, 1921
, “The Phenomena of Rupture and Flow in Solids
,” Philos. Trans. R. Soc. Lond. A
, 221
, pp. 163
–198
. 23.
Huang
, S.
, Zhang
, S.
, Belytschko
, T.
, Terdalkar
, S. S.
, and Zhu
, T.
, 2009
, “Mechanics of Nanocrack: Fracture, Dislocation Emission, and Amorphization
,” J. Mech. Phys. Solids
, 57
(5
), pp. 840
–850
. 24.
Lee
, G. H.
, Shim
, J. S.
, Cui
, C. Y.
, and Beom
, H. G.
, 2019
, “Hydrogen-Induced Cracking of an Aluminum Single Crystal: An Atomistic Simulation
,” Comput. Mater. Sci.
, 169
, p. 109084
. 25.
Sakib
, A. R. N.
, and Adnan
, A.
, 2012
, “On the Size-Dependent Critical Stress Intensity Factor of Confined Brittle Nanofilms
,” Eng. Fract. Mech.
, 86
, pp. 13
–22
. 26.
Dai
, D.
, Hills
, D.
, Härkegard
, G.
, and Pross
, J.
, 1998
, “Simulation of the Growth of Near-Surface Defects
,” Eng. Fract. Mech.
, 59
(4
), pp. 415
–424
. 27.
Sin
, H.-C.
, and Suh
, N. P.
, 1984
, “Subsurface Crack Propagation Due to Surface Traction in Sliding Wear
,” ASME J. Appl. Mech.
, 51
(2
), pp. 317
–323
. 28.
Moftakhar
, A.
, and Glinka
, G.
, 1992
, “Calculation of Stress Intensity Factors by Efficient Integration of Weight Functions
,” Eng. Fract. Mech.
, 43
(5
), pp. 749
–756
. 29.
Beghini
, M.
, Bertini
, L.
, and Fontanari
, V.
, 1999
, “Weight Function for an Inclined Edge Crack in a Semiplane
,” Int. J. Fract.
, 99
(4
), pp. 281
–292
. 30.
Beghini
, M.
, Bertini
, L.
, and Fontanari
, V.
, 2008
, “A Weight Function for 2D Subsurface Cracks Under General Loading Conditions
,” Eng. Fract. Mech.
, 75
(3–4
), pp. 427
–439
. 31.
Höchbauer
, T.
, Misra
, A.
, Nastasi
, M.
, and Mayer
, J. W.
, 2002
, “Physical Mechanisms Behind the Ion-Cut in Hydrogen Implanted Silicon
,” ASME J. Appl. Phys.
, 92
(5
), pp. 2335
–2342
. 32.
Anderson
, T.
, 2017
, Fracture Mechanics: Fundamentals and Applications
, 4th ed., CRC Press
, Boca Raton, FL
.33.
Zhang
, J.
, Ma
, F.
, Xu
, K.
, and Xin
, X.
, 2003
, “Anisotropy Analysis of the Surface Energy of Diamond Cubic Crystals
,” Surf. Interface Anal.
, 35
(10
), pp. 805
–809
. 34.
Shah
, J. S.
, and Straumanis
, M.
, 1972
, “Thermal Expansion Behavior of Silicon at Low Temperatures
,” Solid State Commun.
, 10
(1
), pp. 159
–162
. 35.
Weber
, T. A.
, and Stillinger
, F. H.
, 1985
, “Local Order and Structural Transitions in Amorphous Metal-Metalloid Alloys
,” Phys. Rev. B
, 31
(4
), pp. 1954
–1963
. 36.
Nishinaga
, T.
, 2014
, Handbook of Crystal Growth: Fundamentals
, 2nd ed., Elsevier
, Netherlands
.37.
Arai
, N.
, Takeda
, S.
, and Kohyama
, M.
, 1997
, “Self-Interstitial Clustering in Crystalline Silicon
,” Phys. Rev. Lett.
, 78
(22
), pp. 4265
–4268
. 38.
Alippi
, P.
, and Colombo
, L.
, 2000
, “Lattice-Strain Field Induced by {311} Self-Interstitial Defects in Silicon
,” Phys. Rev. B
, 62
(3
), pp. 1815
–1820
. 39.
Balamane
, H.
