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• Elastic bending modulus of monolayer graphene

  Q. Lu, M. Arroyo, R. Huang

  Journal of physics D: Applied physics (2009). Vol. 42 (10), pp. 1-6

  
  

  Abstract

  An analytic formula is derived for the elastic bending modulus of monolayer graphene based on an empirical potential for solid-state carbon atoms. Two physical origins are identified for the non-vanishing bending stiffness of the atomically thin graphene sheet, one due to the bond-angle effect and the other resulting from the bond-order term associated with the dihedral angles. The analytical prediction compares closely with ab initio energy calculations. Pure bending of graphene monolayers into cylindrical tubes is simulated by a molecular mechanics approach, showing slight nonlinearity and anisotropy in the tangent bending modulus as the bending curvature increases. An intrinsic coupling between bending and in-plane strain is noted for graphene monolayers rolled into carbon nanotubes.

  Abstract
  An analytic formula is derived for the elastic bending modulus of monolayer graphene based on an empirical potential for solid-state carbon atoms. Two physical origins are [...]

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• Smooth, second order, non-negative meshfree approximants selected by maximum entropy

  C. Cyron, M. Arroyo, M. Ortiz

  Int. J. Numer. Meth. Engng. (2009). Vol. 79 (13), pp. 1605-1632

  
  

  Abstract

  We present a family of approximation schemes, which we refer to as second-order maximum-entropy (max-ent) approximation schemes, that extends the first-order local max-ent approximation schemes to second-order consistency. This method retains the fundamental properties of first-order max-ent schemes, namely the shape functions are smooth, non-negative, and satisfy a weak Kronecker-delta property at the boundary. This last property makes the imposition of essential boundary conditions in the numerical solution of partial differential equations trivial. The evaluation of the shape functions is not explicit, but it is very efficient and robust. To our knowledge, the proposed method is the first higher-order scheme for function approximation from unstructured data in arbitrary dimensions with non-negative shape functions. As a consequence, the approximants exhibit variation diminishing properties, as well as an excellent behavior in structural vibrations problems as compared with the Lagrange finite elements, MLS-based meshfree methods and even B-Spline approximations, as shown through numerical experiments. When compared with usual MLS-based second-order meshfree methods, the shape functions presented here are much easier to integrate in a Galerkin approach, as illustrated by the standard benchmark problems.

  Abstract
  We present a family of approximation schemes, which we refer to as second-order maximum-entropy (max-ent) approximation schemes, that extends the first-order local max-ent [...]

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• Relaxation dynamics of fluid membranes

  M. Arroyo, A. DeSimone

  Physical review E: statistical, nonlinear and soft matter physics (2009). Vol. 79 (3), pp. 031915-1/031915-17

  
  

  Abstract

  We study the effect of membrane viscosity in the dynamics of liquid membranes{possibly with free or internal boundaries{ driven by conservative forces (curvature elasticity and line tension) and dragged by the bulk dissipation of the ambient fluid and the friction occurring when the amphiphilic molecules move relative to each other. To this end, we formulate a continuum model which includes a new form of the governing equations for a two-dimensional viscous fluid moving on a curved, time-evolving surface. The effect of membrane viscosity has received very limited attention in previous continuum studies of the dynamics of fluid membranes, although recent coarse-grained discrete simulations suggest its importance. By applying our model to the study of vesiculation and membrane fusion in a simpli ed geometry, we conclude that membrane viscosity plays a dominant role in the relaxation dynamics of fluid membranes of sizes comparable to those found in eukaryotic cells, and is not negligible in many large synthetic systems of current interest.

  Abstract
  We study the effect of membrane viscosity in the dynamics of liquid membranes{possibly with free or internal boundaries{ driven by conservative forces (curvature elasticity [...]

