Nonlinear Mechanics of Monolayer Graphene Rui Huang
|
|
- Dayna Patience Holland
- 6 years ago
- Views:
Transcription
1 Nonlinear Mechanics of Monolayer Graphene Rui Huang Center for Mechanics of Solids, Structures and Materials Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin
2 Acknowledgments Qiang Lu (postdoc) Zachery Aitken (undergraduate student) Prof. Marino Arroyo (Spain) Prof. Li Shi (UT/ME) Funding: DoE and NSF (ARRA)
3 What is special about graphene? So much about carbon nanotubes, let s play with graphene now. Can two-dimensional crystals exist in our 3D space? Rippling or not? Uniqueness of D: electron transport,,q Quantum Hall effect, etc. Any interesting mechanics? Novoselev et al. 005
4 Graphene Based Devices Patterned graphene on oxide for bipolar p-n-p junctions Suspended nanoribbons for NEMS devices Ozyilmaz et al., 007. Garcia-Sanchez, et al., 008.
5 Mechanical properties of monolayer graphene Young s modulus? Ultimate tensile strength? Bending modulus? What is the thickness of a graphene monolayer? Interlayer spacing in graphite: nm C-C covalent bond length: 0.14 nm Mechanically reduced thickness: nm We consider monolayer graphene as a D membrane with no thickness.
6 Nonlinear Continuum Mechanics of D Sheets D-to-3D deformation gradient: F ij X x i J X dx FdX x 3 x X 1 x 1 In-plane deformation: D Green-Lagrange strain tensor E JK 1 ( F F δ ) ij ik JK Bending: D curvature tensor (strain gradient) FiI xi Κ ni ni X X X J I J No need to define any thickness! Strain energy (hyperelasticity): ( E, ) U Φ Κ da A Lu and Huang, Int. J. Applied Mechanics 1, (009).
7 Stresses and Moments in D nd Piola-Kirchoff stress and moment (work conjugates) S Φ E M Φ Κ Tangent moduli: C KL S E KL Φ E E KL D KL M Κ KL Φ Κ Κ KL Λ KL S Κ KL M E KL Φ E Κ KL Intrinsic coupling between tension and bending An incremental form of the generally nonlinear and anisotropic behavior: ds ds ds 11 C C C 11 1 C C C 1 C C C 13 3 de11 Λ11 Λ1 Λ13 dκ de + Λ1 Λ Λ3 dκ de 1 Λ 31 Λ 3 Λ 33 dκ dm11 D11 D1 D13 dκ11 Λ11 Λ1 Λ31 de dm D1 D D3 dκ + Λ1 Λ Λ3 de dm 1 D31 D3 D33 dκ1 Λ13 Λ3 Λ33 de 1 11 Lu and Huang, Int. J. Applied Mechanics 1, (009).
8 Units for D quantities Φ( E,Κ) strain energy density function: J/m S D stress: N/m Φ E C KL E S Φ E E D in-plane modulus: N/m KL KL M Φ Κ moment intensity: (N-m)/m D KL M Κ KL Φ Κ Κ KL bending modulus: N-m Λ KL S M KL Κ KL E E Κ KL p g Φ coupling modulus: N Analogous to the D plate/shell theories.
9 Atomistic Modeling of Graphene Rectangular unit cell in an arbitrary direction, subject to in-plane stretching (, ):a c a and bending. (n,n): armchair nd -generation REBO potential (Brenner et al., 00) Bond angle effect (second-nearest nearest neighbors) Dihedral angle effect (third-nearest neighbors) Radical energetics (defects and edges) α (n,n) (n,0): zigzag Energy minimization (molecular mechanics) for equilibrium states. Stress and moment calculations Energy derivation Virial stress calculations Direct force evaluation
10 Pure Bending of Graphene Roll up a rectangular graphene sheet into a cylindrical tube without in-plane stretch (E 11 0) F 1 πr X cos π L L 0 πr X sin π L L κ E n 11 1 R 1 πr 1 L Str rain energy per atom (ev/atom m) B1, armchair B1, zigzag B, armchair B, zigzag B*, armchair B*, zigzag Bending curvature (1/nm) per length (nn N) Ben nding moment B1, armchair B1, zigzag B, armchair B, zigzag B*, armchair B*, zigzag Bending curvature (1/nm)
11 Bending Modulus of Graphene 0.5 Bending modulus (nn nm) B1, armchair B1, zigzag B, armchair B, zigzag B*, armchair θ jil l i k θ ijk j θ q3 θ q1 θ q4 θq Θ q Θ q1 Θ q3 Θ q4 B*, zigzag Bending curvature (1/nm) Bending moment-curvature is nearly linear, with slight anisotropy. Including the dihedral effect leads to higher bending energy and bending modulus. Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 4, 1000 (009).
12 Physical Origin of Bending Modulus Bending modulus of a thin elastic plate: D dm dκκ 3 ~ Yh h For monolayer graphene, bending moment and bending stiffness result from multibody interatomic interactions (second and third nearest neighbors). D VA( r ) b 14T θijk 3 σ π 0 ij 0 D 0.83 ev (0.133 nn-nm) by REBO-1 D 1.4 ev (0.5 nn-nm) by REBO- D 1.5 ev (0.38 nn-nm) by first principle Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 4, 1000 (009).
