A CONTINUUM MECHANICAL SURROGATE MODEL FOR ATOMIC BEAM STRUCTURES

Starting from a fully atomistic system, we outline a general approach to obtain an approximate continuum surrogate model incorporating specific kinematic state variables. The continuum mechanical system is furnished with a hyperelastic material model. We then adapt the procedure to slender structures with beam-like character, such as Silicon nanowires or carbon nanotubes. The surrogate model can be described as a geometrically exact beam, which can be treated numerically using finite elements. Based on molecular dynamics simulations, we show how to obtain for a given atomistic beam system both a set of suitable deformed states as well as generalized stress and strain measures. Finally, we benchmark the obtained continuum model by assessing its accuracy for a beam coming into contact with an infinite Lennard-Jones wall.

[1]  Stefan Hollerer.  Numerical validation of a concurrent atomistic-continuum multiscale method and its application to the buckling analysis of carbon nanotubes , 2014 .

[2]  Dinar Camotim,et al.  NONLINEAR GENERALIZED BEAM THEORY FOR COLD-FORMED STEEL MEMBERS , 2003 .

[3]  Animesh Agarwal,et al.  Observed mechanism for the breakup of small bundles of cellulose Iα and Iβ in ionic liquids from molecular dynamics simulations. , 2013, The journal of physical chemistry. B.

[4]  P. Lidström.  On the equations of motion for curved slender beams using tubular coordinates , 2014 .

[5]  George C. Schatz,et al.  An atomistic-continuum Cosserat rod model of carbon nanotubes , 2009 .

[6]  W. Duan,et al.  Assessment of continuum mechanics models in predicting buckling strains of single-walled carbon nanotubes , 2009, Nanotechnology.

[7]  Michael Griebel,et al.  Molecular dynamics simulations of the elastic moduli of polymer–carbon nanotube composites , 2004 .

[8]  G. Odegarda,et al.  The stress – strain behavior of polymer – nanotube composites from molecular dynamics simulation , 2003 .

[9]  E Weinan,et al.  Generalized Cauchy-Born rules for elastic deformation of sheets, plates, and rods: Derivation of continuum models from atomistic models , 2006 .

[10]  D. Burton,et al.  A Cosserat rod model with microstructure , 2006 .

[11]  Richard D. James,et al.  A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods , 2000 .

[12]  Y. S. Zhang,et al.  Size dependence of Young's modulus in ZnO nanowires. , 2006, Physical review letters.

[13]  D. M. Duffy,et al.  Including the effects of electronic stopping and electron–ion interactions in radiation damage simulations , 2007 .

[14]  Stuart S. Antman,et al.  The Theory of Rods , 1973 .

[15]  S. Mukherjee,et al.  A Geometrically Exact Rod Model Including In-Plane Cross-Sectional Deformation , 2011 .

[16]  Sanjay Govindjee,et al.  On the use of continuum mechanics to estimate the properties of nanotubes , 1999 .

[17]  Richard B. Lehoucq,et al.  Translation of Walter Noll’s “Derivation of the Fundamental Equations of Continuum Thermodynamics from Statistical Mechanics” , 2010 .

[18]  N. A. Marks,et al.  NanoCap: A framework for generating capped carbon nanotubes and fullerenes , 2014, Comput. Phys. Commun..

[19]  Christian J. Cyron,et al.  A torsion-free non-linear beam model , 2014 .

[20]  Gordan Jelenić,et al.  Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics , 1999 .

[21]  Jacob Fish,et al.  Generalized mathematical homogenization of atomistic media at finite temperatures in three dimensions , 2007 .

[22]  Jacob Fish,et al.  A generalized space–time mathematical homogenization theory for bridging atomistic and continuum scales , 2006 .

[23]  A. Luongo,et al.  A non-linear one-dimensional model of cross-deformable tubular beam , 2014 .

[24]  Robert E. Rudd,et al.  First-principles study of the Young’s modulus of Si ⟨001⟩ nanowires , 2006, cond-mat/0611073.

[25]  J. Tinsley Oden,et al.  Adaptive multiscale modeling of polymeric materials with Arlequin coupling and Goals algorithms , 2009 .

[26]  R. Ansari,et al.  Continuum and molecular dynamics study of C60 fullerene–carbon nanotube oscillators , 2013 .

[27]  K Schulten,et al.  VMD: visual molecular dynamics. , 1996, Journal of molecular graphics.

[28]  J. C. Simo,et al.  A finite strain beam formulation. The three-dimensional dynamic problem. Part I , 1985 .

[29]  S. Sinnott,et al.  Elastic torsional responses of carbon nanotube systems , 2007 .

