Multiscale mass-spring models of carbon nanotube foams

This article is concerned with the mechanical properties of dense, vertically aligned CNT foams subject to one-dimensional compressive loading. We develop a discrete model directly inspired by the micromechanical response reported experimentally for CNT foams, where infinitesimal portions of the tubes are represented by collections of uniform bi-stable springs. Under cyclic loading, the given model predicts an initial elastic deformation, a non-homogeneous buckling regime, and a densification response, accompanied by a hysteretic unloading path. We compute the dynamic dissipation of such a model through an analytic approach. The continuum limit of the microscopic spring chain defines a mesoscopic dissipative element (micro-meso transition) which represents a finite portion of the foam thickness. An upper-scale model formed by a chain of non-uniform mesoscopic springs is employed to describe the entire CNT foam. A numerical approximation illustrates the main features of the proposed multiscale approach. Available experimental results on the compressive response of CNT foams are fitted with excellent agreement

A Tensor-product approach for large scale electronic structure calculations using Kohn–Sham density functional theory

Quantum-mechanical calculations based on Kohn–Sham density functional theory (DFT) played a significant role in accurately predicting various aspects of materials behavior over the past decade. The Kohn–Sham approach to DFT reduces the many-body Schrodinger (eigen value) problem of interacting electrons into an equivalent problem of noninteracting electrons in an effective mean field that is governed by electron-density. Despite the reduced computational complexity of Kohn–Sham DFT, large-scale DFT calculations are still computationally very demanding with the resulting computational complexity scaling cubically with number of atoms in a given materials system. Numerical algorithms with reduced scaling behavior which are robust, computationally efficient and scalable on parallel computing architectures are always desirable to enable simulations at larger scales and on more complex systems. Following this line of thought, this study explores the use of tensor structured methods for ab-initio numerical solution of Kohn–Sham equations arising in DFT calculations. Earlier studies on tensor-structured methods have been quite successful in the accurate calculation of Hartree and the nonlocal exchange operators arising in the Hartree–Fock equations. A recent investigation of low-rank Tucker-type decomposition of the electron-density of large aluminum clusters (obtained from the finite-element discretization of orbital free DFT) shows the exponential decay of approximation error with respect to Tucker rank (number of tensor-basis functions in Tucker type representation). The results also indicate a smaller Tucker rank for the accurate representation of the electron density and is only weakly dependent on the system sizes studied. The promising success of tensor-structured techniques in resolving the electronic structure of material systems has enabled us to take a step further. In this study, we propose a systematic way of computing a globally adapted Tucker-type basis for solving the Kohn–Sham DFT problem by using a separable approximation of the Kohn–Sham Hamiltonian. Further, the resulting Kohn–Sham eigenvalue problem is projected into the aforementioned Tucker basis and is solved for ground-state energy using a self-consistent field iteration. The rank of the resulting Tucker representation and the computational complexity of the calculation are examined on representative benchmark examples involving metallic and insulating systems

Multiscale Mass-Spring Models of Carbon Nanotube Arrays Accounting for Mullins-like Behavior and Permanent Deformation

Based on a one-dimensional discrete system of bistable springs, a mechanical model is introduced to describe plasticity and damage in carbon nanotube (CNT) arrays. The energetics of the mechanical system are investigated analytically, the stress-strain law is derived, and the mechanical dissipation is computed, both for the discrete case as well as for the continuum limit. An information-passing approach is developed that permits the investigation of macroscopic portions of the material. As an application, the simulation of a cyclic compression experiment on real CNT foam is performed, considering both the material response during the primary loading path from the virgin state and the damaged response after preconditioning

Continuum limits of bistable spring models of carbon nanotube arrays accounting for material damage

Using chains of bistable springs, a model is derived to investigate the plastic behavior of carbon nanotube arrays with damage. We study the preconditioning effect due to the loading history by computing analytically the stress-strain pattern corresponding to a fatigue-type damage of the structure. We identify the convergence of the discrete response to the limiting case of infinitely many springs, both analytically in the framework of Gamma-convergence, as well as numerically.Comment: 11 pages, 1 figur

On a non-isothermal model for nematic liquid crystals

A model describing the evolution of a liquid crystal substance in the nematic phase is investigated in terms of three basic state variables: the {\it absolute temperature} \teta, the {\it velocity field} \ub, and the {\it director field} \bd, representing preferred orientation of molecules in a neighborhood of any point of a reference domain. The time evolution of the velocity field is governed by the incompressible Navier-Stokes system, with a non-isotropic stress tensor depending on the gradients of the velocity and of the director field \bd, where the transport (viscosity) coefficients vary with temperature. The dynamics of \bd is described by means of a parabolic equation of Ginzburg-Landau type, with a suitable penalization term to relax the constraint |\bd | = 1. The system is supplemented by a heat equation, where the heat flux is given by a variant of Fourier's law, depending also on the director field \bd. The proposed model is shown compatible with \emph{First and Second laws} of thermodynamics, and the existence of global-in-time weak solutions for the resulting PDE system is established, without any essential restriction on the size of the data

Stress- and diffusion-induced interface motion: Modelling and numerical simulations

Accepted versio

Lagrangian and Hamiltonian two-scale reduction

Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system. In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure.Comment: 40 page