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• Effective coarse-grained simulations of super-thick multi-walled carbon nanotubes under torsion

  Z. Jian, X. Huang, M. Arroyo, S. Zhang

  Journal of Applied Physics (2009). Vol. 105 (3), pp. 33516/33516-8

  
  

  Abstract

  Under torsion and beyond the buckling point, multi-walled carbon nanotubes (MWCNTs) develop a periodic wave-like rippling morphology. Here, we show that torsional rippling deformations can be accurately described by a simple sinusoidal shape function. Combining this observation with the geometry optimization, we develop an effective coarse-grained model that reproduces the complex nonlinear mechanical responses of thick MWCNTs under torsion predicted by large-scale atomistic simulations. Furthermore, the model allows us to simulate super-thick tubes, inaccessible by other coarse-grained methods. With this effective coarse-grained model, we show from an energetic analysis that the rippling deformation is a result of in-plane strain energy relaxation, penalized by the increase in the interlayer van der Waals interaction energy. Our simulations reveal that the torsional response of MWCNTs with up to 100 layers approximately follows a simple bilinear law, and the ratio of the torsional rigidities in the pre- and post-buckling regimes is nearly a constant, independent of the tube radius. In contrast, the bifurcation torsional strain powerly scales with the tube radius. We also find that the wave number in the circumferential direction linearly increases with tube radius, while the wavelength monotonically increases with tube radius, and approaches a constant in the limit of bulk graphite. The bilinear constitutive relation, together with the scaling law of the bifurcation torsional strain, furnishes a simple nonlinear beam theory, which facilitates the analysis of MWCNT bundles and networks.

  Abstract
  Under torsion and beyond the buckling point, multi-walled carbon nanotubes (MWCNTs) develop a periodic wave-like rippling morphology. Here, we show that torsional rippling [...]

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• Optimization of a novel micro-opto-X-ray imaging lens

  H. Ostadi, M. Arroyo, P. Prewett, K. Jiang, S. Huq

  Microelestronic engineering (2008). Vol. 85 (5-6), pp. 1086-1088

  
  

  Abstract

  A mechanically deformable reflection transmission MOEMS system is capable of focusing X-rays to sub micron spots. This paper considers the geometry of the proposed deformed slotted microcantilever lens element under thermally derived strain, using finite element analysis. The work shows that an optimized MOEMS design using a slotted polyimide/gold thermal bimorph cantilever is capable of achieving ideal geometry with a larger than expected number of focusing slots – up to 111 at 2 μm width.

  Abstract
  A mechanically deformable reflection transmission MOEMS system is capable of focusing X-rays to sub micron spots. This paper considers the geometry of the proposed deformed [...]

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• Rippling and a phase-transforming mesoscopic model for multiwalled carbon nanotubes

  M. Arroyo, I. Arias

  Journal of the Mechanics and Physics of Solids (2008). Vol. 56 (4), pp. 1124-1244

  
  

  Abstract

  We propose to model thick multiwalled carbon nanotubes as beams with non-convex curvature energy. Such models develop stressed phase mixtures composed of smoothly bent sections and rippled sections. This model is motivated by experimental observations and large-scale atomistic-based simulations. The model is analyzed, validated against large-scale simulations, and exercised in examples of interest. It is shown that modelling MWCNTs as linear elastic beams can result in poor approximations that overestimate the elastic restoring force considerably, particularly for thick tubes. In contrast, the proposed model produces very accurate predictions both of the restoring force and of the phase pattern. The size effect in the bending response of MWCNTs is also discussed.

  Abstract
  We propose to model thick multiwalled carbon nanotubes as beams with non-convex curvature energy. Such models develop stressed phase mixtures composed of smoothly bent sections [...]