13 Coupling between bending and stretching Strain energy per atom (ev V/atom) W ( κ, ε ) Dκ + M ( κ κ ) + σε + D( κ κ ) Cε R nm R Strain ε R/R 0 1 The tube radius increases upon relaxation, leading to simultaneous bending and stretching. Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 4, 1000 (009).
14 Uniaxial Stretch of Monolayer Graphene Uniaxial i stretch: t λ L L 0 Nominal strain: ε λ 11 Green-Lagrange strain: E ( 1) 1 11 λ E E1 0 per atom (ev) Energy (ev) Energy A A B B Nominal strain ε C C D D Lu and Huang, Int. J. Applied Mechanics 1, (009).
15 40 Biaxial Stresses under Uniaxial Stretch (Poisson s effect) 30 Nominal stress (N N/m) 35 S P S 10 P S P 1 and P Nominal strain ε Green Lagrange strain E 11 (N/m) Membrane stress ( Energy derivation: S 11 dφ de 11 P 11 dφ dε ( ( n) ( m) ) Virial stress calculation for nominal stress: P X X ij 1 A m n J J F ( mn) i Relationship between nominal stresses and nd P-K stress: P11 P1 S 11, S P, S1, S1 1+ ε 1+ ε P 1
16 Anisotropic tangent moduli ds11 C11 C1 C13 de ds C1 C C3 de ds1 C31 C3 C33 de 1 (n,n): armchair 11 C 11 (N/m) α 0 (zigzag) α o α o α 30 o (armchair) C 1 (N/m) 50 α (n,n) (n,0): zigzag Graphene is linear and isotropic under infinitesimal deformation, but becomes nonlinear and anisotropic under finite deformation. Lu and Huang, Int. J. Applied Mechanics 1, (009). C 31 (N/m) Coupling between tension and shear Green Lagrange strain E 11
17 Fracture strength under uniaxial stretch (N/m) No ominal stress P α 0 (zigzag) α o Nominal strain ε α o α 30 o (armchair) Nominal fracture strain Chiral angle (degree) 3 Nom minal fracture stre ess (N/m) Fracture occurs as a result of intrinsic instability of the homogeneous deformation: P P ε Φ 0 ε The nominal stress and strain to fracture depend on the direction of uniaxial stretch.
18 Graphene Nanoribbons: Edge Effects Garcia-Sanchez, et al. (008). Koskinen, et al. (008) Various edge states: zigzag/armchair/mixed, H-terminated, reconstructed Edge effect leads to size-dependent electronic properties of GNRs. Are mechanical properties of GNRs size dependent?
19 Excess Edge Energy and Edge Force Zigzag edge: f Z f Z atom (ev) Energy per armchair, un relaxed armchair, 1 D relaxation zigzag, un relaxed zigzag, 1 D relaxation Eq. (6), γ 11.1 ev/nm Eq. (6), γ 10.6 ev/nm Armchair edge: f A Lu and Huang, arxiv: f A /W (nm 1 ) Edge energy (ev/nm ) Edge force (ev/nm) r 0 Armchair Zigzag Armchair Zigzag (nm) DFT [17] (GPAW) DFT [18] (VASP) DFT [] (SIESTA) MM [0] (AIREBO) MD [1] MM (REBO)
20 Edge buckling of GNRs Zigzag GNR Intrinsic wavelength ~ 6. nm Excess en nergy (ev/nm) y e 05*x *x *x 0.054*x Buckle wavelength λ (nm) Armchair GNR Intrinsic wavelength ~ 8.0 nm Ex xcess energy (ev/n nm) y.e 06*x *x *x 0.014*x Buckle wavelength λ (nm) The wavelengths for edge buckling do not scale with D/f. Lu and Huang, arxiv:
21 F GNRs under Uniaxial Tension U ( ε ) Φ ( ε ) WL + γ ( ε )L F δu FLδε σ F 1 du dφ + W WL dε dε W dγ dε Zigzag GNRs Armchair GNRs
22 Φ Interior Energy Function ( ε ) a 0 + a 1ε + a ε + a 3ε + a 4ε + a 5ε + a 6ε + a 7ε + a 8ε Zigzag Armchair a Φ( (ε ) a a a a a a a a dφ dε d Φ dε
23 γ Edge energy function ( ε ) b 0 + b1ε + bε + b3ε + b4ε + b5ε + b6ε + b7ε + b8ε Zigzag Armchair b b γ (ε ) b b b b b b b dγγ dε d γ dε
24 D Young s Moduli of GNRs σ ( ε ) dφ + dε W dγ dε d Φ E + dε W d γ dε ( ε ) Young s modulus under ifiit infinitesimal i strain: 4b E 0 a + W
25 Fracture of graphene ribbons Zigzag edge Armchair edge
26 MD simulations of fracture (300 K) Zigzag edge Armchair edge Zigzag GNRs: fracture starts away from the edges; fracture strain same as that for an infinite graphene (PBC); Armchair GNRs: fracture starts near the edges; fracture strain lower than that for an infinite graphene (PBC).
