Vincent Monchiet

Inversion of higher order isotropic tensors with minor symmetries and solution of higher order heterogeneity problems

In this paper, the derivation of irreducible bases for a class of isotropic 2nth-order tensors having particular ‘minor symmetries’ is presented. The methodology used for obtaining these bases consists of extending the concept of deviatoric and spherical parts, commonly used for second-order tensors, to the case of an nth-order tensor. It is shown that these bases are useful for effecting the classical tensorial operations and especially the inversion of a 2nth-order tensor. Finally, the formalism introduced in this study is applied for obtaining the closed-form expression of the strain field within a spherical inclusion embedded in an infinite elastic matrix and subjected to linear or quadratic polynomial remote strain fields.

Low frequency locally resonant metamaterials containing composite inclusions

One main feature of metamaterials is the occurrence of a negative dynamic mass density that is produced when an inner local resonance is present. The inner resonance can be obtained in composite materials containing composite inclusions. For suitable ratios of the physical properties of the constituting materials, the composite inclusions act as spring-mass systems. The scaling of physical properties leading to such an inner resonance and the associated effective dynamic properties of materials containing composite inclusions are briefly recalled. The resonance frequencies and dynamic mass densities are obtained in a closed form for materials containing cylindrical composite fibers or spherical composite inclusions, after solving the related boundary value elasticity problems.

A polarization-based fast numerical method for computing the effective conductivity of composites

Purpose The paper deals with the development of an improved Fast Fourier Transform (FFT) –based numerical method for computing the effective properties of composite conductors. The convergence of the basic FFT-based methods is recognized to depend drastically on the contrast between the phases. For instance, the primal formulation is not suited for solving the problems with high conductivity whereas the dual formulation is computationally costly for problems with high resistivity. Consequently, it raises the problem of computing the properties of composites containing both highly conductive and resistive inclusions. Design/methodology/approach In the present work, the authors propose a new iterative shceme for solving that kind of problems which is formulated in terms of the polarization. Findings The capability and relevance of this iterative scheme is illustrated through numerical implementation in the case of composites containing suared inclusions. It is shown that the rate of convergence is increased and thus, particularly when the case of high contrasts is considered. The predominance of the polarization bazed iterative scheme over existing ones is also illustrated in the case of a composite containing both highly conductive and highly resistive inclusions. Originality/value The method is easy to implement and uses the same ingredients as the basic scheme: the FFT and the exact expression of the Green tensor in the Fourier space. Moreover, its convergence conditions do not depend on the conductivity properties of the constituents, which then constitutes the main difference with other existing iterative schemes. The method can then be applied for computing the effective properties of composite conductors with arbitrary contrasts.

Macroscopic Yield Criterion for Ductile Materials Containing Randomly Oriented Spheroidal Cavities

This study is devoted to a micromechanical approach of the macroscopic yield criterion of ductile porous materials made up of a perfectly plastic von Mises matrix and randomly oriented spheroidal voids (with a same shape ratio, prolate or oblate including penny-shaped crack). The approach is based on recent results established by Monchiet et al. [Monchiet, V., Charkaluk, E. and Kondo, D. (2007). An Improvement of Gurson-type Models of Porous Materials by using Eshelby-like Trial Velocity Fields, Comptes Rendus Mecanique, 335: 32–41.] for a unit cell containing a single family of spheroidal cavities. By adopting an approximation introduced by previous authors and which consists in embedding each void family in a medium submitted to the macroscopic stress, we provide for the studied class of materials closed-form expressions of the isotropic macroscopic yield function. The established results are compared with existing ones, and their interest is clearly shown.

Conductivity estimates of spherical-particle suspensions based on triplet structure factors.

In this paper, we present an estimation of the conductivity of composites constituted of identical spheres embedded in a host material. A family of polarization integral equations for the localization problem is constructed and the operator is then minimized to yield an optimal integral equation. As a result, the corresponding Neumann series converges with the fastest rate and can be used to estimate the effective conductivity. By combining this series and integral approximation, one can derive explicit expressions for the overall property using expansions in Fourier domain. For random hard-sphere systems, relations to structure factors and triplet structure factors have been made and Kirkwood superposition approximation is used to evaluate the effective conductivity, taking into account third-order correlations. This presents an original means to account for the statistical information up to third-order correlation when determining the effective properties of composite materials.