[30]  R. James,et al.  Objective quasicontinuum approach for rod problems , 2012 .

[31]  K. M. Liew,et al.  Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction , 2005 .

[32]  Wei Lu,et al.  Mechanical properties of vapor-liquid-solid synthesized silicon nanowires. , 2009, Nano letters.

[33]  C. Bingham,et al.  Single-walled carbon nanotube modelling based on Cosserat surface theory , 2013 .

[34]  B. Schmidt.  On the Passage from Atomic to Continuum Theory for Thin Films , 2008 .

[35]  Ignacio Romero,et al.  The interpolation of rotations and its application to finite element models of geometrically exact rods , 2004 .

[36]  Phaedon Avouris,et al.  Deformation of carbon nanotubes by surface van der Waals forces , 1998 .

[37]  Dong Qian,et al.  Mechanics of carbon nanotubes , 2002 .

[38]  Ignacio Romero,et al.  An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy–momentum conserving scheme in dynamics , 2002 .

[39]  Weber,et al.  Computer simulation of local order in condensed phases of silicon. , 1985, Physical review. B, Condensed matter.

[40]  E. Tadmor,et al.  Stress and heat flux for arbitrary multibody potentials: a unified framework. , 2011, The Journal of chemical physics.

[41]  S. Leyendecker,et al.  Objective energy-momentum conserving integration for the constrained dynamics of geometrically exact beams , 2006 .

[42]  M. Hodak,et al.  Carbon nanotubes, buckyballs, ropes, and a universal graphitic potential , 2000 .

[43]  W. Cai,et al.  Size and temperature effects on the fracture mechanisms of silicon nanowires: Molecular dynamics simulations , 2010 .

[44]  R. Sauer,et al.  Multiscale treatment of mechanical contact problems involving thin polymeric layers , 2014 .

[45]  C. Hui,et al.  Elastica solution for a nanotube formed by self-adhesion of a folded thin film , 2004 .

[46]  P. Steinmann,et al.  Reviewing the roots of continuum formulations in molecular systems. Part III: Stresses, couple stresses, heat fluxes , 2015 .

[47]  Ted Belytschko,et al.  Mechanics of defects in carbon nanotubes: Atomistic and multiscale simulations , 2005 .

[48]  Hongzhi Zhong,et al.  A quaternion-based weak form quadrature element formulation for spatial geometrically exact beams , 2014 .

[49]  R. Hardy,et al.  Formulas for determining local properties in molecular‐dynamics simulations: Shock waves , 1982 .

[50]  Miran Saje,et al.  The quaternion-based three-dimensional beam theory , 2009 .

[51]  Huajian Gao,et al.  Fracture Nucleation in Single-Wall Carbon Nanotubes Under Tension: A Continuum Analysis Incorporating Interatomic Potentials , 2002 .

[52]  Youping Chen,et al.  Connecting molecular dynamics to micromorphic theory. (I). Instantaneous and averaged mechanical variables , 2003 .

[53]  Andreas Öchsner,et al.  On the determination of the shear modulus of carbon nanotubes , 2013 .

[54]  A. Masud,et al.  A multiscale framework for computational nanomechanics: Application to the modeling of carbon nanotubes , 2009 .

[55]  Amir R. Khoei,et al.  Effects of temperature and torsion speed on torsional properties of single-walled carbon nanotubes , 2011 .

[56]  Ted Belytschko,et al.  Finite element methods for the non‐linear mechanics of crystalline sheets and nanotubes , 2004 .

[57]  Jian Ping Lu.  Elastic Properties of Carbon Nanotubes and Nanoropes , 1997 .

[58]  Wolfgang A Wall,et al.  Consistent finite-element approach to Brownian polymer dynamics with anisotropic friction. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[59]  Benjamin S. Kirk,et al.  Library for Parallel Adaptive Mesh Refinement / Coarsening Simulations , 2006 .

[60]  Youping Chen,et al.  Connecting molecular dynamics to micromorphic theory. (II). Balance laws , 2003 .

[61]  Peter Betsch,et al.  Director‐based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates , 2014 .

[62]  Steve Plimpton,et al.  Fast parallel algorithms for short-range molecular dynamics , 1993 .

[63]  T A Arias,et al.  Modeling a suspended nanotube oscillator. , 2005, Nano letters.

[64]  Ellad B. Tadmor,et al.  A Unified Interpretation of Stress in Molecular Systems , 2010, 1008.4819.

[65]  Mark F. Horstemeyer,et al.  Strain Tensors at the Atomic Scale , 1999 .