27 Graphene on Oxide Substrates HR-STM image (Stolyarova a et al., 007) RMS Ozyilmaz et al., 007. The 3D morphology is important for the transport properties of graphene-based devices. Correlation length Ishigami et al., 007.
28 Van der Waals Interaction Lennard-Jones potential for particle-particle interactions: 6 ( ) r0 r0 V r V0 r r 1 Monolayer-substrate interaction (energy per unit area): U U ( h) U h 0 1 h0 h h r h 0 h 0 h -U 0
29 Van der Waals Thickness and Energy Gupta et al., 006. Sonde et al., 009. Interlayer spacing in graphite ~ 0.34 nm; AFM measurements of h 0 for graphene on oxide range from 0.4 to 0.9 nm; The adhesion energy (U 0 )h has not tbeen measured ddirectly; Theoretically estimated values for U 0 range from 0.6 to 0.8 ev/nm.
30 Strain-Induced Corrugation h h 0 + Asin kx U U + U total vdw graphene Linear perturbation analysis: U vdw U graphene 1 λ A U + f 0 1 h h π π 4D + Cε A λ λ ε c ~ λ ~ 3 nm
31 Graphene on Rough Substrates Conformal or non-conformal?
32 Summary Nonlinear continuum mechanics for D graphene monolayer Atomistic modeling of graphene under bending and stretching Excess edge energy, edge forces, and induced edge buckling Graphene nanoribbons under uniaxial tension: edge effects on elastic modulus and fracture Graphene on oxide: van der Waals interaction p and corrugation
Mechanical Interactions at the Interfaces of Atomically Thin Materials (Graphene)
Mechanical Interactions at the Interfaces of Atomically Thin Materials (Graphene) Rui Huang Center for Mechanics of Solids, Structures and Materials Department of Aerospace Engineering and Engineering
More informationShear Properties and Wrinkling Behaviors of Finite Sized Graphene
Shear Properties and Wrinkling Behaviors of Finite Sized Graphene Kyoungmin Min, Namjung Kim and Ravi Bhadauria May 10, 2010 Abstract In this project, we investigate the shear properties of finite sized
More informationMolecular Dynamics Simulation of Fracture of Graphene
Molecular Dynamics Simulation of Fracture of Graphene Dewapriya M. A. N. 1, Rajapakse R. K. N. D. 1,*, Srikantha Phani A. 2 1 School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada
More informationEdge-stress induced warping of graphene sheets and nanoribbons
University of Massachusetts Amherst From the SelectedWorks of Ashwin Ramasubramaniam December, 2008 Edge-stress induced warping of graphene sheets and nanoribbons Ashwin Ramasubramaniam, University of
More informationSupplementary Information for Saddles, Twists, and Curls: Shape Transitions in Freestanding Nanoribbons
Supplementary Information for Saddles, Twists, and Curls: Shape Transitions in Freestanding Nanoribbons Hailong Wang and Moneesh Upmanyu Group for Simulation and Theory of Atomic-Scale Material Phenomena
More informationNonlinear mechanics of single-atomic-layer graphene sheets
Nonlinear mechanics of single-atomic-layer graphene sheets Qiang Lu and Rui Huang Department of Aerospace Engineering and Engineering mechanics, University of Texas, Austin, TX 7871, USA Abstract The unique
More informationA New Extension of Cauchy Born Rule for Monolayer Crystal Films
Nanoscale Res Lett (2010) 5:863 867 DOI 10.1007/s11671-010-9576-3 NANO EXPRESS A New Extension of Cauchy Born Rule for Monolayer Crystal Films Sheng Lu Chongdu Cho Received: 23 February 2010 / Accepted:
More informationQuasi-continuum Non-local Plate and Shell Models of Carbon-Based 2D Nanomaterials
Clemson University TigerPrints All Dissertations Dissertations 5-2018 Quasi-continuum Non-local Plate and Shell Models of Carbon-Based 2D Nanomaterials Jixuan Gong Clemson University, jixuang@clemson.edu
More informationMOLECULAR SIMULATION FOR PREDICTING MECHANICAL STRENGTH OF 3-D JUNCTIONED CARBON NANOSTRUCTURES
ECCM16-16 TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS, Seville, Spain, 22-26 June 214 MOLECULAR SIMULATION FOR PREDICTING MECHANICAL STRENGTH OF 3-D JUNCTIONED CARBON NANOSTRUCTURES S. Sihn a,b*, V.
More informationInternal lattice relaxation of single-layer graphene under in-plane deformation. Jun Zhou and Rui Huang
Internal lattice relaxation of single-layer graphene under in-plane deformation Jun Zhou and Rui Huang Department of Aerospace Engineering and Engineering Mechanics, University of Texas, Austin, TX 7871
More informationSupplementary Information. for. Origami based Mechanical Metamaterials
Supplementary Information for Origami based Mechanical Metamaterials By Cheng Lv, Deepakshyam Krishnaraju, Goran Konjevod, Hongyu Yu, and Hanqing Jiang* [*] Prof. H. Jiang, C. Lv, D. Krishnaraju, Dr. G.
