Overall viscoelastic properties of 2D and two-phase periodic composites constituted of elliptical and rectangular heterogeneities

International audienceThis paper presents analytical solutions for the effective rheological viscoelastic properties of 2D periodic structures. The solutions, based on Fourier series analysis, are derived first in the Laplace-Carson (LC) space for different inclusion shapes (rectangle or ellipse) and arrangements. The effective results are obtained in the form of rational functions of the LC transform variable. Two inversion methods are used to find the relaxation behavior. The first one is based on the exact inverse of the LC transform while the second approximates the overall behavior by using a Standard Linear Solid model, which yields very simple analytical formulas for the coefficients entering the constitutive equations. Results of the two methods are compared in the case of an application to real materials

Elastic properties of two phase composites from optimal Neumann series and structure factors

We consider the problem of finding the effective stiffness tensor \mathbb{C}^e$of periodic heterogeneous martrix-inclusion materials. Given the distribution of the constituents, the cell problem must be solved first and the linear relation between average stress and strain is then established. Estimates can be obtained by making relevant approximation to the ingredients constituting the effective tensor. Although the present contribution concerns the theory of optimally estimating \mathbb{C}^e$ from the microstructure, it is closely related to FFT numerical homogenization methods. By introducing a reference material \mathbb{C}^0$, the heterogeneity effect can be viewed as a distribution of eigenstrains within an homogeneous material. Using the related Green tensor, our problem can be formulated as a Lippmann-Schwinger equation for eigenstrain. The integral equation is the origin of resolution methods based on iteration and Fast Fourier Transform (FFT) techniques (Michel et al., 1999; Bhattacharya and Suquet, 2005). Significant progresses have been made regarding the improvement of convergence rate (Michel et al., 1999; Eyre and Milton, 1999; Milton, 2002; Monchiet and Bonnet, 2012; Brisard and Dormieux, 2010). The study of convergence rate in those works will be extended in the present contribution in the case of new integral equations. The iteration scheme used to solve the Lippmann-Schwinger equation corresponds to the Neumann series summation. The latter can be used to derive exact theoretical relations and estimates. In this paper, we propose a new estimate based on series expansion that works at all contrast ratio, while using the matrix as a reference material. Additionally, we can control and optimize the convergence rate so that the series converges in the quickest way, and therefore produces the best estimates when using a finite sum in the series expansion. A class of integral equations for eigenstrain depending on two parameters \alpha,\beta$ is first derived. The spectral radius and norm of the corresponding operators are bounded by analytical expressions. Different optimization methods are proposed to find the fastest series convergence and the associated estimates. Similarly to the estimations of the effective elasticity tensor using correlation functions, the new method presented in this paper allows to estimate the effective elasticity tensor using the n-$order structure factors, which represent the counterpart in Fourier space of correlation functions. As an example, a direct connection of the effective elasticity tensor to n-$th order structure factors is given in the case of randomly distributed spheres. Numerical applications for cubic arrays and random distribution of spheres yield very good results in comparison with FFT based methods and other results from the literature

Properties of water confined in hydroxyapatite nanopores as derived from molecular dynamics simulations

Bone tissue is characterized by nanopores inside the collagen-apatite matrix where fluid can exist and flow. The description of the fluid flow within the bone has however mostly relied on a macroscopic continuum mechanical treatment of the system, and, for this reason, the role of these nanopores has been largely overlooked. However, neglecting the nanoscopic behaviour of fluid within the bone volume could result in large errors in the overall description of the dynamics of fluid. In this work, we have investigated the nanoscopic origin of fluid motion by conducting atomistic molecular dynamics simulations of water confined between two parallel surfaces of hydroxyapatite (HAP), which is the main mineral phase of mammalian bone. The polarizable core–shell interatomic potential model used in this work to simulate the HAP–water system has been extensively assessed with respect to ab initio calculations and experimental data. The structural (pair distribution functions), dynamical (self-diffusion coefficients) and transport (shear viscosity coefficients) properties of confined water have been computed as a function of the size of the nanopore and the temperature of the system. Analysis of the results shows that the dynamical and transport properties of water are significantly affected by the confinement, which is explained in terms of the layering of water on the surface of HAP as a consequence of the molecular interactions between the water molecules and the calcium and phosphate ions at the surface. Using molecular dynamics simulations, we have also computed the slip length of water on the surface of HAP, the value of which has never been reported before

Stress concentration and surface instability of anisotropic solids with slightly wavy boundary

International audienceThis paper presents a first order perturbation analysis of stress concentration and surface morphology instability of elastically anisotropic solids. The boundary of the solids under consideration is periodic along two orthogonal directions. The magnitude of the undulation is sufficiently small so that a half-space model can be used for simplification. We derive expressions for the stress concentration factors and the critical wavelength of the perturbation in terms of the remote stresses, surface energy anisotropy and the elastic anisotropy of the solid. Numerical applications to cubic materials using Barnett-Lothe integrals are also given. (C) 2011 Elsevier Ltd. All rights reserved