, Halicioglu
, T.
, and Tiller
, W. A.
, 1992
, “Comparative Study of Silicon Empirical Interatomic Potentials
,” Phys. Rev. B
, 46
(4
), pp. 2250
–2279
. 40.
Tersoff
, J.
, 1989
, “Modeling Solid-State Chemistry: Interatomic Potentials for Multicomponent Systems
,” Phys. Rev. B
, 39
(8
), pp. 5566
–5568
. 41.
Stillinger
, F. H.
, and Weber
, T. A.
, 1985
, “Computer Simulation of Local Order in Condensed Phases of Silicon
,” Phys. Rev. B
, 31
(8
), pp. 5262
–5271
. 42.
Yoo
, S. H.
, Lee
, B.
, and Kang
, K.
, 2021
, “Density Functional Theory Study of the Mechanical Behavior of Silicene and Development of a Tersoff Interatomic Potential Model Tailored for Elastic Behavior
,” Nanotechnol.
, 32
(29
), p. 295702
. 43.
Thompson
, A. P.
, Aktulga
, H. M.
, Berger
, R.
, Bolintineanu
, D. S.
, Brown
, W. M.
, Crozier
, P. S.
, Kohlmeyer
, A.
, Moore
, S. G.
, et al., 2022
, “LAMMPS - A Flexible Simulation Tool for Particle-Based Materials Modeling at the Atomic, Meso, and Continuum Scales
,” Comput. Phys. Commun.
, 271
, p. 108171
. 44.
Stukowski
, A.
, 2009
, “Visualization and Analysis of Atomistic Simulation Data With Ovito—The Open Visualization Tool
,” Modell. Simul. Mater. Sci. Eng.
, 18
(1
), p. 015012
. 45.
Brantley
, W. A.
, 1973
, “Calculated Elastic Constants for Stress Problems Associated With Semiconductor Devices
,” ASME J. Appl. Phys.
, 44
(1
), pp. 534
–535
. 46.
Sahmani
, S.
, Aghdam
, M. M.
, and Bahrami
, M.
, 2017
, “An Efficient Size-Dependent Shear Deformable Shell Model and Molecular Dynamics Simulation for Axial Instability Analysis of Silicon Nanoshells
,” J. Mole. Graphics Modell.
, 77
, pp. 263
–279
. 47.
Alekseev
, P. A.
, Borodin
, B. R.
, Geydt
, P.
, Khayrudinov
, V.
, Bespalova
, K.
, Kirilenko
, D. A.
, and Reznik
, R. R.
, 2021
, “Effect of Crystal Structure on the Young’s Modulus of GaP Nanowires
,” Nanotechnol.
, 32
(38
), p. 385706
. 48.
Lekhnitskii
, S. G.
, Fern
, P.
, Brandstatter
, J. J.
, and Dill
, E. H.
, 1964
, “Theory of Elasticity of an Anisotropic Elastic Body
,” Phys. Today
, 17
(1
), pp. 84
–84
. 49.
Kittel
, C.
, 2005
, Introduction to Solid State Physics
, 8th ed., Wiley
, New York
.50.
Huang
, D.
, Zhang
, Q.
, and Qiao
, P.
, 2011
, “Molecular Dynamics Evaluation of Strain Rate and Size Effects on Mechanical Properties of FCC Nickel Nanowires
,” Comput. Mater. Sci.
, 50
(3
), pp. 903
–910
. 51.
Hestenes
, M.
, and Stiefel
, E.
, 1952
, “Methods of Conjugate Gradients for Solving Linear Systems
,” J. Res. Nat. Bureau Stand.
, 49
(6
), p. 409
. 52.
Ray
, J. R.
, 1988
, “Elastic Constants and Statistical Ensembles in Molecular Dynamics
,” Comput. Phys. Rep.
, 8
(3
), pp. 109
–151
. 53.
Kumagai
, T.
, Izumi
, S.
, Hara
, S.
, and Sakai
, S.
, 2007
, “Development of Bond-Order Potentials That Can Reproduce the Elastic Constants and Melting Point of Silicon for Classical Molecular Dynamics Simulation
,” Comput. Mater. Sci.
, 39
(2
), pp. 457
–464
. 54.
Hall
, J. J.