More informationCritical Strain of Carbon Nanotubes: An Atomic-Scale Finite Element Study
X. Guo A. Y. T. Leung 1 e-mail: bcaleung@cityu.edu.hk Department of Building and Construction, City University of Hong Kong, Hong Kong, China H. Jiang Department of Mechanical and Aerospace Engineering,
More informationEffects of mismatch strain and substrate surface corrugation on morphology of supported monolayer graphene
JOURNAL OF APPLIED PHYSICS 7, 353 Effects of mismatch strain and substrate surface corrugation on morphology of supported monolayer graphene Zachary H. Aitken and Rui Huang a Department of Aerospace Engineering
More informationIntensity (a.u.) Intensity (a.u.) Raman Shift (cm -1 ) Oxygen plasma. 6 cm. 9 cm. 1mm. Single-layer graphene sheet. 10mm. 14 cm
Intensity (a.u.) Intensity (a.u.) a Oxygen plasma b 6 cm 1mm 10mm Single-layer graphene sheet 14 cm 9 cm Flipped Si/SiO 2 Patterned chip Plasma-cleaned glass slides c d After 1 sec normal Oxygen plasma
More informationMECHANICS OF 2D MATERIALS
MECHANICS OF 2D MATERIALS Nicola Pugno Cambridge February 23 rd, 2015 2 Outline Stretching Stress Strain Stress-Strain curve Mechanical Properties Young s modulus Strength Ultimate strain Toughness modulus
More informationA short review of continuum mechanics
A short review of continuum mechanics Professor Anette M. Karlsson, Department of Mechanical ngineering, UD September, 006 This is a short and arbitrary review of continuum mechanics. Most of this material
More informationDislocation network structures in 2D bilayer system
Dislocation network structures in 2D bilayer system Shuyang DAI School of Mathematics and Statistics Wuhan University Joint work with: Prof. Yang XIANG, HKUST Prof. David SROLOVITZ, UPENN S. Dai IMS Workshop,
More informationNonlinear Dynamics of Wrinkle Growth and
Nonlinear Dynamics of Wrinkle Growth and Pattern Formation in Stressed Elastic Thin Films Se Hyuk Im and Rui Huang Center for Mechanics of Solids, Structures and Materials Department of Aerospace Engineering
More informationStability analysis of atomic structures
University of Iowa Iowa Research Online Theses and Dissertations 2006 Stability analysis of atomic structures Liang Zhang University of Iowa Copyright 2006 Liang Zhang This dissertation is available at
More informationA Review on Mechanics and Mechanical Properties of 2D Materials Graphene and Beyond
A Review on Mechanics and Mechanical Properties of 2D Materials Graphene and Beyond Deji Akinwande 1, Christopher J. Brennan 1, J. Scott Bunch 2, Philip Egberts 3, Jonathan R. Felts 4, Huajian Gao 5, Rui
More informationFundamentals of Linear Elasticity
Fundamentals of Linear Elasticity Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy
More informationPrediction of Young s Modulus of Graphene Sheets by the Finite Element Method
American Journal of Mechanical Engineering, 15, Vol. 3, No. 6, 5-9 Available online at http://pubs.sciepub.com/ajme/3/6/14 Science and Education Publishing DOI:1.1691/ajme-3-6-14 Prediction of Young s
More informationSmall-Scale Effect on the Static Deflection of a Clamped Graphene Sheet
Copyright 05 Tech Science Press CMC, vol.8, no., pp.03-7, 05 Small-Scale Effect on the Static Deflection of a Clamped Graphene Sheet G. Q. Xie, J. P. Wang, Q. L. Zhang Abstract: Small-scale effect on the
More informationTinselenidene: a Two-dimensional Auxetic Material with Ultralow Lattice Thermal Conductivity and Ultrahigh Hole Mobility
Tinselenidene: a Two-dimensional Auxetic Material with Ultralow Lattice Thermal Conductivity and Ultrahigh Hole Mobility Li-Chuan Zhang, Guangzhao Qin, Wu-Zhang Fang, Hui-Juan Cui, Qing-Rong Zheng, Qing-Bo
More informationBasic 8 Micro-Nano Materials Science. and engineering
Basic 8 Micro-Nano Materials Science and Analysis Atomistic simulations in materials science and engineering Assistant Prof. Y. Kinoshita and Prof. N. Ohno Dept. of Comp. Sci. Eng. and Dept. of Mech. Sci.
