An atomistic-based finite deformation membrane for single layer crystalline films

By Marino Arroyo, Ted Belytschko. 2002

367 citations recorded by the source corpus.

A general methodology to develop hyper-elastic membrane models applicable to crystalline films one-atom thick is presented. In this method, an extension of the Born rule based on the exponential map is proposed. The exponential map accounts for the fact that the lattice vectors of the crystal lie along the chords of the curved membrane, and consequently a tangent map like the standard Born rule is inadequate. In order to obtain practical methods, the exponential map is locally approximated. The effectiveness of our approach is demonstrated by numerical studies of carbon nanotubes. Deformed configurations as well as equilibrium energies of atomistic simulations are compared with those provided by the continuum membrane resulting from this method discretized by finite elements.

Subfields: Biomedical Engineering, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/de05020875f95ac4b37143ee855008c7c8e613a5

How researchers see this paper

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In Biomedical Engineering

1. Chunyu Li
2. Rui Huang
3. Ted Belytschko
4. Harold S. Park
5. T. Chou
6. Shaoping Xiao
7. Jun Zhou
8. Marino Arroyo

In Materials Chemistry

1. Rui Huang
2. Ju Li
3. Ting Zhu
4. R. Rafiee
5. Ted Belytschko
6. Jun Zhou
7. Wei Gao
8. Harold S. Park

Citations in the corpus

• Mechanical properties of single-walled carbon nanotubes based on higher order Cauchy–Born rule (2006, methodology): “(2000) and the exponential Cauchy–Born rule based results given by Arroyo and Belytschko (2002b) are also shown in Fig.”
• Finite element methods for the non‐linear mechanics of crystalline sheets and nanotubes (2004, methodology): “Inspired by the geometric structure of the Cauchy–Born rule in the context of the nite deformation kinematics of surfaces, an extension of the Cauchy–Born rule to account for the curvature of the lm was proposed in Reference [44].”
• The effect of nanotube radius on the constitutive model for carbon nanotubes (2003, background): “There are some recent efforts to develop continuum theories directly from atomistic models for bulk materials [28–32], nanoscale thin films [33], and nanoscale membranes [34].”
• A surface Cauchy–Born model for nanoscale materials (2006, methodology): “The approach can also be applied to more complicated interatomic interactions such as embedded atom (EAM) potentials [46], silicon [47], or carbon nanotubes [45, 48].”
• Shape control of active surfaces inspired by the movement of euglenids (2014, background): “By N we denote a unit normal to and by K the pull-back of the second fundamental form bIJ = x,IJ · N to 0, which is the curvature measure compatible with C (Arroyo and Belytschko, 2002).”
• On the Cauchy—Born Rule (2008, background): “Then, there are the different ideas proposed a bit later by Arroyo and Belytschko [44, 45] on films that I will describe, for the case of one involving a single layer of atoms.”
• Predictive modeling of nanoindentation-induced homogeneous dislocation nucleation in copper (2004, background): “Further extension of interatomic potential-based hyperelasticity has been made to other type of lattices, e.g., Tadmor et al. (1999b) and Smith et al. (2001) for nonsymmorphic crystal lattices such as diamond cubic Si, Arroyo and Belytschko (2002) for a one-atom thick carbon nanotubes, etc.”
• Thermomechanics of monolayer graphene: Rippling, thermal expansion and elasticity (2014, background): “Moreover, it is known in solid state physics that quantum effect dominates the low-temperature behavior for specific heat and thermal expansion (Ashcroft and Mermin, 1976), which is not accounted for in the classical MD simulations.”