, 1967
, “Electronic Effects in the Elastic Constants of N-Type Silicon
,” Phys. Rev.
, 161
(3
), pp. 756
–761
. 55.
Lee
, B.
, and Rudd
, R. E.
, 2007
, “First-Principles Calculation of Mechanical Properties of Si001 Nanowires and Comparison to Nanomechanical Theory
,” Phys. Rev. B - Conden. Matter Mater. Phys.
, 75
(3
), pp. 1
–13
. 56.
Cowley
, E. R.
, 1988
, “Lattice Dynamics of Silicon With Empirical Many-Body Potentials
,” Phys. Rev. Lett.
, 60
(23
), pp. 2379
–2381
. 57.
Hopcroft
, M. A.
, Nix
, W. D.
, and Kenny
, T. W.
, 2010
, “What is the Young’s Modulus of Silicon?
,” J. Microelectromech. Syst.
, 19
(2
), pp. 229
–238
. 58.
Lemak
, A. S.
, and Balabaev
, N. K.
, 1994
, “On the Berendsen Thermostat
,” Mol. Simul.
, 13
(3
), pp. 177
–187
. 59.
Martín Pendás
, A.
, 2002
, “Stress, Virial, and Pressure in the Theory of Atoms in Molecules
,” J. Chem. Phys.
, 117
(3
), pp. 965
–979
. 60.
Singh
, A.
, and Singh
, G.
, 2023
, “A Localized Near Crack Tip Stress Field Approach for Calculating the Fracture Strength of an Anisotropic Solid at an Atomistic Scale
,” Mech. Mater.
, 181
, p. 104632
. 61.
Ebrahimi
, F.
, and Kalwani
, L.
, 1999
, “Fracture Anisotropy in Silicon Single Crystal
,” Mater. Sci. Eng. A
, 268
(1–2
), pp. 116
–126
. 62.
Fitzgerald
, A. M.
, Iyer
, R. S.
, Dauskardt
, R. H.
, and Kenny
, T. W.
, 2002
, “Subcritical Crack Growth in Single-Crystal Silicon Using Micromachined Specimens
,” J. Mater. Res.
, 17
(3
), pp. 683
–692
. 63.
Sumigawa
, T.
, Ashida
, S.
, Tanaka
, S.
, Sanada
, K.
, and Kitamura
, T.
, 2015
, “Fracture Toughness of Silicon in Nanometer-Scale Singular Stress Field
,” Eng. Fract. Mech.
, 150
, pp. 161
–167
. 64.
Zhou
, M.
, 2003
, “A New Look at the Atomic Level Virial Stress: On Continuum-Molecular System Equivalence
,” Proc. R. Soc. Lond. A: Math., Phys. Eng. Sci.
, 459
(2037
), pp. 2347
–2392
. 65.
Pizzagalli
, L.
, Godet
, J.
, Guénolé
, J.
, Brochard
, S.
, Holmstrom
, E.
, Nordlund
, K.
, and Albaret
, T.
, 2013
, “A New Parametrization of the Stillinger-Weber Potential for an Improved Description of Defects and Plasticity of Silicon
,” J. Phys. Conden. Matter
, 25
(5
), p. 055801
. 66.
Smith
, M.
, 2009
, ABAQUS/Standard User’s Manual, Version 6.9
, Dassault Systèmes Simulia Corp
, Johnston, RI
.67.
Huang
, Z.
, Wagner
, D.
, Bathias
, C.
, and Paris
, P. C.
, 2010
, “Subsurface Crack Initiation and Propagation Mechanisms in Gigacycle Fatigue
,” Acta Mater.
, 58
(18
), pp. 6046
–6054
. 68.
Shiozawa
, K.
, Morii
, Y.
, Nishino
, S.
, and Lu
, L.
, 2006
, “Subsurface Crack Initiation and Propagation Mechanism in High-Strength Steel in a Very High Cycle Fatigue Regime
,” Int. J. Fatigue
, 28
(11
), pp. 1521
–1532
. 69.
Starker
, P.
, Wohlfahrt
, H.
, and Macherauch
, E.
, 1979
, “Subsurface Crack Initiation During Fatigueas a Result of Residual Stresses
,” Fatigue Fract. Eng. Mater. Struct.
, 1
(3
), pp. 319
–327
. 