More informationLarge Deformation of Hydrogels Coupled with Solvent Diffusion Rui Huang
Large Deformation of Hydrogels Coupled with Solvent Diffusion Rui Huang Center for Mechanics of Solids, Structures and Materials Department of Aerospace Engineering and Engineering Mechanics The University
More informationSupporting information for Polymer interactions with Reduced Graphene Oxide: Van der Waals binding energies of Benzene on defected Graphene
Supporting information for Polymer interactions with Reduced Graphene Oxide: Van der Waals binding energies of Benzene on defected Graphene Mohamed Hassan, Michael Walter *,,, and Michael Moseler, Freiburg
More informationXI. NANOMECHANICS OF GRAPHENE
XI. NANOMECHANICS OF GRAPHENE Carbon is an element of extraordinary properties. The carbon-carbon bond possesses large magnitude cohesive strength through its covalent bonds. Elemental carbon appears in
More informationMechanical Stabilities and Properties of Graphene-like Two-Dimensional. III-Nitrides
Mechanical Stabilities and Properties of Graphene-like Two-Dimensional III-Nitrides Lai Jiang, Zehao Yang, and Qing Peng* Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic
More informationLecture contents. Stress and strain Deformation potential. NNSE 618 Lecture #23
1 Lecture contents Stress and strain Deformation potential Few concepts from linear elasticity theory : Stress and Strain 6 independent components 2 Stress = force/area ( 3x3 symmetric tensor! ) ij ji
More informationContinuum Mechanics. Continuum Mechanics and Constitutive Equations
Continuum Mechanics Continuum Mechanics and Constitutive Equations Continuum mechanics pertains to the description of mechanical behavior of materials under the assumption that the material is a uniform
More informationMECHANICAL PROPERTIES OF GRAPHENE NANORIBBONS: A SELECTIVE REVIEW OF COMPUTER SIMULATIONS
Mechanical Rev. Adv. Mater. properties Sci. 40 of (2015) graphene 249-256 nanoribbons: a selective review of computer simulations 249 MECHANICAL PROPERTIES OF GRAPHENE NANORIBBONS: A SELECTIVE REVIEW OF
More informationBasic concepts to start Mechanics of Materials
Basic concepts to start Mechanics of Materials Georges Cailletaud Centre des Matériaux Ecole des Mines de Paris/CNRS Notations Notations (maths) (1/2) A vector v (element of a vectorial space) can be seen
More information3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1
Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is
More informationEffects of Free Edges and Vacancy Defects on the Mechanical Properties of Graphene
Proceedings of the 14th IEEE International Conference on Nanotechnology Toronto, Canada, August 18-21, 214 Effects of Free Edges and Vacancy Defects on the Mechanical Properties of Graphene M. A. N. Dewapriya
More informationTorsional Buckling of Double-Walled Carbon Nanotubes
Torsional Buckling of Double-Walled Carbon Nanotubes S. H. Soong Engineering Science Programme, National University of Singapore Kent Ridge, Singapore 119260 ABSTRACT This paper is concerned with the torsional
More informationRanges of Applicability for the Continuum-beam Model in the Constitutive Analysis of Carbon Nanotubes: Nanotubes or Nano-beams?
NASA/CR-2001-211013 ICASE Report No. 2001-16 Ranges of Applicability for the Continuum-beam Model in the Constitutive Analysis of Carbon Nanotubes: Nanotubes or Nano-beams? Vasyl Michael Harik ICASE, Hampton,
More informationFE modelling of multi-walled carbon nanotubes
Estonian Journal of Engineering, 2009, 15, 2, 77 86 doi: 10.3176/eng.2009.2.01 FE modelling of multi-walled carbon nanotubes Marino Brcic, Marko Canadija, Josip Brnic, Domagoj Lanc, Sanjin Krscanski and
More informationCHARACTERIZING MECHANICAL PROPERTIES OF GRAPHITE USING MOLECULAR DYNAMICS SIMULATION
CHARACTERIZING MECHANICAL PROPERTIES OF GRAPHITE USING MOLECULAR DYNAMICS SIMULATION Jia-Lin Tsai and Jie-Feng Tu Department of Mechanical Engineering, National Chiao Tung University 1001 University Road,
More informationBasic Energy Principles in Stiffness Analysis
Basic Energy Principles in Stiffness Analysis Stress-Strain Relations The application of any theory requires knowledge of the physical properties of the material(s) comprising the structure. We are limiting
More informationAnalysis of the Monotonic and Cyclic Behaviour of Graphene
Analysis of the Monotonic and Cyclic Behaviour of Graphene Diogo Manuel Nunes Galhofo diogogalhofo@tecnico.ulisboa.pt Instituto Superior Técnico, Universidade de Lisboa, Portugal April 217 Abstract The
More informationEngineering Sciences 241 Advanced Elasticity, Spring Distributed Thursday 8 February.
Engineering Sciences 241 Advanced Elasticity, Spring 2001 J. R. Rice Homework Problems / Class Notes Mechanics of finite deformation (list of references at end) Distributed Thursday 8 February. Problems
More informationCellular solid structures with unbounded thermal expansion. Roderic Lakes. Journal of Materials Science Letters, 15, (1996).
1 Cellular solid structures with unbounded thermal expansion Roderic Lakes Journal of Materials Science Letters, 15, 475-477 (1996). Abstract Material microstructures are presented which can exhibit coefficients
More information**********************************************************************
Department of Civil and Environmental Engineering School of Mining and Petroleum Engineering 3-33 Markin/CNRL Natural Resources Engineering Facility www.engineering.ualberta.ca/civil Tel: 780.492.4235
More informationIAP 2006: From nano to macro: Introduction to atomistic modeling techniques and application in a case study of modeling fracture of copper (1.