[66]  J. C. Simo,et al.  A Geometrically-exact rod model incorporating shear and torsion-warping deformation , 1991 .

[67]  J. C. Simo,et al.  On the dynamics of finite-strain rods undergoing large motions a geometrically exact approach , 1988 .

[68]  Hanqing Jiang,et al.  A Finite-Temperature Continuum Theory Based on Interatomic , 2005 .

[69]  E. M.,et al.  Statistical Mechanics , 2021, Manual for Theoretical Chemistry.

[70]  J. C. Simo,et al.  A three-dimensional finite-strain rod model. Part II: Computational aspects , 1986 .

[71]  Erik Dujardin,et al.  Young's modulus of single-walled nanotubes , 1998 .

[72]  Shaker A. Meguid,et al.  Nanomechanics of single and multiwalled carbon nanotubes , 2004 .

[73]  Ellad B. Tadmor,et al.  Modeling Materials: Continuum, Atomistic and Multiscale Techniques , 2011 .

[74]  Huajian Gao,et al.  Deformation gradients for continuum mechanical analysis of atomistic simulations , 2009 .

[75]  Tudor S. Ratiu,et al.  Asynchronous variational Lie group integration for geometrically exact beam dynamics , 2013 .

[76]  G. A. D. Briggs,et al.  Elastic and shear moduli of single-walled carbon nanotube ropes , 1999 .

[77]  S. K. Georgantzinos,et al.  Vibration analysis of multi-walled carbon nanotubes using a spring–mass based finite element model , 2009 .

[78]  Reese E. Jones,et al.  A material frame approach for evaluating continuum variables in atomistic simulations , 2008, J. Comput. Phys..

[79]  Gregory J. Wagner,et al.  A multiscale projection method for the analysis of carbon nanotubes , 2004 .

[80]  Qing Chen,et al.  The Very‐Low Shear Modulus of Multi‐Walled Carbon Nanotubes Determined Simultaneously with the Axial Young's Modulus via in situ Experiments , 2008 .

[81]  Peter Wriggers,et al.  Formulation and analysis of a three-dimensional finite element implementation for adhesive contact at the nanoscale , 2009 .

[82]  Angus I Kirkland,et al.  Resolving strain in carbon nanotubes at the atomic level. , 2011, Nature materials.

[83]  Elena Celledoni,et al.  A Hamiltonian and multi-Hamiltonian formulation of a rod model using quaternions , 2010 .

[84]  Wolfgang A. Wall,et al.  Micromechanical simulations of biopolymer networks with finite elements , 2013, J. Comput. Phys..

[85]  John R. Ray,et al.  Statistical ensembles and molecular dynamics studies of anisotropic solids , 1984 .

[86]  J. Tersoff,et al.  Modeling solid-state chemistry: Interatomic potentials for multicomponent systems. , 1989, Physical review. B, Condensed matter.

[87]  R. Jones,et al.  An atomistic-to-continuum coupling method for heat transfer in solids , 2008 .

[88]  S. Antman.  Nonlinear problems of elasticity , 1994 .

[89]  Roger A. Sauer,et al.  A geometrically exact finite beam element formulation for thin film adhesion and debonding , 2014 .

[90]  J. Kovac,et al.  Polymer conformational statistics. III. Modified Gaussian models of stiff chains , 1973 .

[91]  Vijay B. Shenoy,et al.  Size-dependent elastic properties of nanosized structural elements , 2000 .

[92]  Chang Liu,et al.  Mechanical properties of Si nanowires as revealed by in situ transmission electron microscopy and molecular dynamics simulations. , 2012, Nano letters.

[93]  Nicola Rizzi,et al.  The effects of warping constraints on the buckling of thin-walled structures , 2010 .

[94]  Dong Qian,et al.  Mechanical properties of carbon nanotubes: theoretical predictions and experimental measurements , 2003 .

[95]  S. Mukherjee,et al.  Finite element analysis of single-walled carbon nanotubes based on a rod model including in-plane cross-sectional deformation , 2013 .

[96]  Yong-gang Wang,et al.  Nonlinear buckling analyses of a small-radius carbon nanotube , 2014 .

[97]  Nicolas Triantafyllidis,et al.  Stability of crystalline solids - I: Continuum and atomic lattice considerations , 2006 .

[98]  Tienchong Chang,et al.  A molecular based anisotropic shell model for single-walled carbon nanotubes , 2010 .

[99]  D. Pinisetty,et al.  Molecular Dynamics Simulation of FCC Metallic Nanowires: A Review , 2012, JOM.

[100]  Joan Lasenby,et al.  Simo-Vu Quoc rods using Clifford algebra , 1999 .