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• Size-dependent nonlinear elastic scaling of multiwalled carbon nanotubes

  I. Arias, M. Arroyo

  Physical review letters (2008). Vol. 100 (8), pp. 085503-1/085503-4

  
  

  Abstract

  We characterize through large-scale simulations the nonlinear elastic response of multi-walled carbon nanotubes (MWNCNTs) in torsion and bending. We identify a unified law consisting of two distinct power-law regimes in the energy-deformation relation. This law encapsulates the complex mechanics of rippling and is described in terms of elastic constants, a critical length-scale and an anharmonic energy-deformation exponent. The mechanical response of MWCNTs is found to be strongly size-dependent, in that the critical strain beyond which they behave nonlinearly scales as the inverse of their diameter. These predictions are consistent with available experimental observations.

  Abstract
  We characterize through large-scale simulations the nonlinear elastic response of multi-walled carbon nanotubes (MWNCNTs) in torsion and bending. We identify a unified law [...]

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• On the geometric character of stress in continuum mechanics

  E. Kanso, M. Arroyo, Y. Tong, J. Marsden, M. Desbrun

  Zeitschrift für angewandte Mathematik und Physik (2007). Vol. 58 (5), pp. 843-856

  
  

  Abstract

  This paper shows that the stress field in the classical theory of continuum mechanics may be taken to be a covector-valued differential two-form. The balance laws and other fundamental laws of continuum mechanics may be neatly rewritten in terms of this geometric stress. A geometrically attractive and covariant derivation of the balance laws from the principle of energy balance in terms of this stress is presented.

  Abstract
  This paper shows that the stress field in the classical theory of continuum mechanics may be taken to be a covector-valued differential two-form. The balance laws and other [...]

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• Local maximum‐entropy approximation schemes: a seamless bridge between finite elements and meshfree methods

  M. Arroyo, M. Ortiz

  Int. J. Numer. Meth. Engng. (2006). Vol. 65 (13), pp. 2167-2202

  
  

  Abstract

  We present a one‐parameter family of approximation schemes, which we refer to as local maximum‐entropy approximation schemes, that bridges continuously two important limits: Delaunay triangulation and maximum‐entropy (max‐ent) statistical inference. Local max‐ent approximation schemes represent a compromise—in the sense of Pareto optimality—between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width. Local max‐ent approximation schemes are entirely defined by the node set and the domain of analysis, and the shape functions are positive, interpolate affine functions exactly, and have a weak Kronecker‐delta property at the boundary. Local max‐ent approximation may be regarded as a regularization, or thermalization, of Delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation. Local max‐ent approximation schemes can be taken as a convenient basis for the numerical solution of PDEs in the style of meshfree Galerkin methods. In test cases characterized by smooth solutions we find that the accuracy of local max‐ent approximation schemes is vastly superior to that of finite elements.

  Abstract
  We present a one‐parameter family of approximation schemes, which we refer to as local maximum‐entropy approximation schemes, that bridges continuously two important [...]

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• Continuum mechanics modeling and simulation of carbon nanotubes

  M. Arroyo, T. Belytschko

  Meccanica (2005). Vol. 40 (4-6), pp. 455-469

  
  

  Abstract

  The understanding of the mechanics of atomistic systems greatly benefits from continuum mechanics. One appealing approach aims at deductively constructing continuum theories starting from models of the interatomic interactions. This viewpoint has become extremely popular with the quasicontinuum method. The application of these ideas to carbon nanotubes presents a peculiarity with respect to usual crystalline materials: their structure relies on a two-dimensional curved lattice. This renders the cornerstone of crystal elasticity, the Cauchy–Born rule, insufficient to describe the effect of curvature. We discuss the application of a theory which corrects this deficiency to the mechanics of carbon nanotubes (CNTs). We review recent developments of this theory, which include the study of the convergence characteristics of the proposed continuum models to the parent atomistic models, as well as large scale simulations based on this theory. The latter have unveiled the complex nonlinear elastic response of thick multiwalled carbon nanotubes (MWCNTs), with an anomalous elastic regime following an almost absent harmonic range.

  Abstract
  The understanding of the mechanics of atomistic systems greatly benefits from continuum mechanics. One appealing approach aims at deductively constructing continuum theories [...]

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