IAP 2006: From nano to macro: Introduction to atomistic modeling techniques and application in a case study of modeling fracture of copper (1.978 PDF) http://web.mit.edu/mbuehler/www/teaching/iap2006/intro.htm
More informationTransversal electric field effect in multilayer graphene nanoribbon
Transversal electric field effect in multilayer graphene nanoribbon S. Bala kumar and Jing Guo a) Department of Electrical and Computer Engineering, University of Florida, Gainesville, Florida 32608, USA
More informationUnit 18 Other Issues In Buckling/Structural Instability
Unit 18 Other Issues In Buckling/Structural Instability Readings: Rivello Timoshenko Jones 14.3, 14.5, 14.6, 14.7 (read these at least, others at your leisure ) Ch. 15, Ch. 16 Theory of Elastic Stability
More informationSupplementary Figures
Supplementary Figures 8 6 Energy (ev 4 2 2 4 Γ M K Γ Supplementary Figure : Energy bands of antimonene along a high-symmetry path in the Brillouin zone, including spin-orbit coupling effects. Empty circles
More informationTwo-dimensional Phosphorus Carbide as Promising Anode Materials for Lithium-ion Batteries
Electronic Supplementary Material (ESI) for Journal of Materials Chemistry A. This journal is The Royal Society of Chemistry 2018 Supplementary Material for Two-dimensional Phosphorus Carbide as Promising
More informationNanomechanics of carbon nanotubes and composites
Nanomechanics of carbon nanotubes and composites Deepak Srivastava and Chenyu Wei Computational Nanotechnology, NASA Ames Research Center, Moffett Field, California 94035-1000; deepak@nas.nasa.gov Kyeongjae
More informationMechanical Engineering Ph.D. Preliminary Qualifying Examination Solid Mechanics February 25, 2002
student personal identification (ID) number on each sheet. Do not write your name on any sheet. #1. A homogeneous, isotropic, linear elastic bar has rectangular cross sectional area A, modulus of elasticity
More informationMHA042 - Material mechanics: Duggafrågor
MHA042 - Material mechanics: Duggafrågor 1) For a static uniaxial bar problem at isothermal (Θ const.) conditions, state principle of energy conservation (first law of thermodynamics). On the basis of
More informationRippling and Ripping Graphene
Rippling and Ripping Graphene Michael Marder Professor of Physics Center for Nonlinear Dynamics and Department of Physics The University of Texas at Austin Clifton Symposium, June 2012, Symi Marder (UT
More informationLecture 8. Stress Strain in Multi-dimension
Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]
More informationSupporting Information
Supporting Information Failure Processes in Embedded Monolayer Graphene under Axial Compression Charalampos Androulidakis, Emmanuel N. Koukaras, Otakar Frank, Georgia Tsoukleri, Dimitris Sfyris, John Parthenios,
More informationHomogenized Elastic Properties of Graphene for Small Deformations
Materials 2013, 6, 3764-3782; doi:10.3390/ma6093764 Article OPEN ACCESS materials ISSN 1996-1944 www.mdpi.com/journal/materials Homogenized Elastic Properties of Graphene for Small Deformations Eduard
More informationNanoscale Mechanics: A Quantum-Continuum Approach
Nanoscale Mechanics: A Quantum-Continuum Approach V.Venkatasubramanian a and S.Bharath b a Department of Mechanical Engineering, IIT Madras b Department of Chemical Engineering, IIT Madras 1. Introduction
More informationComposite materials: mechanical properties
Composite materials: mechanical properties A computational lattice model describing scale effects @ nano-scale Luciano Colombo In collaboration with: Pier Luca Palla and Stefano Giordano Department of
More informationσ) 6] (1) ( r i Mechanics of C 60 in Nanotubes Dong Qian, Wing Kam Liu, and Rodney S. Ruoff*
J. Phys. Chem. B 2001, 105, 10753-10758 10753 Mechanics of C 60 in Nanotubes Dong Qian, Wing Kam Liu, and Rodney S. Ruoff* Northwestern UniVersity, Department of Mechanical Engineering, EVanston, Illinois
More information(MPa) compute (a) The traction vector acting on an internal material plane with normal n ( e1 e
EN10: Continuum Mechanics Homework : Kinetics Due 1:00 noon Friday February 4th School of Engineering Brown University 1. For the Cauchy stress tensor with components 100 5 50 0 00 (MPa) compute (a) The
More informationMECHANICS OF CARBON NANOTUBE BASED COMPOSITES WITH MOLECULAR DYNAMICS AND MORI TANAKA METHODS. Vinu Unnithan and J. N. Reddy
MECHANICS OF CARBON NANOTUBE BASED COMPOSITES WITH MOLECULAR DYNAMICS AND MORI TANAKA METHODS Vinu Unnithan and J. N. Reddy US-South American Workshop: Mechanics and Advanced Materials Research and Education
More informationChapter 2: Elasticity
OHP 1 Mechanical Properties of Materials Chapter 2: lasticity Prof. Wenjea J. Tseng ( 曾文甲 ) Department of Materials ngineering National Chung Hsing University wenjea@dragon.nchu.edu.tw Reference: W.F.