A numerical-analytical coupling computational method for homogenization of effective thermal conductivity of periodic composites

International audienceBackground : In the framework of periodic homogenization, the conduction problem can be formulated as an integral equation whose solution can be represented by a eumann series. From the theory, many efficient computational methods and analytical estimations have been proposed to compute the effective conductivity of composites.Methods: We combine a Fast Fourier Transform (FFT) numerical method based on the Neumann series and analytical estimation based on the integral equation to solve the problem. Specifically, the analytical approximation is used to estimate the remainder of the series.Results: From some numerical examples, the coupling method has shown to improve significantly the original FFT iteration scheme and results are also superior to the analytical estimation.Conclusions: We have proposed a new efficient computation method to determine the effective conductivity of composites. This method combines the advantages of the FFT methods and the analytical estimation based on integral equation

A numerical-analytical coupling computational method for homogenization of effective thermal conductivity of periodic composites

International audienceBackground : In the framework of periodic homogenization, the conduction problem can be formulated as an integral equation whose solution can be represented by a eumann series. From the theory, many efficient computational methods and analytical estimations have been proposed to compute the effective conductivity of composites.Methods: We combine a Fast Fourier Transform (FFT) numerical method based on the Neumann series and analytical estimation based on the integral equation to solve the problem. Specifically, the analytical approximation is used to estimate the remainder of the series.Results: From some numerical examples, the coupling method has shown to improve significantly the original FFT iteration scheme and results are also superior to the analytical estimation.Conclusions: We have proposed a new efficient computation method to determine the effective conductivity of composites. This method combines the advantages of the FFT methods and the analytical estimation based on integral equation

Collisions, adsorption and self diffusion of gas in nanometric channels by Molecular Dynamics and stochastic simulation and the case of Helium gas in graphitic slit pore

International audienceWe investigate and model the collisions and the self diffusion processes of dilute Helium gas in nanometric graphite channels using molecular dynamics. At high temperature, collisions are mostly specular with short resident time. At temperature as low as 50 K-75 K, the gas atoms stay longer near the surface and the surface diffusion becomes dominant. Both ballistic and diffusive transport regimes are present before the desorption. A waiting time model based on the residence time distribution and coupled with ballistic-diffusive surface motion of atoms and with Cercignani-Lampis scattering model is proposed. The stochastic simulation of self diffusion based on the waiting time model agrees with the MD simulations and theoretical results in literature. The Arrhenius law is used to model the variation of the obtained Knudsen diffusivity as functions of temperature

FFT based numerical homogenization method for porous conductive materials

International audienceThe Fourier series method is used to solve the periodic homogenization problem for conductive materials containing voids. The problems involving voids are special cases of infinite contrast whose full field solution is not unique, causing convergence issues when iteration schemes are used. In this paper, we reformulate the problem based on the temperature field in the skeleton and derive an equation where the temperature field is connected to values on the pore boundary. Iteration schemes based on the new equation show that the convergence is fast, yielding good results both in terms of local fields and effective conductive properties

Conductivity of periodic composites made of matrix and polydispersed composites

International audienceWe present closed-form expressions of the effective conductivity of composites made of matrix and impenetrable polydispersed particles. The approach developed in a recent work has been generalized to treat periodic multiphase composite materials. Particular cases including orthorhombic arrangements of spherical or spheroidal inclusions and random distributions of spheres are considered where simple closed-form expressions are obtained. The case of random distributions allows us to recover Mori-Tanaka estimation after introducing partial structure factors as a statistical information on the microstructure

An FFT method for the computation of thermal diffusivity of porous periodic media

International audienceIn this paper, we provide Fast Fourier Transform iterative schemes to compute the thermal diffusivity of periodic porous medium. We consider the fluid flow through a porous rigid solid due to a prescribed macroscopic gradient of pressure and a macroscopic gradient of temperature. As already proved in the literature, the asymptotic homogenization procedure is reduced to the resolution of two separated problems for the unit cell: (i) the fluid flow governed by the Stokes equations with an applied gradient of pressure, (ii) the heat transfer by both convection and conduction due to an applied macroscopic gradient of temperature. We develop new numerical approaches based on fast Fourier transform for the implementation of the cell problems. In a first approach, a simple iterative based on the primal variable (gradient of temperature) is provided to solve the heat transfer problem. In order to improve the convergence in the range of high values of the prescribed gradient of pressure, we propose a more sophisticated iterative scheme based on the polarization. In order to evaluate their capacities, these FFT algorithms are applied to some specific microstructures of interest including flows past parallel pores (Poiseuille flows) and periodically or randomly distributed cylinders