Computation of permeability with Fast Fourier Transform from 3-D digital images of porous microstructures

Purpose – The purpose of this paper is to present a fully automated numerical tool for computing the effective permeability of porous media from digital images which come from the modern imagery technique. Design/methodology/approach – The permeability is obtained by the homogenization process applied to a periodic rigid solid in which the fluid flow is described by the Stokes equations. The unit cell problem is solved by using the Fast Fourier Transform (FFT) algorithm, well adapted for the microstructures defined by voxels. Findings – Various 3-D examples are considered to show the capacity of the method. First, the case of flow through regular arrays of aligned cylinders or spheres are considered as benchmark problems. Next, the method is applied to some more complex and realistic porous solids obtained with assemblies of overlapping spherical pores having identical or different radii, regularly or randomly distributed within the unit cell. Originality/value – The use of FFT allows the resolution of hi...

Non-parametric wall model and methods of identifying boundary conditions for moments in gas flow equations

In this paper, we use Molecular Dynamics (MD) simulation method to study gas-wall boundary conditions. Discrete scattering information of gas molecules at the wall surface are obtained from collision simulations. The collision data can be used to identify the accommodation coefficients for parametric wall models such as Maxwell, Cercignani-Lampis scattering kernels. Since these scattering kernels are based on a limited number of accommodation coefficients, we adopt non-parametric statistical methods to construct the kernel to overcome these issues. Different from parametric kernels, the non-parametric kernels require no parameter (i.e accommodation coefficients) and no predefined distribution. We also propose approaches to derive directly the Navier friction and Kapitza thermal resistance coefficients as well as other interface coefficients associated to moment equations from the non-parametric kernels. The methods are applied successfully to systems composed of CH 4 or CO 2 and graphite, which are of interest to the petroleum industry.

Dynamic mass density of resonant metamaterials with homogeneous inclusions

The occurrence of a negative dynamic mass density is a striking property of metamaterials. It appears when an inner local resonance is present. Results coming from an asymptotic theory are recalled briefly, showing the scaling of physical properties leading to inner resonance in elastic composites containing homogeneous soft inclusions, with negligible scattering of waves traveling through the matrix. This appears for a large contrast of elastic properties between matrix and inclusion. The frequency-dependent dynamic mass density depends on the resonance frequencies of the inner inclusions and on their related participation factors. Having solved the dynamic elasticity problem, these physical quantities are provided in the case of homogeneous cylindrical and spherical inclusions. It is shown that numerous resonance frequencies do not contribute to the dynamic mass density or have small participation factors, which simplifies significantly the physics involved in the concerned inner resonance phenomena. Finally, non-dimensional resonance frequencies and participation factors are given for both cases of inclusions as functions of the Poissons ratio, defining completely the dynamic mass density.

Strain-induced friction anisotropy between graphene and molecular liquids

In this paper, we study the friction behavior of molecular liquids with anisotropically strained graphene. Due to the changes of lattice and the potential energy surface, the friction is orientation dependent and can be computed by tensorial Green-Kubo formula. Simple quantitative estimations are also proposed for the zero-time response and agree reasonably well with the molecular dynamics results. From simulations, we can obtain the information of structures, dynamics of the system, and study the influence of strain and molecular shapes on the anisotropy degree. It is found that unilateral strain can increase friction in all directions but the strain direction is privileged. Numerical evidences also show that nonspherical molecules are more sensitive to strain and give rise to more pronounced anisotropy effects.

On the Inertia Effects on the Darcy Law: Numerical Implementation and Confrontation of Micromechanics-Based Approaches

In this paper, we investigate the nonlinear deviation of the Darcy law in the domain of high pressure gradient. Classically, the (linear) Darcy law can be deduced from asymptotic homogenization approaches and the numerical resolution of the Stokes flow problem on the unit cell of the porous medium. At high-speed steady flow of a fluid, nonlinear effects on the macroscopic filtration law arise and are accounted by considering the convection term in the Navier–Stokes equation. These nonlinear effects has been often studied in asymptotic homogenization framework by expanding the solution in power series at low Reynolds number. This has two advantages: (i) The Navier–Stokes problems are replaced by a chain of linear problems with source terms which depend on the solution at lower order, and (ii) the macroscopic nonlinear filtration law is derived in the form of a polynom. We develop a Fast Fourier Transform (FFT)-based numerical algorithm to compute the solution of this elementary problems and to compute the higher-order permeability tensors in connection with the morphology of the porous medium. The results are then compared to the solution of the full Navier–Stokes problem by means of finite element method (FEM) which allows evaluating the capacity of the expansion method to account for the nonlinear effects. We determine the convergence radius of the polynomial series, and we give the limit of the series expansion method in terms of the Reynolds number.