More informationA finite deformation membrane based on inter-atomic potentials for single atomic layer films Application to the mechanics of carbon nanotubes
A finite deformation membrane based on inter-atomic potentials for single atomic layer films Application to the mechanics of carbon nanotubes Marino Arroyo and Ted Belytschko Department of Mechanical Engineering
More informationMethods of Continual Modeling for Graphitic Systems: Scrolling at Nanoscale
SV Rotkin 1 Methods of Continual Modeling for Graphitic Systems: Scrolling at Nanoscale SV Rotkin 2 Scrolling at the Nanoscale ~2-4 nm Material properties of the layered lattice of the graphite define
More informationStrain-engineered artificial atom as a broad-spectrum solar energy funnel
Supplementary Information Strain-engineered artificial atom as a broad-spectrum solar energy funnel Ji Feng 1,, Xiaofeng Qian 2,, Cheng-Wei Huang 2 and Ju Li 2,3, * 1 International Center for Quantum Materials,
More informationCHAPTER 5. Beam Theory
CHPTER 5. Beam Theory SangJoon Shin School of Mechanical and erospace Engineering Seoul National University ctive eroelasticity and Rotorcraft Lab. 5. The Euler-Bernoulli assumptions One of its dimensions
More informationSEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by
SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e
More informationA Phenomenological Model for Self-rippling Energy of Free Graphene Monolayer and Its Application
A Phenomenological Model for Self-rippling Energy of Free Graphene Monolayer and Its Application By Bing Jie Wu A thesis submitted in partial fulfillment of the requirements for the degree of Master of
More informationSTANDARD SAMPLE. Reduced section " Diameter. Diameter. 2" Gauge length. Radius
MATERIAL PROPERTIES TENSILE MEASUREMENT F l l 0 A 0 F STANDARD SAMPLE Reduced section 2 " 1 4 0.505" Diameter 3 4 " Diameter 2" Gauge length 3 8 " Radius TYPICAL APPARATUS Load cell Extensometer Specimen
More informationInternal lattice relaxation of single-layer graphene under in-plane deformation
Journal of the Mechanics and Physics of Solids 56 (2008) 1609 1623 www.elsevier.com/locate/jmps Internal lattice relaxation of single-layer graphene under in-plane deformation Jun Zhou, Rui Huang Department
More informationIntroduction to Aerospace Engineering
Introduction to Aerospace Engineering Lecture slides Challenge the future 1 Aircraft & spacecraft loads Translating loads to stresses Faculty of Aerospace Engineering 29-11-2011 Delft University of Technology
More informationSurface stress and relaxation in metals
J. Phys.: Condens. Matter 12 (2000) 5541 5550. Printed in the UK PII: S0953-8984(00)11386-4 Surface stress and relaxation in metals P M Marcus, Xianghong Qian and Wolfgang Hübner IBM Research Center, Yorktown
More information3.22 Mechanical Properties of Materials Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 3.22 Mechanical Properties of Materials Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Problem Set
More informationTwo problems in finite elasticity
University of Wollongong Research Online University of Wollongong Thesis Collection 1954-2016 University of Wollongong Thesis Collections 2009 Two problems in finite elasticity Himanshuki Nilmini Padukka
More informationPURE BENDING. If a simply supported beam carries two point loads of 10 kn as shown in the following figure, pure bending occurs at segment BC.
BENDING STRESS The effect of a bending moment applied to a cross-section of a beam is to induce a state of stress across that section. These stresses are known as bending stresses and they act normally
More informationAn Atomistic-based Cohesive Zone Model for Quasi-continua
An Atomistic-based Cohesive Zone Model for Quasi-continua By Xiaowei Zeng and Shaofan Li Department of Civil and Environmental Engineering, University of California, Berkeley, CA94720, USA Extended Abstract
More informationFree Vibrations of Carbon Nanotubes with Defects
Mechanics and Mechanical Engineering Vol. 17, No. 2 (2013) 157 166 c Lodz University of Technology Free Vibrations of Carbon Nanotubes with Defects Aleksander Muc Aleksander Banaś Ma lgorzata Chwa l Institute
More informationConstitutive Relations
Constitutive Relations Dr. Andri Andriyana Centre de Mise en Forme des Matériaux, CEMEF UMR CNRS 7635 École des Mines de Paris, 06904 Sophia Antipolis, France Spring, 2008 Outline Outline 1 Review of field
More informationPhysics of Continuous media
Physics of Continuous media Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2012 October 26, 2012 Deformations of continuous media If a body is deformed, we say that the point which originally had
More informationElements of Rock Mechanics
Elements of Rock Mechanics Stress and strain Creep Constitutive equation Hooke's law Empirical relations Effects of porosity and fluids Anelasticity and viscoelasticity Reading: Shearer, 3 Stress Consider
More informationLectures on. Constitutive Modelling of Arteries. Ray Ogden
Lectures on Constitutive Modelling of Arteries Ray Ogden University of Aberdeen Xi an Jiaotong University April 2011 Overview of the Ingredients of Continuum Mechanics needed in Soft Tissue Biomechanics
More informationChapter 7. Highlights:
Chapter 7 Highlights: 1. Understand the basic concepts of engineering stress and strain, yield strength, tensile strength, Young's(elastic) modulus, ductility, toughness, resilience, true stress and true
More informationApplication to modeling brittle materials
1.01, 3.01, 10.333,.00 Introduction to Modeling and Simulation Spring 011 Part I Continuum and particle methods Application to modeling brittle materials Lecture 7 Markus J. Buehler Laboratory for Atomistic
More informationMechanics PhD Preliminary Spring 2017
Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n
More informationheight trace of a 2L BN mechanically exfoliated on SiO 2 /Si with pre-fabricated micro-wells. Scale bar 2 µm.
Supplementary Figure 1. Few-layer BN nanosheets. AFM image and the corresponding height trace of a 2L BN mechanically exfoliated on SiO 2 /Si with pre-fabricated micro-wells. Scale bar 2 µm. Supplementary
More informationBending of Simply Supported Isotropic and Composite Laminate Plates
Bending of Simply Supported Isotropic and Composite Laminate Plates Ernesto Gutierrez-Miravete 1 Isotropic Plates Consider simply a supported rectangular plate of isotropic material (length a, width b,
More informationarxiv:cond-mat/ v1 [cond-mat.soft] 29 May 2002
Stretching Instability of Helical Springs David A. Kessler and Yitzhak Rabin Dept. of Physics, Bar-Ilan University, Ramat-Gan, Israel (Dated: October 31, 18) arxiv:cond-mat/05612v1 [cond-mat.soft] 29 May
More informationLinear Constitutive Relations in Isotropic Finite Viscoelasticity
Journal of Elasticity 55: 73 77, 1999. 1999 Kluwer Academic Publishers. Printed in the Netherlands. 73 Linear Constitutive Relations in Isotropic Finite Viscoelasticity R.C. BATRA and JANG-HORNG YU Department
More informationHomework Problems. ( σ 11 + σ 22 ) 2. cos (θ /2), ( σ θθ σ rr ) 2. ( σ 22 σ 11 ) 2
Engineering Sciences 47: Fracture Mechanics J. R. Rice, 1991 Homework Problems 1) Assuming that the stress field near a crack tip in a linear elastic solid is singular in the form σ ij = rλ Σ ij (θ), it
More informationClar Sextet Theory for low-dimensional carbon nanostructures: an efficient approach based on chemical criteria
Clar Sextet Theory for low-dimensional carbon nanostructures: an efficient approach based on chemical criteria Matteo Baldoni Fachbereich Chemie, Technische Universität Dresden, Germany Department of Chemistry
More information1 Nonlinear deformation
NONLINEAR TRUSS 1 Nonlinear deformation When deformation and/or rotation of the truss are large, various strains and stresses can be defined and related by material laws. The material behavior can be expected
More informationBuckling of Double-walled Carbon Nanotubes
Buckling o Double-walled Carbon anotubes Y. H. Teo Engineering Science Programme ational University o Singapore Kent idge Singapore 960 Abstract This paper is concerned with the buckling o double-walled
More informationStrength and Stability Analysis of a Single Walled Black Phosphorus Tube under Axial Compression
Strength and Stability Analysis of a Single Walled Black Phosphorus Tube under Axial Compression Kun Cai 1, 2, Jing Wan 1, Ning Wei 1, Qinghua Qin 2* 1 College of Water Resources and Architectural Engineering,
More informationStability analysis of graphite crystal lattice with moment interactions
Proc. of XXXIV Summer School "Advanced Problems in Mechanics", St.-Petersburg, Russia. 2007. (Accepted) Stability analysis of graphite crystal lattice with moment interactions Igor E. Berinskiy berigor@mail.ru
More informationElastic properties of graphene
Elastic properties of graphene M. I. Katsnelson P. Le Doussal B. Horowitz K. Wiese J. Gonzalez P. San-Jose V. Parente B. Amorim R. Roldan C. Gomez-Navarro J. Gomez G. Lopez-Polin F. Perez-Murano A. Morpurgo
More informationExercise: concepts from chapter 8
Reading: Fundamentals of Structural Geology, Ch 8 1) The following exercises explore elementary concepts associated with a linear elastic material that is isotropic and homogeneous with respect to elastic
More informationThe Rotating Inhomogeneous Elastic Cylinders of. Variable-Thickness and Density
Applied Mathematics & Information Sciences 23 2008, 237 257 An International Journal c 2008 Dixie W Publishing Corporation, U. S. A. The Rotating Inhomogeneous Elastic Cylinders of Variable-Thickness and
More informationA FAILURE CRITERION FOR POLYMERS AND SOFT BIOLOGICAL MATERIALS
Material Technology A FALURE CRTERON FOR POLYMERS AND SOFT BOLOGCAL MATERALS Authors: William W. Feng John O. Hallquist Livermore Software Technology Corp. 7374 Las Positas Road Livermore, CA 94550 USA
More informationBand structure engineering of graphene by strain: First-principles calculations
Band structure engineering of graphene by strain: First-principles calculations Gui Gui, Jin Li, and Jianxin Zhong* Laboratory for Quantum Engineering and Micro-Nano Energy Technology, Xiangtan University,
More informationMechanical Properties of Materials
Mechanical Properties of Materials Strains Material Model Stresses Learning objectives Understand the qualitative and quantitative description of mechanical properties of materials. Learn the logic of
More information